s_euclidean_plane	axiom	Euclidean plane ℝ²		The classical 2D Euclidean plane with its metric.
s_euclidean_3_space	axiom	Euclidean 3-space (sphere-packing context)	s_euclidean_solid_geometry	ℝ³ with unit-ball packing viewpoint.
s_real_numbers	axiom	Real numbers ℝ	s_real_line	The complete ordered field.
s_real_line_or_circle	axiom	ℝ or 𝕋¹		Real line or circle; underlying group for Fourier analysis.
s_complex_numbers	axiom	Complex numbers ℂ		The complex field.
s_integers	axiom	Integers ℤ		The ring of integers.
s_naturals_with_multiplication	axiom	Naturals ℕ (with multiplication)		Natural numbers under multiplication.
s_divisibility_definition	axiom	Divisibility relation		a | b iff ∃c. b = ac.
s_euclid_lemma	axiom	Euclid's Lemma		p | ab ⇒ p | a or p | b for prime p.
s_zfc_axioms	axiom	ZFC axioms		Zermelo–Fraenkel set theory with choice.
s_first_order_peano_arithmetic	axiom	Peano arithmetic (first-order)		First-order Peano axioms.
s_turing_machine_model	axiom	Turing machine model		Tape, head, finite control, transition function.
s_probability_axioms	axiom	Probability axioms (Kolmogorov)		(Ω, F, P) with P(Ω)=1.
s_newtonian_inverse_square_force	axiom	Newtonian inverse-square force law		F = -GMm/r². Physical axiom.
s_conic_sections	axiom	Conic sections		Ellipse, parabola, hyperbola.
s_graph_definition	axiom	Graph (V, E)		Vertex set and edge relation.
s_projective_plane	axiom	Projective plane ℙ²		Lines and points with projective axioms.
s_projective_space_axioms	axiom	Projective space axioms		Axioms of projective geometry in any dim.
s_area_additivity	axiom	Area additivity		Area of disjoint union = sum of areas.
s_circle_definition	axiom	Circle (as locus)		{x : |x − c| = r}.
s_polygon_area_formula	axiom	Polygon area formula		Area of polygon via triangulation.
s_isosceles_triangle_base_angles_equal	axiom	Pons asinorum		Base angles of isosceles triangle are equal.
s_similar_triangle_criterion	axiom	AA similarity		Equal angles ⇒ similar triangles.
s_coprime_pair	axiom	Coprime pair		gcd(m,n)=1.
s_prime_p	axiom	Prime p		Arbitrary prime.
s_prime_p_equiv_1_mod_4	axiom	Prime p ≡ 1 (mod 4)		Prime in a particular residue class.
s_prime_power_divisor_p_n	axiom	Prime-power divisor pⁿ		pⁿ | |G|.
s_pell_equation_x2_minus_N_y2	axiom	Pell equation x² − Ny² = 1		Indefinite binary quadratic form.
s_radical_extension_tower	axiom	Radical extension tower		Kₒ ⊂ K₁ ⊂ … each obtained by adjoining nth roots.
s_field_extension_L_over_K	axiom	Field extension L/K		L ⊇ K with compatible ring ops.
s_group_action	axiom	Group action		G acts on X.
s_differential_form	axiom	Differential form		Alternating k-tensor field.
s_divisor_on_curve	axiom	Divisor on curve		ℤ-linear combination of points on a curve.
s_sublinear_functional_p	axiom	Sublinear functional p		p(x+y) ≤ p(x)+p(y), p(αx)=αp(x) for α≥0.
s_linear_functional_on_subspace	axiom	Linear functional on subspace		Bounded linear functional defined on subspace.
s_family_of_compact_spaces	axiom	Family of compact spaces		{Xᵢ}ᵢ∈I each compact.
s_product_topology	axiom	Product topology		Coarsest topology making projections continuous.
s_iid_sequence_finite_variance	axiom	IID sequence (finite variance)		X₁, X₂, … iid with E|X|² < ∞.
s_smooth_function	axiom	Smooth function		Infinitely differentiable.
s_analytic_exponential_series	axiom	Exponential series Σ zⁿ/n!		Analytic series for exp.
s_unit_circle_in_C	axiom	Unit circle in ℂ		{z : |z|=1}.
s_convex_polyhedron	axiom	Convex polyhedron		Bounded intersection of half-spaces.
s_topological_sphere_S2	axiom	Topological sphere S²		The 2-sphere.
s_sine_function	axiom	Sine function sin(x)		Standard trigonometric function.
s_real_analysis	axiom	Real analysis framework		ε–δ analysis on ℝ.
s_continuous_function_on_interval	axiom	Continuous function on [a,b]		Continuous real-valued function on compact interval.
s_mean_value_theorem	axiom	Mean value theorem (lemma)		Lemma used in FTC assembly.
s_euler_four_square_identity	axiom	Euler four-square identity		Product of two sums of 4 squares is a sum of 4 squares.
s_prime_pair_p_q	axiom	Prime pair (p,q)		Two odd primes for reciprocity.
s_smooth_surface_in_R3	axiom	Smooth surface in ℝ³		2D submanifold of ℝ³.
s_first_fundamental_form	axiom	First fundamental form		Induced Riemannian metric on a surface.
s_holomorphic_function_on_domain	axiom	Holomorphic function on domain		Complex-differentiable function.
s_polynomial_ring_over_Q	axiom	Polynomial ring ℚ[x]		Polynomials with rational coefficients.
s_simply_connected_proper_domain_in_C	axiom	Simply connected proper domain in ℂ		Simply connected open ≠ ℂ.
s_compact_riemann_surface	axiom	Compact Riemann surface		Complex 1-manifold, compact.
s_primes_in_naturals	axiom	Primes in ℕ	s_prime_numbers	Set of prime numbers.
s_riemann_zeta_function	axiom	Riemann zeta function ζ(s)		Σ 1/nˢ analytically continued.
s_finite_group	axiom	Finite group G		Arbitrary finite group.
s_closed_ball_D_n	axiom	Closed ball Dⁿ		{x ∈ ℝⁿ : |x|≤1}.
s_continuous_self_map	axiom	Continuous self-map		f : X → X continuous.
s_lagrangian_action_integral	axiom	Lagrangian action integral		S[γ] = ∫ L(γ, γ̇) dt.
s_lie_group	axiom	Lie group		Smooth group.
s_complete_metric_space	axiom	Complete metric space		Cauchy sequences converge.
s_strict_contraction	axiom	Strict contraction		d(f(x),f(y)) ≤ k·d(x,y), k<1.
s_real_vector_space	axiom	Real vector space		Vector space over ℝ.
s_noetherian_ring_R	axiom	Noetherian ring R		Every ascending chain of ideals stabilizes.
s_polynomial_ring	axiom	Polynomial ring R[x]		Polynomials over R.
s_algebraically_closed_field_k	axiom	Algebraically closed field k		Every nonconstant poly in k[x] has a root.
s_polynomial_ring_in_n_vars	axiom	Polynomial ring k[x₁,…,xₙ]		Multivariate polynomial ring.
s_infinite_set	axiom	Infinite set		Countably or larger infinite set.
s_k_coloring_of_pairs	axiom	k-coloring of pairs		Edge coloring with k colors.
s_measure_preserving_transformation	axiom	Measure-preserving transformation		T : (X,μ) → (X,μ) with μ(T⁻¹A)=μ(A).
s_L2_function_space	axiom	L² function space		Square-integrable functions.
s_compact_smooth_manifold	axiom	Compact smooth manifold		Compact boundaryless smooth manifold.
s_elliptic_operator_D	axiom	Elliptic operator D		Elliptic differential operator between vector bundles.
s_finite_simple_group	axiom	Finite simple group		Nontrivial, no nontrivial normal subgroup.
s_planar_graph	axiom	Planar graph		Graph embeddable in the plane.
s_proper_vertex_coloring	axiom	Proper vertex coloring		Adjacent vertices get different colors.
s_smooth_projective_curve_over_Q	axiom	Smooth projective curve over ℚ		Smooth projective curve defined over ℚ.
s_jacobian_variety	axiom	Jacobian variety		Pic⁰(C) for a curve.
s_elliptic_curve_over_Q	axiom	Elliptic curve over ℚ		Genus-1 curve with rational point.
s_modular_form	axiom	Modular form		Weight-k modular form.
s_galois_representation	axiom	Galois representation		Continuous ℓ-adic representation.
s_closed_3_manifold	axiom	Closed 3-manifold		Compact boundaryless 3-manifold.
s_riemannian_metric	axiom	Riemannian metric		Positive-definite symmetric (0,2)-tensor.
s_unit_balls_sphere_packing	axiom	Unit ball sphere packing		Arrangement of unit balls maximizing density.
s_szemeredi_theorem	axiom	Szemerédi's theorem (input to Green–Tao)		Positive-density subsets of ℤ contain arbitrarily long APs. (Also appears as terminal s_szemeredi_theorem.)
s_admissible_k_tuple	axiom	Admissible k-tuple		No prime forbids all residues.
s_minor_ordering	axiom	Minor ordering on graphs		H minor of G relation.
s_positive_density_subset	axiom	Positive-density subset (of ℤ)		Upper density bounded below.
s_smooth_manifold_with_boundary	axiom	Smooth manifold with boundary		Smooth manifold-with-boundary.
s_right_triangle_in_plane	state	Right triangle in plane		Triangle with a right angle.
s_two_similar_subtriangles	state	Two similar subtriangles		Partition of right triangle by altitude.
s_segment_length_identity_on_hypotenuse	state	Segment-length identity on hypotenuse		Hypotenuse segments ratios.
s_triangle_inscribed_in_semicircle	state	Triangle inscribed in semicircle		Triangle with hypotenuse as diameter.
s_two_isosceles_subtriangles	state	Two isosceles subtriangles		From center to vertex lines.
s_finite_list_of_primes	state	Finite list of primes {p₁,…,pₖ}		Assumed finite enumeration.
s_new_number_N_coprime_to_all_primes_in_list	state	N = p₁⋯pₖ + 1		Constructed number coprime to list.
s_uniqueness_of_prime_factorization	state	Uniqueness of prime factorization		Intermediate lemma.
s_circle	state	Circle (as 2D object)		Concrete circle for Archimedes' argument.
s_inscribed_circumscribed_96_gons	state	Inscribed/circumscribed 96-gons		Bounding polygons.
s_sphere	state	Sphere (as 3D object)		Concrete sphere.
s_sphere_as_solid_of_revolution	state	Sphere as solid of revolution		Generated by rotating a semicircle.
s_cyclic_quadrilateral	state	Cyclic quadrilateral		Vertices on a circle.
s_pair_of_similar_triangles_on_diagonal	state	Pair of similar triangles on diagonal		From auxiliary point K.
s_pair_of_coprime_moduli	state	Pair of coprime moduli (m,n)		gcd(m,n)=1.
s_idempotent_pair_mod_mn	state	Idempotent pair mod mn		From Bezout au+bv=1.
s_near_solution_triple_a_b_k	state	Near-solution triple (a,b,k)		a² − Nb² = k.
s_composed_triple_mod_scaling	state	Composed triple (after bhāvanā)		Result of Brahmagupta composition.
s_general_cubic_ax3_plus_bx2_plus_cx_plus_d	state	General cubic ax³+bx²+cx+d		Arbitrary cubic polynomial.
s_depressed_cubic_t3_plus_pt_plus_q	state	Depressed cubic t³+pt+q		No quadratic term.
s_system_sum_and_product_of_cubes	state	System u+v=?, uv=?		From t = u+v substitution.
s_general_quartic	state	General quartic		Arbitrary quartic polynomial.
s_depressed_quartic	state	Depressed quartic		No cubic term.
s_resolvent_cubic	state	Resolvent cubic		Auxiliary cubic from quartic.
s_two_triangles_in_perspective_in_plane	state	Two triangles in perspective (plane)		Common point of concurrence.
s_two_triangles_in_perspective_in_space	state	Two triangles in perspective (space)		Lifted to 3D.
s_axis_of_perspective	state	Axis of perspective		Common line of perspectivity.
s_multiplicative_group_mod_p	state	(ℤ/p)*		Multiplicative group mod p.
s_orbit_of_a_mod_p	state	Orbit of a mod p		{a, 2a, …, (p-1)a} mod p.
s_existence_of_x_with_x2_equiv_minus1	state	x with x² ≡ -1 (mod p)		Quadratic residue of -1.
s_small_a_b_with_a2_plus_b2_equiv_0_mod_p	state	(a,b) with a²+b² ≡ 0 (mod p)		From Thue-style lattice.
s_integral_as_limit_of_sums	state	Integral as limit of Riemann sums		∫f = lim Σ f(ξ)Δx.
s_tycho_brahe_observation_table	state	Tycho Brahe observation table		Empirical orbital data.
s_T_squared_prop_a_cubed_conjecture	state	T² ∝ a³ conjecture		Kepler's 3rd law as conjecture.
s_third_law	state	Third law (verified)		Empirically verified on all 6 planets.
s_polynomial_approximation_of_order_n	state	Order-n polynomial approximation		Taylor polynomial.
s_complex_number_cos_theta_plus_i_sin_theta	state	cos θ + i sin θ		Point on unit circle.
s_multiplicative_law_on_unit_circle	state	Multiplicative law on S¹		Angle-addition via multiplication.
s_series_for_e_i_theta	state	Series for e^{iθ}		Σ (iθ)ⁿ/n!.
s_triangulated_sphere_S2	state	Triangulated S²		After triangulation of polyhedron.
s_sin_x_as_infinite_product	state	sin(x) = x ∏(1 - x²/n²π²)		Euler's product expansion.
s_konigsberg_bridge_configuration	state	Königsberg bridge configuration		7 bridges of Königsberg.
s_abstract_multigraph	state	Abstract multigraph		After axiomatization.
s_complex_polynomial_p_z	state	Complex polynomial p(z)		Arbitrary complex polynomial.
s_two_real_algebraic_curves_in_plane	state	Two real algebraic curves in ℝ²		Re p = 0 and Im p = 0.
s_intersection_exists	state	Intersection exists (lemma)		From large-circle behavior.
s_auxiliary_mp_equals_sum_of_four_squares	state	mp = a²+b²+c²+d² (with small m)		Auxiliary identity.
s_prime_as_sum_of_four_squares	state	p = sum of four squares		Descent endpoint.
s_characteristic_function_of_sum	state	Characteristic function of sum		φ_{S_n}(t) = φ_X(t/√n)ⁿ.
s_limit_characteristic_function_equals_gaussian	state	Limit characteristic function = e^{-t²/2}		From log-expansion.
s_legendre_symbol_table	state	Legendre symbol table		Tabulated quadratic residues.
s_reciprocity_conjecture	state	Reciprocity conjecture		Gauss's conjectured relation.
s_refined_reciprocity_conjecture	state	Refined reciprocity conjecture		After special-case verification.
s_gauss_curvature_K	state	Gaussian curvature K		Intrinsic curvature function.
s_compact_oriented_surface_without_boundary	state	Compact oriented surface ∂=∅		2D closed orientable manifold.
s_geodesic_triangulation	state	Geodesic triangulation		Triangulation by geodesic arcs.
s_local_angle_defect_identity	state	Local angle-defect identity		Per-triangle angle deficit.
s_cauchy_integral_theorem	state	Cauchy integral theorem (lemma)		∮ f dz = 0 on closed curves.
s_quintic_polynomial	state	Generic quintic polynomial		General degree-5 polynomial.
s_galois_group_S5	state	Galois group S₅		Gal of generic quintic.
s_A5_is_simple_non_abelian	state	A₅ is simple non-abelian		Key obstruction.
s_finite_normal_separable_extension_L_over_K	state	Finite normal separable extension L/K		Galois extension.
s_galois_group	state	Galois group Gal(L/K)		Automorphism group fixing K.
s_intermediate_fields_of_L	state	Lattice of intermediate fields		Subfields between K and L.
s_galois_correspondence	state	Galois correspondence		Subgroups ↔ subfields.
s_heat_conduction_on_rod	state	Heat conduction on rod (physical)		Fourier's physical setup.
s_heat_equation_PDE	state	Heat equation PDE ∂u/∂t = α∂²u/∂x²		Parabolic heat equation.
s_mode_by_mode_ODE_system	state	Mode-by-mode ODE system		Fourier-decomposed heat eq.
s_differential_form_omega_on_manifold	state	Differential form ω on M		Specific form to integrate.
s_d_omega_in_local_coords	state	dω in local coordinates		Exterior derivative.
s_family_F_of_injective_holomorphic_maps_to_unit_disk	state	Family F of injective hol maps to 𝔻		Univalent maps.
s_extremal_map_f_star	state	Extremal map f*		Maximizer of |f'(0)|.
s_sheaf_O_D_on_curve	state	Sheaf O(D) on curve		Line bundle for divisor D.
s_euler_characteristic_chi_O_D	state	Euler char χ(O(D))		h⁰ − h¹.
s_divisor_D	state	Divisor D (specific)		Specific divisor input.
s_euler_product_zeta	state	Euler product ζ(s) = ∏(1-p⁻ˢ)⁻¹		ζ as product over primes.
s_meromorphic_zeta_on_plane	state	Meromorphic ζ on ℂ \ {1}		After analytic continuation.
s_zeta_nonvanishing_on_line_Re_1	state	ζ(1+it) ≠ 0		Key non-vanishing lemma.
s_finite_group_G_with_order_divisible_by_p_n	state	Finite G with pⁿ | |G|		Setup for Sylow.
s_set_of_p_subgroups_with_G_action	state	p-subgroups with G-conjugation action		Conjugation class.
s_alleged_enumeration_of_reals_in_0_1	state	Alleged enumeration of ℝ ∩ [0,1]		Assumed countable listing.
s_real_not_in_list	state	Real not in list		Diagonal construction.
s_continuous_function_on_closed_interval	state	Continuous f on closed interval		Input for Weierstrass.
s_smooth_approximant_f_epsilon	state	Smooth approximant f_ε		Heat-kernel smoothing.
s_ideal_in_R_x	state	Ideal in R[x]		Arbitrary ideal.
s_finite_generating_set_of_ideal	state	Finite generating set of ideal		From ACC.
s_maximal_ideal_m_in_k_x_1_x_n	state	Maximal ideal m ⊂ k[x₁,…,xₙ]		In polynomial ring.
s_point_in_affine_n_space	state	Point in affine n-space		Geometric counterpart of m.
s_ideal_I	state	Ideal I		General ideal in Nullstellensatz.
s_vanishing_variety_V_I	state	Vanishing variety V(I)		V(I) = {x : f(x)=0 ∀f∈I}.
s_continuous_self_map_of_disk_without_fixed_point	state	Continuous self-map of Dⁿ (no fixed pt)		Alleged fixed-point-free map.
s_retraction_to_boundary_sphere	state	Retraction Dⁿ → ∂Dⁿ		Built from fixed-pt-free map.
s_contradiction_no_retraction	state	No retraction Dⁿ → ∂Dⁿ (contradiction)		Degree obstruction.
s_infinitesimal_action_variation	state	Infinitesimal action variation		δS under 1-param group.
s_syntactic_predicates_as_arithmetic_predicates	state	Syntactic predicates arithmetized		Prov, Ded etc. as arithmetic.
s_self_referential_godel_sentence_G	state	Gödel sentence G		G ↔ ¬Prov(⌜G⌝).
s_alleged_decider_H_for_halting	state	Alleged halting decider H		Assumed total decider.
s_encoding_of_machines_as_data	state	Encoding of TMs as data		Gödel-style encoding.
s_self_contradictory_machine_D	state	Self-contradictory machine D		D halts iff it doesn't.
s_model_L_of_ZFC_plus_GCH	state	Model L (constructible universe)		Gödel's L with GCH.
s_godel_L_model	state	Gödel L model (alias)	s_model_L_of_ZFC_plus_GCH	Alias for L.
s_nested_monochromatic_sequence	state	Nested monochromatic sequence		From pigeonhole iterates.
s_ultrafilter_limit_in_product	state	Ultrafilter limit in ∏Xᵢ		Limit along ultrafilter.
s_invariant_subspace_decomposition	state	Invariant subspace decomposition		Spectral decomp of Koopman U.
s_elliptic_operator_D_on_manifold	state	Elliptic operator D on M		Applied elliptic operator.
s_principal_symbol_in_K_theory_of_TM	state	Principal symbol in K(T*M)		σ(D) ∈ K⁰(T*M).
s_topological_index_class_in_K_of_point	state	Topological index in K(pt)=ℤ		After pushforward.
s_component_type_and_characteristic_p_type_cases	state	Component/char-p type cases		Local-analysis dichotomy.
s_case_exhaustion_complete	state	Case exhaustion complete		All cases handled across papers.
s_model_of_ZFC_plus_not_CH	state	Model of ZFC + ¬CH		Cohen's generic extension.
s_finite_list_of_1500_configurations	state	Unavoidable set of ~1500 configs		AHK configurations.
s_four_color_theorem_machine_certified	state	Four Color Theorem (machine-certified)		Gonthier's Coq formalization.
s_curve_of_genus_at_least_2	state	Curve C, genus ≥ 2		Input for Faltings.
s_curve_inside_abelian_variety	state	C ↪ Jacobian J_C		Abel–Jacobi embedding.
s_finiteness_of_isogeny_class	state	Finiteness of isogeny class		Shafarevich-type finiteness.
s_hypothetical_FLT_solution_a_n_plus_b_n_equals_c_n	state	Hypothetical FLT solution aⁿ+bⁿ=cⁿ		Assumed nontrivial solution.
s_frey_elliptic_curve	state	Frey elliptic curve y²=x(x-aⁿ)(x+bⁿ)		Constructed from FLT soln.
s_non_modular_galois_representation_required	state	Non-modular mod-ℓ rep (required)		Ribet's level-lowering output.
s_semistable_modularity_theorem	state	Semistable modularity theorem		R = T theorem output.
s_ricci_flow_equation	state	Ricci flow eq. ∂g/∂t = -2Ric(g)		Hamilton's Ricci flow.
s_long_time_decomposition_into_geometric_pieces	state	Long-time decomposition into geometric pieces		After surgery.
s_thurston_eight_geometries_classification	state	Thurston 8 geometries		8 model geometries.
s_sphere_packing_density_functional	state	Sphere-packing density functional		Density to maximize.
s_finite_family_of_local_star_configurations	state	Finite family of local star configs		Voronoi-based reduction.
s_kepler_conjecture_machine_certified	state	Kepler conjecture (Flyspeck-verified)		Flyspeck HOL Light formalization.
s_primes_with_density_zero	state	Primes with density 0		Density 0 but unbounded.
s_relative_szemeredi_for_pseudorandom_majorants	state	Relative Szemerédi (pseudorandom majorants)		Green–Tao transference result.
s_aps_in_pseudorandom_dense_subset	state	APs in pseudorandom dense subset		APs in dense subset of majorant.
s_goldston_yildirim_sieve_majorant	state	Goldston–Yıldırım sieve majorant		Pseudorandom majorant via sieve.
s_positive_lower_bound_on_prime_pairs_in_tuple	state	Positive lower bound on prime pairs in tuple		From Selberg sieve weights.
s_gap_bound_246	state	Gap bound 246 (Polymath8)		Numeric gap bound after optimization.
s_integer_N_odd_and_larger_than_5	state	Odd integer N > 5		Input for ternary Goldbach.
s_circle_integral_for_r3_N	state	Circle integral for r₃(N)		∫₀¹ F(α)³ e^{-2πiNα} dα.
s_major_arc_asymptotic_plus_minor_arc_error	state	Major/minor arc decomposition		Hardy–Littlewood dissection.
s_infinite_sequence_of_graphs	state	Infinite sequence of graphs		Input for Robertson–Seymour.
s_tree_width_decomposition	state	Tree-width decomposition		3-connected parts.
s_positive_density_subset_of_integers	state	Positive-density subset of ℤ		Density > 0.
s_furstenberg_system_with_positive_measure_A	state	Furstenberg system with μ(A)>0		Correspondence output.
s_multiple_recurrence_for_A	state	Multiple recurrence for A		Ergodic multiple recurrence.
s_one_dim_extension_step	state	One-dim extension step (Hahn–Banach)		Extension of linear functional by one dimension.
s_pythagorean_theorem	theorem	Pythagorean theorem		a² + b² = c² for right triangles.
s_thales_theorem	theorem	Thales's theorem		Inscribed angle in semicircle is right.
s_infinitude_of_primes	theorem	Infinitude of primes		There are infinitely many primes.
s_fundamental_theorem_of_arithmetic	theorem	Fundamental theorem of arithmetic		Unique prime factorization.
s_area_of_circle	theorem	Area of circle = πr²		Archimedes' quadrature.
s_volume_of_sphere	theorem	Volume of sphere = 4/3·πr³		Archimedes' volume.
s_ptolemys_theorem	theorem	Ptolemy's theorem		AC·BD = AB·CD + AD·BC for cyclic quadrilateral.
s_chinese_remainder_theorem	theorem	Chinese remainder theorem		ℤ/mn ≅ ℤ/m × ℤ/n for coprime m,n.
s_solvability_of_pell_equation	theorem	Solvability of Pell's equation (Chakravāla)		Pell equation always solvable.
s_cardano_cubic_formula	theorem	Cardano's cubic formula		Closed form for cubic roots.
s_ferrari_quartic_formula	theorem	Ferrari's quartic formula		Closed form for quartic roots.
s_desargues_theorem	theorem	Desargues's theorem		Perspective triangles have collinear axis.
s_fermat_little_theorem	theorem	Fermat's little theorem		a^p ≡ a (mod p).
s_fermat_two_squares	theorem	Fermat's sum of two squares		p ≡ 1 (mod 4) ⇒ p = a²+b².
s_fundamental_theorem_of_calculus	theorem	Fundamental theorem of calculus		F'(x) = f(x) ⇒ ∫ₐᵇ f = F(b)-F(a).
s_kepler_three_laws	theorem	Kepler's three laws		Ellipse / equal-area / T²∝a³.
s_taylor_theorem	theorem	Taylor's theorem		f(x) = Σ f^(k)(a)(x-a)ᵏ/k! + remainder.
s_de_moivre_formula	theorem	De Moivre's formula		(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ.
s_euler_formula	theorem	Euler's formula		e^{iθ} = cos θ + i sin θ.
s_euler_polyhedron_formula	theorem	Euler polyhedron formula V-E+F=2		Euler characteristic of S².
s_basel_identity	theorem	Basel identity Σ1/n² = π²/6		Euler's Basel solution.
s_eulerian_path_criterion	theorem	Eulerian path criterion		Graph has Eulerian circuit iff all vertices even degree.
s_fundamental_theorem_of_algebra	theorem	Fundamental theorem of algebra		Every nonconstant p ∈ ℂ[z] has a root.
s_lagrange_four_squares	theorem	Lagrange four-square theorem		Every ℕ is a sum of ≤4 squares.
s_central_limit_theorem	theorem	Central limit theorem		Sum of iid tends to normal.
s_quadratic_reciprocity	theorem	Quadratic reciprocity		(p/q)(q/p) = (-1)^{…} for odd primes.
s_theorema_egregium	theorem	Theorema Egregium		K is an isometric invariant.
s_gauss_bonnet_theorem	theorem	Gauss–Bonnet theorem		∫∫ K dA = 2πχ(M).
s_cauchy_integral_formula	theorem	Cauchy integral formula		f(a) = (1/2πi)∮f(z)/(z-a)dz.
s_abel_ruffini	theorem	Abel–Ruffini theorem		Quintic not solvable by radicals.
s_fundamental_theorem_of_galois_theory	theorem	Fundamental theorem of Galois theory		Subgroups ↔ subfields.
s_fourier_theorem_heat	theorem	Fourier's theorem on the heat equation		Heat eq. solved by Fourier series.
s_stokes_theorem	theorem	Stokes's theorem		∫_M dω = ∫_{∂M} ω.
s_riemann_mapping_theorem	theorem	Riemann mapping theorem		Simply connected domain ≠ ℂ is biholomorphic to 𝔻.
s_riemann_roch_theorem	theorem	Riemann–Roch theorem		ℓ(D) − ℓ(K-D) = deg D + 1 − g.
s_prime_number_theorem	theorem	Prime number theorem		π(x) ∼ x / ln x.
s_sylow_theorems	theorem	Sylow theorems		Existence, conjugacy, count of p-Sylows.
s_uncountability_of_reals	theorem	Uncountability of ℝ		|ℝ| > |ℕ|.
s_weierstrass_approximation	theorem	Weierstrass approximation		C[a,b] uniformly approximated by polynomials.
s_hilbert_basis_theorem	theorem	Hilbert basis theorem		R Noetherian ⇒ R[x] Noetherian.
s_nullstellensatz	theorem	Hilbert's Nullstellensatz		I(V(J)) = √J over algebraically closed k.
s_brouwer_fpt	theorem	Brouwer fixed-point theorem		Continuous Dⁿ→Dⁿ has fixed point.
s_noether_theorem	theorem	Noether's theorem		Symmetry ⇒ conservation law.
s_godel_incompleteness	theorem	Gödel's incompleteness theorems		Consistent PA-extensions are incomplete.
s_banach_fpt	theorem	Banach fixed-point theorem		Contraction on complete metric space has unique fixed point.
s_hahn_banach	theorem	Hahn–Banach theorem		Extend bounded linear functional.
s_tychonoff_theorem	theorem	Tychonoff's theorem		Product of compact spaces is compact.
s_undecidability_of_halting	theorem	Undecidability of halting		No TM decides halting.
s_con_zfc_gch	theorem	Con(ZFC) ⇒ Con(ZFC+GCH) (Gödel)		GCH consistent with ZFC via L.
s_ramsey_theorem_infinite	theorem	Infinite Ramsey theorem		Every edge-coloring of Kω contains monochromatic infinite clique.
s_birkhoff_ergodic_theorem	theorem	Birkhoff ergodic theorem		Time averages = space averages a.e.
s_atiyah_singer_index_theorem	theorem	Atiyah–Singer index theorem		Analytic index = topological index.
s_cfsg	theorem	Classification of finite simple groups		List of all finite simple groups.
s_ch_independent_of_zfc	theorem	CH independent of ZFC		Neither CH nor ¬CH provable from ZFC.
s_four_color_theorem	theorem	Four color theorem		Planar graph is 4-colorable.
s_mordell_faltings	theorem	Mordell–Faltings theorem		Genus ≥ 2 curves over ℚ have finitely many rational points.
s_flt	theorem	Fermat's Last Theorem		No nontrivial aⁿ+bⁿ=cⁿ for n≥3.
s_poincare_conjecture	theorem	Poincaré conjecture		Simply connected closed 3-manifold is S³.
s_geometrization_theorem	theorem	Geometrization theorem		3-manifolds decompose into 8 geometric pieces.
s_kepler_conjecture	theorem	Kepler conjecture		Face-centered cubic packing maximizes density.
s_green_tao	theorem	Green–Tao theorem		Primes contain arbitrarily long APs.
s_bounded_gaps_between_primes	theorem	Bounded gaps between primes		Exists constant C with infinitely many prime gaps < C.
s_ternary_goldbach	theorem	Ternary Goldbach (Helfgott)		Every odd N>5 is sum of three primes.
s_graph_minor_theorem	theorem	Robertson–Seymour graph minor theorem		Graphs well-quasi-ordered by minors.
s_szemeredi_theorem_terminal	theorem	Szemerédi's theorem (as terminal)		Positive-density sets contain long APs.
t_spot_pattern_in_table	technique	Spot pattern in table		
t_verify_on_special_cases	technique	Verify on special cases		
t_complete_the_square	technique	Complete the square		
t_reduce_to_canonical_form	technique	Reduce to canonical form		
t_compose_with_identity	technique	Compose with identity		
t_symmetry_reduction	technique	Symmetry reduction		
t_conserved_quantity	technique	Conserved quantity		
t_duality	technique	Duality		
t_character_decomposition_count	technique	Character decomposition count		
t_exhaustion_squeeze	technique	Exhaustion / squeeze		
t_interpolate_and_continue	technique	Interpolate and continue		
t_frequency_decomposition	technique	Frequency decomposition		
t_axiomatize_from_instances	technique	Axiomatize from instances		
t_structural_isomorphism	technique	Structural isomorphism		
t_ultraproduct_transfer	technique	Ultraproduct transfer		
t_raise_dimension	technique	Raise dimension		
t_obstruction_class	technique	Obstruction class		
t_compactness_argument	technique	Compactness argument		
t_deformation_cohomology	technique	Deformation cohomology		
t_rescale_for_asymptotic_geometry	technique	Rescale for asymptotic geometry		
t_diagonalize	technique	Diagonalize		
t_arithmetize_syntax	technique	Arithmetize syntax		
t_force_independence	technique	Force independence		
t_contraction_fixed_point	technique	Contraction fixed point		
t_infinite_descent	technique	Infinite descent		
t_flow_with_surgery	technique	Flow with surgery		
t_physics_to_pde	technique	Physics to PDE		
t_complex_analysis_to_integers	technique	Complex analysis to integers		
t_analysis_algebra_topology_bridge	technique	Analysis–algebra–topology bridge		
t_major_minor_arc_decomposition	technique	Major/minor arc decomposition		
t_ergodic_correspondence	technique	Ergodic correspondence		
t_finite_case_check	technique	Finite case check		
t_formal_verify	technique	Formal verify		
t_distributed_collaboration	technique	Distributed collaboration		
t_probabilistic_existence	technique	Probabilistic existence		
t_pigeonhole_collision	technique	Pigeonhole collision		
t_sieve_by_optimized_quadratic	technique	Sieve by optimized quadratic		
t_group_complete_exact_category	technique	Group-complete exact category (K-theory)		
t_sheafify_on_grothendieck_topology	technique	Sheafify on Grothendieck topology		
t_representable_functor_trick	technique	Representable functor trick		
t_polynomial_method	technique	Polynomial method		
t_double_centralizer_decompose	technique	Double centralizer decompose		
t_fourier_transform	technique	Fourier transform (umbrella)		
t_svd_and_spectral_decomposition	technique	SVD / spectral decomposition		
t_galois_correspondence	technique	Galois correspondence (composite)		
t_ricci_flow_with_surgery	technique	Ricci flow with surgery (composite)		
t_wiles_modularity	technique	Wiles modularity (composite)		
t_godel_numbering	technique	Gödel numbering (composite)		
t_atiyah_singer_index_machinery	technique	Atiyah–Singer index machinery (composite)		
t_selberg_sieve_method	technique	Selberg sieve method (composite)		
t_circle_method	technique	Circle method (composite)		
t_furstenberg_correspondence_principle	technique	Furstenberg correspondence (composite)		
t_category_theoretic_colimits_and_adjoints	technique	Category-theoretic colimits/adjoints (composite)		
t_auxiliary_construction	technique	Auxiliary construction		Introduce a helper object whose structure forces the desired conclusion. Distinct from reduce-to-canonical-form (which simplifies) because this ADDS structure.
t_conjecture_refinement	technique	Conjecture refinement		Modify a conjecture to exclude known failure modes or add missing preconditions. Distinct from verify-on-special-cases — that tests, this refines.
t_reductio_ad_absurdum	technique	Reductio ad absurdum		Assume the statement's negation; derive a contradiction; conclude. Distinct from infinite descent (which is number-theoretic minimal-counterexample).
t_projection_to_subspace	technique	Projection to subspace		Map from a higher-dimensional space to a lower-dimensional subspace along a specified direction. Inverse / cousin of t_raise_dimension.
t_sheaf_cohomology_bridge	technique	Sheaf cohomology bridge		Specialization of the analysis-algebra-topology bridge: compute H^i(X, F) and extract Euler characteristic or related invariants.
t_k_theoretic_index_bridge	technique	K-theoretic index bridge		Lift an elliptic operator to a K-theory class and recover the analytic index via pushforward.
t_heights_and_galois_rep_bridge	technique	Heights / Galois representation bridge		Bridge Diophantine geometry to Galois theory via ell-adic representations and canonical heights.
t_level_lowering_bridge	technique	Level-lowering bridge (Ribet)		Given a Galois rep coming from a modular form at level N with extra congruence, show it comes from level N/p.
t_transference_bridge	technique	Transference bridge		Transfer a structural theorem (e.g., Szemerédi) from dense integers to a pseudorandom sparse subset via relative-density machinery.
s_heron_formula	theorem	Heron's Formula		Derivation chain ingested from Phase A (mathematician_relationships_r1.md).
s_incircle_tangent_decomposition	state	incircle tangent decomposition		Imported from Phase A as intermediate for 'Heron's Formula'.
s_area_squared_equals_s_times_product	state	area squared equals s times product		Imported from Phase A as intermediate for 'Heron's Formula'.
s_menelaus_theorem	theorem	Menelaus's Theorem		Derivation chain ingested from Phase A (mathematician_relationships_r1.md).
s_perpendiculars_from_vertices_to_transversal	state	perpendiculars from vertices to transversal		Imported from Phase A as intermediate for 'Menelaus's Theorem'.
s_ratio_product_cancels_heights	state	ratio product cancels heights		Imported from Phase A as intermediate for 'Menelaus's Theorem'.
s_brahmagupta_formula	theorem	Brahmagupta's Formula		Derivation chain ingested from Phase A (mathematician_relationships_r1.md).
s_two_triangles_with_supplementary_angles	state	two triangles with supplementary angles		Imported from Phase A as intermediate for 'Brahmagupta's Formula'.
s_area_squared_symmetric_in_four_sides	state	area squared symmetric in four sides		Imported from Phase A as intermediate for 'Brahmagupta's Formula'.
s_brahmagupta_midpoint_theorem	theorem	Brahmagupta's Theorem		Derivation chain ingested from Phase A (mathematician_relationships_r1.md).
s_orthodiagonal_cyclic_quadrilateral	state	orthodiagonal cyclic quadrilateral		Imported from Phase A as intermediate for 'Brahmagupta's Theorem'.
s_perpendicular_from_diagonal_intersection_to_side	state	perpendicular from diagonal intersection to side		Imported from Phase A as intermediate for 'Brahmagupta's Theorem'.
s_aryabhata_sine_pi_kuttaka	theorem	Aryabhata's Theorems		Derivation chain ingested from Phase A (mathematician_relationships_r1.md).
s_sine_second_difference_recurrence	state	sine second difference recurrence		Imported from Phase A as intermediate for 'Aryabhata's Theorems'.
s_aryabhata_sine_table	state	aryabhata sine table		Imported from Phase A as intermediate for 'Aryabhata's Theorems'.
s_aryabhata_pi_3_1416	state	aryabhata pi 3 1416		Imported from Phase A as intermediate for 'Aryabhata's Theorems'.
s_kuttaka_extended_euclid	state	kuttaka extended euclid		Imported from Phase A as intermediate for 'Aryabhata's Theorems'.
s_alkhwarizmi_six_quadratic_types	theorem	Al-Khwārizmī's Quadratic Framework		Derivation chain ingested from Phase A (mathematician_relationships_r1.md).
s_six_canonical_quadratic_forms	state	six canonical quadratic forms		Imported from Phase A as intermediate for 'Al-Khwārizmī's Quadratic Framework'.
s_geometric_completion_of_square	state	geometric completion of square		Imported from Phase A as intermediate for 'Al-Khwārizmī's Quadratic Framework'.
s_khayyam_cubic_geometric_solution	theorem	Omar Khayyām's Geometric Solution of Cubics		Derivation chain ingested from Phase A (mathematician_relationships_r1.md).
s_fourteen_irreducible_cubic_types	state	fourteen irreducible cubic types		Imported from Phase A as intermediate for 'Omar Khayyām's Geometric Solution of Cubics'.
s_cubic_as_intersection_of_two_conics	state	cubic as intersection of two conics		Imported from Phase A as intermediate for 'Omar Khayyām's Geometric Solution of Cubics'.
s_viete_formulas	theorem	Viète's Formulas		Derivation chain ingested from Phase A (mathematician_relationships_r1.md).
s_polynomial_as_product_of_root_factors	state	polynomial as product of root factors		Imported from Phase A as intermediate for 'Viète's Formulas'.
s_elementary_symmetric_polynomials	state	elementary symmetric polynomials		Imported from Phase A as intermediate for 'Viète's Formulas'.
s_flt_n_equals_4	theorem	Fermat's Last Theorem, Origin — n = 4 descent		Derivation chain ingested from Phase A (mathematician_relationships_r1.md).
s_flt_general_conjecture	state	flt general conjecture		Imported from Phase A as intermediate for 'Fermat's Last Theorem, Origin — n = 4 descent'.
s_flt_n_4_strengthened_claim	state	flt n 4 strengthened claim		Imported from Phase A as intermediate for 'Fermat's Last Theorem, Origin — n = 4 descent'.
s_descartes_rule_of_signs	theorem	Descartes' Rule of Signs		Derivation chain ingested from Phase A (mathematician_relationships_r1.md).
s_sign_change_parity_conjecture	state	sign change parity conjecture		Imported from Phase A as intermediate for 'Descartes' Rule of Signs'.
s_sign_change_increment_under_positive_factor	state	sign change increment under positive factor		Imported from Phase A as intermediate for 'Descartes' Rule of Signs'.
s_descartes_angular_defect	theorem	Descartes' Theorem on Total Angular Defect		Derivation chain ingested from Phase A (mathematician_relationships_r1.md).
s_vertex_angular_defect	state	vertex angular defect		Imported from Phase A as intermediate for 'Descartes' Theorem on Total Angular Defect'.
s_total_defect_equals_2pi_chi	state	total defect equals 2pi chi		Imported from Phase A as intermediate for 'Descartes' Theorem on Total Angular Defect'.
s_pascal_mystic_hexagram	theorem	Pascal's Theorem on Hexagons in a Conic / Mystic Hexagram		Derivation chain ingested from Phase A (mathematician_relationships_r1.md).
s_inscribed_hexagon_in_circle	state	inscribed hexagon in circle		Imported from Phase A as intermediate for 'Pascal's Theorem on Hexagons in a Conic / Mystic Hexagram'.
s_three_opposite_side_intersection_points	state	three opposite side intersection points		Imported from Phase A as intermediate for 'Pascal's Theorem on Hexagons in a Conic / Mystic Hexagram'.
s_pascal_line	state	pascal line		Imported from Phase A as intermediate for 'Pascal's Theorem on Hexagons in a Conic / Mystic Hexagram'.
s_pascal_triangle_identity	theorem	Pascal's Triangle Identity		Derivation chain ingested from Phase A (mathematician_relationships_r1.md).
s_binomial_recurrence_conjecture	state	binomial recurrence conjecture		Imported from Phase A as intermediate for 'Pascal's Triangle Identity'.
s_distinguished_element_bijection	state	distinguished element bijection		Imported from Phase A as intermediate for 'Pascal's Triangle Identity'.
s_newton_binomial_theorem	theorem	Newton's Generalized Binomial Theorem		Derivation chain ingested from Phase A (mathematician_relationships_r1.md).
s_generalized_binomial_coefficient	state	generalized binomial coefficient		Imported from Phase A as intermediate for 'Newton's Generalized Binomial Theorem'.
s_vandermonde_convolution_extended	state	vandermonde convolution extended		Imported from Phase A as intermediate for 'Newton's Generalized Binomial Theorem'.
s_newton_identities	theorem	Newton's Identities		Derivation chain ingested from Phase A (mathematician_relationships_r1.md).
s_log_derivative_generating_function	state	log derivative generating function		Imported from Phase A as intermediate for 'Newton's Identities'.
s_power_sum_series	state	power sum series		Imported from Phase A as intermediate for 'Newton's Identities'.
s_rolle_theorem	theorem	Rolle's Theorem		Derivation chain ingested from Phase A (mathematician_relationships_r1.md).
s_attained_extrema_on_closed_interval	state	attained extrema on closed interval		Imported from Phase A as intermediate for 'Rolle's Theorem'.
s_interior_extremum_or_constant	state	interior extremum or constant		Imported from Phase A as intermediate for 'Rolle's Theorem'.
s_torricelli_law	theorem	Torricelli's Law		Derivation chain ingested from Phase A (mathematician_relationships_r1.md).
s_galilean_free_fall_speed	state	galilean free fall speed		Imported from Phase A as intermediate for 'Torricelli's Law'.
s_fluid_column_energy_balance	state	fluid column energy balance		Imported from Phase A as intermediate for 'Torricelli's Law'.
s_wallis_product	theorem	Wallis Product for π		Derivation chain ingested from Phase A (mathematician_relationships_r1.md).
s_wallis_integer_quadrature_table	state	wallis integer quadrature table		Imported from Phase A as intermediate for 'Wallis Product for π'.
s_wallis_half_integer_interpolation	state	wallis half integer interpolation		Imported from Phase A as intermediate for 'Wallis Product for π'.
s_bernoulli_lln	theorem	Bernoulli's Law of Large Numbers		Derivation chain ingested from Phase A (mathematician_relationships_r2.md).
s_binomial_tail_ratios	state	binomial tail ratios		Imported from Phase A as intermediate for 'Bernoulli's Law of Large Numbers'.
s_binomial_tail_bound	state	binomial tail bound		Imported from Phase A as intermediate for 'Bernoulli's Law of Large Numbers'.
s_de_moivre_laplace	theorem	De Moivre–Laplace Theorem		Derivation chain ingested from Phase A (mathematician_relationships_r2.md).
s_stirling_central_term_asymptotic	state	stirling central term asymptotic		Imported from Phase A as intermediate for 'De Moivre–Laplace Theorem'.
s_gaussian_density_as_limit	state	gaussian density as limit		Imported from Phase A as intermediate for 'De Moivre–Laplace Theorem'.
s_euler_totient_theorem	theorem	Euler's Totient Theorem		Derivation chain ingested from Phase A (mathematician_relationships_r2.md).
s_unit_group_mod_n	state	unit group mod n		Imported from Phase A as intermediate for 'Euler's Totient Theorem'.
s_orbit_of_a_mod_n	state	orbit of a mod n		Imported from Phase A as intermediate for 'Euler's Totient Theorem'.
s_euler_homogeneous_functions	theorem	Euler's Theorem on Homogeneous Functions		Derivation chain ingested from Phase A (mathematician_relationships_r2.md).
s_homogeneous_function_definition	state	homogeneous function definition		Imported from Phase A as intermediate for 'Euler's Theorem on Homogeneous Functions'.
s_scaling_vector_field_identity	state	scaling vector field identity		Imported from Phase A as intermediate for 'Euler's Theorem on Homogeneous Functions'.
s_lagrange_mvt	theorem	Lagrange's Mean Value Theorem		Derivation chain ingested from Phase A (mathematician_relationships_r2.md).
s_auxiliary_function_with_equal_endpoints	state	auxiliary function with equal endpoints		Imported from Phase A as intermediate for 'Lagrange's Mean Value Theorem'.
s_rolle_interior_critical_point	state	rolle interior critical point		Imported from Phase A as intermediate for 'Lagrange's Mean Value Theorem'.
s_wilson_theorem	theorem	Wilson's Theorem		Derivation chain ingested from Phase A (mathematician_relationships_r2.md).
s_polynomial_x_p_minus_1_factored_mod_p	state	polynomial x p minus 1 factored mod p		Imported from Phase A as intermediate for 'Wilson's Theorem'.
s_factored_form_of_cyclic_unit_polynomial	state	factored form of cyclic unit polynomial		Imported from Phase A as intermediate for 'Wilson's Theorem'.
s_bayes_theorem	theorem	Bayes's Theorem		Derivation chain ingested from Phase A (mathematician_relationships_r2.md).
s_conditional_probability_definition	state	conditional probability definition		Imported from Phase A as intermediate for 'Bayes's Theorem'.
s_symmetric_chain_rule_for_intersection	state	symmetric chain rule for intersection		Imported from Phase A as intermediate for 'Bayes's Theorem'.
s_cauchy_schwarz	theorem	Cauchy–Schwarz Inequality		Derivation chain ingested from Phase A (mathematician_relationships_r2.md).
s_inner_product_space	state	inner product space		Imported from Phase A as intermediate for 'Cauchy–Schwarz Inequality'.
s_non_negative_quadratic_in_t	state	non negative quadratic in t		Imported from Phase A as intermediate for 'Cauchy–Schwarz Inequality'.
s_bolzano_ivt	theorem	Bolzano's Intermediate Value Theorem		Derivation chain ingested from Phase A (mathematician_relationships_r2.md).
s_supremum_set_S	state	supremum set S		Imported from Phase A as intermediate for 'Bolzano's Intermediate Value Theorem'.
s_candidate_root_c_as_sup	state	candidate root c as sup		Imported from Phase A as intermediate for 'Bolzano's Intermediate Value Theorem'.
s_bolzano_weierstrass	theorem	Bolzano–Weierstrass Theorem		Derivation chain ingested from Phase A (mathematician_relationships_r2.md).
s_bounded_sequence_in_Rn	state	bounded sequence in Rn		Imported from Phase A as starting state for 'Bolzano–Weierstrass Theorem'.
s_bounded_sequence_in_closed_box	state	bounded sequence in closed box		Imported from Phase A as intermediate for 'Bolzano–Weierstrass Theorem'.
s_nested_intervals_with_infinitely_many_terms	state	nested intervals with infinitely many terms		Imported from Phase A as intermediate for 'Bolzano–Weierstrass Theorem'.
s_greens_theorem	theorem	Green's Theorem		Derivation chain ingested from Phase A (mathematician_relationships_r2.md).
s_simple_region_decomposition_in_R2	state	simple region decomposition in R2		Imported from Phase A as intermediate for 'Green's Theorem'.
s_per_slice_ftoc_identity	state	per slice ftoc identity		Imported from Phase A as intermediate for 'Green's Theorem'.
s_divergence_theorem	theorem	Divergence Theorem / Gauss–Ostrogradsky		Derivation chain ingested from Phase A (mathematician_relationships_r2.md).
s_z_simple_region_decomposition_in_R3	state	z simple region decomposition in R3		Imported from Phase A as intermediate for 'Divergence Theorem / Gauss–Ostrogradsky'.
s_per_axis_flux_identity	state	per axis flux identity		Imported from Phase A as intermediate for 'Divergence Theorem / Gauss–Ostrogradsky'.
s_liouville_bounded_entire	theorem	Liouville's Theorem (complex analysis)		Derivation chain ingested from Phase A (mathematician_relationships_r2.md).
s_cauchy_derivative_bound	state	cauchy derivative bound		Imported from Phase A as intermediate for 'Liouville's Theorem (complex analysis)'.
s_vanishing_derivative_on_all_of_C	state	vanishing derivative on all of C		Imported from Phase A as intermediate for 'Liouville's Theorem (complex analysis)'.
s_cayley_hamilton	theorem	Cayley–Hamilton Theorem		Derivation chain ingested from Phase A (mathematician_relationships_r2.md).
s_square_matrix_with_char_poly	state	square matrix with char poly		Imported from Phase A as intermediate for 'Cayley–Hamilton Theorem'.
s_adjugate_matrix_polynomial_identity	state	adjugate matrix polynomial identity		Imported from Phase A as intermediate for 'Cayley–Hamilton Theorem'.
s_cantor_bernstein_schroeder	theorem	Cantor–Bernstein–Schroeder Theorem		Derivation chain ingested from Phase A (mathematician_relationships_r2.md).
s_pair_of_injections_with_orphans	state	pair of injections with orphans		Imported from Phase A as intermediate for 'Cantor–Bernstein–Schroeder Theorem'.
s_trajectory_partition_of_A	state	trajectory partition of A		Imported from Phase A as intermediate for 'Cantor–Bernstein–Schroeder Theorem'.
s_poincare_recurrence	theorem	Poincaré Recurrence Theorem		Derivation chain ingested from Phase A (mathematician_relationships_r2.md).
s_measurable_set_of_positive_measure	state	measurable set of positive measure		Imported from Phase A as intermediate for 'Poincaré Recurrence Theorem'.
s_non_returning_subset_B	state	non returning subset B		Imported from Phase A as intermediate for 'Poincaré Recurrence Theorem'.
s_disjoint_preimages_of_B_sum_bounded	state	disjoint preimages of B sum bounded		Imported from Phase A as intermediate for 'Poincaré Recurrence Theorem'.
s_zermelo_well_ordering	theorem	Zermelo's Well-Ordering Theorem		Derivation chain ingested from Phase A (mathematician_relationships_r3.md).
s_axiom_of_choice_function_phi	state	axiom of choice function phi		Imported from Phase A as starting state for 'Zermelo's Well-Ordering Theorem'.
s_maximal_gamma_set_on_M	state	maximal gamma set on M		Imported from Phase A as intermediate for 'Zermelo's Well-Ordering Theorem'.
s_noether_isomorphism_theorems	theorem	Noether's Isomorphism Theorems		Derivation chain ingested from Phase A (mathematician_relationships_r3.md).
s_normal_subgroup_N_in_G	state	normal subgroup N in G		Imported from Phase A as starting state for 'Noether's Isomorphism Theorems'.
s_group_homomorphism_f	state	group homomorphism f		Imported from Phase A as starting state for 'Noether's Isomorphism Theorems'.
s_abstract_subobject_quotient_morphism_pattern	state	abstract subobject quotient morphism pattern		Imported from Phase A as intermediate for 'Noether's Isomorphism Theorems'.
s_canonical_quotient_by_kernel_map	state	canonical quotient by kernel map		Imported from Phase A as intermediate for 'Noether's Isomorphism Theorems'.
s_godel_completeness_theorem	theorem	Gödel's Completeness Theorem		Derivation chain ingested from Phase A (mathematician_relationships_r3.md).
s_first_order_logic_language	axiom	first order logic language		Imported from Phase A as starting axiom for 'Gödel's Completeness Theorem'.
s_consistent_first_order_theory_T	state	consistent first order theory T		Imported from Phase A as starting state for 'Gödel's Completeness Theorem'.
s_consistent_theory_with_witnesses	state	consistent theory with witnesses		Imported from Phase A as intermediate for 'Gödel's Completeness Theorem'.
s_maximal_consistent_henkin_theory_T_star	state	maximal consistent henkin theory T star		Imported from Phase A as intermediate for 'Gödel's Completeness Theorem'.
s_banach_tarski_paradox	theorem	Banach–Tarski Paradox		Derivation chain ingested from Phase A (mathematician_relationships_r3.md).
s_free_group_F2_on_two_generators	state	free group F2 on two generators		Imported from Phase A as starting state for 'Banach–Tarski Paradox'.
s_paradoxical_decomposition_of_F2	state	paradoxical decomposition of F2		Imported from Phase A as intermediate for 'Banach–Tarski Paradox'.
s_F2_as_subgroup_of_SO3	state	F2 as subgroup of SO3		Imported from Phase A as intermediate for 'Banach–Tarski Paradox'.
s_paradoxical_sphere_decomposition	state	paradoxical sphere decomposition		Imported from Phase A as intermediate for 'Banach–Tarski Paradox'.
s_stone_weierstrass_theorem	theorem	Stone–Weierstrass Theorem		Derivation chain ingested from Phase A (mathematician_relationships_r3.md).
s_compact_hausdorff_space	axiom	compact hausdorff space		Imported from Phase A as starting axiom for 'Stone–Weierstrass Theorem'.
s_point_separating_unital_subalgebra_A_of_C_X_R	state	point separating unital subalgebra A of C X R		Imported from Phase A as starting state for 'Stone–Weierstrass Theorem'.
s_closure_of_A_is_a_lattice	state	closure of A is a lattice		Imported from Phase A as intermediate for 'Stone–Weierstrass Theorem'.
s_pointwise_matching_family_f_xy	state	pointwise matching family f xy		Imported from Phase A as intermediate for 'Stone–Weierstrass Theorem'.
s_stone_representation_theorem	theorem	Stone Representation Theorem		Derivation chain ingested from Phase A (mathematician_relationships_r3.md).
s_boolean_algebra_B	state	boolean algebra B		Imported from Phase A as starting state for 'Stone Representation Theorem'.
s_ultrafilter_spectrum_Spec_B	state	ultrafilter spectrum Spec B		Imported from Phase A as intermediate for 'Stone Representation Theorem'.
s_compact_totally_disconnected_stone_space	state	compact totally disconnected stone space		Imported from Phase A as intermediate for 'Stone Representation Theorem'.
s_full_modularity_theorem_BCDT	theorem	Modularity Theorem — full, Breuil–Conrad–Diamond–Taylor 2001		Derivation chain ingested from Phase A (mathematician_relationships_r3.md).
s_ramification_stratification_of_elliptic_curves	state	ramification stratification of elliptic curves		Imported from Phase A as intermediate for 'Modularity Theorem — full, Breuil–Conrad–Diamond–Taylor 2001'.
s_non_semistable_modularity_cases_handled	state	non semistable modularity cases handled		Imported from Phase A as intermediate for 'Modularity Theorem — full, Breuil–Conrad–Diamond–Taylor 2001'.
s_abc_conjecture_mochizuki_claimed	theorem	abc Conjecture — Mochizuki's disputed proof		Derivation chain ingested from Phase A (mathematician_relationships_r3.md).
s_coprime_triple_a_b_c_with_a_plus_b_equals_c	state	coprime triple a b c with a plus b equals c		Imported from Phase A as starting state for 'abc Conjecture — Mochizuki's disputed proof'.
s_mochizuki_IUT_framework	state	mochizuki IUT framework		Imported from Phase A as starting state for 'abc Conjecture — Mochizuki's disputed proof'.
s_anabelian_reconstruction_of_E	state	anabelian reconstruction of E		Imported from Phase A as intermediate for 'abc Conjecture — Mochizuki's disputed proof'.
s_theta_link_between_hodge_theaters	state	theta link between hodge theaters		Imported from Phase A as intermediate for 'abc Conjecture — Mochizuki's disputed proof'.
s_log_volume_inequality	state	log volume inequality		Imported from Phase A as intermediate for 'abc Conjecture — Mochizuki's disputed proof'.
s_compactness_theorem_fol	theorem	Compactness theorem		Imported from Phase B brief catalog area D.
s_lowenheim_skolem	theorem	Löwenheim–Skolem theorem		Imported from Phase B brief catalog area D.
s_church_rosser	theorem	Church–Rosser theorem		Imported from Phase B brief catalog area D.
s_deduction_theorem	theorem	Deduction theorem		Imported from Phase B brief catalog area D.
s_los_theorem	theorem	Łoś's theorem		Imported from Phase B brief catalog area D.
s_craig_interpolation	theorem	Craig's interpolation theorem		Imported from Phase B brief catalog area D.
s_beth_definability	theorem	Beth's definability theorem		Imported from Phase B brief catalog area D.
s_lindstrom_theorem	theorem	Lindström's theorem		Imported from Phase B brief catalog area D.
s_morley_categoricity	theorem	Morley's categoricity theorem		Imported from Phase B brief catalog area D.
s_martin_borel_determinacy	theorem	Martin's Borel determinacy theorem		Imported from Phase B brief catalog area D.
s_cohen_forcing	theorem	Cohen's forcing theorems		Imported from Phase B brief catalog area D.
s_mrdp_theorem	theorem	Matiyasevich's theorem (MRDP)		Imported from Phase B brief catalog area D.
s_paris_harrington	theorem	Paris–Harrington theorem		Imported from Phase B brief catalog area D.
s_gentzen_consistency	theorem	Gentzen's consistency proof		Imported from Phase B brief catalog area D.
s_ac_equivalents	theorem	Axiom of choice equivalents		Imported from Phase B brief catalog area D.
s_schroder_bernstein	theorem	Schröder–Bernstein theorem		Imported from Phase B brief catalog area D.
s_hartogs_theorem	theorem	Hartogs's theorem		Imported from Phase B brief catalog area D.
s_konig_cardinal_theorem	theorem	König's theorem (cardinal)		Imported from Phase B brief catalog area D.
s_reflection_principle	theorem	Reflection principle		Imported from Phase B brief catalog area D.
s_silver_theorem	theorem	Silver's theorem		Imported from Phase B brief catalog area D.
s_hall_marriage	theorem	Hall's marriage theorem		Imported from Phase B brief catalog area D.
s_konig_lemma	theorem	König's lemma		Imported from Phase B brief catalog area D.
s_konig_bipartite	theorem	Kőnig's theorem (bipartite)		Imported from Phase B brief catalog area D.
s_dilworth_theorem	theorem	Dilworth's theorem		Imported from Phase B brief catalog area D.
s_mirsky_theorem	theorem	Mirsky's theorem		Imported from Phase B brief catalog area D.
s_turan_theorem	theorem	Turán's theorem		Imported from Phase B brief catalog area D.
s_erdos_ko_rado	theorem	Erdős–Ko–Rado theorem		Imported from Phase B brief catalog area D.
s_menger_theorem	theorem	Menger's theorem		Imported from Phase B brief catalog area D.
s_maxflow_mincut	theorem	Max-flow min-cut theorem		Imported from Phase B brief catalog area D.
s_kuratowski_planarity	theorem	Kuratowski's theorem		Imported from Phase B brief catalog area D.
s_wagner_planarity	theorem	Wagner's theorem		Imported from Phase B brief catalog area D.
s_schur_ramsey	theorem	Schur's theorem (Ramsey)		Imported from Phase B brief catalog area D.
s_van_der_waerden	theorem	Van der Waerden's theorem		Imported from Phase B brief catalog area D.
s_hales_jewett	theorem	Hales–Jewett theorem		Imported from Phase B brief catalog area D.
s_hindman_theorem	theorem	Hindman's theorem		Imported from Phase B brief catalog area D.
s_cayley_formula	theorem	Cayley's formula		Imported from Phase B brief catalog area D.
s_matrix_tree	theorem	Matrix-tree theorem		Imported from Phase B brief catalog area D.
s_best_theorem	theorem	BEST theorem		Imported from Phase B brief catalog area D.
s_burnside_lemma	theorem	Burnside's lemma		Imported from Phase B brief catalog area D.
s_polya_enumeration	theorem	Pólya enumeration theorem		Imported from Phase B brief catalog area D.
s_pigeonhole_principle	theorem	Pigeonhole principle		Imported from Phase B brief catalog area D.
s_fermat_two_square_theorem	theorem	Fermat's two-square theorem		Imported from Phase B brief catalog area E.
s_lagrange_four_square_theorem	theorem	Lagrange's four-square theorem		Imported from Phase B brief catalog area E.
s_legendre_three_square_theorem	theorem	Legendre's three-square theorem		Imported from Phase B brief catalog area E.
s_bertrand_postulate	theorem	Bertrand's postulate		Imported from Phase B brief catalog area E.
s_dirichlet_primes_in_ap	theorem	Dirichlet's theorem on primes in arithmetic progressions		Imported from Phase B brief catalog area E.
s_dirichlet_unit_theorem	theorem	Dirichlet's unit theorem		Imported from Phase B brief catalog area E.
s_minkowski_convex_body_theorem	theorem	Minkowski's theorem on convex bodies		Imported from Phase B brief catalog area E.
s_hilbert_waring_theorem	theorem	Hilbert–Waring theorem		Imported from Phase B brief catalog area E.
s_thue_theorem	theorem	Thue's theorem		Imported from Phase B brief catalog area E.
s_roth_approximation_theorem	theorem	Roth's theorem		Imported from Phase B brief catalog area E.
s_siegel_integral_points_theorem	theorem	Siegel's theorem on integral points		Imported from Phase B brief catalog area E.
s_mordell_weil_theorem	theorem	Mordell–Weil theorem		Imported from Phase B brief catalog area E.
s_hasse_minkowski_theorem	theorem	Hasse–Minkowski theorem		Imported from Phase B brief catalog area E.
s_hasse_elliptic_bound	theorem	Hasse's theorem on elliptic curves		Imported from Phase B brief catalog area E.
s_weil_conjectures_deligne	theorem	Weil conjectures (Deligne's theorem)		Imported from Phase B brief catalog area E.
s_chebotarev_density_theorem	theorem	Chebotarev density theorem		Imported from Phase B brief catalog area E.
s_class_field_theory_main	theorem	Main theorem of class field theory		Imported from Phase B brief catalog area E.
s_artin_reciprocity_law	theorem	Artin reciprocity law		Imported from Phase B brief catalog area E.
s_erdos_ginzburg_ziv_theorem	theorem	Erdős–Ginzburg–Ziv theorem		Imported from Phase B brief catalog area E.
s_chen_theorem	theorem	Chen's theorem		Imported from Phase B brief catalog area E.
s_vinogradov_three_primes_theorem	theorem	Vinogradov's theorem		Imported from Phase B brief catalog area E.
s_eisenstein_criterion	theorem	Eisenstein's criterion		Imported from Phase B brief catalog area E.
s_gauss_lemma_polynomials	theorem	Gauss's lemma (polynomials)		Imported from Phase B brief catalog area E.
s_primitive_element_theorem	theorem	Primitive element theorem		Imported from Phase B brief catalog area E.
s_artin_schreier_theorem	theorem	Artin–Schreier theorem		Imported from Phase B brief catalog area E.
s_hilbert_theorem_90	theorem	Hilbert's theorem 90		Imported from Phase B brief catalog area E.
s_luroth_theorem	theorem	Lüroth's theorem		Imported from Phase B brief catalog area E.
s_nakayama_lemma	theorem	Nakayama's lemma		Imported from Phase B brief catalog area E.
s_krull_hauptidealsatz	theorem	Krull's principal ideal theorem (Hauptidealsatz)		Imported from Phase B brief catalog area E.
s_krull_intersection_theorem	theorem	Krull intersection theorem		Imported from Phase B brief catalog area E.
s_krull_akizuki_theorem	theorem	Krull–Akizuki theorem		Imported from Phase B brief catalog area E.
s_cohen_structure_theorem	theorem	Cohen structure theorem		Imported from Phase B brief catalog area E.
s_hilbert_syzygy_theorem	theorem	Hilbert's syzygy theorem		Imported from Phase B brief catalog area E.
s_auslander_buchsbaum_formula	theorem	Auslander–Buchsbaum formula		Imported from Phase B brief catalog area E.
s_going_up_going_down_theorem	theorem	Going-up and going-down theorems		Imported from Phase B brief catalog area E.
s_bezout_theorem	theorem	Bézout's theorem		Imported from Phase B brief catalog area E.
s_chow_theorem	theorem	Chow's theorem		Imported from Phase B brief catalog area E.
s_hironaka_resolution	theorem	Hironaka's resolution of singularities		Imported from Phase B brief catalog area E.
s_serre_duality	theorem	Serre duality		Imported from Phase B brief catalog area E.
s_grothendieck_riemann_roch	theorem	Grothendieck–Riemann–Roch		Imported from Phase B brief catalog area E.
s_zariski_main_theorem	theorem	Zariski's main theorem		Imported from Phase B brief catalog area E.
s_kodaira_vanishing_theorem	theorem	Kodaira vanishing theorem		Imported from Phase B brief catalog area E.
s_lefschetz_hyperplane_theorem	theorem	Lefschetz hyperplane theorem		Imported from Phase B brief catalog area E.
s_rank_nullity_theorem	theorem	Rank–nullity theorem		Imported from Phase B brief catalog area E.
s_spectral_theorem_finite_dim	theorem	Spectral theorem (finite dim)		Imported from Phase B brief catalog area E.
s_schur_decomposition	theorem	Schur decomposition		Imported from Phase B brief catalog area E.
s_jordan_normal_form	theorem	Jordan normal form		Imported from Phase B brief catalog area E.
s_singular_value_decomposition	theorem	Singular value decomposition		Imported from Phase B brief catalog area E.
s_perron_frobenius_theorem	theorem	Perron–Frobenius theorem		Imported from Phase B brief catalog area E.
s_gershgorin_circle_theorem	theorem	Gershgorin circle theorem		Imported from Phase B brief catalog area E.
s_sylvester_law_of_inertia	theorem	Sylvester's law of inertia		Imported from Phase B brief catalog area E.
s_gram_schmidt_process	theorem	Gram–Schmidt process		Imported from Phase B brief catalog area E.
s_cramer_rule	theorem	Cramer's rule		Imported from Phase B brief catalog area E.
s_cauchy_group_theorem	theorem	Cauchy's theorem (group theory)		Imported from Phase B brief catalog area E.
s_jordan_holder_theorem	theorem	Jordan–Hölder theorem		Imported from Phase B brief catalog area E.
s_krull_schmidt_theorem	theorem	Krull–Schmidt theorem		Imported from Phase B brief catalog area E.
s_feit_thompson_theorem	theorem	Feit–Thompson theorem		Imported from Phase B brief catalog area E.
s_schur_zassenhaus_theorem	theorem	Schur–Zassenhaus theorem		Imported from Phase B brief catalog area E.
s_fg_abelian_structure_theorem	theorem	Structure theorem for finitely generated abelian groups		Imported from Phase B brief catalog area E.
s_frattini_argument	theorem	Frattini's argument		Imported from Phase B brief catalog area E.
s_jordan_finite_linear_theorem	theorem	Jordan's theorem on finite linear groups		Imported from Phase B brief catalog area E.
s_burnside_paqb_theorem	theorem	Burnside's p^a q^b theorem		Imported from Phase B brief catalog area E.
s_hnn_theorem	theorem	Higman–Neumann–Neumann theorem		Imported from Phase B brief catalog area E.
s_maschke_theorem	theorem	Maschke's theorem		Imported from Phase B brief catalog area E.
s_schur_lemma	theorem	Schur's lemma		Imported from Phase B brief catalog area E.
s_wedderburn_finite_division_ring	theorem	Wedderburn's theorem (finite division rings)		Imported from Phase B brief catalog area E.
s_artin_wedderburn_theorem	theorem	Artin–Wedderburn theorem		Imported from Phase B brief catalog area E.
s_peter_weyl_theorem	theorem	Peter–Weyl theorem		Imported from Phase B brief catalog area E.
s_frobenius_reciprocity	theorem	Frobenius reciprocity		Imported from Phase B brief catalog area E.
s_heine_borel	theorem	Heine–Borel theorem		Imported from Phase B brief catalog area F.
s_urysohn_lemma	theorem	Urysohn's lemma		Imported from Phase B brief catalog area F.
s_normal_hausdorff_space	axiom	normal hausdorff space		Imported from Phase B brief catalog area F as axiom.
s_tietze_extension	theorem	Tietze extension theorem		Imported from Phase B brief catalog area F.
s_urysohn_metrization	theorem	Urysohn metrization theorem		Imported from Phase B brief catalog area F.
s_second_countable_regular_hausdorff_space	axiom	second countable regular hausdorff space		Imported from Phase B brief catalog area F as axiom.
s_alexandroff_compactification	theorem	Alexandroff compactification (one-point)		Imported from Phase B brief catalog area F.
s_locally_compact_hausdorff_space	axiom	locally compact hausdorff space		Imported from Phase B brief catalog area F as axiom.
s_borsuk_ulam	theorem	Borsuk–Ulam theorem		Imported from Phase B brief catalog area F.
s_continuous_map_Sn_to_Rn	axiom	continuous map Sn to Rn		Imported from Phase B brief catalog area F as axiom.
s_ham_sandwich	theorem	Ham sandwich theorem		Imported from Phase B brief catalog area F.
s_n_measurable_bodies_in_Rn	axiom	n measurable bodies in Rn		Imported from Phase B brief catalog area F as axiom.
s_jordan_curve_theorem	theorem	Jordan curve theorem		Imported from Phase B brief catalog area F.
s_simple_closed_curve_in_plane	axiom	simple closed curve in plane		Imported from Phase B brief catalog area F as axiom.
s_hairy_ball	theorem	Hairy ball theorem		Imported from Phase B brief catalog area F.
s_continuous_tangent_vector_field	axiom	continuous tangent vector field		Imported from Phase B brief catalog area F as axiom.
s_poincare_hopf	theorem	Poincaré–Hopf theorem		Imported from Phase B brief catalog area F.
s_invariance_of_domain	theorem	Brouwer's invariance of domain		Imported from Phase B brief catalog area F.
s_continuous_injection_between_Rn_opens	axiom	continuous injection between Rn opens		Imported from Phase B brief catalog area F as axiom.
s_de_rham_theorem	theorem	de Rham's theorem		Imported from Phase B brief catalog area F.
s_whitney_embedding	theorem	Whitney embedding theorem		Imported from Phase B brief catalog area F.
s_nash_embedding	theorem	Nash embedding theorem		Imported from Phase B brief catalog area F.
s_mostow_rigidity	theorem	Mostow rigidity theorem		Imported from Phase B brief catalog area F.
s_closed_hyperbolic_manifold_dim_ge_3	axiom	closed hyperbolic manifold dim ge 3		Imported from Phase B brief catalog area F as axiom.
s_h_cobordism	theorem	Smale's h-cobordism theorem		Imported from Phase B brief catalog area F.
s_simply_connected_manifold_dim_ge_5	axiom	simply connected manifold dim ge 5		Imported from Phase B brief catalog area F as axiom.
s_h_cobordism_between_manifolds	axiom	h cobordism between manifolds		Imported from Phase B brief catalog area F as axiom.
s_freedman_theorem	theorem	Freedman's theorem		Imported from Phase B brief catalog area F.
s_topological_4_manifold	axiom	topological 4 manifold		Imported from Phase B brief catalog area F as axiom.
s_donaldson_theorem	theorem	Donaldson's theorem		Imported from Phase B brief catalog area F.
s_smooth_4_manifold	axiom	smooth 4 manifold		Imported from Phase B brief catalog area F as axiom.
s_alexander_duality	theorem	Alexander duality		Imported from Phase B brief catalog area F.
s_subspace_of_Sn	axiom	subspace of Sn		Imported from Phase B brief catalog area F as axiom.
s_poincare_duality	theorem	Poincaré duality		Imported from Phase B brief catalog area F.
s_closed_oriented_n_manifold	axiom	closed oriented n manifold		Imported from Phase B brief catalog area F as axiom.
s_hopf_rinow	theorem	Hopf–Rinow theorem		Imported from Phase B brief catalog area F.
s_connected_riemannian_manifold	axiom	connected riemannian manifold		Imported from Phase B brief catalog area F as axiom.
s_cartan_hadamard	theorem	Cartan–Hadamard theorem		Imported from Phase B brief catalog area F.
s_complete_simply_connected_nonpositive_curvature_manifold	axiom	complete simply connected nonpositive curvature manifold		Imported from Phase B brief catalog area F as axiom.
s_myers_theorem	theorem	Myers's theorem		Imported from Phase B brief catalog area F.
s_ricci_curvature_positive_lower_bound	axiom	ricci curvature positive lower bound		Imported from Phase B brief catalog area F as axiom.
s_hodge_theorem	theorem	Hodge theorem		Imported from Phase B brief catalog area F.
s_compact_riemannian_manifold	axiom	compact riemannian manifold		Imported from Phase B brief catalog area F as axiom.
s_chern_gauss_bonnet	theorem	Chern–Gauss–Bonnet theorem		Imported from Phase B brief catalog area F.
s_uniformization_theorem	theorem	Uniformization theorem		Imported from Phase B brief catalog area F.
s_simply_connected_riemann_surface	axiom	simply connected riemann surface		Imported from Phase B brief catalog area F as axiom.
s_monotone_convergence_theorem	theorem	Monotone convergence theorem		Imported from Phase B brief catalog area G.
s_fatou_lemma	theorem	Fatou's lemma		Imported from Phase B brief catalog area G.
s_dominated_convergence_theorem	theorem	Dominated convergence theorem		Imported from Phase B brief catalog area G.
s_fubini_theorem	theorem	Fubini's theorem		Imported from Phase B brief catalog area G.
s_tonelli_theorem	theorem	Tonelli's theorem		Imported from Phase B brief catalog area G.
s_radon_nikodym_theorem	theorem	Radon–Nikodym theorem		Imported from Phase B brief catalog area G.
s_lebesgue_differentiation_theorem	theorem	Lebesgue differentiation theorem		Imported from Phase B brief catalog area G.
s_vitali_covering_lemma	theorem	Vitali covering lemma		Imported from Phase B brief catalog area G.
s_egorov_theorem	theorem	Egorov's theorem		Imported from Phase B brief catalog area G.
s_lusin_theorem	theorem	Lusin's theorem		Imported from Phase B brief catalog area G.
s_rademacher_theorem	theorem	Rademacher's theorem		Imported from Phase B brief catalog area G.
s_sard_theorem	theorem	Sard's theorem		Imported from Phase B brief catalog area G.
s_baire_category_theorem	theorem	Baire category theorem		Imported from Phase B brief catalog area G.
s_arzela_ascoli_theorem	theorem	Arzelà–Ascoli theorem		Imported from Phase B brief catalog area G.
s_picard_little_theorem	theorem	Picard's little theorem		Imported from Phase B brief catalog area G.
s_picard_great_theorem	theorem	Picard's great theorem		Imported from Phase B brief catalog area G.
s_casorati_weierstrass_theorem	theorem	Casorati–Weierstrass theorem		Imported from Phase B brief catalog area G.
s_residue_theorem	theorem	Residue theorem		Imported from Phase B brief catalog area G.
s_argument_principle	theorem	Argument principle		Imported from Phase B brief catalog area G.
s_rouche_theorem	theorem	Rouché's theorem		Imported from Phase B brief catalog area G.
s_morera_theorem	theorem	Morera's theorem		Imported from Phase B brief catalog area G.
s_schwarz_lemma	theorem	Schwarz lemma		Imported from Phase B brief catalog area G.
s_phragmen_lindelof_principle	theorem	Phragmén–Lindelöf principle		Imported from Phase B brief catalog area G.
s_mittag_leffler_theorem	theorem	Mittag-Leffler theorem		Imported from Phase B brief catalog area G.
s_weierstrass_factorization_theorem	theorem	Weierstrass factorization theorem		Imported from Phase B brief catalog area G.
s_montel_theorem	theorem	Montel's theorem		Imported from Phase B brief catalog area G.
s_hurwitz_theorem	theorem	Hurwitz's theorem		Imported from Phase B brief catalog area G.
s_bloch_theorem	theorem	Bloch's theorem		Imported from Phase B brief catalog area G.
s_gauss_lucas_theorem	theorem	Gauss–Lucas theorem		Imported from Phase B brief catalog area G.
s_banach_steinhaus_theorem	theorem	Banach–Steinhaus theorem		Imported from Phase B brief catalog area G.
s_open_mapping_theorem	theorem	Open mapping theorem		Imported from Phase B brief catalog area G.
s_closed_graph_theorem	theorem	Closed graph theorem		Imported from Phase B brief catalog area G.
s_banach_alaoglu_theorem	theorem	Banach–Alaoglu theorem		Imported from Phase B brief catalog area G.
s_krein_milman_theorem	theorem	Kreĭn–Milman theorem		Imported from Phase B brief catalog area G.
s_riesz_representation_theorems	theorem	Riesz representation theorems		Imported from Phase B brief catalog area G.
s_spectral_theorem_self_adjoint	theorem	Spectral theorem for self-adjoint operators		Imported from Phase B brief catalog area G.
s_gelfand_naimark_theorem	theorem	Gelfand–Naimark theorem		Imported from Phase B brief catalog area G.
s_lax_milgram_lemma	theorem	Lax–Milgram lemma		Imported from Phase B brief catalog area G.
s_sobolev_embedding_theorem	theorem	Sobolev embedding theorem		Imported from Phase B brief catalog area G.
s_rellich_kondrachov_theorem	theorem	Rellich–Kondrachov theorem		Imported from Phase B brief catalog area G.
s_stone_one_parameter_unitary_theorem	theorem	Stone's theorem on one-parameter unitary groups		Imported from Phase B brief catalog area G.
s_riesz_fischer_theorem	theorem	Riesz–Fischer theorem		Imported from Phase B brief catalog area G.
s_plancherel_theorem	theorem	Plancherel theorem		Imported from Phase B brief catalog area G.
s_riesz_thorin_interpolation_theorem	theorem	Riesz–Thorin interpolation theorem		Imported from Phase B brief catalog area G.
s_marcinkiewicz_interpolation_theorem	theorem	Marcinkiewicz interpolation theorem		Imported from Phase B brief catalog area G.
s_carleson_theorem	theorem	Carleson's theorem		Imported from Phase B brief catalog area G.
s_hardy_littlewood_maximal_inequality	theorem	Hardy–Littlewood maximal inequality		Imported from Phase B brief catalog area G.
s_picard_lindelof_theorem	theorem	Picard–Lindelöf theorem		Imported from Phase B brief catalog area G.
s_peano_existence_theorem	theorem	Peano existence theorem		Imported from Phase B brief catalog area G.
s_cauchy_kovalevskaya_theorem	theorem	Cauchy–Kovalevskaya theorem		Imported from Phase B brief catalog area G.
s_frobenius_theorem_distributions	theorem	Frobenius theorem (DE / distributions)		Imported from Phase B brief catalog area G.
s_poincare_bendixson_theorem	theorem	Poincaré–Bendixson theorem		Imported from Phase B brief catalog area G.
s_hartman_grobman_theorem	theorem	Hartman–Grobman theorem		Imported from Phase B brief catalog area G.
s_kam_theorem	theorem	KAM theorem		Imported from Phase B brief catalog area G.
s_kolmogorov_zero_one_law	theorem	Kolmogorov's 0–1 law		Imported from Phase B brief catalog area H.
s_kolmogorov_extension_theorem	theorem	Kolmogorov's extension theorem		Imported from Phase B brief catalog area H.
s_borel_cantelli_lemma	theorem	Borel–Cantelli lemma		Imported from Phase B brief catalog area H.
s_doob_martingale_convergence	theorem	Doob's martingale convergence theorem		Imported from Phase B brief catalog area H.
s_doob_optional_stopping	theorem	Doob's optional stopping theorem		Imported from Phase B brief catalog area H.
s_ito_lemma	theorem	Itô's lemma		Imported from Phase B brief catalog area H.
s_feynman_kac_formula	theorem	Feynman–Kac formula		Imported from Phase B brief catalog area H.
s_girsanov_theorem	theorem	Girsanov's theorem		Imported from Phase B brief catalog area H.
s_glivenko_cantelli	theorem	Glivenko–Cantelli theorem		Imported from Phase B brief catalog area H.
s_donsker_theorem	theorem	Donsker's theorem		Imported from Phase B brief catalog area H.
s_lindeberg_feller_clt	theorem	Lindeberg–Feller central limit theorem		Imported from Phase B brief catalog area H.
s_sharkovskii_theorem	theorem	Sharkovskii's theorem		Imported from Phase B brief catalog area H.
s_oseledets_multiplicative_ergodic	theorem	Oseledets multiplicative ergodic theorem		Imported from Phase B brief catalog area H.
s_yoneda_lemma	theorem	Yoneda lemma		Imported from Phase B brief catalog area H.
s_diagram_in_C	axiom	diagram in C		Imported from Phase B brief catalog area H as axiom.
s_adjoint_functor_theorem	theorem	Adjoint functor theorem		Imported from Phase B brief catalog area H.
s_rice_theorem	theorem	Rice's theorem		Imported from Phase B brief catalog area H.
s_cook_levin_theorem	theorem	Cook–Levin theorem		Imported from Phase B brief catalog area H.
s_time_hierarchy_theorem	theorem	Time hierarchy theorem		Imported from Phase B brief catalog area H.
s_space_hierarchy_theorem	theorem	Space hierarchy theorem		Imported from Phase B brief catalog area H.
s_ladner_theorem	theorem	Ladner's theorem		Imported from Phase B brief catalog area H.
s_immerman_szelepcsenyi	theorem	Immerman–Szelepcsényi theorem		Imported from Phase B brief catalog area H.
s_pcp_theorem	theorem	PCP theorem		Imported from Phase B brief catalog area H.
s_godel_speed_up_theorem	theorem	Gödel speed-up theorem		Imported from Phase B brief catalog area H.
s_savitch_theorem	theorem	Savitch's theorem		Imported from Phase B brief catalog area H.
s_shannon_source_coding	theorem	Shannon's source coding theorem		Imported from Phase B brief catalog area H.
s_shannon_noisy_channel	theorem	Shannon's noisy-channel coding theorem		Imported from Phase B brief catalog area H.
s_shannon_hartley_theorem	theorem	Shannon–Hartley theorem		Imported from Phase B brief catalog area H.
s_kraft_inequality	theorem	Kraft's inequality		Imported from Phase B brief catalog area H.
s_cpt_theorem	theorem	CPT theorem		Imported from Phase B brief catalog area H.
s_spin_statistics_theorem	theorem	Spin-statistics theorem		Imported from Phase B brief catalog area H.
s_haag_theorem	theorem	Haag's theorem		Imported from Phase B brief catalog area H.
s_noether_second_theorem	theorem	Noether's second theorem		Imported from Phase B brief catalog area H.
s_reeh_schlieder_theorem	theorem	Reeh–Schlieder theorem		Imported from Phase B brief catalog area H.
s_bell_theorem	theorem	Bell's theorem		Imported from Phase B brief catalog area H.
s_kochen_specker_theorem	theorem	Kochen–Specker theorem		Imported from Phase B brief catalog area H.
s_von_neumann_minimax	theorem	Von Neumann's minimax theorem		Imported from Phase B brief catalog area H.
s_nash_existence_theorem	theorem	Nash's existence theorem		Imported from Phase B brief catalog area H.
s_arrow_impossibility	theorem	Arrow's impossibility theorem		Imported from Phase B brief catalog area H.
s_gibbard_satterthwaite	theorem	Gibbard–Satterthwaite theorem		Imported from Phase B brief catalog area H.
s_finite_voting_profiles	axiom	Finite voting profiles		Ballots: each voter supplies a total order over a finite candidate set.
s_cauchy_theorem	theorem	Cauchy's theorem		Iter-3 imported theorem (s_cauchy_theorem).
s_cauchy_p_tuple_set	state	cauchy p tuple set		Iter-3 imported state (s_cauchy_p_tuple_set).
s_cyclic_action_on_p_tuples	state	cyclic action on p tuples		Iter-3 imported state (s_cyclic_action_on_p_tuples).
s_fixed_point_count_divisible_by_p	state	fixed point count divisible by p		Iter-3 imported state (s_fixed_point_count_divisible_by_p).
s_class_equation	theorem	Class equation		Iter-3 imported theorem (s_class_equation).
s_conjugation_action_on_G	state	conjugation action on G		Iter-3 imported state (s_conjugation_action_on_G).
s_orbit_stabilizer_for_conjugation	state	orbit stabilizer for conjugation		Iter-3 imported state (s_orbit_stabilizer_for_conjugation).
s_lagrange_theorem	theorem	Lagrange's theorem		Iter-3 imported theorem (s_lagrange_theorem).
s_subgroup_H	axiom	subgroup H		Iter-3 imported axiom (s_subgroup_H).
s_coset_partition_of_G	state	coset partition of G		Iter-3 imported state (s_coset_partition_of_G).
s_cosets_equinumerous_to_H	state	cosets equinumerous to H		Iter-3 imported state (s_cosets_equinumerous_to_H).
s_orbit_stabilizer_theorem	theorem	Orbit-stabilizer theorem		Iter-3 imported theorem (s_orbit_stabilizer_theorem).
s_coset_to_orbit_map	state	coset to orbit map		Iter-3 imported state (s_coset_to_orbit_map).
s_orbit_isomorphic_to_coset_space	state	orbit isomorphic to coset space		Iter-3 imported state (s_orbit_isomorphic_to_coset_space).
s_burnside_counting_lemma	theorem	Burnside's lemma		Iter-3 imported theorem (s_burnside_counting_lemma).
s_fix_incidence_set	state	fix incidence set		Iter-3 imported state (s_fix_incidence_set).
s_double_count_fix_stab	state	double count fix stab		Iter-3 imported state (s_double_count_fix_stab).
s_burnside_paqb_solvable	theorem	Burnside's p^a q^b solvability theorem		Iter-3 imported theorem (s_burnside_paqb_solvable).
s_finite_group_with_order_paqb	axiom	finite group with order paqb		Iter-3 imported axiom (s_finite_group_with_order_paqb).
s_hypothetical_simple_paqb_group	state	hypothetical simple paqb group		Iter-3 imported state (s_hypothetical_simple_paqb_group).
s_character_table_of_G	state	character table of G		Iter-3 imported state (s_character_table_of_G).
s_algebraic_integer_relation	state	algebraic integer relation		Iter-3 imported state (s_algebraic_integer_relation).
s_central_element_exists	state	central element exists		Iter-3 imported state (s_central_element_exists).
s_schreier_refinement_theorem	theorem	Schreier refinement theorem		Iter-3 imported theorem (s_schreier_refinement_theorem).
s_group_with_two_subnormal_series	axiom	group with two subnormal series		Iter-3 imported axiom (s_group_with_two_subnormal_series).
s_zassenhaus_refined_terms	state	zassenhaus refined terms		Iter-3 imported state (s_zassenhaus_refined_terms).
s_isomorphic_factor_pairs	state	isomorphic factor pairs		Iter-3 imported state (s_isomorphic_factor_pairs).
s_fundamental_theorem_fg_abelian	theorem	Fundamental theorem of finitely generated abelian groups		Iter-3 imported theorem (s_fundamental_theorem_fg_abelian).
s_finitely_generated_abelian_group	axiom	finitely generated abelian group		Iter-3 imported axiom (s_finitely_generated_abelian_group).
s_presentation_matrix_A	state	presentation matrix A		Iter-3 imported state (s_presentation_matrix_A).
s_smith_normal_form_of_A	state	smith normal form of A		Iter-3 imported state (s_smith_normal_form_of_A).
s_invariant_factor_decomposition	state	invariant factor decomposition		Iter-3 imported state (s_invariant_factor_decomposition).
s_pid_structure_theorem	theorem	Structure theorem for f.g. modules over a PID		Iter-3 imported theorem (s_pid_structure_theorem).
s_principal_ideal_domain	axiom	principal ideal domain		Iter-3 imported axiom (s_principal_ideal_domain).
s_finitely_generated_module_over_pid	axiom	finitely generated module over pid		Iter-3 imported axiom (s_finitely_generated_module_over_pid).
s_module_presentation_over_R	state	module presentation over R		Iter-3 imported state (s_module_presentation_over_R).
s_smith_form_module_presentation	state	smith form module presentation		Iter-3 imported state (s_smith_form_module_presentation).
s_invariant_factor_module_decomposition	state	invariant factor module decomposition		Iter-3 imported state (s_invariant_factor_module_decomposition).
s_smith_normal_form_theorem	theorem	Smith normal form		Iter-3 imported theorem (s_smith_normal_form_theorem).
s_matrix_over_pid	axiom	matrix over pid		Iter-3 imported axiom (s_matrix_over_pid).
s_pivot_reduction_sequence	state	pivot reduction sequence		Iter-3 imported state (s_pivot_reduction_sequence).
s_descent_on_ideal_chain	state	descent on ideal chain		Iter-3 imported state (s_descent_on_ideal_chain).
s_cayley_theorem	theorem	Cayley's theorem		Iter-3 imported theorem (s_cayley_theorem).
s_symmetric_group_Sn	axiom	symmetric group Sn		Iter-3 imported axiom (s_symmetric_group_Sn).
s_left_regular_representation	state	left regular representation		Iter-3 imported state (s_left_regular_representation).
s_nielsen_schreier_theorem	theorem	Nielsen–Schreier theorem		Iter-3 imported theorem (s_nielsen_schreier_theorem).
s_free_group_F	axiom	free group F		Iter-3 imported axiom (s_free_group_F).
s_cayley_tree_of_F	state	cayley tree of F		Iter-3 imported state (s_cayley_tree_of_F).
s_quotient_graph_by_H_action	state	quotient graph by H action		Iter-3 imported state (s_quotient_graph_by_H_action).
s_pi1_of_quotient_is_free	state	pi1 of quotient is free		Iter-3 imported state (s_pi1_of_quotient_is_free).
s_kurosh_subgroup_theorem	theorem	Kurosh subgroup theorem		Iter-3 imported theorem (s_kurosh_subgroup_theorem).
s_free_product_of_groups	axiom	free product of groups		Iter-3 imported axiom (s_free_product_of_groups).
s_bass_serre_tree_for_free_product	state	bass serre tree for free product		Iter-3 imported state (s_bass_serre_tree_for_free_product).
s_H_action_on_bass_serre_tree	state	H action on bass serre tree		Iter-3 imported state (s_H_action_on_bass_serre_tree).
s_grushko_decomposition_theorem	theorem	Grushko decomposition theorem		Iter-3 imported theorem (s_grushko_decomposition_theorem).
s_finitely_generated_group_G	axiom	finitely generated group G		Iter-3 imported axiom (s_finitely_generated_group_G).
s_generating_set_of_free_product	state	generating set of free product		Iter-3 imported state (s_generating_set_of_free_product).
s_folded_graph_for_generators	state	folded graph for generators		Iter-3 imported state (s_folded_graph_for_generators).
s_stallings_ends_theorem	theorem	Stallings theorem on ends of groups		Iter-3 imported theorem (s_stallings_ends_theorem).
s_cayley_graph_of_G	axiom	cayley graph of G		Iter-3 imported axiom (s_cayley_graph_of_G).
s_ends_invariant_of_G	state	ends invariant of G		Iter-3 imported state (s_ends_invariant_of_G).
s_end_count_classification	state	end count classification		Iter-3 imported state (s_end_count_classification).
s_G_acts_on_tree_with_finite_edge_stabilizers	state	G acts on tree with finite edge stabilizers		Iter-3 imported state (s_G_acts_on_tree_with_finite_edge_stabilizers).
s_gromov_polynomial_growth_theorem	theorem	Gromov's theorem on groups of polynomial growth		Iter-3 imported theorem (s_gromov_polynomial_growth_theorem).
s_polynomial_growth_function	axiom	polynomial growth function		Iter-3 imported axiom (s_polynomial_growth_function).
s_asymptotic_cone_of_G	state	asymptotic cone of G		Iter-3 imported state (s_asymptotic_cone_of_G).
s_lie_group_acting_on_cone	state	lie group acting on cone		Iter-3 imported state (s_lie_group_acting_on_cone).
s_tits_alternative_theorem	theorem	Tits alternative		Iter-3 imported theorem (s_tits_alternative_theorem).
s_finitely_generated_linear_group	axiom	finitely generated linear group		Iter-3 imported axiom (s_finitely_generated_linear_group).
s_field_of_characteristic_zero	axiom	field of characteristic zero		Iter-3 imported axiom (s_field_of_characteristic_zero).
s_zariski_closure_G	state	zariski closure G		Iter-3 imported state (s_zariski_closure_G).
s_semisimple_or_solvable_part	state	semisimple or solvable part		Iter-3 imported state (s_semisimple_or_solvable_part).
s_free_subgroup_via_ping_pong	state	free subgroup via ping pong		Iter-3 imported state (s_free_subgroup_via_ping_pong).
s_fitting_theorem	theorem	Fitting's theorem		Iter-3 imported theorem (s_fitting_theorem).
s_nilpotent_normal_subgroup	axiom	nilpotent normal subgroup		Iter-3 imported axiom (s_nilpotent_normal_subgroup).
s_product_of_two_nilpotent_normals	state	product of two nilpotent normals		Iter-3 imported state (s_product_of_two_nilpotent_normals).
s_nilpotency_class_bound	state	nilpotency class bound		Iter-3 imported state (s_nilpotency_class_bound).
s_first_isomorphism_theorem	theorem	Fundamental theorem on homomorphisms / first isomorphism theorem		Iter-3 imported theorem (s_first_isomorphism_theorem).
s_group_homomorphism	axiom	group homomorphism		Iter-3 imported axiom (s_group_homomorphism).
s_kernel_image_of_phi	state	kernel image of phi		Iter-3 imported state (s_kernel_image_of_phi).
s_induced_quotient_map	state	induced quotient map		Iter-3 imported state (s_induced_quotient_map).
s_second_isomorphism_theorem	theorem	Second isomorphism theorem (diamond)		Iter-3 imported theorem (s_second_isomorphism_theorem).
s_normal_subgroup_N	axiom	normal subgroup N		Iter-3 imported axiom (s_normal_subgroup_N).
s_HN_and_intersection	state	HN and intersection		Iter-3 imported state (s_HN_and_intersection).
s_diamond_iso_map	state	diamond iso map		Iter-3 imported state (s_diamond_iso_map).
s_third_isomorphism_theorem	theorem	Third isomorphism theorem		Iter-3 imported theorem (s_third_isomorphism_theorem).
s_normal_subgroup_K	axiom	normal subgroup K		Iter-3 imported axiom (s_normal_subgroup_K).
s_nested_normal_chain	state	nested normal chain		Iter-3 imported state (s_nested_normal_chain).
s_correspondence_theorem	theorem	Correspondence theorem (lattice isomorphism)		Iter-3 imported theorem (s_correspondence_theorem).
s_quotient_group_G_mod_N	axiom	quotient group G mod N		Iter-3 imported axiom (s_quotient_group_G_mod_N).
s_subgroup_correspondence_map	state	subgroup correspondence map		Iter-3 imported state (s_subgroup_correspondence_map).
s_schurs_lemma	theorem	Schur's lemma		Iter-3 imported theorem (s_schurs_lemma).
s_simple_module	axiom	simple module		Iter-3 imported axiom (s_simple_module).
s_module_homomorphism	axiom	module homomorphism		Iter-3 imported axiom (s_module_homomorphism).
s_kernel_image_submodules	state	kernel image submodules		Iter-3 imported state (s_kernel_image_submodules).
s_simplicity_dichotomy	state	simplicity dichotomy		Iter-3 imported state (s_simplicity_dichotomy).
s_wedderburn_artin_theorem	theorem	Wedderburn–Artin theorem		Iter-3 imported theorem (s_wedderburn_artin_theorem).
s_semisimple_artinian_ring	axiom	semisimple artinian ring		Iter-3 imported axiom (s_semisimple_artinian_ring).
s_division_ring	axiom	division ring		Iter-3 imported axiom (s_division_ring).
s_simple_module_decomp_of_R	state	simple module decomp of R		Iter-3 imported state (s_simple_module_decomp_of_R).
s_endomorphism_ring_product	state	endomorphism ring product		Iter-3 imported state (s_endomorphism_ring_product).
s_matrix_rings_over_division_rings	state	matrix rings over division rings		Iter-3 imported state (s_matrix_rings_over_division_rings).
s_jacobson_density_theorem	theorem	Jacobson density theorem		Iter-3 imported theorem (s_jacobson_density_theorem).
s_M_as_D_vector_space	state	M as D vector space		Iter-3 imported state (s_M_as_D_vector_space).
s_interpolation_problem_in_M	state	interpolation problem in M		Iter-3 imported state (s_interpolation_problem_in_M).
s_double_centralizer_theorem	theorem	Double centralizer theorem		Iter-3 imported theorem (s_double_centralizer_theorem).
s_central_simple_algebra	axiom	central simple algebra		Iter-3 imported axiom (s_central_simple_algebra).
s_simple_subalgebra_B	axiom	simple subalgebra B		Iter-3 imported axiom (s_simple_subalgebra_B).
s_B_otimes_Aop_action	state	B otimes Aop action		Iter-3 imported state (s_B_otimes_Aop_action).
s_centralizer_dimension_formula	state	centralizer dimension formula		Iter-3 imported state (s_centralizer_dimension_formula).
s_skolem_noether_theorem	theorem	Skolem–Noether theorem		Iter-3 imported theorem (s_skolem_noether_theorem).
s_pair_of_B_embeddings	state	pair of B embeddings		Iter-3 imported state (s_pair_of_B_embeddings).
s_bimodule_isomorphism	state	bimodule isomorphism		Iter-3 imported state (s_bimodule_isomorphism).
s_wedderburn_little_theorem	theorem	Wedderburn's little theorem		Iter-3 imported theorem (s_wedderburn_little_theorem).
s_finite_division_ring	axiom	finite division ring		Iter-3 imported axiom (s_finite_division_ring).
s_assumed_noncommutative_D	state	assumed noncommutative D		Iter-3 imported state (s_assumed_noncommutative_D).
s_class_equation_for_D_units	state	class equation for D units		Iter-3 imported state (s_class_equation_for_D_units).
s_cyclotomic_divisibility_obstruction	state	cyclotomic divisibility obstruction		Iter-3 imported state (s_cyclotomic_divisibility_obstruction).
s_brauer_group_theorem	theorem	Brauer group periodicity		Iter-3 imported theorem (s_brauer_group_theorem).
s_field_K	axiom	field K		Iter-3 imported axiom (s_field_K).
s_brauer_equivalence_classes	state	brauer equivalence classes		Iter-3 imported state (s_brauer_equivalence_classes).
s_brauer_group_structure	state	brauer group structure		Iter-3 imported state (s_brauer_group_structure).
s_hilbert_90_theorem	theorem	Hilbert's theorem 90		Iter-3 imported theorem (s_hilbert_90_theorem).
s_cyclic_galois_extension	axiom	cyclic galois extension		Iter-3 imported axiom (s_cyclic_galois_extension).
s_norm_one_element	state	norm one element		Iter-3 imported state (s_norm_one_element).
s_h1_vanishing	state	h1 vanishing		Iter-3 imported state (s_h1_vanishing).
s_kummer_theory_theorem	theorem	Kummer theory		Iter-3 imported theorem (s_kummer_theory_theorem).
s_field_containing_nth_roots_of_unity	axiom	field containing nth roots of unity		Iter-3 imported axiom (s_field_containing_nth_roots_of_unity).
s_cyclic_extension_of_degree_n	axiom	cyclic extension of degree n		Iter-3 imported axiom (s_cyclic_extension_of_degree_n).
s_kummer_pairing	state	kummer pairing		Iter-3 imported state (s_kummer_pairing).
s_kummer_cohomology_iso	state	kummer cohomology iso		Iter-3 imported state (s_kummer_cohomology_iso).
s_isomorphism_extension_theorem	theorem	Isomorphism extension theorem		Iter-3 imported theorem (s_isomorphism_extension_theorem).
s_partial_extension_family	state	partial extension family		Iter-3 imported state (s_partial_extension_family).
s_maximal_extension_via_zorn	state	maximal extension via zorn		Iter-3 imported state (s_maximal_extension_via_zorn).
s_steinitz_algebraic_closure_theorem	theorem	Steinitz's theorem on field extensions		Iter-3 imported theorem (s_steinitz_algebraic_closure_theorem).
s_algebraic_closure	axiom	algebraic closure		Iter-3 imported axiom (s_algebraic_closure).
s_algebraic_extensions_poset	state	algebraic extensions poset		Iter-3 imported state (s_algebraic_extensions_poset).
s_maximal_algebraic_extension	state	maximal algebraic extension		Iter-3 imported state (s_maximal_algebraic_extension).
s_frobenius_reciprocity_theorem	theorem	Frobenius reciprocity		Iter-3 imported theorem (s_frobenius_reciprocity_theorem).
s_ind_res_functor_pair	state	ind res functor pair		Iter-3 imported state (s_ind_res_functor_pair).
s_ind_res_adjunction	state	ind res adjunction		Iter-3 imported state (s_ind_res_adjunction).
s_mackey_decomposition_theorem	theorem	Mackey decomposition / double coset formula		Iter-3 imported theorem (s_mackey_decomposition_theorem).
s_double_coset_decomposition	state	double coset decomposition		Iter-3 imported state (s_double_coset_decomposition).
s_restriction_per_double_coset	state	restriction per double coset		Iter-3 imported state (s_restriction_per_double_coset).
s_character_orthogonality_theorem	theorem	Character orthogonality relations		Iter-3 imported theorem (s_character_orthogonality_theorem).
s_irreducible_representation	axiom	irreducible representation		Iter-3 imported axiom (s_irreducible_representation).
s_intertwiner_operator	state	intertwiner operator		Iter-3 imported state (s_intertwiner_operator).
s_g_equivariant_operator	state	g equivariant operator		Iter-3 imported state (s_g_equivariant_operator).
s_schur_scalar_evaluation	state	schur scalar evaluation		Iter-3 imported state (s_schur_scalar_evaluation).
s_burnside_character_vanish_theorem	theorem	Burnside's theorem on character vanishing (zeros in character table)		Iter-3 imported theorem (s_burnside_character_vanish_theorem).
s_chi_is_sum_of_roots_of_unity	state	chi is sum of roots of unity		Iter-3 imported state (s_chi_is_sum_of_roots_of_unity).
s_size_bound_on_chi_ratio	state	size bound on chi ratio		Iter-3 imported state (s_size_bound_on_chi_ratio).
s_brauer_induced_character_theorem	theorem	Brauer's theorem on induced characters		Iter-3 imported theorem (s_brauer_induced_character_theorem).
s_irreducible_character	axiom	irreducible character		Iter-3 imported axiom (s_irreducible_character).
s_virtual_character_ring	state	virtual character ring		Iter-3 imported state (s_virtual_character_ring).
s_elementary_subgroup_generators	state	elementary subgroup generators		Iter-3 imported state (s_elementary_subgroup_generators).
s_artin_induction_theorem	theorem	Artin's induction theorem		Iter-3 imported theorem (s_artin_induction_theorem).
s_cyclic_subgroup	axiom	cyclic subgroup		Iter-3 imported axiom (s_cyclic_subgroup).
s_induced_from_cyclic_subgroups	state	induced from cyclic subgroups		Iter-3 imported state (s_induced_from_cyclic_subgroups).
s_rational_combination_from_cyclics	state	rational combination from cyclics		Iter-3 imported state (s_rational_combination_from_cyclics).
s_weyl_complete_reducibility_theorem	theorem	Weyl's theorem on complete reducibility		Iter-3 imported theorem (s_weyl_complete_reducibility_theorem).
s_semisimple_lie_algebra	axiom	semisimple lie algebra		Iter-3 imported axiom (s_semisimple_lie_algebra).
s_finite_dim_representation	axiom	finite dim representation		Iter-3 imported axiom (s_finite_dim_representation).
s_casimir_central_element	state	casimir central element		Iter-3 imported state (s_casimir_central_element).
s_casimir_eigenspace_decomposition	state	casimir eigenspace decomposition		Iter-3 imported state (s_casimir_eigenspace_decomposition).
s_highest_weight_theorem	theorem	Theorem of the highest weight		Iter-3 imported theorem (s_highest_weight_theorem).
s_dominant_integral_weight	axiom	dominant integral weight		Iter-3 imported axiom (s_dominant_integral_weight).
s_borel_subalgebra_b	state	borel subalgebra b		Iter-3 imported state (s_borel_subalgebra_b).
s_verma_module_M_lambda	state	verma module M lambda		Iter-3 imported state (s_verma_module_M_lambda).
s_unique_irreducible_quotient_L_lambda	state	unique irreducible quotient L lambda		Iter-3 imported state (s_unique_irreducible_quotient_L_lambda).
s_borel_weil_bott_theorem	theorem	Borel–Weil–Bott theorem		Iter-3 imported theorem (s_borel_weil_bott_theorem).
s_semisimple_complex_lie_group	axiom	semisimple complex lie group		Iter-3 imported axiom (s_semisimple_complex_lie_group).
s_flag_variety_G_over_B	axiom	flag variety G over B		Iter-3 imported axiom (s_flag_variety_G_over_B).
s_line_bundle_L_lambda	state	line bundle L lambda		Iter-3 imported state (s_line_bundle_L_lambda).
s_sheaf_cohomology_of_L_lambda	state	sheaf cohomology of L lambda		Iter-3 imported state (s_sheaf_cohomology_of_L_lambda).
s_weyl_shift_to_dominant	state	weyl shift to dominant		Iter-3 imported state (s_weyl_shift_to_dominant).
s_lie_kolchin_theorem	theorem	Lie–Kolchin theorem		Iter-3 imported theorem (s_lie_kolchin_theorem).
s_connected_solvable_linear_algebraic_group	axiom	connected solvable linear algebraic group		Iter-3 imported axiom (s_connected_solvable_linear_algebraic_group).
s_g_action_on_flags	state	g action on flags		Iter-3 imported state (s_g_action_on_flags).
s_invariant_complete_flag	state	invariant complete flag		Iter-3 imported state (s_invariant_complete_flag).
s_engel_theorem	theorem	Engel's theorem		Iter-3 imported theorem (s_engel_theorem).
s_lie_algebra_over_field	axiom	lie algebra over field		Iter-3 imported axiom (s_lie_algebra_over_field).
s_nilpotent_endomorphism	axiom	nilpotent endomorphism		Iter-3 imported axiom (s_nilpotent_endomorphism).
s_all_adjoints_nilpotent	state	all adjoints nilpotent		Iter-3 imported state (s_all_adjoints_nilpotent).
s_common_zero_vector_constructed	state	common zero vector constructed		Iter-3 imported state (s_common_zero_vector_constructed).
s_cartan_dieudonne_theorem	theorem	Cartan–Dieudonné theorem		Iter-3 imported theorem (s_cartan_dieudonne_theorem).
s_quadratic_space_over_field	axiom	quadratic space over field		Iter-3 imported axiom (s_quadratic_space_over_field).
s_orthogonal_group_On	axiom	orthogonal group On		Iter-3 imported axiom (s_orthogonal_group_On).
s_reflection_generators	state	reflection generators		Iter-3 imported state (s_reflection_generators).
s_descent_on_fixed_vectors	state	descent on fixed vectors		Iter-3 imported state (s_descent_on_fixed_vectors).
s_crystallographic_restriction_theorem	theorem	Crystallographic restriction theorem		Iter-3 imported theorem (s_crystallographic_restriction_theorem).
s_lattice_in_euclidean_space	axiom	lattice in euclidean space		Iter-3 imported axiom (s_lattice_in_euclidean_space).
s_finite_rotation_subgroup	axiom	finite rotation subgroup		Iter-3 imported axiom (s_finite_rotation_subgroup).
s_integer_trace_constraint	state	integer trace constraint		Iter-3 imported state (s_integer_trace_constraint).
s_allowed_rotation_orders	state	allowed rotation orders		Iter-3 imported state (s_allowed_rotation_orders).
s_artin_rees_lemma	theorem	Artin–Rees lemma		Iter-3 imported theorem (s_artin_rees_lemma).
s_finitely_generated_module	axiom	finitely generated module		Iter-3 imported axiom (s_finitely_generated_module).
s_rees_algebra_construction	state	rees algebra construction		Iter-3 imported state (s_rees_algebra_construction).
s_rees_module_fg	state	rees module fg		Iter-3 imported state (s_rees_module_fg).
s_going_up_theorem	theorem	Going-up theorem (Cohen–Seidenberg)		Iter-3 imported theorem (s_going_up_theorem).
s_integral_ring_extension	axiom	integral ring extension		Iter-3 imported axiom (s_integral_ring_extension).
s_prime_ideal_chain	axiom	prime ideal chain		Iter-3 imported axiom (s_prime_ideal_chain).
s_partial_lift_to_B	state	partial lift to B		Iter-3 imported state (s_partial_lift_to_B).
s_integral_quotient_extension	state	integral quotient extension		Iter-3 imported state (s_integral_quotient_extension).
s_going_down_theorem	theorem	Going-down theorem		Iter-3 imported theorem (s_going_down_theorem).
s_integral_extension_of_integrally_closed_domains	axiom	integral extension of integrally closed domains		Iter-3 imported axiom (s_integral_extension_of_integrally_closed_domains).
s_localized_extension	state	localized extension		Iter-3 imported state (s_localized_extension).
s_galois_action_on_primes	state	galois action on primes		Iter-3 imported state (s_galois_action_on_primes).
s_krull_pit_theorem	theorem	Krull's principal ideal theorem (Hauptidealsatz)		Iter-3 imported theorem (s_krull_pit_theorem).
s_principal_ideal	axiom	principal ideal		Iter-3 imported axiom (s_principal_ideal).
s_minimal_prime_over_a	state	minimal prime over a		Iter-3 imported state (s_minimal_prime_over_a).
s_localized_at_p_setup	state	localized at p setup		Iter-3 imported state (s_localized_at_p_setup).
s_chain_stabilizes	state	chain stabilizes		Iter-3 imported state (s_chain_stabilizes).
s_krull_height_theorem	theorem	Krull dimension theorem / height theorem		Iter-3 imported theorem (s_krull_height_theorem).
s_ideal_with_n_generators	axiom	ideal with n generators		Iter-3 imported axiom (s_ideal_with_n_generators).
s_induction_setup_on_generators	state	induction setup on generators		Iter-3 imported state (s_induction_setup_on_generators).
s_pit_as_base_case	state	pit as base case		Iter-3 imported state (s_pit_as_base_case).
s_quillen_suslin_theorem	theorem	Quillen–Suslin theorem (Serre's conjecture)		Iter-3 imported theorem (s_quillen_suslin_theorem).
s_finitely_generated_projective_module	axiom	finitely generated projective module		Iter-3 imported axiom (s_finitely_generated_projective_module).
s_local_freeness_of_P	state	local freeness of P		Iter-3 imported state (s_local_freeness_of_P).
s_extended_module_certified	state	extended module certified		Iter-3 imported state (s_extended_module_certified).
s_dimension_reduction_via_suslin	state	dimension reduction via suslin		Iter-3 imported state (s_dimension_reduction_via_suslin).
s_auslander_buchsbaum_ufd_theorem	theorem	Auslander–Buchsbaum theorem (regular local ⇒ UFD)		Iter-3 imported theorem (s_auslander_buchsbaum_ufd_theorem).
s_regular_local_ring	axiom	regular local ring		Iter-3 imported axiom (s_regular_local_ring).
s_unique_factorization_domain	axiom	unique factorization domain		Iter-3 imported axiom (s_unique_factorization_domain).
s_height_one_prime_p	state	height one prime p		Iter-3 imported state (s_height_one_prime_p).
s_ext_class_vanishes	state	ext class vanishes		Iter-3 imported state (s_ext_class_vanishes).
s_height_one_principal	state	height one principal		Iter-3 imported state (s_height_one_principal).
s_hopkins_levitzki_theorem	theorem	Hopkins–Levitzki theorem		Iter-3 imported theorem (s_hopkins_levitzki_theorem).
s_artinian_ring	axiom	artinian ring		Iter-3 imported axiom (s_artinian_ring).
s_nilpotent_jacobson_radical	state	nilpotent jacobson radical		Iter-3 imported state (s_nilpotent_jacobson_radical).
s_semisimple_filtration_quotients	state	semisimple filtration quotients		Iter-3 imported state (s_semisimple_filtration_quotients).
s_goldie_theorem	theorem	Goldie's theorem		Iter-3 imported theorem (s_goldie_theorem).
s_semiprime_right_goldie_ring	axiom	semiprime right goldie ring		Iter-3 imported axiom (s_semiprime_right_goldie_ring).
s_classical_ring_of_quotients	axiom	classical ring of quotients		Iter-3 imported axiom (s_classical_ring_of_quotients).
s_ore_set_of_regular_elements	state	ore set of regular elements		Iter-3 imported state (s_ore_set_of_regular_elements).
s_classical_quotient_ring	state	classical quotient ring		Iter-3 imported state (s_classical_quotient_ring).
s_lasker_noether_theorem	theorem	Primary decomposition (Lasker–Noether)		Iter-3 imported theorem (s_lasker_noether_theorem).
s_irreducible_intersection_decomp	state	irreducible intersection decomp		Iter-3 imported state (s_irreducible_intersection_decomp).
s_primary_intersection_decomp	state	primary intersection decomp		Iter-3 imported state (s_primary_intersection_decomp).
s_hom_tensor_adjunction_theorem	theorem	Hom-tensor adjunction		Iter-3 imported theorem (s_hom_tensor_adjunction_theorem).
s_ring_R	axiom	ring R		Iter-3 imported axiom (s_ring_R).
s_module_category_R_Mod	axiom	module category R Mod		Iter-3 imported axiom (s_module_category_R_Mod).
s_tensor_hom_functor_pair	state	tensor hom functor pair		Iter-3 imported state (s_tensor_hom_functor_pair).
s_flatness_tor_criterion_theorem	theorem	Flatness via Tor vanishing (Lazard's theorem flavor)		Iter-3 imported theorem (s_flatness_tor_criterion_theorem).
s_module_M	axiom	module M		Iter-3 imported axiom (s_module_M).
s_tor_with_quotient_modules	state	tor with quotient modules		Iter-3 imported state (s_tor_with_quotient_modules).
s_tor_vanishing_equivalence	state	tor vanishing equivalence		Iter-3 imported state (s_tor_vanishing_equivalence).
s_eilenberg_watts_theorem	theorem	Eilenberg–Watts theorem		Iter-3 imported theorem (s_eilenberg_watts_theorem).
s_colimit_preserving_functor	axiom	colimit preserving functor		Iter-3 imported axiom (s_colimit_preserving_functor).
s_colim_preserving_F	state	colim preserving F		Iter-3 imported state (s_colim_preserving_F).
s_bimodule_B_extracted	state	bimodule B extracted		Iter-3 imported state (s_bimodule_B_extracted).
s_mitchell_embedding_theorem	theorem	Mitchell's embedding theorem		Iter-3 imported theorem (s_mitchell_embedding_theorem).
s_abelian_category	axiom	abelian category		Iter-3 imported axiom (s_abelian_category).
s_small_abelian_category	state	small abelian category		Iter-3 imported state (s_small_abelian_category).
s_yoneda_embedding_setup	state	yoneda embedding setup		Iter-3 imported state (s_yoneda_embedding_setup).
s_snake_lemma	theorem	Snake lemma		Iter-3 imported theorem (s_snake_lemma).
s_short_exact_sequence_diagram	axiom	short exact sequence diagram		Iter-3 imported axiom (s_short_exact_sequence_diagram).
s_connecting_homomorphism_delta	state	connecting homomorphism delta		Iter-3 imported state (s_connecting_homomorphism_delta).
s_six_term_long_sequence	state	six term long sequence		Iter-3 imported state (s_six_term_long_sequence).
s_five_lemma	theorem	Five lemma		Iter-3 imported theorem (s_five_lemma).
s_commutative_diagram_5x2	axiom	commutative diagram 5x2		Iter-3 imported axiom (s_commutative_diagram_5x2).
s_diagram_chase_argument	state	diagram chase argument		Iter-3 imported state (s_diagram_chase_argument).
s_long_exact_sequence_theorem	theorem	Long exact sequence in cohomology		Iter-3 imported theorem (s_long_exact_sequence_theorem).
s_short_exact_sequence_of_chain_complexes	axiom	short exact sequence of chain complexes		Iter-3 imported axiom (s_short_exact_sequence_of_chain_complexes).
s_chain_complex	axiom	chain complex		Iter-3 imported axiom (s_chain_complex).
s_connecting_boundary_map	state	connecting boundary map		Iter-3 imported state (s_connecting_boundary_map).
s_snake_lemma_applied_to_homology	state	snake lemma applied to homology		Iter-3 imported state (s_snake_lemma_applied_to_homology).
s_universal_coefficient_theorem	theorem	Universal coefficient theorem		Iter-3 imported theorem (s_universal_coefficient_theorem).
s_chain_complex_over_pid	axiom	chain complex over pid		Iter-3 imported axiom (s_chain_complex_over_pid).
s_abelian_group_coefficients	axiom	abelian group coefficients		Iter-3 imported axiom (s_abelian_group_coefficients).
s_split_resolution_of_homology	state	split resolution of homology		Iter-3 imported state (s_split_resolution_of_homology).
s_tor_ext_terms_computed	state	tor ext terms computed		Iter-3 imported state (s_tor_ext_terms_computed).
s_kunneth_formula_theorem	theorem	Künneth formula		Iter-3 imported theorem (s_kunneth_formula_theorem).
s_tensor_product_of_complexes	axiom	tensor product of complexes		Iter-3 imported axiom (s_tensor_product_of_complexes).
s_external_product_map	state	external product map		Iter-3 imported state (s_external_product_map).
s_tor_correction_term	state	tor correction term		Iter-3 imported state (s_tor_correction_term).
s_hilbert_serre_theorem	theorem	Hilbert–Serre theorem on Hilbert series rationality		Iter-3 imported theorem (s_hilbert_serre_theorem).
s_graded_noetherian_ring	axiom	graded noetherian ring		Iter-3 imported axiom (s_graded_noetherian_ring).
s_finitely_generated_graded_module	axiom	finitely generated graded module		Iter-3 imported axiom (s_finitely_generated_graded_module).
s_hilbert_series_of_M	state	hilbert series of M		Iter-3 imported state (s_hilbert_series_of_M).
s_induction_on_generators	state	induction on generators		Iter-3 imported state (s_induction_on_generators).
s_cayley_hamilton_theorem	theorem	Cayley–Hamilton theorem		Iter-3 imported theorem (s_cayley_hamilton_theorem).
s_commutative_ring	axiom	commutative ring		Iter-3 imported axiom (s_commutative_ring).
s_square_matrix_over_ring	axiom	square matrix over ring		Iter-3 imported axiom (s_square_matrix_over_ring).
s_adjugate_identity	state	adjugate identity		Iter-3 imported state (s_adjugate_identity).
s_substitution_x_eq_A	state	substitution x eq A		Iter-3 imported state (s_substitution_x_eq_A).
s_jordan_normal_form_theorem	theorem	Jordan normal form theorem		Iter-3 imported theorem (s_jordan_normal_form_theorem).
s_finite_dim_vector_space	axiom	finite dim vector space		Iter-3 imported axiom (s_finite_dim_vector_space).
s_v_as_kt_module	state	v as kt module		Iter-3 imported state (s_v_as_kt_module).
s_primary_cyclic_decomposition	state	primary cyclic decomposition		Iter-3 imported state (s_primary_cyclic_decomposition).
s_rational_canonical_form_theorem	theorem	Rational canonical form		Iter-3 imported theorem (s_rational_canonical_form_theorem).
s_v_as_kt_module_general	state	v as kt module general		Iter-3 imported state (s_v_as_kt_module_general).
s_invariant_factor_decomp_for_V	state	invariant factor decomp for V		Iter-3 imported state (s_invariant_factor_decomp_for_V).
s_chevalley_warning_theorem	theorem	Chevalley–Warning theorem		Iter-3 imported theorem (s_chevalley_warning_theorem).
s_finite_field_Fq	axiom	finite field Fq		Iter-3 imported axiom (s_finite_field_Fq).
s_polynomial_system	axiom	polynomial system		Iter-3 imported axiom (s_polynomial_system).
s_indicator_via_fermat	state	indicator via fermat		Iter-3 imported state (s_indicator_via_fermat).
s_indicator_sum_evaluation	state	indicator sum evaluation		Iter-3 imported state (s_indicator_sum_evaluation).
s_ax_grothendieck_theorem	theorem	Ax–Grothendieck theorem		Iter-3 imported theorem (s_ax_grothendieck_theorem).
s_polynomial_map_C_n_to_C_n	axiom	polynomial map C n to C n		Iter-3 imported axiom (s_polynomial_map_C_n_to_C_n).
s_finite_field_case	state	finite field case		Iter-3 imported state (s_finite_field_case).
s_finite_field_injective_implies_surjective	state	finite field injective implies surjective		Iter-3 imported state (s_finite_field_injective_implies_surjective).
s_amitsur_levitzki_theorem	theorem	Amitsur–Levitzki theorem		Iter-3 imported theorem (s_amitsur_levitzki_theorem).
s_matrix_algebra_M_n	axiom	matrix algebra M n		Iter-3 imported axiom (s_matrix_algebra_M_n).
s_polynomial_identity	axiom	polynomial identity		Iter-3 imported axiom (s_polynomial_identity).
s_standard_polynomial_2n	state	standard polynomial 2n		Iter-3 imported state (s_standard_polynomial_2n).
s_symmetric_trace_relations	state	symmetric trace relations		Iter-3 imported state (s_symmetric_trace_relations).
s_hochster_roberts_theorem	theorem	Hochster–Roberts theorem		Iter-3 imported theorem (s_hochster_roberts_theorem).
s_reductive_group_action	axiom	reductive group action		Iter-3 imported axiom (s_reductive_group_action).
s_invariant_subring_R_G	state	invariant subring R G		Iter-3 imported state (s_invariant_subring_R_G).
s_reynolds_splitting	state	reynolds splitting		Iter-3 imported state (s_reynolds_splitting).
s_eakin_nagata_theorem	theorem	Eakin–Nagata theorem		Iter-3 imported theorem (s_eakin_nagata_theorem).
s_finite_ring_extension	axiom	finite ring extension		Iter-3 imported axiom (s_finite_ring_extension).
s_ascending_chain_in_S	state	ascending chain in S		Iter-3 imported state (s_ascending_chain_in_S).
s_chain_pulled_back_to_R	state	chain pulled back to R		Iter-3 imported state (s_chain_pulled_back_to_R).
s_schur_commutator_theorem	theorem	Schur's theorem on commutator subgroup		Iter-3 imported theorem (s_schur_commutator_theorem).
s_finite_index_center	axiom	finite index center		Iter-3 imported axiom (s_finite_index_center).
s_commutator_value_count	state	commutator value count		Iter-3 imported state (s_commutator_value_count).
s_commutator_subgroup_size_bound	state	commutator subgroup size bound		Iter-3 imported state (s_commutator_subgroup_size_bound).
s_frobenius_real_division_theorem	theorem	Frobenius's theorem on real division algebras		Iter-3 imported theorem (s_frobenius_real_division_theorem).
s_finite_dim_associative_real_division_algebra	axiom	finite dim associative real division algebra		Iter-3 imported axiom (s_finite_dim_associative_real_division_algebra).
s_pure_imaginary_subspace	state	pure imaginary subspace		Iter-3 imported state (s_pure_imaginary_subspace).
s_dimension_dichotomy	state	dimension dichotomy		Iter-3 imported state (s_dimension_dichotomy).
s_hurwitz_composition_theorem	theorem	Hurwitz's theorem on composition algebras		Iter-3 imported theorem (s_hurwitz_composition_theorem).
s_normed_division_algebra_over_R	axiom	normed division algebra over R		Iter-3 imported axiom (s_normed_division_algebra_over_R).
s_cayley_dickson_tower	state	cayley dickson tower		Iter-3 imported state (s_cayley_dickson_tower).
s_structural_property_loss	state	structural property loss		Iter-3 imported state (s_structural_property_loss).
s_fundamental_theorem_symmetric_polynomials	theorem	Fundamental theorem of symmetric polynomials		Iter-3 imported theorem (s_fundamental_theorem_symmetric_polynomials).
s_symmetric_group_Sn_action	axiom	symmetric group Sn action		Iter-3 imported axiom (s_symmetric_group_Sn_action).
s_lex_reduction_algorithm	state	lex reduction algorithm		Iter-3 imported state (s_lex_reduction_algorithm).
s_newton_identities_theorem	theorem	Newton's identities		Iter-3 imported theorem (s_newton_identities_theorem).
s_generating_series_setup	state	generating series setup		Iter-3 imported state (s_generating_series_setup).
s_log_derivative_relation	state	log derivative relation		Iter-3 imported state (s_log_derivative_relation).
s_bezout_identity_theorem	theorem	Bezout's identity in PIDs		Iter-3 imported theorem (s_bezout_identity_theorem).
s_two_ring_elements	axiom	two ring elements		Iter-3 imported axiom (s_two_ring_elements).
s_ideal_a_b_in_R	state	ideal a b in R		Iter-3 imported state (s_ideal_a_b_in_R).
s_ideal_principal_d	state	ideal principal d		Iter-3 imported state (s_ideal_principal_d).
s_euclidean_algorithm_theorem	theorem	Euclid's algorithm in Euclidean domains		Iter-3 imported theorem (s_euclidean_algorithm_theorem).
s_euclidean_domain	axiom	euclidean domain		Iter-3 imported axiom (s_euclidean_domain).
s_euclidean_division_step	state	euclidean division step		Iter-3 imported state (s_euclidean_division_step).
s_termination_via_norm_descent	state	termination via norm descent		Iter-3 imported state (s_termination_via_norm_descent).
s_solvability_by_radicals_theorem	theorem	Galois solvability ⇔ solvable Galois group		Iter-3 imported theorem (s_solvability_by_radicals_theorem).
s_subgroup_chain_for_tower	state	subgroup chain for tower		Iter-3 imported state (s_subgroup_chain_for_tower).
s_abelian_quotient_chain	state	abelian quotient chain		Iter-3 imported state (s_abelian_quotient_chain).
s_cyclotomic_galois_theorem	theorem	Galois theorem on cyclotomic extensions		Iter-3 imported theorem (s_cyclotomic_galois_theorem).
s_cyclotomic_field_Q_zeta_n	axiom	cyclotomic field Q zeta n		Iter-3 imported axiom (s_cyclotomic_field_Q_zeta_n).
s_cyclotomic_minimal_polynomial	state	cyclotomic minimal polynomial		Iter-3 imported state (s_cyclotomic_minimal_polynomial).
s_phi_n_irreducible_over_Q	state	phi n irreducible over Q		Iter-3 imported state (s_phi_n_irreducible_over_Q).
s_dedekind_splitting_theorem	theorem	Dedekind's theorem on splitting of primes		Iter-3 imported theorem (s_dedekind_splitting_theorem).
s_galois_extension_of_number_fields	axiom	galois extension of number fields		Iter-3 imported axiom (s_galois_extension_of_number_fields).
s_decomposition_inertia_groups	state	decomposition inertia groups		Iter-3 imported state (s_decomposition_inertia_groups).
s_galois_transitive_on_primes	state	galois transitive on primes		Iter-3 imported state (s_galois_transitive_on_primes).
s_hasse_arf_theorem	theorem	Hasse–Arf theorem		Iter-3 imported theorem (s_hasse_arf_theorem).
s_local_field	axiom	local field		Iter-3 imported axiom (s_local_field).
s_abelian_extension	axiom	abelian extension		Iter-3 imported axiom (s_abelian_extension).
s_ramification_filtration	state	ramification filtration		Iter-3 imported state (s_ramification_filtration).
s_herbrand_psi_function	state	herbrand psi function		Iter-3 imported state (s_herbrand_psi_function).
s_local_tate_duality_theorem	theorem	Local Tate duality		Iter-3 imported theorem (s_local_tate_duality_theorem).
s_cup_product_pairing	state	cup product pairing		Iter-3 imported state (s_cup_product_pairing).
s_finiteness_of_galois_cohomology	state	finiteness of galois cohomology		Iter-3 imported state (s_finiteness_of_galois_cohomology).
s_norm_residue_isomorphism_theorem	theorem	Norm residue isomorphism (Milnor / Bloch–Kato)		Iter-3 imported theorem (s_norm_residue_isomorphism_theorem).
s_milnor_k_theory	axiom	milnor k theory		Iter-3 imported axiom (s_milnor_k_theory).
s_milnor_K_mod_ell	state	milnor K mod ell		Iter-3 imported state (s_milnor_K_mod_ell).
s_norm_residue_map	state	norm residue map		Iter-3 imported state (s_norm_residue_map).
s_motivic_cohomology_comparison	state	motivic cohomology comparison		Iter-3 imported state (s_motivic_cohomology_comparison).
s_brauer_three_main_theorems	theorem	Brauer's three main theorems (block theory)		Iter-3 imported theorem (s_brauer_three_main_theorems).
s_p_block_of_kG	axiom	p block of kG		Iter-3 imported axiom (s_p_block_of_kG).
s_brauer_block_correspondence	state	brauer block correspondence		Iter-3 imported state (s_brauer_block_correspondence).
s_brauer_first_main	state	brauer first main		Iter-3 imported state (s_brauer_first_main).
s_frobenius_determinant_theorem	theorem	Frobenius determinant theorem		Iter-3 imported theorem (s_frobenius_determinant_theorem).
s_group_determinant	axiom	group determinant		Iter-3 imported axiom (s_group_determinant).
s_group_determinant_polynomial	state	group determinant polynomial		Iter-3 imported state (s_group_determinant_polynomial).
s_factorization_into_irreducible_factors	state	factorization into irreducible factors		Iter-3 imported state (s_factorization_into_irreducible_factors).
s_hilbert_burch_theorem	theorem	Hilbert–Burch theorem		Iter-3 imported theorem (s_hilbert_burch_theorem).
s_noetherian_local_ring	axiom	noetherian local ring		Iter-3 imported axiom (s_noetherian_local_ring).
s_perfect_ideal_of_height_2	axiom	perfect ideal of height 2		Iter-3 imported axiom (s_perfect_ideal_of_height_2).
s_two_step_resolution	state	two step resolution		Iter-3 imported state (s_two_step_resolution).
s_ideal_as_maximal_minors	state	ideal as maximal minors		Iter-3 imported state (s_ideal_as_maximal_minors).
s_jacobson_bourbaki_theorem	theorem	Jacobson–Bourbaki theorem		Iter-3 imported theorem (s_jacobson_bourbaki_theorem).
s_field_L	axiom	field L		Iter-3 imported axiom (s_field_L).
s_subring_of_endomorphisms	axiom	subring of endomorphisms		Iter-3 imported axiom (s_subring_of_endomorphisms).
s_subring_subfield_pairing	state	subring subfield pairing		Iter-3 imported state (s_subring_subfield_pairing).
s_double_centralizer_field_endring	state	double centralizer field endring		Iter-3 imported state (s_double_centralizer_field_endring).
s_gabriel_quiver_theorem	theorem	Gabriel's theorem on quivers		Iter-3 imported theorem (s_gabriel_quiver_theorem).
s_quiver_Q	axiom	quiver Q		Iter-3 imported axiom (s_quiver_Q).
s_tits_quadratic_form	state	tits quadratic form		Iter-3 imported state (s_tits_quadratic_form).
s_ade_dynkin_classification	state	ade dynkin classification		Iter-3 imported state (s_ade_dynkin_classification).
s_hurwitz_automorphism_theorem	theorem	Hurwitz's automorphism theorem		Iter-3 imported theorem (s_hurwitz_automorphism_theorem).
s_automorphism_group	axiom	automorphism group		Iter-3 imported axiom (s_automorphism_group).
s_quotient_riemann_surface	state	quotient riemann surface		Iter-3 imported state (s_quotient_riemann_surface).
s_riemann_hurwitz_relation	state	riemann hurwitz relation		Iter-3 imported state (s_riemann_hurwitz_relation).
s_burnside_basis_theorem	theorem	Burnside's basis theorem		Iter-3 imported theorem (s_burnside_basis_theorem).
s_finite_p_group	axiom	finite p group		Iter-3 imported axiom (s_finite_p_group).
s_frattini_subgroup	axiom	frattini subgroup		Iter-3 imported axiom (s_frattini_subgroup).
s_frattini_subgroup_definition	state	frattini subgroup definition		Iter-3 imported state (s_frattini_subgroup_definition).
s_quotient_is_fp_vsp	state	quotient is fp vsp		Iter-3 imported state (s_quotient_is_fp_vsp).
s_higman_embedding_theorem	theorem	Higman's embedding theorem		Iter-3 imported theorem (s_higman_embedding_theorem).
s_recursively_presented_group	axiom	recursively presented group		Iter-3 imported axiom (s_recursively_presented_group).
s_finitely_presented_group	axiom	finitely presented group		Iter-3 imported axiom (s_finitely_presented_group).
s_arithmetized_relator_set	state	arithmetized relator set		Iter-3 imported state (s_arithmetized_relator_set).
s_hnn_machine_simulation	state	hnn machine simulation		Iter-3 imported state (s_hnn_machine_simulation).
s_adian_rabin_theorem	theorem	Adian–Rabin theorem on Markov properties		Iter-3 imported theorem (s_adian_rabin_theorem).
s_markov_property_of_groups	axiom	markov property of groups		Iter-3 imported axiom (s_markov_property_of_groups).
s_word_problem_encoded	state	word problem encoded		Iter-3 imported state (s_word_problem_encoded).
s_diagonal_reduction_to_word_problem	state	diagonal reduction to word problem		Iter-3 imported state (s_diagonal_reduction_to_word_problem).
s_cartan_brauer_hua_theorem	theorem	Cartan–Brauer–Hua theorem		Iter-3 imported theorem (s_cartan_brauer_hua_theorem).
s_subdivision_ring_K	axiom	subdivision ring K		Iter-3 imported axiom (s_subdivision_ring_K).
s_conjugation_invariant_subring	state	conjugation invariant subring		Iter-3 imported state (s_conjugation_invariant_subring).
s_commutator_argument	state	commutator argument		Iter-3 imported state (s_commutator_argument).
s_maclane_coherence_theorem	theorem	Mac Lane's coherence theorem		Iter-3 imported theorem (s_maclane_coherence_theorem).
s_monoidal_category	axiom	monoidal category		Iter-3 imported axiom (s_monoidal_category).
s_associator_unitor_isomorphisms	axiom	associator unitor isomorphisms		Iter-3 imported axiom (s_associator_unitor_isomorphisms).
s_pentagon_and_triangle_axioms	state	pentagon and triangle axioms		Iter-3 imported state (s_pentagon_and_triangle_axioms).
s_coherence_diagram_setup	state	coherence diagram setup		Iter-3 imported state (s_coherence_diagram_setup).
s_finite_field_classification_theorem	theorem	Fundamental theorem of finite division rings of squares (Frobenius–Stickelberger / classification of finite fields)		Iter-3 imported theorem (s_finite_field_classification_theorem).
s_prime_power_q	axiom	prime power q		Iter-3 imported axiom (s_prime_power_q).
s_splitting_field_of_xq_minus_x	state	splitting field of xq minus x		Iter-3 imported state (s_splitting_field_of_xq_minus_x).
s_q_roots_form_subfield	state	q roots form subfield		Iter-3 imported state (s_q_roots_form_subfield).
s_tensor_product_universal_property	theorem	Tensor product universal property		Iter-3 imported theorem (s_tensor_product_universal_property).
s_bilinear_map	axiom	bilinear map		Iter-3 imported axiom (s_bilinear_map).
s_tensor_product_construction	state	tensor product construction		Iter-3 imported state (s_tensor_product_construction).
s_universal_property_via_hom	state	universal property via hom		Iter-3 imported state (s_universal_property_via_hom).
s_cohn_irreducibility_criterion	theorem	Cohn's irreducibility criterion		Iter-3 imported theorem (s_cohn_irreducibility_criterion).
s_base_b_digits_of_prime	state	base b digits of prime		Iter-3 imported state (s_base_b_digits_of_prime).
s_root_location_bound	state	root location bound		Iter-3 imported state (s_root_location_bound).
s_hua_identity_theorem	theorem	Hua's identity		Iter-3 imported theorem (s_hua_identity_theorem).
s_hua_candidate_identity	state	hua candidate identity		Iter-3 imported state (s_hua_candidate_identity).
s_inverse_manipulation	state	inverse manipulation		Iter-3 imported state (s_inverse_manipulation).
s_ito_character_degree_theorem	theorem	Itô's theorem on character degrees		Iter-3 imported theorem (s_ito_character_degree_theorem).
s_abelian_normal_subgroup	axiom	abelian normal subgroup		Iter-3 imported axiom (s_abelian_normal_subgroup).
s_clifford_induced_characters	state	clifford induced characters		Iter-3 imported state (s_clifford_induced_characters).
s_orbit_sum_on_A	state	orbit sum on A		Iter-3 imported state (s_orbit_sum_on_A).
s_localization_universal_property	theorem	Universal property of localization		Iter-3 imported theorem (s_localization_universal_property).
s_multiplicative_subset	axiom	multiplicative subset		Iter-3 imported axiom (s_multiplicative_subset).
s_construction_of_localization	state	construction of localization		Iter-3 imported state (s_construction_of_localization).
s_splitting_field_theorem	theorem	Splitting field existence and uniqueness		Iter-3 imported theorem (s_splitting_field_theorem).
s_iterated_root_adjunction	state	iterated root adjunction		Iter-3 imported state (s_iterated_root_adjunction).
s_terminating_construction	state	terminating construction		Iter-3 imported state (s_terminating_construction).
s_normal_basis_theorem	theorem	Normal basis theorem		Iter-3 imported theorem (s_normal_basis_theorem).
s_L_as_kG_module	state	L as kG module		Iter-3 imported state (s_L_as_kG_module).
s_dedekind_linear_independence	state	dedekind linear independence		Iter-3 imported state (s_dedekind_linear_independence).
s_stickelberger_theorem	theorem	Stickelberger's theorem on Gauss sums		Iter-3 imported theorem (s_stickelberger_theorem).
s_gauss_sum_definition	state	gauss sum definition		Iter-3 imported state (s_gauss_sum_definition).
s_gauss_sum_prime_factorization	state	gauss sum prime factorization		Iter-3 imported state (s_gauss_sum_prime_factorization).
s_tate_cohomology_cyclic_theorem	theorem	Mac Lane's theorem on cohomology of cyclic groups		Iter-3 imported theorem (s_tate_cohomology_cyclic_theorem).
s_finite_cyclic_group	axiom	finite cyclic group		Iter-3 imported axiom (s_finite_cyclic_group).
s_g_module_M	axiom	g module M		Iter-3 imported axiom (s_g_module_M).
s_periodic_resolution	state	periodic resolution		Iter-3 imported state (s_periodic_resolution).
s_two_periodic_cohomology	state	two periodic cohomology		Iter-3 imported state (s_two_periodic_cohomology).
s_schur_multiplier_theorem	theorem	Schur multiplier and central extensions		Iter-3 imported theorem (s_schur_multiplier_theorem).
s_central_extension	axiom	central extension		Iter-3 imported axiom (s_central_extension).
s_h2_classifies_projective_reps	state	h2 classifies projective reps		Iter-3 imported state (s_h2_classifies_projective_reps).
s_schur_cover_construction	state	schur cover construction		Iter-3 imported state (s_schur_cover_construction).
s_wigner_eckart_theorem	theorem	Wigner–Eckart theorem		Iter-3 imported theorem (s_wigner_eckart_theorem).
s_compact_lie_group	axiom	compact lie group		Iter-3 imported axiom (s_compact_lie_group).
s_tensor_operator	axiom	tensor operator		Iter-3 imported axiom (s_tensor_operator).
s_irreducible_tensor_operator	state	irreducible tensor operator		Iter-3 imported state (s_irreducible_tensor_operator).
s_clebsch_gordan_decomposition	state	clebsch gordan decomposition		Iter-3 imported state (s_clebsch_gordan_decomposition).
s_multiplicity_one_theorem	theorem	Multiplicity-one theorem		Iter-3 imported theorem (s_multiplicity_one_theorem).
s_reductive_group_GL_n	axiom	reductive group GL n		Iter-3 imported axiom (s_reductive_group_GL_n).
s_irreducible_automorphic_representation	axiom	irreducible automorphic representation		Iter-3 imported axiom (s_irreducible_automorphic_representation).
s_whittaker_model_for_pi	state	whittaker model for pi		Iter-3 imported state (s_whittaker_model_for_pi).
s_local_whittaker_uniqueness	state	local whittaker uniqueness		Iter-3 imported state (s_local_whittaker_uniqueness).
s_galois_descent_theorem	theorem	Galois descent		Iter-3 imported theorem (s_galois_descent_theorem).
s_galois_extension_L_over_K	axiom	galois extension L over K		Iter-3 imported axiom (s_galois_extension_L_over_K).
s_object_over_L_with_descent_data	axiom	object over L with descent data		Iter-3 imported axiom (s_object_over_L_with_descent_data).
s_descent_cocycle	state	descent cocycle		Iter-3 imported state (s_descent_cocycle).
s_h1_classification_of_forms	state	h1 classification of forms		Iter-3 imported state (s_h1_classification_of_forms).
s_intermediate_value_theorem	theorem	Intermediate Value theorem		Iter-3 imported theorem (s_intermediate_value_theorem).
s_normalized_sign_change_setup	state	normalized sign change setup		Iter-3 imported state (s_normalized_sign_change_setup).
s_nested_intervals_with_sign_change	state	nested intervals with sign change		Iter-3 imported state (s_nested_intervals_with_sign_change).
s_extreme_value_theorem	theorem	Extreme Value theorem		Iter-3 imported theorem (s_extreme_value_theorem).
s_image_is_bounded	state	image is bounded		Iter-3 imported state (s_image_is_bounded).
s_maximizing_sequence_in_domain	state	maximizing sequence in domain		Iter-3 imported state (s_maximizing_sequence_in_domain).
s_mean_value_theorem_terminal	theorem	Mean Value theorem		Iter-3 imported theorem (s_mean_value_theorem_terminal).
s_differentiable_on_open_interval	axiom	differentiable on open interval		Iter-3 imported axiom (s_differentiable_on_open_interval).
s_tilted_function_with_equal_endpoints	state	tilted function with equal endpoints		Iter-3 imported state (s_tilted_function_with_equal_endpoints).
s_rolle_theorem_point	state	rolle theorem point		Iter-3 imported state (s_rolle_theorem_point).
s_cauchy_mean_value_theorem	theorem	Cauchy mean value theorem		Iter-3 imported theorem (s_cauchy_mean_value_theorem).
s_paired_function_with_equal_endpoints	state	paired function with equal endpoints		Iter-3 imported state (s_paired_function_with_equal_endpoints).
s_lhopital_rule	theorem	L'Hôpital's rule		Iter-3 imported theorem (s_lhopital_rule).
s_indeterminate_form_zero_over_zero	axiom	indeterminate form zero over zero		Iter-3 imported axiom (s_indeterminate_form_zero_over_zero).
s_normalized_0_over_0_pair	state	normalized 0 over 0 pair		Iter-3 imported state (s_normalized_0_over_0_pair).
s_ratio_equals_derivative_ratio_at_xi	state	ratio equals derivative ratio at xi		Iter-3 imported state (s_ratio_equals_derivative_ratio_at_xi).
s_darboux_theorem_analysis	theorem	Darboux's theorem (intermediate value for derivatives)		Iter-3 imported theorem (s_darboux_theorem_analysis).
s_auxiliary_g_with_critical_point	state	auxiliary g with critical point		Iter-3 imported state (s_auxiliary_g_with_critical_point).
s_interior_critical_point_of_g	state	interior critical point of g		Iter-3 imported state (s_interior_critical_point_of_g).
s_cantor_intersection_theorem	theorem	Cantor's intersection theorem		Iter-3 imported theorem (s_cantor_intersection_theorem).
s_nested_nonempty_closed_sets_shrinking_diameter	axiom	nested nonempty closed sets shrinking diameter		Iter-3 imported axiom (s_nested_nonempty_closed_sets_shrinking_diameter).
s_cauchy_sequence_from_nested_sets	state	cauchy sequence from nested sets		Iter-3 imported state (s_cauchy_sequence_from_nested_sets).
s_limit_point_in_each_Fn	state	limit point in each Fn		Iter-3 imported state (s_limit_point_in_each_Fn).
s_banach_steinhaus	theorem	Banach–Steinhaus uniform boundedness principle		Iter-3 imported theorem (s_banach_steinhaus).
s_banach_space	axiom	banach space		Iter-3 imported axiom (s_banach_space).
s_family_of_pointwise_bounded_operators	axiom	family of pointwise bounded operators		Iter-3 imported axiom (s_family_of_pointwise_bounded_operators).
s_closed_covering_of_banach_space	state	closed covering of banach space		Iter-3 imported state (s_closed_covering_of_banach_space).
s_E_n_with_interior_ball	state	E n with interior ball		Iter-3 imported state (s_E_n_with_interior_ball).
s_arzela_ascoli	theorem	Arzelà–Ascoli theorem		Iter-3 imported theorem (s_arzela_ascoli).
s_compact_metric_space	axiom	compact metric space		Iter-3 imported axiom (s_compact_metric_space).
s_uniformly_bounded_equicontinuous_family	axiom	uniformly bounded equicontinuous family		Iter-3 imported axiom (s_uniformly_bounded_equicontinuous_family).
s_family_indexed_by_dense_subset_D	state	family indexed by dense subset D		Iter-3 imported state (s_family_indexed_by_dense_subset_D).
s_pointwise_convergent_on_D_subsequence	state	pointwise convergent on D subsequence		Iter-3 imported state (s_pointwise_convergent_on_D_subsequence).
s_stone_weierstrass	theorem	Stone–Weierstrass theorem		Iter-3 imported theorem (s_stone_weierstrass).
s_closure_is_lattice	state	closure is lattice		Iter-3 imported state (s_closure_is_lattice).
s_two_point_interpolation_property	state	two point interpolation property		Iter-3 imported state (s_two_point_interpolation_property).
s_tietze_extension_theorem	theorem	Tietze extension theorem		Iter-3 imported theorem (s_tietze_extension_theorem).
s_normal_topological_space	axiom	normal topological space		Iter-3 imported axiom (s_normal_topological_space).
s_bounded_continuous_function_on_closed_subset	axiom	bounded continuous function on closed subset		Iter-3 imported axiom (s_bounded_continuous_function_on_closed_subset).
s_urysohn_function_for_each_level	state	urysohn function for each level		Iter-3 imported state (s_urysohn_function_for_each_level).
s_uniformly_convergent_extension_series	state	uniformly convergent extension series		Iter-3 imported state (s_uniformly_convergent_extension_series).
s_dini_theorem	theorem	Dini's theorem		Iter-3 imported theorem (s_dini_theorem).
s_monotone_pointwise_convergent_continuous_sequence	axiom	monotone pointwise convergent continuous sequence		Iter-3 imported axiom (s_monotone_pointwise_convergent_continuous_sequence).
s_open_cover_from_pointwise_bound	state	open cover from pointwise bound		Iter-3 imported state (s_open_cover_from_pointwise_bound).
s_uniform_bound_on_X	state	uniform bound on X		Iter-3 imported state (s_uniform_bound_on_X).
s_caratheodory_extension	theorem	Carathéodory extension theorem		Iter-3 imported theorem (s_caratheodory_extension).
s_premeasure_on_algebra_of_sets	axiom	premeasure on algebra of sets		Iter-3 imported axiom (s_premeasure_on_algebra_of_sets).
s_sigma_algebra_generated	axiom	sigma algebra generated		Iter-3 imported axiom (s_sigma_algebra_generated).
s_outer_measure_on_powerset	state	outer measure on powerset		Iter-3 imported state (s_outer_measure_on_powerset).
s_sigma_algebra_of_measurable_sets	state	sigma algebra of measurable sets		Iter-3 imported state (s_sigma_algebra_of_measurable_sets).
s_riesz_markov_representation	theorem	Riesz–Markov representation theorem		Iter-3 imported theorem (s_riesz_markov_representation).
s_positive_linear_functional_on_Cc_X	axiom	positive linear functional on Cc X		Iter-3 imported axiom (s_positive_linear_functional_on_Cc_X).
s_outer_premeasure_on_opens	state	outer premeasure on opens		Iter-3 imported state (s_outer_premeasure_on_opens).
s_borel_measure_candidate	state	borel measure candidate		Iter-3 imported state (s_borel_measure_candidate).
s_vitali_hahn_saks	theorem	Vitali–Hahn–Saks theorem		Iter-3 imported theorem (s_vitali_hahn_saks).
s_sequence_of_uniformly_absolutely_continuous_measures	axiom	sequence of uniformly absolutely continuous measures		Iter-3 imported axiom (s_sequence_of_uniformly_absolutely_continuous_measures).
s_finite_measure_space	axiom	finite measure space		Iter-3 imported axiom (s_finite_measure_space).
s_complete_metric_space_of_classes	state	complete metric space of classes		Iter-3 imported state (s_complete_metric_space_of_classes).
s_equicontinuous_at_some_set	state	equicontinuous at some set		Iter-3 imported state (s_equicontinuous_at_some_set).
s_lebesgue_monotone_differentiation	theorem	Lebesgue's monotone differentiation theorem		Iter-3 imported theorem (s_lebesgue_monotone_differentiation).
s_monotone_function_on_interval	axiom	monotone function on interval		Iter-3 imported axiom (s_monotone_function_on_interval).
s_lebesgue_measure_on_R	axiom	lebesgue measure on R		Iter-3 imported axiom (s_lebesgue_measure_on_R).
s_dini_derivate_quadruple	state	dini derivate quadruple		Iter-3 imported state (s_dini_derivate_quadruple).
s_zero_measure_disagreement_set	state	zero measure disagreement set		Iter-3 imported state (s_zero_measure_disagreement_set).
s_riesz_representation_Lp	theorem	Riesz representation theorem for L^p		Iter-3 imported theorem (s_riesz_representation_Lp).
s_sigma_finite_measure_space	axiom	sigma finite measure space		Iter-3 imported axiom (s_sigma_finite_measure_space).
s_bounded_linear_functional_on_Lp	axiom	bounded linear functional on Lp		Iter-3 imported axiom (s_bounded_linear_functional_on_Lp).
s_finite_signed_measure_nu_absolutely_continuous	state	finite signed measure nu absolutely continuous		Iter-3 imported state (s_finite_signed_measure_nu_absolutely_continuous).
s_fefferman_stein_vector_valued	theorem	Fefferman–Stein vector-valued maximal inequality		Iter-3 imported theorem (s_fefferman_stein_vector_valued).
s_lebesgue_measure_on_Rn	axiom	lebesgue measure on Rn		Iter-3 imported axiom (s_lebesgue_measure_on_Rn).
s_lp_sequence_of_l1_functions	axiom	lp sequence of l1 functions		Iter-3 imported axiom (s_lp_sequence_of_l1_functions).
s_randomized_linear_combination	state	randomized linear combination		Iter-3 imported state (s_randomized_linear_combination).
s_dual_pairing_to_scalar_maximal	state	dual pairing to scalar maximal		Iter-3 imported state (s_dual_pairing_to_scalar_maximal).
s_calderon_zygmund_decomposition	theorem	Calderón–Zygmund decomposition		Iter-3 imported theorem (s_calderon_zygmund_decomposition).
s_l1_function_on_Rn	axiom	l1 function on Rn		Iter-3 imported axiom (s_l1_function_on_Rn).
s_stopping_cubes_with_average_exceeding_lambda	state	stopping cubes with average exceeding lambda		Iter-3 imported state (s_stopping_cubes_with_average_exceeding_lambda).
s_good_bad_decomposition	state	good bad decomposition		Iter-3 imported state (s_good_bad_decomposition).
s_calderon_zygmund_Lp_bound	theorem	Calderón–Zygmund singular integral L^p bound		Iter-3 imported theorem (s_calderon_zygmund_Lp_bound).
s_calderon_zygmund_kernel_on_Rn	axiom	calderon zygmund kernel on Rn		Iter-3 imported axiom (s_calderon_zygmund_kernel_on_Rn).
s_lp_function_on_Rn	axiom	lp function on Rn		Iter-3 imported axiom (s_lp_function_on_Rn).
s_L2_bound_for_singular_integral	state	L2 bound for singular integral		Iter-3 imported state (s_L2_bound_for_singular_integral).
s_weak_11_bound_via_CZ_decomp	state	weak 11 bound via CZ decomp		Iter-3 imported state (s_weak_11_bound_via_CZ_decomp).
s_ramsey_theorem_finite	theorem	Ramsey's theorem (finite version)		Iter-3 imported theorem (s_ramsey_theorem_finite).
s_finite_ramsey_conjecture	state	finite ramsey conjecture		Iter-3 imported state (s_finite_ramsey_conjecture).
s_recursive_majority_color_neighbourhood	state	recursive majority color neighbourhood		Iter-3 imported state (s_recursive_majority_color_neighbourhood).
s_ramsey_number_recurrence_R_r_s	state	ramsey number recurrence R r s		Iter-3 imported state (s_ramsey_number_recurrence_R_r_s).
s_schur_theorem	theorem	Schur's theorem		Iter-3 imported theorem (s_schur_theorem).
s_edge_colored_complete_graph_from_schur_coloring	state	edge colored complete graph from schur coloring		Iter-3 imported state (s_edge_colored_complete_graph_from_schur_coloring).
s_monochromatic_triangle_to_schur_triple	state	monochromatic triangle to schur triple		Iter-3 imported state (s_monochromatic_triangle_to_schur_triple).
s_van_der_waerden_theorem	theorem	Van der Waerden's theorem		Iter-3 imported theorem (s_van_der_waerden_theorem).
s_vdw_conjecture_W_r_k	state	vdw conjecture W r k		Iter-3 imported state (s_vdw_conjecture_W_r_k).
s_color_focused_progressions	state	color focused progressions		Iter-3 imported state (s_color_focused_progressions).
s_vdw_finite_intersection_lemma	state	vdw finite intersection lemma		Iter-3 imported state (s_vdw_finite_intersection_lemma).
s_hales_jewett_theorem	theorem	Hales–Jewett theorem		Iter-3 imported theorem (s_hales_jewett_theorem).
s_finite_combinatorial_cube_word_space	axiom	finite combinatorial cube word space		Iter-3 imported axiom (s_finite_combinatorial_cube_word_space).
s_combinatorial_line_target	state	combinatorial line target		Iter-3 imported state (s_combinatorial_line_target).
s_shelah_cube_lemma	state	shelah cube lemma		Iter-3 imported state (s_shelah_cube_lemma).
s_graham_rothschild_theorem	theorem	Graham–Rothschild theorem		Iter-3 imported theorem (s_graham_rothschild_theorem).
s_parameter_word_partition_setup	state	parameter word partition setup		Iter-3 imported state (s_parameter_word_partition_setup).
s_iterated_hj_combinatorial_subspace	state	iterated hj combinatorial subspace		Iter-3 imported state (s_iterated_hj_combinatorial_subspace).
s_folkman_theorem	theorem	Folkman's theorem		Iter-3 imported theorem (s_folkman_theorem).
s_folkman_finite_target	state	folkman finite target		Iter-3 imported state (s_folkman_finite_target).
s_folkman_via_hj_reduction	state	folkman via hj reduction		Iter-3 imported state (s_folkman_via_hj_reduction).
s_gallai_theorem_multidim_vdw	theorem	Gallai's theorem (multidimensional VdW)		Iter-3 imported theorem (s_gallai_theorem_multidim_vdw).
s_homothetic_copy_target	state	homothetic copy target		Iter-3 imported state (s_homothetic_copy_target).
s_gallai_via_hj_reduction	state	gallai via hj reduction		Iter-3 imported state (s_gallai_via_hj_reduction).
s_erdos_rado_theorem	theorem	Erdős–Rado theorem		Iter-3 imported theorem (s_erdos_rado_theorem).
s_uncountable_partition_target	state	uncountable partition target		Iter-3 imported state (s_uncountable_partition_target).
s_transfinite_color_canonization	state	transfinite color canonization		Iter-3 imported state (s_transfinite_color_canonization).
s_carlson_simpson_theorem	theorem	Carlson–Simpson theorem		Iter-3 imported theorem (s_carlson_simpson_theorem).
s_infinitary_hj_target	state	infinitary hj target		Iter-3 imported state (s_infinitary_hj_target).
s_ultrafilter_on_variable_words	state	ultrafilter on variable words		Iter-3 imported state (s_ultrafilter_on_variable_words).
s_erdos_ko_rado_theorem	theorem	Erdős–Ko–Rado theorem		Iter-3 imported theorem (s_erdos_ko_rado_theorem).
s_finite_set_family	axiom	finite set family		Iter-3 imported axiom (s_finite_set_family).
s_intersecting_family_property	axiom	intersecting family property		Iter-3 imported axiom (s_intersecting_family_property).
s_max_intersecting_k_uniform_problem	state	max intersecting k uniform problem		Iter-3 imported state (s_max_intersecting_k_uniform_problem).
s_cyclic_arc_counting_bound	state	cyclic arc counting bound		Iter-3 imported state (s_cyclic_arc_counting_bound).
s_double_counting_arc_intersection	state	double counting arc intersection		Iter-3 imported state (s_double_counting_arc_intersection).
s_kruskal_katona_theorem	theorem	Kruskal–Katona theorem		Iter-3 imported theorem (s_kruskal_katona_theorem).
s_shadow_operator_on_set_family	axiom	shadow operator on set family		Iter-3 imported axiom (s_shadow_operator_on_set_family).
s_shadow_minimisation_problem	state	shadow minimisation problem		Iter-3 imported state (s_shadow_minimisation_problem).
s_colex_initial_family_extremum	state	colex initial family extremum		Iter-3 imported state (s_colex_initial_family_extremum).
s_compression_preserves_size_reduces_shadow	state	compression preserves size reduces shadow		Iter-3 imported state (s_compression_preserves_size_reduces_shadow).
s_lym_inequality	theorem	LYM inequality		Iter-3 imported theorem (s_lym_inequality).
s_antichain_in_boolean_lattice	axiom	antichain in boolean lattice		Iter-3 imported axiom (s_antichain_in_boolean_lattice).
s_random_maximal_chain_on_boolean_lattice	state	random maximal chain on boolean lattice		Iter-3 imported state (s_random_maximal_chain_on_boolean_lattice).
s_chain_meets_antichain_at_most_once	state	chain meets antichain at most once		Iter-3 imported state (s_chain_meets_antichain_at_most_once).
s_sperner_theorem	theorem	Sperner's theorem		Iter-3 imported theorem (s_sperner_theorem).
s_middle_layer_bound_for_antichain	state	middle layer bound for antichain		Iter-3 imported state (s_middle_layer_bound_for_antichain).
s_bollobas_set_pair_inequality	theorem	Bollobás set-pair inequality		Iter-3 imported theorem (s_bollobas_set_pair_inequality).
s_cross_intersecting_pair_system	axiom	cross intersecting pair system		Iter-3 imported axiom (s_cross_intersecting_pair_system).
s_random_linear_order_on_ground_set	state	random linear order on ground set		Iter-3 imported state (s_random_linear_order_on_ground_set).
s_probability_event_disjoint_pairs	state	probability event disjoint pairs		Iter-3 imported state (s_probability_event_disjoint_pairs).
s_sauer_shelah_lemma	theorem	Sauer–Shelah lemma (VC dimension)		Iter-3 imported theorem (s_sauer_shelah_lemma).
s_vc_dimension_definition	axiom	vc dimension definition		Iter-3 imported axiom (s_vc_dimension_definition).
s_vc_count_bound_target	state	vc count bound target		Iter-3 imported state (s_vc_count_bound_target).
s_down_compressed_shattering_preserved	state	down compressed shattering preserved		Iter-3 imported state (s_down_compressed_shattering_preserved).
s_shattering_bound_sum_binomials	state	shattering bound sum binomials		Iter-3 imported state (s_shattering_bound_sum_binomials).
s_frankl_wilson_theorem	theorem	Frankl–Wilson theorem		Iter-3 imported theorem (s_frankl_wilson_theorem).
s_mod_p_intersection_pattern_target	state	mod p intersection pattern target		Iter-3 imported state (s_mod_p_intersection_pattern_target).
s_incidence_polynomial_basis_in_F_p	state	incidence polynomial basis in F p		Iter-3 imported state (s_incidence_polynomial_basis_in_F_p).
s_ahlswede_khachatrian_theorem	theorem	Ahlswede–Khachatrian theorem		Iter-3 imported theorem (s_ahlswede_khachatrian_theorem).
s_t_intersecting_family_property	axiom	t intersecting family property		Iter-3 imported axiom (s_t_intersecting_family_property).
s_complete_intersection_problem	state	complete intersection problem		Iter-3 imported state (s_complete_intersection_problem).
s_frankl_family_F_r_candidate	state	frankl family F r candidate		Iter-3 imported state (s_frankl_family_F_r_candidate).
s_johnson_scheme_eigenvalue_optimum	state	johnson scheme eigenvalue optimum		Iter-3 imported state (s_johnson_scheme_eigenvalue_optimum).
s_hall_marriage_theorem	theorem	Hall's marriage theorem		Iter-3 imported theorem (s_hall_marriage_theorem).
s_bipartite_graph	axiom	bipartite graph		Iter-3 imported axiom (s_bipartite_graph).
s_perfect_matching_definition	axiom	perfect matching definition		Iter-3 imported axiom (s_perfect_matching_definition).
s_hall_condition_statement	state	hall condition statement		Iter-3 imported state (s_hall_condition_statement).
s_minimum_vertex_cover_smaller_than_A	state	minimum vertex cover smaller than A		Iter-3 imported state (s_minimum_vertex_cover_smaller_than_A).
s_augmenting_path_yields_violator_S	state	augmenting path yields violator S		Iter-3 imported state (s_augmenting_path_yields_violator_S).
s_konig_theorem	theorem	König's theorem		Iter-3 imported theorem (s_konig_theorem).
s_min_vertex_cover_definition	axiom	min vertex cover definition		Iter-3 imported axiom (s_min_vertex_cover_definition).
s_lp_relaxation_matching_vs_cover	state	lp relaxation matching vs cover		Iter-3 imported state (s_lp_relaxation_matching_vs_cover).
s_konig_cover_constructed_from_max_matching	state	konig cover constructed from max matching		Iter-3 imported state (s_konig_cover_constructed_from_max_matching).
s_max_flow_min_cut_theorem	theorem	Max-flow min-cut theorem		Iter-3 imported theorem (s_max_flow_min_cut_theorem).
s_directed_graph_with_capacities	axiom	directed graph with capacities		Iter-3 imported axiom (s_directed_graph_with_capacities).
s_flow_conservation_axiom	axiom	flow conservation axiom		Iter-3 imported axiom (s_flow_conservation_axiom).
s_max_flow_lp_dual	state	max flow lp dual		Iter-3 imported state (s_max_flow_lp_dual).
s_residual_graph_no_augmenting_path_yields_cut	state	residual graph no augmenting path yields cut		Iter-3 imported state (s_residual_graph_no_augmenting_path_yields_cut).
s_berge_augmenting_theorem	theorem	Berge's theorem (augmenting path)		Iter-3 imported theorem (s_berge_augmenting_theorem).
s_matching_in_graph	axiom	matching in graph		Iter-3 imported axiom (s_matching_in_graph).
s_berge_target	state	berge target		Iter-3 imported state (s_berge_target).
s_symmetric_difference_M_Mprime_decomposes_into_paths	state	symmetric difference M Mprime decomposes into paths		Iter-3 imported state (s_symmetric_difference_M_Mprime_decomposes_into_paths).
s_petersen_theorem	theorem	Petersen's theorem (3-regular bridgeless)		Iter-3 imported theorem (s_petersen_theorem).
s_three_regular_bridgeless_graph	axiom	three regular bridgeless graph		Iter-3 imported axiom (s_three_regular_bridgeless_graph).
s_petersen_target	state	petersen target		Iter-3 imported state (s_petersen_target).
s_tutte_odd_component_count_under_3regular_bridgeless	state	tutte odd component count under 3regular bridgeless		Iter-3 imported state (s_tutte_odd_component_count_under_3regular_bridgeless).
s_tutte_perfect_matching_theorem	theorem	Tutte's theorem (perfect matching)		Iter-3 imported theorem (s_tutte_perfect_matching_theorem).
s_tutte_condition_target	state	tutte condition target		Iter-3 imported state (s_tutte_condition_target).
s_maximal_counterexample_extremal_S	state	maximal counterexample extremal S		Iter-3 imported state (s_maximal_counterexample_extremal_S).
s_block_structure_contradicts_tutte_count	state	block structure contradicts tutte count		Iter-3 imported state (s_block_structure_contradicts_tutte_count).
s_gale_ryser_theorem	theorem	Gale–Ryser theorem (degree sequence)		Iter-3 imported theorem (s_gale_ryser_theorem).
s_degree_sequence	axiom	degree sequence		Iter-3 imported axiom (s_degree_sequence).
s_gale_ryser_target	state	gale ryser target		Iter-3 imported state (s_gale_ryser_target).
s_transportation_polytope_integrality_certificate	state	transportation polytope integrality certificate		Iter-3 imported state (s_transportation_polytope_integrality_certificate).
s_erdos_gallai_theorem	theorem	Erdős–Gallai theorem (graphic sequence)		Iter-3 imported theorem (s_erdos_gallai_theorem).
s_graphic_sequence_target	state	graphic sequence target		Iter-3 imported state (s_graphic_sequence_target).
s_sorted_degree_partial_sums_inequality	state	sorted degree partial sums inequality		Iter-3 imported state (s_sorted_degree_partial_sums_inequality).
s_erdos_stone_theorem	theorem	Erdős–Stone theorem		Iter-3 imported theorem (s_erdos_stone_theorem).
s_chromatic_number_definition	axiom	chromatic number definition		Iter-3 imported axiom (s_chromatic_number_definition).
s_erdos_stone_target	state	erdos stone target		Iter-3 imported state (s_erdos_stone_target).
s_blow_up_complete_multipartite_found	state	blow up complete multipartite found		Iter-3 imported state (s_blow_up_complete_multipartite_found).
s_kovari_sos_turan_theorem	theorem	Kővári–Sós–Turán theorem		Iter-3 imported theorem (s_kovari_sos_turan_theorem).
s_forbidden_subgraph_K_s_t	axiom	forbidden subgraph K s t		Iter-3 imported axiom (s_forbidden_subgraph_K_s_t).
s_kst_extremal_target	state	kst extremal target		Iter-3 imported state (s_kst_extremal_target).
s_jensen_lower_bound_on_K_s_copies	state	jensen lower bound on K s copies		Iter-3 imported state (s_jensen_lower_bound_on_K_s_copies).
s_bondy_simonovits_theorem	theorem	Bondy–Simonovits theorem (even cycles)		Iter-3 imported theorem (s_bondy_simonovits_theorem).
s_forbidden_even_cycle_C_2k	axiom	forbidden even cycle C 2k		Iter-3 imported axiom (s_forbidden_even_cycle_C_2k).
s_even_cycle_extremal_target	state	even cycle extremal target		Iter-3 imported state (s_even_cycle_extremal_target).
s_bfs_layer_count_doubling	state	bfs layer count doubling		Iter-3 imported state (s_bfs_layer_count_doubling).
s_szemeredi_regularity_lemma	theorem	Szemerédi regularity lemma		Iter-3 imported theorem (s_szemeredi_regularity_lemma).
s_density_between_vertex_sets	axiom	density between vertex sets		Iter-3 imported axiom (s_density_between_vertex_sets).
s_regularity_target	state	regularity target		Iter-3 imported state (s_regularity_target).
s_mean_square_density_index_q	state	mean square density index q		Iter-3 imported state (s_mean_square_density_index_q).
s_graph_removal_lemma	theorem	Graph removal lemma		Iter-3 imported theorem (s_graph_removal_lemma).
s_cleaned_reduced_graph_H_free_or_dense	state	cleaned reduced graph H free or dense		Iter-3 imported state (s_cleaned_reduced_graph_H_free_or_dense).
s_counting_lemma_dense_H_implication	state	counting lemma dense H implication		Iter-3 imported state (s_counting_lemma_dense_H_implication).
s_corners_theorem	theorem	Corners theorem (Ajtai–Szemerédi)		Iter-3 imported theorem (s_corners_theorem).
s_two_dimensional_lattice	axiom	two dimensional lattice		Iter-3 imported axiom (s_two_dimensional_lattice).
s_corner_target	state	corner target		Iter-3 imported state (s_corner_target).
s_triangle_removal_yields_corner	state	triangle removal yields corner		Iter-3 imported state (s_triangle_removal_yields_corner).
s_szemeredi_trotter_theorem	theorem	Szemerédi–Trotter theorem (incidence)		Iter-3 imported theorem (s_szemeredi_trotter_theorem).
s_finite_point_set_in_plane	axiom	finite point set in plane		Iter-3 imported axiom (s_finite_point_set_in_plane).
s_finite_line_set_in_plane	axiom	finite line set in plane		Iter-3 imported axiom (s_finite_line_set_in_plane).
s_szemeredi_trotter_target	state	szemeredi trotter target		Iter-3 imported state (s_szemeredi_trotter_target).
s_cutting_into_cells_count_incidences_per_cell	state	cutting into cells count incidences per cell		Iter-3 imported state (s_cutting_into_cells_count_incidences_per_cell).
s_erdos_posa_theorem	theorem	Erdős–Pósa theorem		Iter-3 imported theorem (s_erdos_posa_theorem).
s_disjoint_cycle_packing_vs_cover	axiom	disjoint cycle packing vs cover		Iter-3 imported axiom (s_disjoint_cycle_packing_vs_cover).
s_erdos_posa_dual_target	state	erdos posa dual target		Iter-3 imported state (s_erdos_posa_dual_target).
s_dfs_back_edges_yield_disjoint_cycles_or_small_FVS	state	dfs back edges yield disjoint cycles or small FVS		Iter-3 imported state (s_dfs_back_edges_yield_disjoint_cycles_or_small_FVS).
s_q_analog_ekr_theorem	theorem	Erdős–Ko–Rado for vector spaces (q-analog)		Iter-3 imported theorem (s_q_analog_ekr_theorem).
s_finite_dim_vector_space_over_F_q	axiom	finite dim vector space over F q		Iter-3 imported axiom (s_finite_dim_vector_space_over_F_q).
s_q_ekr_target	state	q ekr target		Iter-3 imported state (s_q_ekr_target).
s_grassmann_scheme_eigen_optimum	state	grassmann scheme eigen optimum		Iter-3 imported state (s_grassmann_scheme_eigen_optimum).
s_matrix_tree_theorem	theorem	Kirchhoff's matrix-tree theorem		Iter-3 imported theorem (s_matrix_tree_theorem).
s_laplacian_matrix_of_graph	axiom	laplacian matrix of graph		Iter-3 imported axiom (s_laplacian_matrix_of_graph).
s_incidence_matrix_factorisation_L_eq_B_B_T	state	incidence matrix factorisation L eq B B T		Iter-3 imported state (s_incidence_matrix_factorisation_L_eq_B_B_T).
s_cauchy_binet_expansion_of_principal_minor	state	cauchy binet expansion of principal minor		Iter-3 imported state (s_cauchy_binet_expansion_of_principal_minor).
s_lgv_lemma	theorem	Lindström–Gessel–Viennot lemma		Iter-3 imported theorem (s_lgv_lemma).
s_acyclic_directed_graph_with_weights	axiom	acyclic directed graph with weights		Iter-3 imported axiom (s_acyclic_directed_graph_with_weights).
s_non_intersecting_lattice_paths_problem	axiom	non intersecting lattice paths problem		Iter-3 imported axiom (s_non_intersecting_lattice_paths_problem).
s_path_matrix_M_setup	state	path matrix M setup		Iter-3 imported state (s_path_matrix_M_setup).
s_sign_reversing_involution_on_crossing_path_systems	state	sign reversing involution on crossing path systems		Iter-3 imported state (s_sign_reversing_involution_on_crossing_path_systems).
s_rsk_correspondence	theorem	RSK correspondence		Iter-3 imported theorem (s_rsk_correspondence).
s_finite_matrix_nonneg_integer_entries	axiom	finite matrix nonneg integer entries		Iter-3 imported axiom (s_finite_matrix_nonneg_integer_entries).
s_pair_of_young_tableaux	axiom	pair of young tableaux		Iter-3 imported axiom (s_pair_of_young_tableaux).
s_rsk_target	state	rsk target		Iter-3 imported state (s_rsk_target).
s_row_insertion_algorithm_well_defined	state	row insertion algorithm well defined		Iter-3 imported state (s_row_insertion_algorithm_well_defined).
s_hook_length_formula	theorem	Hook-length formula		Iter-3 imported theorem (s_hook_length_formula).
s_partition_lambda	axiom	partition lambda		Iter-3 imported axiom (s_partition_lambda).
s_standard_young_tableau	axiom	standard young tableau		Iter-3 imported axiom (s_standard_young_tableau).
s_hook_formula_conjecture	state	hook formula conjecture		Iter-3 imported state (s_hook_formula_conjecture).
s_hook_walk_bijection_setup	state	hook walk bijection setup		Iter-3 imported state (s_hook_walk_bijection_setup).
s_uniform_random_tableau_distribution_argument	state	uniform random tableau distribution argument		Iter-3 imported state (s_uniform_random_tableau_distribution_argument).
s_polya_enumeration_theorem	theorem	Pólya enumeration theorem		Iter-3 imported theorem (s_polya_enumeration_theorem).
s_group_action_on_set_of_colorings	axiom	group action on set of colorings		Iter-3 imported axiom (s_group_action_on_set_of_colorings).
s_cycle_index_polynomial	state	cycle index polynomial		Iter-3 imported state (s_cycle_index_polynomial).
s_burnside_orbit_counting	theorem	Burnside's lemma (orbit counting)		Iter-3 imported theorem (s_burnside_orbit_counting).
s_incidence_relation_R	state	incidence relation R		Iter-3 imported state (s_incidence_relation_R).
s_double_count_fix_stabilizer_identity	state	double count fix stabilizer identity		Iter-3 imported state (s_double_count_fix_stabilizer_identity).
s_lagrange_inversion_theorem	theorem	Lagrange inversion theorem		Iter-3 imported theorem (s_lagrange_inversion_theorem).
s_formal_power_series_ring	axiom	formal power series ring		Iter-3 imported axiom (s_formal_power_series_ring).
s_compositional_inverse_problem	axiom	compositional inverse problem		Iter-3 imported axiom (s_compositional_inverse_problem).
s_lagrange_inversion_target	state	lagrange inversion target		Iter-3 imported state (s_lagrange_inversion_target).
s_residue_calculation_of_coefficient	state	residue calculation of coefficient		Iter-3 imported state (s_residue_calculation_of_coefficient).
s_macmahon_master_theorem	theorem	MacMahon's master theorem		Iter-3 imported theorem (s_macmahon_master_theorem).
s_macmahon_target	state	macmahon target		Iter-3 imported state (s_macmahon_target).
s_residue_extraction_diagonal_coefficient	state	residue extraction diagonal coefficient		Iter-3 imported state (s_residue_extraction_diagonal_coefficient).
s_stanley_reciprocity_theorem	theorem	Stanley's reciprocity theorem		Iter-3 imported theorem (s_stanley_reciprocity_theorem).
s_rational_cone_in_R_n	axiom	rational cone in R n		Iter-3 imported axiom (s_rational_cone_in_R_n).
s_lattice_point_enumerator	axiom	lattice point enumerator		Iter-3 imported axiom (s_lattice_point_enumerator).
s_reciprocity_target	state	reciprocity target		Iter-3 imported state (s_reciprocity_target).
s_polar_duality_on_cones	state	polar duality on cones		Iter-3 imported state (s_polar_duality_on_cones).
s_five_color_theorem	theorem	Five color theorem		Iter-3 imported theorem (s_five_color_theorem).
s_5_color_target	state	5 color target		Iter-3 imported state (s_5_color_target).
s_min_degree_at_most_five_lemma	state	min degree at most five lemma		Iter-3 imported state (s_min_degree_at_most_five_lemma).
s_kempe_chain_recoloring_argument	state	kempe chain recoloring argument		Iter-3 imported state (s_kempe_chain_recoloring_argument).
s_brooks_theorem	theorem	Brooks' theorem		Iter-3 imported theorem (s_brooks_theorem).
s_brooks_target	state	brooks target		Iter-3 imported state (s_brooks_target).
s_good_vertex_ordering_with_two_nonadjacent_priors	state	good vertex ordering with two nonadjacent priors		Iter-3 imported state (s_good_vertex_ordering_with_two_nonadjacent_priors).
s_vizing_theorem	theorem	Vizing's theorem		Iter-3 imported theorem (s_vizing_theorem).
s_edge_chromatic_number	axiom	edge chromatic number		Iter-3 imported axiom (s_edge_chromatic_number).
s_vizing_target	state	vizing target		Iter-3 imported state (s_vizing_target).
s_vizing_fan_recoloring_setup	state	vizing fan recoloring setup		Iter-3 imported state (s_vizing_fan_recoloring_setup).
s_grotzsch_theorem	theorem	Grötzsch's theorem		Iter-3 imported theorem (s_grotzsch_theorem).
s_triangle_free_graph_property	axiom	triangle free graph property		Iter-3 imported axiom (s_triangle_free_graph_property).
s_grotzsch_target	state	grotzsch target		Iter-3 imported state (s_grotzsch_target).
s_reducible_configurations_for_3_coloring	state	reducible configurations for 3 coloring		Iter-3 imported state (s_reducible_configurations_for_3_coloring).
s_kuratowski_theorem	theorem	Kuratowski's theorem (planarity)		Iter-3 imported theorem (s_kuratowski_theorem).
s_planar_embedding_definition	axiom	planar embedding definition		Iter-3 imported axiom (s_planar_embedding_definition).
s_kuratowski_target	state	kuratowski target		Iter-3 imported state (s_kuratowski_target).
s_minimal_non_planar_obstructions_K5_K33	state	minimal non planar obstructions K5 K33		Iter-3 imported state (s_minimal_non_planar_obstructions_K5_K33).
s_wagner_theorem	theorem	Wagner's theorem		Iter-3 imported theorem (s_wagner_theorem).
s_wagner_target	state	wagner target		Iter-3 imported state (s_wagner_target).
s_minor_vs_topological_minor_equivalence	state	minor vs topological minor equivalence		Iter-3 imported state (s_minor_vs_topological_minor_equivalence).
s_fary_theorem	theorem	Fáry's theorem		Iter-3 imported theorem (s_fary_theorem).
s_straight_line_embedding_definition	axiom	straight line embedding definition		Iter-3 imported axiom (s_straight_line_embedding_definition).
s_fary_target	state	fary target		Iter-3 imported state (s_fary_target).
s_inductive_straightening_of_edges	state	inductive straightening of edges		Iter-3 imported state (s_inductive_straightening_of_edges).
s_steinitz_theorem	theorem	Steinitz's theorem (polytopes ↔ planar 3-connected)		Iter-3 imported theorem (s_steinitz_theorem).
s_steinitz_target	state	steinitz target		Iter-3 imported state (s_steinitz_target).
s_tutte_rubber_band_realisation	state	tutte rubber band realisation		Iter-3 imported state (s_tutte_rubber_band_realisation).
s_lovasz_perfect_graph_theorem	theorem	Perfect graph theorem (Lovász)		Iter-3 imported theorem (s_lovasz_perfect_graph_theorem).
s_perfect_graph_definition	axiom	perfect graph definition		Iter-3 imported axiom (s_perfect_graph_definition).
s_perfect_complement_target	state	perfect complement target		Iter-3 imported state (s_perfect_complement_target).
s_complement_duality_on_perfection	state	complement duality on perfection		Iter-3 imported state (s_complement_duality_on_perfection).
s_replication_lemma	state	replication lemma		Iter-3 imported state (s_replication_lemma).
s_strong_perfect_graph_theorem	theorem	Strong perfect graph theorem		Iter-3 imported theorem (s_strong_perfect_graph_theorem).
s_spgt_target	state	spgt target		Iter-3 imported state (s_spgt_target).
s_berge_graph_decomposition_skeleton	state	berge graph decomposition skeleton		Iter-3 imported state (s_berge_graph_decomposition_skeleton).
s_lovasz_local_lemma	theorem	Lovász local lemma		Iter-3 imported theorem (s_lovasz_local_lemma).
s_dependency_graph_of_events	axiom	dependency graph of events		Iter-3 imported axiom (s_dependency_graph_of_events).
s_lll_target	state	lll target		Iter-3 imported state (s_lll_target).
s_weight_certificate_for_lll	state	weight certificate for lll		Iter-3 imported state (s_weight_certificate_for_lll).
s_erdos_ramsey_lower_bound	theorem	Erdős probabilistic lower bound on R(k,k)		Iter-3 imported theorem (s_erdos_ramsey_lower_bound).
s_uniform_random_two_coloring_of_K_n	state	uniform random two coloring of K n		Iter-3 imported state (s_uniform_random_two_coloring_of_K_n).
s_expected_count_less_than_one	state	expected count less than one		Iter-3 imported state (s_expected_count_less_than_one).
s_spencer_six_sigma_theorem	theorem	Spencer's six standard deviations		Iter-3 imported theorem (s_spencer_six_sigma_theorem).
s_set_system_on_n_points	axiom	set system on n points		Iter-3 imported axiom (s_set_system_on_n_points).
s_signed_coloring_problem	axiom	signed coloring problem		Iter-3 imported axiom (s_signed_coloring_problem).
s_spencer_target	state	spencer target		Iter-3 imported state (s_spencer_target).
s_partial_coloring_lemma_entropy_bucket	state	partial coloring lemma entropy bucket		Iter-3 imported state (s_partial_coloring_lemma_entropy_bucket).
s_beck_fiala_theorem	theorem	Beck–Fiala theorem		Iter-3 imported theorem (s_beck_fiala_theorem).
s_beck_fiala_target	state	beck fiala target		Iter-3 imported state (s_beck_fiala_target).
s_floating_lp_iteration	state	floating lp iteration		Iter-3 imported state (s_floating_lp_iteration).
s_beck_three_coloring_theorem	theorem	Beck's three-coloring theorem		Iter-3 imported theorem (s_beck_three_coloring_theorem).
s_beck_three_color_target	state	beck three color target		Iter-3 imported state (s_beck_three_color_target).
s_random_two_coloring_and_local_correction	state	random two coloring and local correction		Iter-3 imported state (s_random_two_coloring_and_local_correction).
s_erdos_renyi_connectivity_threshold	theorem	Erdős–Rényi threshold for connectivity		Iter-3 imported theorem (s_erdos_renyi_connectivity_threshold).
s_random_graph_G_n_p_model	axiom	random graph G n p model		Iter-3 imported axiom (s_random_graph_G_n_p_model).
s_threshold_target	state	threshold target		Iter-3 imported state (s_threshold_target).
s_first_moment_isolated_vertices	state	first moment isolated vertices		Iter-3 imported state (s_first_moment_isolated_vertices).
s_second_moment_estimate	state	second moment estimate		Iter-3 imported state (s_second_moment_estimate).
s_erdos_szekeres_theorem	theorem	Erdős–Szekeres monotone subsequence theorem		Iter-3 imported theorem (s_erdos_szekeres_theorem).
s_finite_sequence_of_reals	axiom	finite sequence of reals		Iter-3 imported axiom (s_finite_sequence_of_reals).
s_monotone_subsequence_problem	axiom	monotone subsequence problem		Iter-3 imported axiom (s_monotone_subsequence_problem).
s_es_target	state	es target		Iter-3 imported state (s_es_target).
s_pair_labels_distinct	state	pair labels distinct		Iter-3 imported state (s_pair_labels_distinct).
s_friendship_theorem	theorem	Friendship theorem (Erdős–Rényi–Sós)		Iter-3 imported theorem (s_friendship_theorem).
s_friendship_condition_every_pair_unique_common_neighbor	axiom	friendship condition every pair unique common neighbor		Iter-3 imported axiom (s_friendship_condition_every_pair_unique_common_neighbor).
s_friendship_target	state	friendship target		Iter-3 imported state (s_friendship_target).
s_adjacency_spectral_argument	state	adjacency spectral argument		Iter-3 imported state (s_adjacency_spectral_argument).
s_ore_theorem	theorem	Ore's theorem		Iter-3 imported theorem (s_ore_theorem).
s_hamilton_cycle_problem	axiom	hamilton cycle problem		Iter-3 imported axiom (s_hamilton_cycle_problem).
s_ore_target	state	ore target		Iter-3 imported state (s_ore_target).
s_longest_path_endpoint_degree_sum_bound	state	longest path endpoint degree sum bound		Iter-3 imported state (s_longest_path_endpoint_degree_sum_bound).
s_bondy_chvatal_closure_theorem	theorem	Bondy–Chvátal closure theorem		Iter-3 imported theorem (s_bondy_chvatal_closure_theorem).
s_closure_target	state	closure target		Iter-3 imported state (s_closure_target).
s_closure_operator_well_defined	state	closure operator well defined		Iter-3 imported state (s_closure_operator_well_defined).
s_chvatal_erdos_theorem	theorem	Chvátal–Erdős theorem		Iter-3 imported theorem (s_chvatal_erdos_theorem).
s_chvatal_erdos_target	state	chvatal erdos target		Iter-3 imported state (s_chvatal_erdos_target).
s_disjoint_paths_argument	state	disjoint paths argument		Iter-3 imported state (s_disjoint_paths_argument).
s_posa_rotation_extension_theorem	theorem	Pósa's rotation-extension lemma		Iter-3 imported theorem (s_posa_rotation_extension_theorem).
s_posa_target	state	posa target		Iter-3 imported state (s_posa_target).
s_endpoint_set_grows_under_rotation	state	endpoint set grows under rotation		Iter-3 imported state (s_endpoint_set_grows_under_rotation).
s_tutte_hamilton_4connected_planar	theorem	Tutte's theorem on Hamilton cycles in 4-connected planar		Iter-3 imported theorem (s_tutte_hamilton_4connected_planar).
s_tutte_4conn_target	state	tutte 4conn target		Iter-3 imported state (s_tutte_4conn_target).
s_tutte_path_construction	state	tutte path construction		Iter-3 imported state (s_tutte_path_construction).
s_fleischner_theorem	theorem	Fleischner's theorem		Iter-3 imported theorem (s_fleischner_theorem).
s_fleischner_target	state	fleischner target		Iter-3 imported state (s_fleischner_target).
s_dfs_ear_traversal_yields_hamilton_cycle_in_square	state	dfs ear traversal yields hamilton cycle in square		Iter-3 imported state (s_dfs_ear_traversal_yields_hamilton_cycle_in_square).
s_cauchy_davenport_theorem	theorem	Cauchy–Davenport theorem		Iter-3 imported theorem (s_cauchy_davenport_theorem).
s_subsets_of_Z_p	axiom	subsets of Z p		Iter-3 imported axiom (s_subsets_of_Z_p).
s_cauchy_davenport_target	state	cauchy davenport target		Iter-3 imported state (s_cauchy_davenport_target).
s_nullstellensatz_certificate_for_sumset	state	nullstellensatz certificate for sumset		Iter-3 imported state (s_nullstellensatz_certificate_for_sumset).
s_davenport_constant_theorem_for_p_groups	theorem	Davenport constant for abelian groups		Iter-3 imported theorem (s_davenport_constant_theorem_for_p_groups).
s_finite_abelian_group	axiom	finite abelian group		Iter-3 imported axiom (s_finite_abelian_group).
s_zero_sum_sequence_problem	axiom	zero sum sequence problem		Iter-3 imported axiom (s_zero_sum_sequence_problem).
s_davenport_target	state	davenport target		Iter-3 imported state (s_davenport_target).
s_chevalley_warning_for_abelian_p_group	state	chevalley warning for abelian p group		Iter-3 imported state (s_chevalley_warning_for_abelian_p_group).
s_kneser_addition_theorem	theorem	Kneser's addition theorem		Iter-3 imported theorem (s_kneser_addition_theorem).
s_sumset_definition	axiom	sumset definition		Iter-3 imported axiom (s_sumset_definition).
s_kneser_target	state	kneser target		Iter-3 imported state (s_kneser_target).
s_quotient_by_stabilizer_gives_strict_sumset_growth	state	quotient by stabilizer gives strict sumset growth		Iter-3 imported state (s_quotient_by_stabilizer_gives_strict_sumset_growth).
s_plunnecke_ruzsa_inequality	theorem	Plünnecke–Ruzsa inequality		Iter-3 imported theorem (s_plunnecke_ruzsa_inequality).
s_plunnecke_target	state	plunnecke target		Iter-3 imported state (s_plunnecke_target).
s_petridis_magnification_argument	state	petridis magnification argument		Iter-3 imported state (s_petridis_magnification_argument).
s_freiman_theorem	theorem	Freiman's theorem		Iter-3 imported theorem (s_freiman_theorem).
s_freiman_target	state	freiman target		Iter-3 imported state (s_freiman_target).
s_ruzsa_covering_applied	state	ruzsa covering applied		Iter-3 imported state (s_ruzsa_covering_applied).
s_bogolyubov_bohr_set_inside_2A_minus_2A	state	bogolyubov bohr set inside 2A minus 2A		Iter-3 imported state (s_bogolyubov_bohr_set_inside_2A_minus_2A).
s_behrend_construction_theorem	theorem	Behrend's construction (large AP-free set)		Iter-3 imported theorem (s_behrend_construction_theorem).
s_three_term_AP_free_property	axiom	three term AP free property		Iter-3 imported axiom (s_three_term_AP_free_property).
s_behrend_target	state	behrend target		Iter-3 imported state (s_behrend_target).
s_sphere_in_Z_m_d_is_AP_free	state	sphere in Z m d is AP free		Iter-3 imported state (s_sphere_in_Z_m_d_is_AP_free).
s_roth_theorem_3AP	theorem	Roth's theorem on 3-APs		Iter-3 imported theorem (s_roth_theorem_3AP).
s_roth_target	state	roth target		Iter-3 imported state (s_roth_target).
s_large_fourier_coefficient_dichotomy	state	large fourier coefficient dichotomy		Iter-3 imported state (s_large_fourier_coefficient_dichotomy).
s_density_increment_iteration	state	density increment iteration		Iter-3 imported state (s_density_increment_iteration).
s_kleene_recursion_theorem	theorem	Recursion theorem (Kleene's second)		Iter-3 imported theorem (s_kleene_recursion_theorem).
s_godel_numbering_of_programs	state	godel numbering of programs		Iter-3 imported state (s_godel_numbering_of_programs).
s_diagonal_program_construction	state	diagonal program construction		Iter-3 imported state (s_diagonal_program_construction).
s_smn_theorem	theorem	Smn theorem (parameter theorem)		Iter-3 imported theorem (s_smn_theorem).
s_uniform_program_index	state	uniform program index		Iter-3 imported state (s_uniform_program_index).
s_re_not_recursive	theorem	Existence of RE but non-recursive sets		Iter-3 imported theorem (s_re_not_recursive).
s_halting_set_K	state	halting set K		Iter-3 imported state (s_halting_set_K).
s_K_is_re	state	K is re		Iter-3 imported state (s_K_is_re).
s_karp_lipton_theorem	theorem	Karp–Lipton theorem (NP ⊆ P/poly ⇒ PH = Σ₂)		Iter-3 imported theorem (s_karp_lipton_theorem).
s_hypothetical_sat_circuit_family	state	hypothetical sat circuit family		Iter-3 imported state (s_hypothetical_sat_circuit_family).
s_self_reducible_circuit_finder	state	self reducible circuit finder		Iter-3 imported state (s_self_reducible_circuit_finder).
s_toda_theorem	theorem	Toda's theorem (PH ⊆ P^#P)		Iter-3 imported theorem (s_toda_theorem).
s_isolation_lemma_witness	state	isolation lemma witness		Iter-3 imported state (s_isolation_lemma_witness).
s_ph_inside_bpp_parity_p	state	ph inside bpp parity p		Iter-3 imported state (s_ph_inside_bpp_parity_p).
s_parity_p_in_p_sharp_p	state	parity p in p sharp p		Iter-3 imported state (s_parity_p_in_p_sharp_p).
s_sipser_gacs_lautemann	theorem	Sipser–Gács–Lautemann theorem (BPP ⊆ Σ₂)		Iter-3 imported theorem (s_sipser_gacs_lautemann).
s_amplified_bpp_acceptance_set	state	amplified bpp acceptance set		Iter-3 imported state (s_amplified_bpp_acceptance_set).
s_translation_cover_witness	state	translation cover witness		Iter-3 imported state (s_translation_cover_witness).
s_adleman_theorem	theorem	Adleman's theorem (BPP ⊆ P/poly)		Iter-3 imported theorem (s_adleman_theorem).
s_amplified_bpp_machine	state	amplified bpp machine		Iter-3 imported state (s_amplified_bpp_machine).
s_universal_advice_string	state	universal advice string		Iter-3 imported state (s_universal_advice_string).
s_yao_minimax_principle	theorem	Yao's minimax principle		Iter-3 imported theorem (s_yao_minimax_principle).
s_randomized_alg_as_mixed_strategy	state	randomized alg as mixed strategy		Iter-3 imported state (s_randomized_alg_as_mixed_strategy).
s_razborov_smolensky	theorem	Razborov–Smolensky lower bound on AC⁰[p]		Iter-3 imported theorem (s_razborov_smolensky).
s_acc0_p_circuit_model	state	acc0 p circuit model		Iter-3 imported state (s_acc0_p_circuit_model).
s_low_degree_polynomial_approximation	state	low degree polynomial approximation		Iter-3 imported state (s_low_degree_polynomial_approximation).
s_parity_not_in_ac0	theorem	Furst–Saxe–Sipser (parity ∉ AC⁰)		Iter-3 imported theorem (s_parity_not_in_ac0).
s_ac0_circuit_model	state	ac0 circuit model		Iter-3 imported state (s_ac0_circuit_model).
s_random_restriction_argument	state	random restriction argument		Iter-3 imported state (s_random_restriction_argument).
s_depth_reduction_to_constant	state	depth reduction to constant		Iter-3 imported state (s_depth_reduction_to_constant).
s_hastad_switching_lemma	theorem	Håstad's switching lemma		Iter-3 imported theorem (s_hastad_switching_lemma).
s_random_restriction_on_dnf	state	random restriction on dnf		Iter-3 imported state (s_random_restriction_on_dnf).
s_encoding_of_bad_paths	state	encoding of bad paths		Iter-3 imported state (s_encoding_of_bad_paths).
s_isolation_lemma_mvv	theorem	Isolation lemma (Mulmuley–Vazirani–Vazirani)		Iter-3 imported theorem (s_isolation_lemma_mvv).
s_random_weighted_set_system	state	random weighted set system		Iter-3 imported state (s_random_weighted_set_system).
s_schwartz_zippel_lemma	theorem	Schwartz–Zippel lemma		Iter-3 imported theorem (s_schwartz_zippel_lemma).
s_polynomial_and_sample_set	state	polynomial and sample set		Iter-3 imported state (s_polynomial_and_sample_set).
s_inductive_root_bound	state	inductive root bound		Iter-3 imported state (s_inductive_root_bound).
s_valiant_vazirani	theorem	Valiant–Vazirani theorem (UNIQUE-SAT randomized hardness)		Iter-3 imported theorem (s_valiant_vazirani).
s_sat_instance_with_unknown_count	state	sat instance with unknown count		Iter-3 imported state (s_sat_instance_with_unknown_count).
s_hashed_sat_instance	state	hashed sat instance		Iter-3 imported state (s_hashed_sat_instance).
s_ip_equals_pspace	theorem	IP = PSPACE (Shamir)		Iter-3 imported theorem (s_ip_equals_pspace).
s_tqbf_canonical_form	state	tqbf canonical form		Iter-3 imported state (s_tqbf_canonical_form).
s_arithmetized_tqbf	state	arithmetized tqbf		Iter-3 imported state (s_arithmetized_tqbf).
s_sum_check_protocol	state	sum check protocol		Iter-3 imported state (s_sum_check_protocol).
s_mip_equals_nexp	theorem	MIP = NEXP (Babai–Fortnow–Lund)		Iter-3 imported theorem (s_mip_equals_nexp).
s_multi_prover_model	state	multi prover model		Iter-3 imported state (s_multi_prover_model).
s_scaled_arithmetization_nexp	state	scaled arithmetization nexp		Iter-3 imported state (s_scaled_arithmetization_nexp).
s_two_prover_oracle_protocol	state	two prover oracle protocol		Iter-3 imported state (s_two_prover_oracle_protocol).
s_gmw_zk_for_np	theorem	GMW zero-knowledge for NP		Iter-3 imported theorem (s_gmw_zk_for_np).
s_three_coloring_canonical_target	state	three coloring canonical target		Iter-3 imported state (s_three_coloring_canonical_target).
s_commit_and_challenge_protocol	state	commit and challenge protocol		Iter-3 imported state (s_commit_and_challenge_protocol).
s_shannon_channel_coding	theorem	Shannon noisy channel coding theorem		Iter-3 imported theorem (s_shannon_channel_coding).
s_dmc_capacity_definition	state	dmc capacity definition		Iter-3 imported state (s_dmc_capacity_definition).
s_random_codebook_achievability	state	random codebook achievability		Iter-3 imported state (s_random_codebook_achievability).
s_kraft_mcmillan_inequality	theorem	Kraft–McMillan inequality		Iter-3 imported theorem (s_kraft_mcmillan_inequality).
s_uniquely_decodable_code_model	state	uniquely decodable code model		Iter-3 imported state (s_uniquely_decodable_code_model).
s_extension_counting_bound	state	extension counting bound		Iter-3 imported state (s_extension_counting_bound).
s_slepian_wolf_theorem	theorem	Slepian–Wolf theorem		Iter-3 imported theorem (s_slepian_wolf_theorem).
s_correlated_source_model	state	correlated source model		Iter-3 imported state (s_correlated_source_model).
s_random_binning_scheme	state	random binning scheme		Iter-3 imported state (s_random_binning_scheme).
s_wyner_ziv_theorem	theorem	Wyner–Ziv theorem		Iter-3 imported theorem (s_wyner_ziv_theorem).
s_side_info_source_model	state	side info source model		Iter-3 imported state (s_side_info_source_model).
s_auxiliary_u_construction	state	auxiliary u construction		Iter-3 imported state (s_auxiliary_u_construction).
s_singleton_bound	theorem	Singleton bound		Iter-3 imported theorem (s_singleton_bound).
s_block_code_model	state	block code model		Iter-3 imported state (s_block_code_model).
s_punctured_code_injection	state	punctured code injection		Iter-3 imported state (s_punctured_code_injection).
s_plotkin_bound	theorem	Plotkin bound		Iter-3 imported theorem (s_plotkin_bound).
s_high_distance_binary_code	state	high distance binary code		Iter-3 imported state (s_high_distance_binary_code).
s_double_count_pairwise_distances	state	double count pairwise distances		Iter-3 imported state (s_double_count_pairwise_distances).
s_hamming_bound	theorem	Hamming (sphere-packing) bound		Iter-3 imported theorem (s_hamming_bound).
s_t_error_correcting_code	state	t error correcting code		Iter-3 imported state (s_t_error_correcting_code).
s_disjoint_hamming_balls	state	disjoint hamming balls		Iter-3 imported state (s_disjoint_hamming_balls).
s_gilbert_varshamov_bound	theorem	Gilbert–Varshamov bound		Iter-3 imported theorem (s_gilbert_varshamov_bound).
s_greedy_code_construction	state	greedy code construction		Iter-3 imported state (s_greedy_code_construction).
s_mrrw_lp_bound	theorem	MRRW (linear programming) bound		Iter-3 imported theorem (s_mrrw_lp_bound).
s_distance_distribution_polynomials	state	distance distribution polynomials		Iter-3 imported state (s_distance_distribution_polynomials).
s_delsarte_linear_program	state	delsarte linear program		Iter-3 imported state (s_delsarte_linear_program).
s_mrrw_dual_polynomial	state	mrrw dual polynomial		Iter-3 imported state (s_mrrw_dual_polynomial).
s_reed_solomon_distance	theorem	Reed–Solomon distance / BCH bound		Iter-3 imported theorem (s_reed_solomon_distance).
s_rs_code_definition	state	rs code definition		Iter-3 imported state (s_rs_code_definition).
s_root_count_argument	state	root count argument		Iter-3 imported state (s_root_count_argument).
s_helly_theorem	theorem	Helly's theorem		Iter-3 imported theorem (s_helly_theorem).
s_helly_local_hypothesis	state	helly local hypothesis		Iter-3 imported state (s_helly_local_hypothesis).
s_radon_partition_step	state	radon partition step		Iter-3 imported state (s_radon_partition_step).
s_caratheodory_convex	theorem	Carathéodory's theorem (convex hull)		Iter-3 imported theorem (s_caratheodory_convex).
s_convex_combination_with_k_points	state	convex combination with k points		Iter-3 imported state (s_convex_combination_with_k_points).
s_affine_dependence_relation	state	affine dependence relation		Iter-3 imported state (s_affine_dependence_relation).
s_radon_theorem	theorem	Radon's theorem		Iter-3 imported theorem (s_radon_theorem).
s_d_plus_two_points	state	d plus two points		Iter-3 imported state (s_d_plus_two_points).
s_linear_dependence_in_d_plus_one	state	linear dependence in d plus one		Iter-3 imported state (s_linear_dependence_in_d_plus_one).
s_tverberg_theorem	theorem	Tverberg's theorem		Iter-3 imported theorem (s_tverberg_theorem).
s_tverberg_input_count	state	tverberg input count		Iter-3 imported state (s_tverberg_input_count).
s_deleted_join_configuration_space	state	deleted join configuration space		Iter-3 imported state (s_deleted_join_configuration_space).
s_colored_tverberg	theorem	Colored Tverberg (Bárány–Larman / Blagojević–Matschke–Ziegler)		Iter-3 imported theorem (s_colored_tverberg).
s_rainbow_partition_problem	state	rainbow partition problem		Iter-3 imported state (s_rainbow_partition_problem).
s_chessboard_complex	state	chessboard complex		Iter-3 imported state (s_chessboard_complex).
s_barany_point_selection	theorem	Bárány's point-selection theorem		Iter-3 imported theorem (s_barany_point_selection).
s_simplex_family_on_n_points	state	simplex family on n points		Iter-3 imported state (s_simplex_family_on_n_points).
s_lp_strong_duality	theorem	LP strong duality		Iter-3 imported theorem (s_lp_strong_duality).
s_primal_dual_lp_pair	state	primal dual lp pair		Iter-3 imported state (s_primal_dual_lp_pair).
s_lp_weak_duality	state	lp weak duality		Iter-3 imported state (s_lp_weak_duality).
s_farkas_lemma	theorem	Farkas's lemma		Iter-3 imported theorem (s_farkas_lemma).
s_farkas_alternatives	state	farkas alternatives		Iter-3 imported state (s_farkas_alternatives).
s_separating_hyperplane_for_cone	state	separating hyperplane for cone		Iter-3 imported state (s_separating_hyperplane_for_cone).
s_gordan_theorem	theorem	Gordan's theorem of alternatives		Iter-3 imported theorem (s_gordan_theorem).
s_gordan_alternatives	state	gordan alternatives		Iter-3 imported state (s_gordan_alternatives).
s_stiemke_theorem	theorem	Stiemke's theorem		Iter-3 imported theorem (s_stiemke_theorem).
s_stiemke_alternatives	state	stiemke alternatives		Iter-3 imported state (s_stiemke_alternatives).
s_khachiyan_ellipsoid	theorem	Ellipsoid method polynomial-time (Khachiyan)		Iter-3 imported theorem (s_khachiyan_ellipsoid).
s_separation_oracle_lp	state	separation oracle lp		Iter-3 imported state (s_separation_oracle_lp).
s_ellipsoid_volume_shrinkage_iteration	state	ellipsoid volume shrinkage iteration		Iter-3 imported state (s_ellipsoid_volume_shrinkage_iteration).
s_karmarkar_polynomial_lp	theorem	Karmarkar interior-point polynomial-time		Iter-3 imported theorem (s_karmarkar_polynomial_lp).
s_projective_lp_form	state	projective lp form		Iter-3 imported state (s_projective_lp_form).
s_karmarkar_potential_decrease	state	karmarkar potential decrease		Iter-3 imported state (s_karmarkar_potential_decrease).
s_max_flow_min_cut	theorem	Max-flow min-cut theorem (Ford–Fulkerson)		Iter-3 imported theorem (s_max_flow_min_cut).
s_network_flow_model	state	network flow model		Iter-3 imported state (s_network_flow_model).
s_residual_graph_construction	state	residual graph construction		Iter-3 imported state (s_residual_graph_construction).
s_no_augmenting_path_implies_min_cut	state	no augmenting path implies min cut		Iter-3 imported state (s_no_augmenting_path_implies_min_cut).
s_edmonds_matroid_intersection	theorem	Edmonds' matroid intersection theorem		Iter-3 imported theorem (s_edmonds_matroid_intersection).
s_two_matroids_on_e	state	two matroids on e		Iter-3 imported state (s_two_matroids_on_e).
s_exchange_digraph_construction	state	exchange digraph construction		Iter-3 imported state (s_exchange_digraph_construction).
s_aks_primality_theorem	theorem	AKS primality test (deterministic polynomial-time)		Iter-3 imported theorem (s_aks_primality_theorem).
s_aks_polynomial_identity	state	aks polynomial identity		Iter-3 imported state (s_aks_polynomial_identity).
s_aks_reduced_identity_modulo_x_r_minus_1	state	aks reduced identity modulo x r minus 1		Iter-3 imported state (s_aks_reduced_identity_modulo_x_r_minus_1).
s_introspective_set_bound	state	introspective set bound		Iter-3 imported state (s_introspective_set_bound).
s_cantor_zassenhaus	theorem	Cantor–Zassenhaus factorization		Iter-3 imported theorem (s_cantor_zassenhaus).
s_equal_degree_factor_setting	state	equal degree factor setting		Iter-3 imported state (s_equal_degree_factor_setting).
s_random_h_power_construction	state	random h power construction		Iter-3 imported state (s_random_h_power_construction).
s_berlekamp_factorization	theorem	Berlekamp factorization		Iter-3 imported theorem (s_berlekamp_factorization).
s_quotient_ring_for_f	state	quotient ring for f		Iter-3 imported state (s_quotient_ring_for_f).
s_berlekamp_subalgebra	state	berlekamp subalgebra		Iter-3 imported state (s_berlekamp_subalgebra).
s_subalgebra_crt_decomposition	state	subalgebra crt decomposition		Iter-3 imported state (s_subalgebra_crt_decomposition).
s_lll_algorithm_theorem	theorem	LLL lattice basis reduction		Iter-3 imported theorem (s_lll_algorithm_theorem).
s_lattice_basis_with_gso	state	lattice basis with gso		Iter-3 imported state (s_lattice_basis_with_gso).
s_lll_swap_and_reduce_step	state	lll swap and reduce step		Iter-3 imported state (s_lll_swap_and_reduce_step).
s_lll_potential_decrease	state	lll potential decrease		Iter-3 imported state (s_lll_potential_decrease).
s_master_theorem	theorem	Master theorem for divide-and-conquer recurrences		Iter-3 imported theorem (s_master_theorem).
s_divide_and_conquer_recurrence	state	divide and conquer recurrence		Iter-3 imported state (s_divide_and_conquer_recurrence).
s_recursion_tree_decomposition	state	recursion tree decomposition		Iter-3 imported state (s_recursion_tree_decomposition).
s_pumping_lemma_regular	theorem	Pumping lemma for regular languages		Iter-3 imported theorem (s_pumping_lemma_regular).
s_dfa_with_p_states	state	dfa with p states		Iter-3 imported state (s_dfa_with_p_states).
s_revisited_state_loop	state	revisited state loop		Iter-3 imported state (s_revisited_state_loop).
s_myhill_nerode_theorem	theorem	Myhill–Nerode theorem		Iter-3 imported theorem (s_myhill_nerode_theorem).
s_nerode_equivalence_relation	state	nerode equivalence relation		Iter-3 imported state (s_nerode_equivalence_relation).
s_minimal_dfa_isomorphism	state	minimal dfa isomorphism		Iter-3 imported state (s_minimal_dfa_isomorphism).
s_kleene_regex_dfa	theorem	Kleene's theorem (regex = DFA)		Iter-3 imported theorem (s_kleene_regex_dfa).
s_regex_syntax	state	regex syntax		Iter-3 imported state (s_regex_syntax).
s_thompson_nfa_construction	state	thompson nfa construction		Iter-3 imported state (s_thompson_nfa_construction).
s_chomsky_schutzenberger	theorem	Chomsky–Schützenberger representation theorem		Iter-3 imported theorem (s_chomsky_schutzenberger).
s_cfl_definition	state	cfl definition		Iter-3 imported state (s_cfl_definition).
s_dyck_language_universal	state	dyck language universal		Iter-3 imported state (s_dyck_language_universal).
s_parikh_theorem	theorem	Parikh's theorem		Iter-3 imported theorem (s_parikh_theorem).
s_parikh_map	state	parikh map		Iter-3 imported state (s_parikh_map).
s_cf_parse_tree_pumping_step	state	cf parse tree pumping step		Iter-3 imported state (s_cf_parse_tree_pumping_step).
s_ogden_lemma	theorem	Ogden's lemma		Iter-3 imported theorem (s_ogden_lemma).
s_marked_string_in_cfl	state	marked string in cfl		Iter-3 imported state (s_marked_string_in_cfl).
s_marked_parse_path_pumping	state	marked parse path pumping		Iter-3 imported state (s_marked_parse_path_pumping).
s_rabin_scott_subset_construction	theorem	Rabin–Scott subset construction (NFA = DFA)		Iter-3 imported theorem (s_rabin_scott_subset_construction).
s_nfa_model	state	nfa model		Iter-3 imported state (s_nfa_model).
s_subset_dfa_construction	state	subset dfa construction		Iter-3 imported state (s_subset_dfa_construction).
s_trakhtenbrot_theorem	theorem	Trakhtenbrot's theorem (finite-model satisfiability undecidable)		Iter-3 imported theorem (s_trakhtenbrot_theorem).
s_fo_encoding_of_tm_computation	state	fo encoding of tm computation		Iter-3 imported state (s_fo_encoding_of_tm_computation).
s_finite_model_iff_halting	state	finite model iff halting		Iter-3 imported state (s_finite_model_iff_halting).
s_khinchin_recurrence	theorem	Khinchin recurrence theorem		Iter-3 imported theorem (s_khinchin_recurrence).
s_set_of_positive_measure_A	axiom	set of positive measure A		Iter-3 imported axiom (s_set_of_positive_measure_A).
s_ergodic_average_of_indicator	state	ergodic average of indicator		Iter-3 imported state (s_ergodic_average_of_indicator).
s_orthogonal_projection_onto_invariants	state	orthogonal projection onto invariants		Iter-3 imported state (s_orthogonal_projection_onto_invariants).
s_density_lower_bound_on_return_times	state	density lower bound on return times		Iter-3 imported state (s_density_lower_bound_on_return_times).
s_furstenberg_multiple_recurrence	theorem	Furstenberg multiple recurrence theorem		Iter-3 imported theorem (s_furstenberg_multiple_recurrence).
s_furstenberg_structure_tower	state	furstenberg structure tower		Iter-3 imported state (s_furstenberg_structure_tower).
s_distal_factor_with_AP_returns	state	distal factor with AP returns		Iter-3 imported state (s_distal_factor_with_AP_returns).
s_von_neumann_mean_ergodic	theorem	von Neumann mean ergodic theorem		Iter-3 imported theorem (s_von_neumann_mean_ergodic).
s_unitary_operator_on_hilbert_space	axiom	unitary operator on hilbert space		Iter-3 imported axiom (s_unitary_operator_on_hilbert_space).
s_spectral_measure_on_unit_circle	state	spectral measure on unit circle		Iter-3 imported state (s_spectral_measure_on_unit_circle).
s_orthogonal_decomposition_fixed_plus_coboundary	state	orthogonal decomposition fixed plus coboundary		Iter-3 imported state (s_orthogonal_decomposition_fixed_plus_coboundary).
s_wiener_ergodic_l1	theorem	Wiener ergodic theorem (L¹)		Iter-3 imported theorem (s_wiener_ergodic_l1).
s_L1_function	axiom	L1 function		Iter-3 imported axiom (s_L1_function).
s_hardy_littlewood_maximal_for_T	state	hardy littlewood maximal for T		Iter-3 imported state (s_hardy_littlewood_maximal_for_T).
s_weak_type_inequality_for_averages	state	weak type inequality for averages		Iter-3 imported state (s_weak_type_inequality_for_averages).
s_kingman_subadditive_ergodic	theorem	Kingman subadditive ergodic theorem		Iter-3 imported theorem (s_kingman_subadditive_ergodic).
s_subadditive_cocycle	axiom	subadditive cocycle		Iter-3 imported axiom (s_subadditive_cocycle).
s_invariant_limsup_liminf_pair	state	invariant limsup liminf pair		Iter-3 imported state (s_invariant_limsup_liminf_pair).
s_block_decomposition_with_controlled_error	state	block decomposition with controlled error		Iter-3 imported state (s_block_decomposition_with_controlled_error).
s_oseledec_multiplicative_ergodic	theorem	Oseledec multiplicative ergodic theorem		Iter-3 imported theorem (s_oseledec_multiplicative_ergodic).
s_matrix_cocycle_over_T	axiom	matrix cocycle over T		Iter-3 imported axiom (s_matrix_cocycle_over_T).
s_lyapunov_growth_function	state	lyapunov growth function		Iter-3 imported state (s_lyapunov_growth_function).
s_singular_value_filtration	state	singular value filtration		Iter-3 imported state (s_singular_value_filtration).
s_lyapunov_spectrum_a_e	state	lyapunov spectrum a e		Iter-3 imported state (s_lyapunov_spectrum_a_e).
s_pesin_entropy_formula	theorem	Pesin entropy formula		Iter-3 imported theorem (s_pesin_entropy_formula).
s_smooth_volume_preserving_diffeomorphism	axiom	smooth volume preserving diffeomorphism		Iter-3 imported axiom (s_smooth_volume_preserving_diffeomorphism).
s_oseledec_splitting_for_f	state	oseledec splitting for f		Iter-3 imported state (s_oseledec_splitting_for_f).
s_local_unstable_manifold_family	state	local unstable manifold family		Iter-3 imported state (s_local_unstable_manifold_family).
s_conditional_entropy_along_unstable	state	conditional entropy along unstable		Iter-3 imported state (s_conditional_entropy_along_unstable).
s_margulis_ruelle_inequality	theorem	Margulis–Ruelle inequality		Iter-3 imported theorem (s_margulis_ruelle_inequality).
s_C1_diffeomorphism_of_compact_manifold	axiom	C1 diffeomorphism of compact manifold		Iter-3 imported axiom (s_C1_diffeomorphism_of_compact_manifold).
s_invariant_borel_probability	axiom	invariant borel probability		Iter-3 imported axiom (s_invariant_borel_probability).
s_lyapunov_exponents_ae	state	lyapunov exponents ae		Iter-3 imported state (s_lyapunov_exponents_ae).
s_partition_subordinate_to_expanding	state	partition subordinate to expanding		Iter-3 imported state (s_partition_subordinate_to_expanding).
s_variational_principle_entropy	theorem	Variational principle for topological entropy		Iter-3 imported theorem (s_variational_principle_entropy).
s_continuous_self_map_of_compact_space	axiom	continuous self map of compact space		Iter-3 imported axiom (s_continuous_self_map_of_compact_space).
s_topological_entropy_h_top	state	topological entropy h top		Iter-3 imported state (s_topological_entropy_h_top).
s_measure_entropy_bounded_by_h_top	state	measure entropy bounded by h top		Iter-3 imported state (s_measure_entropy_bounded_by_h_top).
s_supremum_attained_by_invariant_measure	state	supremum attained by invariant measure		Iter-3 imported state (s_supremum_attained_by_invariant_measure).
s_krylov_bogolyubov_invariant_measure	theorem	Krylov–Bogolyubov existence of invariant measures		Iter-3 imported theorem (s_krylov_bogolyubov_invariant_measure).
s_cesaro_average_of_dirac_orbits	state	cesaro average of dirac orbits		Iter-3 imported state (s_cesaro_average_of_dirac_orbits).
s_weak_star_limit_point_of_averages	state	weak star limit point of averages		Iter-3 imported state (s_weak_star_limit_point_of_averages).
s_hopf_decomposition	theorem	Hopf decomposition theorem		Iter-3 imported theorem (s_hopf_decomposition).
s_conservative_dissipative_split	state	conservative dissipative split		Iter-3 imported state (s_conservative_dissipative_split).
s_invariant_partition_C_D	state	invariant partition C D		Iter-3 imported state (s_invariant_partition_C_D).
s_ergodic_decomposition_theorem	theorem	Ergodic decomposition theorem		Iter-3 imported theorem (s_ergodic_decomposition_theorem).
s_invariant_probability_measure	axiom	invariant probability measure		Iter-3 imported axiom (s_invariant_probability_measure).
s_conditional_measure_family	state	conditional measure family		Iter-3 imported state (s_conditional_measure_family).
s_ergodic_components_disintegration	state	ergodic components disintegration		Iter-3 imported state (s_ergodic_components_disintegration).
s_bowen_specification_unique_equilibrium	theorem	Bowen's specification property implies unique equilibrium		Iter-3 imported theorem (s_bowen_specification_unique_equilibrium).
s_expansive_homeomorphism_with_specification	axiom	expansive homeomorphism with specification		Iter-3 imported axiom (s_expansive_homeomorphism_with_specification).
s_holder_potential_phi	axiom	holder potential phi		Iter-3 imported axiom (s_holder_potential_phi).
s_bowen_partition_function_Z_n	state	bowen partition function Z n		Iter-3 imported state (s_bowen_partition_function_Z_n).
s_topological_pressure_P_phi	state	topological pressure P phi		Iter-3 imported state (s_topological_pressure_P_phi).
s_eigenmeasure_for_L_phi	state	eigenmeasure for L phi		Iter-3 imported state (s_eigenmeasure_for_L_phi).
s_bowen_dimension_formula	theorem	Bowen formula for Hausdorff dimension of repellers		Iter-3 imported theorem (s_bowen_dimension_formula).
s_conformal_expanding_repeller_J	axiom	conformal expanding repeller J		Iter-3 imported axiom (s_conformal_expanding_repeller_J).
s_geometric_pressure_function_P_s	state	geometric pressure function P s		Iter-3 imported state (s_geometric_pressure_function_P_s).
s_bowen_root_s_star	state	bowen root s star		Iter-3 imported state (s_bowen_root_s_star).
s_curtis_hedlund_lyndon	theorem	Curtis–Hedlund–Lyndon theorem		Iter-3 imported theorem (s_curtis_hedlund_lyndon).
s_shift_space_A_Z	axiom	shift space A Z		Iter-3 imported axiom (s_shift_space_A_Z).
s_continuous_shift_commuting_map	axiom	continuous shift commuting map		Iter-3 imported axiom (s_continuous_shift_commuting_map).
s_uniform_continuity_via_compactness	state	uniform continuity via compactness		Iter-3 imported state (s_uniform_continuity_via_compactness).
s_local_rule_of_radius_r	state	local rule of radius r		Iter-3 imported state (s_local_rule_of_radius_r).
s_williams_shift_equivalence	theorem	Williams classification of SFTs up to shift equivalence		Iter-3 imported theorem (s_williams_shift_equivalence).
s_subshift_of_finite_type	axiom	subshift of finite type		Iter-3 imported axiom (s_subshift_of_finite_type).
s_transition_matrix_presentation_A	state	transition matrix presentation A		Iter-3 imported state (s_transition_matrix_presentation_A).
s_strong_shift_equivalence_chain	state	strong shift equivalence chain		Iter-3 imported state (s_strong_shift_equivalence_chain).
s_singer_theorem	theorem	Singer's theorem (negative Schwarzian implies bounded attractors)		Iter-3 imported theorem (s_singer_theorem).
s_interval_map_with_negative_schwarzian	axiom	interval map with negative schwarzian		Iter-3 imported axiom (s_interval_map_with_negative_schwarzian).
s_negative_schwarzian_preserved_under_iteration	state	negative schwarzian preserved under iteration		Iter-3 imported state (s_negative_schwarzian_preserved_under_iteration).
s_basin_with_no_interior_critical_point_or_boundary	state	basin with no interior critical point or boundary		Iter-3 imported state (s_basin_with_no_interior_critical_point_or_boundary).
s_thurston_pcf_rigidity	theorem	Misiurewicz–Thurston rigidity for postcritically finite maps		Iter-3 imported theorem (s_thurston_pcf_rigidity).
s_postcritically_finite_branched_cover_of_S2	axiom	postcritically finite branched cover of S2		Iter-3 imported axiom (s_postcritically_finite_branched_cover_of_S2).
s_teichmuller_pullback_map_sigma_f	state	teichmuller pullback map sigma f		Iter-3 imported state (s_teichmuller_pullback_map_sigma_f).
s_fixed_point_or_obstruction_dichotomy	state	fixed point or obstruction dichotomy		Iter-3 imported state (s_fixed_point_or_obstruction_dichotomy).
s_hadamard_perron_stable_manifold	theorem	Hadamard–Perron stable/unstable manifold theorem		Iter-3 imported theorem (s_hadamard_perron_stable_manifold).
s_hyperbolic_fixed_point_of_diffeomorphism	axiom	hyperbolic fixed point of diffeomorphism		Iter-3 imported axiom (s_hyperbolic_fixed_point_of_diffeomorphism).
s_graph_transform_operator	state	graph transform operator		Iter-3 imported state (s_graph_transform_operator).
s_invariant_lipschitz_graph_W_s	state	invariant lipschitz graph W s		Iter-3 imported state (s_invariant_lipschitz_graph_W_s).
s_anosov_closing_lemma	theorem	Anosov closing lemma		Iter-3 imported theorem (s_anosov_closing_lemma).
s_anosov_diffeomorphism	axiom	anosov diffeomorphism		Iter-3 imported axiom (s_anosov_diffeomorphism).
s_epsilon_pseudo_periodic_orbit	axiom	epsilon pseudo periodic orbit		Iter-3 imported axiom (s_epsilon_pseudo_periodic_orbit).
s_local_product_box	state	local product box		Iter-3 imported state (s_local_product_box).
s_unique_periodic_orbit_near_pseudo	state	unique periodic orbit near pseudo		Iter-3 imported state (s_unique_periodic_orbit_near_pseudo).
s_anosov_shadowing_lemma	theorem	Anosov shadowing lemma		Iter-3 imported theorem (s_anosov_shadowing_lemma).
s_delta_pseudo_orbit	axiom	delta pseudo orbit		Iter-3 imported axiom (s_delta_pseudo_orbit).
s_correction_sequence_space	state	correction sequence space		Iter-3 imported state (s_correction_sequence_space).
s_unique_shadowing_orbit	state	unique shadowing orbit		Iter-3 imported state (s_unique_shadowing_orbit).
s_smale_horseshoe	theorem	Smale horseshoe		Iter-3 imported theorem (s_smale_horseshoe).
s_diffeomorphism_with_transverse_homoclinic_point	axiom	diffeomorphism with transverse homoclinic point		Iter-3 imported axiom (s_diffeomorphism_with_transverse_homoclinic_point).
s_markov_rectangle_R	state	markov rectangle R		Iter-3 imported state (s_markov_rectangle_R).
s_smale_birkhoff_homoclinic	theorem	Smale–Birkhoff homoclinic theorem		Iter-3 imported theorem (s_smale_birkhoff_homoclinic).
s_iterated_neighborhood_of_homoclinic	state	iterated neighborhood of homoclinic		Iter-3 imported state (s_iterated_neighborhood_of_homoclinic).
s_horseshoe_in_f_N	state	horseshoe in f N		Iter-3 imported state (s_horseshoe_in_f_N).
s_smale_spectral_decomposition	theorem	Smale spectral decomposition theorem		Iter-3 imported theorem (s_smale_spectral_decomposition).
s_axiom_A_diffeomorphism	axiom	axiom A diffeomorphism		Iter-3 imported axiom (s_axiom_A_diffeomorphism).
s_non_wandering_set_Omega	state	non wandering set Omega		Iter-3 imported state (s_non_wandering_set_Omega).
s_basic_set_partition	state	basic set partition		Iter-3 imported state (s_basic_set_partition).
s_srb_measure_existence	theorem	Sinai–Ruelle–Bowen (SRB) measure existence for Axiom A		Iter-3 imported theorem (s_srb_measure_existence).
s_axiom_A_attractor	axiom	axiom A attractor		Iter-3 imported axiom (s_axiom_A_attractor).
s_markov_partition_of_attractor	state	markov partition of attractor		Iter-3 imported state (s_markov_partition_of_attractor).
s_sft_conjugate_to_attractor	state	sft conjugate to attractor		Iter-3 imported state (s_sft_conjugate_to_attractor).
s_gibbs_state_for_unstable_potential	state	gibbs state for unstable potential		Iter-3 imported state (s_gibbs_state_for_unstable_potential).
s_manning_volume_entropy	theorem	Manning's theorem (volume entropy = topological entropy)		Iter-3 imported theorem (s_manning_volume_entropy).
s_compact_negatively_curved_riemannian_manifold	axiom	compact negatively curved riemannian manifold		Iter-3 imported axiom (s_compact_negatively_curved_riemannian_manifold).
s_volume_growth_function	state	volume growth function		Iter-3 imported state (s_volume_growth_function).
s_volume_entropy_h_vol	state	volume entropy h vol		Iter-3 imported state (s_volume_entropy_h_vol).
s_topological_entropy_of_geodesic_flow	state	topological entropy of geodesic flow		Iter-3 imported state (s_topological_entropy_of_geodesic_flow).
s_pesin_set_theorem	theorem	Pesin set / Pesin block existence		Iter-3 imported theorem (s_pesin_set_theorem).
s_smooth_diffeomorphism_with_nonzero_lyapunov_spectrum	axiom	smooth diffeomorphism with nonzero lyapunov spectrum		Iter-3 imported axiom (s_smooth_diffeomorphism_with_nonzero_lyapunov_spectrum).
s_lyapunov_charts	state	lyapunov charts		Iter-3 imported state (s_lyapunov_charts).
s_pesin_block_Lambda_ell	state	pesin block Lambda ell		Iter-3 imported state (s_pesin_block_Lambda_ell).
s_mane_bowen_formula	theorem	Mañé–Bowen formula for repellers		Iter-3 imported theorem (s_mane_bowen_formula).
s_conformal_repeller_with_holder_jacobian	axiom	conformal repeller with holder jacobian		Iter-3 imported axiom (s_conformal_repeller_with_holder_jacobian).
s_pressure_function_of_s	state	pressure function of s		Iter-3 imported state (s_pressure_function_of_s).
s_bowen_root_for_repeller	state	bowen root for repeller		Iter-3 imported state (s_bowen_root_for_repeller).
s_bowen_periodic_equidistribution	theorem	Bowen's equidistribution of periodic orbits		Iter-3 imported theorem (s_bowen_periodic_equidistribution).
s_axiom_A_basic_set	axiom	axiom A basic set		Iter-3 imported axiom (s_axiom_A_basic_set).
s_weighted_periodic_orbit_measure	state	weighted periodic orbit measure		Iter-3 imported state (s_weighted_periodic_orbit_measure).
s_convergence_to_equilibrium_state	state	convergence to equilibrium state		Iter-3 imported state (s_convergence_to_equilibrium_state).
s_julia_fatou_dichotomy	theorem	Julia–Fatou dichotomy		Iter-3 imported theorem (s_julia_fatou_dichotomy).
s_rational_map_on_riemann_sphere	axiom	rational map on riemann sphere		Iter-3 imported axiom (s_rational_map_on_riemann_sphere).
s_normality_locus_F_f	state	normality locus F f		Iter-3 imported state (s_normality_locus_F_f).
s_montel_normal_family_criterion	state	montel normal family criterion		Iter-3 imported state (s_montel_normal_family_criterion).
s_sullivan_no_wandering	theorem	Sullivan no-wandering-domains theorem		Iter-3 imported theorem (s_sullivan_no_wandering).
s_teichmuller_space_of_lamination	state	teichmuller space of lamination		Iter-3 imported state (s_teichmuller_space_of_lamination).
s_finite_dimensional_deformation_space	state	finite dimensional deformation space		Iter-3 imported state (s_finite_dimensional_deformation_space).
s_mane_sad_sullivan_stability	theorem	Mañé–Sad–Sullivan structural stability		Iter-3 imported theorem (s_mane_sad_sullivan_stability).
s_lambda_lemma_extension	state	lambda lemma extension		Iter-3 imported state (s_lambda_lemma_extension).
s_douady_hubbard_straightening	theorem	Douady–Hubbard straightening theorem		Iter-3 imported theorem (s_douady_hubbard_straightening).
s_polynomial_like_map_of_degree_d	axiom	polynomial like map of degree d		Iter-3 imported axiom (s_polynomial_like_map_of_degree_d).
s_invariant_beltrami_coefficient	state	invariant beltrami coefficient		Iter-3 imported state (s_invariant_beltrami_coefficient).
s_quasiconformal_conjugacy_to_polynomial	state	quasiconformal conjugacy to polynomial		Iter-3 imported state (s_quasiconformal_conjugacy_to_polynomial).
s_yoccoz_local_connectivity	theorem	Yoccoz's theorem on local connectivity of Mandelbrot at finitely renormalizable points		Iter-3 imported theorem (s_yoccoz_local_connectivity).
s_finitely_renormalizable_quadratic	axiom	finitely renormalizable quadratic		Iter-3 imported axiom (s_finitely_renormalizable_quadratic).
s_yoccoz_puzzle_at_level_n	state	yoccoz puzzle at level n		Iter-3 imported state (s_yoccoz_puzzle_at_level_n).
s_a_priori_moduli_bound	state	a priori moduli bound		Iter-3 imported state (s_a_priori_moduli_bound).
s_mane_hyperbolicity_theorem	theorem	Misiurewicz–Mañé theorem on hyperbolicity from non-recurrent critical orbits		Iter-3 imported theorem (s_mane_hyperbolicity_theorem).
s_rational_map_with_nonrecurrent_critical_orbits	axiom	rational map with nonrecurrent critical orbits		Iter-3 imported axiom (s_rational_map_with_nonrecurrent_critical_orbits).
s_local_expansion_estimate	state	local expansion estimate		Iter-3 imported state (s_local_expansion_estimate).
s_uniform_expansion_on_J	state	uniform expansion on J		Iter-3 imported state (s_uniform_expansion_on_J).
s_brolin_lyubich_measure	theorem	Brolin–Lyubich measure of maximal entropy for rational maps		Iter-3 imported theorem (s_brolin_lyubich_measure).
s_rational_map_of_degree_d_at_least_2	axiom	rational map of degree d at least 2		Iter-3 imported axiom (s_rational_map_of_degree_d_at_least_2).
s_pullback_point_measure_sequence	state	pullback point measure sequence		Iter-3 imported state (s_pullback_point_measure_sequence).
s_potential_theoretic_limit	state	potential theoretic limit		Iter-3 imported state (s_potential_theoretic_limit).
s_ratner_measure_rigidity	theorem	Ratner measure rigidity theorem		Iter-3 imported theorem (s_ratner_measure_rigidity).
s_unipotent_one_parameter_subgroup_u_t	axiom	unipotent one parameter subgroup u t		Iter-3 imported axiom (s_unipotent_one_parameter_subgroup_u_t).
s_lattice_subgroup_Gamma	axiom	lattice subgroup Gamma		Iter-3 imported axiom (s_lattice_subgroup_Gamma).
s_unipotent_q_shift_invariance	state	unipotent q shift invariance		Iter-3 imported state (s_unipotent_q_shift_invariance).
s_extra_invariance_under_subgroup_H	state	extra invariance under subgroup H		Iter-3 imported state (s_extra_invariance_under_subgroup_H).
s_invariant_measure_is_homogeneous	state	invariant measure is homogeneous		Iter-3 imported state (s_invariant_measure_is_homogeneous).
s_ratner_orbit_closure	theorem	Ratner orbit closure theorem		Iter-3 imported theorem (s_ratner_orbit_closure).
s_orbit_closure_candidate	state	orbit closure candidate		Iter-3 imported state (s_orbit_closure_candidate).
s_homogeneous_subset_closure	state	homogeneous subset closure		Iter-3 imported state (s_homogeneous_subset_closure).
s_ratner_equidistribution	theorem	Ratner equidistribution theorem		Iter-3 imported theorem (s_ratner_equidistribution).
s_unipotent_time_average	state	unipotent time average		Iter-3 imported state (s_unipotent_time_average).
s_unique_limit_is_homogeneous_haar	state	unique limit is homogeneous haar		Iter-3 imported state (s_unique_limit_is_homogeneous_haar).
s_margulis_arithmeticity	theorem	Margulis arithmeticity theorem		Iter-3 imported theorem (s_margulis_arithmeticity).
s_semisimple_lie_group_of_higher_rank	axiom	semisimple lie group of higher rank		Iter-3 imported axiom (s_semisimple_lie_group_of_higher_rank).
s_irreducible_lattice_subgroup	axiom	irreducible lattice subgroup		Iter-3 imported axiom (s_irreducible_lattice_subgroup).
s_finite_dim_linear_rep_of_lattice	state	finite dim linear rep of lattice		Iter-3 imported state (s_finite_dim_linear_rep_of_lattice).
s_extension_to_algebraic_rep_of_G	state	extension to algebraic rep of G		Iter-3 imported state (s_extension_to_algebraic_rep_of_G).
s_q_structure_yielding_arithmetic_lattice	state	q structure yielding arithmetic lattice		Iter-3 imported state (s_q_structure_yielding_arithmetic_lattice).
s_margulis_super_rigidity	theorem	Margulis super-rigidity		Iter-3 imported theorem (s_margulis_super_rigidity).
s_lattice_representation_rho	state	lattice representation rho		Iter-3 imported state (s_lattice_representation_rho).
s_boundary_map_phi	state	boundary map phi		Iter-3 imported state (s_boundary_map_phi).
s_equivariant_boundary_value	state	equivariant boundary value		Iter-3 imported state (s_equivariant_boundary_value).
s_margulis_normal_subgroup	theorem	Margulis normal subgroup theorem		Iter-3 imported theorem (s_margulis_normal_subgroup).
s_amenable_property_T_clash	state	amenable property T clash		Iter-3 imported state (s_amenable_property_T_clash).
s_rudolph_measure_rigidity	theorem	Furstenberg ×2 ×3 ergodic measure rigidity (Rudolph)		Iter-3 imported theorem (s_rudolph_measure_rigidity).
s_circle_R_mod_Z	axiom	circle R mod Z		Iter-3 imported axiom (s_circle_R_mod_Z).
s_jointly_invariant_ergodic_measure_for_x2_x3	axiom	jointly invariant ergodic measure for x2 x3		Iter-3 imported axiom (s_jointly_invariant_ergodic_measure_for_x2_x3).
s_conditional_entropy_for_x2_given_x3	state	conditional entropy for x2 given x3		Iter-3 imported state (s_conditional_entropy_for_x2_given_x3).
s_positive_entropy_implies_full_support_lebesgue	state	positive entropy implies full support lebesgue		Iter-3 imported state (s_positive_entropy_implies_full_support_lebesgue).
s_host_normal_numbers	theorem	Host's theorem on normal numbers in base 2 and 3		Iter-3 imported theorem (s_host_normal_numbers).
s_x2_invariant_ergodic_measure	axiom	x2 invariant ergodic measure		Iter-3 imported axiom (s_x2_invariant_ergodic_measure).
s_x3_orbit_fourier_correlation	state	x3 orbit fourier correlation		Iter-3 imported state (s_x3_orbit_fourier_correlation).
s_decay_of_dual_iterates	state	decay of dual iterates		Iter-3 imported state (s_decay_of_dual_iterates).
s_mahler_compactness_criterion	theorem	Mahler compactness criterion		Iter-3 imported theorem (s_mahler_compactness_criterion).
s_space_of_unimodular_lattices_SL_n_R_over_SL_n_Z	axiom	space of unimodular lattices SL n R over SL n Z		Iter-3 imported axiom (s_space_of_unimodular_lattices_SL_n_R_over_SL_n_Z).
s_shortest_vector_function	state	shortest vector function		Iter-3 imported state (s_shortest_vector_function).
s_compact_subset_via_lambda1_bound	state	compact subset via lambda1 bound		Iter-3 imported state (s_compact_subset_via_lambda1_bound).
s_dani_correspondence	theorem	Dani correspondence for Diophantine approximation		Iter-3 imported theorem (s_dani_correspondence).
s_irrational_vector_alpha_in_R_n	axiom	irrational vector alpha in R n		Iter-3 imported axiom (s_irrational_vector_alpha_in_R_n).
s_lattice_associated_to_alpha	state	lattice associated to alpha		Iter-3 imported state (s_lattice_associated_to_alpha).
s_diagonal_flow_orbit	state	diagonal flow orbit		Iter-3 imported state (s_diagonal_flow_orbit).
s_oppenheim_conjecture	theorem	Oppenheim conjecture (Margulis)		Iter-3 imported theorem (s_oppenheim_conjecture).
s_indefinite_irrational_quadratic_form_Q_in_n_at_least_3	axiom	indefinite irrational quadratic form Q in n at least 3		Iter-3 imported axiom (s_indefinite_irrational_quadratic_form_Q_in_n_at_least_3).
s_SO_Q_orbit_in_homogeneous_space	state	SO Q orbit in homogeneous space		Iter-3 imported state (s_SO_Q_orbit_in_homogeneous_space).
s_orbit_dense_in_full_space	state	orbit dense in full space		Iter-3 imported state (s_orbit_dense_in_full_space).
s_linnik_equidistribution_spheres	theorem	Linnik's equidistribution of integer points on spheres		Iter-3 imported theorem (s_linnik_equidistribution_spheres).
s_integer_points_on_sphere_x2_plus_y2_plus_z2_equals_N	axiom	integer points on sphere x2 plus y2 plus z2 equals N		Iter-3 imported axiom (s_integer_points_on_sphere_x2_plus_y2_plus_z2_equals_N).
s_hecke_orbit_on_homogeneous_space	state	hecke orbit on homogeneous space		Iter-3 imported state (s_hecke_orbit_on_homogeneous_space).
s_torus_orbit_equidistribution	state	torus orbit equidistribution		Iter-3 imported state (s_torus_orbit_equidistribution).
s_selberg_trace_formula	theorem	Selberg trace formula		Iter-3 imported theorem (s_selberg_trace_formula).
s_compact_hyperbolic_surface_Gamma_minus_H2	axiom	compact hyperbolic surface Gamma minus H2		Iter-3 imported axiom (s_compact_hyperbolic_surface_Gamma_minus_H2).
s_convolution_kernel_operator	state	convolution kernel operator		Iter-3 imported state (s_convolution_kernel_operator).
s_spectral_side_of_trace	state	spectral side of trace		Iter-3 imported state (s_spectral_side_of_trace).
s_geometric_side_of_trace	state	geometric side of trace		Iter-3 imported state (s_geometric_side_of_trace).
s_lindenstrauss_quantum_unique_ergodicity	theorem	Lindenstrauss arithmetic quantum unique ergodicity		Iter-3 imported theorem (s_lindenstrauss_quantum_unique_ergodicity).
s_hecke_eigenfunctions_on_arithmetic_surface	axiom	hecke eigenfunctions on arithmetic surface		Iter-3 imported axiom (s_hecke_eigenfunctions_on_arithmetic_surface).
s_microlocal_lift_measure_mu_inf	state	microlocal lift measure mu inf		Iter-3 imported state (s_microlocal_lift_measure_mu_inf).
s_diagonal_plus_hecke_invariant_measure	state	diagonal plus hecke invariant measure		Iter-3 imported state (s_diagonal_plus_hecke_invariant_measure).
s_only_invariant_measure_is_haar	state	only invariant measure is haar		Iter-3 imported state (s_only_invariant_measure_is_haar).
s_ekl_littlewood_partial	theorem	Einsiedler–Katok–Lindenstrauss theorem on Littlewood (partial)		Iter-3 imported theorem (s_ekl_littlewood_partial).
s_diagonal_action_on_SL_3_R_over_SL_3_Z	axiom	diagonal action on SL 3 R over SL 3 Z		Iter-3 imported axiom (s_diagonal_action_on_SL_3_R_over_SL_3_Z).
s_A_invariant_ergodic_measure	state	A invariant ergodic measure		Iter-3 imported state (s_A_invariant_ergodic_measure).
s_positive_entropy_under_some_one_parameter_flow	state	positive entropy under some one parameter flow		Iter-3 imported state (s_positive_entropy_under_some_one_parameter_flow).
s_measure_classification	state	measure classification		Iter-3 imported state (s_measure_classification).
s_poincare_linearization_theorem	theorem	Lyapunov–Poincaré linearization theorem		Iter-3 imported theorem (s_poincare_linearization_theorem).
s_analytic_vector_field_with_hyperbolic_singular_point	axiom	analytic vector field with hyperbolic singular point		Iter-3 imported axiom (s_analytic_vector_field_with_hyperbolic_singular_point).
s_formal_conjugacy_series_h_z	state	formal conjugacy series h z		Iter-3 imported state (s_formal_conjugacy_series_h_z).
s_non_resonance_condition_on_eigenvalues	state	non resonance condition on eigenvalues		Iter-3 imported state (s_non_resonance_condition_on_eigenvalues).
s_siegel_linearization	theorem	Siegel linearization theorem for irrational rotation numbers		Iter-3 imported theorem (s_siegel_linearization).
s_holomorphic_germ_with_diophantine_rotation_number	axiom	holomorphic germ with diophantine rotation number		Iter-3 imported axiom (s_holomorphic_germ_with_diophantine_rotation_number).
s_newton_iteration_of_conjugacies	state	newton iteration of conjugacies		Iter-3 imported state (s_newton_iteration_of_conjugacies).
s_small_divisor_bound	state	small divisor bound		Iter-3 imported state (s_small_divisor_bound).
s_floquet_theorem	theorem	Floquet theorem for periodic linear ODE		Iter-3 imported theorem (s_floquet_theorem).
s_linear_ode_with_t_periodic_coefficient_A_t	axiom	linear ode with t periodic coefficient A t		Iter-3 imported axiom (s_linear_ode_with_t_periodic_coefficient_A_t).
s_monodromy_matrix_M	state	monodromy matrix M		Iter-3 imported state (s_monodromy_matrix_M).
s_floquet_exponent_matrix_B	state	floquet exponent matrix B		Iter-3 imported state (s_floquet_exponent_matrix_B).
s_lyapunov_stability_theorem	theorem	Lyapunov stability theorem		Iter-3 imported theorem (s_lyapunov_stability_theorem).
s_ode_with_equilibrium_x_star	axiom	ode with equilibrium x star		Iter-3 imported axiom (s_ode_with_equilibrium_x_star).
s_lyapunov_function_V	axiom	lyapunov function V		Iter-3 imported axiom (s_lyapunov_function_V).
s_lyapunov_function_decreasing	state	lyapunov function decreasing		Iter-3 imported state (s_lyapunov_function_decreasing).
s_trapped_in_level_set	state	trapped in level set		Iter-3 imported state (s_trapped_in_level_set).
s_sternberg_linearization	theorem	Hartman's theorem on smoothness of conjugacies		Iter-3 imported theorem (s_sternberg_linearization).
s_smooth_vector_field_with_nonresonant_hyperbolic_point	axiom	smooth vector field with nonresonant hyperbolic point		Iter-3 imported axiom (s_smooth_vector_field_with_nonresonant_hyperbolic_point).
s_normal_form_at_jet_k	state	normal form at jet k		Iter-3 imported state (s_normal_form_at_jet_k).
s_sternberg_normal_form	state	sternberg normal form		Iter-3 imported state (s_sternberg_normal_form).
s_center_manifold_theorem	theorem	Center manifold theorem		Iter-3 imported theorem (s_center_manifold_theorem).
s_smooth_vector_field_with_partially_hyperbolic_fixed_point	axiom	smooth vector field with partially hyperbolic fixed point		Iter-3 imported axiom (s_smooth_vector_field_with_partially_hyperbolic_fixed_point).
s_graph_transform_on_center	state	graph transform on center		Iter-3 imported state (s_graph_transform_on_center).
s_invariant_center_graph	state	invariant center graph		Iter-3 imported state (s_invariant_center_graph).
s_peixoto_structural_stability	theorem	Peixoto's theorem on structural stability in dimension 2		Iter-3 imported theorem (s_peixoto_structural_stability).
s_smooth_vector_field_on_compact_2_manifold	axiom	smooth vector field on compact 2 manifold		Iter-3 imported axiom (s_smooth_vector_field_on_compact_2_manifold).
s_morse_smale_condition	state	morse smale condition		Iter-3 imported state (s_morse_smale_condition).
s_structural_stability_for_morse_smale	state	structural stability for morse smale		Iter-3 imported state (s_structural_stability_for_morse_smale).
s_aubry_mather_theorem	theorem	Aubry–Mather theory		Iter-3 imported theorem (s_aubry_mather_theorem).
s_monotone_twist_map_of_annulus	axiom	monotone twist map of annulus		Iter-3 imported axiom (s_monotone_twist_map_of_annulus).
s_action_minimizing_configurations	state	action minimizing configurations		Iter-3 imported state (s_action_minimizing_configurations).
s_minimizing_set_M_omega	state	minimizing set M omega		Iter-3 imported state (s_minimizing_set_M_omega).
s_mather_connecting_orbits	theorem	Mather's variational principle for connecting orbits		Iter-3 imported theorem (s_mather_connecting_orbits).
s_tonelli_lagrangian_on_T_M	axiom	tonelli lagrangian on T M		Iter-3 imported axiom (s_tonelli_lagrangian_on_T_M).
s_alpha_beta_function_pair	state	alpha beta function pair		Iter-3 imported state (s_alpha_beta_function_pair).
s_dual_pair_alpha_beta	state	dual pair alpha beta		Iter-3 imported state (s_dual_pair_alpha_beta).
s_denjoy_theorem	theorem	Denjoy's theorem on circle diffeomorphisms		Iter-3 imported theorem (s_denjoy_theorem).
s_C2_circle_diffeomorphism_with_irrational_rotation_number	axiom	C2 circle diffeomorphism with irrational rotation number		Iter-3 imported axiom (s_C2_circle_diffeomorphism_with_irrational_rotation_number).
s_distortion_control_via_BV_log_derivative	state	distortion control via BV log derivative		Iter-3 imported state (s_distortion_control_via_BV_log_derivative).
s_no_wandering_intervals	state	no wandering intervals		Iter-3 imported state (s_no_wandering_intervals).
s_herman_smooth_conjugacy	theorem	Herman's theorem on smooth conjugacy of circle diffeomorphisms		Iter-3 imported theorem (s_herman_smooth_conjugacy).
s_smooth_circle_diffeomorphism_with_diophantine_rotation_number	axiom	smooth circle diffeomorphism with diophantine rotation number		Iter-3 imported axiom (s_smooth_circle_diffeomorphism_with_diophantine_rotation_number).
s_herman_renormalization_scheme	state	herman renormalization scheme		Iter-3 imported state (s_herman_renormalization_scheme).
s_small_divisor_estimate_circle_case	state	small divisor estimate circle case		Iter-3 imported state (s_small_divisor_estimate_circle_case).
s_sinai_billiard_ergodicity	theorem	Sinai dispersing billiards ergodicity		Iter-3 imported theorem (s_sinai_billiard_ergodicity).
s_sinai_billiard_table_with_dispersing_walls	axiom	sinai billiard table with dispersing walls		Iter-3 imported axiom (s_sinai_billiard_table_with_dispersing_walls).
s_invariant_cone_fields_for_billiard_map	state	invariant cone fields for billiard map		Iter-3 imported state (s_invariant_cone_fields_for_billiard_map).
s_pesin_theory_for_billiards	state	pesin theory for billiards		Iter-3 imported state (s_pesin_theory_for_billiards).
s_bunimovich_stadium_ergodicity	theorem	Bunimovich stadium ergodicity		Iter-3 imported theorem (s_bunimovich_stadium_ergodicity).
s_bunimovich_stadium_table	axiom	bunimovich stadium table		Iter-3 imported axiom (s_bunimovich_stadium_table).
s_defocusing_property_on_arcs	state	defocusing property on arcs		Iter-3 imported state (s_defocusing_property_on_arcs).
s_hyperbolicity_for_stadium	state	hyperbolicity for stadium		Iter-3 imported state (s_hyperbolicity_for_stadium).
s_boltzmann_grad_limit_chaos	theorem	Bunimovich–Spohn–Boldrighini Boltzmann-Grad limit		Iter-3 imported theorem (s_boltzmann_grad_limit_chaos).
s_finitely_many_hard_disks_in_box	axiom	finitely many hard disks in box		Iter-3 imported axiom (s_finitely_many_hard_disks_in_box).
s_bbgky_hierarchy	state	bbgky hierarchy		Iter-3 imported state (s_bbgky_hierarchy).
s_boltzmann_grad_scaling_limit	state	boltzmann grad scaling limit		Iter-3 imported state (s_boltzmann_grad_scaling_limit).
s_veech_dichotomy	theorem	Veech dichotomy for translation surfaces		Iter-3 imported theorem (s_veech_dichotomy).
s_veech_translation_surface	axiom	veech translation surface		Iter-3 imported axiom (s_veech_translation_surface).
s_teichmuller_geodesic	state	teichmuller geodesic		Iter-3 imported state (s_teichmuller_geodesic).
s_closed_sl2_orbit_in_strata	state	closed sl2 orbit in strata		Iter-3 imported state (s_closed_sl2_orbit_in_strata).
s_masur_veech_ergodicity	theorem	Masur–Veech ergodicity of Teichmüller flow on moduli of abelian differentials		Iter-3 imported theorem (s_masur_veech_ergodicity).
s_stratum_of_abelian_differentials	axiom	stratum of abelian differentials		Iter-3 imported axiom (s_stratum_of_abelian_differentials).
s_masur_veech_measure	state	masur veech measure		Iter-3 imported state (s_masur_veech_measure).
s_finite_volume_normalization	state	finite volume normalization		Iter-3 imported state (s_finite_volume_normalization).
s_conley_index_theorem	theorem	Morse–Conley index theorem		Iter-3 imported theorem (s_conley_index_theorem).
s_continuous_dynamical_system_on_locally_compact_space	axiom	continuous dynamical system on locally compact space		Iter-3 imported axiom (s_continuous_dynamical_system_on_locally_compact_space).
s_isolated_invariant_set_S	axiom	isolated invariant set S		Iter-3 imported axiom (s_isolated_invariant_set_S).
s_conley_index_pair	state	conley index pair		Iter-3 imported state (s_conley_index_pair).
s_conley_index_class_h_S	state	conley index class h S		Iter-3 imported state (s_conley_index_class_h_S).
s_morse_inequalities	theorem	Morse inequalities (Smale gradient version)		Iter-3 imported theorem (s_morse_inequalities).
s_morse_smale_gradient_flow_on_compact_manifold	axiom	morse smale gradient flow on compact manifold		Iter-3 imported axiom (s_morse_smale_gradient_flow_on_compact_manifold).
s_morse_chain_complex	state	morse chain complex		Iter-3 imported state (s_morse_chain_complex).
s_morse_homology_equals_h_star	state	morse homology equals h star		Iter-3 imported state (s_morse_homology_equals_h_star).
s_hopf_bifurcation_theorem	theorem	Andronov–Hopf bifurcation theorem		Iter-3 imported theorem (s_hopf_bifurcation_theorem).
s_one_parameter_family_of_vector_fields_with_pair_of_eigenvalues_crossing_imaginary_axis	axiom	one parameter family of vector fields with pair of eigenvalues crossing imaginary axis		Iter-3 imported axiom (s_one_parameter_family_of_vector_fields_with_pair_of_eigenvalues_crossing_imaginary_axis).
s_normal_form_for_hopf	state	normal form for hopf		Iter-3 imported state (s_normal_form_for_hopf).
s_first_lyapunov_coefficient	state	first lyapunov coefficient		Iter-3 imported state (s_first_lyapunov_coefficient).
s_saddle_node_bifurcation	theorem	Saddle-node (fold) bifurcation theorem		Iter-3 imported theorem (s_saddle_node_bifurcation).
s_one_parameter_family_with_zero_eigenvalue_at_mu_zero	axiom	one parameter family with zero eigenvalue at mu zero		Iter-3 imported axiom (s_one_parameter_family_with_zero_eigenvalue_at_mu_zero).
s_saddle_node_normal_form	state	saddle node normal form		Iter-3 imported state (s_saddle_node_normal_form).
s_equilibrium_curve_in_mu_x_plane	state	equilibrium curve in mu x plane		Iter-3 imported state (s_equilibrium_curve_in_mu_x_plane).
s_feigenbaum_universality	theorem	Period-doubling cascade (Feigenbaum universality)		Iter-3 imported theorem (s_feigenbaum_universality).
s_unimodal_family_of_interval_maps	axiom	unimodal family of interval maps		Iter-3 imported axiom (s_unimodal_family_of_interval_maps).
s_renormalization_operator_R	state	renormalization operator R		Iter-3 imported state (s_renormalization_operator_R).
s_feigenbaum_fixed_point_g_star	state	feigenbaum fixed point g star		Iter-3 imported state (s_feigenbaum_fixed_point_g_star).
s_feigenbaum_constant_delta	state	feigenbaum constant delta		Iter-3 imported state (s_feigenbaum_constant_delta).
s_avila_reducibility	theorem	Avila's reducibility theorem for quasi-periodic cocycles		Iter-3 imported theorem (s_avila_reducibility).
s_quasi_periodic_SL_2_R_cocycle_with_diophantine_frequency	axiom	quasi periodic SL 2 R cocycle with diophantine frequency		Iter-3 imported axiom (s_quasi_periodic_SL_2_R_cocycle_with_diophantine_frequency).
s_cocycle_renormalization_orbit	state	cocycle renormalization orbit		Iter-3 imported state (s_cocycle_renormalization_orbit).
s_small_divisor_estimate_for_cocycle	state	small divisor estimate for cocycle		Iter-3 imported state (s_small_divisor_estimate_for_cocycle).
s_ten_martini_theorem	theorem	Avila–Damanik almost Mathieu spectrum (Ten Martini Problem)		Iter-3 imported theorem (s_ten_martini_theorem).
s_almost_mathieu_operator_with_irrational_frequency	axiom	almost mathieu operator with irrational frequency		Iter-3 imported axiom (s_almost_mathieu_operator_with_irrational_frequency).
s_lyapunov_exponent_along_E	state	lyapunov exponent along E		Iter-3 imported state (s_lyapunov_exponent_along_E).
s_aubry_dual_operator	state	aubry dual operator		Iter-3 imported state (s_aubry_dual_operator).
s_global_phase_picture_of_amo	state	global phase picture of amo		Iter-3 imported state (s_global_phase_picture_of_amo).
s_avila_krikorian_theorem	theorem	Avila–Krikorian renormalization theorem		Iter-3 imported theorem (s_avila_krikorian_theorem).
s_schrodinger_cocycle_with_irrational_frequency	axiom	schrodinger cocycle with irrational frequency		Iter-3 imported axiom (s_schrodinger_cocycle_with_irrational_frequency).
s_renormalization_fixed_point_for_cocycle	state	renormalization fixed point for cocycle		Iter-3 imported state (s_renormalization_fixed_point_for_cocycle).
s_positive_measure_of_regular_energies	state	positive measure of regular energies		Iter-3 imported state (s_positive_measure_of_regular_energies).
s_avila_global_theory	theorem	Avila global theory of one-frequency Schrödinger operators		Iter-3 imported theorem (s_avila_global_theory).
s_one_frequency_quasi_periodic_schrodinger_operator	axiom	one frequency quasi periodic schrodinger operator		Iter-3 imported axiom (s_one_frequency_quasi_periodic_schrodinger_operator).
s_complexified_lyapunov_function	state	complexified lyapunov function		Iter-3 imported state (s_complexified_lyapunov_function).
s_piecewise_linear_lyapunov_in_strip	state	piecewise linear lyapunov in strip		Iter-3 imported state (s_piecewise_linear_lyapunov_in_strip).
s_birkhoff_minimal_recurrence	theorem	Birkhoff's theorem on transitive points (recurrence)		Iter-3 imported theorem (s_birkhoff_minimal_recurrence).
s_minimal_subsystem	axiom	minimal subsystem		Iter-3 imported axiom (s_minimal_subsystem).
s_minimal_invariant_subset_M	state	minimal invariant subset M		Iter-3 imported state (s_minimal_invariant_subset_M).
s_uniform_recurrence_in_M	state	uniform recurrence in M		Iter-3 imported state (s_uniform_recurrence_in_M).
s_weyl_equidistribution	theorem	Weyl equidistribution theorem		Iter-3 imported theorem (s_weyl_equidistribution).
s_irrational_real_alpha	axiom	irrational real alpha		Iter-3 imported axiom (s_irrational_real_alpha).
s_weyl_exponential_sum	state	weyl exponential sum		Iter-3 imported state (s_weyl_exponential_sum).
s_decay_of_exponential_sum	state	decay of exponential sum		Iter-3 imported state (s_decay_of_exponential_sum).
s_furstenberg_kesten_theorem	theorem	Furstenberg–Kesten theorem (Lyapunov exponents from cocycles)		Iter-3 imported theorem (s_furstenberg_kesten_theorem).
s_log_integrable_matrix_cocycle	axiom	log integrable matrix cocycle		Iter-3 imported axiom (s_log_integrable_matrix_cocycle).
s_subadditive_log_norm	state	subadditive log norm		Iter-3 imported state (s_subadditive_log_norm).
s_pointwise_limit_lambda_top	state	pointwise limit lambda top		Iter-3 imported state (s_pointwise_limit_lambda_top).
s_glasner_weiss_structure	theorem	Glasner–Weiss strong proximality / structure theorem		Iter-3 imported theorem (s_glasner_weiss_structure).
s_compact_metric_dynamical_system_with_group_action	axiom	compact metric dynamical system with group action		Iter-3 imported axiom (s_compact_metric_dynamical_system_with_group_action).
s_proximal_equivalence_relation	state	proximal equivalence relation		Iter-3 imported state (s_proximal_equivalence_relation).
s_maximal_equicontinuous_factor	state	maximal equicontinuous factor		Iter-3 imported state (s_maximal_equicontinuous_factor).
s_hindman_finite_sums	theorem	Hindman's finite-sums theorem		Iter-3 imported theorem (s_hindman_finite_sums).
s_finite_coloring_of_naturals	axiom	finite coloring of naturals		Iter-3 imported axiom (s_finite_coloring_of_naturals).
s_beta_n_compactification	state	beta n compactification		Iter-3 imported state (s_beta_n_compactification).
s_minimal_idempotent_p	state	minimal idempotent p		Iter-3 imported state (s_minimal_idempotent_p).
s_furstenberg_distal_structure	theorem	Furstenberg's structure theorem for distal systems		Iter-3 imported theorem (s_furstenberg_distal_structure).
s_minimal_distal_topological_dynamical_system	axiom	minimal distal topological dynamical system		Iter-3 imported axiom (s_minimal_distal_topological_dynamical_system).
s_equicontinuous_tower	state	equicontinuous tower		Iter-3 imported state (s_equicontinuous_tower).
s_transfinite_isometric_tower	state	transfinite isometric tower		Iter-3 imported state (s_transfinite_isometric_tower).
s_ornstein_isomorphism	theorem	Bernoulli isomorphism theorem (Ornstein)		Iter-3 imported theorem (s_ornstein_isomorphism).
s_two_bernoulli_shifts_of_equal_entropy	axiom	two bernoulli shifts of equal entropy		Iter-3 imported axiom (s_two_bernoulli_shifts_of_equal_entropy).
s_vwb_tower_decomposition	state	vwb tower decomposition		Iter-3 imported state (s_vwb_tower_decomposition).
s_finitely_determined_invariant	state	finitely determined invariant		Iter-3 imported state (s_finitely_determined_invariant).
s_hopf_geodesic_flow_ergodicity	theorem	Hopf ergodicity of geodesic flow on negative-curvature surfaces		Iter-3 imported theorem (s_hopf_geodesic_flow_ergodicity).
s_horocycle_foliations	state	horocycle foliations		Iter-3 imported state (s_horocycle_foliations).
s_invariant_function_is_constant	state	invariant function is constant		Iter-3 imported state (s_invariant_function_is_constant).
s_pesin_stable_manifold_theorem	theorem	Smale's stable manifold theorem for non-uniformly hyperbolic systems (Pesin)		Iter-3 imported theorem (s_pesin_stable_manifold_theorem).
s_smooth_measure_preserving_diffeomorphism_with_nonzero_lyapunov_exponents	axiom	smooth measure preserving diffeomorphism with nonzero lyapunov exponents		Iter-3 imported axiom (s_smooth_measure_preserving_diffeomorphism_with_nonzero_lyapunov_exponents).
s_anisotropic_lyapunov_chart	state	anisotropic lyapunov chart		Iter-3 imported state (s_anisotropic_lyapunov_chart).
s_pesin_local_stable_graph	state	pesin local stable graph		Iter-3 imported state (s_pesin_local_stable_graph).
s_hopf_ratio_ergodic	theorem	Hopf ratio ergodic theorem		Iter-3 imported theorem (s_hopf_ratio_ergodic).
s_two_L1_functions_f_g	axiom	two L1 functions f g		Iter-3 imported axiom (s_two_L1_functions_f_g).
s_birkhoff_ratio_sequence	state	birkhoff ratio sequence		Iter-3 imported state (s_birkhoff_ratio_sequence).
s_hopf_maximal_inequality	state	hopf maximal inequality		Iter-3 imported state (s_hopf_maximal_inequality).
s_furstenberg_zimmer_structure	theorem	Furstenberg–Zimmer structure theorem		Iter-3 imported theorem (s_furstenberg_zimmer_structure).
s_ergodic_measure_preserving_system	axiom	ergodic measure preserving system		Iter-3 imported axiom (s_ergodic_measure_preserving_system).
s_hilbert_module_decomposition	state	hilbert module decomposition		Iter-3 imported state (s_hilbert_module_decomposition).
s_tower_of_compact_extensions	state	tower of compact extensions		Iter-3 imported state (s_tower_of_compact_extensions).
s_host_kra_nilfactor_structure	theorem	Host–Kra structure theorem (nilfactors)		Iter-3 imported theorem (s_host_kra_nilfactor_structure).
s_uniformity_seminorms_U_k	state	uniformity seminorms U k		Iter-3 imported state (s_uniformity_seminorms_U_k).
s_nilfactor_Z_k	state	nilfactor Z k		Iter-3 imported state (s_nilfactor_Z_k).
s_tao_ziegler_nilequidistribution	theorem	Tao–Ziegler equidistribution of nilsequences		Iter-3 imported theorem (s_tao_ziegler_nilequidistribution).
s_polynomial_nilorbit_in_nilmanifold_G_over_Gamma	axiom	polynomial nilorbit in nilmanifold G over Gamma		Iter-3 imported axiom (s_polynomial_nilorbit_in_nilmanifold_G_over_Gamma).
s_green_tao_factorization	state	green tao factorization		Iter-3 imported state (s_green_tao_factorization).
s_subnilmanifold_descent_step	state	subnilmanifold descent step		Iter-3 imported state (s_subnilmanifold_descent_step).
s_pugh_shub_stable_ergodicity	theorem	Stable ergodicity of partially hyperbolic systems (Pugh–Shub)		Iter-3 imported theorem (s_pugh_shub_stable_ergodicity).
s_partially_hyperbolic_diffeomorphism_with_accessible_center	axiom	partially hyperbolic diffeomorphism with accessible center		Iter-3 imported axiom (s_partially_hyperbolic_diffeomorphism_with_accessible_center).
s_accessibility_class_decomposition	state	accessibility class decomposition		Iter-3 imported state (s_accessibility_class_decomposition).
s_invariant_functions_constant_on_classes	state	invariant functions constant on classes		Iter-3 imported state (s_invariant_functions_constant_on_classes).
s_brin_katok_local_entropy	theorem	Brin–Katok local entropy formula		Iter-3 imported theorem (s_brin_katok_local_entropy).
s_measure_preserving_homeomorphism_of_compact_metric_space	axiom	measure preserving homeomorphism of compact metric space		Iter-3 imported axiom (s_measure_preserving_homeomorphism_of_compact_metric_space).
s_bowen_ball_measure_decay	state	bowen ball measure decay		Iter-3 imported state (s_bowen_ball_measure_decay).
s_pointwise_entropy_function	state	pointwise entropy function		Iter-3 imported state (s_pointwise_entropy_function).
s_bowen_limit_set_dimension	theorem	Bowen 1979 dimension formula (Hausdorff dim of limit sets of Kleinian groups)		Iter-3 imported theorem (s_bowen_limit_set_dimension).
s_convex_cocompact_kleinian_group_Gamma	axiom	convex cocompact kleinian group Gamma		Iter-3 imported axiom (s_convex_cocompact_kleinian_group_Gamma).
s_poincare_series_critical_exponent_delta	state	poincare series critical exponent delta		Iter-3 imported state (s_poincare_series_critical_exponent_delta).
s_transfer_operator_fixed_point	state	transfer operator fixed point		Iter-3 imported state (s_transfer_operator_fixed_point).
s_sarnak_mobius_horocycle	theorem	Sarnak's Möbius randomness in horocycle flows		Iter-3 imported theorem (s_sarnak_mobius_horocycle).
s_unipotent_horocycle_flow_on_modular_surface	axiom	unipotent horocycle flow on modular surface		Iter-3 imported axiom (s_unipotent_horocycle_flow_on_modular_surface).
s_mobius_weighted_average	state	mobius weighted average		Iter-3 imported state (s_mobius_weighted_average).
s_disjointness_with_mobius	state	disjointness with mobius		Iter-3 imported state (s_disjointness_with_mobius).
s_sarig_symbolic_extension	theorem	Buzzi–Pollicott–Sarig classification of measures for surface diffeomorphisms		Iter-3 imported theorem (s_sarig_symbolic_extension).
s_C_inf_surface_diffeomorphism_with_positive_topological_entropy	axiom	C inf surface diffeomorphism with positive topological entropy		Iter-3 imported axiom (s_C_inf_surface_diffeomorphism_with_positive_topological_entropy).
s_countable_markov_partition	state	countable markov partition		Iter-3 imported state (s_countable_markov_partition).
s_countable_sft_extension	state	countable sft extension		Iter-3 imported state (s_countable_sft_extension).
s_eskin_mirzakhani_mohammadi	theorem	Eskin–Mirzakhani–Mohammadi rigidity for SL(2,ℝ) action on moduli space		Iter-3 imported theorem (s_eskin_mirzakhani_mohammadi).
s_stratum_of_translation_surfaces_with_SL2R_action	axiom	stratum of translation surfaces with SL2R action		Iter-3 imported axiom (s_stratum_of_translation_surfaces_with_SL2R_action).
s_P_invariant_ergodic_measure	state	P invariant ergodic measure		Iter-3 imported state (s_P_invariant_ergodic_measure).
s_full_SL2R_invariance	state	full SL2R invariance		Iter-3 imported state (s_full_SL2R_invariance).
s_affine_invariant_submanifold_closure	state	affine invariant submanifold closure		Iter-3 imported state (s_affine_invariant_submanifold_closure).
s_mcmullen_renormalization	theorem	McMullen's renormalization theorem for complex dynamics		Iter-3 imported theorem (s_mcmullen_renormalization).
s_infinitely_renormalizable_quadratic_with_bounded_combinatorics	axiom	infinitely renormalizable quadratic with bounded combinatorics		Iter-3 imported axiom (s_infinitely_renormalizable_quadratic_with_bounded_combinatorics).
s_renormalization_operator_on_germs	state	renormalization operator on germs		Iter-3 imported state (s_renormalization_operator_on_germs).
s_renormalization_fixed_point_or_attractor	state	renormalization fixed point or attractor		Iter-3 imported state (s_renormalization_fixed_point_or_attractor).
s_sullivan_conformal_expanding_classification	theorem	Sullivan's classification of conformal expanding repellers		Iter-3 imported theorem (s_sullivan_conformal_expanding_classification).
s_conformal_expanding_repeller_in_complex_plane	axiom	conformal expanding repeller in complex plane		Iter-3 imported axiom (s_conformal_expanding_repeller_in_complex_plane).
s_sullivan_conformal_measure	state	sullivan conformal measure		Iter-3 imported state (s_sullivan_conformal_measure).
s_unique_eigenmeasure_for_T_delta	state	unique eigenmeasure for T delta		Iter-3 imported state (s_unique_eigenmeasure_for_T_delta).
s_pesin_sinai_srb_existence	theorem	Pesin–Sinai SRB physical-measure existence		Iter-3 imported theorem (s_pesin_sinai_srb_existence).
s_dissipative_attractor_with_one_positive_lyapunov_exponent	axiom	dissipative attractor with one positive lyapunov exponent		Iter-3 imported axiom (s_dissipative_attractor_with_one_positive_lyapunov_exponent).
s_ac_conditional_on_unstable_manifold	state	ac conditional on unstable manifold		Iter-3 imported state (s_ac_conditional_on_unstable_manifold).
s_cesaro_limit_along_unstable	state	cesaro limit along unstable		Iter-3 imported state (s_cesaro_limit_along_unstable).
s_newhouse_phenomenon	theorem	Newhouse phenomenon (persistence of homoclinic tangencies)		Iter-3 imported theorem (s_newhouse_phenomenon).
s_C2_diffeomorphism_with_homoclinic_tangency	axiom	C2 diffeomorphism with homoclinic tangency		Iter-3 imported axiom (s_C2_diffeomorphism_with_homoclinic_tangency).
s_thick_horseshoe_Lambda	state	thick horseshoe Lambda		Iter-3 imported state (s_thick_horseshoe_Lambda).
s_residual_intersection_of_cantor_sets	state	residual intersection of cantor sets		Iter-3 imported state (s_residual_intersection_of_cantor_sets).
s_palis_takens_density	theorem	Palis–Takens density of hyperbolicity for one-dim endomorphisms		Iter-3 imported theorem (s_palis_takens_density).
s_C2_family_of_unimodal_interval_maps	axiom	C2 family of unimodal interval maps		Iter-3 imported axiom (s_C2_family_of_unimodal_interval_maps).
s_hyperbolic_or_stochastic_parameter_set	state	hyperbolic or stochastic parameter set		Iter-3 imported state (s_hyperbolic_or_stochastic_parameter_set).
s_full_measure_parameters	state	full measure parameters		Iter-3 imported state (s_full_measure_parameters).
s_krieger_generator_theorem	theorem	Krieger's generator theorem		Iter-3 imported theorem (s_krieger_generator_theorem).
s_ergodic_measure_preserving_transformation_with_finite_entropy	axiom	ergodic measure preserving transformation with finite entropy		Iter-3 imported axiom (s_ergodic_measure_preserving_transformation_with_finite_entropy).
s_candidate_generating_partition	state	candidate generating partition		Iter-3 imported state (s_candidate_generating_partition).
s_partition_size_upper_bound	state	partition size upper bound		Iter-3 imported state (s_partition_size_upper_bound).
s_rohlin_lemma	theorem	Rohlin lemma (Rokhlin tower)		Iter-3 imported theorem (s_rohlin_lemma).
s_aperiodic_measure_preserving_transformation	axiom	aperiodic measure preserving transformation		Iter-3 imported axiom (s_aperiodic_measure_preserving_transformation).
s_candidate_tower_base_B	state	candidate tower base B		Iter-3 imported state (s_candidate_tower_base_B).
s_exhaustion_of_X_by_tower	state	exhaustion of X by tower		Iter-3 imported state (s_exhaustion_of_X_by_tower).
s_ks_entropy_invariance	theorem	Kolmogorov–Sinai entropy invariant theorem		Iter-3 imported theorem (s_ks_entropy_invariance).
s_partition_entropy_function	state	partition entropy function		Iter-3 imported state (s_partition_entropy_function).
s_ks_entropy_h_T	state	ks entropy h T		Iter-3 imported state (s_ks_entropy_h_T).
s_mautner_phenomenon	theorem	Mautner phenomenon		Iter-3 imported theorem (s_mautner_phenomenon).
s_unitary_representation_of_simple_lie_group_G	axiom	unitary representation of simple lie group G		Iter-3 imported axiom (s_unitary_representation_of_simple_lie_group_G).
s_contracting_decomposition_in_G	state	contracting decomposition in G		Iter-3 imported state (s_contracting_decomposition_in_G).
s_extra_invariance_propagation	state	extra invariance propagation		Iter-3 imported state (s_extra_invariance_propagation).
s_howe_moore_mixing	theorem	Howe–Moore mixing for semisimple groups		Iter-3 imported theorem (s_howe_moore_mixing).
s_unitary_representation_of_semisimple_lie_group_no_invariant_vectors	axiom	unitary representation of semisimple lie group no invariant vectors		Iter-3 imported axiom (s_unitary_representation_of_semisimple_lie_group_no_invariant_vectors).
s_matrix_coefficient_function	state	matrix coefficient function		Iter-3 imported state (s_matrix_coefficient_function).
s_decay_estimate_on_K_a_K_decomposition	state	decay estimate on K a K decomposition		Iter-3 imported state (s_decay_estimate_on_K_a_K_decomposition).
s_hahn_banach_separation	theorem	Hahn–Banach separation theorem		Iter-3 imported theorem (s_hahn_banach_separation).
s_disjoint_convex_sets_A_B	axiom	disjoint convex sets A B		Iter-3 imported axiom (s_disjoint_convex_sets_A_B).
s_sublinear_gauge_with_zero_in_interior	state	sublinear gauge with zero in interior		Iter-3 imported state (s_sublinear_gauge_with_zero_in_interior).
s_extended_functional_separating_A_from_B	state	extended functional separating A from B		Iter-3 imported state (s_extended_functional_separating_A_from_B).
s_mazur_lemma	theorem	Mazur's lemma (weak closure equals norm closure for convex sets)		Iter-3 imported theorem (s_mazur_lemma).
s_weakly_convergent_sequence	axiom	weakly convergent sequence		Iter-3 imported axiom (s_weakly_convergent_sequence).
s_separating_hyperplane_required	state	separating hyperplane required		Iter-3 imported state (s_separating_hyperplane_required).
s_hahn_banach_separates_limit_from_convex_hull	state	hahn banach separates limit from convex hull		Iter-3 imported state (s_hahn_banach_separates_limit_from_convex_hull).
s_banach_alaoglu	theorem	Banach–Alaoglu theorem		Iter-3 imported theorem (s_banach_alaoglu).
s_normed_space_X	axiom	normed space X		Iter-3 imported axiom (s_normed_space_X).
s_dual_space_X_star	axiom	dual space X star		Iter-3 imported axiom (s_dual_space_X_star).
s_weak_star_topology_on_X_star	state	weak star topology on X star		Iter-3 imported state (s_weak_star_topology_on_X_star).
s_embedding_into_product_of_closed_intervals	state	embedding into product of closed intervals		Iter-3 imported state (s_embedding_into_product_of_closed_intervals).
s_product_compact_target_space	state	product compact target space		Iter-3 imported state (s_product_compact_target_space).
s_goldstine_theorem	theorem	Goldstine's theorem		Iter-3 imported theorem (s_goldstine_theorem).
s_bidual_X_double_star	axiom	bidual X double star		Iter-3 imported axiom (s_bidual_X_double_star).
s_canonical_embedding_into_bidual	state	canonical embedding into bidual		Iter-3 imported state (s_canonical_embedding_into_bidual).
s_unit_ball_of_X_in_weak_star_X_double_star	state	unit ball of X in weak star X double star		Iter-3 imported state (s_unit_ball_of_X_in_weak_star_X_double_star).
s_krein_milman	theorem	Krein–Milman theorem		Iter-3 imported theorem (s_krein_milman).
s_locally_convex_topological_vector_space	axiom	locally convex topological vector space		Iter-3 imported axiom (s_locally_convex_topological_vector_space).
s_compact_convex_subset_K	axiom	compact convex subset K		Iter-3 imported axiom (s_compact_convex_subset_K).
s_family_of_extremal_faces	state	family of extremal faces		Iter-3 imported state (s_family_of_extremal_faces).
s_minimal_face_is_singleton_extreme_point	state	minimal face is singleton extreme point		Iter-3 imported state (s_minimal_face_is_singleton_extreme_point).
s_choquet_theorem	theorem	Choquet's theorem		Iter-3 imported theorem (s_choquet_theorem).
s_metrizable_compact_convex_set	axiom	metrizable compact convex set		Iter-3 imported axiom (s_metrizable_compact_convex_set).
s_evaluation_functional_at_x	state	evaluation functional at x		Iter-3 imported state (s_evaluation_functional_at_x).
s_representing_probability_measure_on_K	state	representing probability measure on K		Iter-3 imported state (s_representing_probability_measure_on_K).
s_maximal_boundary_measure	state	maximal boundary measure		Iter-3 imported state (s_maximal_boundary_measure).
s_bauer_maximum_principle	theorem	Bauer maximum principle		Iter-3 imported theorem (s_bauer_maximum_principle).
s_upper_semicontinuous_convex_function	axiom	upper semicontinuous convex function		Iter-3 imported axiom (s_upper_semicontinuous_convex_function).
s_maximum_attained_on_K	state	maximum attained on K		Iter-3 imported state (s_maximum_attained_on_K).
s_argmax_face_of_K	state	argmax face of K		Iter-3 imported state (s_argmax_face_of_K).
s_uniform_boundedness_principle	theorem	Uniform boundedness principle (Banach–Steinhaus)		Iter-3 imported theorem (s_uniform_boundedness_principle).
s_family_of_bounded_operators_pointwise_bounded	axiom	family of bounded operators pointwise bounded		Iter-3 imported axiom (s_family_of_bounded_operators_pointwise_bounded).
s_closed_sublevel_sets_E_n	state	closed sublevel sets E n		Iter-3 imported state (s_closed_sublevel_sets_E_n).
s_some_E_n_has_interior	state	some E n has interior		Iter-3 imported state (s_some_E_n_has_interior).
s_hellinger_toeplitz	theorem	Hellinger–Toeplitz theorem		Iter-3 imported theorem (s_hellinger_toeplitz).
s_hilbert_space	axiom	hilbert space		Iter-3 imported axiom (s_hilbert_space).
s_everywhere_defined_symmetric_operator	axiom	everywhere defined symmetric operator		Iter-3 imported axiom (s_everywhere_defined_symmetric_operator).
s_closed_graph_of_T	state	closed graph of T		Iter-3 imported state (s_closed_graph_of_T).
s_hilbert_projection_theorem	theorem	Hilbert projection theorem		Iter-3 imported theorem (s_hilbert_projection_theorem).
s_closed_convex_subset_C	axiom	closed convex subset C		Iter-3 imported axiom (s_closed_convex_subset_C).
s_minimizing_sequence_in_C	state	minimizing sequence in C		Iter-3 imported state (s_minimizing_sequence_in_C).
s_cauchy_minimizing_sequence	state	cauchy minimizing sequence		Iter-3 imported state (s_cauchy_minimizing_sequence).
s_riesz_repr_hilbert	theorem	Riesz representation theorem (Hilbert space)		Iter-3 imported theorem (s_riesz_repr_hilbert).
s_continuous_linear_functional	axiom	continuous linear functional		Iter-3 imported axiom (s_continuous_linear_functional).
s_closed_kernel_hyperplane	state	closed kernel hyperplane		Iter-3 imported state (s_closed_kernel_hyperplane).
s_one_dim_orthogonal_complement	state	one dim orthogonal complement		Iter-3 imported state (s_one_dim_orthogonal_complement).
s_riesz_markov_kakutani	theorem	Riesz–Markov–Kakutani representation theorem		Iter-3 imported theorem (s_riesz_markov_kakutani).
s_outer_measure_from_functional	state	outer measure from functional		Iter-3 imported state (s_outer_measure_from_functional).
s_riesz_repr_Lp	theorem	Riesz representation for L^p		Iter-3 imported theorem (s_riesz_repr_Lp).
s_continuous_linear_functional_on_Lp	axiom	continuous linear functional on Lp		Iter-3 imported axiom (s_continuous_linear_functional_on_Lp).
s_density_g_in_Lq	state	density g in Lq		Iter-3 imported state (s_density_g_in_Lq).
s_lax_milgram	theorem	Lax–Milgram theorem		Iter-3 imported theorem (s_lax_milgram).
s_continuous_coercive_bilinear_form	axiom	continuous coercive bilinear form		Iter-3 imported axiom (s_continuous_coercive_bilinear_form).
s_bounded_operator_A_associated_to_form	state	bounded operator A associated to form		Iter-3 imported state (s_bounded_operator_A_associated_to_form).
s_A_bounded_below_with_closed_range	state	A bounded below with closed range		Iter-3 imported state (s_A_bounded_below_with_closed_range).
s_A_bijective	state	A bijective		Iter-3 imported state (s_A_bijective).
s_stampacchia_theorem	theorem	Stampacchia theorem (variational inequality)		Iter-3 imported theorem (s_stampacchia_theorem).
s_closed_convex_subset_C_and_coercive_form	axiom	closed convex subset C and coercive form		Iter-3 imported axiom (s_closed_convex_subset_C_and_coercive_form).
s_projection_iteration_map_T	state	projection iteration map T		Iter-3 imported state (s_projection_iteration_map_T).
s_unique_fixed_point_u_in_C	state	unique fixed point u in C		Iter-3 imported state (s_unique_fixed_point_u_in_C).
s_babuska_lax_milgram	theorem	Babuška–Lax–Milgram theorem		Iter-3 imported theorem (s_babuska_lax_milgram).
s_pair_of_hilbert_spaces_U_V	axiom	pair of hilbert spaces U V		Iter-3 imported axiom (s_pair_of_hilbert_spaces_U_V).
s_inf_sup_stable_bilinear_form_b	axiom	inf sup stable bilinear form b		Iter-3 imported axiom (s_inf_sup_stable_bilinear_form_b).
s_operator_B_with_inf_sup_bound	state	operator B with inf sup bound		Iter-3 imported state (s_operator_B_with_inf_sup_bound).
s_B_injective_closed_range	state	B injective closed range		Iter-3 imported state (s_B_injective_closed_range).
s_B_bijective_between_U_V	state	B bijective between U V		Iter-3 imported state (s_B_bijective_between_U_V).
s_spectral_theorem_compact_sa	theorem	Spectral theorem for compact self-adjoint operators		Iter-3 imported theorem (s_spectral_theorem_compact_sa).
s_compact_self_adjoint_operator_T	axiom	compact self adjoint operator T		Iter-3 imported axiom (s_compact_self_adjoint_operator_T).
s_maximizer_x_1_yields_eigenpair	state	maximizer x 1 yields eigenpair		Iter-3 imported state (s_maximizer_x_1_yields_eigenpair).
s_first_eigenvector_extracted	state	first eigenvector extracted		Iter-3 imported state (s_first_eigenvector_extracted).
s_iterative_orthogonal_eigenbasis	state	iterative orthogonal eigenbasis		Iter-3 imported state (s_iterative_orthogonal_eigenbasis).
s_spectral_theorem_bounded_sa	theorem	Spectral theorem for bounded self-adjoint operators		Iter-3 imported theorem (s_spectral_theorem_bounded_sa).
s_bounded_self_adjoint_operator_A	axiom	bounded self adjoint operator A		Iter-3 imported axiom (s_bounded_self_adjoint_operator_A).
s_polynomial_functional_calculus	state	polynomial functional calculus		Iter-3 imported state (s_polynomial_functional_calculus).
s_spectral_measures_for_vector_pairs	state	spectral measures for vector pairs		Iter-3 imported state (s_spectral_measures_for_vector_pairs).
s_spectral_theorem_unbounded_sa	theorem	Spectral theorem for unbounded self-adjoint operators		Iter-3 imported theorem (s_spectral_theorem_unbounded_sa).
s_unbounded_self_adjoint_operator_T	axiom	unbounded self adjoint operator T		Iter-3 imported axiom (s_unbounded_self_adjoint_operator_T).
s_unitary_cayley_transform	state	unitary cayley transform		Iter-3 imported state (s_unitary_cayley_transform).
s_spectral_resolution_of_U	state	spectral resolution of U		Iter-3 imported state (s_spectral_resolution_of_U).
s_spectral_measure_E_on_R_from_F	state	spectral measure E on R from F		Iter-3 imported state (s_spectral_measure_E_on_R_from_F).
s_spectral_theorem_normal	theorem	Spectral theorem for normal operators		Iter-3 imported theorem (s_spectral_theorem_normal).
s_bounded_normal_operator_N	axiom	bounded normal operator N		Iter-3 imported axiom (s_bounded_normal_operator_N).
s_real_imag_self_adjoint_pair	state	real imag self adjoint pair		Iter-3 imported state (s_real_imag_self_adjoint_pair).
s_joint_spectral_measure_on_C	state	joint spectral measure on C		Iter-3 imported state (s_joint_spectral_measure_on_C).
s_stone_theorem	theorem	Stone's theorem on one-parameter unitary groups		Iter-3 imported theorem (s_stone_theorem).
s_strongly_continuous_one_param_unitary_group	axiom	strongly continuous one param unitary group		Iter-3 imported axiom (s_strongly_continuous_one_param_unitary_group).
s_skew_adjoint_generator_iA	state	skew adjoint generator iA		Iter-3 imported state (s_skew_adjoint_generator_iA).
s_self_adjoint_generator_A	state	self adjoint generator A		Iter-3 imported state (s_self_adjoint_generator_A).
s_spectral_measure_for_A	state	spectral measure for A		Iter-3 imported state (s_spectral_measure_for_A).
s_hille_yosida	theorem	Hille–Yosida theorem		Iter-3 imported theorem (s_hille_yosida).
s_densely_defined_closed_operator_A	axiom	densely defined closed operator A		Iter-3 imported axiom (s_densely_defined_closed_operator_A).
s_yosida_bounded_approximants	state	yosida bounded approximants		Iter-3 imported state (s_yosida_bounded_approximants).
s_resolvent_bounded_uniformly	state	resolvent bounded uniformly		Iter-3 imported state (s_resolvent_bounded_uniformly).
s_lumer_phillips	theorem	Lumer–Phillips theorem		Iter-3 imported theorem (s_lumer_phillips).
s_densely_defined_dissipative_operator_A	axiom	densely defined dissipative operator A		Iter-3 imported axiom (s_densely_defined_dissipative_operator_A).
s_dissipativity_in_semi_inner_product	state	dissipativity in semi inner product		Iter-3 imported state (s_dissipativity_in_semi_inner_product).
s_trotter_product_formula	theorem	Trotter product formula		Iter-3 imported theorem (s_trotter_product_formula).
s_pair_of_semigroup_generators_A_B	axiom	pair of semigroup generators A B		Iter-3 imported axiom (s_pair_of_semigroup_generators_A_B).
s_sum_closure_is_generator	axiom	sum closure is generator		Iter-3 imported axiom (s_sum_closure_is_generator).
s_split_step_approximant	state	split step approximant		Iter-3 imported state (s_split_step_approximant).
s_chernoff_limit_yields_semigroup	state	chernoff limit yields semigroup		Iter-3 imported state (s_chernoff_limit_yields_semigroup).
s_gelfand_mazur	theorem	Gelfand–Mazur theorem		Iter-3 imported theorem (s_gelfand_mazur).
s_complex_banach_division_algebra	axiom	complex banach division algebra		Iter-3 imported axiom (s_complex_banach_division_algebra).
s_spectrum_nonempty	axiom	spectrum nonempty		Iter-3 imported axiom (s_spectrum_nonempty).
s_entire_bounded_resolvent_for_division_algebra	state	entire bounded resolvent for division algebra		Iter-3 imported state (s_entire_bounded_resolvent_for_division_algebra).
s_gelfand_representation	theorem	Gelfand representation (commutative Banach algebra)		Iter-3 imported theorem (s_gelfand_representation).
s_commutative_unital_banach_algebra	axiom	commutative unital banach algebra		Iter-3 imported axiom (s_commutative_unital_banach_algebra).
s_gelfand_transform_continuous_homomorphism	state	gelfand transform continuous homomorphism		Iter-3 imported state (s_gelfand_transform_continuous_homomorphism).
s_gelfand_naimark_commutative	theorem	Gelfand–Naimark theorem for commutative C*-algebras		Iter-3 imported theorem (s_gelfand_naimark_commutative).
s_commutative_unital_c_star_algebra	axiom	commutative unital c star algebra		Iter-3 imported axiom (s_commutative_unital_c_star_algebra).
s_gelfand_transform_to_continuous_functions	state	gelfand transform to continuous functions		Iter-3 imported state (s_gelfand_transform_to_continuous_functions).
s_star_homomorphism_preserving_involution	state	star homomorphism preserving involution		Iter-3 imported state (s_star_homomorphism_preserving_involution).
s_isometric_star_homomorphism	state	isometric star homomorphism		Iter-3 imported state (s_isometric_star_homomorphism).
s_gns_construction	theorem	GNS construction (Gelfand–Naimark–Segal)		Iter-3 imported theorem (s_gns_construction).
s_unital_c_star_algebra	axiom	unital c star algebra		Iter-3 imported axiom (s_unital_c_star_algebra).
s_pre_inner_product_on_A	state	pre inner product on A		Iter-3 imported state (s_pre_inner_product_on_A).
s_gelfand_naimark_general	theorem	Gelfand–Naimark theorem (every C*-algebra embeds in B(H))		Iter-3 imported theorem (s_gelfand_naimark_general).
s_general_c_star_algebra	axiom	general c star algebra		Iter-3 imported axiom (s_general_c_star_algebra).
s_family_of_pure_states	state	family of pure states		Iter-3 imported state (s_family_of_pure_states).
s_family_of_irreducible_representations	state	family of irreducible representations		Iter-3 imported state (s_family_of_irreducible_representations).
s_universal_faithful_representation	state	universal faithful representation		Iter-3 imported state (s_universal_faithful_representation).
s_stinespring_dilation	theorem	Stinespring dilation theorem		Iter-3 imported theorem (s_stinespring_dilation).
s_pre_inner_product_on_A_otimes_H	state	pre inner product on A otimes H		Iter-3 imported state (s_pre_inner_product_on_A_otimes_H).
s_naimark_dilation	theorem	Naimark dilation theorem		Iter-3 imported theorem (s_naimark_dilation).
s_hilbert_space_H	axiom	hilbert space H		Iter-3 imported axiom (s_hilbert_space_H).
s_positive_operator_valued_measure_POVM	axiom	positive operator valued measure POVM		Iter-3 imported axiom (s_positive_operator_valued_measure_POVM).
s_stinespring_for_commutative_target	state	stinespring for commutative target		Iter-3 imported state (s_stinespring_for_commutative_target).
s_sz_nagy_dilation	theorem	Sz.-Nagy dilation theorem		Iter-3 imported theorem (s_sz_nagy_dilation).
s_contraction_T_on_H	axiom	contraction T on H		Iter-3 imported axiom (s_contraction_T_on_H).
s_defect_operator_and_defect_space	state	defect operator and defect space		Iter-3 imported state (s_defect_operator_and_defect_space).
s_enlarged_hilbert_space_K	state	enlarged hilbert space K		Iter-3 imported state (s_enlarged_hilbert_space_K).
s_unitary_dilation_of_T	state	unitary dilation of T		Iter-3 imported state (s_unitary_dilation_of_T).
s_choi_theorem	theorem	Choi's theorem on completely positive maps		Iter-3 imported theorem (s_choi_theorem).
s_choi_matrix	state	choi matrix		Iter-3 imported state (s_choi_matrix).
s_positive_choi_matrix_characterization	state	positive choi matrix characterization		Iter-3 imported state (s_positive_choi_matrix_characterization).
s_kraus_operators_V_k_from_eigenvectors	state	kraus operators V k from eigenvectors		Iter-3 imported state (s_kraus_operators_V_k_from_eigenvectors).
s_kadison_transitivity	theorem	Kadison transitivity theorem		Iter-3 imported theorem (s_kadison_transitivity).
s_finite_set_of_target_vectors	axiom	finite set of target vectors		Iter-3 imported axiom (s_finite_set_of_target_vectors).
s_strong_density_in_unit_ball	state	strong density in unit ball		Iter-3 imported state (s_strong_density_in_unit_ball).
s_kaplansky_density	theorem	Kaplansky density theorem		Iter-3 imported theorem (s_kaplansky_density).
s_double_commutant_A_double_prime	axiom	double commutant A double prime		Iter-3 imported axiom (s_double_commutant_A_double_prime).
s_sa_part_dense_via_functional_calculus	state	sa part dense via functional calculus		Iter-3 imported state (s_sa_part_dense_via_functional_calculus).
s_unit_ball_approximation_by_truncation	state	unit ball approximation by truncation		Iter-3 imported state (s_unit_ball_approximation_by_truncation).
s_bicommutant_theorem	theorem	Von Neumann bicommutant theorem		Iter-3 imported theorem (s_bicommutant_theorem).
s_commutant_M_prime	axiom	commutant M prime		Iter-3 imported axiom (s_commutant_M_prime).
s_cyclic_subspaces_with_projections_in_M_prime	state	cyclic subspaces with projections in M prime		Iter-3 imported state (s_cyclic_subspaces_with_projections_in_M_prime).
s_strong_approximation_of_T_by_M	state	strong approximation of T by M		Iter-3 imported state (s_strong_approximation_of_T_by_M).
s_mvn_factor_classification	theorem	Murray–von Neumann classification of factors		Iter-3 imported theorem (s_mvn_factor_classification).
s_factor_M_trivial_center	axiom	factor M trivial center		Iter-3 imported axiom (s_factor_M_trivial_center).
s_dimension_function_d	state	dimension function d		Iter-3 imported state (s_dimension_function_d).
s_three_dimension_patterns_observed	state	three dimension patterns observed		Iter-3 imported state (s_three_dimension_patterns_observed).
s_type_axioms_I_II_III	state	type axioms I II III		Iter-3 imported state (s_type_axioms_I_II_III).
s_tomita_takesaki	theorem	Tomita–Takesaki modular theory		Iter-3 imported theorem (s_tomita_takesaki).
s_von_neumann_algebra_M	axiom	von neumann algebra M		Iter-3 imported axiom (s_von_neumann_algebra_M).
s_antilinear_operator_S_closable	state	antilinear operator S closable		Iter-3 imported state (s_antilinear_operator_S_closable).
s_connes_injective_classification	theorem	Connes' classification of injective factors		Iter-3 imported theorem (s_connes_injective_classification).
s_injective_factor_M	axiom	injective factor M		Iter-3 imported axiom (s_injective_factor_M).
s_modular_flow_on_M	state	modular flow on M		Iter-3 imported state (s_modular_flow_on_M).
s_connes_invariants_S_T	state	connes invariants S T		Iter-3 imported state (s_connes_invariants_S_T).
s_atkinson_theorem	theorem	Atkinson's theorem (Fredholm operators)		Iter-3 imported theorem (s_atkinson_theorem).
s_bounded_operator_T_on_banach_space	axiom	bounded operator T on banach space		Iter-3 imported axiom (s_bounded_operator_T_on_banach_space).
s_calkin_algebra_quotient	state	calkin algebra quotient		Iter-3 imported state (s_calkin_algebra_quotient).
s_fredholm_iff_invertible_modulo_compacts	state	fredholm iff invertible modulo compacts		Iter-3 imported state (s_fredholm_iff_invertible_modulo_compacts).
s_toeplitz_index_formula	theorem	Index of Toeplitz operators on the circle		Iter-3 imported theorem (s_toeplitz_index_formula).
s_hardy_space_H2	axiom	hardy space H2		Iter-3 imported axiom (s_hardy_space_H2).
s_calkin_index_map	theorem	Calkin algebra index map		Iter-3 imported theorem (s_calkin_index_map).
s_six_term_K_theory_sequence	state	six term K theory sequence		Iter-3 imported state (s_six_term_K_theory_sequence).
s_fuglede_theorem	theorem	Fuglede's theorem		Iter-3 imported theorem (s_fuglede_theorem).
s_normal_operator_N	axiom	normal operator N		Iter-3 imported axiom (s_normal_operator_N).
s_entire_bounded_operator_valued_function	state	entire bounded operator valued function		Iter-3 imported state (s_entire_bounded_operator_valued_function).
s_putnam_fuglede	theorem	Putnam–Fuglede theorem		Iter-3 imported theorem (s_putnam_fuglede).
s_pair_of_normal_operators_M_N	axiom	pair of normal operators M N		Iter-3 imported axiom (s_pair_of_normal_operators_M_N).
s_block_normal_operator	state	block normal operator		Iter-3 imported state (s_block_normal_operator).
s_block_intertwiner_commutes_with_adjoint	state	block intertwiner commutes with adjoint		Iter-3 imported state (s_block_intertwiner_commutes_with_adjoint).
s_banach_stone	theorem	Banach–Stone theorem		Iter-3 imported theorem (s_banach_stone).
s_compact_hausdorff_pair_K_L	axiom	compact hausdorff pair K L		Iter-3 imported axiom (s_compact_hausdorff_pair_K_L).
s_isometry_sends_point_masses_to_point_masses	state	isometry sends point masses to point masses		Iter-3 imported state (s_isometry_sends_point_masses_to_point_masses).
s_bochner_theorem	theorem	Bochner's theorem		Iter-3 imported theorem (s_bochner_theorem).
s_continuous_positive_definite_function_on_R	axiom	continuous positive definite function on R		Iter-3 imported axiom (s_continuous_positive_definite_function_on_R).
s_fourier_inversion_program	axiom	fourier inversion program		Iter-3 imported axiom (s_fourier_inversion_program).
s_state_on_commutative_group_C_star	state	state on commutative group C star		Iter-3 imported state (s_state_on_commutative_group_C_star).
s_peter_weyl	theorem	Peter–Weyl theorem		Iter-3 imported theorem (s_peter_weyl).
s_compact_topological_group_G	axiom	compact topological group G		Iter-3 imported axiom (s_compact_topological_group_G).
s_haar_measure_on_G	axiom	haar measure on G		Iter-3 imported axiom (s_haar_measure_on_G).
s_compact_self_adjoint_convolution_operators	state	compact self adjoint convolution operators		Iter-3 imported state (s_compact_self_adjoint_convolution_operators).
s_stone_von_neumann	theorem	Stone–von Neumann theorem		Iter-3 imported theorem (s_stone_von_neumann).
s_irreducibility_assumption	axiom	irreducibility assumption		Iter-3 imported axiom (s_irreducibility_assumption).
s_weyl_form_of_CCR	state	weyl form of CCR		Iter-3 imported state (s_weyl_form_of_CCR).
s_heisenberg_group_representation	state	heisenberg group representation		Iter-3 imported state (s_heisenberg_group_representation).
s_fixed_central_character_irreducible	state	fixed central character irreducible		Iter-3 imported state (s_fixed_central_character_irreducible).
s_schwartz_kernel_theorem	theorem	Schwartz kernel theorem		Iter-3 imported theorem (s_schwartz_kernel_theorem).
s_continuous_bilinear_form_on_test_functions	axiom	continuous bilinear form on test functions		Iter-3 imported axiom (s_continuous_bilinear_form_on_test_functions).
s_schwartz_space_distributions_pair	axiom	schwartz space distributions pair		Iter-3 imported axiom (s_schwartz_space_distributions_pair).
s_jointly_continuous_bilinear_form	state	jointly continuous bilinear form		Iter-3 imported state (s_jointly_continuous_bilinear_form).
s_nuclear_space_grothendieck	theorem	Nuclear space characterization (Grothendieck)		Iter-3 imported theorem (s_nuclear_space_grothendieck).
s_locally_convex_space_X	axiom	locally convex space X		Iter-3 imported axiom (s_locally_convex_space_X).
s_canonical_maps_between_local_banach_spaces	state	canonical maps between local banach spaces		Iter-3 imported state (s_canonical_maps_between_local_banach_spaces).
s_nuclear_canonical_maps	state	nuclear canonical maps		Iter-3 imported state (s_nuclear_canonical_maps).
s_uniqueness_of_tensor_product_topology	state	uniqueness of tensor product topology		Iter-3 imported state (s_uniqueness_of_tensor_product_topology).
s_sobolev_embedding	theorem	Sobolev embedding theorem		Iter-3 imported theorem (s_sobolev_embedding).
s_riesz_potential_estimate	axiom	riesz potential estimate		Iter-3 imported axiom (s_riesz_potential_estimate).
s_bessel_potential_representation	state	bessel potential representation		Iter-3 imported state (s_bessel_potential_representation).
s_dyadic_block_decomposition_of_u	state	dyadic block decomposition of u		Iter-3 imported state (s_dyadic_block_decomposition_of_u).
s_hardy_littlewood_sobolev_inequality	state	hardy littlewood sobolev inequality		Iter-3 imported state (s_hardy_littlewood_sobolev_inequality).
s_rellich_kondrachov	theorem	Rellich–Kondrachov compactness theorem		Iter-3 imported theorem (s_rellich_kondrachov).
s_frechet_kolmogorov	theorem	Fréchet–Kolmogorov compactness theorem		Iter-3 imported theorem (s_frechet_kolmogorov).
s_subset_F_of_Lp	axiom	subset F of Lp		Iter-3 imported axiom (s_subset_F_of_Lp).
s_translation_continuity_uniform	axiom	translation continuity uniform		Iter-3 imported axiom (s_translation_continuity_uniform).
s_mollified_family_uniformly_close_to_F	state	mollified family uniformly close to F		Iter-3 imported state (s_mollified_family_uniformly_close_to_F).
s_precompact_after_mollification	state	precompact after mollification		Iter-3 imported state (s_precompact_after_mollification).
s_trace_theorem	theorem	Trace theorem for Sobolev spaces		Iter-3 imported theorem (s_trace_theorem).
s_smooth_functions_dense_in_W1p	state	smooth functions dense in W1p		Iter-3 imported state (s_smooth_functions_dense_in_W1p).
s_boundary_trace_estimate	state	boundary trace estimate		Iter-3 imported state (s_boundary_trace_estimate).
s_calderon_extension	theorem	Calderón extension theorem		Iter-3 imported theorem (s_calderon_extension).
s_local_half_space_reduction	state	local half space reduction		Iter-3 imported state (s_local_half_space_reduction).
s_reflection_extension_in_each_chart	state	reflection extension in each chart		Iter-3 imported state (s_reflection_extension_in_each_chart).
s_gagliardo_nirenberg	theorem	Gagliardo–Nirenberg interpolation inequality		Iter-3 imported theorem (s_gagliardo_nirenberg).
s_smooth_compactly_supported_function	axiom	smooth compactly supported function		Iter-3 imported axiom (s_smooth_compactly_supported_function).
s_lebesgue_index_triple_p_q_r	axiom	lebesgue index triple p q r		Iter-3 imported axiom (s_lebesgue_index_triple_p_q_r).
s_scaling_dimensional_balance	state	scaling dimensional balance		Iter-3 imported state (s_scaling_dimensional_balance).
s_interpolation_inequality_on_dyadic_blocks	state	interpolation inequality on dyadic blocks		Iter-3 imported state (s_interpolation_inequality_on_dyadic_blocks).
s_full_interpolation_estimate	state	full interpolation estimate		Iter-3 imported state (s_full_interpolation_estimate).
s_riesz_thorin	theorem	Riesz–Thorin interpolation theorem		Iter-3 imported theorem (s_riesz_thorin).
s_linear_operator_bounded_at_two_endpoints	axiom	linear operator bounded at two endpoints		Iter-3 imported axiom (s_linear_operator_bounded_at_two_endpoints).
s_lebesgue_endpoint_pairs	axiom	lebesgue endpoint pairs		Iter-3 imported axiom (s_lebesgue_endpoint_pairs).
s_analytic_family_in_complex_strip	state	analytic family in complex strip		Iter-3 imported state (s_analytic_family_in_complex_strip).
s_marcinkiewicz_interpolation	theorem	Marcinkiewicz interpolation theorem		Iter-3 imported theorem (s_marcinkiewicz_interpolation).
s_sublinear_operator_weak_type_at_endpoints	axiom	sublinear operator weak type at endpoints		Iter-3 imported axiom (s_sublinear_operator_weak_type_at_endpoints).
s_lorentz_spaces_Lp_q	axiom	lorentz spaces Lp q		Iter-3 imported axiom (s_lorentz_spaces_Lp_q).
s_layer_cake_decomposition	state	layer cake decomposition		Iter-3 imported state (s_layer_cake_decomposition).
s_distribution_function_bounds	state	distribution function bounds		Iter-3 imported state (s_distribution_function_bounds).
s_strong_type_estimate	state	strong type estimate		Iter-3 imported state (s_strong_type_estimate).
s_hormander_mikhlin	theorem	Hörmander–Mikhlin multiplier theorem		Iter-3 imported theorem (s_hormander_mikhlin).
s_bounded_fourier_multiplier_m	axiom	bounded fourier multiplier m		Iter-3 imported axiom (s_bounded_fourier_multiplier_m).
s_mikhlin_derivative_estimates	axiom	mikhlin derivative estimates		Iter-3 imported axiom (s_mikhlin_derivative_estimates).
s_convolution_operator_with_kernel_K_m	state	convolution operator with kernel K m		Iter-3 imported state (s_convolution_operator_with_kernel_K_m).
s_dyadic_pieces_with_bounded_derivatives	state	dyadic pieces with bounded derivatives		Iter-3 imported state (s_dyadic_pieces_with_bounded_derivatives).
s_CZ_singular_integral_bounds	state	CZ singular integral bounds		Iter-3 imported state (s_CZ_singular_integral_bounds).
s_lebesgue_differentiation	theorem	Lebesgue differentiation theorem		Iter-3 imported theorem (s_lebesgue_differentiation).
s_locally_integrable_function_on_Rn	axiom	locally integrable function on Rn		Iter-3 imported axiom (s_locally_integrable_function_on_Rn).
s_eberlein_smulian	theorem	Eberlein–Šmulian theorem		Iter-3 imported theorem (s_eberlein_smulian).
s_weakly_compact_subset_K	axiom	weakly compact subset K		Iter-3 imported axiom (s_weakly_compact_subset_K).
s_K_in_double_dual_with_weak_star_compact_image	state	K in double dual with weak star compact image		Iter-3 imported state (s_K_in_double_dual_with_weak_star_compact_image).
s_metrizable_weak_star_compact_image	state	metrizable weak star compact image		Iter-3 imported state (s_metrizable_weak_star_compact_image).
s_james_theorem	theorem	James's theorem		Iter-3 imported theorem (s_james_theorem).
s_banach_space_X	axiom	banach space X		Iter-3 imported axiom (s_banach_space_X).
s_every_continuous_linear_functional_attains_norm_on_B_X	axiom	every continuous linear functional attains norm on B X		Iter-3 imported axiom (s_every_continuous_linear_functional_attains_norm_on_B_X).
s_non_attainment_in_some_functional_construction	state	non attainment in some functional construction		Iter-3 imported state (s_non_attainment_in_some_functional_construction).
s_non_norm_attaining_functional	state	non norm attaining functional		Iter-3 imported state (s_non_norm_attaining_functional).
s_milman_pettis	theorem	Milman–Pettis theorem		Iter-3 imported theorem (s_milman_pettis).
s_uniformly_convex_banach_space	axiom	uniformly convex banach space		Iter-3 imported axiom (s_uniformly_convex_banach_space).
s_quantitative_strict_convexity	state	quantitative strict convexity		Iter-3 imported state (s_quantitative_strict_convexity).
s_canonical_embedding_surjective_on_unit_balls	state	canonical embedding surjective on unit balls		Iter-3 imported state (s_canonical_embedding_surjective_on_unit_balls).
s_bishop_phelps	theorem	Bishop–Phelps theorem		Iter-3 imported theorem (s_bishop_phelps).
s_support_cone_partial_order	state	support cone partial order		Iter-3 imported state (s_support_cone_partial_order).
s_maximal_element_in_cone	state	maximal element in cone		Iter-3 imported state (s_maximal_element_in_cone).
s_krein_smulian	theorem	Krein–Smulian theorem		Iter-3 imported theorem (s_krein_smulian).
s_dual_banach_space_X_star	axiom	dual banach space X star		Iter-3 imported axiom (s_dual_banach_space_X_star).
s_polar_set_in_X	state	polar set in X		Iter-3 imported state (s_polar_set_in_X).
s_weak_star_closure_recoverable_levelwise	state	weak star closure recoverable levelwise		Iter-3 imported state (s_weak_star_closure_recoverable_levelwise).
s_krein_rutman	theorem	Krein–Rutman theorem		Iter-3 imported theorem (s_krein_rutman).
s_compact_positive_operator_on_cone	axiom	compact positive operator on cone		Iter-3 imported axiom (s_compact_positive_operator_on_cone).
s_solid_pointed_cone_K	axiom	solid pointed cone K		Iter-3 imported axiom (s_solid_pointed_cone_K).
s_rescaled_iteration_on_cone	state	rescaled iteration on cone		Iter-3 imported state (s_rescaled_iteration_on_cone).
s_fixed_point_in_cone	state	fixed point in cone		Iter-3 imported state (s_fixed_point_in_cone).
s_schauder_fpt	theorem	Schauder fixed-point theorem		Iter-3 imported theorem (s_schauder_fpt).
s_compact_convex_subset_of_banach	axiom	compact convex subset of banach		Iter-3 imported axiom (s_compact_convex_subset_of_banach).
s_finite_dim_approximation_of_K	state	finite dim approximation of K		Iter-3 imported state (s_finite_dim_approximation_of_K).
s_markov_kakutani	theorem	Markov–Kakutani fixed-point theorem		Iter-3 imported theorem (s_markov_kakutani).
s_compact_convex_subset_in_locally_convex_space	axiom	compact convex subset in locally convex space		Iter-3 imported axiom (s_compact_convex_subset_in_locally_convex_space).
s_commuting_family_of_continuous_affine_maps	axiom	commuting family of continuous affine maps		Iter-3 imported axiom (s_commuting_family_of_continuous_affine_maps).
s_cesaro_averages_of_maps	state	cesaro averages of maps		Iter-3 imported state (s_cesaro_averages_of_maps).
s_nonempty_intersection_per_map	state	nonempty intersection per map		Iter-3 imported state (s_nonempty_intersection_per_map).
s_ryll_nardzewski	theorem	Ryll-Nardzewski fixed-point theorem		Iter-3 imported theorem (s_ryll_nardzewski).
s_weakly_compact_convex_set_K	axiom	weakly compact convex set K		Iter-3 imported axiom (s_weakly_compact_convex_set_K).
s_noncontracting_semigroup_of_isometries	axiom	noncontracting semigroup of isometries		Iter-3 imported axiom (s_noncontracting_semigroup_of_isometries).
s_minimal_invariant_set_K_0	state	minimal invariant set K 0		Iter-3 imported state (s_minimal_invariant_set_K_0).
s_K_0_singleton_via_contradiction	state	K 0 singleton via contradiction		Iter-3 imported state (s_K_0_singleton_via_contradiction).
s_ekeland_variational_principle	theorem	Ekeland's variational principle		Iter-3 imported theorem (s_ekeland_variational_principle).
s_lower_semicontinuous_bounded_below_function	axiom	lower semicontinuous bounded below function		Iter-3 imported axiom (s_lower_semicontinuous_bounded_below_function).
s_partial_order_on_X	state	partial order on X		Iter-3 imported state (s_partial_order_on_X).
s_browder_minty	theorem	Browder–Minty theorem (monotone operators)		Iter-3 imported theorem (s_browder_minty).
s_reflexive_banach_space	axiom	reflexive banach space		Iter-3 imported axiom (s_reflexive_banach_space).
s_monotone_coercive_hemicontinuous_operator	axiom	monotone coercive hemicontinuous operator		Iter-3 imported axiom (s_monotone_coercive_hemicontinuous_operator).
s_finite_dim_galerkin_approximants	state	finite dim galerkin approximants		Iter-3 imported state (s_finite_dim_galerkin_approximants).
s_finite_dim_solutions_x_n	state	finite dim solutions x n		Iter-3 imported state (s_finite_dim_solutions_x_n).
s_mercer_theorem	theorem	Mercer's theorem		Iter-3 imported theorem (s_mercer_theorem).
s_continuous_symmetric_positive_kernel	axiom	continuous symmetric positive kernel		Iter-3 imported axiom (s_continuous_symmetric_positive_kernel).
s_compact_domain_X	axiom	compact domain X		Iter-3 imported axiom (s_compact_domain_X).
s_integral_operator_compact_self_adjoint	state	integral operator compact self adjoint		Iter-3 imported state (s_integral_operator_compact_self_adjoint).
s_spectral_decomposition_of_T_K	state	spectral decomposition of T K		Iter-3 imported state (s_spectral_decomposition_of_T_K).
s_uniformly_convergent_kernel_series	state	uniformly convergent kernel series		Iter-3 imported state (s_uniformly_convergent_kernel_series).
s_hilbert_schmidt_theorem	theorem	Hilbert–Schmidt theorem		Iter-3 imported theorem (s_hilbert_schmidt_theorem).
s_hilbert_schmidt_integral_operator	axiom	hilbert schmidt integral operator		Iter-3 imported axiom (s_hilbert_schmidt_integral_operator).
s_min_max_theorem	theorem	Min-max theorem (Courant–Fischer–Weyl)		Iter-3 imported theorem (s_min_max_theorem).
s_eigenvalues_ordered_decreasing	axiom	eigenvalues ordered decreasing		Iter-3 imported axiom (s_eigenvalues_ordered_decreasing).
s_eigenpair_sequence	state	eigenpair sequence		Iter-3 imported state (s_eigenpair_sequence).
s_rayleigh_quotient_with_extremal_property	state	rayleigh quotient with extremal property		Iter-3 imported state (s_rayleigh_quotient_with_extremal_property).
s_weyl_essential_spectrum	theorem	Weyl's theorem on essential spectrum		Iter-3 imported theorem (s_weyl_essential_spectrum).
s_self_adjoint_operator_A	axiom	self adjoint operator A		Iter-3 imported axiom (s_self_adjoint_operator_A).
s_compact_perturbation_K	axiom	compact perturbation K		Iter-3 imported axiom (s_compact_perturbation_K).
s_weyl_sequence_preserved_under_compact_perturbation	state	weyl sequence preserved under compact perturbation		Iter-3 imported state (s_weyl_sequence_preserved_under_compact_perturbation).
s_lomonosov_invariant_subspace	theorem	Lomonosov's invariant subspace theorem		Iter-3 imported theorem (s_lomonosov_invariant_subspace).
s_bounded_operator_commuting_with_nonzero_compact	axiom	bounded operator commuting with nonzero compact		Iter-3 imported axiom (s_bounded_operator_commuting_with_nonzero_compact).
s_banach_space_infinite_dim	axiom	banach space infinite dim		Iter-3 imported axiom (s_banach_space_infinite_dim).
s_orbit_density_assumption	state	orbit density assumption		Iter-3 imported state (s_orbit_density_assumption).
s_self_map_with_compact_image	state	self map with compact image		Iter-3 imported state (s_self_map_with_compact_image).
s_compact_eigenvector_yields_invariant_subspace	state	compact eigenvector yields invariant subspace		Iter-3 imported state (s_compact_eigenvector_yields_invariant_subspace).
s_closed_range_theorem	theorem	Closed range theorem		Iter-3 imported theorem (s_closed_range_theorem).
s_densely_defined_closed_operator_T	axiom	densely defined closed operator T		Iter-3 imported axiom (s_densely_defined_closed_operator_T).
s_adjoint_T_star	axiom	adjoint T star		Iter-3 imported axiom (s_adjoint_T_star).
s_orthogonality_relation_range_kernel	state	orthogonality relation range kernel		Iter-3 imported state (s_orthogonality_relation_range_kernel).
s_equivalent_closedness_conditions	state	equivalent closedness conditions		Iter-3 imported state (s_equivalent_closedness_conditions).
s_fredholm_alternative	theorem	Fredholm alternative		Iter-3 imported theorem (s_fredholm_alternative).
s_compact_operator_K_on_banach	axiom	compact operator K on banach		Iter-3 imported axiom (s_compact_operator_K_on_banach).
s_operator_I_minus_K	axiom	operator I minus K		Iter-3 imported axiom (s_operator_I_minus_K).
s_finite_dim_kernel	state	finite dim kernel		Iter-3 imported state (s_finite_dim_kernel).
s_index_zero_fredholm_operator	state	index zero fredholm operator		Iter-3 imported state (s_index_zero_fredholm_operator).
s_analytic_fredholm	theorem	Analytic Fredholm theorem		Iter-3 imported theorem (s_analytic_fredholm).
s_connected_domain_D	axiom	connected domain D		Iter-3 imported axiom (s_connected_domain_D).
s_meromorphic_resolvent	state	meromorphic resolvent		Iter-3 imported state (s_meromorphic_resolvent).
s_local_invertibility_or_finite_kernel	state	local invertibility or finite kernel		Iter-3 imported state (s_local_invertibility_or_finite_kernel).
s_cotlar_stein	theorem	Cotlar–Stein lemma		Iter-3 imported theorem (s_cotlar_stein).
s_almost_orthogonal_family_of_operators_T_j	axiom	almost orthogonal family of operators T j		Iter-3 imported axiom (s_almost_orthogonal_family_of_operators_T_j).
s_l1_summable_correlation_estimates	axiom	l1 summable correlation estimates		Iter-3 imported axiom (s_l1_summable_correlation_estimates).
s_expanded_2n_th_power	state	expanded 2n th power		Iter-3 imported state (s_expanded_2n_th_power).
s_chain_correlation_bound	state	chain correlation bound		Iter-3 imported state (s_chain_correlation_bound).
s_t1_theorem	theorem	T(1) theorem (David–Journé)		Iter-3 imported theorem (s_t1_theorem).
s_singular_integral_operator_T_with_CZ_kernel	axiom	singular integral operator T with CZ kernel		Iter-3 imported axiom (s_singular_integral_operator_T_with_CZ_kernel).
s_paraproduct_corrected_operator_T_0	state	paraproduct corrected operator T 0		Iter-3 imported state (s_paraproduct_corrected_operator_T_0).
s_almost_orthogonal_pieces_T_jk	state	almost orthogonal pieces T jk		Iter-3 imported state (s_almost_orthogonal_pieces_T_jk).
s_hormander_symbol_class	theorem	Hörmander's pseudodifferential symbol class theorem		Iter-3 imported theorem (s_hormander_symbol_class).
s_oscillatory_integral_definition	axiom	oscillatory integral definition		Iter-3 imported axiom (s_oscillatory_integral_definition).
s_dyadic_oscillatory_pieces	state	dyadic oscillatory pieces		Iter-3 imported state (s_dyadic_oscillatory_pieces).
s_symbol_calculus_modulo_lower_order	state	symbol calculus modulo lower order		Iter-3 imported state (s_symbol_calculus_modulo_lower_order).
s_hormander_l2_dbar	theorem	Hörmander L² ∂̄ estimates		Iter-3 imported theorem (s_hormander_l2_dbar).
s_pseudoconvex_domain_in_Cn	axiom	pseudoconvex domain in Cn		Iter-3 imported axiom (s_pseudoconvex_domain_in_Cn).
s_bochner_kodaira_morrey_identity	state	bochner kodaira morrey identity		Iter-3 imported state (s_bochner_kodaira_morrey_identity).
s_garding_inequality	theorem	Garding's inequality		Iter-3 imported theorem (s_garding_inequality).
s_pointwise_positivity_of_symbol	state	pointwise positivity of symbol		Iter-3 imported state (s_pointwise_positivity_of_symbol).
s_local_constant_coefficient_pieces	state	local constant coefficient pieces		Iter-3 imported state (s_local_constant_coefficient_pieces).
s_aubin_lions	theorem	Aubin–Lions lemma		Iter-3 imported theorem (s_aubin_lions).
s_ehrling_interpolation_inequality	state	ehrling interpolation inequality		Iter-3 imported state (s_ehrling_interpolation_inequality).
s_equicontinuity_in_time_in_B	state	equicontinuity in time in B		Iter-3 imported state (s_equicontinuity_in_time_in_B).
s_meyers_serrin	theorem	Meyers–Serrin theorem (H = W)		Iter-3 imported theorem (s_meyers_serrin).
s_locally_finite_cover_with_pou	state	locally finite cover with pou		Iter-3 imported state (s_locally_finite_cover_with_pou).
s_local_mollification_summed	state	local mollification summed		Iter-3 imported state (s_local_mollification_summed).
s_mazur_ulam	theorem	Mazur–Ulam theorem		Iter-3 imported theorem (s_mazur_ulam).
s_real_normed_spaces	axiom	real normed spaces		Iter-3 imported axiom (s_real_normed_spaces).
s_midpoint_set_characterization	state	midpoint set characterization		Iter-3 imported state (s_midpoint_set_characterization).
s_midpoint_preserved_under_reflection	state	midpoint preserved under reflection		Iter-3 imported state (s_midpoint_preserved_under_reflection).
s_kirszbraun_theorem	theorem	Kirszbraun theorem (Lipschitz extension)		Iter-3 imported theorem (s_kirszbraun_theorem).
s_subset_S_of_hilbert_space	axiom	subset S of hilbert space		Iter-3 imported axiom (s_subset_S_of_hilbert_space).
s_intersection_of_balls_at_each_x	state	intersection of balls at each x		Iter-3 imported state (s_intersection_of_balls_at_each_x).
s_nonempty_intersection_for_each_x	state	nonempty intersection for each x		Iter-3 imported state (s_nonempty_intersection_for_each_x).
s_michael_selection	theorem	Michael selection theorem		Iter-3 imported theorem (s_michael_selection).
s_paracompact_space	axiom	paracompact space		Iter-3 imported axiom (s_paracompact_space).
s_lower_semicontinuous_correspondence_with_convex_values_in_banach	axiom	lower semicontinuous correspondence with convex values in banach		Iter-3 imported axiom (s_lower_semicontinuous_correspondence_with_convex_values_in_banach).
s_cauchy_sequence_of_approximate_selections	state	cauchy sequence of approximate selections		Iter-3 imported state (s_cauchy_sequence_of_approximate_selections).
s_dvoretzky_theorem	theorem	Dvoretzky's theorem		Iter-3 imported theorem (s_dvoretzky_theorem).
s_infinite_dim_banach_space	axiom	infinite dim banach space		Iter-3 imported axiom (s_infinite_dim_banach_space).
s_concentration_of_measure_on_sphere	axiom	concentration of measure on sphere		Iter-3 imported axiom (s_concentration_of_measure_on_sphere).
s_random_high_dim_subspace_selection	state	random high dim subspace selection		Iter-3 imported state (s_random_high_dim_subspace_selection).
s_concentration_of_norm_on_sphere	state	concentration of norm on sphere		Iter-3 imported state (s_concentration_of_norm_on_sphere).
s_almost_euclidean_section_existence	state	almost euclidean section existence		Iter-3 imported state (s_almost_euclidean_section_existence).
s_grothendieck_inequality	theorem	Grothendieck inequality		Iter-3 imported theorem (s_grothendieck_inequality).
s_unit_vectors_x_i_y_j_in_hilbert	axiom	unit vectors x i y j in hilbert		Iter-3 imported axiom (s_unit_vectors_x_i_y_j_in_hilbert).
s_gaussian_rounding_estimator	state	gaussian rounding estimator		Iter-3 imported state (s_gaussian_rounding_estimator).
s_arcsin_kernel_relation	state	arcsin kernel relation		Iter-3 imported state (s_arcsin_kernel_relation).
s_connes_embedding_statement	theorem	Connes embedding problem / theorem statement		Iter-3 imported theorem (s_connes_embedding_statement).
s_separable_II_1_factor_M	axiom	separable II 1 factor M		Iter-3 imported axiom (s_separable_II_1_factor_M).
s_ultrapower_of_hyperfinite_R_omega	axiom	ultrapower of hyperfinite R omega		Iter-3 imported axiom (s_ultrapower_of_hyperfinite_R_omega).
s_tracial_ultrapower_target	state	tracial ultrapower target		Iter-3 imported state (s_tracial_ultrapower_target).
s_finite_dim_matricial_microstates	state	finite dim matricial microstates		Iter-3 imported state (s_finite_dim_matricial_microstates).
s_bessel_inequality	theorem	Bessel's inequality		Iter-3 imported theorem (s_bessel_inequality).
s_orthonormal_sequence_e_n	axiom	orthonormal sequence e n		Iter-3 imported axiom (s_orthonormal_sequence_e_n).
s_partial_orthogonal_projection	state	partial orthogonal projection		Iter-3 imported state (s_partial_orthogonal_projection).
s_parseval_identity	theorem	Parseval's identity		Iter-3 imported theorem (s_parseval_identity).
s_complete_orthonormal_basis_e_n	axiom	complete orthonormal basis e n		Iter-3 imported axiom (s_complete_orthonormal_basis_e_n).
s_bessel_becomes_equality_for_complete_basis	state	bessel becomes equality for complete basis		Iter-3 imported state (s_bessel_becomes_equality_for_complete_basis).
s_holder_inequality	theorem	Hölder's inequality		Iter-3 imported theorem (s_holder_inequality).
s_measure_space	axiom	measure space		Iter-3 imported axiom (s_measure_space).
s_young_pointwise_inequality	state	young pointwise inequality		Iter-3 imported state (s_young_pointwise_inequality).
s_propagation_of_singularities	theorem	Propagation of singularities theorem		Iter-3 imported theorem (s_propagation_of_singularities).
s_pseudodifferential_operator_P_real_principal_symbol	axiom	pseudodifferential operator P real principal symbol		Iter-3 imported axiom (s_pseudodifferential_operator_P_real_principal_symbol).
s_hamiltonian_flow_bicharacteristics	state	hamiltonian flow bicharacteristics		Iter-3 imported state (s_hamiltonian_flow_bicharacteristics).
s_microlocal_propagation_estimate	state	microlocal propagation estimate		Iter-3 imported state (s_microlocal_propagation_estimate).
s_commutator_inequality_yields_regularity_transport	state	commutator inequality yields regularity transport		Iter-3 imported state (s_commutator_inequality_yields_regularity_transport).
s_open_mapping_frechet	theorem	Open mapping theorem (Fréchet space version)		Iter-3 imported theorem (s_open_mapping_frechet).
s_pair_of_frechet_spaces	axiom	pair of frechet spaces		Iter-3 imported axiom (s_pair_of_frechet_spaces).
s_continuous_surjective_linear_map_T	axiom	continuous surjective linear map T		Iter-3 imported axiom (s_continuous_surjective_linear_map_T).
s_translation_invariant_metric_setup	state	translation invariant metric setup		Iter-3 imported state (s_translation_invariant_metric_setup).
s_wold_decomposition	theorem	Wold decomposition		Iter-3 imported theorem (s_wold_decomposition).
s_hilbert_space_with_isometry_V	axiom	hilbert space with isometry V		Iter-3 imported axiom (s_hilbert_space_with_isometry_V).
s_unitary_part_H_u	state	unitary part H u		Iter-3 imported state (s_unitary_part_H_u).
s_shift_part_H_s_as_orthogonal_sum	state	shift part H s as orthogonal sum		Iter-3 imported state (s_shift_part_H_s_as_orthogonal_sum).
s_commutant_lifting	theorem	Commutant lifting theorem		Iter-3 imported theorem (s_commutant_lifting).
s_contraction_T_with_isometric_dilation_V	axiom	contraction T with isometric dilation V		Iter-3 imported axiom (s_contraction_T_with_isometric_dilation_V).
s_intertwining_operator_X_commuting_with_T	axiom	intertwining operator X commuting with T		Iter-3 imported axiom (s_intertwining_operator_X_commuting_with_T).
s_minimal_isometric_dilation_space	state	minimal isometric dilation space		Iter-3 imported state (s_minimal_isometric_dilation_space).
s_norm_preserving_extension_problem	state	norm preserving extension problem		Iter-3 imported state (s_norm_preserving_extension_problem).
s_parrott_completion_yields_Y	state	parrott completion yields Y		Iter-3 imported state (s_parrott_completion_yields_Y).
s_russo_dye	theorem	Russo–Dye theorem		Iter-3 imported theorem (s_russo_dye).
s_convex_hull_of_unitaries	axiom	convex hull of unitaries		Iter-3 imported axiom (s_convex_hull_of_unitaries).
s_two_unitary_decomposition_of_strict_contraction	state	two unitary decomposition of strict contraction		Iter-3 imported state (s_two_unitary_decomposition_of_strict_contraction).
s_explicit_unitary_pair	state	explicit unitary pair		Iter-3 imported state (s_explicit_unitary_pair).
s_crossed_product_construction	theorem	Crossed product construction (von Neumann)		Iter-3 imported theorem (s_crossed_product_construction).
s_l2_G_with_M_valued_functions	axiom	l2 G with M valued functions		Iter-3 imported axiom (s_l2_G_with_M_valued_functions).
s_covariant_representation	state	covariant representation		Iter-3 imported state (s_covariant_representation).
s_continuous_functional_calculus	theorem	Continuous functional calculus		Iter-3 imported theorem (s_continuous_functional_calculus).
s_normal_element_a_in_c_star_algebra	axiom	normal element a in c star algebra		Iter-3 imported axiom (s_normal_element_a_in_c_star_algebra).
s_polynomial_star_calculus	state	polynomial star calculus		Iter-3 imported state (s_polynomial_star_calculus).
s_isometry_to_polynomials_on_spectrum	state	isometry to polynomials on spectrum		Iter-3 imported state (s_isometry_to_polynomials_on_spectrum).
s_borel_functional_calculus	theorem	Borel functional calculus		Iter-3 imported theorem (s_borel_functional_calculus).
s_normal_operator_on_hilbert	axiom	normal operator on hilbert		Iter-3 imported axiom (s_normal_operator_on_hilbert).
s_spectral_measure_per_vector_pair	state	spectral measure per vector pair		Iter-3 imported state (s_spectral_measure_per_vector_pair).
s_borel_functional_calculus_per_pair	state	borel functional calculus per pair		Iter-3 imported state (s_borel_functional_calculus_per_pair).
s_direct_integral_decomposition	theorem	Direct integral decomposition (von Neumann)		Iter-3 imported theorem (s_direct_integral_decomposition).
s_separable_hilbert_space	axiom	separable hilbert space		Iter-3 imported axiom (s_separable_hilbert_space).
s_abelian_von_neumann_subalgebra_A	axiom	abelian von neumann subalgebra A		Iter-3 imported axiom (s_abelian_von_neumann_subalgebra_A).
s_a_acts_as_multiplication_on_measure_space_X	state	a acts as multiplication on measure space X		Iter-3 imported state (s_a_acts_as_multiplication_on_measure_space_X).
s_measurable_hilbert_field	state	measurable hilbert field		Iter-3 imported state (s_measurable_hilbert_field).
s_direct_integral_isomorphism	state	direct integral isomorphism		Iter-3 imported state (s_direct_integral_isomorphism).
s_banach_mazur_embedding	theorem	Banach–Mazur theorem (separable Banach spaces embed in C[0,1])		Iter-3 imported theorem (s_banach_mazur_embedding).
s_separable_banach_space_X	axiom	separable banach space X		Iter-3 imported axiom (s_separable_banach_space_X).
s_unit_ball_of_X_star	axiom	unit ball of X star		Iter-3 imported axiom (s_unit_ball_of_X_star).
s_compact_metrizable_dual_unit_ball_K	state	compact metrizable dual unit ball K		Iter-3 imported state (s_compact_metrizable_dual_unit_ball_K).
s_plancherel_nonabelian	theorem	Plancherel for non-abelian compact groups		Iter-3 imported theorem (s_plancherel_nonabelian).
s_compact_lie_group_G	axiom	compact lie group G		Iter-3 imported axiom (s_compact_lie_group_G).
s_peter_weyl_decomposition	state	peter weyl decomposition		Iter-3 imported state (s_peter_weyl_decomposition).
s_matrix_coefficient_orthogonality	state	matrix coefficient orthogonality		Iter-3 imported state (s_matrix_coefficient_orthogonality).
s_operator_valued_fourier_transform	state	operator valued fourier transform		Iter-3 imported state (s_operator_valued_fourier_transform).
s_pontryagin_duality	theorem	Pontryagin duality		Iter-3 imported theorem (s_pontryagin_duality).
s_locally_compact_abelian_group_G	axiom	locally compact abelian group G		Iter-3 imported axiom (s_locally_compact_abelian_group_G).
s_canonical_double_dual_map	state	canonical double dual map		Iter-3 imported state (s_canonical_double_dual_map).
s_choi_effros_lifting	theorem	Aubert–Choi continuity / Choi–Effros lifting theorem		Iter-3 imported theorem (s_choi_effros_lifting).
s_nuclear_separable_c_star_algebra	axiom	nuclear separable c star algebra		Iter-3 imported axiom (s_nuclear_separable_c_star_algebra).
s_cp_approximation_diagrams	state	cp approximation diagrams		Iter-3 imported state (s_cp_approximation_diagrams).
s_diagram_lifted_into_B	state	diagram lifted into B		Iter-3 imported state (s_diagram_lifted_into_B).
s_coherent_lift_of_CP_diagrams	state	coherent lift of CP diagrams		Iter-3 imported state (s_coherent_lift_of_CP_diagrams).
s_glimm_dichotomy	theorem	Glimm's dichotomy / type I vs non-type-I		Iter-3 imported theorem (s_glimm_dichotomy).
s_separable_c_star_algebra	axiom	separable c star algebra		Iter-3 imported axiom (s_separable_c_star_algebra).
s_irreducible_representation_class	axiom	irreducible representation class		Iter-3 imported axiom (s_irreducible_representation_class).
s_pattern_T_0_dual_indicates_type_I	state	pattern T 0 dual indicates type I		Iter-3 imported state (s_pattern_T_0_dual_indicates_type_I).
s_non_type_I_dichotomy_witness	state	non type I dichotomy witness		Iter-3 imported state (s_non_type_I_dichotomy_witness).
s_connes_noncommutative_index	theorem	Connes' noncommutative index formula		Iter-3 imported theorem (s_connes_noncommutative_index).
s_chern_connes_pairing	axiom	chern connes pairing		Iter-3 imported axiom (s_chern_connes_pairing).
s_k_homology_class_of_D	state	k homology class of D		Iter-3 imported state (s_k_homology_class_of_D).
s_pairing_with_K_theory_of_A	state	pairing with K theory of A		Iter-3 imported state (s_pairing_with_K_theory_of_A).
s_chern_connes_character_in_periodic_cyclic	state	chern connes character in periodic cyclic		Iter-3 imported state (s_chern_connes_character_in_periodic_cyclic).
s_pettis_measurability	theorem	Pettis theorem (weakly measurable = strongly measurable for separable values)		Iter-3 imported theorem (s_pettis_measurability).
s_simple_function_approximant_pointwise	state	simple function approximant pointwise		Iter-3 imported state (s_simple_function_approximant_pointwise).
s_borel_measurability_in_weak_topology	state	borel measurability in weak topology		Iter-3 imported state (s_borel_measurability_in_weak_topology).
s_eidelheit_separation	theorem	Eidelheit separation theorem		Iter-3 imported theorem (s_eidelheit_separation).
s_convex_set_with_nonempty_interior_in_topological_vector_space	axiom	convex set with nonempty interior in topological vector space		Iter-3 imported axiom (s_convex_set_with_nonempty_interior_in_topological_vector_space).
s_disjoint_convex_set	axiom	disjoint convex set		Iter-3 imported axiom (s_disjoint_convex_set).
s_minkowski_gauge_continuous	state	minkowski gauge continuous		Iter-3 imported state (s_minkowski_gauge_continuous).
s_continuous_separating_functional	state	continuous separating functional		Iter-3 imported state (s_continuous_separating_functional).
s_nash_moser	theorem	Nash–Moser implicit function theorem		Iter-3 imported theorem (s_nash_moser).
s_tame_frechet_space_pair	axiom	tame frechet space pair		Iter-3 imported axiom (s_tame_frechet_space_pair).
s_smoothing_operators_S_t	axiom	smoothing operators S t		Iter-3 imported axiom (s_smoothing_operators_S_t).
s_smoothing_family_S_t	state	smoothing family S t		Iter-3 imported state (s_smoothing_family_S_t).
s_modified_newton_iteration	state	modified newton iteration		Iter-3 imported state (s_modified_newton_iteration).
s_convergent_quadratic_iteration	state	convergent quadratic iteration		Iter-3 imported state (s_convergent_quadratic_iteration).
s_weak_formulation_elliptic	theorem	Weak formulation of elliptic BVP / Lax–Milgram applied		Iter-3 imported theorem (s_weak_formulation_elliptic).
s_sobolev_space_H1_0	axiom	sobolev space H1 0		Iter-3 imported axiom (s_sobolev_space_H1_0).
s_coercive_continuous_bilinear_form	state	coercive continuous bilinear form		Iter-3 imported state (s_coercive_continuous_bilinear_form).
s_levi_civita_connection	theorem	Fundamental theorem of Riemannian geometry (Levi-Civita)		Iter-3 imported theorem (s_levi_civita_connection).
s_smooth_manifold	axiom	smooth manifold		Iter-3 imported axiom (s_smooth_manifold).
s_connection_axioms_metric_torsion_free	state	connection axioms metric torsion free		Iter-3 imported state (s_connection_axioms_metric_torsion_free).
s_koszul_formula_for_connection	state	koszul formula for connection		Iter-3 imported state (s_koszul_formula_for_connection).
s_hopf_rinow_theorem	theorem	Hopf–Rinow theorem		Iter-3 imported theorem (s_hopf_rinow_theorem).
s_completeness_notions_for_riemannian_manifold	state	completeness notions for riemannian manifold		Iter-3 imported state (s_completeness_notions_for_riemannian_manifold).
s_closed_bounded_sets_compact	state	closed bounded sets compact		Iter-3 imported state (s_closed_bounded_sets_compact).
s_minimizing_geodesic_between_any_two_points	state	minimizing geodesic between any two points		Iter-3 imported state (s_minimizing_geodesic_between_any_two_points).
s_cartan_hadamard_theorem	theorem	Cartan–Hadamard theorem		Iter-3 imported theorem (s_cartan_hadamard_theorem).
s_complete_nonpositively_curved_manifold	state	complete nonpositively curved manifold		Iter-3 imported state (s_complete_nonpositively_curved_manifold).
s_no_conjugate_points	state	no conjugate points		Iter-3 imported state (s_no_conjugate_points).
s_exp_map_is_covering	state	exp map is covering		Iter-3 imported state (s_exp_map_is_covering).
s_bonnet_myers_theorem	theorem	Bonnet–Myers theorem		Iter-3 imported theorem (s_bonnet_myers_theorem).
s_complete_manifold_with_ricci_lower_bound	state	complete manifold with ricci lower bound		Iter-3 imported state (s_complete_manifold_with_ricci_lower_bound).
s_second_variation_index_form	state	second variation index form		Iter-3 imported state (s_second_variation_index_form).
s_diameter_bound_pi_over_sqrt_k	state	diameter bound pi over sqrt k		Iter-3 imported state (s_diameter_bound_pi_over_sqrt_k).
s_synge_theorem	theorem	Synge's theorem		Iter-3 imported theorem (s_synge_theorem).
s_compact_even_dim_orientable_positively_curved	state	compact even dim orientable positively curved		Iter-3 imported state (s_compact_even_dim_orientable_positively_curved).
s_closed_geodesic_minimizer	state	closed geodesic minimizer		Iter-3 imported state (s_closed_geodesic_minimizer).
s_parallel_vector_field_along_geodesic	state	parallel vector field along geodesic		Iter-3 imported state (s_parallel_vector_field_along_geodesic).
s_cheng_diameter_rigidity	theorem	Cheng's diameter rigidity theorem		Iter-3 imported theorem (s_cheng_diameter_rigidity).
s_extremal_diameter_case_of_bonnet_myers	state	extremal diameter case of bonnet myers		Iter-3 imported state (s_extremal_diameter_case_of_bonnet_myers).
s_volume_equality_with_round_sphere	state	volume equality with round sphere		Iter-3 imported state (s_volume_equality_with_round_sphere).
s_cheeger_gromoll_splitting	theorem	Cheeger–Gromoll splitting theorem		Iter-3 imported theorem (s_cheeger_gromoll_splitting).
s_complete_nonneg_ricci_with_line	state	complete nonneg ricci with line		Iter-3 imported state (s_complete_nonneg_ricci_with_line).
s_pair_of_busemann_functions	state	pair of busemann functions		Iter-3 imported state (s_pair_of_busemann_functions).
s_harmonic_busemann_function_pair	state	harmonic busemann function pair		Iter-3 imported state (s_harmonic_busemann_function_pair).
s_soul_theorem	theorem	Soul theorem		Iter-3 imported theorem (s_soul_theorem).
s_open_nonneg_curved_manifold	state	open nonneg curved manifold		Iter-3 imported state (s_open_nonneg_curved_manifold).
s_nested_totally_convex_sets	state	nested totally convex sets		Iter-3 imported state (s_nested_totally_convex_sets).
s_soul_submanifold	state	soul submanifold		Iter-3 imported state (s_soul_submanifold).
s_rauch_comparison_theorem	theorem	Rauch comparison theorem		Iter-3 imported theorem (s_rauch_comparison_theorem).
s_jacobi_equation_along_geodesic	state	jacobi equation along geodesic		Iter-3 imported state (s_jacobi_equation_along_geodesic).
s_sturm_comparison_for_jacobi_norm	state	sturm comparison for jacobi norm		Iter-3 imported state (s_sturm_comparison_for_jacobi_norm).
s_toponogov_theorem	theorem	Toponogov comparison theorem		Iter-3 imported theorem (s_toponogov_theorem).
s_lower_curvature_bounded_manifold	state	lower curvature bounded manifold		Iter-3 imported state (s_lower_curvature_bounded_manifold).
s_geodesic_triangle_pair_M_and_Mk	state	geodesic triangle pair M and Mk		Iter-3 imported state (s_geodesic_triangle_pair_M_and_Mk).
s_angle_inequality_M_geq_Mk	state	angle inequality M geq Mk		Iter-3 imported state (s_angle_inequality_M_geq_Mk).
s_bishop_gromov_comparison	theorem	Bishop–Gromov volume comparison		Iter-3 imported theorem (s_bishop_gromov_comparison).
s_ricci_lower_bounded_manifold	state	ricci lower bounded manifold		Iter-3 imported state (s_ricci_lower_bounded_manifold).
s_jacobian_of_exp_map_in_polar	state	jacobian of exp map in polar		Iter-3 imported state (s_jacobian_of_exp_map_in_polar).
s_pointwise_jacobian_bound	state	pointwise jacobian bound		Iter-3 imported state (s_pointwise_jacobian_bound).
s_gromov_compactness_riemannian	theorem	Gromov compactness for Riemannian manifolds		Iter-3 imported theorem (s_gromov_compactness_riemannian).
s_class_of_manifolds_with_uniform_geometry	state	class of manifolds with uniform geometry		Iter-3 imported state (s_class_of_manifolds_with_uniform_geometry).
s_gromov_hausdorff_distance_on_class	state	gromov hausdorff distance on class		Iter-3 imported state (s_gromov_hausdorff_distance_on_class).
s_total_boundedness_of_class	state	total boundedness of class		Iter-3 imported state (s_total_boundedness_of_class).
s_cheeger_finiteness	theorem	Cheeger finiteness theorem		Iter-3 imported theorem (s_cheeger_finiteness).
s_uniformly_bounded_geometry_class	state	uniformly bounded geometry class		Iter-3 imported state (s_uniformly_bounded_geometry_class).
s_uniform_injectivity_radius	state	uniform injectivity radius		Iter-3 imported state (s_uniform_injectivity_radius).
s_finite_diffeomorphism_types	state	finite diffeomorphism types		Iter-3 imported state (s_finite_diffeomorphism_types).
s_sphere_theorem_quarter_pinched	theorem	Sphere theorem (1/4-pinched)		Iter-3 imported theorem (s_sphere_theorem_quarter_pinched).
s_strictly_quarter_pinched_manifold	state	strictly quarter pinched manifold		Iter-3 imported state (s_strictly_quarter_pinched_manifold).
s_lower_injectivity_radius_pi	state	lower injectivity radius pi		Iter-3 imported state (s_lower_injectivity_radius_pi).
s_two_exponential_charts_cover_M	state	two exponential charts cover M		Iter-3 imported state (s_two_exponential_charts_cover_M).
s_killing_hopf_theorem	theorem	Killing–Hopf theorem		Iter-3 imported theorem (s_killing_hopf_theorem).
s_space_form_class	state	space form class		Iter-3 imported state (s_space_form_class).
s_universal_cover_is_model_space	state	universal cover is model space		Iter-3 imported state (s_universal_cover_is_model_space).
s_myers_steenrod_theorem	theorem	Myers–Steenrod theorem		Iter-3 imported theorem (s_myers_steenrod_theorem).
s_isometry_group_topological	state	isometry group topological		Iter-3 imported state (s_isometry_group_topological).
s_isometry_group_locally_compact	state	isometry group locally compact		Iter-3 imported state (s_isometry_group_locally_compact).
s_nash_embedding_theorem	theorem	Nash embedding theorem		Iter-3 imported theorem (s_nash_embedding_theorem).
s_free_embedding_existence	state	free embedding existence		Iter-3 imported state (s_free_embedding_existence).
s_perturbative_isometric_embedding	state	perturbative isometric embedding		Iter-3 imported state (s_perturbative_isometric_embedding).
s_preissmann_theorem	theorem	Preissmann's theorem		Iter-3 imported theorem (s_preissmann_theorem).
s_compact_negatively_curved_manifold	state	compact negatively curved manifold		Iter-3 imported state (s_compact_negatively_curved_manifold).
s_pi1_action_on_Cartan_Hadamard_space	state	pi1 action on Cartan Hadamard space		Iter-3 imported state (s_pi1_action_on_Cartan_Hadamard_space).
s_flat_strip_contradiction	state	flat strip contradiction		Iter-3 imported state (s_flat_strip_contradiction).
s_mostow_rigidity_theorem	theorem	Mostow rigidity theorem		Iter-3 imported theorem (s_mostow_rigidity_theorem).
s_finite_volume_hyperbolic_manifold	state	finite volume hyperbolic manifold		Iter-3 imported state (s_finite_volume_hyperbolic_manifold).
s_quasiconformal_boundary_map	state	quasiconformal boundary map		Iter-3 imported state (s_quasiconformal_boundary_map).
s_boundary_map_is_conformal	state	boundary map is conformal		Iter-3 imported state (s_boundary_map_is_conformal).
s_hodge_decomposition	theorem	Hodge decomposition theorem		Iter-3 imported theorem (s_hodge_decomposition).
s_hodge_laplacian	state	hodge laplacian		Iter-3 imported state (s_hodge_laplacian).
s_eigenform_orthogonal_decomp	state	eigenform orthogonal decomp		Iter-3 imported state (s_eigenform_orthogonal_decomp).
s_orthogonal_decomp_im_d_im_dstar_harmonic	state	orthogonal decomp im d im dstar harmonic		Iter-3 imported state (s_orthogonal_decomp_im_d_im_dstar_harmonic).
s_frobenius_integrability	theorem	Frobenius theorem (integrable distributions)		Iter-3 imported theorem (s_frobenius_integrability).
s_smooth_distribution	state	smooth distribution		Iter-3 imported state (s_smooth_distribution).
s_involutive_distribution	state	involutive distribution		Iter-3 imported state (s_involutive_distribution).
s_local_foliation_chart	state	local foliation chart		Iter-3 imported state (s_local_foliation_chart).
s_generalized_stokes_theorem	theorem	Generalized Stokes theorem		Iter-3 imported theorem (s_generalized_stokes_theorem).
s_FTC_and_Greens_theorem_special_cases	state	FTC and Greens theorem special cases		Iter-3 imported state (s_FTC_and_Greens_theorem_special_cases).
s_local_to_global_integration_identity	state	local to global integration identity		Iter-3 imported state (s_local_to_global_integration_identity).
s_cartan_formula	theorem	Cartan formula for Lie derivative		Iter-3 imported theorem (s_cartan_formula).
s_three_derivations_on_omega_star	state	three derivations on omega star		Iter-3 imported state (s_three_derivations_on_omega_star).
s_cartan_identity_on_low_degree	state	cartan identity on low degree		Iter-3 imported state (s_cartan_identity_on_low_degree).
s_fundamental_theorem_of_curves	theorem	Fundamental theorem of curves		Iter-3 imported theorem (s_fundamental_theorem_of_curves).
s_frenet_frame_ODE	state	frenet frame ODE		Iter-3 imported state (s_frenet_frame_ODE).
s_curve_recovered_from_kappa_tau	state	curve recovered from kappa tau		Iter-3 imported state (s_curve_recovered_from_kappa_tau).
s_four_vertex_theorem	theorem	Four vertex theorem		Iter-3 imported theorem (s_four_vertex_theorem).
s_smooth_convex_closed_curve	state	smooth convex closed curve		Iter-3 imported state (s_smooth_convex_closed_curve).
s_kappa_prime_sign_change_contradiction	state	kappa prime sign change contradiction		Iter-3 imported state (s_kappa_prime_sign_change_contradiction).
s_fenchel_theorem	theorem	Fenchel's theorem		Iter-3 imported theorem (s_fenchel_theorem).
s_tantrix_spherical_curve	state	tantrix spherical curve		Iter-3 imported state (s_tantrix_spherical_curve).
s_total_curvature_equals_tantrix_length	state	total curvature equals tantrix length		Iter-3 imported state (s_total_curvature_equals_tantrix_length).
s_hilbert_no_isometric_immersion_hyperbolic_plane	theorem	Hilbert's theorem on hyperbolic plane		Iter-3 imported theorem (s_hilbert_no_isometric_immersion_hyperbolic_plane).
s_hyperbolic_plane_as_abstract_surface	state	hyperbolic plane as abstract surface		Iter-3 imported state (s_hyperbolic_plane_as_abstract_surface).
s_chebyshev_net_with_angle_omega	state	chebyshev net with angle omega		Iter-3 imported state (s_chebyshev_net_with_angle_omega).
s_sine_gordon_equation	state	sine gordon equation		Iter-3 imported state (s_sine_gordon_equation).
s_bertrand_diguet_puiseux	theorem	Bertrand–Diguet–Puiseux theorem		Iter-3 imported theorem (s_bertrand_diguet_puiseux).
s_geodesic_disk_circumference_C_r	state	geodesic disk circumference C r		Iter-3 imported state (s_geodesic_disk_circumference_C_r).
s_taylor_expansion_of_circumference	state	taylor expansion of circumference		Iter-3 imported state (s_taylor_expansion_of_circumference).
s_tennis_ball_theorem	theorem	Tennis ball theorem		Iter-3 imported theorem (s_tennis_ball_theorem).
s_area_bisecting_curve_on_sphere	state	area bisecting curve on sphere		Iter-3 imported state (s_area_bisecting_curve_on_sphere).
s_zero_signed_geodesic_curvature_integral	state	zero signed geodesic curvature integral		Iter-3 imported state (s_zero_signed_geodesic_curvature_integral).
s_three_geodesics_theorem	theorem	Theorem of the three geodesics		Iter-3 imported theorem (s_three_geodesics_theorem).
s_loop_space_energy_functional	state	loop space energy functional		Iter-3 imported state (s_loop_space_energy_functional).
s_min_max_critical_values	state	min max critical values		Iter-3 imported state (s_min_max_critical_values).
s_three_distinct_simple_closed_geodesics	state	three distinct simple closed geodesics		Iter-3 imported state (s_three_distinct_simple_closed_geodesics).
s_lyusternik_fet_theorem	theorem	Lyusternik–Fet theorem		Iter-3 imported theorem (s_lyusternik_fet_theorem).
s_free_loop_space_energy	state	free loop space energy		Iter-3 imported state (s_free_loop_space_energy).
s_nontrivial_loop_homotopy_class	state	nontrivial loop homotopy class		Iter-3 imported state (s_nontrivial_loop_homotopy_class).
s_gage_hamilton_grayson	theorem	Gage–Hamilton–Grayson theorem		Iter-3 imported theorem (s_gage_hamilton_grayson).
s_curve_shortening_flow_eqn	state	curve shortening flow eqn		Iter-3 imported state (s_curve_shortening_flow_eqn).
s_convexification_of_embedded_curve	state	convexification of embedded curve		Iter-3 imported state (s_convexification_of_embedded_curve).
s_round_point_limit	state	round point limit		Iter-3 imported state (s_round_point_limit).
s_willmore_conjecture	theorem	Willmore conjecture (Marques–Neves)		Iter-3 imported theorem (s_willmore_conjecture).
s_willmore_energy_functional	state	willmore energy functional		Iter-3 imported state (s_willmore_energy_functional).
s_conformally_invariant_energy	state	conformally invariant energy		Iter-3 imported state (s_conformally_invariant_energy).
s_min_max_widths_of_torus_class	state	min max widths of torus class		Iter-3 imported state (s_min_max_widths_of_torus_class).
s_bernstein_problem	theorem	Bernstein's problem		Iter-3 imported theorem (s_bernstein_problem).
s_minimal_surface_equation	state	minimal surface equation		Iter-3 imported state (s_minimal_surface_equation).
s_entire_minimal_graph_in_dim_n	state	entire minimal graph in dim n		Iter-3 imported state (s_entire_minimal_graph_in_dim_n).
s_tangent_cone_minimal_cone	state	tangent cone minimal cone		Iter-3 imported state (s_tangent_cone_minimal_cone).
s_tennenbaum_theorem	theorem	Tennenbaum's theorem		Iter-3 imported theorem (s_tennenbaum_theorem).
s_countable_nonstandard_model_of_PA	state	countable nonstandard model of PA		Iter-3 imported state (s_countable_nonstandard_model_of_PA).
s_re_inseparable_pair_coded_in_M	state	re inseparable pair coded in M		Iter-3 imported state (s_re_inseparable_pair_coded_in_M).
s_recursive_separation_contradiction	state	recursive separation contradiction		Iter-3 imported state (s_recursive_separation_contradiction).
s_goodstein_theorem	theorem	Goodstein's theorem		Iter-3 imported theorem (s_goodstein_theorem).
s_ordinal_arithmetic_below_epsilon_0	axiom	ordinal arithmetic below epsilon 0		Iter-3 imported axiom (s_ordinal_arithmetic_below_epsilon_0).
s_natural_number_n	state	natural number n		Iter-3 imported state (s_natural_number_n).
s_goodstein_sequence_definition	state	goodstein sequence definition		Iter-3 imported state (s_goodstein_sequence_definition).
s_ordinal_majorant_sequence_below_epsilon_0	state	ordinal majorant sequence below epsilon 0		Iter-3 imported state (s_ordinal_majorant_sequence_below_epsilon_0).
s_ordinal_sequence_must_hit_zero	state	ordinal sequence must hit zero		Iter-3 imported state (s_ordinal_sequence_must_hit_zero).
s_kirby_paris_independence	theorem	Goodstein independence from PA (Kirby–Paris)		Iter-3 imported theorem (s_kirby_paris_independence).
s_total_function_growth_of_Goodstein	state	total function growth of Goodstein		Iter-3 imported state (s_total_function_growth_of_Goodstein).
s_hardy_hierarchy_majorant_at_epsilon_0	state	hardy hierarchy majorant at epsilon 0		Iter-3 imported state (s_hardy_hierarchy_majorant_at_epsilon_0).
s_hydra_theorem	theorem	Kirby–Paris hydra theorem		Iter-3 imported theorem (s_hydra_theorem).
s_finite_rooted_tree	axiom	finite rooted tree		Iter-3 imported axiom (s_finite_rooted_tree).
s_hydra_game_state	state	hydra game state		Iter-3 imported state (s_hydra_game_state).
s_hydra_ordinal_invariant	state	hydra ordinal invariant		Iter-3 imported state (s_hydra_ordinal_invariant).
s_hydra_terminates	state	hydra terminates		Iter-3 imported state (s_hydra_terminates).
s_kanamori_mcaloon_theorem	theorem	Kanamori–McAloon theorem		Iter-3 imported theorem (s_kanamori_mcaloon_theorem).
s_regressive_finite_ramsey_statement	state	regressive finite ramsey statement		Iter-3 imported state (s_regressive_finite_ramsey_statement).
s_indicator_for_PA_models	state	indicator for PA models		Iter-3 imported state (s_indicator_for_PA_models).
s_kreisel_basis_theorem	theorem	Kreisel's basis theorem		Iter-3 imported theorem (s_kreisel_basis_theorem).
s_pi_0_1_class_of_reals	axiom	pi 0 1 class of reals		Iter-3 imported axiom (s_pi_0_1_class_of_reals).
s_recursive_binary_tree_T	state	recursive binary tree T		Iter-3 imported state (s_recursive_binary_tree_T).
s_existence_of_infinite_branch	state	existence of infinite branch		Iter-3 imported state (s_existence_of_infinite_branch).
s_leftmost_branch_recursive_in_0_prime	state	leftmost branch recursive in 0 prime		Iter-3 imported state (s_leftmost_branch_recursive_in_0_prime).
s_hyperarithmetic_hierarchy	theorem	Hyperarithmetic hierarchy theorem		Iter-3 imported theorem (s_hyperarithmetic_hierarchy).
s_turing_jump_operator	axiom	turing jump operator		Iter-3 imported axiom (s_turing_jump_operator).
s_recursive_ordinal	axiom	recursive ordinal		Iter-3 imported axiom (s_recursive_ordinal).
s_transfinite_jump_hierarchy_H_a	state	transfinite jump hierarchy H a		Iter-3 imported state (s_transfinite_jump_hierarchy_H_a).
s_delta_1_1_equals_hyperarithmetic	state	delta 1 1 equals hyperarithmetic		Iter-3 imported state (s_delta_1_1_equals_hyperarithmetic).
s_strict_hyperarithmetic_hierarchy	state	strict hyperarithmetic hierarchy		Iter-3 imported state (s_strict_hyperarithmetic_hierarchy).
s_friedman_finite_form_kruskal	theorem	Friedman's finite gap (Kruskal-style) theorem		Iter-3 imported theorem (s_friedman_finite_form_kruskal).
s_finite_labelled_trees	axiom	finite labelled trees		Iter-3 imported axiom (s_finite_labelled_trees).
s_wqo_axioms	axiom	wqo axioms		Iter-3 imported axiom (s_wqo_axioms).
s_kruskal_wqo_statement	state	kruskal wqo statement		Iter-3 imported state (s_kruskal_wqo_statement).
s_finitary_TREE_function_statement	state	finitary TREE function statement		Iter-3 imported state (s_finitary_TREE_function_statement).
s_mostowski_collapse	theorem	Mostowski collapse lemma		Iter-3 imported theorem (s_mostowski_collapse).
s_wellfounded_extensional_relation	axiom	wellfounded extensional relation		Iter-3 imported axiom (s_wellfounded_extensional_relation).
s_collapse_function_pi	state	collapse function pi		Iter-3 imported state (s_collapse_function_pi).
s_pi_total_and_injective	state	pi total and injective		Iter-3 imported state (s_pi_total_and_injective).
s_mostowski_absoluteness	theorem	Mostowski absoluteness		Iter-3 imported theorem (s_mostowski_absoluteness).
s_transitive_model_of_ZF	axiom	transitive model of ZF		Iter-3 imported axiom (s_transitive_model_of_ZF).
s_sigma_1_1_as_wellfoundedness	state	sigma 1 1 as wellfoundedness		Iter-3 imported state (s_sigma_1_1_as_wellfoundedness).
s_wellfoundedness_absolute_for_transitive_models	state	wellfoundedness absolute for transitive models		Iter-3 imported state (s_wellfoundedness_absolute_for_transitive_models).
s_shoenfield_absoluteness	theorem	Shoenfield absoluteness theorem		Iter-3 imported theorem (s_shoenfield_absoluteness).
s_sigma_1_2_formula	state	sigma 1 2 formula		Iter-3 imported state (s_sigma_1_2_formula).
s_sigma_1_2_normal_form	state	sigma 1 2 normal form		Iter-3 imported state (s_sigma_1_2_normal_form).
s_shoenfield_tree_with_absolute_wellfoundedness	state	shoenfield tree with absolute wellfoundedness		Iter-3 imported state (s_shoenfield_tree_with_absolute_wellfoundedness).
s_absoluteness_to_L	state	absoluteness to L		Iter-3 imported state (s_absoluteness_to_L).
s_levy_reflection	theorem	Levy reflection principle		Iter-3 imported theorem (s_levy_reflection).
s_cumulative_hierarchy_V_alpha	axiom	cumulative hierarchy V alpha		Iter-3 imported axiom (s_cumulative_hierarchy_V_alpha).
s_reflection_schema_for_finite_subtheory	state	reflection schema for finite subtheory		Iter-3 imported state (s_reflection_schema_for_finite_subtheory).
s_closure_under_skolem_functions	state	closure under skolem functions		Iter-3 imported state (s_closure_under_skolem_functions).
s_constructible_universe_L	theorem	Constructible universe L (Gödel)		Iter-3 imported theorem (s_constructible_universe_L).
s_definable_powerset_operator	axiom	definable powerset operator		Iter-3 imported axiom (s_definable_powerset_operator).
s_L_hierarchy_L_alpha	state	L hierarchy L alpha		Iter-3 imported state (s_L_hierarchy_L_alpha).
s_L_models_ZF	state	L models ZF		Iter-3 imported state (s_L_models_ZF).
s_canonical_wellorder_of_L	state	canonical wellorder of L		Iter-3 imported state (s_canonical_wellorder_of_L).
s_GCH_in_L	theorem	V=L implies GCH in L		Iter-3 imported theorem (s_GCH_in_L).
s_condensation_lemma	axiom	condensation lemma		Iter-3 imported axiom (s_condensation_lemma).
s_countable_elementary_substructure	state	countable elementary substructure		Iter-3 imported state (s_countable_elementary_substructure).
s_collapsed_substructure_is_L_alpha	state	collapsed substructure is L alpha		Iter-3 imported state (s_collapsed_substructure_is_L_alpha).
s_definability_in_L_alpha	axiom	definability in L alpha		Iter-3 imported axiom (s_definability_in_L_alpha).
s_elementary_substructure_of_L_alpha	state	elementary substructure of L alpha		Iter-3 imported state (s_elementary_substructure_of_L_alpha).
s_transitive_isomorph_M_bar	state	transitive isomorph M bar		Iter-3 imported state (s_transitive_isomorph_M_bar).
s_M_bar_equals_L_beta	state	M bar equals L beta		Iter-3 imported state (s_M_bar_equals_L_beta).
s_jensen_covering	theorem	Jensen's covering lemma		Iter-3 imported theorem (s_jensen_covering).
s_zero_sharp_does_not_exist	axiom	zero sharp does not exist		Iter-3 imported axiom (s_zero_sharp_does_not_exist).
s_uncovered_set_hypothesis	state	uncovered set hypothesis		Iter-3 imported state (s_uncovered_set_hypothesis).
s_fine_structure_witness_in_L	state	fine structure witness in L		Iter-3 imported state (s_fine_structure_witness_in_L).
s_construction_of_zero_sharp	state	construction of zero sharp		Iter-3 imported state (s_construction_of_zero_sharp).
s_zero_sharp_existence	theorem	Zero sharp (0#) existence		Iter-3 imported theorem (s_zero_sharp_existence).
s_measurable_cardinal_axiom	axiom	measurable cardinal axiom		Iter-3 imported axiom (s_measurable_cardinal_axiom).
s_ultrapower_embedding_j	state	ultrapower embedding j		Iter-3 imported state (s_ultrapower_embedding_j).
s_nontrivial_elementary_j_restricted_to_L	state	nontrivial elementary j restricted to L		Iter-3 imported state (s_nontrivial_elementary_j_restricted_to_L).
s_silver_indiscernibles_in_L	state	silver indiscernibles in L		Iter-3 imported state (s_silver_indiscernibles_in_L).
s_silver_indiscernibles_theorem	theorem	Silver indiscernibles theorem		Iter-3 imported theorem (s_silver_indiscernibles_theorem).
s_EM_blueprint_for_L	state	EM blueprint for L		Iter-3 imported state (s_EM_blueprint_for_L).
s_indiscernibility_of_uncountable_cardinals	state	indiscernibility of uncountable cardinals		Iter-3 imported state (s_indiscernibility_of_uncountable_cardinals).
s_solovay_model	theorem	Solovay's model (every set Lebesgue-measurable)		Iter-3 imported theorem (s_solovay_model).
s_inaccessible_cardinal_axiom	axiom	inaccessible cardinal axiom		Iter-3 imported axiom (s_inaccessible_cardinal_axiom).
s_levy_collapse_model	state	levy collapse model		Iter-3 imported state (s_levy_collapse_model).
s_HOD_R_ordinals_inner_model	state	HOD R ordinals inner model		Iter-3 imported state (s_HOD_R_ordinals_inner_model).
s_every_set_definable_from_real	state	every set definable from real		Iter-3 imported state (s_every_set_definable_from_real).
s_random_real_forcing	theorem	Solovay random real forcing		Iter-3 imported theorem (s_random_real_forcing).
s_borel_measure_algebra	axiom	borel measure algebra		Iter-3 imported axiom (s_borel_measure_algebra).
s_complete_BA_for_random_forcing	state	complete BA for random forcing		Iter-3 imported state (s_complete_BA_for_random_forcing).
s_generic_random_real	state	generic random real		Iter-3 imported state (s_generic_random_real).
s_random_real_avoids_null_sets	state	random real avoids null sets		Iter-3 imported state (s_random_real_avoids_null_sets).
s_mansfield_solovay	theorem	Mansfield–Solovay theorem		Iter-3 imported theorem (s_mansfield_solovay).
s_sigma_1_2_set_of_reals	axiom	sigma 1 2 set of reals		Iter-3 imported axiom (s_sigma_1_2_set_of_reals).
s_sigma_1_2_as_tree_projection	state	sigma 1 2 as tree projection		Iter-3 imported state (s_sigma_1_2_as_tree_projection).
s_perfect_subtree_case	state	perfect subtree case		Iter-3 imported state (s_perfect_subtree_case).
s_dichotomy_perfect_or_inside_L	state	dichotomy perfect or inside L		Iter-3 imported state (s_dichotomy_perfect_or_inside_L).
s_vaught_two_cardinal	theorem	Vaught's two-cardinal theorem		Iter-3 imported theorem (s_vaught_two_cardinal).
s_first_order_theory_T	axiom	first order theory T		Iter-3 imported axiom (s_first_order_theory_T).
s_elementary_substructure_property	axiom	elementary substructure property		Iter-3 imported axiom (s_elementary_substructure_property).
s_two_cardinal_model_M	state	two cardinal model M		Iter-3 imported state (s_two_cardinal_model_M).
s_elementary_chain_template	state	elementary chain template		Iter-3 imported state (s_elementary_chain_template).
s_vaught_never_22_status	theorem	Vaught's conjecture (status / never-22)		Iter-3 imported theorem (s_vaught_never_22_status).
s_complete_countable_first_order_theory	axiom	complete countable first order theory		Iter-3 imported axiom (s_complete_countable_first_order_theory).
s_morley_analysis_of_countable_models	axiom	morley analysis of countable models		Iter-3 imported axiom (s_morley_analysis_of_countable_models).
s_isomorphism_class_count_I_T	state	isomorphism class count I T		Iter-3 imported state (s_isomorphism_class_count_I_T).
s_morley_tree_analysis	state	morley tree analysis		Iter-3 imported state (s_morley_tree_analysis).
s_morley_dichotomy_via_scott_analysis	state	morley dichotomy via scott analysis		Iter-3 imported state (s_morley_dichotomy_via_scott_analysis).
s_shelah_main_gap	theorem	Shelah's main gap theorem		Iter-3 imported theorem (s_shelah_main_gap).
s_complete_first_order_theory	axiom	complete first order theory		Iter-3 imported axiom (s_complete_first_order_theory).
s_classification_invariants_dop_otop	axiom	classification invariants dop otop		Iter-3 imported axiom (s_classification_invariants_dop_otop).
s_classification_dichotomy_invariants	state	classification dichotomy invariants		Iter-3 imported state (s_classification_dichotomy_invariants).
s_tree_of_models_decomposition	state	tree of models decomposition		Iter-3 imported state (s_tree_of_models_decomposition).
s_dimensional_decomposition_in_classifiable_case	state	dimensional decomposition in classifiable case		Iter-3 imported state (s_dimensional_decomposition_in_classifiable_case).
s_stability_spectrum_theorem	theorem	Shelah stability spectrum		Iter-3 imported theorem (s_stability_spectrum_theorem).
s_type_space_S_n_T	axiom	type space S n T		Iter-3 imported axiom (s_type_space_S_n_T).
s_order_property_dichotomy	state	order property dichotomy		Iter-3 imported state (s_order_property_dichotomy).
s_stability_classification	state	stability classification		Iter-3 imported state (s_stability_classification).
s_omitting_types_theorem	theorem	Morley's theorem on ω-stable theories (omitted-types)		Iter-3 imported theorem (s_omitting_types_theorem).
s_countable_language	axiom	countable language		Iter-3 imported axiom (s_countable_language).
s_henkin_construction_with_omitting	state	henkin construction with omitting		Iter-3 imported state (s_henkin_construction_with_omitting).
s_at_each_step_omit_type	state	at each step omit type		Iter-3 imported state (s_at_each_step_omit_type).
s_pillay_steinhorn_o_minimal_cell_decomposition	theorem	Pillay–Steinhorn o-minimality theorem		Iter-3 imported theorem (s_pillay_steinhorn_o_minimal_cell_decomposition).
s_ordered_field_structure	axiom	ordered field structure		Iter-3 imported axiom (s_ordered_field_structure).
s_o_minimal_definability	axiom	o minimal definability		Iter-3 imported axiom (s_o_minimal_definability).
s_monotonicity_lemma	state	monotonicity lemma		Iter-3 imported state (s_monotonicity_lemma).
s_cell_decomposition_in_R_n	state	cell decomposition in R n		Iter-3 imported state (s_cell_decomposition_in_R_n).
s_hrushovski_construction	theorem	Hrushovski's predimension construction		Iter-3 imported theorem (s_hrushovski_construction).
s_class_of_finite_structures_K	axiom	class of finite structures K		Iter-3 imported axiom (s_class_of_finite_structures_K).
s_predimension_function_delta	axiom	predimension function delta		Iter-3 imported axiom (s_predimension_function_delta).
s_predimension_paired_class	state	predimension paired class		Iter-3 imported state (s_predimension_paired_class).
s_fraisse_amalgamation_with_predimension	state	fraisse amalgamation with predimension		Iter-3 imported state (s_fraisse_amalgamation_with_predimension).
s_hrushovski_generic_structure	state	hrushovski generic structure		Iter-3 imported state (s_hrushovski_generic_structure).
s_tarski_RCF_quantifier_elimination	theorem	Tarski's quantifier elimination for RCF		Iter-3 imported theorem (s_tarski_RCF_quantifier_elimination).
s_real_closed_field	axiom	real closed field		Iter-3 imported axiom (s_real_closed_field).
s_polynomial_ring_over_RCF	axiom	polynomial ring over RCF		Iter-3 imported axiom (s_polynomial_ring_over_RCF).
s_sturm_sign_change_count	state	sturm sign change count		Iter-3 imported state (s_sturm_sign_change_count).
s_one_step_quantifier_elimination	state	one step quantifier elimination		Iter-3 imported state (s_one_step_quantifier_elimination).
s_quantifier_free_equivalent	state	quantifier free equivalent		Iter-3 imported state (s_quantifier_free_equivalent).
s_ACF_quantifier_elimination	theorem	Tarski's quantifier elimination for ACF		Iter-3 imported theorem (s_ACF_quantifier_elimination).
s_polynomial_solvability_predicate	state	polynomial solvability predicate		Iter-3 imported state (s_polynomial_solvability_predicate).
s_resultant_eliminates_existential_quantifier	state	resultant eliminates existential quantifier		Iter-3 imported state (s_resultant_eliminates_existential_quantifier).
s_ax_kochen_theorem	theorem	Ax–Kochen theorem		Iter-3 imported theorem (s_ax_kochen_theorem).
s_henselian_valued_field	axiom	henselian valued field		Iter-3 imported axiom (s_henselian_valued_field).
s_residue_field_and_value_group	axiom	residue field and value group		Iter-3 imported axiom (s_residue_field_and_value_group).
s_ax_kochen_ershov_axioms	state	ax kochen ershov axioms		Iter-3 imported state (s_ax_kochen_ershov_axioms).
s_elementary_equivalence_of_ultraproducts	state	elementary equivalence of ultraproducts		Iter-3 imported state (s_elementary_equivalence_of_ultraproducts).
s_ax_kochen_transfer	state	ax kochen transfer		Iter-3 imported state (s_ax_kochen_transfer).
s_presburger_decidability	theorem	Decidability of Presburger arithmetic		Iter-3 imported theorem (s_presburger_decidability).
s_naturals_with_addition	axiom	naturals with addition		Iter-3 imported axiom (s_naturals_with_addition).
s_first_order_logic_with_equality	axiom	first order logic with equality		Iter-3 imported axiom (s_first_order_logic_with_equality).
s_presburger_with_congruences	state	presburger with congruences		Iter-3 imported state (s_presburger_with_congruences).
s_qe_procedure_for_presburger	state	qe procedure for presburger		Iter-3 imported state (s_qe_procedure_for_presburger).
s_quantifier_free_form	state	quantifier free form		Iter-3 imported state (s_quantifier_free_form).
s_rice_shapiro_theorem	theorem	Rice–Shapiro theorem		Iter-3 imported theorem (s_rice_shapiro_theorem).
s_re_set_of_partial_recursive_functions	axiom	re set of partial recursive functions		Iter-3 imported axiom (s_re_set_of_partial_recursive_functions).
s_re_index_set_property	state	re index set property		Iter-3 imported state (s_re_index_set_property).
s_finite_witness_property	state	finite witness property		Iter-3 imported state (s_finite_witness_property).
s_kleene_second_recursion_theorem	theorem	Kleene's second recursion theorem		Iter-3 imported theorem (s_kleene_second_recursion_theorem).
s_s_m_n_theorem	axiom	s m n theorem		Iter-3 imported axiom (s_s_m_n_theorem).
s_diagonal_program_d_x	state	diagonal program d x		Iter-3 imported state (s_diagonal_program_d_x).
s_fixed_point_index_e	state	fixed point index e		Iter-3 imported state (s_fixed_point_index_e).
s_universal_turing_machine	axiom	universal turing machine		Iter-3 imported axiom (s_universal_turing_machine).
s_program_with_hardcoded_input	state	program with hardcoded input		Iter-3 imported state (s_program_with_hardcoded_input).
s_index_function_s_e_x	state	index function s e x		Iter-3 imported state (s_index_function_s_e_x).
s_friedberg_muchnik_theorem	theorem	Friedberg–Muchnik theorem		Iter-3 imported theorem (s_friedberg_muchnik_theorem).
s_re_sets	axiom	re sets		Iter-3 imported axiom (s_re_sets).
s_turing_reducibility	axiom	turing reducibility		Iter-3 imported axiom (s_turing_reducibility).
s_priority_requirement_list	state	priority requirement list		Iter-3 imported state (s_priority_requirement_list).
s_finite_injury_construction	state	finite injury construction		Iter-3 imported state (s_finite_injury_construction).
s_each_requirement_satisfied	state	each requirement satisfied		Iter-3 imported state (s_each_requirement_satisfied).
s_sacks_density_theorem	theorem	Sacks density theorem		Iter-3 imported theorem (s_sacks_density_theorem).
s_re_turing_degrees	axiom	re turing degrees		Iter-3 imported axiom (s_re_turing_degrees).
s_priority_method	axiom	priority method		Iter-3 imported axiom (s_priority_method).
s_intermediate_degree_requirements	state	intermediate degree requirements		Iter-3 imported state (s_intermediate_degree_requirements).
s_infinite_injury_construction	state	infinite injury construction		Iter-3 imported state (s_infinite_injury_construction).
s_true_path_argument_succeeds	state	true path argument succeeds		Iter-3 imported state (s_true_path_argument_succeeds).
s_sacks_splitting_theorem	theorem	Sacks splitting theorem		Iter-3 imported theorem (s_sacks_splitting_theorem).
s_re_set_A_nonrecursive	axiom	re set A nonrecursive		Iter-3 imported axiom (s_re_set_A_nonrecursive).
s_splitting_requirement_list	state	splitting requirement list		Iter-3 imported state (s_splitting_requirement_list).
s_splitting_construction	state	splitting construction		Iter-3 imported state (s_splitting_construction).
s_sacks_jump_inversion	theorem	Sacks jump inversion theorem		Iter-3 imported theorem (s_sacks_jump_inversion).
s_re_degree_zero_prime	axiom	re degree zero prime		Iter-3 imported axiom (s_re_degree_zero_prime).
s_jump_inversion_requirements	state	jump inversion requirements		Iter-3 imported state (s_jump_inversion_requirements).
s_jump_inversion_construction	state	jump inversion construction		Iter-3 imported state (s_jump_inversion_construction).
s_jump_target_achieved	state	jump target achieved		Iter-3 imported state (s_jump_target_achieved).
s_wadge_lemma	theorem	Wadge's lemma		Iter-3 imported theorem (s_wadge_lemma).
s_borel_sets_in_baire_space	axiom	borel sets in baire space		Iter-3 imported axiom (s_borel_sets_in_baire_space).
s_axiom_of_determinacy_for_borel_games	axiom	axiom of determinacy for borel games		Iter-3 imported axiom (s_axiom_of_determinacy_for_borel_games).
s_wadge_game	state	wadge game		Iter-3 imported state (s_wadge_game).
s_wadge_game_determined	state	wadge game determined		Iter-3 imported state (s_wadge_game_determined).
s_wadge_hierarchy_theorem	theorem	Wadge hierarchy / antichain structure		Iter-3 imported theorem (s_wadge_hierarchy_theorem).
s_wadge_ordering	state	wadge ordering		Iter-3 imported state (s_wadge_ordering).
s_wadge_wellfounded	state	wadge wellfounded		Iter-3 imported state (s_wadge_wellfounded).
s_martin_analytic_determinacy	theorem	Martin's theorem: Σ¹₁-determinacy from measurable		Iter-3 imported theorem (s_martin_analytic_determinacy).
s_analytic_set_in_baire_space	axiom	analytic set in baire space		Iter-3 imported axiom (s_analytic_set_in_baire_space).
s_tree_representation_of_analytic_set	state	tree representation of analytic set		Iter-3 imported state (s_tree_representation_of_analytic_set).
s_homogeneous_tree_for_T	state	homogeneous tree for T		Iter-3 imported state (s_homogeneous_tree_for_T).
s_tower_argument_winning_strategy	state	tower argument winning strategy		Iter-3 imported state (s_tower_argument_winning_strategy).
s_AD_in_L_R	theorem	AD in L(ℝ) from large cardinals (Woodin)		Iter-3 imported theorem (s_AD_in_L_R).
s_infinitely_many_woodin_cardinals	axiom	infinitely many woodin cardinals		Iter-3 imported axiom (s_infinitely_many_woodin_cardinals).
s_constructible_closure_L_R	axiom	constructible closure L R		Iter-3 imported axiom (s_constructible_closure_L_R).
s_iteration_trees_on_V	state	iteration trees on V		Iter-3 imported state (s_iteration_trees_on_V).
s_universally_baire_pointclass	state	universally baire pointclass		Iter-3 imported state (s_universally_baire_pointclass).
s_projective_determinacy	state	projective determinacy		Iter-3 imported state (s_projective_determinacy).
s_souslin_theorem	theorem	Souslin's theorem (Σ¹₁ ∩ Π¹₁ = Borel)		Iter-3 imported theorem (s_souslin_theorem).
s_polish_space	axiom	polish space		Iter-3 imported axiom (s_polish_space).
s_lusin_separation_for_disjoint_analytics	state	lusin separation for disjoint analytics		Iter-3 imported state (s_lusin_separation_for_disjoint_analytics).
s_separating_borel_set_for_A_and_comp_A	state	separating borel set for A and comp A		Iter-3 imported state (s_separating_borel_set_for_A_and_comp_A).
s_kondo_uniformization	theorem	Kondô's uniformization theorem		Iter-3 imported theorem (s_kondo_uniformization).
s_pi_1_1_set_in_product_space	axiom	pi 1 1 set in product space		Iter-3 imported axiom (s_pi_1_1_set_in_product_space).
s_wellfounded_tree_T_x	state	wellfounded tree T x		Iter-3 imported state (s_wellfounded_tree_T_x).
s_canonical_section_selector	state	canonical section selector		Iter-3 imported state (s_canonical_section_selector).
s_transfinite_recursion_theorem	theorem	Transfinite recursion theorem		Iter-3 imported theorem (s_transfinite_recursion_theorem).
s_wellordered_class_ON	axiom	wellordered class ON		Iter-3 imported axiom (s_wellordered_class_ON).
s_set_of_attempts	state	set of attempts		Iter-3 imported state (s_set_of_attempts).
s_uniqueness_of_attempts	state	uniqueness of attempts		Iter-3 imported state (s_uniqueness_of_attempts).
s_erdos_tarski_theorem	theorem	Erdős–Tarski theorem on weakly compact cardinals		Iter-3 imported theorem (s_erdos_tarski_theorem).
s_partition_relation_kappa_to_kappa_2_2	axiom	partition relation kappa to kappa 2 2		Iter-3 imported axiom (s_partition_relation_kappa_to_kappa_2_2).
s_tree_property_for_kappa	state	tree property for kappa		Iter-3 imported state (s_tree_property_for_kappa).
s_pi_1_1_indescribability	state	pi 1 1 indescribability		Iter-3 imported state (s_pi_1_1_indescribability).
s_weak_compactness_characterization	state	weak compactness characterization		Iter-3 imported state (s_weak_compactness_characterization).
s_scott_theorem_V_not_L	theorem	Scott's theorem (no measurable cardinal in L)		Iter-3 imported theorem (s_scott_theorem_V_not_L).
s_kunen_inconsistency_in_L	state	kunen inconsistency in L		Iter-3 imported state (s_kunen_inconsistency_in_L).
s_kunen_inconsistency	theorem	Kunen inconsistency theorem		Iter-3 imported theorem (s_kunen_inconsistency).
s_elementary_embedding_j_V_to_V	axiom	elementary embedding j V to V		Iter-3 imported axiom (s_elementary_embedding_j_V_to_V).
s_iterated_critical_points	state	iterated critical points		Iter-3 imported state (s_iterated_critical_points).
s_problematic_definable_subset	state	problematic definable subset		Iter-3 imported state (s_problematic_definable_subset).
s_reflection_in_L	theorem	Reflection theorem in L		Iter-3 imported theorem (s_reflection_in_L).
s_phi_reflection_schema_in_L	state	phi reflection schema in L		Iter-3 imported state (s_phi_reflection_schema_in_L).
s_levy_reflection_in_L	state	levy reflection in L		Iter-3 imported state (s_levy_reflection_in_L).
s_mostowski_rank_function	theorem	Mostowski rank theorem		Iter-3 imported theorem (s_mostowski_rank_function).
s_wellfounded_relation_R	axiom	wellfounded relation R		Iter-3 imported axiom (s_wellfounded_relation_R).
s_rank_recursive_definition	state	rank recursive definition		Iter-3 imported state (s_rank_recursive_definition).
s_rank_total_and_uniquely_defined	state	rank total and uniquely defined		Iter-3 imported state (s_rank_total_and_uniquely_defined).
s_friedman_borel_diagonalization	theorem	Friedman's harmonious functor theorem		Iter-3 imported theorem (s_friedman_borel_diagonalization).
s_subsystem_RCA_0	axiom	subsystem RCA 0		Iter-3 imported axiom (s_subsystem_RCA_0).
s_class_of_borel_functors	axiom	class of borel functors		Iter-3 imported axiom (s_class_of_borel_functors).
s_BDT_statement	state	BDT statement		Iter-3 imported state (s_BDT_statement).
s_BDT_not_in_ATR_0	state	BDT not in ATR 0		Iter-3 imported state (s_BDT_not_in_ATR_0).
s_BDT_provable_in_pi_1_1_CA_0	state	BDT provable in pi 1 1 CA 0		Iter-3 imported state (s_BDT_provable_in_pi_1_1_CA_0).
s_strong_ramsey_independence	theorem	Ramsey-type Paris–Harrington-style independence (general PA-unprovable)		Iter-3 imported theorem (s_strong_ramsey_independence).
s_relatively_large_ramsey_statement	state	relatively large ramsey statement		Iter-3 imported state (s_relatively_large_ramsey_statement).
s_indicator_for_PA_growth	state	indicator for PA growth		Iter-3 imported state (s_indicator_for_PA_growth).
s_skolem_paradox	theorem	Skolem's paradox		Iter-3 imported theorem (s_skolem_paradox).
s_downward_lowenheim_skolem	axiom	downward lowenheim skolem		Iter-3 imported axiom (s_downward_lowenheim_skolem).
s_countable_elementary_substructure_of_V	state	countable elementary substructure of V		Iter-3 imported state (s_countable_elementary_substructure_of_V).
s_countable_transitive_model	state	countable transitive model		Iter-3 imported state (s_countable_transitive_model).
s_PA_proof_theoretic_ordinal_epsilon_0	theorem	Specker–Ackermann ordinal of PA / Gentzen analysis		Iter-3 imported theorem (s_PA_proof_theoretic_ordinal_epsilon_0).
s_proof_ordinal_assignment	state	proof ordinal assignment		Iter-3 imported state (s_proof_ordinal_assignment).
s_cut_elimination_bound	state	cut elimination bound		Iter-3 imported state (s_cut_elimination_bound).
s_consistency_via_descent	state	consistency via descent		Iter-3 imported state (s_consistency_via_descent).
s_cut_elimination_theorem	theorem	Cut-elimination theorem (Gentzen Hauptsatz)		Iter-3 imported theorem (s_cut_elimination_theorem).
s_sequent_calculus_LK	axiom	sequent calculus LK		Iter-3 imported axiom (s_sequent_calculus_LK).
s_proof_tree	axiom	proof tree		Iter-3 imported axiom (s_proof_tree).
s_local_cut_reduction_rules	state	local cut reduction rules		Iter-3 imported state (s_local_cut_reduction_rules).
s_double_induction_termination	state	double induction termination		Iter-3 imported state (s_double_induction_termination).
s_curry_howard_correspondence	theorem	Curry–Howard correspondence		Iter-3 imported theorem (s_curry_howard_correspondence).
s_simply_typed_lambda_calculus	axiom	simply typed lambda calculus		Iter-3 imported axiom (s_simply_typed_lambda_calculus).
s_intuitionistic_propositional_logic	axiom	intuitionistic propositional logic		Iter-3 imported axiom (s_intuitionistic_propositional_logic).
s_types_as_propositions_map	state	types as propositions map		Iter-3 imported state (s_types_as_propositions_map).
s_proofs_as_terms_map	state	proofs as terms map		Iter-3 imported state (s_proofs_as_terms_map).
s_iso_STLC_IPL_proofs	state	iso STLC IPL proofs		Iter-3 imported state (s_iso_STLC_IPL_proofs).
s_strong_normalization_STLC	theorem	Strong normalization for STLC		Iter-3 imported theorem (s_strong_normalization_STLC).
s_type_assignment	axiom	type assignment		Iter-3 imported axiom (s_type_assignment).
s_reducibility_candidates	state	reducibility candidates		Iter-3 imported state (s_reducibility_candidates).
s_candidate_closure_properties	state	candidate closure properties		Iter-3 imported state (s_candidate_closure_properties).
s_lob_theorem	theorem	Löb's theorem		Iter-3 imported theorem (s_lob_theorem).
s_provability_predicate_box	axiom	provability predicate box		Iter-3 imported axiom (s_provability_predicate_box).
s_HB_provability_axioms	state	HB provability axioms		Iter-3 imported state (s_HB_provability_axioms).
s_lob_diagonal_sentence	state	lob diagonal sentence		Iter-3 imported state (s_lob_diagonal_sentence).
s_tarski_undefinability	theorem	Tarski's undefinability of truth		Iter-3 imported theorem (s_tarski_undefinability).
s_arithmetical_definability	axiom	arithmetical definability		Iter-3 imported axiom (s_arithmetical_definability).
s_godel_numbering_of_arithmetic	state	godel numbering of arithmetic		Iter-3 imported state (s_godel_numbering_of_arithmetic).
s_liar_via_truth_predicate	state	liar via truth predicate		Iter-3 imported state (s_liar_via_truth_predicate).
s_robinson_joint_consistency	theorem	Robinson's joint consistency theorem		Iter-3 imported theorem (s_robinson_joint_consistency).
s_two_consistent_first_order_theories_T1_T2	axiom	two consistent first order theories T1 T2		Iter-3 imported axiom (s_two_consistent_first_order_theories_T1_T2).
s_common_subtheory_T_0	axiom	common subtheory T 0		Iter-3 imported axiom (s_common_subtheory_T_0).
s_common_complete_subtheory	state	common complete subtheory		Iter-3 imported state (s_common_complete_subtheory).
s_no_disagreement_via_interpolant	state	no disagreement via interpolant		Iter-3 imported state (s_no_disagreement_via_interpolant).
s_keisler_shelah_theorem	theorem	Keisler–Shelah ultrapower characterization of elementary equivalence		Iter-3 imported theorem (s_keisler_shelah_theorem).
s_two_first_order_structures_M_N	axiom	two first order structures M N		Iter-3 imported axiom (s_two_first_order_structures_M_N).
s_ultrafilter_on_index_set	axiom	ultrafilter on index set		Iter-3 imported axiom (s_ultrafilter_on_index_set).
s_ultrapowers_M_U_N_U	state	ultrapowers M U N U		Iter-3 imported state (s_ultrapowers_M_U_N_U).
s_saturated_ultrapowers	state	saturated ultrapowers		Iter-3 imported state (s_saturated_ultrapowers).
s_isomorphism_of_saturated_models	state	isomorphism of saturated models		Iter-3 imported state (s_isomorphism_of_saturated_models).
s_saturated_model_existence	theorem	Existence of saturated models		Iter-3 imported theorem (s_saturated_model_existence).
s_continuum_hypothesis_or_GCH	axiom	continuum hypothesis or GCH		Iter-3 imported axiom (s_continuum_hypothesis_or_GCH).
s_elementary_chain_realizing_types	state	elementary chain realizing types		Iter-3 imported state (s_elementary_chain_realizing_types).
s_type_realization_succeeds	state	type realization succeeds		Iter-3 imported state (s_type_realization_succeeds).
s_vaught_test	theorem	Vaught's test		Iter-3 imported theorem (s_vaught_test).
s_first_order_theory_T_no_finite_models	axiom	first order theory T no finite models		Iter-3 imported axiom (s_first_order_theory_T_no_finite_models).
s_kappa_categoricity	axiom	kappa categoricity		Iter-3 imported axiom (s_kappa_categoricity).
s_two_distinct_completions_T1_T2	state	two distinct completions T1 T2		Iter-3 imported state (s_two_distinct_completions_T1_T2).
s_two_kappa_sized_models_distinct	state	two kappa sized models distinct		Iter-3 imported state (s_two_kappa_sized_models_distinct).
s_ryll_nardzewski_theorem	theorem	Ryll-Nardzewski theorem		Iter-3 imported theorem (s_ryll_nardzewski_theorem).
s_complete_countable_theory_T	axiom	complete countable theory T		Iter-3 imported axiom (s_complete_countable_theory_T).
s_stone_space_of_T	state	stone space of T		Iter-3 imported state (s_stone_space_of_T).
s_isolated_types_dense_iff_omega_categorical	state	isolated types dense iff omega categorical		Iter-3 imported state (s_isolated_types_dense_iff_omega_categorical).
s_DCF_0_model_complete	theorem	Robinson's theorem on differentially closed fields		Iter-3 imported theorem (s_DCF_0_model_complete).
s_differential_field	axiom	differential field		Iter-3 imported axiom (s_differential_field).
s_existential_closure	axiom	existential closure		Iter-3 imported axiom (s_existential_closure).
s_DCF_0_axioms	state	DCF 0 axioms		Iter-3 imported state (s_DCF_0_axioms).
s_DCF_0_qe	state	DCF 0 qe		Iter-3 imported state (s_DCF_0_qe).
s_DCF_0_omega_stable	state	DCF 0 omega stable		Iter-3 imported state (s_DCF_0_omega_stable).
s_hrushovski_lascar_group_configuration	theorem	Hrushovski–Lascar group configuration / definable groups in stable theories		Iter-3 imported theorem (s_hrushovski_lascar_group_configuration).
s_stable_theory_T	axiom	stable theory T		Iter-3 imported axiom (s_stable_theory_T).
s_group_configuration_in_types	axiom	group configuration in types		Iter-3 imported axiom (s_group_configuration_in_types).
s_associativity_diagram_of_types	state	associativity diagram of types		Iter-3 imported state (s_associativity_diagram_of_types).
s_germ_of_group_multiplication	state	germ of group multiplication		Iter-3 imported state (s_germ_of_group_multiplication).
s_hrushovski_zilber_trichotomy	theorem	Trichotomy in Zariski geometries / Hrushovski–Zilber		Iter-3 imported theorem (s_hrushovski_zilber_trichotomy).
s_strongly_minimal_set	axiom	strongly minimal set		Iter-3 imported axiom (s_strongly_minimal_set).
s_geometry_of_algebraic_closure	axiom	geometry of algebraic closure		Iter-3 imported axiom (s_geometry_of_algebraic_closure).
s_zariski_geometry_axioms	state	zariski geometry axioms		Iter-3 imported state (s_zariski_geometry_axioms).
s_local_modularity_dichotomy	state	local modularity dichotomy		Iter-3 imported state (s_local_modularity_dichotomy).
s_ACF_interpretation_in_non_modular_case	state	ACF interpretation in non modular case		Iter-3 imported state (s_ACF_interpretation_in_non_modular_case).
s_levy_solovay_theorem	theorem	Lévy–Solovay theorem on indestructibility of large cardinals under small forcing		Iter-3 imported theorem (s_levy_solovay_theorem).
s_small_forcing_P_below_kappa	axiom	small forcing P below kappa		Iter-3 imported axiom (s_small_forcing_P_below_kappa).
s_ground_normal_measure_U	state	ground normal measure U		Iter-3 imported state (s_ground_normal_measure_U).
s_generic_extension_V_G	state	generic extension V G		Iter-3 imported state (s_generic_extension_V_G).
s_lifted_measure_U_star	state	lifted measure U star		Iter-3 imported state (s_lifted_measure_U_star).
s_easton_theorem	theorem	Easton's theorem on continuum function		Iter-3 imported theorem (s_easton_theorem).
s_regular_cardinal_arithmetic	axiom	regular cardinal arithmetic		Iter-3 imported axiom (s_regular_cardinal_arithmetic).
s_easton_admissible_F	state	easton admissible F		Iter-3 imported state (s_easton_admissible_F).
s_easton_class_forcing	state	easton class forcing		Iter-3 imported state (s_easton_class_forcing).
s_generic_extension_with_F	state	generic extension with F		Iter-3 imported state (s_generic_extension_with_F).
s_silver_singular_cardinals_theorem	theorem	Silver's theorem on singular cardinals		Iter-3 imported theorem (s_silver_singular_cardinals_theorem).
s_singular_cardinal_kappa_of_uncountable_cofinality	axiom	singular cardinal kappa of uncountable cofinality		Iter-3 imported axiom (s_singular_cardinal_kappa_of_uncountable_cofinality).
s_stationary_GCH_carrier	state	stationary GCH carrier		Iter-3 imported state (s_stationary_GCH_carrier).
s_generic_ultrapower	state	generic ultrapower		Iter-3 imported state (s_generic_ultrapower).
s_2_kappa_bounded_in_quotient	state	2 kappa bounded in quotient		Iter-3 imported state (s_2_kappa_bounded_in_quotient).
s_solovay_stationary_splitting	theorem	Solovay's stationary splitting theorem		Iter-3 imported theorem (s_solovay_stationary_splitting).
s_regular_uncountable_cardinal_kappa	axiom	regular uncountable cardinal kappa		Iter-3 imported axiom (s_regular_uncountable_cardinal_kappa).
s_stationary_subset_of_kappa	axiom	stationary subset of kappa		Iter-3 imported axiom (s_stationary_subset_of_kappa).
s_regressive_function_on_S	state	regressive function on S		Iter-3 imported state (s_regressive_function_on_S).
s_fodor_constancy_on_stationary_subset	state	fodor constancy on stationary subset		Iter-3 imported state (s_fodor_constancy_on_stationary_subset).
s_fodor_lemma	theorem	Fodor's pressing-down lemma		Iter-3 imported theorem (s_fodor_lemma).
s_regressive_function_on_stationary_set	axiom	regressive function on stationary set		Iter-3 imported axiom (s_regressive_function_on_stationary_set).
s_each_fibre_nonstationary	state	each fibre nonstationary		Iter-3 imported state (s_each_fibre_nonstationary).
s_diagonal_intersection_club	state	diagonal intersection club		Iter-3 imported state (s_diagonal_intersection_club).
s_solovay_RVM_theorem	theorem	Solovay's theorem on real-valued measurable cardinals		Iter-3 imported theorem (s_solovay_RVM_theorem).
s_real_valued_measurable_cardinal	axiom	real valued measurable cardinal		Iter-3 imported axiom (s_real_valued_measurable_cardinal).
s_measure_extension_problem	axiom	measure extension problem		Iter-3 imported axiom (s_measure_extension_problem).
s_random_real_extension_over_measurable	state	random real extension over measurable		Iter-3 imported state (s_random_real_extension_over_measurable).
s_RVM_continuum_witnessed	state	RVM continuum witnessed		Iter-3 imported state (s_RVM_continuum_witnessed).
s_diamond_implies_suslin_tree	theorem	Suslin's hypothesis / ◊ implies a Suslin tree (Jensen)		Iter-3 imported theorem (s_diamond_implies_suslin_tree).
s_diamond_principle_on_omega_1	axiom	diamond principle on omega 1		Iter-3 imported axiom (s_diamond_principle_on_omega_1).
s_tree_of_height_omega_1	axiom	tree of height omega 1		Iter-3 imported axiom (s_tree_of_height_omega_1).
s_diamond_anticipates_antichains	state	diamond anticipates antichains		Iter-3 imported state (s_diamond_anticipates_antichains).
s_construct_tree_killing_each_predicted_antichain	state	construct tree killing each predicted antichain		Iter-3 imported state (s_construct_tree_killing_each_predicted_antichain).
s_MA_implies_SH	theorem	Martin's axiom and Suslin's hypothesis		Iter-3 imported theorem (s_MA_implies_SH).
s_martins_axiom_MA_omega_1	axiom	martins axiom MA omega 1		Iter-3 imported axiom (s_martins_axiom_MA_omega_1).
s_ccc_partial_order	axiom	ccc partial order		Iter-3 imported axiom (s_ccc_partial_order).
s_hypothetical_suslin_tree	state	hypothetical suslin tree		Iter-3 imported state (s_hypothetical_suslin_tree).
s_T_as_ccc_poset	state	T as ccc poset		Iter-3 imported state (s_T_as_ccc_poset).
s_omega_1_branch_kills_suslin	state	omega 1 branch kills suslin		Iter-3 imported state (s_omega_1_branch_kills_suslin).
s_PFA_consequences	theorem	Proper Forcing Axiom consequences (Todorčević)		Iter-3 imported theorem (s_PFA_consequences).
s_proper_forcing_axiom	axiom	proper forcing axiom		Iter-3 imported axiom (s_proper_forcing_axiom).
s_countable_proper_forcing_class	axiom	countable proper forcing class		Iter-3 imported axiom (s_countable_proper_forcing_class).
s_OCA_from_PFA	state	OCA from PFA		Iter-3 imported state (s_OCA_from_PFA).
s_structural_dichotomy_on_separable_metric_spaces	state	structural dichotomy on separable metric spaces		Iter-3 imported state (s_structural_dichotomy_on_separable_metric_spaces).
s_halpern_lauchli_theorem	theorem	Halpern–Läuchli theorem		Iter-3 imported theorem (s_halpern_lauchli_theorem).
s_finite_product_of_finitely_branching_trees	axiom	finite product of finitely branching trees		Iter-3 imported axiom (s_finite_product_of_finitely_branching_trees).
s_finite_coloring	axiom	finite coloring		Iter-3 imported axiom (s_finite_coloring).
s_strong_subtree_template	state	strong subtree template		Iter-3 imported state (s_strong_subtree_template).
s_per_level_monochromatic_choice	state	per level monochromatic choice		Iter-3 imported state (s_per_level_monochromatic_choice).
s_galvin_prikry_theorem	theorem	Galvin–Prikry theorem		Iter-3 imported theorem (s_galvin_prikry_theorem).
s_borel_subset_of_omega_omega	axiom	borel subset of omega omega		Iter-3 imported axiom (s_borel_subset_of_omega_omega).
s_infinite_subsets_of_omega	axiom	infinite subsets of omega		Iter-3 imported axiom (s_infinite_subsets_of_omega).
s_open_case_reduction	state	open case reduction		Iter-3 imported state (s_open_case_reduction).
s_galvin_prikry_game	state	galvin prikry game		Iter-3 imported state (s_galvin_prikry_game).
s_winning_strategy_yields_homogeneous_A	state	winning strategy yields homogeneous A		Iter-3 imported state (s_winning_strategy_yields_homogeneous_A).
s_novikov_boone_theorem	theorem	Word problem for groups (Novikov–Boone)		Iter-3 imported theorem (s_novikov_boone_theorem).
s_semigroup_encoding_of_TM	state	semigroup encoding of TM		Iter-3 imported state (s_semigroup_encoding_of_TM).
s_group_simulating_TM	state	group simulating TM		Iter-3 imported state (s_group_simulating_TM).
s_liouville_phase_volume_theorem	theorem	Liouville's theorem on phase-space volume		Iter-3 imported theorem (s_liouville_phase_volume_theorem).
s_hamiltonian_phase_space	axiom	hamiltonian phase space		Iter-3 imported axiom (s_hamiltonian_phase_space).
s_hamilton_canonical_equations	axiom	hamilton canonical equations		Iter-3 imported axiom (s_hamilton_canonical_equations).
s_liouville_equation	state	liouville equation		Iter-3 imported state (s_liouville_equation).
s_divergence_free_hamiltonian_flow	state	divergence free hamiltonian flow		Iter-3 imported state (s_divergence_free_hamiltonian_flow).
s_hamilton_jacobi_theorem	theorem	Hamilton–Jacobi theorem		Iter-3 imported theorem (s_hamilton_jacobi_theorem).
s_hamilton_jacobi_pde	state	hamilton jacobi pde		Iter-3 imported state (s_hamilton_jacobi_pde).
s_characteristics_match_hamilton_equations	state	characteristics match hamilton equations		Iter-3 imported state (s_characteristics_match_hamilton_equations).
s_complete_integral_generating_canonical_transformation	state	complete integral generating canonical transformation		Iter-3 imported state (s_complete_integral_generating_canonical_transformation).
s_arnold_liouville_theorem	theorem	Arnold–Liouville theorem (commuting integrals ⇒ integrability)		Iter-3 imported theorem (s_arnold_liouville_theorem).
s_n_commuting_independent_first_integrals	axiom	n commuting independent first integrals		Iter-3 imported axiom (s_n_commuting_independent_first_integrals).
s_invariant_lagrangian_submanifold	state	invariant lagrangian submanifold		Iter-3 imported state (s_invariant_lagrangian_submanifold).
s_level_set_is_n_torus	state	level set is n torus		Iter-3 imported state (s_level_set_is_n_torus).
s_action_angle_coordinates_exist	state	action angle coordinates exist		Iter-3 imported state (s_action_angle_coordinates_exist).
s_bertrand_theorem	theorem	Bertrand's theorem (only 1/r and r² have closed orbits)		Iter-3 imported theorem (s_bertrand_theorem).
s_central_force_problem	axiom	central force problem		Iter-3 imported axiom (s_central_force_problem).
s_effective_radial_potential	state	effective radial potential		Iter-3 imported state (s_effective_radial_potential).
s_apsidal_angle_perturbation	state	apsidal angle perturbation		Iter-3 imported state (s_apsidal_angle_perturbation).
s_only_inverse_square_and_harmonic_close_at_first_order	state	only inverse square and harmonic close at first order		Iter-3 imported state (s_only_inverse_square_and_harmonic_close_at_first_order).
s_poincare_recurrence_theorem	theorem	Poincaré recurrence theorem		Iter-3 imported theorem (s_poincare_recurrence_theorem).
s_some_iterates_must_overlap_in_measure	state	some iterates must overlap in measure		Iter-3 imported state (s_some_iterates_must_overlap_in_measure).
s_almost_every_point_returns	state	almost every point returns		Iter-3 imported state (s_almost_every_point_returns).
s_gromov_nonsqueezing_theorem	theorem	Symplectic non-squeezing (Gromov)		Iter-3 imported theorem (s_gromov_nonsqueezing_theorem).
s_symplectic_manifold	axiom	symplectic manifold		Iter-3 imported axiom (s_symplectic_manifold).
s_symplectic_embedding	axiom	symplectic embedding		Iter-3 imported axiom (s_symplectic_embedding).
s_pseudoholomorphic_disks_in_target	state	pseudoholomorphic disks in target		Iter-3 imported state (s_pseudoholomorphic_disks_in_target).
s_existence_of_area_bounded_disk_through_image	state	existence of area bounded disk through image		Iter-3 imported state (s_existence_of_area_bounded_disk_through_image).
s_marsden_weinstein_reduction_theorem	theorem	Marsden–Weinstein symplectic reduction		Iter-3 imported theorem (s_marsden_weinstein_reduction_theorem).
s_hamiltonian_lie_group_action_with_moment_map	axiom	hamiltonian lie group action with moment map		Iter-3 imported axiom (s_hamiltonian_lie_group_action_with_moment_map).
s_moment_map_fiber_mu_inverse_zero	state	moment map fiber mu inverse zero		Iter-3 imported state (s_moment_map_fiber_mu_inverse_zero).
s_quotient_manifold_M_red	state	quotient manifold M red		Iter-3 imported state (s_quotient_manifold_M_red).
s_darboux_theorem	theorem	Darboux theorem (local symplectic uniqueness)		Iter-3 imported theorem (s_darboux_theorem).
s_closed_nondegenerate_2_form	axiom	closed nondegenerate 2 form		Iter-3 imported axiom (s_closed_nondegenerate_2_form).
s_moser_homotopy_of_symplectic_forms	state	moser homotopy of symplectic forms		Iter-3 imported state (s_moser_homotopy_of_symplectic_forms).
s_diffeomorphism_pulling_back_to_standard_form	state	diffeomorphism pulling back to standard form		Iter-3 imported state (s_diffeomorphism_pulling_back_to_standard_form).
s_stone_von_neumann_theorem	theorem	Stone–von Neumann uniqueness theorem		Iter-3 imported theorem (s_stone_von_neumann_theorem).
s_canonical_commutation_relations_weyl_form	axiom	canonical commutation relations weyl form		Iter-3 imported axiom (s_canonical_commutation_relations_weyl_form).
s_irreducible_unitary_representation	axiom	irreducible unitary representation		Iter-3 imported axiom (s_irreducible_unitary_representation).
s_weyl_unitary_ccr_algebra	state	weyl unitary ccr algebra		Iter-3 imported state (s_weyl_unitary_ccr_algebra).
s_schrodinger_representation	state	schrodinger representation		Iter-3 imported state (s_schrodinger_representation).
s_wigner_symmetry_theorem	theorem	Wigner's theorem (symmetries of quantum states)		Iter-3 imported theorem (s_wigner_symmetry_theorem).
s_projective_hilbert_space	axiom	projective hilbert space		Iter-3 imported axiom (s_projective_hilbert_space).
s_ray_preserving_transition_probability	axiom	ray preserving transition probability		Iter-3 imported axiom (s_ray_preserving_transition_probability).
s_inner_product_preserved_up_to_phase	state	inner product preserved up to phase		Iter-3 imported state (s_inner_product_preserved_up_to_phase).
s_unique_lift_to_U_or_anti_U	state	unique lift to U or anti U		Iter-3 imported state (s_unique_lift_to_U_or_anti_U).
s_gleason_theorem	theorem	Gleason's theorem		Iter-3 imported theorem (s_gleason_theorem).
s_hilbert_space_dim_geq_3	axiom	hilbert space dim geq 3		Iter-3 imported axiom (s_hilbert_space_dim_geq_3).
s_frame_function_on_projections	axiom	frame function on projections		Iter-3 imported axiom (s_frame_function_on_projections).
s_continuous_frame_function	state	continuous frame function		Iter-3 imported state (s_continuous_frame_function).
s_positive_trace_class_operator_rho	state	positive trace class operator rho		Iter-3 imported state (s_positive_trace_class_operator_rho).
s_robertson_schrodinger_uncertainty	theorem	Heisenberg uncertainty (Robertson–Schrödinger inequality)		Iter-3 imported theorem (s_robertson_schrodinger_uncertainty).
s_hilbert_space_with_self_adjoint_operators	axiom	hilbert space with self adjoint operators		Iter-3 imported axiom (s_hilbert_space_with_self_adjoint_operators).
s_centered_observables	state	centered observables		Iter-3 imported state (s_centered_observables).
s_cauchy_schwarz_variance_bound	state	cauchy schwarz variance bound		Iter-3 imported state (s_cauchy_schwarz_variance_bound).
s_ehrenfest_theorem	theorem	Ehrenfest theorem		Iter-3 imported theorem (s_ehrenfest_theorem).
s_schrodinger_equation	axiom	schrodinger equation		Iter-3 imported axiom (s_schrodinger_equation).
s_heisenberg_equation_of_motion	state	heisenberg equation of motion		Iter-3 imported state (s_heisenberg_equation_of_motion).
s_d_dt_expectation_equals_i_hbar_commutator	state	d dt expectation equals i hbar commutator		Iter-3 imported state (s_d_dt_expectation_equals_i_hbar_commutator).
s_rage_theorem	theorem	RAGE theorem (spectral decomposition of dynamics)		Iter-3 imported theorem (s_rage_theorem).
s_self_adjoint_hamiltonian_H	axiom	self adjoint hamiltonian H		Iter-3 imported axiom (s_self_adjoint_hamiltonian_H).
s_unitary_schrodinger_flow	axiom	unitary schrodinger flow		Iter-3 imported axiom (s_unitary_schrodinger_flow).
s_lebesgue_spectral_decomposition_of_H	state	lebesgue spectral decomposition of H		Iter-3 imported state (s_lebesgue_spectral_decomposition_of_H).
s_decay_of_continuous_part_in_compact_set	state	decay of continuous part in compact set		Iter-3 imported state (s_decay_of_continuous_part_in_compact_set).
s_wightman_reconstruction_theorem	theorem	Wightman reconstruction theorem		Iter-3 imported theorem (s_wightman_reconstruction_theorem).
s_wightman_axioms	axiom	wightman axioms		Iter-3 imported axiom (s_wightman_axioms).
s_tempered_distribution_n_point_functions	axiom	tempered distribution n point functions		Iter-3 imported axiom (s_tempered_distribution_n_point_functions).
s_wightman_functional_satisfies_GNS_positivity	state	wightman functional satisfies GNS positivity		Iter-3 imported state (s_wightman_functional_satisfies_GNS_positivity).
s_borchers_algebra_GNS_representation	state	borchers algebra GNS representation		Iter-3 imported state (s_borchers_algebra_GNS_representation).
s_osterwalder_schrader_theorem	theorem	Osterwalder–Schrader reconstruction		Iter-3 imported theorem (s_osterwalder_schrader_theorem).
s_euclidean_schwinger_functions	axiom	euclidean schwinger functions		Iter-3 imported axiom (s_euclidean_schwinger_functions).
s_OS_axioms_reflection_positivity	axiom	OS axioms reflection positivity		Iter-3 imported axiom (s_OS_axioms_reflection_positivity).
s_euclidean_schwinger_data	state	euclidean schwinger data		Iter-3 imported state (s_euclidean_schwinger_data).
s_physical_hilbert_space_from_RP	state	physical hilbert space from RP		Iter-3 imported state (s_physical_hilbert_space_from_RP).
s_wightman_functions_recovered	state	wightman functions recovered		Iter-3 imported state (s_wightman_functions_recovered).
s_bisognano_wichmann_theorem	theorem	Bisognano–Wichmann theorem		Iter-3 imported theorem (s_bisognano_wichmann_theorem).
s_wedge_local_algebra_R_W	axiom	wedge local algebra R W		Iter-3 imported axiom (s_wedge_local_algebra_R_W).
s_modular_operator_for_wedge	state	modular operator for wedge		Iter-3 imported state (s_modular_operator_for_wedge).
s_modular_flow_equals_boost	state	modular flow equals boost		Iter-3 imported state (s_modular_flow_equals_boost).
s_coleman_mandula_theorem	theorem	Coleman–Mandula theorem		Iter-3 imported theorem (s_coleman_mandula_theorem).
s_relativistic_s_matrix_axioms	axiom	relativistic s matrix axioms		Iter-3 imported axiom (s_relativistic_s_matrix_axioms).
s_internal_symmetry_lie_algebra_G	axiom	internal symmetry lie algebra G		Iter-3 imported axiom (s_internal_symmetry_lie_algebra_G).
s_bosonic_charge_algebra_constraints	state	bosonic charge algebra constraints		Iter-3 imported state (s_bosonic_charge_algebra_constraints).
s_charges_commute_with_translation_generators	state	charges commute with translation generators		Iter-3 imported state (s_charges_commute_with_translation_generators).
s_HLS_theorem	theorem	Haag–Łopuszański–Sohnius theorem		Iter-3 imported theorem (s_HLS_theorem).
s_graded_lie_algebra_extension	axiom	graded lie algebra extension		Iter-3 imported axiom (s_graded_lie_algebra_extension).
s_super_charge_anticommutator_structure	state	super charge anticommutator structure		Iter-3 imported state (s_super_charge_anticommutator_structure).
s_unique_super_poincare_extension	state	unique super poincare extension		Iter-3 imported state (s_unique_super_poincare_extension).
s_glimm_jaffe_phi4_construction	theorem	Glimm–Jaffe construction of φ⁴ in 2 and 3 dimensions		Iter-3 imported theorem (s_glimm_jaffe_phi4_construction).
s_phi_4_lagrangian_lattice_regularization	axiom	phi 4 lagrangian lattice regularization		Iter-3 imported axiom (s_phi_4_lagrangian_lattice_regularization).
s_lattice_phi_4_probability_measure	state	lattice phi 4 probability measure		Iter-3 imported state (s_lattice_phi_4_probability_measure).
s_renormalization_group_flow_to_continuum	state	renormalization group flow to continuum		Iter-3 imported state (s_renormalization_group_flow_to_continuum).
s_limit_measure_exists_in_d2_d3	state	limit measure exists in d2 d3		Iter-3 imported state (s_limit_measure_exists_in_d2_d3).
s_mermin_wagner_theorem	theorem	Mermin–Wagner theorem		Iter-3 imported theorem (s_mermin_wagner_theorem).
s_lattice_spin_model_continuous_symmetry	axiom	lattice spin model continuous symmetry		Iter-3 imported axiom (s_lattice_spin_model_continuous_symmetry).
s_short_range_interaction_in_d_le_2	axiom	short range interaction in d le 2		Iter-3 imported axiom (s_short_range_interaction_in_d_le_2).
s_putative_order_parameter	state	putative order parameter		Iter-3 imported state (s_putative_order_parameter).
s_bogoliubov_inequality_for_correlations	state	bogoliubov inequality for correlations		Iter-3 imported state (s_bogoliubov_inequality_for_correlations).
s_order_parameter_must_vanish	state	order parameter must vanish		Iter-3 imported state (s_order_parameter_must_vanish).
s_lee_yang_theorem	theorem	Lee–Yang circle theorem		Iter-3 imported theorem (s_lee_yang_theorem).
s_ferromagnetic_ising_partition_function	axiom	ferromagnetic ising partition function		Iter-3 imported axiom (s_ferromagnetic_ising_partition_function).
s_polynomial_in_fugacity_z	axiom	polynomial in fugacity z		Iter-3 imported axiom (s_polynomial_in_fugacity_z).
s_partition_function_polynomial_P_z	state	partition function polynomial P z		Iter-3 imported state (s_partition_function_polynomial_P_z).
s_inductive_zero_locus_constraint	state	inductive zero locus constraint		Iter-3 imported state (s_inductive_zero_locus_constraint).
s_onsager_2d_ising_solution	theorem	Onsager exact solution of 2D Ising		Iter-3 imported theorem (s_onsager_2d_ising_solution).
s_2d_ising_lattice_square	axiom	2d ising lattice square		Iter-3 imported axiom (s_2d_ising_lattice_square).
s_nearest_neighbor_coupling	axiom	nearest neighbor coupling		Iter-3 imported axiom (s_nearest_neighbor_coupling).
s_transfer_matrix_formulation	state	transfer matrix formulation		Iter-3 imported state (s_transfer_matrix_formulation).
s_jordan_wigner_fermionization	state	jordan wigner fermionization		Iter-3 imported state (s_jordan_wigner_fermionization).
s_eigenvalues_in_closed_form	state	eigenvalues in closed form		Iter-3 imported state (s_eigenvalues_in_closed_form).
s_peierls_phase_transition	theorem	Peierls argument for spontaneous magnetization		Iter-3 imported theorem (s_peierls_phase_transition).
s_low_temperature_regime	axiom	low temperature regime		Iter-3 imported axiom (s_low_temperature_regime).
s_peierls_contour_ensemble	state	peierls contour ensemble		Iter-3 imported state (s_peierls_contour_ensemble).
s_summable_contour_weight_bound	state	summable contour weight bound		Iter-3 imported state (s_summable_contour_weight_bound).
s_lieb_robinson_bound	theorem	Lieb–Robinson bound		Iter-3 imported theorem (s_lieb_robinson_bound).
s_lattice_local_hamiltonian	axiom	lattice local hamiltonian		Iter-3 imported axiom (s_lattice_local_hamiltonian).
s_bounded_local_interaction_norm	axiom	bounded local interaction norm		Iter-3 imported axiom (s_bounded_local_interaction_norm).
s_heisenberg_evolution_operator	state	heisenberg evolution operator		Iter-3 imported state (s_heisenberg_evolution_operator).
s_commutator_norm_grows_at_most_exponentially_in_distance_minus_v_t	state	commutator norm grows at most exponentially in distance minus v t		Iter-3 imported state (s_commutator_norm_grows_at_most_exponentially_in_distance_minus_v_t).
s_lieb_thirring_inequality	theorem	Lieb–Thirring inequalities		Iter-3 imported theorem (s_lieb_thirring_inequality).
s_schrodinger_operator_minus_laplacian_plus_V	axiom	schrodinger operator minus laplacian plus V		Iter-3 imported axiom (s_schrodinger_operator_minus_laplacian_plus_V).
s_lp_potential_V	axiom	lp potential V		Iter-3 imported axiom (s_lp_potential_V).
s_sum_over_negative_spectrum	state	sum over negative spectrum		Iter-3 imported state (s_sum_over_negative_spectrum).
s_birman_schwinger_kernel_bound	state	birman schwinger kernel bound		Iter-3 imported state (s_birman_schwinger_kernel_bound).
s_stability_of_matter_theorem	theorem	Lieb's stability of matter		Iter-3 imported theorem (s_stability_of_matter_theorem).
s_many_body_coulomb_hamiltonian	axiom	many body coulomb hamiltonian		Iter-3 imported axiom (s_many_body_coulomb_hamiltonian).
s_fermionic_antisymmetry	axiom	fermionic antisymmetry		Iter-3 imported axiom (s_fermionic_antisymmetry).
s_thomas_fermi_lower_bound	state	thomas fermi lower bound		Iter-3 imported state (s_thomas_fermi_lower_bound).
s_extensive_kinetic_energy_bound	state	extensive kinetic energy bound		Iter-3 imported state (s_extensive_kinetic_energy_bound).
s_dlr_theorem	theorem	Dobrushin–Lanford–Ruelle (DLR) characterization of Gibbs states		Iter-3 imported theorem (s_dlr_theorem).
s_specifications_family_of_conditional_distributions	axiom	specifications family of conditional distributions		Iter-3 imported axiom (s_specifications_family_of_conditional_distributions).
s_consistent_specification	state	consistent specification		Iter-3 imported state (s_consistent_specification).
s_set_of_DLR_solutions_nonempty_compact_convex	state	set of DLR solutions nonempty compact convex		Iter-3 imported state (s_set_of_DLR_solutions_nonempty_compact_convex).
s_kramers_wannier_duality_theorem	theorem	Kramers–Wannier duality		Iter-3 imported theorem (s_kramers_wannier_duality_theorem).
s_low_temperature_high_temperature_expansions	axiom	low temperature high temperature expansions		Iter-3 imported axiom (s_low_temperature_high_temperature_expansions).
s_partition_function_duality_relation	state	partition function duality relation		Iter-3 imported state (s_partition_function_duality_relation).
s_self_dual_temperature_equation	state	self dual temperature equation		Iter-3 imported state (s_self_dual_temperature_equation).
s_gibbs_variational_principle	theorem	Gibbs variational principle		Iter-3 imported theorem (s_gibbs_variational_principle).
s_relative_entropy_functional	axiom	relative entropy functional		Iter-3 imported axiom (s_relative_entropy_functional).
s_free_energy_legendre_dual	state	free energy legendre dual		Iter-3 imported state (s_free_energy_legendre_dual).
s_pressure_attains_supremum_at_gibbs_state	state	pressure attains supremum at gibbs state		Iter-3 imported state (s_pressure_attains_supremum_at_gibbs_state).
s_birkhoff_gr_theorem	theorem	Birkhoff's theorem (GR)		Iter-3 imported theorem (s_birkhoff_gr_theorem).
s_einstein_field_equations_vacuum	axiom	einstein field equations vacuum		Iter-3 imported axiom (s_einstein_field_equations_vacuum).
s_spherical_symmetry_ansatz	axiom	spherical symmetry ansatz		Iter-3 imported axiom (s_spherical_symmetry_ansatz).
s_spherically_symmetric_metric_ansatz	state	spherically symmetric metric ansatz		Iter-3 imported state (s_spherically_symmetric_metric_ansatz).
s_radial_einstein_equations	state	radial einstein equations		Iter-3 imported state (s_radial_einstein_equations).
s_staticity_emerges	state	staticity emerges		Iter-3 imported state (s_staticity_emerges).
s_penrose_singularity_theorem	theorem	Penrose singularity theorem		Iter-3 imported theorem (s_penrose_singularity_theorem).
s_einstein_field_equations	axiom	einstein field equations		Iter-3 imported axiom (s_einstein_field_equations).
s_null_energy_condition	axiom	null energy condition		Iter-3 imported axiom (s_null_energy_condition).
s_trapped_surface_exists	axiom	trapped surface exists		Iter-3 imported axiom (s_trapped_surface_exists).
s_raychaudhuri_focusing_inequality	state	raychaudhuri focusing inequality		Iter-3 imported state (s_raychaudhuri_focusing_inequality).
s_null_geodesic_incompleteness_obstruction	state	null geodesic incompleteness obstruction		Iter-3 imported state (s_null_geodesic_incompleteness_obstruction).
s_hawking_singularity_theorem	theorem	Hawking singularity theorem		Iter-3 imported theorem (s_hawking_singularity_theorem).
s_strong_energy_condition	axiom	strong energy condition		Iter-3 imported axiom (s_strong_energy_condition).
s_globally_hyperbolic_spacetime	axiom	globally hyperbolic spacetime		Iter-3 imported axiom (s_globally_hyperbolic_spacetime).
s_timelike_focusing_inequality	state	timelike focusing inequality		Iter-3 imported state (s_timelike_focusing_inequality).
s_finite_proper_time_to_focal_point	state	finite proper time to focal point		Iter-3 imported state (s_finite_proper_time_to_focal_point).
s_hawking_area_theorem	theorem	Hawking's area theorem		Iter-3 imported theorem (s_hawking_area_theorem).
s_cosmic_censorship	axiom	cosmic censorship		Iter-3 imported axiom (s_cosmic_censorship).
s_horizon_generator_focusing	state	horizon generator focusing		Iter-3 imported state (s_horizon_generator_focusing).
s_nonnegative_expansion_on_horizon	state	nonnegative expansion on horizon		Iter-3 imported state (s_nonnegative_expansion_on_horizon).
s_schoen_yau_positive_mass_theorem	theorem	Schoen–Yau positive mass theorem		Iter-3 imported theorem (s_schoen_yau_positive_mass_theorem).
s_asymptotically_flat_initial_data_set	axiom	asymptotically flat initial data set		Iter-3 imported axiom (s_asymptotically_flat_initial_data_set).
s_dominant_energy_condition	axiom	dominant energy condition		Iter-3 imported axiom (s_dominant_energy_condition).
s_adm_mass_definition	state	adm mass definition		Iter-3 imported state (s_adm_mass_definition).
s_minimal_surface_reduction	state	minimal surface reduction		Iter-3 imported state (s_minimal_surface_reduction).
s_no_zero_mass_initial_data_unless_minkowski	state	no zero mass initial data unless minkowski		Iter-3 imported state (s_no_zero_mass_initial_data_unless_minkowski).
s_bekenstein_hawking_entropy	theorem	Bekenstein–Hawking entropy formula		Iter-3 imported theorem (s_bekenstein_hawking_entropy).
s_black_hole_thermodynamics_axioms	axiom	black hole thermodynamics axioms		Iter-3 imported axiom (s_black_hole_thermodynamics_axioms).
s_area_acts_as_entropy_proxy	state	area acts as entropy proxy		Iter-3 imported state (s_area_acts_as_entropy_proxy).
s_hawking_temperature_kappa_over_2pi	state	hawking temperature kappa over 2pi		Iter-3 imported state (s_hawking_temperature_kappa_over_2pi).
s_cosmic_no_hair_theorem	theorem	Cosmic no-hair theorem (de Sitter attractor)		Iter-3 imported theorem (s_cosmic_no_hair_theorem).
s_einstein_field_equations_with_positive_lambda	axiom	einstein field equations with positive lambda		Iter-3 imported axiom (s_einstein_field_equations_with_positive_lambda).
s_initially_inhomogeneous_data	axiom	initially inhomogeneous data		Iter-3 imported axiom (s_initially_inhomogeneous_data).
s_constraint_equations_with_cosmological_constant	state	constraint equations with cosmological constant		Iter-3 imported state (s_constraint_equations_with_cosmological_constant).
s_anisotropy_decays_like_e_minus_Ht	state	anisotropy decays like e minus Ht		Iter-3 imported state (s_anisotropy_decays_like_e_minus_Ht).
s_geroch_spinor_obstruction_theorem	theorem	Geroch's no-go for global spinor structure		Iter-3 imported theorem (s_geroch_spinor_obstruction_theorem).
s_orientable_lorentzian_manifold	axiom	orientable lorentzian manifold		Iter-3 imported axiom (s_orientable_lorentzian_manifold).
s_spin_lift_existence_question	axiom	spin lift existence question		Iter-3 imported axiom (s_spin_lift_existence_question).
s_w2_vanishing_required	state	w2 vanishing required		Iter-3 imported state (s_w2_vanishing_required).
s_spin_lift_exists_iff_w2_zero	state	spin lift exists iff w2 zero		Iter-3 imported state (s_spin_lift_exists_iff_w2_zero).
s_israel_uniqueness_theorem	theorem	Israel uniqueness of Schwarzschild		Iter-3 imported theorem (s_israel_uniqueness_theorem).
s_static_asymptotically_flat_black_hole	axiom	static asymptotically flat black hole		Iter-3 imported axiom (s_static_asymptotically_flat_black_hole).
s_static_metric_ansatz	state	static metric ansatz		Iter-3 imported state (s_static_metric_ansatz).
s_elliptic_system_on_riemannian_slice	state	elliptic system on riemannian slice		Iter-3 imported state (s_elliptic_system_on_riemannian_slice).
s_newman_janis_algorithm_theorem	theorem	Newman–Janis algorithm		Iter-3 imported theorem (s_newman_janis_algorithm_theorem).
s_schwarzschild_metric_in_null_coordinates	axiom	schwarzschild metric in null coordinates		Iter-3 imported axiom (s_schwarzschild_metric_in_null_coordinates).
s_complex_coordinate_shift_recipe	axiom	complex coordinate shift recipe		Iter-3 imported axiom (s_complex_coordinate_shift_recipe).
s_complexified_null_tetrad	state	complexified null tetrad		Iter-3 imported state (s_complexified_null_tetrad).
s_shifted_metric_in_real_form	state	shifted metric in real form		Iter-3 imported state (s_shifted_metric_in_real_form).
s_choquet_bruhat_well_posedness	theorem	Choquet–Bruhat well-posedness of Einstein vacuum		Iter-3 imported theorem (s_choquet_bruhat_well_posedness).
s_initial_data_satisfying_constraints	axiom	initial data satisfying constraints		Iter-3 imported axiom (s_initial_data_satisfying_constraints).
s_harmonic_gauge_reduces_to_quasilinear_wave	state	harmonic gauge reduces to quasilinear wave		Iter-3 imported state (s_harmonic_gauge_reduces_to_quasilinear_wave).
s_hyperbolic_cauchy_problem	state	hyperbolic cauchy problem		Iter-3 imported state (s_hyperbolic_cauchy_problem).
s_local_existence_and_uniqueness	state	local existence and uniqueness		Iter-3 imported state (s_local_existence_and_uniqueness).
s_raychaudhuri_equation_theorem	theorem	Raychaudhuri equation		Iter-3 imported theorem (s_raychaudhuri_equation_theorem).
s_lorentzian_manifold	axiom	lorentzian manifold		Iter-3 imported axiom (s_lorentzian_manifold).
s_geodesic_congruence	axiom	geodesic congruence		Iter-3 imported axiom (s_geodesic_congruence).
s_kinematic_decomposition_of_congruence	state	kinematic decomposition of congruence		Iter-3 imported state (s_kinematic_decomposition_of_congruence).
s_dtheta_ds_equation	state	dtheta ds equation		Iter-3 imported state (s_dtheta_ds_equation).
s_faddeev_popov_partition_function_theorem	theorem	Faddeev–Popov gauge-fixing procedure		Iter-3 imported theorem (s_faddeev_popov_partition_function_theorem).
s_yang_mills_action	axiom	yang mills action		Iter-3 imported axiom (s_yang_mills_action).
s_path_integral_with_gauge_redundancy	axiom	path integral with gauge redundancy		Iter-3 imported axiom (s_path_integral_with_gauge_redundancy).
s_gauge_slice_with_jacobian	state	gauge slice with jacobian		Iter-3 imported state (s_gauge_slice_with_jacobian).
s_grassmann_representation_of_FP_determinant	state	grassmann representation of FP determinant		Iter-3 imported state (s_grassmann_representation_of_FP_determinant).
s_bps_bound_and_saturation_theorem	theorem	BPS bound and BPS saturation		Iter-3 imported theorem (s_bps_bound_and_saturation_theorem).
s_supersymmetry_algebra_with_central_charge	axiom	supersymmetry algebra with central charge		Iter-3 imported axiom (s_supersymmetry_algebra_with_central_charge).
s_massive_state_in_rep_of_algebra	axiom	massive state in rep of algebra		Iter-3 imported axiom (s_massive_state_in_rep_of_algebra).
s_positivity_of_Q_dagger_Q	state	positivity of Q dagger Q		Iter-3 imported state (s_positivity_of_Q_dagger_Q).
s_bps_mass_inequality	state	bps mass inequality		Iter-3 imported state (s_bps_mass_inequality).
s_witten_index_theorem	theorem	Witten index		Iter-3 imported theorem (s_witten_index_theorem).
s_supersymmetric_quantum_mechanics	axiom	supersymmetric quantum mechanics		Iter-3 imported axiom (s_supersymmetric_quantum_mechanics).
s_supercharge_Q_with_Q_squared_equals_H	axiom	supercharge Q with Q squared equals H		Iter-3 imported axiom (s_supercharge_Q_with_Q_squared_equals_H).
s_graded_trace_definition	state	graded trace definition		Iter-3 imported state (s_graded_trace_definition).
s_only_zero_modes_contribute	state	only zero modes contribute		Iter-3 imported state (s_only_zero_modes_contribute).
s_wess_zumino_descent_theorem	theorem	Wess–Zumino consistency / descent equations		Iter-3 imported theorem (s_wess_zumino_descent_theorem).
s_gauge_symmetry_with_anomaly_functional	axiom	gauge symmetry with anomaly functional		Iter-3 imported axiom (s_gauge_symmetry_with_anomaly_functional).
s_brst_cohomology	axiom	brst cohomology		Iter-3 imported axiom (s_brst_cohomology).
s_anomaly_satisfies_cocycle_condition	state	anomaly satisfies cocycle condition		Iter-3 imported state (s_anomaly_satisfies_cocycle_condition).
s_anomaly_class_in_local_BRST_cohomology	state	anomaly class in local BRST cohomology		Iter-3 imported state (s_anomaly_class_in_local_BRST_cohomology).
s_atiyah_bott_fixed_point_theorem	theorem	Atiyah–Bott Lefschetz fixed point (equivariant)		Iter-3 imported theorem (s_atiyah_bott_fixed_point_theorem).
s_smooth_self_map_with_isolated_fixed_points	axiom	smooth self map with isolated fixed points		Iter-3 imported axiom (s_smooth_self_map_with_isolated_fixed_points).
s_equivariant_index_localized_to_fixed_points	state	equivariant index localized to fixed points		Iter-3 imported state (s_equivariant_index_localized_to_fixed_points).
s_localization_to_sum_over_fixed_points	state	localization to sum over fixed points		Iter-3 imported state (s_localization_to_sum_over_fixed_points).
s_chiral_anomaly_from_index_theorem	theorem	Atiyah–Singer applied to anomalies (Stora–Zumino)		Iter-3 imported theorem (s_chiral_anomaly_from_index_theorem).
s_chiral_dirac_operator	axiom	chiral dirac operator		Iter-3 imported axiom (s_chiral_dirac_operator).
s_gauge_bundle_with_connection	axiom	gauge bundle with connection		Iter-3 imported axiom (s_gauge_bundle_with_connection).
s_index_of_chiral_dirac_equals_A_roof_ch_E	state	index of chiral dirac equals A roof ch E		Iter-3 imported state (s_index_of_chiral_dirac_equals_A_roof_ch_E).
s_perturbative_anomaly_coefficient	state	perturbative anomaly coefficient		Iter-3 imported state (s_perturbative_anomaly_coefficient).
s_donaldson_intersection_form_theorem	theorem	Donaldson's theorem on smooth 4-manifolds		Iter-3 imported theorem (s_donaldson_intersection_form_theorem).
s_smooth_simply_connected_4_manifold	axiom	smooth simply connected 4 manifold		Iter-3 imported axiom (s_smooth_simply_connected_4_manifold).
s_definite_intersection_form	axiom	definite intersection form		Iter-3 imported axiom (s_definite_intersection_form).
s_ASD_moduli_space	state	ASD moduli space		Iter-3 imported state (s_ASD_moduli_space).
s_dimension_of_moduli_via_index	state	dimension of moduli via index		Iter-3 imported state (s_dimension_of_moduli_via_index).
s_seiberg_witten_invariants_theorem	theorem	Seiberg–Witten invariants existence		Iter-3 imported theorem (s_seiberg_witten_invariants_theorem).
s_smooth_4_manifold_with_spin_c_structure	axiom	smooth 4 manifold with spin c structure		Iter-3 imported axiom (s_smooth_4_manifold_with_spin_c_structure).
s_seiberg_witten_equations	axiom	seiberg witten equations		Iter-3 imported axiom (s_seiberg_witten_equations).
s_monopole_moduli_space	state	monopole moduli space		Iter-3 imported state (s_monopole_moduli_space).
s_compact_smooth_moduli_under_generic_metric	state	compact smooth moduli under generic metric		Iter-3 imported state (s_compact_smooth_moduli_under_generic_metric).
s_dimension_of_SW_moduli_via_index	state	dimension of SW moduli via index		Iter-3 imported state (s_dimension_of_SW_moduli_via_index).
s_verlinde_formula_theorem	theorem	Verlinde formula for SU(N) WZW		Iter-3 imported theorem (s_verlinde_formula_theorem).
s_WZW_model_at_level_k_for_SU_N	axiom	WZW model at level k for SU N		Iter-3 imported axiom (s_WZW_model_at_level_k_for_SU_N).
s_modular_S_matrix_on_characters	axiom	modular S matrix on characters		Iter-3 imported axiom (s_modular_S_matrix_on_characters).
s_fusion_rules_as_tensor_product_decomposition	state	fusion rules as tensor product decomposition		Iter-3 imported state (s_fusion_rules_as_tensor_product_decomposition).
s_S_matrix_diagonalizes_fusion	state	S matrix diagonalizes fusion		Iter-3 imported state (s_S_matrix_diagonalizes_fusion).
s_cft_modular_invariance_theorem	theorem	Modular invariance of CFT partition function (S, T action)		Iter-3 imported theorem (s_cft_modular_invariance_theorem).
s_2d_cft_on_torus	axiom	2d cft on torus		Iter-3 imported axiom (s_2d_cft_on_torus).
s_partition_function_Z_tau	axiom	partition function Z tau		Iter-3 imported axiom (s_partition_function_Z_tau).
s_partition_function_depends_only_on_modulus	state	partition function depends only on modulus		Iter-3 imported state (s_partition_function_depends_only_on_modulus).
s_Z_as_quadratic_form_in_characters	state	Z as quadratic form in characters		Iter-3 imported state (s_Z_as_quadratic_form_in_characters).
s_lax_pair_conserved_quantities_theorem	theorem	Lax pair representation yields conserved quantities		Iter-3 imported theorem (s_lax_pair_conserved_quantities_theorem).
s_evolution_equation_with_lax_pair_L_M	axiom	evolution equation with lax pair L M		Iter-3 imported axiom (s_evolution_equation_with_lax_pair_L_M).
s_isospectral_flow	state	isospectral flow		Iter-3 imported state (s_isospectral_flow).
s_spectrum_independent_of_time	state	spectrum independent of time		Iter-3 imported state (s_spectrum_independent_of_time).
s_kdv_inverse_scattering_theorem	theorem	Inverse scattering transform for KdV		Iter-3 imported theorem (s_kdv_inverse_scattering_theorem).
s_kdv_equation	axiom	kdv equation		Iter-3 imported axiom (s_kdv_equation).
s_schrodinger_scattering_operator_with_potential_u	axiom	schrodinger scattering operator with potential u		Iter-3 imported axiom (s_schrodinger_scattering_operator_with_potential_u).
s_lax_pair_for_kdv	state	lax pair for kdv		Iter-3 imported state (s_lax_pair_for_kdv).
s_scattering_data_evolve_linearly	state	scattering data evolve linearly		Iter-3 imported state (s_scattering_data_evolve_linearly).
s_gelfand_levitan_marchenko_reconstruction	state	gelfand levitan marchenko reconstruction		Iter-3 imported state (s_gelfand_levitan_marchenko_reconstruction).
s_yang_baxter_integrability_theorem	theorem	Yang–Baxter equation as integrability condition		Iter-3 imported theorem (s_yang_baxter_integrability_theorem).
s_two_particle_scattering_matrix_R	axiom	two particle scattering matrix R		Iter-3 imported axiom (s_two_particle_scattering_matrix_R).
s_factorized_n_particle_S_matrix	axiom	factorized n particle S matrix		Iter-3 imported axiom (s_factorized_n_particle_S_matrix).
s_triangle_consistency_condition	state	triangle consistency condition		Iter-3 imported state (s_triangle_consistency_condition).
s_yang_baxter_equation	state	yang baxter equation		Iter-3 imported state (s_yang_baxter_equation).
s_bethe_ansatz_xxx_theorem	theorem	Bethe ansatz solution of the Heisenberg XXX chain		Iter-3 imported theorem (s_bethe_ansatz_xxx_theorem).
s_heisenberg_xxx_spin_chain	axiom	heisenberg xxx spin chain		Iter-3 imported axiom (s_heisenberg_xxx_spin_chain).
s_plane_wave_ansatz_with_rapidities	axiom	plane wave ansatz with rapidities		Iter-3 imported axiom (s_plane_wave_ansatz_with_rapidities).
s_magnon_sector_decomposition	state	magnon sector decomposition		Iter-3 imported state (s_magnon_sector_decomposition).
s_two_particle_S_matrix_phase	state	two particle S matrix phase		Iter-3 imported state (s_two_particle_S_matrix_phase).
s_bethe_equations	state	bethe equations		Iter-3 imported state (s_bethe_equations).
s_hirota_soliton_theorem	theorem	Hirota bilinear form ⇒ soliton solutions		Iter-3 imported theorem (s_hirota_soliton_theorem).
s_integrable_pde_eg_kdv_or_KP	axiom	integrable pde eg kdv or KP		Iter-3 imported axiom (s_integrable_pde_eg_kdv_or_KP).
s_tau_function_substitution	state	tau function substitution		Iter-3 imported state (s_tau_function_substitution).
s_bilinear_pde_in_tau	state	bilinear pde in tau		Iter-3 imported state (s_bilinear_pde_in_tau).
s_riemann_hilbert_method_theorem	theorem	Riemann–Hilbert reformulation of integrable PDE		Iter-3 imported theorem (s_riemann_hilbert_method_theorem).
s_integrable_pde_with_lax_pair	axiom	integrable pde with lax pair		Iter-3 imported axiom (s_integrable_pde_with_lax_pair).
s_jump_matrix_on_contour	axiom	jump matrix on contour		Iter-3 imported axiom (s_jump_matrix_on_contour).
s_jump_data_on_spectral_curve	state	jump data on spectral curve		Iter-3 imported state (s_jump_data_on_spectral_curve).
s_matrix_RH_problem	state	matrix RH problem		Iter-3 imported state (s_matrix_RH_problem).
s_calogero_moser_integrability_theorem	theorem	Calogero–Moser integrability		Iter-3 imported theorem (s_calogero_moser_integrability_theorem).
s_calogero_moser_hamiltonian_with_1_over_x_sq_pair	axiom	calogero moser hamiltonian with 1 over x sq pair		Iter-3 imported axiom (s_calogero_moser_hamiltonian_with_1_over_x_sq_pair).
s_lax_pair_for_calogero_moser	state	lax pair for calogero moser		Iter-3 imported state (s_lax_pair_for_calogero_moser).
s_n_commuting_first_integrals_constructed	state	n commuting first integrals constructed		Iter-3 imported state (s_n_commuting_first_integrals_constructed).
s_toda_lattice_integrability_theorem	theorem	Toda lattice integrability		Iter-3 imported theorem (s_toda_lattice_integrability_theorem).
s_toda_lattice_hamiltonian_exponential_potential	axiom	toda lattice hamiltonian exponential potential		Iter-3 imported axiom (s_toda_lattice_hamiltonian_exponential_potential).
s_flaschka_variable_form	state	flaschka variable form		Iter-3 imported state (s_flaschka_variable_form).
s_jacobi_matrix_isospectral_flow	state	jacobi matrix isospectral flow		Iter-3 imported state (s_jacobi_matrix_isospectral_flow).
s_sklyanin_sov_theorem	theorem	Sklyanin separation of variables		Iter-3 imported theorem (s_sklyanin_sov_theorem).
s_quantum_integrable_system_with_transfer_matrix	axiom	quantum integrable system with transfer matrix		Iter-3 imported axiom (s_quantum_integrable_system_with_transfer_matrix).
s_separation_coordinates_from_B_zeros	state	separation coordinates from B zeros		Iter-3 imported state (s_separation_coordinates_from_B_zeros).
s_baxter_TQ_relation	state	baxter TQ relation		Iter-3 imported state (s_baxter_TQ_relation).
s_painleve_classification_theorem	theorem	Painlevé property and isomonodromy		Iter-3 imported theorem (s_painleve_classification_theorem).
s_second_order_ode_with_movable_singularities	axiom	second order ode with movable singularities		Iter-3 imported axiom (s_second_order_ode_with_movable_singularities).
s_painleve_property_definition	state	painleve property definition		Iter-3 imported state (s_painleve_property_definition).
s_50_canonical_ODEs_reduced_to_6_irreducible	state	50 canonical ODEs reduced to 6 irreducible		Iter-3 imported state (s_50_canonical_ODEs_reduced_to_6_irreducible).
s_wolstenholme_theorem	theorem	Wolstenholme's theorem		Iter-3 imported theorem (s_wolstenholme_theorem).
s_normalized_central_binomial_mod_p4	state	normalized central binomial mod p4		Iter-3 imported state (s_normalized_central_binomial_mod_p4).
s_harmonic_congruence_mod_p3	state	harmonic congruence mod p3		Iter-3 imported state (s_harmonic_congruence_mod_p3).
s_euclid_euler_perfect_numbers	theorem	Euclid–Euler theorem (even perfect numbers)		Iter-3 imported theorem (s_euclid_euler_perfect_numbers).
s_mersenne_prime_2p_minus_1	axiom	mersenne prime 2p minus 1		Iter-3 imported axiom (s_mersenne_prime_2p_minus_1).
s_sigma_multiplicative	state	sigma multiplicative		Iter-3 imported state (s_sigma_multiplicative).
s_euclid_perfect_sufficient	state	euclid perfect sufficient		Iter-3 imported state (s_euclid_perfect_sufficient).
s_pentagonal_number_theorem	theorem	Pentagonal number theorem (Euler)		Iter-3 imported theorem (s_pentagonal_number_theorem).
s_formal_power_series_over_z	axiom	formal power series over z		Iter-3 imported axiom (s_formal_power_series_over_z).
s_partition_generating_function_q_pochhammer	axiom	partition generating function q pochhammer		Iter-3 imported axiom (s_partition_generating_function_q_pochhammer).
s_sparse_pentagonal_signature	state	sparse pentagonal signature		Iter-3 imported state (s_sparse_pentagonal_signature).
s_franklin_involution_cancellation	state	franklin involution cancellation		Iter-3 imported state (s_franklin_involution_cancellation).
s_zeckendorf_theorem	theorem	Zeckendorf's theorem		Iter-3 imported theorem (s_zeckendorf_theorem).
s_fibonacci_numbers	axiom	fibonacci numbers		Iter-3 imported axiom (s_fibonacci_numbers).
s_greedy_fibonacci_decomposition	state	greedy fibonacci decomposition		Iter-3 imported state (s_greedy_fibonacci_decomposition).
s_fermat_right_triangle_no_rational_square_area	theorem	Fermat's right-triangle theorem		Iter-3 imported theorem (s_fermat_right_triangle_no_rational_square_area).
s_pythagorean_triple_primitive_parametrization	axiom	pythagorean triple primitive parametrization		Iter-3 imported axiom (s_pythagorean_triple_primitive_parametrization).
s_smallest_squared_area_triple	state	smallest squared area triple		Iter-3 imported state (s_smallest_squared_area_triple).
s_descended_smaller_triple	state	descended smaller triple		Iter-3 imported state (s_descended_smaller_triple).
s_sophie_germain_theorem	theorem	Sophie Germain's theorem		Iter-3 imported theorem (s_sophie_germain_theorem).
s_flt_exponent_p_first_case	axiom	flt exponent p first case		Iter-3 imported axiom (s_flt_exponent_p_first_case).
s_aux_sophie_germain_prime_q	state	aux sophie germain prime q		Iter-3 imported state (s_aux_sophie_germain_prime_q).
s_flt_residue_obstruction_mod_q	state	flt residue obstruction mod q		Iter-3 imported state (s_flt_residue_obstruction_mod_q).
s_fermat_polygonal_number_theorem	theorem	Fermat polygonal number theorem		Iter-3 imported theorem (s_fermat_polygonal_number_theorem).
s_gauss_triangular_eureka	state	gauss triangular eureka		Iter-3 imported state (s_gauss_triangular_eureka).
s_jacobi_four_square_formula	theorem	Jacobi's four-square theorem		Iter-3 imported theorem (s_jacobi_four_square_formula).
s_r4_formula_8_times_sigma	state	r4 formula 8 times sigma		Iter-3 imported state (s_r4_formula_8_times_sigma).
s_three_gap_theorem	theorem	Three-gap theorem (Steinhaus)		Iter-3 imported theorem (s_three_gap_theorem).
s_reals_with_topology	axiom	reals with topology		Iter-3 imported axiom (s_reals_with_topology).
s_circle_t_1	axiom	circle t 1		Iter-3 imported axiom (s_circle_t_1).
s_orbit_of_n_alpha_mod_1	state	orbit of n alpha mod 1		Iter-3 imported state (s_orbit_of_n_alpha_mod_1).
s_at_most_three_gap_lengths	state	at most three gap lengths		Iter-3 imported state (s_at_most_three_gap_lengths).
s_hurwitz_diophantine_approximation	theorem	Hurwitz's theorem on Diophantine approximation		Iter-3 imported theorem (s_hurwitz_diophantine_approximation).
s_continued_fraction_expansion	axiom	continued fraction expansion		Iter-3 imported axiom (s_continued_fraction_expansion).
s_convergent_best_approximation	state	convergent best approximation		Iter-3 imported state (s_convergent_best_approximation).
s_markov_spectrum_minimal_gap_sqrt5	state	markov spectrum minimal gap sqrt5		Iter-3 imported state (s_markov_spectrum_minimal_gap_sqrt5).
s_dirichlet_approximation	theorem	Dirichlet's approximation theorem		Iter-3 imported theorem (s_dirichlet_approximation).
s_unit_interval	axiom	unit interval		Iter-3 imported axiom (s_unit_interval).
s_two_close_kalpha_points	state	two close kalpha points		Iter-3 imported state (s_two_close_kalpha_points).
s_mihailescu_catalan_theorem	theorem	Mihăilescu's theorem (Catalan)		Iter-3 imported theorem (s_mihailescu_catalan_theorem).
s_cyclotomic_field_q_zeta_p	axiom	cyclotomic field q zeta p		Iter-3 imported axiom (s_cyclotomic_field_q_zeta_p).
s_catalan_equation_canonical	state	catalan equation canonical		Iter-3 imported state (s_catalan_equation_canonical).
s_class_group_divisibility_obstruction	state	class group divisibility obstruction		Iter-3 imported state (s_class_group_divisibility_obstruction).
s_cubic_reciprocity	theorem	Cubic reciprocity (Eisenstein)		Iter-3 imported theorem (s_cubic_reciprocity).
s_eisenstein_integers_z_omega	axiom	eisenstein integers z omega		Iter-3 imported axiom (s_eisenstein_integers_z_omega).
s_primary_eisenstein_primes	state	primary eisenstein primes		Iter-3 imported state (s_primary_eisenstein_primes).
s_cubic_residue_symbol_reciprocal_law	state	cubic residue symbol reciprocal law		Iter-3 imported state (s_cubic_residue_symbol_reciprocal_law).
s_quartic_reciprocity	theorem	Quartic / biquadratic reciprocity		Iter-3 imported theorem (s_quartic_reciprocity).
s_gaussian_integers_z_i	axiom	gaussian integers z i		Iter-3 imported axiom (s_gaussian_integers_z_i).
s_primary_gaussian_primes	state	primary gaussian primes		Iter-3 imported state (s_primary_gaussian_primes).
s_quartic_residue_symbol	state	quartic residue symbol		Iter-3 imported state (s_quartic_residue_symbol).
s_jacobi_triple_product	theorem	Jacobi triple product		Iter-3 imported theorem (s_jacobi_triple_product).
s_q_pochhammer_symbol	axiom	q pochhammer symbol		Iter-3 imported axiom (s_q_pochhammer_symbol).
s_quasiperiodic_shift_z_qz	state	quasiperiodic shift z qz		Iter-3 imported state (s_quasiperiodic_shift_z_qz).
s_hasse_minkowski	theorem	Hasse–Minkowski theorem		Iter-3 imported theorem (s_hasse_minkowski).
s_rationals_q	axiom	rationals q		Iter-3 imported axiom (s_rationals_q).
s_p_adic_completions_q_p	axiom	p adic completions q p		Iter-3 imported axiom (s_p_adic_completions_q_p).
s_quadratic_form_over_q	axiom	quadratic form over q		Iter-3 imported axiom (s_quadratic_form_over_q).
s_diagonal_quadratic_form	state	diagonal quadratic form		Iter-3 imported state (s_diagonal_quadratic_form).
s_hilbert_symbol_product_formula	state	hilbert symbol product formula		Iter-3 imported state (s_hilbert_symbol_product_formula).
s_meyer_theorem_indefinite	theorem	Meyer's theorem (indefinite quadratic in ≥5 variables)		Iter-3 imported theorem (s_meyer_theorem_indefinite).
s_indefinite_form_rank_ge_5	state	indefinite form rank ge 5		Iter-3 imported state (s_indefinite_form_rank_ge_5).
s_brauer_forms_theorem	theorem	Brauer's theorem on forms		Iter-3 imported theorem (s_brauer_forms_theorem).
s_finite_field_f_q	axiom	finite field f q		Iter-3 imported axiom (s_finite_field_f_q).
s_homogeneous_form_degree_d	axiom	homogeneous form degree d		Iter-3 imported axiom (s_homogeneous_form_degree_d).
s_chevalley_warning_zero_count	state	chevalley warning zero count		Iter-3 imported state (s_chevalley_warning_zero_count).
s_bhargava_290_theorem	theorem	Bhargava–Hanke 15 and 290 theorems		Iter-3 imported theorem (s_bhargava_290_theorem).
s_positive_definite_integral_quadratic_form	axiom	positive definite integral quadratic form		Iter-3 imported axiom (s_positive_definite_integral_quadratic_form).
s_universality_witness_set_290	state	universality witness set 290		Iter-3 imported state (s_universality_witness_set_290).
s_hilbert_irreducibility	theorem	Hilbert's irreducibility theorem		Iter-3 imported theorem (s_hilbert_irreducibility).
s_polynomial_ring_in_two_vars_q_t_x	axiom	polynomial ring in two vars q t x		Iter-3 imported axiom (s_polynomial_ring_in_two_vars_q_t_x).
s_thin_set_in_an	axiom	thin set in an		Iter-3 imported axiom (s_thin_set_in_an).
s_generic_specialization_remains_irreducible	state	generic specialization remains irreducible		Iter-3 imported state (s_generic_specialization_remains_irreducible).
s_thin_exceptional_set	state	thin exceptional set		Iter-3 imported state (s_thin_exceptional_set).
s_dedekind_discriminant_theorem	theorem	Dedekind discriminant theorem		Iter-3 imported theorem (s_dedekind_discriminant_theorem).
s_number_field_k	axiom	number field k		Iter-3 imported axiom (s_number_field_k).
s_ring_of_integers_o_k	axiom	ring of integers o k		Iter-3 imported axiom (s_ring_of_integers_o_k).
s_different_ideal_definition	state	different ideal definition		Iter-3 imported state (s_different_ideal_definition).
s_dedekind_kummer_theorem	theorem	Dedekind–Kummer theorem		Iter-3 imported theorem (s_dedekind_kummer_theorem).
s_minimal_polynomial_theta	axiom	minimal polynomial theta		Iter-3 imported axiom (s_minimal_polynomial_theta).
s_minimal_poly_factorization_mod_p	state	minimal poly factorization mod p		Iter-3 imported state (s_minimal_poly_factorization_mod_p).
s_kummer_cyclotomic_factorization	theorem	Kummer's theorem on cyclotomic ideal factorization		Iter-3 imported theorem (s_kummer_cyclotomic_factorization).
s_cyclotomic_galois_structure	state	cyclotomic galois structure		Iter-3 imported state (s_cyclotomic_galois_structure).
s_kummer_congruence	theorem	Kummer's congruence (Bernoulli)		Iter-3 imported theorem (s_kummer_congruence).
s_bernoulli_numbers_b_n	axiom	bernoulli numbers b n		Iter-3 imported axiom (s_bernoulli_numbers_b_n).
s_p_adic_l_function	state	p adic l function		Iter-3 imported state (s_p_adic_l_function).
s_von_staudt_clausen_theorem	theorem	Von Staudt–Clausen theorem		Iter-3 imported theorem (s_von_staudt_clausen_theorem).
s_bernoulli_integer_part_formula	state	bernoulli integer part formula		Iter-3 imported state (s_bernoulli_integer_part_formula).
s_apery_zeta3_irrational	theorem	Apéry's theorem (ζ(3) irrational)		Iter-3 imported theorem (s_apery_zeta3_irrational).
s_apery_recurrence_sequences	state	apery recurrence sequences		Iter-3 imported state (s_apery_recurrence_sequences).
s_super_dirichlet_approximation_of_zeta3	state	super dirichlet approximation of zeta3		Iter-3 imported state (s_super_dirichlet_approximation_of_zeta3).
s_chebotarev_density	theorem	Chebotarev density theorem		Iter-3 imported theorem (s_chebotarev_density).
s_galois_extension_l_over_k	axiom	galois extension l over k		Iter-3 imported axiom (s_galois_extension_l_over_k).
s_frobenius_conjugacy_class_distribution	state	frobenius conjugacy class distribution		Iter-3 imported state (s_frobenius_conjugacy_class_distribution).
s_density_via_character_sums	state	density via character sums		Iter-3 imported state (s_density_via_character_sums).
s_chebyshev_pi_bounds	theorem	Chebyshev's bounds on π(x)		Iter-3 imported theorem (s_chebyshev_pi_bounds).
s_chebyshev_function_theta_psi	axiom	chebyshev function theta psi		Iter-3 imported axiom (s_chebyshev_function_theta_psi).
s_psi_squeeze_via_central_binomial	state	psi squeeze via central binomial		Iter-3 imported state (s_psi_squeeze_via_central_binomial).
s_mertens_theorems	theorem	Mertens' theorems		Iter-3 imported theorem (s_mertens_theorems).
s_log_p_over_p_partial_sums	state	log p over p partial sums		Iter-3 imported state (s_log_p_over_p_partial_sums).
s_mertens_constant_M	state	mertens constant M		Iter-3 imported state (s_mertens_constant_M).
s_brun_twin_prime_constant_finite	theorem	Brun's theorem on twin primes		Iter-3 imported theorem (s_brun_twin_prime_constant_finite).
s_twin_prime_set	axiom	twin prime set		Iter-3 imported axiom (s_twin_prime_set).
s_pi_2_x_upper_bound	state	pi 2 x upper bound		Iter-3 imported state (s_pi_2_x_upper_bound).
s_brun_titchmarsh_inequality	theorem	Brun–Titchmarsh inequality		Iter-3 imported theorem (s_brun_titchmarsh_inequality).
s_dirichlet_character_mod_q	axiom	dirichlet character mod q		Iter-3 imported axiom (s_dirichlet_character_mod_q).
s_sifted_count_primes_in_ap	state	sifted count primes in ap		Iter-3 imported state (s_sifted_count_primes_in_ap).
s_selberg_erdos_elementary_pnt	theorem	Selberg–Erdős elementary proof of PNT		Iter-3 imported theorem (s_selberg_erdos_elementary_pnt).
s_selberg_symmetry_formula	state	selberg symmetry formula		Iter-3 imported state (s_selberg_symmetry_formula).
s_psi_minus_x_decay	state	psi minus x decay		Iter-3 imported state (s_psi_minus_x_decay).
s_wiener_ikehara_theorem	theorem	Wiener–Ikehara Tauberian theorem		Iter-3 imported theorem (s_wiener_ikehara_theorem).
s_dirichlet_series	axiom	dirichlet series		Iter-3 imported axiom (s_dirichlet_series).
s_simple_pole_at_s_eq_1	state	simple pole at s eq 1		Iter-3 imported state (s_simple_pole_at_s_eq_1).
s_tauberian_fourier_input	state	tauberian fourier input		Iter-3 imported state (s_tauberian_fourier_input).
s_vinogradov_three_prime_theorem	theorem	Vinogradov's three-prime theorem		Iter-3 imported theorem (s_vinogradov_three_prime_theorem).
s_major_minor_arc_decomposition_for_3p	state	major minor arc decomposition for 3p		Iter-3 imported state (s_major_minor_arc_decomposition_for_3p).
s_minor_arc_bound	state	minor arc bound		Iter-3 imported state (s_minor_arc_bound).
s_vinogradov_mean_value_theorem	theorem	Vinogradov mean-value theorem		Iter-3 imported theorem (s_vinogradov_mean_value_theorem).
s_polynomial_ring_z_x	axiom	polynomial ring z x		Iter-3 imported axiom (s_polynomial_ring_z_x).
s_exponential_sum	axiom	exponential sum		Iter-3 imported axiom (s_exponential_sum).
s_vinogradov_system_count	state	vinogradov system count		Iter-3 imported state (s_vinogradov_system_count).
s_efficient_congruencing_bound	state	efficient congruencing bound		Iter-3 imported state (s_efficient_congruencing_bound).
s_bombieri_vinogradov	theorem	Bombieri–Vinogradov theorem		Iter-3 imported theorem (s_bombieri_vinogradov).
s_large_sieve_inequality	state	large sieve inequality		Iter-3 imported state (s_large_sieve_inequality).
s_zero_density_combined	state	zero density combined		Iter-3 imported state (s_zero_density_combined).
s_linnik_smallest_prime_in_ap	theorem	Linnik's theorem on smallest prime in AP		Iter-3 imported theorem (s_linnik_smallest_prime_in_ap).
s_dirichlet_l_function	axiom	dirichlet l function		Iter-3 imported axiom (s_dirichlet_l_function).
s_zero_density_and_repulsion	state	zero density and repulsion		Iter-3 imported state (s_zero_density_and_repulsion).
s_polya_vinogradov_inequality	theorem	Pólya–Vinogradov inequality		Iter-3 imported theorem (s_polya_vinogradov_inequality).
s_character_in_additive_frequencies	state	character in additive frequencies		Iter-3 imported state (s_character_in_additive_frequencies).
s_burgess_bound	theorem	Burgess bound on character sums		Iter-3 imported theorem (s_burgess_bound).
s_shifted_character_moment	state	shifted character moment		Iter-3 imported state (s_shifted_character_moment).
s_short_character_sum_bound	state	short character sum bound		Iter-3 imported state (s_short_character_sum_bound).
s_kim_sarnak_bound	theorem	Selberg's 1/4 conjecture status / Kim–Sarnak bound		Iter-3 imported theorem (s_kim_sarnak_bound).
s_modular_forms_holomorphic_or_maass	axiom	modular forms holomorphic or maass		Iter-3 imported axiom (s_modular_forms_holomorphic_or_maass).
s_symmetric_power_l_functions	state	symmetric power l functions		Iter-3 imported state (s_symmetric_power_l_functions).
s_hardy_ramanujan_partition_asymptotic	theorem	Hardy–Ramanujan partition asymptotic		Iter-3 imported theorem (s_hardy_ramanujan_partition_asymptotic).
s_integer_partitions	axiom	integer partitions		Iter-3 imported axiom (s_integer_partitions).
s_eta_function_dedekind	axiom	eta function dedekind		Iter-3 imported axiom (s_eta_function_dedekind).
s_modular_transformation_of_eta	state	modular transformation of eta		Iter-3 imported state (s_modular_transformation_of_eta).
s_farey_arc_contributions	state	farey arc contributions		Iter-3 imported state (s_farey_arc_contributions).
s_rademacher_exact_formula_partition	theorem	Rademacher exact formula for p(n)		Iter-3 imported theorem (s_rademacher_exact_formula_partition).
s_rademacher_contour_integral	state	rademacher contour integral		Iter-3 imported state (s_rademacher_contour_integral).
s_erdos_kac_theorem	theorem	Erdős–Kac theorem		Iter-3 imported theorem (s_erdos_kac_theorem).
s_additive_function_omega	axiom	additive function omega		Iter-3 imported axiom (s_additive_function_omega).
s_quasi_independence_decomposition	state	quasi independence decomposition		Iter-3 imported state (s_quasi_independence_decomposition).
s_clt_for_omega	state	clt for omega		Iter-3 imported state (s_clt_for_omega).
s_erdos_wintner_theorem	theorem	Erdős–Wintner theorem		Iter-3 imported theorem (s_erdos_wintner_theorem).
s_additive_function_f	axiom	additive function f		Iter-3 imported axiom (s_additive_function_f).
s_three_series_criterion	state	three series criterion		Iter-3 imported state (s_three_series_criterion).
s_turan_kubilius_inequality	theorem	Turán–Kubilius inequality		Iter-3 imported theorem (s_turan_kubilius_inequality).
s_second_moment_bound	state	second moment bound		Iter-3 imported state (s_second_moment_bound).
s_erdos_fuchs_theorem	theorem	Erdős–Fuchs theorem		Iter-3 imported theorem (s_erdos_fuchs_theorem).
s_additive_basis	axiom	additive basis		Iter-3 imported axiom (s_additive_basis).
s_proposed_smooth_error_term	state	proposed smooth error term		Iter-3 imported state (s_proposed_smooth_error_term).
s_fourier_error_obstruction	state	fourier error obstruction		Iter-3 imported state (s_fourier_error_obstruction).
s_roth_theorem_3_aps	theorem	Roth's theorem on 3-APs		Iter-3 imported theorem (s_roth_theorem_3_aps).
s_finite_subset_a_in_z	axiom	finite subset a in z		Iter-3 imported axiom (s_finite_subset_a_in_z).
s_fourier_density_increment	state	fourier density increment		Iter-3 imported state (s_fourier_density_increment).
s_roth_diophantine_approximation	theorem	Roth's theorem on Diophantine approximation		Iter-3 imported theorem (s_roth_diophantine_approximation).
s_algebraic_number_alpha	axiom	algebraic number alpha		Iter-3 imported axiom (s_algebraic_number_alpha).
s_aux_multivariate_polynomial	state	aux multivariate polynomial		Iter-3 imported state (s_aux_multivariate_polynomial).
s_index_height_obstruction	state	index height obstruction		Iter-3 imported state (s_index_height_obstruction).
s_schmidt_subspace_theorem	theorem	Subspace theorem (Schmidt)		Iter-3 imported theorem (s_schmidt_subspace_theorem).
s_linear_form_l_i	axiom	linear form l i		Iter-3 imported axiom (s_linear_form_l_i).
s_higher_dim_diophantine_inequality	state	higher dim diophantine inequality		Iter-3 imported state (s_higher_dim_diophantine_inequality).
s_lindemann_weierstrass	theorem	Lindemann–Weierstrass theorem		Iter-3 imported theorem (s_lindemann_weierstrass).
s_exponential_function_complex	axiom	exponential function complex		Iter-3 imported axiom (s_exponential_function_complex).
s_hermite_integral	state	hermite integral		Iter-3 imported state (s_hermite_integral).
s_integer_vs_small_obstruction	state	integer vs small obstruction		Iter-3 imported state (s_integer_vs_small_obstruction).
s_gelfond_schneider_theorem	theorem	Gelfond–Schneider theorem		Iter-3 imported theorem (s_gelfond_schneider_theorem).
s_logarithm_function_complex	axiom	logarithm function complex		Iter-3 imported axiom (s_logarithm_function_complex).
s_gelfond_aux_function	state	gelfond aux function		Iter-3 imported state (s_gelfond_aux_function).
s_extrapolation_zeroes	state	extrapolation zeroes		Iter-3 imported state (s_extrapolation_zeroes).
s_baker_theorem	theorem	Baker's theorem on linear forms in logarithms		Iter-3 imported theorem (s_baker_theorem).
s_baker_auxiliary_many_vars	state	baker auxiliary many vars		Iter-3 imported state (s_baker_auxiliary_many_vars).
s_baker_nonzero_integer_too_small	state	baker nonzero integer too small		Iter-3 imported state (s_baker_nonzero_integer_too_small).
s_six_exponentials_theorem	theorem	Six exponentials theorem		Iter-3 imported theorem (s_six_exponentials_theorem).
s_2x3_interpolation_determinant	state	2x3 interpolation determinant		Iter-3 imported state (s_2x3_interpolation_determinant).
s_vanishing_determinant_pigeonhole	state	vanishing determinant pigeonhole		Iter-3 imported state (s_vanishing_determinant_pigeonhole).
s_schneider_lang_theorem	theorem	Schneider–Lang theorem		Iter-3 imported theorem (s_schneider_lang_theorem).
s_meromorphic_function_finite_order	axiom	meromorphic function finite order		Iter-3 imported axiom (s_meromorphic_function_finite_order).
s_meromorphic_algebraic_ring	state	meromorphic algebraic ring		Iter-3 imported state (s_meromorphic_algebraic_ring).
s_schneider_lang_aux	state	schneider lang aux		Iter-3 imported state (s_schneider_lang_aux).
s_wustholz_analytic_subgroup_theorem	theorem	Wüstholz analytic subgroup theorem		Iter-3 imported theorem (s_wustholz_analytic_subgroup_theorem).
s_commutative_algebraic_group_g_over_q_bar	axiom	commutative algebraic group g over q bar		Iter-3 imported axiom (s_commutative_algebraic_group_g_over_q_bar).
s_wustholz_analytic_subgroup_statement	state	wustholz analytic subgroup statement		Iter-3 imported state (s_wustholz_analytic_subgroup_statement).
s_weil_conjectures_complete	theorem	Weil conjectures (over finite fields)		Iter-3 imported theorem (s_weil_conjectures_complete).
s_smooth_projective_variety_over_fq	axiom	smooth projective variety over fq		Iter-3 imported axiom (s_smooth_projective_variety_over_fq).
s_etale_cohomology	axiom	etale cohomology		Iter-3 imported axiom (s_etale_cohomology).
s_zeta_function_of_variety	state	zeta function of variety		Iter-3 imported state (s_zeta_function_of_variety).
s_etale_trace_formula	state	etale trace formula		Iter-3 imported state (s_etale_trace_formula).
s_hasse_weil_bound	theorem	Hasse–Weil bound for curves over F_q		Iter-3 imported theorem (s_hasse_weil_bound).
s_smooth_projective_curve_over_fq	axiom	smooth projective curve over fq		Iter-3 imported axiom (s_smooth_projective_curve_over_fq).
s_riemann_hypothesis_for_curves	axiom	riemann hypothesis for curves		Iter-3 imported axiom (s_riemann_hypothesis_for_curves).
s_curve_l_function_eigenvalues	state	curve l function eigenvalues		Iter-3 imported state (s_curve_l_function_eigenvalues).
s_honda_tate_theorem	theorem	Honda–Tate theorem		Iter-3 imported theorem (s_honda_tate_theorem).
s_abelian_variety_over_fq	axiom	abelian variety over fq		Iter-3 imported axiom (s_abelian_variety_over_fq).
s_weil_q_numbers	axiom	weil q numbers		Iter-3 imported axiom (s_weil_q_numbers).
s_frobenius_to_weil_number	state	frobenius to weil number		Iter-3 imported state (s_frobenius_to_weil_number).
s_tate_isogeny_theorem	theorem	Tate's isogeny theorem		Iter-3 imported theorem (s_tate_isogeny_theorem).
s_l_adic_tate_module	axiom	l adic tate module		Iter-3 imported axiom (s_l_adic_tate_module).
s_galois_action_on_tate_module	state	galois action on tate module		Iter-3 imported state (s_galois_action_on_tate_module).
s_nagell_lutz_theorem	theorem	Nagell–Lutz theorem		Iter-3 imported theorem (s_nagell_lutz_theorem).
s_elliptic_curve_over_q	axiom	elliptic curve over q		Iter-3 imported axiom (s_elliptic_curve_over_q).
s_torsion_subgroup	axiom	torsion subgroup		Iter-3 imported axiom (s_torsion_subgroup).
s_minimal_weierstrass_form	state	minimal weierstrass form		Iter-3 imported state (s_minimal_weierstrass_form).
s_mazur_torsion_theorem	theorem	Mazur's torsion theorem		Iter-3 imported theorem (s_mazur_torsion_theorem).
s_modular_curve_x0_n	axiom	modular curve x0 n		Iter-3 imported axiom (s_modular_curve_x0_n).
s_rational_points_on_x1_n	state	rational points on x1 n		Iter-3 imported state (s_rational_points_on_x1_n).
s_modular_curves_no_extra_rational_points	state	modular curves no extra rational points		Iter-3 imported state (s_modular_curves_no_extra_rational_points).
s_merel_uniform_boundedness	theorem	Merel's uniform boundedness		Iter-3 imported theorem (s_merel_uniform_boundedness).
s_modular_curve_x1_n	axiom	modular curve x1 n		Iter-3 imported axiom (s_modular_curve_x1_n).
s_eisenstein_quotient_bound	state	eisenstein quotient bound		Iter-3 imported state (s_eisenstein_quotient_bound).
s_manin_drinfeld_theorem	theorem	Manin–Drinfeld theorem		Iter-3 imported theorem (s_manin_drinfeld_theorem).
s_cusp_divisors	axiom	cusp divisors		Iter-3 imported axiom (s_cusp_divisors).
s_eisenstein_divisor_rationality	state	eisenstein divisor rationality		Iter-3 imported state (s_eisenstein_divisor_rationality).
s_tunnell_theorem	theorem	Tunnell's theorem (congruent numbers)		Iter-3 imported theorem (s_tunnell_theorem).
s_weight_three_halves_form	state	weight three halves form		Iter-3 imported state (s_weight_three_halves_form).
s_modularity_theorem_full	theorem	Modularity theorem (full, Breuil–Conrad–Diamond–Taylor)		Iter-3 imported theorem (s_modularity_theorem_full).
s_l_adic_galois_representation_of_E	state	l adic galois representation of E		Iter-3 imported state (s_l_adic_galois_representation_of_E).
s_full_modularity_lift_machine	state	full modularity lift machine		Iter-3 imported state (s_full_modularity_lift_machine).
s_ribet_level_lowering	theorem	Ribet's level-lowering theorem		Iter-3 imported theorem (s_ribet_level_lowering).
s_galois_representation_residual	axiom	galois representation residual		Iter-3 imported axiom (s_galois_representation_residual).
s_residual_unramified_at_p_consequence	state	residual unramified at p consequence		Iter-3 imported state (s_residual_unramified_at_p_consequence).
s_eichler_shimura_congruence	theorem	Eichler–Shimura congruence		Iter-3 imported theorem (s_eichler_shimura_congruence).
s_hecke_operator	axiom	hecke operator		Iter-3 imported axiom (s_hecke_operator).
s_hecke_mod_p_decomposition	state	hecke mod p decomposition		Iter-3 imported state (s_hecke_mod_p_decomposition).
s_khare_wintenberger_serre_modularity	theorem	Khare–Wintenberger (Serre's modularity conjecture)		Iter-3 imported theorem (s_khare_wintenberger_serre_modularity).
s_modularity_lifting_for_residual_rep	state	modularity lifting for residual rep		Iter-3 imported state (s_modularity_lifting_for_residual_rep).
s_waldspurger_theorem	theorem	Waldspurger's theorem		Iter-3 imported theorem (s_waldspurger_theorem).
s_l_function_central_value	axiom	l function central value		Iter-3 imported axiom (s_l_function_central_value).
s_shimura_correspondence_pair	state	shimura correspondence pair		Iter-3 imported state (s_shimura_correspondence_pair).
s_gross_zagier_formula	theorem	Gross–Zagier formula		Iter-3 imported theorem (s_gross_zagier_formula).
s_heegner_points	axiom	heegner points		Iter-3 imported axiom (s_heegner_points).
s_heegner_height	state	heegner height		Iter-3 imported state (s_heegner_height).
s_kolyvagin_theorem	theorem	Kolyvagin's theorem		Iter-3 imported theorem (s_kolyvagin_theorem).
s_euler_system_kolyvagin	axiom	euler system kolyvagin		Iter-3 imported axiom (s_euler_system_kolyvagin).
s_anti_cyclotomic_compatible_system	state	anti cyclotomic compatible system		Iter-3 imported state (s_anti_cyclotomic_compatible_system).
s_iwasawa_main_conjecture	theorem	Iwasawa main conjecture (Mazur–Wiles, total)		Iter-3 imported theorem (s_iwasawa_main_conjecture).
s_cyclotomic_z_p_extension	axiom	cyclotomic z p extension		Iter-3 imported axiom (s_cyclotomic_z_p_extension).
s_class_group_inverse_limit_module	state	class group inverse limit module		Iter-3 imported state (s_class_group_inverse_limit_module).
s_class_number_formula	theorem	Class number formula (Dirichlet)		Iter-3 imported theorem (s_class_number_formula).
s_dedekind_zeta_function	axiom	dedekind zeta function		Iter-3 imported axiom (s_dedekind_zeta_function).
s_residue_of_dedekind_zeta	state	residue of dedekind zeta		Iter-3 imported state (s_residue_of_dedekind_zeta).
s_minkowski_lattice_theorem	theorem	Minkowski's theorem on lattices		Iter-3 imported theorem (s_minkowski_lattice_theorem).
s_lattice_in_rn	axiom	lattice in rn		Iter-3 imported axiom (s_lattice_in_rn).
s_convex_symmetric_body	axiom	convex symmetric body		Iter-3 imported axiom (s_convex_symmetric_body).
s_volume_dominance_condition	state	volume dominance condition		Iter-3 imported state (s_volume_dominance_condition).
s_minkowski_class_number_bound	theorem	Minkowski bound for class group		Iter-3 imported theorem (s_minkowski_class_number_bound).
s_class_rep_below_minkowski_bound	state	class rep below minkowski bound		Iter-3 imported state (s_class_rep_below_minkowski_bound).
s_stark_rank_0_abelian	theorem	Stark conjectures (rank-0 abelian case, proven)		Iter-3 imported theorem (s_stark_rank_0_abelian).
s_artin_l_function	axiom	artin l function		Iter-3 imported axiom (s_artin_l_function).
s_abelian_extension_of_number_field	axiom	abelian extension of number field		Iter-3 imported axiom (s_abelian_extension_of_number_field).
s_stark_unit_candidate	state	stark unit candidate		Iter-3 imported state (s_stark_unit_candidate).
s_brauer_siegel_theorem	theorem	Brauer–Siegel theorem		Iter-3 imported theorem (s_brauer_siegel_theorem).
s_log_hr_asymptotic	state	log hr asymptotic		Iter-3 imported state (s_log_hr_asymptotic).
s_heegner_class_number_one	theorem	Heegner's theorem (class number 1)		Iter-3 imported theorem (s_heegner_class_number_one).
s_imaginary_quadratic_field	axiom	imaginary quadratic field		Iter-3 imported axiom (s_imaginary_quadratic_field).
s_class_number	axiom	class number		Iter-3 imported axiom (s_class_number).
s_hilbert_class_polynomial	state	hilbert class polynomial		Iter-3 imported state (s_hilbert_class_polynomial).
s_goldfeld_effective_class_number	theorem	Goldfeld's effective lower bound on class number		Iter-3 imported theorem (s_goldfeld_effective_class_number).
s_gross_zagier_input	state	gross zagier input		Iter-3 imported state (s_gross_zagier_input).
s_faltings_product_theorem	theorem	Faltings' product theorem		Iter-3 imported theorem (s_faltings_product_theorem).
s_smooth_projective_variety_over_q	axiom	smooth projective variety over q		Iter-3 imported axiom (s_smooth_projective_variety_over_q).
s_arakelov_geometry	axiom	arakelov geometry		Iter-3 imported axiom (s_arakelov_geometry).
s_high_index_arakelov_section	state	high index arakelov section		Iter-3 imported state (s_high_index_arakelov_section).
s_vojta_height_inequality_statement	theorem	Lang–Vojta conjecture (status / Vojta height inequality)		Iter-3 imported theorem (s_vojta_height_inequality_statement).
s_naive_height_on_an	axiom	naive height on an		Iter-3 imported axiom (s_naive_height_on_an).
s_vojta_diophantine_dictionary	state	vojta diophantine dictionary		Iter-3 imported state (s_vojta_diophantine_dictionary).
s_maynard_bounded_gaps_improved	theorem	Bounded gaps between primes (Zhang / Maynard) (cite: https://en.wikipedia.org/wiki/Bounded_gaps_between_primes) — extension of `s_bounded_gaps_between_primes`		Iter-3 imported theorem (s_maynard_bounded_gaps_improved).
s_multidim_selberg_weight	state	multidim selberg weight		Iter-3 imported state (s_multidim_selberg_weight).
s_heath_brown_three_cubes_density	theorem	Heath-Brown three-cubes representation		Iter-3 imported theorem (s_heath_brown_three_cubes_density).
s_three_cubes_minor_arc_bound	state	three cubes minor arc bound		Iter-3 imported state (s_three_cubes_minor_arc_bound).
s_hooley_artin_primitive_root_grh	theorem	Hooley on Artin's primitive-root conjecture (under GRH)		Iter-3 imported theorem (s_hooley_artin_primitive_root_grh).
s_grh_for_dedekind_zeta	axiom	grh for dedekind zeta		Iter-3 imported axiom (s_grh_for_dedekind_zeta).
s_grh_chebotarev_explicit	state	grh chebotarev explicit		Iter-3 imported state (s_grh_chebotarev_explicit).
s_hardy_littlewood_singular_series_predictor	theorem	Hardy–Littlewood prime k-tuples conjecture (status / circle-method input) (cite: https://en.wikipedia.org/wiki/First_Hardy%E2%80%93Littlewood_conjecture) — note: a conjecture, but the circle-method singular-series predictor is a theorem		Iter-3 imported theorem (s_hardy_littlewood_singular_series_predictor).
s_singular_series_major_arc_count	state	singular series major arc count		Iter-3 imported state (s_singular_series_major_arc_count).
s_fundamental_lemma_of_sieve	theorem	Brun–Hooley sieve / linear sieve fundamental lemma		Iter-3 imported theorem (s_fundamental_lemma_of_sieve).
s_sieve_problem_axiomatic	axiom	sieve problem axiomatic		Iter-3 imported axiom (s_sieve_problem_axiomatic).
s_selberg_upper_lower_bounds	state	selberg upper lower bounds		Iter-3 imported state (s_selberg_upper_lower_bounds).
s_hasse_principle_failure_genus_one	theorem	Hasse principle failure / Reichardt–Lind counterexample		Iter-3 imported theorem (s_hasse_principle_failure_genus_one).
s_smooth_projective_curve_over_q	axiom	smooth projective curve over q		Iter-3 imported axiom (s_smooth_projective_curve_over_q).
s_reichardt_lind_curve_specific	state	reichardt lind curve specific		Iter-3 imported state (s_reichardt_lind_curve_specific).
s_birch_theorem_systems_of_forms	theorem	Birch's theorem on systems of forms		Iter-3 imported theorem (s_birch_theorem_systems_of_forms).
s_birch_circle_method_setup	state	birch circle method setup		Iter-3 imported state (s_birch_circle_method_setup).
s_mahler_compactness_theorem	theorem	Mahler's compactness theorem		Iter-3 imported theorem (s_mahler_compactness_theorem).
s_space_of_unimodular_lattices	axiom	space of unimodular lattices		Iter-3 imported axiom (s_space_of_unimodular_lattices).
s_mahler_pre_compact_criterion	state	mahler pre compact criterion		Iter-3 imported state (s_mahler_pre_compact_criterion).
s_skolem_mahler_lech_theorem	theorem	Skolem–Mahler–Lech theorem		Iter-3 imported theorem (s_skolem_mahler_lech_theorem).
s_linear_recurrence_sequence	axiom	linear recurrence sequence		Iter-3 imported axiom (s_linear_recurrence_sequence).
s_p_adic_analytic_extension	state	p adic analytic extension		Iter-3 imported state (s_p_adic_analytic_extension).
s_mahler_continuous_p_adic	theorem	Mahler's theorem on continuous p-adic functions		Iter-3 imported theorem (s_mahler_continuous_p_adic).
s_continuous_function_qp_to_qp	axiom	continuous function qp to qp		Iter-3 imported axiom (s_continuous_function_qp_to_qp).
s_mahler_expansion_basis	state	mahler expansion basis		Iter-3 imported state (s_mahler_expansion_basis).
s_e_transcendental	theorem	Mahler's theorem on transcendence of e (special case)		Iter-3 imported theorem (s_e_transcendental).
s_hermite_integral_for_e	state	hermite integral for e		Iter-3 imported state (s_hermite_integral_for_e).
s_pi_transcendental	theorem	Transcendence of π (Lindemann)		Iter-3 imported theorem (s_pi_transcendental).
s_prime_local_factor_geometric	state	prime local factor geometric		Iter-3 imported state (s_prime_local_factor_geometric).
s_zeta_functional_equation	theorem	Functional equation of ζ(s) (Riemann)		Iter-3 imported theorem (s_zeta_functional_equation).
s_theta_modular_inversion	state	theta modular inversion		Iter-3 imported state (s_theta_modular_inversion).
s_completed_xi_function	state	completed xi function		Iter-3 imported state (s_completed_xi_function).
s_rh_equivalents_robin	theorem	Riemann hypothesis equivalents (Robin, Mertens-type)		Iter-3 imported theorem (s_rh_equivalents_robin).
s_mertens_and_sigma_translations	state	mertens and sigma translations		Iter-3 imported state (s_mertens_and_sigma_translations).
s_riemann_von_mangoldt_explicit_formula	theorem	Riemann–von Mangoldt explicit formula		Iter-3 imported theorem (s_riemann_von_mangoldt_explicit_formula).
s_perron_for_psi	state	perron for psi		Iter-3 imported state (s_perron_for_psi).
s_de_la_vallee_poussin_zero_free_region	theorem	Zero-free region (de la Vallée Poussin)		Iter-3 imported theorem (s_de_la_vallee_poussin_zero_free_region).
s_3_4_cos_trick	state	3 4 cos trick		Iter-3 imported state (s_3_4_cos_trick).
s_riemann_pi_x_formula	theorem	Riemann's prime-counting formula		Iter-3 imported theorem (s_riemann_pi_x_formula).
s_j_x_inversion	state	j x inversion		Iter-3 imported state (s_j_x_inversion).
s_ramanujan_partition_congruences	theorem	Ramanujan's congruences		Iter-3 imported theorem (s_ramanujan_partition_congruences).
s_partition_generating_modular	state	partition generating modular		Iter-3 imported state (s_partition_generating_modular).
s_deligne_tau_bound	theorem	Deligne's bound for τ (Ramanujan conjecture)		Iter-3 imported theorem (s_deligne_tau_bound).
s_modular_galois_rep	state	modular galois rep		Iter-3 imported state (s_modular_galois_rep).
s_frobenius_eigenvalues_modular	state	frobenius eigenvalues modular		Iter-3 imported state (s_frobenius_eigenvalues_modular).
s_arthur_selberg_trace_formula	theorem	Arthur–Selberg trace formula (overview)		Iter-3 imported theorem (s_arthur_selberg_trace_formula).
s_reductive_algebraic_group	axiom	reductive algebraic group		Iter-3 imported axiom (s_reductive_algebraic_group).
s_adeles_a	axiom	adeles a		Iter-3 imported axiom (s_adeles_a).
s_automorphic_spectral_side	state	automorphic spectral side		Iter-3 imported state (s_automorphic_spectral_side).
s_geometric_side_decomposition	state	geometric side decomposition		Iter-3 imported state (s_geometric_side_decomposition).
s_langlands_functoriality_known_cases	theorem	Langlands functoriality (transferring from one group to another — known cases, e.g. Arthur for classical groups)		Iter-3 imported theorem (s_langlands_functoriality_known_cases).
s_automorphic_representation	axiom	automorphic representation		Iter-3 imported axiom (s_automorphic_representation).
s_l_group_functoriality_setup	state	l group functoriality setup		Iter-3 imported state (s_l_group_functoriality_setup).
s_local_langlands_gl_n	theorem	Local Langlands for GL_n		Iter-3 imported theorem (s_local_langlands_gl_n).
s_local_field_k	axiom	local field k		Iter-3 imported axiom (s_local_field_k).
s_weil_deligne_rep_side	state	weil deligne rep side		Iter-3 imported state (s_weil_deligne_rep_side).
s_lafforgue_global_langlands_function_field	theorem	Global Langlands for GL_n over function fields (Lafforgue)		Iter-3 imported theorem (s_lafforgue_global_langlands_function_field).
s_function_field_over_fq	axiom	function field over fq		Iter-3 imported axiom (s_function_field_over_fq).
s_shtuka_cohomology_galois_reps	state	shtuka cohomology galois reps		Iter-3 imported state (s_shtuka_cohomology_galois_reps).
s_carmichael_theorem_orders	theorem	Carmichael's theorem (orders)		Iter-3 imported theorem (s_carmichael_theorem_orders).
s_lucas_sequence	axiom	lucas sequence		Iter-3 imported axiom (s_lucas_sequence).
s_primitive_divisor_existence	state	primitive divisor existence		Iter-3 imported state (s_primitive_divisor_existence).
s_zsigmondy_theorem	theorem	Zsigmondy's theorem		Iter-3 imported theorem (s_zsigmondy_theorem).
s_cyclotomic_polynomial_phi_n	axiom	cyclotomic polynomial phi n		Iter-3 imported axiom (s_cyclotomic_polynomial_phi_n).
s_lifting_exponent_factor_decomposition	state	lifting exponent factor decomposition		Iter-3 imported state (s_lifting_exponent_factor_decomposition).
s_lame_euclidean_algorithm_bound	theorem	Lamé's theorem on Euclidean algorithm		Iter-3 imported theorem (s_lame_euclidean_algorithm_bound).
s_worst_case_consecutive_fibs	state	worst case consecutive fibs		Iter-3 imported state (s_worst_case_consecutive_fibs).
s_midy_theorem	theorem	Midy's theorem		Iter-3 imported theorem (s_midy_theorem).
s_decimal_expansion_repetend	axiom	decimal expansion repetend		Iter-3 imported axiom (s_decimal_expansion_repetend).
s_two_halves_of_repetend	state	two halves of repetend		Iter-3 imported state (s_two_halves_of_repetend).
s_beatty_theorem	theorem	Beatty's theorem		Iter-3 imported theorem (s_beatty_theorem).
s_complementary_irrationals_pair	state	complementary irrationals pair		Iter-3 imported state (s_complementary_irrationals_pair).
s_beatty_partition_count_match	state	beatty partition count match		Iter-3 imported state (s_beatty_partition_count_match).
s_davenport_schmidt_theorem	theorem	Davenport–Schmidt theorem		Iter-3 imported theorem (s_davenport_schmidt_theorem).
s_quadratic_approximation_bounds	state	quadratic approximation bounds		Iter-3 imported state (s_quadratic_approximation_bounds).
s_smyth_lower_bound_non_reciprocal	theorem	Lehmer pair / Lehmer's conjecture (Mahler measure) — Smyth's theorem on non-reciprocal polynomials		Iter-3 imported theorem (s_smyth_lower_bound_non_reciprocal).
s_mahler_measure_function	axiom	mahler measure function		Iter-3 imported axiom (s_mahler_measure_function).
s_hilbert_90	theorem	Hilbert's Theorem 90		Iter-3 imported theorem (s_hilbert_90).
s_cyclic_galois_group	axiom	cyclic galois group		Iter-3 imported axiom (s_cyclic_galois_group).
s_galois_action_on_units	state	galois action on units		Iter-3 imported state (s_galois_action_on_units).
s_local_class_field_theory	theorem	Local class field theory		Iter-3 imported theorem (s_local_class_field_theory).
s_lubin_tate_formal_group	axiom	lubin tate formal group		Iter-3 imported axiom (s_lubin_tate_formal_group).
s_local_reciprocity_isomorphism	state	local reciprocity isomorphism		Iter-3 imported state (s_local_reciprocity_isomorphism).
s_kronecker_weber_theorem	theorem	Kronecker–Weber theorem		Iter-3 imported theorem (s_kronecker_weber_theorem).
s_local_kronecker_weber_components	state	local kronecker weber components		Iter-3 imported state (s_local_kronecker_weber_components).
s_chebotarev_effective_grh	theorem	Chebotarev with effective bound under GRH (cite: https://en.wikipedia.org/wiki/Chebotarev%27s_density_theorem#Effective_version) — extends `s_chebotarev_density`		Iter-3 imported theorem (s_chebotarev_effective_grh).
s_grh_zero_density_input	state	grh zero density input		Iter-3 imported state (s_grh_zero_density_input).
s_weyl_polynomial_equidistribution	theorem	Weyl's polynomial equidistribution		Iter-3 imported theorem (s_weyl_polynomial_equidistribution).
s_weyl_differencing_reduction	state	weyl differencing reduction		Iter-3 imported state (s_weyl_differencing_reduction).
s_sarkozy_theorem_squares	theorem	Three-distance / Sárközy-on-squares		Iter-3 imported theorem (s_sarkozy_theorem_squares).
s_squares_fourier_input	state	squares fourier input		Iter-3 imported state (s_squares_fourier_input).
s_behrend_3_ap_free_set	theorem	Behrend's construction of dense 3-AP-free set		Iter-3 imported theorem (s_behrend_3_ap_free_set).
s_high_dimensional_sphere	axiom	high dimensional sphere		Iter-3 imported axiom (s_high_dimensional_sphere).
s_high_dim_sphere_lattice_layer	state	high dim sphere lattice layer		Iter-3 imported state (s_high_dim_sphere_lattice_layer).
s_bloom_sisask_roth_bound	theorem	Bloom–Sisask on Roth		Iter-3 imported theorem (s_bloom_sisask_roth_bound).
s_bohr_density_increment	state	bohr density increment		Iter-3 imported state (s_bohr_density_increment).
s_gowers_inverse_theorem	theorem	Gowers norms and inverse theorem		Iter-3 imported theorem (s_gowers_inverse_theorem).
s_finite_abelian_group_zn	axiom	finite abelian group zn		Iter-3 imported axiom (s_finite_abelian_group_zn).
s_function_f_g_to_c	axiom	function f g to c		Iter-3 imported axiom (s_function_f_g_to_c).
s_gowers_uk_definition	state	gowers uk definition		Iter-3 imported state (s_gowers_uk_definition).
s_szemeredi_trotter_incidences	theorem	Szemerédi–Trotter incidence theorem		Iter-3 imported theorem (s_szemeredi_trotter_incidences).
s_points_in_plane	axiom	points in plane		Iter-3 imported axiom (s_points_in_plane).
s_lines_in_plane	axiom	lines in plane		Iter-3 imported axiom (s_lines_in_plane).
s_cell_decomposition_of_plane	state	cell decomposition of plane		Iter-3 imported state (s_cell_decomposition_of_plane).
s_guth_katz_distinct_distances	theorem	Erdős distinct-distances (Guth–Katz)		Iter-3 imported theorem (s_guth_katz_distinct_distances).
s_lines_in_r3_as_ruled_surfaces	axiom	lines in r3 as ruled surfaces		Iter-3 imported axiom (s_lines_in_r3_as_ruled_surfaces).
s_elekes_sharir_lines_in_r3	state	elekes sharir lines in r3		Iter-3 imported state (s_elekes_sharir_lines_in_r3).
s_erdos_szemeredi_sum_product	theorem	Erdős–Szemerédi sum–product		Iter-3 imported theorem (s_erdos_szemeredi_sum_product).
s_sum_product_basic_bound	state	sum product basic bound		Iter-3 imported state (s_sum_product_basic_bound).
s_three_distance_theorem	theorem	Three-distance theorem (Steinhaus, distinct from three-gap)		Iter-3 imported theorem (s_three_distance_theorem).
s_circle_orbit_partition	state	circle orbit partition		Iter-3 imported state (s_circle_orbit_partition).
s_khintchine_diophantine_theorem	theorem	Khintchine's theorem on Diophantine approximation		Iter-3 imported theorem (s_khintchine_diophantine_theorem).
s_random_alpha_uniform	state	random alpha uniform		Iter-3 imported state (s_random_alpha_uniform).
s_davenport_erdos_density_multiples	theorem	Davenport–Erdős theorem on multiples		Iter-3 imported theorem (s_davenport_erdos_density_multiples).
s_density_equality_for_multiples_sets	state	density equality for multiples sets		Iter-3 imported state (s_density_equality_for_multiples_sets).
s_cauchy_lipschitz_local_existence	theorem	Cauchy–Lipschitz local existence		Iter-3 imported theorem (s_cauchy_lipschitz_local_existence).
s_lipschitz_vector_field	axiom	lipschitz vector field		Iter-3 imported axiom (s_lipschitz_vector_field).
s_initial_condition	axiom	initial condition		Iter-3 imported axiom (s_initial_condition).
s_canonical_ivp	state	canonical ivp		Iter-3 imported state (s_canonical_ivp).
s_local_unique_solution	state	local unique solution		Iter-3 imported state (s_local_unique_solution).
s_gronwall_inequality	theorem	Gronwall's inequality (differential form)		Iter-3 imported theorem (s_gronwall_inequality).
s_nonneg_continuous_function	axiom	nonneg continuous function		Iter-3 imported axiom (s_nonneg_continuous_function).
s_linear_integral_majorant	axiom	linear integral majorant		Iter-3 imported axiom (s_linear_integral_majorant).
s_multiplied_inequality_for_integrating_factor	state	multiplied inequality for integrating factor		Iter-3 imported state (s_multiplied_inequality_for_integrating_factor).
s_monotone_decreasing_witness	state	monotone decreasing witness		Iter-3 imported state (s_monotone_decreasing_witness).
s_ode_comparison_principle	theorem	Comparison principle for scalar ODE		Iter-3 imported theorem (s_ode_comparison_principle).
s_scalar_ode_with_monotone_rhs	axiom	scalar ode with monotone rhs		Iter-3 imported axiom (s_scalar_ode_with_monotone_rhs).
s_two_initial_values_ordered	axiom	two initial values ordered		Iter-3 imported axiom (s_two_initial_values_ordered).
s_difference_satisfies_linear_majorant	state	difference satisfies linear majorant		Iter-3 imported state (s_difference_satisfies_linear_majorant).
s_sign_persistence_of_difference	state	sign persistence of difference		Iter-3 imported state (s_sign_persistence_of_difference).
s_sturm_comparison_theorem	theorem	Sturm comparison theorem		Iter-3 imported theorem (s_sturm_comparison_theorem).
s_two_sturm_liouville_problems	axiom	two sturm liouville problems		Iter-3 imported axiom (s_two_sturm_liouville_problems).
s_potential_ordering	axiom	potential ordering		Iter-3 imported axiom (s_potential_ordering).
s_wronskian_witness	state	wronskian witness		Iter-3 imported state (s_wronskian_witness).
s_interleaving_witness	state	interleaving witness		Iter-3 imported state (s_interleaving_witness).
s_sturm_liouville_eigenvalue_theorem	theorem	Sturm–Liouville eigenvalue theorem		Iter-3 imported theorem (s_sturm_liouville_eigenvalue_theorem).
s_regular_sturm_liouville_problem	axiom	regular sturm liouville problem		Iter-3 imported axiom (s_regular_sturm_liouville_problem).
s_self_adjoint_operator_L	state	self adjoint operator L		Iter-3 imported state (s_self_adjoint_operator_L).
s_compact_self_adjoint_inverse	state	compact self adjoint inverse		Iter-3 imported state (s_compact_self_adjoint_inverse).
s_orthonormal_eigenbasis_with_discrete_spectrum	state	orthonormal eigenbasis with discrete spectrum		Iter-3 imported state (s_orthonormal_eigenbasis_with_discrete_spectrum).
s_stable_manifold_theorem	theorem	Stable manifold theorem		Iter-3 imported theorem (s_stable_manifold_theorem).
s_hyperbolic_equilibrium_of_smooth_vector_field	axiom	hyperbolic equilibrium of smooth vector field		Iter-3 imported axiom (s_hyperbolic_equilibrium_of_smooth_vector_field).
s_stable_subspace_E_s	axiom	stable subspace E s		Iter-3 imported axiom (s_stable_subspace_E_s).
s_dulac_criterion	theorem	Dulac's criterion		Iter-3 imported theorem (s_dulac_criterion).
s_planar_smooth_flow	axiom	planar smooth flow		Iter-3 imported axiom (s_planar_smooth_flow).
s_simply_connected_region	axiom	simply connected region		Iter-3 imported axiom (s_simply_connected_region).
s_signed_divergence_of_BX	state	signed divergence of BX		Iter-3 imported state (s_signed_divergence_of_BX).
s_green_theorem_contradiction	state	green theorem contradiction		Iter-3 imported state (s_green_theorem_contradiction).
s_mather_theorem_twist_maps	theorem	Mather's theorem for twist maps		Iter-3 imported theorem (s_mather_theorem_twist_maps).
s_monotone_twist_map_on_cylinder	axiom	monotone twist map on cylinder		Iter-3 imported axiom (s_monotone_twist_map_on_cylinder).
s_generating_function	axiom	generating function		Iter-3 imported axiom (s_generating_function).
s_action_functional_on_configurations	state	action functional on configurations		Iter-3 imported state (s_action_functional_on_configurations).
s_minimal_configurations	state	minimal configurations		Iter-3 imported state (s_minimal_configurations).
s_aubry_mather_invariant_set	state	aubry mather invariant set		Iter-3 imported state (s_aubry_mather_invariant_set).
s_liouville_volume_preservation	theorem	Liouville's theorem on Hamiltonian flow (volume preservation)		Iter-3 imported theorem (s_liouville_volume_preservation).
s_hamiltonian_vector_field	axiom	hamiltonian vector field		Iter-3 imported axiom (s_hamiltonian_vector_field).
s_symplectic_form_on_phase_space	axiom	symplectic form on phase space		Iter-3 imported axiom (s_symplectic_form_on_phase_space).
s_zero_lie_derivative_of_symplectic_form	state	zero lie derivative of symplectic form		Iter-3 imported state (s_zero_lie_derivative_of_symplectic_form).
s_cauchy_kowalevskaya_theorem	theorem	Cauchy–Kowalevskaya theorem		Iter-3 imported theorem (s_cauchy_kowalevskaya_theorem).
s_real_analytic_PDE_system	axiom	real analytic PDE system		Iter-3 imported axiom (s_real_analytic_PDE_system).
s_non_characteristic_cauchy_data	axiom	non characteristic cauchy data		Iter-3 imported axiom (s_non_characteristic_cauchy_data).
s_normal_form_analytic_pde	state	normal form analytic pde		Iter-3 imported state (s_normal_form_analytic_pde).
s_formal_power_series_solution	state	formal power series solution		Iter-3 imported state (s_formal_power_series_solution).
s_convergent_power_series_solution	state	convergent power series solution		Iter-3 imported state (s_convergent_power_series_solution).
s_holmgren_uniqueness_theorem	theorem	Holmgren's uniqueness theorem		Iter-3 imported theorem (s_holmgren_uniqueness_theorem).
s_linear_pde_with_analytic_coefficients	axiom	linear pde with analytic coefficients		Iter-3 imported axiom (s_linear_pde_with_analytic_coefficients).
s_non_characteristic_hypersurface	axiom	non characteristic hypersurface		Iter-3 imported axiom (s_non_characteristic_hypersurface).
s_formal_adjoint_operator_L_star	state	formal adjoint operator L star		Iter-3 imported state (s_formal_adjoint_operator_L_star).
s_dense_family_of_analytic_test_functions	state	dense family of analytic test functions		Iter-3 imported state (s_dense_family_of_analytic_test_functions).
s_ck_holmgren_combined	theorem	Cauchy–Kowalevskaya–Holmgren combined statement		Iter-3 imported theorem (s_ck_holmgren_combined).
s_analytic_local_existence_witness	state	analytic local existence witness		Iter-3 imported state (s_analytic_local_existence_witness).
s_C_infty_uniqueness_witness	state	C infty uniqueness witness		Iter-3 imported state (s_C_infty_uniqueness_witness).
s_hadamard_well_posedness_criterion	theorem	Hadamard well-posedness criterion		Iter-3 imported theorem (s_hadamard_well_posedness_criterion).
s_pde_with_data	axiom	pde with data		Iter-3 imported axiom (s_pde_with_data).
s_continuous_dependence_norm	axiom	continuous dependence norm		Iter-3 imported axiom (s_continuous_dependence_norm).
s_three_axiom_definition_of_well_posedness	state	three axiom definition of well posedness		Iter-3 imported state (s_three_axiom_definition_of_well_posedness).
s_separation_of_well_posed_from_ill_posed_PDE	state	separation of well posed from ill posed PDE		Iter-3 imported state (s_separation_of_well_posed_from_ill_posed_PDE).
s_backward_heat_ill_posed	theorem	Hadamard ill-posedness of backward heat		Iter-3 imported theorem (s_backward_heat_ill_posed).
s_high_frequency_fourier_modes	axiom	high frequency fourier modes		Iter-3 imported axiom (s_high_frequency_fourier_modes).
s_mode_decay_factor_exp_minus_k2_t	state	mode decay factor exp minus k2 t		Iter-3 imported state (s_mode_decay_factor_exp_minus_k2_t).
s_mode_growth_factor_exp_plus_k2_t_backward	state	mode growth factor exp plus k2 t backward		Iter-3 imported state (s_mode_growth_factor_exp_plus_k2_t_backward).
s_weak_maximum_principle_laplace	theorem	Maximum principle for Laplace equation (weak)		Iter-3 imported theorem (s_weak_maximum_principle_laplace).
s_harmonic_function_on_bounded_domain	axiom	harmonic function on bounded domain		Iter-3 imported axiom (s_harmonic_function_on_bounded_domain).
s_continuity_to_boundary	axiom	continuity to boundary		Iter-3 imported axiom (s_continuity_to_boundary).
s_strict_subharmonic_perturbation	state	strict subharmonic perturbation		Iter-3 imported state (s_strict_subharmonic_perturbation).
s_no_interior_max_for_perturbation	state	no interior max for perturbation		Iter-3 imported state (s_no_interior_max_for_perturbation).
s_hopf_strong_maximum_principle	theorem	Hopf strong maximum principle		Iter-3 imported theorem (s_hopf_strong_maximum_principle).
s_elliptic_operator_L	axiom	elliptic operator L		Iter-3 imported axiom (s_elliptic_operator_L).
s_connected_domain	axiom	connected domain		Iter-3 imported axiom (s_connected_domain).
s_hopf_barrier_function	state	hopf barrier function		Iter-3 imported state (s_hopf_barrier_function).
s_propagation_of_max_to_boundary_of_ball	state	propagation of max to boundary of ball		Iter-3 imported state (s_propagation_of_max_to_boundary_of_ball).
s_hopf_boundary_point_lemma	theorem	Hopf boundary point lemma		Iter-3 imported theorem (s_hopf_boundary_point_lemma).
s_C2_domain_with_interior_ball_condition	axiom	C2 domain with interior ball condition		Iter-3 imported axiom (s_C2_domain_with_interior_ball_condition).
s_radial_comparison_function_w	state	radial comparison function w		Iter-3 imported state (s_radial_comparison_function_w).
s_subsolution_inside_ball	state	subsolution inside ball		Iter-3 imported state (s_subsolution_inside_ball).
s_mean_value_property_harmonic	theorem	Mean value property for harmonic functions		Iter-3 imported theorem (s_mean_value_property_harmonic).
s_ball_in_R_n	axiom	ball in R n		Iter-3 imported axiom (s_ball_in_R_n).
s_spherical_average_function_phi_of_r	state	spherical average function phi of r		Iter-3 imported state (s_spherical_average_function_phi_of_r).
s_constant_spherical_average	state	constant spherical average		Iter-3 imported state (s_constant_spherical_average).
s_liouville_harmonic_theorem	theorem	Liouville's theorem for harmonic functions		Iter-3 imported theorem (s_liouville_harmonic_theorem).
s_harmonic_function_on_R_n	axiom	harmonic function on R n		Iter-3 imported axiom (s_harmonic_function_on_R_n).
s_uniform_boundedness	axiom	uniform boundedness		Iter-3 imported axiom (s_uniform_boundedness).
s_two_ball_mean_value_comparison	state	two ball mean value comparison		Iter-3 imported state (s_two_ball_mean_value_comparison).
s_zero_difference_between_two_points	state	zero difference between two points		Iter-3 imported state (s_zero_difference_between_two_points).
s_harnack_inequality_classical	theorem	Harnack's inequality (classical)		Iter-3 imported theorem (s_harnack_inequality_classical).
s_nonnegative_harmonic_function	axiom	nonnegative harmonic function		Iter-3 imported axiom (s_nonnegative_harmonic_function).
s_compact_subset_of_open_set	axiom	compact subset of open set		Iter-3 imported axiom (s_compact_subset_of_open_set).
s_poisson_kernel_pointwise_bound	state	poisson kernel pointwise bound		Iter-3 imported state (s_poisson_kernel_pointwise_bound).
s_chain_of_local_ratio_estimates	state	chain of local ratio estimates		Iter-3 imported state (s_chain_of_local_ratio_estimates).
s_de_giorgi_nash_moser_regularity	theorem	De Giorgi–Nash–Moser regularity theorem		Iter-3 imported theorem (s_de_giorgi_nash_moser_regularity).
s_divergence_form_elliptic_with_L_infty_coefficients	axiom	divergence form elliptic with L infty coefficients		Iter-3 imported axiom (s_divergence_form_elliptic_with_L_infty_coefficients).
s_weak_solution_in_H_1	axiom	weak solution in H 1		Iter-3 imported axiom (s_weak_solution_in_H_1).
s_caccioppoli_inequality	state	caccioppoli inequality		Iter-3 imported state (s_caccioppoli_inequality).
s_local_boundedness_L_infty	state	local boundedness L infty		Iter-3 imported state (s_local_boundedness_L_infty).
s_parabolic_harnack_for_general_coefficients	state	parabolic harnack for general coefficients		Iter-3 imported state (s_parabolic_harnack_for_general_coefficients).
s_schauder_estimates	theorem	Schauder estimates		Iter-3 imported theorem (s_schauder_estimates).
s_elliptic_operator_L_with_C_alpha_coefficients	axiom	elliptic operator L with C alpha coefficients		Iter-3 imported axiom (s_elliptic_operator_L_with_C_alpha_coefficients).
s_holder_space_C_2_alpha	axiom	holder space C 2 alpha		Iter-3 imported axiom (s_holder_space_C_2_alpha).
s_frozen_constant_coefficient_problem	state	frozen constant coefficient problem		Iter-3 imported state (s_frozen_constant_coefficient_problem).
s_constant_coefficient_holder_estimate	state	constant coefficient holder estimate		Iter-3 imported state (s_constant_coefficient_holder_estimate).
s_local_schauder_estimate	state	local schauder estimate		Iter-3 imported state (s_local_schauder_estimate).
s_calderon_zygmund_Lp_regularity	theorem	Calderón–Zygmund Lp elliptic regularity		Iter-3 imported theorem (s_calderon_zygmund_Lp_regularity).
s_elliptic_operator_L_constant_coeff	axiom	elliptic operator L constant coeff		Iter-3 imported axiom (s_elliptic_operator_L_constant_coeff).
s_singular_integral_operator_with_calderon_zygmund_kernel	axiom	singular integral operator with calderon zygmund kernel		Iter-3 imported axiom (s_singular_integral_operator_with_calderon_zygmund_kernel).
s_L2_boundedness_of_singular_integral	state	L2 boundedness of singular integral		Iter-3 imported state (s_L2_boundedness_of_singular_integral).
s_cz_good_lambda_decomposition	state	cz good lambda decomposition		Iter-3 imported state (s_cz_good_lambda_decomposition).
s_Lp_boundedness_of_singular_integral	state	Lp boundedness of singular integral		Iter-3 imported state (s_Lp_boundedness_of_singular_integral).
s_krylov_safonov_theorem	theorem	Krylov–Safonov Harnack for nondivergence-form		Iter-3 imported theorem (s_krylov_safonov_theorem).
s_nondivergence_elliptic_with_L_infty_coeff	axiom	nondivergence elliptic with L infty coeff		Iter-3 imported axiom (s_nondivergence_elliptic_with_L_infty_coeff).
s_measurable_coefficient_uniform_ellipticity	axiom	measurable coefficient uniform ellipticity		Iter-3 imported axiom (s_measurable_coefficient_uniform_ellipticity).
s_abp_maximum_principle	state	abp maximum principle		Iter-3 imported state (s_abp_maximum_principle).
s_measure_density_lemma_nondivergence	state	measure density lemma nondivergence		Iter-3 imported state (s_measure_density_lemma_nondivergence).
s_harnack_inequality_nondivergence	state	harnack inequality nondivergence		Iter-3 imported state (s_harnack_inequality_nondivergence).
s_fredholm_alternative_elliptic	theorem	Fredholm alternative for elliptic operators		Iter-3 imported theorem (s_fredholm_alternative_elliptic).
s_elliptic_operator_L_self_adjoint	axiom	elliptic operator L self adjoint		Iter-3 imported axiom (s_elliptic_operator_L_self_adjoint).
s_compact_resolvent	axiom	compact resolvent		Iter-3 imported axiom (s_compact_resolvent).
s_riesz_schauder_compact_operator_framework	state	riesz schauder compact operator framework		Iter-3 imported state (s_riesz_schauder_compact_operator_framework).
s_orthogonality_range_perp_kernel_adjoint	state	orthogonality range perp kernel adjoint		Iter-3 imported state (s_orthogonality_range_perp_kernel_adjoint).
s_lax_milgram_theorem	theorem	Lax–Milgram theorem		Iter-3 imported theorem (s_lax_milgram_theorem).
s_coercive_bilinear_form_on_hilbert_space	axiom	coercive bilinear form on hilbert space		Iter-3 imported axiom (s_coercive_bilinear_form_on_hilbert_space).
s_bounded_linear_functional	axiom	bounded linear functional		Iter-3 imported axiom (s_bounded_linear_functional).
s_unique_solution_of_Au_equals_f	state	unique solution of Au equals f		Iter-3 imported state (s_unique_solution_of_Au_equals_f).
s_weak_solution_dirichlet_existence	theorem	Existence of weak solutions for Dirichlet problem		Iter-3 imported theorem (s_weak_solution_dirichlet_existence).
s_uniformly_elliptic_operator_in_divergence_form	axiom	uniformly elliptic operator in divergence form		Iter-3 imported axiom (s_uniformly_elliptic_operator_in_divergence_form).
s_H_1_0_sobolev_space	axiom	H 1 0 sobolev space		Iter-3 imported axiom (s_H_1_0_sobolev_space).
s_bilinear_form_a_continuous_coercive	state	bilinear form a continuous coercive		Iter-3 imported state (s_bilinear_form_a_continuous_coercive).
s_weak_solution_in_H_1_0	state	weak solution in H 1 0		Iter-3 imported state (s_weak_solution_in_H_1_0).
s_perron_method_for_dirichlet_problem	theorem	Perron's method for Laplace–Dirichlet		Iter-3 imported theorem (s_perron_method_for_dirichlet_problem).
s_continuous_boundary_data	axiom	continuous boundary data		Iter-3 imported axiom (s_continuous_boundary_data).
s_bounded_domain_in_R_n	axiom	bounded domain in R n		Iter-3 imported axiom (s_bounded_domain_in_R_n).
s_family_of_subsolutions_S_phi	state	family of subsolutions S phi		Iter-3 imported state (s_family_of_subsolutions_S_phi).
s_perron_harmonic_envelope_u_star	state	perron harmonic envelope u star		Iter-3 imported state (s_perron_harmonic_envelope_u_star).
s_wiener_criterion	theorem	Wiener criterion for regular boundary points		Iter-3 imported theorem (s_wiener_criterion).
s_potential_theory_capacity	axiom	potential theory capacity		Iter-3 imported axiom (s_potential_theory_capacity).
s_dyadic_capacity_series	state	dyadic capacity series		Iter-3 imported state (s_dyadic_capacity_series).
s_harmonic_measure_representation	theorem	Riesz representation for harmonic measure		Iter-3 imported theorem (s_harmonic_measure_representation).
s_linear_functional_phi_to_u_phi_at_x	state	linear functional phi to u phi at x		Iter-3 imported state (s_linear_functional_phi_to_u_phi_at_x).
s_hormander_L2_dbar_existence	theorem	Hörmander L² existence for ∂̄		Iter-3 imported theorem (s_hormander_L2_dbar_existence).
s_pseudoconvex_domain_in_C_n	axiom	pseudoconvex domain in C n		Iter-3 imported axiom (s_pseudoconvex_domain_in_C_n).
s_dbar_complex	axiom	dbar complex		Iter-3 imported axiom (s_dbar_complex).
s_weighted_L2_space_with_phi	state	weighted L2 space with phi		Iter-3 imported state (s_weighted_L2_space_with_phi).
s_a_priori_L2_estimate_for_dbar_star	state	a priori L2 estimate for dbar star		Iter-3 imported state (s_a_priori_L2_estimate_for_dbar_star).
s_solvability_of_dbar_u_equals_f	state	solvability of dbar u equals f		Iter-3 imported state (s_solvability_of_dbar_u_equals_f).
s_hormander_sum_of_squares_theorem	theorem	Hörmander sum-of-squares hypoellipticity		Iter-3 imported theorem (s_hormander_sum_of_squares_theorem).
s_smooth_vector_fields_X_1_X_k	axiom	smooth vector fields X 1 X k		Iter-3 imported axiom (s_smooth_vector_fields_X_1_X_k).
s_hormander_bracket_generating_condition	axiom	hormander bracket generating condition		Iter-3 imported axiom (s_hormander_bracket_generating_condition).
s_subelliptic_a_priori_estimate	state	subelliptic a priori estimate		Iter-3 imported state (s_subelliptic_a_priori_estimate).
s_microlocal_smoothing_witness	state	microlocal smoothing witness		Iter-3 imported state (s_microlocal_smoothing_witness).
s_bony_paradifferential_calculus	theorem	Bony's paradifferential calculus theorem		Iter-3 imported theorem (s_bony_paradifferential_calculus).
s_littlewood_paley_decomposition	axiom	littlewood paley decomposition		Iter-3 imported axiom (s_littlewood_paley_decomposition).
s_product_of_two_distributions	axiom	product of two distributions		Iter-3 imported axiom (s_product_of_two_distributions).
s_paraproduct_decomposition_T_u_T_v_R	state	paraproduct decomposition T u T v R		Iter-3 imported state (s_paraproduct_decomposition_T_u_T_v_R).
s_paraproduct_symbol_algebra	state	paraproduct symbol algebra		Iter-3 imported state (s_paraproduct_symbol_algebra).
s_coifman_meyer_theorem	theorem	Coifman–Meyer multilinear multiplier theorem		Iter-3 imported theorem (s_coifman_meyer_theorem).
s_multilinear_fourier_multiplier_symbol	axiom	multilinear fourier multiplier symbol		Iter-3 imported axiom (s_multilinear_fourier_multiplier_symbol).
s_calderon_zygmund_condition_on_symbol	axiom	calderon zygmund condition on symbol		Iter-3 imported axiom (s_calderon_zygmund_condition_on_symbol).
s_multilinear_operator_T_m	state	multilinear operator T m		Iter-3 imported state (s_multilinear_operator_T_m).
s_multilinear_calderon_zygmund_kernel_estimate	state	multilinear calderon zygmund kernel estimate		Iter-3 imported state (s_multilinear_calderon_zygmund_kernel_estimate).
s_gagliardo_nirenberg_interpolation	theorem	Gagliardo–Nirenberg interpolation inequality		Iter-3 imported theorem (s_gagliardo_nirenberg_interpolation).
s_lebesgue_norms_at_two_exponents	axiom	lebesgue norms at two exponents		Iter-3 imported axiom (s_lebesgue_norms_at_two_exponents).
s_scaling_invariance_constraint	state	scaling invariance constraint		Iter-3 imported state (s_scaling_invariance_constraint).
s_two_parameter_interpolation_inequality	state	two parameter interpolation inequality		Iter-3 imported state (s_two_parameter_interpolation_inequality).
s_riesz_potential_kernel	axiom	riesz potential kernel		Iter-3 imported axiom (s_riesz_potential_kernel).
s_lebesgue_spaces_L_p_L_q	axiom	lebesgue spaces L p L q		Iter-3 imported axiom (s_lebesgue_spaces_L_p_L_q).
s_dyadic_layer_decomposition	state	dyadic layer decomposition		Iter-3 imported state (s_dyadic_layer_decomposition).
s_weak_type_estimate_at_endpoints	state	weak type estimate at endpoints		Iter-3 imported state (s_weak_type_estimate_at_endpoints).
s_aubin_talenti_sharp_sobolev	theorem	Aubin–Talenti sharp Sobolev constant		Iter-3 imported theorem (s_aubin_talenti_sharp_sobolev).
s_radial_decreasing_rearrangement	axiom	radial decreasing rearrangement		Iter-3 imported axiom (s_radial_decreasing_rearrangement).
s_reduction_to_radial_decreasing_functions	state	reduction to radial decreasing functions		Iter-3 imported state (s_reduction_to_radial_decreasing_functions).
s_talenti_bubble_extremal_functions	state	talenti bubble extremal functions		Iter-3 imported state (s_talenti_bubble_extremal_functions).
s_brezis_lieb_lemma	theorem	Brezis–Lieb lemma		Iter-3 imported theorem (s_brezis_lieb_lemma).
s_a_e_convergent_bounded_L_p_sequence	axiom	a e convergent bounded L p sequence		Iter-3 imported axiom (s_a_e_convergent_bounded_L_p_sequence).
s_lebesgue_space_L_p	axiom	lebesgue space L p		Iter-3 imported axiom (s_lebesgue_space_L_p).
s_pointwise_split_with_w_n_to_0_ae	state	pointwise split with w n to 0 ae		Iter-3 imported state (s_pointwise_split_with_w_n_to_0_ae).
s_norm_convergence_of_difference_term	state	norm convergence of difference term		Iter-3 imported state (s_norm_convergence_of_difference_term).
s_concentration_compactness_principle	theorem	Lions concentration-compactness principle		Iter-3 imported theorem (s_concentration_compactness_principle).
s_bounded_sequence_in_sobolev_space	axiom	bounded sequence in sobolev space		Iter-3 imported axiom (s_bounded_sequence_in_sobolev_space).
s_levy_concentration_function	axiom	levy concentration function		Iter-3 imported axiom (s_levy_concentration_function).
s_levy_concentration_function_witness	state	levy concentration function witness		Iter-3 imported state (s_levy_concentration_function_witness).
s_concentration_trichotomy	state	concentration trichotomy		Iter-3 imported state (s_concentration_trichotomy).
s_trace_theorem_sobolev	theorem	Trace theorem for Sobolev spaces		Iter-3 imported theorem (s_trace_theorem_sobolev).
s_sobolev_space_W_1_p_on_domain	axiom	sobolev space W 1 p on domain		Iter-3 imported axiom (s_sobolev_space_W_1_p_on_domain).
s_lipschitz_boundary	axiom	lipschitz boundary		Iter-3 imported axiom (s_lipschitz_boundary).
s_trace_on_smooth_dense_subset	state	trace on smooth dense subset		Iter-3 imported state (s_trace_on_smooth_dense_subset).
s_trace_operator_to_W_1_minus_1_p_p	state	trace operator to W 1 minus 1 p p		Iter-3 imported state (s_trace_operator_to_W_1_minus_1_p_p).
s_poincare_inequality	theorem	Poincaré inequality on bounded domain		Iter-3 imported theorem (s_poincare_inequality).
s_sobolev_space_W_1_p_zero_boundary_value	axiom	sobolev space W 1 p zero boundary value		Iter-3 imported axiom (s_sobolev_space_W_1_p_zero_boundary_value).
s_pointwise_bound_in_terms_of_gradient	state	pointwise bound in terms of gradient		Iter-3 imported state (s_pointwise_bound_in_terms_of_gradient).
s_korn_inequality	theorem	Korn's inequality		Iter-3 imported theorem (s_korn_inequality).
s_vector_valued_sobolev_space	axiom	vector valued sobolev space		Iter-3 imported axiom (s_vector_valued_sobolev_space).
s_korn_identity_at_full_space	state	korn identity at full space		Iter-3 imported state (s_korn_identity_at_full_space).
s_korn_on_bounded_domain	state	korn on bounded domain		Iter-3 imported state (s_korn_on_bounded_domain).
s_strichartz_estimates_schrodinger	theorem	Strichartz estimates (Schrödinger)		Iter-3 imported theorem (s_strichartz_estimates_schrodinger).
s_free_schrodinger_propagator_e_it_delta	axiom	free schrodinger propagator e it delta		Iter-3 imported axiom (s_free_schrodinger_propagator_e_it_delta).
s_admissible_pair_q_r	axiom	admissible pair q r		Iter-3 imported axiom (s_admissible_pair_q_r).
s_dispersive_L_infty_to_L_1_estimate	state	dispersive L infty to L 1 estimate		Iter-3 imported state (s_dispersive_L_infty_to_L_1_estimate).
s_TT_star_bilinear_estimate	state	TT star bilinear estimate		Iter-3 imported state (s_TT_star_bilinear_estimate).
s_space_time_L_q_t_L_r_x_bound	state	space time L q t L r x bound		Iter-3 imported state (s_space_time_L_q_t_L_r_x_bound).
s_strichartz_estimates_wave	theorem	Strichartz estimates (wave)		Iter-3 imported theorem (s_strichartz_estimates_wave).
s_free_wave_propagator_cos_t_sqrt_minus_delta	axiom	free wave propagator cos t sqrt minus delta		Iter-3 imported axiom (s_free_wave_propagator_cos_t_sqrt_minus_delta).
s_wave_admissible_pair	axiom	wave admissible pair		Iter-3 imported axiom (s_wave_admissible_pair).
s_wave_dispersive_decay_estimate	state	wave dispersive decay estimate		Iter-3 imported state (s_wave_dispersive_decay_estimate).
s_wave_TT_star_bilinear_estimate	state	wave TT star bilinear estimate		Iter-3 imported state (s_wave_TT_star_bilinear_estimate).
s_kato_strichartz_non_trapping	theorem	Kato non-trapping Strichartz		Iter-3 imported theorem (s_kato_strichartz_non_trapping).
s_non_trapping_metric_perturbation	axiom	non trapping metric perturbation		Iter-3 imported axiom (s_non_trapping_metric_perturbation).
s_local_smoothing_estimate_kato_type	state	local smoothing estimate kato type		Iter-3 imported state (s_local_smoothing_estimate_kato_type).
s_perturbed_strichartz	state	perturbed strichartz		Iter-3 imported state (s_perturbed_strichartz).
s_nls_local_well_posedness	theorem	Local well-posedness of nonlinear Schrödinger (NLS)		Iter-3 imported theorem (s_nls_local_well_posedness).
s_duhamel_form_of_NLS	state	duhamel form of NLS		Iter-3 imported state (s_duhamel_form_of_NLS).
s_unique_local_solution_in_strichartz_space	state	unique local solution in strichartz space		Iter-3 imported state (s_unique_local_solution_in_strichartz_space).
s_nls_global_subcritical	theorem	Global existence of energy-subcritical NLS via energy conservation		Iter-3 imported theorem (s_nls_global_subcritical).
s_defocusing_subcritical_nonlinearity	axiom	defocusing subcritical nonlinearity		Iter-3 imported axiom (s_defocusing_subcritical_nonlinearity).
s_energy_conservation_for_NLS	state	energy conservation for NLS		Iter-3 imported state (s_energy_conservation_for_NLS).
s_a_priori_global_H_1_bound	state	a priori global H 1 bound		Iter-3 imported state (s_a_priori_global_H_1_bound).
s_lax_pair_for_KdV	theorem	Lax pair / integrable structure for KdV		Iter-3 imported theorem (s_lax_pair_for_KdV).
s_schrodinger_operator_L_with_potential_u	axiom	schrodinger operator L with potential u		Iter-3 imported axiom (s_schrodinger_operator_L_with_potential_u).
s_lax_pair_LP	state	lax pair LP		Iter-3 imported state (s_lax_pair_LP).
s_isospectral_evolution_of_L	state	isospectral evolution of L		Iter-3 imported state (s_isospectral_evolution_of_L).
s_inverse_scattering_transform_KdV	theorem	Inverse scattering transform for KdV		Iter-3 imported theorem (s_inverse_scattering_transform_KdV).
s_decaying_initial_data	axiom	decaying initial data		Iter-3 imported axiom (s_decaying_initial_data).
s_time_evolution_of_scattering_data	state	time evolution of scattering data		Iter-3 imported state (s_time_evolution_of_scattering_data).
s_potential_reconstruction_via_GLM	state	potential reconstruction via GLM		Iter-3 imported state (s_potential_reconstruction_via_GLM).
s_kdv_local_well_posedness_bourgain	theorem	Local well-posedness of KdV in low regularity (Bourgain spaces)		Iter-3 imported theorem (s_kdv_local_well_posedness_bourgain).
s_bourgain_space_X_s_b	axiom	bourgain space X s b		Iter-3 imported axiom (s_bourgain_space_X_s_b).
s_dyadic_bilinear_estimates_in_X_s_b	state	dyadic bilinear estimates in X s b		Iter-3 imported state (s_dyadic_bilinear_estimates_in_X_s_b).
s_duhamel_form_in_bourgain_space	state	duhamel form in bourgain space		Iter-3 imported state (s_duhamel_form_in_bourgain_space).
s_beale_kato_majda_criterion	theorem	Beale–Kato–Majda blow-up criterion		Iter-3 imported theorem (s_beale_kato_majda_criterion).
s_local_smooth_solution_of_euler	axiom	local smooth solution of euler		Iter-3 imported axiom (s_local_smooth_solution_of_euler).
s_vorticity_omega	axiom	vorticity omega		Iter-3 imported axiom (s_vorticity_omega).
s_log_sobolev_bound_on_velocity_gradient	state	log sobolev bound on velocity gradient		Iter-3 imported state (s_log_sobolev_bound_on_velocity_gradient).
s_double_exponential_growth_bound	state	double exponential growth bound		Iter-3 imported state (s_double_exponential_growth_bound).
s_local_existence_euler_3d	theorem	Local existence for incompressible Euler in 3D		Iter-3 imported theorem (s_local_existence_euler_3d).
s_incompressible_euler_equations	axiom	incompressible euler equations		Iter-3 imported axiom (s_incompressible_euler_equations).
s_sobolev_space_H_s_with_s_gt_5_2	axiom	sobolev space H s with s gt 5 2		Iter-3 imported axiom (s_sobolev_space_H_s_with_s_gt_5_2).
s_helmholtz_projected_euler_equation	state	helmholtz projected euler equation		Iter-3 imported state (s_helmholtz_projected_euler_equation).
s_approximate_smooth_solution	state	approximate smooth solution		Iter-3 imported state (s_approximate_smooth_solution).
s_uniform_H_s_bound_short_time	state	uniform H s bound short time		Iter-3 imported state (s_uniform_H_s_bound_short_time).
s_leray_hopf_weak_solutions	theorem	Leray–Hopf weak solutions of Navier–Stokes		Iter-3 imported theorem (s_leray_hopf_weak_solutions).
s_navier_stokes_equations	axiom	navier stokes equations		Iter-3 imported axiom (s_navier_stokes_equations).
s_L2_initial_data	axiom	L2 initial data		Iter-3 imported axiom (s_L2_initial_data).
s_galerkin_ode_for_first_N_modes	state	galerkin ode for first N modes		Iter-3 imported state (s_galerkin_ode_for_first_N_modes).
s_uniform_energy_bound_on_galerkin	state	uniform energy bound on galerkin		Iter-3 imported state (s_uniform_energy_bound_on_galerkin).
s_convergent_subsequence_to_weak_solution	state	convergent subsequence to weak solution		Iter-3 imported state (s_convergent_subsequence_to_weak_solution).
s_ckn_partial_regularity	theorem	Caffarelli–Kohn–Nirenberg partial regularity for Navier–Stokes		Iter-3 imported theorem (s_ckn_partial_regularity).
s_suitable_weak_solution	axiom	suitable weak solution		Iter-3 imported axiom (s_suitable_weak_solution).
s_local_energy_inequality	state	local energy inequality		Iter-3 imported state (s_local_energy_inequality).
s_epsilon_regularity_criterion	state	epsilon regularity criterion		Iter-3 imported state (s_epsilon_regularity_criterion).
s_vlasov_poisson_global_existence	theorem	Vlasov–Poisson global existence (Pfaffelmoser)		Iter-3 imported theorem (s_vlasov_poisson_global_existence).
s_vlasov_poisson_system	axiom	vlasov poisson system		Iter-3 imported axiom (s_vlasov_poisson_system).
s_compactly_supported_smooth_initial_data	axiom	compactly supported smooth initial data		Iter-3 imported axiom (s_compactly_supported_smooth_initial_data).
s_velocity_support_evolution_inequality	state	velocity support evolution inequality		Iter-3 imported state (s_velocity_support_evolution_inequality).
s_double_exponential_growth_bound_on_Q	state	double exponential growth bound on Q		Iter-3 imported state (s_double_exponential_growth_bound_on_Q).
s_lions_perthame_vlasov	theorem	Lions–Perthame global existence for Vlasov–Poisson		Iter-3 imported theorem (s_lions_perthame_vlasov).
s_finite_velocity_moment_initial_data	axiom	finite velocity moment initial data		Iter-3 imported axiom (s_finite_velocity_moment_initial_data).
s_evolution_inequality_for_velocity_moments	state	evolution inequality for velocity moments		Iter-3 imported state (s_evolution_inequality_for_velocity_moments).
s_global_bound_on_velocity_moments	state	global bound on velocity moments		Iter-3 imported state (s_global_bound_on_velocity_moments).
s_galerkin_parabolic_existence	theorem	Galerkin existence for parabolic PDE		Iter-3 imported theorem (s_galerkin_parabolic_existence).
s_parabolic_pde_in_divergence_form	axiom	parabolic pde in divergence form		Iter-3 imported axiom (s_parabolic_pde_in_divergence_form).
s_finite_dim_ode_approximation	state	finite dim ode approximation		Iter-3 imported state (s_finite_dim_ode_approximation).
s_uniform_a_priori_bounds_in_L2_H_1	state	uniform a priori bounds in L2 H 1		Iter-3 imported state (s_uniform_a_priori_bounds_in_L2_H_1).
s_lions_monotone_operator_theorem	theorem	Lions monotone-operator existence theorem		Iter-3 imported theorem (s_lions_monotone_operator_theorem).
s_monotone_hemicontinuous_coercive_operator	axiom	monotone hemicontinuous coercive operator		Iter-3 imported axiom (s_monotone_hemicontinuous_coercive_operator).
s_finite_dim_brouwer_existence	state	finite dim brouwer existence		Iter-3 imported state (s_finite_dim_brouwer_existence).
s_weak_limit_in_reflexive_space	state	weak limit in reflexive space		Iter-3 imported state (s_weak_limit_in_reflexive_space).
s_minty_lemma	theorem	Minty's lemma on monotone operators		Iter-3 imported theorem (s_minty_lemma).
s_monotone_operator_on_hilbert_space	axiom	monotone operator on hilbert space		Iter-3 imported axiom (s_monotone_operator_on_hilbert_space).
s_hemicontinuity	axiom	hemicontinuity		Iter-3 imported axiom (s_hemicontinuity).
s_pairing_against_test_in_monotone_form	state	pairing against test in monotone form		Iter-3 imported state (s_pairing_against_test_in_monotone_form).
s_hamilton_short_time_ricci_flow	theorem	Hamilton short-time existence for Ricci flow		Iter-3 imported theorem (s_hamilton_short_time_ricci_flow).
s_deturck_strictly_parabolic_system	state	deturck strictly parabolic system		Iter-3 imported state (s_deturck_strictly_parabolic_system).
s_short_time_solution_of_deturck_system	state	short time solution of deturck system		Iter-3 imported state (s_short_time_solution_of_deturck_system).
s_deturck_trick	theorem	DeTurck trick		Iter-3 imported theorem (s_deturck_trick).
s_diffeomorphism_gauge_field	axiom	diffeomorphism gauge field		Iter-3 imported axiom (s_diffeomorphism_gauge_field).
s_gauge_orbit_of_metrics	state	gauge orbit of metrics		Iter-3 imported state (s_gauge_orbit_of_metrics).
s_deturck_modified_ricci_flow	state	deturck modified ricci flow		Iter-3 imported state (s_deturck_modified_ricci_flow).
s_eells_sampson_harmonic_map_flow	theorem	Eells–Sampson harmonic map heat flow existence		Iter-3 imported theorem (s_eells_sampson_harmonic_map_flow).
s_smooth_map_between_riemannian_manifolds	axiom	smooth map between riemannian manifolds		Iter-3 imported axiom (s_smooth_map_between_riemannian_manifolds).
s_target_with_nonpositive_sectional_curvature	axiom	target with nonpositive sectional curvature		Iter-3 imported axiom (s_target_with_nonpositive_sectional_curvature).
s_dirichlet_energy_for_maps	state	dirichlet energy for maps		Iter-3 imported state (s_dirichlet_energy_for_maps).
s_bochner_subharmonicity_of_energy_density	state	bochner subharmonicity of energy density		Iter-3 imported state (s_bochner_subharmonicity_of_energy_density).
s_long_time_existence_of_harmonic_heat_flow	state	long time existence of harmonic heat flow		Iter-3 imported state (s_long_time_existence_of_harmonic_heat_flow).
s_schoen_uhlenbeck_regularity	theorem	Schoen–Uhlenbeck regularity for harmonic maps		Iter-3 imported theorem (s_schoen_uhlenbeck_regularity).
s_minimizing_harmonic_map	axiom	minimizing harmonic map		Iter-3 imported axiom (s_minimizing_harmonic_map).
s_target_riemannian_manifold	axiom	target riemannian manifold		Iter-3 imported axiom (s_target_riemannian_manifold).
s_tangent_map_homogeneous_minimizer	state	tangent map homogeneous minimizer		Iter-3 imported state (s_tangent_map_homogeneous_minimizer).
s_epsilon_regularity_lemma	state	epsilon regularity lemma		Iter-3 imported state (s_epsilon_regularity_lemma).
s_mcf_short_time_existence	theorem	Mean curvature flow short-time existence		Iter-3 imported theorem (s_mcf_short_time_existence).
s_smooth_hypersurface_in_R_n_plus_1	axiom	smooth hypersurface in R n plus 1		Iter-3 imported axiom (s_smooth_hypersurface_in_R_n_plus_1).
s_mean_curvature_vector_H	axiom	mean curvature vector H		Iter-3 imported axiom (s_mean_curvature_vector_H).
s_graphical_mcf_quasilinear_parabolic_equation	state	graphical mcf quasilinear parabolic equation		Iter-3 imported state (s_graphical_mcf_quasilinear_parabolic_equation).
s_short_time_graphical_solution	state	short time graphical solution		Iter-3 imported state (s_short_time_graphical_solution).
s_huisken_monotonicity_formula	theorem	Huisken's monotonicity formula for MCF		Iter-3 imported theorem (s_huisken_monotonicity_formula).
s_backward_heat_kernel_in_R_n_plus_1	axiom	backward heat kernel in R n plus 1		Iter-3 imported axiom (s_backward_heat_kernel_in_R_n_plus_1).
s_gaussian_density_function_theta	state	gaussian density function theta		Iter-3 imported state (s_gaussian_density_function_theta).
s_schoen_yau_positive_mass	theorem	Schoen–Yau positive mass theorem		Iter-3 imported theorem (s_schoen_yau_positive_mass).
s_asymptotically_flat_3_manifold	axiom	asymptotically flat 3 manifold		Iter-3 imported axiom (s_asymptotically_flat_3_manifold).
s_nonnegative_scalar_curvature	axiom	nonnegative scalar curvature		Iter-3 imported axiom (s_nonnegative_scalar_curvature).
s_minimal_surface_argument	state	minimal surface argument		Iter-3 imported state (s_minimal_surface_argument).
s_contradiction_via_stable_minimal_surface	state	contradiction via stable minimal surface		Iter-3 imported state (s_contradiction_via_stable_minimal_surface).
s_yang_mills_heat_flow_existence	theorem	Yang–Mills heat flow existence		Iter-3 imported theorem (s_yang_mills_heat_flow_existence).
s_yang_mills_functional	axiom	yang mills functional		Iter-3 imported axiom (s_yang_mills_functional).
s_principal_bundle_over_compact_4_manifold	axiom	principal bundle over compact 4 manifold		Iter-3 imported axiom (s_principal_bundle_over_compact_4_manifold).
s_coulomb_gauge_yang_mills_flow	state	coulomb gauge yang mills flow		Iter-3 imported state (s_coulomb_gauge_yang_mills_flow).
s_yang_mills_energy_a_priori_bound	state	yang mills energy a priori bound		Iter-3 imported state (s_yang_mills_energy_a_priori_bound).
s_uhlenbeck_compactness_yang_mills	theorem	Uhlenbeck compactness for Yang–Mills		Iter-3 imported theorem (s_uhlenbeck_compactness_yang_mills).
s_sequence_of_connections_with_L_2_bounded_curvature	axiom	sequence of connections with L 2 bounded curvature		Iter-3 imported axiom (s_sequence_of_connections_with_L_2_bounded_curvature).
s_coulomb_gauge_representatives	state	coulomb gauge representatives		Iter-3 imported state (s_coulomb_gauge_representatives).
s_weak_limit_of_connections_modulo_gauge	state	weak limit of connections modulo gauge		Iter-3 imported state (s_weak_limit_of_connections_modulo_gauge).
s_lewy_unsolvability	theorem	Lewy's example of unsolvable linear PDE		Iter-3 imported theorem (s_lewy_unsolvability).
s_lewy_operator_d_dz_minus_2i_z_bar_d_dt	axiom	lewy operator d dz minus 2i z bar d dt		Iter-3 imported axiom (s_lewy_operator_d_dz_minus_2i_z_bar_d_dt).
s_smooth_inhomogeneity	axiom	smooth inhomogeneity		Iter-3 imported axiom (s_smooth_inhomogeneity).
s_cr_operator_realization	state	cr operator realization		Iter-3 imported state (s_cr_operator_realization).
s_contradiction_via_hartogs_extension	state	contradiction via hartogs extension		Iter-3 imported state (s_contradiction_via_hartogs_extension).
s_malgrange_ehrenpreis_theorem	theorem	Malgrange–Ehrenpreis theorem		Iter-3 imported theorem (s_malgrange_ehrenpreis_theorem).
s_constant_coefficient_linear_pde_operator	axiom	constant coefficient linear pde operator		Iter-3 imported axiom (s_constant_coefficient_linear_pde_operator).
s_compactly_supported_distribution	axiom	compactly supported distribution		Iter-3 imported axiom (s_compactly_supported_distribution).
s_division_in_paley_wiener_space	state	division in paley wiener space		Iter-3 imported state (s_division_in_paley_wiener_space).
s_fundamental_solution_E	state	fundamental solution E		Iter-3 imported state (s_fundamental_solution_E).
s_poincare_hopf_theorem	theorem	Poincaré–Hopf for vector fields		Iter-3 imported theorem (s_poincare_hopf_theorem).
s_smooth_vector_field_with_isolated_zeros	axiom	smooth vector field with isolated zeros		Iter-3 imported axiom (s_smooth_vector_field_with_isolated_zeros).
s_local_index_count	state	local index count		Iter-3 imported state (s_local_index_count).
s_global_index_equals_euler_characteristic	state	global index equals euler characteristic		Iter-3 imported state (s_global_index_equals_euler_characteristic).
s_direct_method_calculus_of_variations	theorem	Direct method of calculus of variations		Iter-3 imported theorem (s_direct_method_calculus_of_variations).
s_coercive_lower_semicontinuous_functional	axiom	coercive lower semicontinuous functional		Iter-3 imported axiom (s_coercive_lower_semicontinuous_functional).
s_bounded_minimizing_sequence	state	bounded minimizing sequence		Iter-3 imported state (s_bounded_minimizing_sequence).
s_weak_limit_candidate_minimizer	state	weak limit candidate minimizer		Iter-3 imported state (s_weak_limit_candidate_minimizer).
s_euler_lagrange_equation	theorem	Euler–Lagrange equation derivation		Iter-3 imported theorem (s_euler_lagrange_equation).
s_compactly_supported_variation	axiom	compactly supported variation		Iter-3 imported axiom (s_compactly_supported_variation).
s_first_variation_formula	state	first variation formula		Iter-3 imported state (s_first_variation_formula).
s_weak_form_with_arbitrary_test_function	state	weak form with arbitrary test function		Iter-3 imported state (s_weak_form_with_arbitrary_test_function).
s_pohozaev_identity	theorem	Pohozaev identity		Iter-3 imported theorem (s_pohozaev_identity).
s_semilinear_elliptic_equation	axiom	semilinear elliptic equation		Iter-3 imported axiom (s_semilinear_elliptic_equation).
s_starshaped_bounded_domain	axiom	starshaped bounded domain		Iter-3 imported axiom (s_starshaped_bounded_domain).
s_multiplied_equation_by_x_dot_grad_u	state	multiplied equation by x dot grad u		Iter-3 imported state (s_multiplied_equation_by_x_dot_grad_u).
s_boundary_volume_identity	state	boundary volume identity		Iter-3 imported state (s_boundary_volume_identity).
s_mountain_pass_theorem	theorem	Mountain pass theorem		Iter-3 imported theorem (s_mountain_pass_theorem).
s_C_1_functional_with_palais_smale	axiom	C 1 functional with palais smale		Iter-3 imported axiom (s_C_1_functional_with_palais_smale).
s_two_separated_local_minima	axiom	two separated local minima		Iter-3 imported axiom (s_two_separated_local_minima).
s_minimax_value_c_equals_inf_max_J_on_gamma	state	minimax value c equals inf max J on gamma		Iter-3 imported state (s_minimax_value_c_equals_inf_max_J_on_gamma).
s_palais_smale_sequence_at_level_c	state	palais smale sequence at level c		Iter-3 imported state (s_palais_smale_sequence_at_level_c).
s_pohozaev_nonexistence_supercritical	theorem	Pohozaev nonexistence for supercritical Yamabe-type equations		Iter-3 imported theorem (s_pohozaev_nonexistence_supercritical).
s_supercritical_semilinear_elliptic	axiom	supercritical semilinear elliptic		Iter-3 imported axiom (s_supercritical_semilinear_elliptic).
s_signed_boundary_integral	state	signed boundary integral		Iter-3 imported state (s_signed_boundary_integral).
s_yamabe_problem_solution	theorem	Sobolev–Talenti–Aubin Yamabe problem solution		Iter-3 imported theorem (s_yamabe_problem_solution).
s_subcritical_minimizers_u_p	state	subcritical minimizers u p		Iter-3 imported state (s_subcritical_minimizers_u_p).
s_gidas_ni_nirenberg_liouville	theorem	Liouville theorem for nonnegative elliptic solutions in R^n		Iter-3 imported theorem (s_gidas_ni_nirenberg_liouville).
s_semilinear_elliptic_in_R_n	axiom	semilinear elliptic in R n		Iter-3 imported axiom (s_semilinear_elliptic_in_R_n).
s_nonnegative_classical_solution	axiom	nonnegative classical solution		Iter-3 imported axiom (s_nonnegative_classical_solution).
s_radial_symmetry_via_moving_planes	state	radial symmetry via moving planes		Iter-3 imported state (s_radial_symmetry_via_moving_planes).
s_radial_ode_problem	state	radial ode problem		Iter-3 imported state (s_radial_ode_problem).
s_gidas_ni_nirenberg_symmetry	theorem	Gidas–Ni–Nirenberg radial symmetry		Iter-3 imported theorem (s_gidas_ni_nirenberg_symmetry).
s_positive_solution_in_ball	axiom	positive solution in ball		Iter-3 imported axiom (s_positive_solution_in_ball).
s_reflected_solution_u_lambda	state	reflected solution u lambda		Iter-3 imported state (s_reflected_solution_u_lambda).
s_solution_symmetric_about_critical_plane	state	solution symmetric about critical plane		Iter-3 imported state (s_solution_symmetric_about_critical_plane).
s_lax_equivalence_theorem	theorem	Lax equivalence theorem		Iter-3 imported theorem (s_lax_equivalence_theorem).
s_well_posed_linear_initial_value_problem	axiom	well posed linear initial value problem		Iter-3 imported axiom (s_well_posed_linear_initial_value_problem).
s_consistent_finite_difference_scheme	axiom	consistent finite difference scheme		Iter-3 imported axiom (s_consistent_finite_difference_scheme).
s_three_property_axiom_definition	state	three property axiom definition		Iter-3 imported state (s_three_property_axiom_definition).
s_stability_implies_uniform_boundedness	state	stability implies uniform boundedness		Iter-3 imported state (s_stability_implies_uniform_boundedness).
s_von_neumann_stability_criterion	theorem	Von Neumann stability analysis		Iter-3 imported theorem (s_von_neumann_stability_criterion).
s_constant_coefficient_finite_difference_scheme	axiom	constant coefficient finite difference scheme		Iter-3 imported axiom (s_constant_coefficient_finite_difference_scheme).
s_periodic_grid	axiom	periodic grid		Iter-3 imported axiom (s_periodic_grid).
s_cfl_criterion	theorem	CFL convergence criterion for hyperbolic schemes		Iter-3 imported theorem (s_cfl_criterion).
s_explicit_finite_difference_for_hyperbolic_pde	axiom	explicit finite difference for hyperbolic pde		Iter-3 imported axiom (s_explicit_finite_difference_for_hyperbolic_pde).
s_characteristic_propagation_cone	axiom	characteristic propagation cone		Iter-3 imported axiom (s_characteristic_propagation_cone).
s_numerical_domain_of_dependence	state	numerical domain of dependence		Iter-3 imported state (s_numerical_domain_of_dependence).
s_lax_wendroff_convergence	theorem	Lax–Wendroff convergence theorem for conservation laws		Iter-3 imported theorem (s_lax_wendroff_convergence).
s_consistent_conservative_finite_difference_scheme	axiom	consistent conservative finite difference scheme		Iter-3 imported axiom (s_consistent_conservative_finite_difference_scheme).
s_a_priori_BV_bound	axiom	a priori BV bound		Iter-3 imported axiom (s_a_priori_BV_bound).
s_convergent_subsequence_to_function_v	state	convergent subsequence to function v		Iter-3 imported state (s_convergent_subsequence_to_function_v).
s_v_is_weak_entropy_solution	state	v is weak entropy solution		Iter-3 imported state (s_v_is_weak_entropy_solution).
s_kruzhkov_uniqueness_theorem	theorem	Kruzhkov uniqueness for scalar conservation laws		Iter-3 imported theorem (s_kruzhkov_uniqueness_theorem).
s_kruzhkov_entropy_pair	axiom	kruzhkov entropy pair		Iter-3 imported axiom (s_kruzhkov_entropy_pair).
s_kruzhkov_doubling_variables_technique	state	kruzhkov doubling variables technique		Iter-3 imported state (s_kruzhkov_doubling_variables_technique).
s_L1_contraction_for_entropy_solutions	state	L1 contraction for entropy solutions		Iter-3 imported state (s_L1_contraction_for_entropy_solutions).
s_vanishing_viscosity_convergence	theorem	Vanishing viscosity convergence		Iter-3 imported theorem (s_vanishing_viscosity_convergence).
s_viscous_approximation_u_epsilon	axiom	viscous approximation u epsilon		Iter-3 imported axiom (s_viscous_approximation_u_epsilon).
s_BV_uniform_bound	state	BV uniform bound		Iter-3 imported state (s_BV_uniform_bound).
s_convergent_subsequence	state	convergent subsequence		Iter-3 imported state (s_convergent_subsequence).
s_hille_yosida_theorem	theorem	Hille–Yosida theorem for semigroups		Iter-3 imported theorem (s_hille_yosida_theorem).
s_densely_defined_closed_linear_operator	axiom	densely defined closed linear operator		Iter-3 imported axiom (s_densely_defined_closed_linear_operator).
s_resolvent_bound_on_real_axis	axiom	resolvent bound on real axis		Iter-3 imported axiom (s_resolvent_bound_on_real_axis).
s_generated_C_0_semigroup	state	generated C 0 semigroup		Iter-3 imported state (s_generated_C_0_semigroup).
s_lumer_phillips_theorem	theorem	Lumer–Phillips theorem for dissipative operators		Iter-3 imported theorem (s_lumer_phillips_theorem).
s_dissipative_operator_on_hilbert_space	axiom	dissipative operator on hilbert space		Iter-3 imported axiom (s_dissipative_operator_on_hilbert_space).
s_range_condition_lambda_A_surjective	axiom	range condition lambda A surjective		Iter-3 imported axiom (s_range_condition_lambda_A_surjective).
s_contraction_property_of_resolvent	state	contraction property of resolvent		Iter-3 imported state (s_contraction_property_of_resolvent).
s_contraction_semigroup_generation	state	contraction semigroup generation		Iter-3 imported state (s_contraction_semigroup_generation).
s_trotter_kato_product_formula	theorem	Trotter–Kato product formula		Iter-3 imported theorem (s_trotter_kato_product_formula).
s_two_semigroup_generators_A_B	axiom	two semigroup generators A B		Iter-3 imported axiom (s_two_semigroup_generators_A_B).
s_sum_A_plus_B_generates_semigroup	axiom	sum A plus B generates semigroup		Iter-3 imported axiom (s_sum_A_plus_B_generates_semigroup).
s_trotter_splitting_approximant	state	trotter splitting approximant		Iter-3 imported state (s_trotter_splitting_approximant).
s_kato_rellich_theorem	theorem	Kato–Rellich self-adjointness theorem		Iter-3 imported theorem (s_kato_rellich_theorem).
s_symmetric_relatively_bounded_perturbation_B	axiom	symmetric relatively bounded perturbation B		Iter-3 imported axiom (s_symmetric_relatively_bounded_perturbation_B).
s_relatively_bounded_sum	state	relatively bounded sum		Iter-3 imported state (s_relatively_bounded_sum).
s_essentially_self_adjointness_of_sum	state	essentially self adjointness of sum		Iter-3 imported state (s_essentially_self_adjointness_of_sum).
s_weyl_law	theorem	Weyl's law for eigenvalue counting		Iter-3 imported theorem (s_weyl_law).
s_dirichlet_laplacian_on_bounded_domain	axiom	dirichlet laplacian on bounded domain		Iter-3 imported axiom (s_dirichlet_laplacian_on_bounded_domain).
s_heat_kernel_trace_asymptotic	state	heat kernel trace asymptotic		Iter-3 imported state (s_heat_kernel_trace_asymptotic).
s_agmon_decay_theorem	theorem	Agmon decay of eigenfunctions		Iter-3 imported theorem (s_agmon_decay_theorem).
s_schrodinger_operator_minus_delta_plus_V	axiom	schrodinger operator minus delta plus V		Iter-3 imported axiom (s_schrodinger_operator_minus_delta_plus_V).
s_eigenfunction_below_essential_spectrum	axiom	eigenfunction below essential spectrum		Iter-3 imported axiom (s_eigenfunction_below_essential_spectrum).
s_agmon_riemannian_distance	state	agmon riemannian distance		Iter-3 imported state (s_agmon_riemannian_distance).
s_weighted_energy_estimate	state	weighted energy estimate		Iter-3 imported state (s_weighted_energy_estimate).
s_lax_phillips_scattering	theorem	Lax–Phillips scattering theory		Iter-3 imported theorem (s_lax_phillips_scattering).
s_unperturbed_unitary_group	axiom	unperturbed unitary group		Iter-3 imported axiom (s_unperturbed_unitary_group).
s_perturbed_unitary_group_with_compactly_supported_obstacle	axiom	perturbed unitary group with compactly supported obstacle		Iter-3 imported axiom (s_perturbed_unitary_group_with_compactly_supported_obstacle).
s_translation_representation_via_D_minus_D_plus	state	translation representation via D minus D plus		Iter-3 imported state (s_translation_representation_via_D_minus_D_plus).
s_wave_operators_intertwining	state	wave operators intertwining		Iter-3 imported state (s_wave_operators_intertwining).
s_birman_schwinger_principle	theorem	Birman–Schwinger principle		Iter-3 imported theorem (s_birman_schwinger_principle).
s_negative_eigenvalue_minus_E	axiom	negative eigenvalue minus E		Iter-3 imported axiom (s_negative_eigenvalue_minus_E).
s_birman_schwinger_operator_K_E	state	birman schwinger operator K E		Iter-3 imported state (s_birman_schwinger_operator_K_E).
s_clr_bound	theorem	Cwikel–Lieb–Rozenblum bound on bound states		Iter-3 imported theorem (s_clr_bound).
s_negative_part_V_minus_in_L_n_2	axiom	negative part V minus in L n 2		Iter-3 imported axiom (s_negative_part_V_minus_in_L_n_2).
s_negative_eigenvalues_via_birman_schwinger	state	negative eigenvalues via birman schwinger		Iter-3 imported state (s_negative_eigenvalues_via_birman_schwinger).
s_trace_norm_bound_on_K_0	state	trace norm bound on K 0		Iter-3 imported state (s_trace_norm_bound_on_K_0).
s_aronson_gaussian_bounds	theorem	Aronson Gaussian bounds for fundamental solution		Iter-3 imported theorem (s_aronson_gaussian_bounds).
s_divergence_form_parabolic_with_L_infty_coeff	axiom	divergence form parabolic with L infty coeff		Iter-3 imported axiom (s_divergence_form_parabolic_with_L_infty_coeff).
s_holder_regularity_of_fundamental_solution	state	holder regularity of fundamental solution		Iter-3 imported state (s_holder_regularity_of_fundamental_solution).
s_two_sided_gaussian_estimate	state	two sided gaussian estimate		Iter-3 imported state (s_two_sided_gaussian_estimate).
s_li_yau_differential_harnack	theorem	Li–Yau differential Harnack inequality		Iter-3 imported theorem (s_li_yau_differential_harnack).
s_positive_solution_of_heat_equation_on_riemannian_manifold	axiom	positive solution of heat equation on riemannian manifold		Iter-3 imported axiom (s_positive_solution_of_heat_equation_on_riemannian_manifold).
s_nonneg_ricci_curvature	axiom	nonneg ricci curvature		Iter-3 imported axiom (s_nonneg_ricci_curvature).
s_li_yau_gradient_quantity_F	state	li yau gradient quantity F		Iter-3 imported state (s_li_yau_gradient_quantity_F).
s_max_principle_bound_on_F	state	max principle bound on F		Iter-3 imported state (s_max_principle_bound_on_F).
s_bakry_emery_criterion	theorem	Bakry–Émery curvature-dimension criterion		Iter-3 imported theorem (s_bakry_emery_criterion).
s_diffusion_generator_L_with_carre_du_champ	axiom	diffusion generator L with carre du champ		Iter-3 imported axiom (s_diffusion_generator_L_with_carre_du_champ).
s_gamma_2_operator	state	gamma 2 operator		Iter-3 imported state (s_gamma_2_operator).
s_CD_K_N_inequality	state	CD K N inequality		Iter-3 imported state (s_CD_K_N_inequality).
s_otto_gradient_flow_heat	theorem	Otto's gradient flow interpretation of heat equation		Iter-3 imported theorem (s_otto_gradient_flow_heat).
s_wasserstein_2_metric_on_probability_measures	axiom	wasserstein 2 metric on probability measures		Iter-3 imported axiom (s_wasserstein_2_metric_on_probability_measures).
s_jko_time_discretization	state	jko time discretization		Iter-3 imported state (s_jko_time_discretization).
s_continuous_time_gradient_flow	state	continuous time gradient flow		Iter-3 imported state (s_continuous_time_gradient_flow).
s_viscosity_solutions_theory	theorem	Crandall–Lions viscosity solutions of Hamilton–Jacobi		Iter-3 imported theorem (s_viscosity_solutions_theory).
s_continuous_hamiltonian_H	axiom	continuous hamiltonian H		Iter-3 imported axiom (s_continuous_hamiltonian_H).
s_definition_of_viscosity_sub_super_solution	state	definition of viscosity sub super solution		Iter-3 imported state (s_definition_of_viscosity_sub_super_solution).
s_doubling_variables_comparison	state	doubling variables comparison		Iter-3 imported state (s_doubling_variables_comparison).
s_evans_krylov_regularity	theorem	Evans–Krylov C^{2,α} regularity for fully nonlinear PDE		Iter-3 imported theorem (s_evans_krylov_regularity).
s_concave_fully_nonlinear_uniformly_elliptic_pde	axiom	concave fully nonlinear uniformly elliptic pde		Iter-3 imported axiom (s_concave_fully_nonlinear_uniformly_elliptic_pde).
s_continuous_viscosity_solution	axiom	continuous viscosity solution		Iter-3 imported axiom (s_continuous_viscosity_solution).
s_second_difference_subsolution_property	state	second difference subsolution property		Iter-3 imported state (s_second_difference_subsolution_property).
s_holder_continuity_of_D_2_u	state	holder continuity of D 2 u		Iter-3 imported state (s_holder_continuity_of_D_2_u).
s_caffarelli_monge_ampere_regularity	theorem	Caffarelli C^{2,α} regularity for Monge–Ampère		Iter-3 imported theorem (s_caffarelli_monge_ampere_regularity).
s_strictly_convex_alexandrov_solution	axiom	strictly convex alexandrov solution		Iter-3 imported axiom (s_strictly_convex_alexandrov_solution).
s_C_alpha_right_hand_side_f	axiom	C alpha right hand side f		Iter-3 imported axiom (s_C_alpha_right_hand_side_f).
s_normalized_section_of_solution	state	normalized section of solution		Iter-3 imported state (s_normalized_section_of_solution).
s_C2_alpha_estimate_on_sections	state	C2 alpha estimate on sections		Iter-3 imported state (s_C2_alpha_estimate_on_sections).
s_allard_regularity_theorem	theorem	Allard regularity theorem for varifolds		Iter-3 imported theorem (s_allard_regularity_theorem).
s_integral_varifold_with_bounded_mean_curvature	axiom	integral varifold with bounded mean curvature		Iter-3 imported axiom (s_integral_varifold_with_bounded_mean_curvature).
s_small_density_excess_at_point	axiom	small density excess at point		Iter-3 imported axiom (s_small_density_excess_at_point).
s_flat_tangent_cone	state	flat tangent cone		Iter-3 imported state (s_flat_tangent_cone).
s_lipschitz_graph_approximation	state	lipschitz graph approximation		Iter-3 imported state (s_lipschitz_graph_approximation).
s_toponogov_comparison_theorem	theorem	Toponogov triangle comparison		Iter-3 imported theorem (s_toponogov_comparison_theorem).
s_complete_riemannian_manifold_with_lower_sectional_curvature_bound	axiom	complete riemannian manifold with lower sectional curvature bound		Iter-3 imported axiom (s_complete_riemannian_manifold_with_lower_sectional_curvature_bound).
s_geodesic_triangle	axiom	geodesic triangle		Iter-3 imported axiom (s_geodesic_triangle).
s_geodesic_length_comparison	state	geodesic length comparison		Iter-3 imported state (s_geodesic_length_comparison).
s_cheeger_inequality	theorem	Cheeger isoperimetric–spectral gap inequality		Iter-3 imported theorem (s_cheeger_inequality).
s_coarea_inequality	state	coarea inequality		Iter-3 imported state (s_coarea_inequality).
s_lambda_1_geq_h_squared_over_4	state	lambda 1 geq h squared over 4		Iter-3 imported state (s_lambda_1_geq_h_squared_over_4).
s_cheeger_gromov_compactness	theorem	Cheeger–Gromov compactness theorem		Iter-3 imported theorem (s_cheeger_gromov_compactness).
s_sequence_of_riemannian_manifolds_with_curvature_volume_bounds	axiom	sequence of riemannian manifolds with curvature volume bounds		Iter-3 imported axiom (s_sequence_of_riemannian_manifolds_with_curvature_volume_bounds).
s_pointed_riemannian_manifolds	axiom	pointed riemannian manifolds		Iter-3 imported axiom (s_pointed_riemannian_manifolds).
s_uniform_harmonic_atlas	state	uniform harmonic atlas		Iter-3 imported state (s_uniform_harmonic_atlas).
s_subsequential_C_1_alpha_convergence	state	subsequential C 1 alpha convergence		Iter-3 imported state (s_subsequential_C_1_alpha_convergence).
s_hamilton_ricci_flow_compactness	theorem	Hamilton's compactness theorem for Ricci flows		Iter-3 imported theorem (s_hamilton_ricci_flow_compactness).
s_sequence_of_pointed_ricci_flows_with_uniform_curvature_bound	axiom	sequence of pointed ricci flows with uniform curvature bound		Iter-3 imported axiom (s_sequence_of_pointed_ricci_flows_with_uniform_curvature_bound).
s_injectivity_radius_lower_bound	axiom	injectivity radius lower bound		Iter-3 imported axiom (s_injectivity_radius_lower_bound).
s_C_infty_subsequential_metric_limit	state	C infty subsequential metric limit		Iter-3 imported state (s_C_infty_subsequential_metric_limit).
s_C_infty_smooth_limit_flow	state	C infty smooth limit flow		Iter-3 imported state (s_C_infty_smooth_limit_flow).
s_perelman_W_monotonicity	theorem	Perelman entropy monotonicity (W-functional)		Iter-3 imported theorem (s_perelman_W_monotonicity).
s_perelman_W_functional	axiom	perelman W functional		Iter-3 imported axiom (s_perelman_W_functional).
s_coupled_flow_with_conjugate_heat	state	coupled flow with conjugate heat		Iter-3 imported state (s_coupled_flow_with_conjugate_heat).
s_perelman_no_local_collapsing	theorem	Perelman no-local-collapsing theorem		Iter-3 imported theorem (s_perelman_no_local_collapsing).
s_W_lower_bound_in_terms_of_initial_data	state	W lower bound in terms of initial data		Iter-3 imported state (s_W_lower_bound_in_terms_of_initial_data).
s_hamilton_tensor_max_principle	theorem	Hamilton tensor maximum principle		Iter-3 imported theorem (s_hamilton_tensor_max_principle).
s_symmetric_tensor_satisfying_parabolic_inequality	axiom	symmetric tensor satisfying parabolic inequality		Iter-3 imported axiom (s_symmetric_tensor_satisfying_parabolic_inequality).
s_convex_invariant_K	state	convex invariant K		Iter-3 imported state (s_convex_invariant_K).
s_brakke_flow_existence	theorem	Brakke mean-curvature flow weak existence		Iter-3 imported theorem (s_brakke_flow_existence).
s_initial_integral_varifold	axiom	initial integral varifold		Iter-3 imported axiom (s_initial_integral_varifold).
s_brakke_inequality_for_varifolds	axiom	brakke inequality for varifolds		Iter-3 imported axiom (s_brakke_inequality_for_varifolds).
s_elliptic_regularization_approximation	state	elliptic regularization approximation		Iter-3 imported state (s_elliptic_regularization_approximation).
s_subsequential_varifold_limit	state	subsequential varifold limit		Iter-3 imported state (s_subsequential_varifold_limit).
s_implicit_function_theorem_banach	theorem	Implicit function theorem (Banach version)		Iter-3 imported theorem (s_implicit_function_theorem_banach).
s_C1_map_between_banach_spaces	axiom	C1 map between banach spaces		Iter-3 imported axiom (s_C1_map_between_banach_spaces).
s_invertible_partial_derivative_at_a_point	axiom	invertible partial derivative at a point		Iter-3 imported axiom (s_invertible_partial_derivative_at_a_point).
s_fixed_point_equation_for_y	state	fixed point equation for y		Iter-3 imported state (s_fixed_point_equation_for_y).
s_nash_moser_inverse_function_theorem	theorem	Nash–Moser inverse function theorem		Iter-3 imported theorem (s_nash_moser_inverse_function_theorem).
s_tame_frechet_space_map	axiom	tame frechet space map		Iter-3 imported axiom (s_tame_frechet_space_map).
s_tame_estimate_for_inverse_of_linearization	axiom	tame estimate for inverse of linearization		Iter-3 imported axiom (s_tame_estimate_for_inverse_of_linearization).
s_smoothing_operator_family	state	smoothing operator family		Iter-3 imported state (s_smoothing_operator_family).
s_quadratically_convergent_iteration	state	quadratically convergent iteration		Iter-3 imported state (s_quadratically_convergent_iteration).
s_lasalle_invariance_principle	theorem	Krasovskii–LaSalle invariance principle		Iter-3 imported theorem (s_lasalle_invariance_principle).
s_smooth_dynamical_system	axiom	smooth dynamical system		Iter-3 imported axiom (s_smooth_dynamical_system).
s_lyapunov_function_V_with_V_dot_le_0	axiom	lyapunov function V with V dot le 0		Iter-3 imported axiom (s_lyapunov_function_V_with_V_dot_le_0).
s_V_monotonically_nonincreasing	state	V monotonically nonincreasing		Iter-3 imported state (s_V_monotonically_nonincreasing).
s_omega_limit_inside_set_V_dot_equal_0	state	omega limit inside set V dot equal 0		Iter-3 imported state (s_omega_limit_inside_set_V_dot_equal_0).
s_massera_converse_lyapunov	theorem	Massera converse Lyapunov theorem		Iter-3 imported theorem (s_massera_converse_lyapunov).
s_uniformly_asymptotically_stable_equilibrium	axiom	uniformly asymptotically stable equilibrium		Iter-3 imported axiom (s_uniformly_asymptotically_stable_equilibrium).
s_integral_lyapunov_function_candidate	state	integral lyapunov function candidate		Iter-3 imported state (s_integral_lyapunov_function_candidate).
s_smale_horseshoe_theorem	theorem	Smale horseshoe / chaos theorem		Iter-3 imported theorem (s_smale_horseshoe_theorem).
s_transverse_homoclinic_orbit	axiom	transverse homoclinic orbit		Iter-3 imported axiom (s_transverse_homoclinic_orbit).
s_smooth_diffeomorphism	axiom	smooth diffeomorphism		Iter-3 imported axiom (s_smooth_diffeomorphism).
s_horseshoe_invariant_set	state	horseshoe invariant set		Iter-3 imported state (s_horseshoe_invariant_set).
s_topological_conjugacy_to_bernoulli_shift	state	topological conjugacy to bernoulli shift		Iter-3 imported state (s_topological_conjugacy_to_bernoulli_shift).
s_arnold_tongues_theorem	theorem	Arnold tongues / mode-locking theorem		Iter-3 imported theorem (s_arnold_tongues_theorem).
s_one_parameter_family_of_circle_maps	axiom	one parameter family of circle maps		Iter-3 imported axiom (s_one_parameter_family_of_circle_maps).
s_resonant_rotation_number	axiom	resonant rotation number		Iter-3 imported axiom (s_resonant_rotation_number).
s_resonance_tongue_in_parameter_space	state	resonance tongue in parameter space		Iter-3 imported state (s_resonance_tongue_in_parameter_space).
s_tongue_structure_with_devils_staircase	state	tongue structure with devils staircase		Iter-3 imported state (s_tongue_structure_with_devils_staircase).
s_weak_law_large_numbers	theorem	Weak Law of Large Numbers (Khintchine)		Iter-3 imported theorem (s_weak_law_large_numbers).
s_iid_sequence_finite_mean	axiom	iid sequence finite mean		Iter-3 imported axiom (s_iid_sequence_finite_mean).
s_characteristic_function_of_sample_mean	state	characteristic function of sample mean		Iter-3 imported state (s_characteristic_function_of_sample_mean).
s_phi_n_to_exp_itmu	state	phi n to exp itmu		Iter-3 imported state (s_phi_n_to_exp_itmu).
s_strong_law_large_numbers	theorem	Strong Law of Large Numbers (Kolmogorov)		Iter-3 imported theorem (s_strong_law_large_numbers).
s_truncated_iid_sequence	state	truncated iid sequence		Iter-3 imported state (s_truncated_iid_sequence).
s_partial_sum_variance_bound	state	partial sum variance bound		Iter-3 imported state (s_partial_sum_variance_bound).
s_subsequence_a_s_convergence	state	subsequence a s convergence		Iter-3 imported state (s_subsequence_a_s_convergence).
s_etemadi_slln	theorem	Etemadi's Strong Law		Iter-3 imported theorem (s_etemadi_slln).
s_pairwise_iid_finite_mean	axiom	pairwise iid finite mean		Iter-3 imported axiom (s_pairwise_iid_finite_mean).
s_nonnegative_pairwise_iid	state	nonnegative pairwise iid		Iter-3 imported state (s_nonnegative_pairwise_iid).
s_subseq_partial_sums	state	subseq partial sums		Iter-3 imported state (s_subseq_partial_sums).
s_subseq_a_s_convergence	state	subseq a s convergence		Iter-3 imported state (s_subseq_a_s_convergence).
s_multivariate_clt	theorem	Multivariate CLT		Iter-3 imported theorem (s_multivariate_clt).
s_iid_random_vectors_finite_cov	axiom	iid random vectors finite cov		Iter-3 imported axiom (s_iid_random_vectors_finite_cov).
s_one_d_projection_iid	state	one d projection iid		Iter-3 imported state (s_one_d_projection_iid).
s_clt_along_every_direction	state	clt along every direction		Iter-3 imported state (s_clt_along_every_direction).
s_berry_esseen	theorem	Berry–Esseen Theorem		Iter-3 imported theorem (s_berry_esseen).
s_iid_sequence_finite_third_moment	axiom	iid sequence finite third moment		Iter-3 imported axiom (s_iid_sequence_finite_third_moment).
s_characteristic_function_with_third_moment_bound	state	characteristic function with third moment bound		Iter-3 imported state (s_characteristic_function_with_third_moment_bound).
s_phi_minus_gaussian_pointwise_bound	state	phi minus gaussian pointwise bound		Iter-3 imported state (s_phi_minus_gaussian_pointwise_bound).
s_kolmogorov_distance_bound_C_rho_over_sqrtn	state	kolmogorov distance bound C rho over sqrtn		Iter-3 imported state (s_kolmogorov_distance_bound_C_rho_over_sqrtn).
s_donsker_invariance	theorem	Donsker's Invariance Principle		Iter-3 imported theorem (s_donsker_invariance).
s_skorokhod_space_DC01	axiom	skorokhod space DC01		Iter-3 imported axiom (s_skorokhod_space_DC01).
s_polygonal_partial_sum_process	state	polygonal partial sum process		Iter-3 imported state (s_polygonal_partial_sum_process).
s_finite_dim_marginals_gaussian	state	finite dim marginals gaussian		Iter-3 imported state (s_finite_dim_marginals_gaussian).
s_tightness_in_C01	state	tightness in C01		Iter-3 imported state (s_tightness_in_C01).
s_skorokhod_embedding	theorem	Skorokhod Embedding		Iter-3 imported theorem (s_skorokhod_embedding).
s_mean_zero_distribution_finite_variance	axiom	mean zero distribution finite variance		Iter-3 imported axiom (s_mean_zero_distribution_finite_variance).
s_brownian_motion	axiom	brownian motion		Iter-3 imported axiom (s_brownian_motion).
s_two_point_optional_law	state	two point optional law		Iter-3 imported state (s_two_point_optional_law).
s_brownian_stopping_time_matches_two_point	state	brownian stopping time matches two point		Iter-3 imported state (s_brownian_stopping_time_matches_two_point).
s_skorokhod_representation	theorem	Skorokhod Representation Theorem		Iter-3 imported theorem (s_skorokhod_representation).
s_weakly_convergent_sequence_of_laws	axiom	weakly convergent sequence of laws		Iter-3 imported axiom (s_weakly_convergent_sequence_of_laws).
s_inverse_cdf_couplings	state	inverse cdf couplings		Iter-3 imported state (s_inverse_cdf_couplings).
s_uniform_coupling_existence	state	uniform coupling existence		Iter-3 imported state (s_uniform_coupling_existence).
s_kolmogorov_three_series	theorem	Kolmogorov Three-Series Theorem		Iter-3 imported theorem (s_kolmogorov_three_series).
s_independent_real_random_variables	axiom	independent real random variables		Iter-3 imported axiom (s_independent_real_random_variables).
s_truncated_independent_series	state	truncated independent series		Iter-3 imported state (s_truncated_independent_series).
s_variance_sum_finite_iff_a_s_conv	state	variance sum finite iff a s conv		Iter-3 imported state (s_variance_sum_finite_iff_a_s_conv).
s_hewitt_savage_zero_one_law	theorem	Hewitt–Savage 0–1 Law		Iter-3 imported theorem (s_hewitt_savage_zero_one_law).
s_iid_sequence	axiom	iid sequence		Iter-3 imported axiom (s_iid_sequence).
s_exchangeable_sigma_algebra	axiom	exchangeable sigma algebra		Iter-3 imported axiom (s_exchangeable_sigma_algebra).
s_exchangeable_events_under_permutation	state	exchangeable events under permutation		Iter-3 imported state (s_exchangeable_events_under_permutation).
s_cylinder_approximation_of_exchangeable_event	state	cylinder approximation of exchangeable event		Iter-3 imported state (s_cylinder_approximation_of_exchangeable_event).
s_borel_cantelli_first	theorem	Borel–Cantelli Lemma (First)		Iter-3 imported theorem (s_borel_cantelli_first).
s_summable_event_probabilities	axiom	summable event probabilities		Iter-3 imported axiom (s_summable_event_probabilities).
s_tail_union_bound_goes_to_zero	state	tail union bound goes to zero		Iter-3 imported state (s_tail_union_bound_goes_to_zero).
s_borel_cantelli_second	theorem	Borel–Cantelli Lemma (Second)		Iter-3 imported theorem (s_borel_cantelli_second).
s_independent_events_divergent_sum	axiom	independent events divergent sum		Iter-3 imported axiom (s_independent_events_divergent_sum).
s_product_of_complements_bound	state	product of complements bound		Iter-3 imported state (s_product_of_complements_bound).
s_finite_intersections_of_complements_to_zero	state	finite intersections of complements to zero		Iter-3 imported state (s_finite_intersections_of_complements_to_zero).
s_conditional_borel_cantelli	theorem	Conditional Borel–Cantelli (Lévy)		Iter-3 imported theorem (s_conditional_borel_cantelli).
s_filtration_F_n	axiom	filtration F n		Iter-3 imported axiom (s_filtration_F_n).
s_adapted_event_sequence_A_n	axiom	adapted event sequence A n		Iter-3 imported axiom (s_adapted_event_sequence_A_n).
s_compensated_martingale	state	compensated martingale		Iter-3 imported state (s_compensated_martingale).
s_a_s_limit_of_M_n	state	a s limit of M n		Iter-3 imported state (s_a_s_limit_of_M_n).
s_doob_martingale_convergence_as	theorem	Doob's Martingale Convergence (a.s.)		Iter-3 imported theorem (s_doob_martingale_convergence_as).
s_L1_bounded_martingale	axiom	L1 bounded martingale		Iter-3 imported axiom (s_L1_bounded_martingale).
s_filtered_probability_space	axiom	filtered probability space		Iter-3 imported axiom (s_filtered_probability_space).
s_upcrossing_inequality	state	upcrossing inequality		Iter-3 imported state (s_upcrossing_inequality).
s_finite_upcrossings_a_s	state	finite upcrossings a s		Iter-3 imported state (s_finite_upcrossings_a_s).
s_doob_Lp_convergence	theorem	Doob L^p Martingale Convergence		Iter-3 imported theorem (s_doob_Lp_convergence).
s_Lp_bounded_martingale	axiom	Lp bounded martingale		Iter-3 imported axiom (s_Lp_bounded_martingale).
s_a_s_limit_X_infty	state	a s limit X infty		Iter-3 imported state (s_a_s_limit_X_infty).
s_uniform_integrability_of_Lp_martingale	state	uniform integrability of Lp martingale		Iter-3 imported state (s_uniform_integrability_of_Lp_martingale).
s_doob_maximal_inequality	theorem	Doob's Maximal Inequality		Iter-3 imported theorem (s_doob_maximal_inequality).
s_nonnegative_submartingale	axiom	nonnegative submartingale		Iter-3 imported axiom (s_nonnegative_submartingale).
s_stopped_submartingale_at_tau	state	stopped submartingale at tau		Iter-3 imported state (s_stopped_submartingale_at_tau).
s_lambda_P_max_le_E_X_n_1_event	state	lambda P max le E X n 1 event		Iter-3 imported state (s_lambda_P_max_le_E_X_n_1_event).
s_doob_meyer_decomposition	theorem	Doob–Meyer Decomposition		Iter-3 imported theorem (s_doob_meyer_decomposition).
s_class_D_submartingale	axiom	class D submartingale		Iter-3 imported axiom (s_class_D_submartingale).
s_discrete_doob_decomposition_X_n	state	discrete doob decomposition X n		Iter-3 imported state (s_discrete_doob_decomposition_X_n).
s_weak_limit_predictable_A_t	state	weak limit predictable A t		Iter-3 imported state (s_weak_limit_predictable_A_t).
s_optional_stopping_theorem	theorem	Optional Stopping Theorem		Iter-3 imported theorem (s_optional_stopping_theorem).
s_uniformly_integrable_martingale	axiom	uniformly integrable martingale		Iter-3 imported axiom (s_uniformly_integrable_martingale).
s_bounded_stopping_time_tau	axiom	bounded stopping time tau		Iter-3 imported axiom (s_bounded_stopping_time_tau).
s_stopped_process_X_tau_wedge_n	state	stopped process X tau wedge n		Iter-3 imported state (s_stopped_process_X_tau_wedge_n).
s_constant_expectation_along_stop	state	constant expectation along stop		Iter-3 imported state (s_constant_expectation_along_stop).
s_doob_upcrossing_inequality	theorem	Doob's Upcrossing Inequality		Iter-3 imported theorem (s_doob_upcrossing_inequality).
s_submartingale_X_n	axiom	submartingale X n		Iter-3 imported axiom (s_submartingale_X_n).
s_betting_strategy_C_n	state	betting strategy C n		Iter-3 imported state (s_betting_strategy_C_n).
s_gambler_winnings_lower_bound_b_minus_a_times_U	state	gambler winnings lower bound b minus a times U		Iter-3 imported state (s_gambler_winnings_lower_bound_b_minus_a_times_U).
s_wald_identity	theorem	Wald's Identity		Iter-3 imported theorem (s_wald_identity).
s_stopping_time_finite_mean	axiom	stopping time finite mean		Iter-3 imported axiom (s_stopping_time_finite_mean).
s_centered_random_walk_martingale	state	centered random walk martingale		Iter-3 imported state (s_centered_random_walk_martingale).
s_E_S_tau_minus_mu_tau_eq_0	state	E S tau minus mu tau eq 0		Iter-3 imported state (s_E_S_tau_minus_mu_tau_eq_0).
s_wald_variance_identity	theorem	Wald's Second Identity (variance)		Iter-3 imported theorem (s_wald_variance_identity).
s_second_order_martingale	state	second order martingale		Iter-3 imported state (s_second_order_martingale).
s_E_centered_sq_eq_sigma2_E_tau	state	E centered sq eq sigma2 E tau		Iter-3 imported state (s_E_centered_sq_eq_sigma2_E_tau).
s_wald_likelihood_ratio_identity	theorem	Wald's Likelihood-Ratio Identity (SPRT)		Iter-3 imported theorem (s_wald_likelihood_ratio_identity).
s_iid_likelihood_ratio_sequence	axiom	iid likelihood ratio sequence		Iter-3 imported axiom (s_iid_likelihood_ratio_sequence).
s_stopping_rule_for_sprt	axiom	stopping rule for sprt		Iter-3 imported axiom (s_stopping_rule_for_sprt).
s_lr_martingale_under_null	state	lr martingale under null		Iter-3 imported state (s_lr_martingale_under_null).
s_E_lr_at_tau_eq_1	state	E lr at tau eq 1		Iter-3 imported state (s_E_lr_at_tau_eq_1).
s_burkholder_davis_gundy	theorem	Burkholder–Davis–Gundy Inequality		Iter-3 imported theorem (s_burkholder_davis_gundy).
s_local_martingale_M_t	axiom	local martingale M t		Iter-3 imported axiom (s_local_martingale_M_t).
s_quadratic_variation_M	axiom	quadratic variation M		Iter-3 imported axiom (s_quadratic_variation_M).
s_good_lambda_inequality_pair	state	good lambda inequality pair		Iter-3 imported state (s_good_lambda_inequality_pair).
s_distribution_function_comparison	state	distribution function comparison		Iter-3 imported state (s_distribution_function_comparison).
s_khintchine_inequality	theorem	Khintchine Inequality		Iter-3 imported theorem (s_khintchine_inequality).
s_rademacher_random_signs_epsilon_i	axiom	rademacher random signs epsilon i		Iter-3 imported axiom (s_rademacher_random_signs_epsilon_i).
s_real_sequence_a_i	axiom	real sequence a i		Iter-3 imported axiom (s_real_sequence_a_i).
s_phi_sum_eps_i_a_i_explicit	state	phi sum eps i a i explicit		Iter-3 imported state (s_phi_sum_eps_i_a_i_explicit).
s_subgaussian_moment_bound	state	subgaussian moment bound		Iter-3 imported state (s_subgaussian_moment_bound).
s_levy_characterization_bm	theorem	Lévy's Characterization of Brownian Motion		Iter-3 imported theorem (s_levy_characterization_bm).
s_continuous_local_martingale_M_t	axiom	continuous local martingale M t		Iter-3 imported axiom (s_continuous_local_martingale_M_t).
s_quadratic_variation_equal_t	axiom	quadratic variation equal t		Iter-3 imported axiom (s_quadratic_variation_equal_t).
s_exponential_local_martingale_e_ixiM	state	exponential local martingale e ixiM		Iter-3 imported state (s_exponential_local_martingale_e_ixiM).
s_phi_t_evolves_as_minus_xi_squared_over_2	state	phi t evolves as minus xi squared over 2		Iter-3 imported state (s_phi_t_evolves_as_minus_xi_squared_over_2).
s_levy_modulus_continuity	theorem	Lévy Modulus of Continuity for BM		Iter-3 imported theorem (s_levy_modulus_continuity).
s_dyadic_increment_array	state	dyadic increment array		Iter-3 imported state (s_dyadic_increment_array).
s_borel_cantelli_bound_on_increments	state	borel cantelli bound on increments		Iter-3 imported state (s_borel_cantelli_bound_on_increments).
s_lil_hartman_wintner	theorem	Law of Iterated Logarithm (Hartman–Wintner)		Iter-3 imported theorem (s_lil_hartman_wintner).
s_random_walk_as_BM_at_stopping_times	state	random walk as BM at stopping times		Iter-3 imported state (s_random_walk_as_BM_at_stopping_times).
s_rescaled_random_walk	state	rescaled random walk		Iter-3 imported state (s_rescaled_random_walk).
s_lil_for_BM	state	lil for BM		Iter-3 imported state (s_lil_for_BM).
s_ito_isometry	theorem	Itô Isometry		Iter-3 imported theorem (s_ito_isometry).
s_progressive_L2_integrand_H	axiom	progressive L2 integrand H		Iter-3 imported axiom (s_progressive_L2_integrand_H).
s_simple_predictable_step_process	state	simple predictable step process		Iter-3 imported state (s_simple_predictable_step_process).
s_step_integral_L2_norm_eq_integral_H_squared	state	step integral L2 norm eq integral H squared		Iter-3 imported state (s_step_integral_L2_norm_eq_integral_H_squared).
s_cameron_martin_theorem	theorem	Cameron–Martin Theorem		Iter-3 imported theorem (s_cameron_martin_theorem).
s_wiener_measure_on_C0	axiom	wiener measure on C0		Iter-3 imported axiom (s_wiener_measure_on_C0).
s_cameron_martin_space_H1	axiom	cameron martin space H1		Iter-3 imported axiom (s_cameron_martin_space_H1).
s_translation_of_wiener_path	state	translation of wiener path		Iter-3 imported state (s_translation_of_wiener_path).
s_radon_nikodym_density_explicit	state	radon nikodym density explicit		Iter-3 imported state (s_radon_nikodym_density_explicit).
s_kolmogorov_backward_equation	theorem	Kolmogorov Backward Equation		Iter-3 imported theorem (s_kolmogorov_backward_equation).
s_markov_diffusion_X_t	axiom	markov diffusion X t		Iter-3 imported axiom (s_markov_diffusion_X_t).
s_smooth_terminal_payoff_f	axiom	smooth terminal payoff f		Iter-3 imported axiom (s_smooth_terminal_payoff_f).
s_expected_payoff_function_u	state	expected payoff function u		Iter-3 imported state (s_expected_payoff_function_u).
s_martingale_in_initial_time	state	martingale in initial time		Iter-3 imported state (s_martingale_in_initial_time).
s_fokker_planck_equation	theorem	Kolmogorov Forward (Fokker–Planck) Equation		Iter-3 imported theorem (s_fokker_planck_equation).
s_density_p_t_x	axiom	density p t x		Iter-3 imported axiom (s_density_p_t_x).
s_adjoint_operator_L_star	state	adjoint operator L star		Iter-3 imported state (s_adjoint_operator_L_star).
s_weak_form_for_density	state	weak form for density		Iter-3 imported state (s_weak_form_for_density).
s_stratonovich_ito_conversion	theorem	Stratonovich–Itô Conversion		Iter-3 imported theorem (s_stratonovich_ito_conversion).
s_ito_integral_definition	axiom	ito integral definition		Iter-3 imported axiom (s_ito_integral_definition).
s_stratonovich_integral_definition	axiom	stratonovich integral definition		Iter-3 imported axiom (s_stratonovich_integral_definition).
s_midpoint_correction_term	state	midpoint correction term		Iter-3 imported state (s_midpoint_correction_term).
s_half_partial_f_partial_x_times_dX_squared	state	half partial f partial x times dX squared		Iter-3 imported state (s_half_partial_f_partial_x_times_dX_squared).
s_sarkozy_theorem	theorem	Sárközy's Theorem (square-difference)		Iter-3 imported theorem (s_sarkozy_theorem).
s_square_step_set	axiom	square step set		Iter-3 imported axiom (s_square_step_set).
s_furstenberg_system_with_n_squared_action	state	furstenberg system with n squared action		Iter-3 imported state (s_furstenberg_system_with_n_squared_action).
s_spectral_resolution_for_polynomial_returns	state	spectral resolution for polynomial returns		Iter-3 imported state (s_spectral_resolution_for_polynomial_returns).
s_cramer_large_deviations	theorem	Cramér's Theorem (large deviations)		Iter-3 imported theorem (s_cramer_large_deviations).
s_iid_sequence_finite_mgf	axiom	iid sequence finite mgf		Iter-3 imported axiom (s_iid_sequence_finite_mgf).
s_cumulant_generating_function	state	cumulant generating function		Iter-3 imported state (s_cumulant_generating_function).
s_rate_function_Lambda_star	state	rate function Lambda star		Iter-3 imported state (s_rate_function_Lambda_star).
s_upper_bound_LDP	state	upper bound LDP		Iter-3 imported state (s_upper_bound_LDP).
s_gartner_ellis_theorem	theorem	Gärtner–Ellis Theorem		Iter-3 imported theorem (s_gartner_ellis_theorem).
s_sequence_random_variables_with_log_mgf_limit	axiom	sequence random variables with log mgf limit		Iter-3 imported axiom (s_sequence_random_variables_with_log_mgf_limit).
s_essential_smoothness_of_limit	axiom	essential smoothness of limit		Iter-3 imported axiom (s_essential_smoothness_of_limit).
s_abstract_cumulant_generator_Lambda	state	abstract cumulant generator Lambda		Iter-3 imported state (s_abstract_cumulant_generator_Lambda).
s_rate_function_Lambda_star_general	state	rate function Lambda star general		Iter-3 imported state (s_rate_function_Lambda_star_general).
s_schilder_theorem	theorem	Schilder's Theorem		Iter-3 imported theorem (s_schilder_theorem).
s_small_noise_brownian_path	state	small noise brownian path		Iter-3 imported state (s_small_noise_brownian_path).
s_path_density_under_shift	state	path density under shift		Iter-3 imported state (s_path_density_under_shift).
s_freidlin_wentzell_theorem	theorem	Freidlin–Wentzell Theorem		Iter-3 imported theorem (s_freidlin_wentzell_theorem).
s_sde_with_small_noise_eps_sigma	axiom	sde with small noise eps sigma		Iter-3 imported axiom (s_sde_with_small_noise_eps_sigma).
s_drift_vector_field_b	axiom	drift vector field b		Iter-3 imported axiom (s_drift_vector_field_b).
s_brownian_LDP_lifted	state	brownian LDP lifted		Iter-3 imported state (s_brownian_LDP_lifted).
s_continuous_image_LDP	state	continuous image LDP		Iter-3 imported state (s_continuous_image_LDP).
s_sanov_theorem	theorem	Sanov's Theorem		Iter-3 imported theorem (s_sanov_theorem).
s_iid_sequence_on_polish_space	axiom	iid sequence on polish space		Iter-3 imported axiom (s_iid_sequence_on_polish_space).
s_empirical_measure_L_n	axiom	empirical measure L n		Iter-3 imported axiom (s_empirical_measure_L_n).
s_type_class_partition	state	type class partition		Iter-3 imported state (s_type_class_partition).
s_KL_divergence_emerges_as_rate	state	KL divergence emerges as rate		Iter-3 imported state (s_KL_divergence_emerges_as_rate).
s_azuma_hoeffding	theorem	Azuma–Hoeffding Inequality		Iter-3 imported theorem (s_azuma_hoeffding).
s_martingale_with_bounded_differences	axiom	martingale with bounded differences		Iter-3 imported axiom (s_martingale_with_bounded_differences).
s_per_step_subgaussian_bound	state	per step subgaussian bound		Iter-3 imported state (s_per_step_subgaussian_bound).
s_exp_lambda_M_n_is_supermartingale	state	exp lambda M n is supermartingale		Iter-3 imported state (s_exp_lambda_M_n_is_supermartingale).
s_mcdiarmid_inequality	theorem	McDiarmid's Inequality		Iter-3 imported theorem (s_mcdiarmid_inequality).
s_independent_inputs_X_i	axiom	independent inputs X i		Iter-3 imported axiom (s_independent_inputs_X_i).
s_function_with_bounded_differences_c_i	axiom	function with bounded differences c i		Iter-3 imported axiom (s_function_with_bounded_differences_c_i).
s_doob_filtration_martingale	state	doob filtration martingale		Iter-3 imported state (s_doob_filtration_martingale).
s_bounded_differences_for_martingale	state	bounded differences for martingale		Iter-3 imported state (s_bounded_differences_for_martingale).
s_talagrand_concentration	theorem	Talagrand's Concentration Inequality		Iter-3 imported theorem (s_talagrand_concentration).
s_product_probability_space	axiom	product probability space		Iter-3 imported axiom (s_product_probability_space).
s_convex_lipschitz_function_F	axiom	convex lipschitz function F		Iter-3 imported axiom (s_convex_lipschitz_function_F).
s_talagrand_convex_distance	state	talagrand convex distance		Iter-3 imported state (s_talagrand_convex_distance).
s_isoperimetric_inequality_for_product	state	isoperimetric inequality for product		Iter-3 imported state (s_isoperimetric_inequality_for_product).
s_gaussian_log_sobolev	theorem	Logarithmic Sobolev Inequality (Gaussian)		Iter-3 imported theorem (s_gaussian_log_sobolev).
s_standard_gaussian_measure_gamma	axiom	standard gaussian measure gamma		Iter-3 imported axiom (s_standard_gaussian_measure_gamma).
s_smooth_function_f	axiom	smooth function f		Iter-3 imported axiom (s_smooth_function_f).
s_ornstein_uhlenbeck_semigroup	state	ornstein uhlenbeck semigroup		Iter-3 imported state (s_ornstein_uhlenbeck_semigroup).
s_bakry_emery_CD_inequality_for_OU	state	bakry emery CD inequality for OU		Iter-3 imported state (s_bakry_emery_CD_inequality_for_OU).
s_erdos_renyi_giant_component	theorem	Erdős–Rényi Giant Component Threshold		Iter-3 imported theorem (s_erdos_renyi_giant_component).
s_erdos_renyi_graph_G_n_p	axiom	erdos renyi graph G n p		Iter-3 imported axiom (s_erdos_renyi_graph_G_n_p).
s_branching_approximation_of_local_neighborhood	state	branching approximation of local neighborhood		Iter-3 imported state (s_branching_approximation_of_local_neighborhood).
s_phase_transition_at_lambda_eq_1	state	phase transition at lambda eq 1		Iter-3 imported state (s_phase_transition_at_lambda_eq_1).
s_erdos_renyi_connectedness_threshold	theorem	Erdős–Rényi Connectedness Threshold		Iter-3 imported theorem (s_erdos_renyi_connectedness_threshold).
s_isolated_vertex_count_X_n	state	isolated vertex count X n		Iter-3 imported state (s_isolated_vertex_count_X_n).
s_poisson_limit_for_isolated_vertices	state	poisson limit for isolated vertices		Iter-3 imported state (s_poisson_limit_for_isolated_vertices).
s_bollobas_zero_one_law	theorem	Bollobás 0–1 Law for Random Graphs		Iter-3 imported theorem (s_bollobas_zero_one_law).
s_first_order_graph_property_phi	axiom	first order graph property phi		Iter-3 imported axiom (s_first_order_graph_property_phi).
s_almost_sure_extension_axioms	state	almost sure extension axioms		Iter-3 imported state (s_almost_sure_extension_axioms).
s_limit_is_complete_theory_of_rado	state	limit is complete theory of rado		Iter-3 imported state (s_limit_is_complete_theory_of_rado).
s_wigner_semicircle_law	theorem	Wigner Semicircle Law		Iter-3 imported theorem (s_wigner_semicircle_law).
s_wigner_matrix	axiom	wigner matrix		Iter-3 imported axiom (s_wigner_matrix).
s_empirical_spectral_distribution	state	empirical spectral distribution		Iter-3 imported state (s_empirical_spectral_distribution).
s_moments_match_catalan_numbers	state	moments match catalan numbers		Iter-3 imported state (s_moments_match_catalan_numbers).
s_marchenko_pastur_law	theorem	Marchenko–Pastur Law		Iter-3 imported theorem (s_marchenko_pastur_law).
s_iid_rectangular_matrix_X	axiom	iid rectangular matrix X		Iter-3 imported axiom (s_iid_rectangular_matrix_X).
s_aspect_ratio_lambda	axiom	aspect ratio lambda		Iter-3 imported axiom (s_aspect_ratio_lambda).
s_sample_covariance_matrix_W	state	sample covariance matrix W		Iter-3 imported state (s_sample_covariance_matrix_W).
s_self_consistent_equation_for_m_z	state	self consistent equation for m z		Iter-3 imported state (s_self_consistent_equation_for_m_z).
s_tracy_widom_distribution	theorem	Tracy–Widom Distribution (largest eigenvalue)		Iter-3 imported theorem (s_tracy_widom_distribution).
s_gaussian_unitary_ensemble	axiom	gaussian unitary ensemble		Iter-3 imported axiom (s_gaussian_unitary_ensemble).
s_largest_eigenvalue_lambda_max	axiom	largest eigenvalue lambda max		Iter-3 imported axiom (s_largest_eigenvalue_lambda_max).
s_determinantal_point_process_at_edge	state	determinantal point process at edge		Iter-3 imported state (s_determinantal_point_process_at_edge).
s_rescaled_edge_point_process	state	rescaled edge point process		Iter-3 imported state (s_rescaled_edge_point_process).
s_voiculescu_free_clt	theorem	Voiculescu's Free CLT		Iter-3 imported theorem (s_voiculescu_free_clt).
s_freely_independent_random_variables	axiom	freely independent random variables		Iter-3 imported axiom (s_freely_independent_random_variables).
s_free_probability_space	axiom	free probability space		Iter-3 imported axiom (s_free_probability_space).
s_R_transform_of_a_i	state	R transform of a i		Iter-3 imported state (s_R_transform_of_a_i).
s_additivity_of_R_under_freeness	state	additivity of R under freeness		Iter-3 imported state (s_additivity_of_R_under_freeness).
s_R_transform_limit_eq_z	state	R transform limit eq z		Iter-3 imported state (s_R_transform_limit_eq_z).
s_asymptotic_freeness_GUE	theorem	Asymptotic Freeness of Independent GUE		Iter-3 imported theorem (s_asymptotic_freeness_GUE).
s_independent_GUE_pair_X_N_Y_N	axiom	independent GUE pair X N Y N		Iter-3 imported axiom (s_independent_GUE_pair_X_N_Y_N).
s_normalized_trace_tau	axiom	normalized trace tau		Iter-3 imported axiom (s_normalized_trace_tau).
s_mixed_moment_diagrams	state	mixed moment diagrams		Iter-3 imported state (s_mixed_moment_diagrams).
s_planar_diagrams_only	state	planar diagrams only		Iter-3 imported state (s_planar_diagrams_only).
s_yamada_watanabe_uniqueness	theorem	Yamada–Watanabe Pathwise Uniqueness Theorem		Iter-3 imported theorem (s_yamada_watanabe_uniqueness).
s_sde_with_1_d_diffusion_coefficient	axiom	sde with 1 d diffusion coefficient		Iter-3 imported axiom (s_sde_with_1_d_diffusion_coefficient).
s_holder_half_diffusion_coeff	axiom	holder half diffusion coeff		Iter-3 imported axiom (s_holder_half_diffusion_coeff).
s_smooth_approximations_to_abs_value	state	smooth approximations to abs value		Iter-3 imported state (s_smooth_approximations_to_abs_value).
s_local_time_at_zero_vanishes	state	local time at zero vanishes		Iter-3 imported state (s_local_time_at_zero_vanishes).
s_krylov_estimate	theorem	Krylov's Estimate for SDEs		Iter-3 imported theorem (s_krylov_estimate).
s_uniformly_elliptic_diffusion	axiom	uniformly elliptic diffusion		Iter-3 imported axiom (s_uniformly_elliptic_diffusion).
s_borel_function_f	axiom	borel function f		Iter-3 imported axiom (s_borel_function_f).
s_PDE_solution_to_obstacle_problem	state	PDE solution to obstacle problem		Iter-3 imported state (s_PDE_solution_to_obstacle_problem).
s_expectation_le_norm_f_Ld_bound	state	expectation le norm f Ld bound		Iter-3 imported state (s_expectation_le_norm_f_Ld_bound).
s_dynkin_formula	theorem	Dynkin's Formula		Iter-3 imported theorem (s_dynkin_formula).
s_markov_process_X_t_with_generator_A	axiom	markov process X t with generator A		Iter-3 imported axiom (s_markov_process_X_t_with_generator_A).
s_smooth_compactly_supported_f	axiom	smooth compactly supported f		Iter-3 imported axiom (s_smooth_compactly_supported_f).
s_dynkin_martingale	state	dynkin martingale		Iter-3 imported state (s_dynkin_martingale).
s_E_f_X_tau_minus_E_int_A_f_eq_f_x	state	E f X tau minus E int A f eq f x		Iter-3 imported state (s_E_f_X_tau_minus_E_int_A_f_eq_f_x).
s_hunt_theorem_potential	theorem	Hunt's Theorem on Potential		Iter-3 imported theorem (s_hunt_theorem_potential).
s_transient_markov_process	axiom	transient markov process		Iter-3 imported axiom (s_transient_markov_process).
s_potential_kernel_U	axiom	potential kernel U		Iter-3 imported axiom (s_potential_kernel_U).
s_capacity_via_hitting_probability	state	capacity via hitting probability		Iter-3 imported state (s_capacity_via_hitting_probability).
s_dual_potential_kernel_U_hat	state	dual potential kernel U hat		Iter-3 imported state (s_dual_potential_kernel_U_hat).
s_spitzer_identity	theorem	Spitzer's Identity		Iter-3 imported theorem (s_spitzer_identity).
s_iid_step_distribution_F	axiom	iid step distribution F		Iter-3 imported axiom (s_iid_step_distribution_F).
s_random_walk_S_n	axiom	random walk S n		Iter-3 imported axiom (s_random_walk_S_n).
s_running_maxima_M_n	state	running maxima M n		Iter-3 imported state (s_running_maxima_M_n).
s_wiener_hopf_factorization	state	wiener hopf factorization		Iter-3 imported state (s_wiener_hopf_factorization).
s_polya_recurrence	theorem	Pólya's Recurrence Theorem		Iter-3 imported theorem (s_polya_recurrence).
s_simple_random_walk_on_Z_d	axiom	simple random walk on Z d		Iter-3 imported axiom (s_simple_random_walk_on_Z_d).
s_characteristic_function_of_step	state	characteristic function of step		Iter-3 imported state (s_characteristic_function_of_step).
s_greens_function_at_origin_integral	state	greens function at origin integral		Iter-3 imported state (s_greens_function_at_origin_integral).
s_ballot_theorem	theorem	Ballot Theorem (Bertrand)		Iter-3 imported theorem (s_ballot_theorem).
s_two_candidate_election_count_p_q	axiom	two candidate election count p q		Iter-3 imported axiom (s_two_candidate_election_count_p_q).
s_sequential_vote_counts	axiom	sequential vote counts		Iter-3 imported axiom (s_sequential_vote_counts).
s_ballot_path_encoding	state	ballot path encoding		Iter-3 imported state (s_ballot_path_encoding).
s_cycle_lemma_count	state	cycle lemma count		Iter-3 imported state (s_cycle_lemma_count).
s_catalan_via_bertrand	theorem	Catalan Numbers via Bertrand		Iter-3 imported theorem (s_catalan_via_bertrand).
s_dyck_paths_2n	axiom	dyck paths 2n		Iter-3 imported axiom (s_dyck_paths_2n).
s_combinatorial_axioms	axiom	combinatorial axioms		Iter-3 imported axiom (s_combinatorial_axioms).
s_set_of_bad_paths	state	set of bad paths		Iter-3 imported state (s_set_of_bad_paths).
s_bijection_bad_to_unconstrained	state	bijection bad to unconstrained		Iter-3 imported state (s_bijection_bad_to_unconstrained).
s_stein_chen_poisson	theorem	Stein–Chen Poisson Approximation		Iter-3 imported theorem (s_stein_chen_poisson).
s_sum_of_dependent_indicators_W	axiom	sum of dependent indicators W		Iter-3 imported axiom (s_sum_of_dependent_indicators_W).
s_target_poisson_mean_lambda	axiom	target poisson mean lambda		Iter-3 imported axiom (s_target_poisson_mean_lambda).
s_stein_equation_for_poisson	state	stein equation for poisson		Iter-3 imported state (s_stein_equation_for_poisson).
s_lipschitz_bounds_on_stein_solution	state	lipschitz bounds on stein solution		Iter-3 imported state (s_lipschitz_bounds_on_stein_solution).
s_stein_method_gaussian	theorem	Stein's Method for Gaussian Approximation		Iter-3 imported theorem (s_stein_method_gaussian).
s_sum_of_dependent_random_variables_W	axiom	sum of dependent random variables W		Iter-3 imported axiom (s_sum_of_dependent_random_variables_W).
s_normal_target_N_0_1	axiom	normal target N 0 1		Iter-3 imported axiom (s_normal_target_N_0_1).
s_gaussian_stein_equation	state	gaussian stein equation		Iter-3 imported state (s_gaussian_stein_equation).
s_test_function_bounds	state	test function bounds		Iter-3 imported state (s_test_function_bounds).
s_stein_coupling_yields_error_terms	state	stein coupling yields error terms		Iter-3 imported state (s_stein_coupling_yields_error_terms).
s_chebyshev_inequality	theorem	Chebyshev's Inequality		Iter-3 imported theorem (s_chebyshev_inequality).
s_random_variable_finite_variance	axiom	random variable finite variance		Iter-3 imported axiom (s_random_variable_finite_variance).
s_indicator_event_upper_bound	state	indicator event upper bound		Iter-3 imported state (s_indicator_event_upper_bound).
s_pointwise_bound_for_indicator	state	pointwise bound for indicator		Iter-3 imported state (s_pointwise_bound_for_indicator).
s_markov_inequality	theorem	Markov's Inequality		Iter-3 imported theorem (s_markov_inequality).
s_nonnegative_random_variable	axiom	nonnegative random variable		Iter-3 imported axiom (s_nonnegative_random_variable).
s_indicator_dominated_by_X_over_a	state	indicator dominated by X over a		Iter-3 imported state (s_indicator_dominated_by_X_over_a).
s_jensen_inequality	theorem	Jensen's Inequality		Iter-3 imported theorem (s_jensen_inequality).
s_convex_function_phi	axiom	convex function phi		Iter-3 imported axiom (s_convex_function_phi).
s_integrable_random_variable_X	axiom	integrable random variable X		Iter-3 imported axiom (s_integrable_random_variable_X).
s_tangent_line_minorant_at_mean	state	tangent line minorant at mean		Iter-3 imported state (s_tangent_line_minorant_at_mean).
s_pointwise_tangent_bound	state	pointwise tangent bound		Iter-3 imported state (s_pointwise_tangent_bound).
s_levy_continuity_theorem	theorem	Lévy's Continuity Theorem		Iter-3 imported theorem (s_levy_continuity_theorem).
s_sequence_of_probability_measures_mu_n	axiom	sequence of probability measures mu n		Iter-3 imported axiom (s_sequence_of_probability_measures_mu_n).
s_characteristic_functions_phi_n	axiom	characteristic functions phi n		Iter-3 imported axiom (s_characteristic_functions_phi_n).
s_pointwise_phi_limit_phi	state	pointwise phi limit phi		Iter-3 imported state (s_pointwise_phi_limit_phi).
s_tightness_of_mu_n	state	tightness of mu n		Iter-3 imported state (s_tightness_of_mu_n).
s_helly_selection_theorem	theorem	Helly's Selection Theorem		Iter-3 imported theorem (s_helly_selection_theorem).
s_uniformly_bounded_monotone_function_sequence	axiom	uniformly bounded monotone function sequence		Iter-3 imported axiom (s_uniformly_bounded_monotone_function_sequence).
s_pointwise_limit_on_dense_set	state	pointwise limit on dense set		Iter-3 imported state (s_pointwise_limit_on_dense_set).
s_prokhorov_theorem	theorem	Prokhorov's Theorem		Iter-3 imported theorem (s_prokhorov_theorem).
s_tight_family_of_probability_measures	axiom	tight family of probability measures		Iter-3 imported axiom (s_tight_family_of_probability_measures).
s_tightness_compact_set_family	state	tightness compact set family		Iter-3 imported state (s_tightness_compact_set_family).
s_weakly_convergent_subsequence	state	weakly convergent subsequence		Iter-3 imported state (s_weakly_convergent_subsequence).
s_portmanteau_theorem	theorem	Portmanteau Theorem		Iter-3 imported theorem (s_portmanteau_theorem).
s_metric_space	axiom	metric space		Iter-3 imported axiom (s_metric_space).
s_weak_convergence_definition	axiom	weak convergence definition		Iter-3 imported axiom (s_weak_convergence_definition).
s_equivalent_test_classes	state	equivalent test classes		Iter-3 imported state (s_equivalent_test_classes).
s_limsup_le_open_liminf_ge_closed	state	limsup le open liminf ge closed		Iter-3 imported state (s_limsup_le_open_liminf_ge_closed).
s_dkw_inequality	theorem	Dvoretzky–Kiefer–Wolfowitz Inequality		Iter-3 imported theorem (s_dkw_inequality).
s_iid_real_random_variables_with_cdf_F	axiom	iid real random variables with cdf F		Iter-3 imported axiom (s_iid_real_random_variables_with_cdf_F).
s_empirical_cdf_F_n	axiom	empirical cdf F n		Iter-3 imported axiom (s_empirical_cdf_F_n).
s_kmt_strong_approximation	state	kmt strong approximation		Iter-3 imported state (s_kmt_strong_approximation).
s_subgaussian_tail_for_KS_statistic	state	subgaussian tail for KS statistic		Iter-3 imported state (s_subgaussian_tail_for_KS_statistic).
s_kolmogorov_smirnov_law	theorem	Kolmogorov–Smirnov Test Statistic Law		Iter-3 imported theorem (s_kolmogorov_smirnov_law).
s_empirical_process	state	empirical process		Iter-3 imported state (s_empirical_process).
s_brownian_bridge_limit	state	brownian bridge limit		Iter-3 imported state (s_brownian_bridge_limit).
s_cochran_theorem	theorem	Cochran's Theorem		Iter-3 imported theorem (s_cochran_theorem).
s_iid_gaussian_vector_n	axiom	iid gaussian vector n		Iter-3 imported axiom (s_iid_gaussian_vector_n).
s_orthogonal_decomposition_of_quadratic_forms	axiom	orthogonal decomposition of quadratic forms		Iter-3 imported axiom (s_orthogonal_decomposition_of_quadratic_forms).
s_orthogonal_idempotents_summing_to_I	state	orthogonal idempotents summing to I		Iter-3 imported state (s_orthogonal_idempotents_summing_to_I).
s_each_quadratic_form_is_chi_squared	state	each quadratic form is chi squared		Iter-3 imported state (s_each_quadratic_form_is_chi_squared).
s_slutsky_theorem	theorem	Slutsky's Theorem		Iter-3 imported theorem (s_slutsky_theorem).
s_X_n_weak_to_X	axiom	X n weak to X		Iter-3 imported axiom (s_X_n_weak_to_X).
s_Y_n_to_constant_c	axiom	Y n to constant c		Iter-3 imported axiom (s_Y_n_to_constant_c).
s_joint_convergence_pair	state	joint convergence pair		Iter-3 imported state (s_joint_convergence_pair).
s_apply_continuous_mapping_theorem	state	apply continuous mapping theorem		Iter-3 imported state (s_apply_continuous_mapping_theorem).
s_continuous_mapping_theorem	theorem	Continuous Mapping Theorem		Iter-3 imported theorem (s_continuous_mapping_theorem).
s_X_n_to_X_in_some_mode	axiom	X n to X in some mode		Iter-3 imported axiom (s_X_n_to_X_in_some_mode).
s_g_continuous_on_support_of_X	axiom	g continuous on support of X		Iter-3 imported axiom (s_g_continuous_on_support_of_X).
s_portmanteau_test_function_class	state	portmanteau test function class		Iter-3 imported state (s_portmanteau_test_function_class).
s_h_circ_g_bounded_continuous	state	h circ g bounded continuous		Iter-3 imported state (s_h_circ_g_bounded_continuous).
s_wiener_existence	theorem	Daniell–Kolmogorov Construction of Brownian Motion		Iter-3 imported theorem (s_wiener_existence).
s_gaussian_finite_dim_marginals_with_BM_covariance	axiom	gaussian finite dim marginals with BM covariance		Iter-3 imported axiom (s_gaussian_finite_dim_marginals_with_BM_covariance).
s_continuity_modulus_estimates	axiom	continuity modulus estimates		Iter-3 imported axiom (s_continuity_modulus_estimates).
s_law_on_path_space_without_continuity	state	law on path space without continuity		Iter-3 imported state (s_law_on_path_space_without_continuity).
s_holder_continuous_version	state	holder continuous version		Iter-3 imported state (s_holder_continuous_version).
s_markov_chain_ergodic_theorem	theorem	Markov Chain Ergodic Theorem		Iter-3 imported theorem (s_markov_chain_ergodic_theorem).
s_irreducible_aperiodic_positive_recurrent_markov_chain	axiom	irreducible aperiodic positive recurrent markov chain		Iter-3 imported axiom (s_irreducible_aperiodic_positive_recurrent_markov_chain).
s_invariant_distribution_pi	axiom	invariant distribution pi		Iter-3 imported axiom (s_invariant_distribution_pi).
s_regeneration_at_atom_via_nummelin_splitting	state	regeneration at atom via nummelin splitting		Iter-3 imported state (s_regeneration_at_atom_via_nummelin_splitting).
s_iid_excursions_yield_LLN	state	iid excursions yield LLN		Iter-3 imported state (s_iid_excursions_yield_LLN).
s_perron_frobenius_stochastic	theorem	Perron–Frobenius for Stochastic Matrices		Iter-3 imported theorem (s_perron_frobenius_stochastic).
s_finite_irreducible_stochastic_matrix_P	axiom	finite irreducible stochastic matrix P		Iter-3 imported axiom (s_finite_irreducible_stochastic_matrix_P).
s_probability_simplex	axiom	probability simplex		Iter-3 imported axiom (s_probability_simplex).
s_invariant_distribution_existence	state	invariant distribution existence		Iter-3 imported state (s_invariant_distribution_existence).
s_largest_eigenvalue_one_simple	state	largest eigenvalue one simple		Iter-3 imported state (s_largest_eigenvalue_one_simple).
s_de_finetti_theorem	theorem	De Finetti's Exchangeability Theorem		Iter-3 imported theorem (s_de_finetti_theorem).
s_infinite_exchangeable_sequence	axiom	infinite exchangeable sequence		Iter-3 imported axiom (s_infinite_exchangeable_sequence).
s_polish_state_space	axiom	polish state space		Iter-3 imported axiom (s_polish_state_space).
s_symmetric_law_under_permutations	state	symmetric law under permutations		Iter-3 imported state (s_symmetric_law_under_permutations).
s_random_empirical_limit_M	state	random empirical limit M		Iter-3 imported state (s_random_empirical_limit_M).
s_poisson_process_characterization	theorem	Poisson Process Existence and Characterization		Iter-3 imported theorem (s_poisson_process_characterization).
s_rate_measure_lambda_on_sigma_finite_space	axiom	rate measure lambda on sigma finite space		Iter-3 imported axiom (s_rate_measure_lambda_on_sigma_finite_space).
s_independence_and_poisson_count_axioms	axiom	independence and poisson count axioms		Iter-3 imported axiom (s_independence_and_poisson_count_axioms).
s_atomic_partition_construction	state	atomic partition construction		Iter-3 imported state (s_atomic_partition_construction).
s_finite_dim_distributions_match	state	finite dim distributions match		Iter-3 imported state (s_finite_dim_distributions_match).
s_levy_khintchine_representation	theorem	Lévy–Khintchine Representation		Iter-3 imported theorem (s_levy_khintchine_representation).
s_levy_process_X_t	axiom	levy process X t		Iter-3 imported axiom (s_levy_process_X_t).
s_characteristic_function_phi_t	axiom	characteristic function phi t		Iter-3 imported axiom (s_characteristic_function_phi_t).
s_infinitely_divisible_law	state	infinitely divisible law		Iter-3 imported state (s_infinitely_divisible_law).
s_jump_compensated_decomposition	state	jump compensated decomposition		Iter-3 imported state (s_jump_compensated_decomposition).
s_ito_representation_theorem	theorem	Itô Representation Theorem (martingale representation)		Iter-3 imported theorem (s_ito_representation_theorem).
s_brownian_filtration_F_t	axiom	brownian filtration F t		Iter-3 imported axiom (s_brownian_filtration_F_t).
s_L2_F_T_random_variable	axiom	L2 F T random variable		Iter-3 imported axiom (s_L2_F_T_random_variable).
s_dense_set_of_explicit_stochastic_integrals	state	dense set of explicit stochastic integrals		Iter-3 imported state (s_dense_set_of_explicit_stochastic_integrals).
s_density_in_L2_F_T	state	density in L2 F T		Iter-3 imported state (s_density_in_L2_F_T).
s_clark_ocone_formula	theorem	Clark–Ocone Formula		Iter-3 imported theorem (s_clark_ocone_formula).
s_malliavin_differentiable_random_variable_F	axiom	malliavin differentiable random variable F		Iter-3 imported axiom (s_malliavin_differentiable_random_variable_F).
s_malliavin_derivative_D_F	state	malliavin derivative D F		Iter-3 imported state (s_malliavin_derivative_D_F).
s_predictable_projection_of_D_F	state	predictable projection of D F		Iter-3 imported state (s_predictable_projection_of_D_F).
s_hoeffding_inequality	theorem	Hoeffding's Inequality		Iter-3 imported theorem (s_hoeffding_inequality).
s_independent_bounded_random_variables	axiom	independent bounded random variables		Iter-3 imported axiom (s_independent_bounded_random_variables).
s_per_variable_subgaussian_mgf_bound	state	per variable subgaussian mgf bound		Iter-3 imported state (s_per_variable_subgaussian_mgf_bound).
s_mgf_of_sum_bound	state	mgf of sum bound		Iter-3 imported state (s_mgf_of_sum_bound).
s_bernstein_inequality	theorem	Bernstein's Inequality		Iter-3 imported theorem (s_bernstein_inequality).
s_finite_variance_sum_sigma2	axiom	finite variance sum sigma2		Iter-3 imported axiom (s_finite_variance_sum_sigma2).
s_per_variable_bernstein_mgf_bound	state	per variable bernstein mgf bound		Iter-3 imported state (s_per_variable_bernstein_mgf_bound).
s_sum_log_mgf_bound	state	sum log mgf bound		Iter-3 imported state (s_sum_log_mgf_bound).
s_chernoff_bound	theorem	Chernoff Bound		Iter-3 imported theorem (s_chernoff_bound).
s_random_variable_with_mgf	axiom	random variable with mgf		Iter-3 imported axiom (s_random_variable_with_mgf).
s_exponential_markov_bound	state	exponential markov bound		Iter-3 imported state (s_exponential_markov_bound).
s_doob_Lp_maximal_inequality	theorem	Doob's L^p Maximal Inequality		Iter-3 imported theorem (s_doob_Lp_maximal_inequality).
s_p_strictly_greater_than_1	axiom	p strictly greater than 1		Iter-3 imported axiom (s_p_strictly_greater_than_1).
s_weak_type_1_1_for_maximum	state	weak type 1 1 for maximum		Iter-3 imported state (s_weak_type_1_1_for_maximum).
s_layer_cake_to_Lp_norm	state	layer cake to Lp norm		Iter-3 imported state (s_layer_cake_to_Lp_norm).
s_coupling_inequality	theorem	Lindvall's Coupling Inequality		Iter-3 imported theorem (s_coupling_inequality).
s_two_random_variables_on_common_space	axiom	two random variables on common space		Iter-3 imported axiom (s_two_random_variables_on_common_space).
s_coupling_distribution	axiom	coupling distribution		Iter-3 imported axiom (s_coupling_distribution).
s_coupling_time_tau	state	coupling time tau		Iter-3 imported state (s_coupling_time_tau).
s_tv_bound_via_coupling_time	state	tv bound via coupling time		Iter-3 imported state (s_tv_bound_via_coupling_time).
s_kingman_coalescent	theorem	Kingman's Coalescent Existence		Iter-3 imported theorem (s_kingman_coalescent).
s_genealogy_under_neutral_evolution	axiom	genealogy under neutral evolution		Iter-3 imported axiom (s_genealogy_under_neutral_evolution).
s_haploid_population_model_N_haploids	axiom	haploid population model N haploids		Iter-3 imported axiom (s_haploid_population_model_N_haploids).
s_rescaled_coalescence_time	state	rescaled coalescence time		Iter-3 imported state (s_rescaled_coalescence_time).
s_pairwise_rate_in_limit	state	pairwise rate in limit		Iter-3 imported state (s_pairwise_rate_in_limit).
s_mixing_time_spectral_gap	theorem	Mixing Time and Spectral Gap		Iter-3 imported theorem (s_mixing_time_spectral_gap).
s_reversible_finite_markov_chain_P	axiom	reversible finite markov chain P		Iter-3 imported axiom (s_reversible_finite_markov_chain_P).
s_spectral_gap_gamma	axiom	spectral gap gamma		Iter-3 imported axiom (s_spectral_gap_gamma).
s_spectral_decomposition_of_P_in_L2_pi	state	spectral decomposition of P in L2 pi		Iter-3 imported state (s_spectral_decomposition_of_P_in_L2_pi).
s_L2_decay_bound	state	L2 decay bound		Iter-3 imported state (s_L2_decay_bound).
s_aldous_hoover_theorem	theorem	Aldous–Hoover Theorem (exchangeable arrays)		Iter-3 imported theorem (s_aldous_hoover_theorem).
s_jointly_exchangeable_random_array	axiom	jointly exchangeable random array		Iter-3 imported axiom (s_jointly_exchangeable_random_array).
s_double_index_exchangeability	state	double index exchangeability		Iter-3 imported state (s_double_index_exchangeability).
s_latent_variable_representation	state	latent variable representation		Iter-3 imported state (s_latent_variable_representation).
s_strassen_theorem	theorem	Strassen's Theorem (coupling characterization)		Iter-3 imported theorem (s_strassen_theorem).
s_two_probability_measures_mu_nu	axiom	two probability measures mu nu		Iter-3 imported axiom (s_two_probability_measures_mu_nu).
s_metric_or_closed_relation	axiom	metric or closed relation		Iter-3 imported axiom (s_metric_or_closed_relation).
s_lp_duality_for_optimal_transport_form	state	lp duality for optimal transport form		Iter-3 imported state (s_lp_duality_for_optimal_transport_form).
s_optimal_coupling_exists	state	optimal coupling exists		Iter-3 imported state (s_optimal_coupling_exists).
s_kantorovich_rubinstein_duality	theorem	Kantorovich–Rubinstein Duality		Iter-3 imported theorem (s_kantorovich_rubinstein_duality).
s_two_probability_measures_with_finite_first_moment	axiom	two probability measures with finite first moment		Iter-3 imported axiom (s_two_probability_measures_with_finite_first_moment).
s_polish_metric_space	axiom	polish metric space		Iter-3 imported axiom (s_polish_metric_space).
s_W1_as_primal_optimal_transport	state	W1 as primal optimal transport		Iter-3 imported state (s_W1_as_primal_optimal_transport).
s_dual_problem_over_1_lipschitz	state	dual problem over 1 lipschitz		Iter-3 imported state (s_dual_problem_over_1_lipschitz).
s_ehrhard_inequality	theorem	Brunn–Minkowski for Gaussian (Ehrhard inequality)		Iter-3 imported theorem (s_ehrhard_inequality).
s_standard_gaussian_measure_in_R_n	axiom	standard gaussian measure in R n		Iter-3 imported axiom (s_standard_gaussian_measure_in_R_n).
s_two_borel_sets_A_B	axiom	two borel sets A B		Iter-3 imported axiom (s_two_borel_sets_A_B).
s_gaussian_isoperimetric_profile	state	gaussian isoperimetric profile		Iter-3 imported state (s_gaussian_isoperimetric_profile).
s_one_dim_ehrhard_reduction	state	one dim ehrhard reduction		Iter-3 imported state (s_one_dim_ehrhard_reduction).
s_sle_existence	theorem	Stochastic Loewner Evolution Existence (SLE)		Iter-3 imported theorem (s_sle_existence).
s_loewner_chordal_ODE	axiom	loewner chordal ODE		Iter-3 imported axiom (s_loewner_chordal_ODE).
s_sle_kappa_driver	state	sle kappa driver		Iter-3 imported state (s_sle_kappa_driver).
s_conformal_maps_g_t	state	conformal maps g t		Iter-3 imported state (s_conformal_maps_g_t).
s_kolmogorov_two_series_theorem	theorem	Two-Series Theorem (Kolmogorov)		Iter-3 imported theorem (s_kolmogorov_two_series_theorem).
s_independent_centered_random_variables	axiom	independent centered random variables		Iter-3 imported axiom (s_independent_centered_random_variables).
s_uniform_partial_sum_bound	state	uniform partial sum bound		Iter-3 imported state (s_uniform_partial_sum_bound).
s_a_s_convergence_via_levy_equivalence	state	a s convergence via levy equivalence		Iter-3 imported state (s_a_s_convergence_via_levy_equivalence).
s_levy_equivalence_theorem	theorem	Lévy's Equivalence Theorem		Iter-3 imported theorem (s_levy_equivalence_theorem).
s_sum_of_independent_random_variables	axiom	sum of independent random variables		Iter-3 imported axiom (s_sum_of_independent_random_variables).
s_symmetrized_independent_sum	state	symmetrized independent sum		Iter-3 imported state (s_symmetrized_independent_sum).
s_max_partial_sum_concentration	state	max partial sum concentration		Iter-3 imported state (s_max_partial_sum_concentration).
s_stein_lemma_gaussian	theorem	Stein's Lemma (Gaussian integration by parts)		Iter-3 imported theorem (s_stein_lemma_gaussian).
s_standard_gaussian_Z	axiom	standard gaussian Z		Iter-3 imported axiom (s_standard_gaussian_Z).
s_absolutely_continuous_function_g	axiom	absolutely continuous function g		Iter-3 imported axiom (s_absolutely_continuous_function_g).
s_gaussian_density_ODE_identity	state	gaussian density ODE identity		Iter-3 imported state (s_gaussian_density_ODE_identity).
s_IBP_with_gaussian_density	state	IBP with gaussian density		Iter-3 imported state (s_IBP_with_gaussian_density).
s_slepian_inequality	theorem	Slepian's Inequality (Gaussian comparison)		Iter-3 imported theorem (s_slepian_inequality).
s_two_centered_gaussian_vectors_X_Y	axiom	two centered gaussian vectors X Y		Iter-3 imported axiom (s_two_centered_gaussian_vectors_X_Y).
s_pairwise_covariance_inequality	axiom	pairwise covariance inequality		Iter-3 imported axiom (s_pairwise_covariance_inequality).
s_gaussian_interpolation_path	state	gaussian interpolation path		Iter-3 imported state (s_gaussian_interpolation_path).
s_derivative_sign_via_covariance_difference	state	derivative sign via covariance difference		Iter-3 imported state (s_derivative_sign_via_covariance_difference).
s_borell_tis_inequality	theorem	Borell–TIS Inequality (Gaussian concentration of maxima)		Iter-3 imported theorem (s_borell_tis_inequality).
s_centered_gaussian_process_X_t_separable	axiom	centered gaussian process X t separable		Iter-3 imported axiom (s_centered_gaussian_process_X_t_separable).
s_almost_sure_finite_sup	axiom	almost sure finite sup		Iter-3 imported axiom (s_almost_sure_finite_sup).
s_sup_as_lipschitz_function_of_iid_gaussians	state	sup as lipschitz function of iid gaussians		Iter-3 imported state (s_sup_as_lipschitz_function_of_iid_gaussians).
s_lipschitz_gaussian_concentration_bound	state	lipschitz gaussian concentration bound		Iter-3 imported state (s_lipschitz_gaussian_concentration_bound).
s_dudley_entropy_bound	theorem	Dudley's Entropy Bound		Iter-3 imported theorem (s_dudley_entropy_bound).
s_subgaussian_process_on_metric_space_T	axiom	subgaussian process on metric space T		Iter-3 imported axiom (s_subgaussian_process_on_metric_space_T).
s_metric_entropy_log_N_eps	axiom	metric entropy log N eps		Iter-3 imported axiom (s_metric_entropy_log_N_eps).
s_dyadic_chaining_nets	state	dyadic chaining nets		Iter-3 imported state (s_dyadic_chaining_nets).
s_chaining_subgaussian_bound	state	chaining subgaussian bound		Iter-3 imported state (s_chaining_subgaussian_bound).
s_donsker_varadhan_LDP	theorem	Donsker–Varadhan Variational Formula		Iter-3 imported theorem (s_donsker_varadhan_LDP).
s_ergodic_markov_process	axiom	ergodic markov process		Iter-3 imported axiom (s_ergodic_markov_process).
s_empirical_occupation_measure	axiom	empirical occupation measure		Iter-3 imported axiom (s_empirical_occupation_measure).
s_variational_rate_function	state	variational rate function		Iter-3 imported state (s_variational_rate_function).
s_DV_rate_function_eq_dirichlet_form	state	DV rate function eq dirichlet form		Iter-3 imported state (s_DV_rate_function_eq_dirichlet_form).
s_varadhan_lemma	theorem	Varadhan's Lemma		Iter-3 imported theorem (s_varadhan_lemma).
s_family_satisfying_LDP_with_rate_I	axiom	family satisfying LDP with rate I		Iter-3 imported axiom (s_family_satisfying_LDP_with_rate_I).
s_continuous_bounded_function_F	axiom	continuous bounded function F		Iter-3 imported axiom (s_continuous_bounded_function_F).
s_laplace_upper_bound	state	laplace upper bound		Iter-3 imported state (s_laplace_upper_bound).
s_laplace_lower_bound	state	laplace lower bound		Iter-3 imported state (s_laplace_lower_bound).
s_contraction_principle_LDP	theorem	Contraction Principle (large deviations)		Iter-3 imported theorem (s_contraction_principle_LDP).
s_family_satisfying_LDP_with_rate_I_on_X	axiom	family satisfying LDP with rate I on X		Iter-3 imported axiom (s_family_satisfying_LDP_with_rate_I_on_X).
s_continuous_map_F_X_to_Y	axiom	continuous map F X to Y		Iter-3 imported axiom (s_continuous_map_F_X_to_Y).
s_push_forward_family_on_Y	state	push forward family on Y		Iter-3 imported state (s_push_forward_family_on_Y).
s_LDP_with_inverse_image_rate	state	LDP with inverse image rate		Iter-3 imported state (s_LDP_with_inverse_image_rate).
s_blackwell_renewal_theorem	theorem	Renewal Theorem (Blackwell)		Iter-3 imported theorem (s_blackwell_renewal_theorem).
s_iid_positive_inter_arrival_times	axiom	iid positive inter arrival times		Iter-3 imported axiom (s_iid_positive_inter_arrival_times).
s_renewal_function_U_t	axiom	renewal function U t		Iter-3 imported axiom (s_renewal_function_U_t).
s_stationary_renewal_coupling	state	stationary renewal coupling		Iter-3 imported state (s_stationary_renewal_coupling).
s_TV_distance_to_stationary_renewal_zero	state	TV distance to stationary renewal zero		Iter-3 imported state (s_TV_distance_to_stationary_renewal_zero).
s_kesten_stigum_theorem	theorem	Kesten–Stigum Theorem (Branching processes)		Iter-3 imported theorem (s_kesten_stigum_theorem).
s_supercritical_galton_watson_process	axiom	supercritical galton watson process		Iter-3 imported axiom (s_supercritical_galton_watson_process).
s_x_log_x_moment_condition	axiom	x log x moment condition		Iter-3 imported axiom (s_x_log_x_moment_condition).
s_normalized_population_martingale_W_n	state	normalized population martingale W n		Iter-3 imported state (s_normalized_population_martingale_W_n).
s_W_infty_nondegenerate	state	W infty nondegenerate		Iter-3 imported state (s_W_infty_nondegenerate).
s_robbins_monro_convergence	theorem	Stochastic Approximation (Robbins–Monro) Convergence		Iter-3 imported theorem (s_robbins_monro_convergence).
s_root_finding_problem_h_x_eq_0	axiom	root finding problem h x eq 0		Iter-3 imported axiom (s_root_finding_problem_h_x_eq_0).
s_step_size_sequence_alpha_n_with_robbins_monro_conditions	axiom	step size sequence alpha n with robbins monro conditions		Iter-3 imported axiom (s_step_size_sequence_alpha_n_with_robbins_monro_conditions).
s_lyapunov_for_robbins_monro	state	lyapunov for robbins monro		Iter-3 imported state (s_lyapunov_for_robbins_monro).
s_supermartingale_convergence_for_V_n	state	supermartingale convergence for V n		Iter-3 imported state (s_supermartingale_convergence_for_V_n).
s_strassen_functional_lil	theorem	Functional Law of the Iterated Logarithm (Strassen)		Iter-3 imported theorem (s_strassen_functional_lil).
s_brownian_embedding_of_random_walk	state	brownian embedding of random walk		Iter-3 imported state (s_brownian_embedding_of_random_walk).
s_rescaled_brownian_paths	state	rescaled brownian paths		Iter-3 imported state (s_rescaled_brownian_paths).
s_cluster_set_is_strassen_ball	state	cluster set is strassen ball		Iter-3 imported state (s_cluster_set_is_strassen_ball).
s_karhunen_loeve_expansion	theorem	Karhunen–Loève Expansion		Iter-3 imported theorem (s_karhunen_loeve_expansion).
s_L2_centered_stochastic_process_with_continuous_covariance	axiom	L2 centered stochastic process with continuous covariance		Iter-3 imported axiom (s_L2_centered_stochastic_process_with_continuous_covariance).
s_hilbert_schmidt_kernel	axiom	hilbert schmidt kernel		Iter-3 imported axiom (s_hilbert_schmidt_kernel).
s_eigenfunction_eigenvalue_decomposition	state	eigenfunction eigenvalue decomposition		Iter-3 imported state (s_eigenfunction_eigenvalue_decomposition).
s_uncorrelated_coefficient_sequence	state	uncorrelated coefficient sequence		Iter-3 imported state (s_uncorrelated_coefficient_sequence).
s_urysohn_metrization_theorem	theorem	Urysohn metrization theorem		Iter-3 imported theorem (s_urysohn_metrization_theorem).
s_countable_separating_family	state	countable separating family		Iter-3 imported state (s_countable_separating_family).
s_embedding_into_hilbert_cube	state	embedding into hilbert cube		Iter-3 imported state (s_embedding_into_hilbert_cube).
s_nagata_smirnov_metrization	theorem	Nagata–Smirnov metrization theorem		Iter-3 imported theorem (s_nagata_smirnov_metrization).
s_regular_hausdorff_space	axiom	regular hausdorff space		Iter-3 imported axiom (s_regular_hausdorff_space).
s_sigma_locally_finite_basis	axiom	sigma locally finite basis		Iter-3 imported axiom (s_sigma_locally_finite_basis).
s_sigma_locally_finite_basis_present	state	sigma locally finite basis present		Iter-3 imported state (s_sigma_locally_finite_basis_present).
s_countable_family_of_pseudometrics	state	countable family of pseudometrics		Iter-3 imported state (s_countable_family_of_pseudometrics).
s_compatible_metric	state	compatible metric		Iter-3 imported state (s_compatible_metric).
s_bing_metrization	theorem	Bing metrization theorem		Iter-3 imported theorem (s_bing_metrization).
s_sigma_discrete_basis_present	state	sigma discrete basis present		Iter-3 imported state (s_sigma_discrete_basis_present).
s_nagata_smirnov_form	state	nagata smirnov form		Iter-3 imported state (s_nagata_smirnov_form).
s_stone_cech_compactification	theorem	Stone–Čech compactification		Iter-3 imported theorem (s_stone_cech_compactification).
s_tychonoff_space	axiom	tychonoff space		Iter-3 imported axiom (s_tychonoff_space).
s_evaluation_into_product_of_intervals	state	evaluation into product of intervals		Iter-3 imported state (s_evaluation_into_product_of_intervals).
s_compact_hausdorff_closure_beta_X	state	compact hausdorff closure beta X		Iter-3 imported state (s_compact_hausdorff_closure_beta_X).
s_alexandroff_one_point_compactification	theorem	Alexandroff one-point compactification		Iter-3 imported theorem (s_alexandroff_one_point_compactification).
s_x_plus_infinity_set	state	x plus infinity set		Iter-3 imported state (s_x_plus_infinity_set).
s_one_point_topology	state	one point topology		Iter-3 imported state (s_one_point_topology).
s_lebesgue_number_lemma	theorem	Lebesgue number lemma		Iter-3 imported theorem (s_lebesgue_number_lemma).
s_open_cover_of_compact_set	axiom	open cover of compact set		Iter-3 imported axiom (s_open_cover_of_compact_set).
s_finite_subcover_with_radii	state	finite subcover with radii		Iter-3 imported state (s_finite_subcover_with_radii).
s_positive_lebesgue_number_delta	state	positive lebesgue number delta		Iter-3 imported state (s_positive_lebesgue_number_delta).
s_pasting_lemma	theorem	Pasting lemma		Iter-3 imported theorem (s_pasting_lemma).
s_topological_space	axiom	topological space		Iter-3 imported axiom (s_topological_space).
s_finite_closed_cover_with_continuous_pieces	axiom	finite closed cover with continuous pieces		Iter-3 imported axiom (s_finite_closed_cover_with_continuous_pieces).
s_preimage_of_closed_is_closed_per_piece	state	preimage of closed is closed per piece		Iter-3 imported state (s_preimage_of_closed_is_closed_per_piece).
s_heine_cantor_theorem	theorem	Heine–Cantor theorem (uniform continuity on compacts)		Iter-3 imported theorem (s_heine_cantor_theorem).
s_continuous_function_on_compact_metric_space	axiom	continuous function on compact metric space		Iter-3 imported axiom (s_continuous_function_on_compact_metric_space).
s_epsilon_continuity_finite_subcover	state	epsilon continuity finite subcover		Iter-3 imported state (s_epsilon_continuity_finite_subcover).
s_alexander_subbase_lemma	theorem	Alexander subbase lemma		Iter-3 imported theorem (s_alexander_subbase_lemma).
s_subbase_of_topology	axiom	subbase of topology		Iter-3 imported axiom (s_subbase_of_topology).
s_maximal_subbase_anticover_via_zorn	state	maximal subbase anticover via zorn		Iter-3 imported state (s_maximal_subbase_anticover_via_zorn).
s_michael_selection_theorem	theorem	Michael selection theorem		Iter-3 imported theorem (s_michael_selection_theorem).
s_paracompact_hausdorff_space	axiom	paracompact hausdorff space		Iter-3 imported axiom (s_paracompact_hausdorff_space).
s_lower_semicontinuous_convex_valued_correspondence	axiom	lower semicontinuous convex valued correspondence		Iter-3 imported axiom (s_lower_semicontinuous_convex_valued_correspondence).
s_epsilon_approximate_selection	state	epsilon approximate selection		Iter-3 imported state (s_epsilon_approximate_selection).
s_cauchy_sequence_of_selections	state	cauchy sequence of selections		Iter-3 imported state (s_cauchy_sequence_of_selections).
s_sorgenfrey_not_metrizable	theorem	Sorgenfrey line not metrizable		Iter-3 imported theorem (s_sorgenfrey_not_metrizable).
s_sorgenfrey_line	axiom	sorgenfrey line		Iter-3 imported axiom (s_sorgenfrey_line).
s_sorgenfrey_squared_not_lindelof	state	sorgenfrey squared not lindelof		Iter-3 imported state (s_sorgenfrey_squared_not_lindelof).
s_hilbert_cube_embedding	theorem	Embedding into Hilbert cube		Iter-3 imported theorem (s_hilbert_cube_embedding).
s_compact_metrizable_space	axiom	compact metrizable space		Iter-3 imported axiom (s_compact_metrizable_space).
s_countable_dense_subset_of_X	state	countable dense subset of X		Iter-3 imported state (s_countable_dense_subset_of_X).
s_distance_to_dense_coords	state	distance to dense coords		Iter-3 imported state (s_distance_to_dense_coords).
s_compact_iff_sequentially_compact_metric	theorem	Compactness ⇔ sequential compactness in metric spaces		Iter-3 imported theorem (s_compact_iff_sequentially_compact_metric).
s_totally_bounded_and_complete	state	totally bounded and complete		Iter-3 imported state (s_totally_bounded_and_complete).
s_cauchy_subsequence_extraction	state	cauchy subsequence extraction		Iter-3 imported state (s_cauchy_subsequence_extraction).
s_cantor_bendixson_theorem	theorem	Cantor–Bendixson theorem		Iter-3 imported theorem (s_cantor_bendixson_theorem).
s_cantor_bendixson_derivatives	state	cantor bendixson derivatives		Iter-3 imported state (s_cantor_bendixson_derivatives).
s_perfect_kernel_plus_countable_scattered	state	perfect kernel plus countable scattered		Iter-3 imported state (s_perfect_kernel_plus_countable_scattered).
s_lusin_separation_theorem	theorem	Lusin separation theorem		Iter-3 imported theorem (s_lusin_separation_theorem).
s_pair_of_disjoint_analytic_sets	axiom	pair of disjoint analytic sets		Iter-3 imported axiom (s_pair_of_disjoint_analytic_sets).
s_souslin_scheme_separation	state	souslin scheme separation		Iter-3 imported state (s_souslin_scheme_separation).
s_borel_separating_set	state	borel separating set		Iter-3 imported state (s_borel_separating_set).
s_sierpinski_continuum_theorem	theorem	Sierpiński theorem on continuum connectedness		Iter-3 imported theorem (s_sierpinski_continuum_theorem).
s_compact_connected_hausdorff_space	axiom	compact connected hausdorff space		Iter-3 imported axiom (s_compact_connected_hausdorff_space).
s_countable_closed_partition_hypothesis	state	countable closed partition hypothesis		Iter-3 imported state (s_countable_closed_partition_hypothesis).
s_contradiction_with_compactness	state	contradiction with compactness		Iter-3 imported state (s_contradiction_with_compactness).
s_borsuk_ulam_theorem	theorem	Borsuk–Ulam theorem		Iter-3 imported theorem (s_borsuk_ulam_theorem).
s_sphere_S_n	axiom	sphere S n		Iter-3 imported axiom (s_sphere_S_n).
s_continuous_map_S_n_to_R_n	axiom	continuous map S n to R n		Iter-3 imported axiom (s_continuous_map_S_n_to_R_n).
s_antipodal_free_map_to_S_n_minus_1	state	antipodal free map to S n minus 1		Iter-3 imported state (s_antipodal_free_map_to_S_n_minus_1).
s_Z2_equivariant_map_to_lower_sphere	state	Z2 equivariant map to lower sphere		Iter-3 imported state (s_Z2_equivariant_map_to_lower_sphere).
s_ham_sandwich_theorem	theorem	Ham sandwich theorem		Iter-3 imported theorem (s_ham_sandwich_theorem).
s_n_finite_measures_in_R_n	axiom	n finite measures in R n		Iter-3 imported axiom (s_n_finite_measures_in_R_n).
s_signed_half_volume_map_on_S_n	state	signed half volume map on S n		Iter-3 imported state (s_signed_half_volume_map_on_S_n).
s_odd_map_S_n_to_R_n	state	odd map S n to R n		Iter-3 imported state (s_odd_map_S_n_to_R_n).
s_lyusternik_schnirelmann_theorem	theorem	Lyusternik–Schnirelmann theorem		Iter-3 imported theorem (s_lyusternik_schnirelmann_theorem).
s_closed_cover_S_n_by_n_plus_1_sets	axiom	closed cover S n by n plus 1 sets		Iter-3 imported axiom (s_closed_cover_S_n_by_n_plus_1_sets).
s_antipodal_separated_cover_hypothesis	state	antipodal separated cover hypothesis		Iter-3 imported state (s_antipodal_separated_cover_hypothesis).
s_distance_vector_map_S_n_to_R_n	state	distance vector map S n to R n		Iter-3 imported state (s_distance_vector_map_S_n_to_R_n).
s_lefschetz_fixed_point_theorem	theorem	Lefschetz fixed-point theorem		Iter-3 imported theorem (s_lefschetz_fixed_point_theorem).
s_compact_polyhedron	axiom	compact polyhedron		Iter-3 imported axiom (s_compact_polyhedron).
s_continuous_self_map_of_polyhedron	axiom	continuous self map of polyhedron		Iter-3 imported axiom (s_continuous_self_map_of_polyhedron).
s_induced_map_on_H_star_Q	state	induced map on H star Q		Iter-3 imported state (s_induced_map_on_H_star_Q).
s_lefschetz_number_L_f	state	lefschetz number L f		Iter-3 imported state (s_lefschetz_number_L_f).
s_schauder_fixed_point_theorem	theorem	Schauder fixed-point theorem		Iter-3 imported theorem (s_schauder_fixed_point_theorem).
s_compact_convex_subset_banach_space	axiom	compact convex subset banach space		Iter-3 imported axiom (s_compact_convex_subset_banach_space).
s_continuous_self_map_on_convex_compact	axiom	continuous self map on convex compact		Iter-3 imported axiom (s_continuous_self_map_on_convex_compact).
s_finite_dim_approx_self_map	state	finite dim approx self map		Iter-3 imported state (s_finite_dim_approx_self_map).
s_finite_dim_approx_fixed_points	state	finite dim approx fixed points		Iter-3 imported state (s_finite_dim_approx_fixed_points).
s_kakutani_fixed_point_theorem	theorem	Kakutani fixed-point theorem		Iter-3 imported theorem (s_kakutani_fixed_point_theorem).
s_compact_convex_subset_R_n	axiom	compact convex subset R n		Iter-3 imported axiom (s_compact_convex_subset_R_n).
s_upper_hemicontinuous_convex_valued_correspondence	axiom	upper hemicontinuous convex valued correspondence		Iter-3 imported axiom (s_upper_hemicontinuous_convex_valued_correspondence).
s_continuous_selection_on_simplex	state	continuous selection on simplex		Iter-3 imported state (s_continuous_selection_on_simplex).
s_approximate_fixed_points	state	approximate fixed points		Iter-3 imported state (s_approximate_fixed_points).
s_hairy_ball_theorem	theorem	Hairy ball theorem		Iter-3 imported theorem (s_hairy_ball_theorem).
s_even_sphere_S_2n	axiom	even sphere S 2n		Iter-3 imported axiom (s_even_sphere_S_2n).
s_tangent_vector_field	axiom	tangent vector field		Iter-3 imported axiom (s_tangent_vector_field).
s_nonvanishing_field_hypothesis	state	nonvanishing field hypothesis		Iter-3 imported state (s_nonvanishing_field_hypothesis).
s_homotopy_id_to_antipodal	state	homotopy id to antipodal		Iter-3 imported state (s_homotopy_id_to_antipodal).
s_jordan_brouwer_separation	theorem	Jordan–Brouwer separation theorem		Iter-3 imported theorem (s_jordan_brouwer_separation).
s_topological_n_minus_1_sphere_in_R_n	axiom	topological n minus 1 sphere in R n		Iter-3 imported axiom (s_topological_n_minus_1_sphere_in_R_n).
s_n_dim_complement_of_sphere	state	n dim complement of sphere		Iter-3 imported state (s_n_dim_complement_of_sphere).
s_two_components_via_duality	state	two components via duality		Iter-3 imported state (s_two_components_via_duality).
s_schoenflies_theorem	theorem	Schoenflies theorem		Iter-3 imported theorem (s_schoenflies_theorem).
s_conformal_map_disk_to_inside	state	conformal map disk to inside		Iter-3 imported state (s_conformal_map_disk_to_inside).
s_extension_to_boundary_homeomorphism	state	extension to boundary homeomorphism		Iter-3 imported state (s_extension_to_boundary_homeomorphism).
s_hex_theorem	theorem	Hex theorem ⇒ Brouwer		Iter-3 imported theorem (s_hex_theorem).
s_hex_board_game	axiom	hex board game		Iter-3 imported axiom (s_hex_board_game).
s_winning_chain_existence	state	winning chain existence		Iter-3 imported state (s_winning_chain_existence).
s_sperner_lemma	theorem	Sperner's lemma		Iter-3 imported theorem (s_sperner_lemma).
s_simplex_triangulation_with_sperner_labeling	axiom	simplex triangulation with sperner labeling		Iter-3 imported axiom (s_simplex_triangulation_with_sperner_labeling).
s_door_counting_argument	state	door counting argument		Iter-3 imported state (s_door_counting_argument).
s_odd_number_of_full_simplices	state	odd number of full simplices		Iter-3 imported state (s_odd_number_of_full_simplices).
s_kkm_theorem	theorem	KKM theorem (Knaster–Kuratowski–Mazurkiewicz)		Iter-3 imported theorem (s_kkm_theorem).
s_simplex_with_KKM_cover	axiom	simplex with KKM cover		Iter-3 imported axiom (s_simplex_with_KKM_cover).
s_sperner_labeled_triangulation_from_cover	state	sperner labeled triangulation from cover		Iter-3 imported state (s_sperner_labeled_triangulation_from_cover).
s_fully_labeled_small_simplex	state	fully labeled small simplex		Iter-3 imported state (s_fully_labeled_small_simplex).
s_lebesgue_covering_dimension_R_n_equals_n	theorem	Topological dimension equals covering dimension		Iter-3 imported theorem (s_lebesgue_covering_dimension_R_n_equals_n).
s_open_cover_with_orders	axiom	open cover with orders		Iter-3 imported axiom (s_open_cover_with_orders).
s_nerve_of_cover_simplicial_complex	state	nerve of cover simplicial complex		Iter-3 imported state (s_nerve_of_cover_simplicial_complex).
s_no_refinement_below_order_n	state	no refinement below order n		Iter-3 imported state (s_no_refinement_below_order_n).
s_pi_1_S1_equals_Z	theorem	Fundamental group of S¹ is ℤ		Iter-3 imported theorem (s_pi_1_S1_equals_Z).
s_circle_S_1	axiom	circle S 1		Iter-3 imported axiom (s_circle_S_1).
s_universal_cover_R_to_S_1	axiom	universal cover R to S 1		Iter-3 imported axiom (s_universal_cover_R_to_S_1).
s_lifted_path_with_integer_endpoint	state	lifted path with integer endpoint		Iter-3 imported state (s_lifted_path_with_integer_endpoint).
s_winding_number_map_pi_1_to_Z	state	winding number map pi 1 to Z		Iter-3 imported state (s_winding_number_map_pi_1_to_Z).
s_pi_1_S_n_trivial	theorem	Fundamental group of S^n trivial for n ≥ 2		Iter-3 imported theorem (s_pi_1_S_n_trivial).
s_sphere_S_n_n_geq_2	axiom	sphere S n n geq 2		Iter-3 imported axiom (s_sphere_S_n_n_geq_2).
s_two_hemispheres_cover	state	two hemispheres cover		Iter-3 imported state (s_two_hemispheres_cover).
s_seifert_van_kampen_theorem	theorem	Seifert–van Kampen theorem		Iter-3 imported theorem (s_seifert_van_kampen_theorem).
s_path_connected_space	axiom	path connected space		Iter-3 imported axiom (s_path_connected_space).
s_open_cover_by_path_connected_opens_with_path_connected_intersection	axiom	open cover by path connected opens with path connected intersection		Iter-3 imported axiom (s_open_cover_by_path_connected_opens_with_path_connected_intersection).
s_pushout_diagram_of_pi_1	state	pushout diagram of pi 1		Iter-3 imported state (s_pushout_diagram_of_pi_1).
s_loop_decomposition_into_segments_in_pieces	state	loop decomposition into segments in pieces		Iter-3 imported state (s_loop_decomposition_into_segments_in_pieces).
s_universal_cover_existence	theorem	Universal cover existence		Iter-3 imported theorem (s_universal_cover_existence).
s_path_connected_locally_simply_connected_space	axiom	path connected locally simply connected space		Iter-3 imported axiom (s_path_connected_locally_simply_connected_space).
s_homotopy_class_path_space	state	homotopy class path space		Iter-3 imported state (s_homotopy_class_path_space).
s_topologized_path_space_with_projection	state	topologized path space with projection		Iter-3 imported state (s_topologized_path_space_with_projection).
s_covering_space_galois_correspondence	theorem	Galois correspondence of covering spaces		Iter-3 imported theorem (s_covering_space_galois_correspondence).
s_fundamental_group	axiom	fundamental group		Iter-3 imported axiom (s_fundamental_group).
s_deck_group_action_on_universal_cover	state	deck group action on universal cover		Iter-3 imported state (s_deck_group_action_on_universal_cover).
s_classification_of_compact_surfaces	theorem	Classification of surfaces (Smale / Brahana)		Iter-3 imported theorem (s_classification_of_compact_surfaces).
s_compact_surface_without_boundary	axiom	compact surface without boundary		Iter-3 imported axiom (s_compact_surface_without_boundary).
s_triangulated_compact_surface	state	triangulated compact surface		Iter-3 imported state (s_triangulated_compact_surface).
s_polygon_identification_form	state	polygon identification form		Iter-3 imported state (s_polygon_identification_form).
s_eilenberg_steenrod_uniqueness	theorem	Eilenberg–Steenrod axioms uniqueness		Iter-3 imported theorem (s_eilenberg_steenrod_uniqueness).
s_cw_complex_category	axiom	cw complex category		Iter-3 imported axiom (s_cw_complex_category).
s_homotopy_excision_dimension_axioms	axiom	homotopy excision dimension axioms		Iter-3 imported axiom (s_homotopy_excision_dimension_axioms).
s_axiom_system_on_finite_cw	state	axiom system on finite cw		Iter-3 imported state (s_axiom_system_on_finite_cw).
s_inductive_determination_by_cells	state	inductive determination by cells		Iter-3 imported state (s_inductive_determination_by_cells).
s_mayer_vietoris_sequence	theorem	Mayer–Vietoris sequence		Iter-3 imported theorem (s_mayer_vietoris_sequence).
s_open_cover_X_equals_A_union_B	axiom	open cover X equals A union B		Iter-3 imported axiom (s_open_cover_X_equals_A_union_B).
s_short_exact_chain_sequence	state	short exact chain sequence		Iter-3 imported state (s_short_exact_chain_sequence).
s_long_exact_sequence_in_homology	state	long exact sequence in homology		Iter-3 imported state (s_long_exact_sequence_in_homology).
s_excision_theorem	theorem	Excision theorem		Iter-3 imported theorem (s_excision_theorem).
s_topological_pair_X_A	axiom	topological pair X A		Iter-3 imported axiom (s_topological_pair_X_A).
s_subset_U_with_closure_in_interior_A	axiom	subset U with closure in interior A		Iter-3 imported axiom (s_subset_U_with_closure_in_interior_A).
s_small_chains_subdivision	state	small chains subdivision		Iter-3 imported state (s_small_chains_subdivision).
s_subdivision_operator_chain_homotopy	state	subdivision operator chain homotopy		Iter-3 imported state (s_subdivision_operator_chain_homotopy).
s_lefschetz_duality	theorem	Lefschetz duality		Iter-3 imported theorem (s_lefschetz_duality).
s_compact_oriented_manifold_with_boundary	axiom	compact oriented manifold with boundary		Iter-3 imported axiom (s_compact_oriented_manifold_with_boundary).
s_double_M_along_boundary	state	double M along boundary		Iter-3 imported state (s_double_M_along_boundary).
s_duality_on_double	state	duality on double		Iter-3 imported state (s_duality_on_double).
s_kunneth_formula	theorem	Künneth formula		Iter-3 imported theorem (s_kunneth_formula).
s_topological_spaces_X_Y	axiom	topological spaces X Y		Iter-3 imported axiom (s_topological_spaces_X_Y).
s_pid_coefficients	axiom	pid coefficients		Iter-3 imported axiom (s_pid_coefficients).
s_eilenberg_zilber_cross_product	state	eilenberg zilber cross product		Iter-3 imported state (s_eilenberg_zilber_cross_product).
s_tor_short_exact_sequence	state	tor short exact sequence		Iter-3 imported state (s_tor_short_exact_sequence).
s_eilenberg_zilber_theorem	theorem	Eilenberg–Zilber theorem		Iter-3 imported theorem (s_eilenberg_zilber_theorem).
s_AW_EZ_chain_maps	state	AW EZ chain maps		Iter-3 imported state (s_AW_EZ_chain_maps).
s_chain_homotopy_equivalence	state	chain homotopy equivalence		Iter-3 imported state (s_chain_homotopy_equivalence).
s_sheaf_de_rham_isomorphism	theorem	Sheaf-theoretic de Rham (Poincaré lemma + sheafification)		Iter-3 imported theorem (s_sheaf_de_rham_isomorphism).
s_de_rham_complex	axiom	de rham complex		Iter-3 imported axiom (s_de_rham_complex).
s_poincare_lemma_local_resolution	state	poincare lemma local resolution		Iter-3 imported state (s_poincare_lemma_local_resolution).
s_sheaf_resolution_of_constant_sheaf	state	sheaf resolution of constant sheaf		Iter-3 imported state (s_sheaf_resolution_of_constant_sheaf).
s_cup_product_associative_graded_commutative	theorem	Künneth in cohomology / cup product associativity		Iter-3 imported theorem (s_cup_product_associative_graded_commutative).
s_singular_cochain_complex	axiom	singular cochain complex		Iter-3 imported axiom (s_singular_cochain_complex).
s_cup_product_definition	state	cup product definition		Iter-3 imported state (s_cup_product_definition).
s_kunneth_for_sheaves	theorem	Künneth for sheaf cohomology		Iter-3 imported theorem (s_kunneth_for_sheaves).
s_compact_spaces_with_field_coefficients	axiom	compact spaces with field coefficients		Iter-3 imported axiom (s_compact_spaces_with_field_coefficients).
s_sheafified_external_product	state	sheafified external product		Iter-3 imported state (s_sheafified_external_product).
s_cech_singular_iso_good_cover	theorem	Čech-to-singular isomorphism on good covers		Iter-3 imported theorem (s_cech_singular_iso_good_cover).
s_good_cover_with_contractible_intersections	axiom	good cover with contractible intersections		Iter-3 imported axiom (s_good_cover_with_contractible_intersections).
s_nerve_simplicial_model	state	nerve simplicial model		Iter-3 imported state (s_nerve_simplicial_model).
s_cech_double_complex	state	cech double complex		Iter-3 imported state (s_cech_double_complex).
s_hurewicz_theorem	theorem	Hurewicz theorem		Iter-3 imported theorem (s_hurewicz_theorem).
s_n_connected_space_n_geq_1	axiom	n connected space n geq 1		Iter-3 imported axiom (s_n_connected_space_n_geq_1).
s_hurewicz_homomorphism	state	hurewicz homomorphism		Iter-3 imported state (s_hurewicz_homomorphism).
s_cellular_representatives	state	cellular representatives		Iter-3 imported state (s_cellular_representatives).
s_whitehead_theorem	theorem	Whitehead theorem		Iter-3 imported theorem (s_whitehead_theorem).
s_cw_complexes_X_Y	axiom	cw complexes X Y		Iter-3 imported axiom (s_cw_complexes_X_Y).
s_weak_homotopy_equivalence_f	axiom	weak homotopy equivalence f		Iter-3 imported axiom (s_weak_homotopy_equivalence_f).
s_vanishing_obstructions_in_relative_pi_n	state	vanishing obstructions in relative pi n		Iter-3 imported state (s_vanishing_obstructions_in_relative_pi_n).
s_homotopy_inverse_on_skeleta	state	homotopy inverse on skeleta		Iter-3 imported state (s_homotopy_inverse_on_skeleta).
s_freudenthal_suspension_theorem	theorem	Freudenthal suspension theorem		Iter-3 imported theorem (s_freudenthal_suspension_theorem).
s_n_connected_space_X	axiom	n connected space X		Iter-3 imported axiom (s_n_connected_space_X).
s_suspension_map_on_pi_k	state	suspension map on pi k		Iter-3 imported state (s_suspension_map_on_pi_k).
s_homotopy_excision_range	state	homotopy excision range		Iter-3 imported state (s_homotopy_excision_range).
s_blakers_massey_theorem	theorem	Blakers–Massey theorem		Iter-3 imported theorem (s_blakers_massey_theorem).
s_pushout_square_in_spaces	axiom	pushout square in spaces		Iter-3 imported axiom (s_pushout_square_in_spaces).
s_connectivity_hypotheses	axiom	connectivity hypotheses		Iter-3 imported axiom (s_connectivity_hypotheses).
s_homotopy_pushout_comparison	state	homotopy pushout comparison		Iter-3 imported state (s_homotopy_pushout_comparison).
s_total_fiber_connectivity_bound	state	total fiber connectivity bound		Iter-3 imported state (s_total_fiber_connectivity_bound).
s_cellular_approximation	theorem	Cellular approximation theorem		Iter-3 imported theorem (s_cellular_approximation).
s_continuous_map_f	axiom	continuous map f		Iter-3 imported axiom (s_continuous_map_f).
s_image_meets_finitely_many_cells	state	image meets finitely many cells		Iter-3 imported state (s_image_meets_finitely_many_cells).
s_homotopy_to_lower_skeleton	state	homotopy to lower skeleton		Iter-3 imported state (s_homotopy_to_lower_skeleton).
s_simplicial_approximation	theorem	Simplicial approximation theorem		Iter-3 imported theorem (s_simplicial_approximation).
s_simplicial_complexes_K_L	axiom	simplicial complexes K L		Iter-3 imported axiom (s_simplicial_complexes_K_L).
s_continuous_map_K_to_L	axiom	continuous map K to L		Iter-3 imported axiom (s_continuous_map_K_to_L).
s_lebesgue_number_for_open_stars	state	lebesgue number for open stars		Iter-3 imported state (s_lebesgue_number_for_open_stars).
s_finely_subdivided_K	state	finely subdivided K		Iter-3 imported state (s_finely_subdivided_K).
s_brown_representability	theorem	Brown representability theorem		Iter-3 imported theorem (s_brown_representability).
s_homotopy_invariant_functor_on_pointed_cw	axiom	homotopy invariant functor on pointed cw		Iter-3 imported axiom (s_homotopy_invariant_functor_on_pointed_cw).
s_wedge_and_mayer_vietoris_axioms	axiom	wedge and mayer vietoris axioms		Iter-3 imported axiom (s_wedge_and_mayer_vietoris_axioms).
s_half_exact_functor	state	half exact functor		Iter-3 imported state (s_half_exact_functor).
s_classifying_space_Y_universal_element	state	classifying space Y universal element		Iter-3 imported state (s_classifying_space_Y_universal_element).
s_postnikov_tower_existence	theorem	Postnikov tower existence		Iter-3 imported theorem (s_postnikov_tower_existence).
s_connected_cw_complex_X	axiom	connected cw complex X		Iter-3 imported axiom (s_connected_cw_complex_X).
s_X_n_with_pi_k_killed_above_n	state	X n with pi k killed above n		Iter-3 imported state (s_X_n_with_pi_k_killed_above_n).
s_k_invariant_obstruction	state	k invariant obstruction		Iter-3 imported state (s_k_invariant_obstruction).
s_K_G_n_represents_H_n_G	theorem	Eilenberg–MacLane spaces represent cohomology		Iter-3 imported theorem (s_K_G_n_represents_H_n_G).
s_abelian_group_G	axiom	abelian group G		Iter-3 imported axiom (s_abelian_group_G).
s_cw_complex_X	axiom	cw complex X		Iter-3 imported axiom (s_cw_complex_X).
s_K_G_n_space	state	K G n space		Iter-3 imported state (s_K_G_n_space).
s_serre_spectral_sequence	theorem	Serre spectral sequence		Iter-3 imported theorem (s_serre_spectral_sequence).
s_serre_fibration	axiom	serre fibration		Iter-3 imported axiom (s_serre_fibration).
s_simply_connected_base	axiom	simply connected base		Iter-3 imported axiom (s_simply_connected_base).
s_skeleton_filtration_of_total_space	state	skeleton filtration of total space		Iter-3 imported state (s_skeleton_filtration_of_total_space).
s_exact_couple_E_2_to_E_infinity	state	exact couple E 2 to E infinity		Iter-3 imported state (s_exact_couple_E_2_to_E_infinity).
s_leray_hirsch_theorem	theorem	Leray–Hirsch theorem		Iter-3 imported theorem (s_leray_hirsch_theorem).
s_fiber_bundle_F_to_E_to_B	axiom	fiber bundle F to E to B		Iter-3 imported axiom (s_fiber_bundle_F_to_E_to_B).
s_classes_restrict_to_basis_on_fiber	axiom	classes restrict to basis on fiber		Iter-3 imported axiom (s_classes_restrict_to_basis_on_fiber).
s_external_module_map	state	external module map		Iter-3 imported state (s_external_module_map).
s_dold_thom_theorem	theorem	Dold–Thom theorem		Iter-3 imported theorem (s_dold_thom_theorem).
s_pointed_cw_complex_X	axiom	pointed cw complex X		Iter-3 imported axiom (s_pointed_cw_complex_X).
s_infinite_symmetric_product_SPX	state	infinite symmetric product SPX		Iter-3 imported state (s_infinite_symmetric_product_SPX).
s_SP_preserves_quasifibrations	state	SP preserves quasifibrations		Iter-3 imported state (s_SP_preserves_quasifibrations).
s_eilenberg_ganea_theorem	theorem	Eilenberg–Ganea theorem		Iter-3 imported theorem (s_eilenberg_ganea_theorem).
s_discrete_group_G	axiom	discrete group G		Iter-3 imported axiom (s_discrete_group_G).
s_cohomological_dimension_n_geq_3	axiom	cohomological dimension n geq 3		Iter-3 imported axiom (s_cohomological_dimension_n_geq_3).
s_low_dim_skeleton_of_K_G_1	state	low dim skeleton of K G 1		Iter-3 imported state (s_low_dim_skeleton_of_K_G_1).
s_vanishing_obstruction_in_high_dim	state	vanishing obstruction in high dim		Iter-3 imported state (s_vanishing_obstruction_in_high_dim).
s_barratt_priddy_theorem	theorem	Barratt–Priddy theorem		Iter-3 imported theorem (s_barratt_priddy_theorem).
s_symmetric_groups_Sigma_n	axiom	symmetric groups Sigma n		Iter-3 imported axiom (s_symmetric_groups_Sigma_n).
s_classifying_space_B_sigma_infty	state	classifying space B sigma infty		Iter-3 imported state (s_classifying_space_B_sigma_infty).
s_plus_construction_B_sigma_infty_plus	state	plus construction B sigma infty plus		Iter-3 imported state (s_plus_construction_B_sigma_infty_plus).
s_milnor_lim_one_exact_sequence	theorem	Milnor exact sequence (lim¹)		Iter-3 imported theorem (s_milnor_lim_one_exact_sequence).
s_inverse_system_of_groups	axiom	inverse system of groups		Iter-3 imported axiom (s_inverse_system_of_groups).
s_tower_of_fibrations	axiom	tower of fibrations		Iter-3 imported axiom (s_tower_of_fibrations).
s_homotopy_inverse_limit_construction	state	homotopy inverse limit construction		Iter-3 imported state (s_homotopy_inverse_limit_construction).
s_quillen_theorem_A	theorem	Quillen's theorem A		Iter-3 imported theorem (s_quillen_theorem_A).
s_functor_F_between_small_categories	axiom	functor F between small categories		Iter-3 imported axiom (s_functor_F_between_small_categories).
s_comma_categories_contractible	axiom	comma categories contractible		Iter-3 imported axiom (s_comma_categories_contractible).
s_bisimplicial_replacement	state	bisimplicial replacement		Iter-3 imported state (s_bisimplicial_replacement).
s_realization_homotopy_equivalence	state	realization homotopy equivalence		Iter-3 imported state (s_realization_homotopy_equivalence).
s_adams_spectral_sequence	theorem	Adams spectral sequence		Iter-3 imported theorem (s_adams_spectral_sequence).
s_spectrum_X	axiom	spectrum X		Iter-3 imported axiom (s_spectrum_X).
s_mod_p_cohomology_with_steenrod_action	axiom	mod p cohomology with steenrod action		Iter-3 imported axiom (s_mod_p_cohomology_with_steenrod_action).
s_adams_resolution_tower	state	adams resolution tower		Iter-3 imported state (s_adams_resolution_tower).
s_E_2_equals_ext_over_steenrod_algebra	state	E 2 equals ext over steenrod algebra		Iter-3 imported state (s_E_2_equals_ext_over_steenrod_algebra).
s_nishida_nilpotence	theorem	Nishida nilpotence theorem		Iter-3 imported theorem (s_nishida_nilpotence).
s_stable_homotopy_groups_of_spheres	axiom	stable homotopy groups of spheres		Iter-3 imported axiom (s_stable_homotopy_groups_of_spheres).
s_positive_degree_element	axiom	positive degree element		Iter-3 imported axiom (s_positive_degree_element).
s_dyer_lashof_action_on_powers	state	dyer lashof action on powers		Iter-3 imported state (s_dyer_lashof_action_on_powers).
s_x_to_the_N_vanishes	state	x to the N vanishes		Iter-3 imported state (s_x_to_the_N_vanishes).
s_hopf_invariant_one_theorem	theorem	Hopf invariant one (Adams)		Iter-3 imported theorem (s_hopf_invariant_one_theorem).
s_continuous_map_S_2n_minus_1_to_S_n	axiom	continuous map S 2n minus 1 to S n		Iter-3 imported axiom (s_continuous_map_S_2n_minus_1_to_S_n).
s_hopf_invariant_H_f	state	hopf invariant H f		Iter-3 imported state (s_hopf_invariant_H_f).
s_secondary_operation_obstruction	state	secondary operation obstruction		Iter-3 imported state (s_secondary_operation_obstruction).
s_adams_vector_fields_theorem	theorem	Vector fields on spheres (Adams)		Iter-3 imported theorem (s_adams_vector_fields_theorem).
s_sphere_S_n_minus_1	axiom	sphere S n minus 1		Iter-3 imported axiom (s_sphere_S_n_minus_1).
s_max_linearly_independent_vector_fields	axiom	max linearly independent vector fields		Iter-3 imported axiom (s_max_linearly_independent_vector_fields).
s_stiefel_manifold_K_theory	state	stiefel manifold K theory		Iter-3 imported state (s_stiefel_manifold_K_theory).
s_adams_operation_obstruction_psi_k	state	adams operation obstruction psi k		Iter-3 imported state (s_adams_operation_obstruction_psi_k).
s_adams_conjecture_image_of_J	theorem	J-homomorphism image (Adams conjecture / Quillen)		Iter-3 imported theorem (s_adams_conjecture_image_of_J).
s_real_vector_bundle_xi	axiom	real vector bundle xi		Iter-3 imported axiom (s_real_vector_bundle_xi).
s_adams_operation_psi_k	axiom	adams operation psi k		Iter-3 imported axiom (s_adams_operation_psi_k).
s_J_homomorphism_image	state	J homomorphism image		Iter-3 imported state (s_J_homomorphism_image).
s_psi_k_minus_1_action	state	psi k minus 1 action		Iter-3 imported state (s_psi_k_minus_1_action).
s_transversality_theorem	theorem	Transversality theorem		Iter-3 imported theorem (s_transversality_theorem).
s_smooth_map_f_X_to_Y	axiom	smooth map f X to Y		Iter-3 imported axiom (s_smooth_map_f_X_to_Y).
s_submanifold_Z_of_Y	axiom	submanifold Z of Y		Iter-3 imported axiom (s_submanifold_Z_of_Y).
s_parametrized_family_F	state	parametrized family F		Iter-3 imported state (s_parametrized_family_F).
s_generic_parameter_makes_f_transverse	state	generic parameter makes f transverse		Iter-3 imported state (s_generic_parameter_makes_f_transverse).
s_whitney_embedding_theorem	theorem	Whitney embedding theorem		Iter-3 imported theorem (s_whitney_embedding_theorem).
s_smooth_n_manifold_M	axiom	smooth n manifold M		Iter-3 imported axiom (s_smooth_n_manifold_M).
s_initial_immersion_into_high_R_N	state	initial immersion into high R N		Iter-3 imported state (s_initial_immersion_into_high_R_N).
s_generic_projection_remains_immersion	state	generic projection remains immersion		Iter-3 imported state (s_generic_projection_remains_immersion).
s_whitney_immersion_theorem	theorem	Whitney immersion theorem		Iter-3 imported theorem (s_whitney_immersion_theorem).
s_normal_bundle_obstruction_class	state	normal bundle obstruction class		Iter-3 imported state (s_normal_bundle_obstruction_class).
s_obstruction_vanishes	state	obstruction vanishes		Iter-3 imported state (s_obstruction_vanishes).
s_ehresmann_lemma	theorem	Ehresmann's lemma		Iter-3 imported theorem (s_ehresmann_lemma).
s_proper_smooth_submersion_f_M_to_N	axiom	proper smooth submersion f M to N		Iter-3 imported axiom (s_proper_smooth_submersion_f_M_to_N).
s_horizontal_distribution	state	horizontal distribution		Iter-3 imported state (s_horizontal_distribution).
s_flow_lifts_local_trivialization	state	flow lifts local trivialization		Iter-3 imported state (s_flow_lifts_local_trivialization).
s_morse_lemma	theorem	Morse lemma		Iter-3 imported theorem (s_morse_lemma).
s_smooth_function_with_nondegenerate_critical_point	axiom	smooth function with nondegenerate critical point		Iter-3 imported axiom (s_smooth_function_with_nondegenerate_critical_point).
s_quadratic_part_plus_remainder	state	quadratic part plus remainder		Iter-3 imported state (s_quadratic_part_plus_remainder).
s_canonical_quadratic_form_in_charts	state	canonical quadratic form in charts		Iter-3 imported state (s_canonical_quadratic_form_in_charts).
s_h_cobordism_theorem	theorem	h-cobordism theorem (Smale)		Iter-3 imported theorem (s_h_cobordism_theorem).
s_simply_connected_h_cobordism_dim_geq_6	axiom	simply connected h cobordism dim geq 6		Iter-3 imported axiom (s_simply_connected_h_cobordism_dim_geq_6).
s_handle_decomposition_of_cobordism	state	handle decomposition of cobordism		Iter-3 imported state (s_handle_decomposition_of_cobordism).
s_handles_cancelled	state	handles cancelled		Iter-3 imported state (s_handles_cancelled).
s_s_cobordism_theorem	theorem	s-cobordism theorem		Iter-3 imported theorem (s_s_cobordism_theorem).
s_h_cobordism_with_pi_1_nontrivial	axiom	h cobordism with pi 1 nontrivial		Iter-3 imported axiom (s_h_cobordism_with_pi_1_nontrivial).
s_whitehead_torsion_tau	axiom	whitehead torsion tau		Iter-3 imported axiom (s_whitehead_torsion_tau).
s_whitehead_torsion_obstruction	state	whitehead torsion obstruction		Iter-3 imported state (s_whitehead_torsion_obstruction).
s_handles_cancellable_when_tau_zero	state	handles cancellable when tau zero		Iter-3 imported state (s_handles_cancellable_when_tau_zero).
s_annulus_theorem	theorem	Annulus theorem (high-dim)		Iter-3 imported theorem (s_annulus_theorem).
s_two_locally_flat_embedded_spheres_in_R_n_n_geq_5	axiom	two locally flat embedded spheres in R n n geq 5		Iter-3 imported axiom (s_two_locally_flat_embedded_spheres_in_R_n_n_geq_5).
s_region_as_h_cobordism	state	region as h cobordism		Iter-3 imported state (s_region_as_h_cobordism).
s_disc_theorem	theorem	Disc theorem (Palais)		Iter-3 imported theorem (s_disc_theorem).
s_connected_smooth_manifold	axiom	connected smooth manifold		Iter-3 imported axiom (s_connected_smooth_manifold).
s_two_smooth_embeddings_of_disk_D_n	axiom	two smooth embeddings of disk D n		Iter-3 imported axiom (s_two_smooth_embeddings_of_disk_D_n).
s_isotopy_aligning_basepoints	state	isotopy aligning basepoints		Iter-3 imported state (s_isotopy_aligning_basepoints).
s_pontryagin_thom_iso	theorem	Pontryagin–Thom construction		Iter-3 imported theorem (s_pontryagin_thom_iso).
s_framed_cobordism_class	axiom	framed cobordism class		Iter-3 imported axiom (s_framed_cobordism_class).
s_collapse_map_to_thom_space	state	collapse map to thom space		Iter-3 imported state (s_collapse_map_to_thom_space).
s_stable_map_to_sphere_spectrum	state	stable map to sphere spectrum		Iter-3 imported state (s_stable_map_to_sphere_spectrum).
s_thom_transversality	theorem	Thom transversality theorem		Iter-3 imported theorem (s_thom_transversality).
s_smooth_map_with_jet_extension	axiom	smooth map with jet extension		Iter-3 imported axiom (s_smooth_map_with_jet_extension).
s_submanifold_W_in_jet_space	axiom	submanifold W in jet space		Iter-3 imported axiom (s_submanifold_W_in_jet_space).
s_jet_section_construction	state	jet section construction		Iter-3 imported state (s_jet_section_construction).
s_cerf_pseudoisotopy_theorem	theorem	Cerf's theorem (pseudoisotopy in high dim)		Iter-3 imported theorem (s_cerf_pseudoisotopy_theorem).
s_simply_connected_smooth_manifold_dim_geq_5	axiom	simply connected smooth manifold dim geq 5		Iter-3 imported axiom (s_simply_connected_smooth_manifold_dim_geq_5).
s_pseudoisotopy_F	axiom	pseudoisotopy F		Iter-3 imported axiom (s_pseudoisotopy_F).
s_cerf_graphic_of_critical_points	state	cerf graphic of critical points		Iter-3 imported state (s_cerf_graphic_of_critical_points).
s_simplified_to_no_critical_points	state	simplified to no critical points		Iter-3 imported state (s_simplified_to_no_critical_points).
s_donaldson_diagonalizability	theorem	Donaldson's theorem (intersection forms of smooth 4-manifolds)		Iter-3 imported theorem (s_donaldson_diagonalizability).
s_closed_simply_connected_smooth_4_manifold	axiom	closed simply connected smooth 4 manifold		Iter-3 imported axiom (s_closed_simply_connected_smooth_4_manifold).
s_instanton_moduli_space_M	state	instanton moduli space M		Iter-3 imported state (s_instanton_moduli_space_M).
s_compactified_5_dim_cobordism	state	compactified 5 dim cobordism		Iter-3 imported state (s_compactified_5_dim_cobordism).
s_exotic_R_4_existence	theorem	Exotic R^4 (Freedman+Donaldson)		Iter-3 imported theorem (s_exotic_R_4_existence).
s_freedman_topological_4_manifolds	axiom	freedman topological 4 manifolds		Iter-3 imported axiom (s_freedman_topological_4_manifolds).
s_topological_E8_manifold	state	topological E8 manifold		Iter-3 imported state (s_topological_E8_manifold).
s_no_smooth_E8_via_donaldson	state	no smooth E8 via donaldson		Iter-3 imported state (s_no_smooth_E8_via_donaldson).
s_rokhlin_theorem	theorem	Rokhlin's theorem		Iter-3 imported theorem (s_rokhlin_theorem).
s_smooth_closed_oriented_spin_4_manifold	axiom	smooth closed oriented spin 4 manifold		Iter-3 imported axiom (s_smooth_closed_oriented_spin_4_manifold).
s_dirac_operator_index_A_hat	state	dirac operator index A hat		Iter-3 imported state (s_dirac_operator_index_A_hat).
s_index_equals_signature_over_16_with_quaternion_correction	state	index equals signature over 16 with quaternion correction		Iter-3 imported state (s_index_equals_signature_over_16_with_quaternion_correction).
s_hirzebruch_signature_theorem	theorem	Hirzebruch signature theorem		Iter-3 imported theorem (s_hirzebruch_signature_theorem).
s_closed_oriented_4k_manifold	axiom	closed oriented 4k manifold		Iter-3 imported axiom (s_closed_oriented_4k_manifold).
s_signature_operator_on_forms	state	signature operator on forms		Iter-3 imported state (s_signature_operator_on_forms).
s_index_equals_L_genus_pairing	state	index equals L genus pairing		Iter-3 imported state (s_index_equals_L_genus_pairing).
s_atiyah_bott_fixed_point	theorem	Atiyah–Bott fixed-point theorem		Iter-3 imported theorem (s_atiyah_bott_fixed_point).
s_elliptic_complex	axiom	elliptic complex		Iter-3 imported axiom (s_elliptic_complex).
s_smooth_endomorphism_with_simple_fixed_points	axiom	smooth endomorphism with simple fixed points		Iter-3 imported axiom (s_smooth_endomorphism_with_simple_fixed_points).
s_elliptic_lefschetz_number	state	elliptic lefschetz number		Iter-3 imported state (s_elliptic_lefschetz_number).
s_local_contribution_at_fixed_points	state	local contribution at fixed points		Iter-3 imported state (s_local_contribution_at_fixed_points).
s_atiyah_hirzebruch_spectral_sequence	theorem	Atiyah–Hirzebruch spectral sequence		Iter-3 imported theorem (s_atiyah_hirzebruch_spectral_sequence).
s_generalized_cohomology_theory_E	axiom	generalized cohomology theory E		Iter-3 imported axiom (s_generalized_cohomology_theory_E).
s_skeleton_filtration_E_cohomology	state	skeleton filtration E cohomology		Iter-3 imported state (s_skeleton_filtration_E_cohomology).
s_exact_couple_for_E	state	exact couple for E		Iter-3 imported state (s_exact_couple_for_E).
s_bott_periodicity_complex	theorem	Bott periodicity (complex K-theory)		Iter-3 imported theorem (s_bott_periodicity_complex).
s_unitary_group_U_infinity	axiom	unitary group U infinity		Iter-3 imported axiom (s_unitary_group_U_infinity).
s_morse_theory_on_loop_space_U	state	morse theory on loop space U		Iter-3 imported state (s_morse_theory_on_loop_space_U).
s_handle_decomposition_BU_times_Z	state	handle decomposition BU times Z		Iter-3 imported state (s_handle_decomposition_BU_times_Z).
s_bott_periodicity_real	theorem	Bott periodicity (real K-theory, period 8)		Iter-3 imported theorem (s_bott_periodicity_real).
s_orthogonal_group_O_infinity	axiom	orthogonal group O infinity		Iter-3 imported axiom (s_orthogonal_group_O_infinity).
s_clifford_algebra_Cl_k	axiom	clifford algebra Cl k		Iter-3 imported axiom (s_clifford_algebra_Cl_k).
s_clifford_8_periodicity	state	clifford 8 periodicity		Iter-3 imported state (s_clifford_8_periodicity).
s_homotopy_of_loop_O	state	homotopy of loop O		Iter-3 imported state (s_homotopy_of_loop_O).
s_atiyah_segal_completion	theorem	Atiyah–Segal completion theorem		Iter-3 imported theorem (s_atiyah_segal_completion).
s_representation_ring_R_G	axiom	representation ring R G		Iter-3 imported axiom (s_representation_ring_R_G).
s_equivariant_K_theory_EG	state	equivariant K theory EG		Iter-3 imported state (s_equivariant_K_theory_EG).
s_I_adic_completion_K_G	state	I adic completion K G		Iter-3 imported state (s_I_adic_completion_K_G).
s_kuiper_theorem	theorem	Kuiper's theorem		Iter-3 imported theorem (s_kuiper_theorem).
s_infinite_dimensional_hilbert_space	axiom	infinite dimensional hilbert space		Iter-3 imported axiom (s_infinite_dimensional_hilbert_space).
s_unitary_group_U_H	axiom	unitary group U H		Iter-3 imported axiom (s_unitary_group_U_H).
s_eilenberg_swindle_homotopy	state	eilenberg swindle homotopy		Iter-3 imported state (s_eilenberg_swindle_homotopy).
s_U_H_contractible	state	U H contractible		Iter-3 imported state (s_U_H_contractible).
s_loop_theorem	theorem	Loop theorem (Papakyriakopoulos)		Iter-3 imported theorem (s_loop_theorem).
s_3_manifold_M_with_boundary	axiom	3 manifold M with boundary		Iter-3 imported axiom (s_3_manifold_M_with_boundary).
s_essential_loop_in_boundary	axiom	essential loop in boundary		Iter-3 imported axiom (s_essential_loop_in_boundary).
s_tower_of_branched_covers	state	tower of branched covers		Iter-3 imported state (s_tower_of_branched_covers).
s_terminal_tower_with_embedded_disk	state	terminal tower with embedded disk		Iter-3 imported state (s_terminal_tower_with_embedded_disk).
s_sphere_theorem_3_manifolds	theorem	Sphere theorem (3-manifolds)		Iter-3 imported theorem (s_sphere_theorem_3_manifolds).
s_orientable_3_manifold_M_with_pi_2_nontrivial	axiom	orientable 3 manifold M with pi 2 nontrivial		Iter-3 imported axiom (s_orientable_3_manifold_M_with_pi_2_nontrivial).
s_least_area_sphere_representative	state	least area sphere representative		Iter-3 imported state (s_least_area_sphere_representative).
s_embedded_essential_sphere	state	embedded essential sphere		Iter-3 imported state (s_embedded_essential_sphere).
s_lickorish_wallace_theorem	theorem	Lickorish–Wallace theorem		Iter-3 imported theorem (s_lickorish_wallace_theorem).
s_closed_orientable_3_manifold	axiom	closed orientable 3 manifold		Iter-3 imported axiom (s_closed_orientable_3_manifold).
s_heegaard_splitting	state	heegaard splitting		Iter-3 imported state (s_heegaard_splitting).
s_dehn_surgery_description	state	dehn surgery description		Iter-3 imported state (s_dehn_surgery_description).
s_gordon_luecke_theorem	theorem	Gordon–Luecke theorem (knot complements determine knots)		Iter-3 imported theorem (s_gordon_luecke_theorem).
s_knot_K_in_S_3	axiom	knot K in S 3		Iter-3 imported axiom (s_knot_K_in_S_3).
s_knot_complement_S_3_minus_K	axiom	knot complement S 3 minus K		Iter-3 imported axiom (s_knot_complement_S_3_minus_K).
s_hypothetical_nontrivial_surgery_giving_S_3	state	hypothetical nontrivial surgery giving S 3		Iter-3 imported state (s_hypothetical_nontrivial_surgery_giving_S_3).
s_planar_surface_obstruction	state	planar surface obstruction		Iter-3 imported state (s_planar_surface_obstruction).
s_nielsen_thurston_classification	theorem	Nielsen–Thurston classification		Iter-3 imported theorem (s_nielsen_thurston_classification).
s_mapping_class_of_closed_surface_genus_geq_2	axiom	mapping class of closed surface genus geq 2		Iter-3 imported axiom (s_mapping_class_of_closed_surface_genus_geq_2).
s_action_on_teichmuller_space	state	action on teichmuller space		Iter-3 imported state (s_action_on_teichmuller_space).
s_trichotomy_of_actions	state	trichotomy of actions		Iter-3 imported state (s_trichotomy_of_actions).
s_smale_diff_S_2_homotopy_equiv_O_3	theorem	Smale's theorem on 2-sphere diffeomorphisms		Iter-3 imported theorem (s_smale_diff_S_2_homotopy_equiv_O_3).
s_diffeomorphism_group_of_S_2	axiom	diffeomorphism group of S 2		Iter-3 imported axiom (s_diffeomorphism_group_of_S_2).
s_diff_S2_as_fiber_bundle	state	diff S2 as fiber bundle		Iter-3 imported state (s_diff_S2_as_fiber_bundle).
s_hatcher_smale_conjecture_S_3	theorem	Smale conjecture for S³ (Hatcher)		Iter-3 imported theorem (s_hatcher_smale_conjecture_S_3).
s_diffeomorphism_group_of_S_3	axiom	diffeomorphism group of S 3		Iter-3 imported axiom (s_diffeomorphism_group_of_S_3).
s_space_of_unknotted_2_spheres	state	space of unknotted 2 spheres		Iter-3 imported state (s_space_of_unknotted_2_spheres).
s_contractibility_of_space_of_spheres	state	contractibility of space of spheres		Iter-3 imported state (s_contractibility_of_space_of_spheres).
s_double_suspension_theorem	theorem	Double suspension theorem (Edwards–Cannon)		Iter-3 imported theorem (s_double_suspension_theorem).
s_homology_3_sphere_H	axiom	homology 3 sphere H		Iter-3 imported axiom (s_homology_3_sphere_H).
s_single_suspension_with_two_singular_points	state	single suspension with two singular points		Iter-3 imported state (s_single_suspension_with_two_singular_points).
s_double_suspension_S_5_topologically	state	double suspension S 5 topologically		Iter-3 imported state (s_double_suspension_S_5_topologically).
s_bing_recognition_theorem	theorem	Bing's recognition theorem for S³		Iter-3 imported theorem (s_bing_recognition_theorem).
s_compact_3_manifold_with_simply_connected_sphere_property	axiom	compact 3 manifold with simply connected sphere property		Iter-3 imported axiom (s_compact_3_manifold_with_simply_connected_sphere_property).
s_curve_in_3_cell_condition	state	curve in 3 cell condition		Iter-3 imported state (s_curve_in_3_cell_condition).
s_exhaustion_by_3_cells	state	exhaustion by 3 cells		Iter-3 imported state (s_exhaustion_by_3_cells).
s_fary_milnor_theorem	theorem	Fáry–Milnor theorem		Iter-3 imported theorem (s_fary_milnor_theorem).
s_smooth_closed_curve_in_R_3_with_total_curvature_leq_4pi	axiom	smooth closed curve in R 3 with total curvature leq 4pi		Iter-3 imported axiom (s_smooth_closed_curve_in_R_3_with_total_curvature_leq_4pi).
s_average_number_of_crossings_per_projection	state	average number of crossings per projection		Iter-3 imported state (s_average_number_of_crossings_per_projection).
s_projection_with_few_extrema	state	projection with few extrema		Iter-3 imported state (s_projection_with_few_extrema).
s_reeb_sphere_theorem	theorem	Reeb sphere theorem		Iter-3 imported theorem (s_reeb_sphere_theorem).
s_closed_n_manifold_with_morse_function_with_exactly_2_critical_points	axiom	closed n manifold with morse function with exactly 2 critical points		Iter-3 imported axiom (s_closed_n_manifold_with_morse_function_with_exactly_2_critical_points).
s_two_disk_decomposition	state	two disk decomposition		Iter-3 imported state (s_two_disk_decomposition).
s_phragmen_brouwer_theorem	theorem	Phragmén–Brouwer theorem		Iter-3 imported theorem (s_phragmen_brouwer_theorem).
s_connected_locally_connected_normal_space	axiom	connected locally connected normal space		Iter-3 imported axiom (s_connected_locally_connected_normal_space).
s_two_separating_continua	axiom	two separating continua		Iter-3 imported axiom (s_two_separating_continua).
s_irreducible_separator	state	irreducible separator		Iter-3 imported state (s_irreducible_separator).
s_contradicting_pair_of_continua	state	contradicting pair of continua		Iter-3 imported state (s_contradicting_pair_of_continua).
s_thurston_geometrization_haken	theorem	Geometrization / Thurston for Haken manifolds		Iter-3 imported theorem (s_thurston_geometrization_haken).
s_haken_3_manifold	axiom	haken 3 manifold		Iter-3 imported axiom (s_haken_3_manifold).
s_jsj_decomposition	state	jsj decomposition		Iter-3 imported state (s_jsj_decomposition).
s_hyperbolic_structures_on_atoroidal_pieces	state	hyperbolic structures on atoroidal pieces		Iter-3 imported state (s_hyperbolic_structures_on_atoroidal_pieces).
s_smith_conjecture	theorem	Smith conjecture		Iter-3 imported theorem (s_smith_conjecture).
s_finite_cyclic_smooth_action_on_S_3_with_nonempty_fixed_set	axiom	finite cyclic smooth action on S 3 with nonempty fixed set		Iter-3 imported axiom (s_finite_cyclic_smooth_action_on_S_3_with_nonempty_fixed_set).
s_fixed_circle_K	state	fixed circle K		Iter-3 imported state (s_fixed_circle_K).
s_quotient_orbifold_geometric	state	quotient orbifold geometric		Iter-3 imported state (s_quotient_orbifold_geometric).
s_tameness_theorem	theorem	Tameness theorem (Agol–Calegari–Gabai)		Iter-3 imported theorem (s_tameness_theorem).
s_hyperbolic_3_manifold_with_finitely_generated_pi_1	axiom	hyperbolic 3 manifold with finitely generated pi 1		Iter-3 imported axiom (s_hyperbolic_3_manifold_with_finitely_generated_pi_1).
s_shrinkwrapped_minimal_surfaces	state	shrinkwrapped minimal surfaces		Iter-3 imported state (s_shrinkwrapped_minimal_surfaces).
s_exhaustion_by_compact_cores	state	exhaustion by compact cores		Iter-3 imported state (s_exhaustion_by_compact_cores).
s_kahn_markovic_surface_subgroup	theorem	Surface subgroup theorem (Kahn–Markovic)		Iter-3 imported theorem (s_kahn_markovic_surface_subgroup).
s_closed_hyperbolic_3_manifold	axiom	closed hyperbolic 3 manifold		Iter-3 imported axiom (s_closed_hyperbolic_3_manifold).
s_collection_of_pants	state	collection of pants		Iter-3 imported state (s_collection_of_pants).
s_pants_pair_gluing_compatible	state	pants pair gluing compatible		Iter-3 imported state (s_pants_pair_gluing_compatible).
s_virtually_haken_theorem	theorem	Virtually Haken theorem (Agol)		Iter-3 imported theorem (s_virtually_haken_theorem).
s_dual_cube_complex_action	state	dual cube complex action		Iter-3 imported state (s_dual_cube_complex_action).
s_finite_index_special_cover	state	finite index special cover		Iter-3 imported state (s_finite_index_special_cover).
s_sullivan_minimal_model_theorem	theorem	Sullivan's structure theorem on rational homotopy		Iter-3 imported theorem (s_sullivan_minimal_model_theorem).
s_simply_connected_space_X_finite_type	axiom	simply connected space X finite type		Iter-3 imported axiom (s_simply_connected_space_X_finite_type).
s_PL_de_rham_algebra	state	PL de rham algebra		Iter-3 imported state (s_PL_de_rham_algebra).
s_sullivan_minimal_model	state	sullivan minimal model		Iter-3 imported state (s_sullivan_minimal_model).
s_andreotti_frankel_theorem	theorem	Andreotti–Frankel theorem		Iter-3 imported theorem (s_andreotti_frankel_theorem).
s_smooth_affine_complex_variety_complex_dimension_n	axiom	smooth affine complex variety complex dimension n		Iter-3 imported axiom (s_smooth_affine_complex_variety_complex_dimension_n).
s_real_distance_squared_morse_function	state	real distance squared morse function		Iter-3 imported state (s_real_distance_squared_morse_function).
s_index_bound_on_critical_points	state	index bound on critical points		Iter-3 imported state (s_index_bound_on_critical_points).
s_hodge_decomposition_theorem	theorem	Hodge decomposition theorem		Iter-3 imported theorem (s_hodge_decomposition_theorem).
s_compact_kahler_manifold	axiom	compact kahler manifold		Iter-3 imported axiom (s_compact_kahler_manifold).
s_harmonic_forms_kernel_of_Laplacian	state	harmonic forms kernel of Laplacian		Iter-3 imported state (s_harmonic_forms_kernel_of_Laplacian).
s_hodge_orthogonal_decomposition	state	hodge orthogonal decomposition		Iter-3 imported state (s_hodge_orthogonal_decomposition).
s_hodge_index_theorem	theorem	Hodge index theorem		Iter-3 imported theorem (s_hodge_index_theorem).
s_smooth_projective_surface_X	axiom	smooth projective surface X		Iter-3 imported axiom (s_smooth_projective_surface_X).
s_intersection_pairing_on_H_2	state	intersection pairing on H 2		Iter-3 imported state (s_intersection_pairing_on_H_2).
s_signature_of_intersection_pairing	state	signature of intersection pairing		Iter-3 imported state (s_signature_of_intersection_pairing).
s_smale_2_manifold_classification	theorem	Smale's classification of compact 2-manifolds (gradient flow)		Iter-3 imported theorem (s_smale_2_manifold_classification).
s_closed_surface_with_morse_function	axiom	closed surface with morse function		Iter-3 imported axiom (s_closed_surface_with_morse_function).
s_handle_decomposition_of_surface	state	handle decomposition of surface		Iter-3 imported state (s_handle_decomposition_of_surface).
s_canonical_genus_handle_form	state	canonical genus handle form		Iter-3 imported state (s_canonical_genus_handle_form).
s_quillen_plus_construction_K_theory	theorem	Quillen +-construction and algebraic K-theory		Iter-3 imported theorem (s_quillen_plus_construction_K_theory).
s_associative_ring_R	axiom	associative ring R		Iter-3 imported axiom (s_associative_ring_R).
s_classifying_space_BGL_R	axiom	classifying space BGL R		Iter-3 imported axiom (s_classifying_space_BGL_R).
s_BGL_R_plus_simply_connected_in_pi_1_quotient	state	BGL R plus simply connected in pi 1 quotient		Iter-3 imported state (s_BGL_R_plus_simply_connected_in_pi_1_quotient).
s_group_completion_yields_K_R_spectrum	state	group completion yields K R spectrum		Iter-3 imported state (s_group_completion_yields_K_R_spectrum).
s_snaith_theorem	theorem	Snaith's theorem		Iter-3 imported theorem (s_snaith_theorem).
s_infinite_loop_space_CP_infinity	axiom	infinite loop space CP infinity		Iter-3 imported axiom (s_infinite_loop_space_CP_infinity).
s_bott_element_beta_in_K_0	axiom	bott element beta in K 0		Iter-3 imported axiom (s_bott_element_beta_in_K_0).
s_sigma_infty_CP_infty_bott_inverted	state	sigma infty CP infty bott inverted		Iter-3 imported state (s_sigma_infty_CP_infty_bott_inverted).
s_anderson_kadec_theorem	theorem	Anderson–Kadec theorem		Iter-3 imported theorem (s_anderson_kadec_theorem).
s_infinite_dimensional_separable_frechet_space	axiom	infinite dimensional separable frechet space		Iter-3 imported axiom (s_infinite_dimensional_separable_frechet_space).
s_coordinate_maps_to_R_infty	state	coordinate maps to R infty		Iter-3 imported state (s_coordinate_maps_to_R_infty).
s_kuratowski_14_set_theorem	theorem	Kuratowski 14-set theorem		Iter-3 imported theorem (s_kuratowski_14_set_theorem).
s_subset_A_of_X	axiom	subset A of X		Iter-3 imported axiom (s_subset_A_of_X).
s_free_monoid_closure_complement	state	free monoid closure complement		Iter-3 imported state (s_free_monoid_closure_complement).
s_relations_collapse_to_14	state	relations collapse to 14		Iter-3 imported state (s_relations_collapse_to_14).
s_topological_invariance_of_dimension	theorem	Topological invariance of Lebesgue covering dimension		Iter-3 imported theorem (s_topological_invariance_of_dimension).
s_homeomorphic_topological_spaces	axiom	homeomorphic topological spaces		Iter-3 imported axiom (s_homeomorphic_topological_spaces).
s_covering_dimension_definition	state	covering dimension definition		Iter-3 imported state (s_covering_dimension_definition).
s_dim_preserved_under_homeomorphism	state	dim preserved under homeomorphism		Iter-3 imported state (s_dim_preserved_under_homeomorphism).
s_suspension_isomorphism_H_tilde	theorem	Suspension isomorphism (reduced homology)		Iter-3 imported theorem (s_suspension_isomorphism_H_tilde).
s_pointed_space_X	axiom	pointed space X		Iter-3 imported axiom (s_pointed_space_X).
s_reduced_suspension_sigma_X	axiom	reduced suspension sigma X		Iter-3 imported axiom (s_reduced_suspension_sigma_X).
s_LES_cone_pair	state	LES cone pair		Iter-3 imported state (s_LES_cone_pair).
s_eilenberg_swindle_K_theory_vanishing	theorem	Eilenberg swindle / vanishing of K-theory of "infinite sum" rings		Iter-3 imported theorem (s_eilenberg_swindle_K_theory_vanishing).
s_ring_admitting_countable_infinite_direct_sums	axiom	ring admitting countable infinite direct sums		Iter-3 imported axiom (s_ring_admitting_countable_infinite_direct_sums).
s_swindle_isomorphism	state	swindle isomorphism		Iter-3 imported state (s_swindle_isomorphism).
s_K_0_vanishes_via_swindle	state	K 0 vanishes via swindle		Iter-3 imported state (s_K_0_vanishes_via_swindle).
s_whitney_graustein_theorem	theorem	Whitney–Graustein theorem		Iter-3 imported theorem (s_whitney_graustein_theorem).
s_immersion_of_circle_in_plane	axiom	immersion of circle in plane		Iter-3 imported axiom (s_immersion_of_circle_in_plane).
s_winding_number_of_tangent	axiom	winding number of tangent		Iter-3 imported axiom (s_winding_number_of_tangent).
s_rotation_number_of_immersion	state	rotation number of immersion		Iter-3 imported state (s_rotation_number_of_immersion).
s_complete_invariant_for_regular_homotopy	state	complete invariant for regular homotopy		Iter-3 imported state (s_complete_invariant_for_regular_homotopy).
s_smale_sphere_eversion	theorem	Smale's sphere eversion		Iter-3 imported theorem (s_smale_sphere_eversion).
s_immersion_S_2_in_R_3	axiom	immersion S 2 in R 3		Iter-3 imported axiom (s_immersion_S_2_in_R_3).
s_immersions_classified_by_homotopy_of_tangent_map	state	immersions classified by homotopy of tangent map		Iter-3 imported state (s_immersions_classified_by_homotopy_of_tangent_map).
s_no_obstruction_to_homotopy_id_to_minus_id	state	no obstruction to homotopy id to minus id		Iter-3 imported state (s_no_obstruction_to_homotopy_id_to_minus_id).
s_hopf_classification_theorem	theorem	Hopf classification theorem		Iter-3 imported theorem (s_hopf_classification_theorem).
s_compact_connected_oriented_n_manifold	axiom	compact connected oriented n manifold		Iter-3 imported axiom (s_compact_connected_oriented_n_manifold).
s_continuous_map_to_S_n	axiom	continuous map to S n		Iter-3 imported axiom (s_continuous_map_to_S_n).
s_degree_of_map	state	degree of map		Iter-3 imported state (s_degree_of_map).
s_degree_complete_invariant	state	degree complete invariant		Iter-3 imported state (s_degree_complete_invariant).
s_preimage_theorem	theorem	Preimage theorem (regular value theorem)		Iter-3 imported theorem (s_preimage_theorem).
s_smooth_map_f_M_to_N	axiom	smooth map f M to N		Iter-3 imported axiom (s_smooth_map_f_M_to_N).
s_regular_value_q_in_N	axiom	regular value q in N		Iter-3 imported axiom (s_regular_value_q_in_N).
s_local_submersion_form	state	local submersion form		Iter-3 imported state (s_local_submersion_form).
s_tubular_neighborhood_theorem	theorem	Tubular neighborhood theorem		Iter-3 imported theorem (s_tubular_neighborhood_theorem).
s_smooth_submanifold_S_of_M	axiom	smooth submanifold S of M		Iter-3 imported axiom (s_smooth_submanifold_S_of_M).
s_normal_bundle_NS	axiom	normal bundle NS		Iter-3 imported axiom (s_normal_bundle_NS).
s_exp_normal_bundle_map	state	exp normal bundle map		Iter-3 imported state (s_exp_normal_bundle_map).
s_local_diffeomorphism_on_small_disk_bundle	state	local diffeomorphism on small disk bundle		Iter-3 imported state (s_local_diffeomorphism_on_small_disk_bundle).
s_stiefel_whitney_classes_axioms	theorem	Stiefel–Whitney class characterization (axioms)		Iter-3 imported theorem (s_stiefel_whitney_classes_axioms).
s_axiom_list_for_w_i	state	axiom list for w i		Iter-3 imported state (s_axiom_list_for_w_i).
s_classes_in_H_i_M_F_2	state	classes in H i M F 2		Iter-3 imported state (s_classes_in_H_i_M_F_2).
s_chern_classes_axioms	theorem	Chern class characterization (axioms)		Iter-3 imported theorem (s_chern_classes_axioms).
s_complex_vector_bundle_eta	axiom	complex vector bundle eta		Iter-3 imported axiom (s_complex_vector_bundle_eta).
s_axiom_list_for_c_i	state	axiom list for c i		Iter-3 imported state (s_axiom_list_for_c_i).
s_classes_in_H_2i_M_Z	state	classes in H 2i M Z		Iter-3 imported state (s_classes_in_H_2i_M_Z).
s_pontryagin_classes_definition_theorem	theorem	Pontryagin classes from Chern		Iter-3 imported theorem (s_pontryagin_classes_definition_theorem).
s_complexification_xi_otimes_C	axiom	complexification xi otimes C		Iter-3 imported axiom (s_complexification_xi_otimes_C).
s_chern_classes_of_complexification	state	chern classes of complexification		Iter-3 imported state (s_chern_classes_of_complexification).
s_pontryagin_class_p_k	state	pontryagin class p k		Iter-3 imported state (s_pontryagin_class_p_k).
s_chern_weil_theorem	theorem	Chern–Weil theorem		Iter-3 imported theorem (s_chern_weil_theorem).
s_principal_G_bundle_with_connection	axiom	principal G bundle with connection		Iter-3 imported axiom (s_principal_G_bundle_with_connection).
s_invariant_polynomial_in_curvature	state	invariant polynomial in curvature		Iter-3 imported state (s_invariant_polynomial_in_curvature).
s_closed_de_rham_class_of_invariant_polynomial	state	closed de rham class of invariant polynomial		Iter-3 imported state (s_closed_de_rham_class_of_invariant_polynomial).
s_gauss_bonnet_chern_theorem	theorem	Gauss–Bonnet–Chern theorem		Iter-3 imported theorem (s_gauss_bonnet_chern_theorem).
s_closed_oriented_riemannian_2n_manifold	axiom	closed oriented riemannian 2n manifold		Iter-3 imported axiom (s_closed_oriented_riemannian_2n_manifold).
s_pfaffian_of_curvature	state	pfaffian of curvature		Iter-3 imported state (s_pfaffian_of_curvature).
s_pfaffian_represents_euler_class	state	pfaffian represents euler class		Iter-3 imported state (s_pfaffian_represents_euler_class).
s_unoriented_bordism_via_thom_spectrum_MO	theorem	Bordism groups via Pontryagin–Thom		Iter-3 imported theorem (s_unoriented_bordism_via_thom_spectrum_MO).
s_smooth_n_manifolds_modulo_cobordism	axiom	smooth n manifolds modulo cobordism		Iter-3 imported axiom (s_smooth_n_manifolds_modulo_cobordism).
s_collapse_to_MO_k_thom_space	state	collapse to MO k thom space		Iter-3 imported state (s_collapse_to_MO_k_thom_space).
s_stable_map_to_MO_spectrum	state	stable map to MO spectrum		Iter-3 imported state (s_stable_map_to_MO_spectrum).
s_thom_unoriented_bordism_calculation	theorem	Thom's calculation of unoriented bordism		Iter-3 imported theorem (s_thom_unoriented_bordism_calculation).
s_MO_splits_as_HF_2_wedges	state	MO splits as HF 2 wedges		Iter-3 imported state (s_MO_splits_as_HF_2_wedges).
s_adem_relations_steenrod_algebra	theorem	Adem relations / Steenrod algebra structure		Iter-3 imported theorem (s_adem_relations_steenrod_algebra).
s_mod_p_cohomology_with_steenrod_operations	axiom	mod p cohomology with steenrod operations		Iter-3 imported axiom (s_mod_p_cohomology_with_steenrod_operations).
s_sq_i_via_equivariant_diagonal	state	sq i via equivariant diagonal		Iter-3 imported state (s_sq_i_via_equivariant_diagonal).
s_relations_from_double_cyclic_structure	state	relations from double cyclic structure		Iter-3 imported state (s_relations_from_double_cyclic_structure).
s_spanier_whitehead_duality	theorem	Spanier–Whitehead duality		Iter-3 imported theorem (s_spanier_whitehead_duality).
s_finite_cw_complex_X_embedded_in_S_n	axiom	finite cw complex X embedded in S n		Iter-3 imported axiom (s_finite_cw_complex_X_embedded_in_S_n).
s_spanier_whitehead_dual_DX	state	spanier whitehead dual DX		Iter-3 imported state (s_spanier_whitehead_dual_DX).
s_thom_isomorphism	theorem	Thom isomorphism theorem		Iter-3 imported theorem (s_thom_isomorphism).
s_oriented_real_vector_bundle_xi_rank_n	axiom	oriented real vector bundle xi rank n		Iter-3 imported axiom (s_oriented_real_vector_bundle_xi_rank_n).
s_thom_class_U	state	thom class U		Iter-3 imported state (s_thom_class_U).
s_cup_with_U_iso	state	cup with U iso		Iter-3 imported state (s_cup_with_U_iso).
s_gysin_sequence	theorem	Gysin sequence		Iter-3 imported theorem (s_gysin_sequence).
s_oriented_sphere_bundle_S_n_to_E_to_B	axiom	oriented sphere bundle S n to E to B		Iter-3 imported axiom (s_oriented_sphere_bundle_S_n_to_E_to_B).
s_disk_sphere_pair_with_thom_iso	state	disk sphere pair with thom iso		Iter-3 imported state (s_disk_sphere_pair_with_thom_iso).
s_LES_of_pair	state	LES of pair		Iter-3 imported state (s_LES_of_pair).
s_wang_sequence	theorem	Wang sequence		Iter-3 imported theorem (s_wang_sequence).
s_fiber_bundle_with_base_S_n	axiom	fiber bundle with base S n		Iter-3 imported axiom (s_fiber_bundle_with_base_S_n).
s_two_hemispheres_MV	state	two hemispheres MV		Iter-3 imported state (s_two_hemispheres_MV).
s_borel_equivariant_cohomology	theorem	Borel construction / equivariant cohomology		Iter-3 imported theorem (s_borel_equivariant_cohomology).
s_topological_group_G_acting_on_X	axiom	topological group G acting on X		Iter-3 imported axiom (s_topological_group_G_acting_on_X).
s_borel_construction_EG_times_X	state	borel construction EG times X		Iter-3 imported state (s_borel_construction_EG_times_X).
s_equivariant_cohomology_is_module_over_H_BG	state	equivariant cohomology is module over H BG		Iter-3 imported state (s_equivariant_cohomology_is_module_over_H_BG).
s_lefschetz_coincidence_formula	theorem	Lefschetz coincidence / fixed-point trace formula (general)		Iter-3 imported theorem (s_lefschetz_coincidence_formula).
s_continuous_map_pair_f_g_M_to_N	axiom	continuous map pair f g M to N		Iter-3 imported axiom (s_continuous_map_pair_f_g_M_to_N).
s_coincidence_intersection_number	state	coincidence intersection number		Iter-3 imported state (s_coincidence_intersection_number).
s_cauchy_riemann_characterization	theorem	Cauchy–Riemann equations characterize holomorphy		Iter-3 imported theorem (s_cauchy_riemann_characterization).
s_differentiable_complex_function	axiom	differentiable complex function		Iter-3 imported axiom (s_differentiable_complex_function).
s_real_C1_function_pair_u_v	axiom	real C1 function pair u v		Iter-3 imported axiom (s_real_C1_function_pair_u_v).
s_two_directional_limit_equality	state	two directional limit equality		Iter-3 imported state (s_two_directional_limit_equality).
s_cauchy_riemann_system	state	cauchy riemann system		Iter-3 imported state (s_cauchy_riemann_system).
s_cauchy_integral_theorem_general	theorem	Cauchy's theorem on simply connected domains (Goursat)		Iter-3 imported theorem (s_cauchy_integral_theorem_general).
s_nested_shrinking_triangle_with_integral_bound	state	nested shrinking triangle with integral bound		Iter-3 imported state (s_nested_shrinking_triangle_with_integral_bound).
s_integral_over_triangle_vanishes	state	integral over triangle vanishes		Iter-3 imported state (s_integral_over_triangle_vanishes).
s_liouville_theorem	theorem	Liouville's theorem		Iter-3 imported theorem (s_liouville_theorem).
s_bounded_entire_function	axiom	bounded entire function		Iter-3 imported axiom (s_bounded_entire_function).
s_cauchy_estimate_f_prime_bounded_by_M_over_R	state	cauchy estimate f prime bounded by M over R		Iter-3 imported state (s_cauchy_estimate_f_prime_bounded_by_M_over_R).
s_f_prime_vanishes_everywhere	state	f prime vanishes everywhere		Iter-3 imported state (s_f_prime_vanishes_everywhere).
s_identity_theorem	theorem	Identity theorem		Iter-3 imported theorem (s_identity_theorem).
s_set_with_accumulation_point_in_domain	axiom	set with accumulation point in domain		Iter-3 imported axiom (s_set_with_accumulation_point_in_domain).
s_power_series_with_zero_coefficients	state	power series with zero coefficients		Iter-3 imported state (s_power_series_with_zero_coefficients).
s_open_subset_where_f_zero	state	open subset where f zero		Iter-3 imported state (s_open_subset_where_f_zero).
s_maximum_modulus_principle	theorem	Maximum modulus principle		Iter-3 imported theorem (s_maximum_modulus_principle).
s_connected_open_subset_of_C	axiom	connected open subset of C		Iter-3 imported axiom (s_connected_open_subset_of_C).
s_mean_value_property_for_f	state	mean value property for f		Iter-3 imported state (s_mean_value_property_for_f).
s_constant_on_neighborhood_of_max	state	constant on neighborhood of max		Iter-3 imported state (s_constant_on_neighborhood_of_max).
s_complex_open_mapping_theorem	theorem	Open mapping theorem (complex analysis)		Iter-3 imported theorem (s_complex_open_mapping_theorem).
s_nonconstant_holomorphic_function	axiom	nonconstant holomorphic function		Iter-3 imported axiom (s_nonconstant_holomorphic_function).
s_local_normal_form_n_to_1	state	local normal form n to 1		Iter-3 imported state (s_local_normal_form_n_to_1).
s_winding_number_n_for_small_disk_image	state	winding number n for small disk image		Iter-3 imported state (s_winding_number_n_for_small_disk_image).
s_schwarz_pick_theorem	theorem	Schwarz–Pick theorem		Iter-3 imported theorem (s_schwarz_pick_theorem).
s_holomorphic_self_map_of_unit_disk	axiom	holomorphic self map of unit disk		Iter-3 imported axiom (s_holomorphic_self_map_of_unit_disk).
s_hyperbolic_metric_on_disk	axiom	hyperbolic metric on disk		Iter-3 imported axiom (s_hyperbolic_metric_on_disk).
s_normalized_problem_at_origin	state	normalized problem at origin		Iter-3 imported state (s_normalized_problem_at_origin).
s_bound_at_origin	state	bound at origin		Iter-3 imported state (s_bound_at_origin).
s_casorati_weierstrass	theorem	Casorati–Weierstrass theorem		Iter-3 imported theorem (s_casorati_weierstrass).
s_holomorphic_function_with_essential_singularity	axiom	holomorphic function with essential singularity		Iter-3 imported axiom (s_holomorphic_function_with_essential_singularity).
s_punctured_disk_neighborhood	axiom	punctured disk neighborhood		Iter-3 imported axiom (s_punctured_disk_neighborhood).
s_1_over_f_minus_w_bounded_near_singularity	state	1 over f minus w bounded near singularity		Iter-3 imported state (s_1_over_f_minus_w_bounded_near_singularity).
s_pole_or_removable_contradiction	state	pole or removable contradiction		Iter-3 imported state (s_pole_or_removable_contradiction).
s_jensen_formula	theorem	Jensen's formula		Iter-3 imported theorem (s_jensen_formula).
s_holomorphic_function_on_closed_disk	axiom	holomorphic function on closed disk		Iter-3 imported axiom (s_holomorphic_function_on_closed_disk).
s_zeros_inside_disk	axiom	zeros inside disk		Iter-3 imported axiom (s_zeros_inside_disk).
s_zero_free_factor_g	state	zero free factor g		Iter-3 imported state (s_zero_free_factor_g).
s_mean_value_for_log_abs_g	state	mean value for log abs g		Iter-3 imported state (s_mean_value_for_log_abs_g).
s_hadamard_three_lines_theorem	theorem	Hadamard three-lines theorem		Iter-3 imported theorem (s_hadamard_three_lines_theorem).
s_holomorphic_function_on_vertical_strip	axiom	holomorphic function on vertical strip		Iter-3 imported axiom (s_holomorphic_function_on_vertical_strip).
s_bounded_function	axiom	bounded function		Iter-3 imported axiom (s_bounded_function).
s_modified_function_with_uniform_bound_on_boundary	state	modified function with uniform bound on boundary		Iter-3 imported state (s_modified_function_with_uniform_bound_on_boundary).
s_max_modulus_bound_on_strip	state	max modulus bound on strip		Iter-3 imported state (s_max_modulus_bound_on_strip).
s_hadamard_three_circles_theorem	theorem	Hadamard three-circles theorem		Iter-3 imported theorem (s_hadamard_three_circles_theorem).
s_holomorphic_function_on_annulus	axiom	holomorphic function on annulus		Iter-3 imported axiom (s_holomorphic_function_on_annulus).
s_log_radius_substitution	axiom	log radius substitution		Iter-3 imported axiom (s_log_radius_substitution).
s_strip_version_of_problem	state	strip version of problem		Iter-3 imported state (s_strip_version_of_problem).
s_borel_caratheodory_theorem	theorem	Borel–Carathéodory theorem		Iter-3 imported theorem (s_borel_caratheodory_theorem).
s_real_part_bound	axiom	real part bound		Iter-3 imported axiom (s_real_part_bound).
s_holomorphic_self_map_of_disk	state	holomorphic self map of disk		Iter-3 imported state (s_holomorphic_self_map_of_disk).
s_bound_on_phi_implies_bound_on_f	state	bound on phi implies bound on f		Iter-3 imported state (s_bound_on_phi_implies_bound_on_f).
s_hadamard_factorization_theorem	theorem	Hadamard factorization theorem		Iter-3 imported theorem (s_hadamard_factorization_theorem).
s_entire_function_of_finite_order	axiom	entire function of finite order		Iter-3 imported axiom (s_entire_function_of_finite_order).
s_zero_distribution_with_genus_p	axiom	zero distribution with genus p		Iter-3 imported axiom (s_zero_distribution_with_genus_p).
s_canonical_product_with_zero_set	state	canonical product with zero set		Iter-3 imported state (s_canonical_product_with_zero_set).
s_zero_free_entire_with_growth_order	state	zero free entire with growth order		Iter-3 imported state (s_zero_free_entire_with_growth_order).
s_log_f_is_polynomial_of_degree_leq_rho	state	log f is polynomial of degree leq rho		Iter-3 imported state (s_log_f_is_polynomial_of_degree_leq_rho).
s_runge_theorem	theorem	Runge's theorem		Iter-3 imported theorem (s_runge_theorem).
s_holomorphic_function_on_open_set	axiom	holomorphic function on open set		Iter-3 imported axiom (s_holomorphic_function_on_open_set).
s_compact_subset_with_connected_complement	axiom	compact subset with connected complement		Iter-3 imported axiom (s_compact_subset_with_connected_complement).
s_integral_representation_with_pole_at_each_point	state	integral representation with pole at each point		Iter-3 imported state (s_integral_representation_with_pole_at_each_point).
s_approximation_by_rational_functions_with_poles_in_complement	state	approximation by rational functions with poles in complement		Iter-3 imported state (s_approximation_by_rational_functions_with_poles_in_complement).
s_koebe_quarter_theorem	theorem	Koebe 1/4 theorem		Iter-3 imported theorem (s_koebe_quarter_theorem).
s_univalent_function_on_unit_disk	axiom	univalent function on unit disk		Iter-3 imported axiom (s_univalent_function_on_unit_disk).
s_auxiliary_univalent_g_in_class_S	state	auxiliary univalent g in class S		Iter-3 imported state (s_auxiliary_univalent_g_in_class_S).
s_coefficient_bound_abs_a2_leq_2	state	coefficient bound abs a2 leq 2		Iter-3 imported state (s_coefficient_bound_abs_a2_leq_2).
s_schwarz_reflection_principle	theorem	Schwarz reflection principle		Iter-3 imported theorem (s_schwarz_reflection_principle).
s_holomorphic_on_upper_half_disk_continuous_to_real_axis	axiom	holomorphic on upper half disk continuous to real axis		Iter-3 imported axiom (s_holomorphic_on_upper_half_disk_continuous_to_real_axis).
s_real_boundary_values	axiom	real boundary values		Iter-3 imported axiom (s_real_boundary_values).
s_piecewise_function_on_full_disk	state	piecewise function on full disk		Iter-3 imported state (s_piecewise_function_on_full_disk).
s_holomorphic_F_on_full_disk	state	holomorphic F on full disk		Iter-3 imported state (s_holomorphic_F_on_full_disk).
s_schwarz_christoffel_mapping	theorem	Schwarz–Christoffel mapping		Iter-3 imported theorem (s_schwarz_christoffel_mapping).
s_simply_connected_polygonal_domain	axiom	simply connected polygonal domain		Iter-3 imported axiom (s_simply_connected_polygonal_domain).
s_upper_half_plane	axiom	upper half plane		Iter-3 imported axiom (s_upper_half_plane).
s_abstract_conformal_map_to_polygon	state	abstract conformal map to polygon		Iter-3 imported state (s_abstract_conformal_map_to_polygon).
s_derivative_form_product_w_minus_xk_alpha_k	state	derivative form product w minus xk alpha k		Iter-3 imported state (s_derivative_form_product_w_minus_xk_alpha_k).
s_de_branges_theorem	theorem	Bieberbach conjecture / de Branges theorem		Iter-3 imported theorem (s_de_branges_theorem).
s_univalent_function_in_class_S	axiom	univalent function in class S		Iter-3 imported axiom (s_univalent_function_in_class_S).
s_taylor_coefficients_a_n	axiom	taylor coefficients a n		Iter-3 imported axiom (s_taylor_coefficients_a_n).
s_Loewner_ODE_for_f_t	state	Loewner ODE for f t		Iter-3 imported state (s_Loewner_ODE_for_f_t).
s_milin_inequality_on_log_derivatives	state	milin inequality on log derivatives		Iter-3 imported state (s_milin_inequality_on_log_derivatives).
s_riemann_removable_singularity	theorem	Riemann's removable singularity theorem		Iter-3 imported theorem (s_riemann_removable_singularity).
s_holomorphic_function_on_punctured_disk	axiom	holomorphic function on punctured disk		Iter-3 imported axiom (s_holomorphic_function_on_punctured_disk).
s_bounded_near_puncture	axiom	bounded near puncture		Iter-3 imported axiom (s_bounded_near_puncture).
s_holomorphic_extension_h_with_zero_of_order_2	state	holomorphic extension h with zero of order 2		Iter-3 imported state (s_holomorphic_extension_h_with_zero_of_order_2).
s_holomorphic_recovery_of_f	state	holomorphic recovery of f		Iter-3 imported state (s_holomorphic_recovery_of_f).
s_riemann_lebesgue_lemma	theorem	Riemann–Lebesgue lemma		Iter-3 imported theorem (s_riemann_lebesgue_lemma).
s_l1_function_on_R	axiom	l1 function on R		Iter-3 imported axiom (s_l1_function_on_R).
s_fourier_transform_definition	axiom	fourier transform definition		Iter-3 imported axiom (s_fourier_transform_definition).
s_decay_for_indicator_of_interval	state	decay for indicator of interval		Iter-3 imported state (s_decay_for_indicator_of_interval).
s_decay_for_step_functions	state	decay for step functions		Iter-3 imported state (s_decay_for_step_functions).
s_fejer_theorem	theorem	Fejér's theorem		Iter-3 imported theorem (s_fejer_theorem).
s_continuous_function_on_torus	axiom	continuous function on torus		Iter-3 imported axiom (s_continuous_function_on_torus).
s_cesaro_means_of_fourier_series	axiom	cesaro means of fourier series		Iter-3 imported axiom (s_cesaro_means_of_fourier_series).
s_positive_approximate_identity	state	positive approximate identity		Iter-3 imported state (s_positive_approximate_identity).
s_cesaro_average_as_convolution	state	cesaro average as convolution		Iter-3 imported state (s_cesaro_average_as_convolution).
s_dirichlet_pointwise_convergence	theorem	Dirichlet's theorem on pointwise Fourier convergence		Iter-3 imported theorem (s_dirichlet_pointwise_convergence).
s_piecewise_C1_function_on_torus	axiom	piecewise C1 function on torus		Iter-3 imported axiom (s_piecewise_C1_function_on_torus).
s_partial_sums_of_fourier_series	axiom	partial sums of fourier series		Iter-3 imported axiom (s_partial_sums_of_fourier_series).
s_integral_representation_S_N_f	state	integral representation S N f		Iter-3 imported state (s_integral_representation_S_N_f).
s_localized_integrand_with_smooth_factor	state	localized integrand with smooth factor		Iter-3 imported state (s_localized_integrand_with_smooth_factor).
s_hausdorff_young_inequality	theorem	Hausdorff–Young inequality		Iter-3 imported theorem (s_hausdorff_young_inequality).
s_lp_function_on_R	axiom	lp function on R		Iter-3 imported axiom (s_lp_function_on_R).
s_L1_L_inf_endpoint	state	L1 L inf endpoint		Iter-3 imported state (s_L1_L_inf_endpoint).
s_L2_L2_endpoint	state	L2 L2 endpoint		Iter-3 imported state (s_L2_L2_endpoint).
s_riesz_thorin_theorem	theorem	Riesz–Thorin interpolation theorem		Iter-3 imported theorem (s_riesz_thorin_theorem).
s_linear_operator_bounded_at_endpoints	axiom	linear operator bounded at endpoints		Iter-3 imported axiom (s_linear_operator_bounded_at_endpoints).
s_lp_lq_endpoint_data	axiom	lp lq endpoint data		Iter-3 imported axiom (s_lp_lq_endpoint_data).
s_holomorphic_in_strip_with_endpoint_bounds	state	holomorphic in strip with endpoint bounds		Iter-3 imported state (s_holomorphic_in_strip_with_endpoint_bounds).
s_log_convex_bound_on_strip	state	log convex bound on strip		Iter-3 imported state (s_log_convex_bound_on_strip).
s_paley_wiener_theorem	theorem	Paley–Wiener theorem		Iter-3 imported theorem (s_paley_wiener_theorem).
s_L2_function_with_compactly_supported_fourier_transform	axiom	L2 function with compactly supported fourier transform		Iter-3 imported axiom (s_L2_function_with_compactly_supported_fourier_transform).
s_entire_function_of_exponential_type	axiom	entire function of exponential type		Iter-3 imported axiom (s_entire_function_of_exponential_type).
s_holomorphic_extension_of_f	state	holomorphic extension of f		Iter-3 imported state (s_holomorphic_extension_of_f).
s_exponential_type_bound	state	exponential type bound		Iter-3 imported state (s_exponential_type_bound).
s_poisson_summation_formula	theorem	Poisson summation formula		Iter-3 imported theorem (s_poisson_summation_formula).
s_schwartz_function_on_R	axiom	schwartz function on R		Iter-3 imported axiom (s_schwartz_function_on_R).
s_integer_lattice_in_R	axiom	integer lattice in R		Iter-3 imported axiom (s_integer_lattice_in_R).
s_1_periodic_smooth_function_F	state	1 periodic smooth function F		Iter-3 imported state (s_1_periodic_smooth_function_F).
s_fourier_coefficients_of_F_equal_hat_f_at_integers	state	fourier coefficients of F equal hat f at integers		Iter-3 imported state (s_fourier_coefficients_of_F_equal_hat_f_at_integers).
s_heisenberg_uncertainty_principle	theorem	Heisenberg uncertainty principle		Iter-3 imported theorem (s_heisenberg_uncertainty_principle).
s_L2_normalization	axiom	L2 normalization		Iter-3 imported axiom (s_L2_normalization).
s_canonical_commutator_identity	state	canonical commutator identity		Iter-3 imported state (s_canonical_commutator_identity).
s_product_of_variances_bound	state	product of variances bound		Iter-3 imported state (s_product_of_variances_bound).
s_hardy_inequality	theorem	Hardy's inequality		Iter-3 imported theorem (s_hardy_inequality).
s_nonneg_function_on_positive_reals	axiom	nonneg function on positive reals		Iter-3 imported axiom (s_nonneg_function_on_positive_reals).
s_lp_norm_definition	axiom	lp norm definition		Iter-3 imported axiom (s_lp_norm_definition).
s_hardy_averaging_operator	state	hardy averaging operator		Iter-3 imported state (s_hardy_averaging_operator).
s_lp_norm_inequality_for_A_f	state	lp norm inequality for A f		Iter-3 imported state (s_lp_norm_inequality_for_A_f).
s_hilbert_inequality	theorem	Hilbert's inequality		Iter-3 imported theorem (s_hilbert_inequality).
s_nonneg_sequences_a_m_b_n_in_l2	axiom	nonneg sequences a m b n in l2		Iter-3 imported axiom (s_nonneg_sequences_a_m_b_n_in_l2).
s_double_sum_kernel_1_over_m_plus_n	axiom	double sum kernel 1 over m plus n		Iter-3 imported axiom (s_double_sum_kernel_1_over_m_plus_n).
s_bilinear_form_with_homogeneous_kernel	state	bilinear form with homogeneous kernel		Iter-3 imported state (s_bilinear_form_with_homogeneous_kernel).
s_Schur_test_satisfied	state	Schur test satisfied		Iter-3 imported state (s_Schur_test_satisfied).
s_hilbert_transform_Lp_bound	theorem	Hilbert transform L^p boundedness		Iter-3 imported theorem (s_hilbert_transform_Lp_bound).
s_principal_value_convolution_with_1_over_x	axiom	principal value convolution with 1 over x		Iter-3 imported axiom (s_principal_value_convolution_with_1_over_x).
s_L2_isometry_up_to_sign	state	L2 isometry up to sign		Iter-3 imported state (s_L2_isometry_up_to_sign).
s_CZ_kernel_class_membership	state	CZ kernel class membership		Iter-3 imported state (s_CZ_kernel_class_membership).
s_littlewood_paley_square_function_theorem	theorem	Littlewood–Paley decomposition		Iter-3 imported theorem (s_littlewood_paley_square_function_theorem).
s_dyadic_frequency_annulus_decomposition	axiom	dyadic frequency annulus decomposition		Iter-3 imported axiom (s_dyadic_frequency_annulus_decomposition).
s_dyadic_blocks_Delta_k_f	state	dyadic blocks Delta k f		Iter-3 imported state (s_dyadic_blocks_Delta_k_f).
s_square_function_S_f	state	square function S f		Iter-3 imported state (s_square_function_S_f).
s_lp_norm_equivalence_with_square_function	state	lp norm equivalence with square function		Iter-3 imported state (s_lp_norm_equivalence_with_square_function).
s_stone_theorem_unitary_groups	theorem	Stone's theorem on one-parameter unitary groups		Iter-3 imported theorem (s_stone_theorem_unitary_groups).
s_strongly_continuous_one_parameter_unitary_group	axiom	strongly continuous one parameter unitary group		Iter-3 imported axiom (s_strongly_continuous_one_parameter_unitary_group).
s_dense_domain_with_skew_adjoint_generator	state	dense domain with skew adjoint generator		Iter-3 imported state (s_dense_domain_with_skew_adjoint_generator).
s_spectral_resolution_of_iA	state	spectral resolution of iA		Iter-3 imported state (s_spectral_resolution_of_iA).
s_spectral_theorem_bounded_self_adjoint	theorem	Spectral theorem for bounded self-adjoint operators		Iter-3 imported theorem (s_spectral_theorem_bounded_self_adjoint).
s_bounded_self_adjoint_operator_on_hilbert_space	axiom	bounded self adjoint operator on hilbert space		Iter-3 imported axiom (s_bounded_self_adjoint_operator_on_hilbert_space).
s_polynomial_calculus_with_C_star_norm	state	polynomial calculus with C star norm		Iter-3 imported state (s_polynomial_calculus_with_C_star_norm).
s_continuous_functional_calculus_extension	state	continuous functional calculus extension		Iter-3 imported state (s_continuous_functional_calculus_extension).
s_projection_valued_spectral_measure	state	projection valued spectral measure		Iter-3 imported state (s_projection_valued_spectral_measure).
s_spectral_theorem_compact_self_adjoint	theorem	Spectral theorem for compact self-adjoint operators		Iter-3 imported theorem (s_spectral_theorem_compact_self_adjoint).
s_compact_self_adjoint_operator_on_hilbert_space	axiom	compact self adjoint operator on hilbert space		Iter-3 imported axiom (s_compact_self_adjoint_operator_on_hilbert_space).
s_largest_eigenvalue_lambda_1_attained	state	largest eigenvalue lambda 1 attained		Iter-3 imported state (s_largest_eigenvalue_lambda_1_attained).
s_recurse_on_invariant_subspace	state	recurse on invariant subspace		Iter-3 imported state (s_recurse_on_invariant_subspace).
s_darboux_theorem_symplectic	theorem	Darboux's theorem (symplectic)		Iter-3 imported theorem (s_darboux_theorem_symplectic).
s_symplectic_form_omega	state	symplectic form omega		Iter-3 imported state (s_symplectic_form_omega).
s_linear_symplectic_basis_at_p	state	linear symplectic basis at p		Iter-3 imported state (s_linear_symplectic_basis_at_p).
s_local_diffeomorphism_to_standard_omega	state	local diffeomorphism to standard omega		Iter-3 imported state (s_local_diffeomorphism_to_standard_omega).
s_moser_trick_theorem	theorem	Moser's trick		Iter-3 imported theorem (s_moser_trick_theorem).
s_homotopy_of_forms_omega_t	state	homotopy of forms omega t		Iter-3 imported state (s_homotopy_of_forms_omega_t).
s_moser_vector_field_ODE	state	moser vector field ODE		Iter-3 imported state (s_moser_vector_field_ODE).
s_isotopy_phi_t	state	isotopy phi t		Iter-3 imported state (s_isotopy_phi_t).
s_weinstein_neighbourhood	theorem	Weinstein neighbourhood theorem		Iter-3 imported theorem (s_weinstein_neighbourhood).
s_lagrangian_submanifold	state	lagrangian submanifold		Iter-3 imported state (s_lagrangian_submanifold).
s_normal_bundle_identification	state	normal bundle identification		Iter-3 imported state (s_normal_bundle_identification).
s_gromov_non_squeezing	theorem	Gromov non-squeezing theorem		Iter-3 imported theorem (s_gromov_non_squeezing).
s_alleged_squeezing_phi_B_to_Z	state	alleged squeezing phi B to Z		Iter-3 imported state (s_alleged_squeezing_phi_B_to_Z).
s_j_holomorphic_disk_existence	state	j holomorphic disk existence		Iter-3 imported state (s_j_holomorphic_disk_existence).
s_area_bound_pi_r2_leq_pi_R2	state	area bound pi r2 leq pi R2		Iter-3 imported state (s_area_bound_pi_r2_leq_pi_R2).
s_gromov_compactness_j_holomorphic	theorem	Gromov compactness for J-holomorphic curves		Iter-3 imported theorem (s_gromov_compactness_j_holomorphic).
s_moduli_bounded_energy_curves	state	moduli bounded energy curves		Iter-3 imported state (s_moduli_bounded_energy_curves).
s_bubble_tree_at_singular_points	state	bubble tree at singular points		Iter-3 imported state (s_bubble_tree_at_singular_points).
s_compactified_moduli_of_stable_maps	state	compactified moduli of stable maps		Iter-3 imported state (s_compactified_moduli_of_stable_maps).
s_arnold_conjecture	theorem	Arnold conjecture (Floer)		Iter-3 imported theorem (s_arnold_conjecture).
s_hamiltonian_diffeomorphism_phi_H	state	hamiltonian diffeomorphism phi H		Iter-3 imported state (s_hamiltonian_diffeomorphism_phi_H).
s_floer_pde_for_chords	state	floer pde for chords		Iter-3 imported state (s_floer_pde_for_chords).
s_floer_homology_HF_M	state	floer homology HF M		Iter-3 imported state (s_floer_homology_HF_M).
s_fixed_point_count_geq_Betti_sum	state	fixed point count geq Betti sum		Iter-3 imported state (s_fixed_point_count_geq_Betti_sum).
s_delzant_theorem	theorem	Delzant's theorem		Iter-3 imported theorem (s_delzant_theorem).
s_toric_symplectic_manifold	state	toric symplectic manifold		Iter-3 imported state (s_toric_symplectic_manifold).
s_delzant_polytope	state	delzant polytope		Iter-3 imported state (s_delzant_polytope).
s_duistermaat_heckman	theorem	Duistermaat–Heckman formula		Iter-3 imported theorem (s_duistermaat_heckman).
s_hamiltonian_circle_action	state	hamiltonian circle action		Iter-3 imported state (s_hamiltonian_circle_action).
s_atiyah_bott_localization_formula	state	atiyah bott localization formula		Iter-3 imported state (s_atiyah_bott_localization_formula).
s_eliashberg_contact_3_classification	theorem	Eliashberg classification of contact 3-manifolds		Iter-3 imported theorem (s_eliashberg_contact_3_classification).
s_tight_overtwisted_dichotomy	state	tight overtwisted dichotomy		Iter-3 imported state (s_tight_overtwisted_dichotomy).
s_overtwisted_disk_class	state	overtwisted disk class		Iter-3 imported state (s_overtwisted_disk_class).
s_h_principle_classification_of_overtwisted	state	h principle classification of overtwisted		Iter-3 imported state (s_h_principle_classification_of_overtwisted).
s_weinstein_conjecture_3d	theorem	Weinstein conjecture (3-dim, Taubes)		Iter-3 imported theorem (s_weinstein_conjecture_3d).
s_reeb_vector_field	state	reeb vector field		Iter-3 imported state (s_reeb_vector_field).
s_sw_to_closed_reeb_orbit	state	sw to closed reeb orbit		Iter-3 imported state (s_sw_to_closed_reeb_orbit).
s_existence_of_closed_reeb_orbit	state	existence of closed reeb orbit		Iter-3 imported state (s_existence_of_closed_reeb_orbit).
s_conley_conjecture	theorem	Conley conjecture		Iter-3 imported theorem (s_conley_conjecture).
s_aspherical_symplectic_manifold	state	aspherical symplectic manifold		Iter-3 imported state (s_aspherical_symplectic_manifold).
s_hypothetical_finite_periodic_phi	state	hypothetical finite periodic phi		Iter-3 imported state (s_hypothetical_finite_periodic_phi).
s_local_floer_homology_obstruction	state	local floer homology obstruction		Iter-3 imported state (s_local_floer_homology_obstruction).
s_lagrangian_intersection_floer	theorem	Lagrangian intersection theorem (Floer)		Iter-3 imported theorem (s_lagrangian_intersection_floer).
s_lagrangian_pair_L0_L1	state	lagrangian pair L0 L1		Iter-3 imported state (s_lagrangian_pair_L0_L1).
s_floer_strips_between_L0_L1	state	floer strips between L0 L1		Iter-3 imported state (s_floer_strips_between_L0_L1).
s_lagrangian_floer_homology	state	lagrangian floer homology		Iter-3 imported state (s_lagrangian_floer_homology).
s_caratheodory_jacobi_lie	theorem	Carathéodory–Jacobi–Lie theorem		Iter-3 imported theorem (s_caratheodory_jacobi_lie).
s_involutive_function_system	state	involutive function system		Iter-3 imported state (s_involutive_function_system).
s_commuting_hamiltonian_flows	state	commuting hamiltonian flows		Iter-3 imported state (s_commuting_hamiltonian_flows).
s_liouville_arnold	theorem	Liouville–Arnold integrability theorem		Iter-3 imported theorem (s_liouville_arnold).
s_completely_integrable_hamiltonian_system	state	completely integrable hamiltonian system		Iter-3 imported state (s_completely_integrable_hamiltonian_system).
s_compact_connected_level_set	state	compact connected level set		Iter-3 imported state (s_compact_connected_level_set).
s_lagrangian_torus_fibration	state	lagrangian torus fibration		Iter-3 imported state (s_lagrangian_torus_fibration).
s_marsden_weinstein_reduction	theorem	Marsden–Weinstein symplectic reduction		Iter-3 imported theorem (s_marsden_weinstein_reduction).
s_hamiltonian_G_action_with_mu	state	hamiltonian G action with mu		Iter-3 imported state (s_hamiltonian_G_action_with_mu).
s_zero_level_set_mu	state	zero level set mu		Iter-3 imported state (s_zero_level_set_mu).
s_reduced_space_M_sslash_G	state	reduced space M sslash G		Iter-3 imported state (s_reduced_space_M_sslash_G).
s_atiyah_guillemin_sternberg	theorem	Atiyah–Guillemin–Sternberg convexity theorem		Iter-3 imported theorem (s_atiyah_guillemin_sternberg).
s_compact_torus_hamiltonian_action	state	compact torus hamiltonian action		Iter-3 imported state (s_compact_torus_hamiltonian_action).
s_connected_fibers_of_moment_map	state	connected fibers of moment map		Iter-3 imported state (s_connected_fibers_of_moment_map).
s_newlander_nirenberg	theorem	Newlander–Nirenberg theorem		Iter-3 imported theorem (s_newlander_nirenberg).
s_almost_complex_structure	state	almost complex structure		Iter-3 imported state (s_almost_complex_structure).
s_nijenhuis_obstruction	state	nijenhuis obstruction		Iter-3 imported state (s_nijenhuis_obstruction).
s_dbar_system_solvability	state	dbar system solvability		Iter-3 imported state (s_dbar_system_solvability).
s_kahler_hodge_decomposition	theorem	Hodge decomposition for compact Kähler manifolds		Iter-3 imported theorem (s_kahler_hodge_decomposition).
s_kahler_manifold	state	kahler manifold		Iter-3 imported state (s_kahler_manifold).
s_kahler_identities	state	kahler identities		Iter-3 imported state (s_kahler_identities).
s_p_q_decomposition_of_harmonic_forms	state	p q decomposition of harmonic forms		Iter-3 imported state (s_p_q_decomposition_of_harmonic_forms).
s_kodaira_embedding	theorem	Kodaira embedding theorem		Iter-3 imported theorem (s_kodaira_embedding).
s_hodge_manifold	state	hodge manifold		Iter-3 imported state (s_hodge_manifold).
s_sections_of_high_tensor_power	state	sections of high tensor power		Iter-3 imported state (s_sections_of_high_tensor_power).
s_base_point_free_separating_jets	state	base point free separating jets		Iter-3 imported state (s_base_point_free_separating_jets).
s_kodaira_vanishing	theorem	Kodaira vanishing theorem		Iter-3 imported theorem (s_kodaira_vanishing).
s_positive_line_bundle_L	state	positive line bundle L		Iter-3 imported state (s_positive_line_bundle_L).
s_bochner_kodaira_identity	state	bochner kodaira identity		Iter-3 imported state (s_bochner_kodaira_identity).
s_pointwise_positivity_of_curvature_operator	state	pointwise positivity of curvature operator		Iter-3 imported state (s_pointwise_positivity_of_curvature_operator).
s_akizuki_nakano	theorem	Akizuki–Nakano vanishing theorem		Iter-3 imported theorem (s_akizuki_nakano).
s_bochner_kodaira_nakano_identity_extended	state	bochner kodaira nakano identity extended		Iter-3 imported state (s_bochner_kodaira_nakano_identity_extended).
s_positive_curvature_pairing_on_p_q_forms	state	positive curvature pairing on p q forms		Iter-3 imported state (s_positive_curvature_pairing_on_p_q_forms).
s_yau_calabi	theorem	Calabi conjecture (Yau's theorem)		Iter-3 imported theorem (s_yau_calabi).
s_kahler_class_with_target_Ricci	state	kahler class with target Ricci		Iter-3 imported state (s_kahler_class_with_target_Ricci).
s_complex_monge_ampere_pde	state	complex monge ampere pde		Iter-3 imported state (s_complex_monge_ampere_pde).
s_unique_smooth_solution_phi	state	unique smooth solution phi		Iter-3 imported state (s_unique_smooth_solution_phi).
s_aubin_yau_KE	theorem	Aubin–Yau theorem (negative curvature Kähler–Einstein)		Iter-3 imported theorem (s_aubin_yau_KE).
s_kahler_manifold_with_negative_c1	state	kahler manifold with negative c1		Iter-3 imported state (s_kahler_manifold_with_negative_c1).
s_monge_ampere_negative_c1	state	monge ampere negative c1		Iter-3 imported state (s_monge_ampere_negative_c1).
s_a_priori_estimates_for_KE_metric	state	a priori estimates for KE metric		Iter-3 imported state (s_a_priori_estimates_for_KE_metric).
s_donaldson_uhlenbeck_yau	theorem	Donaldson–Uhlenbeck–Yau theorem		Iter-3 imported theorem (s_donaldson_uhlenbeck_yau).
s_stable_holomorphic_bundle	state	stable holomorphic bundle		Iter-3 imported state (s_stable_holomorphic_bundle).
s_hermitian_yang_mills_eqn	state	hermitian yang mills eqn		Iter-3 imported state (s_hermitian_yang_mills_eqn).
s_long_time_existence_of_metric_flow	state	long time existence of metric flow		Iter-3 imported state (s_long_time_existence_of_metric_flow).
s_seiberg_witten_invariants	theorem	Seiberg–Witten invariants		Iter-3 imported theorem (s_seiberg_witten_invariants).
s_spin_c_4_manifold	state	spin c 4 manifold		Iter-3 imported state (s_spin_c_4_manifold).
s_sw_equations	state	sw equations		Iter-3 imported state (s_sw_equations).
s_compact_sw_moduli_space	state	compact sw moduli space		Iter-3 imported state (s_compact_sw_moduli_space).
s_bertini_smoothness	theorem	Bertini's theorem on smoothness		Iter-3 imported theorem (s_bertini_smoothness).
s_smooth_subvariety_in_PN	state	smooth subvariety in PN		Iter-3 imported state (s_smooth_subvariety_in_PN).
s_hyperplane_family_in_dual_PN	state	hyperplane family in dual PN		Iter-3 imported state (s_hyperplane_family_in_dual_PN).
s_generic_H_meets_X_transversely	state	generic H meets X transversely		Iter-3 imported state (s_generic_H_meets_X_transversely).
s_bertini_irreducibility	theorem	Bertini irreducibility theorem		Iter-3 imported theorem (s_bertini_irreducibility).
s_irreducible_variety_dim_ge_2	state	irreducible variety dim ge 2		Iter-3 imported state (s_irreducible_variety_dim_ge_2).
s_incidence_variety_construction	state	incidence variety construction		Iter-3 imported state (s_incidence_variety_construction).
s_generic_section_connected	state	generic section connected		Iter-3 imported state (s_generic_section_connected).
s_riemann_roch_surfaces	theorem	Riemann–Roch theorem for surfaces		Iter-3 imported theorem (s_riemann_roch_surfaces).
s_smooth_projective_surface_with_divisor	state	smooth projective surface with divisor		Iter-3 imported state (s_smooth_projective_surface_with_divisor).
s_euler_characteristic_of_O_D	state	euler characteristic of O D		Iter-3 imported state (s_euler_characteristic_of_O_D).
s_intersection_with_canonical_class	state	intersection with canonical class		Iter-3 imported state (s_intersection_with_canonical_class).
s_hirzebruch_riemann_roch	theorem	Hirzebruch–Riemann–Roch theorem		Iter-3 imported theorem (s_hirzebruch_riemann_roch).
s_holomorphic_bundle_on_X	state	holomorphic bundle on X		Iter-3 imported state (s_holomorphic_bundle_on_X).
s_ch_and_todd_classes	state	ch and todd classes		Iter-3 imported state (s_ch_and_todd_classes).
s_index_of_dbar_operator	state	index of dbar operator		Iter-3 imported state (s_index_of_dbar_operator).
s_lefschetz_hyperplane	theorem	Lefschetz hyperplane theorem		Iter-3 imported theorem (s_lefschetz_hyperplane).
s_smooth_projective_with_hyperplane_section	state	smooth projective with hyperplane section		Iter-3 imported state (s_smooth_projective_with_hyperplane_section).
s_morse_function_with_indices_geq_n	state	morse function with indices geq n		Iter-3 imported state (s_morse_function_with_indices_geq_n).
s_Y_obtained_from_X_by_cells_of_dim_geq_n	state	Y obtained from X by cells of dim geq n		Iter-3 imported state (s_Y_obtained_from_X_by_cells_of_dim_geq_n).
s_hard_lefschetz	theorem	Hard Lefschetz theorem		Iter-3 imported theorem (s_hard_lefschetz).
s_compact_kahler_with_class	state	compact kahler with class		Iter-3 imported state (s_compact_kahler_with_class).
s_sl2_action_on_H_star_X	state	sl2 action on H star X		Iter-3 imported state (s_sl2_action_on_H_star_X).
s_primitive_decomposition	state	primitive decomposition		Iter-3 imported state (s_primitive_decomposition).
s_lefschetz_1_1_classes	theorem	Lefschetz (1,1)-classes theorem		Iter-3 imported theorem (s_lefschetz_1_1_classes).
s_integral_1_1_class_on_kahler	state	integral 1 1 class on kahler		Iter-3 imported state (s_integral_1_1_class_on_kahler).
s_exponential_exact_sequence_pic	state	exponential exact sequence pic		Iter-3 imported state (s_exponential_exact_sequence_pic).
s_pic_to_NS_via_chern_class	state	pic to NS via chern class		Iter-3 imported state (s_pic_to_NS_via_chern_class).
s_minkowski_inequality	theorem	Minkowski's inequality		Iter-3 imported theorem (s_minkowski_inequality).
s_lp_function_pair	axiom	lp function pair		Iter-3 imported axiom (s_lp_function_pair).
s_pointwise_split_of_pth_power	state	pointwise split of pth power		Iter-3 imported state (s_pointwise_split_of_pth_power).
s_two_holder_bounds_on_each_term	state	two holder bounds on each term		Iter-3 imported state (s_two_holder_bounds_on_each_term).
s_young_convolution_inequality	theorem	Young's convolution inequality		Iter-3 imported theorem (s_young_convolution_inequality).
s_three_factor_decomposition_of_kernel	state	three factor decomposition of kernel		Iter-3 imported state (s_three_factor_decomposition_of_kernel).
s_pointwise_bound_for_convolution	state	pointwise bound for convolution		Iter-3 imported state (s_pointwise_bound_for_convolution).
s_hardy_inequality_continuous	theorem	Hardy's inequality (continuous)		Iter-3 imported theorem (s_hardy_inequality_continuous).
s_lebesgue_measure_on_R_plus	axiom	lebesgue measure on R plus		Iter-3 imported axiom (s_lebesgue_measure_on_R_plus).
s_lp_function_on_R_plus	axiom	lp function on R plus		Iter-3 imported axiom (s_lp_function_on_R_plus).
s_scale_invariant_kernel	state	scale invariant kernel		Iter-3 imported state (s_scale_invariant_kernel).
s_hardy_inequality_discrete	theorem	Hardy's inequality (discrete)		Iter-3 imported theorem (s_hardy_inequality_discrete).
s_lp_sequence	axiom	lp sequence		Iter-3 imported axiom (s_lp_sequence).
s_natural_numbers	axiom	natural numbers		Iter-3 imported axiom (s_natural_numbers).
s_discrete_averaging_sequence	state	discrete averaging sequence		Iter-3 imported state (s_discrete_averaging_sequence).
s_dual_pairing_with_test_sequence	state	dual pairing with test sequence		Iter-3 imported state (s_dual_pairing_with_test_sequence).
s_carleman_inequality	theorem	Carleman's inequality		Iter-3 imported theorem (s_carleman_inequality).
s_l1_positive_sequence	axiom	l1 positive sequence		Iter-3 imported axiom (s_l1_positive_sequence).
s_weighted_substitution_for_GM	state	weighted substitution for GM		Iter-3 imported state (s_weighted_substitution_for_GM).
s_term_by_term_AM_GM_bound	state	term by term AM GM bound		Iter-3 imported state (s_term_by_term_AM_GM_bound).
s_riesz_thorin_interpolation	theorem	Riesz–Thorin interpolation theorem		Iter-3 imported theorem (s_riesz_thorin_interpolation).
s_linear_operator_T_bounded_at_endpoints	axiom	linear operator T bounded at endpoints		Iter-3 imported axiom (s_linear_operator_T_bounded_at_endpoints).
s_complex_strip_0_leq_Re_z_leq_1	axiom	complex strip 0 leq Re z leq 1		Iter-3 imported axiom (s_complex_strip_0_leq_Re_z_leq_1).
s_analytic_function_on_complex_strip	state	analytic function on complex strip		Iter-3 imported state (s_analytic_function_on_complex_strip).
s_log_norm_convex_in_z	state	log norm convex in z		Iter-3 imported state (s_log_norm_convex_in_z).
s_stein_interpolation	theorem	Stein interpolation theorem		Iter-3 imported theorem (s_stein_interpolation).
s_analytic_family_of_operators_T_z	axiom	analytic family of operators T z		Iter-3 imported axiom (s_analytic_family_of_operators_T_z).
s_lp_lq_pair_endpoints	axiom	lp lq pair endpoints		Iter-3 imported axiom (s_lp_lq_pair_endpoints).
s_analytic_F_with_growth_control	state	analytic F with growth control		Iter-3 imported state (s_analytic_F_with_growth_control).
s_hahn_decomposition	theorem	Hahn decomposition theorem		Iter-3 imported theorem (s_hahn_decomposition).
s_signed_measure_on_measurable_space	axiom	signed measure on measurable space		Iter-3 imported axiom (s_signed_measure_on_measurable_space).
s_measurable_space	axiom	measurable space		Iter-3 imported axiom (s_measurable_space).
s_supremum_of_positive_set_values	state	supremum of positive set values		Iter-3 imported state (s_supremum_of_positive_set_values).
s_limit_positive_set_P_star	state	limit positive set P star		Iter-3 imported state (s_limit_positive_set_P_star).
s_jordan_decomposition_measure	theorem	Jordan decomposition of a measure		Iter-3 imported theorem (s_jordan_decomposition_measure).
s_two_positive_measures_mutually_singular	state	two positive measures mutually singular		Iter-3 imported state (s_two_positive_measures_mutually_singular).
s_lebesgue_decomposition	theorem	Lebesgue decomposition theorem		Iter-3 imported theorem (s_lebesgue_decomposition).
s_pair_of_sigma_finite_measures_mu_nu	axiom	pair of sigma finite measures mu nu		Iter-3 imported axiom (s_pair_of_sigma_finite_measures_mu_nu).
s_radon_nikodym_density_against_sum	state	radon nikodym density against sum		Iter-3 imported state (s_radon_nikodym_density_against_sum).
s_singular_plus_absolutely_continuous_split	state	singular plus absolutely continuous split		Iter-3 imported state (s_singular_plus_absolutely_continuous_split).
s_riesz_representation_C_K	theorem	Riesz representation theorem for C(K)		Iter-3 imported theorem (s_riesz_representation_C_K).
s_bounded_linear_functional_on_C_K	axiom	bounded linear functional on C K		Iter-3 imported axiom (s_bounded_linear_functional_on_C_K).
s_pair_of_positive_functionals	state	pair of positive functionals		Iter-3 imported state (s_pair_of_positive_functionals).
s_pair_of_radon_measures	state	pair of radon measures		Iter-3 imported state (s_pair_of_radon_measures).
s_lebesgue_stieltjes_existence	theorem	Lebesgue–Stieltjes integral existence		Iter-3 imported theorem (s_lebesgue_stieltjes_existence).
s_right_continuous_nondecreasing_function_on_R	axiom	right continuous nondecreasing function on R		Iter-3 imported axiom (s_right_continuous_nondecreasing_function_on_R).
s_borel_sigma_algebra_on_R	axiom	borel sigma algebra on R		Iter-3 imported axiom (s_borel_sigma_algebra_on_R).
s_premeasure_on_half_open_intervals	state	premeasure on half open intervals		Iter-3 imported state (s_premeasure_on_half_open_intervals).
s_borel_cantelli_lemmas	theorem	Borel–Cantelli lemmas		Iter-3 imported theorem (s_borel_cantelli_lemmas).
s_sequence_of_measurable_events	axiom	sequence of measurable events		Iter-3 imported axiom (s_sequence_of_measurable_events).
s_limsup_event_definition	state	limsup event definition		Iter-3 imported state (s_limsup_event_definition).
s_first_borel_cantelli	state	first borel cantelli		Iter-3 imported state (s_first_borel_cantelli).
s_lebesgue_density_theorem	theorem	Lebesgue density theorem		Iter-3 imported theorem (s_lebesgue_density_theorem).
s_lebesgue_measurable_set_in_Rn	axiom	lebesgue measurable set in Rn		Iter-3 imported axiom (s_lebesgue_measurable_set_in_Rn).
s_indicator_function_of_E	state	indicator function of E		Iter-3 imported state (s_indicator_function_of_E).
s_ae_average_indicator_converges_to_indicator	state	ae average indicator converges to indicator		Iter-3 imported state (s_ae_average_indicator_converges_to_indicator).
s_cauchy_riemann_equations	theorem	Cauchy–Riemann equations		Iter-3 imported theorem (s_cauchy_riemann_equations).
s_complex_differentiable_function	axiom	complex differentiable function		Iter-3 imported axiom (s_complex_differentiable_function).
s_open_domain_in_C	axiom	open domain in C		Iter-3 imported axiom (s_open_domain_in_C).
s_real_imaginary_split	state	real imaginary split		Iter-3 imported state (s_real_imaginary_split).
s_two_directional_derivatives_match	state	two directional derivatives match		Iter-3 imported state (s_two_directional_derivatives_match).
s_looman_menchoff_theorem	theorem	Looman–Menchoff theorem		Iter-3 imported theorem (s_looman_menchoff_theorem).
s_continuous_function_satisfying_CR_ae	axiom	continuous function satisfying CR ae		Iter-3 imported axiom (s_continuous_function_satisfying_CR_ae).
s_vanishing_rectangle_integral	state	vanishing rectangle integral		Iter-3 imported state (s_vanishing_rectangle_integral).
s_morera_hypothesis_satisfied	state	morera hypothesis satisfied		Iter-3 imported state (s_morera_hypothesis_satisfied).
s_cauchy_goursat_theorem	theorem	Cauchy's theorem (rectangle / Goursat)		Iter-3 imported theorem (s_cauchy_goursat_theorem).
s_rectangle_in_C	axiom	rectangle in C		Iter-3 imported axiom (s_rectangle_in_C).
s_nested_quartered_rectangle_with_largest_integral	state	nested quartered rectangle with largest integral		Iter-3 imported state (s_nested_quartered_rectangle_with_largest_integral).
s_point_with_local_linear_approximation	state	point with local linear approximation		Iter-3 imported state (s_point_with_local_linear_approximation).
s_cauchy_theorem_homotopy	theorem	Cauchy's theorem (homotopy version)		Iter-3 imported theorem (s_cauchy_theorem_homotopy).
s_two_homotopic_closed_curves	axiom	two homotopic closed curves		Iter-3 imported axiom (s_two_homotopic_closed_curves).
s_homotopy_grid_partition	state	homotopy grid partition		Iter-3 imported state (s_homotopy_grid_partition).
s_finite_grid_with_goursat_per_cell	state	finite grid with goursat per cell		Iter-3 imported state (s_finite_grid_with_goursat_per_cell).
s_liouville_theorem_complex	theorem	Liouville's theorem		Iter-3 imported theorem (s_liouville_theorem_complex).
s_cauchy_estimate_for_derivative	state	cauchy estimate for derivative		Iter-3 imported state (s_cauchy_estimate_for_derivative).
s_f_prime_identically_zero	state	f prime identically zero		Iter-3 imported state (s_f_prime_identically_zero).
s_open_mapping_theorem_complex	theorem	Open mapping theorem (complex)		Iter-3 imported theorem (s_open_mapping_theorem_complex).
s_shifted_function_with_zero_at_z_0	state	shifted function with zero at z 0		Iter-3 imported state (s_shifted_function_with_zero_at_z_0).
s_neighborhood_of_w_0_in_image	state	neighborhood of w 0 in image		Iter-3 imported state (s_neighborhood_of_w_0_in_image).
s_phragmen_lindelof	theorem	Phragmén–Lindelöf principle		Iter-3 imported theorem (s_phragmen_lindelof).
s_holomorphic_function_on_sector_with_growth_bound	axiom	holomorphic function on sector with growth bound		Iter-3 imported axiom (s_holomorphic_function_on_sector_with_growth_bound).
s_unbounded_sector_in_C	axiom	unbounded sector in C		Iter-3 imported axiom (s_unbounded_sector_in_C).
s_auxiliary_decaying_function_g	state	auxiliary decaying function g		Iter-3 imported state (s_auxiliary_decaying_function_g).
s_g_bounded_on_bounded_part	state	g bounded on bounded part		Iter-3 imported state (s_g_bounded_on_bounded_part).
s_hadamard_three_lines	theorem	Hadamard three-lines theorem		Iter-3 imported theorem (s_hadamard_three_lines).
s_holomorphic_bounded_function_on_strip	axiom	holomorphic bounded function on strip		Iter-3 imported axiom (s_holomorphic_bounded_function_on_strip).
s_vertical_strip_0_leq_Re_z_leq_1	axiom	vertical strip 0 leq Re z leq 1		Iter-3 imported axiom (s_vertical_strip_0_leq_Re_z_leq_1).
s_normalized_function_with_boundary_modulus_one	state	normalized function with boundary modulus one		Iter-3 imported state (s_normalized_function_with_boundary_modulus_one).
s_modulus_g_leq_1_in_strip	state	modulus g leq 1 in strip		Iter-3 imported state (s_modulus_g_leq_1_in_strip).
s_hadamard_three_circles	theorem	Hadamard three-circles theorem		Iter-3 imported theorem (s_hadamard_three_circles).
s_pullback_to_vertical_strip	state	pullback to vertical strip		Iter-3 imported state (s_pullback_to_vertical_strip).
s_log_max_convex_in_log_r	state	log max convex in log r		Iter-3 imported state (s_log_max_convex_in_log_r).
s_hadamard_factorization	theorem	Hadamard factorization theorem		Iter-3 imported theorem (s_hadamard_factorization).
s_zero_sequence_with_density_bound	state	zero sequence with density bound		Iter-3 imported state (s_zero_sequence_with_density_bound).
s_canonical_weierstrass_product	state	canonical weierstrass product		Iter-3 imported state (s_canonical_weierstrass_product).
s_weierstrass_factorization	theorem	Weierstrass factorization theorem		Iter-3 imported theorem (s_weierstrass_factorization).
s_sequence_a_n_to_infinity_with_multiplicities	axiom	sequence a n to infinity with multiplicities		Iter-3 imported axiom (s_sequence_a_n_to_infinity_with_multiplicities).
s_elementary_factors_with_convergence_exponents	state	elementary factors with convergence exponents		Iter-3 imported state (s_elementary_factors_with_convergence_exponents).
s_convergent_infinite_product	state	convergent infinite product		Iter-3 imported state (s_convergent_infinite_product).
s_vitali_convergence_complex	theorem	Vitali's convergence theorem (complex)		Iter-3 imported theorem (s_vitali_convergence_complex).
s_locally_uniformly_bounded_holomorphic_sequence	axiom	locally uniformly bounded holomorphic sequence		Iter-3 imported axiom (s_locally_uniformly_bounded_holomorphic_sequence).
s_open_connected_domain_in_C	axiom	open connected domain in C		Iter-3 imported axiom (s_open_connected_domain_in_C).
s_normal_family_with_subsequence_limit	state	normal family with subsequence limit		Iter-3 imported state (s_normal_family_with_subsequence_limit).
s_limit_function_determined_by_pointwise_set	state	limit function determined by pointwise set		Iter-3 imported state (s_limit_function_determined_by_pointwise_set).
s_mergelyan_theorem	theorem	Mergelyan's theorem		Iter-3 imported theorem (s_mergelyan_theorem).
s_continuous_function_on_compact_K_holomorphic_in_interior	axiom	continuous function on compact K holomorphic in interior		Iter-3 imported axiom (s_continuous_function_on_compact_K_holomorphic_in_interior).
s_compact_subset_K_of_C_with_connected_complement	axiom	compact subset K of C with connected complement		Iter-3 imported axiom (s_compact_subset_K_of_C_with_connected_complement).
s_smooth_approximant_of_f	state	smooth approximant of f		Iter-3 imported state (s_smooth_approximant_of_f).
s_rational_approximation_uniform_on_K	state	rational approximation uniform on K		Iter-3 imported state (s_rational_approximation_uniform_on_K).
s_carleman_approximation	theorem	Carleman approximation theorem		Iter-3 imported theorem (s_carleman_approximation).
s_continuous_function_on_real_line	axiom	continuous function on real line		Iter-3 imported axiom (s_continuous_function_on_real_line).
s_positive_continuous_error_function_eps	axiom	positive continuous error function eps		Iter-3 imported axiom (s_positive_continuous_error_function_eps).
s_sequence_of_entire_partial_approximants	state	sequence of entire partial approximants		Iter-3 imported state (s_sequence_of_entire_partial_approximants).
s_uniform_approximation_with_error_eps_on_R	state	uniform approximation with error eps on R		Iter-3 imported state (s_uniform_approximation_with_error_eps_on_R).
s_bernstein_theorem_monotonic	theorem	Bernstein's theorem on absolutely monotonic functions		Iter-3 imported theorem (s_bernstein_theorem_monotonic).
s_completely_monotonic_function_on_R_plus	axiom	completely monotonic function on R plus		Iter-3 imported axiom (s_completely_monotonic_function_on_R_plus).
s_positive_real_line	axiom	positive real line		Iter-3 imported axiom (s_positive_real_line).
s_nonneg_finite_differences_at_grid	state	nonneg finite differences at grid		Iter-3 imported state (s_nonneg_finite_differences_at_grid).
s_limit_measure_dmu_on_R_plus	state	limit measure dmu on R plus		Iter-3 imported state (s_limit_measure_dmu_on_R_plus).
s_kahane_katznelson_de_leeuw	theorem	Kahane–Katznelson–de Leeuw theorem		Iter-3 imported theorem (s_kahane_katznelson_de_leeuw).
s_continuous_function_on_circle	axiom	continuous function on circle		Iter-3 imported axiom (s_continuous_function_on_circle).
s_l2_fourier_coefficient_sequence	axiom	l2 fourier coefficient sequence		Iter-3 imported axiom (s_l2_fourier_coefficient_sequence).
s_random_sign_modification_of_coefficients	state	random sign modification of coefficients		Iter-3 imported state (s_random_sign_modification_of_coefficients).
s_existence_of_realization_with_bounded_sup_norm	state	existence of realization with bounded sup norm		Iter-3 imported state (s_existence_of_realization_with_bounded_sup_norm).
s_hormander_mikhlin_multiplier	theorem	Hörmander–Mikhlin multiplier theorem		Iter-3 imported theorem (s_hormander_mikhlin_multiplier).
s_symbol_m_smooth_with_homogeneous_derivative_bounds	axiom	symbol m smooth with homogeneous derivative bounds		Iter-3 imported axiom (s_symbol_m_smooth_with_homogeneous_derivative_bounds).
s_dyadic_decomposition_of_symbol	state	dyadic decomposition of symbol		Iter-3 imported state (s_dyadic_decomposition_of_symbol).
s_kernel_satisfies_CZ_conditions	state	kernel satisfies CZ conditions		Iter-3 imported state (s_kernel_satisfies_CZ_conditions).
s_cotlar_stein_lemma	theorem	Cotlar–Stein lemma		Iter-3 imported theorem (s_cotlar_stein_lemma).
s_family_of_bounded_operators_T_j_on_Hilbert_space	axiom	family of bounded operators T j on Hilbert space		Iter-3 imported axiom (s_family_of_bounded_operators_T_j_on_Hilbert_space).
s_almost_orthogonality_estimates	axiom	almost orthogonality estimates		Iter-3 imported axiom (s_almost_orthogonality_estimates).
s_pair_of_positive_operator_sums	state	pair of positive operator sums		Iter-3 imported state (s_pair_of_positive_operator_sums).
s_norm_bound_on_sum_via_schur	state	norm bound on sum via schur		Iter-3 imported state (s_norm_bound_on_sum_via_schur).
s_T_one_theorem	theorem	T(1) theorem (David–Journé)		Iter-3 imported theorem (s_T_one_theorem).
s_singular_integral_operator_with_CZ_kernel	axiom	singular integral operator with CZ kernel		Iter-3 imported axiom (s_singular_integral_operator_with_CZ_kernel).
s_modified_operator_with_T_one_zero	state	modified operator with T one zero		Iter-3 imported state (s_modified_operator_with_T_one_zero).
s_L2_bounded_modified_operator	state	L2 bounded modified operator		Iter-3 imported state (s_L2_bounded_modified_operator).
s_rellich_kondrachov_compactness	theorem	Rellich–Kondrachov compactness theorem		Iter-3 imported theorem (s_rellich_kondrachov_compactness).
s_bounded_lipschitz_domain_in_Rn	axiom	bounded lipschitz domain in Rn		Iter-3 imported axiom (s_bounded_lipschitz_domain_in_Rn).
s_sobolev_space_W_kp_on_domain	axiom	sobolev space W kp on domain		Iter-3 imported axiom (s_sobolev_space_W_kp_on_domain).
s_mollification_with_uniform_Lp_control	state	mollification with uniform Lp control		Iter-3 imported state (s_mollification_with_uniform_Lp_control).
s_relatively_compact_in_Lp_after_mollification	state	relatively compact in Lp after mollification		Iter-3 imported state (s_relatively_compact_in_Lp_after_mollification).
s_symmetrized_radial_problem	state	symmetrized radial problem		Iter-3 imported state (s_symmetrized_radial_problem).
s_scale_critical_setup	state	scale critical setup		Iter-3 imported state (s_scale_critical_setup).
s_gagliardo_nirenberg_inequality	theorem	Gagliardo–Nirenberg interpolation inequality		Iter-3 imported theorem (s_gagliardo_nirenberg_inequality).
s_smooth_compactly_supported_function_on_Rn	axiom	smooth compactly supported function on Rn		Iter-3 imported axiom (s_smooth_compactly_supported_function_on_Rn).
s_sobolev_norms_with_mixed_orders	axiom	sobolev norms with mixed orders		Iter-3 imported axiom (s_sobolev_norms_with_mixed_orders).
s_dyadic_pieces_with_localized_frequency	state	dyadic pieces with localized frequency		Iter-3 imported state (s_dyadic_pieces_with_localized_frequency).
s_pointwise_interpolation_inequality	state	pointwise interpolation inequality		Iter-3 imported state (s_pointwise_interpolation_inequality).
s_john_nirenberg_inequality	theorem	John–Nirenberg inequality		Iter-3 imported theorem (s_john_nirenberg_inequality).
s_bmo_function_on_Rn	axiom	bmo function on Rn		Iter-3 imported axiom (s_bmo_function_on_Rn).
s_cube_in_Rn	axiom	cube in Rn		Iter-3 imported axiom (s_cube_in_Rn).
s_stopping_cubes_at_threshold_lambda	state	stopping cubes at threshold lambda		Iter-3 imported state (s_stopping_cubes_at_threshold_lambda).
s_exponential_decay_of_distribution	state	exponential decay of distribution		Iter-3 imported state (s_exponential_decay_of_distribution).
s_fefferman_duality_H1_BMO	theorem	Fefferman duality H¹–BMO		Iter-3 imported theorem (s_fefferman_duality_H1_BMO).
s_hardy_space_H1_on_Rn	axiom	hardy space H1 on Rn		Iter-3 imported axiom (s_hardy_space_H1_on_Rn).
s_bmo_space_on_Rn	axiom	bmo space on Rn		Iter-3 imported axiom (s_bmo_space_on_Rn).
s_atomic_decomposition_of_H1	state	atomic decomposition of H1		Iter-3 imported state (s_atomic_decomposition_of_H1).
s_bounded_bilinear_pairing_BMO_times_H1	state	bounded bilinear pairing BMO times H1		Iter-3 imported state (s_bounded_bilinear_pairing_BMO_times_H1).
s_cauchy_schwarz_inequality	theorem	Cauchy–Schwarz inequality		Iter-3 imported theorem (s_cauchy_schwarz_inequality).
s_two_vectors_in_space	axiom	two vectors in space		Iter-3 imported axiom (s_two_vectors_in_space).
s_quadratic_in_t_nonneg	state	quadratic in t nonneg		Iter-3 imported state (s_quadratic_in_t_nonneg).
s_discriminant_inequality	state	discriminant inequality		Iter-3 imported state (s_discriminant_inequality).
s_gagliardo_nirenberg_sobolev_inequality	theorem	Gagliardo–Nirenberg–Sobolev inequality		Iter-3 imported theorem (s_gagliardo_nirenberg_sobolev_inequality).
s_lp_norm_interpolation_relation	axiom	lp norm interpolation relation		Iter-3 imported axiom (s_lp_norm_interpolation_relation).
s_one_derivative_L_n_over_n_minus_1_bound	state	one derivative L n over n minus 1 bound		Iter-3 imported state (s_one_derivative_L_n_over_n_minus_1_bound).
s_full_GN_interpolation_chain	state	full GN interpolation chain		Iter-3 imported state (s_full_GN_interpolation_chain).
s_sobolev_trace_theorem	theorem	Trace theorem for Sobolev spaces		Iter-3 imported theorem (s_sobolev_trace_theorem).
s_W_1_p_function_on_lipschitz_domain	axiom	W 1 p function on lipschitz domain		Iter-3 imported axiom (s_W_1_p_function_on_lipschitz_domain).
s_boundary_of_domain	axiom	boundary of domain		Iter-3 imported axiom (s_boundary_of_domain).
s_half_space_local_chart	state	half space local chart		Iter-3 imported state (s_half_space_local_chart).
s_pointwise_boundary_value_bound	state	pointwise boundary value bound		Iter-3 imported state (s_pointwise_boundary_value_bound).
s_riesz_potential_kernel_abs_x_minus_lambda	axiom	riesz potential kernel abs x minus lambda		Iter-3 imported axiom (s_riesz_potential_kernel_abs_x_minus_lambda).
s_dyadic_kernel_pieces	state	dyadic kernel pieces		Iter-3 imported state (s_dyadic_kernel_pieces).
s_bound_via_maximal_function_majorant	state	bound via maximal function majorant		Iter-3 imported state (s_bound_via_maximal_function_majorant).
s_choquet_representation_theorem	theorem	Choquet representation theorem		Iter-3 imported theorem (s_choquet_representation_theorem).
s_point_in_convex_set	axiom	point in convex set		Iter-3 imported axiom (s_point_in_convex_set).
s_positive_linear_functional_on_continuous_functions	state	positive linear functional on continuous functions		Iter-3 imported state (s_positive_linear_functional_on_continuous_functions).
s_probability_measure_on_K	state	probability measure on K		Iter-3 imported state (s_probability_measure_on_K).
s_riesz_lemma	theorem	Riesz lemma (almost-perpendicular vector)		Iter-3 imported theorem (s_riesz_lemma).
s_normed_space_with_proper_closed_subspace_Y	axiom	normed space with proper closed subspace Y		Iter-3 imported axiom (s_normed_space_with_proper_closed_subspace_Y).
s_quantity_alpha_in_0_1	axiom	quantity alpha in 0 1		Iter-3 imported axiom (s_quantity_alpha_in_0_1).
s_near_extremal_x_outside_Y	state	near extremal x outside Y		Iter-3 imported state (s_near_extremal_x_outside_Y).
s_unit_vector_alpha_far_from_Y	state	unit vector alpha far from Y		Iter-3 imported state (s_unit_vector_alpha_far_from_Y).
s_riesz_compactness_unit_ball	theorem	Riesz's theorem on compactness of unit ball		Iter-3 imported theorem (s_riesz_compactness_unit_ball).
s_normed_space	axiom	normed space		Iter-3 imported axiom (s_normed_space).
s_unit_ball_compactness	axiom	unit ball compactness		Iter-3 imported axiom (s_unit_ball_compactness).
s_compact_unit_ball_assumption	state	compact unit ball assumption		Iter-3 imported state (s_compact_unit_ball_assumption).
s_sequence_with_pairwise_distance_geq_half	state	sequence with pairwise distance geq half		Iter-3 imported state (s_sequence_with_pairwise_distance_geq_half).
s_hahn_decomposition_theorem	theorem	Hahn decomposition theorem		Iter-3 imported theorem (s_hahn_decomposition_theorem).
s_supremum_achieved_via_chain	state	supremum achieved via chain		Iter-3 imported state (s_supremum_achieved_via_chain).
s_positive_set_P_with_sup	state	positive set P with sup		Iter-3 imported state (s_positive_set_P_with_sup).
s_jordan_decomposition_theorem	theorem	Jordan decomposition theorem		Iter-3 imported theorem (s_jordan_decomposition_theorem).
s_hahn_decomposition_pair	axiom	hahn decomposition pair		Iter-3 imported axiom (s_hahn_decomposition_pair).
s_two_positive_measures_nu_plus_nu_minus	state	two positive measures nu plus nu minus		Iter-3 imported state (s_two_positive_measures_nu_plus_nu_minus).
s_mutually_singular_decomposition	state	mutually singular decomposition		Iter-3 imported state (s_mutually_singular_decomposition).
s_lebesgue_decomposition_theorem	theorem	Lebesgue decomposition theorem		Iter-3 imported theorem (s_lebesgue_decomposition_theorem).
s_sigma_finite_signed_measure	axiom	sigma finite signed measure		Iter-3 imported axiom (s_sigma_finite_signed_measure).
s_reference_sigma_finite_measure	axiom	reference sigma finite measure		Iter-3 imported axiom (s_reference_sigma_finite_measure).
s_maximal_singular_carrier_A	state	maximal singular carrier A		Iter-3 imported state (s_maximal_singular_carrier_A).
s_two_part_decomposition	state	two part decomposition		Iter-3 imported state (s_two_part_decomposition).
s_riesz_representation_hilbert	theorem	Riesz representation theorem on Hilbert space		Iter-3 imported theorem (s_riesz_representation_hilbert).
s_bounded_linear_functional_on_hilbert	axiom	bounded linear functional on hilbert		Iter-3 imported axiom (s_bounded_linear_functional_on_hilbert).
s_closed_subspace_kernel_of_L	state	closed subspace kernel of L		Iter-3 imported state (s_closed_subspace_kernel_of_L).
s_orthogonal_complement_spanned_by_z_0	state	orthogonal complement spanned by z 0		Iter-3 imported state (s_orthogonal_complement_spanned_by_z_0).
s_mertens_cauchy_product_theorem	theorem	Mertens's theorem on convolution of series		Iter-3 imported theorem (s_mertens_cauchy_product_theorem).
s_two_real_series_one_absolutely_convergent	axiom	two real series one absolutely convergent		Iter-3 imported axiom (s_two_real_series_one_absolutely_convergent).
s_cauchy_product	axiom	cauchy product		Iter-3 imported axiom (s_cauchy_product).
s_partial_sum_split_main_tail	state	partial sum split main tail		Iter-3 imported state (s_partial_sum_split_main_tail).
s_tail_estimate_uniform	state	tail estimate uniform		Iter-3 imported state (s_tail_estimate_uniform).
s_abel_summation_theorem	theorem	Abel's summation theorem		Iter-3 imported theorem (s_abel_summation_theorem).
s_real_power_series_with_radius_of_convergence_1	axiom	real power series with radius of convergence 1		Iter-3 imported axiom (s_real_power_series_with_radius_of_convergence_1).
s_convergent_series_at_endpoint_x_eq_1	axiom	convergent series at endpoint x eq 1		Iter-3 imported axiom (s_convergent_series_at_endpoint_x_eq_1).
s_partial_summation_identity	state	partial summation identity		Iter-3 imported state (s_partial_summation_identity).
s_uniform_convergence_on_segment_to_1	state	uniform convergence on segment to 1		Iter-3 imported state (s_uniform_convergence_on_segment_to_1).
s_hardy_littlewood_tauberian_theorem	theorem	Tauberian theorem (Hardy–Littlewood)		Iter-3 imported theorem (s_hardy_littlewood_tauberian_theorem).
s_series_Abel_summable_to_s	axiom	series Abel summable to s		Iter-3 imported axiom (s_series_Abel_summable_to_s).
s_tauberian_growth_condition_on_a_n	axiom	tauberian growth condition on a n		Iter-3 imported axiom (s_tauberian_growth_condition_on_a_n).
s_polynomial_smoothed_partial_sum	state	polynomial smoothed partial sum		Iter-3 imported state (s_polynomial_smoothed_partial_sum).
s_indicator_approximation_via_polynomial	state	indicator approximation via polynomial		Iter-3 imported state (s_indicator_approximation_via_polynomial).
s_wiener_tauberian_theorem	theorem	Wiener Tauberian theorem		Iter-3 imported theorem (s_wiener_tauberian_theorem).
s_l1_function_on_R_with_nonvanishing_fourier_transform	axiom	l1 function on R with nonvanishing fourier transform		Iter-3 imported axiom (s_l1_function_on_R_with_nonvanishing_fourier_transform).
s_translation_invariant_subspace_of_L1	axiom	translation invariant subspace of L1		Iter-3 imported axiom (s_translation_invariant_subspace_of_L1).
s_closed_ideal_in_Wiener_algebra	state	closed ideal in Wiener algebra		Iter-3 imported state (s_closed_ideal_in_Wiener_algebra).
s_no_common_zero_in_spectrum	state	no common zero in spectrum		Iter-3 imported state (s_no_common_zero_in_spectrum).
s_polynomial_continuity_theorem	theorem	Bolzano's theorem on continuity of polynomials		Iter-3 imported theorem (s_polynomial_continuity_theorem).
s_polynomial_function_on_R	axiom	polynomial function on R		Iter-3 imported axiom (s_polynomial_function_on_R).
s_epsilon_delta_definition	axiom	epsilon delta definition		Iter-3 imported axiom (s_epsilon_delta_definition).
s_monomial_continuity	state	monomial continuity		Iter-3 imported state (s_monomial_continuity).
s_stone_weierstrass_complex	theorem	Stone–Weierstrass complex version		Iter-3 imported theorem (s_stone_weierstrass_complex).
s_self_adjoint_separating_unital_subalgebra_of_complex_C_X	axiom	self adjoint separating unital subalgebra of complex C X		Iter-3 imported axiom (s_self_adjoint_separating_unital_subalgebra_of_complex_C_X).
s_real_subalgebra_separating	state	real subalgebra separating		Iter-3 imported state (s_real_subalgebra_separating).
s_real_subalgebra_dense	state	real subalgebra dense		Iter-3 imported state (s_real_subalgebra_dense).
s_gelfand_representation_theorem	theorem	Gelfand representation theorem		Iter-3 imported theorem (s_gelfand_representation_theorem).
s_space_of_characters_with_weak_star_topology	axiom	space of characters with weak star topology		Iter-3 imported axiom (s_space_of_characters_with_weak_star_topology).
s_compact_hausdorff_character_space	state	compact hausdorff character space		Iter-3 imported state (s_compact_hausdorff_character_space).
s_continuous_algebra_homomorphism_A_to_C_Delta_A	state	continuous algebra homomorphism A to C Delta A		Iter-3 imported state (s_continuous_algebra_homomorphism_A_to_C_Delta_A).
s_eberlein_smulian_theorem	theorem	Eberlein–Šmulian theorem		Iter-3 imported theorem (s_eberlein_smulian_theorem).
s_subset_of_banach_space	axiom	subset of banach space		Iter-3 imported axiom (s_subset_of_banach_space).
s_net_based_weak_compactness	state	net based weak compactness		Iter-3 imported state (s_net_based_weak_compactness).
s_metrizable_weak_topology_on_separable_part	state	metrizable weak topology on separable part		Iter-3 imported state (s_metrizable_weak_topology_on_separable_part).
s_mackey_arens_theorem	theorem	Mackey–Arens theorem		Iter-3 imported theorem (s_mackey_arens_theorem).
s_dual_pair_of_vector_spaces	axiom	dual pair of vector spaces		Iter-3 imported axiom (s_dual_pair_of_vector_spaces).
s_locally_convex_topology_compatible_with_duality	axiom	locally convex topology compatible with duality		Iter-3 imported axiom (s_locally_convex_topology_compatible_with_duality).
s_polar_topology_family	state	polar topology family		Iter-3 imported state (s_polar_topology_family).
s_compatible_topology_characterization	state	compatible topology characterization		Iter-3 imported state (s_compatible_topology_characterization).
s_banach_mazur_theorem	theorem	Banach–Mazur theorem		Iter-3 imported theorem (s_banach_mazur_theorem).
s_separable_banach_space	axiom	separable banach space		Iter-3 imported axiom (s_separable_banach_space).
s_continuous_function_space_C_0_1	axiom	continuous function space C 0 1		Iter-3 imported axiom (s_continuous_function_space_C_0_1).
s_countable_dense_sequence_in_unit_ball	state	countable dense sequence in unit ball		Iter-3 imported state (s_countable_dense_sequence_in_unit_ball).
s_isometric_embedding_via_Auerbach_type_basis	state	isometric embedding via Auerbach type basis		Iter-3 imported state (s_isometric_embedding_via_Auerbach_type_basis).
s_krein_smulian_theorem	theorem	Krein–Šmulian theorem		Iter-3 imported theorem (s_krein_smulian_theorem).
s_convex_subset_of_dual_with_weak_star_closed_intersections_with_bounded_sets	axiom	convex subset of dual with weak star closed intersections with bounded sets		Iter-3 imported axiom (s_convex_subset_of_dual_with_weak_star_closed_intersections_with_bounded_sets).
s_compatible_topology_setup	state	compatible topology setup		Iter-3 imported state (s_compatible_topology_setup).
s_weak_star_closed_globally	state	weak star closed globally		Iter-3 imported state (s_weak_star_closed_globally).
s_rearrangement_inequality	theorem	Hardy–Littlewood–Polya rearrangement inequality		Iter-3 imported theorem (s_rearrangement_inequality).
s_two_finite_sequences_of_real_numbers	axiom	two finite sequences of real numbers		Iter-3 imported axiom (s_two_finite_sequences_of_real_numbers).
s_permutation_group_on_indices	axiom	permutation group on indices		Iter-3 imported axiom (s_permutation_group_on_indices).
s_pairwise_swap_decreases_sum_if_inversion	state	pairwise swap decreases sum if inversion		Iter-3 imported state (s_pairwise_swap_decreases_sum_if_inversion).
s_sorted_pairing_extremal	state	sorted pairing extremal		Iter-3 imported state (s_sorted_pairing_extremal).
s_brunn_minkowski_inequality	theorem	Brunn–Minkowski inequality		Iter-3 imported theorem (s_brunn_minkowski_inequality).
s_two_nonempty_measurable_subsets_of_Rn	axiom	two nonempty measurable subsets of Rn		Iter-3 imported axiom (s_two_nonempty_measurable_subsets_of_Rn).
s_Minkowski_sum_A_plus_B	axiom	Minkowski sum A plus B		Iter-3 imported axiom (s_Minkowski_sum_A_plus_B).
s_brunn_minkowski_for_boxes_via_AM_GM	state	brunn minkowski for boxes via AM GM		Iter-3 imported state (s_brunn_minkowski_for_boxes_via_AM_GM).
s_brunn_minkowski_for_finite_unions_of_boxes	state	brunn minkowski for finite unions of boxes		Iter-3 imported state (s_brunn_minkowski_for_finite_unions_of_boxes).
s_isoperimetric_inequality_Rn	theorem	Isoperimetric inequality in R^n		Iter-3 imported theorem (s_isoperimetric_inequality_Rn).
s_bounded_open_subset_of_Rn_with_smooth_boundary	axiom	bounded open subset of Rn with smooth boundary		Iter-3 imported axiom (s_bounded_open_subset_of_Rn_with_smooth_boundary).
s_volume_and_surface_area	axiom	volume and surface area		Iter-3 imported axiom (s_volume_and_surface_area).
s_surface_area_as_eps_limit_of_volumes	state	surface area as eps limit of volumes		Iter-3 imported state (s_surface_area_as_eps_limit_of_volumes).
s_first_variation_inequality	state	first variation inequality		Iter-3 imported state (s_first_variation_inequality).
s_prekopa_leindler_inequality	theorem	Prékopa–Leindler inequality		Iter-3 imported theorem (s_prekopa_leindler_inequality).
s_three_nonneg_measurable_functions_with_log_concave_constraint	axiom	three nonneg measurable functions with log concave constraint		Iter-3 imported axiom (s_three_nonneg_measurable_functions_with_log_concave_constraint).
s_indicator_case_equivalent_to_brunn_minkowski	state	indicator case equivalent to brunn minkowski		Iter-3 imported state (s_indicator_case_equivalent_to_brunn_minkowski).
s_layer_cake_lift_to_general_functions	state	layer cake lift to general functions		Iter-3 imported state (s_layer_cake_lift_to_general_functions).
s_loomis_whitney_inequality	theorem	Loomis–Whitney inequality		Iter-3 imported theorem (s_loomis_whitney_inequality).
s_measurable_subset_of_Rn	axiom	measurable subset of Rn		Iter-3 imported axiom (s_measurable_subset_of_Rn).
s_n_coordinate_projections	axiom	n coordinate projections		Iter-3 imported axiom (s_n_coordinate_projections).
s_iterated_holder_on_indicator	state	iterated holder on indicator		Iter-3 imported state (s_iterated_holder_on_indicator).
s_projection_product_bound	state	projection product bound		Iter-3 imported state (s_projection_product_bound).
s_hausdorff_moment_problem	theorem	Hausdorff moment problem		Iter-3 imported theorem (s_hausdorff_moment_problem).
s_real_sequence_m_n	axiom	real sequence m n		Iter-3 imported axiom (s_real_sequence_m_n).
s_uniqueness_of_measure_on_unit_interval	axiom	uniqueness of measure on unit interval		Iter-3 imported axiom (s_uniqueness_of_measure_on_unit_interval).
s_complete_monotonicity_criterion	state	complete monotonicity criterion		Iter-3 imported state (s_complete_monotonicity_criterion).
s_positive_functional_on_C_0_1_via_density	state	positive functional on C 0 1 via density		Iter-3 imported state (s_positive_functional_on_C_0_1_via_density).
s_bernstein_widder_theorem	theorem	Bernstein–Widder theorem on completely monotone functions		Iter-3 imported theorem (s_bernstein_widder_theorem).
s_completely_monotone_function_on_positive_reals	axiom	completely monotone function on positive reals		Iter-3 imported axiom (s_completely_monotone_function_on_positive_reals).
s_laplace_transform_of_positive_measure	axiom	laplace transform of positive measure		Iter-3 imported axiom (s_laplace_transform_of_positive_measure).
s_finite_difference_quadrature_approximation	state	finite difference quadrature approximation		Iter-3 imported state (s_finite_difference_quadrature_approximation).
s_representing_measures_on_intervals_converging	state	representing measures on intervals converging		Iter-3 imported state (s_representing_measures_on_intervals_converging).
s_herglotz_representation	theorem	Riesz–Herglotz representation		Iter-3 imported theorem (s_herglotz_representation).
s_holomorphic_function_on_unit_disk_with_nonneg_real_part	axiom	holomorphic function on unit disk with nonneg real part		Iter-3 imported axiom (s_holomorphic_function_on_unit_disk_with_nonneg_real_part).
s_positive_borel_measure_on_circle	axiom	positive borel measure on circle		Iter-3 imported axiom (s_positive_borel_measure_on_circle).
s_family_of_finite_positive_measures_mu_r	state	family of finite positive measures mu r		Iter-3 imported state (s_family_of_finite_positive_measures_mu_r).
s_weak_star_limit_measure_mu	state	weak star limit measure mu		Iter-3 imported state (s_weak_star_limit_measure_mu).
s_vitali_convergence_theorem	theorem	Vitali convergence theorem		Iter-3 imported theorem (s_vitali_convergence_theorem).
s_uniformly_integrable_sequence	axiom	uniformly integrable sequence		Iter-3 imported axiom (s_uniformly_integrable_sequence).
s_ae_convergent_sequence_on_finite_measure_space	axiom	ae convergent sequence on finite measure space		Iter-3 imported axiom (s_ae_convergent_sequence_on_finite_measure_space).
s_uniform_tail_smallness_in_n	state	uniform tail smallness in n		Iter-3 imported state (s_uniform_tail_smallness_in_n).
s_split_integral_main_plus_tail	state	split integral main plus tail		Iter-3 imported state (s_split_integral_main_plus_tail).
s_dunford_pettis_theorem	theorem	Eberlein theorem on weak compactness in L^1 (Dunford–Pettis)		Iter-3 imported theorem (s_dunford_pettis_theorem).
s_subset_of_L1_finite_measure_space	axiom	subset of L1 finite measure space		Iter-3 imported axiom (s_subset_of_L1_finite_measure_space).
s_weak_compactness_in_L1	axiom	weak compactness in L1		Iter-3 imported axiom (s_weak_compactness_in_L1).
s_uniform_integrability_criterion	state	uniform integrability criterion		Iter-3 imported state (s_uniform_integrability_criterion).
s_weak_compactness_via_alaoglu_and_eberlein	state	weak compactness via alaoglu and eberlein		Iter-3 imported state (s_weak_compactness_via_alaoglu_and_eberlein).
s_markov_kakutani_fpt	theorem	Markov–Kakutani fixed-point theorem		Iter-3 imported theorem (s_markov_kakutani_fpt).
s_commuting_family_of_continuous_affine_self_maps	axiom	commuting family of continuous affine self maps		Iter-3 imported axiom (s_commuting_family_of_continuous_affine_self_maps).
s_compact_convex_subset_of_locally_convex_space	axiom	compact convex subset of locally convex space		Iter-3 imported axiom (s_compact_convex_subset_of_locally_convex_space).
s_cesaro_average_iterates_in_K	state	cesaro average iterates in K		Iter-3 imported state (s_cesaro_average_iterates_in_K).
s_limit_point_almost_fixed_for_T	state	limit point almost fixed for T		Iter-3 imported state (s_limit_point_almost_fixed_for_T).
s_bv_sobolev_endpoint_inequality	theorem	Sobolev–Gagliardo–Nirenberg inequality endpoint (BV–L^{n/(n-1)})		Iter-3 imported theorem (s_bv_sobolev_endpoint_inequality).
s_BV_function_on_Rn	axiom	BV function on Rn		Iter-3 imported axiom (s_BV_function_on_Rn).
s_total_variation_and_perimeter	axiom	total variation and perimeter		Iter-3 imported axiom (s_total_variation_and_perimeter).
s_coarea_formula_for_BV	state	coarea formula for BV		Iter-3 imported state (s_coarea_formula_for_BV).
s_per_level_isoperimetric_bound	state	per level isoperimetric bound		Iter-3 imported state (s_per_level_isoperimetric_bound).
s_levy_khintchine_formula	theorem	Lévy–Khintchine formula		Iter-3 imported theorem (s_levy_khintchine_formula).
s_infinitely_divisible_distribution_on_R	axiom	infinitely divisible distribution on R		Iter-3 imported axiom (s_infinitely_divisible_distribution_on_R).
s_characteristic_function_log_psi	axiom	characteristic function log psi		Iter-3 imported axiom (s_characteristic_function_log_psi).
s_decomposition_drift_plus_brownian_plus_jumps	state	decomposition drift plus brownian plus jumps		Iter-3 imported state (s_decomposition_drift_plus_brownian_plus_jumps).
s_levy_measure_integral_existence	state	levy measure integral existence		Iter-3 imported state (s_levy_measure_integral_existence).
s_kolmogorov_three_series_theorem	theorem	Three-series theorem (Kolmogorov)		Iter-3 imported theorem (s_kolmogorov_three_series_theorem).
s_sequence_of_independent_random_variables_X_n	axiom	sequence of independent random variables X n		Iter-3 imported axiom (s_sequence_of_independent_random_variables_X_n).
s_truncation_at_level_A	axiom	truncation at level A		Iter-3 imported axiom (s_truncation_at_level_A).
s_three_series_summability_condition	state	three series summability condition		Iter-3 imported state (s_three_series_summability_condition).
s_ae_convergence_of_centered_truncations	state	ae convergence of centered truncations		Iter-3 imported state (s_ae_convergence_of_centered_truncations).
s_strong_law_of_large_numbers	theorem	Strong law of large numbers (Kolmogorov)		Iter-3 imported theorem (s_strong_law_of_large_numbers).
s_iid_sequence_with_finite_mean	axiom	iid sequence with finite mean		Iter-3 imported axiom (s_iid_sequence_with_finite_mean).
s_centered_truncated_sequence	state	centered truncated sequence		Iter-3 imported state (s_centered_truncated_sequence).
s_partial_sum_concentration	state	partial sum concentration		Iter-3 imported state (s_partial_sum_concentration).
s_donsker_invariance_principle	theorem	Donsker's invariance principle		Iter-3 imported theorem (s_donsker_invariance_principle).
s_iid_sequence_with_finite_variance_unit_normalized	axiom	iid sequence with finite variance unit normalized		Iter-3 imported axiom (s_iid_sequence_with_finite_variance_unit_normalized).
s_brownian_motion_on_0_1	axiom	brownian motion on 0 1		Iter-3 imported axiom (s_brownian_motion_on_0_1).
s_polygonal_random_walk_process	state	polygonal random walk process		Iter-3 imported state (s_polygonal_random_walk_process).
s_finite_dim_marginals_converge_to_BM	state	finite dim marginals converge to BM		Iter-3 imported state (s_finite_dim_marginals_converge_to_BM).
s_wiener_inversion_theorem	theorem	Wiener's theorem on Fourier series (absolute convergence)		Iter-3 imported theorem (s_wiener_inversion_theorem).
s_function_with_absolutely_convergent_fourier_series_nonvanishing	axiom	function with absolutely convergent fourier series nonvanishing		Iter-3 imported axiom (s_function_with_absolutely_convergent_fourier_series_nonvanishing).
s_inverse_function_1_over_f	axiom	inverse function 1 over f		Iter-3 imported axiom (s_inverse_function_1_over_f).
s_banach_algebra_A_T	state	banach algebra A T		Iter-3 imported state (s_banach_algebra_A_T).
s_spectrum_of_f_in_A_T_equals_image	state	spectrum of f in A T equals image		Iter-3 imported state (s_spectrum_of_f_in_A_T_equals_image).
s_helly_bray_theorem	theorem	Helly–Bray theorem (weak convergence of distribution functions)		Iter-3 imported theorem (s_helly_bray_theorem).
s_bounded_continuous_function_on_R	axiom	bounded continuous function on R		Iter-3 imported axiom (s_bounded_continuous_function_on_R).
s_weakly_convergent_sequence_of_distribution_functions	axiom	weakly convergent sequence of distribution functions		Iter-3 imported axiom (s_weakly_convergent_sequence_of_distribution_functions).
s_truncated_integrand_on_compact	state	truncated integrand on compact		Iter-3 imported state (s_truncated_integrand_on_compact).
s_convergence_on_compact_window	state	convergence on compact window		Iter-3 imported state (s_convergence_on_compact_window).
s_helly_theorem_convex	theorem	Helly's theorem on convex sets		Iter-3 imported theorem (s_helly_theorem_convex).
s_finite_family_of_convex_subsets_of_Rd	axiom	finite family of convex subsets of Rd		Iter-3 imported axiom (s_finite_family_of_convex_subsets_of_Rd).
s_d_plus_1_intersection_condition	axiom	d plus 1 intersection condition		Iter-3 imported axiom (s_d_plus_1_intersection_condition).
s_induction_base_at_d_plus_1	state	induction base at d plus 1		Iter-3 imported state (s_induction_base_at_d_plus_1).
s_radon_partition_giving_common_intersection	state	radon partition giving common intersection		Iter-3 imported state (s_radon_partition_giving_common_intersection).
s_caratheodory_convex_hull_theorem	theorem	Carathéodory's convex hull theorem		Iter-3 imported theorem (s_caratheodory_convex_hull_theorem).
s_subset_of_Rd	axiom	subset of Rd		Iter-3 imported axiom (s_subset_of_Rd).
s_convex_hull_with_convex_combinations	axiom	convex hull with convex combinations		Iter-3 imported axiom (s_convex_hull_with_convex_combinations).
s_redundant_convex_combination_assumption	state	redundant convex combination assumption		Iter-3 imported state (s_redundant_convex_combination_assumption).
s_linear_dependence_extracts_smaller_combination	state	linear dependence extracts smaller combination		Iter-3 imported state (s_linear_dependence_extracts_smaller_combination).
s_lindelof_principle_subharmonic	theorem	Lindelöf principle (subharmonic version)		Iter-3 imported theorem (s_lindelof_principle_subharmonic).
s_subharmonic_function_on_unbounded_domain	axiom	subharmonic function on unbounded domain		Iter-3 imported axiom (s_subharmonic_function_on_unbounded_domain).
s_growth_bound_at_infinity	axiom	growth bound at infinity		Iter-3 imported axiom (s_growth_bound_at_infinity).
s_harmonic_comparison_function	state	harmonic comparison function		Iter-3 imported state (s_harmonic_comparison_function).
s_bound_on_truncations	state	bound on truncations		Iter-3 imported state (s_bound_on_truncations).
s_perron_method_dirichlet	theorem	Perron method for Dirichlet problem		Iter-3 imported theorem (s_perron_method_dirichlet).
s_bounded_domain_with_boundary_data_in_C_boundary	axiom	bounded domain with boundary data in C boundary		Iter-3 imported axiom (s_bounded_domain_with_boundary_data_in_C_boundary).
s_family_of_subharmonic_subsolutions	axiom	family of subharmonic subsolutions		Iter-3 imported axiom (s_family_of_subharmonic_subsolutions).
s_perron_solution_candidate_u	state	perron solution candidate u		Iter-3 imported state (s_perron_solution_candidate_u).
s_harmonic_interior_function	state	harmonic interior function		Iter-3 imported state (s_harmonic_interior_function).
s_harmonic_mvp_characterization	theorem	Mean value property characterizes harmonic functions		Iter-3 imported theorem (s_harmonic_mvp_characterization).
s_continuous_function_on_open_subset_of_Rn	axiom	continuous function on open subset of Rn		Iter-3 imported axiom (s_continuous_function_on_open_subset_of_Rn).
s_mean_value_property_on_balls	axiom	mean value property on balls		Iter-3 imported axiom (s_mean_value_property_on_balls).
s_smooth_mollification_equals_f	state	smooth mollification equals f		Iter-3 imported state (s_smooth_mollification_equals_f).
s_laplacian_zero_almost_everywhere	state	laplacian zero almost everywhere		Iter-3 imported state (s_laplacian_zero_almost_everywhere).
s_harnack_inequality	theorem	Harnack's inequality		Iter-3 imported theorem (s_harnack_inequality).
s_nonneg_harmonic_function_on_ball	axiom	nonneg harmonic function on ball		Iter-3 imported axiom (s_nonneg_harmonic_function_on_ball).
s_compactly_contained_subdomain	axiom	compactly contained subdomain		Iter-3 imported axiom (s_compactly_contained_subdomain).
s_poisson_kernel_explicit_bounds	state	poisson kernel explicit bounds		Iter-3 imported state (s_poisson_kernel_explicit_bounds).
s_two_point_ratio_inequality	state	two point ratio inequality		Iter-3 imported state (s_two_point_ratio_inequality).
s_carleson_embedding_theorem	theorem	Carleson measure characterization (Carleson embedding)		Iter-3 imported theorem (s_carleson_embedding_theorem).
s_positive_measure_mu_on_upper_half_plane	axiom	positive measure mu on upper half plane		Iter-3 imported axiom (s_positive_measure_mu_on_upper_half_plane).
s_lp_function_on_real_line	axiom	lp function on real line		Iter-3 imported axiom (s_lp_function_on_real_line).
s_Poisson_extension_to_half_plane	state	Poisson extension to half plane		Iter-3 imported state (s_Poisson_extension_to_half_plane).
s_tent_decomposition_estimate	state	tent decomposition estimate		Iter-3 imported state (s_tent_decomposition_estimate).
s_fefferman_BMO_H1_duality	theorem	BMO–H^1 duality (Fefferman)		Iter-3 imported theorem (s_fefferman_BMO_H1_duality).
s_real_hardy_space_H1_on_Rn	axiom	real hardy space H1 on Rn		Iter-3 imported axiom (s_real_hardy_space_H1_on_Rn).
s_BMO_space_on_Rn	axiom	BMO space on Rn		Iter-3 imported axiom (s_BMO_space_on_Rn).
s_atomic_decomposition_for_H1	state	atomic decomposition for H1		Iter-3 imported state (s_atomic_decomposition_for_H1).
s_dual_pairing_estimate_per_atom	state	dual pairing estimate per atom		Iter-3 imported state (s_dual_pairing_estimate_per_atom).
s_muckenhoupt_Ap_theorem	theorem	A_p weight theorem (Muckenhoupt)		Iter-3 imported theorem (s_muckenhoupt_Ap_theorem).
s_locally_integrable_nonneg_weight_w_on_Rn	axiom	locally integrable nonneg weight w on Rn		Iter-3 imported axiom (s_locally_integrable_nonneg_weight_w_on_Rn).
s_Hardy_Littlewood_maximal_operator	axiom	Hardy Littlewood maximal operator		Iter-3 imported axiom (s_Hardy_Littlewood_maximal_operator).
s_Ap_class_definition	state	Ap class definition		Iter-3 imported state (s_Ap_class_definition).
s_weak_p_p_bound_on_M_with_weight_w	state	weak p p bound on M with weight w		Iter-3 imported state (s_weak_p_p_bound_on_M_with_weight_w).
s_stein_interpolation_theorem	theorem	Stein interpolation theorem		Iter-3 imported theorem (s_stein_interpolation_theorem).
s_analytic_family_of_operators_T_z_on_strip	axiom	analytic family of operators T z on strip		Iter-3 imported axiom (s_analytic_family_of_operators_T_z_on_strip).
s_lp_lq_endpoint_bounds_with_admissible_growth	axiom	lp lq endpoint bounds with admissible growth		Iter-3 imported axiom (s_lp_lq_endpoint_bounds_with_admissible_growth).
s_holomorphic_in_strip_with_admissible_growth	state	holomorphic in strip with admissible growth		Iter-3 imported state (s_holomorphic_in_strip_with_admissible_growth).
s_three_lines_with_admissible_growth	state	three lines with admissible growth		Iter-3 imported state (s_three_lines_with_admissible_growth).
s_fractional_sobolev_embedding	theorem	Sobolev–Slobodeckij embedding (fractional Sobolev)		Iter-3 imported theorem (s_fractional_sobolev_embedding).
s_fractional_sobolev_space_W_s_p_Rn	axiom	fractional sobolev space W s p Rn		Iter-3 imported axiom (s_fractional_sobolev_space_W_s_p_Rn).
s_lq_target_with_critical_exponent	axiom	lq target with critical exponent		Iter-3 imported axiom (s_lq_target_with_critical_exponent).
s_HLS_lp_to_lq_bound	state	HLS lp to lq bound		Iter-3 imported state (s_HLS_lp_to_lq_bound).
s_polya_szego_rearrangement_inequality	theorem	Pólya–Szegő rearrangement inequality		Iter-3 imported theorem (s_polya_szego_rearrangement_inequality).
s_nonneg_measurable_function_on_Rn	axiom	nonneg measurable function on Rn		Iter-3 imported axiom (s_nonneg_measurable_function_on_Rn).
s_symmetric_decreasing_rearrangement_f_star	axiom	symmetric decreasing rearrangement f star		Iter-3 imported axiom (s_symmetric_decreasing_rearrangement_f_star).
s_layer_cake_representation_of_f	state	layer cake representation of f		Iter-3 imported state (s_layer_cake_representation_of_f).
s_level_set_isoperimetric_bound	state	level set isoperimetric bound		Iter-3 imported state (s_level_set_isoperimetric_bound).
s_strichartz_estimate	theorem	Strichartz estimates		Iter-3 imported theorem (s_strichartz_estimate).
s_schrodinger_propagator_e_it_delta	axiom	schrodinger propagator e it delta		Iter-3 imported axiom (s_schrodinger_propagator_e_it_delta).
s_dispersive_decay_estimate	state	dispersive decay estimate		Iter-3 imported state (s_dispersive_decay_estimate).
s_marcinkiewicz_zygmund_inequality	theorem	Riesz–Thorin endpoint Marcinkiewicz–Zygmund		Iter-3 imported theorem (s_marcinkiewicz_zygmund_inequality).
s_iid_random_variable_sequence	axiom	iid random variable sequence		Iter-3 imported axiom (s_iid_random_variable_sequence).
s_lp_norm_of_partial_sums	axiom	lp norm of partial sums		Iter-3 imported axiom (s_lp_norm_of_partial_sums).
s_symmetrized_partial_sum_via_random_signs	state	symmetrized partial sum via random signs		Iter-3 imported state (s_symmetrized_partial_sum_via_random_signs).
s_conditional_khintchine_bound	state	conditional khintchine bound		Iter-3 imported state (s_conditional_khintchine_bound).
s_bochner_riesz_lp_bound	theorem	Bochner–Riesz mean theorem		Iter-3 imported theorem (s_bochner_riesz_lp_bound).
s_bochner_riesz_multiplier_1_minus_abs_xi_2_to_alpha	axiom	bochner riesz multiplier 1 minus abs xi 2 to alpha		Iter-3 imported axiom (s_bochner_riesz_multiplier_1_minus_abs_xi_2_to_alpha).
s_decomposition_into_annular_pieces	state	decomposition into annular pieces		Iter-3 imported state (s_decomposition_into_annular_pieces).
s_oscillatory_kernel_estimate	state	oscillatory kernel estimate		Iter-3 imported state (s_oscillatory_kernel_estimate).
s_tomas_stein_restriction_theorem	theorem	Tomas–Stein restriction theorem		Iter-3 imported theorem (s_tomas_stein_restriction_theorem).
s_smooth_curved_hypersurface_in_Rn	axiom	smooth curved hypersurface in Rn		Iter-3 imported axiom (s_smooth_curved_hypersurface_in_Rn).
s_lp_function_with_fourier_restriction	axiom	lp function with fourier restriction		Iter-3 imported axiom (s_lp_function_with_fourier_restriction).
s_extension_operator_setup	state	extension operator setup		Iter-3 imported state (s_extension_operator_setup).
s_surface_measure_decay_estimate	state	surface measure decay estimate		Iter-3 imported state (s_surface_measure_decay_estimate).
s_kakeya_maximal_inequality_planar	theorem	Kakeya maximal inequality (n=2)		Iter-3 imported theorem (s_kakeya_maximal_inequality_planar).
s_family_of_unit_segments_in_R2_with_delta_separated_directions	axiom	family of unit segments in R2 with delta separated directions		Iter-3 imported axiom (s_family_of_unit_segments_in_R2_with_delta_separated_directions).
s_kakeya_maximal_operator	axiom	kakeya maximal operator		Iter-3 imported axiom (s_kakeya_maximal_operator).
s_combinatorial_incidence_setup	state	combinatorial incidence setup		Iter-3 imported state (s_combinatorial_incidence_setup).
s_log_loss_bound_on_kakeya_maximal	state	log loss bound on kakeya maximal		Iter-3 imported state (s_log_loss_bound_on_kakeya_maximal).
s_calderon_reproducing_formula	theorem	Calderón reproducing formula (continuous wavelet)		Iter-3 imported theorem (s_calderon_reproducing_formula).
s_admissible_mother_wavelet_psi_on_R	axiom	admissible mother wavelet psi on R		Iter-3 imported axiom (s_admissible_mother_wavelet_psi_on_R).
s_L2_function_f	axiom	L2 function f		Iter-3 imported axiom (s_L2_function_f).
s_admissibility_constant_C_psi_finite	state	admissibility constant C psi finite		Iter-3 imported state (s_admissibility_constant_C_psi_finite).
s_parseval_identity_for_wavelet_pairing	state	parseval identity for wavelet pairing		Iter-3 imported state (s_parseval_identity_for_wavelet_pairing).
s_frostman_lemma	theorem	Frostman's lemma		Iter-3 imported theorem (s_frostman_lemma).
s_compact_subset_K_of_Rn	axiom	compact subset K of Rn		Iter-3 imported axiom (s_compact_subset_K_of_Rn).
s_hausdorff_dimension_s_lower_bound	axiom	hausdorff dimension s lower bound		Iter-3 imported axiom (s_hausdorff_dimension_s_lower_bound).
s_balanced_dyadic_mass	state	balanced dyadic mass		Iter-3 imported state (s_balanced_dyadic_mass).
s_borel_measure_with_growth_bound	state	borel measure with growth bound		Iter-3 imported state (s_borel_measure_with_growth_bound).
s_besicovitch_covering_theorem	theorem	Besicovitch covering theorem		Iter-3 imported theorem (s_besicovitch_covering_theorem).
s_collection_of_balls_in_Rn_with_bounded_radii_centered_at_set_A	axiom	collection of balls in Rn with bounded radii centered at set A		Iter-3 imported axiom (s_collection_of_balls_in_Rn_with_bounded_radii_centered_at_set_A).
s_sorted_ball_sequence	state	sorted ball sequence		Iter-3 imported state (s_sorted_ball_sequence).
s_finite_number_of_disjoint_subfamilies_covering_A	state	finite number of disjoint subfamilies covering A		Iter-3 imported state (s_finite_number_of_disjoint_subfamilies_covering_A).
s_whitney_extension_theorem	theorem	Whitney extension theorem		Iter-3 imported theorem (s_whitney_extension_theorem).
s_closed_subset_of_Rn_with_jet_data_satisfying_compatibility	axiom	closed subset of Rn with jet data satisfying compatibility		Iter-3 imported axiom (s_closed_subset_of_Rn_with_jet_data_satisfying_compatibility).
s_target_class_C_m	axiom	target class C m		Iter-3 imported axiom (s_target_class_C_m).
s_whitney_decomposition_of_complement	state	whitney decomposition of complement		Iter-3 imported state (s_whitney_decomposition_of_complement).
s_locally_polynomial_extension	state	locally polynomial extension		Iter-3 imported state (s_locally_polynomial_extension).
s_borel_theorem_taylor_series	theorem	Borel's theorem on Taylor series		Iter-3 imported theorem (s_borel_theorem_taylor_series).
s_arbitrary_real_sequence_a_n	axiom	arbitrary real sequence a n		Iter-3 imported axiom (s_arbitrary_real_sequence_a_n).
s_smooth_function_on_R	axiom	smooth function on R		Iter-3 imported axiom (s_smooth_function_on_R).
s_scaled_bump_functions_with_decreasing_support	state	scaled bump functions with decreasing support		Iter-3 imported state (s_scaled_bump_functions_with_decreasing_support).
s_smooth_function_with_prescribed_jet_at_0	state	smooth function with prescribed jet at 0		Iter-3 imported state (s_smooth_function_with_prescribed_jet_at_0).
s_whitney_approximation_theorem	theorem	Whitney C^k approximation theorem		Iter-3 imported theorem (s_whitney_approximation_theorem).
s_continuous_function_on_smooth_manifold	axiom	continuous function on smooth manifold		Iter-3 imported axiom (s_continuous_function_on_smooth_manifold).
s_target_smooth_function_class	axiom	target smooth function class		Iter-3 imported axiom (s_target_smooth_function_class).
s_partition_of_unity_chart_decomposition	state	partition of unity chart decomposition		Iter-3 imported state (s_partition_of_unity_chart_decomposition).
s_locally_smooth_approximation	state	locally smooth approximation		Iter-3 imported state (s_locally_smooth_approximation).
s_torelli_curves	theorem	Torelli theorem for curves		Iter-3 imported theorem (s_torelli_curves).
s_smooth_curve_C	state	smooth curve C		Iter-3 imported state (s_smooth_curve_C).
s_polarized_jacobian	state	polarized jacobian		Iter-3 imported state (s_polarized_jacobian).
s_theta_divisor_singularities_recover_C	state	theta divisor singularities recover C		Iter-3 imported state (s_theta_divisor_singularities_recover_C).
s_torelli_K3	theorem	Torelli theorem for K3 surfaces		Iter-3 imported theorem (s_torelli_K3).
s_K3_surface_with_hodge_structure	state	K3 surface with hodge structure		Iter-3 imported state (s_K3_surface_with_hodge_structure).
s_period_map_for_K3	state	period map for K3		Iter-3 imported state (s_period_map_for_K3).
s_fourier_mukai_transform	theorem	Fourier–Mukai transform		Iter-3 imported theorem (s_fourier_mukai_transform).
s_abelian_variety_with_dual	state	abelian variety with dual		Iter-3 imported state (s_abelian_variety_with_dual).
s_poincare_bundle	state	poincare bundle		Iter-3 imported state (s_poincare_bundle).
s_integral_transform_on_derived_category	state	integral transform on derived category		Iter-3 imported state (s_integral_transform_on_derived_category).
s_castelnuovo_mumford_regularity	theorem	Castelnuovo–Mumford regularity		Iter-3 imported theorem (s_castelnuovo_mumford_regularity).
s_coherent_sheaf_F_on_PN	state	coherent sheaf F on PN		Iter-3 imported state (s_coherent_sheaf_F_on_PN).
s_regularity_definition	state	regularity definition		Iter-3 imported state (s_regularity_definition).
s_vanishing_propagation	state	vanishing propagation		Iter-3 imported state (s_vanishing_propagation).
s_castelnuovo_contractibility	theorem	Castelnuovo's contractibility criterion		Iter-3 imported theorem (s_castelnuovo_contractibility).
s_minus_one_curve_on_surface	state	minus one curve on surface		Iter-3 imported state (s_minus_one_curve_on_surface).
s_morphism_contracting_E	state	morphism contracting E		Iter-3 imported state (s_morphism_contracting_E).
s_mori_cone_theorem	theorem	Mori cone theorem		Iter-3 imported theorem (s_mori_cone_theorem).
s_smooth_X_with_canonical	state	smooth X with canonical		Iter-3 imported state (s_smooth_X_with_canonical).
s_cone_NE_decomposition	state	cone NE decomposition		Iter-3 imported state (s_cone_NE_decomposition).
s_extremal_rays_spanned_by_rational_curves	state	extremal rays spanned by rational curves		Iter-3 imported state (s_extremal_rays_spanned_by_rational_curves).
s_mori_extremal_contraction	theorem	Mori extremal contraction		Iter-3 imported theorem (s_mori_extremal_contraction).
s_extremal_ray_R	state	extremal ray R		Iter-3 imported state (s_extremal_ray_R).
s_supporting_nef_divisor	state	supporting nef divisor		Iter-3 imported state (s_supporting_nef_divisor).
s_semi_ample_supporting_divisor	state	semi ample supporting divisor		Iter-3 imported state (s_semi_ample_supporting_divisor).
s_base_point_free_theorem	theorem	Kawamata–Shokurov base-point-free theorem		Iter-3 imported theorem (s_base_point_free_theorem).
s_nef_big_divisor_setup	state	nef big divisor setup		Iter-3 imported state (s_nef_big_divisor_setup).
s_KV_vanishing_application	state	KV vanishing application		Iter-3 imported state (s_KV_vanishing_application).
s_inductive_section_lifting	state	inductive section lifting		Iter-3 imported state (s_inductive_section_lifting).
s_kawamata_viehweg_vanishing	theorem	Kawamata–Viehweg vanishing theorem		Iter-3 imported theorem (s_kawamata_viehweg_vanishing).
s_big_nef_q_divisor	state	big nef q divisor		Iter-3 imported state (s_big_nef_q_divisor).
s_cyclic_cover_construction	state	cyclic cover construction		Iter-3 imported state (s_cyclic_cover_construction).
s_descended_vanishing_statement	state	descended vanishing statement		Iter-3 imported state (s_descended_vanishing_statement).
s_cartan_A_B	theorem	Cartan's theorems A and B		Iter-3 imported theorem (s_cartan_A_B).
s_stein_manifold_with_coherent_F	state	stein manifold with coherent F		Iter-3 imported state (s_stein_manifold_with_coherent_F).
s_psh_exhaustion_of_X	state	psh exhaustion of X		Iter-3 imported state (s_psh_exhaustion_of_X).
s_H_i_X_F_vanish_for_i_geq_1	state	H i X F vanish for i geq 1		Iter-3 imported state (s_H_i_X_F_vanish_for_i_geq_1).
s_hurwitz_automorphisms	theorem	Hurwitz's automorphisms theorem		Iter-3 imported theorem (s_hurwitz_automorphisms).
s_curve_with_automorphism_group	state	curve with automorphism group		Iter-3 imported state (s_curve_with_automorphism_group).
s_belyi_theorem	theorem	Belyi's theorem		Iter-3 imported theorem (s_belyi_theorem).
s_curve_over_algebraic_closure_of_Q	state	curve over algebraic closure of Q		Iter-3 imported state (s_curve_over_algebraic_closure_of_Q).
s_belyi_map_construction	state	belyi map construction		Iter-3 imported state (s_belyi_map_construction).
s_ramification_over_3_points_only	state	ramification over 3 points only		Iter-3 imported state (s_ramification_over_3_points_only).
s_gaga_theorem	theorem	GAGA principle (Serre)		Iter-3 imported theorem (s_gaga_theorem).
s_proper_X_with_coherent_sheaf	state	proper X with coherent sheaf		Iter-3 imported state (s_proper_X_with_coherent_sheaf).
s_analytification_functor	state	analytification functor		Iter-3 imported state (s_analytification_functor).
s_cohomology_isomorphism	state	cohomology isomorphism		Iter-3 imported state (s_cohomology_isomorphism).
s_zariski_connectedness	theorem	Zariski's connectedness theorem		Iter-3 imported theorem (s_zariski_connectedness).
s_proper_birational_to_normal	state	proper birational to normal		Iter-3 imported state (s_proper_birational_to_normal).
s_pushforward_of_structure_sheaf_iso	state	pushforward of structure sheaf iso		Iter-3 imported state (s_pushforward_of_structure_sheaf_iso).
s_tarski_seidenberg	theorem	Tarski–Seidenberg theorem		Iter-3 imported theorem (s_tarski_seidenberg).
s_semialgebraic_set	state	semialgebraic set		Iter-3 imported state (s_semialgebraic_set).
s_CAD_of_S	state	CAD of S		Iter-3 imported state (s_CAD_of_S).
s_projection_is_semialgebraic	state	projection is semialgebraic		Iter-3 imported state (s_projection_is_semialgebraic).
s_chevalley_constructibility	theorem	Chevalley constructibility theorem		Iter-3 imported theorem (s_chevalley_constructibility).
s_finite_type_morphism	state	finite type morphism		Iter-3 imported state (s_finite_type_morphism).
s_noetherian_induction_setup	state	noetherian induction setup		Iter-3 imported state (s_noetherian_induction_setup).
s_borel_fixed_point	theorem	Borel fixed-point theorem		Iter-3 imported theorem (s_borel_fixed_point).
s_solvable_action_on_proper	state	solvable action on proper		Iter-3 imported state (s_solvable_action_on_proper).
s_minimal_orbit_closure	state	minimal orbit closure		Iter-3 imported state (s_minimal_orbit_closure).
s_fixed_point_exists	state	fixed point exists		Iter-3 imported state (s_fixed_point_exists).
s_chow_lemma	theorem	Chow's lemma		Iter-3 imported theorem (s_chow_lemma).
s_proper_morphism_finite_type	state	proper morphism finite type		Iter-3 imported state (s_proper_morphism_finite_type).
s_blow_up_yielding_projective_birational_X_tilde	state	blow up yielding projective birational X tilde		Iter-3 imported state (s_blow_up_yielding_projective_birational_X_tilde).
s_nagata_compactification	theorem	Nagata compactification theorem		Iter-3 imported theorem (s_nagata_compactification).
s_separated_finite_type	state	separated finite type		Iter-3 imported state (s_separated_finite_type).
s_local_compactifications	state	local compactifications		Iter-3 imported state (s_local_compactifications).
s_glued_proper_overscheme_X_bar	state	glued proper overscheme X bar		Iter-3 imported state (s_glued_proper_overscheme_X_bar).
s_modularity_theorem	theorem	Modularity theorem (Taylor–Wiles–Breuil–Conrad–Diamond)		Iter-3 imported theorem (s_modularity_theorem).
s_galois_rep_of_elliptic_curve	state	galois rep of elliptic curve		Iter-3 imported state (s_galois_rep_of_elliptic_curve).
s_deformation_ring_R_to_T_map	state	deformation ring R to T map		Iter-3 imported state (s_deformation_ring_R_to_T_map).
s_R_equals_T_for_E	state	R equals T for E		Iter-3 imported state (s_R_equals_T_for_E).
s_tate_isogeny	theorem	Tate isogeny theorem		Iter-3 imported theorem (s_tate_isogeny).
s_abelian_variety_over_Fq	state	abelian variety over Fq		Iter-3 imported state (s_abelian_variety_over_Fq).
s_tate_module_with_frobenius	state	tate module with frobenius		Iter-3 imported state (s_tate_module_with_frobenius).
s_hom_iso_via_galois_action	state	hom iso via galois action		Iter-3 imported state (s_hom_iso_via_galois_action).
s_chow_rashevskii	theorem	Chow–Rashevskii theorem		Iter-3 imported theorem (s_chow_rashevskii).
s_bracket_generating_distribution	state	bracket generating distribution		Iter-3 imported state (s_bracket_generating_distribution).
s_lie_bracket_filtration	state	lie bracket filtration		Iter-3 imported state (s_lie_bracket_filtration).
s_horizontal_path_reaches_neighborhood	state	horizontal path reaches neighborhood		Iter-3 imported state (s_horizontal_path_reaches_neighborhood).
s_pansu_differentiation	theorem	Pansu differentiation theorem		Iter-3 imported theorem (s_pansu_differentiation).
s_carnot_group_setting	state	carnot group setting		Iter-3 imported state (s_carnot_group_setting).
s_blowup_limit_of_lipschitz	state	blowup limit of lipschitz		Iter-3 imported state (s_blowup_limit_of_lipschitz).
s_cartan_dieudonne	theorem	Cartan–Dieudonné theorem		Iter-3 imported theorem (s_cartan_dieudonne).
s_orthogonal_transformation	state	orthogonal transformation		Iter-3 imported state (s_orthogonal_transformation).
s_canonical_reflection	state	canonical reflection		Iter-3 imported state (s_canonical_reflection).
s_reduction_to_identity_after_n_reflections	state	reduction to identity after n reflections		Iter-3 imported state (s_reduction_to_identity_after_n_reflections).
s_beltrami_theorem	theorem	Beltrami theorem		Iter-3 imported theorem (s_beltrami_theorem).
s_projectively_flat_metric	state	projectively flat metric		Iter-3 imported state (s_projectively_flat_metric).
s_constant_sectional_curvature	state	constant sectional curvature		Iter-3 imported state (s_constant_sectional_curvature).
s_schur_riemannian	theorem	Schur's theorem (Riemannian, isotropic implies constant curvature)		Iter-3 imported theorem (s_schur_riemannian).
s_isotropic_pointwise_curvature	state	isotropic pointwise curvature		Iter-3 imported state (s_isotropic_pointwise_curvature).
s_bianchi_identity_constraint	state	bianchi identity constraint		Iter-3 imported state (s_bianchi_identity_constraint).
s_schwarz_ahlfors_pick	theorem	Schwarz–Ahlfors–Pick theorem		Iter-3 imported theorem (s_schwarz_ahlfors_pick).
s_holomorphic_map_of_hyperbolic_surfaces	state	holomorphic map of hyperbolic surfaces		Iter-3 imported state (s_holomorphic_map_of_hyperbolic_surfaces).
s_poincare_metric_pair	state	poincare metric pair		Iter-3 imported state (s_poincare_metric_pair).
s_distance_decreasing_property	state	distance decreasing property		Iter-3 imported state (s_distance_decreasing_property).
s_williamson_normal_form	theorem	Williamson normal form theorem		Iter-3 imported theorem (s_williamson_normal_form).
s_positive_quadratic_form_on_symplectic_space	state	positive quadratic form on symplectic space		Iter-3 imported state (s_positive_quadratic_form_on_symplectic_space).
s_symplectic_spectrum	state	symplectic spectrum		Iter-3 imported state (s_symplectic_spectrum).
s_birkhoff_grothendieck	theorem	Birkhoff–Grothendieck theorem		Iter-3 imported theorem (s_birkhoff_grothendieck).
s_holomorphic_bundle_on_P1	state	holomorphic bundle on P1		Iter-3 imported state (s_holomorphic_bundle_on_P1).
s_maximal_subbundle	state	maximal subbundle		Iter-3 imported state (s_maximal_subbundle).
s_inductive_splitting_into_line_bundles	state	inductive splitting into line bundles		Iter-3 imported state (s_inductive_splitting_into_line_bundles).
s_clifford_theorem	theorem	Clifford's theorem on special divisors		Iter-3 imported theorem (s_clifford_theorem).
s_special_divisor_setup	state	special divisor setup		Iter-3 imported state (s_special_divisor_setup).
s_multiplication_map_of_linear_systems	state	multiplication map of linear systems		Iter-3 imported state (s_multiplication_map_of_linear_systems).
s_clifford_inequality	state	clifford inequality		Iter-3 imported state (s_clifford_inequality).
s_bernoulli_number	state	Bernoulli number		A sequence of rational numbers arising in the Taylor expansion of t/(e^t − 1) and connected to values of the Riemann zeta function at negative integers.
s_brauer_group	state	Brauer group		The group of equivalence classes of central simple algebras over a field, under tensor product.
s_composition_series	state	Composition series		A maximal chain of subnormal subgroups from the trivial subgroup to the whole group, with each factor being simple.
s_dedekind_domain	axiom	Dedekind domain		An integral domain that is Noetherian, integrally closed, and in which every nonzero prime ideal is maximal.
s_derivation	axiom	Derivation (algebra)		A linear map D on an algebra satisfying the Leibniz rule D(ab) = D(a)b + aD(b).
s_determinant	state	Determinant		The unique alternating multilinear function on the columns of a square matrix that equals one on the identity, measuring signed volume scaling.
s_discriminant	state	Discriminant		An algebraic invariant of a polynomial or number field encoding ramification information and separability of roots.
s_endomorphism_ring	state	Endomorphism ring		The ring of all endomorphisms of an abelian group or module, with addition and composition as operations.
s_ext_functor	state	Ext functor		The right derived functors of Hom, classifying extensions of modules and measuring failure of exactness.
s_exterior_algebra	axiom	Exterior algebra		The quotient of the tensor algebra by the ideal generated by v ⊗ v, yielding an algebra with anticommutative multiplication.
s_flat_module	axiom	Flat module		A module M over a ring such that tensoring with M preserves exact sequences.
s_free_group	axiom	Free group		The group on a set of generators with no relations other than the group axioms.
s_free_module	axiom	Free module		A module isomorphic to a direct sum of copies of the base ring, possessing a basis.
s_frobenius_endomorphism	state	Frobenius endomorphism		The ring endomorphism x → x^p in characteristic p, fundamental to finite field and algebraic geometry structure theory.
s_global_field	axiom	Global field		Either a number field or a function field of a curve over a finite field.
s_graded_ring	axiom	Graded ring		A ring decomposed as a direct sum of abelian groups indexed by integers, with multiplication respecting the grading.
s_grobner_basis	state	Gröbner basis		A generating set of a polynomial ideal enabling algorithmic solution of the ideal membership problem.
s_grothendieck_group	state	Grothendieck group		The universal abelian group K₀ associated to a commutative monoid by formally adjoining inverses.
s_group_algebra	axiom	Group algebra		The algebra over a field whose basis elements are the group elements and whose multiplication extends the group operation linearly.
s_hensels_lemma	theorem	Hensel's lemma		A simple root of a polynomial modulo a prime can be lifted uniquely to a root in the p-adic integers.
s_hilbert_class_field	state	Hilbert class field		The maximal unramified abelian extension of a number field, whose Galois group is isomorphic to the ideal class group.
s_hopf_algebra	axiom	Hopf algebra		A bialgebra equipped with an antipode map, axiomatizing group-like objects in a symmetric monoidal category.
s_ideal_class_group	state	Ideal class group		The quotient of fractional ideals by principal ideals of a number field, measuring deviation from unique factorization.
s_injective_module	axiom	Injective module		A module I such that every homomorphism from a submodule extends to the whole module.
s_integral_closure	state	Integral closure		The set of elements in a ring extension satisfying a monic polynomial over the base ring, forming a subring.
s_integral_domain	axiom	Integral domain		A nonzero commutative ring with no zero divisors.
s_koszul_complex	state	Koszul complex		A chain complex associated to a sequence of ring elements, used to compute Tor/Ext and characterize regular sequences.
s_local_ring	axiom	Local ring		A ring with a unique maximal ideal.
s_localization	state	Localization (ring theory)		The construction inverting a multiplicative subset of a commutative ring, generalizing passage from ℤ to ℚ.
s_morita_equivalence	theorem	Morita equivalence		Two rings are Morita equivalent iff their module categories are equivalent, realized by tensoring with a bimodule.
s_nilpotent_group	axiom	Nilpotent group		A group whose lower central series terminates at the trivial subgroup in finitely many steps.
s_p_adic_number	axiom	p-adic number		An element of the completion of ℚ with respect to the p-adic absolute value, forming a locally compact non-archimedean field.
s_permutation_group	state	Permutation group		A group acting faithfully on a set, realized as a subgroup of the symmetric group.
s_picard_group	state	Picard group		The group of isomorphism classes of invertible sheaves on an algebraic variety, under tensor product.
s_radical	state	Radical (Jacobson)		The intersection of all maximal left ideals of a ring, consisting of elements annihilating all simple modules.
s_ring_of_integers	state	Ring of integers		The integral closure of ℤ in a number field, a Dedekind domain serving as the arithmetic foundation of the field.
s_root_system	axiom	Root system		A finite set of vectors in a Euclidean space satisfying reflection and integrality axioms, classifying semisimple Lie algebras.
s_semi_direct_product	state	Semi-direct product		A group constructed from a normal subgroup and a complement with a specified conjugation action.
s_solvable_group	axiom	Solvable group		A group possessing a subnormal series whose successive quotients are all abelian.
s_tor_functor	state	Tor functor		The left derived functors of the tensor product, measuring failure of tensoring to preserve exactness.
s_transcendence_degree	state	Transcendence degree		The cardinality of a maximal algebraically independent subset of a field extension.
s_universal_enveloping_algebra	state	Universal enveloping algebra		The associative algebra U(𝔤) associated to a Lie algebra through which all representations factor.
s_witt_vectors	state	Witt vectors		A functorial construction lifting a perfect field of characteristic p to a complete DVR of characteristic zero.
s_adele_ring	axiom	Adèle ring		The restricted product of all completions of a global field, providing a single topological ring unifying all local data.
s_idele_group	state	Idèle group		The group of units of the adèle ring, whose quotient by global units governs class field theory.
s_noether_normalization_lemma	theorem	Noether normalization lemma		Every finitely generated algebra over a field is a finite extension of a polynomial subring.
s_cohen_macaulay_ring	axiom	Cohen–Macaulay ring		A Noetherian local ring in which the depth equals the Krull dimension.
s_dynkin_diagram	axiom	Dynkin diagram		A graph encoding the simple roots and their angles in a root system, classifying simple Lie algebras and finite reflection groups.
s_weyl_group	state	Weyl group		The finite group generated by reflections associated to the root system of a semisimple Lie algebra.
s_cartan_subalgebra	axiom	Cartan subalgebra		A maximal abelian self-normalizing subalgebra of a semisimple Lie algebra, whose simultaneous eigenspaces yield the root space decomposition.
s_algebra	axiom	σ-algebra		A collection of subsets closed under complementation and countable unions, serving as the domain of a measure.
s_borel_algebra	axiom	Borel σ-algebra		The smallest σ-algebra containing all open sets of a topological space.
s_hausdorff_measure	axiom	Hausdorff measure		A family of outer measures parametrized by dimension, generalizing length/area/volume to fractional dimensions.
s_radon_measure	axiom	Radon measure		A locally finite inner-regular Borel measure on a locally compact Hausdorff space.
s_frechet_space	axiom	Fréchet space		A complete metrizable locally convex space whose topology is induced by a countable family of seminorms.
s_dual_space	state	Dual space		The space of all continuous linear functionals on a topological vector space.
s_c_algebra	axiom	C*-algebra		A Banach algebra with involution satisfying ‖a*a‖ = ‖a‖², abstractly characterizing norm-closed *-subalgebras of B(H).
s_spectrum	state	Spectrum (operator)		The set of λ ∈ ℂ for which T − λI lacks a bounded inverse, decomposing into point, continuous, and residual spectrum.
s_schwartz_space	axiom	Schwartz space		The Fréchet space of rapidly decreasing smooth functions on ℝⁿ, the natural domain for the Fourier transform.
s_greens_function	state	Green's function		The integral kernel of the inverse of a linear differential operator, encoding the response to a point source.
s_gamma_function	state	Gamma function		The meromorphic function Γ(s) = ∫₀^∞ t^{s−1}e^{−t}dt extending the factorial to complex numbers, with poles at non-positive integers.
s_analytic_continuation	state	Analytic continuation		The unique extension of a holomorphic function beyond its original domain of definition, following from the identity theorem.
s_normal_distribution	axiom	Normal distribution		The probability distribution with density (2πσ²)^{−1/2}exp(−(x−μ)²/(2σ²)), characterized by the central limit theorem.
s_bessel_function	state	Bessel function		Solutions of Bessel's differential equation x²y″+xy′+(x²−ν²)y=0 arising in cylindrical and spherical wave problems.
s_legendre_polynomial	state	Legendre polynomial		Orthogonal polynomials on [−1,1] with weight 1 satisfying Legendre's ODE, arising as zonal spherical harmonics.
s_hermite_polynomial	state	Hermite polynomial		Orthogonal polynomials on ℝ with Gaussian weight e^{−x²}, eigenfunctions of the quantum harmonic oscillator.
s_orthogonal_polynomial	state	Orthogonal polynomial		A sequence of polynomials orthogonal with respect to a weight function, satisfying a three-term recurrence relation.
s_bernstein_polynomial	state	Bernstein polynomial		The polynomial B_n(f;x) = Σ f(k/n)C(n,k)x^k(1−x)^{n−k} giving a constructive proof of the Weierstrass approximation theorem.
s_stirlings_formula	theorem	Stirling's formula		n! ~ √(2πn)(n/e)ⁿ, the fundamental asymptotic approximation for the factorial function.
s_elliptic_integral	state	Elliptic integral		An integral of the form ∫R(x,√P(x))dx where P has degree 3 or 4, not expressible in elementary functions.
s_normal_space	axiom	Normal space		A topological space in which any two disjoint closed sets can be separated by disjoint open neighborhoods.
s_metrization_theorem	theorem	Metrization theorem (Nagata–Smirnov)		A topological space is metrizable iff it is regular, Hausdorff, and has a countably locally finite basis.
s_quotient_space	state	Quotient space (topology)		The space obtained by identifying points via an equivalence relation, with the finest topology making the projection continuous.
s_cofibration	axiom	Cofibration		A continuous map satisfying the homotopy extension property.
s_fibration	axiom	Fibration		A continuous map satisfying the homotopy lifting property, yielding a long exact sequence of homotopy groups.
s_cellular_homology	state	Cellular homology		Homology computed from the chain complex of relative homology of successive skeleta of a CW complex.
s_cohomology_ring	state	Cohomology ring		The graded ring structure on cohomology given by the cup product, a fundamental invariant of spaces.
s_cech_cohomology	state	Čech cohomology		Cohomology defined via open covers and nerve simplicial complexes, agreeing with singular cohomology on paracompact spaces.
s_classifying_space	state	Classifying space		A space BG whose principal G-bundles over X are classified by homotopy classes [X, BG].
s_characteristic_class	state	Characteristic class		A cohomology class functorially associated to a vector bundle measuring non-triviality, e.g. Chern, Pontryagin, Stiefel–Whitney.
s_obstruction_theory	state	Obstruction theory		A systematic use of cohomology classes to determine extendability of maps on CW skeleta.
s_k_theory	state	K-theory (topological)		Generalized cohomology theory from stable isomorphism classes of vector bundles, with Bott periodicity of period 2 or 8.
s_cobordism	state	Cobordism		The equivalence relation on manifolds where two are cobordant if their disjoint union bounds a manifold of one higher dimension.
s_knot_theory	axiom	Knot theory		The study of embeddings of circles in 3-space up to ambient isotopy, classified by polynomial and group invariants.
s_jones_polynomial	state	Jones polynomial		A Laurent polynomial knot invariant from Hecke algebra representations, distinguishing knots the Alexander polynomial cannot.
s_connection	axiom	Connection (differential geometry)		A structure on a bundle specifying parallel transport and covariant differentiation of sections along curves.
s_levi_civita_connection_hzw	state	Levi-Civita connection		The unique torsion-free metric-compatible connection on a Riemannian manifold.
s_holonomy_group	state	Holonomy group		The group of parallel transport maps around loops at a point, classifying connections by the Ambrose–Singer theorem.
s_riemann_curvature_tensor	state	Riemann curvature tensor		The (1,3)-tensor measuring failure of second covariant derivatives to commute, encoding all intrinsic curvature.
s_exponential_map	state	Exponential map (Riemannian)		The map from the tangent space sending a vector to the geodesic endpoint, a local diffeomorphism near the origin.
s_jacobi_field	state	Jacobi field		A vector field along a geodesic satisfying the Jacobi equation, detecting conjugate points.
s_stokes_theorem_hzw	theorem	Stokes' theorem		The integral of a differential form over the boundary of an oriented manifold equals the integral of its exterior derivative.
s_exterior_derivative	axiom	Exterior derivative		The unique linear operator on forms satisfying the Leibniz rule and d² = 0, generalizing grad, curl, div.
s_hodge_theory	state	Hodge theory		Every de Rham cohomology class on a compact Riemannian manifold has a unique harmonic representative.
s_hodge_star_operator	state	Hodge star operator		The linear map on k-forms to (n−k)-forms induced by the metric and orientation.
s_immersion	axiom	Immersion		A differentiable map whose differential is injective at every point, a local embedding allowing self-intersections.
s_submersion	axiom	Submersion		A differentiable map whose differential is surjective at every point.
s_transversality	axiom	Transversality		Two submanifolds intersect transversally when their tangent spaces span the ambient tangent space at each intersection point.
s_tangent_bundle	state	Tangent bundle		The vector bundle over a smooth manifold whose fiber at each point is the tangent space.
s_cotangent_bundle	state	Cotangent bundle		The dual of the tangent bundle, whose sections are differential 1-forms and which carries a canonical symplectic structure.
s_foliation	axiom	Foliation		A decomposition of a manifold into connected immersed submanifolds of constant dimension.
s_algebraic_variety	axiom	Algebraic variety		The zero locus of a set of polynomial equations in affine or projective space.
s_zariski_topology	axiom	Zariski topology		The topology on varieties/schemes whose closed sets are algebraic subsets, coarser than the Euclidean topology.
s_ample_line_bundle	axiom	Ample line bundle		A line bundle some power of which embeds the variety into projective space.
s_etale_cohomology_hzw	state	Étale cohomology		Cohomology using the étale topology, providing ℓ-adic groups as an algebraic analogue of singular cohomology.
s_intersection_theory	state	Intersection theory		The formalism assigning intersection multiplicities to algebraic cycles, yielding a ring structure on the Chow group.
s_chow_group	state	Chow group		The group of algebraic cycles modulo rational equivalence, the algebro-geometric analogue of homology.
s_kodaira_dimension	state	Kodaira dimension		An integer invariant measuring growth of pluricanonical sections, providing the coarsest birational classification.
s_canonical_bundle	state	Canonical bundle		The line bundle of top-degree holomorphic forms, governing birational classification of varieties.
s_calabi_yau_manifold	axiom	Calabi–Yau manifold		A compact Kähler manifold with trivial canonical bundle admitting Ricci-flat metrics by Yau's theorem.
t_maximum_likelihood_estimation	technique	Maximum likelihood estimation		Estimating parameters by maximizing the likelihood function over the parameter space.
s_neyman_pearson_lemma	theorem	Neyman–Pearson lemma		The likelihood ratio test is the most powerful test at any given significance level for simple hypotheses.
s_fisher_information	state	Fisher information		The variance of the score function, measuring information an observable carries about an unknown parameter.
s_cramer_rao_inequality	theorem	Cramér–Rao inequality		The variance of any unbiased estimator is bounded below by the reciprocal of the Fisher information.
s_conditional_expectation	axiom	Conditional expectation		The essentially unique measurable function satisfying the tower property, giving the expected value given a sub-σ-algebra.
s_kolmogorov_axioms	axiom	Kolmogorov axioms		The foundational axioms defining probability as a countably additive non-negative normalized measure.
s_wiener_process	axiom	Wiener process		A continuous-time process with independent stationary Gaussian increments and continuous paths, the rigorous Brownian motion.
t_it_calculus	technique	Itô calculus		Stochastic calculus for semimartingales with the Itô integral and formula featuring a second-order correction term.
s_stochastic_differential_equation	axiom	Stochastic differential equation		A differential equation driven by Brownian motion, with solutions interpreted via Itô or Stratonovich integration.
s_levy_process	axiom	Lévy process		A process with independent stationary increments continuous in probability, characterized by the Lévy–Khintchine formula.
t_kalman_filter	technique	Kalman filter		A recursive algorithm producing optimal linear state estimates for linear dynamical systems from noisy observations.
s_propositional_calculus	axiom	Propositional calculus		The formal system of logic dealing with propositions connected by logical connectives.
s_predicate_calculus	axiom	Predicate calculus		First-order logic extending propositional calculus with quantifiers and predicates over a domain.
t_transfinite_induction	technique	Transfinite induction		Proof method extending induction to well-ordered sets via successor and limit ordinal cases.
s_cardinal_number	axiom	Cardinal number		An equivalence class of sets under bijection measuring size, extended to the transfinite via the aleph hierarchy.
s_ordinal_number	axiom	Ordinal number		A transitive set well-ordered by membership, extending the natural numbers to represent order types of well-orderings.
s_zorns_lemma	theorem	Zorn's lemma		Every partially ordered set in which every chain has an upper bound contains a maximal element.
s_wave_equation	axiom	Wave equation		The hyperbolic PDE u_tt = c²Δu describing wave propagation, solved by d'Alembert (1D) or Kirchhoff (3D) formulas.
t_finite_element_method	technique	Finite element method		Numerical PDE solving by discretizing into elements and approximating with piecewise polynomial basis functions.
t_runge_kutta_method	technique	Runge–Kutta method		Iterative ODE solvers evaluating the vector field at intermediate stages within each time step.
t_galerkin_method	technique	Galerkin method		Converting a continuous operator equation to a discrete problem by projecting onto a finite-dimensional subspace.
s_pontryagin_maximum_principle	theorem	Pontryagin maximum principle		The optimal control maximizes the Hamiltonian along the optimal trajectory with respect to adjoint variables.
s_linear_programming	axiom	Linear programming		Optimization of a linear objective subject to linear constraints, solvable in polynomial time.
t_simplex_method	technique	Simplex method		An LP algorithm traversing vertices of the feasible polytope along improving edges.
s_hamilton_jacobi_equation	axiom	Hamilton–Jacobi equation		A first-order nonlinear PDE linking Hamiltonian mechanics to the calculus of variations.
s_catalan_number	state	Catalan number		The sequence C_n = C(2n,n)/(n+1) counting parenthesizations, binary trees, triangulations, and 200+ structures.
s_monad	axiom	Monad (category theory)		An endofunctor with unit and multiplication natural transformations satisfying associativity and unit axioms.
s_limit	axiom	Limit (category theory)		The universal cone over a diagram, generalizing products, equalizers, and pullbacks.
s_kan_extension	state	Kan extension		The universal construction extending a functor along another, subsuming limits, colimits, and adjunctions.
s_equivalence_of_categories	axiom	Equivalence of categories		A pair of mutually pseudo-inverse functors up to natural isomorphism.
s_triangulated_category	axiom	Triangulated category		An additive category with translation functor and distinguished triangles axiomatizing derived and stable homotopy categories.
s_group_cohomology	state	Group cohomology		Derived functors of the fixed-point functor, classifying group extensions and connecting algebra to topology.
s_partially_ordered_set	axiom	Partially ordered set		A set with a reflexive, antisymmetric, transitive binary relation.
t_proof_by_contradiction	technique	Proof by Contradiction		Assume the negation of a statement and derive a logical impossibility.
t_proof_by_induction	technique	Proof by Induction		Prove a base case and an inductive step to establish a statement for all natural numbers.
s_total_order	axiom	Total Order		A partial order in which every pair of elements is comparable.
s_cantor_diagonal_argument	theorem	Cantor Diagonal Argument		The real numbers are uncountable: no surjection ℕ→ℝ exists (Cantor, 1891)
s_subfield	axiom	Subfield		A subset of a field that is itself a field under the inherited operations.
s_finite_field_f_p	state	Finite Field F_p		The field of integers modulo a prime p.
s_basis	axiom	Basis		A linearly independent spanning set for a vector space.
s_dimension	axiom	Dimension		The cardinality of any basis of a vector space (well-defined by basis invariance)
s_ring	axiom	Ring		A set R with two binary operations (+,·) where (R,+) is an abelian group and · is associative and distributes over +.
s_ring_with_unity	axiom	Ring with Unity		A ring possessing a multiplicative identity element 1.
s_ring_homomorphism	axiom	Ring Homomorphism		A function φ:R→S between rings preserving both addition and multiplication.
s_field_of_fractions	state	Field of Fractions		The localization of an integral domain at its nonzero elements, giving the smallest containing field.
s_open_ball	axiom	Open Ball		The set B(x,r) = {y∈X : d(x,y)<r} in a metric space.
s_open_set_metric	axiom	Open Set (metric)		A set U in a metric space where every point has an open ball contained in U.
s_open_set_topological	axiom	Open Set (topological)		A member of the topology τ on a topological space.
s_closed_set	axiom	Closed Set		The complement of an open set in a topological space.
s_neighborhood	axiom	Neighborhood		An open set containing a given point.
s_homeomorphism	axiom	Homeomorphism		A continuous bijection whose inverse is also continuous.
s_closure	state	Closure		The smallest closed set containing a given subset of a topological space.
s_interior	state	Interior		The largest open set contained in a given subset of a topological space.
s_boundary	state	Boundary		The set of points in the closure but not the interior of a given set.
s_basis_for_a_topology	axiom	Basis for a Topology		A collection of open sets such that every open set is a union of basis elements.
s_measure	axiom	Measure		A countably additive function μ from a σ-algebra to [0,∞] with μ(∅)=0.
s_measurable_set	axiom	Measurable Set		A member of a σ-algebra.
s_orbit	state	Orbit		The set of all images of a point x under a group action.
s_stabilizer	state	Stabilizer		The subgroup of elements in G fixing a given point x under a group action.
s_limit_of_a_sequence	axiom	Limit of a Sequence		A value L that a sequence approaches: for every ε>0 there exists N such that.
s_continuity	axiom	Continuity (ε-δ)		A function f is continuous at a if for every ε>0 there exists δ>0 such that d(x,a)<δ implies d(f(x),f(a))<ε.
s_derivative	axiom	Derivative		The limit f'(x) = lim_{h→0} [f(x+h)−f(x)]/h measuring instantaneous rate of change.
s_gradient	state	Gradient		The vector of all partial derivatives of a scalar-valued function.
s_series	state	Series		A formal sum ∑a_n; its value is the limit of partial sums if convergent.
s_event	axiom	Event		A measurable subset of the sample space Ω in a probability space.
s_expectation	state	Expectation		The Lebesgue integral E[X]=∫X dP of a random variable.
s_variance	state	Variance		Var(X)=E[(X−E[X])²], measuring the spread of a random variable.
s_independence_events	axiom	Independence (events)		Events A and B are independent if P(A∩B)=P(A)P(B)
s_probability_distribution	state	Probability Distribution		The pushforward measure P∘X⁻¹ on ℝ induced by a random variable X.
s_martingale	state	Martingale		A sequence of random variables where the conditional expectation of the next value given the past equals the current value.
s_chart	axiom	Chart		A homeomorphism from an open subset of a manifold to an open subset of ℝⁿ.
s_atlas	axiom	Atlas		A collection of charts covering the entire manifold with compatible transition maps.
s_kform	state	k-Form		A differential form of degree k, i.e., a section of Λ^k(T*M)
s_geodesic	axiom	geodesic		A curve gamma(t) that locally minimizes length, equivalently satisfying the equation nabla_{gamma'} gamma' = 0.
s_complex_manifold	axiom	complex manifold		A manifold with holomorphic transition functions, carrying a complex structure.
s_volume_form	state	Volume Form		A nowhere-vanishing n-form on an n-dimensional orientable manifold.
s_embedding	axiom	Embedding		A smooth injective immersion that is a homeomorphism onto its image.
s_genus	state	Genus		The topological invariant counting the number of handles of a compact Riemann surface.
s_torus_complex	state	Torus (complex)		A Riemann surface of genus 1, realized as ℂ/Λ for a lattice Λ.
s_orbifold	axiom	Orbifold		A generalization of a manifold locally modeled on quotients ℝⁿ/G by finite group actions.
s_orbifold_point	state	Orbifold Point		A point in an orbifold whose local model has a nontrivial isotropy group.
s_fiber	axiom	Fiber		The space F that each fiber π⁻¹(b) is homeomorphic to.
s_section	axiom	Section		A continuous (or smooth) map s:B→E with π∘s=id_B, choosing an element in each fiber.
s_curvature_of_a_connection	state	Curvature of a Connection		The 2-form measuring the failure of a connection's horizontal distribution to be integrable.
s_flat_connection	state	Flat Connection		A connection whose curvature vanishes identically.
s_covariant_derivative	axiom	Covariant Derivative		The derivative operator ∇ along a vector field, induced by a connection on a vector bundle.
s_reynolds_number	state	Reynolds Number		The dimensionless ratio Re=UL/ν characterizing the relative importance of inertial to viscous forces.
s_knot	axiom	Knot		A smooth embedding of S¹ into ℝ³ (or S³), considered up to ambient isotopy.
s_knot_invariant	axiom	Knot Invariant		A quantity assigned to a knot that is unchanged by ambient isotopy.
s_unknot	state	Unknot		The trivial knot, equivalent to a standard circle in ℝ³.
s_trefoil_knot	state	Trefoil Knot		The simplest nontrivial knot, with three crossings.
s_knot_diagram	state	Knot Diagram		A planar projection of a knot with over/under crossing information.
s_link	state	Link		A smooth embedding of several disjoint copies of S¹ into ℝ³.
s_reidemeister_moves	axiom	Reidemeister Moves		Three local moves on knot diagrams that generate all ambient isotopies between diagrams.
s_matroid	axiom	Matroid		A finite set E with a collection of independent subsets satisfying the hereditary and augmentation axioms, abstracting linear independence.
s_matroid_circuit	state	Matroid Circuit		A minimal dependent set in a matroid.
s_dual_matroid	state	Dual Matroid		The matroid M* on the same ground set where bases are complements of bases of M.
s_graphic_matroid	state	Graphic Matroid		The matroid on the edges of a graph where independent sets are acyclic edge sets (forests)
s_representable_matroid	state	Representable Matroid		A matroid realizable as the column matroid of a matrix over some field.
s_morphism	axiom	Morphism		An arrow f:A→B between objects in a category.
s_isomorphism_category	state	Isomorphism (category)		A morphism with a two-sided inverse in a category.
s_product_categorical	state	Product (categorical)		An object A×B with projections satisfying the universal property of products.
s_coproduct	state	Coproduct		An object A⊔B with inclusions satisfying the universal property dual to products.
s_colimit	state	Colimit		A universal cocone under a diagram, dual to limit, generalizing coproducts, coequalizers, and pushouts.
s_schwartz_distribution	axiom	Schwartz Distribution		A continuous linear functional on the space of test functions.
s_dirac_delta_distribution	state	Dirac Delta Distribution		The distribution δ_a defined by δ_a(φ)=φ(a), generalizing the point mass.
s_distributional_derivative	state	Distributional Derivative		The derivative of a distribution T defined by T'(φ)=−T(φ'), extending differentiation to non-differentiable objects.
s_convolution	state	Convolution		The operation (f*g)(x)=∫f(x−y)g(y)dy combining two functions.
s_tensor_product_of_modules	state	Tensor Product of Modules		For R-modules M,N: M⊗_R N, the universal object for R-bilinear maps.
s_tensor	axiom	Tensor		An element of a tensor product; a multilinear functional on products of vector spaces and their duals.
s_tensor_field	state	Tensor Field		A smooth assignment of a tensor to each point of a manifold.
s_spinor	state	Spinor		An element of a representation of the Spin group; a square root of a geometry.
s_quaternions	state	Quaternions ℍ		The 4-dimensional real Clifford algebra with basis {1,i,j,k} and i²=j²=k²=ijk=−1.
s_grassmannian_grkn	state	Grassmannian Gr(k,n)		The space of all k-dimensional linear subspaces of an n-dimensional vector space.
s_plucker_embedding	state	Plücker Embedding		The embedding of Gr(k,n) into projective space via exterior products.
s_mirror_symmetry_calabiyau	state	Mirror Symmetry (Calabi-Yau)		A duality pairing Calabi-Yau manifolds in which the complex and symplectic geometries are exchanged.
s_modular_arithmetic	axiom	Modular Arithmetic		The arithmetic of integers modulo n, forming the ring ℤ/nℤ.
s_quadratic_residue	axiom	Quadratic Residue		An integer a is a quadratic residue mod p if a≡x² (mod p) for some x.
s_jinvariant	state	j-Invariant		The invariant j(E) classifying elliptic curves over algebraically closed fields up to isomorphism.
s_lattice_in	axiom	Lattice (in ℂ)		A discrete subgroup Λ=ℤω₁+ℤω₂ of ℂ with ℂ/Λ a complex torus.
s_weierstrass_function	state	Weierstrass ℘-function		The doubly periodic meromorphic function ℘(z;Λ) uniformizing an elliptic curve.
s_mandelbrot_set	state	Mandelbrot set		M = {c ∈ C : the orbit of 0 under z² + c is bounded}, the parameter space of quadratic polynomials.
s_julia_set	state	Julia set		J(f) = ∂{z : f^n(z) → ∞} for a rational map f: C → C, the boundary of the basin of attraction of infinity.
s_filled_julia_set_k_c	state	Filled Julia Set K_c		The set of all z_0∈ℂ for which the orbit of z under z↦z²+c is bounded.
s_ordinary_differential_equation_ode	axiom	Ordinary Differential Equation (ODE)		An equation involving a function of one variable and its derivatives: F(x,y,y',…,y^{(n)})=0.
s_linear_ode	state	Linear ODE		An ODE where the unknown function and its derivatives appear linearly.
s_autonomous_ode	state	Autonomous ODE		An ODE dy/dt=f(y) where f does not depend explicitly on t.
s_flow	state	Flow		The one-parameter family φ_t of diffeomorphisms generated by the solutions of an autonomous ODE.
s_equilibrium_point	state	Equilibrium Point		A point y_0 where f(y_0)=0 in dy/dt=f(y), so the constant function y(t)=y_0 is a solution.
s_partial_differential_equation_pde	axiom	Partial Differential Equation (PDE)		An equation involving partial derivatives of a function of several variables.
s_elliptic_pde	state	Elliptic PDE		A PDE whose principal symbol is positive-definite (e.g., Laplace's equation Δu=0)
s_parabolic_pde	state	Parabolic PDE		A PDE whose principal symbol is degenerate in one direction (e.g., heat equation u_t=Δu)
s_hyperbolic_pde	state	Hyperbolic PDE		A PDE whose principal symbol has signature (n−1,1) (e.g., wave equation u_{tt}=c²Δu)
t_lagrange_multipliers	technique	Lagrange Multipliers		A method for finding extrema of f(x) subject to constraints g(x)=0 by solving ∇f=λ∇g.
t_convex_optimization	technique	Convex Optimization		Minimization of a convex function over a convex set, where every local minimum is global.
s_bernoulli_distribution	state	Bernoulli Distribution		The distribution of a binary random variable taking value 1 with probability p and 0 with probability 1−p.
s_binomial_distribution	state	Binomial Distribution		The distribution of the number of successes in n independent Bernoulli trials.
s_poisson_distribution	state	Poisson Distribution		The distribution P(X=k)=λ^k e^{−λ}/k! modeling rare events with rate λ.
s_geometric_distribution	state	Geometric Distribution		The distribution of the number of trials until the first success in Bernoulli trials.
s_chisquared_distribution	state	Chi-Squared Distribution		The distribution of ∑X_i² where X_i are i.i.d. standard normal random variables.
s_students_tdistribution	state	Student's t-Distribution		The distribution of Z/√(V/k) where Z~N(0,1) and V~χ²(k) are independent.
s_multivariate_normal_distribution	state	Multivariate Normal Distribution		The joint distribution N(μ,Σ) of a random vector with density determined by mean μ and covariance matrix Σ.
s_prior_distribution	axiom	Prior Distribution		A probability distribution encoding beliefs about a parameter before observing data.
s_posterior_distribution	state	Posterior Distribution		The updated probability distribution of a parameter after incorporating observed data via Bayes' theorem.
t_bayesian_inference	technique	Bayesian Inference		Updating beliefs about parameters via posterior ∝ likelihood × prior.
s_conjugate_prior	state	Conjugate Prior		A prior distribution whose posterior under a given likelihood belongs to the same family.
s_wavelet	state	Wavelet		A function ψ∈L²(ℝ) with zero mean, used to generate a basis via translations and dilations.
s_haar_wavelet	state	Haar Wavelet		The simplest wavelet, a piecewise constant function taking values ±1 on subintervals of [0,1].
s_multiresolution_analysis	state	Multiresolution Analysis		A nested sequence of closed subspaces V_j of L²(ℝ) with scaling and density properties.
s_daubechies_wavelet	state	Daubechies Wavelet		A family of compactly supported orthonormal wavelets with maximal smoothness for their support size (Daubechies, 1988)
s_block_design	axiom	Block Design		A collection of subsets (blocks) of a finite set satisfying prescribed intersection properties.
s_tdesign	state	t-Design		A block design where every t-element subset is contained in the same number of blocks.
s_steiner_system_stkn	state	Steiner System S(t,k,n)		A t-design with block size k on n points where every t-subset appears in exactly one block.
s_linear_code	state	Linear Code		A code that is a linear subspace of F_q^n.
s_hamming_distance	state	Hamming Distance		The number of coordinates in which two codewords differ, defining a metric on F^n.
s_hamming_code	state	Hamming Code		A perfect single-error-correcting linear code with parameters [2^r−1, 2^r−r−1, 3].
s_minimum_distance	axiom	Minimum Distance		The smallest Hamming distance between any two distinct codewords in a code.
s_natural_logarithm	state	Natural Logarithm		The inverse function ln(x)=∫_1^x dt/t, the unique antiderivative of 1/x with ln(1)=0.
s_matrix_exponential	state	Matrix Exponential		exp(A)=∑A^n/n! for a square matrix A, generating the solution to dX/dt=AX.
s_spherical_harmonics_y_lm	state	Spherical Harmonics Y_l^m		The eigenfunctions of the Laplacian on the 2-sphere S², forming an orthonormal basis for L²(S²)
s_stirlings_approximation	theorem	Stirling's Approximation		Γ(n+1)=n!∼√(2πn)(n/e)^n as n→∞ (Stirling)
s_transcendental_number	axiom	Transcendental Number		A complex number that is not algebraic, i.e., not a root of any integer polynomial.
s_irrationality_of_2	theorem	Irrationality of √2		The square root of 2 cannot be expressed as a ratio of integers (Pythagoreans, ~500 BCE)
s_transcendence_of	theorem	Transcendence of π		π is transcendental (Lindemann, 1882), implying the impossibility of squaring the circle.
s_transcendence_of_e	theorem	Transcendence of e		e is transcendental (Hermite, 1873)
s_liouville_numbers	state	Liouville Numbers		Numbers with irrationality measure ∞, the first explicit examples of transcendental numbers (Liouville, 1844)
s_degree_of_a_field_extension	state	Degree of a Field Extension		The dimension [F:K]=dim_K F of F as a vector space over K.
s_maximal_ideal	axiom	Maximal Ideal		An ideal M such that no ideal lies strictly between M and R.
s_radical_of_an_ideal	state	Radical of an Ideal		√I = {r∈R : r^n∈I for some n}, the set of elements with a power in I.
s_affine_scheme_specr	state	Affine Scheme Spec(R)		The space of prime ideals of a commutative ring R with the Zariski topology and structure sheaf.
s_scheme	axiom	scheme		A locally ringed space locally isomorphic to an affine scheme, the fundamental object of modern algebraic geometry (Grothendieck)
s_sheaf	axiom	sheaf		A functor assigning to each open set a set/group/ring of sections with compatible restriction maps and gluing.
s_morphism_of_schemes	state	Morphism of Schemes		A continuous map f:X→Y of underlying spaces with a compatible map of structure sheaves.
s_homotopy	state	Homotopy		A continuous deformation of one map into another; maps f,g:X→Y are homotopic if connected by a continuous family.
s_eigenvalue	state	Eigenvalue		A scalar λ such that Tv=λv for a nonzero v; an element of the point spectrum of T.
s_eigenvector	state	Eigenvector		A nonzero vector v satisfying Tv=λv for an eigenvalue λ of a linear map T.
s_number_field	axiom	number_field		A finite field extension of the rational numbers Q.
s_field_extension_degree	axiom	field_extension_degree		The dimension [K:Q] of a number field K as a vector space over Q.
s_trace_of_element	axiom	trace_of_element		The sum of all Galois conjugates of an element of a number field, Tr_{K/Q}(alpha)
s_discriminant_of_number_field	axiom	discriminant_of_number_field		The determinant of the matrix of traces of products of basis elements of O_K, measuring ramification.
s_prime_ideal	axiom	prime_ideal		A proper ideal P of O_K such that if ab is in P then a or b is in P.
s_fractional_ideal	axiom	fractional_ideal		An O_K-submodule of K that becomes an ideal after multiplication by some nonzero element of O_K.
s_ideal_class_group_2	axiom	ideal_class_group		The quotient group of fractional ideals modulo principal ideals, measuring failure of unique factorization.
s_class_number_2	axiom	class_number		The order of the ideal class group of a number field, h_K.
s_root_of_unity	axiom	root_of_unity		An algebraic integer zeta such that zeta^n = 1 for some positive integer n.
s_real_embedding	axiom	real_embedding		A field homomorphism from K into the real numbers R.
s_complex_embedding	axiom	complex_embedding		A field homomorphism from K into the complex numbers C that is not real.
s_signature_of_number_field	axiom	signature_of_number_field		The pair (r_1, r_2) where r_1 is the number of real embeddings and r_2 the number of conjugate pairs of complex embeddings.
s_ramification_index	axiom	ramification_index		The exponent e_i in the factorization pO_K = P_1^{e_1}...P_g^{e_g} of a rational prime p in O_K.
s_inertia_degree	axiom	inertia_degree		The degree [O_K/P_i : Z/pZ] of the residue field extension for a prime ideal P_i above p.
s_ramified_prime	axiom	ramified_prime		A rational prime p such that some prime ideal above it in O_K has ramification index greater than 1.
s_unramified_prime	axiom	unramified_prime		A rational prime p such that every prime ideal above it in O_K has ramification index 1.
s_split_prime	axiom	split_prime		A rational prime p that factors into the maximum number of distinct prime ideals in O_K.
s_inert_prime	axiom	inert_prime		A rational prime p that remains prime in O_K, i.e., pO_K is a prime ideal.
s_decomposition_group	axiom	decomposition_group		The stabilizer subgroup D(P.
s_inertia_group	axiom	inertia_group		The subgroup I(P.
s_frobenius_element	axiom	frobenius_element		The canonical generator of the decomposition group modulo inertia for an unramified prime, acting as x -> x^p on residues.
s_p_adic_numbers	axiom	p_adic_numbers		The completion Q_p of Q with respect to the p-adic absolute value.
s_p_adic_valuation	axiom	p_adic_valuation		The function v_p assigning to each rational number the exponent of p in its prime factorization.
s_p_adic_absolute_value	axiom	p_adic_absolute_value		The absolute value.
s_p_adic_integers	axiom	p_adic_integers		The ring Z_p = {x in Q_p :.
s_completion_of_number_field	axiom	completion_of_number_field		The field K_v obtained by completing a number field K at a place v.
s_place_of_number_field	axiom	place_of_number_field		An equivalence class of absolute values on a number field, either archimedean or non-archimedean.
s_adele_ring_2	axiom	adele_ring		The restricted direct product of all completions K_v of K with respect to the rings of integers O_{K_v}.
s_idele_group_2	axiom	idele_group		The group of invertible elements (units) of the adele ring.
s_idele_class_group	state	idele_class_group		The quotient C_K = 𝔸_K*/K* of the idèles by the principal idèles; the fundamental object of class field theory.
s_absolute_value_product_formula	axiom	absolute_value_product_formula		The identity that the product of.
s_integral_basis	state	integral_basis		A Z-basis for the ring of integers O_K as a free abelian group of rank [K:Q].
s_regulator_of_number_field	state	regulator_of_number_field		The absolute value of the determinant of the matrix of logarithms of fundamental units, measuring the lattice of units.
s_fundamental_units	state	fundamental_units		A maximal set of multiplicatively independent units of O_K, forming a basis for the free part of O_K^*.
s_class_field	state	class_field		An abelian extension of K corresponding to a given open subgroup of the idele class group via class field theory.
s_ray_class_group	state	ray_class_group		A generalization of the ideal class group incorporating a modulus, classifying abelian extensions with prescribed ramification.
s_lattice_in_r_n	state	lattice_in_R_n		A discrete subgroup of R^n isomorphic to Z^n, used in the geometry of numbers applied to number fields.
s_class_number_formula_2	theorem	class_number_formula		Relates the residue of the Dedekind zeta function at s=1 to class number, regulator, discriminant, and number of roots of unity (Dedekind, Dirichlet)
s_ostrowski_theorem	theorem	ostrowski_theorem		Every nontrivial absolute value on Q is equivalent to either the usual absolute value or a p-adic absolute value (Ostrowski)
s_hensels_lemma_2	theorem	hensels_lemma		A simple root of a polynomial modulo p lifts uniquely to a root in the p-adic integers Z_p (Hensel)
s_fundamental_identity_efg	theorem	fundamental_identity_efg		For a prime p in a number field K of degree n, the sum of e_i * f_i over all primes above p equals n (Dedekind)
t_geometry_of_numbers	technique	geometry_of_numbers		Using geometric properties of lattices and convex bodies to prove existence results about algebraic integers.
t_localization_at_prime	technique	localization_at_prime		Studying a number field one prime at a time by passing to completions K_v.
t_ideal_arithmetic	technique	ideal_arithmetic		Computing with ideals (multiplication, factorization, class representatives) as a substitute for element factorization.
s_von_mangoldt_function	axiom	von_mangoldt_function		The function Lambda(n) = log p if n = p^k and 0 otherwise, weighting primes and prime powers.
s_chebyshev_functions	axiom	chebyshev_functions		The functions theta(x) = sum_{p<=x} log p and psi(x) = sum_{n<=x} Lambda(n)
s_mobius_function	axiom	mobius_function		The function mu(n) = (-1)^k if n is a product of k distinct primes, 0 if n has a squared factor.
s_principal_character	state	principal_character		The Dirichlet character χ₀ with χ₀(n)=1 for gcd(n,d)=1; its L-function has a pole at s=1.
s_conductor_of_character	axiom	conductor_of_character		The minimal modulus q_0 such that a Dirichlet character factors through (Z/q_0 Z)^*.
s_primitive_character	axiom	primitive_character		A Dirichlet character whose conductor equals its modulus.
s_trivial_zeros_of_zeta	axiom	trivial_zeros_of_zeta		The zeros of zeta(s) at s = -2, -4, -6, ..., arising from the functional equation.
s_critical_strip	axiom	critical_strip		The region 0 < Re(s) < 1 in the complex plane containing all nontrivial zeros of zeta(s)
s_critical_line	axiom	critical_line		The line Re(s) = 1/2, conjectured to contain all nontrivial zeros of the Riemann zeta function.
s_arithmetic_function	axiom	arithmetic_function		A function f: N -> C, the basic object of study in multiplicative and additive number theory.
s_multiplicative_function	axiom	multiplicative_function		An arithmetic function f with f(mn) = f(m)f(n) whenever gcd(m,n)=1.
s_dirichlet_convolution	axiom	dirichlet_convolution		The product (f*g)(n) = sum_{d.
s_sieve_of_eratosthenes	axiom	sieve_of_eratosthenes		The classical method of finding primes by successively removing multiples of each prime.
s_siegel_zero	state	siegel_zero		A hypothetical real zero of an L-function L(s,chi) very close to s=1 for a real primitive character chi.
s_li_function	state	li_function		The logarithmic integral Li(x) = integral from 2 to x of dt/log t, the main term in the prime number theorem.
s_prime_gaps	state	prime_gaps		The differences p_{n+1} - p_n between consecutive primes.
s_major_arcs	state	major_arcs		The subsets of [0,1) near rationals a/q with small q, on which generating functions are well approximated.
s_minor_arcs	state	minor_arcs		The complement of the major arcs in [0,1), where exponential sum estimates are needed.
s_gaussian_sum	state	gaussian_sum		The sum G(chi) = sum_{a mod q} chi(a) e^{2pi i a/q} associated to a Dirichlet character.
s_kloosterman_sum	state	kloosterman_sum		The exponential sum S(m,n;c) = sum_{x mod c, gcd(x,c)=1} e^{2pi i(mx+n x^{-1})/c}.
s_ramanujan_sum	state	ramanujan_sum		The sum c_q(n) = sum_{a mod q, gcd(a,q)=1} e^{2pi i an/q}.
s_density_estimate_for_zeros	state	density_estimate_for_zeros		Upper bounds for the number of zeros of zeta(s) or L(s,chi) in rectangles within the critical strip.
s_prime_number_theorem_2	theorem	prime_number_theorem		π(x) ~ x/ln(x), or equivalently π(x) ~ Li(x); proved independently by Hadamard and de la Vallée Poussin (1896)
s_chebyshev_estimates	theorem	chebyshev_estimates		The bounds c_1 x/log x < pi(x) < c_2 x/log x for explicit constants c_1 < 1 < c_2 (Chebyshev)
s_dirichlet_l_nonvanishing_at_1	theorem	dirichlet_l_nonvanishing_at_1		L(1,chi) != 0 for any nonprincipal Dirichlet character chi, key to primes in arithmetic progressions (Dirichlet)
s_vinogradovs_theorem	theorem	vinogradovs_theorem		Every sufficiently large odd integer is the sum of three primes (Vinogradov, 1937)
s_warings_problem	theorem	warings_problem		Every natural number is the sum of a bounded number g(k) of k-th powers (Hilbert; Hardy-Littlewood asymptotics via circle method)
s_landau_prime_ideal_theorem	theorem	landau_prime_ideal_theorem		The number of prime ideals of norm at most x in a number field K is asymptotic to x/log x (Landau)
t_exponential_sum_technique	technique	exponential_sum_technique		Bounding sums sum e^{2pi i f(n)} using van der Corput, Weyl differencing, or Vinogradov's method.
t_selberg_sieve	technique	selberg_sieve		Selberg's upper-bound sieve choosing optimal weights to minimize the main term, yielding sharp upper bounds for prime-counting sums.
t_bruns_sieve	technique	bruns_sieve		Brun's combinatorial sieve using inclusion-exclusion truncation to give upper and lower bounds for sifted sets.
t_perron_formula	technique	perron_formula		The expression (1/2pi i) integral c-iT to c+iT F(s) x^s/s ds recovering partial sums of Dirichlet series coefficients.
t_mobius_inversion	technique	mobius_inversion		The formula f(n) = sum_{d.
t_contour_integration_in_ant	technique	contour_integration_in_ant		Shifting contours of integration to extract main terms and error estimates from Dirichlet series.
s_computational_complexity_class	axiom	computational_complexity_class		A class of decision problems (P, NP, BPP, etc.) defined by resource bounds on Turing machines.
s_probabilistic_algorithm	axiom	probabilistic_algorithm		An algorithm that uses random bits and may give incorrect answers with bounded probability.
s_deterministic_algorithm	axiom	deterministic_algorithm		An algorithm that uses no randomness and always produces the correct output.
s_integer_factorization_problem	axiom	integer_factorization_problem		The problem of decomposing a composite integer into its prime factors.
s_primality_testing_problem	axiom	primality_testing_problem		The problem of determining whether a given integer is prime or composite.
s_discrete_logarithm_problem	axiom	discrete_logarithm_problem		Given g, h in a group G, find the integer n such that g^n = h, believed to be computationally hard in certain groups.
s_rsa_cryptosystem	axiom	rsa_cryptosystem		Public-key encryption scheme whose security rests on the difficulty of factoring the product of two large primes (Rivest, Shamir, Adleman)
s_lattice_in_z_n	axiom	lattice_in_Z_n		A discrete subgroup of Z^n, the setting for lattice-based computational problems.
s_smooth_number	axiom	smooth_number		An integer all of whose prime factors are below a given bound B, central to subexponential factoring algorithms.
s_b_smooth	axiom	b_smooth		An integer n is B-smooth if all prime factors of n are at most B.
s_miller_rabin_witness	state	miller_rabin_witness		An integer a that certifies the compositeness of n via the strong pseudoprime test.
s_pseudoprime	state	pseudoprime		A composite number n that passes a primality test for a given base a, behaving like a prime.
s_carmichael_number	state	carmichael_number		A composite number n that is a Fermat pseudoprime to every base coprime to n.
s_factor_base	state	factor_base		A set of small primes used to express relations among elements in factoring algorithms.
s_closest_vector_problem	state	closest_vector_problem		The problem of finding the lattice point closest to a given target point.
s_aks_primality_theorem_2	theorem	aks_primality_theorem		There exists a deterministic polynomial-time algorithm for primality testing (Agrawal, Kayal, Saxena, 2002)
s_miller_rabin_correctness	theorem	miller_rabin_correctness		If n is an odd composite, then at least 3/4 of all bases a in [2,n-2] are Miller-Rabin witnesses for n (Rabin)
s_lenstra_ecm_expected_time	theorem	lenstra_ecm_expected_time		Lenstra's elliptic curve method finds a prime factor p of n in expected time exp(O(sqrt(log p log log p))) (Lenstra)
s_number_field_sieve_complexity	theorem	number_field_sieve_complexity		The general number field sieve factors an integer n in time exp(O((log n)^{1/3} (log log n)^{2/3})), the fastest known general factoring algorithm.
s_lll_algorithm_properties	theorem	lll_algorithm_properties		The LLL algorithm finds a short vector within a factor of 2^{n/2} of the shortest vector in polynomial time (Lenstra, Lenstra, Lovasz)
t_trial_division	technique	trial_division		Testing divisibility of n by each prime up to sqrt(n), the simplest factoring method.
t_miller_rabin_test	technique	miller_rabin_test		Probabilistic primality test refining the Fermat test using the factorization n-1 = 2^s * d.
t_aks_algorithm	technique	aks_algorithm		Deterministic polynomial-time primality test based on the polynomial identity (X+a)^n = X^n + a in Z_n[X]/(X^r-1)
t_sieve_of_eratosthenes_computational	technique	sieve_of_eratosthenes_computational		Enumerating primes up to N in O(N log log N) time by iteratively marking multiples.
t_quadratic_sieve	technique	quadratic_sieve		Factoring method finding smooth values of (x^2 - n) to build congruences of squares, subexponential time.
t_number_field_sieve	technique	number_field_sieve		The asymptotically fastest known general-purpose factoring algorithm using algebraic number fields to find smooth relations.
t_pollard_rho_method	technique	pollard_rho_method		Probabilistic factoring algorithm using a pseudo-random sequence and cycle detection, expected time O(p^{1/2}) for smallest factor p.
t_pollard_p_minus_1	technique	pollard_p_minus_1		Factoring method exploiting smoothness of p-1 for a prime factor p of n.
t_baby_step_giant_step	technique	baby_step_giant_step		Algorithm computing discrete logarithms in O(sqrt(.
t_index_calculus	technique	index_calculus		Subexponential algorithm for discrete logarithms in (Z/pZ)^* using a factor base of small primes.
t_lll_algorithm	technique	lll_algorithm		Polynomial-time lattice basis reduction algorithm producing a nearly orthogonal basis (Lenstra, Lenstra, Lovasz)
t_schoof_algorithm	technique	schoof_algorithm		Polynomial-time algorithm for counting points on an elliptic curve over F_p by computing the Frobenius trace modulo small primes (Schoof)
s_projective_space	axiom	projective_space		The space P^n(k) of lines through the origin in k^{n+1}, parametrizing directions.
s_coordinate_ring	axiom	coordinate_ring		The quotient ring k[x_1,...,x_n]/I(V), the ring of regular functions on an affine variety V.
s_function_field	axiom	function_field		The field of fractions of the coordinate ring of an irreducible variety, the field of rational functions.
s_morphism_of_varieties	axiom	morphism_of_varieties		A map between varieties given locally by polynomial (regular) functions.
s_rational_map	axiom	rational_map		A map between varieties defined by ratios of polynomials, possibly undefined on a subvariety.
s_birational_equivalence	axiom	birational_equivalence		Two varieties are birational if they have isomorphic function fields, equivalent to being isomorphic on dense open subsets.
s_affine_scheme	axiom	affine_scheme		Spec(R) for a commutative ring R, the set of prime ideals with the Zariski topology and structure sheaf.
s_stalk_of_sheaf	axiom	stalk_of_sheaf		The direct limit of sections over all open sets containing a point, giving the local ring at that point.
s_morphism_of_schemes_2	axiom	morphism_of_schemes		A pair (f, f^#) of a continuous map and a sheaf map between locally ringed spaces compatible with the ring structure.
s_fiber_of_morphism	axiom	fiber_of_morphism		The preimage of a point under a morphism of schemes, obtained by base change to the residue field.
s_base_change	axiom	base_change		The fiber product X x_S T for a morphism X -> S and T -> S, changing the base scheme.
s_reduced_scheme	axiom	reduced_scheme		A scheme whose local rings have no nilpotent elements.
s_integral_scheme	axiom	integral_scheme		A scheme that is both reduced and irreducible.
s_noetherian_scheme	axiom	noetherian_scheme		A scheme admitting a finite covering by spectra of noetherian rings.
s_locally_free_sheaf	axiom	locally_free_sheaf		A coherent sheaf locally isomorphic to O_X^n, corresponding to a vector bundle.
s_divisor	axiom	divisor		A formal integer linear combination of codimension-1 subvarieties (Weil divisor) or local equations (Cartier divisor)
s_weil_divisor	axiom	weil_divisor		A formal Z-linear combination of irreducible codimension-1 subvarieties of a normal variety.
s_cartier_divisor	axiom	cartier_divisor		A divisor given locally by a single equation, corresponding to a line bundle with a rational section.
s_linear_equivalence_of_divisors	axiom	linear_equivalence_of_divisors		Two divisors are linearly equivalent if their difference is the divisor of a rational function.
s_canonical_divisor	axiom	canonical_divisor		A divisor K_X in the linear equivalence class of the sheaf of top differential forms Omega^n_X.
s_cotangent_sheaf	axiom	cotangent_sheaf		The sheaf Omega^1_{X/k} of Kahler differentials on a smooth variety X.
s_tangent_sheaf	axiom	tangent_sheaf		The dual of the cotangent sheaf, the sheaf of derivations.
s_cech_cohomology_2	state	cech_cohomology		A computational approach to sheaf cohomology using open covers and combinatorial cochains.
s_chow_ring	state	chow_ring		The ring A^*(X) of algebraic cycles modulo rational equivalence with the intersection product.
s_hilbert_scheme	state	hilbert_scheme		The scheme parametrizing closed subschemes of projective space with a given Hilbert polynomial (Grothendieck)
s_grassmannian	state	grassmannian		The variety G(k,n) parametrizing k-dimensional linear subspaces of an n-dimensional vector space.
s_blowup	state	blowup		The birational transformation replacing a subvariety Z of X by the projectivized normal cone, resolving singularities.
s_exceptional_divisor	state	exceptional_divisor		The preimage of the blown-up center; a projective space introduced by a blowup.
s_nef_divisor	state	nef_divisor		A divisor D such that D.C >= 0 for every curve C on X, the closure of the ample cone.
s_canonical_ring	state	canonical_ring		The graded ring R(X, K_X) = direct sum of H^0(X, mK_X) over m >= 0.
s_variety_of_general_type	state	variety_of_general_type		A variety with maximal Kodaira dimension kappa(X) = dim X, having many pluricanonical forms.
s_fano_variety	state	fano_variety		A smooth projective variety whose anticanonical divisor -K_X is ample.
s_calabi_yau_variety	state	calabi_yau_variety		A smooth projective variety with trivial canonical bundle K_X = O_X and vanishing intermediate cohomology.
s_jacobian_of_curve	state	jacobian_of_curve		The abelian variety J(C) parametrizing degree-0 line bundles on a smooth curve C.
s_neron_model	state	neron_model		A smooth group scheme over a Dedekind domain extending an abelian variety over the generic fiber (Neron)
s_serres_gaga	theorem	serres_gaga		For a projective variety X over C, the categories of algebraic and analytic coherent sheaves are equivalent (Serre)
s_bezouts_theorem	theorem	bezouts_theorem		Two projective plane curves of degrees d and e with no common component meet in exactly de points counted with multiplicity (Bezout)
s_finite_generation_of_canonical_ring	theorem	finite_generation_of_canonical_ring		The canonical ring R(X, K_X) is finitely generated for any smooth projective variety of general type (Birkar, Cascini, Hacon, McKernan)
s_existence_of_hilbert_schemes	theorem	existence_of_hilbert_schemes		Hilbert schemes exist as projective schemes and represent the Hilbert functor (Grothendieck)
s_deligne_mumford_irreducibility	theorem	deligne_mumford_irreducibility		The moduli space M_g is irreducible of dimension 3g-3 for g >= 2 (Deligne, Mumford)
t_birational_geometry_program	technique	birational_geometry_program		The study of varieties up to birational equivalence, including the minimal model program and classification.
t_grothendieck_cohomology_methods	technique	grothendieck_cohomology_methods		Using sheaf cohomology and derived categories as the primary computational tool in algebraic geometry.
t_deformation_theory	technique	deformation_theory		Studying families of algebraic structures by analyzing infinitesimal deformations and obstruction classes.
t_grobner_basis_computation	technique	grobner_basis_computation		Using monomial orderings to compute in polynomial rings and solve systems of polynomial equations algorithmically.
s_rational_point	axiom	rational_point		A point on a variety V defined over a field K with all coordinates in K.
s_diophantine_equation	axiom	diophantine_equation		A polynomial equation f(x_1,...,x_n) = 0 to be solved in integers or rationals.
s_height_of_rational_point	axiom	height_of_rational_point		A measure of the arithmetic complexity of a rational point, H(p/q) = max(.
s_weil_height	axiom	weil_height		The logarithmic height h(P) defined using the product formula over all places, well-defined on projective space.
s_neron_tate_height	axiom	neron_tate_height		The canonical height hat{h} on an abelian variety, a quadratic form refining the Weil height (Neron, Tate)
s_good_reduction	axiom	good_reduction		An elliptic curve E/Q has good reduction at p if E mod p is still smooth.
s_bad_reduction	axiom	bad_reduction		An elliptic curve E/Q has bad reduction at p if E mod p acquires a singularity.
s_etale_topology	axiom	etale_topology		A Grothendieck topology on schemes where covering maps are etale morphisms (local isomorphisms) rather than open inclusions.
s_etale_morphism	axiom	etale_morphism		A flat unramified morphism of finite type between schemes, the algebraic analogue of a local diffeomorphism.
s_etale_fundamental_group	axiom	etale_fundamental_group		The profinite group pi_1^et(X, x) classifying finite etale covers of a scheme, generalizing the topological fundamental group.
s_shimura_variety	axiom	shimura_variety		A higher-dimensional generalization of modular curves arising from quotients of symmetric domains by arithmetic groups.
s_newform	axiom	newform		A normalized eigenform for all Hecke operators in the new subspace of S_k(Gamma_0(N))
s_automorphic_form	axiom	automorphic_form		A generalization of modular forms to reductive groups over adeles, the central object of the Langlands program.
s_selmer_group	axiom	selmer_group		A subgroup of H^1(Gal(Q-bar/Q), E[n]) controlling the n-descent on an elliptic curve, containing the image of E(Q)/nE(Q)
s_bsd_order_of_vanishing	state	bsd_order_of_vanishing		The order of vanishing of L(E,s) at s=1, conjectured to equal the rank of E(Q)
s_motive	state	motive		A conjectural universal cohomological object associated to a variety, from which all cohomology theories are derived (Grothendieck)
s_l_function_of_motive	state	l_function_of_motive		The L-function associated to a motive, unifying Hasse-Weil L-functions, Artin L-functions, and automorphic L-functions.
s_brauer_manin_obstruction	state	brauer_manin_obstruction		The obstruction to the Hasse principle arising from elements of the Brauer group of a variety (Manin)
s_tamagawa_number	state	tamagawa_number		The volume of G(A)/G(K) with respect to canonical measures on adelic points of an algebraic group G.
s_birch_swinnerton_dyer_conjecture	theorem	birch_swinnerton_dyer_conjecture		The rank of E(Q) equals the order of vanishing of L(E,s) at s=1, with an explicit formula for the leading coefficient (Birch, Swinnerton-Dyer; largely open)
s_tate_conjecture	theorem	tate_conjecture		The rank of the group of algebraic cycles modulo homological equivalence equals the order of the pole of the L-function of H^{2i} (Tate; open in general)
s_finiteness_of_sha_conjecture	theorem	finiteness_of_sha_conjecture		The Tate-Shafarevich group Sha(E/K) is finite for every elliptic curve over a number field (widely believed, open in general)
t_descent_method	technique	descent_method		Bounding the rank of E(K) by computing the Selmer group through Galois cohomology and local analysis.
t_height_machine	technique	height_machine		Using height functions and their properties (Northcott, descent) to prove finiteness and boundedness results for rational points.
t_deformation_of_galois_representations	technique	deformation_of_galois_representations		Taylor-Wiles method of studying the moduli of Galois representations to prove modularity lifting theorems.
t_iwasawa_theory	technique	iwasawa_theory		Studying the growth of arithmetic invariants (class groups, Selmer groups) in towers of number fields using p-adic L-functions.
s_group_representation	axiom	group_representation		A homomorphism rho: G -> GL(V) from a group G to the group of invertible linear maps on a vector space V.
s_degree_of_representation	axiom	degree_of_representation		The dimension of the vector space V on which G acts.
s_subrepresentation	axiom	subrepresentation		A G-invariant subspace W of V with the restricted action.
s_direct_sum_of_representations	axiom	direct_sum_of_representations		The representation on V + W with G acting on each summand independently.
s_tensor_product_of_representations	axiom	tensor_product_of_representations		The representation on V tensor W with g acting as rho_1(g) tensor rho_2(g)
s_character_of_representation	axiom	character_of_representation		The function chi_rho(g) = Tr(rho(g)), a class function on G that determines the representation up to isomorphism.
s_class_function	axiom	class_function		A function f: G -> C constant on conjugacy classes.
s_trivial_representation	axiom	trivial_representation		The one-dimensional representation sending every group element to the identity.
s_representation_ring	axiom	representation_ring		The Grothendieck ring R(G) of virtual representations under direct sum and tensor product.
s_induced_representation	axiom	induced_representation		The representation Ind_H^G(sigma) = C[G] tensor_{C[H]} V for a representation sigma of a subgroup H.
s_restricted_representation	axiom	restricted_representation		The representation Res_H^G(rho) obtained by restricting a G-representation to a subgroup H.
s_lie_algebra_representation	axiom	lie_algebra_representation		A Lie algebra homomorphism rho: g -> gl(V), equivalently a g-module structure on V.
s_weight_of_representation	axiom	weight_of_representation		An eigenvalue lambda in h^* for the action of a Cartan subalgebra h on a representation.
s_weight_space	axiom	weight_space		The eigenspace V_lambda = {v in V : h.v = lambda(h)v for all h in h} for a weight lambda.
s_fundamental_weight	axiom	fundamental_weight		The weight omega_i dual to the simple coroots: omega_i(h_j) = delta_{ij}.
s_positive_root	axiom	positive_root		A root that is a non-negative integer linear combination of simple roots.
s_simple_root	axiom	simple_root		An indecomposable positive root, forming a basis for the root system.
s_category_o	axiom	category_O		The category of finitely generated g-modules that are h-semisimple and locally n^+-finite (Bernstein-Gelfand-Gelfand)
s_isotypic_component	state	isotypic_component		The sum of all irreducible subrepresentations isomorphic to a given irreducible, the multiplicity space times the irreducible.
s_multiplicity_of_irreducible	state	multiplicity_of_irreducible		The number of times an irreducible representation appears in the decomposition of a given representation.
s_weyl_character	state	weyl_character		The character of the irreducible highest-weight representation L(lambda), given by the Weyl character formula.
s_formal_character	state	formal_character		The element sum_{mu} dim(V_mu) e^mu in the group ring of the weight lattice, encoding the weight space dimensions.
s_jordan_holder_multiplicity	state	jordan_holder_multiplicity		The multiplicity [M:L] of a simple module L in a composition series of M, independent of the series chosen.
s_kazhdan_lusztig_polynomial	state	kazhdan_lusztig_polynomial		The polynomial P_{y,w}(q) for pairs of Weyl group elements encoding multiplicities in the BGG resolution and intersection cohomology (Kazhdan, Lusztig)
s_bgg_resolution	state	bgg_resolution		The resolution of a finite-dimensional irreducible module by Verma modules, with maps indexed by the Bruhat order (Bernstein, Gelfand, Gelfand)
s_harish_chandra_module	state	harish_chandra_module		A (g, K)-module with finite K-multiplicities, the algebraic counterpart of a unitary representation of a real reductive group.
s_langlands_parameter	state	langlands_parameter		The data (M, sigma, nu) classifying irreducible admissible representations of a reductive group via the Langlands classification.
s_schurs_lemma_2	theorem	schurs_lemma		Any G-equivariant map between irreducible representations is either zero or an isomorphism; over C, the endomorphism ring is C (Schur)
s_maschkes_theorem	theorem	maschkes_theorem		Every finite-dimensional representation of a finite group over a field of characteristic not dividing.
s_number_of_irreducibles_equals_conjugacy_classes	theorem	number_of_irreducibles_equals_conjugacy_classes		The number of irreducible representations of a finite group equals the number of its conjugacy classes.
s_weyl_character_formula	theorem	weyl_character_formula		The character of the irreducible representation L(lambda) is sum_{w in W} (-1)^{l(w)} e^{w(lambda+rho)} / sum_{w in W} (-1)^{l(w)} e^{w(rho)} (Weyl)
s_kazhdan_lusztig_conjecture	theorem	kazhdan_lusztig_conjecture		The multiplicity [M(w.lambda):L(y.lambda)] equals P_{y,w}(1), proved by Beilinson-Bernstein and Brylinski-Kashiwara using D-modules.
t_character_methods	technique	character_methods		Using characters to decompose representations, compute multiplicities, and prove structural theorems about groups.
t_induction_restriction_technique	technique	induction_restriction_technique		Systematically using Ind and Res to relate representations of a group to those of subgroups.
t_geometric_representation_theory	technique	geometric_representation_theory		Realizing representations via geometric objects: D-modules, perverse sheaves, intersection cohomology on flag varieties.
t_crystal_base_method	technique	crystal_base_method		Kashiwara's approach to representation theory using crystal bases and combinatorial structures at q=0.
s_presentation_of_group	axiom	presentation_of_group		A description G = <S.
s_word_metric	axiom	word_metric		The metric d(g,h) on a finitely generated group defined as the minimal length of a word representing g^{-1}h in the generators.
s_conjugacy_problem	axiom	conjugacy_problem		The decision problem of determining whether two given elements of a group are conjugate.
s_isomorphism_problem	axiom	isomorphism_problem		The decision problem of determining whether two finitely presented groups are isomorphic.
s_free_group_2	state	free_group		A group with no relations among generators; SO(3) contains a free subgroup on two generators.
s_free_product	axiom	free_product		The group G * H formed by alternating words from G and H with no additional relations.
s_amalgamated_free_product	axiom	amalgamated_free_product		The group G *_A H formed by identifying a common subgroup A in G and H.
s_hnn_extension	axiom	hnn_extension		The group G*_A = <G, t.
t_quasi_isometry	technique	quasi_isometry		A map between metric spaces that distorts distances by at most multiplicative and additive constants; boundary maps from q.i.'s are key to Mostow's proof.
s_quasi_isometry_invariant	axiom	quasi_isometry_invariant		A property of groups preserved under quasi-isometry, hence independent of generating set.
s_dehn_function	axiom	dehn_function		The function delta(n) giving the maximum area needed to fill a loop of length n in the Cayley complex, measuring difficulty of the word problem.
s_hyperbolic_group	axiom	hyperbolic_group		A finitely generated group whose Cayley graph is delta-hyperbolic (triangles are delta-thin) in the sense of Gromov.
s_boundary_at_infinity	axiom	boundary_at_infinity		The Gromov boundary of a hyperbolic group, consisting of equivalence classes of geodesic rays.
s_cat_0_space	axiom	cat_0_space		A geodesic metric space where triangles are at least as thin as comparison triangles in Euclidean space.
s_cat_0_group	axiom	cat_0_group		A group acting properly and cocompactly on a CAT(0) space by isometries.
s_amenable_group	axiom	amenable_group		A group admitting a finitely additive left-invariant probability measure on all subsets.
s_residually_finite_group	axiom	residually_finite_group		A group where for every nontrivial element g, there exists a finite quotient in which g maps nontrivially.
s_linear_group	axiom	linear_group		A group admitting a faithful representation into GL_n(k) for some field k.
s_automatic_group	axiom	automatic_group		A group with a regular language of normal forms and synchronous fellow-traveling for multiplication by generators.
s_growth_rate	state	growth_rate		The exponential growth rate lim_{n->infty} gamma_G(n)^{1/n}, distinguishing polynomial, intermediate, and exponential growth.
s_ends_of_group	state	ends_of_group		The number of ends of the Cayley graph, a quasi-isometry invariant taking values 0, 1, 2, or infinity.
s_dehn_presentation	state	dehn_presentation		A presentation where the Dehn algorithm (greedy relator replacement) solves the word problem in linear time.
s_rips_complex	state	rips_complex		The simplicial complex R_d(G,S) with a simplex for each subset of diameter at most d in the Cayley graph.
s_hyperbolic_dehn_function_is_linear	theorem	hyperbolic_dehn_function_is_linear		Hyperbolic groups have linear Dehn function: delta(n) is O(n), and conversely (Gromov)
s_milnor_svarc_lemma	theorem	milnor_svarc_lemma		If G acts properly and cocompactly by isometries on a geodesic metric space X, then G is quasi-isometric to X (Milnor, Svarc)
s_flat_torus_theorem	theorem	flat_torus_theorem		A CAT(0) group containing Z^n contains a flat: an isometrically embedded copy of R^n (Bridson-Haefliger)
t_quasi_isometric_classification	technique	quasi_isometric_classification		Classifying groups up to quasi-isometry using invariants like growth, ends, hyperbolicity, and asymptotic dimension.
t_van_kampen_diagram	technique	van_kampen_diagram		A planar diagram whose boundary represents a word equal to 1, with 2-cells labeled by relators, used to bound Dehn functions.
t_small_cancellation_theory	technique	small_cancellation_theory		Studying groups given by presentations where relators have small overlap, yielding hyperbolic groups and solvable word problems.
t_geometric_group_actions	technique	geometric_group_actions		Studying algebraic properties of groups via their isometric actions on metric spaces, trees, buildings, and CAT(0) complexes.
s_christoffel_symbols	axiom	christoffel_symbols		The coefficients Gamma^k_{ij} expressing the Levi-Civita connection in local coordinates.
s_exponential_map_diff_geom	axiom	exponential_map_diff_geom		The map exp_p: T_pM -> M sending v to gamma_v(1) where gamma_v is the geodesic with initial velocity v.
s_parallel_transport	axiom	parallel_transport		Moving a vector along a curve while keeping it "constant" with respect to the connection: nabla_{gamma'} V = 0.
s_covariant_derivative_2	axiom	covariant_derivative		The operator nabla_X Y giving the derivative of a vector field Y in the direction X using a connection.
s_differential_form_2	axiom	differential_form		A section of the exterior power Lambda^k T^*M, the natural objects for integration on manifolds.
s_de_rham_cohomology	axiom	de_rham_cohomology		The cohomology H^k_{dR}(M) = ker d / im d of the exterior derivative complex.
s_connection_form	axiom	connection_form		The 1-form omega on a principal bundle encoding the connection, valued in the Lie algebra of the structure group.
s_curvature_form	axiom	curvature_form		The 2-form Omega = d omega + omega wedge omega on a principal bundle, measuring the connection's curvature.
s_todd_class	state	todd_class		The Todd class Td(E) of a complex vector bundle, a characteristic class appearing in index formulas.
s_cut_locus	state	cut_locus		The set of points where geodesics from a given point p cease to be minimizing.
s_volume_form_2	state	volume_form		The canonical n-form on an oriented Riemannian manifold vol_g = sqrt(det g) dx^1 wedge ... wedge dx^n.
s_laplacian_on_manifold	state	laplacian_on_manifold		The Laplace-Beltrami operator Delta f = div(grad f) on functions, or the Hodge Laplacian on forms.
s_harmonic_form	state	harmonic_form		A differential form alpha with Delta alpha = 0, representing a de Rham cohomology class.
s_einstein_metric	state	einstein_metric		A Riemannian metric satisfying Ric = lambda g for a constant lambda, the vacuum field equation of general relativity.
s_kahler_metric	state	kahler_metric		A Riemannian metric on a complex manifold that is simultaneously Hermitian and symplectic, with parallel complex structure.
s_bochner_weitzenboeck_formula	theorem	bochner_weitzenboeck_formula		Delta alpha = nabla^* nabla alpha + curvature terms, relating the Hodge Laplacian to the connection Laplacian and curvature.
s_berger_classification	theorem	berger_classification		The possible holonomy groups of irreducible simply connected non-symmetric Riemannian manifolds: SO(n), U(n), SU(n), Sp(n), Sp(n)Sp(1), G_2, Spin(7) (Berger)
s_hamilton_three_manifolds	theorem	hamilton_three_manifolds		A compact 3-manifold with positive Ricci curvature converges under Ricci flow to a space of constant positive curvature (Hamilton)
t_comparison_geometry	technique	comparison_geometry		Using sectional or Ricci curvature bounds to compare a manifold's geometry with model spaces of constant curvature.
t_bochner_technique	technique	bochner_technique		Using the Bochner-Weitzenbock formula to deduce vanishing of harmonic forms from curvature positivity.
t_moving_frames_method	technique	moving_frames_method		Cartan's method of choosing adapted frames to compute curvature and connection forms efficiently.
s_trigonometric_system	axiom	trigonometric system		The orthonormal system {e^{2πinx} : n ∈ Z} on the circle.
s_lp_space_on_the_circle	axiom	L^p space on the circle		Space of measurable functions with finite p-th power integral on T.
s_dirichlet_kernel	state	Dirichlet kernel		D_N(x) = sum_{.
s_fejer_kernel	state	Fejér kernel		F_N(x) = (1/N) sum_{k=0}^{N-1} D_k(x), averaging kernel for Cesàro means.
s_plancherel_identity	state	Plancherel identity		
s_maximal_partial_sum_operator	state	maximal partial sum operator		S* f(x) = sup_N.
s_riesz_transform	state	Riesz transform		R_j f(x) = c_n p.v. integral (x_j - y_j)/.
s_multiplier_operator	state	multiplier operator		T_m f = (m · f-hat)^∨ for a bounded function m (Fourier multiplier)
s_stationary_phase_point	state	stationary phase point		A point x_0 where ∇φ(x_0) = 0, dominating oscillatory integral asymptotics.
s_kakeya_set	state	Kakeya set		A compact set in R^n containing a unit line segment in every direction.
s_bmo_space	state	BMO space		Space of functions of bounded mean oscillation,.
s_a_p_weight	state	A_p weight		A nonnegative locally integrable function w satisfying the Muckenhoupt A_p condition.
s_stationary_phase_method	theorem	stationary phase method		Asymptotic expansion I(λ) ~ λ^{-n/2} sum a_k λ^{-k} when phase has nondegenerate critical point.
s_van_der_corput_lemma	theorem	van der Corput lemma		Bounds on oscillatory integrals in one dimension:.
s_h1bmo_duality	theorem	H^1-BMO duality		(H^1)* = BMO, the dual of the real Hardy space H^1 is the space of bounded mean oscillation.
s_kakeya_conjecture	theorem	Kakeya conjecture		Kakeya sets in R^n have Hausdorff dimension n (known for n=2, open for n ≥ 3)
t_tt_method	technique	TT* method		Prove L^p → L^q bounds by analyzing the self-adjoint operator TT* instead of T directly.
s_weak_derivative	state	weak derivative		A locally integrable function g such that integral f ∂φ = -integral g φ for all test functions φ.
s_neumann_problem	state	Neumann problem		Find u with Δu = f in Ω and ∂u/∂n = g on ∂Ω.
s_conservation_law	state	conservation law		∂u/∂t + div F(u) = 0, a first-order hyperbolic system from physics.
s_shock_wave	state	shock wave		A discontinuous weak solution of a conservation law across a hypersurface.
s_rankinehugoniot_condition	state	Rankine-Hugoniot condition		Jump condition [F(u)]·n = s[u] relating shock speed s to jumps in u and F(u)
s_characteristics_of_a_pde	state	characteristics of a PDE		Curves or surfaces along which information propagates for first-order or hyperbolic equations.
s_riemann_problem	state	Riemann problem		An initial value problem for a conservation law with piecewise constant initial data.
s_leray_existence_of_weak_solutions_for_navierstokes	theorem	Leray existence of weak solutions for Navier-Stokes		For any L^2 initial data there exists a global weak solution of 3D Navier-Stokes.
s_hopf_maximum_principle_boundary_point_lemma	theorem	Hopf maximum principle / boundary point lemma		If Lu ≥ 0, u achieves its max on the boundary, and at that boundary point the outward normal derivative is strictly positive.
t_energy_method_for_pde	technique	energy method for PDE		Multiply the PDE by u or its derivatives, integrate by parts to obtain a priori estimates.
t_method_of_characteristics	technique	method of characteristics		Solve a first-order PDE by following characteristic curves along which the PDE becomes an ODE.
t_galerkin_approximation	technique	Galerkin approximation		Approximate weak solutions in finite-dimensional subspaces and pass to the limit using compactness.
s_stressenergy_tensor	axiom	stress-energy tensor		A symmetric (0,2)-tensor T_{μν} describing the density and flux of energy-momentum in spacetime.
s_causal_structure	axiom	causal structure		The partition of tangent vectors into timelike, null, and spacelike types determined by the Lorentzian metric.
s_einstein_tensor	state	Einstein tensor		G_{μν} = R_{μν} - (1/2)Rg_{μν}, the divergence-free combination entering the field equations.
s_schwarzschild_solution	state	Schwarzschild solution		The unique spherically symmetric vacuum solution ds² = -(1-2M/r)dt² + (1-2M/r)^{-1}dr² + r²dΩ².
s_kerr_solution	state	Kerr solution		The rotating black hole solution, a stationary axisymmetric vacuum spacetime parametrized by mass M and angular momentum a.
s_friedmannlematrerobertsonwalker_metric	state	Friedmann-Lemaître-Robertson-Walker metric		Homogeneous isotropic cosmological solutions ds² = -dt² + a(t)²[dr²/(1-kr²) + r²dΩ²].
s_event_horizon	state	event horizon		A null hypersurface in spacetime beyond which events cannot send signals to infinity.
s_singularity_in_spacetime	state	singularity in spacetime		A geodesic incompleteness: timelike or null geodesics that cannot be extended to infinite affine parameter.
s_penrose_diagram	state	Penrose diagram		A conformal compactification of spacetime mapping the causal structure to a bounded region.
s_adm_mass	state	ADM mass		The total mass-energy of an asymptotically flat spacetime measured at spatial infinity.
s_bondi_mass	state	Bondi mass		Mass measured at null infinity, accounting for energy lost to gravitational radiation.
s_cauchy_surface	state	Cauchy surface		A spacelike hypersurface intersected exactly once by every inextendible timelike curve — an initial data surface.
s_weyl_tensor	state	Weyl tensor		C_{μνρσ}, the trace-free part of the Riemann tensor, encoding gravitational radiation.
s_penrose_inequality	theorem	Penrose inequality		The ADM mass M ≥ sqrt(A/16π) where A is the area of the outermost apparent horizon.
s_orbit_of_a_point	state	orbit of a point		{f^n(x) : n ≥ 0} for a discrete system, or {φ_t(x) : t ∈ R} for a flow.
s_attractor	state	attractor		A compact invariant set A that attracts all nearby orbits: d(f^n(x), A) → 0.
s_strange_attractor	state	strange attractor		An attractor with sensitive dependence on initial conditions and fractal structure.
s_basin_of_attraction	state	basin of attraction		The set of all initial conditions whose orbits converge to a given attractor.
s_sensitive_dependence_on_initial_conditions	state	sensitive dependence on initial conditions		Nearby orbits diverge exponentially: the hallmark of chaos, quantified by positive Lyapunov exponents.
s_symbolic_dynamics	state	symbolic dynamics		Representing orbits as sequences in a shift space {0,1,...,k-1}^Z with the shift map σ.
s_hyperbolic_set	state	hyperbolic set		A compact f-invariant set Λ where the tangent bundle splits TΛ M = E^s ⊕ E^u with exponential contraction on E^s and expansion on E^u.
s_fatou_set	state	Fatou set		The complement of the Julia set: the maximal open set where iterates form a normal family.
s_bifurcation	state	bifurcation		A qualitative change in the dynamics (e.g., creation/destruction of fixed points) as a parameter varies.
s_logistic_map	state	logistic map		f(x) = rx(1-x) on [0,1], the simplest model exhibiting period-doubling cascade to chaos.
s_henon_map	state	Hénon map		(x,y) ↦ (1-ax² + y, bx), a two-dimensional map exhibiting a strange attractor.
s_lorenz_attractor	state	Lorenz attractor		The strange attractor of the Lorenz system dx/dt = σ(y-x), dy/dt = rx-y-xz, dz/dt = xy-bz.
s_kam_torus	state	KAM torus		An invariant torus in a nearly integrable Hamiltonian system surviving small perturbation (Diophantine frequency)
s_jakobsons_theorem	theorem	Jakobson's theorem		For a positive measure set of parameters, the logistic map has an absolutely continuous invariant measure.
t_poincare_section	technique	Poincaré section		Reduce a flow to a discrete map by recording intersections with a transversal hypersurface.
s_involution_on_an_algebra	axiom	involution on an algebra		An antilinear map a ↦ a* satisfying (ab)* = b*a* and (a*)* = a.
s_spectrum_of_an_element	state	spectrum of an element		σ(a) = {λ ∈ C : a - λ1 is not invertible}, the set of spectral values.
s_gns_representation	state	GNS representation		The *-representation π_φ: A → B(H_φ) on a Hilbert space H_φ constructed from a state φ via the GNS construction.
s_factor	state	factor		A von Neumann algebra M whose center Z(M) = M ∩ M' consists only of scalar multiples of the identity.
s_type_i_factor	state	Type I factor		A factor isomorphic to B(H) for some Hilbert space H.
s_type_ii_1_factor	state	Type II_1 factor		An infinite-dimensional factor admitting a faithful normal tracial state (unique up to scalar)
s_type_ii_factor	state	Type II_∞ factor		A factor isomorphic to a Type II_1 factor tensored with B(H) for infinite-dimensional H.
s_type_iii_factor	state	Type III factor		A factor with no nonzero finite projections (all nonzero projections are Murray-von Neumann equivalent)
s_hyperfinite_ii_1_factor_r	state	hyperfinite II_1 factor R		The unique (up to isomorphism) approximately finite-dimensional Type II_1 factor.
s_kms_condition	state	KMS condition		A state φ is KMS at inverse temperature β if for all a,b ∈ M the function F_{a,b}(t) = φ(a σ_t(b)) extends analytically with F(t+iβ) = φ(σ_t(b)a)
s_subfactor	state	subfactor		An inclusion N ⊂ M of Type II_1 factors.
s_jones_index_mn	state	Jones index [M:N]		A real number ≥ 1 measuring the relative size of a subfactor inclusion N ⊂ M.
s_standard_invariant_of_a_subfactor	state	standard invariant of a subfactor		The tower of relative commutants (the Bratteli diagram / paragroup / planar algebra) capturing the combinatorial data of N ⊂ M.
s_k_0_group_of_a_calgebra	state	K_0 group of a C*-algebra		The Grothendieck group of Murray-von Neumann equivalence classes of projections in matrix algebras over A.
s_k_1_group_of_a_calgebra	state	K_1 group of a C*-algebra		The group of connected components of the unitary group of the stabilization of A.
s_gns_theorem	theorem	GNS theorem		Every state on a C*-algebra arises from a cyclic vector in a *-representation; the GNS representation is unique up to unitary equivalence.
s_jones_index_theorem	theorem	Jones index theorem		The index [M:N] for a subfactor N ⊂ M takes values in {4cos²(π/n) : n ≥ 3} ∪ [4,∞)
s_elliott_classification_of_af_algebras	theorem	Elliott classification of AF algebras		Approximately finite-dimensional C*-algebras are classified up to isomorphism by their ordered K_0 group with distinguished order unit.
s_takesaki_duality_theorem	theorem	Takesaki duality theorem		For a crossed product M ⋊_α G, the double crossed product (M ⋊_α G) ⋊_{α-hat} G-hat is isomorphic to M ⊗ B(L²(G))
t_gns_construction_method	technique	GNS construction method		Construct a Hilbert space representation from a state by quotienting out the null space and completing.
t_jones_basic_construction	technique	Jones basic construction		From N ⊂ M, construct M_1 = <M, e_N> where e_N is the Jones projection onto L²(N), iterating to build the Jones tower.
s_string_theory_compactification	axiom	string theory compactification		In string theory, extra dimensions are compactified on a Calabi-Yau 3-fold to produce a 4D effective theory.
s_hodge_numbers_hpq	state	Hodge numbers h^{p,q}		Dimensions of Dolbeault cohomology H^{p,q}(X) = H^q(X, Ω^p), refined Betti numbers of a Kähler manifold.
s_hodge_diamond	state	Hodge diamond		The array of Hodge numbers h^{p,q} arranged symmetrically, encoding the cohomological structure of a Kähler manifold.
s_period_integral	state	period integral		integral_γ Ω for a holomorphic n-form Ω and a cycle γ ∈ H_n(X), encoding variation of complex structure.
s_yukawa_coupling	state	Yukawa coupling		A three-point function on the moduli space computed from a cubic form on cohomology, with physical meaning as a coupling constant.
s_gromovwitten_invariant	state	Gromov-Witten invariant		A virtual count of holomorphic curves of genus g and degree β in a target manifold X.
s_quantum_cohomology	state	quantum cohomology		A deformation of the ordinary cohomology ring of X where the product is corrected by Gromov-Witten invariants.
s_prepotential	state	prepotential		The generating function F(t) = sum_{β} N_β e^{2πi<β,t>} encoding genus-0 Gromov-Witten invariants (instanton corrections)
s_fukaya_category	state	Fukaya category		An A_∞-category whose objects are Lagrangian submanifolds of a symplectic manifold with morphisms given by Floer cohomology.
s_amodel	state	A-model		The topological sigma model capturing symplectic geometry (Gromov-Witten invariants and the Fukaya category)
s_bmodel	state	B-model		The topological sigma model capturing complex geometry (variations of Hodge structure and derived category of coherent sheaves)
s_picardfuchs_equation	state	Picard-Fuchs equation		An ODE satisfied by period integrals as the complex structure varies, encoding monodromy of the variation of Hodge structure.
s_mirror_symmetry_for_the_quintic_candelasde_la_ossagreenparkes	theorem	mirror symmetry for the quintic (Candelas-de la Ossa-Green-Parkes)		The genus-0 Gromov-Witten invariants of the quintic 3-fold can be computed from the periods of its mirror via the Picard-Fuchs equation.
s_kontsevich_homological_mirror_symmetry_conjecture	theorem	Kontsevich homological mirror symmetry conjecture		For a mirror pair (X, X̌), the derived Fukaya category of X is equivalent to the derived category of coherent sheaves on X̌.
s_syz_conjecture_stromingeryauzaslow	theorem	SYZ conjecture (Strominger-Yau-Zaslow)		Mirror Calabi-Yau manifolds are T-dual special Lagrangian torus fibrations over the same base.
s_batyrev_mirror_construction	theorem	Batyrev mirror construction		For a Calabi-Yau hypersurface in a toric variety associated to a reflexive polytope Δ, the mirror is the CY hypersurface of the dual polytope Δ*.
t_localization_on_moduli_spaces	technique	localization on moduli spaces		Use torus actions and virtual localization to compute Gromov-Witten invariants.
s_vertex_algebra	axiom	vertex algebra		A vector space V with a vacuum vector, a translation operator T, and a vertex operator map Y: V → End(V)[[z,z^{-1}]] satisfying locality.
s_vertex_operator	state	vertex operator		Y(a,z) = sum a_{(n)} z^{-n-1}, a formal Laurent series of operators associated to a state a ∈ V.
s_virasoro_algebra	state	Virasoro algebra		The infinite-dimensional Lie algebra spanned by {L_n}_{n ∈ Z} and central element c, with [L_m,L_n] = (m-n)L_{m+n} + (c/12)(m³-m)δ_{m+n,0}.
s_conformal_vertex_algebra	state	conformal vertex algebra		A vertex algebra with a Virasoro element ω ∈ V such that Y(ω,z) = sum L_n z^{-n-2} gives V a grading by L_0-eigenvalues.
s_monster_group_m	state	monster group M		The largest sporadic finite simple group, of order ~8 × 10^53.
s_moonshine_module_v	state	moonshine module V^♮		A vertex operator algebra of central charge 24 whose automorphism group is the monster group M.
s_jfunction	state	j-function		The modular invariant j(τ) = q^{-1} + 744 + 196884q + ··· whose Fourier coefficients are related to representations of the monster.
s_mckaythompson_series	state	McKay-Thompson series		T_g(τ) = sum_{n} Tr(g.
s_monstrous_moonshine	state	monstrous moonshine		The observation that Fourier coefficients of the j-function are sums of dimensions of irreducible representations of the monster group.
s_affine_lie_algebra_kacmoody	state	affine Lie algebra (Kac-Moody)		The central extension of the loop algebra g ⊗ C[t,t^{-1}], an infinite-dimensional Lie algebra.
s_wzw_model	state	WZW model		The Wess-Zumino-Witten conformal field theory associated to a compact Lie group and a level k.
s_rational_vertex_algebra	state	rational vertex algebra		A vertex algebra with finitely many irreducible modules and complete reducibility (semisimple representation theory)
s_conformal_field_theory_2d	state	conformal field theory (2D)		A quantum field theory on a surface invariant under conformal transformations, axiomatized by vertex algebras.
s_operator_product_expansion	state	operator product expansion		The expansion Y(a,z)Y(b,w) ~ sum c_k (z-w)^{-k-1} encoding singular behavior as z → w.
s_modular_tensor_category	state	modular tensor category		A braided tensor category arising from the representation category of a rational vertex algebra, yielding a modular functor.
s_borcherds_theorem_monstrous_moonshine_conjecture	theorem	Borcherds' theorem (monstrous moonshine conjecture)		The McKay-Thompson series T_g(τ) for each g ∈ M is a principal modulus (Hauptmodul) for a genus-zero subgroup of SL_2(R)
s_frenkellepowskymeurman_construction	theorem	Frenkel-Lepowsky-Meurman construction		The moonshine module V^♮ is constructed as an orbifold of the Leech lattice vertex algebra, proving the monster acts on it.
s_goddardkentolive_coset_construction	theorem	Goddard-Kent-Olive coset construction		Given a vertex subalgebra W ⊂ V, the commutant (coset) C(V,W) is again a vertex algebra, providing new conformal field theories.
s_singular_homology_h_nxz	state	singular homology H_n(X;Z)		H_n(X) = ker ∂_n / im ∂_{n+1}, the n-th homology group of the singular chain complex.
s_singular_cohomology_hnxr	state	singular cohomology H^n(X;R)		H^n(X;R) = Hom(H_n(X), R) (approximately), dual to homology, with a ring structure via cup product.
s_cup_product	state	cup product		A graded ring structure on H*(X;R): α ∪ β ∈ H^{p+q}(X;R) for α ∈ H^p and β ∈ H^q.
s_relative_homology_h_nxa	state	relative homology H_n(X,A)		Homology of the quotient complex C_*(X)/C_*(A), measuring the topology of X relative to a subspace A.
s_excision_in_homology	state	excision in homology		H_n(X,A) ≅ H_n(X\U, A\U) when the closure of U is contained in the interior of A.
s_reduced_ktheory_kx	state	reduced K-theory K̃(X)		K̃(X) = ker(K(X) → K(pt)), the reduced K-theory group.
s_cobordism_group_n	state	cobordism group Ω_n		The group of closed n-manifolds up to cobordism equivalence (M ~ N if there exists W with ∂W = M ⊔ N)
s_thom_spectrum	state	Thom spectrum		The spectrum MO (or MU, MSO, etc.) whose homotopy groups are cobordism groups.
s_splitting_principle	theorem	splitting principle		To prove an identity about characteristic classes of vector bundles, it suffices to prove it for direct sums of line bundles.
s_smooth_map	axiom	smooth map		A map between smooth manifolds that is infinitely differentiable in local coordinates.
s_smooth_structure	state	smooth structure		An equivalence class of smooth atlases on a topological manifold, not always unique.
s_surgery_on_a_manifold	state	surgery on a manifold		Removing S^k × D^{n-k} from an n-manifold and replacing it with D^{k+1} × S^{n-k-1}, changing the topology.
s_connected_sum	state	connected sum		M₁ # M₂: remove a small ball from each manifold and glue along the resulting boundary spheres.
s_exotic_sphere	state	exotic sphere		A smooth manifold homeomorphic but not diffeomorphic to the standard sphere S^n.
s_milnors_exotic_spheres	theorem	Milnor's exotic spheres		There exist smooth structures on S^7 not diffeomorphic to the standard one; specifically.
s_kervairemilnor_classification	theorem	Kervaire-Milnor classification		The group Θ_n of exotic spheres is finite for n ≥ 5 and computable via the surgery exact sequence.
s_exotic_r4	theorem	exotic R^4		There exist uncountably many pairwise non-diffeomorphic smooth structures on R^4.
s_rohlins_theorem	theorem	Rohlin's theorem		The signature of a smooth closed spin 4-manifold is divisible by 16.
t_surgery_theory	technique	surgery theory		Systematically modify manifolds by cutting and pasting handles to achieve desired homotopy type.
s_delignemumford_compactification_m_g	state	Deligne-Mumford compactification M̄_g		The compactification of M_g by stable curves (nodal curves with finite automorphism groups)
s_stable_curve	state	stable curve		A connected nodal curve of arithmetic genus g with only nodes as singularities and finite automorphism group.
s_teichmuller_space_t_g	state	Teichmüller space T_g		The universal covering of M_g, parametrizing marked Riemann surfaces (a complex manifold diffeomorphic to R^{6g-6})
s_moduli_of_vector_bundles_on_a_curve	state	moduli of vector bundles on a curve		The moduli space M(r,d) parametrizing semistable vector bundles of rank r and degree d on a fixed curve.
s_sequivalence	state	S-equivalence		Two semistable bundles are S-equivalent if they have the same associated graded object; points of the moduli space represent S-equivalence classes.
s_period_matrix	state	period matrix		A g × 2g matrix of integrals of holomorphic 1-forms over cycles, mapping a curve to its Jacobian in A_g.
s_torelli_map	state	Torelli map		The map M_g → A_g sending a curve C to its Jacobian variety Jac(C)
s_stability_in_git	state	stability in GIT		A point x is stable if its orbit is closed in the semistable locus and its stabilizer is finite.
s_algebraic_stack	state	algebraic stack		A generalization of a scheme allowing objects with nontrivial automorphisms to be parametrized, e.g. [X/G].
s_delignemumford_stack	state	Deligne-Mumford stack		An algebraic stack with finite, reduced automorphism groups at every point.
s_tautological_classes_on_m_g	state	tautological classes on M̄_g		The classes ψ_i, κ_j, λ ∈ H*(M̄_{g,n}) generating the tautological ring.
s_mumfords_git_construction	theorem	Mumford's GIT construction		For a reductive group G acting on a projective variety X with linearization, the GIT quotient X//G exists as a projective variety.
s_narasimhanseshadri_theorem	theorem	Narasimhan-Seshadri theorem		Stable vector bundles on a Riemann surface of degree 0 correspond to irreducible unitary representations of the fundamental group.
s_harer_stability	theorem	Harer stability		H_k(Mod_g; Z) is independent of g for g >> k; the homology of the mapping class group stabilizes.
s_witten_conjecture_kontsevichs_theorem	theorem	Witten conjecture (Kontsevich's theorem)		The generating function for intersection numbers of ψ-classes on M̄_{g,n} satisfies the KdV hierarchy.
s_partition_of_an_integer	axiom	partition of an integer		A way of writing n as an unordered sum of positive integers: n = λ_1 + λ_2 + ··· + λ_k with λ_1 ≥ ··· ≥ λ_k ≥ 1.
s_young_diagram	axiom	Young diagram		A left-justified array of boxes with λ_i boxes in row i, the visual representation of a partition λ.
s_symmetric_function	state	symmetric function		A formal power series f(x_1,x_2,...) invariant under all permutations of the variables.
s_schur_polynomial_s	state	Schur polynomial s_λ		The symmetric function s_λ = sum_T x^T summed over semistandard Young tableaux of shape λ.
s_kostka_number_k	state	Kostka number K_{λ,μ}		The number of semistandard Young tableaux of shape λ and weight μ; the transition matrix between Schur and monomial symmetric functions.
s_littlewoodrichardson_coefficient_c	state	Littlewood-Richardson coefficient c^ν_{λ,μ}		The structure constant s_λ · s_μ = sum c^ν_{λ,μ} s_ν for multiplication of Schur polynomials.
s_littlewoodrichardson_rule	state	Littlewood-Richardson rule		c^ν_{λ,μ} counts the number of LR tableaux (skew SSYT of shape ν/λ and weight μ satisfying the lattice word condition)
s_robinsonschensted_correspondence	state	Robinson-Schensted correspondence		A bijection between permutations of {1,...,n} and pairs (P,Q) of standard Young tableaux of the same shape.
s_mobius_function_of_a_poset	state	Möbius function of a poset		μ: P × P → Z defined by sum_{z: x≤z≤y} μ(x,z) = δ_{x,y}, the combinatorial analogue of inclusion-exclusion.
s_mobius_inversion_on_posets	state	Möbius inversion on posets		If g(x) = sum_{y≤x} f(y), then f(x) = sum_{y≤x} μ(y,x)g(y)
s_zeta_polynomial_of_a_poset	state	zeta polynomial of a poset		Z(P,n) = number of multichains x_0 ≤ x_1 ≤ ··· ≤ x_n in P, a polynomial in n.
s_incidence_algebra_of_a_poset	state	incidence algebra of a poset		The algebra of functions f: {(x,y) : x ≤ y} → k with convolution product (f*g)(x,y) = sum_{x≤z≤y} f(x,z)g(z,y)
s_ppartition	state	P-partition		An order-preserving map σ: P → {1,2,...} from a labeled poset P to the positive integers.
s_jacobitrudi_formula	state	Jacobi-Trudi formula		s_λ = det(h_{λ_i - i + j})_{1≤i,j≤k}, expressing Schur functions as determinants of complete homogeneous symmetric functions.
s_dual_jacobitrudi_formula	state	dual Jacobi-Trudi formula		s_λ = det(e_{λ'_i - i + j}), expressing Schur functions via elementary symmetric functions and the conjugate partition.
s_cauchy_identity	state	Cauchy identity		sum_λ s_λ(x) s_λ(y) = prod_{i,j} 1/(1 - x_i y_j), a fundamental identity for Schur functions.
s_pieris_rule	state	Pieri's rule		s_λ · h_k = sum s_μ summed over μ obtained from λ by adding k boxes with no two in the same column.
s_frobenius_characteristic_map	state	Frobenius characteristic map		A ring isomorphism from the representation ring of all symmetric groups to the ring of symmetric functions, sending irreps to Schur functions.
s_cauchy_identity_proof_and_generalizations	theorem	Cauchy identity proof and generalizations		The Cauchy identity and its dual encode the RSK correspondence bijectively and underlie much of symmetric function theory.
s_simple_graph	axiom	simple graph		A graph G = (V,E) with no loops or multiple edges.
s_turan_graph_tnr	state	Turán graph T(n,r)		The complete r-partite graph on n vertices with part sizes as equal as possible, the extremal graph for K_{r+1}-free graphs.
s_graph_density	state	graph density		The ratio.
s_clique_number_g	state	clique number ω(G)		The size of the largest complete subgraph of G.
s_independence_number_g	state	independence number α(G)		The size of the largest independent set (no two adjacent) in G.
s_regularity_partition	state	regularity partition		A partition of V(G) into clusters such that most pairs of clusters have edge density close to the overall density (ε-regular)
s_regular_pair	state	ε-regular pair		A pair (A,B) of vertex sets where for all large subsets A' ⊂ A, B' ⊂ B, the edge density d(A',B') ≈ d(A,B)
s_arithmetic_progression_of_length_k	state	arithmetic progression of length k		A set {a, a+d, a+2d, ..., a+(k-1)d} in the integers.
s_graph_ramsey_theory	state	graph Ramsey theory		The study of which monochromatic substructures must appear in any coloring of a large complete graph.
s_sunflower_system	state	sunflower (Δ-system)		A family of sets {A_1,...,A_k} whose pairwise intersections are all equal: A_i ∩ A_j = K for all i ≠ j.
s_extremal_number_exnh	state	extremal number ex(n,H)		The maximum number of edges in an H-free graph on n vertices.
s_sunflower_lemma_erdsko	theorem	Sunflower lemma (Erdős-Ko)		A family of more than (p-1)^k k! sets of size k contains a sunflower with p petals.
t_probabilistic_method	technique	probabilistic method		Prove existence of combinatorial structures by showing a random object has the desired property with positive probability.
t_regularity_method	technique	regularity method		Apply Szemerédi regularity lemma then counting/embedding lemma to find substructures in dense graphs.
t_alteration_method	technique	alteration method		Start with a random structure, then deterministically fix a small number of violations.
s_alphabet_and_strings	axiom	alphabet and strings		A finite alphabet Σ and the set Σ* of all finite strings over Σ.
s_boolean_circuit	axiom	Boolean circuit		A directed acyclic graph of AND, OR, NOT gates computing a Boolean function.
s_decision_problem_language	state	decision problem / language		A subset L ⊂ Σ* to be decided (accepted or rejected) by a computational model.
s_complexity_class_p	state	complexity class P		The class of decision problems solvable by a deterministic Turing machine in polynomial time.
s_complexity_class_np	state	complexity class NP		The class of decision problems for which a YES certificate can be verified in polynomial time.
s_npcompleteness	state	NP-completeness		A language L is NP-complete if L ∈ NP and every NP problem reduces to L in polynomial time.
s_sat_boolean_satisfiability	state	SAT (Boolean satisfiability)		Given a Boolean formula, is there a truth assignment making it true? The canonical NP-complete problem.
s_3sat	state	3-SAT		SAT restricted to CNF formulas with exactly 3 literals per clause, still NP-complete.
s_complexity_class_conp	state	complexity class coNP		The class of languages whose complements are in NP.
s_complexity_class_pspace	state	complexity class PSPACE		Problems solvable using polynomial space (regardless of time)
s_complexity_class_l_logspace	state	complexity class L (logspace)		Problems solvable using O(log n) bits of working memory.
s_complexity_class_nl_nondeterministic_logspace	state	complexity class NL (nondeterministic logspace)		Problems solvable by a nondeterministic TM using O(log n) space.
s_complexity_class_bpp	state	complexity class BPP		Problems solvable by a randomized TM in polynomial time with error probability < 1/3.
s_complexity_class_rp	state	complexity class RP		Problems solvable by a randomized TM in polynomial time: no-instances always rejected, yes-instances accepted with probability ≥ 1/2.
s_complexity_class_ip	state	complexity class IP		Problems solvable by an interactive proof system: a polynomial-time verifier interacting with an all-powerful prover.
s_circuit_complexity	state	circuit complexity		The minimum size (number of gates) of a Boolean circuit computing a given function.
s_circuit_class_nc	state	circuit class NC		Problems solvable by polylog-depth polynomial-size circuits (efficiently parallelizable problems)
s_circuit_class_ac0	state	circuit class AC^0		Problems solvable by constant-depth polynomial-size circuits with unbounded fan-in AND/OR gates.
s_polynomial_hierarchy_ph	state	polynomial hierarchy PH		PH = ∪_k Σ^P_k, the hierarchy of alternating quantifier classes generalizing NP and coNP.
s_space_complexity_class	state	space complexity class		DSPACE(f(n)) and NSPACE(f(n)): problems solvable in deterministic/nondeterministic f(n) space.
s_approximation_algorithm	state	approximation algorithm		An algorithm that finds a solution within a guaranteed factor of the optimal for an NP-hard optimization problem.
s_pcp_verifier	state	PCP verifier		A probabilistic verifier that checks a proof by reading only O(1) random bits and O(1) proof bits.
s_p_np_conjecture	theorem	P ≠ NP conjecture		The (unresolved) conjecture that not every problem with polynomial-time verifiable solutions has a polynomial-time algorithm.
s_ip_pspace	theorem	IP = PSPACE		Every PSPACE language has an interactive proof, and conversely every IP language is in PSPACE.
s_pcp_theorem_and_inapproximability	theorem	PCP theorem and inapproximability		The PCP theorem implies that MAX-3SAT cannot be approximated within a factor better than 7/8 unless P = NP.
s_npcompleteness_of_3sat	theorem	NP-completeness of 3-SAT		Reduction from SAT to 3-SAT proving that the restriction remains NP-complete.
s_bakergillsolovay_relativization_barrier	theorem	Baker-Gill-Solovay relativization barrier		There exist oracles A,B such that P^A = NP^A and P^B ≠ NP^B, so P vs NP cannot be resolved by relativizing techniques.
t_diagonalization	technique	diagonalization		Construct a language differing from every machine in a list on at least one input, proving hierarchy separations.
t_arithmetization	technique	arithmetization		Convert a Boolean formula to a polynomial over a finite field to design interactive proofs.
s_floatingpoint_number_system	axiom	floating-point number system		A finite subset F ⊂ R representing real numbers as ±d × β^e with precision t and exponent range.
s_mesh_grid	axiom	mesh / grid		A discretization of a domain Ω into finitely many elements (triangles, rectangles, tetrahedra)
s_interpolation	state	interpolation		Constructing a function p from a finite-dimensional space that matches given data values at specified nodes.
s_polynomial_interpolation	state	polynomial interpolation		Finding the unique polynomial of degree ≤ n passing through n+1 given data points.
s_lagrange_interpolation	state	Lagrange interpolation		The polynomial p(x) = sum f(x_i) L_i(x) where L_i are the Lagrange basis polynomials.
s_newton_interpolation	state	Newton interpolation		Interpolation using the Newton divided-difference form with the forward-difference table.
s_spline_interpolation	state	spline interpolation		Piecewise polynomial interpolation with specified smoothness at the knots.
s_numerical_quadrature	state	numerical quadrature		Approximating integral_a^b f(x) dx by a weighted sum sum w_i f(x_i)
s_gaussian_quadrature	state	Gaussian quadrature		Quadrature with nodes and weights chosen to be exact for polynomials of degree ≤ 2n-1 using n nodes.
s_newtoncotes_formula	state	Newton-Cotes formula		Quadrature using equally spaced nodes (trapezoidal rule, Simpson's rule, etc.)
s_condition_number_a	state	condition number κ(A)		κ(A) =.
s_finite_element_space	state	finite element space		A space of piecewise polynomial functions on a triangulation, e.g. continuous piecewise linear (P1) functions.
s_stiffness_matrix	state	stiffness matrix		The matrix K_{ij} = a(φ_j, φ_i) arising from the bilinear form of a finite element discretization.
s_spectral_method	state	spectral method		Approximating a PDE solution by a truncated expansion in orthogonal polynomials or Fourier modes.
s_linear_multistep_method	state	linear multistep method		An ODE solver using a linear combination of previous steps: sum a_j y_{n-j} = h sum b_j f_{n-j}.
s_truncation_error	state	truncation error		The difference between the exact and numerical operator applied to the exact solution, measuring discretization quality.
s_stability_of_a_numerical_method	state	stability of a numerical method		The property that small perturbations in data produce small perturbations in the numerical solution.
s_convergence_of_a_numerical_method	state	convergence of a numerical method		The numerical solution approaches the exact solution as the mesh is refined (h → 0)
s_cfl_condition	state	CFL condition		Courant-Friedrichs-Lewy condition: Δt/Δx ≤ C for stability of explicit time-stepping for hyperbolic PDEs.
s_conjugate_gradient_method	state	conjugate gradient method		An iterative method for symmetric positive definite systems achieving the minimum error in the A-norm over Krylov subspaces.
s_multigrid_method	state	multigrid method		An iterative method using a hierarchy of grids: smooth the error on a fine grid, restrict to a coarse grid, solve, and interpolate back.
s_krylov_subspace	state	Krylov subspace		K_k(A,b) = span{b, Ab, A²b, ..., A^{k-1}b}, the subspace explored by iterative methods like CG and GMRES.
s_ceas_lemma	theorem	Céa's lemma		The finite element solution u_h satisfies.
s_bramblehilbert_lemma	theorem	Bramble-Hilbert lemma		An approximation theory result bounding the error of polynomial interpolation on a reference element.
s_runge_phenomenon	theorem	Runge phenomenon		Polynomial interpolation on equally spaced nodes can diverge for smooth functions as the degree increases.
s_laxrichtmyer_theorem	theorem	Lax-Richtmyer theorem		A consistent numerical scheme is convergent if and only if it is stable (for well-posed linear initial value problems)
s_dahlquist_barrier_theorems	theorem	Dahlquist barrier theorems		No linear multistep method of order > 2 is A-stable (second Dahlquist barrier); stable methods have order ≤ p+2 where p is the number of steps.
t_richardson_extrapolation	technique	Richardson extrapolation		Combine numerical solutions at different step sizes to cancel leading error terms and obtain higher-order accuracy.
t_adaptive_mesh_refinement	technique	adaptive mesh refinement		Locally refine the mesh where the estimated error is large, achieving efficiency for problems with varying regularity.
s_axiom_of_extensionality	axiom	axiom of extensionality		Two sets are equal iff they have the same elements: ∀x(x∈A ↔ x∈B) → A=B.
s_axiom_of_infinity	axiom	axiom of infinity		There exists an infinite set (containing ∅ and closed under successor)
s_axiom_of_replacement	axiom	axiom of replacement		The image of a set under a definable function is a set.
s_axiom_of_power_set	axiom	axiom of power set		For any set A, the collection P(A) of all subsets of A is a set.
s_axiom_of_regularity_foundation	axiom	axiom of regularity / foundation		Every nonempty set x contains an element y with x ∩ y = ∅ (no infinite descending ∈-chains)
s_successor_ordinal	state	successor ordinal		An ordinal of the form α + 1 = α ∪ {α}.
s_limit_ordinal	state	limit ordinal		An ordinal that is neither 0 nor a successor, e.g. ω, the first infinite ordinal.
s_aleph_numbers	state	aleph numbers ℵ_α		The α-th infinite cardinal: ℵ_0 =.
s_v_l_axiom_of_constructibility	state	V = L (axiom of constructibility)		The assertion that every set is constructible, a restrictive axiom implying GCH and many combinatorial principles.
s_forcing	state	forcing		Cohen's technique for constructing models of set theory by adding generic filters over a poset to a ground model.
s_generic_filter	state	generic filter		A filter G on a partial order P meeting every dense set in the ground model, used in forcing extensions.
s_forcing_extension_mg	state	forcing extension M[G]		The smallest model of ZFC containing M and the generic filter G, where new sets are interpreted via names.
s_booleanvalued_model	state	Boolean-valued model		A model of set theory where truth values come from a complete Boolean algebra rather than {0,1}.
s_large_cardinal	state	large cardinal		A cardinal number whose existence transcends ZFC provability, providing a hierarchy of consistency strength.
s_supercompact_cardinal	state	supercompact cardinal		A cardinal κ such that for every λ ≥ κ there is an elementary embedding j: V → M with critical point κ and M^λ ⊂ M.
s_woodin_cardinal	state	Woodin cardinal		A cardinal δ such that for every A ⊂ V_δ there exists an elementary embedding with critical point below δ.
s_determinacy_ad	state	determinacy (AD)		Every two-player infinite game on ω with a Borel (or more complex) payoff set is determined.
s_borel_set	state	Borel set		A set in the σ-algebra generated by open sets in a topological space.
s_analytic_1_1_set	state	analytic / Σ^1_1 set		The continuous image of a Borel set, first level of the projective hierarchy.
s_descriptive_set_theory	state	descriptive set theory		The study of definable subsets of Polish spaces (Borel, analytic, projective sets) using set-theoretic methods.
s_godels_consistency_of_ch	theorem	Gödel's consistency of CH		The continuum hypothesis and the axiom of choice hold in the constructible universe L, so Con(ZFC) → Con(ZFC + CH)
s_cohens_independence_of_ch	theorem	Cohen's independence of CH		By forcing, there exists a model of ZFC where CH fails, so CH is independent of ZFC.
s_large_cardinal_consistency_hierarchy	theorem	large cardinal consistency hierarchy		The consistency strength ordering: inaccessible < measurable < Woodin < supercompact < ...
t_forcing_technique	technique	forcing technique		Add a generic object to a model of ZFC by choosing an appropriate partial order and interpreting names.
t_iterated_forcing	technique	iterated forcing		Perform forcing at successive stages along an ordinal, supporting at limit stages with direct/inverse limits.
s_model_of_a_theory	state	model of a theory		A structure M satisfying all sentences in T, written M ⊨ T.
s_complete_theory	state	complete theory		A consistent theory T such that for every sentence φ, either T ⊢ φ or T ⊢ ¬φ.
s_type_over_a	state	type over A		A maximal consistent collection of formulas in variables x with parameters from A, an element of the Stone space S_n(A)
s_definable_set	state	definable set		A subset of M^n defined by a first-order formula: {a ∈ M^n : M ⊨ φ(a)}.
s_ultrafilter	state	ultrafilter		A maximal proper filter on a Boolean algebra (or index set), used to form ultraproducts.
s_ultraproduct	state	ultraproduct		∏_I M_i / U, the quotient of a product of structures by an ultrafilter U with Łoś's theorem determining truth.
s_saturated_model	state	saturated model		A model M that realizes every type over every subset A ⊂ M of cardinality <.
s_homogeneous_model	state	homogeneous model		A model where every elementary map between small subsets extends to an automorphism.
s_model_completeness	state	model completeness		A theory T such that every embedding between models is elementary.
s_morley_rank	state	Morley rank		An ordinal-valued rank on definable sets in an ω-stable theory, generalizing dimension.
s_ominimal_structure	state	o-minimal structure		An ordered structure (M, <, ...) where every definable subset of M is a finite union of intervals and points.
s_categoricity_in_power	state	categoricity in power κ		A theory T is κ-categorical if all models of T of cardinality κ are isomorphic.
s_baldwinlachlan_theorem	theorem	Baldwin-Lachlan theorem		An uncountably categorical theory that is not totally categorical has exactly two countable models (up to isomorphism)
s_shelahs_classification_theory	theorem	Shelah's classification theory		A complete theory is classifiable (its models are determined by cardinal invariants) iff it is superstable with NDOP and NOTOP.
t_ultraproduct_construction	technique	ultraproduct construction		Form new models by taking products modulo an ultrafilter, transferring first-order properties.
t_backandforth_method	technique	back-and-forth method		Construct isomorphisms or elementary maps between countable structures by alternately extending partial maps.
s_filtration_f_t	axiom	filtration {F_t}		An increasing family of σ-algebras F_s ⊂ F_t for s ≤ t, representing information available at time t.
s_adapted_process	axiom	adapted process		A stochastic process X_t where X_t is F_t-measurable for each t.
s_stationary_distribution	state	stationary distribution		A probability vector π with πP = π, the long-run distribution of an ergodic Markov chain.
s_supermartingale	state	supermartingale		An adapted process with E[X_t.
s_it_integral	state	Itô integral		integral_0^t H_s dW_s for adapted processes H, extending integration to Brownian motion as integrator.
s_it_process	state	Itô process		X_t = X_0 + integral_0^t μ_s ds + integral_0^t σ_s dW_s, a process driven by drift and Brownian noise.
s_its_formula	state	Itô's formula		d f(X_t) = f'(X_t) dX_t + (1/2) f''(X_t) (dX_t)² with (dW_t)² = dt, the chain rule of stochastic calculus.
s_gaussian_free_field	state	Gaussian free field		The canonical Gaussian process indexed by functions on a domain, with covariance given by the Green's function.
s_percolation_model	state	percolation model		On Z^d or another lattice, each edge (bond) or vertex (site) is open with probability p, independently.
s_critical_probability_p_c	state	critical probability p_c		The threshold p_c such that an infinite open cluster exists a.s. for p > p_c and not for p < p_c.
s_harrisfkg_inequality	theorem	Harris-FKG inequality		For increasing events A, B on a product probability space, P(A ∩ B) ≥ P(A)P(B)
t_coupling_of_markov_chains	technique	coupling of Markov chains		Construct two chains on the same probability space so they meet, bounding mixing times and distances between distributions.
s_levys_lemma_on_the_sphere	state	Lévy's lemma on the sphere		A 1-Lipschitz function on S^{n-1} is within ε of its median except on a set of measure ≤ C e^{-cεn}.
s_johnsonlindenstrauss_projection	state	Johnson-Lindenstrauss projection		A random linear projection from R^n to R^k with k = O(ε^{-2} log N) preserves pairwise distances of N points within (1±ε)
s_rademacher_average	state	Rademacher average		E.
s_volume_ratio	state	volume ratio		vr(K) = (vol(K)/vol(E))^{1/n} where E is the maximal volume ellipsoid in K, measuring roundness.
s_entropy_and_covering_numbers	state	entropy and covering numbers		N(K,εB) = the minimum number of translates of εB needed to cover K, measuring metric complexity.
s_sudakov_minoration	state	Sudakov minoration		For a Gaussian process indexed by T ⊂ S^{n-1}: E sup_{t∈T} <g,t> ≥ c ε √(log N(T,ε))
s_restricted_invertibility	state	restricted invertibility		Given vectors in a high-dimensional space, one can find a proportional-size subset on which the associated operator is well-conditioned.
s_johnsonlindenstrauss_lemma	theorem	Johnson-Lindenstrauss lemma		For any ε > 0 and N points in R^n, there exists a linear map into R^k with k = O(ε^{-2} log N) preserving all pairwise distances within (1±ε)
s_gaussian_isoperimetric_theorem_borell_sudakovtsirelson	theorem	Gaussian isoperimetric theorem (Borell, Sudakov-Tsirelson)		Half-spaces are the isoperimetric extremizers for Gaussian measure: they minimize boundary measure for given Gaussian volume.
s_bourgainmilman_inverse_santal_inequality	theorem	Bourgain-Milman inverse Santaló inequality		vol(K) · vol(K°) ≥ c^n vol(B_2^n)² for centrally symmetric convex bodies K.
s_talagrands_majorizing_measures_theorem	theorem	Talagrand's majorizing measures theorem		E sup_{t∈T} X_t ~ γ_2(T,d) where γ_2 is the generic chaining functional, characterizing the expected supremum of a Gaussian process.
s_schechtmans_theorem_on_random_sections	theorem	Schechtman's theorem on random sections		A random k-dimensional section of ℓ_1^n is (1+ε)-Euclidean for k ≤ c ε² n / log n.
s_bourgaintzafriri_restricted_invertibility_theorem	theorem	Bourgain-Tzafriri restricted invertibility theorem		Given vectors v_1,...,v_n with.
t_random_projection_sketching	technique	random projection / sketching		Project high-dimensional data onto a random low-dimensional subspace to approximately preserve geometry.
t_generic_chaining	technique	generic chaining		Bound the supremum of a stochastic process by a multi-scale entropy integral summing over dyadic scales.
s_radical_of_integer	state	radical_of_integer		The radical rad(n) of a positive integer n is the product of the distinct prime factors of n.
s_abc_triple	state	abc_triple		A triple (a,b,c) of coprime positive integers satisfying a + b = c.
s_abc_conjecture	state	abc_conjecture		For every ε > 0 there exist finitely many abc-triples with c > rad(abc)^(1+ε)
s_quality_of_abc_triple	state	quality_of_abc_triple		The quality q(a,b,c) = log(c)/log(rad(abc)) measuring how exceptional an abc-triple is.
s_mason_stothers_theorem	theorem	mason_stothers_theorem		For coprime polynomials a(t)+b(t)=c(t), max(deg a, deg b, deg c) ≤ deg(rad(abc)) - 1; the polynomial analogue of ABC.
s_szpiros_conjecture	state	szpiros_conjecture		For elliptic curves E/ℚ with conductor N and minimal discriminant Δ,.
s_halls_conjecture	state	halls_conjecture		If x³ − y² ≠ 0 for integers x,y then.
s_erdos_woods_conjecture_consequence	theorem	erdos_woods_conjecture_consequence		ABC implies that for large enough k, there is no pair m < n with rad(m+i) = rad(n+i) for i=0,…,k.
s_weak_abc_conjecture	state	weak_abc_conjecture		There exists a constant C > 0 such that c < rad(abc)^C for all abc-triples.
s_abc_implies_fermat_asymptotic	theorem	abc_implies_fermat_asymptotic		The ABC conjecture implies FLT for sufficiently large exponents.
s_abc_implies_mordell	state	abc_implies_mordell		ABC conjecture implies Mordell conjecture (Faltings' theorem) via Elkies' result.
s_langevin_result	theorem	langevin_result		Effective bounds on number of solutions to S-unit equations follow from ABC.
s_analytic_index	state	analytic_index		The analytic index of an elliptic operator D is dim ker D − dim coker D.
s_k_theory_2	state	k_theory		The Grothendieck group K(X) of vector bundles on X, the natural home for the index.
t_k_theory_proof	technique	k_theory_proof		Atiyah-Singer's original proof via cobordism invariance and K-theory.
s_families_index_theorem	state	families_index_theorem		A generalization of the index theorem to continuous families of elliptic operators, yielding a K-theory class.
s_eta_invariant	state	eta_invariant		A spectral invariant measuring the asymmetry of the spectrum of a self-adjoint elliptic operator on the boundary.
s_hausdorff_paradox	theorem	hausdorff_paradox		The sphere S² minus countably many points is paradoxical under rotations; precursor to Banach-Tarski.
s_tarski_alternative	theorem	tarski_alternative		A group G is non-amenable if and only if it is paradoxical.
s_finitely_additive_measure	state	finitely_additive_measure		A measure where μ(A ∪ B) = μ(A) + μ(B) for disjoint sets, but not necessarily for countable unions.
s_von_neumann_conjecture	state	von_neumann_conjecture		A group is non-amenable if and only if it contains a free subgroup of rank 2 (now known to be false)
s_no_paradox_in_plane	theorem	no_paradox_in_plane		The Banach-Tarski paradox does not occur in ℝ¹ or ℝ², because the isometry groups are amenable (solvable)
s_equidecomposable	state	equidecomposable		Two sets A,B are equidecomposable if A can be partitioned into pieces rearranged by group elements to form B.
s_algebraic_rank	state	algebraic_rank		The rank of the Mordell-Weil group E(ℚ), i.e., the number of independent points of infinite order.
s_analytic_rank	state	analytic_rank		The order of vanishing of L(E,s) at s = 1.
s_bsd_conjecture	state	bsd_conjecture		The algebraic rank of E equals the analytic rank of L(E,s), plus an exact formula for the leading coefficient.
s_bsd_leading_coefficient	state	bsd_leading_coefficient		The leading term of L(E,s) at s=1 involves Ш, the regulator, Tamagawa numbers, and the torsion.
s_neron_tate_height_pairing	state	neron_tate_height_pairing		A canonical quadratic form on E(ℚ) used to define the regulator appearing in BSD.
s_carlesons_theorem	theorem	carlesons_theorem		The Fourier series of every L² function converges pointwise almost everywhere.
s_carleson_operator	state	carleson_operator		The maximal operator C*f(x) = sup_N.
t_time_frequency_analysis	technique	time_frequency_analysis		Decomposition of functions simultaneously in time and frequency, underlying modern proofs of Carleson's theorem.
s_luzin_conjecture	theorem	luzin_conjecture		Carleson's theorem resolved Luzin's 1915 conjecture that L² Fourier series converge a.e.
s_du_bois_reymond_example	state	du_bois_reymond_example		A continuous function whose Fourier series diverges at a point; shows continuity alone is insufficient.
s_normal_distribution_2	state	normal_distribution		The Gaussian distribution N(μ,σ²) with density (2πσ²)^{-1/2} exp(−(x−μ)²/(2σ²))
s_lindeberg_clt	theorem	lindeberg_clt		CLT holds for independent (non-identically distributed) variables if Lindeberg's condition is satisfied.
s_lindeberg_condition	state	lindeberg_condition		For each ε > 0, the sum of truncated variances is negligible relative to total variance; sufficient for CLT.
s_lyapunov_clt	theorem	lyapunov_clt		CLT holds if a Lyapunov condition on moments of order 2+δ is satisfied; implies Lindeberg.
s_multivariate_clt_2	state	multivariate_clt		The CLT extends to random vectors, with convergence to a multivariate normal distribution.
s_simple_group	state	simple_group		A group G ≠ {e} with no proper normal subgroups; the atoms of group theory.
s_alternating_group	state	alternating_group		The group A_n of even permutations on n letters, simple for n ≥ 5.
s_groups_of_lie_type	state	groups_of_lie_type		Finite analogues of simple Lie groups, including classical groups (PSL, PSp, PSO, PSU) and exceptional types.
s_chevalley_groups	state	chevalley_groups		Groups of Lie type over finite fields constructed by Chevalley's uniform method.
s_sporadic_groups	state	sporadic_groups		The 26 finite simple groups that do not belong to any infinite family.
s_monster_group	state	monster_group		The largest sporadic simple group, of order approximately 8 × 10⁵³.
s_mathieu_groups	state	mathieu_groups		Five sporadic simple groups M₁₁, M₁₂, M₂₂, M₂₃, M₂₄ discovered by Mathieu; the first sporadic groups.
s_baby_monster	state	baby_monster		The second-largest sporadic group, a subquotient of the Monster.
s_thompson_classification	theorem	thompson_classification		Thompson's classification of minimal simple groups (simple groups all of whose proper subgroups are solvable)
s_bn_pair	state	bn_pair		A pair of subgroups (B,N) in a group G satisfying Tits' axioms; the structural basis for groups of Lie type.
t_local_analysis	technique	local_analysis		Studying a finite group via the structure of normalizers of p-subgroups.
t_character_theory_in_cfsg	technique	character_theory_in_cfsg		Use of ordinary and modular character theory to constrain the structure of simple groups.
s_quasithin_groups	state	quasithin_groups		Simple groups of characteristic-2 type with e(G) ≤ 2; the last major case completed by Aschbacher-Smith.
s_monstrous_moonshine_2	state	monstrous_moonshine		Unexpected connection between the Monster group and modular functions (j-invariant)
s_griess_construction	theorem	griess_construction		Construction of the Monster group as the automorphism group of a 196884-dimensional commutative algebra.
s_arithmetic_progression	state	arithmetic_progression		A sequence a, a+d, a+2d, … with gcd(a,d)=1.
s_non_vanishing_of_l1_chi	theorem	non_vanishing_of_l1_chi		L(1,χ) ≠ 0 for all non-principal Dirichlet characters χ; the key step in Dirichlet's proof.
t_orthogonality_of_characters	technique	orthogonality_of_characters		The relation Σ_χ χ(n) = φ(d) if n ≡ 1 mod d, 0 otherwise; used to isolate primes in a progression.
s_dirichlet_density	state	dirichlet_density		The Dirichlet density of a set S of primes is lim_{s→1+} Σ_{p∈S} p^{−s} / log(1/(s−1))
s_equidistribution_of_primes	theorem	equidistribution_of_primes		Primes are equidistributed among the φ(d) residue classes coprime to d, each having Dirichlet density 1/φ(d)
t_logarithmic_derivative_of_l	technique	logarithmic_derivative_of_l		Using −L'/L(s,χ) = Σ Λ(n)χ(n)n^{−s} to connect primes to L-functions.
s_furstenberg_correspondence_principle	theorem	furstenberg_correspondence_principle		Any set of integers with positive upper density corresponds to a set in an ergodic system, translating combinatorial problems to dynamics.
s_furstenberg_szemeredi	theorem	furstenberg_szemeredi		Ergodic-theoretic proof that sets of positive upper density contain arbitrarily long arithmetic progressions (Szemerédi's theorem)
s_mixing_transformation	state	mixing_transformation		A measure-preserving T where μ(A ∩ T⁻ⁿB) → μ(A)μ(B); stronger than ergodicity.
s_upper_density	state	upper_density		The upper density d*(A) = lim sup.
s_ribets_theorem	theorem	ribets_theorem		The Frey curve (if it exists) is not modular; proved by Ribet, reducing FLT to the modularity conjecture.
s_taniyama_shimura_conjecture	state	taniyama_shimura_conjecture		Every elliptic curve over ℚ is modular (now the modularity theorem); the key conjecture linking FLT to modular forms.
s_hecke_algebra	state	hecke_algebra		The algebra T of Hecke operators acting on modular forms; the modular deformation ring.
s_r_equals_t	theorem	r_equals_t		Wiles proved R = T (deformation ring equals Hecke algebra) for semistable curves, establishing modularity.
s_langlands_tunnell	theorem	langlands_tunnell		The modularity of GL₂ Galois representations with solvable image, providing the base case for Wiles' induction.
s_contraction_mapping	state	contraction_mapping		A map T on a metric space with d(Tx,Ty) ≤ c·d(x,y) for some c < 1.
t_picard_iteration	technique	picard_iteration		The iterative scheme x_{n+1} = T(x_n) converging to the unique fixed point of a contraction.
s_knaster_tarski	theorem	knaster_tarski		Every monotone function on a complete lattice has a least and greatest fixed point.
s_poincare_birkhoff	theorem	poincare_birkhoff		An area-preserving twist map of an annulus has at least two fixed points.
t_discharging_method	technique	discharging_method		Assigning charges to vertices/faces and redistributing to derive structural properties of minimal counterexamples.
s_unavoidable_set	state	unavoidable_set		A set of configurations such that every planar graph must contain at least one; combined with reducibility.
s_fundamental_theorem_of_algebra_2	theorem	fundamental_theorem_of_algebra		Every non-constant polynomial with complex coefficients has at least one root in ℂ.
s_fta_topological_proof	theorem	fta_topological_proof		Proof of FTA using the fact that a polynomial map ℂ → ℂ of degree n has topological degree n, hence is surjective.
s_fta_analytic_proof	theorem	fta_analytic_proof		Proof using Liouville's theorem: if p has no root, then 1/p is entire and bounded, hence constant.
s_splitting_of_polynomials	state	splitting_of_polynomials		Over ℂ, every degree-n polynomial factors completely as c(z−α₁)⋯(z−αₙ)
s_fta_algebraic_proof	theorem	fta_algebraic_proof		Proof using Galois theory and the fact that ℝ has no odd-degree extensions, plus every element of ℂ has a square root.
s_euclids_lemma	theorem	euclids_lemma		If a prime p divides ab then p divides a or p divides b; the key step for uniqueness.
s_bezouts_identity	theorem	bezouts_identity		For integers a,b, there exist x,y with ax + by = gcd(a,b)
s_formal_system	state	formal_system		A system of axioms and inference rules for deriving theorems in a formal language.
s_goedel_numbering	state	goedel_numbering		An encoding of formulas and proofs as natural numbers, allowing self-reference within arithmetic.
s_omega_consistency	state	omega_consistency		A stronger condition than consistency: the system does not prove ∃x¬P(x) while proving P(0), P(1), P(2),….
s_rossers_theorem	theorem	rossers_theorem		Strengthening of Gödel's first theorem: simple consistency (not ω-consistency) suffices.
t_diagonalization_in_arithmetic	technique	diagonalization_in_arithmetic		Constructing a sentence G that asserts "G is not provable," the self-referential heart of Gödel's proof.
s_primitive_recursive_function	state	primitive_recursive_function		A function computable by bounded recursion; the provability predicate is primitive recursive.
s_representability	state	representability		A formal system S represents a function f if for each n, S proves f(n̄) = f(n)̄; ensures arithmetic encodes its own proof theory.
s_consistency_statement	state	consistency_statement		The arithmetic sentence Con(S) formalizing "S does not prove 0=1".
s_bass_guivarch_formula	theorem	bass_guivarch_formula		For a finitely generated nilpotent group, the degree of polynomial growth is Σ i·rank(Gᵢ/Gᵢ₊₁)
s_montgomery_zippin	theorem	montgomery_zippin		A locally compact group that is a topological manifold is a Lie group; used in Gromov's proof.
s_vanishing_ideal	state	vanishing_ideal		The ideal I(V) of all polynomials vanishing on a variety V.
s_weak_nullstellensatz	theorem	weak_nullstellensatz		If I is a proper ideal in k[x₁,…,xₙ] (k algebraically closed), then V(I) ≠ ∅.
s_strong_nullstellensatz	theorem	strong_nullstellensatz		I(V(J)) = √J for any ideal J in k[x₁,…,xₙ] (k algebraically closed); a polynomial vanishing on V(J) has a power in J.
s_radical_ideal	state	radical_ideal		The radical √I = {f : fⁿ ∈ I for some n}; the Nullstellensatz says vanishing ideals are always radical.
t_rabinowitsch_trick	technique	rabinowitsch_trick		Introducing a new variable to reduce the strong Nullstellensatz to the weak form: if fg = 0 on V(I) use I + (1−yg)
s_zariski_correspondence	state	zariski_correspondence		The bijection between radical ideals in k[x₁,…,xₙ] and algebraic subsets of kⁿ via V and I.
s_maximal_ideals_are_points	state	maximal_ideals_are_points		Under the Nullstellensatz, the maximal ideals of k[x₁,…,xₙ] are exactly (x₁−a₁,…,xₙ−aₙ) for (a₁,…,aₙ) ∈ kⁿ.
s_zfc_axioms_2	axiom	zfc_axioms		Zermelo-Fraenkel set theory with the Axiom of Choice; the standard axioms of set theory.
s_goedel_constructibility	theorem	goedel_constructibility		GCH holds in L; hence if ZFC is consistent, so is ZFC + GCH.
s_generic_filter_2	state	generic_filter		A filter meeting all dense sets in a partial order; the object adjoined in forcing constructions.
s_cohen_independence_of_ch	theorem	cohen_independence_of_ch		CH is independent of ZFC: Cohen constructed a model where CH fails by adding ℵ₂ many reals via forcing.
s_partial_order_for_forcing	state	partial_order_for_forcing		A poset (P,≤) whose conditions approximate the generic object; the structure driving a forcing extension.
s_boolean_valued_models	state	boolean_valued_models		An alternative formulation of forcing using complete Boolean algebras instead of partial orders.
s_large_cardinal_axioms	state	large_cardinal_axioms		Axioms asserting the existence of very large infinite cardinals; often interact with CH and forcing.
s_am_gm_inequality	theorem	am_gm_inequality		The arithmetic mean is at least the geometric mean: (Σaᵢ)/n ≥ (∏aᵢ)^{1/n} for non-negative reals.
s_holders_inequality	theorem	holders_inequality		‖fg‖₁ ≤ ‖f‖_p ‖g‖_q for 1/p + 1/q = 1; generalizes Cauchy-Schwarz.
s_minkowskis_inequality	theorem	minkowskis_inequality		‖f+g‖_p ≤ ‖f‖_p + ‖g‖_p for p ≥ 1; the triangle inequality for Lᵖ spaces.
s_jensens_inequality	theorem	jensens_inequality		For a convex function φ, φ(E[X]) ≤ E[φ(X)]; underlies AM-GM and many other inequalities.
s_gronwalls_inequality	theorem	gronwalls_inequality		If u'(t) ≤ β(t)u(t), then u(t) ≤ u(0)exp(∫β); key for uniqueness and stability in ODEs.
s_schur_inequality	theorem	schur_inequality		For non-negative reals and t > 0: xᵗ(x−y)(x−z) + yᵗ(y−x)(y−z) + zᵗ(z−x)(z−y) ≥ 0.
s_poincare_inequality_2	theorem	poincare_inequality		‖f − f̄‖_{L²} ≤ C‖∇f‖_{L²} on a bounded domain; controls oscillation by gradient.
s_halting_problem	state	halting_problem		The problem of determining whether a given Turing machine halts on a given input.
t_diagonalization_argument	technique	diagonalization_argument		Assuming a halting decider exists leads to a contradiction by constructing a machine that does the opposite of what is predicted.
s_decidable_set	state	decidable_set		A set (language) for which there is a Turing machine that always halts and correctly accepts/rejects.
s_computably_enumerable_set	state	computably_enumerable_set		A set for which there is a Turing machine that halts and accepts exactly the elements of the set (may not halt on non-elements)
s_rices_theorem	theorem	rices_theorem		Every non-trivial semantic property of programs (partial functions) is undecidable.
t_reduction_between_problems	technique	reduction_between_problems		Showing problem A is undecidable by constructing a computable transformation taking instances of the halting problem to instances of A.
s_halting_not_co_ce	theorem	halting_not_co_ce		The complement of the halting set is not computably enumerable; the halting set is c.e. but not decidable.
s_polynomial_equation_by_radicals	state	polynomial_equation_by_radicals		Expressing the roots of a polynomial using only +, −, ×, ÷ and nth roots applied to the coefficients.
s_galois_criterion	theorem	galois_criterion		A polynomial is solvable by radicals if and only if its Galois group is a solvable group.
s_sn_not_solvable	theorem	sn_not_solvable		The symmetric group S_n is not solvable for n ≥ 5 because A_n is simple and non-abelian.
s_quadratic_cubic_quartic_formulas	theorem	quadratic_cubic_quartic_formulas		Polynomials of degree 2, 3, 4 are solvable by radicals because S₂, S₃, S₄ are solvable groups.
s_transcendental_number_2	state	transcendental_number		A complex number that is not algebraic; e.g., e and π.
s_liouville_number	state	liouville_number		A real number with extraordinarily good rational approximations; specifically, for each n there exist p,q with.
s_liouville_transcendence	theorem	liouville_transcendence		Liouville numbers are transcendental, providing the first explicit examples of transcendental numbers.
s_irrationality_measure	state	irrationality_measure		The infimum μ of exponents w such that.
t_thue_siegel_method	technique	thue_siegel_method		A method using auxiliary polynomials and the pigeonhole principle, refined through Thue, Siegel, and Roth.
s_lattice_in_lie_group	state	lattice_in_lie_group		A discrete subgroup Γ of a Lie group G with finite covolume G/Γ.
s_margulis_superrigidity	theorem	margulis_superrigidity		For lattices in higher-rank semisimple groups, homomorphisms to algebraic groups extend to the ambient group; generalizes Mostow.
s_no_rigidity_in_dimension_2	state	no_rigidity_in_dimension_2		Closed surfaces of genus g ≥ 2 have a (6g−6)-dimensional moduli space of hyperbolic metrics; rigidity fails in dimension 2.
s_complexity_class_p_2	state	complexity_class_p		The class of decision problems solvable by a deterministic Turing machine in polynomial time.
s_complexity_class_np_2	state	complexity_class_np		The class of decision problems for which a YES answer has a polynomial-time verifiable certificate.
s_p_vs_np_problem	state	p_vs_np_problem		The open question of whether P = NP; one of the Clay Millennium Problems.
s_boolean_satisfiability	state	boolean_satisfiability		The problem of determining whether a Boolean formula in CNF has a satisfying assignment.
s_np_hard	state	np_hard		A problem to which every NP problem reduces in polynomial time (may or may not be in NP)
s_co_np	state	co_np		The class of problems whose complement is in NP; if P ≠ NP then NP ≠ co-NP.
s_ladners_theorem	theorem	ladners_theorem		If P ≠ NP then there exist NP-intermediate problems (in NP but neither in P nor NP-complete)
s_three_sat	state	three_sat		The satisfiability problem restricted to clauses of size 3; still NP-complete.
s_traveling_salesman_problem	state	traveling_salesman_problem		Given cities and distances, find the shortest tour visiting all cities; NP-hard optimization problem.
s_graph_isomorphism	state	graph_isomorphism		Determining whether two graphs are isomorphic; in NP, not known to be NP-complete or in P.
s_poincare_conjecture_statement	state	poincare_conjecture_statement		Every simply connected closed 3-manifold is homeomorphic to S³.
s_poincare_conjecture_2	theorem	poincare_conjecture		Perelman's proof (2003) that every simply connected closed 3-manifold is homeomorphic to S³, via Ricci flow.
t_hamiltons_program	technique	hamiltons_program		Hamilton's strategy of using Ricci flow to deform a 3-manifold metric toward constant curvature.
s_perelman_geometrization	theorem	perelman_geometrization		Perelman proved the full Thurston geometrization conjecture using Ricci flow with surgery.
s_neckpinch_singularity	state	neckpinch_singularity		A singularity in Ricci flow where the manifold develops a long thin neck approaching a cylinder.
s_freedman_4d_poincare	theorem	freedman_4d_poincare		The topological Poincaré conjecture in dimension 4, proved by Freedman (1982)
s_non_trivial_zeros	state	non_trivial_zeros		The zeros of ζ(s) in the critical strip, as opposed to the trivial zeros at s = −2, −4, −6, ….
s_logarithmic_integral	state	logarithmic_integral		Li(x) = ∫₂ˣ dt/ln(t); the better approximation to π(x) in the PNT.
s_goldbach_conjecture	state	goldbach_conjecture		Every even integer greater than 2 is the sum of two primes (unproven)
s_schnirelmann_density	state	schnirelmann_density		The density σ(A) = inf_{n≥1} A(n)/n of a set A of non-negative integers, where A(n) =.
s_sumset	state	sumset		The sumset A + B = {a + b : a ∈ A, b ∈ B} of two sets of integers.
s_freimans_theorem	theorem	freimans_theorem		A finite set A with.
s_szemeredis_theorem	theorem	szemeredis_theorem		Every subset of ℤ with positive upper density contains arbitrarily long arithmetic progressions.
s_goldbach_weak_conjecture	state	goldbach_weak_conjecture		Every odd integer greater than 5 is the sum of three primes; proved by Helfgott (2013)
s_conductor_of_abelian_extension	state	conductor_of_abelian_extension		The modulus measuring the ramification of an abelian extension; determines which primes ramify.
s_langlands_program	state	langlands_program		A vast generalization of class field theory to non-abelian extensions, relating automorphic forms to Galois representations.
s_mordell_conjecture_statement	state	mordell_conjecture_statement		A curve of genus ≥ 2 over a number field has only finitely many rational points (conjecture)
t_chabauty_method	technique	chabauty_method		A p-adic method for determining rational points on curves of genus g when the Mordell-Weil rank is less than g.
t_heights_in_number_theory	technique	heights_in_number_theory		A function measuring the arithmetic complexity of rational points; Faltings' proof uses height bounds.
s_tate_conjecture_for_abelian_varieties	theorem	tate_conjecture_for_abelian_varieties		Faltings proved: Hom(A,B) ⊗ ℤₗ ≅ Hom_{Gal}(T_ℓA, T_ℓB) for abelian varieties over number fields.
s_siegel_finiteness	theorem	siegel_finiteness		There are finitely many integer points on affine curves of genus ≥ 1; precursor to Mordell.
s_hironaka_theorem	theorem	hironaka_theorem		Every algebraic variety over a field of characteristic zero has a resolution of singularities.
t_blowing_up	technique	blowing_up		Replacing a point (or subvariety) by a projective space (exceptional divisor) to separate tangent directions; the basic operation in resolution.
s_embedded_resolution	state	embedded_resolution		A resolution where X ⊂ Y and we resolve by blowing up the ambient smooth variety Y.
s_normal_crossings_divisor	state	normal_crossings_divisor		A divisor whose components are smooth and meet transversally; the goal of embedded resolution.
s_resolution_of_curves	theorem	resolution_of_curves		Every singular curve can be resolved by normalization; the classical case.
s_resolution_of_surfaces	theorem	resolution_of_surfaces		Every singular surface (in any characteristic) can be resolved; proved by various methods.
s_nash_problem	state	nash_problem		A question relating the arc space of a singularity to its resolution; still partially open.
s_resolution_in_positive_characteristic	state	resolution_in_positive_characteristic		Resolution of singularities for varieties in characteristic p > 0 remains open in dimensions ≥ 4.
t_induction_on_invariants	technique	induction_on_invariants		Hironaka's strategy of defining a numerical invariant of a singularity and showing blowups decrease it.
s_linear_system_of_divisor	state	linear_system_of_divisor		The space L(D) = {f : div(f) + D ≥ 0} of meromorphic functions with poles bounded by D, with dimension ℓ(D)
s_riemann_inequality	theorem	riemann_inequality		ℓ(D) ≥ deg(D) − g + 1 (Riemann's original result, before Roch completed it)
s_minor_closed_class	state	minor_closed_class		A class of graphs closed under taking minors; e.g., planar graphs, graphs embeddable on a fixed surface.
s_wagners_conjecture	state	wagners_conjecture		The Robertson-Seymour theorem was originally known as Wagner's conjecture.
s_kuratowski_wagner	theorem	kuratowski_wagner		A graph is planar if and only if it has no K₅ or K₃,₃ minor (Wagner) / subdivision (Kuratowski)
s_excluded_minor_characterization	state	excluded_minor_characterization		The finite list of forbidden minors characterizing a minor-closed class; existence guaranteed but may be hard to determine.
s_n_body_problem	state	n_body_problem		The generalization to n point masses under gravitational interaction.
s_poincare_nonintegrability	theorem	poincare_nonintegrability		Poincaré proved the three-body problem has no additional analytic first integrals beyond energy and momentum; it is not integrable.
s_sundmans_theorem	theorem	sundmans_theorem		The three-body problem can be solved by convergent power series in t^{1/3} (after regularizing collisions), but the series converge too slowly to be practical.
s_kam_theorem_2	theorem	kam_theorem		Kolmogorov-Arnold-Moser theorem: for small perturbations of integrable Hamiltonian systems, most invariant tori survive.
s_lagrange_points	state	lagrange_points		The five equilibrium configurations L₁–L₅ in the restricted three-body problem.
s_restricted_three_body_problem	state	restricted_three_body_problem		The case where one mass is negligible and moves in the gravitational field of two massive bodies.
t_perturbation_theory	technique	perturbation_theory		Expanding solutions in powers of a small parameter (mass ratio, eccentricity) around a known solution.
s_poincare_homoclinic_tangle	state	poincare_homoclinic_tangle		Transverse intersections of stable and unstable manifolds in the restricted three-body problem; the origin of chaos theory.
s_chaos_in_celestial_mechanics	state	chaos_in_celestial_mechanics		The sensitive dependence on initial conditions discovered by Poincaré in the three-body problem.
s_invariant_torus	state	invariant_torus		A torus in phase space preserved by the Hamiltonian flow, carrying quasi-periodic motion.
s_koebe_uniformization	theorem	koebe_uniformization		Koebe's proof of the uniformization theorem using normal families and the Dirichlet problem.
s_fuchsian_group	state	fuchsian_group		A discrete subgroup of PSL₂(ℝ) = Aut(𝔻); Riemann surfaces of genus ≥ 2 are quotients 𝔻/Γ.
s_weil_conjecture_rationality	theorem	weil_conjecture_rationality		Z(V,t) is a rational function of t; proved by Dwork (1960) using p-adic methods.
s_deligne_proof	theorem	deligne_proof		Deligne's proof of the Riemann hypothesis for varieties over finite fields, using étale cohomology and monodromy.
s_grothendieck_standard_conjectures	state	grothendieck_standard_conjectures		A set of conjectures on algebraic cycles that would give a more conceptual proof of the Weil conjectures; still mostly open.
s_l_adic_cohomology	state	l_adic_cohomology		ℓ-adic étale cohomology Hⁱ(V, ℚ_ℓ), providing the ℓ-adic Weil cohomology theory.
s_weak_mixing	axiom	Weak mixing		A measure-preserving system whose Cartesian square is ergodic, equivalent to having no non-trivial eigenvalues.
s_k_system	axiom	K-system (Kolmogorov system)		A measure-preserving system possessing a sub-σ-algebra whose forward iterates generate and backward iterates are trivial, implying mixing of all orders.
s_shannon_entropy_of_partition	state	Shannon entropy of a partition		The quantity H(α) = −Σ μ(Aᵢ) log μ(Aᵢ) measuring the information content of a measurable partition α.
s_pesin_stable_unstable_manifold_theorem	theorem	Pesin stable/unstable manifold theorem		For a C^{1+α} diffeomorphism preserving a hyperbolic measure, almost every point admits smooth local stable and unstable manifolds tangent to the Oseledets subs
s_arnold_diffusion	state	Arnold diffusion		The phenomenon in Hamiltonian systems with three or more degrees of freedom where orbits drift along resonances through gaps between KAM tori.
s_nekhoroshev_theorem	theorem	Nekhoroshev theorem		For a steep (or convex) nearly integrable Hamiltonian, action variables remain close to their initial values for exponentially long times in 1/ε.
s_pitchfork_bifurcation	state	Pitchfork bifurcation		A local bifurcation in which a symmetric equilibrium loses stability and two new equilibria appear (supercritical) or an unstable pair merges (subcritical).
t_birkhoff_normal_form	technique	Birkhoff normal form		A canonical near-identity transformation that reduces a Hamiltonian near an elliptic fixed point to a function of action variables alone up to a prescribed orde
s_rossler_attractor	state	Rössler attractor		A chaotic attractor of a three-dimensional ODE system with a single quadratic nonlinearity exhibiting a folded-band structure.
s_benedicks_carleson_theorem	theorem	Benedicks–Carleson theorem		For a positive-measure set of parameters near a = 2, the quadratic map x ↦ 1 − ax² has an absolutely continuous invariant measure and a positive Lyapunov expone
t_thermodynamic_formalism	technique	Thermodynamic formalism		A framework adapting statistical mechanics (pressure, partition functions, Gibbs measures) to study invariant measures and entropy of dynamical systems.
s_ruelle_perron_frobenius_theorem	theorem	Ruelle–Perron–Frobenius theorem		For an expanding or hyperbolic system with a Hölder potential, the transfer operator has a simple maximal eigenvalue whose eigenfunction determines the unique e
s_domain_of_holomorphy	axiom	Domain of holomorphy		An open set in ℂⁿ that is the maximal domain of existence of some holomorphic function, equivalently a pseudoconvex domain.
s_holomorphically_convex	axiom	Holomorphically convex domain		An open set Ω ⊂ ℂⁿ such that for every compact K ⊂ Ω the holomorphically convex hull K̂_Ω is again compact in Ω.
s_plurisubharmonic_function	axiom	Plurisubharmonic function		An upper semicontinuous function φ: Ω → [−∞,∞) whose restriction to every complex line is subharmonic, the fundamental convexity notion in several complex varia
s_levi_problem	theorem	Levi problem (solution)		A domain in ℂⁿ is a domain of holomorphy if and only if it is pseudoconvex, solved by Oka, Bremermann, and Norguet.
s_cousin_problem	state	Cousin problem		The additive (Cousin I) and multiplicative (Cousin II) problems of constructing global meromorphic functions from local data, generalizing Mittag-Leffler and We
s_oka_cartan_theorem	theorem	Oka–Cartan theorem		On a Stein manifold, every coherent analytic sheaf is acyclic and globally generated (Cartan's Theorems A and B for Stein manifolds).
s_grauert_direct_image_theorem	theorem	Grauert direct image theorem		The higher direct images of a coherent analytic sheaf under a proper holomorphic map are coherent.
s_remmert_proper_mapping_theorem	theorem	Remmert proper mapping theorem		The image of an analytic set under a proper holomorphic map is again an analytic set.
s_bergman_kernel	state	Bergman kernel		The reproducing kernel K(z,w) of the Hilbert space of square-integrable holomorphic functions on a bounded domain, encoding its biholomorphic geometry.
s_szego_kernel	state	Szegő kernel		The reproducing kernel of the Hardy space H²(∂Ω) on the boundary of a domain, projecting L² boundary data onto boundary values of holomorphic functions.
s_fefferman_mapping_theorem	theorem	Fefferman mapping theorem		A biholomorphism between smooth strictly pseudoconvex bounded domains in ℂⁿ extends to a smooth diffeomorphism of the closures.
s_tangential_cauchy_riemann_equation	axiom	Tangential Cauchy–Riemann equation		The system ∂̄_b u = f on a CR manifold, obtained by restricting the Cauchy–Riemann equations to the tangential directions of a real hypersurface.
s_baouendi_treves_approximation	theorem	Baouendi–Trèves approximation theorem		CR functions on a CR manifold can be locally uniformly approximated by restrictions of holomorphic functions from ambient space.
s_bochner_martinelli_kernel	state	Bochner–Martinelli kernel		The explicit integral kernel in ℂⁿ generalizing the Cauchy kernel, used to represent holomorphic functions and solve ∂̄ in convex domains.
t_henkin_ramirez_integral	technique	Henkin–Ramirez integral representation		An integral formula using holomorphic support functions that provides explicit solutions to ∂̄ with optimal Hölder estimates on strictly pseudoconvex domains.
s_koppelman_formula	theorem	Koppelman formula		An integral representation formula for (0,q)-forms in ℂⁿ that simultaneously generalizes Bochner–Martinelli and provides homotopy operators for ∂̄.
s_kohn_laplacian	state	Kohn Laplacian □_b		The second-order operator □_b = ∂̄_b ∂̄_b* + ∂̄_b* ∂̄_b on a CR manifold, whose regularity theory underlies solvability of the tangential Cauchy–Riemann equatio
s_edge_of_the_wedge_theorem	theorem	Edge-of-the-wedge theorem		If two holomorphic functions defined in wedge-shaped domains on opposite sides of a real hyperplane agree on the common edge, they extend to a single holomorphi
s_analytic_sheaf	axiom	Analytic sheaf		A sheaf of modules over the sheaf of germs of holomorphic functions on a complex analytic space, foundational to the Oka–Cartan theory.
s_pluripotential_theory	state	Pluripotential theory		The study of plurisubharmonic functions and the complex Monge–Ampère operator (dd^c)ⁿ as a nonlinear potential theory in several complex variables.
s_proper_holomorphic_mapping_theorem	theorem	Proper holomorphic mapping theorem		A proper holomorphic map between bounded domains is a finite branched covering, and between equidimensional balls it is an automorphism (Alexander's theorem).
s_oka_principle	theorem	Oka principle		On a Stein manifold, the inclusion of holomorphic sections into continuous sections of an elliptic submersion is a weak homotopy equivalence.
s_weierstrass_preparation_theorem_scv	theorem	Weierstrass preparation theorem (several variables)		A holomorphic function vanishing to order k in one variable can be uniquely factored near the origin as a unit times a Weierstrass polynomial of degree k.
s_fiber_product_of_schemes	state	Fiber product of schemes		The categorical pullback X ×_S Y in the category of schemes, representing the functor of pairs of compatible morphisms, fundamental to base change.
s_smooth_morphism	axiom	Smooth morphism		A morphism of schemes that is flat, locally of finite presentation, and has geometrically regular fibers, generalizing submersion to algebraic geometry.
s_verdier_duality	theorem	Verdier duality		A duality in the derived category of sheaves on a locally compact space giving an adjunction between f_! and f^! functors, generalizing Poincaré and Serre duali
s_perverse_sheaf	axiom	Perverse sheaf		An object of the heart of the perverse t-structure on the bounded derived category of constructible sheaves, satisfying support and cosupport conditions.
s_bbd_decomposition_theorem	theorem	BBD decomposition theorem		The direct image of an intersection cohomology complex under a proper map decomposes as a direct sum of shifted semisimple perverse sheaves (Beilinson–Bernstein
s_mixed_hodge_structure	axiom	Mixed Hodge structure		A finitely generated abelian group with an increasing weight filtration over ℚ and a decreasing Hodge filtration over ℂ satisfying compatibility conditions, pre
s_higher_algebraic_k_theory	state	Higher algebraic K-theory (Quillen)		The higher K-groups Kₙ(R) for n ≥ 0 defined via Quillen's Q-construction or +-construction on the classifying space of the category of projective modules.
s_flip	state	Flip (algebraic geometry)		A birational surgery in the minimal model program replacing a small extremal contraction by its flip, increasing the intersection with the canonical divisor.
s_abundance_conjecture	state	Abundance conjecture		On a minimal projective variety with nef canonical divisor, the canonical divisor is semiample, i.e., some multiple is base-point free.
s_zariski_decomposition	state	Zariski decomposition		A pseudo-effective divisor on a smooth surface decomposes uniquely as D = P + N where P is nef and N is an effective ℚ-divisor with negative-definite intersecti
s_bloch_higher_chow_group	state	Bloch higher Chow group		The group CH^p(X, n) defined as the homology of a complex of algebraic cycles on X × Δⁿ, providing a cycle-theoretic model for motivic cohomology.
s_quillen_q_construction	state	Quillen Q-construction		A categorical construction that produces the algebraic K-theory space BQC from an exact category C by adding extra morphisms encoding short exact sequences.
s_riemann_roch_chern_character	theorem	Riemann–Roch via Chern character		The Chern character ch: K₀(X) → CH*(X)_ℚ becomes a ring isomorphism after tensoring with ℚ, with the Todd class correction giving Grothendieck–Riemann–Roch.
s_rigid_analytic_space	axiom	Rigid analytic space		A space obtained by gluing affinoid algebras over a non-archimedean field with a Grothendieck topology, providing p-adic analytic geometry.
s_nakai_moishezon_criterion	theorem	Nakai–Moishezon criterion		A Cartier divisor D on a proper scheme is ample if and only if D^dim(V) · V > 0 for every positive-dimensional subvariety V.
s_quantum_group	state	Quantum group		A Hopf algebra deformation U_q(𝔤) of the universal enveloping algebra of a semisimple Lie algebra, introduced by Drinfeld and Jimbo.
s_gorenstein_ring	axiom	Gorenstein ring		A commutative Noetherian local ring of finite injective dimension, equivalently a Cohen–Macaulay ring whose canonical module is free of rank one.
s_local_cohomology	state	Local cohomology		The right derived functors H^i_I(M) of the I-torsion functor Γ_I on modules over a ring, measuring the failure of extending sections across the subscheme V(I).
s_associated_prime	axiom	Associated prime		A prime ideal p of a ring R is associated to an R-module M if p = Ann(m) for some m ∈ M; the set Ass(M) controls zero-divisors and supports primary decompositio
s_hilbert_samuel_multiplicity	state	Hilbert–Samuel multiplicity		The leading coefficient (times d!) of the Hilbert–Samuel polynomial measuring the asymptotic growth of lengths ℓ(M/I^n M), a fundamental multiplicity in local a
s_hilbert_polynomial	state	Hilbert polynomial		The polynomial P(n) agreeing with the Hilbert function dim_k(M_n) for large n, encoding the degree and dimension of a projective variety or graded module.
s_discrete_valuation_ring	axiom	Discrete valuation ring		A principal ideal domain with exactly one non-zero maximal ideal, equivalently a regular local ring of dimension one, fundamental in arithmetic geometry.
s_i_adic_completion	state	I-adic completion of a ring		The inverse limit R̂ = lim←(R/Iⁿ) completing a ring R with respect to an ideal I, producing a complete local ring when I is maximal.
s_global_class_field_theory	theorem	Global class field theory		The Artin map induces an isomorphism between the idèle class group modulo norms and the Galois group of the maximal abelian extension of a number field.
s_b_spline	state	B-spline		A piecewise polynomial basis function with minimal support defined by a recurrence on knot vectors, fundamental to computational approximation theory and CAGD.
t_calderon_zygmund_operator	technique	Calderón–Zygmund singular integral operator		A principal-value convolution operator with kernel satisfying size and smoothness conditions away from the diagonal, bounded on Lᵖ by the Calderón–Zygmund theor
s_wave_front_set	state	Wave front set		The subset WF(u) ⊂ T*X \ 0 of the cotangent bundle encoding both the singular support and the codirections of non-smoothness of a distribution u, central to mic
s_killing_form	state	Killing form		The symmetric bilinear form B(X,Y) = tr(ad(X) ad(Y)) on a Lie algebra, whose nondegeneracy characterizes semisimplicity.
a_abstract_root_system	axiom	Abstract root system		A finite set of vectors in a Euclidean space satisfying axioms of spanning, integrality, and reflection closure, abstracting root structure from Lie algebras.
s_classification_of_simple_lie_algebras	theorem	Classification of simple Lie algebras		Every simple complex Lie algebra is isomorphic to one of the classical series A_n, B_n, C_n, D_n or the exceptional algebras G_2, F_4, E_6, E_7, E_8.
s_weyl_dimension_formula	theorem	Weyl dimension formula		Expresses the dimension of an irreducible highest-weight representation as a product over positive roots of inner-product ratios.
s_cartan_decomposition	theorem	Cartan decomposition		Decomposes a semisimple Lie algebra g = k + p into eigenspaces of a Cartan involution, yielding the global decomposition G = K exp(p).
a_symmetric_space	axiom	Symmetric space		A Riemannian manifold where every point is an isolated fixed point of an involutive isometry, equivalently a quotient G/K of a Lie group by the fixed-point set 
s_cartan_classification_of_symmetric_spaces	theorem	Cartan classification of symmetric spaces		Classifies irreducible Riemannian symmetric spaces into types I-IV corresponding to compact/non-compact forms and their duality.
s_invariant_theory_hilbert_finiteness	theorem	Invariant theory (Hilbert finiteness)		Hilbert's theorem that the ring of polynomial invariants of a reductive group acting on a finite-dimensional vector space is finitely generated.
s_lie_group_exponential_map	state	Lie group exponential map		The map exp: g -> G from a Lie algebra to its Lie group defined by the time-one flow of left-invariant vector fields, providing a local diffeomorphism near the 
s_adjoint_representation	state	Adjoint representation		The representation Ad: G -> GL(g) of a Lie group on its Lie algebra by conjugation, with differential ad: g -> End(g) given by the Lie bracket.
s_harish_chandra_isomorphism	theorem	Harish-Chandra isomorphism		Identifies the center of the universal enveloping algebra Z(U(g)) with the Weyl-group invariants of the symmetric algebra S(h)^W via a shift by the half-sum of 
s_bruhat_decomposition	theorem	Bruhat decomposition		Decomposes a reductive group G into double cosets G = ∐_{w in W} BwB indexed by the Weyl group, stratifying the flag variety into Schubert cells.
s_r_matrix_yang_baxter	state	R-matrix (Yang-Baxter)		A solution R of the Yang-Baxter equation R₁₂R₁₃R₂₃ = R₂₃R₁₃R₁₂ in End(V⊗V), governing integrability of quantum systems and providing structure for quantum group
s_iwasawa_decomposition	theorem	Iwasawa decomposition		Decomposes a semisimple Lie group as G = KAN where K is maximal compact, A is abelian, and N is nilpotent, generalizing the QR decomposition.
a_quantum_group_drinfeld_jimbo	axiom	Quantum group (Drinfeld-Jimbo)		A one-parameter deformation U_q(g) of the universal enveloping algebra of a semisimple Lie algebra, forming a quasitriangular Hopf algebra.
s_demazure_character_formula	theorem	Demazure character formula		Expresses the character of a Demazure module (B-submodule of an irreducible G-module) via iterated applications of Demazure operators.
s_duflo_theorem	theorem	Duflo theorem		Establishes that the Duflo map, a symmetrization twisted by the square root of the Jacobian of the exponential, is an algebra isomorphism from S(g)^g to Z(U(g))
s_weyl_integration_formula	theorem	Weyl integration formula		Reduces integration of class functions on a compact Lie group to integration over a maximal torus weighted by the square of the Weyl denominator.
s_harish_chandra_character_formula	theorem	Harish-Chandra character formula		Gives an explicit formula for the distribution character of an irreducible admissible representation of a real reductive group in terms of orbital integrals.
s_plancherel_for_semisimple_groups	theorem	Plancherel theorem for semisimple groups		Decomposes L²(G) for a real semisimple Lie group into a direct integral of irreducible unitary representations with an explicit Plancherel measure.
s_langlands_classification	theorem	Langlands classification		Parametrizes irreducible admissible representations of a real reductive group by triples (M,sigma,nu) consisting of Levi subgroup, tempered representation, and 
s_steinberg_representation	state	Steinberg representation		The irreducible representation of a reductive group over a finite field whose dimension equals the largest power of p dividing |G(F_q)|, appearing in the cohomo
s_jackson_theorem	theorem	Jackson's theorem		Gives upper bounds on the best polynomial approximation error E_n(f) in terms of the modulus of continuity, establishing direct theorems in approximation theory
s_bernstein_inverse_theorem	theorem	Bernstein inverse theorem		The converse of Jackson's theorem: if E_n(f) = O(n^{-r-alpha}) then f has r continuous derivatives with the r-th derivative in Lip(alpha).
s_best_approximation_chebyshev	state	Best approximation (Chebyshev)		The element of a finite-dimensional subspace minimizing the supremum-norm distance to a given continuous function, whose existence is guaranteed by compactness.
s_chebyshev_equioscillation_theorem	theorem	Chebyshev equioscillation theorem		A polynomial of degree n is the best uniform approximation to a continuous function if and only if the error equioscillates at least n+2 times.
a_spline_function	axiom	Spline function		A piecewise polynomial of specified degree with prescribed smoothness conditions at the breakpoints (knots).
s_schoenberg_whitney_theorem	theorem	Schoenberg-Whitney theorem		B-spline interpolation at a set of data sites has a unique solution if and only if each B-spline is nonzero at at least one data site (the interlacing condition
s_scaling_equation	state	Scaling equation		The two-scale relation phi(x) = sum_k h_k phi(2x - k) satisfied by the scaling function in a multiresolution analysis, determining the associated wavelet.
s_calderon_zygmund_singular_integral_operator	state	Calderón-Zygmund singular integral operator		An L²-bounded operator whose distributional kernel satisfies standard size and regularity estimates away from the diagonal, generalizing the Hilbert transform.
t_calderon_zygmund_decomposition_technique	technique	Calderón-Zygmund decomposition (technique)		Decomposes an integrable function at height lambda into a bounded good part and a bad part concentrated on cubes where the average exceeds lambda.
s_parametrix	state	Parametrix		An approximate inverse of an elliptic (pseudo)differential operator, differing from the true inverse by a smoothing operator.
s_fourier_integral_operator	state	Fourier integral operator		A generalization of pseudodifferential operators defined by oscillatory integrals with a phase function parametrizing a Lagrangian submanifold, used to propagat
s_hormander_propagation_of_singularities	theorem	Hörmander propagation of singularities		For a principal-type operator P, the wave front set of a solution Pu = f propagates along the null bicharacteristic flow of the principal symbol.
a_besov_space	axiom	Besov space		The function space B^s_{p,q} defined by Littlewood-Paley decomposition with l^q summability of frequency-localized pieces, interpolating between Sobolev and Höl
s_mikhlin_multiplier_theorem	theorem	Mikhlin multiplier theorem		A Fourier multiplier m(xi) defines an L^p-bounded operator for 1 < p < infty if m satisfies |xi^alpha D^alpha m| <= C for |alpha| <= [n/2] + 1.
s_kolmogorov_superposition_theorem	theorem	Kolmogorov superposition theorem		Every continuous function of n variables can be represented as a finite superposition of continuous functions of one variable and addition.
s_shannon_sampling_theorem	theorem	Shannon sampling theorem		A bandlimited function with frequencies in [-W,W] is completely determined by and reconstructible from its samples taken at rate 2W.
t_kotelnikov_shannon_whittaker_interpolation	technique	Kotelnikov-Shannon-Whittaker interpolation		Reconstructs a bandlimited function from uniform samples using the sinc interpolation formula f(t) = sum_n f(n/2W) sinc(2Wt - n).
t_symbol_calculus	technique	Symbol calculus		The algebraic framework for computing compositions, adjoints, and parametrices of pseudodifferential operators via asymptotic expansion of symbols.
s_leray_spectral_sequence	theorem	Leray spectral sequence		A spectral sequence E_2^{p,q} = H^p(B, R^q f_* F) converging to H^{p+q}(X, F) for a continuous map f: X -> B, relating sheaf cohomology of total and base spaces
s_grothendieck_spectral_sequence	theorem	Grothendieck spectral sequence		For composable left-exact functors F, G with G sending injectives to F-acyclics, a spectral sequence E_2^{p,q} = R^p F(R^q G(-)) converging to R^{p+q}(F ∘ G)(-)
s_riemann_hilbert_correspondence	theorem	Riemann-Hilbert correspondence		An equivalence between the category of regular holonomic D-modules and the category of perverse sheaves (constructible sheaves), generalizing the monodromy corr
s_wall_groups	state	Wall groups L_n(π)		Algebraic L-groups of a group ring that classify surgery obstructions for manifolds with fundamental group π.
t_surgery_exact_sequence	technique	Surgery exact sequence		Long exact sequence relating the structure set of a manifold to normal invariants and surgery obstruction groups.
s_surgery_obstruction	state	Surgery obstruction		Element in a Wall L-group that vanishes if and only if a normal map is normally cobordant to a homotopy equivalence.
s_normal_invariant	state	Normal invariant		Element of [X, G/O] classifying lifts of the Spivak normal fibration to a vector bundle, the first step in the surgery program.
s_poincare_duality_complex	state	Poincaré duality complex		Finite CW complex satisfying Poincaré duality with a fundamental class, serving as the homotopy-theoretic model of a closed manifold.
s_spivak_normal_fibration	state	Spivak normal fibration		Stable spherical fibration over a Poincaré duality complex that plays the role of the stable normal bundle for manifolds.
t_assembly_map	technique	Assembly map		Map from the homology of the classifying space with L-theory coefficients to the L-groups, whose injectivity is the Novikov conjecture.
s_novikov_conjecture	axiom	Novikov conjecture		Conjecture that higher signatures of a closed oriented manifold are homotopy invariants, equivalent to rational injectivity of the assembly map.
s_borel_conjecture	axiom	Borel conjecture		Conjecture that every homotopy equivalence between closed aspherical manifolds is homotopic to a homeomorphism.
s_oriented_cobordism_ring	state	Oriented cobordism ring Ω^SO_*		Graded ring of cobordism classes of oriented manifolds, computed by Thom, Wall, and others as a polynomial ring over ℤ tensored with 2-torsion.
s_complex_cobordism_ring	state	Complex cobordism ring MU_*		Coefficient ring of complex cobordism, isomorphic to the Lazard ring ℤ[x_1, x_2, …] classifying formal group laws.
s_thom_cobordism_theorem	theorem	Thom's cobordism theorem		Identifies unoriented cobordism groups π_*(MO) with homology of BO, proving Ω^O_n ≅ H_n(BO; ℤ/2) and computing the unoriented cobordism ring.
s_formal_group_law_of_MU	state	Formal group law of MU		Universal formal group law arising from complex cobordism via the first Chern class of the tensor product of complex line bundles.
s_quillen_theorem_MU_lazard	theorem	Quillen's theorem (MU_* ≅ Lazard ring)		The coefficient ring of complex cobordism MU is canonically isomorphic to the Lazard ring classifying one-dimensional commutative formal group laws.
t_landweber_exact_functor	technique	Landweber exact functor theorem		Produces a cohomology theory from an MU_*-module by tensoring with MU_*(X), provided the sequence (p, v_1, v_2, …) acts regularly.
s_adams_e_invariant	state	Adams e-invariant		Invariant e: π_{2k-1}(S^0) → ℚ/ℤ detecting the image of the J-homomorphism via Chern character calculations in K-theory.
s_chromatic_homotopy_theory	state	Chromatic homotopy theory		Framework organizing stable homotopy theory by height of formal group laws, decomposing the stable category into chromatic layers via Morava K-theories.
s_morava_k_theory	state	Morava K-theory K(n)		Family of generalized cohomology theories K(n) at a prime p, with coefficient ring 𝔽_p[v_n, v_n⁻¹], detecting height-n periodic phenomena in stable homotopy.
s_nilpotence_theorem	theorem	Nilpotence theorem (Devinatz–Hopkins–Smith)		A self-map of a finite spectrum is nilpotent if and only if it induces zero on all Morava K-theories, establishing chromatic detection.
s_periodicity_theorem	theorem	Periodicity theorem (Hopkins–Smith)		Every type-n finite p-local spectrum admits a v_n-self-map, producing periodic families in stable homotopy groups.
s_thick_subcategory_theorem	theorem	Thick subcategory theorem (Hopkins–Smith)		The thick subcategories of finite p-local spectra form a chain C_0 ⊃ C_1 ⊃ ⋯ indexed by chromatic height.
s_smash_product_theorem	theorem	Smash product theorem (Hopkins–Ravenel)		The smash product of L_n-local spectra is L_n-local, establishing good monoidal properties for chromatic localization.
s_segal_conjecture	theorem	Segal conjecture (Carlsson's theorem)		The Burnside ring of a finite group G maps isomorphically (after completion) to the zeroth stable cohomotopy of BG.
s_eta_invariant_ems	state	Atiyah–Patodi–Singer eta invariant		Spectral invariant η(D) = Σ sign(λ)|λ|^{-s}|_{s=0} measuring spectral asymmetry of a self-adjoint elliptic operator on a closed manifold.
s_aps_index_theorem	theorem	Atiyah–Patodi–Singer index theorem		Index formula for elliptic operators on manifolds with boundary, involving the eta invariant as a boundary correction to the Atiyah–Singer integrand.
s_families_index_theorem_ems	theorem	Families index theorem		Atiyah–Singer index theorem for families of elliptic operators parametrized by a base space, giving an element in the K-theory of the parameter space.
s_chern_character	state	Chern character		Ring homomorphism ch: K^0(X) → H^{even}(X; ℚ) that becomes an isomorphism after tensoring with ℚ, computed via Chern classes.
s_l_class	state	Hirzebruch L-class		Characteristic class L(p_1, p_2, …) of a real vector bundle appearing in the Hirzebruch signature theorem: σ(M) = ⟨L(M), [M]⟩.
s_a_hat_genus	state	Â-genus		Multiplicative genus associated to the power series x/2 / sinh(x/2), equal to the index of the Dirac operator on a spin manifold by Atiyah–Singer.
s_associated_bundle	state	Associated bundle		Fiber bundle P ×_G F obtained from a principal G-bundle P by replacing the fiber with a G-space F via the diagonal action.
t_milnor_construction	technique	Milnor's construction of classifying spaces		Constructs the classifying space BG as the infinite join G * G * ⋯ , providing a contractible free G-space.
t_clutching_construction	technique	Clutching construction		Builds a bundle over a sphere by gluing trivial bundles over hemispheres using a transition function on the equator.
s_homflypt_polynomial	state	HOMFLY-PT polynomial		Two-variable knot invariant P(l,m) generalizing both the Jones and Alexander polynomials, defined by a skein relation.
s_vassiliev_invariants	state	Vassiliev (finite-type) invariants		Knot invariants vanishing on singular knots with more than n double points, forming a filtered space whose graded quotients are chord diagram functionals.
t_kontsevich_integral	technique	Kontsevich integral		Universal Vassiliev invariant given by an iterated integral formula that dominates all finite-type knot invariants.
s_kauffman_bracket	state	Kauffman bracket		State-sum polynomial ⟨D⟩ of unoriented link diagrams defined by local smoothing rules, related to the Jones polynomial by a writhe normalization.
s_alexander_polynomial	state	Alexander polynomial		Classical knot invariant Δ_K(t) computed from the infinite cyclic cover of the knot complement, equal to the order of the Alexander module.
s_seifert_surface	state	Seifert surface		Compact oriented surface embedded in S^3 whose boundary is a given knot or link, used to compute the Alexander polynomial via the Seifert matrix.
s_khovanov_homology	state	Khovanov homology		Bigraded homology theory categorifying the Jones polynomial, whose graded Euler characteristic recovers the Jones polynomial.
s_eight_thurston_geometries	state	Eight Thurston geometries		The eight maximal homogeneous Riemannian 3-dimensional geometries (E^3, S^3, H^3, S^2×ℝ, H^2×ℝ, Nil, Sol, SL₂(ℝ)̃) underlying Thurston's geometrization.
s_kneser_milnor_theorem	theorem	Kneser–Milnor decomposition		Every closed oriented 3-manifold decomposes uniquely as a connected sum of prime 3-manifolds.
s_mapping_class_group	state	Mapping class group		Group Mod(Σ) = π_0(Diff^+(Σ)) of isotopy classes of orientation-preserving diffeomorphisms of a surface Σ.
s_dehn_twist	state	Dehn twist		Diffeomorphism of a surface obtained by cutting along a simple closed curve, twisting one side by 2π, and regluing; generates the mapping class group.
t_kirby_calculus	technique	Kirby calculus		Combinatorial method for studying 3- and 4-manifolds through handle decompositions, using Kirby moves on framed link diagrams.
s_cuntz_algebra	state	Cuntz algebra O_n		Simple purely infinite C*-algebra generated by n isometries with orthogonal ranges summing to the identity, with K_0(O_n) ≅ ℤ/(n-1).
s_toeplitz_algebra	state	Toeplitz algebra		C*-algebra generated by the unilateral shift on ℓ^2(ℕ), fitting in a short exact sequence 0 → K → T → C(S^1) → 0.
s_irrational_rotation_algebra	state	Irrational rotation algebra A_θ		C*-algebra generated by two unitaries U, V with VU = e^{2πiθ}UV, the prototypical noncommutative torus with K_0 ≅ ℤ^2.
s_modular_automorphism_group	state	Modular automorphism group		One-parameter group σ^φ_t of *-automorphisms of a von Neumann algebra canonically associated to a faithful normal state φ by Tomita–Takesaki theory.
s_kirchberg_phillips_theorem	theorem	Kirchberg–Phillips theorem		Purely infinite simple separable nuclear C*-algebras satisfying the UCT are classified up to isomorphism by their K-theory.
s_spectral_triple	state	Spectral triple (Connes)		Triple (A, H, D) of a *-algebra, Hilbert space, and self-adjoint operator with compact resolvent encoding noncommutative Riemannian geometry.
s_cyclic_cohomology	state	Cyclic cohomology		Cohomology theory HC^*(A) for associative algebras introduced by Connes, serving as the noncommutative analogue of de Rham cohomology.
s_connes_chern_character	state	Connes–Chern character		Map from K-theory to cyclic cohomology that extends the classical Chern character to the noncommutative setting.
s_hochschild_cohomology	state	Hochschild cohomology		Cohomology HH^*(A, M) of an associative algebra A with coefficients in a bimodule M, related to cyclic cohomology by the SBI exact sequence.
s_noncommutative_torus	state	Noncommutative torus		Smooth noncommutative manifold (A_θ^∞, H, D) obtained by equipping the irrational rotation algebra with a canonical spectral triple.
s_connes_index_foliations	theorem	Connes index theorem for foliations		Index theorem for longitudinal elliptic operators along the leaves of a foliation, taking values in the K-theory of the foliation C*-algebra.
s_kasparov_kk_theory	state	Kasparov KK-theory		Bivariant K-theory KK(A, B) for C*-algebras unifying K-theory and K-homology, with a composition product generalizing index pairings.
t_kasparov_product	technique	Kasparov product		Associative composition product in KK-theory that encodes index pairings, boundary maps, and functorial operations on C*-algebras.
s_uct_for_kk	theorem	Universal coefficient theorem for KK-theory		Short exact sequence 0 → Ext(K_*(A), K_*(B)) → KK(A,B) → Hom(K_*(A), K_*(B)) → 0 for A in the UCT class.
s_baum_connes_conjecture	axiom	Baum–Connes conjecture		Conjecture that the assembly map μ: K^G_*(EG) → K_*(C*_r(G)) from equivariant K-homology to C*-algebra K-theory is an isomorphism.
s_planar_algebra	state	Planar algebra		Algebraic structure introduced by Jones axiomatizing the standard invariant of a subfactor via planar tangles and their composition.
s_free_independence	axiom	Free independence		Noncommutative analogue of classical independence where alternating centered products vanish, the foundational axiom of free probability.
s_r_transform	state	R-transform (Voiculescu)		Analytic transform linearizing free additive convolution: R_{a+b} = R_a + R_b for freely independent a, b in a noncommutative probability space.
s_free_entropy	state	Free entropy (Voiculescu)		Noncommutative analogue χ(X_1,…,X_n) of Shannon entropy for tuples of self-adjoint operators, defined via microstate approximation by random matrices.
s_s_transform	state	S-transform (Voiculescu)		Analytic transform linearizing free multiplicative convolution: S_{ab} = S_a · S_b for freely independent positive a, b.
s_kpz_equation	state	KPZ equation		Stochastic PDE ∂_t h = ν∂²_x h + λ(∂_x h)² + ξ for random interface growth, with characteristic 1:2:3 scaling exponents.
s_kpz_universality	state	KPZ universality class		Universality class of random growth models in 1+1 dimensions sharing Tracy–Widom fluctuations and 1:2:3 scaling exponents.
s_stochastic_heat_equation	state	Stochastic heat equation		Linear SPDE ∂_t u = Δu + σ(u)ξ driven by space-time white noise, whose solution via Itô integration is the Hopf–Cole transform of KPZ.
s_hairer_regularity_structures	state	Hairer's regularity structures		Framework for giving meaning to singular stochastic PDEs by replacing Taylor polynomials with model-dependent local approximations and renormalization.
s_paracontrolled_distributions	state	Paracontrolled distributions		Approach by Gubinelli–Imkeller–Perkowski to singular SPDEs using Bony paraproduct decomposition to define products of distributions.
s_random_matrix_theory	state	Random matrix theory		Study of eigenvalue distributions of large random matrices, connecting probability, mathematical physics, and number theory via universality phenomena.
s_sine_kernel_universality	theorem	Sine kernel universality (bulk)		Universal local eigenvalue statistics in the bulk of the spectrum of Wigner matrices, with correlation kernel converging to sin(π(x-y))/(π(x-y)).
s_dyson_brownian_motion	state	Dyson Brownian motion		Stochastic process of eigenvalues of a matrix-valued Ornstein–Uhlenbeck process, with repulsive log-gas interaction between particles.
s_rsw_estimates	theorem	Russo–Seymour–Welsh estimates		Scale-invariant lower bounds on crossing probabilities for rectangles in critical percolation, key to proving conformal invariance.
s_smirnov_percolation_theorem	theorem	Smirnov's theorem (critical site percolation)		Conformal invariance of crossing probabilities for critical site percolation on the triangular lattice, proved via discrete holomorphicity.
s_contact_process	state	Contact process		Interacting particle system modeling infection spread where occupied sites infect neighbors at rate λ and recover at rate 1, exhibiting a phase transition.
s_voter_model	state	Voter model		Interacting particle system where each site adopts the opinion of a random neighbor at rate 1, with clustering behavior depending on spatial dimension.
s_exclusion_process	state	Exclusion process		Interacting particle system where particles perform random walks subject to the exclusion rule of at most one particle per site.
s_hydrodynamic_limit	state	Hydrodynamic limit		Scaling limit of interacting particle systems where the empirical density profile converges to the solution of a macroscopic PDE.
s_liggett_ergodic_theorem	theorem	Liggett ergodic theorem for IPS		Characterization of the ergodic behavior of attractive interacting particle systems via comparison with product measures.
s_malliavin_hormander_proof	theorem	Malliavin's probabilistic proof of Hörmander's theorem		Proof that hypoellipticity of Hörmander-type operators follows from the non-degeneracy of the Malliavin covariance matrix of the associated SDE.
s_rough_path_theory	state	Rough path theory (Lyons)		Extension of classical integration to paths of low regularity by enriching paths with iterated integral data (signature), enabling pathwise stochastic calculus.
t_lyons_universal_limit_theorem	theorem	Lyons universal limit theorem		Continuity of the Itô map from rough paths to solution paths in p-variation topology, providing pathwise stability for SDEs.
t_lyons_extension_theorem	theorem	Lyons extension theorem		A rough path defined up to level ⌊p⌋ uniquely determines the full signature and hence all rough integrals against it.
t_gubinelli_controlled_rough_paths	technique	Gubinelli controlled rough paths		Simplified approach to rough integration where integrands controlled by a reference rough path can be integrated via a sewing-type argument.
t_sewing_lemma	technique	Sewing lemma		Key technical lemma in rough path theory providing existence and uniqueness of a limit for approximately additive functionals on partitions.
s_sle_phases	state	SLE phases (κ ≤ 4, 4 < κ < 8, κ ≥ 8)		Phase diagram of SLE_κ: simple curves for κ ≤ 4, self-touching curves for 4 < κ < 8, and space-filling curves for κ ≥ 8.
s_sle_cft_correspondence	state	SLE/CFT correspondence		Identification of SLE_κ partition functions with conformal field theory correlators at central charge c = (6-κ)(3κ-8)/(2κ).
s_yang_mills_connection	state	Yang–Mills connection		Connection on a principal G-bundle that is a critical point of the Yang–Mills functional ∫|F_A|^2, generalizing Maxwell's equations to nonabelian gauge groups.
s_bpst_instanton	state	BPST instanton		Explicit self-dual SU(2) connection on S^4 with instanton number 1, discovered by Belavin–Polyakov–Schwarz–Tyupkin, minimizing Yang–Mills energy in its topologi
s_uhlenbeck_removable_singularity	theorem	Uhlenbeck removable singularity theorem		A Yang–Mills connection on a punctured 4-ball with finite energy extends smoothly across the puncture after a gauge transformation.
s_chern_simons_functional	state	Chern–Simons functional		Secondary characteristic class functional CS(A) = ∫ tr(A∧dA + ⅔A∧A∧A) on connections over a 3-manifold, whose critical points are flat connections.
s_sugawara_construction	state	Sugawara construction		Realization of the Virasoro algebra as quadratic expressions in affine Lie algebra currents, yielding the energy-momentum tensor for WZW models.
s_modular_functor	state	Modular functor		Functor from the category of surfaces with boundary to vector spaces, satisfying gluing axioms, encoding the algebraic structure of a 2D conformal field theory.
s_kdv_soliton	state	KdV soliton		Solitary wave solution u(x,t) = -c/2 · sech²(√c/2 · (x - ct)) of the KdV equation, preserving shape after interaction with other solitons.
s_zakharov_shabat_system	state	Zakharov–Shabat system		Generalized eigenvalue problem (Lax pair) for the AKNS hierarchy, extending inverse scattering to the nonlinear Schrödinger and sine-Gordon equations.
s_hitchin_system	state	Hitchin system		Algebraically completely integrable Hamiltonian system on the moduli space of Higgs bundles over a Riemann surface, with spectral curve fibration.
s_kp_hierarchy	state	KP hierarchy		Infinite system of integrable PDEs in 2+1 dimensions generalizing KdV, with solutions parametrized by points of the Sato Grassmannian.
s_drinfeld_jimbo_quantum_group	state	Drinfeld–Jimbo quantum group U_q(𝔤)		Hopf algebra deformation of the universal enveloping algebra of a semisimple Lie algebra, with generators and q-deformed Serre relations.
s_quantum_r_matrix	state	Quantum R-matrix		Solution R ∈ End(V⊗V) of the quantum Yang–Baxter equation providing braiding in the category of quantum group representations.
t_reshetikhin_turaev_invariants	technique	Reshetikhin–Turaev invariants		Topological invariants of 3-manifolds constructed from surgery presentations using the representation theory of quantum groups.
s_atiyah_tqft_axioms	axiom	Atiyah's TQFT axioms		Axioms defining a topological quantum field theory as a symmetric monoidal functor from the cobordism category to the category of vector spaces.
s_witten_chern_simons_tqft	state	Witten–Chern–Simons TQFT		Three-dimensional TQFT arising from Chern–Simons gauge theory with compact gauge group, producing the Jones polynomial and its generalizations.
s_turaev_viro_invariant	state	Turaev–Viro invariant		State-sum invariant of 3-manifolds defined from a triangulation using quantum 6j-symbols, equivalent to the square modulus of the Reshetikhin–Turaev invariant.
s_dijkgraaf_witten_invariant	state	Dijkgraaf–Witten invariant		Finite-group TQFT invariant of 3-manifolds obtained by summing a cocycle weight over gauge equivalence classes of principal G-bundles.
s_ising_model	state	Ising model		Statistical mechanics lattice model with spins σ_i = ±1 interacting via nearest-neighbor coupling H = -J Σ σ_iσ_j, exhibiting a ferromagnetic phase transition.
s_gopakumar_vafa_invariants	state	Gopakumar–Vafa invariants		Integer-valued curve-counting invariants of a Calabi–Yau threefold, conjecturally equivalent to Gromov–Witten invariants via a BPS state reorganization.
s_donaldson_thomas_invariants	state	Donaldson–Thomas invariants		Virtual counts of ideal sheaves on a Calabi–Yau threefold, related to Gromov–Witten invariants by the MNOP conjecture.
s_supermanifold	state	Supermanifold		Ringed space locally modeled on (ℝ^m|n, C^∞ ⊗ ∧ℝ^n) with commuting and anticommuting coordinates, the geometric setting of supersymmetry.
s_super_lie_algebra	state	Super Lie algebra		ℤ/2-graded algebra with a bracket satisfying graded antisymmetry and the graded Jacobi identity, generalizing Lie algebras to include fermionic generators.
s_spectral_zeta_function	state	Spectral zeta function		Zeta function ζ_D(s) = Σ λ_n^{-s} formed from eigenvalues of an elliptic operator, whose special values encode spectral invariants via zeta regularization.
s_heat_kernel_expansion	state	Heat kernel asymptotic expansion		Small-time expansion tr(e^{-tΔ}) ~ (4πt)^{-d/2} Σ a_k t^k as t→0, with coefficients a_k encoding local geometric invariants of the manifold.
s_spectral_invariants	state	Spectral invariants		Geometric quantities determined by the spectrum of the Laplacian, including dimension, volume, and total scalar curvature, via the heat kernel expansion.
s_scattering_matrix	state	Scattering matrix (PDE)		Unitary operator S = W_+^* W_- relating incoming and outgoing asymptotic states in quantum scattering theory via the Møller wave operators.
s_moller_wave_operators	state	Møller wave operators		Strong limits W_± = s-lim_{t→±∞} e^{iHt} e^{-iH_0t} intertwining the free and full Hamiltonians in scattering theory.
s_asymptotic_completeness	theorem	Asymptotic completeness		Statement that the ranges of the Møller wave operators span the absolutely continuous subspace of the full Hamiltonian, so all scattering states scatter.
s_limiting_absorption_principle	theorem	Limiting absorption principle		Boundary values of the resolvent (H - λ ± iε)^{-1} exist as bounded operators between weighted spaces as ε → 0, yielding unique outgoing solutions.
s_mourre_estimate	theorem	Mourre estimate		Positive commutator estimate E_Δ i[H,A] E_Δ ≥ θ E_Δ implying the limiting absorption principle and absence of singular continuous spectrum on Δ.
t_compensated_compactness	technique	Compensated compactness		Method of Murat–Tartar showing that certain nonlinear expressions pass to the weak limit when the sequences satisfy differential constraints.
s_div_curl_lemma	theorem	Div-curl lemma		If E_n ⇀ E with curl E_n bounded and B_n ⇀ B with div B_n bounded, then E_n · B_n ⇀ E · B in the sense of distributions.
s_crandall_ishii_lemma	theorem	Crandall–Ishii lemma		Maximum principle for semicontinuous functions providing second-order jet estimates at points where a difference touches its maximum, fundamental for viscosity 
s_stefan_problem	state	Stefan problem		Free boundary problem modeling phase transitions (melting/solidification) where the interface velocity equals the jump in heat flux across the phase boundary.
s_caffarelli_free_boundary_regularity	theorem	Caffarelli free boundary regularity		Regularity theory for the free boundary in the obstacle problem: the free boundary is C^{1,α} near regular points where the contact set has positive density.
s_alt_caffarelli_friedman	theorem	Alt–Caffarelli–Friedman monotonicity formula		Monotonicity formula for pairs of nonnegative subharmonic functions with disjoint supports, a key tool for free boundary regularity in two-phase problems.
s_homogenization_theorem	theorem	Homogenization theorem (elliptic)		Solutions u^ε of -div(A(x/ε)∇u^ε) = f converge as ε → 0 to the solution of -div(A^{hom}∇u) = f with constant effective coefficients A^{hom}.
s_cell_problem	state	Cell problem (homogenization)		Periodic boundary value problem on the unit cell whose solution determines the effective homogenized coefficients A^{hom} via averaging.
s_two_scale_convergence	state	Two-scale convergence		Convergence notion where u^ε(x) ⇀ u_0(x,y) in the sense of testing against oscillatory functions φ(x, x/ε), capturing both macroscopic and microscopic behavior.
s_gamma_convergence	state	Γ-convergence		Variational convergence of functionals ensuring that minimizers converge to minimizers of the limit functional, fundamental in homogenization and phase field mo
s_matroid_rank_function	state	Matroid rank function		Function r: 2^E → ℤ satisfying r(∅) = 0, monotonicity, and submodularity, providing a cryptomorphic axiomatization of matroids.
s_tutte_polynomial	state	Tutte polynomial		Two-variable polynomial T_M(x,y) of a matroid M encoding deletion-contraction invariants, specializing to chromatic, flow, and reliability polynomials.
s_matroid_union_theorem	theorem	Matroid union theorem		The union of k matroids on E is again a matroid, with rank function r(A) = min_{A₁∪⋯∪A_k=A} Σ r_i(A_i).
s_rados_theorem	theorem	Rado's theorem (matroid transversals)		A family of sets (A_i) admits a system of distinct representatives belonging to an independent set of a matroid if and only if the rank condition r(∪_{i∈J} A_i)
s_matroid_duality	state	Matroid duality		Every matroid M on ground set E has a dual matroid M* whose bases are complements of bases of M, satisfying (M*)* = M.
s_mds_codes	state	Maximum distance separable (MDS) codes		Linear codes meeting the Singleton bound d = n - k + 1 with equality, including Reed–Solomon codes as the primary example.
s_bch_code	state	BCH code		Cyclic error-correcting code defined by roots that are consecutive powers of a primitive element, with designed distance guaranteeing error-correction capabilit
s_turbo_ldpc_codes	state	Turbo and LDPC codes		Capacity-approaching codes using iterative decoding: turbo codes via parallel concatenation with interleaving, LDPC codes via sparse parity-check matrices.
s_elliptic_curve_cryptography	state	Elliptic curve cryptography		Public-key cryptography based on the discrete logarithm problem in the group of rational points on an elliptic curve over a finite field.
s_lattice_based_cryptography	state	Lattice-based cryptography		Cryptographic constructions whose security relies on the hardness of lattice problems such as LWE and SVP, conjectured to be post-quantum secure.
s_lwe	state	Learning with errors (LWE)		Computational problem of distinguishing noisy linear equations mod q from uniform, forming the basis of lattice-based cryptographic schemes.
s_bibd	state	Balanced incomplete block design (BIBD)		Collection of b blocks of size k from v points such that every pair of points appears in exactly λ blocks, satisfying bk = vr and r(k-1) = λ(v-1).
s_fisher_inequality	theorem	Fisher's inequality		In a non-trivial BIBD, the number of blocks b is at least the number of points v, proved using the rank of the incidence matrix.
s_latin_square	state	Latin square		An n × n array filled with n different symbols, each occurring exactly once in each row and column, equivalent to the Cayley table of a quasigroup.
s_witt_design	state	Witt design		The unique 5-(24,8,1) Steiner system S(5,8,24) associated to the Mathieu group M₂₄, with remarkable combinatorial and group-theoretic properties.
s_graphon	state	Graphon		Measurable symmetric function W: [0,1]² → [0,1] arising as the limit of dense graph sequences in the cut metric, the basic object of graph limit theory.
s_lovasz_szegedy_theorem	theorem	Lovász–Szegedy theorem		A sequence of dense graphs converges in the cut metric if and only if all subgraph densities converge, and the limit is a graphon.
s_density_hales_jewett	theorem	Density Hales–Jewett theorem		For any δ > 0 and alphabet size k, sufficiently high-dimensional combinatorial subspaces of [k]^n must contain a combinatorial line in any set of density δ.
s_intersection_cohomology	state	Intersection cohomology		Cohomology theory IH^*(X) for singular spaces introduced by Goresky–MacPherson, restoring Poincaré duality by imposing allowability conditions on chains.
s_bbdg_decomposition_theorem	theorem	BBD decomposition theorem		For a proper morphism f: X → Y of varieties, the direct image Rf_*IC_X decomposes as a direct sum of shifted simple perverse sheaves on Y.
s_d_module	state	D-module		Module over the sheaf of algebraic differential operators on a variety, providing an algebraic framework for systems of linear PDEs and the Riemann–Hilbert corr
s_beilinson_bernstein_localization	theorem	Beilinson–Bernstein localization		Equivalence between the category of D-modules on the flag variety G/B and modules over U(𝔤) with a fixed central character, algebraizing representation theory.
s_springer_correspondence	state	Springer correspondence		Bijection between irreducible representations of the Weyl group and pairs (nilpotent orbit, local system) via the cohomology of Springer fibers.
s_geometric_satake	theorem	Geometric Satake equivalence		Equivalence between the category of perverse sheaves on the affine Grassmannian of G and the representation category of the Langlands dual group G^∨.
s_weyl_module	state	Weyl module		Universal highest weight module V(λ) for a reductive algebraic group in positive characteristic, the characteristic-p analogue of a finite-dimensional simple mo
s_lusztig_conjecture	axiom	Lusztig conjecture		Conjecture expressing characters of simple modules for reductive groups in characteristic p in terms of Kazhdan–Lusztig polynomials, proved for large p.
s_steinberg_tensor_product	theorem	Steinberg tensor product theorem		Decomposes simple modules for reductive groups in characteristic p as tensor products of Frobenius twists of restricted simple modules.
s_jantzen_filtration	state	Jantzen filtration		Canonical filtration of a Weyl module by submodules with character sum given by an alternating sum formula involving Kazhdan–Lusztig polynomials.
s_langlands_dual_group	state	Langlands dual group G^∨		Reductive group whose root datum is dual to that of G (interchanging roots and coroots), fundamental to the Langlands correspondence.
s_langlands_l_function	state	Langlands L-function		L-function L(s, π, r) attached to an automorphic representation π and a representation r of the L-group, generalizing Hecke and Artin L-functions.
s_arthur_endoscopic_classification	theorem	Arthur's endoscopic classification		Classification of automorphic representations of classical groups via Arthur parameters and endoscopic transfer from GL(N).
s_fundamental_lemma_ngo	theorem	Fundamental lemma (Ngô)		Identity of orbital integrals for a reductive group and its endoscopic groups, proved by Ngô using the geometry of the Hitchin fibration.
s_geometric_langlands	axiom	Geometric Langlands conjecture		Conjectural equivalence between D-modules on Bun_G and quasi-coherent sheaves on Loc_{G^∨} for a smooth projective curve, geometrizing the Langlands program.
s_hecke_eigensheaf	state	Hecke eigensheaf		D-module on Bun_G that is an eigenvector for all Hecke operators, the geometric analogue of a Hecke eigenform in the geometric Langlands program.
s_reynolds_operator	state	Reynolds operator		Projection R: k[V] → k[V]^G onto the ring of invariants by averaging over the group action, existing when the group is linearly reductive.
s_molien_formula	theorem	Molien's formula		Hilbert series of the ring of invariants k[V]^G equals 1/|G| Σ_{g∈G} 1/det(I - tg), computing the dimensions of graded components.
s_chevalley_shephard_todd	theorem	Chevalley–Shephard–Todd theorem		The ring of invariants k[V]^G is a polynomial ring if and only if G is generated by pseudo-reflections.
s_hilbert_mumford_criterion	theorem	Hilbert–Mumford criterion		A point is GIT-(semi)stable if and only if it is (semi)stable for all one-parameter subgroups, reducing stability to a numerical check on weights.
s_first_fundamental_theorem	theorem	First fundamental theorem of invariant theory		For the classical groups (GL, O, Sp), the ring of polynomial invariants on V^⊕n is generated by specific contractions (inner products, determinants).
s_second_fundamental_theorem	theorem	Second fundamental theorem of invariant theory		Describes the ideal of relations among the generators given by the first fundamental theorem for the classical groups.
s_pontryagin_thom_framed	state	Pontryagin–Thom (framed cobordism)		Bijection between framed cobordism classes of framed submanifolds in S^{n+k} and π_{n+k}(S^n), the original formulation of the Pontryagin–Thom construction.
s_cobordism_hypothesis	axiom	Cobordism hypothesis (Baez–Dolan)		A fully extended n-dimensional TQFT is determined by its value on a point, which must be a fully dualizable object in the target (∞,n)-category.
s_nuclear_c_star_algebra	state	Nuclear C*-algebra		C*-algebra A for which every C*-tensor product with another C*-algebra is unique, equivalently admitting a completely positive approximation by finite-rank maps
s_exact_c_star_algebra	state	Exact C*-algebra		C*-algebra A such that the reduced cross product by A preserves short exact sequences of C*-algebras, equivalently embeddable in a nuclear C*-algebra.
s_formal_language	axiom	Formal language		A precisely specified system of signs, assemblies (terms and relations), and syntactic rules for constructing well-formed expressions.
s_quantifier	axiom	Quantifier (universal, existential)		The operators for-all and exists that bind a variable in a relation to express universal or existential properties.
s_axiom_scheme_of_substitution	axiom	Axiom scheme of substitution (logic)		The logical scheme permitting replacement of a free variable by a term in any well-formed relation, subject to no variable capture.
t_modus_ponens	technique	Modus ponens		The fundamental rule of inference: from A and A => B, deduce B.
s_deductive_system	axiom	Deductive system		The formal framework of proofs as finite sequences of relations, each an axiom or derived by rules of inference; a theorem is the final relation.
s_hilbert_epsilon_operator	axiom	Hilbert epsilon operator (tau-operator)		A term-forming operator tau_x(R) selecting an object satisfying relation R if one exists, used to eliminate quantifiers and formalize choice at the logical leve
s_axiom_of_the_empty_set	axiom	Axiom of the empty set		There exists a set having no elements, denoted empty-set.
s_axiom_of_pairing	axiom	Axiom of pairing		For any two objects a, b, there exists a set {a, b} whose only elements are a and b.
s_axiom_of_separation	axiom	Axiom scheme of separation (comprehension)		For any set A and property P(x), there exists the set {x in A : P(x)} of exactly those elements of A satisfying P.
s_axiom_of_union	axiom	Axiom of union		For any set of sets F, there exists a set whose elements are exactly those belonging to at least one member of F.
s_set_membership	axiom	Set membership (element-of relation)		The primitive binary relation x in A asserting that x is an element of the set A.
s_subset_inclusion	axiom	Subset (inclusion)		A subset-of B means every element of A is an element of B, defining the inclusion partial order on sets.
s_set_intersection	state	Set intersection		A cap B = {x : x in A and x in B}, the set of elements common to both A and B.
s_set_union	state	Set union (binary)		A cup B = {x : x in A or x in B}, the set of elements belonging to at least one of A, B.
s_set_complement	state	Set complement (difference)		A \ B = {x in A : x not in B}, the set of elements in A but not in B.
s_symmetric_difference	state	Symmetric difference		A triangle B = (A \ B) u (B \ A), the set of elements belonging to exactly one of A, B.
s_ordered_pair	axiom	Ordered pair		The object (a, b) defined via Kuratowski encoding such that (a,b) = (c,d) iff a = c and b = d.
s_cartesian_product	state	Cartesian product		A x B = {(a, b) : a in A, b in B}, the set of all ordered pairs with first component in A and second in B.
s_relation	axiom	Relation (graph of a relation)		A subset R of A x B; one writes xRy to mean (x, y) in R.
s_function	axiom	Function (map, mapping)		A relation f subset A x B such that for every a in A there exists exactly one b in B with (a, b) in f, written f : A -> B.
s_injection	axiom	Injection (one-to-one map)		A function f : A -> B such that f(a1) = f(a2) implies a1 = a2.
s_surjection	axiom	Surjection (onto map)		A function f : A -> B such that for every b in B there exists a in A with f(a) = b.
s_bijection	axiom	Bijection		A function that is both injective and surjective, establishing a one-to-one correspondence.
s_composition_of_functions	state	Composition of functions		Given f : A -> B and g : B -> C, the composite g o f : A -> C defined by (g o f)(a) = g(f(a)).
s_inverse_function	state	Inverse function		For a bijection f : A -> B, the unique function f^{-1} : B -> A satisfying f^{-1} o f = id_A and f o f^{-1} = id_B.
s_image_and_preimage	state	Image and preimage		For f : A -> B, the direct image f(S) = {f(x) : x in S} and the inverse image f^{-1}(T) = {x in A : f(x) in T}.
s_restriction_of_function	state	Restriction and extension of a function		The restriction f|_S of f : A -> B to S subset A; an extension of g : S -> B is any f : A -> B with f|_S = g.
s_family_of_sets	axiom	Family of sets (indexed family)		A function i -> A_i from an index set I to a collection of sets, written (A_i)_{i in I}.
s_generalized_union_intersection	state	Generalized union and intersection		For a family (A_i)_{i in I}, the union U_{i in I} A_i and intersection cap_{i in I} A_i, generalizing binary operations to arbitrary index sets.
s_generalized_cartesian_product	state	Generalized Cartesian product (product of a family)		Product_{i in I} A_i, the set of all choice functions on the family, each selecting an element from A_i for every i.
s_equivalence_relation	axiom	Equivalence relation		A relation R on a set A that is reflexive, symmetric, and transitive.
s_equivalence_class	state	Equivalence class		For an equivalence relation R on A, the class [a] = {x in A : xRa}.
s_quotient_set	state	Quotient set		A/R = {[a] : a in A}, the set of all equivalence classes, with canonical surjection pi : A -> A/R.
s_canonical_decomposition_of_map	theorem	Canonical decomposition of a map		Every function f : A -> B factors uniquely as A -> A/~ -> f(A) -> B (surjection, bijection, inclusion).
s_partition_of_a_set	state	Partition of a set		A family of pairwise disjoint nonempty subsets whose union is A, in canonical bijection with equivalence relations on A.
s_diagonal_of_a_product	state	Diagonal of a product		Delta_A = {(a, a) : a in A} subset A x A, the graph of the identity function.
s_projection_map	state	Projection map		The canonical surjections pr_1 : A x B -> A and pr_2 : A x B -> B extracting components of an ordered pair.
s_preorder	axiom	Preorder (quasi-order)		A reflexive and transitive relation, not necessarily antisymmetric; the quotient by the associated equivalence is a partial order.
s_strict_partial_order	state	Strict partial order		The irreflexive and transitive relation < derived from <= by x < y iff x <= y and x != y.
s_greatest_least_element	state	Greatest element and least element		In a poset, a greatest (resp. least) element a satisfies x <= a (resp. a <= x) for all x.
s_supremum_infimum	state	Supremum and infimum		sup S is the least upper bound of S; inf S is the greatest lower bound of S, when they exist.
s_lattice	axiom	Lattice		A poset in which every pair of elements has both a supremum (join) and an infimum (meet).
s_complete_lattice	axiom	Complete lattice		A poset in which every subset has a supremum and an infimum.
s_directed_set	axiom	Directed set		A nonempty preordered set in which every finite subset has an upper bound.
s_order_preserving_map	axiom	Increasing (order-preserving) map		A function f between posets such that x <= y implies f(x) <= f(y).
s_order_isomorphism	axiom	Order isomorphism		A bijection f between posets such that both f and f^{-1} are order-preserving.
s_chain	state	Chain (totally ordered subset)		A subset of a poset that is totally ordered under the induced order.
s_well_ordered_set	axiom	Well-ordered set		A totally ordered set in which every nonempty subset has a least element.
s_hausdorff_maximality_principle	theorem	Hausdorff maximality principle		Every partially ordered set contains a maximal chain; equivalent to axiom of choice.
s_initial_segment	state	Initial segment (section)		In a well-ordered set W, the set S_a = {x in W : x < a} of all elements strictly preceding a.
s_comparability_well_ordered	theorem	Comparability of well-ordered sets		Any two well-ordered sets are comparable: one is order-isomorphic to an initial segment of the other.
s_cantor_theorem_power_set	theorem	Cantor's theorem		For any set A, there is no surjection from A onto P(A); equivalently |A| < |P(A)|.
s_equipotence	state	Equipotence (equinumerosity)		Two sets are equipotent if there exists a bijection between them.
s_finite_set	axiom	Finite set		A set equipotent to some natural number {0, 1, ..., n-1}.
s_countable_set	axiom	Countable set (denumerable)		A set equipotent to a subset of the natural numbers; countably infinite if equipotent to N.
s_disjoint_union	state	Disjoint union (coproduct of sets)		A disjoint-union B = (A x {0}) u (B x {1}), a union keeping copies distinct even when they overlap.
s_absorption_infinite_cardinals	theorem	Absorption law for infinite cardinals		For an infinite cardinal kappa, kappa + kappa = kappa * kappa = kappa.
t_mathematical_induction	technique	Mathematical induction		If P(0) holds and P(n) => P(n+1), then P(n) holds for all natural numbers n.
s_arithmetic_of_naturals	state	Arithmetic of natural numbers		Addition and multiplication on N defined by recursion, satisfying commutativity, associativity, distributivity.
s_order_on_naturals	state	Order on natural numbers		The total order on N where m <= n iff m subset n (as sets), making (N, <=) a well-ordered set.
s_species_of_structures	axiom	Species of structures		Bourbaki's formal specification: base sets, auxiliary sets, typification via echelon construction, and an axiom invariant under transport.
s_echelon_construction_type	state	Echelon construction type		A set obtained from base sets by iterated application of power set and Cartesian product, providing the type of a structure.
s_isomorphism_of_structures	axiom	Isomorphism of structures		A bijection between base sets of two structures of the same species that transports one structure to the other.
t_transport_of_structures	technique	Transport of structures		Transferring a structure from one set to another via a bijection, preserving all structural axioms.
s_induced_structure	state	Induced structure (substructure)		The structure inherited by a subset of the base set, obtained by restricting the structural data.
s_product_structure	state	Product structure		The structure on a Cartesian product obtained by combining the structures on each factor componentwise.
s_quotient_structure	state	Quotient structure		The structure induced on a quotient set A/R when the equivalence relation R is compatible with the given structure.
s_initial_structure	state	Initial structure		The coarsest structure on a set making a given family of maps into structured sets into morphisms.
s_final_structure	state	Final structure		The finest structure on a set making a given family of maps from structured sets into morphisms.
s_morphism_of_structures	axiom	Morphism (of a species of structures)		A map between base sets that respects the structural data; the precise definition depends on the species.
s_neighborhood_filter	state	Neighborhood filter		The filter V(x) of all neighborhoods of a point x in a topological space, whose properties characterize the topology.
s_filter	axiom	Filter		A nonempty collection of subsets of a set X, closed under finite intersection and supersets, and not containing the empty set.
s_filter_base	axiom	Filter base (basis of a filter)		A nonempty collection of nonempty subsets such that any two members contain a common refinement; it generates a filter.
s_convergence_of_filter	axiom	Convergence of a filter		A filter F on a topological space converges to x if F is finer than the neighborhood filter of x, i.e., V(x) subset F.
s_cluster_point_of_filter	state	Cluster point of a filter		A point x such that every neighborhood of x meets every set in the filter.
s_continuous_map	axiom	Continuous map		A function f : X -> Y between topological spaces such that the preimage of every open set in Y is open in X.
s_open_map	axiom	Open map and closed map		A map sending open (resp. closed) sets to open (resp. closed) sets.
s_subspace_topology	axiom	Subspace topology (induced topology)		The topology on a subset A of X whose open sets are intersections of open sets of X with A.
s_initial_topology	axiom	Initial topology		The coarsest topology on X making a given family of maps f_i : X -> Y_i continuous, with subbase {f_i^{-1}(U_i)}.
s_final_topology	axiom	Final topology		The finest topology on Y making a given family of maps g_i : X_i -> Y continuous.
s_hausdorff_space	axiom	Hausdorff space (T2)		A topological space in which any two distinct points have disjoint neighborhoods.
s_t1_space	axiom	T1 space		A topological space in which every singleton {x} is closed.
s_regular_space	axiom	Regular space (T3)		A T1 space in which each point and disjoint closed set can be separated by disjoint open sets.
s_completely_regular_space	axiom	Completely regular space (Tychonoff, T3.5)		A T1 space in which points and disjoint closed sets can be separated by continuous real-valued functions.
s_limit_of_function	axiom	Limit of a function		f(x) -> l as x -> a along a filter means the image filter converges to l, unifying all notions of limit.
s_net	axiom	Net (Moore-Smith sequence)		A function from a directed set into a topological space, generalizing sequences.
s_connected_space	axiom	Connected space		A topological space that cannot be expressed as the union of two disjoint nonempty open sets.
s_connected_component	state	Connected component		The maximal connected subset containing a given point; connected components partition the space into closed sets.
s_locally_connected_space	axiom	Locally connected space		A space in which every point has a neighborhood base consisting of connected open sets.
s_second_countable_space	axiom	Second-countable space		A topological space possessing a countable base for its topology.
s_first_countable_space	axiom	First-countable space		A topological space in which every point has a countable neighborhood base.
s_separable_space	axiom	Separable space		A topological space containing a countable dense subset.
s_uniform_space	axiom	Uniform space (uniformity)		A set X equipped with a filter on X x X (entourages) containing the diagonal, symmetric, and satisfying the triangle condition.
s_entourage	axiom	Entourage (surrounding)		A member V of the uniformity; V[x] = {y : (x,y) in V} is the V-neighborhood of x.
s_uniform_topology	state	Uniform topology		The topology on a uniform space where U is open iff for each x in U there exists entourage V with V[x] subset U.
s_uniformly_continuous_map	axiom	Uniformly continuous map		A function f between uniform spaces such that for every entourage W of Y, (f x f)^{-1}(W) is an entourage of X.
s_cauchy_filter	axiom	Cauchy filter		A filter on a uniform space such that for every entourage V, there exists F in the filter with F x F subset V.
s_complete_uniform_space	axiom	Complete uniform space		A uniform space in which every Cauchy filter converges.
s_completion_of_uniform_space	state	Completion of a uniform space		A complete Hausdorff uniform space X-hat with uniformly continuous dense embedding X -> X-hat, unique up to uniform isomorphism.
s_uniform_isomorphism	axiom	Uniform isomorphism		A bijection f between uniform spaces such that both f and f^{-1} are uniformly continuous.
s_product_uniformity	state	Product uniformity		The coarsest uniformity on the product making all projections uniformly continuous.
s_induced_uniformity	state	Induced uniformity (subspace uniformity)		The uniformity on a subset A induced by restricting entourages to A x A.
s_minimal_cauchy_filter	state	Minimal Cauchy filter		A Cauchy filter minimal among all Cauchy filters, used in the construction of completions.
s_left_uniform_structure_topgrp	state	Left uniform structure on a topological group		The uniformity on G with entourage base {(x,y) : x^{-1}y in V} where V ranges over identity neighborhoods.
s_subgroup_topology	state	Subgroup topology		The subspace topology on a subgroup H of a topological group G, making H a topological group.
s_quotient_group_topology	state	Quotient group topology		The quotient topology on G/H for a normal subgroup H, making G/H a topological group and projection open.
s_closure_of_subgroup	theorem	Closure of a subgroup		The closure of a subgroup in a topological group is again a subgroup; the closure of a normal subgroup is normal.
s_neighborhood_base_identity	state	Neighborhood base at the identity		A topological group's topology is completely determined by a neighborhood base at identity e, since translation is a homeomorphism.
s_open_subgroup_theorem	theorem	Open subgroup theorem		Every open subgroup of a topological group is also closed; every subgroup containing an open subgroup is open.
s_completion_of_topological_group	state	Completion of a topological group		The Hausdorff completion of a topological group carries a canonical topological group structure.
s_completeness_of_R	axiom	Completeness of R (least upper bound property)		Every nonempty subset of R bounded above has a least upper bound (supremum) in R.
s_archimedean_property	theorem	Archimedean property		For any real x > 0 and y, there exists natural n with nx > y; equivalently, Z is unbounded in R.
s_density_of_Q_in_R	theorem	Density of Q in R		Between any two distinct real numbers there exists a rational number; Q is dense in R.
s_order_topology_on_R	state	Order topology on R		The topology on R generated by open intervals (a, b), coinciding with the metric topology from |x - y|.
s_extended_real_line	state	Extended real line		R-bar = R u {-inf, +inf}, a compact totally ordered space extending R with two points at infinity.
s_absolute_value_on_R	state	Absolute value (on R)		The function |x| = max(x, -x) on R, defining the standard metric d(x,y) = |x - y|.
s_monotone_convergence_sequences	theorem	Monotone convergence for sequences in R		Every bounded monotone sequence in R converges to its supremum (if increasing) or infimum (if decreasing).
s_one_sided_limits	state	Right and left limits		For a real function, f(a+) = lim_{x->a, x>a} f(x) and f(a-) = lim_{x->a, x<a} f(x).
s_power_function	state	Power function		x^alpha = exp(alpha ln x) for x > 0, extending integer powers to arbitrary real exponents.
s_euclidean_topology_Rn	axiom	Euclidean topology on R^n		The product topology on R^n, equivalently the metric topology from the Euclidean norm.
s_normed_vector_space	axiom	Normed vector space (over R)		A real vector space equipped with a norm satisfying positive definiteness, homogeneity, and the triangle inequality.
s_metric_uniformity	state	Metric uniformity		The uniformity on a metric space with entourage base {(x,y) : d(x,y) < epsilon} for epsilon > 0.
s_equivalent_norms_theorem	theorem	Equivalent norms on finite-dimensional spaces		On a finite-dimensional real vector space, all norms are equivalent, inducing the same topology.
s_projective_limit	state	Projective limit (inverse limit)		For a directed inverse system with connecting morphisms, the subspace of the product consisting of compatible threads.
s_inductive_limit	state	Inductive limit (direct limit)		For a directed system with connecting morphisms, the quotient of the disjoint union identifying elements related by morphisms.
s_solenoid	state	Solenoid		The projective limit of the inverse system ... -> S^1 ->^{xn} S^1, a compact connected abelian group not locally connected.
s_profinite_group	state	Profinite group		A topological group that is a projective limit of finite discrete groups; equivalently, compact totally disconnected Hausdorff.
s_quasi_compact_space	axiom	Quasi-compact space		A topological space (not necessarily Hausdorff) in which every open cover has a finite subcover; Bourbaki reserves 'compact' for Hausdorff.
s_finite_intersection_property	theorem	Finite intersection property characterization of compactness		A space is compact iff every collection of closed sets with the finite intersection property has nonempty intersection.
s_continuous_image_compact	theorem	Continuous image of compact is compact		The continuous image of a compact space is compact.
s_closed_subset_compact	theorem	Closed subsets of compact spaces are compact		Every closed subset of a compact space is compact; every compact subset of a Hausdorff space is closed.
s_compact_hausdorff_normal	theorem	Compact Hausdorff implies normal		Every compact Hausdorff space is normal (T4).
s_compact_to_hausdorff_homeomorphism	theorem	Continuous bijection from compact to Hausdorff is homeomorphism		A continuous bijection from a compact space to a Hausdorff space is automatically a homeomorphism.
s_proper_map	axiom	Proper map		A continuous map f : X -> Y such that preimages of compact sets are compact (equivalently, f x id_Z is closed for every Z).
s_locally_finite_family	axiom	Locally finite family		A family of subsets such that every point has a neighborhood meeting only finitely many members.
s_refinement_of_cover	axiom	Refinement of a cover		A cover {V_j} refines {U_i} if every V_j is contained in some U_i.
s_metrizable_paracompact	theorem	Paracompactness of metrizable spaces (Stone's theorem)		Every metrizable space is paracompact; every open cover has a sigma-discrete open refinement.
s_nowhere_dense_set	axiom	Nowhere dense set		A set whose closure has empty interior.
s_meager_set	state	Meager set (first category)		A countable union of nowhere dense sets; by Baire's theorem, a complete metric space is not meager in itself.
s_semicontinuity_liminf_limsup	theorem	Characterization of semicontinuity via liminf/limsup		f is lower semicontinuous at x iff f(x) <= lim inf_{y->x} f(y); upper iff f(x) >= lim sup_{y->x} f(y).
s_sup_lsc_functions	theorem	Supremum of lower semicontinuous functions		The pointwise supremum of any family of lower semicontinuous functions is lower semicontinuous.
s_paracompact_partition_unity	theorem	Paracompactness and partitions of unity		A Hausdorff space is paracompact iff every open cover admits a subordinate partition of unity.
s_lindelof_space	axiom	Lindelof space		A topological space in which every open cover has a countable subcover.
s_second_countable_lindelof	theorem	Second-countable implies Lindelof		Every second-countable space is Lindelof, separable, and hereditarily Lindelof.
s_compact_open_topology	axiom	Compact-open topology		The topology on C(X,Y) generated by subbasic sets {f : f(K) subset U} where K compact and U open.
s_topology_uniform_convergence	axiom	Topology of uniform convergence		The topology on function spaces defined by the uniformity: f and g are V-close if (f(x),g(x)) in V for all x.
s_topology_compact_convergence	state	Topology of compact convergence		The topology on C(X,Y) defined by uniform convergence on each compact subset of X.
s_pointwise_convergence_topology	state	Pointwise convergence topology		The topology on Y^X induced by the product topology; convergence means convergence at each point.
s_evaluation_map	state	Evaluation map		The map ev : C(X,Y) x X -> Y defined by ev(f,x) = f(x); continuous when C(X,Y) has the compact-open topology and X is locally compact.
s_exponential_law_function_spaces	theorem	Exponential law for function spaces		For locally compact Hausdorff X, C(X x Y, Z) -> C(Y, C(X,Z)) by currying is a homeomorphism under compact-open topology.
s_uniform_limit_continuity	theorem	Uniform limit of continuous functions is continuous		C(X,Y) is closed in Y^X under the topology of uniform convergence when Y is Hausdorff.
s_completeness_function_spaces	theorem	Completeness of function spaces		If Y is a complete uniform space, the space of bounded uniformly continuous functions X -> Y is complete under uniform convergence.
s_filter_generated_by_base	state	Filter generated by a base		The unique smallest filter containing a given filter base, consisting of all supersets of members of the base.
s_finer_coarser_topologies	axiom	Finer and coarser topologies		Topology T1 is finer than T2 if T2 subset T1; the identity (X,T1) -> (X,T2) is then continuous.
s_discrete_indiscrete_topology	state	Discrete and indiscrete topologies		The finest topology (every subset open) and the coarsest topology ({empty, X} only) on a set X.
s_clopen_set	state	Clopen set		A set that is simultaneously open and closed; a space is connected iff its only clopen sets are empty and X.
s_sigma_compact_space	axiom	Sigma-compact space		A topological space that is a countable union of compact subsets.
s_cauchy_sequence	state	Cauchy sequence		A sequence (x_n) in a metric/uniform space such that for every entourage V, there exists N with (x_m, x_n) in V for all m, n >= N.
s_pseudometric_uniformity	state	Pseudometric uniformity		A pseudometric (allowing d(x,y)=0 for x != y) generates a uniformity; the associated topology need not be Hausdorff.
s_continuity_via_filters	theorem	Continuity via filters		A map f : X -> Y is continuous at x iff the image filter of the neighborhood filter of x converges to f(x).
s_universal_property_completion	theorem	Universal property of completion		Every uniformly continuous map from X to a complete Hausdorff uniform space extends uniquely through X -> X-hat.
s_compact_complete_totally_bounded	theorem	Compactness equals complete and totally bounded		A uniform space is compact iff it is complete and totally bounded.
s_lebesgue_covering_lemma	theorem	Lebesgue covering lemma		For every open cover of a compact metric space, there exists delta > 0 such that every subset of diameter < delta lies in some member.
s_topological_embedding	axiom	Topological embedding		An injective continuous map f : X -> Y that is a homeomorphism onto its image f(X) with the subspace topology.
s_monoid	axiom	Monoid		A set equipped with an associative binary operation possessing an identity element.
s_group	axiom	Group		A monoid in which every element is invertible.
s_characteristic_of_a_ring	state	Characteristic of a ring		The nonnegative generator of the kernel of the unique ring homomorphism Z -> A.
s_opposite_ring	state	Opposite ring		For a ring A, the ring A^op with the same additive group but reversed multiplication a *' b = ba.
s_left_right_module	axiom	Left module / right module		A module where scalars act on the left (resp. right), with axiom (ab)x = a(bx) for left modules and x(ab) = (xa)b for right modules.
s_basis_of_a_module	axiom	Basis of a module		A family of elements such that every element of the module can be written uniquely as a finite linear combination of them.
s_rank_of_a_linear_map	state	Rank of a linear map		The dimension of the image of a linear map between vector spaces.
s_direct_sum_decomposition	state	Direct sum decomposition		A module M is a direct sum of submodules M_1,...,M_n if the canonical map M_1 + ... + M_n -> M is an isomorphism.
s_noetherian_module	axiom	Noetherian module		A module satisfying the ascending chain condition on submodules, equivalently every submodule is finitely generated.
s_artinian_module	axiom	Artinian module		A module satisfying the descending chain condition on submodules.
s_annihilator	state	Annihilator		For a submodule N of M, the ideal Ann(N) = {a in A : aN = 0}; dually for an ideal I, Ann_M(I) = {x in M : Ix = 0}.
s_multilinear_map	axiom	Multilinear map		A map f: M_1 x ... x M_n -> N that is linear in each variable when the others are held fixed.
s_right_exactness_of_tensor_product	theorem	Right exactness of tensor product		The functor M tensor_A (-) is right exact: it preserves cokernels and direct sums but not injections in general.
s_tensor_algebra	state	Tensor algebra		The graded algebra T(M) = direct_sum_{n>=0} M^{tensor n} with concatenation multiplication, universal for linear maps from M to associative algebras.
s_exterior_product	state	Exterior product (wedge product)		The product in the exterior algebra, denoted x wedge y, satisfying x wedge x = 0 and x wedge y = -y wedge x.
s_exterior_power	state	Exterior power		The p-th component Lambda^p(M) of the exterior algebra, the quotient of M^{tensor p} by tensors with a repeated factor.
s_symmetric_algebra	state	Symmetric algebra		The quotient S(M) = T(M)/<x tensor y - y tensor x>, a graded commutative algebra universal for linear maps from M to commutative algebras.
s_symmetric_power	state	Symmetric power		The p-th component S^p(M) of the symmetric algebra, the quotient of M^{tensor p} by differences of permuted tensors.
s_graded_algebra	axiom	Graded algebra		An algebra A with a direct sum decomposition A = direct_sum_{n>=0} A_n such that A_m * A_n subset A_{m+n}.
t_extension_of_scalars	technique	Extension of scalars		Constructs a B-module from an A-module M and a ring homomorphism A -> B via the tensor product M tensor_A B.
s_polynomial_ring_several_variables	state	Polynomial ring in several variables		The ring A[X_1,...,X_n] defined inductively as A[X_1,...,X_{n-1}][X_n], or as formal sums over multi-indices.
s_euclidean_division_polynomials	theorem	Euclidean division for polynomials		For f, g in K[X] with g nonzero and K a field, there exist unique q, r with f = qg + r and deg(r) < deg(g).
s_root_of_a_polynomial	axiom	Root of a polynomial		An element alpha in an extension of the coefficient ring such that f(alpha) = 0.
s_factor_theorem	theorem	Factor theorem		An element alpha is a root of f in K[X] if and only if (X - alpha) divides f.
s_irreducible_polynomial	axiom	Irreducible polynomial		A nonconstant polynomial that cannot be written as a product of two polynomials of lower degree.
s_unique_factorization_in_kx	theorem	Unique factorization in K[X]		The polynomial ring K[X] over a field K is a UFD: every nonconstant polynomial factors uniquely into irreducibles up to units.
s_gcd_of_polynomials	state	Greatest common divisor of polynomials		The monic polynomial of greatest degree dividing two given polynomials, computable by the Euclidean algorithm.
t_partial_fraction_decomposition	technique	Partial fraction decomposition		Expresses a rational fraction as a polynomial plus a sum of terms with prime-power denominators via the Chinese remainder theorem.
s_algebraic_element	axiom	Algebraic element		An element alpha in L that is a root of some nonzero polynomial in K[X].
s_transcendental_element	axiom	Transcendental element		An element alpha in L that is not algebraic over K.
s_simple_algebraic_extension	state	Simple algebraic extension		An extension K(alpha)/K generated by a single algebraic element alpha, isomorphic to K[X]/(minimal polynomial of alpha).
s_tower_law	theorem	Tower law (multiplicativity of degrees)		If K subset L subset M are field extensions, then [M:K] = [M:L]*[L:K].
s_existence_uniqueness_algebraic_closure	theorem	Existence and uniqueness of algebraic closure		Every field admits an algebraic closure, and any two algebraic closures are isomorphic (though not canonically).
s_separable_polynomial	axiom	Separable polynomial		A polynomial whose roots in an algebraic closure are all distinct, equivalently gcd(f, f') = 1.
s_inseparable_extension	axiom	Inseparable extension / purely inseparable extension		An algebraic extension that is not separable; purely inseparable if every element has minimal polynomial X^{p^n} - a in characteristic p.
s_perfect_field	axiom	Perfect field		A field of characteristic 0, or of characteristic p in which the Frobenius endomorphism x -> x^p is surjective.
s_normal_extension	axiom	Normal extension		An algebraic extension L/K such that every irreducible polynomial in K[X] with a root in L splits completely in L.
s_transcendence_basis	state	Transcendence basis		A maximal algebraically independent subset of L over K, such that L is algebraic over K(B).
s_norm_and_trace_of_field_extension	state	Norm and trace of a field extension		For a finite extension L/K and alpha in L, N_{L/K}(alpha) = det(mult. by alpha) and Tr_{L/K}(alpha) = trace(mult. by alpha) as K-linear endomorphisms.
s_artin_theorem_on_characters	theorem	Artin's theorem on characters		Distinct characters (homomorphisms) from a group to the multiplicative group of a field are linearly independent over that field.
s_totally_ordered_group	axiom	Totally ordered group		A group equipped with a total order compatible with the group operation: x <= y implies xz <= yz and zx <= zy.
s_archimedean_ordered_group	axiom	Archimedean ordered group		A totally ordered group in which for any positive a, b there exists n in Z with na > b, equivalently embeddable in (R,+).
s_krull_valuation	axiom	Valuation (Krull valuation)		A map v: K* -> Gamma to a totally ordered abelian group satisfying v(xy) = v(x) + v(y) and v(x+y) >= min(v(x),v(y)).
s_absolute_value_on_a_field	axiom	Absolute value on a field		A map |*|: K -> R_{>=0} satisfying |x| = 0 iff x = 0, |xy| = |x||y|, and |x+y| <= |x|+|y|.
s_convex_subgroup	axiom	Convex subgroup		A subgroup H of a totally ordered group G such that if a <= x <= b with a,b in H then x in H.
s_elementary_divisors	state	Elementary divisors		The prime power factors in the primary decomposition of a finitely generated module over a PID, determining its isomorphism class.
s_characteristic_polynomial	state	Characteristic polynomial		For a matrix A or endomorphism u, the polynomial chi(X) = det(XI - A) whose roots are the eigenvalues with algebraic multiplicities.
s_minimal_polynomial_of_endomorphism	state	Minimal polynomial of an endomorphism		The monic generator of the annihilator ideal of the K[X]-module defined by the endomorphism, dividing the characteristic polynomial.
s_torsion_submodule	state	Torsion submodule		For a module M over an integral domain A, the submodule Tors(M) = {m in M : am = 0 for some nonzero a in A}.
s_torsion_free_module	axiom	Torsion-free module		A module M over an integral domain such that Tors(M) = 0, i.e., am = 0 with a nonzero implies m = 0.
s_semisimple_module	axiom	Semisimple module		A module that is a direct sum of simple modules, equivalently every submodule is a direct summand.
s_semisimple_ring	axiom	Semisimple ring		A ring that is semisimple as a module over itself, equivalently an Artinian ring with zero Jacobson radical.
s_simple_ring	axiom	Simple ring		A ring with no two-sided ideals other than 0 and itself (and with R^2 nonzero).
s_symmetric_bilinear_form	axiom	Symmetric bilinear form		A bilinear form Phi with Phi(x,y) = Phi(y,x) for all x,y.
s_alternating_bilinear_form	axiom	Alternating bilinear form		A bilinear form Phi with Phi(x,x) = 0 for all x, hence Phi(x,y) = -Phi(y,x).
s_sesquilinear_form	axiom	Sesquilinear form		A map Phi: M x M -> K that is linear in one variable and conjugate-linear with respect to an involution in the other.
s_hermitian_form	axiom	Hermitian form		A sesquilinear form Phi satisfying Phi(y,x) = sigma(Phi(x,y)) where sigma is the given involution.
s_non_degenerate_form	axiom	Non-degenerate form		A bilinear or quadratic form for which the canonical map M -> M* is injective, equivalently the radical is zero.
s_radical_of_a_form	state	Radical of a form		The submodule rad(Phi) = {x in M : Phi(x,y) = 0 for all y in M}, the kernel of the map M -> M* induced by the form.
s_orthogonal_complement	state	Orthogonal complement		For a subset S of M with bilinear form, the submodule S^perp = {y in M : Phi(x,y) = 0 for all x in S}.
s_witt_theorem	theorem	Witt's theorem		An isometry between non-degenerate subspaces of a regular quadratic space extends to an isometry of the whole space.
s_witt_group	state	Witt group		The group W(K) of isometry classes of non-degenerate quadratic forms over K modulo hyperbolic forms, with orthogonal sum.
s_hyperbolic_plane_quadratic	state	Hyperbolic plane (quadratic forms)		A non-degenerate quadratic module of dimension 2 with basis {e,f} satisfying q(e) = q(f) = 0 and Phi(e,f) = 1.
s_symplectic_group	state	Symplectic group		The group Sp(M,Phi) of linear maps preserving a non-degenerate alternating bilinear form.
s_spin_group	state	Spin group		The double cover Spin(n) of SO(n) constructed inside the even part of the Clifford algebra, kernel of the spinor norm on the special Clifford group.
s_spinor_norm	state	Spinor norm		A homomorphism from the orthogonal group to K*/K*^2 defined via the Clifford algebra, measuring the signed length of reflections composing an isometry.
s_associative_algebra	axiom	Associative algebra		A ring A that is also a module over a commutative ring K, with K acting centrally: k(ab) = (ka)b = a(kb).
s_lie_subalgebra	axiom	Lie subalgebra		A submodule of a Lie algebra closed under the Lie bracket.
s_ideal_of_lie_algebra	axiom	Ideal of a Lie algebra		A Lie subalgebra h of g such that [g,h] subset h.
s_commutator_lie_algebra	state	Commutator Lie algebra		For an associative algebra A, the Lie algebra A^- with bracket [a,b] = ab - ba.
s_poincare_birkhoff_witt_theorem	theorem	Poincare-Birkhoff-Witt theorem		The canonical map g -> U(g) is injective, and ordered monomials of a basis form a basis of the universal enveloping algebra.
s_free_lie_algebra	state	Free Lie algebra		The Lie algebra L(X) on a set X satisfying the universal property that every map X -> g extends uniquely to a homomorphism L(X) -> g.
s_nilpotent_lie_algebra	axiom	Nilpotent Lie algebra		A Lie algebra whose lower central series g supset [g,g] supset [g,[g,g]] supset ... terminates at 0 in finitely many steps.
s_solvable_lie_algebra	axiom	Solvable Lie algebra		A Lie algebra whose derived series g supset [g,g] supset [[g,g],[g,g]] supset ... terminates at 0 in finitely many steps.
s_abelian_lie_algebra	axiom	Abelian Lie algebra		A Lie algebra in which [x,y] = 0 for all x,y.
s_faithfully_flat_module	axiom	Faithfully flat module		A flat A-module M such that M tensor_A N = 0 implies N = 0, equivalently M tensor_A (-) reflects exactness.
t_faithfully_flat_descent	technique	Faithfully flat descent		Deduces properties of modules or algebras over A from their base change to a faithfully flat extension B.
t_local_global_principle	technique	Local-global principle		A property of modules holds globally iff it holds after localization at every prime ideal.
s_support_of_a_module	state	Support of a module		The set Supp(M) = {p in Spec(A) : M_p nonzero}, a closed subset of Spec(A) when M is finitely generated.
s_associated_graded	state	Associated graded ring/module		For a filtered ring A, the graded ring gr(A) = direct_sum_n A_n/A_{n+1}, capturing the leading term structure.
s_filtered_ring	axiom	Filtered ring		A ring equipped with a decreasing sequence of ideals ... supset F_n supset F_{n+1} supset ... compatible with multiplication: F_m * F_n subset F_{m+n}.
s_i_adic_filtration	state	I-adic filtration		The filtration of a ring A by powers of an ideal I: F_n = I^n.
s_i_adic_topology	state	I-adic topology		The linear topology on a ring or module defined by powers of an ideal I, where {I^n} forms a neighborhood base of 0.
s_hilbert_function	state	Hilbert function / Hilbert-Samuel function		For a graded module M, H(n) = dim_k(M_n); for I-adic filtrations, H_I(n) = length(M/I^{n+1}M), both eventually polynomial.
s_primary_ideal	axiom	Primary ideal		An ideal q such that if ab in q and a not in q then b^n in q for some n, equivalently A/q has every zero-divisor nilpotent.
s_uniqueness_primary_decomposition	theorem	Uniqueness theorems for primary decomposition		The associated primes and primary components for minimal primes are uniquely determined; embedded components may not be unique.
s_integral_element	axiom	Integral element		An element b of a ring extension B supset A is integral over A if it satisfies a monic polynomial with coefficients in A.
s_lying_over_theorem	theorem	Lying-over theorem		If A subset B is an integral extension, then for every prime ideal p of A there exists a prime ideal q of B with q intersect A = p.
s_valuation_ring	axiom	Valuation ring		An integral domain A such that for every nonzero element x of its fraction field, either x in A or x^{-1} in A.
s_place_of_a_valuation	state	Place		An equivalence class of valuations on a field, or a homomorphism from a valuation ring to a field together with infinity.
s_uniformizer	state	Uniformizer		A generator of the maximal ideal of a DVR, equivalently an element of valuation 1.
s_rank_of_a_valuation	state	Rank of a valuation		The Krull dimension of the associated valuation ring, equivalently the order type of convex subgroups of the value group.
s_henselian_ring	axiom	Henselian ring		A local ring satisfying the conclusion of Hensel's lemma, equivalently every finite algebra over it is a product of local algebras.
s_ramification_index_and_residue_degree	state	Ramification index and residue degree		For a valued field extension L/K, e = [v(L*):v(K*)] and f = [kappa_L:kappa_K]; the fundamental inequality gives ef <= [L:K].
s_extension_of_valuations	theorem	Extension of valuations		Every valuation on a field K extends to any algebraic extension L of K, with at most [L:K] extensions.
s_principal_divisor	state	Principal divisor		The divisor div(f) associated to a nonzero element f of the fraction field, defined as sum_v v(f)*[v] over height-1 primes.
s_factorial_ring	axiom	Factorial ring (UFD)		An integral domain with unique factorization into irreducibles; equivalently a Noetherian integrally closed domain with trivial divisor class group.
s_invertible_ideal	state	Invertible ideal		A fractional ideal I such that I*I^{-1} = A where I^{-1} = {x in K : xI subset A}; equivalently a locally principal fractional ideal.
s_unique_factorization_ideals_dedekind	theorem	Unique factorization of ideals in Dedekind domains		In a Dedekind domain, every nonzero ideal factors uniquely as a product of prime ideals.
s_approximation_theorem_valuations	theorem	Approximation theorem for valuations		Given finitely many independent valuations and target elements, one can find x simultaneously approximating each target.
s_system_of_parameters	state	System of parameters		In a Noetherian local ring of dimension d, a sequence of d elements whose generated ideal is m-primary.
s_dimension_theorem_noetherian_local	theorem	Dimension theorem for Noetherian local rings		The Krull dimension equals the degree of the Hilbert-Samuel polynomial, equals the minimum number of generators of an m-primary ideal.
s_dimension_of_polynomial_rings	theorem	Dimension of polynomial rings		If A is a Noetherian ring, then dim(A[X]) = dim(A) + 1.
s_catenary_ring	axiom	Catenary ring		A Noetherian ring in which all maximal chains of primes between any pair p subset q have the same length.
s_universally_catenary_ring	axiom	Universally catenary ring		A Noetherian ring A such that every finitely generated A-algebra is catenary.
s_unmixedness_theorem	theorem	Unmixedness theorem		In a Cohen-Macaulay ring, every ideal generated by height-many elements is unmixed (all associated primes are minimal of equal height).
s_multiplicity_of_local_ring	state	Multiplicity of a local ring		The normalized leading coefficient of the Hilbert-Samuel polynomial, measuring algebraic multiplicity; equals 1 for regular local rings.
s_coefficient_field	state	Coefficient field / coefficient ring		A subfield or complete DVR of a complete local ring mapping isomorphically to the residue field, guaranteed by the Cohen structure theorem.
s_weierstrass_division_theorem	theorem	Weierstrass division theorem		Division with remainder by a Weierstrass polynomial in k[[X_1,...,X_n]]: unique q and r with deg_{X_n}(r) < d.
s_nagata_ring	axiom	Nagata ring (Japanese ring)		A Noetherian domain whose integral closure in any finite extension of its fraction field is a finite module.
s_excellent_ring	axiom	Excellent ring		A universally catenary Noetherian ring with geometrically regular formal fibers and open regular locus; complete local rings are excellent.
s_depth	state	Depth		The length of a maximal regular sequence in m on a finitely generated module over a Noetherian local ring, equivalently the smallest i with Ext^i(k,M) nonzero.
s_regular_sequence	axiom	Regular sequence		A sequence x_1,...,x_n such that x_i is a non-zero-divisor on M/(x_1,...,x_{i-1})M for each i, and the quotient is nonzero.
s_depth_lemma	theorem	Depth lemma		For a short exact sequence 0 -> M' -> M -> M'' -> 0, depth(M) >= min(depth(M'), depth(M'') + 1).
s_injective_hull	state	Injective hull (injective envelope)		The minimal injective module containing M as an essential submodule, unique up to isomorphism.
s_matlis_duality	theorem	Matlis duality		Over a complete Noetherian local ring, Hom(-,E(k)) gives an anti-equivalence between finitely generated and Artinian modules.
s_dualizing_module	state	Dualizing module (canonical module)		A finitely generated module over a Cohen-Macaulay local ring satisfying local duality via Ext and local cohomology.
s_local_duality_theorem	theorem	Local duality theorem		For a complete Noetherian local ring of dimension d with dualizing module omega, H^i_m(M)^v isomorphic to Ext^{d-i}(M, omega).
s_regular_rings_finite_global_dimension	theorem	Regular rings have finite global dimension		A Noetherian local ring is regular iff it has finite global dimension, and then gl.dim(A) = dim(A).
s_bass_theorem	theorem	Bass's theorem		A Noetherian local ring is Gorenstein iff it has finite injective dimension, and then inj.dim(A) = dim(A).
s_derivative_of_a_vectorvalued_function	axiom	Derivative of a vector-valued function		For a function f mapping an interval of R into a normed vector space E, the derivative at a point x₀ is the limit of (f(x) - f(x₀))/(x - x₀) as x → x₀, when thi
s_differentiability	axiom	Differentiability		A function f is differentiable at x₀ if there exists a continuous linear map L such that f(x₀ + h) = f(x₀) + L(h) + o(h) as h → 0.
s_linearity_of_differentiation	theorem	Linearity of differentiation		The derivative operator is linear: (αf + βg)' = αf' + βg' for differentiable functions f, g and scalars α, β.
s_product_rule	theorem	Product rule (Leibniz rule)		If f and g are differentiable real-valued functions, then (fg)' = f'g + fg'.
s_chain_rule	theorem	Chain rule		If g is differentiable at x₀ and f is differentiable at g(x₀), then f ∘ g is differentiable at x₀ with derivative f'(g(x₀)) · g'(x₀).
s_derivative_of_the_inverse_function	theorem	Derivative of the inverse function		If f is a continuous bijection differentiable at x₀ with f'(x₀) ≠ 0, then f⁻¹ is differentiable at f(x₀) with derivative 1/f'(x₀).
s_monotonicity_criterion_via_derivative	theorem	Monotonicity criterion via derivative		A differentiable function f on an interval is non-decreasing if and only if f'(x) ≥ 0 for all x in the interval.
s_convexity_criterion_via_second_derivative	theorem	Convexity criterion via second derivative		A twice-differentiable function f on an interval is convex if and only if f''(x) ≥ 0 for all x in the interval.
s_higherorder_derivative	axiom	Higher-order derivative		The n-th derivative f⁽ⁿ⁾ is defined inductively as the derivative of f⁽ⁿ⁻¹⁾, when it exists.
s_leibniz_formula_for_nth_derivative_of_a_product	theorem	Leibniz formula for n-th derivative of a product		The n-th derivative of a product fg equals Σₖ₌₀ⁿ C(n,k) f⁽ᵏ⁾g⁽ⁿ⁻ᵏ⁾.
s_regulated_function	axiom	Regulated function		A function f from an interval I into a Banach space E is regulated if it has a left limit and a right limit at every point of I, equivalently if it is a uniform
s_step_function	axiom	Step function		A function on an interval that takes only finitely many values, each on a sub-interval of a finite partition.
s_integral_of_a_regulated_function	axiom	Integral of a regulated function		The integral of a regulated function f is defined as the limit of integrals of step functions converging uniformly to f, and this limit is independent of the ap
s_primitive	axiom	Primitive (antiderivative)		A primitive of a function f on an interval I is a function F such that F' = f on I.
s_integration_by_parts	theorem	Integration by parts		If f and g are continuously differentiable on [a,b], then ∫ₐᵇ f'(t)g(t) dt = f(b)g(b) - f(a)g(a) - ∫ₐᵇ f(t)g'(t) dt.
s_change_of_variables_in_integrals	theorem	Change of variables in integrals		If φ is a C¹ diffeomorphism and f is regulated, then ∫_{φ(a)}^{φ(b)} f(t) dt = ∫ₐᵇ f(φ(u)) φ'(u) du.
s_integral_remainder_for_taylors_formula	theorem	Integral remainder for Taylor's formula		If f is of class Cⁿ⁺¹, the remainder in the n-th order Taylor expansion equals (1/n!) ∫₀ʰ (h-t)ⁿ f⁽ⁿ⁺¹⁾(x₀+t) dt.
s_uniform_convergence_and_integration	theorem	Uniform convergence and integration		If a sequence of regulated functions converges uniformly to f, then f is regulated and the integral of the limit equals the limit of the integrals.
s_improper_integral	axiom	Improper integral (convergent integral on unbounded interval)		The integral ∫ₐ^∞ f(t) dt is defined as limₓ→∞ ∫ₐˣ f(t) dt when this limit exists.
s_absolute_convergence_of_improper_integrals	theorem	Absolute convergence of improper integrals		If ∫ₐ^∞ ‖f(t)‖ dt converges, then ∫ₐ^∞ f(t) dt converges and ‖∫ₐ^∞ f(t) dt‖ ≤ ∫ₐ^∞ ‖f(t)‖ dt.
s_power_function_x	axiom	Power function x^α		For x > 0 and α ∈ R, defined as x^α = exp(α log x), extending integer powers to all real exponents.
s_number	axiom	Number π		Defined as twice the smallest positive zero of cosine, establishing the period 2π of the trigonometric functions.
s_inverse_trigonometric_functions	axiom	Inverse trigonometric functions (arcsin, arccos, arctan)		The inverse functions of sin, cos, tan restricted to appropriate intervals where they are bijective, with derivatives 1/√(1-x²), -1/√(1-x²), 1/(1+x²) respective
s_hyperbolic_functions	axiom	Hyperbolic functions (sinh, cosh, tanh)		Defined as sinh(x) = (eˣ - e⁻ˣ)/2, cosh(x) = (eˣ + e⁻ˣ)/2, tanh(x) = sinh(x)/cosh(x), satisfying the identity cosh²(x) - sinh²(x) = 1.
s_inverse_hyperbolic_functions	axiom	Inverse hyperbolic functions		The inverses of sinh, cosh|[0,∞), tanh, expressible in terms of logarithm: arsinh(x) = log(x + √(x²+1)).
s_analyticity_of_elementary_functions	theorem	Analyticity of elementary functions		The functions exp, log, sin, cos, and their inverses are all real-analytic on their domains of definition, represented by convergent power series.
s_growth_comparison_of_elementary_functions	theorem	Growth comparison of elementary functions		As x → +∞, for any α > 0 and β > 0: (log x)^α = o(x^β) and x^β = o(e^{αx}), establishing the asymptotic hierarchy log ≪ polynomial ≪ exponential.
s_lipschitz_condition	axiom	Lipschitz condition		A function f(t, y) satisfies a Lipschitz condition in y if ‖f(t, y₁) - f(t, y₂)‖ ≤ K‖y₁ - y₂‖ for some constant K and all (t, yᵢ) in the domain.
s_integral_equation_formulation_of_ode	state	Integral equation formulation of ODE		The initial value problem y' = f(t,y), y(t₀) = y₀ is equivalent to the integral equation y(t) = y₀ + ∫_{t₀}^t f(s, y(s)) ds.
s_maximal_solution_of_an_ode	axiom	Maximal solution of an ODE		A solution defined on a maximal interval of existence that cannot be extended to a larger interval, whose existence is guaranteed by Zorn's lemma from local sol
s_blowup_criterion	theorem	Blow-up criterion (escape to boundary)		If the maximal interval of existence of a solution is bounded, then the solution leaves every compact subset of the domain of f, i.e., ‖y(t)‖ → ∞ or the solutio
s_linear_ordinary_differential_equation	axiom	Linear ordinary differential equation		An ODE of the form y' = A(t)y + b(t), where A(t) is a continuous matrix-valued function and b(t) is continuous, whose solutions form an affine subspace.
s_homogeneous_linear_ode	axiom	Homogeneous linear ODE		A linear ODE of the form y' = A(t)y (with b = 0) whose solutions form a vector space of dimension equal to the order of the system.
s_fundamental_matrix	state	Fundamental matrix (resolvent)		A matrix-valued function Φ(t) whose columns form a basis of solutions of y' = A(t)y, satisfying Φ' = A(t)Φ with Φ(t₀) = I.
t_variation_of_constants	technique	Variation of constants (variation of parameters)		A method to solve the non-homogeneous equation y' = A(t)y + b(t) by writing y(t) = Φ(t)c(t) where Φ is the fundamental matrix, leading to c'(t) = Φ(t)⁻¹b(t).
s_linear_ode_with_constant_coefficients	state	Linear ODE with constant coefficients		The system y' = Ay with A constant has the explicit solution y(t) = e^{tA}y₀, reducible via Jordan normal form to exponential-polynomial expressions.
s_differentiable_dependence_on_parameters	theorem	Differentiable dependence on parameters		If f(t, y, λ) is Cᵏ in all variables, then the solution y(t, λ) of y' = f(t, y, λ) is Cᵏ in the parameter λ.
s_lagrange_remainder	state	Lagrange remainder		For f of class Cⁿ⁺¹, the remainder Rₙ(h) = f⁽ⁿ⁺¹⁾(x₀ + θh) hⁿ⁺¹/(n+1)! for some θ ∈ (0,1).
s_asymptotic_expansion	axiom	Asymptotic expansion		A formal series Σ aₙφₙ(x) is an asymptotic expansion of f(x) as x → x₀ if f(x) - Σₖ₌₀ⁿ aₖφₖ(x) = o(φₙ(x)) for each n, where (φₙ) is an asymptotic scale.
s_asymptotic_scale	axiom	Asymptotic scale		A sequence of functions (φₙ) such that φₙ₊₁(x) = o(φₙ(x)) as x → x₀, providing a graded comparison framework for the growth or decay of functions.
s_littleo_and_bigo_notation	axiom	Little-o and big-O notation		f = o(g) means f/g → 0 and f = O(g) means |f/g| is bounded, as x approaches a specified limit, providing a calculus of asymptotic comparison.
t_asymptotic_comparison_of_functions	technique	Asymptotic comparison of functions		The systematic use of asymptotic expansions and asymptotic scales to compare functions' growth rates, compute limits of indeterminate forms, and approximate fun
s_bernoulli_numbers	axiom	Bernoulli numbers		The sequence of rational numbers Bₙ defined by the generating function t/(eᵗ - 1) = Σₙ₌₀^∞ Bₙ tⁿ/n!, with B₀ = 1, B₁ = -1/2, and B₂ₖ₊₁ = 0 for k ≥ 1.
s_bernoulli_polynomials	axiom	Bernoulli polynomials		Polynomials Bₙ(x) defined by the generating function te^{xt}/(eᵗ - 1) = Σₙ₌₀^∞ Bₙ(x) tⁿ/n!, satisfying Bₙ'(x) = nBₙ₋₁(x) and Bₙ(0) = Bₙ.
s_eulermaclaurin_summation_formula	theorem	Euler-Maclaurin summation formula		Relates a sum Σₖ₌ₐᵇ f(k) to the integral ∫ₐᵇ f(t) dt plus correction terms involving Bernoulli numbers and derivatives of f at the endpoints, with a remainder e
t_summation_of_divergent_series_by_eulermaclaurin	technique	Summation of divergent series by Euler-Maclaurin		Using the Euler-Maclaurin formula to assign finite values to formally divergent sums by analytic continuation, yielding values for ζ(-n) in terms of Bernoulli n
s_asymptotic_expansion_of_the_gamma_function	state	Asymptotic expansion of the gamma function		The Stirling series Γ(x) ~ √(2π) x^{x-1/2} e^{-x} (1 + Σₖ₌₁^∞ cₖ/x^k), an asymptotic expansion for x → +∞ with coefficients expressible via Bernoulli numbers.
s_beta_function	state	Beta function		Defined as B(p,q) = ∫₀¹ t^{p-1}(1-t)^{q-1} dt for p, q > 0, related to the gamma function by B(p,q) = Γ(p)Γ(q)/Γ(p+q).
s_digamma_function	state	Digamma function		The logarithmic derivative ψ(x) = Γ'(x)/Γ(x) = (d/dx) log Γ(x), whose asymptotic expansion involves Bernoulli numbers: ψ(x) ~ log x - 1/(2x) - Σₖ₌₁^∞ B₂ₖ/(2k·x²
s_abscissa_of_convergence	axiom	Abscissa of convergence		The infimum σ₀ of real numbers σ such that the Laplace integral ∫₀^∞ e^{-σt} f(t) dt converges, establishing a half-plane Re(s) > σ₀ of convergence.
s_abscissa_of_absolute_convergence	axiom	Abscissa of absolute convergence		The infimum σₐ of real numbers σ such that ∫₀^∞ e^{-σt} |f(t)| dt converges, always satisfying σ₀ ≤ σₐ ≤ σ₀ + 1 for locally integrable f.
s_laplace_transform_of_derivatives	theorem	Laplace transform of derivatives		L{f'}(s) = sL{f}(s) - f(0⁺), and more generally L{f⁽ⁿ⁾}(s) = sⁿL{f}(s) - Σₖ₌₀ⁿ⁻¹ sⁿ⁻¹⁻ᵏ f⁽ᵏ⁾(0⁺), converting differential equations to algebraic equations.
s_laplace_transform_of_convolutions	theorem	Laplace transform of convolutions		L{f * g}(s) = L{f}(s) · L{g}(s), where (f * g)(t) = ∫₀ᵗ f(t-u) g(u) du, converting convolution to pointwise multiplication.
s_inversion_formula_for_laplace_transform	theorem	Inversion formula for Laplace transform		Under suitable conditions, f(t) = (1/2πi) ∫_{c-i∞}^{c+i∞} e^{st} L{f}(s) ds for c > σ₀, recovering f from its Laplace transform via a Bromwich contour integral.
s_uniqueness_theorem_for_laplace_transform	theorem	Uniqueness theorem for Laplace transform		If L{f}(s) = L{g}(s) for all s in a half-plane and f, g are continuous, then f = g, establishing injectivity of the Laplace transform on continuous functions.
t_application_of_laplace_transform_to_linear_ode	technique	Application of Laplace transform to linear ODE		Converting a linear ODE with constant coefficients and initial conditions to an algebraic equation via Laplace transform, solving algebraically, then inverting 
s_valued_field	axiom	Valued field		A field K equipped with an absolute value |·|: K → R₊ satisfying |xy| = |x||y| and the triangle inequality, inducing a metric topology on K.
s_neighborhood_of_zero	axiom	Neighborhood of zero (fundamental system)		In a TVS, the topology is entirely determined by a fundamental system of neighborhoods of the origin, which must be absorbing, balanced sets; translations then 
s_absorbing_set	axiom	Absorbing set		A subset A of a vector space E such that for every x ∈ E there exists λ₀ > 0 with x ∈ λA for all |λ| ≥ λ₀.
s_balanced_set	axiom	Balanced set		A subset B of a vector space E such that λB ⊆ B for all scalars λ with |λ| ≤ 1.
s_bounded_set_in_a_tvs	axiom	Bounded set in a TVS		A subset B of a TVS E is bounded if for every neighborhood V of 0, there exists λ > 0 such that B ⊆ λV, equivalently if B is absorbed by every neighborhood of z
s_metrizable_tvs	state	Metrizable TVS		A TVS whose topology is metrizable, which occurs if and only if it has a countable fundamental system of neighborhoods of zero, in which case it can be metrized
s_normable_tvs	state	Normable TVS		A TVS whose topology can be defined by a single norm, characterized by having a bounded convex neighborhood of zero.
s_completeness_of_a_tvs	axiom	Completeness of a TVS		A TVS is complete if every Cauchy net (or Cauchy filter) converges; a complete metrizable locally convex space is a Fréchet space.
s_topological_direct_sum_of_tvs	state	Topological direct sum of TVS		A TVS E is a topological direct sum of subspaces E₁ and E₂ if the map (x₁, x₂) ↦ x₁ + x₂ is a topological isomorphism from E₁ × E₂ to E.
s_quotient_tvs	state	Quotient TVS		For a closed subspace F of a TVS E, the quotient E/F with the quotient topology is again a TVS, and E/F is Hausdorff if and only if F is closed.
s_product_of_tvs	state	Product of TVS		The Cartesian product Πᵢ Eᵢ of TVS equipped with the product topology is again a TVS, and it is locally convex (resp. complete) if each factor is.
s_minkowski_functional	axiom	Minkowski functional (gauge)		For a convex absorbing set C containing 0, the Minkowski functional is p_C(x) = inf{λ > 0 : x ∈ λC}, which is a sublinear function and a seminorm when C is bala
s_strict_separation_theorem	theorem	Strict separation theorem		If A is a closed convex set and B is a compact convex set in a locally convex space with A ∩ B = ∅, then there exists a continuous linear functional strictly se
s_supporting_hyperplane_theorem	theorem	Supporting hyperplane theorem		At every boundary point of a closed convex set in a locally convex space, there exists a closed hyperplane containing the point and having the convex set on one
s_bipolar_theorem	theorem	Bipolar theorem		For a subset A of a locally convex space, the bipolar A°° equals the closed convex balanced hull of A.
s_space_of_continuous_linear_maps_l	axiom	Space of continuous linear maps L(E, F)		The vector space of all continuous linear maps from a TVS E to a TVS F, which can be given various topologies depending on the notion of convergence desired.
s_topology_of_bounded_convergence	axiom	Topology of bounded convergence (strong topology on L(E,F))		The topology on L(E,F) of uniform convergence on bounded subsets of E, making L(E,F) into a locally convex space when F is locally convex.
s_topology_of_pointwise_convergence	axiom	Topology of pointwise convergence (weak topology on L(E,F))		The coarsest topology on L(E,F) making all evaluation maps u ↦ u(x) continuous, also called the simple topology.
s_barrelled_space	axiom	Barrelled space		A locally convex space in which every barrel (closed, convex, balanced, absorbing set) is a neighborhood of zero; every Fréchet space and every Baire locally co
s_barrel	axiom	Barrel		A subset of a TVS that is closed, convex, balanced, and absorbing.
s_weak_topology	axiom	Weak-* topology σ(E', E)		The weak topology on the dual E' of a TVS E induced by the duality ⟨E', E⟩, under which E' is a locally convex space and E embeds canonically into (E')'.
s_mackey_topology	axiom	Mackey topology τ(E, F)		The finest locally convex topology on E consistent with the duality ⟨E, F⟩, defined as the topology of uniform convergence on σ(F, E)-compact convex balanced su
s_semireflexive_space	axiom	Semi-reflexive space		A locally convex space E is semi-reflexive if the canonical map J: E → E'' is surjective (algebraic isomorphism), without requiring it to be a homeomorphism.
s_strong_topology	axiom	Strong topology β(E', E)		The topology on E' of uniform convergence on all bounded subsets of E, making E' a locally convex space called the strong dual.
s_grothendieck_completion_theorem	theorem	Grothendieck completion theorem		Every Hausdorff locally convex space has a completion that is unique up to topological isomorphism, constructed as a subspace of the algebraic dual of E' with t
s_transpose_of_a_continuous_linear_map	state	Transpose (adjoint) of a continuous linear map		For u: E → F continuous linear, the transpose ᵗu: F' → E' is defined by ⟨ᵗu(f), x⟩ = ⟨f, u(x)⟩, and ᵗu is continuous for the weak-* and strong topologies.
s_weak_compactness_and_completeness	theorem	Weak compactness and completeness		In a complete locally convex space, a closed bounded set is weakly compact if and only if every bounded sequence has a weakly convergent subsequence.
s_parallelogram_law	theorem	Parallelogram law		In an inner product space, ‖x+y‖² + ‖x-y‖² = 2(‖x‖² + ‖y‖²), and a normed space satisfying this identity is an inner product space (Jordan-von Neumann theorem).
s_orthogonality	axiom	Orthogonality		Two elements x, y of an inner product space are orthogonal (x ⊥ y) if ⟨x,y⟩ = 0; a set S is orthogonal if its elements are pairwise orthogonal.
s_orthogonal_projection_theorem	theorem	Orthogonal projection theorem		For every closed subspace M of a Hilbert space H, each x ∈ H has a unique decomposition x = m + m⊥ with m ∈ M and m⊥ ∈ M⊥, and m is the unique best approximatio
s_orthonormal_system	axiom	Orthonormal system		A family (eᵢ) in a Hilbert space such that ⟨eᵢ, eⱼ⟩ = δᵢⱼ (Kronecker delta), i.e., pairwise orthogonal unit vectors.
s_normal_operator	axiom	Normal operator		A bounded linear operator T on a Hilbert space satisfying T*T = TT*, implying the spectral theorem applies.
s_direct_sum_of_hilbert_spaces	state	Direct sum of Hilbert spaces		The Hilbert space direct sum ⊕ᵢ Hᵢ consists of families (xᵢ) with xᵢ ∈ Hᵢ and Σᵢ ‖xᵢ‖² < ∞, equipped with the inner product ⟨(xᵢ),(yᵢ)⟩ = Σᵢ ⟨xᵢ,yᵢ⟩.
s_youngs_inequality_for_products	theorem	Young's inequality for products		For a, b >= 0 and conjugate exponents p, q (1/p + 1/q = 1): ab <= a^p/p + b^q/q, with equality iff a^p = b^q.
s_conjugate_exponents	axiom	Conjugate exponents		A pair (p, q) with 1 <= p, q <= infinity satisfying 1/p + 1/q = 1, fundamental to Lp space duality and Holder's inequality.
s_logconvexity_of_lp_norms	theorem	Log-convexity of Lp norms (Riesz interpolation inequality)		For 1 <= p1 <= p <= p2 <= infinity, if f in Lp1 cap Lp2, then f in Lp and ||f||_p <= ||f||_{p1}^theta ||f||_{p2}^{1-theta} where 1/p = theta/p1 + (1-theta)/p2.
s_bounded_radon_measure	axiom	Bounded Radon measure		A Radon measure mu on X that extends to a continuous linear form on C_0(X), equivalently one with finite total variation |mu|(X) < infinity.
s_support_of_a_measure	axiom	Support of a measure		The smallest closed set F such that mu(f) = 0 for every continuous function f with compact support contained in X \ F.
s_total_variation_measure	state	Total variation measure		For a complex Radon measure mu, the positive measure |mu| defined by |mu|(f) = sup{|mu(g)| : |g| <= f, g in K(X)} for f >= 0, making bounded measures a Banach s
s_vague_topology_on_measures	axiom	Vague topology on measures		The weak-* topology sigma(M(X), K(X)) on the space of Radon measures, where mu_n -> mu vaguely iff mu_n(f) -> mu(f) for all f in K(X).
s_dirac_measure	axiom	Dirac measure		The measure delta_x defined by delta_x(f) = f(x) for a point x in X, the atomic probability measure concentrated at a single point.
s_image_measure_pushforward	state	Image measure (pushforward measure)		For a proper continuous map phi: X -> Y and a measure mu on X, the image measure phi_*(mu) on Y defined by phi_*(mu)(f) = mu(f o phi).
s_regularity_of_radon_measures	theorem	Regularity of Radon measures		Every Radon measure on a locally compact space is inner regular (approximable from below by compact sets) and outer regular (approximable from above by open set
s_upper_integral	state	Upper integral		For a positive measure mu and a lower semicontinuous function f >= 0, defined as mu*(f) = sup{mu(g) : g in K(X), 0 <= g <= f}, extending integration beyond cont
s_mu_measurable_function_bourbaki	axiom	Measurable function (Bourbaki, mu-measurable)		A function f: X -> R is mu-measurable if for every compact K and epsilon > 0, there exists compact K' subset K with mu(K \ K') < epsilon such that f|_{K'} is co
s_integrable_function_lebesgue	axiom	Integrable function (Lebesgue integrable)		A measurable function f is mu-integrable if integral |f| dmu < infinity, where the integral is defined via the Daniell construction extending from continuous fu
s_negligible_set	axiom	Negligible set (mu-null set)		A set N subset X is mu-negligible if mu*(chi_N) = 0, equivalently if it is contained in an open set of arbitrarily small measure.
s_lp_space	axiom	Lp space (1 <= p < infinity)		The Banach space of equivalence classes of measurable functions f with integral |f|^p dmu < infinity, modulo equality a.e., with norm ||f||_p = (integral |f|^p 
s_l_infinity_space	axiom	L-infinity space		The Banach space of equivalence classes of essentially bounded measurable functions, with norm ||f||_infinity = ess sup |f| = inf{M : |f| <= M a.e.}.
s_dense_subspaces_of_lp	theorem	Dense subspaces of Lp		For 1 <= p < infinity, the space K(X) of continuous functions with compact support is dense in Lp(mu), as is the space of simple functions.
s_essential_supremum	axiom	Essential supremum		The essential supremum of a measurable function f is ess sup f = inf{M in R : f <= M a.e.}, the smallest a.e. upper bound.
s_convergence_in_measure	axiom	Convergence in measure		A sequence (f_n) converges to f in measure if for every epsilon > 0, mu({x : |f_n(x) - f(x)| > epsilon}) -> 0; Lp convergence implies convergence in measure.
s_product_measure	axiom	Product measure		For sigma-finite measures mu on X and nu on Y, the unique measure mu tensor nu on X x Y satisfying (mu tensor nu)(A x B) = mu(A) nu(B) for measurable rectangles
s_tonellis_theorem	theorem	Tonelli's theorem		If f is a non-negative measurable function on X x Y, then integral f d(mu tensor nu) = integral integral f(x,y) dmu(x) dnu(y), with iterated integrals equal wit
s_change_of_variables_formula_rn	theorem	Change of variables formula (integration on Rn)		If phi: U -> V is a C1 diffeomorphism between open sets in Rn, then integral_V f(y) dy = integral_U f(phi(x)) |det J_phi(x)| dx for integrable f.
s_bochner_integral	axiom	Bochner integral		The integral of a strongly measurable function f: X -> E (E a Banach space) with integral ||f|| dmu < infinity, defined as the limit of integrals of simple func
s_strongly_measurable_function	axiom	Strongly measurable function		A function f: X -> E (E separable Banach space) is strongly measurable if it is the a.e. limit of a sequence of simple (measurable, finitely-valued) functions.
s_vector_valued_measure	axiom	Vector-valued measure (vector measure)		A countably additive map mu: Sigma -> E from a sigma-algebra into a Banach space, generalizing scalar measures.
s_radon_nikodym_derivative	state	Radon-Nikodym derivative (density)		The a.e. unique measurable function dnu/dmu satisfying nu(A) = integral_A (dnu/dmu) dmu for all measurable sets A, existing when nu << mu and both are sigma-fin
s_left_invariance_of_haar_measure	axiom	Left invariance of Haar measure		A measure mu on a topological group G is left-invariant if mu(gA) = mu(A) for all g in G and all measurable A, equivalently integral f(g^{-1}x) dmu(x) = integra
s_right_haar_measure	state	Right Haar measure		A right-invariant positive Radon measure on G, related to left Haar measure via the modular function: d mu_r(x) = Delta(x^{-1}) d mu_l(x).
s_modular_function_of_locally_compact_group	axiom	Modular function		The continuous homomorphism Delta: G -> R_+* defined by mu(Ag) = Delta(g) mu(A) for left Haar measure mu, measuring the failure of left Haar measure to be right
s_unimodular_group	axiom	Unimodular group		A locally compact group whose modular function is identically 1, so left and right Haar measures coincide; includes all compact groups, abelian groups, and semi
s_haar_measure_on_compact_groups	state	Haar measure on compact groups		On a compact group, the Haar measure is both left- and right-invariant (unimodular) and can be normalized to a probability measure with mu(G) = 1.
s_integration_formula_for_quotient_groups	theorem	Integration formula for quotient groups		If H is a closed subgroup of G with Haar measures mu_G, mu_H, and G/H admits an invariant measure, then integral_G f dmu_G = integral_{G/H} (integral_H f(xh) dm
s_weils_formula	theorem	Weil's formula (quotient integral formula)		A G-invariant measure on the quotient space G/H exists if and only if Delta_G|_H = Delta_H, and when it exists it is unique up to a positive scalar.
s_convolution_of_measures	axiom	Convolution of measures		For bounded Radon measures mu and nu on a locally compact group G, the convolution (mu * nu)(f) = integral integral f(xy) dmu(x) dnu(y), making bounded measures
s_convolution_of_functions_on_a_group	axiom	Convolution of functions on a group		For f, g in L1(G), the convolution (f * g)(x) = integral_G f(xy^{-1}) g(y) dmu(y) with respect to left Haar measure, making L1(G) a Banach algebra.
s_approximate_identity_in_group_algebra	state	Approximate identity in L1(G)		A net (e_i) in L1(G) with integral e_i dmu = 1 and support shrinking to {e} such that e_i * f -> f in L1 norm for all f, replacing the absent unit element.
s_character_of_a_representation	axiom	Character of a representation		The character of a finite-dimensional representation pi is the function chi_pi(g) = tr(pi(g)), a continuous class function that determines the representation up
s_convolution_algebra_of_measures	state	Convolution algebra of measures M(G)		The Banach space of bounded Radon measures on G with convolution as multiplication and total variation as norm, forming a unital Banach algebra with unit delta_
s_integrated_representation	state	Integrated representation		For a unitary representation pi of G on H, the integrated form pi~: L1(G) -> B(H) defined by pi~(f) = integral_G f(g) pi(g) dmu(g) is a non-degenerate *-represe
s_fourier_transform_on_lca_groups	state	Fourier transform on locally compact abelian groups		For f in L1(G), the Fourier transform f_hat(chi) = integral_G f(g) overline{chi(g)} dmu(g) for chi in G_hat, mapping L1(G) to C_0(G_hat) and satisfying (f * g)^
s_fourier_inversion_formula_lca	theorem	Fourier inversion formula (LCA groups)		Under appropriate normalization of Haar measures, if f in L1(G) and f_hat in L1(G_hat), then f(g) = integral_{G_hat} f_hat(chi) chi(g) d mu_hat(chi) for a.e. g.
s_tight_measure	axiom	Tight measure		A Borel measure mu on a topological space X is tight if for every epsilon > 0, there exists a compact set K with mu(X \ K) < epsilon.
s_weak_convergence_of_measures	axiom	Weak convergence of measures (narrow convergence)		A sequence of measures mu_n converges weakly to mu if integral f dmu_n -> integral f dmu for all bounded continuous functions f.
s_regular_borel_measure	axiom	Regular Borel measure		A Borel measure that is both inner regular (mu(A) = sup{mu(K) : K subset A compact}) and outer regular (mu(A) = inf{mu(U) : U supset A open}) on all measurable 
s_inner_regularity	axiom	Inner regularity		A measure mu is inner regular on a measurable set A if mu(A) = sup{mu(K) : K subset A, K compact}.
s_outer_regularity	axiom	Outer regularity		A measure mu is outer regular on a measurable set A if mu(A) = inf{mu(U) : A subset U, U open}.
s_suslin_space	axiom	Suslin space (analytic space)		A Hausdorff topological space that is the continuous image of a Polish space; every Borel subset of a Polish space is Suslin, and measures on Suslin spaces have
s_choquet_capacitability_theorem	theorem	Choquet capacitability theorem		In a Suslin space, every analytic set is capacitable, meaning its outer measure equals the supremum of measures of compact subsets, extending inner regularity t
t_daniell_integral_approach	technique	Daniell integral approach		Bourbaki's method of defining integration starting from a positive linear functional on a lattice of functions, then extending via upper integrals and completio
t_approximation_by_continuous_functions	technique	Approximation by continuous functions		The systematic use of Urysohn's lemma and partitions of unity to approximate measurable or integrable functions by continuous functions with compact support.
s_inductive_limit_topology	axiom	Inductive limit topology		The finest locally convex topology on K(X) = union_K C(K) making all inclusion maps C(K) -> K(X) continuous, used in Bourbaki's definition of Radon measures.
s_abels_liouville_formula	theorem	Abel-Liouville formula (Wronskian formula)		The Wronskian of solutions of y' = A(t)y satisfies W(t) = W(t_0) exp(integral_{t_0}^t tr A(s) ds), showing it is either identically zero or never zero.
s_pontryagin_dual_group	axiom	Pontryagin dual group		For a locally compact abelian group G, the dual group G_hat = Hom(G, T) is the group of continuous homomorphisms from G to the circle group T, equipped with the
s_spectral_radius_formula	theorem	Spectral radius formula		For a bounded linear operator T on a Banach space, the spectral radius r(T) = lim ||T^n||^{1/n} = inf ||T^n||^{1/n}.
s_resolvent_of_an_operator	state	Resolvent of an operator		For lambda not in sigma(T), the resolvent R(lambda,T) = (lambda I - T)^{-1} is a bounded operator depending analytically on lambda, satisfying the resolvent ide
s_pettis_integral	axiom	Pettis integral (weak integral)		A vector x in E is the Pettis integral of f: X -> E if <x, x'> = integral <f, x'> dmu for all x' in E'; weaker than Bochner, requiring only scalar measurability
s_convolution_semigroup	state	Convolution semigroup		A family (mu_t)_{t>=0} of probability measures on a locally compact group satisfying mu_s * mu_t = mu_{s+t} and mu_0 = delta_e, modeling diffusion and Markov pr
s_lie_algebra_homomorphism	axiom	Lie algebra homomorphism		A linear map φ: g → h between Lie algebras satisfying φ([x,y]) = [φ(x),φ(y)] for all x,y in g.
s_quotient_lie_algebra	state	Quotient Lie algebra		For an ideal a of g, the vector space g/a equipped with the bracket [x+a, y+a] = [x,y]+a, yielding a well-defined Lie algebra structure.
s_center_of_a_lie_algebra	state	Center of a Lie algebra		The ideal z(g) = {x ∈ g : [x,y] = 0 for all y ∈ g}, consisting of elements that bracket trivially with all of g.
s_derived_subalgebra	state	Derived subalgebra (commutator subalgebra)		The ideal [g,g] = D(g) spanned by all brackets [x,y], measuring the failure of g to be abelian.
s_derived_series	state	Derived series		The descending sequence D⁰(g) = g, Dⁿ⁺¹(g) = [Dⁿ(g), Dⁿ(g)] obtained by iterating the derived subalgebra construction.
s_radical_of_a_lie_algebra	state	Radical of a Lie algebra		The unique maximal solvable ideal rad(g) of a finite-dimensional Lie algebra g, which contains every solvable ideal.
s_simple_lie_algebra	axiom	Simple Lie algebra		A non-abelian Lie algebra whose only ideals are 0 and itself, serving as the irreducible building blocks of semisimple Lie algebras.
s_lie_s_theorem	theorem	Lie's theorem		If g is a solvable Lie algebra over an algebraically closed field of characteristic zero and V is a finite-dimensional g-module, then there exists a common eige
s_cartan_s_criterion_for_solvability	theorem	Cartan's criterion for solvability		A finite-dimensional Lie algebra g over a field of characteristic zero is solvable if and only if B(x,y) = 0 for all x ∈ g and y ∈ [g,g], where B is the Killing
s_cartan_s_criterion_for_semisimplicity	theorem	Cartan's criterion for semisimplicity		A finite-dimensional Lie algebra g over a field of characteristic zero is semisimple if and only if its Killing form is non-degenerate.
s_levi_decomposition	theorem	Levi decomposition		Every finite-dimensional Lie algebra g over a field of characteristic zero is a semidirect product g = rad(g) ⋊ s where s is a semisimple subalgebra (called a L
s_poincar_birkhoff_witt_theorem	theorem	Poincaré-Birkhoff-Witt theorem		If {x₁,...,xₙ} is an ordered basis of g, then the monomials x₁^{a₁}···xₙ^{aₙ} form a basis of U(g), proving the canonical map g → U(g) is injective.
s_normalizer_of_a_subalgebra	state	Normalizer of a subalgebra		For a subalgebra h of g, the subalgebra N_g(h) = {x ∈ g : [x,h] ⊆ h}, the largest subalgebra of g in which h is an ideal.
s_centralizer_in_a_lie_algebra	state	Centralizer in a Lie algebra		For a subset S of g, the subalgebra C_g(S) = {x ∈ g : [x,s] = 0 for all s ∈ S}, consisting of elements commuting with all of S.
s_semidirect_product_of_lie_algebras	state	Semidirect product of Lie algebras		Given Lie algebras h and a with a representation ρ: h → der(a), the vector space h ⊕ a with bracket [(h₁,a₁),(h₂,a₂)] = ([h₁,h₂], [a₁,a₂]+ρ(h₁)a₂−ρ(h₂)a₁).
s_derivation_of_a_lie_algebra	axiom	Derivation of a Lie algebra		A linear map D: g → g satisfying D([x,y]) = [D(x),y] + [x,D(y)], with inner derivations being those of the form ad(x).
s_whitehead_first_lemma	theorem	Whitehead's lemma (first)		For a semisimple Lie algebra g over a field of characteristic zero and a finite-dimensional g-module V, every derivation from g to V is inner, i.e., H¹(g,V) = 0
s_whitehead_second_lemma	theorem	Whitehead's lemma (second)		For a semisimple Lie algebra g over characteristic zero and a finite-dimensional g-module V, every extension of g by V splits, i.e., H²(g,V) = 0.
s_jordan_decomposition_in_a_lie_algebra	state	Jordan decomposition in a Lie algebra		In a semisimple Lie algebra g over a perfect field, each element x has a unique decomposition x = xₛ + xₙ with xₛ semisimple, xₙ nilpotent, and [xₛ,xₙ] = 0, com
s_invariant_bilinear_form	axiom	Invariant bilinear form		A bilinear form B on g satisfying B([x,y],z) = B(x,[y,z]) for all x,y,z ∈ g, with the Killing form being the canonical example.
s_witt_formula	theorem	Witt formula		The dimension of the degree-n homogeneous component of the free Lie algebra on q generators is (1/n)∑_{d|n} μ(d)q^{n/d}, where μ is the Möbius function.
s_baker_campbell_hausdorff_formula	theorem	Baker-Campbell-Hausdorff formula		For elements x,y in a complete filtered Lie algebra (or in a neighborhood of identity in a Lie group), log(eˣeʸ) = x + y + ½[x,y] + 1/12([x,[x,y]]+[y,[y,x]]) + 
s_lie_group_homomorphism	axiom	Lie group homomorphism		A smooth group homomorphism between Lie groups, whose differential at the identity is a Lie algebra homomorphism.
s_lie_algebra_of_a_lie_group	state	Lie algebra of a Lie group		The tangent space at the identity g = T_eG equipped with the Lie bracket inherited from left-invariant vector fields, functorially associated to the Lie group G
s_lie_subgroup	state	Lie subgroup		A subgroup H of a Lie group G that is an immersed submanifold (or, in the closed case, an embedded submanifold) carrying an induced Lie group structure.
s_closed_subgroup_theorem	theorem	Closed subgroup theorem (Cartan's theorem)		Every closed subgroup of a Lie group is an embedded Lie subgroup, i.e., it automatically inherits a smooth manifold structure making it a Lie group in the subsp
s_quotient_of_a_lie_group	state	Quotient of a Lie group		For a closed normal subgroup N of G, the quotient G/N carries a unique Lie group structure making the projection G → G/N a smooth surjective homomorphism with L
t_lie_functor	technique	Lie functor		The functor Lie: LieGrp → LieAlg sending each Lie group to its Lie algebra and each homomorphism to its differential at the identity, providing the algebraic li
t_integration_of_lie_algebra_homomorphisms	technique	Integration of Lie algebra homomorphisms		The process of lifting a Lie algebra homomorphism g → h to a Lie group homomorphism G → H when G is simply connected, by using the exponential map and path-lift
s_coxeter_group	axiom	Coxeter group		A group W with a presentation ⟨s₁,...,sₙ | (sᵢsⱼ)^{mᵢⱼ} = 1⟩ where mᵢᵢ = 1 and mᵢⱼ = mⱼᵢ ≥ 2 (or ∞), abstracting the structure of reflection groups.
s_coxeter_matrix	axiom	Coxeter matrix		A symmetric matrix M = (mᵢⱼ) with mᵢᵢ = 1 and mᵢⱼ ∈ {2,3,...,∞} for i≠j, encoding the orders of products of pairs of generators in a Coxeter group.
s_coxeter_graph	state	Coxeter graph		A labeled graph with vertices indexed by generators of a Coxeter group, edges connecting pairs with mᵢⱼ ≥ 3, and edge labels mᵢⱼ when mᵢⱼ ≥ 4, encoding the Coxe
s_exchange_property	theorem	Exchange property		In a Coxeter group, if ℓ(sw) < ℓ(w) for a simple reflection s, then there exists an index i such that sw = s₁···ŝᵢ···sₖ (omitting sᵢ) for any reduced expression
s_deletion_property	theorem	Deletion property		In a Coxeter group, any non-reduced expression s₁···sₖ can be shortened by deleting an even number of factors to obtain a reduced expression for the same elemen
s_tits_system	axiom	Tits system (BN-pair)		A group G with subgroups B and N such that B ∩ N = T is normal in N, W = N/T is a Coxeter group with generators S, and the axioms BwB·BsB ⊆ BwsB ∪ BwB and sBs⁻¹
s_parabolic_subgroup_of_a_coxeter_group	state	Parabolic subgroup of a Coxeter group		For a subset J ⊆ S of simple reflections, the subgroup W_J generated by J, which is itself a Coxeter group with Coxeter matrix restricted to J.
s_coxeter_complex	state	Coxeter complex		The simplicial complex associated to a Coxeter group W whose simplices are cosets of parabolic subgroups of W, serving as the building block (apartment) of a Ti
s_matsumoto_s_theorem	theorem	Matsumoto's theorem		Any two reduced expressions for the same element of a Coxeter group are related by a sequence of braid relations sᵢsⱼsᵢ··· = sⱼsᵢsⱼ··· (mᵢⱼ factors each side), 
s_strong_exchange_property	theorem	Strong exchange property		In a Coxeter group, if ℓ(tw) < ℓ(w) for a reflection t (not necessarily simple), then t can replace one of the factors in any reduced expression for w.
s_classification_of_finite_coxeter_groups	theorem	Classification of finite Coxeter groups		The irreducible finite Coxeter groups are exactly the types Aₙ, Bₙ, Dₙ, E₆, E₇, E₈, F₄, G₂ (crystallographic, i.e., Weyl groups) together with H₃, H₄, and I₂(m)
s_coroot	state	Coroot		For a root α, the coroot α∨ = 2α/⟨α,α⟩, such that the reflection s_α(β) = β − ⟨β,α∨⟩α, and the coroots form the dual root system.
s_cartan_matrix	state	Cartan matrix		The n×n integer matrix A = (aᵢⱼ) where aᵢⱼ = ⟨αᵢ,αⱼ∨⟩ for simple roots αᵢ,αⱼ, satisfying aᵢᵢ = 2, aᵢⱼ ≤ 0 for i≠j, and aᵢⱼ = 0 iff aⱼᵢ = 0, determining the root
s_classification_of_irreducible_root_systems	theorem	Classification of irreducible root systems		The connected Dynkin diagrams (equivalently irreducible root systems) are exactly Aₙ (n≥1), Bₙ (n≥2), Cₙ (n≥3), Dₙ (n≥4), E₆, E₇, E₈, F₄, and G₂.
s_weyl_chamber	state	Weyl chamber		A connected component of E \ ∪_{α∈Φ} H_α where H_α = ker(α), forming a fundamental domain for the W-action; the dominant chamber is the one defined by ⟨·,αᵢ∨⟩ >
s_root_lattice	state	Root lattice		The Z-span Q = ZΦ of the root system in E, a sublattice of the weight lattice, with the quotient P/Q being a finite group isomorphic to the center of the simply
s_fundamental_weights	state	Fundamental weights		The basis {ω₁,...,ωₙ} of the weight lattice P dual to the simple coroots, defined by ⟨ωᵢ,αⱼ∨⟩ = δᵢⱼ, indexing the vertices of the fundamental Weyl chamber.
s_extended_dynkin_diagram	state	Extended (affine) Dynkin diagram		The Dynkin diagram augmented by an additional node corresponding to the negative of the highest root −θ, classifying affine root systems and appearing in the th
s_affine_root_system	state	Affine root system		An infinite root system on an affine space obtained by adding integer translates α + nδ of roots of a finite root system, whose Weyl group is the affine Weyl gr
s_affine_weyl_group	state	Affine Weyl group		The infinite Coxeter group generated by reflections in affine hyperplanes H_{α,n} = {x : ⟨x,α⟩ = n}, isomorphic to the semidirect product W ⋉ Q∨ of the Weyl gro
s_alcove	state	Alcove		A connected component of the complement of all affine reflecting hyperplanes, forming a fundamental domain for the affine Weyl group; the fundamental alcove is 
s_root_space_decomposition	state	Root space decomposition		For a semisimple Lie algebra g with Cartan subalgebra h, the decomposition g = h ⊕ (⊕_{α∈Φ} g_α) where g_α = {x ∈ g : [h,x] = α(h)x for all h ∈ h} and Φ ⊂ h* is
s_chevalley_basis	state	Chevalley basis		A basis {hᵢ, e_α} of a semisimple Lie algebra with integer structure constants, consisting of coroots hᵢ = αᵢ∨ in h and root vectors e_α ∈ g_α satisfying [e_α, 
s_conjugacy_of_cartan_subalgebras	theorem	Conjugacy of Cartan subalgebras		Any two Cartan subalgebras of a finite-dimensional Lie algebra over an algebraically closed field of characteristic zero are conjugate by an inner automorphism 
s_sl2_triple	state	sl₂-triple		A triple (e,h,f) in a Lie algebra satisfying [h,e] = 2e, [h,f] = −2f, [e,f] = h, spanning a subalgebra isomorphic to sl(2), with one such triple for each root.
s_jacobson_morozov_theorem	theorem	Jacobson-Morozov theorem		In a semisimple Lie algebra over a field of characteristic zero, every nilpotent element e can be embedded in an sl₂-triple (e,h,f), and the sl₂-triples for a g
s_split_semisimple_lie_algebra	axiom	Split semisimple Lie algebra		A semisimple Lie algebra g over a field k possessing a splitting Cartan subalgebra h such that all roots are in k (not just in the algebraic closure), equivalen
s_parabolic_subalgebra	state	Parabolic subalgebra		A subalgebra p of g containing a Borel subalgebra, classified by subsets of simple roots and corresponding to parabolic subgroups in the group setting.
s_kostant_multiplicity_formula	theorem	Kostant multiplicity formula		The multiplicity of weight μ in L(λ) is dim L(λ)_μ = ∑_{w∈W} ε(w) P(w(λ+ρ) − (μ+ρ)) where P is the Kostant partition function counting expressions as sums of po
s_kostant_partition_function	state	Kostant partition function		The function P(ν) counting the number of ways to write ν as an unordered sum of positive roots (with repetition), appearing in weight multiplicity formulas.
s_freudenthal_s_formula	theorem	Freudenthal's formula		A recursive formula computing the multiplicity of a weight μ in L(λ) using the inner product of weights and the multiplicities of higher weights, derived from t
s_existence_theorem_for_semisimple_lie_algebras	theorem	Existence theorem for semisimple Lie algebras		For every Cartan matrix (equivalently every Dynkin diagram), there exists a unique (up to isomorphism) semisimple Lie algebra over an algebraically closed field
s_serre_s_presentation	theorem	Serre's presentation (Serre relations)		A semisimple Lie algebra is the quotient of the free Lie algebra on generators {eᵢ, fᵢ, hᵢ} by the relations [hᵢ,hⱼ]=0, [hᵢ,eⱼ]=aᵢⱼeⱼ, [hᵢ,fⱼ]=−aᵢⱼfⱼ, [eᵢ,fⱼ]=δ
s_maximal_torus	state	Maximal torus		A maximal connected abelian subgroup T of a compact Lie group G, isomorphic to (S¹)ʳ where r = rank(G); every element of G is conjugate to an element of T.
s_conjugacy_of_maximal_tori	theorem	Conjugacy of maximal tori		Any two maximal tori in a compact Lie group are conjugate by an inner automorphism, so the rank and the Weyl group N_G(T)/T are well-defined invariants.
s_compact_real_form	state	Compact real form		A compact real Lie subalgebra g_u of a complex semisimple Lie algebra g_C such that g_C = g_u ⊗_R C, unique up to conjugacy; its Killing form is negative defini
t_weyl_s_unitary_trick	technique	Weyl's unitary trick		The technique of proving results for a complex semisimple Lie algebra by passing to its compact real form, where integration over the compact group via Haar mea
s_banach_algebra	axiom	Banach algebra		A complete normed algebra A over C (or R) satisfying ‖xy‖ ≤ ‖x‖·‖y‖, providing the analytic framework for spectral theory and operator algebras.
s_resolvent_identity	theorem	Resolvent identity		For λ,μ in the resolvent set, R(λ,a) − R(μ,a) = (μ−λ)R(λ,a)R(μ,a), and the resolvent is an analytic function of λ vanishing at infinity.
s_character	axiom	Character (multiplicative linear functional)		A nonzero algebra homomorphism χ: A → C from a commutative Banach algebra A, necessarily continuous with ‖χ‖ = 1 and χ(1) = 1, forming the points of the Gelfand
s_gelfand_spectrum	state	Gelfand spectrum (maximal ideal space)		The set Ω(A) of all characters of a commutative unital Banach algebra A, equipped with the weak-* topology making it a compact Hausdorff space; maximal ideals c
s_c_star_identity	axiom	C*-identity		The norm condition ‖a*a‖ = ‖a‖² characterizing C*-algebras, from which it follows that ‖a*‖ = ‖a‖ and the norm is uniquely determined by the algebraic structure
s_state_on_c_star_algebra	axiom	State on a C*-algebra		A positive linear functional φ: A → C with ‖φ‖ = 1 (or φ(1) = 1 in the unital case), serving as a noncommutative probability measure and providing the input for
s_cyclic_vector	axiom	Cyclic vector		A vector ξ in a Hilbert space H such that π(A)ξ is dense in H for a representation π of A, ensuring the representation is determined by the state φ(a) = ⟨π(a)ξ,
s_spectral_mapping_theorem	theorem	Spectral mapping theorem		For a continuous function f and a normal element a of a C*-algebra, σ(f(a)) = f(σ(a)), so the spectrum of f(a) is the image of the spectrum of a under f.
s_unitization_of_c_star_algebra	state	Unitization of a C*-algebra		For a non-unital C*-algebra A, the C*-algebra Ã = A ⊕ C with multiplication (a,λ)(b,μ) = (ab+λb+μa, λμ) and a suitable norm making it a unital C*-algebra contai
s_locally_compact_group	axiom	Locally compact group		A topological group whose underlying topology is locally compact and Hausdorff, providing the natural setting for abstract harmonic analysis with the existence 
s_convolution_product	state	Convolution product		The binary operation (f*g)(x) = ∫_G f(y)g(y⁻¹x)dy on L¹(G), making it a Banach algebra; for abelian groups, the Gelfand transform of L¹(G) is the Fourier transf
s_group_c_star_algebra	state	Group C*-algebra C*(G)		The completion of L¹(G) with respect to the norm ‖f‖_{C*} = sup_π ‖π(f)‖ where the supremum is over all *-representations of L¹(G), whose representations corres
s_spectral_theorem_for_bounded_normal_operators	theorem	Spectral theorem for bounded normal operators		For a bounded normal operator T on a Hilbert space, there exists a unique spectral measure E on σ(T) such that T = ∫_{σ(T)} λ dE(λ), establishing a bijection be
s_spectral_theorem_for_commuting_normal_operators	theorem	Spectral theorem for commuting normal operators		A finite family of pairwise commuting bounded normal operators on a Hilbert space has a joint spectral measure E on the product of their spectra, simultaneously
s_snag_theorem	theorem	SNAG theorem (Stone-Naimark-Ambrose-Godement)		Every strongly continuous unitary representation of a locally compact abelian group G on a Hilbert space is of the form π(g) = ∫_Ĝ χ(g) dE(χ) for a unique spect
s_type_decomposition_of_von_neumann_algebras	theorem	Type decomposition of von Neumann algebras		Every von Neumann algebra decomposes uniquely as a direct sum of type I, type II₁, type II_∞, and type III factors, classified by the structure of their project
s_holomorphic_functional_calculus	theorem	Holomorphic functional calculus		For an element a of a unital Banach algebra and f holomorphic on a neighborhood of σ(a), f(a) = (1/2πi)∮_Γ f(λ)(λ−a)⁻¹dλ defines an algebra homomorphism from th
s_fell_topology	state	Fell topology		The topology on the set of irreducible representations of a C*-algebra (or locally compact group) defined by convergence of states, making the unitary dual a (g
s_spectral_permanence	theorem	Spectral permanence		If B is a closed *-subalgebra of a C*-algebra A and a ∈ B, then σ_B(a) = σ_A(a); the spectrum of an element does not change when computed in a C*-subalgebra.
s_mellin_transform	axiom	Mellin Transform		The integral transform M{f}(s) = integral_0^infty x^{s-1} f(x) dx, analytic in a vertical strip.
s_faa_di_bruno_formula	theorem	Faa di Bruno's Formula		Explicit formula for the n-th derivative of a composite function f(g(x)) as a sum over partitions involving Bell polynomials.
s_leibniz_rule_integral	theorem	Leibniz Integral Rule (Differentiation Under the Sign)		d/dx integral_{a(x)}^{b(x)} f(x,t) dt = f(x,b)b' - f(x,a)a' + integral partial_x f dt under suitable regularity.
s_poincare_asymptotic_expansion	axiom	Poincare Asymptotic Expansion		A formal series f(z) ~ sum a_n z^{-n} as z -> infty, where for each N the remainder is O(z^{-N}) in a given sector.
s_watsons_lemma	theorem	Watson's Lemma		If f(t) has an asymptotic expansion at t=0, then its Laplace-type integral admits term-by-term asymptotic integration as z -> infty.
s_stokes_phenomenon	state	Stokes Phenomenon		The discontinuous change in asymptotic expansion form as the argument crosses Stokes lines in the complex plane, where subdominant exponentials switch on or off
s_abel_plana_summation_formula	theorem	Abel-Plana Summation Formula		Relates sum f(n) to integral f(t) dt plus a correction term involving f(it)-f(-it), converting sums to integrals without requiring derivatives.
t_laplaces_method	technique	Laplace's Method		Obtains asymptotic expansions of Laplace-type integrals for large z by expanding about the minimum of the exponent.
t_method_of_steepest_descents	technique	Method of Steepest Descents		Deforms a contour integral through saddle points along paths of steepest descent to obtain asymptotic expansions.
t_bleisteins_method	technique	Bleistein's Method		Obtains uniform asymptotic expansions when two stationary points coalesce, expressing results in terms of Airy functions.
t_mellin_transform_asymptotics	technique	Mellin Transform Asymptotics		Derives asymptotic expansions by shifting contours in Mellin-Barnes integrals and collecting residues.
t_liouville_green_wkb_approximation	technique	Liouville-Green (WKB) Approximation		Constructs approximate solutions to d^2w/dz^2 = u^2 f(z) w for large u as f^{-1/4} exp(+/-u integral sqrt{f} dz).
t_fuchs_frobenius_theory	technique	Fuchs-Frobenius Theory		Constructs solutions of linear ODEs near regular singular points as z^r sum a_n z^n where r satisfies the indicial equation.
t_darboux_method	technique	Darboux's Method		Extracts asymptotic behavior of Taylor coefficients from the singularity structure on the circle of convergence.
t_exponentially_improved_asymptotics	technique	Exponentially Improved Asymptotics		Re-expands the remainder after optimal truncation to capture exponentially small terms beyond all algebraic orders.
t_hyperasymptotics	technique	Hyperasymptotics		Iterates exponentially improved asymptotics to achieve successively more accurate approximations.
s_borel_summation	axiom	Borel Summation		Method assigning a value to a divergent series via Borel transform and Laplace integral: sum a_n z^n -> integral_0^infty e^{-t} (sum a_n (tz)^n/n!) dt.
t_clenshaw_curtis_quadrature	technique	Clenshaw-Curtis Quadrature		Approximates definite integrals by interpolating at Chebyshev points and integrating the polynomial exactly.
t_romberg_integration	technique	Romberg Integration		Accelerates the trapezoidal rule by systematic Richardson extrapolation on halved step sizes.
t_pade_approximation	technique	Pade Approximation		Approximates a function by a rational function whose Taylor expansion matches the given series to maximum order.
t_chebyshev_expansion	technique	Chebyshev Expansion		Expands a function in Chebyshev polynomials T_n(x) yielding near-optimal uniform polynomial approximations.
t_minimax_rational_approximation	technique	Minimax Rational Approximation		Constructs the rational approximation minimizing maximum absolute error, characterized by equioscillation.
t_aitken_delta_squared_process	technique	Aitken's Delta-Squared Process		Transforms a linearly convergent sequence via s'_n = s_n - (Delta s_n)^2/(Delta^2 s_n) to achieve faster convergence.
t_wynn_epsilon_algorithm	technique	Wynn's Epsilon Algorithm		Computes sequence transformations equivalent to diagonal Pade approximants via a recursive triangular array.
t_levin_transformation	technique	Levin's Transformation		A nonlinear sequence transformation accelerating convergence by using estimates of the general term's behavior.
t_gauss_legendre_quadrature	technique	Gauss-Legendre Quadrature		Evaluates integral with optimal polynomial degree accuracy 2n-1 using zeros of Legendre polynomials as nodes.
t_gauss_laguerre_quadrature	technique	Gauss-Laguerre Quadrature		Optimal quadrature for integrands with Laguerre weight using zeros of Laguerre polynomials.
t_gauss_hermite_quadrature	technique	Gauss-Hermite Quadrature		Optimal quadrature for Gaussian-weighted integrals using zeros of Hermite polynomials.
s_lambert_w_function	axiom	Lambert W-Function		The multivalued inverse of f(w) = we^w, with principal branch W_0 real-valued for x >= -1/e.
s_gudermannian_function	state	Gudermannian Function gd(x)		gd(x) = 2 arctan(tanh(x/2)), connecting circular and hyperbolic functions without complex numbers.
s_euler_mascheroni_constant	axiom	Euler-Mascheroni Constant		The constant gamma = lim(sum 1/k - ln n) approx 0.5772, appearing in the digamma function psi(1) = -gamma.
s_pochhammer_symbol	axiom	Pochhammer Symbol (Rising Factorial)		The rising factorial (a)_n = a(a+1)...(a+n-1) = Gamma(a+n)/Gamma(a), fundamental to hypergeometric series.
s_gamma_recurrence_relation	theorem	Gamma Function Recurrence Relation		The functional equation Gamma(z+1) = z Gamma(z), generalizing n! = n(n-1)!.
s_euler_reflection_formula	theorem	Euler's Reflection Formula		Gamma(z) Gamma(1-z) = pi/sin(pi z), connecting Gamma at complementary arguments.
s_legendre_duplication_formula	theorem	Legendre Duplication Formula		Gamma(z) Gamma(z+1/2) = sqrt(pi) 2^{1-2z} Gamma(2z), relating Gamma at z and 2z.
s_gauss_multiplication_formula	theorem	Gauss Multiplication Formula		Product of Gamma(z+k/n) for k=0..n-1 equals (2pi)^{(n-1)/2} n^{1/2-nz} Gamma(nz).
s_bohr_mollerup_theorem	theorem	Bohr-Mollerup Theorem		Gamma is the unique log-convex function on (0,infty) with f(1)=1 and f(x+1)=xf(x).
s_hankel_contour_integral_reciprocal_gamma	theorem	Hankel's Contour Integral for 1/Gamma(z)		1/Gamma(z) = (1/2pi i) oint (-t)^{-z} e^{-t} dt over a Hankel contour, showing 1/Gamma is entire.
s_binet_formula_log_gamma	theorem	Binet's Formula for ln Gamma		Integral representation expressing the Stirling remainder of ln Gamma(z) as an explicit arctangent integral.
s_polygamma_functions	state	Polygamma Functions		The functions psi^{(n)}(z) = d^{n+1}/dz^{n+1} ln Gamma(z), with psi^{(0)} the digamma function.
s_polygamma_recurrence_relation	theorem	Polygamma Recurrence Relation		psi^{(n)}(z+1) = psi^{(n)}(z) + (-1)^n n!/z^{n+1}, reducing polygamma evaluation to a strip.
s_polygamma_reflection_formula	theorem	Polygamma Reflection Formula		psi^{(n)}(1-z) + (-1)^{n+1} psi^{(n)}(z) = (-1)^n pi d^n/dz^n cot(pi z).
s_barnes_g_function	state	Barnes G-Function		Entire function satisfying G(z+1) = Gamma(z) G(z) with G(1)=1, a double gamma function.
s_q_gamma_function	state	q-Gamma Function		q-analog of Gamma defined via Gamma_q(z) = (q;q)_infty/(q^z;q)_infty (1-q)^{1-z}, reducing to Gamma(z) as q->1.
s_reciprocal_gamma_entire	state	Reciprocal Gamma as Entire Function		1/Gamma(z) is an entire function of order 1 with zeros at z = 0, -1, -2, ... and Weierstrass product representation.
s_beta_integral_representation	theorem	Beta Function Integral Representation		B(a,b) = integral_0^1 t^{a-1}(1-t)^{b-1} dt = Gamma(a)Gamma(b)/Gamma(a+b) for Re a, Re b > 0.
s_exponential_integral_E1	axiom	Exponential Integral E_1(z)		E_1(z) = integral_z^infty e^{-t}/t dt, the principal exponential integral with logarithmic singularity at z=0.
s_exponential_integral_Ei	axiom	Exponential Integral Ei(x)		Ei(x) = P.V. integral_{-infty}^x e^t/t dt for real x, related to E_1 by Ei(x) = -E_1(-x) +/- i pi.
s_sine_integral	axiom	Sine Integral Si(z)		Si(z) = integral_0^z sin(t)/t dt, an entire odd function with Si(+infty) = pi/2.
s_cosine_integral	axiom	Cosine Integral Ci(z)		Ci(z) = -integral_z^infty cos(t)/t dt = gamma + ln z + integral_0^z (cos t -1)/t dt with branch point at origin.
s_error_function	axiom	Error Function erf(z)		erf(z) = (2/sqrt(pi)) integral_0^z e^{-t^2} dt, the normalized incomplete Gaussian integral.
s_complementary_error_function	axiom	Complementary Error Function erfc(z)		erfc(z) = 1 - erf(z) = (2/sqrt(pi)) integral_z^infty e^{-t^2} dt, preferred for large arguments.
s_faddeeva_function	state	Faddeeva Function w(z)		w(z) = e^{-z^2} erfc(-iz), central to Voigt profile computation and plasma dispersion.
s_dawsons_integral	axiom	Dawson's Integral F(z)		F(z) = e^{-z^2} integral_0^z e^{t^2} dt, a one-sided Fourier-Laplace transform of the Gaussian.
s_fresnel_cosine_integral	axiom	Fresnel Cosine Integral C(z)		C(z) = integral_0^z cos(pi t^2/2) dt, an entire function whose plot forms the Cornu spiral.
s_fresnel_sine_integral	axiom	Fresnel Sine Integral S(z)		S(z) = integral_0^z sin(pi t^2/2) dt, companion to C(z) in the Cornu spiral.
s_voigt_profile	state	Voigt Profile		Convolution of Gaussian and Lorentzian line shapes V(x;sigma,gamma) = Re[w(z)]/(sigma sqrt(2pi)), central to spectroscopy.
s_lower_incomplete_gamma	axiom	Lower Incomplete Gamma Function gamma(a,z)		gamma(a,z) = integral_0^z t^{a-1} e^{-t} dt, satisfying gamma(a,z) + Gamma(a,z) = Gamma(a).
s_upper_incomplete_gamma	axiom	Upper Incomplete Gamma Function Gamma(a,z)		Gamma(a,z) = integral_z^infty t^{a-1} e^{-t} dt, related to E_n(z) = z^{n-1} Gamma(1-n,z).
s_regularized_incomplete_gamma	state	Regularized Incomplete Gamma P(a,z) and Q(a,z)		P = gamma(a,z)/Gamma(a) and Q = Gamma(a,z)/Gamma(a) with P+Q=1, giving the chi-squared CDF.
s_incomplete_beta_function	axiom	Incomplete Beta Function B_x(a,b)		B_x(a,b) = integral_0^x t^{a-1}(1-t)^{b-1} dt, whose regularized form I_x gives the Beta CDF.
s_generalized_exponential_integral	state	Generalized Exponential Integral E_p(z)		E_p(z) = integral_1^infty e^{-zt}/t^p dt, extending E_1 to complex order via E_p(z) = z^{p-1} Gamma(1-p,z).
s_airy_differential_equation	axiom	Airy Differential Equation		The ODE w'' - zw = 0, exhibiting a transition from oscillatory to exponential behavior at z=0.
s_airy_function_ai	axiom	Airy Function Ai(z)		The recessive solution of w'' = zw, defined by Ai(z) = (1/pi) integral_0^infty cos(t^3/3 + zt) dt.
s_airy_function_bi	axiom	Airy Function Bi(z)		The dominant solution of w'' = zw, exponentially growing as z -> +infty and linearly independent from Ai.
s_scorer_function_gi	state	Scorer Function Gi(z)		Particular solution of w'' - zw = 1/pi via Gi(z) = (1/pi) integral_0^infty sin(t^3/3 + zt) dt.
s_scorer_function_hi	state	Scorer Function Hi(z)		Particular solution of w'' - zw = 1/pi complementary to Gi, with Hi(z) - Gi(z) = Bi(z).
s_airy_integral_representation	theorem	Airy Integral Representation		Ai(z) = (1/2pi i) integral exp(t^3/3 - zt) dt along a contour from infty e^{-pi i/3} to infty e^{pi i/3}.
s_bessel_differential_equation	axiom	Bessel Differential Equation		The ODE z^2 w'' + z w' + (z^2 - nu^2) w = 0 with regular singularity at z=0 and irregular at infinity.
s_bessel_function_j_nu	axiom	Bessel Function J_nu(z)		Bessel function of the first kind, the solution regular at the origin defined by (z/2)^nu sum (-z^2/4)^k/(k! Gamma(nu+k+1)).
s_bessel_function_y_nu	axiom	Bessel Function Y_nu(z)		Bessel function of the second kind (Neumann function), singular at the origin, linearly independent from J_nu.
s_hankel_functions	state	Hankel Functions H_nu^{(1)}(z) and H_nu^{(2)}(z)		H_nu^{(1)} = J_nu + iY_nu and H_nu^{(2)} = J_nu - iY_nu, representing outgoing and incoming cylindrical waves.
s_modified_bessel_function_i_nu	axiom	Modified Bessel Function I_nu(z)		Modified Bessel function of the first kind, exponentially growing solution of z^2w'' + zw' - (z^2+nu^2)w = 0.
s_modified_bessel_function_k_nu	axiom	Modified Bessel Function K_nu(z)		Modified Bessel function of the second kind, exponentially decaying for large real positive z.
s_spherical_bessel_functions	state	Spherical Bessel Functions		Solutions j_n, y_n, h_n of the radial Helmholtz equation, related to Bessel functions by j_n(z) = sqrt(pi/(2z)) J_{n+1/2}(z).
s_kelvin_functions	state	Kelvin Functions ber, bei, ker, kei		Real and imaginary parts of Bessel functions at complex arguments xe^{3pi i/4}, used in electromagnetic skin-effect calculations.
s_mittag_leffler_function	state	Mittag-Leffler Function E_{a,b}(z)		The entire function sum z^k/Gamma(ak+b) generalizing the exponential, fundamental in fractional calculus.
t_hankel_transform	technique	Hankel Transform		Integral transform with Bessel function kernel, the radial part of the 2D Fourier transform.
s_bessel_addition_theorem	theorem	Bessel Addition Theorem		Expresses J_0(R) in terms of J_n at component distances: J_0(|r-r'|) = sum epsilon_n J_n(r) J_n(r') cos n(theta-theta').
s_nicholson_integral	theorem	Nicholson's Integral		J_nu(z)^2 + Y_nu(z)^2 = (8/pi^2) integral_0^infty K_0(2z sinh t) cosh(2nu t) dt.
s_bessel_integral_representation_poisson	theorem	Poisson Integral for Bessel Functions		J_nu(z) = (z/2)^nu/(sqrt(pi) Gamma(nu+1/2)) integral_0^pi cos(z cos theta) sin^{2nu}(theta) d theta for Re nu > -1/2.
s_struve_function_h_nu	axiom	Struve Function H_nu(z)		Solution of the inhomogeneous Bessel equation with right-hand side (z/2)^{nu-1}/(sqrt(pi) Gamma(nu+1/2)).
s_modified_struve_function_l_nu	state	Modified Struve Function L_nu(z)		L_nu(z) = -ie^{-inu pi/2} H_nu(iz), solving the inhomogeneous modified Bessel equation.
s_lommel_function_s	axiom	Lommel Functions s_{mu,nu} and S_{mu,nu}		Particular solutions of the inhomogeneous Bessel equation z^2y'' + zy' + (z^2 - nu^2)y = z^{mu+1}.
s_anger_function	axiom	Anger Function		(1/pi) integral_0^pi cos(nu theta - z sin theta) d theta, generalizing J_nu to non-integer order.
s_weber_function_e_nu	axiom	Weber Function E_nu(z)		(1/pi) integral_0^pi sin(nu theta - z sin theta) d theta, companion to the Anger function.
s_parabolic_cylinder_equation	axiom	Parabolic Cylinder Differential Equation		The ODE w'' + (nu + 1/2 - z^2/4)w = 0 from separation in parabolic coordinates.
s_parabolic_cylinder_function_u	axiom	Parabolic Cylinder Function U(a,z)		Standard solution decaying as z -> +infty, expressible via confluent hypergeometric functions.
s_parabolic_cylinder_function_v	axiom	Parabolic Cylinder Function V(a,z)		Companion solution growing as z -> +infty, linearly independent from U(a,z).
s_weber_parabolic_cylinder_d_nu	state	Weber Parabolic Cylinder Function D_nu(z)		Classical notation D_nu(z) = U(-nu-1/2, z) for parabolic cylinder functions.
s_kummer_confluent_equation	axiom	Kummer's Confluent Hypergeometric Equation		The ODE zw'' + (b-z)w' - aw = 0 with regular singularity at z=0 and irregular singularity at infinity.
s_kummer_function_m	axiom	Kummer Function M(a,b,z)		Confluent hypergeometric function 1F1(a;b;z) = sum (a)_k z^k/((b)_k k!), entire in z for b not a non-positive integer.
s_tricomi_function_u	axiom	Tricomi Function U(a,b,z)		Second confluent hypergeometric function with U(a,b,z) ~ z^{-a} as z -> infty, unique with this decay.
s_kummer_transformation	theorem	Kummer Transformation		M(a,b,z) = e^z M(b-a,b,-z), the fundamental symmetry of confluent hypergeometric functions.
s_whittaker_differential_equation	axiom	Whittaker Differential Equation		W'' + (-1/4 + kappa/z + (1/4-mu^2)/z^2)W = 0, alternative form of Kummer's equation.
s_whittaker_function_m_kappa_mu	state	Whittaker Function M_{kappa,mu}(z)		Whittaker function related to Kummer M by M_{kappa,mu} = e^{-z/2} z^{mu+1/2} M(mu-kappa+1/2, 2mu+1, z).
s_whittaker_function_w_kappa_mu	state	Whittaker Function W_{kappa,mu}(z)		Whittaker function related to Tricomi U, the recessive solution decaying as z -> infty.
s_confluent_hypergeometric_limit	theorem	Confluent Hypergeometric Limit		1F1(a;b;z/b) -> 0F1(;a;z) as b -> infty, the confluence process merging regular singularities.
s_associated_legendre_equation	axiom	Associated Legendre Differential Equation		(1-x^2)y'' - 2xy' + [nu(nu+1) - mu^2/(1-x^2)]y = 0, generalizing Legendre's equation with order mu.
s_ferrers_function_p	axiom	Ferrers Function P_nu^mu(x)		Associated Legendre function of the first kind on (-1,1), defined via the hypergeometric function.
s_toroidal_functions	state	Toroidal (Ring) Functions		Associated Legendre functions with half-integer degree P_{n-1/2}^m(cosh eta) for potential theory in toroidal geometry.
s_conical_mehler_functions	state	Conical (Mehler) Functions		Associated Legendre functions P_{-1/2+itau}^mu(x) with complex degree, arising in conical boundary problems.
t_mehler_fock_transform	technique	Mehler-Fock Transform		Integral transform with conical function kernel for axially symmetric potential problems.
s_legendre_function_second_kind	axiom	Legendre Function of the Second Kind Q_nu(z)		Q_nu(z) = (pi/2)(cos nu pi P_nu(z) - P_{-nu-1}(z))/sin nu pi, singular at z = +/- 1.
s_associated_legendre_addition_theorem	theorem	Addition Theorem for Legendre Functions		P_n(cos gamma) = sum_{m} epsilon_m P_n^m(cos theta) P_n^m(cos theta') cos m(phi-phi') connecting P_n at composed angles.
s_gauss_hypergeometric_function	axiom	Gauss Hypergeometric Function 2F1(a,b;c;z)		sum (a)_k(b)_k z^k/((c)_k k!) analytic in |z|<1, satisfying a Fuchsian ODE with 3 regular singular points.
s_hypergeometric_differential_equation	axiom	Hypergeometric Differential Equation		z(1-z)w'' + [c-(a+b+1)z]w' - ab w = 0, the Fuchsian equation with regular singularities at 0, 1, infty.
s_gauss_summation_theorem	theorem	Gauss Summation Theorem		2F1(a,b;c;1) = Gamma(c)Gamma(c-a-b)/(Gamma(c-a)Gamma(c-b)) when Re(c-a-b) > 0.
s_chu_vandermonde_identity	theorem	Chu-Vandermonde Identity		2F1(-n,b;c;1) = (c-b)_n/(c)_n, a terminating Gauss summation.
s_euler_transformation_hypergeometric	theorem	Euler Transformation for 2F1		2F1(a,b;c;z) = (1-z)^{c-a-b} 2F1(c-a,c-b;c;z), a fundamental symmetry.
s_pfaff_transformation	theorem	Pfaff Transformation		2F1(a,b;c;z) = (1-z)^{-a} 2F1(a,c-b;c;z/(z-1)), mapping the argument fractionally.
s_riemann_p_symbol	axiom	Riemann Differential Equation and P-symbol		Most general second-order Fuchsian ODE with three regular singularities, encoded by the Riemann P-symbol.
s_euler_integral_representation_2f1	theorem	Euler Integral Representation for 2F1		2F1(a,b;c;z) = B(b,c-b)^{-1} integral_0^1 t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a} dt for Re c > Re b > 0.
s_kummer_24_solutions	theorem	Kummer's 24 Solutions		The 24 solutions of the hypergeometric ODE obtained by transforming z -> z, 1-z, 1/z, 1/(1-z), z/(z-1), (z-1)/z and choosing exponents.
s_generalized_hypergeometric_pfq	axiom	Generalized Hypergeometric Function pFq		pFq(a_1,...,a_p;b_1,...,b_q;z) = sum (a_1)_k...(a_p)_k z^k/((b_1)_k...(b_q)_k k!).
s_pfaff_saalschutz_theorem	theorem	Pfaff-Saalschutz Summation Theorem		Evaluates terminating balanced 3F2 at z=1 as a ratio of Pochhammer symbols.
s_dixon_summation_theorem	theorem	Dixon's Summation Theorem		Evaluates well-poised 3F2(a,b,c;1+a-b,1+a-c;1) as a ratio of Gamma functions.
s_dougall_summation_theorem	theorem	Dougall Summation Theorem		Evaluates terminating very-well-poised 7F6(1) as a product of Pochhammer symbols.
s_meijer_g_function	axiom	Meijer G-Function		Mellin-Barnes contour integral unifying hypergeometric, Bessel, and other special functions as special cases.
s_appell_function_f1	axiom	Appell Function F_1		Two-variable hypergeometric function F_1(a;b,b';c;x,y) with coupled rising factorials.
s_appell_functions_f2_f3_f4	axiom	Appell Functions F_2, F_3, F_4		The remaining three Appell two-variable hypergeometric functions with distinct convergence regions.
s_fox_h_function	axiom	Fox H-Function		Generalization of the Meijer G-function with arbitrary powers in the Gamma factors, used in fractional calculus and statistics.
s_bilateral_hypergeometric_series	axiom	Bilateral Hypergeometric Series		pHq: sum from n=-infty to infty of (a_1)_n...(a_p)_n z^n/((b_1)_n...(b_q)_n), generalizing one-sided series.
s_clausen_formula	theorem	Clausen's Formula		[2F1(a,b;a+b+1/2;z)]^2 = 3F2(2a,2b,a+b;2a+2b,a+b+1/2;z), squaring identity for hypergeometric functions.
s_whipple_transformation	theorem	Whipple's Transformation		Relates a well-poised 7F6(1) to a balanced 4F3(1), connecting different types of hypergeometric series.
s_q_binomial_coefficient	axiom	q-Binomial Coefficient (Gaussian Binomial)		[n choose k]_q = (q;q)_n/((q;q)_k(q;q)_{n-k}), counting subspaces of F_q^n.
s_q_derivative_operator	axiom	q-Derivative Operator D_q		D_q f(z) = (f(z)-f(qz))/((1-q)z), the q-analog of differentiation reducing to f'(z) as q->1.
s_q_integral_jackson	axiom	q-Integral (Jackson Integral)		integral_0^a f(x) d_q x = a(1-q) sum f(aq^k) q^k, the inverse of the q-derivative.
s_q_binomial_theorem	theorem	q-Binomial Theorem		sum (a;q)_k z^k/(q;q)_k = (az;q)_infty/(z;q)_infty, the q-analog of the binomial series.
s_basic_hypergeometric_function	axiom	Basic Hypergeometric Function r+1-phi-s		q-analog of pFq with q-shifted factorials, converging for all z when s>r and |z|<1 when s=r.
s_ramanujan_1_psi_1	theorem	Ramanujan's 1-psi-1 Bilateral Summation		Evaluates the bilateral q-series sum (a;q)_n z^n/(b;q)_n as an infinite product of q-Pochhammer symbols.
t_bailey_pairs_chain	technique	Bailey Pairs and Bailey Chain		Iterative mechanism: the Bailey lemma produces new q-hypergeometric identities from existing Bailey pairs.
s_q_exponential_functions	state	q-Exponential Functions e_q and E_q		q-analogs of exp: e_q(z) = sum z^n/(q;q)_n = 1/(z;q)_infty and E_q(z) = sum q^{n(n-1)/2} z^n/(q;q)_n = (-z;q)_infty.
s_rogers_ramanujan_identities	theorem	Rogers-Ramanujan Identities		sum q^{n^2}/(q;q)_n = prod 1/((1-q^{5n+1})(1-q^{5n+4})) and sum q^{n^2+n}/(q;q)_n = prod 1/((1-q^{5n+2})(1-q^{5n+3})).
s_jacobi_polynomial	axiom	Jacobi Polynomial P_n^{alpha,beta}(x)		Orthogonal polynomials on [-1,1] with weight (1-x)^alpha(1+x)^beta, the most general classical family on a finite interval.
s_gegenbauer_polynomial	state	Gegenbauer (Ultraspherical) Polynomial C_n^lambda(x)		Jacobi polynomials with alpha=beta=lambda-1/2, orthogonal with weight (1-x^2)^{lambda-1/2}.
s_laguerre_polynomial	axiom	Laguerre Polynomial L_n^alpha(x)		Orthogonal polynomials on [0,infty) with weight x^alpha e^{-x}, arising as hydrogen atom radial wavefunctions.
s_christoffel_darboux_formula	theorem	Christoffel-Darboux Formula		Closed form for the reproducing kernel sum p_k(x)p_k(y)/h_k as a ratio of consecutive polynomials.
s_favard_theorem	theorem	Favard's Theorem		Any polynomial sequence satisfying a three-term recurrence with positive coefficients is orthogonal with respect to some positive measure.
s_hahn_polynomial	axiom	Hahn Polynomial		Discrete orthogonal polynomials on {0,...,N} with hypergeometric weight, the discrete analog of Jacobi polynomials.
s_wilson_polynomial	axiom	Wilson Polynomial		The most general hypergeometric orthogonal polynomials (continuous variable), expressed as 4F3 series atop the Askey scheme.
s_racah_polynomial	axiom	Racah Polynomial		The most general discrete hypergeometric orthogonal polynomials, the discrete companion of Wilson polynomials.
s_askey_scheme	state	Askey Scheme		Hierarchical classification of hypergeometric orthogonal polynomials by limit relations, with Wilson and Racah at the top.
s_askey_wilson_polynomial	axiom	Askey-Wilson Polynomial		Most general q-hypergeometric orthogonal polynomial family defined as a 4_phi_3, atop the q-Askey scheme.
s_krawtchouk_polynomial	state	Krawtchouk Polynomial		Discrete orthogonal polynomials on {0,...,N} with binomial weight, characters of the Hamming scheme.
s_meixner_polynomial	state	Meixner Polynomial		Discrete orthogonal polynomials on non-negative integers with negative binomial weight.
s_charlier_polynomial	state	Charlier Polynomial		Discrete orthogonal polynomials on non-negative integers with Poisson weight.
s_chebyshev_polynomial_first_kind	axiom	Chebyshev Polynomial of the First Kind T_n(x)		T_n(cos theta) = cos(n theta), orthogonal on [-1,1] with weight (1-x^2)^{-1/2}, optimal for minimax approximation.
s_chebyshev_polynomial_second_kind	axiom	Chebyshev Polynomial of the Second Kind U_n(x)		U_n(cos theta) = sin((n+1)theta)/sin(theta), orthogonal on [-1,1] with weight (1-x^2)^{1/2}.
s_meixner_pollaczek_polynomial	state	Meixner-Pollaczek Polynomial P_n^lambda(x;phi)		Continuous orthogonal polynomials on R with weight |Gamma(lambda+ix)|^2 e^{(2phi-pi)x}, in the Askey scheme.
s_continuous_hahn_polynomial	state	Continuous Hahn Polynomial		Orthogonal polynomials on R with weight |Gamma(a+ix)Gamma(b+ix)|^2, between Wilson and Meixner-Pollaczek.
s_continuous_dual_hahn_polynomial	state	Continuous Dual Hahn Polynomial		Orthogonal polynomials on [0,infty) as a 3F2 at z=-1, dual to continuous Hahn polynomials.
s_dual_hahn_polynomial	state	Dual Hahn Polynomial		Discrete orthogonal polynomials dual to Hahn, related to 6j symbols in angular momentum theory.
s_rodrigues_formula_general	theorem	Rodrigues Formula (General)		p_n(x) = (1/(k_n w(x))) (d/dx)^n [w(x) sigma(x)^n], generating classical orthogonal polynomials from weight and classification.
s_three_term_recurrence	theorem	Three-Term Recurrence Relation		x p_n(x) = a_n p_{n+1}(x) + b_n p_n(x) + c_n p_{n-1}(x), the fundamental structure theorem for orthogonal polynomials.
s_complete_elliptic_integral_k	axiom	Complete Elliptic Integral K(k)		K(k) = integral_0^{pi/2} d theta/sqrt(1-k^2 sin^2 theta), the quarter-period of Jacobian elliptic functions.
s_complete_elliptic_integral_e	axiom	Complete Elliptic Integral E(k)		E(k) = integral_0^{pi/2} sqrt(1-k^2 sin^2 theta) d theta, giving the arc length of an ellipse.
s_complete_elliptic_integral_pi	axiom	Complete Elliptic Integral Pi(n,k)		Pi(n,k) = integral_0^{pi/2} d theta/((1-n sin^2 theta) sqrt(1-k^2 sin^2 theta)), the third-kind integral.
s_carlson_symmetric_rf	axiom	Carlson Symmetric R_F(x,y,z)		R_F = (1/2) integral_0^infty [(t+x)(t+y)(t+z)]^{-1/2} dt, symmetric elliptic integral of the first kind.
s_carlson_symmetric_rj	axiom	Carlson Symmetric R_J(x,y,z,p)		R_J = (3/2) integral_0^infty [(t+x)(t+y)(t+z)]^{-1/2}(t+p)^{-1} dt, symmetric third-kind integral.
s_carlson_symmetric_rd	axiom	Carlson Symmetric R_D(x,y,z)		R_D = R_J(x,y,z,z), special case of R_J with E(k) = R_F(0,k'^2,1) - (k^2/3) R_D(0,k'^2,1).
s_carlson_symmetric_rc	axiom	Carlson Symmetric R_C(x,y)		R_C(x,y) = R_F(x,y,y), degenerate symmetric integral reducing to inverse trig/hyperbolic functions.
t_arithmetic_geometric_mean	technique	Arithmetic-Geometric Mean M(a,g)		Iterates a_{n+1}=(a_n+g_n)/2, g_{n+1}=sqrt(a_n g_n) with quadratic convergence to compute elliptic integrals.
t_landen_transformation	technique	Landen Transformation		Modulus-changing substitution preserving elliptic integral values, enabling iterative computation.
s_legendre_relation_elliptic	theorem	Legendre's Relation for Elliptic Integrals		E(k)K'(k) + E'(k)K(k) - K(k)K'(k) = pi/2, the fundamental relation between complete elliptic integrals.
s_jacobi_theta_1	axiom	Jacobi Theta Function theta_1(z,q)		The odd Jacobi theta function theta_1 = 2 sum (-1)^n q^{(n+1/2)^2} sin((2n+1)z), vanishing at z=0.
s_jacobi_theta_2	axiom	Jacobi Theta Function theta_2(z,q)		The even Jacobi theta function theta_2 = 2 sum q^{(n+1/2)^2} cos((2n+1)z).
s_jacobi_theta_3	axiom	Jacobi Theta Function theta_3(z,q)		theta_3 = 1 + 2 sum q^{n^2} cos(2nz), the basic Jacobi theta function.
s_jacobi_theta_4	axiom	Jacobi Theta Function theta_4(z,q)		theta_4 = 1 + 2 sum (-1)^n q^{n^2} cos(2nz), related to theta_3 by a half-period shift.
s_theta_function_heat_equation	theorem	Theta Function Heat Equation		Jacobi theta functions satisfy 4pi i partial_tau theta = partial_z^2 theta, connecting them to heat kernels.
s_jacobi_imaginary_transformation	theorem	Jacobi Imaginary Transformation		theta_3(z|tau) = (-i tau)^{-1/2} exp(z^2/(pi i tau)) theta_3(z/tau | -1/tau), the modular transformation.
s_riemann_theta_function	axiom	Riemann Theta Function		Theta(z|Omega) = sum_{n in Z^g} exp(pi i n^T Omega n + 2pi i n^T z) for g-dimensional z and g x g Riemann matrix Omega.
s_siegel_modular_group	state	Siegel Modular Group Sp(2g,Z)		The group of transformations of genus-g Riemann matrices under which Riemann theta functions transform covariantly.
s_jacobian_sn	axiom	Jacobian Elliptic Function sn(z,k)		The sine-amplitude function inverting F(phi,k), doubly periodic with periods 4K and 2iK'.
s_jacobian_cn	axiom	Jacobian Elliptic Function cn(z,k)		The cosine-amplitude function satisfying sn^2 + cn^2 = 1, doubly periodic with periods 4K and 2K+2iK'.
s_jacobian_dn	axiom	Jacobian Elliptic Function dn(z,k)		The delta-amplitude function satisfying dn^2 + k^2 sn^2 = 1, doubly periodic with periods 2K and 4iK'.
s_jacobi_amplitude_function	axiom	Jacobi Amplitude Function am(u,k)		am(u,k) = phi where u = F(phi,k), the inverse of the incomplete elliptic integral of the first kind.
s_jacobi_twelve_elliptic_functions	state	Twelve Jacobian Elliptic Functions		The twelve pq(z,k) functions (p,q in {s,c,d,n}) defined as ratios of any two of H, H_1, Theta, Theta_1 theta functions.
s_jacobi_elliptic_addition_theorems	theorem	Jacobi Elliptic Addition Theorems		sn(u+v) = (sn u cn v dn v + sn v cn u dn u)/(1-k^2 sn^2 u sn^2 v) and analogues for cn, dn.
s_weierstrass_p_function	axiom	Weierstrass Elliptic Function wp(z)		wp(z;omega_1,omega_2) = 1/z^2 + sum'_{m,n} [1/(z-Omega_{m,n})^2 - 1/Omega_{m,n}^2], the canonical elliptic function of order 2.
s_weierstrass_sigma_function	axiom	Weierstrass Sigma Function sigma(z)		Entire function with simple zeros at lattice points: sigma(z) = z prod' (1-z/Omega) exp(z/Omega + z^2/(2Omega^2)).
s_weierstrass_zeta_function	axiom	Weierstrass Zeta Function zeta(z)		zeta(z) = sigma'(z)/sigma(z) = 1/z + sum' [1/(z-Omega) + 1/Omega + z/Omega^2], a quasi-periodic meromorphic function.
s_weierstrass_ode	axiom	Weierstrass Differential Equation		(wp')^2 = 4 wp^3 - g_2 wp - g_3, the ODE defining wp in terms of invariants g_2, g_3.
s_modular_discriminant_delta	axiom	Modular Discriminant Delta(tau)		Delta = g_2^3 - 27 g_3^2 = (2pi)^{12} eta(tau)^{24}, a weight-12 cusp form vanishing iff the lattice degenerates.
s_klein_j_invariant	axiom	Klein j-Invariant j(tau)		j = 1728 g_2^3/Delta, the unique modular function for SL(2,Z) classifying isomorphism classes of elliptic curves.
s_eisenstein_series_e2k	axiom	Eisenstein Series E_{2k}(tau)		E_{2k}(tau) = 1 - (4k/B_{2k}) sum sigma_{2k-1}(n) q^n, modular forms of weight 2k for k >= 2.
s_elliptic_modular_function_lambda	axiom	Elliptic Modular Function lambda(tau)		lambda(tau) = theta_2(0,q)^4/theta_3(0,q)^4 = k^2, a modular function for Gamma(2) mapping H to C\{0,1}.
s_bernoulli_polynomial	axiom	Bernoulli Polynomial B_n(x)		Defined by te^{xt}/(e^t-1) = sum B_n(x) t^n/n!, with B_n(0) = B_n the Bernoulli numbers.
s_euler_polynomial	axiom	Euler Polynomial E_n(x)		Defined by 2e^{xt}/(e^t+1) = sum E_n(x) t^n/n!, with E_n(1/2) related to Euler numbers.
s_genocchi_numbers	state	Genocchi Numbers		G_n = 2(1-2^n)B_n, integers related to Bernoulli numbers with generating function 2t/(e^t+1).
s_bernoulli_euler_complement_formula	theorem	Bernoulli-Euler Complement Formula		B_n(1-x) = (-1)^n B_n(x) and E_n(1-x) = (-1)^n E_n(x), reflection symmetries.
s_euler_maclaurin_formula	theorem	Euler-Maclaurin Summation Formula		sum f(k) = integral f(x)dx + f(a)/2 + f(b)/2 + sum B_{2k}/(2k)! (f^{(2k-1)}(b)-f^{(2k-1)}(a)) + R, bridging sums and integrals via Bernoulli numbers.
s_hurwitz_zeta_function	axiom	Hurwitz Zeta Function zeta(s,a)		zeta(s,a) = sum_{n=0}^infty (n+a)^{-s} for Re s > 1, generalizing Riemann zeta with zeta(s,1) = zeta(s).
s_polylogarithm	axiom	Polylogarithm Li_s(z)		Li_s(z) = sum z^n/n^s for |z|<=1, generalizing -log(1-z)=Li_1(z) and extending to all s by analytic continuation.
s_lerch_transcendent	axiom	Lerch Transcendent Phi(z,s,a)		Phi(z,s,a) = sum z^n/(n+a)^s unifying Hurwitz zeta (z=1), polylogarithm (a=1), and Riemann zeta (z=1,a=1).
s_fermi_dirac_integral	state	Fermi-Dirac Integral		F_s(x) = -Li_{s+1}(-e^x), standard integral for fermion energy distributions in quantum statistics.
s_bose_einstein_integral	state	Bose-Einstein Integral		G_s(x) = Li_{s+1}(e^x), the bosonic counterpart to the Fermi-Dirac integral.
s_stieltjes_constants	state	Stieltjes Constants		Coefficients gamma_n in the Laurent expansion zeta(s) = 1/(s-1) + sum (-1)^n gamma_n (s-1)^n/n! near s=1.
s_dirichlet_beta_function	axiom	Dirichlet Beta Function beta(s)		beta(s) = sum (-1)^n/(2n+1)^s = (1/4^s)(zeta(s,1/4) - zeta(s,3/4)), with beta(1) = pi/4.
s_clausen_function	state	Clausen Function Cl_2(theta)		Cl_2(theta) = -integral_0^theta ln|2 sin(t/2)| dt = sum sin(n theta)/n^2, related to the dilogarithm.
s_bell_number	axiom	Bell Number B_n		Number of partitions of {1,...,n}: B_n = sum S(n,k), with e.g.f. exp(e^x-1).
s_bell_polynomial	axiom	Bell Polynomial B_{n,k}		Partial Bell polynomials B_{n,k}(x_1,...,x_{n-k+1}) appearing in Faa di Bruno's formula for composite derivatives.
s_plane_partition	axiom	Plane Partition		A 2D array of nonneg integers weakly decreasing along rows and columns, with MacMahon generating function prod(1-q^k)^{-k}.
s_macmahon_box_formula	theorem	MacMahon Box Formula		Number of plane partitions in an a x b x c box = prod_{i,j,k} (i+j+k-1)/(i+j+k-2).
t_twelvefold_way	technique	Twelvefold Way		Classification of 12 counting problems by distinguishability of objects/boxes and mapping type.
s_derangement_number	axiom	Derangement Number D_n		Number of permutations with no fixed points: D_n = n! sum_{k=0}^n (-1)^k/k!, with D_n/n! -> 1/e.
s_narayana_number	axiom	Narayana Number N(n,k)		N(n,k) = (1/n) C(n,k) C(n,k-1), refining Catalan numbers by counting paths by number of peaks.
s_jordan_totient_function	state	Jordan's Totient Function J_k(n)		J_k(n) = n^k prod_{p|n}(1-p^{-k}), counting k-tuples coprime to n; J_1 = Euler totient phi.
s_ramanujan_tau_function	axiom	Ramanujan Tau Function		tau(n) defined by Delta(q) = q prod(1-q^n)^{24} = sum tau(n)q^n, a multiplicative function.
t_lambert_series	technique	Lambert Series		Converts between arithmetic functions and their summatory transforms via divisor sums.
s_liouville_lambda_function	state	Liouville Lambda Function lambda(n)		lambda(n) = (-1)^{Omega(n)} where Omega counts prime factors with multiplicity; Dirichlet series zeta(2s)/zeta(s).
s_dedekind_sum	state	Dedekind Sum s(h,k)		s(h,k) = sum_{r=1}^{k-1} ((r/k))((hr/k)) with sawtooth function ((x)), appearing in the transformation of log eta(tau).
s_mathieu_equation	axiom	Mathieu Equation		The ODE w'' + (a - 2q cos 2z)w = 0 arising from the wave equation in elliptic cylinder coordinates.
s_mathieu_characteristic_values	axiom	Mathieu Characteristic Values a_n(q), b_n(q)		Eigenvalues of Mathieu's equation for which periodic solutions of period pi or 2pi exist.
s_mathieu_function_ce	axiom	Mathieu Function ce_n(z,q)		Even periodic Mathieu eigenfunction with eigenvalue a_n(q), reducing to cos(nz) when q=0.
s_mathieu_function_se	axiom	Mathieu Function se_n(z,q)		Odd periodic Mathieu eigenfunction with eigenvalue b_n(q), reducing to sin(nz) when q=0.
s_modified_mathieu_equation	axiom	Modified Mathieu Equation		w'' - (a - 2q cosh 2z)w = 0 from Mathieu's equation by z -> iz, for radial separation.
s_hill_equation	axiom	Hill's Equation		The ODE w'' + Q(z)w = 0 with periodic Q, generalizing Mathieu's equation to arbitrary periodic potentials.
s_hill_discriminant	state	Hill Discriminant		Delta(lambda) = w_1(pi) + w_2'(pi), whose values +/-2 locate stability band edges of Hill's equation.
s_mathieu_floquet_solution	state	Mathieu-Floquet Solution		Solution of Mathieu's equation of the form e^{i nu z} P(z) with P periodic, where nu is the Floquet exponent.
s_lame_equation	axiom	Lame Equation		d^2w/dz^2 + (h - nu(nu+1)k^2 sn^2(z,k))w = 0, a doubly-periodic potential eigenvalue problem.
s_lame_polynomials	axiom	Lame Polynomials		Polynomial solutions of the Lame equation for integer nu, classified into eight types by parity structure.
s_ellipsoidal_harmonics	state	Ellipsoidal Harmonics		Products of Lame functions in confocal ellipsoidal coordinates, generalizing spherical harmonics.
s_lame_eigenvalues	state	Lame Eigenvalues		The 2nu+1 eigenvalues h of Lame's equation for integer nu, organized by parity and node structure.
s_spheroidal_equation	axiom	Spheroidal Differential Equation		d/dz[(1-z^2)dw/dz] + [lambda + gamma^2(1-z^2) - mu^2/(1-z^2)]w = 0 from Helmholtz in spheroidal coords.
s_angular_spheroidal_wave_function	axiom	Angular Spheroidal Wave Function Ps_n^m		Eigenfunction on [-1,1] reducing to Ferrers P_n^m when gamma=0, used in antenna and scattering theory.
s_radial_spheroidal_wave_functions	axiom	Radial Spheroidal Wave Functions		Four kinds of radial solutions analogous to spherical Bessel functions, for the exterior region.
s_spheroidal_eigenvalues	state	Spheroidal Eigenvalues lambda_n^m(gamma^2)		Eigenvalues of the angular spheroidal equation, reducing to n(n+1) when gamma=0.
s_prolate_spheroidal_function	state	Prolate Spheroidal Wave Function		Angular spheroidal function with gamma^2 > 0, bandlimited functions with maximal spatial concentration (Slepian).
s_heun_equation	axiom	Heun's Equation		Most general second-order Fuchsian ODE with four regular singular points at {0, 1, a, infty}.
s_heun_function	axiom	Heun Function Hl		Local Frobenius solution Hl(a,q;alpha,beta,gamma,delta;z) analytic at z=0 with accessory parameter q.
s_accessory_parameter	axiom	Accessory Parameter		The parameter q in Heun's equation not determined by exponents, controlling global connection behavior.
s_confluent_heun_equation	axiom	Confluent Heun Equation		Heun equation with two regular singularities merged into one irregular, having three singularities total.
s_biconfluent_heun_equation	state	Biconfluent Heun Equation		Heun equation with three regular singularities merged into one irregular of rank 2 plus one regular.
s_triconfluent_heun_equation	state	Triconfluent Heun Equation		Maximally confluent Heun equation with one irregular singularity at infinity.
s_heun_192_local_solutions	state	Heun's 192 Local Solutions		The 192 = 4 x 48 local Frobenius solutions of Heun's equation at its four singular points, generalizing Kummer's 24.
s_painleve_equation_p1	axiom	First Painleve Equation P_I		y'' = 6y^2 + z, the simplest Painleve transcendent with no free parameters.
s_painleve_equation_p2	axiom	Second Painleve Equation P_II		y'' = 2y^3 + zy + alpha, with Airy function solutions when alpha = n + 1/2.
s_painleve_equation_p3	axiom	Third Painleve Equation P_III		y'' = (y')^2/y - y'/z + (alpha y^2 + beta)/z + gamma y^3 + delta/y, with Bessel function solutions.
s_painleve_equation_p4	axiom	Fourth Painleve Equation P_IV		2yy'' = (y')^2 + 3y^4 + 8zy^3 + 4(z^2-alpha)y^2 + 2beta, with parabolic cylinder function solutions.
s_painleve_equation_p5	axiom	Fifth Painleve Equation P_V		Most complex Painleve equation with four parameters, with Whittaker/confluent hypergeometric solutions.
s_painleve_equation_p6	axiom	Sixth Painleve Equation P_VI		Most general Painleve equation with four parameters, containing P_I-P_V as coalescence limits.
s_painleve_coalescence_cascade	theorem	Painleve Coalescence Cascade		Limiting process P_VI->P_V->P_IV->P_II->P_I and P_V->P_III->P_II relating all six Painleve equations.
t_backlund_transformations_painleve	technique	Backlund Transformations for Painleve		Rational transformations mapping Painleve solutions to solutions with shifted parameter values.
s_isomonodromic_deformation_connection	theorem	Isomonodromic Deformation Connection		Each Painleve equation is the compatibility condition for isomonodromic deformation of an associated linear system.
s_painleve_property	axiom	Painleve Property		An ODE has the Painleve property if all movable singularities of all solutions are poles (no movable branch points or essential singularities).
s_coulomb_wave_equation	axiom	Coulomb Wave Equation		d^2w/drho^2 + (1 - 2eta/rho - l(l+1)/rho^2)w = 0 for scattering in a Coulomb field.
s_regular_coulomb_function	axiom	Regular Coulomb Function F_l(eta,rho)		Regular solution vanishing at origin, normalized by the Gamow factor C_l(eta).
s_irregular_coulomb_functions	axiom	Irregular Coulomb Functions G_l, H_l^{+/-}		Irregular solution G_l and Coulomb-Hankel functions H_l^{+/-} = G_l +/- iF_l for scattering theory.
s_gamow_sommerfeld_factor	state	Gamow-Sommerfeld Factor		C_0^2(eta) = 2pi eta/(e^{2pi eta}-1), the Coulomb barrier penetration probability for nuclear reactions.
s_coulomb_phase_shift	state	Coulomb Phase Shift sigma_l(eta)		sigma_l(eta) = arg Gamma(l+1+i eta), the phase shift from the long-range Coulomb potential.
s_wigner_3j_symbol	axiom	Wigner 3j Symbol		Coupling coefficient (j1 j2 j3; m1 m2 m3) for three angular momenta summing to zero total.
s_6j_symbol	axiom	6j Symbol (Racah Coefficient)		Recoupling coefficient {j1 j2 j3; l1 l2 l3} for three angular momenta, expressible as a 4F3 series.
s_9j_symbol	axiom	9j Symbol		Recoupling coefficient for four angular momenta, a sum of products of three 6j symbols.
s_gaunt_coefficient	state	Gaunt Coefficient		Integral of three spherical harmonics over the sphere, equal to a product of two 3j symbols.
s_biedenharn_elliott_identity	theorem	Biedenharn-Elliott Identity		Pentagon identity: sum of product of three 6j symbols over intermediate j equals a product of two 6j symbols.
s_clebsch_gordan_coefficient	axiom	Clebsch-Gordan Coefficient		C^{j,m}_{j_1,m_1,j_2,m_2} coupling two angular momenta j_1+j_2 to resultant j, related to 3j by phase and sqrt factor.
s_wigner_d_matrix	axiom	Wigner D-Matrix D^j_{m'm}(alpha,beta,gamma)		Matrix elements of rotation operators in angular momentum basis, D^j_{m'm} = e^{-im'alpha} d^j_{m'm}(beta) e^{-im gamma}.
s_multivariate_gamma_function	axiom	Multivariate Gamma Function Gamma_m(a)		Gamma_m(a) = pi^{m(m-1)/4} prod Gamma(a-(j-1)/2), arising in matrix variate distributions.
s_zonal_polynomials	axiom	Zonal Polynomials Z_kappa(T)		Symmetric polynomials in eigenvalues of symmetric matrix T, the spherical functions of GL(m)/O(m).
s_hypergeometric_matrix_argument	axiom	Hypergeometric Function of Matrix Argument		pF_q with matrix argument via sum over partitions of zonal polynomial expansions.
s_wishart_distribution	state	Wishart Distribution		Distribution of S = X^T X for Gaussian X, with density involving |S|^{(n-p-1)/2} exp(-tr Sigma^{-1}S/2) / (Gamma_p(n/2) |Sigma|^{n/2}).
s_jack_polynomial	state	Jack Polynomial P_kappa^{(alpha)}		Symmetric polynomials generalizing Schur polynomials (alpha=1) and zonal polynomials (alpha=2), indexed by partitions.
s_cuspoid_canonical_integral	axiom	Cuspoid Canonical Integral Psi_K		Psi_K(x) = integral exp(i(t^{K+2} + x_K t^K + ... + x_1 t)) dt, the K-th cuspoid diffraction integral.
s_pearcey_integral	state	Pearcey Integral		The K=2 cuspoid integral integral exp(i(t^4 + x_2 t^2 + x_1 t)) dt, governing cusp diffraction.
s_swallowtail_integral	state	Swallowtail Canonical Integral		The K=3 cuspoid integral integral exp(i(t^5 + zt^3 + yt^2 + xt)) dt for swallowtail diffraction.
s_elliptic_umbilic_catastrophe	axiom	Elliptic Umbilic Catastrophe		Catastrophe with phase s^3 - 3st^2 + z(s^2+t^2) + ys + xt, possessing three-fold rotational symmetry.
s_hyperbolic_umbilic_catastrophe	axiom	Hyperbolic Umbilic Catastrophe		Catastrophe with phase s^3 + t^3 + zst + ys + xt, dual to the elliptic umbilic.
s_butterfly_canonical_integral	state	Butterfly Canonical Integral		The K=4 cuspoid integral with phase t^6 + x_4 t^4 + x_3 t^3 + x_2 t^2 + x_1 t, the butterfly catastrophe.
t_hilbert_transform	technique	Hilbert Transform		Singular integral transform producing the analytic signal; maps cos to sin and connects real/imaginary parts of boundary values.
t_abel_transform	technique	Abel Transform		Integral transform for recovering radial profiles from projected data, inverse via Abel inversion formula.
t_hankel_contour_method	technique	Hankel Contour Method		Evaluates integrals and sums using a contour wrapping the negative real axis, fundamental for Gamma and 1/Gamma representations.
s_bateman_function	state	Bateman's Function k_nu(x)		k_nu(x) = (2/pi) integral_0^{pi/2} cos(x tan theta - nu theta) d theta, related to Bessel functions.
s_wright_function	state	Wright Function W_{lambda,mu}(z)		W_{lambda,mu}(z) = sum z^n/(n! Gamma(lambda n + mu)), generalizing Airy functions (lambda = +/-1/3) and Mittag-Leffler.
s_fox_wright_psi_function	state	Fox-Wright Psi Function		p_Psi_q: generalized hypergeometric series with Gamma ratios allowing non-unit increments, interpolating between pFq and H-function.
s_dedekind_eta_function	axiom	Dedekind Eta Function eta(tau)		eta(tau) = q^{1/24} prod_{n=1}^infty (1-q^n) for q=e^{2pi i tau}, a weight-1/2 modular form with multiplier system.
s_weber_modular_function	state	Weber Modular Functions f, f_1, f_2		f(tau) = e^{-pi i/24} eta((tau+1)/2)/eta(tau), algebraic at CM points and used in class field theory.
s_darboux_christoffel_kernel	state	Darboux-Christoffel Kernel		K_n(x,y) = sum_{k=0}^n p_k(x) p_k(y)/h_k, the reproducing kernel whose diagonal gives the density of states.
s_turán_inequality	theorem	Turan Inequality for Orthogonal Polynomials		P_n(x)^2 - P_{n-1}(x) P_{n+1}(x) >= 0 for x in [-1,1], a universal positivity result for Legendre polynomials.
s_dirichlet_character	axiom	Dirichlet Character chi mod q		A completely multiplicative function chi: Z -> C with period q and chi(n) = 0 when gcd(n,q) > 1.
s_gauss_sum	axiom	Gauss Sum g(chi)		g(chi) = sum_{a=1}^q chi(a) e^{2pi ia/q}, with |g(chi)| = sqrt(q) for primitive chi.
s_jacobi_sum	axiom	Jacobi Sum J(chi_1, chi_2)		J(chi_1,chi_2) = sum_{a+b=1} chi_1(a) chi_2(b) = g(chi_1)g(chi_2)/g(chi_1 chi_2), product of Gauss sums.
s_riemann_xi_function	axiom	Riemann Xi Function xi(s)		xi(s) = (1/2)s(s-1) pi^{-s/2} Gamma(s/2) zeta(s), entire and satisfying xi(s) = xi(1-s).
s_functional_equation_hurwitz_zeta	theorem	Functional Equation for Hurwitz Zeta		zeta(1-s,a) = Gamma(s)/(2pi)^s [e^{-pi is/2} F(a,s) + e^{pi is/2} F(-a,s)] with F the periodic zeta function.
s_partition_function_stat_mech	state	Partition Function (Statistical Mechanics) Z		Z = sum_states exp(-E/kT), the normalizing factor in Boltzmann statistics from which thermodynamic quantities derive.
s_hamiltonians_painleve	state	Painleve Hamiltonians		Each Painleve equation admits a Hamiltonian formulation y' = dH/dq, q' = -dH/dy with polynomial Hamiltonian H.
s_painleve_tau_function	state	Painleve Tau Function		tau(z) defined by d/dz ln tau = H(z), entire function whose zeros encode the poles of the Painleve transcendent.
t_separation_of_variables	technique	Separation of Variables in Coordinate Systems		Decomposes PDEs into ODEs by exploiting coordinate symmetry; 11 coordinate systems separate the Helmholtz equation.
s_connection_formulas_hypergeometric	theorem	Connection Formulas for 2F1		Express solutions at one singularity as linear combinations of solutions at another, with coefficients involving Gamma functions.
s_lipschitz_summation_formula	theorem	Lipschitz Summation Formula		sum_{n=0}^infty (n+a)^{-s} e^{2pi in z} = Gamma(s)^{-1} sum_{n=-infty}^infty 1/(z-n+a)^s for Im z > 0, connecting Hurwitz zeta to periodic sums.
s_ramanujan_master_theorem	theorem	Ramanujan's Master Theorem		If f(x) = sum (-1)^n phi(n) x^n/n!, then integral_0^infty x^{s-1} f(x) dx = Gamma(s) phi(-s), a Mellin transform result.
s_big_q_jacobi_polynomial	state	Big q-Jacobi Polynomial		q-analog of Jacobi polynomials on a q-linear lattice, in the q-Askey scheme between Askey-Wilson and q-Hahn.
s_little_q_jacobi_polynomial	state	Little q-Jacobi Polynomial		Simpler q-analog of Jacobi polynomials on {q^k}, a limiting case of big q-Jacobi polynomials.
s_al_salam_chihara_polynomial	state	Al-Salam-Chihara Polynomial		q-orthogonal polynomials on [-1,1] defined as 3_phi_2, in the q-Askey scheme below Askey-Wilson.
s_q_racah_polynomial	state	q-Racah Polynomial		q-analog of Racah polynomials atop the discrete branch of the q-Askey scheme.
s_q_hahn_polynomial	state	q-Hahn Polynomial		q-analog of Hahn polynomials, discrete orthogonal on {1, q, ..., q^N}.
s_volterra_integral_equation	axiom	Volterra Integral Equation		f(x) = g(x) + lambda integral_a^x K(x,t) f(t) dt, with variable upper limit making it uniquely solvable.
s_fredholm_integral_equation	axiom	Fredholm Integral Equation		f(x) = g(x) + lambda integral_a^b K(x,t) f(t) dt, with fixed limits; solvability governed by Fredholm theory.
s_appell_sequence	state	Appell Sequence		A polynomial sequence {p_n} with d/dx p_n = n p_{n-1}, including Bernoulli, Euler, and Hermite polynomials.
s_sheffer_sequence	state	Sheffer Sequence		Polynomial sequence defined by generating function A(t) exp(x B(t)) = sum p_n(x) t^n/n!, generalizing Appell sequences.
s_bessel_zeros	state	Zeros of Bessel Functions		The positive zeros j_{nu,s} of J_nu(z) are real, simple, and interlace with zeros of J_{nu+1}; asymptotically j_{nu,s} ~ (s+nu/2-1/4)pi.
s_debye_expansion_bessel	theorem	Debye Asymptotic Expansion for Bessel Functions		Uniform large-order expansion J_nu(nu z) ~ (phi/sqrt(1-z^2))^{1/2} [Ai(nu^{2/3}phi)/(nu^{1/3})] sum, valid through the transition region.
s_olver_uniform_asymptotics_bessel	theorem	Olver Uniform Asymptotic Expansion		Uniform expansion of J_nu(nu z) valid for all z including the turning point, involving Airy functions and their derivatives.
s_weierstrass_addition_theorem	theorem	Weierstrass Addition Theorem		wp(u+v) = -wp(u) - wp(v) + (1/4)((wp'(u)-wp'(v))/(wp(u)-wp(v)))^2, the algebraic addition law.
s_abel_theorem_elliptic	theorem	Abel's Theorem for Elliptic Functions		For an elliptic function f of order n, the sum of zeros minus sum of poles in a period parallelogram is a lattice point.
t_stationary_phase_method	technique	Method of Stationary Phase		Asymptotic evaluation of oscillatory integrals via contributions from critical points of the phase phi(x).
t_watson_transformation	technique	Watson Transformation		Converts a sum over angular momentum into a contour integral, enabling asymptotic evaluation of scattering amplitudes.
s_perron_formula	theorem	Perron's Formula		sum_{n<=x} a_n = (1/2pi i) integral_{c-iT}^{c+iT} F(s) x^s/s ds + error, recovering partial sums from Dirichlet series.
s_marcenko_pastur_distribution	state	Marchenko-Pastur Distribution		Limiting spectral distribution of Wishart matrices X^TX/n as p/n -> gamma, with density on [(1-sqrt(gamma))^2, (1+sqrt(gamma))^2].
s_motzkin_number	axiom	Motzkin Number M_n		Number of paths from (0,0) to (n,0) with steps (1,1), (1,-1), (1,0) staying nonneg; M_n = sum C(n,2k) C_k.
s_schroder_number	axiom	Schroder Number		Number of lattice paths from (0,0) to (n,n) with steps (1,0), (0,1), (1,1) not crossing the diagonal.
s_greatest_common_divisor	axiom	Greatest common divisor		The greatest common divisor gcd(a,b) is the largest positive integer dividing both a and b, defined for any pair of integers not both zero.
s_least_common_multiple	axiom	Least common multiple		The least common multiple lcm(a,b) is the smallest positive integer divisible by both a and b.
s_gcd_lcm_product_relation	theorem	GCD–LCM product relation		For positive integers a and b, gcd(a,b) · lcm(a,b) = ab.
s_prime_number_definition	axiom	Prime number		A natural number p > 1 whose only positive divisors are 1 and p.
s_composite_number	axiom	Composite number		A natural number n > 1 that is not prime, equivalently n = ab for some integers 1 < a, b < n.
s_canonical_prime_factorization	axiom	Canonical prime factorization		The standard-form representation n = p₁^{a₁} p₂^{a₂} ··· pₖ^{aₖ} with p₁ < p₂ < ··· < pₖ prime and each aᵢ ≥ 1.
s_divisor_characterization_via_factorization	theorem	Divisor characterization via prime factorization		If n = ∏ pᵢ^{aᵢ}, then d | n iff d = ∏ pᵢ^{bᵢ} with 0 ≤ bᵢ ≤ aᵢ for all i.
s_gcd_lcm_via_prime_factorization	theorem	GCD and LCM via prime factorization		For n = ∏ pᵢ^{aᵢ} and m = ∏ pᵢ^{bᵢ}, gcd(n,m) = ∏ pᵢ^{min(aᵢ,bᵢ)} and lcm(n,m) = ∏ pᵢ^{max(aᵢ,bᵢ)}.
s_integrality_of_binomial_coefficients	theorem	Integrality of binomial coefficients		The binomial coefficient C(n,k) = n!/(k!(n−k)!) is an integer for all 0 ≤ k ≤ n.
s_prime_divisibility_of_binomial_coefficients	theorem	Prime divisibility of binomial coefficients		For prime p and 1 ≤ k ≤ p−1, the prime p divides C(p,k).
s_completely_multiplicative_function	axiom	Completely multiplicative function		An arithmetic function f with f(1) = 1 and f(mn) = f(m)f(n) for all positive integers m, n without any coprimality restriction.
s_multiplicative_determination_by_prime_powers	theorem	Multiplicative function determination by prime powers		A multiplicative function f is completely determined by its values f(pᵃ) at prime powers, since f(n) = ∏ f(pᵢ^{aᵢ}).
s_mobius_sum_identity	theorem	Möbius function sum identity		For all positive integers n, ∑_{d|n} μ(d) = [n = 1], equaling 1 if n = 1 and 0 otherwise.
s_euler_totient_divisor_sum	theorem	Euler totient divisor sum		For all positive integers n, ∑_{d|n} φ(d) = n.
s_euler_totient_product_formula	theorem	Euler totient product formula		φ(n) = n ∏_{p|n} (1 − 1/p), expressing the totient as a product over the distinct prime divisors of n.
s_dirichlet_identity_function_epsilon	axiom	Dirichlet identity function ε		The function ε(n) = [n = 1], serving as the multiplicative identity for Dirichlet convolution.
s_dirichlet_inverse_existence	theorem	Dirichlet inverse existence		An arithmetic function f has a unique Dirichlet inverse f⁻¹ (with f ∗ f⁻¹ = ε) if and only if f(1) ≠ 0.
s_dirichlet_product_abelian_group	theorem	Abelian group of arithmetic functions under Dirichlet product		The set of arithmetic functions f with f(1) ≠ 0 forms an abelian group under Dirichlet convolution, with identity ε.
s_unit_function	axiom	Unit function u(n) = 1		The arithmetic function u(n) = 1 for all positive integers n, a completely multiplicative function.
s_number_theoretic_identity_function_N	axiom	Number-theoretic identity function N(n) = n		The arithmetic function N(n) = n, a completely multiplicative function.
s_mobius_dirichlet_inverse_of_unit	theorem	Möbius function is Dirichlet inverse of unit function		μ = u⁻¹ under Dirichlet convolution, equivalently μ ∗ u = ε.
s_von_mangoldt_divisor_sum_identity	theorem	Von Mangoldt divisor sum identity		For all n ≥ 1, ∑_{d|n} Λ(d) = log n, equivalently Λ = μ ∗ log as Dirichlet convolution.
s_liouville_completely_multiplicative	theorem	Liouville function is completely multiplicative		The Liouville function λ(n) = (−1)^{Ω(n)} satisfies λ(mn) = λ(m)λ(n) for all positive integers m, n.
s_liouville_divisor_sum	theorem	Liouville function divisor sum		∑_{d|n} λ(d) = [n is a perfect square], equaling 1 if n is a perfect square and 0 otherwise.
s_divisor_function_sigma_alpha	axiom	Generalized divisor function σ_α		The function σ_α(n) = ∑_{d|n} d^α, a family of arithmetic functions parametrized by α ∈ ℂ.
s_number_of_divisors_function	axiom	Number of divisors function d(n)		The function d(n) = σ₀(n) = |{d > 0 : d | n}|, counting the number of positive divisors of n.
s_sum_of_divisors_function	axiom	Sum of divisors function σ(n)		The function σ(n) = σ₁(n) = ∑_{d|n} d, giving the sum of all positive divisors of n.
s_divisor_function_multiplicativity	theorem	Divisor function multiplicativity		The generalized divisor function σ_α is multiplicative for every α.
s_divisor_function_prime_power_formula	theorem	Divisor function prime power formula		For prime p and α ≠ 0, σ_α(pᵃ) = (p^{α(a+1)} − 1)/(p^α − 1); for α = 0, d(pᵃ) = a + 1.
s_product_multiplicative_is_multiplicative	theorem	Dirichlet product of multiplicative functions is multiplicative		If f and g are multiplicative, their Dirichlet product f ∗ g is also multiplicative.
s_dirichlet_inverse_multiplicative_is_multiplicative	theorem	Dirichlet inverse of multiplicative function is multiplicative		If f is multiplicative, its Dirichlet inverse f⁻¹ is also multiplicative.
s_bell_series	axiom	Bell series		The Bell series of an arithmetic function f at a prime p is the formal power series f_p(x) = ∑_{n=0}^{∞} f(pⁿ) xⁿ.
s_bell_series_multiplication	theorem	Bell series multiplication theorem		If h = f ∗ g (Dirichlet product) and all three are multiplicative, then h_p(x) = f_p(x) · g_p(x) for every prime p.
s_euler_totient_via_mobius	theorem	Euler totient via Möbius inversion		φ = μ ∗ N under Dirichlet convolution, equivalently φ(n) = ∑_{d|n} μ(d)(n/d).
s_euler_totient_multiplicativity	theorem	Euler totient multiplicativity		Euler's totient function φ is multiplicative: φ(mn) = φ(m)φ(n) whenever gcd(m,n) = 1.
s_dirichlet_ring_of_arithmetic_functions	axiom	Dirichlet ring of arithmetic functions		The commutative ring of all arithmetic functions under pointwise addition and Dirichlet convolution, with identity ε.
s_generalized_mobius_inversion	theorem	Generalized Möbius inversion		If F(x) = ∑_{n≤x} f(x/n) for all x ≥ 1, then f(x) = ∑_{n≤x} μ(n)F(x/n), and conversely.
s_completely_multiplicative_inverse_formula	theorem	Completely multiplicative Dirichlet inverse formula		If f is completely multiplicative, then f⁻¹(n) = μ(n)f(n) for all n.
s_asymptotic_equality	axiom	Asymptotic equality		f ∼ g means f(x)/g(x) → 1 as x → ∞.
s_average_order_definition	axiom	Average order of an arithmetic function		A function g(n) is an average order of f(n) if ∑_{n≤x} f(n) ∼ ∑_{n≤x} g(n).
s_euler_summation_formula	theorem	Euler summation formula		∑_{a<n≤b} f(n) = ∫_a^b f(t)dt + ∫_a^b (t − [t])f′(t)dt + f(x)([x] − x) − f(y)([y] − y), for f with continuous derivative.
s_average_order_of_d_n	theorem	Average order of d(n)		∑_{n≤x} d(n) = x log x + (2γ − 1)x + O(√x), where γ is the Euler–Mascheroni constant.
t_dirichlet_hyperbola_method	technique	Dirichlet hyperbola method		Splits the sum over the region {(a,b) : ab ≤ x} into three parts along a hyperbola to improve error terms in divisor-sum estimates.
s_average_order_of_sigma_n	theorem	Average order of σ(n)		∑_{n≤x} σ(n) = (π²/12)x² + O(x log x).
s_average_order_of_phi_n	theorem	Average order of φ(n)		∑_{n≤x} φ(n) = (3/π²)x² + O(x log x).
s_harmonic_sum_estimate	theorem	Harmonic sum estimate		∑_{n≤x} 1/n = log x + γ + O(1/x), where γ is the Euler–Mascheroni constant.
s_average_order_of_mu	theorem	Average order of μ(n) is zero		M(x) = ∑_{n≤x} μ(n) = o(x), equivalently the Möbius function has mean value zero.
s_mertens_function	axiom	Mertens function		M(x) = ∑_{n≤x} μ(n), the summatory function of the Möbius function.
t_lattice_point_counting	technique	Lattice point counting		Estimates arithmetic sums by counting integer lattice points in planar regions, e.g., under the hyperbola ab = x.
s_asymptotic_reciprocal_phi	theorem	Asymptotic average of 1/φ(n)		∑_{n≤x} 1/φ(n) = (315ζ(3))/(2π⁴) · log x + O(1).
s_prime_counting_function	axiom	Prime counting function π(x)		π(x) = |{p ≤ x : p prime}|, counting the number of primes not exceeding x.
s_theta_psi_relation	theorem	Relation between Chebyshev θ and ψ		ψ(x) = θ(x) + O(√x log² x), so θ(x) ∼ x if and only if ψ(x) ∼ x.
s_abel_summation_pi_theta	theorem	Abel summation relating π(x) and θ(x)		θ(x) = π(x) log x − ∫₂ˣ π(t)/t dt, obtained by partial summation.
s_equivalence_of_pnt_forms	theorem	Equivalence of PNT forms		The statements π(x) ∼ x/log x, θ(x) ∼ x, ψ(x) ∼ x, and π(x) ∼ Li(x) are mutually equivalent.
s_shapiro_tauberian_theorem	theorem	Shapiro's Tauberian theorem		If aₙ ≥ 0 and ∑_{n≤x} aₙ [x/n] = x log x + O(x), then ∑_{n≤x} aₙ/n = log x + O(1).
s_selberg_identity	theorem	Selberg's identity		Λ(n) log n + ∑_{d|n} Λ(d)Λ(n/d) = ∑_{d|n} μ(d) log²(n/d), an exact identity for the von Mangoldt function.
s_nth_prime_asymptotic	theorem	Asymptotic formula for the n-th prime		The n-th prime satisfies pₙ ∼ n log n as n → ∞.
s_sum_prime_reciprocals_diverges	theorem	Divergence of the sum of prime reciprocals		∑_{p prime} 1/p = ∞; more precisely ∑_{p≤x} 1/p = log log x + M + O(1/log x) where M is the Meissel–Mertens constant.
s_residue_class	axiom	Residue class		The residue class of a modulo m is the equivalence class a + mℤ = {a + km : k ∈ ℤ}.
s_complete_residue_system	axiom	Complete residue system		A set of m integers containing exactly one representative from each residue class modulo m.
s_reduced_residue_system	axiom	Reduced residue system		A set of φ(m) integers, one from each residue class coprime to m.
s_linear_congruence	axiom	Linear congruence		An equation ax ≡ b (mod m) seeking integer solutions x.
s_linear_congruence_solvability	theorem	Linear congruence solvability		The congruence ax ≡ b (mod m) has a solution iff gcd(a,m) | b, in which case there are exactly gcd(a,m) incongruent solutions.
s_multiplicative_inverse_mod_m	theorem	Multiplicative inverse modulo m		An integer a has a multiplicative inverse modulo m if and only if gcd(a,m) = 1.
s_polynomial_congruence	axiom	Polynomial congruence		An equation f(x) ≡ 0 (mod m) where f is a polynomial with integer coefficients.
s_lagrange_polynomial_congruence_mod_p	theorem	Lagrange's theorem on polynomial congruences mod p		A polynomial of degree n with integer coefficients has at most n incongruent roots modulo a prime p.
s_order_of_group_element	axiom	Order of a group element		The smallest positive integer n such that gⁿ = e, for an element g in a finite group G.
s_order_divides_group_order	theorem	Order of element divides group order		In a finite group G, the order of every element divides |G|.
s_cyclic_group	axiom	Cyclic group		A group generated by a single element, isomorphic to ℤ/nℤ for some positive integer n.
s_subgroups_of_cyclic_are_cyclic	theorem	Subgroups of cyclic groups are cyclic		Every subgroup of a cyclic group is cyclic, and for each divisor d of |G| there is exactly one subgroup of order d.
s_direct_product_of_groups	axiom	Direct product of groups		The direct product G × H of two groups is the set of pairs (g,h) with componentwise multiplication.
s_group_character	axiom	Group character (finite abelian)		A homomorphism χ: G → ℂ* from a finite abelian group G to the multiplicative group of nonzero complex numbers.
s_character_group_isomorphic_to_G	theorem	Character group is isomorphic to G		For a finite abelian group G, its character group Ĝ is (non-canonically) isomorphic to G, so |Ĝ| = |G|.
s_orthogonality_dirichlet_characters	theorem	Orthogonality relations for Dirichlet characters		∑_{χ mod m} χ(a)χ̄(b) = φ(m) if a ≡ b (mod m) with gcd(a,m) = 1, and 0 otherwise.
t_extraction_ap_via_characters	technique	Extraction of arithmetic progressions via characters		Isolates a single residue class using the orthogonality relation: 𝟙_{n≡l} = (1/φ(m)) ∑_χ χ̄(l) χ(n).
s_euler_product_for_l_function	theorem	Euler product for L(s,χ)		For Re(s) > 1, L(s,χ) = ∏_p (1 − χ(p)p⁻ˢ)⁻¹, expressing the Dirichlet L-function as a product over primes.
s_l_chi0_relation_to_zeta	state	L(s,χ₀) relation to Riemann zeta		L(s,χ₀) = ζ(s) ∏_{p|m} (1 − p⁻ˢ) for the principal character χ₀ mod m.
s_log_l_series	state	Logarithm of L(s,χ) series		−log L(s,χ) = ∑_p ∑_{k≥1} χ(pᵏ)/(kpᵏˢ) for Re(s) > 1, from the Euler product.
s_primes_in_ap_dirichlet_series	state	Dirichlet series for primes in arithmetic progressions		∑_{p≡l(m)} p⁻ˢ = (1/φ(m)) ∑_χ χ̄(l) log L(s,χ) + bounded terms for Re(s) > 1.
s_l_chi0_pole_at_1	theorem	Pole of L(s,χ₀) at s = 1		L(s,χ₀) has a simple pole at s = 1 with residue ∏_{p|m}(1 − 1/p), inherited from the pole of ζ(s).
t_product_all_l_functions_mod_m	technique	Product of all L-functions mod m		The product ∏_χ L(s,χ) yields a Dirichlet series with nonnegative coefficients, used to prove L(1,χ) ≠ 0 for complex χ.
t_nonvanishing_real_chi_via_landau	technique	Nonvanishing of L(1,χ) for real χ via Landau's theorem		Proves L(1,χ) ≠ 0 for real χ using Landau's theorem that a Dirichlet series with nonneg coefficients has a singularity at its abscissa of convergence.
s_equidistribution_primes_residue_classes	theorem	Equidistribution of primes in residue classes		The primes are equidistributed among the φ(m) reduced residue classes mod m: ∑_{p≤x, p≡l(m)} 1/p ∼ (1/φ(m)) log log x.
s_convergence_l_nonprincipal_s_gt_0	theorem	Convergence of L(s,χ) for nonprincipal χ at s > 0		For nonprincipal χ, the series L(s,χ) converges (conditionally) for all s with Re(s) > 0.
s_periodic_arithmetical_function	axiom	Periodic arithmetical function		An arithmetical function f satisfying f(n + q) = f(n) for all integers n and some fixed positive integer q.
s_finite_fourier_expansion	theorem	Finite Fourier expansion of periodic functions		Every arithmetical function with period q can be written as f(n) = ∑_{k=1}^q a(k) e^{2πikn/q} with a(k) = (1/q) ∑_{r=1}^q f(r) e^{-2πikr/q}.
s_ramanujan_sum_mobius_formula	theorem	Ramanujan sum via Möbius function		c_q(n) = ∑_{d|gcd(n,q)} μ(q/d) d, expressing the Ramanujan sum as a divisor sum involving μ.
s_ramanujan_sum_multiplicativity	theorem	Multiplicativity of Ramanujan sums		For fixed n, the function q ↦ c_q(n) is multiplicative: c_{q₁q₂}(n) = c_{q₁}(n) c_{q₂}(n) when gcd(q₁,q₂) = 1.
t_ramanujan_expansion	technique	Ramanujan expansion of arithmetical functions		Expresses an arithmetical function as a pointwise convergent series in Ramanujan sums, analogous to Fourier expansion.
s_separability_of_gauss_sums	axiom	Separability of Gauss sums		A character χ mod m is separable if G(n,χ) = χ̄(n) G(1,χ) holds for all integers n.
s_primitive_characters_are_separable	theorem	Primitive characters are separable		Every primitive Dirichlet character mod m is separable: G(n,χ) = χ̄(n) G(1,χ) for all n.
s_gauss_sum_modulus_primitive	theorem	Modulus of Gauss sum for primitive characters		For a primitive character χ mod m, |G(1,χ)|² = m, equivalently |G(1,χ)| = √m.
s_quadratic_gauss_sum	state	Quadratic Gauss sum		The sum G = ∑_{a=0}^{p−1} (a/p) e^{2πia/p} associated to the Legendre symbol, satisfying G² = (−1)^{(p−1)/2} p.
s_sign_of_quadratic_gauss_sum	theorem	Sign of the quadratic Gauss sum		The quadratic Gauss sum equals √p if p ≡ 1 (mod 4) and i√p if p ≡ 3 (mod 4).
s_chi_via_gauss_sum	theorem	Representation of χ(n) via Gauss sums		For primitive χ mod m, χ(n) = (1/G(1,χ̄)) ∑_{a=1}^m χ̄(a) e^{2πian/m}, recovering χ from its Gauss sum.
s_gauss_sum_product_identity	theorem	Gauss sum product identity G(1,χ)G(1,χ̄) = χ(−1)m		For primitive χ mod m, G(1,χ)G(1,χ̄) = χ(−1) · m.
s_von_sterneck_formula	theorem	Von Sterneck's formula for Ramanujan sums		c_q(n) = μ(q/gcd(n,q)) φ(q)/φ(q/gcd(n,q)) when q/gcd(n,q) is squarefree, and 0 otherwise.
s_half_residues_are_qr	theorem	Half of reduced residues are quadratic residues		Among 1, 2, …, p−1, exactly (p−1)/2 are quadratic residues mod p and (p−1)/2 are nonresidues.
s_euler_criterion	theorem	Euler's criterion		For odd prime p and gcd(a,p) = 1, the Legendre symbol (a/p) ≡ a^{(p−1)/2} (mod p).
s_legendre_symbol_completely_multiplicative	theorem	Legendre symbol is completely multiplicative		The Legendre symbol satisfies (ab/p) = (a/p)(b/p) for all integers a, b.
s_first_supplement_qr	theorem	First supplement to quadratic reciprocity		(−1/p) = (−1)^{(p−1)/2}, so −1 is a quadratic residue mod p iff p ≡ 1 (mod 4).
s_gauss_lemma_legendre	theorem	Gauss's lemma for the Legendre symbol		(a/p) = (−1)ⁿ where n counts the least positive residues of a, 2a, …, ((p−1)/2)a exceeding p/2.
s_second_supplement_qr	theorem	Second supplement to quadratic reciprocity		(2/p) = (−1)^{(p²−1)/8}, so 2 is a quadratic residue mod p iff p ≡ ±1 (mod 8).
s_jacobi_symbol	axiom	Jacobi symbol		For odd n = p₁^{a₁}···pₖ^{aₖ}, the Jacobi symbol (a/n) = ∏ (a/pᵢ)^{aᵢ} extends the Legendre symbol.
s_jacobi_reciprocity	theorem	Quadratic reciprocity for Jacobi symbol		For coprime odd positive m, n: (m/n)(n/m) = (−1)^{((m−1)/2)((n−1)/2)}.
s_jacobi_first_supplement	theorem	First supplement for Jacobi symbol		(−1/n) = (−1)^{(n−1)/2} for every odd positive integer n.
s_jacobi_second_supplement	theorem	Second supplement for Jacobi symbol		(2/n) = (−1)^{(n²−1)/8} for every odd positive integer n.
s_eisenstein_lemma	theorem	Eisenstein's lemma		(a/p) = (−1)^{∑_{k=1}^{(p−1)/2} ⌊ka/p⌋}, expressing the Legendre symbol via a sum of floor functions.
t_qr_proof_via_gauss_sums	technique	Proof of quadratic reciprocity via Gauss sums		Raises the quadratic Gauss sum to the q-th power and reduces mod q to derive (p/q)(q/p) = (−1)^{((p−1)/2)((q−1)/2)}.
t_lattice_point_qr_proof	technique	Lattice point counting proof of quadratic reciprocity		Counts lattice points in the rectangle [1,(p−1)/2] × [1,(q−1)/2] above and below the diagonal to prove reciprocity.
s_number_solutions_x2_eq_a_mod_p	theorem	Number of solutions of x² ≡ a (mod p)		The number of solutions of x² ≡ a (mod p) is 1 + (a/p): two if a is a nonzero QR, zero if a is a QNR, one if p | a.
s_order_of_integer_mod_m	axiom	Order of an integer modulo m		For gcd(a,m) = 1, the order ord_m(a) is the smallest positive integer k such that aᵏ ≡ 1 (mod m).
s_order_divides_iff	theorem	Order divides exponent iff congruence holds		aⁿ ≡ 1 (mod m) if and only if ord_m(a) | n.
s_order_of_powers	theorem	Order of a power of an element		ord_m(aʲ) = k/gcd(j,k) where k = ord_m(a).
s_number_of_primitive_roots	theorem	Number of primitive roots modulo m		If primitive roots exist modulo m, there are exactly φ(φ(m)) incongruent primitive roots.
s_primitive_roots_mod_prime_powers	theorem	Existence of primitive roots modulo p^k		For odd prime p and k ≥ 1, the group (ℤ/pᵏℤ)* is cyclic, i.e., primitive roots exist modulo pᵏ.
s_primitive_roots_mod_2pk	theorem	Existence of primitive roots modulo 2p^k		For odd prime p and k ≥ 1, the group (ℤ/2pᵏℤ)* is cyclic.
s_no_primitive_roots_mod_2k	theorem	Nonexistence of primitive roots modulo 2^k for k ≥ 3		For k ≥ 3, (ℤ/2ᵏℤ)* ≅ ℤ/2 × ℤ/2^{k−2} is not cyclic, so no primitive root exists.
s_classification_moduli_primitive_roots	theorem	Classification of moduli having primitive roots		Primitive roots exist modulo m iff m ∈ {1, 2, 4, pᵏ, 2pᵏ} for an odd prime p and k ≥ 1.
s_discrete_logarithm_index	axiom	Index (discrete logarithm) modulo m		For primitive root g mod m, ind_g(a) is the unique integer r mod φ(m) with gʳ ≡ a (mod m).
s_index_homomorphism	theorem	Homomorphism properties of the index function		ind_g(ab) ≡ ind_g(a) + ind_g(b) (mod φ(m)) and ind_g(aⁿ) ≡ n·ind_g(a) (mod φ(m)), giving (ℤ/mℤ)* ≅ ℤ/φ(m)ℤ.
t_solving_congruences_via_indices	technique	Solving polynomial congruences via indices		Reduces xⁿ ≡ a (mod m) to the linear congruence n·ind_g(x) ≡ ind_g(a) (mod φ(m)).
s_power_residue_criterion	theorem	Power residue criterion via indices		xⁿ ≡ a (mod p) is solvable iff a^{(p−1)/d} ≡ 1 (mod p) where d = gcd(n, p−1), with exactly d solutions.
s_characters_mod_p_via_primitive_roots	theorem	Characters mod p via primitive roots		Every Dirichlet character mod p is χⱼ(gᵏ) = e^{2πijk/(p−1)} for unique j ∈ {0,…,p−2}, where g is a primitive root.
s_abscissa_relation	theorem	Abscissa of convergence vs absolute convergence relation		For a Dirichlet series ∑ f(n)/nˢ, σ_c ≤ σ_a ≤ σ_c + 1, bounding the gap between conditional and absolute convergence.
s_analyticity_dirichlet_series	theorem	Analyticity of Dirichlet series		A Dirichlet series ∑ aₙ n⁻ˢ represents a holomorphic function in its half-plane of convergence Re(s) > σ_c, differentiable term by term.
s_uniqueness_dirichlet_series	theorem	Uniqueness theorem for Dirichlet series		If ∑ aₙ/nˢ = ∑ bₙ/nˢ in some half-plane, then aₙ = bₙ for all n.
s_euler_product_multiplicative	theorem	Euler product for multiplicative functions		If f is multiplicative and ∑ f(n)/nˢ converges absolutely, then ∑ f(n)/nˢ = ∏_p (1 + f(p)/pˢ + f(p²)/p²ˢ + ···).
s_euler_product_completely_multiplicative	theorem	Euler product for completely multiplicative functions		If f is completely multiplicative with absolutely convergent Dirichlet series, then ∑ f(n)/nˢ = ∏_p (1 − f(p)/pˢ)⁻¹.
s_dirichlet_series_1_over_zeta	state	Dirichlet series for 1/ζ(s)		1/ζ(s) = ∑ μ(n)/nˢ for Re(s) > 1, expressing the reciprocal of zeta via the Möbius function.
s_dirichlet_series_zeta_prime_over_zeta	state	Dirichlet series for −ζ′(s)/ζ(s)		−ζ′(s)/ζ(s) = ∑ Λ(n)/nˢ for Re(s) > 1, the logarithmic derivative of zeta via the von Mangoldt function.
s_l_function_via_hurwitz_zeta	theorem	L(s,χ) as linear combination of Hurwitz zeta functions		L(s,χ) = k⁻ˢ ∑_{r=1}^k χ(r) ζ(s, r/k) for a character χ mod k.
s_analytic_continuation_zeta_re_gt_0	theorem	Analytic continuation of ζ(s) to Re(s) > 0		ζ(s) extends to a meromorphic function on Re(s) > 0 with a simple pole at s = 1 with residue 1.
s_analytic_continuation_l_re_gt_0	theorem	Analytic continuation of L(s,χ) to Re(s) > 0		For nonprincipal χ, L(s,χ) extends to an analytic function on Re(s) > 0 via partial summation.
s_gamma_analytic_continuation	theorem	Analytic continuation of Γ to meromorphic function on ℂ		Γ(s) extends to a meromorphic function on all of ℂ with simple poles at s = 0, −1, −2, … and residue (−1)ⁿ/n! at s = −n.
s_bernoulli_polynomial_difference	theorem	Difference equation for Bernoulli polynomials		Bₙ(x+1) − Bₙ(x) = nxⁿ⁻¹, characterizing Bernoulli polynomials via their first difference.
s_fourier_expansion_bernoulli_polynomials	theorem	Fourier expansion of Bernoulli polynomials		Bₙ(x) = −n!/(2πi)ⁿ ∑_{k≠0} e^{2πikx}/kⁿ for 0 < x < 1 (n ≥ 2).
s_contour_integral_hurwitz_zeta	theorem	Contour integral representation of Hurwitz zeta		ζ(s,a) = −Γ(1−s)/(2πi) ∮_C (−z)^{s−1} e^{−az}/(1−e^{−z}) dz over a Hankel contour, valid for all s ≠ 1.
s_hurwitz_zeta_continuation_all_c	theorem	Analytic continuation of Hurwitz zeta to all of ℂ		Via the contour integral, ζ(s,a) extends to a meromorphic function on all of ℂ with only a simple pole at s = 1.
s_hurwitz_zeta_negative_integers	theorem	Values of ζ(s,a) at negative integers		ζ(−n, a) = −B_{n+1}(a)/(n+1) for n ≥ 0, expressing Hurwitz zeta at nonpositive integers via Bernoulli polynomials.
s_zeta_at_negative_integers	theorem	Values of ζ(s) at negative integers		ζ(−n) = −B_{n+1}/(n+1) for n ≥ 0, giving ζ(0) = −1/2 and ζ(−2k) = 0 for k ≥ 1.
s_evaluation_zeta_2n	theorem	Evaluation of ζ(2n) via Bernoulli numbers		ζ(2n) = (−1)^{n+1} (2π)^{2n} B_{2n}/(2(2n)!) for n ≥ 1, giving ζ(2) = π²/6, ζ(4) = π⁴/90.
s_functional_equation_l_primitive	theorem	Functional equation of L(s,χ) for primitive characters		For primitive χ mod k with a = (1−χ(−1))/2, the completed L-function Λ(s,χ) = (k/π)^{(s+a)/2} Γ((s+a)/2) L(s,χ) satisfies Λ(s,χ) = ε(χ) Λ(1−s,χ̄).
s_analytic_continuation_l_all_c	theorem	Analytic continuation of L(s,χ) to all of ℂ		For primitive nonprincipal χ, L(s,χ) extends to an entire function on ℂ via the functional equation.
s_integral_representation_zeta_gamma	theorem	Integral representation Γ(s)ζ(s) via Mellin transform		Γ(s)ζ(s) = ∫₀^∞ t^{s−1}/(eᵗ−1) dt for Re(s) > 1.
s_periodic_zeta_function	state	Periodic zeta function F(a,s)		F(a,s) = ∑_{n=1}^∞ e^{2πina}/nˢ for Re(s) > 1 and 0 < a < 1, appearing in the functional equation of Hurwitz zeta.
s_pole_of_zeta_at_1	theorem	Simple pole of ζ(s) at s = 1		ζ(s) has a simple pole at s = 1 with residue 1, equivalently lim_{s→1} (s−1)ζ(s) = 1.
t_mertens_3_4_1_inequality	technique	Mertens 3-4-1 inequality		Uses 3 + 4cos θ + cos 2θ ≥ 0 to show |ζ(σ)³ ζ(σ+it)⁴ ζ(σ+2it)| ≥ 1, proving nonvanishing of ζ on Re(s) = 1.
s_meromorphic_zeta_prime_over_zeta	theorem	Meromorphic structure of ζ′(s)/ζ(s)		−ζ′(s)/ζ(s) is meromorphic with a simple pole at s = 1 (residue 1) and poles at zeros ρ of ζ(s).
t_pnt_from_wiener_ikehara	technique	Derivation of PNT from Wiener–Ikehara theorem		Applies the Wiener–Ikehara Tauberian theorem to −ζ′(s)/ζ(s) with its simple pole at s = 1 to deduce ψ(x) ∼ x.
t_newman_tauberian_pnt	technique	Newman's short Tauberian proof of PNT		Newman's simplified analytic proof of PNT using a direct contour-integral argument avoiding the full Wiener–Ikehara machinery.
s_integral_psi_via_zeta	theorem	Integral representation of ψ(x) via ζ′/ζ		ψ(x) = (1/2πi) ∫_{c−i∞}^{c+i∞} (−ζ′(s)/ζ(s)) xˢ/s ds for c > 1, via Perron's formula.
t_contour_integration_pnt	technique	Contour integration method for PNT		Shifts the Perron-type contour leftward past Re(s) = 1, extracting the main term x from the pole and bounding the remainder.
s_pnt_error_term	theorem	PNT with error term (de la Vallée-Poussin)		ψ(x) = x + O(x exp(−c√(log x))) for some c > 0, the quantitative prime number theorem.
s_pnt_equivalent_to_nonvanishing	theorem	PNT equivalent to nonvanishing of ζ on Re(s) = 1		ψ(x) ∼ x is logically equivalent to ζ(1+it) ≠ 0 for all real t.
s_convexity_bound_zeta	theorem	Convexity bound for ζ(s) in the critical strip		ζ(σ+it) = O(|t|^{(1−σ)/2+ε}) for 0 ≤ σ ≤ 1 and |t| ≥ 1, by the Phragmén–Lindelöf principle.
s_dirichlet_eta_zeta_relation	state	Dirichlet eta function relation to ζ(s)		ζ(s) = (1 − 2^{1−s})⁻¹ η(s) where η(s) = ∑ (−1)^{n−1}/nˢ converges for Re(s) > 0.
s_zeta_squared_divisor_series	state	ζ(s)² = ∑ d(n)/nˢ		ζ(s)² = ∑ d(n)/nˢ for Re(s) > 1, since d = 1 ∗ 1 under Dirichlet convolution.
s_zeta_product_sigma_series	state	ζ(s)ζ(s−1) = ∑ σ(n)/nˢ		ζ(s)ζ(s−1) = ∑ σ(n)/nˢ for Re(s) > 2, from the Dirichlet convolution of N and u.
s_landau_theorem_nonneg_coefficients	theorem	Landau's theorem on Dirichlet series with nonneg coefficients		If aₙ ≥ 0 and ∑ aₙ/nˢ has finite abscissa of convergence σ_c, then s = σ_c is a singularity of the function.
s_psi_pi_relation	theorem	Relation between ψ(x) and π(x) via Abel summation		π(x) = ψ(x)/log x + ∫₂ˣ ψ(t)/(t log² t) dt, so ψ(x) ∼ x implies π(x) ∼ x/log x.
s_euler_totient_function	axiom	Euler totient function φ(n)		φ(n) = |{1 ≤ k ≤ n : gcd(k,n) = 1}|, counting the integers up to n that are coprime to n.
s_asymptotic_euler_totient_sum	theorem	Asymptotic for the summatory Euler totient		∑_{n≤x} φ(n) = (3/π²)x² + O(x log x), following from Möbius inversion and the hyperbola method.
s_density_of_squarefree_integers	theorem	Density of squarefree integers		The proportion of squarefree integers up to x tends to 6/π² = 1/ζ(2); equivalently ∑_{n≤x} |μ(n)| = (6/π²)x + O(√x).
s_big_omega_function	axiom	Big-omega function Ω(n)		Ω(n) = ∑_{p^a‖n} a, the number of prime factors of n counted with multiplicity; a completely additive function.
s_mean_value_of_omega	theorem	Mean value of ω(n)		∑_{n≤x} ω(n) = x log log x + Mx + o(x) where M is the Meissel–Mertens constant; similarly for Ω(n).
s_hardy_ramanujan_normal_order	theorem	Hardy–Ramanujan theorem on normal order of ω(n)		The normal order of ω(n) is log log n: for every ε > 0, the set of n with |ω(n) − log log n| > ε√(log log n) has density zero.
s_turan_inequality_number_theory	theorem	Turán's inequality for ω(n)		∑_{n≤x} (ω(n) − log log x)² = O(x log log x), giving the variance bound that implies the Hardy–Ramanujan normal order result.
s_legendre_sieve_formula	state	Legendre sieve formula		S(A,z) = ∑_{d|P(z)} μ(d)|A_d|, the inclusion-exclusion identity counting elements of A not divisible by primes below z, where P(z) = ∏_{p<z} p.
s_twin_prime_sieve_estimate	state	Twin prime sieve estimate		|{n ≤ x : n and n+2 both prime}| ≪ x/(log x)² by Selberg's or Brun's sieve, consistent with the Hardy–Littlewood conjecture up to the constant.
s_additive_character_mod_q	axiom	Additive character mod q		The function e_q(a) = e^{2πia/q} for a ∈ ℤ/qℤ, forming the character group of (ℤ/qℤ, +).
s_orthogonality_additive_characters	theorem	Orthogonality of additive characters		∑_{a=0}^{q−1} e_q(na) = q if q|n and 0 otherwise, the discrete Fourier orthogonality for ℤ/qℤ.
s_truncated_perron_formula	theorem	Truncated Perron formula		∑_{n≤x} aₙ = (1/2πi) ∫_{c−iT}^{c+iT} F(s)xˢ/s ds + O(xᶜ/(T·min(1,‖x‖))), the effective version of Perron's formula with explicit error from truncation.
s_cesaro_riesz_means	axiom	Cesàro summability and Riesz means		The Riesz mean of order δ of a Dirichlet series: (1/Γ(δ+1)) ∑_{n≤x} aₙ(1−n/x)^δ, a smoothed partial sum that improves convergence properties.
s_landau_coefficient_formula	theorem	Landau's formula for Dirichlet series coefficients		∑_{n≤x} aₙ = (1/2πi) ∮ F(s)xˢ/s ds with the sum over residues giving the main terms, relating singularities of F(s) to the average behavior of aₙ.
s_three_four_one_inequality	state	3-4-1 trigonometric inequality		3 + 4cos θ + cos 2θ ≥ 0 for all real θ, the key inequality used to prove ζ(1+it) ≠ 0 via |ζ(σ)³ ζ(σ+it)⁴ ζ(σ+2it)| ≥ 1.
s_smooth_number_definition	axiom	y-smooth integer (friable number)		A positive integer n is y-smooth if all its prime factors satisfy p ≤ y; Ψ(x,y) = |{n ≤ x : P⁺(n) ≤ y}| counts them.
s_dickman_rho_function	axiom	Dickman rho function ρ(u)		The continuous function satisfying ρ(u) = 1 for 0 ≤ u ≤ 1 and uρ′(u) = −ρ(u−1) for u > 1, governing the density of smooth numbers: Ψ(x, x^{1/u}) ∼ ρ(u)x.
s_smooth_number_asymptotic	theorem	Smooth number asymptotic (de Bruijn / Hildebrand–Tenenbaum)		Ψ(x,y) ∼ xρ(u) uniformly for u = log x/log y in a wide range, with Hildebrand's refinement extending this to exp((log y)^{3/5−ε}) ≤ y ≤ x.
s_buchstab_function	axiom	Buchstab function ω(u)		The continuous function with ω(u) = 1/u for 1 ≤ u ≤ 2 and (uω(u))′ = ω(u−1) for u > 2, arising in the Buchstab identity for Φ(x,y) = |{n ≤ x : P⁻(n) > y}|.
s_primes_in_short_intervals	theorem	Primes in short intervals		π(x+h) − π(x) ∼ h/log x for h = x^θ with θ > 7/12 (Huxley), showing primes exist in intervals [x, x + x^θ].
s_sathe_selberg_theorem	theorem	Sathe–Selberg theorem on integers with k prime factors		|{n ≤ x : ω(n) = k}| ∼ (x/log x)(log log x)^{k−1}/((k−1)!) uniformly for 1 ≤ k ≤ (2−ε)log log x.
s_beurling_generalized_primes	axiom	Beurling generalized primes		A sequence 1 < p₁ ≤ p₂ ≤ ⋯ of reals (generalized primes) generating by multiplicative closure a set of generalized integers; the framework for studying PNT-type
s_beurling_pnt	theorem	Beurling PNT for generalized primes		If N(x) = Ax + O(x/(log x)^γ) with γ > 3/2 for the counting function of Beurling integers, then π_P(x) ∼ x/log x.
s_induced_dirichlet_character	axiom	Induced Dirichlet character		Given χ a character mod q and q|Q, the character χ* mod Q defined by χ*(n) = χ(n) if gcd(n,Q) = 1 and 0 otherwise; every character is induced from a unique prim
s_quadratic_dirichlet_character	axiom	Quadratic Dirichlet character		A real-valued Dirichlet character of order 2, given by the Kronecker symbol χ_D(n) = (D/n) for a fundamental discriminant D.
s_riemann_hypothesis	state	Riemann hypothesis		All nontrivial zeros of ζ(s) lie on the critical line Re(s) = 1/2; equivalently ψ(x) = x + O(x^{1/2+ε}) for every ε > 0.
s_generalized_riemann_hypothesis	state	Generalized Riemann hypothesis (GRH)		All nontrivial zeros of every Dirichlet L-function L(s,χ) lie on the critical line Re(s) = 1/2.
s_hadamard_product_for_xi	theorem	Hadamard product for ξ(s)		ξ(s) = ξ(0) ∏_ρ (1 − s/ρ), the product over nontrivial zeros ρ of ζ(s), obtained by applying the Hadamard factorization theorem to the entire function ξ(s) of o
s_zero_counting_function_N_T	state	Zero-counting function N(T)		N(T) = |{ρ : ζ(ρ) = 0, 0 < Im(ρ) ≤ T}| = (T/2π)log(T/2πe) + O(log T), counting nontrivial zeros of ζ up to height T.
s_zero_free_region_l_functions	theorem	Classical zero-free region for L-functions		L(s,χ) ≠ 0 for σ > 1 − c/log(q(|t|+2)) with at most one exceptional real zero for real χ, extending de la Vallée Poussin's region to Dirichlet L-functions.
s_siegel_theorem_l_functions	theorem	Siegel's theorem on L(1,χ)		For every ε > 0 there exists c(ε) > 0 (ineffective) such that L(1,χ) > c(ε)q^{−ε} for real primitive χ mod q, ruling out Siegel zeros in an ineffective sense.
s_page_theorem	theorem	Page's theorem		Among all real primitive characters χ mod q with q ≤ Q, at most one L(s,χ) can have a real zero β > 1 − c/log Q, and that exceptional modulus q₁ satisfies q₁ > 
s_pnt_arithmetic_progressions	theorem	Prime number theorem for arithmetic progressions		π(x;q,a) ∼ Li(x)/φ(q) for fixed q with gcd(a,q) = 1, with an error term depending on the zero-free region for L-functions mod q.
s_siegel_walfisz_theorem	theorem	Siegel–Walfisz theorem		For any A > 0, ψ(x;q,a) = x/φ(q) + O(x exp(−c√(log x))) uniformly for q ≤ (log x)^A with gcd(a,q) = 1, using Siegel's ineffective zero-free region.
t_vaughan_identity	technique	Vaughan's identity		Decomposes Λ(n) into Type I and Type II sums via Λ = μ_{≤U} ∗ log + μ_{≤U} ∗ Λ_{>V} ∗ 1 + remainder, enabling bilinear sum estimates in the Bombieri–Vinogradov 
s_explicit_formula_psi_q_a	theorem	Explicit formula for ψ(x;q,a)		ψ(x;q,a) = x/φ(q) − (1/φ(q)) ∑_χ ∑_ρ x^ρ/ρ + lower order terms, where ρ runs over nontrivial zeros of L(s,χ) for all χ mod q.
s_weil_explicit_formula	theorem	Weil's explicit formula		∑_ρ h(ρ) = h(1) + h(0) − ∑_p ∑_k (log p / p^{k/2}) ĝ(k log p) + integral terms, relating a test function h evaluated at zeros of ζ to sums over primes.
s_guinand_weil_formula	theorem	Guinand–Weil explicit formula		A distributional identity ∑_ρ ĥ(γ_ρ) = ĥ(i/2) + ĥ(−i/2) − ∑_p log p ∑_k p^{−k/2}(h(k log p) + h(−k log p)) + integral terms, the symmetric form of Weil's explic
s_von_koch_rh_prime_estimate	theorem	Von Koch's theorem: RH implies sharp prime counting		Under RH, π(x) = Li(x) + O(√x log x), equivalently ψ(x) = x + O(x^{1/2} log²x), the best possible error term up to logarithmic factors.
s_rh_sharp_error_ap	theorem	RH implies sharp error in arithmetic progressions		Under GRH, ψ(x;q,a) = x/φ(q) + O(x^{1/2} log²x) uniformly in q, giving the conditional analogue of von Koch's theorem for primes in AP.
s_lindelof_hypothesis	state	Lindelöf hypothesis		ζ(1/2 + it) = O(|t|^ε) for every ε > 0, equivalent to μ(1/2) = 0 where μ(σ) is the infimum of exponents bounding |ζ(σ+it)|; implied by RH.
s_conditional_bounds_on_zeta	theorem	Conditional bounds on ζ(s) under RH		Under RH, |ζ(σ+it)| ≪ |t|^{(1−σ)/2+ε} for σ ≥ 1/2 and log|ζ(1/2+it)| ≪ log t/log log t, improving the unconditional convexity bound.
s_conditional_estimate_mertens_function	theorem	Conditional estimate for M(x) under RH		Under RH, M(x) = ∑_{n≤x} μ(n) = O(x^{1/2+ε}) for every ε > 0, the best conditional bound on the Mertens function.
s_conditional_prime_gaps	theorem	Conditional bounds on gaps between primes under RH		Under RH, p_{n+1} − p_n ≪ √p_n log p_n, showing that consecutive prime gaps are O(√p log p).
s_conditional_pi_large_q	theorem	Conditional estimate for π(x;q,a) with large q under GRH		Under GRH, π(x;q,a) = Li(x)/φ(q) + O(√x log x) for q ≤ x, effective for much larger moduli than unconditional results.
s_zero_density_estimate	theorem	Zero density estimate N(σ,T)		N(σ,T) = |{ρ = β+iγ : L(ρ,χ) = 0, β ≥ σ, |γ| ≤ T}| ≪ T^{A(σ)(1−σ)+ε} for σ > 1/2, bounding zeros of L-functions away from the critical line.
s_ingham_density_estimate	theorem	Ingham density estimate		N(σ,T) ≪ T^{3(1−σ)/(2−σ)+ε} for 1/2 ≤ σ ≤ 1, Ingham's classical zero density estimate for the Riemann zeta function.
s_hardy_theorem_zeros_critical_line	theorem	Hardy's theorem on zeros on the critical line		ζ(s) has infinitely many zeros on the critical line Re(s) = 1/2 (Hardy, 1914).
s_proportion_zeros_critical_line	theorem	Positive proportion of zeros on the critical line		At least 41.05% of nontrivial zeros of ζ(s) lie on the critical line (Conrey, 1989), improving Selberg's result that a positive proportion do.
s_density_hypothesis	state	Density hypothesis		N(σ,T) ≪ T^{2(1−σ)+ε} for σ ≥ 1/2, a conjecture on the optimal exponent in zero density estimates, weaker than RH but sufficient for many applications.
s_landau_oscillation_theorem	theorem	Landau's oscillation theorem		If a Dirichlet series F(s) = ∑ aₙ/nˢ (aₙ real) has a singularity on the real axis at its abscissa of convergence σ_c, then ∑_{n≤x} aₙ oscillates in sign.
s_omega_plus_minus_notation	axiom	Ω₊/Ω₋ notation for oscillation		f(x) = Ω₊(g(x)) means lim sup f(x)/g(x) > 0; f(x) = Ω₋(g(x)) means lim inf f(x)/g(x) < 0; f(x) = Ω±(g(x)) means both hold.
s_littlewood_omega_theorem	theorem	Littlewood's omega theorem for π(x) − Li(x)		π(x) − Li(x) = Ω±(√x log log log x / log x), proving that Li(x) is not always an overestimate of π(x), despite being so for all computationally accessible x.
s_oscillation_psi_minus_x	theorem	Oscillation of ψ(x) − x		ψ(x) − x = Ω±(√x), i.e. ψ(x) − x changes sign infinitely often and achieves magnitude at least c√x for some c > 0.
s_omega_result_mertens_function	theorem	Omega result for M(x)		M(x) = Ω±(x^{1/2}), the Mertens function oscillates with amplitude at least c√x infinitely often.
s_ingham_omega_theorem	theorem	Ingham's theorem on omega results from zeros		A general method producing Ω-results for summatory functions from knowledge of zeros of the associated Dirichlet series on the critical line.
s_skewes_number	state	Skewes number and sign changes of π(x) − Li(x)		The first x where π(x) > Li(x) exists (Littlewood); Skewes gave the first explicit upper bound, later improved to around e^{727.95} unconditionally.
s_riemann_stieltjes_integral	axiom	Riemann–Stieltjes integral		∫ₐᵇ f dα = lim ∑ f(tₖ)(α(xₖ) − α(xₖ₋₁)), generalizing the Riemann integral by integrating against a monotone or BV function α.
t_integration_by_parts_stieltjes	technique	Integration by parts for Stieltjes integrals		∫ₐᵇ f dα = fα|ₐᵇ − ∫ₐᵇ α df, the analogue of integration by parts for Riemann–Stieltjes integrals, fundamental for partial summation.
s_zeta_at_positive_even_integers	theorem	Values of ζ(s) at positive even integers		ζ(2n) = (−1)^{n+1}(2π)^{2n}B_{2n}/(2(2n)!) for n ≥ 1, giving ζ(2) = π²/6, ζ(4) = π⁴/90, etc.
s_weierstrass_product_gamma	theorem	Weierstrass product for 1/Γ(z)		1/Γ(z) = ze^{γz} ∏_{n=1}^∞ (1 + z/n)e^{−z/n}, the infinite product representation showing 1/Γ is entire of order 1.
s_theta_function_number_theory	axiom	Theta function (number-theoretic)		ϑ(t) = ∑_{n=−∞}^{∞} e^{−πn²t} for t > 0, equal to θ₃(0, e^{−πt}) in Jacobi notation, central to the proof of the functional equation of ζ(s).
s_theta_transformation_formula	theorem	Theta function transformation formula		ϑ(1/t) = √t · ϑ(t) for t > 0, a consequence of Poisson summation, used to derive the functional equation of the Riemann zeta function.
t_rankin_trick	technique	Rankin's trick		Bounding a counting function ∑_{n≤x} aₙ by (x/y)^δ ∑_{n≤y} aₙ(n/y)^{−δ} for a free parameter δ > 0, converting a sharp cutoff into a smooth weight.
s_dirichlet_divisor_problem	state	Dirichlet's divisor problem		Determine the infimum α such that ∑_{n≤x} d(n) = x log x + (2γ−1)x + O(x^α); conjectured α = 1/4, best known α ≈ 0.314 (Kolesnik, Huxley).
s_selberg_class	axiom	Selberg class		An axiomatic class S of Dirichlet series satisfying: (i) analytic continuation, (ii) Ramanujan bound on coefficients, (iii) functional equation, (iv) Euler prod
s_vinogradov_korobov_zero_free_region	theorem	Vinogradov–Korobov zero-free region		ζ(σ+it) ≠ 0 for σ > 1 − c/(log|t|)^{2/3}(log log|t|)^{1/3}, a wider zero-free region than de la Vallée Poussin's, obtained via exponential sum estimates.
s_mertens_conjecture_disproved	state	Mertens conjecture (disproved)		|M(x)| < √x for all x > 1 was conjectured by Mertens; disproved by Odlyzko and te Riele (1985) using computations of zeros of ζ(s).
s_twin_prime_conjecture	state	Twin prime conjecture		There are infinitely many primes p such that p+2 is also prime; unproven, though Zhang (2013) proved bounded gaps between primes.
s_hardy_littlewood_twin_prime_conjecture	state	Hardy–Littlewood twin prime conjecture		|{p ≤ x : p+2 prime}| ∼ 2C₂ x/(log x)² where C₂ = ∏_{p≥3}(1−1/(p−1)²) ≈ 0.6602 is the twin prime constant.
s_dirichlet_series_liouville	state	Dirichlet series for the Liouville function		∑_{n=1}^∞ λ(n)/nˢ = ζ(2s)/ζ(s) for Re(s) > 1, expressing the Dirichlet series of Liouville's function via zeta.
s_ramanujan_sum_formula_sigma	theorem	Ramanujan's sum formula for σ_s(n)		σ_s(n)/n^s = ζ(s+1) ∑_{q=1}^∞ c_q(n)/q^{s+1}, expressing the divisor function as an absolutely convergent series of Ramanujan sums.
s_vinogradov_korobov_pnt_error	theorem	Vinogradov–Korobov error term in PNT		ψ(x) = x + O(x exp(−c(log x)^{3/5}(log log x)^{−1/5})), the best known unconditional error term in the PNT.
s_montgomery_pair_correlation	state	Montgomery's pair correlation conjecture		The pair correlation of normalized spacings of nontrivial zeros of ζ(s) follows 1 − (sin πu/(πu))² (GUE statistics), connecting zeros of zeta to random matrix t
t_explicit_formula_test_function	technique	Explicit formula with general test function		A general identity ∑_ρ h(ρ) = δ + ∫ h·K − ∑_p ∑_k g(k log p) log p / p^{k/2} for suitable test functions h with Fourier transform g, unifying various explicit f
s_halasz_theorem	theorem	Halász's theorem		For multiplicative f with |f(n)| ≤ 1, (1/x)∑_{n≤x} f(n) = O(exp(−c min_|t|≤T ∑_{p≤x}(1−Re f(p)p^{−it})/p)) + O(1/T), characterizing when mean values of multipli
s_generalized_von_mangoldt_lambda_k	axiom	Generalized von Mangoldt function Λ_k		Λ_k = μ ∗ log^k, the k-fold Dirichlet convolution giving Λ₁ = Λ and Λ_k(n) supported on integers with at most k prime factors; −ζ^(k)(s)/ζ(s) = ∑ Λ_k(n)/nˢ.
s_grand_riemann_hypothesis	state	Grand Riemann hypothesis		All nontrivial zeros of all automorphic L-functions lie on Re(s) = 1/2, the strongest form of RH encompassing Dedekind zeta, Artin L-functions, and all automorp
s_effective_vs_ineffective_constants	axiom	Effective vs ineffective constants		A constant is effective if it can in principle be computed; ineffective if its existence is proved indirectly (e.g., via Siegel's theorem). The distinction is c
t_riemann_stieltjes_partial_summation	technique	Riemann–Stieltjes partial summation technique		Converting ∑_{n≤x} f(n)g(n) into ∫₁ˣ g(t) dA(t) where A(t) = ∑_{n≤t} f(n), then integrating by parts to separate smooth and arithmetic parts.
s_dirichlet_series_multiplication_theorem	theorem	Dirichlet series multiplication theorem		If F(s) = ∑ f(n)/nˢ and G(s) = ∑ g(n)/nˢ converge absolutely, then F(s)G(s) = ∑ (f∗g)(n)/nˢ where f∗g is the Dirichlet convolution.
s_explicit_zero_free_region_improvements	state	Explicit zero-free region constant improvements		Numerical improvements to the classical zero-free region σ > 1 − 1/(R log t) for ζ(s), with the best explicit R values due to Ford, Kadiri, and others.
s_completely_additive_function	axiom	Completely additive function	totally additive function	An arithmetic function f satisfying f(mn) = f(m) + f(n) for all m, n ≥ 1; equivalently f(p^a) = a·f(p), e.g. Ω(n) and log n.
s_generalized_divisor_function_tau_k	axiom	Generalized divisor function τ_k(n)	Piltz divisor function | d_k(n) | τ_k	τ_k(n) = ∑_{n=n₁⋯nₖ} 1, the number of ordered k-fold factorizations of n; its Dirichlet series is ζ(s)^k.
t_selberg_delange_method	technique	Selberg–Delange method	Selberg-Delange method	Obtains asymptotics for partial sums of Dirichlet series behaving like ζ(s)^z near s = 1 via contour integration around a Hankel contour.
s_smoothed_perron_formula	theorem	Smoothed Perron formula	smooth Perron formula	A variant of Perron's formula using a smooth weight function to replace the sharp cutoff, yielding faster-decaying error terms via rapid decay of the Mellin tra
t_dyadic_decomposition	technique	Dyadic decomposition	dyadic subdivision	Partitioning a sum over [1, X] into O(log X) blocks of the form (N, 2N], reducing estimates to those uniform in a parameter N.
t_smooth_partition_of_unity	technique	Smooth partition of unity	smooth dyadic partition	A collection of smooth bump functions {ψ_j} with ∑_j ψ_j ≡ 1, used to decompose sums into pieces with controlled Mellin or Fourier transforms.
t_heath_brown_identity	technique	Heath-Brown's identity	Heath-Brown decomposition	Decomposes the von Mangoldt function into O(k) multilinear sums using k-fold applications of an identity involving μ and log, yielding flexible Type I and Type 
t_linnik_identity	technique	Linnik's identity	Linnik decomposition	Expresses Λ(n) as Λ = ∑_{k≥1} (−1)^{k−1}/k · (1∗⋯∗1 − ε)^{∗k} · log, converting prime sums into iterated convolution sums.
s_type_i_sum	axiom	Type I sum	Type I estimate | linear sum	A sum of the form ∑_{m≤M} a_m ∑_{n: mn∈A} 1 where a_m are general coefficients and the inner sum is over a well-understood sequence.
s_type_ii_sum	axiom	Type II sum	Type II estimate | bilinear sum	A bilinear sum ∑_m ∑_n a_m b_n c(mn) where both variables carry nontrivial coefficients, requiring Cauchy–Schwarz or bilinear form techniques.
s_wirsing_theorem	theorem	Wirsing's theorem	Wirsing mean value theorem	For nonnegative multiplicative f with ∑_p f(p)/p ~ κ log log x, gives ∑_{n≤x} f(n) ~ Ce^{-γκ}/Γ(κ) · x(log x)^{κ-1} ∏_p(1+f(p)/p+⋯)(1-1/p)^κ.
s_shiu_bound	theorem	Shiu's bound	Shiu's theorem	For multiplicative f with 0 ≤ f(n) ≤ τ(n)^A, gives ∑_{x<n≤x+y, n≡a(q)} f(n) ≪ (y/φ(q))exp(∑_{p≤x} f(p)/p) for y ≥ x^{1/2+ε}.
s_general_l_function_ik	axiom	General L-function (Iwaniec–Kowalski framework)	standard L-function | IK axiomatic L-function	An L-function satisfying: Euler product of degree d, analytic continuation, functional equation Λ(s) = εΛ̄(1−s) with gamma factors, and the Ramanujan bound |a(p
s_completed_l_function	axiom	Completed L-function	Lambda function | Λ-function	Λ(s, π) = q^{s/2} ∏_j Γ_ℝ(s + μ_j) · L(s, π) incorporating conductor q and gamma factors, satisfying Λ(s) = ε Λ̄(1−s).
s_root_number	axiom	Root number	epsilon factor | sign of functional equation	The complex number ε of absolute value 1 in the functional equation Λ(s) = εΛ̄(1−s); for self-dual L-functions ε = ±1, governing vanishing at the central point.
s_gamma_factors_functional_equation	axiom	Gamma factors in functional equations	archimedean gamma factor	The archimedean part γ(s) = ∏_j Γ_ℝ(s + μ_j) of a completed L-function, where Γ_ℝ(s) = π^{−s/2}Γ(s/2), encoding the archimedean local factors.
s_conductor_l_function	axiom	Conductor of an L-function	arithmetic conductor | q(π)	The positive integer q(π) in the functional equation measuring arithmetic ramification; equals ∏_p p^{f_p} with f_p the local conductor exponents.
s_analytic_conductor	axiom	Analytic conductor	C(π) | effective conductor	C(π) = q(π)∏_j(|μ_j|+3), the quantity governing convexity bounds, zero-free regions, and approximate functional equations of an L-function.
s_degree_l_function	axiom	Degree of an L-function	d(π)	The number d of Γ_ℝ-factors in the completed L-function; equals n for GL(n) automorphic L-functions, governing the growth rate and density of zeros.
s_convexity_bound_l_functions	theorem	Convexity bound for L-functions	convexity estimate	For an L-function of analytic conductor C, L(1/2+it) ≪ C(t)^{1/4+ε}, obtained from the Phragmén–Lindelöf principle applied to the functional equation.
s_subconvexity_bound	theorem	Subconvexity bound	subconvex estimate | breaking convexity	Any bound L(1/2, π) ≪ C(π)^{1/4−δ} for some δ > 0 breaking the convexity barrier; has applications to equidistribution and mass transfer.
s_grand_lindelof_hypothesis	axiom	Grand Lindelöf hypothesis	GLH | generalized Lindelöf hypothesis	The conjecture that L(1/2+it, π) ≪ C(π, t)^ε for every ε > 0 and every automorphic L-function; generalizes the Lindelöf hypothesis.
s_approximate_functional_equation	theorem	Approximate functional equation	AFE	L(s) = ∑_{n≤x} a_n n^{−s} V(n/x) + ε(s)∑_{n≤y} ā_n n^{s−1} W(n/y) + remainder, expressing L(s) on the critical line as two finite sums of length ~√(q|t|).
s_mean_value_theorem_dirichlet_polynomials	theorem	Mean value theorem for Dirichlet polynomials	Montgomery-Vaughan mean value theorem	∫_0^T |∑_{n≤N} a_n n^{−it}|² dt = (T + O(N))∑|a_n|², the Montgomery–Vaughan form equating the L² norm over t to the sum of squared coefficients.
s_montgomery_large_values	theorem	Montgomery's mean value theorem for large values	large values estimate	Bounds the number R of well-spaced t-values where |∑ a_n n^{−it}| ≥ V: R ≪ (N+T)(∑|a_n|²)/V², controlling large values of Dirichlet polynomials.
s_huxley_jutila_large_values	theorem	Huxley–Jutila large values estimate	Jutila large values	Refined bounds on the measure of t-values where a Dirichlet polynomial is large, improving Montgomery's estimate via mean-square and fourth-power moment methods
s_fourth_moment_zeta	theorem	Fourth moment of the Riemann zeta function	Ingham fourth moment	∫_0^T |ζ(1/2+it)|⁴ dt ~ (2π²)⁻¹ T(log T)⁴, proved by Ingham (1926); the highest moment of ζ(1/2+it) for which an asymptotic is known.
s_second_moment_l_functions_characters	theorem	Second moment of L-functions over characters	mean square of L(1/2,χ)	∑_{χ mod q} |L(1/2, χ)|² ~ (φ(q)/q)·q log q, the mean-square for central values of Dirichlet L-functions averaged over characters mod q.
s_moments_l_functions_conjectural	axiom	Moments of L-functions (conjectural asymptotics)	CFKRS moment conjecture | Keating-Snaith conjecture	The conjecture that ∫_0^T |ζ(1/2+it)|^{2k} dt ~ c_k T(log T)^{k²} with c_k predicted by random matrix theory (Keating–Snaith).
s_hecke_l_function_grossencharakter	axiom	Hecke L-function	Hecke Grössencharakter L-function	L(s, λ) = ∑_𝔞 λ(𝔞)N(𝔞)^{−s} attached to a Hecke Grössencharakter λ of a number field, admitting Euler product, analytic continuation, and functional equation.
s_rankin_selberg_l_function	axiom	Rankin–Selberg L-function	Rankin-Selberg convolution	L(s, π × π') = ∏_p det(I − A_π(p) ⊗ A_{π'}(p) p^{−s})^{−1}, a degree-mn L-function for GL(m)×GL(n) forms with analytic continuation and functional equation.
s_kronecker_limit_formula	theorem	Kronecker's limit formula	first Kronecker limit formula	The Laurent expansion of the Epstein zeta function at s = 1 has constant term involving −π(γ + log 2 − log(√y|η(τ)|²)), connecting Eisenstein series to the Dede
s_sifting_function	axiom	Sifting function S(A, P, z)	sieve sum | S(A,z)	S(A, P, z) = |{a ∈ A : gcd(a, P(z)) = 1}| where P(z) = ∏_{p<z, p∈P} p, counting elements not divisible by any sifting prime below z.
s_sieve_dimension	axiom	Sieve dimension	sifting density	The parameter κ such that ∑_{w≤p<z} g(p)log p = κ log(z/w) + O(1), governing sifting prime density; κ = 1 for the linear sieve, κ = 1/2 for binary quadratic for
s_buchstab_identity	theorem	Buchstab's identity	Buchstab recursion	S(A, P, z) = S(A, P, w) − ∑_{w≤p<z} S(A_p, P, p), recursively expressing the sifting function at level z in terms of lower levels.
s_linear_sieve	theorem	Linear sieve (Iwaniec's formulation)	Iwaniec linear sieve	For sieve dimension κ = 1, S(A,P,z) = X·W(z){f(s)+O((log D)^{-1/3})} with s = log D/log z, where f and F solve the linear sieve differential-delay equations.
s_sieve_limit_functions	axiom	Sieve upper/lower bound functions F_κ(s), f_κ(s)	sieve adjoint functions	Continuous functions satisfying differential-delay equations giving best possible sieve bounds: f_κ(s)XV(z) ≤ S ≤ F_κ(s)XV(z) + remainder.
t_rosser_iwaniec_sieve	technique	Rosser–Iwaniec sieve (β-sieve)	beta-sieve | β-sieve	A combinatorial sieve using Rosser's alternating truncation combined with Iwaniec's bilinear remainder estimates, achieving optimal bounds for the linear sieve.
s_parity_problem_sieve	axiom	Parity problem in sieve theory	parity barrier | Selberg parity phenomenon	The obstruction that classical sieve methods cannot distinguish integers with even vs odd number of prime factors, preventing sieves from detecting primes in th
t_semilinear_sieve	technique	Semilinear sieve (κ = 1/2)	half-dimensional sieve	The sieve of dimension κ = 1/2, applicable to primes represented by binary quadratic forms; the lower bound becomes positive for s > 1.
t_bombieri_asymptotic_sieve	technique	Bombieri's asymptotic sieve	Bombieri sieve	A sieve producing asymptotic formulas for sifting functions under strong equidistribution hypotheses, overcoming the parity problem.
t_asymptotic_sieve_for_primes	technique	Asymptotic sieve for primes (Bombieri–Friedlander–Iwaniec)	BFI sieve	Combines Bombieri's asymptotic sieve with bilinear form estimates to produce asymptotic formulas for primes in sequences, circumventing the parity problem.
s_level_of_distribution	axiom	Level of distribution	exponent of distribution	The supremum ϑ such that ∑_{q≤x^ϑ-ε} max_{(a,q)=1} |A(x;q,a) − A(x)/φ(q)| ≪ A(x)/(log x)^B; the primes have ϑ = 1/2 by Bombieri–Vinogradov.
s_well_factorable_function	axiom	Well-factorable function	well-factorable weights	A function λ(d) is well-factorable of level D if for every D = D₁D₂ there exist λ₁, λ₂ with λ = λ₁∗λ₂, |λ_i| ≤ 1, and support at scale D_i; key to Iwaniec's bil
t_switching_principle_sieve	technique	Switching principle in sieve theory	Buchstab switching	Reversing roles of sifted set and sifting primes via Buchstab decomposition, exploiting different distributional properties to convert upper into lower bounds.
t_van_der_corput_a_process	technique	Van der Corput's A-process	van der Corput second derivative test | A-process	The second derivative test: if f'' is monotone and |f''(x)| ≍ λ, then ∑ e(f(n)) ≪ Nλ^{1/2} + λ^{-1/2}.
t_van_der_corput_b_process	technique	Van der Corput's B-process	B-process | Poisson transform for exponential sums	Transforms ∑ e(f(n)) into a dual sum via Poisson summation, converting information about f' into a shorter dual sum.
t_method_of_exponent_pairs	technique	Method of exponent pairs	exponent pair method	A framework for bounding exponential sums by iterating A-process (k,ℓ)→(k/(2k+2),(k+ℓ+1)/(2k+2)) and B-process (k,ℓ)→(ℓ-1/2,k+1/2).
s_exponent_pair	axiom	Exponent pair (k, ℓ)	van der Corput exponent pair	A pair (k,ℓ) such that ∑_{n~N} e(f(n)) ≪ N^ℓ λ^k when |f^(j)| ≍ λ/N^j; the trivial pair is (0,1) and (1/2,1/2) from van der Corput.
t_vinogradov_method_exponential_sums	technique	Vinogradov's method of exponential sums	Vinogradov exponential sum method	Bounds exponential sums using iterated Weyl differencing followed by Hölder's inequality and mean value estimates, producing the Vinogradov–Korobov zero-free re
s_salie_sum	axiom	Salié sum	Salié-Kloosterman sum	S*(a,b;c) = ∑_{x mod c, (x,c)=1} (x/c)e((ax+bx̄)/c), the Kloosterman sum twisted by the Legendre symbol; evaluable in terms of Gauss sums when c is prime.
s_hyper_kloosterman_sum	axiom	Hyper-Kloosterman sum	generalized Kloosterman sum	Kl_k(a;q) = ∑_{x₁⋯x_k ≡ a(q)} e((x₁+⋯+x_k)/q), the k-variable Kloosterman sum satisfying Deligne's bound |Kl_k(a;p)| ≤ k·p^{(k-1)/2}.
t_stepanov_method	technique	Stepanov's method	Stepanov auxiliary polynomial method	An elementary method bounding rational points on curves over finite fields by constructing an auxiliary polynomial vanishing to high order at each rational poin
s_bombieri_proof_rh_curves	theorem	Bombieri's proof of RH for curves	Bombieri-Stepanov proof	A simplified proof of the Riemann hypothesis for curves over finite fields using Stepanov's method and Riemann-Roch, avoiding Weil's algebraic geometry.
t_square_sieve	technique	Square sieve (Heath-Brown)	Heath-Brown square sieve	Detects perfect squares in sequences via n is a square iff (n/p) = 1 for all primes p, applying Legendre symbols and the large sieve.
t_bombieri_iwaniec_method	technique	Bombieri–Iwaniec method	Bombieri-Iwaniec first spacing method	Bounds exponential sums by converting to a lattice point counting problem near a curve, then applying the large sieve to detect coincidences; yields exponent pa
s_duality_principle_bilinear	theorem	Duality principle for bilinear forms	bilinear duality | norm duality	The operator norm of a bilinear form equals its transpose norm; in the large sieve, interchanges the roles of the sequence and test points.
s_dual_large_sieve_inequality	theorem	Dual large sieve inequality	dual form of the large sieve	∑_n |∑_r b_r e(nα_r)|² ≤ (N + δ⁻¹)∑|b_r|² for δ-spaced α_r; the dual form of the large sieve.
s_large_sieve_characters	theorem	Large sieve inequality for Dirichlet characters	multiplicative large sieve	∑_{q≤Q} ∑*_{χ mod q} |∑_{n≤N} a_n χ(n)|² ≤ (N + Q² − 1)∑|a_n|², bounding mean squares of character sums over primitive characters.
t_deshouillers_iwaniec_method	technique	Deshouillers–Iwaniec method	Deshouillers-Iwaniec spectral method	Combines Kloosterman sum estimates, spectral theory, and the Kuznetsov trace formula to bound bilinear forms in Kloosterman sums.
t_linnik_dispersion_method	technique	Linnik's dispersion method	dispersion method	Estimates ∑_q(∑_{n≡a(q)} f(n) − main)² by expanding the square and exchanging summation, reducing to bilinear sums.
s_halasz_montgomery_inequality	theorem	Halász–Montgomery inequality	Montgomery-Halász inequality	For R well-spaced t_r: ∑_r |∑_n a_n n^{-it_r}|² ≤ (N + RT)(∑|a_n|²), a discrete mean value theorem for Dirichlet polynomials.
s_motohashi_formula	theorem	Motohashi's formula	Motohashi spectral formula	An exact identity relating the fourth moment of ζ(1/2+it) to a spectral sum over Maass forms, expressing ∫|ζ|⁴ in terms of Hecke eigenvalues.
s_gallagher_larger_sieve	theorem	Gallagher's larger sieve	Gallagher sieve | the larger sieve	If S ⊂ [1,N] reduces to at most ν_p residue classes mod p for each p ∈ P, then |S| ≤ N(∑ log p)/(∑ ν_p log p/(p−ν_p+1)) − corrections.
s_barban_davenport_halberstam_theorem	theorem	Barban–Davenport–Halberstam theorem	BDH theorem	∑_{q≤Q} ∑_{(a,q)=1} |ψ(x;q,a) − x/φ(q)|² ≪ xQ log x for Q ≤ x, an L² average bound on PNT errors in arithmetic progressions.
s_elliott_halberstam_conjecture	axiom	Elliott–Halberstam conjecture	EH conjecture	The conjecture that primes have level of distribution ϑ = 1 − ε; implies bounded gaps between primes by the Goldston–Pintz–Yıldırım method.
s_modular_group_sl2z	axiom	Modular group SL₂(ℤ)	SL(2,Z) | full modular group	The group of 2×2 integer matrices with determinant 1, acting on the upper half-plane by Möbius transformations; the fundamental discrete symmetry group of modul
s_congruence_subgroup_gamma0	axiom	Congruence subgroup Γ₀(N)	Gamma_0(N) | Hecke congruence subgroup	The subgroup of SL₂(ℤ) consisting of matrices whose lower-left entry is divisible by N, the standard level structure for Hecke theory.
s_congruence_subgroup_gamma1	axiom	Congruence subgroup Γ₁(N)	Gamma_1(N)	The subgroup of SL₂(ℤ) with lower-left entry ≡ 0 mod N and diagonal entries ≡ 1 mod N, parametrizing modular forms with Nebentypus.
s_principal_congruence_subgroup	axiom	Principal congruence subgroup Γ(N)	Gamma(N)	The kernel of SL₂(ℤ) → SL₂(ℤ/Nℤ); a subgroup of SL₂(ℤ) is a congruence subgroup iff it contains some Γ(N).
s_nebentypus_character	axiom	Nebentypus character	Nebentypus	A Dirichlet character χ mod N in the transformation law f(γz) = χ(d)(cz+d)^k f(z) for forms on Γ₁(N), refining the level structure.
s_fundamental_domain_sl2z	axiom	Fundamental domain for SL₂(ℤ)	standard fundamental domain	The region F = {z ∈ H : |Re(z)| ≤ 1/2, |z| ≥ 1} for the action of SL₂(ℤ) on H, with hyperbolic area π/3.
s_cusps_of_congruence_subgroup	axiom	Cusps of a congruence subgroup	cusps | parabolic vertices	The Γ-equivalence classes in ℙ¹(ℚ) = ℚ ∪ {∞}, forming the boundary points where Γ\H is compactified to a Riemann surface.
s_hyperbolic_laplacian	axiom	Hyperbolic Laplacian on H	Laplace-Beltrami operator on H	The Laplace-Beltrami operator Δ = −y²(∂²/∂x² + ∂²/∂y²) on the upper half-plane with Poincaré metric, commuting with SL₂(ℝ).
s_cusp_form	axiom	Cusp form	cuspidal modular form | Spitzenform	A modular form of weight k and level N that vanishes at all cusps, spanning the subspace S_k(Γ) ⊂ M_k(Γ).
s_slash_operator	axiom	Slash operator |_k	slash operator | weight-k action	The weight-k right action (f|_k γ)(z) = (cz+d)^{-k} f(γz) of GL₂⁺(ℝ) on functions on H, encoding the modular transformation law.
s_poincare_series_modular	axiom	Poincaré series P_m(z, k, Γ)	Poincaré series for modular forms	P_m(z) = ∑_{γ ∈ Γ_∞\Γ} e^{2πimγz}(cz+d)^{-k}, spanning cusp forms and satisfying ⟨f, P_m⟩ = c·a_f(m) for cusp forms f.
s_petersson_inner_product	axiom	Petersson inner product	Petersson scalar product	⟨f,g⟩ = ∫_{Γ\H} f(z)ḡ(z) y^k dμ(z) on cusp forms, making S_k(Γ) a Hilbert space with Hecke operators self-adjoint.
s_fourier_expansion_at_cusps	axiom	Fourier expansion at cusps	q-expansion	The q-expansion f(z) = ∑_{n≥0} a_f(n)e^{2πinz/w} of a modular form at a cusp of width w; the expansion at each cusp determines the form.
s_atkin_lehner_theory	theorem	Atkin-Lehner theory	newform-oldform decomposition	S_k(Γ₀(N)) = S_k^{old} ⊕ S_k^{new} with multiplicity-one for newforms, decomposing cusp forms into oldforms from lower levels and primitive newforms.
s_atkin_lehner_involution	axiom	Atkin-Lehner involution W_Q	Atkin-Lehner operator	For Q || N, an involution on S_k(Γ₀(N)) normalizing Γ₀(N); newforms are eigenforms for all W_Q with eigenvalue ±1.
s_fricke_involution	axiom	Fricke involution W_N	Fricke operator	The Atkin-Lehner involution z ↦ −1/(Nz) relating the functional equation of L(f,s) to its Fricke eigenvalue.
s_oldform	axiom	Oldform	old form	A form in S_k(Γ₀(N)) arising from lower level M | N via f(z) ↦ f(dz) for d | (N/M), spanning the old subspace.
s_conductor_of_modular_form	axiom	Conductor of a modular form	level of a newform	The minimal level N such that a newform lives in S_k(Γ₀(N), χ), encoding the precise ramification data of the associated automorphic representation.
s_dimension_formula_mk_sl2z	theorem	Dimension formula for M_k(SL₂(ℤ))	dimension of modular forms for SL₂(ℤ)	dim M_k(SL₂(ℤ)) = ⌊k/12⌋ + 1 for even k ≥ 2 (with correction for k ≡ 2 mod 12), from the Riemann-Roch theorem on the genus-0 modular curve.
s_dimension_formula_sk_gamma0	theorem	Dimension formula for S_k(Γ₀(N))	Riemann-Roch dimension formula for cusp forms	dim S_k(Γ₀(N)) = (k−1)(g−1) + corrections from elliptic and parabolic elements, where g is the genus of X₀(N).
s_theta_function_of_quadratic_form	axiom	Theta function of a quadratic form	theta series of a lattice	θ_Q(z) = ∑_{n ∈ ℤ^m} e^{2πiQ(n)z} for a positive definite quadratic form Q in m variables; a modular form of weight m/2 whose coefficients count representations
s_maass_form	axiom	Maass form	Maass wave form	A smooth Γ-automorphic eigenfunction of the hyperbolic Laplacian Δf = λf on H with at most polynomial growth at cusps.
s_maass_cusp_form	axiom	Maass cusp form	Maass cuspidal form	A Maass form that decays exponentially at all cusps, belonging to the discrete cuspidal spectrum of the Laplacian on L²(Γ\H).
s_spectral_parameter_maass	axiom	Spectral parameter r	Maass spectral parameter	The parameter r such that a Maass form has eigenvalue λ = 1/4 + r²; the Selberg conjecture asserts r ∈ ℝ for congruence subgroups.
s_ramanujan_petersson_conjecture	theorem	Ramanujan-Petersson conjecture	Ramanujan conjecture for modular forms	For holomorphic cuspidal newforms of weight k, |a_p(f)| ≤ 2p^{(k−1)/2} (Deligne); for Maass forms, |a_p| ≤ 2p^{1/2} (open, best bound from Kim-Sarnak).
s_real_analytic_eisenstein_series	axiom	Real-analytic Eisenstein series E(z,s)	non-holomorphic Eisenstein series	E(z,s) = ∑_{γ ∈ Γ_∞\Γ}(Im γz)^s for Re(s) > 1, continued meromorphically to all s, spanning the continuous spectrum of the Laplacian.
s_scattering_matrix_automorphic	axiom	Scattering matrix for Γ\H	scattering matrix | phi function	The matrix Φ(s) relating constant terms of Eisenstein series at different cusps, satisfying Φ(s)Φ(1−s) = I, whose poles give the residual spectrum.
s_spectral_decomposition_l2_gamma_h	theorem	Spectral decomposition of L²(Γ\H)	Roelcke-Selberg spectral decomposition	L²(Γ\H) = ℂ ⊕ V_cusp ⊕ V_res ⊕ V_cont, with V_cusp from Maass cusp forms, V_res from residues of Eisenstein series, and V_cont the continuous Eisenstein spectru
s_discrete_spectrum_gamma_h	axiom	Discrete spectrum of Γ\H	cuspidal plus residual spectrum	The subspace of L²(Γ\H) spanned by eigenfunctions with isolated eigenvalues: constants, Maass cusp forms, and residual Eisenstein contributions.
s_continuous_spectrum_gamma_h	axiom	Continuous spectrum of Γ\H	Eisenstein contribution	The absolutely continuous part of the Laplacian spectrum on L²(Γ\H), parametrized by Eisenstein series E(z, 1/2+it) for t ∈ ℝ, occupying [1/4, ∞).
s_pre_trace_formula	theorem	Pre-trace formula	automorphic kernel identity	∑_j h(r_j)|u_j(z)|² + (cont. part) = ∑_{γ ∈ Γ} k(u(z,γz)), the spectral expansion of an automorphic kernel, preceding the Selberg trace formula.
s_selberg_zeta_function	axiom	Selberg zeta function Z_Γ(s)	Selberg zeta function	Z_Γ(s) = ∏_{γ₀} ∏_{k≥0}(1 − N(γ₀)^{−s−k}) over primitive hyperbolic classes, an entire function whose zeros encode eigenvalues of Δ on Γ\H.
s_maass_selberg_relations	theorem	Maass-Selberg relations	Maass-Selberg formula	Inner product formulas for truncated Eisenstein series in terms of the scattering matrix, controlling the continuous spectrum contribution.
s_petersson_trace_formula	theorem	Petersson trace formula	Petersson-Kuznetsov formula	∑_f ā_f(m)a_f(n)/⟨f,f⟩ = δ_{m,n} + 2πi^{-k}∑_{c≡0(N)} c^{-1}S(m,n;c)J_{k-1}(4π√(mn)/c), relating Fourier coefficients to Kloosterman sums.
s_eichler_selberg_trace_formula	theorem	Eichler-Selberg trace formula	Selberg trace formula for Hecke operators	An explicit formula for tr(T_n | S_k(Γ₀(N))) as a sum over conjugacy classes involving class numbers, totient values, and cusp/elliptic contributions.
s_kuznetsov_trace_formula	theorem	Kuznetsov trace formula	Kuznetsov sum formula | Bruggeman-Kuznetsov formula	A sum formula relating spectral sums over Maass/holomorphic forms weighted by Fourier coefficients to geometric sums of Kloosterman sums via Bessel transforms.
s_voronoi_summation_formula	theorem	Voronoi summation formula (GL(2))	Voronoi formula	Transforms ∑_n a_f(n)e(nα/q)g(n) for a GL(2) form into a dual sum with Kloosterman twists and Bessel integral transforms; the automorphic Poisson summation.
s_l_function_of_modular_form	axiom	L-function of a modular form	standard L-function of f	L(f,s) = ∑ a_f(n)n^{-s} for a Hecke eigenform f, admitting analytic continuation and functional equation Λ(f,s) = ε·Λ(f,k−s).
s_euler_product_hecke_eigenform	theorem	Euler product for Hecke eigenform L-function	Hecke Euler product	L(f,s) = ∏_p(1 − a_p p^{-s} + χ(p)p^{k-1-2s})^{-1} for (p,N) = 1, with modified local factors at primes dividing the level.
t_rankin_selberg_method	technique	Rankin-Selberg method	Rankin-Selberg unfolding	Unfolds ∫_{Γ\H} f(z)ḡ(z)E(z,s)y^k dμ to obtain L(f⊗g,s), yielding analytic continuation and bounds for convolution L-functions.
s_symmetric_square_l_function	axiom	Symmetric square L-function L(sym²f, s)	symmetric square lift	The degree-3 L-function ∏_p(1−α²_p p^{-s})^{-1}(1−α_p β_p p^{-s})^{-1}(1−β²_p p^{-s})^{-1} whose analytic properties yield bounds toward Ramanujan.
s_jacquet_langlands_correspondence	theorem	Jacquet-Langlands correspondence	Jacquet-Langlands transfer	A bijection between automorphic representations of GL₂(𝔸) and those of quaternion algebra D×(𝔸) that are not one-dimensional, preserving L-functions.
s_gelbart_jacquet_lift	theorem	Gelbart-Jacquet lift	symmetric square lift to GL(3)	The functorial lift GL(2)→GL(3) corresponding to the symmetric square, proving L(sym²f,s) is automorphic on GL(3).
s_gl_n_automorphic_form	axiom	GL(n) automorphic form	automorphic form on GL(n)	A smooth function on GL_n(ℚ)\GL_n(𝔸) satisfying moderate growth, K-finiteness, and Z(g_∞)-finiteness, generalizing classical modular forms.
s_kirillov_model	axiom	Kirillov model	Kirillov space	A realization of a generic representation of GL₂(F) on functions on F×, obtained by restricting the Whittaker model to the diagonal torus.
s_strong_multiplicity_one_gl2	theorem	Strong multiplicity one for GL(2)	strong multiplicity one theorem	Two cuspidal automorphic representations of GL(2) with isomorphic local components at all but finitely many places are globally isomorphic.
s_twisted_l_function	axiom	Twisted L-function L(f⊗χ, s)	character twist of L-function	L(f⊗χ,s) = ∑ a_f(n)χ(n)n^{-s} for a Hecke eigenform f and Dirichlet character χ, with functional equation involving the conductor of χ.
s_satake_parameters	axiom	Satake parameters	Langlands parameters at unramified primes	The pair (α_p, β_p) with α_p + β_p = a_f(p) and α_p β_p = χ(p)p^{k-1} parametrizing the unramified local factor at prime p via the Satake isomorphism.
s_hecke_multiplicativity	theorem	Hecke multiplicativity relations	Hecke recursion	a(m)a(n) = ∑_{d|(m,n)} χ(d)d^{k-1} a(mn/d²) for a Hecke eigenform, encoding the ring structure of Hecke operators.
t_amplification_method	technique	Amplification method	amplification technique	Constructs an amplifier large on a target form, applies spectral mean-value estimates to convert spectral information into pointwise L-function bounds.
s_spectral_large_sieve	theorem	Spectral large sieve inequality	large sieve for automorphic forms	∑_j |∑_n a_n ρ_j(n)|² ≤ (N + T²)∑|a_n|², bounding bilinear forms in Fourier coefficients of Maass forms, the spectral analogue of the classical large sieve.
s_subconvexity_gl2	theorem	Subconvexity for GL(2) L-functions	GL(2) subconvexity	L(f, 1/2) ≪ C(f)^{1/4−δ} for some δ > 0, breaking the convexity bound; proved in various aspects by Iwaniec, Duke-Friedlander-Iwaniec, and Michel-Venkatesh.
t_spectral_mean_value_estimates	technique	Mean value estimates via spectral methods	spectral moment method	Uses the spectral decomposition and Kuznetsov formula to evaluate moments of L-functions over spectral families.
s_spectral_reciprocity	theorem	Spectral reciprocity formula	reciprocity formula	An identity relating moments of L-functions over one spectral family to moments over a dual family, transferring subconvexity between aspects.
s_nonvanishing_l_functions_at_center	theorem	Nonvanishing results for central L-values	nonvanishing at the central point	L(f,1/2) ≠ 0 or L(f⊗χ,1/2) ≠ 0 for a positive proportion of forms/characters in a family, proved via mollified moments.
s_singular_integral_circle_method	axiom	Singular integral (circle method)	singular integral in Waring's problem	The archimedean factor J(n) = ∫ δ(x₁^k+…+x_s^k − n) dx in the circle method asymptotic, measuring the real-analytic solution density.
s_hua_inequality	theorem	Hua's inequality	Hua's lemma	∫₀¹ |∑_{n≤N} e(αn^k)|^{2^k} dα ≪ N^{2^k − k + ε}, a mean value estimate for Weyl sums fundamental to the minor arc analysis in Waring's problem.
s_hardy_littlewood_asymptotic_waring	theorem	Hardy-Littlewood asymptotic for Waring's problem	Waring asymptotic formula	R_{s,k}(n) = 𝔖(n)·J(n)·n^{s/k−1} + o(n^{s/k−1}) for s sufficiently large, where 𝔖 is the singular series and J the singular integral.
t_farey_dissection	technique	Farey dissection	Farey arc decomposition	Partition of [0,1) into arcs centered at Farey fractions of order Q, providing finer major/minor arc decomposition than the standard Hardy-Littlewood dissection
t_kloosterman_refinement	technique	Kloosterman refinement (HLK method)	HLK method	The Hardy-Littlewood-Kloosterman method: major arc contributions yield Kloosterman sums whose cancellation (Weil bound) improves the error term.
t_delta_method	technique	Delta method (Duke-Friedlander-Iwaniec)	DFI delta method	Replaces the circle method's major/minor arc dissection with a smooth delta symbol built from Kloosterman sums, effective for binary problems.
s_huxley_zero_density_estimate	theorem	Huxley's zero density estimate	Huxley density estimate	N(σ,T) ≪ T^{12(1−σ)/5+ε} for σ ≥ 1/2, improving Ingham's estimate and implying primes in short intervals [x, x + x^{7/12+ε}].
s_jutila_zero_density_estimate	theorem	Jutila's zero density estimate	Jutila density estimate	Improved zero density bounds for Dirichlet L-functions uniform in the conductor q, with applications to primes in arithmetic progressions.
s_log_free_zero_density_estimate	theorem	Log-free zero density estimate	log-free density estimate	∑_{χ mod q} N(σ,T,χ) ≪ (qT)^{A(1−σ)} without logarithmic factors, critical for the Bombieri-Vinogradov theorem.
t_zero_detecting_polynomial	technique	Zero-detecting polynomial	zero detector	A Dirichlet polynomial M(s) chosen so M(s)L(s,χ) ≈ 1 near zeros, converting zero-counting into mean value estimates.
t_turan_power_sum_method	technique	Turán's power sum method	power sum method	Bounds max_{1≤ν≤N} |∑ b_j z_j^ν| from below for |z_j| ≥ 1, applied to detect zeros of zeta functions via power sum analysis.
t_halasz_montgomery_method	technique	Halász-Montgomery method	Halász-Montgomery zero density method	Combines Halász's theorem on mean values with Montgomery's zero-detection, exploiting duality between zeros and prime distribution.
s_gue_hypothesis	axiom	GUE hypothesis for ζ(s) zeros	random matrix theory connection | Katz-Sarnak philosophy	The conjecture that all local statistics of normalized zeros of ζ(s) match those of eigenvalues of large random matrices from the Gaussian Unitary Ensemble.
t_mollifier_method	technique	Mollifier method	mollification technique	Multiplies ζ(s) by M(s) ≈ 1/ζ(s) to smooth zeros, then analyzes moments of ζ·M on Re(s)=1/2 to bound zeros on the critical line.
t_levinson_method	technique	Levinson's method	Levinson's theorem on zeros	A refinement of the mollifier method using ζ'/ζ to prove that at least 1/3 of nontrivial zeros of ζ(s) lie on the critical line.
s_deuring_heilbronn_repulsion	theorem	Deuring-Heilbronn repulsion phenomenon	Deuring-Heilbronn phenomenon | zero repulsion	A real zero β₁ of L(s,χ₁) near 1 forces all other L-functions to have enlarged zero-free regions near 1, with explicit repulsion depending on 1 − β₁.
s_linnik_constant	axiom	Linnik's constant	Linnik constant L	The infimum L such that the least prime p ≡ a mod q satisfies p ≪ q^L; currently known L ≤ 5 (Xylouris, 2011), and L = 2+ε under GRH.
s_fouvry_theorem	theorem	Fouvry's theorem	Fouvry-Iwaniec level of distribution	A Bombieri-Vinogradov type result with level of distribution beyond x^{1/2}: for a positive proportion of moduli q ≤ x^{4/7−ε}, the PNT error satisfies strong a
s_duke_equidistribution	theorem	Duke's theorem (equidistribution of CM points)	Duke's equidistribution theorem	CM points of discriminant −D and closed geodesics of discriminant D become equidistributed on SL₂(ℤ)\H as D → ∞, proved via subconvexity for half-integral weigh
s_subconvexity_implies_equidistribution	theorem	Subconvexity implies equidistribution	subconvexity-equidistribution paradigm	Subconvexity L(f⊗χ_D, 1/2) ≪ D^{1/4−δ} implies equidistribution of associated arithmetic objects via Weyl's criterion applied to spectral test functions.
s_erdos_turan_inequality	theorem	Erdős-Turán inequality	Erdős-Turán discrepancy inequality	D_N ≤ C(1/H + ∑_{h=1}^H (1/h)|∑_{n≤N} e(hω_n)|/N) for any H ≥ 1, bounding discrepancy by exponential sums.
s_discrepancy_of_sequence	axiom	Discrepancy of a sequence	discrepancy | star discrepancy	D_N = sup_{[a,b)} |#{n ≤ N : ω_n ∈ [a,b)}/N − (b−a)|, the quantitative deviation from uniform distribution.
s_friedlander_iwaniec_theorem	theorem	Friedlander-Iwaniec theorem	primes of the form a²+b⁴	There are infinitely many primes of the form a² + b⁴, proved using the half-dimensional sieve, bilinear forms, and automorphic techniques.
t_half_dimensional_sieve	technique	Iwaniec's half-dimensional sieve	half-dimensional sieve	A sieve achieving dimension κ = 1/2 by exploiting bilinear structure in error terms, detecting primes where the linear sieve has parity obstruction.
s_shifted_convolution_sums	axiom	Shifted convolution sums	shifted convolution problem	Sums ∑_n a_f(n)a_g(n+h) for automorphic coefficients, estimated via spectral methods or the delta method with applications to subconvexity.
s_additive_divisor_problem	axiom	Additive divisor problem	binary additive divisor problem	The estimation of ∑_{n≤x} d(n)d(n+h), a prototype shifted convolution sum controlled by spectral theory of automorphic forms.
s_hecke_converse_theorem	theorem	Hecke's converse theorem	Hecke's inverse Mellin theorem	A Dirichlet series with Euler product and the standard functional equation Λ(s) = εΛ(k−s) necessarily arises from a modular form of weight k for SL₂(ℤ).
s_weil_converse_theorem	theorem	Weil's converse theorem	Weil's converse theorem for GL(2)	If L(s,χ) = ∑ a_n χ(n)n^{-s} satisfies functional equations for sufficiently many twists χ, then ∑ a_n q^n is a modular form of the predicted level and characte
s_holomorphic_modular_form_formal	axiom	Holomorphic modular form (weight k, level N)	classical modular form	A holomorphic f: H → ℂ with f|_k γ = χ(d)f for all γ ∈ Γ₀(N) and holomorphic at all cusps; M_k(Γ₀(N),χ) is finite-dimensional.
s_eisenstein_series_nebentypus	axiom	Eisenstein series with Nebentypus	Eisenstein series with character	Eisenstein series E_k(z,χ,ψ) = ∑ χ(c)ψ(d)(cz+d)^{-k} for characters χ,ψ with χψ(-1) = (-1)^k, spanning the Eisenstein subspace of M_k(Γ₁(N)).
s_ramanujan_conjecture_gl2	axiom	Ramanujan conjecture for GL(2)	Ramanujan-Petersson conjecture	The conjecture that for a cuspidal π of GL(2)/ℚ, all unramified local components are tempered; equivalently |α_p| = |β_p| = 1 at Satake parameters.
s_extremal_order_definition	axiom	Extremal order of an arithmetic function		The maximal (resp. minimal) order of f(n) is a function g(n) bounding f from above (resp. below) up to (1+o(1)) factor, achieved along some subsequence.
s_highly_composite_number	axiom	Highly composite number		An integer n such that d(m) < d(n) for all m < n; has the form ∏ p_i^{a_i} with a_1 ≥ a_2 ≥ … ≥ 1 (Ramanujan).
s_largest_prime_factor	axiom	Largest prime factor P⁺(n)		P⁺(n) = max{p : p | n} for n ≥ 2, with P⁺(1) = 1 by convention; central to the study of smooth/friable integers.
s_smallest_prime_factor_definition	axiom	Smallest prime factor P⁻(n)		P⁻(n) = min{p : p | n} for n ≥ 2, with P⁻(1) = +∞ by convention; central to the study of rough integers.
s_integers_free_of_small_prime_factors	axiom	Rough integers Φ(x, y)		Φ(x, y) = #{n ≤ x : P⁻(n) > y}, counting integers up to x whose smallest prime factor exceeds y.
s_natural_density	axiom	Natural density (asymptotic density)		d(A) = lim_{x→∞} |A ∩ [1,x]| / x for a set A ⊆ ℤ_{>0}, when the limit exists.
s_logarithmic_density	axiom	Logarithmic density		δ(A) = lim_{x→∞} (1/log x) ∑_{n≤x, n∈A} 1/n; if natural density d(A) exists then δ(A) = d(A), but the converse fails.
s_slowly_varying_function	axiom	Slowly varying function		A measurable function L: [1,∞) → (0,∞) with L(tx)/L(x) → 1 for every t > 0 as x → ∞; examples include log x, log log x, constants.
s_generalized_divisor_function_d_z	axiom	Generalized divisor function d_z(n)		For z ∈ ℂ, d_z(n) is the n-th Dirichlet coefficient of ζ(s)^z; multiplicative with d_z(p^k) = z(z+1)⋯(z+k−1)/k!.
s_local_zeta_function_y_smooth	axiom	Local zeta function ζ(s, y)		ζ(s, y) = ∏_{p≤y}(1 − p^{−s})^{−1} = ∑_{P⁺(n)≤y} n^{−s}, the truncated Euler product restricted to y-smooth integers.
s_pretentious_distance_function	axiom	Pretentious distance D(f, g; x)		D(f, g; x)² = ∑_{p≤x} (1 − Re(f(p)g̅(p)))/p, measuring proximity between multiplicative functions f, g of modulus ≤ 1 on primes.
s_hooley_delta_function	axiom	Hooley's Δ-function		Δ(n) = max_{u∈ℝ} #{d | n : e^u < d ≤ e^{u+1}}, measuring the maximum concentration of divisors of n in a dyadic interval.
s_distribution_function_arithmetic_function	axiom	Distribution function of an arithmetic function		F_N(t) = (1/N)#{n ≤ N : f(n) ≤ t}, the empirical cumulative distribution function of f evaluated at the first N integers.
s_limiting_distribution_arithmetic_function	axiom	Limiting distribution of an arithmetic function		An arithmetic function f has limiting distribution F if F_N(t) → F(t) at all continuity points of F as N → ∞.
s_summatory_liouville_function	state	Summatory Liouville function L(x)		L(x) = ∑_{n≤x} λ(n); the PNT is equivalent to L(x) = o(x), and the RH to L(x) = O(x^{1/2+ε}).
s_dickman_function_differential_delay_equation	state	Dickman function differential-delay equation		ρ satisfies uρ′(u) = −ρ(u−1) for u > 1 with ρ(u) = 1 for 0 ≤ u ≤ 1; ρ(u) decreases super-exponentially as u → ∞.
s_buchstab_function_differential_delay_equation	state	Buchstab function differential-delay equation		(uω(u))′ = ω(u−1) for u > 2 with uω(u) = 1 for 1 ≤ u ≤ 2; ω(u) → e^{−γ} as u → ∞.
s_saddle_point_alpha_x_y	state	Saddle-point parameter α(x, y)		The unique positive solution α to ∑_{p≤y} (log p)/(p^α − 1) = log x, the saddle point in the Hildebrand–Tenenbaum estimate for Ψ(x, y).
s_robin_inequality	state	Robin's inequality		σ(n) < e^γ n log log n for all n ≥ 5041; equivalent to the Riemann hypothesis (Robin, 1984).
s_mertens_first_theorem	theorem	Mertens' first theorem		∑_{p≤x} (log p)/p = log x + O(1).
s_mertens_second_theorem	theorem	Mertens' second theorem		∑_{p≤x} 1/p = log log x + M + O(1/log x), where M is the Meissel–Mertens constant.
s_mertens_third_theorem	theorem	Mertens' third theorem		∏_{p≤x}(1 − 1/p) = e^{−γ}/log x · (1 + O(1/log x)), where γ is the Euler–Mascheroni constant.
s_average_order_omega_n	theorem	Average order of ω(n)		∑_{n≤x} ω(n) = x log log x + M₁x + O(x/log x), where M₁ is a constant related to the Meissel–Mertens constant.
s_average_order_big_omega_n	theorem	Average order of Ω(n)		∑_{n≤x} Ω(n) = x log log x + M₂x + O(x/log x), with M₂ = M₁ + ∑_p 1/(p(p−1)).
s_squarefree_counting_asymptotic	theorem	Squarefree counting asymptotic		Q(x) = ∑_{n≤x} μ(n)² = (6/π²)x + O(√x), counting squarefree integers up to x.
s_maximal_order_divisor_function	theorem	Maximal order of d(n)		lim sup_{n→∞} log d(n) · log log n / log n = log 2; equivalently d(n) ≤ 2^{(1+o(1)) log n / log log n} with equality along highly composite numbers.
s_gronwall_theorem_sigma	theorem	Gronwall's theorem (maximal order of σ(n)/n)		lim sup_{n→∞} σ(n)/(n log log n) = e^γ, where γ is the Euler–Mascheroni constant.
s_minimal_order_euler_totient_ratio	theorem	Minimal order of φ(n)/n		lim inf_{n→∞} φ(n) log log n / n = e^{−γ}; achieved along primorial numbers n_k = ∏_{p≤p_k} p.
s_mean_value_zeta_critical_line	theorem	Mean value of |ζ(1/2+it)|²		∫₀ᵀ |ζ(1/2 + it)|² dt ∼ T log T as T → ∞.
s_dickman_theorem	theorem	Dickman's theorem		For fixed u > 0, Ψ(x, x^{1/u})/x → ρ(u) as x → ∞, where ρ is the Dickman function.
s_karamata_tauberian_theorem	theorem	Karamata's Tauberian theorem		If a_n ≥ 0 and ∑a_n n^{−s} ∼ Γ(α+1)(s−α)^{−1}L(1/(s−α)) as s → α⁺ with L slowly varying, then ∑_{n≤x} a_n ∼ x^α L(x).
s_delange_tauberian_theorem	theorem	Delange's Tauberian theorem		If ∑a_n n^{−s} behaves like ζ(s)^z near s = 1 for z ∈ ℂ, then ∑_{n≤x} a_n ∼ Cx(log x)^{z−1}/Γ(z) for an explicit constant C.
s_landau_page_theorem	theorem	Landau–Page theorem		Among all real primitive characters χ mod q with q ≤ Q, at most one L(s, χ) can vanish in the region σ > 1 − c/log Q.
s_dual_turan_kubilius_inequality	theorem	Dual Turán–Kubilius inequality		∑_{p^k≤x} |g(p^k)|²/p^k ≪ (1/x) ∑_{n≤x} |∑_{p^k‖n} g(p^k)|², dual to the Turán–Kubilius inequality.
s_normal_order_log_divisor_function	theorem	Normal order of log d(n)		The normal order of log d(n) is (log 2)(log log n): for almost all n, log d(n) ∼ (log 2) log log n.
s_delange_mean_value_multiplicative	theorem	Delange's theorem on mean values of multiplicative functions		If f is multiplicative with |f(p)| ≤ 1 and ∑(1 − f(p))/p converges, then (1/x)∑_{n≤x} f(n) → ∏_p(1 − 1/p)(1 + f(p)/p + f(p²)/p² + ⋯).
s_equivalence_pnt_mobius_mean_zero	theorem	Equivalence of PNT and ∑μ(n) = o(x)		The prime number theorem is equivalent to ∑_{n≤x} μ(n) = o(x), and also to ∑_{n≤x} μ(n)/n → 0.
s_elliott_multiplicative_in_progressions	theorem	Elliott's theorem on multiplicative functions in progressions		Extension of Halász's theorem to averages of multiplicative functions over arithmetic progressions, using character sum techniques.
s_hildebrand_dickman_extension	theorem	Hildebrand's extension of Dickman's theorem		Ψ(x, y) = xρ(u)(1 + O(log(u+1)/log y)) uniformly for u ≤ exp((log y)^{3/5−ε}), extending Dickman's result to a wide range.
s_hildebrand_tenenbaum_saddle_point_estimate	theorem	Hildebrand–Tenenbaum saddle-point estimate for Ψ(x, y)		Ψ(x, y) = (x^{1−α}ζ(α, y))/(α√(2πφ₂(α, y)))(1 + O(1/u)) where α is the saddle point, valid for exp((log log x)^{5/3+ε}) ≤ y ≤ x.
s_canfield_erdos_pomerance_theorem	theorem	Canfield–Erdős–Pomerance theorem		log Ψ(x, y) ∼ −u log u when u = log x / log y → ∞ with y → ∞, refining the Dickman approximation for large u.
s_alladi_erdos_largest_prime_factor_average	theorem	Alladi–Erdős result on average of log P⁺(n)		∑_{n≤x} log P⁺(n) = cx + o(x) for an explicit constant c = 1 − ∫₀^∞ ρ(t) dt related to the Dickman function.
s_smooth_numbers_short_intervals	theorem	Smooth numbers in short intervals		Ψ(x + h, y) − Ψ(x, y) ≫ hρ(u) under suitable conditions on h and y, showing smooth numbers are well-distributed in short intervals.
s_asymptotic_rough_integers_phi	theorem	Asymptotic for rough integers Φ(x, y)		Φ(x, y) = xω(u)∏_{p≤y}(1 − 1/p)(1 + O(1/log y)) for u = log x / log y, where ω is the Buchstab function.
s_friable_integers_arithmetic_progressions	theorem	Friable integers in arithmetic progressions		Ψ(x, y; q, a) = Ψ(x, y)/φ(q)(1 + O(…)) under suitable conditions on q and y, showing equidistribution of smooth numbers in progressions.
s_distribution_smallest_prime_factor	theorem	Distribution of the smallest prime factor		#{n ≤ x : P⁻(n) > y} = x∏_{p≤y}(1 − 1/p)(ω(u) + O(1/log y)); for fixed u the probability that P⁻(n) > n^{1/u} approaches e^{−γ}.
s_wintner_random_multiplicative_theorem	theorem	Wintner's theorem on random multiplicative functions		If f(p) = ±1 independently with probability 1/2 each, then ∑_{n≤x} f(n) = O(x^{1/2+ε}) almost surely.
s_maier_theorem_short_intervals	theorem	Maier's theorem on short intervals		π(x + (log x)^A) − π(x) is NOT asymptotic to (log x)^{A−1} for any A > 1; short intervals exhibit irregularities beyond PNT predictions.
s_daboussi_theorem	theorem	Daboussi's theorem		If f is multiplicative with |f| ≤ 1, then ∑_{n≤x} f(n)e(nα) = o(x) for every irrational α.
s_schnirelman_additive_basis_theorem	theorem	Schnirelman's additive basis theorem		If A has positive Schnirelman density σ(A) > 0 and 0 ∈ A, then A is an additive basis of finite order: every sufficiently large integer is a sum of at most h el
s_tenenbaum_divisor_distribution	theorem	Tenenbaum's theorem on divisor distribution		For almost all n, the divisors of n are approximately uniformly distributed on a logarithmic scale in [1, n].
s_generalized_divisor_d_z_asymptotic	theorem	Asymptotic for ∑d_z(n) (Selberg–Delange)		∑_{n≤x} d_z(n) ∼ Cx(log x)^{z−1}/Γ(z) for z ∈ ℂ, obtained via the Selberg–Delange method applied to ζ(s)^z.
t_laplace_method_integrals	technique	Laplace method for integrals		Asymptotic evaluation of integrals ∫e^{λφ(t)}g(t)dt as λ → ∞ by expanding around the maximum of φ.
t_kubilius_probabilistic_model	technique	Kubilius probabilistic model		Models events {p | n} for different primes as approximately independent with probability 1/p, making ω(n) behave like a sum of independent Bernoulli variables.
t_turan_second_moment_method	technique	Turán's second moment method		Elementary proof of normal order via ∑_{n≤x}(ω(n) − log log x)² = O(x log log x), yielding the Hardy–Ramanujan theorem without complex analysis.
t_pretentious_distance_method	technique	Pretentious distance method (Halász)		Characterizes when a multiplicative function has non-vanishing mean value: iff D(f, n^{it}; x) is bounded for some real t.
t_effective_tauberian_method	technique	Effective Tauberian method		Quantitative Tauberian theorems providing explicit error terms, typically requiring information about the zero-free region of the associated L-function.
t_counting_primes_smooth_rough_decomposition	technique	Smooth–rough decomposition for counting primes		Decomposes π(x) = π(y) + Φ(x,y) − 1 plus sieve correction terms, linking prime counting to counts of rough and smooth numbers.
t_granville_soundararajan_pretentious	technique	Granville–Soundararajan pretentious number theory		Framework studying multiplicative functions through D(f, χn^{it}; x): f has large partial sums iff it pretends to be χ(n)n^{it} for some character χ and real t.
t_renyi_turan_power_sum_method	technique	Rényi–Turán power sum method		Bounds the discrepancy of a sequence from its power sums (exponential sums), used in proving equidistribution and large sieve results.
t_twisting_by_dirichlet_characters	technique	Twisting by Dirichlet characters		Multiplying a Dirichlet series by a character χ to study its behavior in arithmetic progressions via L(s,χ) = ∑χ(n)n^{−s}.
s_logarithmic_density_primes_in_ap	state	Logarithmic density of primes in arithmetic progressions		The set of primes p ≡ a (mod q) with gcd(a,q) = 1 has logarithmic density 1/φ(q) among all primes, proved as an intermediate step toward Dirichlet's theorem.
s_gaussian_periods	state	Gaussian periods		The sums η_j = ∑_{k} ζ^{g^{ek+j}} where g is a primitive root mod p and e | (p−1), forming the basis for Gauss's theory of cyclotomy and constructibility.
s_gauss_constructible_regular_polygons	theorem	Gauss's theorem on constructible regular polygons		A regular n-gon is constructible by straightedge and compass if and only if n = 2^a p_1 ⋯ p_k where the p_i are distinct Fermat primes.
s_fundamental_discriminant	axiom	Fundamental discriminant		An integer D that is the discriminant of a quadratic field ℚ(√d): either D = d with d ≡ 1 (mod 4) squarefree, or D = 4d with d ≡ 2 or 3 (mod 4) squarefree.
s_binary_quadratic_form	axiom	Binary quadratic form		A function f(x,y) = ax² + bxy + cy² with a,b,c ∈ ℤ and discriminant D = b² − 4ac, central to the theory of class numbers and representations by forms.
s_finite_expression_l1_chi	state	Finite expression for L(1, χ)		For a nonprincipal character χ mod q, L(1,χ) equals a finite sum involving χ(a) and log sin(πa/q) or linear terms, providing computable values of Dirichlet L-fu
s_riemann_approximate_formula_for_N_T	state	Riemann's approximate formula for N(T)		N(T) ≈ (T/2π) log(T/2πe), Riemann's original heuristic approximation for the number of zeros of ζ(s) with imaginary part between 0 and T.
t_contour_integration_arithmetic_functions	technique	Contour integration for arithmetic functions		Recovers partial sums of arithmetic functions from their Dirichlet series via Perron-type contour integrals, shifting contours past poles and zeros to extract m
s_truncated_explicit_formula_psi	state	Truncated explicit formula for ψ(x)		ψ(x) = x − ∑_{|γ|≤T} x^ρ/ρ + O(x log²x / T), a practical form of the explicit formula with controlled truncation error.
s_chebyshev_psi_twisted_by_chi	state	Chebyshev ψ-function twisted by χ		ψ(x, χ) = ∑_{n≤x} Λ(n) χ(n), the Chebyshev function weighted by a Dirichlet character, whose analytic properties are governed by the zeros of L(s, χ).
t_character_sum_completion	technique	Character sum completion technique		Extends an incomplete character sum to a complete sum over a full period by inserting additive characters, enabling evaluation via Gauss sums.
s_further_prime_number_sums	state	Further prime number sums		Estimates for sums ∑_{p≤x} f(p) for various arithmetic functions f derived from the prime number theorem and partial summation.
s_generalized_xi_function	state	Generalized xi-function ξ(s, χ)		The completed L-function ξ(s, χ) = (q/π)^{(s+a)/2} Γ((s+a)/2) L(s, χ) for primitive χ mod q, an entire function of order 1 satisfying ξ(s, χ) = W(χ) ξ(1−s, χ̄).
s_root_number_w_chi	state	Root number W(χ)		The complex constant W(χ) = τ(χ)/(i^a √q) of absolute value 1 appearing in the functional equation of ξ(s, χ), where a = (1−χ(−1))/2.
s_counting_zeros_entire_order_1	state	Counting zeros of entire functions of order 1		An entire function f of order 1 satisfies n(r) = O(r) where n(r) counts zeros with |z| ≤ r, and ∑ 1/|ρ|^{1+ε} converges for every ε > 0.
s_hadamard_product_for_xi_chi	theorem	Hadamard product for ξ(s, χ)		ξ(s, χ) = e^{A(χ)+B(χ)s} ∏_ρ (1 − s/ρ) e^{s/ρ}, the product over nontrivial zeros ρ of L(s, χ), obtained by Hadamard factorization of the entire function ξ(s, χ
s_partial_fraction_l_prime_over_l	state	Partial fraction for −L′/L		−L′(s,χ)/L(s,χ) = −B(χ) − ∑_ρ (1/(s−ρ) + 1/ρ) + (terms from trivial zeros), expressing the logarithmic derivative as a sum over nontrivial zeros.
t_zero_counting_argument_principle	technique	Zero-counting via argument principle		Counts zeros of ζ(s) or L(s,χ) in a rectangle by integrating (1/2πi) ∮ f′/f ds around the boundary.
s_log_zeta_bound_near_sigma_1	state	log ζ(s) bound near σ = 1		log ζ(σ+it) = O(log t) for σ ≥ 1 − c/log t and |t| ≥ 2, a consequence of the zero-free region providing control on ζ near the 1-line.
s_repulsion_of_zeros_by_exceptional_zero	theorem	Repulsion of zeros by exceptional zero		If L(β₁, χ₁) = 0 with β₁ near 1 for a real character χ₁, then all other zeros ρ of L-functions mod q satisfy enlarged zero-free regions, quantifying the Deuring
s_s_t_argument_function	state	S(T) argument function		S(T) = (1/π) arg ζ(1/2 + iT) defined by continuous variation, satisfying S(T) = O(log T) and appearing in the Riemann–von Mangoldt formula as the oscillatory co
s_n_t_chi_zero_counting	state	N(T, χ) zero-counting function		N(T, χ) counts nontrivial zeros of L(s, χ) with imaginary part in [−T, T], generalizing the Riemann–von Mangoldt N(T) to Dirichlet L-functions.
s_asymptotic_n_t_chi	theorem	Asymptotic for N(T, χ)		N(T, χ) = (T/π) log(qT/2πe) + O(log qT) for primitive χ mod q, generalizing the Riemann–von Mangoldt formula to Dirichlet L-functions.
s_exponential_sum_over_primes	state	Exponential sum over primes		S(α) = ∑_{p≤N} (log p) e(pα), the generating function for primes used in the circle method, whose size on major and minor arcs controls additive problems involv
s_vinogradov_estimate_prime_exponential_sums	theorem	Vinogradov's estimate for prime exponential sums		If |α − a/q| ≤ 1/qN with (a,q) = 1, then |∑_{n≤N} Λ(n) e(nα)| ≪ N(log N)^B (q^{−1/2} + N^{−1/2} + q^{1/2} N^{−1/2}).
s_singular_series_three_primes	state	Singular series for three primes		The arithmetic factor 𝔖(N) in the asymptotic for representations of odd N as a sum of three primes, a convergent product over primes encoding local solubility.
t_major_arc_approximation_via_l_functions	technique	Major arc approximation via L-functions		Approximates ∑_{p≤N} (log p) e(pα) on major arcs using Dirichlet characters and L-function zero information, reducing the circle method to analytic properties o
s_large_sieve_as_mean_value_estimate	state	Large sieve as mean-value estimate		The large sieve inequality reinterpreted as a mean-value bound on character sums averaging over all primitive characters to moduli q ≤ Q, often matching GRH-str
t_vaughan_proof_bombieri_vinogradov	technique	Vaughan's proof of Bombieri–Vinogradov		Proves the Bombieri–Vinogradov theorem by decomposing Λ(n) via Vaughan's identity into bilinear forms, then bounding each via the large sieve inequality for cha
t_reduction_to_primitive_characters	technique	Reduction to primitive characters		Replaces sums over all Dirichlet characters mod q by sums over primitive characters of smaller moduli dividing q, exploiting the factorization χ = χ* · χ₀.
s_bombieri_vinogradov_as_grh_substitute	state	Bombieri–Vinogradov as substitute for GRH		The Bombieri–Vinogradov theorem provides, on average over q ≤ x^{1/2−ε}, the same quality of error term in ψ(x;q,a) that GRH would give for each individual q.
s_application_titchmarsh_divisor_problem	state	Application to the Titchmarsh divisor problem		∑_{p≤x} d(p−1) ∼ Cx log x for an explicit constant C, proved using the Bombieri–Vinogradov theorem to handle the sum over shifted primes.
t_de_la_vallee_poussin_proof_strategy	technique	De la Vallée-Poussin's proof strategy		Establishes the prime number theorem by proving ζ(1+it) ≠ 0 via the 3-4-1 inequality, deriving a zero-free region, then extracting ψ(x) ~ x via contour integrat
s_zero_free_region_real_characters_near_t_0	theorem	Zero-free region for real characters near t = 0		For a real primitive character χ mod q, L(s, χ) has at most one real zero β in (1 − c/log q, 1), requiring treatment separate from the 3-4-1 method which fails 
s_lower_bound_l1_chi_via_class_number	state	Lower bound for L(1, χ) via class number		For real primitive χ mod q, the class number formula gives L(1, χ) ≥ c/√q since h(d) ≥ 1, providing an effective but weak lower bound.
s_siegel_lemma_l_function_products	theorem	Siegel's lemma on L-function products		If L(β, χ₁) = 0 with β near 1 for real χ₁, then for any other real character χ₂ the product L(s, χ₁)L(s, χ₂) has no zero near s = 1, forcing L(1, χ₂) to be boun
s_ineffectivity_in_siegel_theorem	state	Ineffectivity in Siegel's theorem		The constant c(ε) in Siegel's bound L(1, χ) > c(ε) q^{−ε} cannot be effectively computed because the proof splits into cases depending on the existence of an ex
s_symmetric_pairing_of_zeros	state	Symmetric pairing of zeros		The functional equation of ξ(s, χ) implies that if ρ is a nontrivial zero of L(s, χ), then 1 − ρ̄ is a zero of L(s, χ̄), and for real χ the zeros come in symmet
s_density_of_zeros_near_sigma_1	state	Density of zeros near σ = 1		The number of zeros ρ of L(s, χ) with Re(ρ) ≥ σ and |Im(ρ)| ≤ T satisfies N(σ, T, χ) = O(T^{A(1−σ)} log T) for σ near 1.
s_contribution_of_major_arcs	state	Contribution of major arcs		The integral of S(α)³ e(−Nα) over the major arcs yields the main term (1/2) 𝔖(N) N²/(log N)³ in Vinogradov's three-primes theorem.
s_contribution_of_minor_arcs	state	Contribution of minor arcs		The integral of S(α)³ e(−Nα) over the minor arcs is O(N²/(log N)^A), shown negligible using Vinogradov's estimate for exponential sums over primes.
s_dirichlet_polynomial	state	Dirichlet polynomial		A finite sum D(s) = ∑_{n≤N} a_n n^{−s}, a fundamental building block in the analytic theory of L-functions, large sieve, and Vaughan's identity.
s_character_table_of_z_qz_star	state	Character table of (ℤ/qℤ)*		The φ(q) × φ(q) matrix (χ_i(a_j)) of Dirichlet characters mod q evaluated at reduced residue representatives, satisfying row and column orthogonality.
s_factorization_characters_via_crt	state	Factorization of characters via CRT		For q = q₁q₂ with gcd(q₁,q₂) = 1, every Dirichlet character χ mod q factors uniquely as χ = χ₁χ₂ via the CRT isomorphism (ℤ/qℤ)* ≅ (ℤ/q₁ℤ)* × (ℤ/q₂ℤ)*.
s_dirichlet_l1_evaluation_quadratic_chi	theorem	Dirichlet's L(1, χ) evaluation for quadratic χ		For real primitive χ_D mod |D|, L(1, χ_D) equals an explicit finite sum involving χ_D(a) and log sin(πa/|D|) or linear terms, depending on the sign of D.
s_estimate_sum_chi_p_over_p	state	Estimate for ∑ χ(p)/p		For nonprincipal χ mod q, ∑_{p≤x} χ(p)/p = O(1) uniformly in x, while for the principal character the sum equals log log x + M + O(1/log x).
s_mertens_type_estimate_in_ap	state	Mertens-type estimate in arithmetic progressions		∑_{p≤x, p≡a mod q} 1/p = (1/φ(q)) log log x + C(q,a) + O(1/log x) for gcd(a,q) = 1, the analogue of Mertens' second theorem for primes in arithmetic progression
s_cyclotomic_polynomial_irreducibility	theorem	Cyclotomic polynomial irreducibility over ℚ		The n-th cyclotomic polynomial Φ_n(x) = ∏_{gcd(k,n)=1} (x − e^{2πik/n}) is irreducible over ℚ, equivalently [ℚ(ζ_n):ℚ] = φ(n).
s_algebraic_integer_neukirch	axiom	Algebraic integer	integral element over ℤ	An element of an algebraic number field that is a root of a monic polynomial with coefficients in ℤ.
s_norm_of_ideal	axiom	Norm of an ideal N(𝔞)		For a nonzero ideal 𝔞 of O_K, the finite cardinality N(𝔞) = |O_K/𝔞|, which is multiplicative: N(𝔞𝔟) = N(𝔞)N(𝔟).
s_minkowski_embedding	axiom	Minkowski embedding (canonical embedding)		The ring homomorphism σ: K → ℝ^{r₁} × ℂ^{r₂} sending α to (σ₁(α),...,σ_{r₁}(α),τ₁(α),...,τ_{r₂}(α)), which embeds O_K as a full-rank lattice.
s_finiteness_of_class_number	theorem	Finiteness of the class number		The ideal class group Cl(K) of any algebraic number field K is finite, proved via the Minkowski bound.
s_transitivity_of_different_discriminant	theorem	Transitivity of different and discriminant		For a tower K ⊂ L ⊂ M, 𝔇_{M/K} = 𝔇_{M/L}·𝔇_{L/K} and d_{M/K} = N_{L/K}(d_{M/L})·d_{L/K}^{[M:L]}.
s_complementary_module	axiom	Complementary module (codifferent)		The fractional ideal 𝔇_{L/K}^{-1} = {x ∈ L : Tr_{L/K}(x·O_L) ⊂ O_K}, the O_L-dual of O_L with respect to the trace pairing.
s_different_discriminant_relation	theorem	Different–discriminant relation		The relative discriminant equals the norm of the different: d_{L/K} = N_{L/K}(𝔇_{L/K}), and a prime ramifies iff it divides d_{L/K}.
s_exponent_of_different_at_prime	theorem	Exponent of the different at a prime		For a prime 𝔓|𝔭 with ramification index e, v_𝔓(𝔇_{L/K}) = e−1 if tamely ramified and ≥ e if wildly ramified.
s_tame_and_wild_ramification	axiom	Tame and wild ramification		A prime is tamely ramified if e is coprime to the residue characteristic p (wild inertia group V₁ is trivial), and wildly ramified otherwise.
s_discriminant_of_cyclotomic_field	theorem	Discriminant of cyclotomic fields		The discriminant of ℚ(ζ_p) for a prime p is (-1)^{(p-1)/2}·p^{p-2}, and p is the only ramified prime, totally ramified as (1−ζ_p)O = 𝔭^{p-1}.
s_crt_dedekind_domains	theorem	Chinese Remainder Theorem for Dedekind domains		For pairwise coprime ideals 𝔞₁,...,𝔞ᵣ in a Dedekind domain, O/𝔞₁···𝔞ᵣ ≅ O/𝔞₁ × ··· × O/𝔞ᵣ.
s_finiteness_of_ramification	theorem	Finiteness of ramification		In any finite extension L/K of number fields, only finitely many primes ramify, namely those dividing the relative discriminant d_{L/K}.
s_discrete_valuation	axiom	Discrete valuation		A surjective map v: K× → ℤ satisfying v(xy) = v(x) + v(y) and v(x+y) ≥ min(v(x),v(y)), whose valuation ring is a DVR.
s_newton_polygon	axiom	Newton polygon		For a polynomial f(x) = Σaᵢxⁱ over a non-archimedean valued field, the lower convex hull of the points (i, v(aᵢ)) in the plane.
s_newton_polygon_theorem	theorem	Newton polygon theorem		The slopes of the Newton polygon of f(x) over a complete non-archimedean field are the negatives of the valuations of the roots, with multiplicities given by ho
s_krasners_lemma	theorem	Krasner's lemma		If α is separable over K(β) and |α−β| < |α−α'| for all conjugates α' ≠ α in a complete non-archimedean field, then K(α) ⊆ K(β).
s_classification_of_local_fields	theorem	Classification of local fields		A non-archimedean local field of characteristic 0 is a finite extension of ℚ_p; of characteristic p > 0 it is isomorphic to 𝔽_q((t)).
s_unramified_extension_local	axiom	Unramified extension of a local field		A finite extension L/K of local fields with e(L/K) = 1, obtained by adjoining roots of unity of order prime to the residue characteristic.
s_totally_ramified_extension_local	axiom	Totally ramified extension of a local field		A finite extension L/K of local fields with f(L/K) = 1 (same residue field), generated by a root of an Eisenstein polynomial.
s_eisenstein_polynomial	axiom	Eisenstein polynomial		A monic polynomial f(x) = xⁿ + aₙ₋₁xⁿ⁻¹ + ··· + a₀ with all aᵢ in the maximal ideal 𝔪 and a₀ ∉ 𝔪², which is irreducible and generates a totally ramified extensi
s_structure_of_unramified_extensions	theorem	Structure of unramified extensions		For each f ≥ 1 there is a unique unramified extension of degree f of a local field K, obtained by adjoining a (q^f−1)-th root of unity, with cyclic Galois group
s_unramified_totally_ramified_decomposition	theorem	Decomposition into unramified and totally ramified parts		Every finite extension L/K of local fields contains a unique maximal unramified subextension T/K with [T:K] = f, and L/T is totally ramified of degree e.
s_teichmuller_representatives	axiom	Teichmüller representatives		The unique system of (q−1)-th roots of unity in a local field K providing a canonical multiplicative section of the reduction map O → κ.
s_product_formula_valuations	theorem	Product formula for valuations		For any nonzero α in a number field K, the product of normalized absolute values over all places equals 1: ∏_v |α|_v = 1.
s_replete_ideal	axiom	Replete ideal (Arakelov ideal)		A formal product ∏_{𝔭 finite} 𝔭^{v_𝔭} · ∏_{σ infinite} σ^{v_σ} with v_𝔭 ∈ ℤ (almost all zero) and v_σ ∈ ℝ, extending fractional ideals to include archimedean da
s_replete_divisor	axiom	Replete divisor		An element of ⊕_{𝔭 finite} ℤ·𝔭 ⊕ ⊕_{σ infinite} ℝ·σ, the Arakelov-theoretic analogue of a Weil divisor on the arithmetic curve Spec(O_K).
s_arakelov_class_group	state	Arakelov class group (replete divisor class group)		The quotient CH¹(O_K) = Div(O_K)/{replete principal divisors}, an extension of the ideal class group by a real torus.
s_metrized_line_bundle	axiom	Metrized line bundle on Spec(O_K)		A projective O_K-module of rank 1 equipped with hermitian metrics at each infinite place, the arithmetic-geometric analogue of a line bundle on a curve.
s_degree_map_replete	state	Degree map on replete divisors		The homomorphism deg: Div(O_K) → ℝ defined by deg(D) = Σ_𝔭 v_𝔭·log N(𝔭) + Σ_σ v_σ, inducing a degree map on the Arakelov class group.
s_euler_minkowski_characteristic	state	Euler–Minkowski characteristic		For a metrized fractional ideal 𝔞, χ(𝔞) = −log(vol(𝔞)) + (n/2)log(π), the arithmetic analogue of the Euler characteristic of a line bundle on a curve.
s_riemann_roch_number_fields	theorem	Riemann–Roch theorem for number fields		For a replete divisor D on Spec(O_K), χ(D) = deg(D) − ½ log|d_K|, analogous to χ(D) = deg(D) − g + 1 for algebraic curves.
s_ideal_lattice	state	Ideal lattice		The image of a fractional ideal 𝔞 under the Minkowski embedding, forming a full lattice in K_ℝ with covolume |d_K|^{1/2}·N(𝔞).
s_tate_cohomology_groups	axiom	Tate cohomology groups Ĥⁿ(G,A)		For a finite group G and G-module A, a modification of group cohomology defined for all n ∈ ℤ, with Ĥ⁰ = A^G/N_G(A) and Ĥ⁻¹ = ker(N_G)/I_G·A.
s_herbrand_quotient	state	Herbrand quotient h(G,A)		For a finite cyclic group G acting on A, the ratio h(G,A) = |Ĥ⁰(G,A)|/|Ĥ⁻¹(G,A)|, which is multiplicative on short exact sequences and equals 1 for finite modul
s_class_formation	axiom	Class formation		A pair (G,A) of a profinite group with closed subgroups and a G-module satisfying H¹ = 0 for all layers and H² ≅ (1/n)ℤ/ℤ canonically for degree-n Galois layers
s_formation_module	axiom	Formation module		The G-module A in a class formation playing the role of the multiplicative group (local) or idèle class group (global), satisfying the class formation axioms.
s_invariant_map	state	Invariant map inv_{L/K}		The canonical isomorphism inv_{L/K}: H²(G_{L/K}, A_L) → (1/[L:K])ℤ/ℤ that is part of the class formation data, generalizing the Hasse invariant for Brauer group
s_fundamental_class	state	Fundamental class u_{L/K}		The unique element of H²(G_{L/K}, A_L) mapping to 1/[L:K] under the invariant map, generating H² as a cyclic group and driving the Nakayama–Tate isomorphism.
s_nakayama_tate_duality	theorem	Nakayama–Tate duality theorem		Cup product with the fundamental class u_{L/K} induces isomorphisms Ĥⁿ(G,ℤ) → Ĥⁿ⁺²(G,A) for all n; at n = −2 this gives the abstract reciprocity map G^{ab} ≅ A^
s_abstract_reciprocity_map	theorem	Abstract reciprocity map		The isomorphism rec_{L/K}: A_K/N_{L/K}(A_L) → G_{L/K}^{ab} from Nakayama–Tate at n = −2, which in concrete cases gives the local or global Artin map.
s_abstract_existence_theorem	theorem	Abstract existence theorem		In a class formation satisfying an existence axiom, there is a bijection between open subgroups of finite index in A_K and finite abelian extensions of K, via n
s_inflation_map	state	Inflation map in group cohomology		For H normal in G, the map inf: Hⁿ(G/H, A^H) → Hⁿ(G,A) induced by the projection G → G/H, injective on H¹ when H¹(H,A) = 0.
s_restriction_map_cohomology	state	Restriction map in group cohomology		For H ≤ G, the map res: Hⁿ(G,A) → Hⁿ(H,A) induced by restricting cochains, satisfying cor ∘ res = [G:H].
s_corestriction_map	state	Corestriction (transfer) map in group cohomology		For H of finite index in G, the map cor: Hⁿ(H,A) → Hⁿ(G,A) satisfying cor ∘ res = [G:H]·id, relating cohomology at different levels.
s_inflation_restriction_exact_sequence	theorem	Inflation–restriction exact sequence		For H normal in G, the sequence 0 → H¹(G/H, A^H) → H¹(G,A) → H¹(H,A)^{G/H} → H²(G/H, A^H) → H²(G,A) is exact.
s_shapiros_lemma	theorem	Shapiro's lemma		For H ≤ G and H-module B, Hⁿ(G, Ind_H^G B) ≅ Hⁿ(H,B) for all n ≥ 0, extending to Tate cohomology when [G:H] is finite.
s_cohomologically_trivial_module	axiom	Cohomologically trivial G-module		A G-module M with Ĥⁿ(H,M) = 0 for all n ∈ ℤ and all subgroups H of G.
s_tate_theorem_cohom_trivial	theorem	Tate's theorem on cohomologically trivial modules		A G-module M is cohomologically trivial iff Ĥ⁰(G_p,M) = 0 and Ĥ¹(G_p,M) = 0 for every Sylow p-subgroup G_p of G.
s_norm_map_g_modules	state	Norm map N_G for G-modules		For a finite group G acting on A, the map N_G: A → A^G defined by N_G(a) = Σ_{g∈G} g·a, whose image and kernel define Tate cohomology.
s_augmentation_ideal	axiom	Augmentation ideal I_G		The kernel of the augmentation map ε: ℤ[G] → ℤ, generated by elements g−1 for g ∈ G, with I_G·A being the 'co-norms' in A.
s_two_periodicity_tate	theorem	2-periodicity of Tate cohomology for cyclic groups		For a finite cyclic group G, Ĥⁿ(G,A) ≅ Ĥⁿ⁺²(G,A) for all n and all G-modules A, via cup product with a generator of Ĥ²(G,ℤ).
s_profinite_completion	state	Profinite completion		For a group G, the profinite completion Ĝ = lim_{←} G/N over normal subgroups N of finite index, with dense image of the canonical map G → Ĝ.
s_g_module_discrete	axiom	G-module (discrete)		An abelian group A with continuous action of a profinite group G (A discrete), i.e., each element is fixed by some open subgroup of G.
s_cohomological_dimension	axiom	Cohomological dimension		A profinite group G has cd(G) ≤ n if Hⁱ(G,A) = 0 for all i > n and all torsion discrete G-modules A; e.g., absolute Galois groups of local fields have cd = 2.
s_brauer_group_as_h2	theorem	Brauer group as H²		The Brauer group Br(K) is canonically isomorphic to H²(Gal(K^{sep}/K), (K^{sep})×); for local fields, inv: Br(K) → ℚ/ℤ is an isomorphism.
s_long_exact_cohomology_sequence	theorem	Long exact cohomology sequence (Tate)		A short exact sequence 0 → A → B → C → 0 of G-modules induces an infinite long exact sequence ··· → Ĥⁿ(G,A) → Ĥⁿ(G,B) → Ĥⁿ(G,C) → Ĥⁿ⁺¹(G,A) → ···.
s_hochschild_serre_exact_sequence	theorem	Hochschild–Serre five-term exact sequence		For H normal in G: 0 → H¹(G/H, A^H) → H¹(G,A) → H¹(H,A)^{G/H} → H²(G/H, A^H) → H²(G,A), a special case of the Hochschild–Serre spectral sequence.
s_hilbert_90_abstract	theorem	Hilbert 90 (abstract version)		The vanishing H¹(G_{L/K}, A_L) = 0 (Formation Axiom I), generalizing Hilbert's Theorem 90 for multiplicative groups and for idèle class groups.
s_higher_unit_groups	axiom	Higher unit groups U^{(n)}		The filtration U^{(n)} = 1 + 𝔪ⁿ of the unit group U of the ring of integers O of a local field, with U^{(0)}/U^{(1)} ≅ κ× and U^{(n)}/U^{(n+1)} ≅ κ.
s_formal_group_law	axiom	Formal group law		A power series F(X,Y) ∈ O[[X,Y]] satisfying the axioms of a commutative group law (associativity, commutativity, identity at 0, existence of inverses).
s_lubin_tate_polynomial	axiom	Lubin–Tate polynomial		A polynomial f(X) ∈ O_K[X] with f(X) ≡ πX (mod X²) and f(X) ≡ X^q (mod π), serving as the endomorphism [π] of the associated Lubin–Tate formal group.
s_lubin_tate_extension	state	Lubin–Tate extension		The totally ramified abelian extension K_{π,n} = K(λ) where λ ranges over πⁿ-torsion of the Lubin–Tate group, with Gal(K_{π,n}/K) ≅ (O_K/πⁿO_K)×.
s_maximal_abelian_extension_local	state	Maximal abelian extension of a local field		The compositum K^{ab} = K^{ur}·K_π, independent of the choice of uniformizer π, where K^{ur} is the maximal unramified extension and K_π = ∪ K_{π,n}.
s_local_reciprocity_law	theorem	Local reciprocity law		For every finite abelian extension L/K of local fields, the local Artin map induces an isomorphism K×/N_{L/K}(L×) → Gal(L/K).
s_local_existence_theorem	theorem	Local existence theorem		Every open subgroup of finite index in K× is the norm group N_{L/K}(L×) for a unique finite abelian extension L/K of local fields.
s_hilbert_symbol	axiom	Hilbert symbol		For a local field K with n-th roots of unity, the pairing (a,b)_n: K×/K×ⁿ × K×/K×ⁿ → μ_n satisfying bimultiplicativity and (a,1−a)_n = 1.
s_ramification_upper_numbering	axiom	Ramification filtration (upper numbering)		The filtration Gal(L/K)^s (s ∈ ℝ, s ≥ −1) of the Galois group by higher ramification groups in upper numbering, compatible with quotients via the Herbrand funct
s_herbrand_function	state	Herbrand function		The function φ_{L/K}(s) = ∫₀ˢ dt/[G₀:G_t] converting lower numbering to upper numbering of ramification groups, with inverse ψ_{L/K}.
s_maximal_unramified_extension	state	Maximal unramified extension		The extension K^{ur} = ∪ K(ζ_{q^n−1}) of a local field K, with Gal(K^{ur}/K) ≅ Ẑ via the Frobenius automorphism.
s_local_norm_residue_symbol	state	Local norm residue symbol (local Artin map)		The continuous homomorphism r_K: K× → Gal(K^{ab}/K) sending a uniformizer to Frobenius on K^{ur} and acting on torsion points via [a⁻¹].
s_local_kronecker_weber	theorem	Local Kronecker–Weber theorem		Every finite abelian extension of ℚ_p is contained in a cyclotomic field ℚ_p(ζ_n) for some n.
s_strong_approximation_theorem	theorem	Strong approximation theorem		For a number field K and a place v₀, the image of K in ∏_{v≠v₀} K_v is dense; equivalently J_K = K×·J_K^{v₀}.
s_global_artin_map	state	Global Artin map		The surjective continuous homomorphism r_K: C_K → Gal(K^{ab}/K) whose restriction to each local factor K_v× is the local Artin map.
s_artin_reciprocity_global	theorem	Artin reciprocity law (global)		For every finite abelian extension L/K, the global Artin map induces an isomorphism C_K/N_{L/K}(C_L) → Gal(L/K).
s_global_existence_theorem	theorem	Global existence theorem		Every open subgroup of finite index in C_K is the norm group N_{L/K}(C_L) for a unique finite abelian extension L/K.
s_ray_class_field	state	Ray class field		The unique abelian extension K(𝔪)/K corresponding to the ray class group Cl_𝔪(K), with Gal(K(𝔪)/K) ≅ Cl_𝔪(K); a prime is unramified iff it does not divide 𝔪.
s_modulus_cycle	axiom	Modulus (cycle)		A formal product 𝔪 = ∏_v 𝔭_v^{n_v} over places with n_v ≥ 0 for finite places and n_v ∈ {0,1} for real places, defining congruence conditions for ray class grou
s_conductor_of_extension	state	Conductor of an abelian extension		The smallest modulus 𝔣 = 𝔣(L/K) such that the Artin map factors through Cl_𝔣(K), equal to the product of local conductors over all places.
s_principal_ideal_theorem_cft	theorem	Principal ideal theorem (Hauptidealsatz, class field theory)		Every fractional ideal of K becomes principal in the Hilbert class field H, proved via the triviality of the transfer map Gal(K₂/K) → Gal(K₂/H).
s_hasse_norm_theorem	theorem	Hasse norm theorem		An element a ∈ K× is a norm from a cyclic extension L/K if and only if a is a local norm at every place v.
s_grunwald_wang_theorem	theorem	Grunwald–Wang theorem		Given local conditions at finitely many places compatible with a global degree, there exists a cyclic extension realizing them, subject to a specific exception 
s_class_field_tower	state	Class field tower		The tower K = K₀ ⊂ K₁ ⊂ K₂ ⊂ ··· where Kᵢ₊₁ is the Hilbert class field of Kᵢ; by Golod–Shafarevich, infinite towers exist.
s_norm_limitation_theorem	theorem	Norm limitation theorem		For a finite Galois extension L/K, N_{L/K}(C_L) = N_{M/K}(C_M) where M/K is the maximal abelian sub-extension; the norm group depends only on the abelian part.
s_artin_symbol	state	Artin symbol (Frobenius at unramified primes)		For an abelian extension L/K unramified at 𝔭, the element (L/K, 𝔭) ∈ Gal(L/K) acting as x ↦ x^{N𝔭} on the residue field.
s_idelic_norm_map	state	Idelic norm map		For L/K, the map N_{L/K}: J_L → J_K defined component-wise by local norms, continuous and inducing C_L → C_K on idèle class groups.
s_conductor_discriminant_formula	theorem	Conductor–discriminant formula		For a finite abelian extension L/K, d_{L/K} = ∏_χ 𝔣(χ) where χ ranges over characters of Gal(L/K) and 𝔣(χ) is the Artin conductor.
s_decomposition_law_abelian	theorem	Decomposition law in abelian extensions		For an abelian extension L/K, a prime 𝔭 with Artin symbol of order f splits into g = [L:K]/ef primes of residue degree f and ramification index e.
s_content_idele_norm	state	Content (idèle norm)		The continuous homomorphism ‖·‖: J_K → ℝ_{>0} defined by ‖a‖ = ∏_v |a_v|_v, trivial on K× by the product formula, factoring through C_K.
s_analytic_continuation_dedekind_zeta	theorem	Analytic continuation of the Dedekind zeta function		ζ_K(s) extends to a meromorphic function on ℂ with a simple pole at s = 1 of residue (2^{r₁}(2π)^{r₂}h_K R_K)/(w_K√|d_K|).
s_functional_equation_dedekind_zeta	theorem	Functional equation for the Dedekind zeta function		The completed zeta function ξ_K(s) = |d_K|^{s/2}·Γ_ℝ(s)^{r₁}·Γ_ℂ(s)^{r₂}·ζ_K(s) satisfies ξ_K(s) = ξ_K(1−s).
s_hecke_character	axiom	Hecke character (Größencharakter)		A continuous homomorphism χ: C_K = J_K/K× → ℂ×, unifying Dirichlet characters and characters at infinity into a single adelic framework.
s_artin_conductor	state	Artin conductor		For a representation ρ of Gal(L/K), the ideal 𝔣(ρ) = ∏_𝔭 𝔭^{f_𝔭(ρ)} where f_𝔭 = Σᵢ (|Gᵢ|/|G₀|)·codim(V^{Gᵢ}), measuring ramification.
s_artin_conjecture	state	Artin conjecture		Every nontrivial irreducible Artin L-function L(s,ρ,L/K) is entire; known for 1-dimensional representations via class field theory, open in general.
s_meromorphic_continuation_artin	theorem	Meromorphic continuation of Artin L-functions		By Brauer's theorem, every Artin L-function admits meromorphic continuation to ℂ, being expressible as a quotient of products of Hecke L-functions.
s_l_function_nonvanishing_s1	theorem	L-function non-vanishing at s = 1		For any nontrivial character χ of a ray class group, L(1,χ) ≠ 0; the key analytic input for the Chebotarev density theorem and the existence theorem.
s_schwartz_bruhat_function	axiom	Schwartz–Bruhat function		On a local field K_v: a Schwartz function for archimedean v, a locally constant compactly supported function for non-archimedean v; the test functions for adeli
s_local_zeta_integral	state	Local zeta integral (Tate)		Z_v(f_v, χ_v, s) = ∫_{K_v×} f_v(x)χ_v(x)|x|_v^s d×x for a local Schwartz–Bruhat function f_v, whose ratio with the local L-factor gives the epsilon factor.
s_local_epsilon_factor	state	Local epsilon factor		The local constant ε(s, χ_v, ψ_v) appearing in the local functional equation, depending on the character χ_v and additive character ψ_v, whose product gives the
s_adelic_poisson_summation	theorem	Adelic Poisson summation		For a Schwartz–Bruhat function f on 𝔸_K: Σ_{α∈K} f(α) = Σ_{α∈K} f̂(α), with Fourier transform via the self-dual Haar measure on 𝔸_K/K.
s_self_duality_adeles	state	Self-duality of 𝔸_K/K		The compact group 𝔸_K/K is Pontryagin self-dual via the pairing (x,y) ↦ ψ(xy) for a nontrivial additive character ψ, the key ingredient for adelic Poisson summa
s_abelian_l_equals_hecke	theorem	Abelian Artin L-function equals Hecke L-function		For a one-dimensional Artin representation χ of Gal(L/K) with L/K abelian, L(s,χ,L/K) = L(s,χ') where χ' is the Hecke character via the Artin map.
s_inductivity_artin_l	theorem	Inductivity of Artin L-functions		L(s, Ind_H^G σ) = L(s, σ, L/F) for induced representations, meaning Artin L-functions are invariant under induction from subgroups.
s_factorization_dedekind_zeta	theorem	Factorization of the Dedekind zeta function		For a Galois extension L/K, ζ_L(s) = ∏_ρ L(s,ρ)^{dim(ρ)} over irreducible representations ρ of Gal(L/K); for abelian extensions, ζ_L = ∏_χ L(s,χ).
s_functional_equation_hecke_l	theorem	Functional equation for Hecke L-functions		Λ(s,χ) = (|d_K|·N𝔣(χ))^{s/2}·∏_{v|∞} γ_v(s,χ_v)·L(s,χ) satisfies Λ(s,χ) = W(χ)·Λ(1−s,χ̄), proved via Tate's thesis.
s_functional_equation_artin_l	theorem	Functional equation for Artin L-functions		Λ(s,ρ) = (|d_K|^{dim ρ}·N𝔣(ρ))^{s/2}·∏_{v|∞} L_v(s,ρ)·L(s,ρ) satisfies Λ(s,ρ) = W(ρ)·Λ(1−s,ρ̌), via Brauer's theorem.
s_analytic_continuation_hecke_l	theorem	Analytic continuation of Hecke L-functions		Every Hecke L-function for a nontrivial character extends to an entire function on ℂ; for the trivial character it has a simple pole at s = 1.
s_haar_measure_ideles	axiom	Haar measure on idèles		The product measure d×x = ∏_v d×x_v on J_K with vol(O_v×) = 1 for almost all v and d×x_v = dx_v/|x_v|_v, giving J_K a locally compact group structure.
s_inertia_invariant_subspace	state	Inertia-invariant subspace V^{I_𝔭}		For a Galois representation ρ and prime 𝔭 with inertia group I_𝔭, the subspace of V fixed by I_𝔭, on which Frobenius acts to define the local Euler factor at ra
s_dirichlet_theorem_primes_number_fields	theorem	Dirichlet's theorem on primes in number fields		For any nontrivial class c in Cl_𝔪(K), there are infinitely many primes in c, with Dirichlet density 1/|Cl_𝔪(K)|.
t_logarithmic_embedding	technique	Logarithmic embedding (Dirichlet map)		Maps a unit u to (log|σ₁(u)|,...,log|σ_{r₁+r₂}(u)|), whose image is a lattice of rank r₁+r₂−1 in the trace-zero hyperplane.
t_minkowski_lattice_point_argument	technique	Minkowski's lattice point argument		Embeds an ideal as a lattice via the canonical embedding then applies the convex body theorem to find algebraic integers of bounded norm.
t_ideal_norm_reduction	technique	Ideal norm reduction		Shows every ideal class contains an ideal of norm at most the Minkowski bound by applying the convex body theorem to the ideal lattice.
t_hensel_lifting	technique	Hensel lifting		Iterative Newton-like procedure lifting approximate roots modulo higher powers of the maximal ideal, converging in the complete valued field.
t_dimension_shifting_cohomology	technique	Dimension shifting in group cohomology		Embeds A into an induced module I and uses the long exact sequence to relate Hⁿ(G,A) to Hⁿ⁻¹(G,I/A), reducing higher cohomology to lower.
t_herbrand_quotient_computation	technique	Herbrand quotient computation		Exploits multiplicativity of Herbrand quotients on short exact sequences and h(G,A) = 1 for finite A, reducing computations via filtrations.
t_cup_product_technique	technique	Cup product pairing technique		Uses cup product to establish dualities and periodicity in Tate cohomology, notably cupping with a class in H² to produce periodicity isomorphisms.
t_reduction_to_sylow	technique	Reduction to Sylow subgroups in cohomology		Reduces cohomological statements to Sylow subgroups via injectivity of restriction on p-primary parts and cor ∘ res = [G:G_p].
t_lubin_tate_construction	technique	Lubin–Tate formal group construction		Constructs explicit abelian extensions of a local field by adjoining torsion of a formal group, providing a non-cohomological construction of the local Artin ma
t_brauer_induction	technique	Brauer induction		Expresses an arbitrary Galois representation as a virtual sum of induced 1-dimensional characters, reducing Artin L-function properties to abelian L-functions.
t_tate_thesis_adelic	technique	Tate's thesis (adelic zeta integrals)		Computes L-functions as integrals ζ(f,χ,s) = ∫_{J_K} f(x)χ(x)|x|^s d×x, with the functional equation arising from adelic Poisson summation.
t_transfer_homomorphism	technique	Transfer (Verlagerung) homomorphism		The group-theoretic transfer map Ver: G^{ab} → H^{ab} for H ≤ G, used in proving the principal ideal theorem by reducing to a group-theoretic triviality stateme
t_poisson_summation_number_field	technique	Poisson summation in number field setting		Applies Σ_{x∈Λ} f(x) = (1/vol)Σ_{y∈Λ*} f̂(y) over ideal lattices to derive the functional equation of Dedekind zeta functions.
s_pro_p_group	axiom	Pro-p group	pro-p-group | p-profinite group	A profinite group that is an inverse limit of finite p-groups for a fixed prime p.
s_cm_field	axiom	Complex multiplication (CM) field	CM field	A totally imaginary quadratic extension of a totally real number field; the endomorphism algebra of a CM abelian variety.
s_linear_algebraic_group	axiom	Linear algebraic group	algebraic group	An algebraic variety G equipped with a group structure such that multiplication and inversion are morphisms of varieties.
s_semi_simple_algebraic_group	axiom	Semi-simple algebraic group	semisimple algebraic group	A connected linear algebraic group with no nontrivial connected solvable normal subgroup (trivial radical).
s_simply_connected_algebraic_group	axiom	Simply connected algebraic group	simply connected semisimple group	A semi-simple algebraic group that admits no nontrivial central isogeny from another semi-simple group; its own universal cover.
s_restricted_topological_product	axiom	Restricted topological product (restricted direct product)	restricted direct product | restricted product	Given locally compact groups {G_v} with compact open subgroups {H_v} for almost all v, the subgroup of the direct product consisting of tuples (g_v) with g_v ∈ 
s_group_homology	state	Group homology H_n(G, M)	homology of groups	The left derived functors of the G-coinvariants functor M ↦ M_G = M/I_G·M, where I_G is the augmentation ideal of ℤ[G].
s_absolute_galois_group	state	Absolute Galois group Gal(K^sep/K)	absolute Galois group	The Galois group of the separable closure of K over K, equipped with the Krull topology as an inverse limit of finite Galois groups.
s_supernatural_number	state	Supernatural number	Steinitz number	A formal product ∏ p^{n_p} over all primes with exponents n_p ∈ {0, 1, 2, …, ∞}; the order of a profinite group is a supernatural number.
s_free_profinite_group	state	Free profinite group		A profinite group F such that every continuous map from a basis set to a profinite group G extends uniquely to a continuous homomorphism F → G.
s_continuous_cohomology_profinite	state	Continuous cohomology of profinite groups	profinite group cohomology | Galois cohomology	H^n(G, M) = lim_U H^n(G/U, M^U) over open normal subgroups U, the cohomology theory for profinite groups with discrete coefficient modules.
s_cm_type	state	CM type		For a CM field E of degree 2g over ℚ, a set Φ = {φ_1, …, φ_g} of embeddings E ↪ ℂ choosing one from each complex-conjugate pair; determines the complex structur
s_abelian_variety_with_cm	state	Abelian variety with complex multiplication	CM abelian variety	An abelian variety A of dimension g whose endomorphism algebra End(A) ⊗ ℚ contains a CM field of degree 2g; the richest class of abelian varieties for arithmeti
s_reflex_field	state	Reflex field (reflex CM type)	reflex CM type | dual CM type	For a CM type (E, Φ), the subfield E* of ℂ generated over ℚ by {∑_{φ ∈ Φ} φ(a) : a ∈ E}; the field of definition of the CM type up to isogeny.
s_algebraic_hecke_character	state	Algebraic Hecke character (Grössencharakter of type A₀)	algebraic Grössencharakter | type A₀ Hecke character	A Hecke character χ: I_K → ℂ* whose infinity-type is algebraic, given by a product of embeddings raised to integer powers; naturally associated to CM abelian va
s_l_extension	state	ℓ-extension	l-extension | p-extension	A Galois extension whose Galois group is a pro-ℓ group for a fixed prime ℓ; the study of the ℓ-part of class field theory.
s_z_l_extension	state	ℤ_ℓ-extension	Z_l-extension | Z_p-extension	A Galois extension with Galois group isomorphic to ℤ_ℓ (the ℓ-adic integers); the cyclotomic ℤ_p-extension ℚ(μ_{p^∞})/ℚ is the prototypical example.
s_global_zeta_integral_tate	state	Global zeta integral (Tate)	Tate's global zeta function	Z(f, χ, s) = ∫_{I_K} f(x)χ(x)|x|^s d*x where f is a Schwartz–Bruhat function on 𝔸_K and χ is a Hecke character; a single integral computing all Hecke L-function
s_quasi_character	state	Quasi-character of K_v*	quasi-character	A continuous homomorphism χ: K_v* → ℂ*; of the form χ(x) = χ₀(x)|x|^s where χ₀ is a unitary character and s ∈ ℂ.
s_self_dual_measure	state	Self-dual Haar measure	self-dual measure	The unique Haar measure on 𝔸_K (relative to an additive character ψ) such that the Fourier transform is an isometry; equivalently, vol(𝔸_K/K) = 1 with standard 
s_kroneckers_jugendtraum	state	Kronecker's Jugendtraum	Kronecker's youth dream | Hilbert's 12th problem (partial)	The program (partially realized by CM theory) to generate all abelian extensions of imaginary quadratic fields via special values of elliptic functions and modu
s_kneser_tits_conjecture	state	Kneser–Tits conjecture	Kneser–Tits problem	The conjecture that for an isotropic simply connected simple algebraic group G over a field K, the Whitehead group G(K)/G(K)⁺ is trivial, where G(K)⁺ is generat
s_golod_shafarevich_theorem	theorem	Golod–Shafarevich theorem	Golod–Šafarevič theorem | Golod–Shafarevich inequality	If G is a finite p-group with d minimal generators and r relations, then r > d²/4; consequently, certain pro-p groups with enough generators relative to relatio
s_existence_infinite_class_field_towers	theorem	Existence of infinite class field towers		There exist number fields whose p-class field tower K ⊂ K₁ ⊂ K₂ ⊂ ⋯ does not terminate, answering negatively a question of Furtwängler; proved via the Golod–Sha
s_hasse_principle_algebraic_groups	theorem	Hasse principle for algebraic groups (Kneser–Harder)	Kneser–Harder theorem	For simply connected semi-simple groups G over a number field K, the map H¹(K, G) → ∏_v H¹(K_v, G) is injective; the local-global principle holds for principal 
s_strong_approximation_algebraic_groups	theorem	Strong approximation for algebraic groups		For a simply connected semi-simple group G over a number field K and a place v with G(K_v) non-compact, G(K) is dense in ∏'_{w≠v} G(K_w).
s_main_theorem_complex_multiplication	theorem	Main theorem of complex multiplication	Shimura–Taniyama reciprocity for CM	For an abelian variety A with CM type (E, Φ) and σ ∈ Aut(ℂ/E*), the conjugate σ(A) is isogenous to A via an isogeny controlled by the reflex norm and the Artin 
s_serre_tate_good_reduction_cm	theorem	Serre–Tate theorem (good reduction of CM abelian varieties)		An abelian variety with sufficiently many complex multiplications has potentially good reduction at every finite place; it acquires good reduction over a finite
s_iwasawa_class_number_theorem	theorem	Iwasawa's theorem on class numbers in ℤ_p-extensions	Iwasawa class number formula	For a ℤ_p-extension K_∞/K with intermediate fields K_n, the p-part of the class number satisfies v_p(h_n) = μp^n + λn + ν for sufficiently large n, with Iwasawa
s_local_functional_equation_tate	theorem	Local functional equation (Tate)		The local zeta integral satisfies Z(f̂, χ⁻¹, 1−s) = ε(χ, s, ψ)·Z(f, χ, s) where f̂ is the Fourier transform, ψ is an additive character, and ε is the local epsi
s_global_functional_equation_tate	theorem	Global functional equation (Tate's thesis)		The global zeta integral Z(f, χ, s) has meromorphic continuation to all s ∈ ℂ and satisfies Z(f̂, χ⁻¹, 1−s) = Z(f, χ, s); implies the functional equation and an
s_l_function_euler_product_local	theorem	Decomposition of L-functions into local zeta integrals	Euler product from local factors	For factorizable Schwartz–Bruhat functions f = ∏_v f_v, the global zeta integral decomposes as Z(f, χ, s) = ∏_v Z(f_v, χ_v, s); with standard choices, this reco
s_global_local_compatibility_reciprocity	theorem	Global–local compatibility of reciprocity maps	compatibility of local and global Artin maps	The global Artin map r_K restricted to K_v* (via the natural embedding K_v* → I_K → C_K) equals the local Artin map φ_{K_v} for each place v.
s_hilbert_different_formula	theorem	Hilbert's different formula	Hilbert's formula for the different	For a Galois extension of local fields with ramification groups G_i, the different exponent equals ∑_{i=0}^∞ (|G_i| − 1), linking the different ideal to the hig
s_first_inequality_cft	theorem	First inequality (global class field theory)	first inequality	For L/K cyclic of degree n, [C_K : N_{L/K}C_L] ≥ n; proved analytically using the non-vanishing of L(1, χ) for nontrivial characters.
s_second_inequality_cft	theorem	Second inequality (global class field theory)	second inequality	For L/K cyclic of degree n, |H²(Gal(L/K), C_L)| ≤ n; combined with the first inequality yields |Ĥ⁰| = n and |Ĥ⁻¹| = 1 for the Tate cohomology of C_L.
s_krull_theorem_infinite_galois	theorem	Krull's theorem (infinite Galois theory)	fundamental theorem of infinite Galois theory	For an infinite Galois extension L/K, there is an inclusion-reversing bijection between intermediate fields and closed subgroups of Gal(L/K) in the Krull topolo
s_herbrand_theorem_upper_numbering	theorem	Herbrand's theorem (upper numbering of ramification groups)	Herbrand's upper numbering theorem	The ramification groups in upper numbering G^t = G_{ψ(t)} (defined via the Hasse–Arf function φ) are compatible with passage to quotients: for H normal in G, (G
t_bar_resolution	technique	Standard complex (bar resolution)		The explicit free resolution of ℤ over ℤ[G] with B_n = ℤ[G^{n+1}] and boundary maps; produces group (co)homology via cochains/chains and enables concrete comput
t_cohomology_of_ideles	technique	Cohomology of idèles (local-to-global reduction)		Reduces the computation of H^n(Gal(L/K), I_L) to local cohomology computations via the decomposition I_L = ∏'_w L_w* and Shapiro's lemma.
t_lyndon_hochschild_serre_spectral_sequence	technique	Lyndon–Hochschild–Serre spectral sequence		For N normal in G, a spectral sequence converging to H^*(G,M) with E₂ page given by the cohomology of G/N with coefficients in H^*(N,M); the five-term exact seq
s_projective_modules_over_dedekind_domains	theorem	Projective modules over Dedekind domains		Every finitely generated projective module over a Dedekind domain R is isomorphic to a direct sum of ideals; equivalently, it has the form R^{n-1} ⊕ I for some 
s_non_archimedean_absolute_value	axiom	Non-archimedean absolute value		An absolute value |·| on a field satisfying the ultrametric inequality |x+y| ≤ max(|x|,|y|), equivalently one for which |n| ≤ 1 for all integers n.
s_trace_form	axiom	Trace form		The symmetric bilinear form (x,y) ↦ Tr_{L/K}(xy) on a finite separable extension L/K, whose discriminant equals the field discriminant d_{L/K}.
s_different_ramification_theorem	theorem	Different–ramification theorem		A prime P of O_L divides the different D_{L/K} if and only if P is ramified over K; the exponent of P in D_{L/K} satisfies d_P ≥ e_P − 1 with equality iff the r
s_discriminant_ramification_theorem	theorem	Discriminant–ramification theorem		A prime p of O_K ramifies in L/K if and only if p divides the discriminant d_{L/K}; follows from the different–discriminant relation N_{L/K}(D_{L/K}) = d_{L/K}.
s_ring_of_integers_of_cyclotomic_field	theorem	Ring of integers of cyclotomic fields		The ring of integers of Q(ζ_n) is Z[ζ_n], generated as a Z-module by powers of a primitive n-th root of unity.
s_ramification_in_cyclotomic_fields	theorem	Ramification in cyclotomic fields		In Q(ζ_n)/Q, a prime p ramifies if and only if p divides n; specifically pO_{Q(ζ_{p^r})} = (1 − ζ_{p^r})^{φ(p^r)} and the ramification is totally ramified with 
s_gauss_sum_norm_formula	theorem	Gauss sum norm formula		For a primitive Dirichlet character χ mod m, the Gauss sum satisfies |g(χ)|² = m, and more generally g(χ)g(χ̄) = χ(−1)m.
s_stickelberger_element	state	Stickelberger element		The element θ = ∑_{a ∈ (Z/mZ)×} (a/m) σ_a⁻¹ in the group ring Q[Gal(Q(ζ_m)/Q)], whose annihilation properties on the class group are given by Stickelberger’s th
t_volume_computation_fundamental_domain	technique	Volume computation of fundamental domain		Computing the volume of the fundamental parallelotope of the image of O_K under the Minkowski embedding, yielding the key factor in Minkowski’s bound.
s_counting_ideals_of_bounded_norm	theorem	Counting ideals of bounded norm		The number of integral ideals of norm ≤ x in a number field K satisfies #{a ⊂ O_K : N(a) ≤ x} = κ·x + O(x^{1−1/[K:Q]}) where κ = (2^{r_1}(2π)^{r_2}R_K h_K)/(w_K
t_lattice_points_in_expanding_domains	technique	Lattice points in homogeneously expanding domains		Estimating the number of lattice points in a region scaled by parameter t, the fundamental counting technique behind ideal-counting and class number formulas.
s_content_map	state	Content (idèle norm) map		The continuous homomorphism ||·||: J_K → R>0 defined by ||x|| = ∏_v |x_v|_v, whose kernel J_K^1 contains K× by the product formula, yielding the exact sequence 
s_compactness_of_c_k_1	theorem	Compactness of C_K^1		The norm-one idèle class group C_K^1 = J_K^1/K× is compact, encoding both finiteness of the class number and Dirichlet’s unit theorem in a single topological st
s_idele_class_group_ray_class_group_relationship	theorem	Relationship between idèle class group and ray class groups		For a modulus m, the ray class group Cl_m(K) is canonically isomorphic to C_K / U_m K× where U_m is the open subgroup of idèles congruent to 1 mod m, connecting
s_galois_action_on_ideles	state	Galois action on idèles		For a Galois extension L/K, the Galois group Gal(L/K) acts on J_L by permuting components according to its action on places; the fixed subgroup satisfies J_L^{G
s_convergence_and_euler_product_for_dedekind_zeta	theorem	Convergence and Euler product for ζ_K		The Dedekind zeta function ζ_K(s) = ∑_{a ⊂ O_K} N(a)^{−s} converges absolutely for Re(s) > 1 and admits an Euler product ζ_K(s) = ∏_p (1 − N(p)^{−s})^{−1} over 
s_faltings_finiteness_theorem	theorem	Faltings’ finiteness theorem (Shafarevich conjecture)		Up to isomorphism, there are only finitely many abelian varieties of given dimension over a number field K with good reduction outside a fixed finite set of pri
s_local_norm_group	state	Local norm group		For a finite extension L_w/K_v of local fields, the norm group N_{L_w/K_v}(L_w×) is an open subgroup of finite index in K_v×, and the map L ↦ N_{L/K}(L×) gives 
s_local_norm_index_theorem	theorem	Local norm index theorem		[K_v× : N_{L_w/K_v}(L_w×)] = [L_w : K_v] for any finite Galois extension L_w/K_v of local fields; equivalently |Ĥ⁰(G, L_w×)| = n.
t_p_adic_exponential_and_logarithm	technique	p-adic exponential and logarithm		The p-adic exponential exp_p(x) = ∑ x^n/n! converges for v_p(x) > 1/(p−1) and the p-adic logarithm log_p(1+x) = ∑ (−1)^{n+1} x^n/n converges for |x|_p < 1, prov
s_structure_of_local_units	theorem	Structure of local units		For a local field K_v with residue field of order q = p^f, the unit group O_v× ≅ μ_{q−1} × Z_p^f as a topological group, where μ_{q−1} is the group of (q−1)-th 
s_global_cyclic_norm_index	theorem	Global cyclic norm index inequality		For a cyclic extension L/K of number fields of degree n, the norm index [J_K : K× · N_{L/K}(J_L)] divides n, a key step toward proving the first and second ineq
s_takagi_existence_theorem	theorem	Takagi’s existence theorem		Every open subgroup of finite index in the idèle class group C_K (equivalently, every congruence subgroup defined by a modulus m) is the norm group N_{L/K}(C_L)
s_conductor_class_field_relationship	theorem	Conductor–class field relationship		The conductor f(L/K) of an abelian extension L/K is the smallest modulus m such that L is contained in the ray class field K_m; a prime ramifies in L/K if and o
s_existence_theorem_of_cft	theorem	Existence theorem of class field theory		The correspondence L ↦ N_{L/K}(C_L) is an inclusion-reversing bijection between finite abelian extensions of K and open subgroups of finite index in C_K, with [
t_reduction_to_kummer_extensions	technique	Reduction to Kummer extensions		Proving the main theorems of CFT by first adjoining enough roots of unity to reduce to the case where the extension is Kummer, then using explicit norm computat
s_complete_splitting_theorem	theorem	Complete splitting theorem		A prime p of K splits completely in an abelian extension L/K if and only if p lies in the norm group N_{L/K}(C_L); equivalently, the Frobenius at p is trivial i
s_ramification_theorem_cft	theorem	Ramification theorem (class field theory)		A prime p of K is unramified in the abelian extension L/K if and only if the local norm group N_{L_P/K_p}(L_P×) contains the local units O_{K_p}×; the conductor
s_infinite_divisibility_of_universal_norms	theorem	Infinite divisibility of universal norms		The intersection ∩_L N_{L/K}(L×) over all finite extensions L/K equals {1} in K× (or the connected component in the local case), meaning elements that are norms
s_induced_character_l_series_factorization	theorem	Induced character and L-series factorization		Writing ρ = ∑ n_i Ind_{H_i}^G χ_i by Brauer’s theorem gives L(s,ρ,L/K) = ∏ L(s,χ_i,L/L^{H_i})^{n_i}, reducing non-abelian Artin L-functions to products of abeli
s_local_additive_duality	theorem	Local additive duality (self-duality of K_v)		For a local field K_v, the map x ↦ ψ_v(x·) gives an isomorphism K_v ≅ K̂_v between K_v and its Pontryagin dual, making K_v additively self-dual; the self-dual H
t_dirichlet_integral_technique	technique	Dirichlet integral		Recovering partial sums of arithmetic functions from their Dirichlet series via the integral (1/2πi) ∫ f(s) x^s/s ds, the analytic bridge between multiplicative
s_tauberian_theorem_for_dirichlet_series	theorem	Tauberian theorem for Dirichlet series		If f(s) = ∑ a_n n^{−s} with a_n ≥ 0 has a simple pole at s = 1 with residue c, and is otherwise analytic on Re(s) ≥ 1, then ∑_{n≤x} a_n ∼ cx, converting analyti
s_nonvanishing_l1_hecke_characters	theorem	Non-vanishing of L(1, χ) for Hecke characters		For a nontrivial Hecke character χ (Grössencharakter) of a number field K, L(1, χ) ≠ 0; generalizes the classical Dirichlet non-vanishing and is essential for t
s_dirichlet_as_special_case_of_chebotarev	theorem	Dirichlet’s theorem as special case of Chebotarev		Dirichlet’s theorem on primes in arithmetic progressions follows from the Chebotarev density theorem applied to the cyclotomic extension Q(ζ_m)/Q, where Frobeni
s_upper_estimate_residue_zeta_k	theorem	Upper estimate for residue of ζ_K at s = 1		The residue κ_K of ζ_K(s) at s = 1 satisfies κ_K ≤ c(ε) |d_K|^ε for every ε > 0, where c(ε) depends only on the degree [K:Q] and ε; controls the class number ti
s_siegel_lower_bound_residue	theorem	Siegel’s lower bound for the residue		For totally real fields K (and more generally), κ_K ≥ c(ε) |d_K|^{−ε} for every ε > 0; the constant c(ε) is ineffective due to the possible existence of Siegel 
t_comparison_of_residues_in_normal_extensions	technique	Comparison of residues in normal extensions		Comparing the residues of Dedekind zeta functions of a normal extension L/K by factoring ζ_L/ζ_K into Artin L-functions and bounding each factor, a key step in 
s_weierstrass_product_for_completed_l_function	state	Weierstrass product for completed L-function		The completed L-function Λ(s, χ) admits a Hadamard product Λ(s, χ) = e^{A+Bs} ∏_ρ (1 − s/ρ) e^{s/ρ} over its nontrivial zeros ρ, encoding the distribution of ze
t_logarithmic_derivative_xi	technique	Logarithmic derivative ξ’/ξ		Taking the logarithmic derivative of the Hadamard product to express ξ’/ξ(s) = B + ∑_ρ (1/(s−ρ) + 1/ρ), the basic tool for relating zeros to prime-counting func
s_estimate_for_xi_prime_over_xi	theorem	Estimate for ξ’/ξ		The bound ∑_ρ 1/|s−ρ|² = O(log |t|) for |Im(s)| = t → ∞ away from zeros, and the consequence that ξ’/ξ(s) = ∑_{|γ−t|≤1} 1/(s−ρ) + O(log t), localizing the contr
s_weil_positivity_criterion	theorem	Weil’s positivity criterion		The Riemann hypothesis for ζ_K is equivalent to the positivity ∑_{ρ} ĥ(ρ) ≥ 0 for all test functions h in a suitable class, reformulating RH as a positive-defin
t_evaluation_of_basic_sum_part_1	technique	Evaluation of basic sum (Part 1)		First stage of evaluating the prime-side sum in Weil’s explicit formula by Fourier analysis, converting the sum over primes into integrals involving the Fourier
t_evaluation_of_basic_sum_part_2	technique	Evaluation of basic sum (Part 2)		Second stage completing the evaluation by handling archimedean contributions and the integral terms, yielding the full Weil explicit formula relating prime sums
s_characteristic_of_a_field	axiom	Characteristic of a field		The smallest positive integer p such that p·1 = 0 in the field, or 0 if no such integer exists; a field has characteristic 0 or a prime.
s_cyclicity_of_fq_star	theorem	Cyclicity of F_q*		The multiplicative group F_q* of a finite field with q elements is cyclic of order q − 1.
t_character_sum_counting_solutions	technique	Character sum method for counting solutions		Expresses the number of solutions of equations over a finite field as sums involving multiplicative and additive characters, reducing counting to evaluation of 
s_p_adic_units	axiom	Units of Z_p		Z_p* = {x ∈ Z_p : |x|_p = 1} is the group of p-adic integers with p-adic absolute value 1, equivalently those x with x mod p ≠ 0.
s_compactness_of_z_p	theorem	Compactness of Z_p		The ring Z_p of p-adic integers is compact and totally disconnected in the p-adic topology, being the inverse limit of the finite rings Z/p^nZ.
s_structure_of_qp_star	theorem	Structure of Q_p*		Q_p* ≅ p^Z × μ_{p−1} × (1 + pZ_p) for p odd, where μ_{p−1} is the group of Teichmüller representatives and 1 + pZ_p ≅ Z_p as topological groups.
s_squares_in_qp_star	theorem	Squares in Q_p*		For p odd, Q_p*/Q_p*² ≅ (Z/2Z)² has order 4, generated by the classes of p and a non-square unit; for p = 2, Q_2*/Q_2*² ≅ (Z/2Z)³ has order 8.
s_density_of_q_in_qp	theorem	Density of Q in Q_p		The rational numbers Q are dense in Q_p with respect to the p-adic topology, since every p-adic number is a limit of rationals.
s_hilbert_symbol_at_infinity	axiom	Hilbert symbol at the archimedean place		The Hilbert symbol (a,b)_∞ for a, b ∈ R* equals −1 if and only if a < 0 and b < 0, reflecting the sign obstruction for the real norm form.
s_hilbert_symbol_properties	theorem	Bilinearity and non-degeneracy of the Hilbert symbol		The Hilbert symbol (·,·)_v : K_v*/K_v*² × K_v*/K_v*² → {±1} is bimultiplicative, symmetric, satisfies (a,1−a)_v = 1, and is non-degenerate in each variable.
s_hilbert_symbol_computation_odd_p	theorem	Hilbert symbol computation for odd p		For odd p and units u, v ∈ Z_p*, (u,v)_p = 1, (u,p)_p = (u/p), and (p^a u, p^b v)_p = (−1)^{ab(p−1)/2}(u/p)^b(v/p)^a.
s_existence_theorem_prescribed_hilbert_symbols	theorem	Existence theorem for prescribed Hilbert symbols		Given local invariants (ε_v) ∈ {±1} at finitely many places with ∏_v ε_v = 1, there exist a, b ∈ Q* with (a,b)_v = ε_v for all v.
s_discriminant_of_quadratic_form	axiom	Discriminant of a quadratic form		The determinant of the Gram matrix of a non-degenerate quadratic form, viewed as an element of K*/K*², an invariant under equivalence.
s_hasse_witt_invariant	axiom	Hasse–Witt invariant		For a diagonal form ⟨a_1,…,a_n⟩ over a local field K_v, the product ε_v = ∏_{i<j}(a_i,a_j)_v ∈ {±1}, a complete local invariant complementing rank and discrimin
s_isotropic_quadratic_form	axiom	Isotropic quadratic form		A quadratic form Q over a field K is isotropic if there exists a nonzero vector v ∈ K^n with Q(v) = 0; otherwise it is anisotropic.
s_witt_decomposition	theorem	Witt decomposition		Every non-degenerate quadratic form over a field decomposes uniquely as an orthogonal sum of hyperbolic planes and an anisotropic form.
s_classification_quadratic_forms_qp	theorem	Classification of quadratic forms over Q_p		A non-degenerate quadratic form over Q_p is determined up to equivalence by its rank, discriminant in Q_p*/Q_p*², and Hasse–Witt invariant ε_p ∈ {±1}.
s_representation_theorem_qp	theorem	Representation theorem for quadratic forms over Q_p		A non-degenerate quadratic form of rank ≥ 2 over Q_p represents a ∈ Q_p* if and only if the form ⟨Q, −a⟩ has the correct Hasse–Witt invariant.
s_classification_quadratic_forms_q	theorem	Classification of quadratic forms over Q		By Hasse–Minkowski, a non-degenerate quadratic form over Q is determined up to equivalence by its rank, signature, discriminant, and Hasse–Witt invariants at al
t_hasse_invariant_computation	technique	Hasse invariant computation		Computes the Hasse–Witt invariant ε_v = ∏_{i<j}(a_i,a_j)_v by reducing Hilbert symbols to Legendre symbols and explicit formulas.
s_unimodular_lattice	axiom	Unimodular lattice		An integral lattice L whose Gram matrix has determinant ±1, equivalently L = L* (self-dual with respect to the bilinear form).
s_even_type_ii_lattice	axiom	Even (type II) lattice		An integral lattice in which every vector has even norm Q(v) ∈ 2Z for all v ∈ L; if unimodular, its rank must be divisible by 8.
s_odd_type_i_lattice	axiom	Odd (type I) lattice		An integral unimodular lattice that is not even, i.e., some vector v ∈ L satisfies Q(v) ≡ 1 mod 2.
s_signature_of_real_quadratic_form	axiom	Signature of a real quadratic form		The pair (p, q) where p is the number of positive and q the number of negative eigenvalues, with p + q = n by Sylvester's law of inertia.
s_e8_lattice	axiom	E_8 lattice		The unique positive definite even unimodular lattice of rank 8, with root system E_8, minimal norm 2, and 240 roots.
s_classification_even_unimodular_lattices	theorem	Classification of even unimodular lattices		Positive definite even unimodular lattices exist only in ranks divisible by 8; unique in rank 8 (E_8) and rank 16, with 24 in rank 24 (Niemeier lattices).
s_classification_indefinite_unimodular_lattices	theorem	Classification of indefinite unimodular lattices		An indefinite unimodular lattice is determined up to isometry by its rank, signature, and type (even or odd); the even case requires signature ≡ 0 mod 8.
s_genus_of_integral_quadratic_form	axiom	Genus of an integral quadratic form		The equivalence class of an integral quadratic form under Z_p-equivalence at all primes p and R-equivalence; forms in the same genus are locally equivalent ever
s_genus_determines_rational_equivalence	theorem	Genus determines rational equivalence		Two integral quadratic forms in the same genus are equivalent over Q, though they need not be equivalent over Z.
t_local_to_global_lattices	technique	Local-to-global passage for lattices		Constructs or classifies integral quadratic forms by assembling local conditions at each prime and the archimedean place, using strong approximation or mass for
s_character_of_finite_abelian_group	axiom	Character of a finite abelian group		A group homomorphism χ: G → C* from a finite abelian group G to the multiplicative group of nonzero complex numbers; the characters form the dual group Ĝ ≅ G.
s_modular_function	axiom	Modular function		A meromorphic function on the upper half-plane invariant under SL₂(Z) (weight 0) and meromorphic at the cusp; the field of modular functions is C(j).
s_q_expansion_modular_form	axiom	Fourier expansion (q-expansion) of a modular form		The expansion f(τ) = Σ_{n≥0} a_n q^n with q = e^{2πiτ}, expressing a modular form as a power series in q; a cusp form has a_0 = 0.
s_valence_formula_modular_forms	theorem	Valence formula for modular forms		For a nonzero modular form f of weight k on SL₂(Z), v_∞(f) + v_i(f)/2 + v_ρ(f)/3 + Σ'_P v_P(f) = k/12.
s_decomposition_mk_eisenstein_cusp	theorem	Decomposition M_k = C·E_k ⊕ S_k		For even k ≥ 4, M_k(SL₂(Z)) decomposes as the direct sum of the one-dimensional span of the Eisenstein series E_k and the space S_k of cusp forms.
s_algebra_of_modular_forms	theorem	Structure of the algebra of modular forms		The graded algebra M_*(SL₂(Z)) = ⊕_{k≥0} M_k(SL₂(Z)) is the polynomial ring C[E_4, E_6], freely generated by Eisenstein series of weights 4 and 6.
s_delta_eisenstein_relation	theorem	Relation Δ = (E_4³ − E_6²)/1728		The modular discriminant Δ equals (E_4³ − E_6²)/1728 as a weight-12 cusp form, following from the structure of M_12(SL₂(Z)).
s_hecke_operators_self_adjoint	theorem	Hecke operators are self-adjoint		The Hecke operators T_n on S_k(SL₂(Z)) are self-adjoint with respect to the Petersson inner product: ⟨T_n f, g⟩ = ⟨f, T_n g⟩.
s_ramanujan_tau_multiplicativity	theorem	Multiplicativity of the Ramanujan tau function		τ(mn) = τ(m)τ(n) for gcd(m,n) = 1 and τ(p^{a+1}) = τ(p)τ(p^a) − p^{11}τ(p^{a−1}), since Δ is a Hecke eigenform.
s_theta_function_is_modular_form	theorem	Theta function of a quadratic form is a modular form		For a positive definite integral quadratic form Q in m variables, θ_Q(τ) is a modular form of weight m/2 for a congruence subgroup determined by Q.
s_representation_numbers_via_modular_forms	axiom	Representation numbers via modular forms		The number r_Q(n) of representations of n by Q equals the n-th Fourier coefficient of θ_Q, enabling exact formulas by decomposing θ_Q into Eisenstein and cusp p
s_eight_squares_formula	theorem	Eight squares formula		The number of representations of n as a sum of eight squares is r_8(n) = 16Σ_{d|n}(−1)^{n+d}d³, derived from the identity θ⁸ = E_4.
s_modular_form_fourier_coefficient_bound	theorem	Growth bound for Fourier coefficients of modular forms		For a modular form of weight k, a_n = O(n^{k−1}); for a cusp form, a_n = O(n^{k/2}) from the boundedness of y^{k/2}|f(z)| on H.
t_contour_integration_valence	technique	Contour integration on the fundamental domain		Integrates f'/f around the boundary of the fundamental domain, using modular transformation laws to match boundary segments, yielding the valence formula.
t_poisson_summation_for_theta	technique	Poisson summation for theta functions		Applies Poisson summation to the lattice sum defining a theta function to derive its modular transformation property under τ → −1/τ.
t_hecke_eigenform_method	technique	Hecke eigenform method		Simultaneously diagonalizes the commuting self-adjoint Hecke operators on the space of cusp forms to produce eigenforms whose L-functions have Euler products.
s_farey_sequence	axiom	Farey sequence F_n	Farey series	The ascending sequence of all irreducible fractions h/k with 0 ≤ h/k ≤ 1 and 1 ≤ k ≤ n.
s_mediant	axiom	Mediant of two fractions	Farey mediant | freshman sum	The mediant of a/b and c/d is the fraction (a+c)/(b+d).
s_normal_number	axiom	Normal number		A real number x is normal in base r if every block of k consecutive digits appears in its base-r expansion with asymptotic frequency r⁻ᵏ.
s_simply_normal_number	axiom	Simply normal number		A real number x is simply normal in base r if each digit 0, 1, …, r−1 occurs with frequency 1/r in its base-r expansion.
s_absolutely_normal_number	axiom	Absolutely normal number		A real number that is normal in every integer base r ≥ 2.
s_badly_approximable_number	axiom	Badly approximable number	bounded partial quotients	An irrational α with inf_{p,q} q²|α − p/q| > 0, equivalently one whose continued fraction partial quotients are bounded.
s_durfee_square	axiom	Durfee square		The side length of the largest square fitting in the upper-left corner of the Ferrers diagram of a partition.
s_stern_brocot_tree	axiom	Stern–Brocot tree	Stern-Brocot tree | Calkin-Wilf tree	The complete infinite binary tree of all positive rationals constructed by iterated mediant insertion between 0/1 and 1/0.
s_visible_lattice_point_density	state	Density of visible lattice points	coprimality probability	The number of lattice points (x,y) visible from the origin with |x|,|y| ≤ n is asymptotically 6n²/π², equivalently the probability two random integers are copri
s_fibonacci_golden_ratio_convergents	state	Fibonacci numbers as convergents of the golden ratio		The convergents of the continued fraction [1; 1, 1, 1, …] for the golden ratio φ = (1+√5)/2 are F_{n+1}/F_n, yielding the slowest-converging simple continued fr
s_failure_of_ufd_in_z_sqrt_neg5	state	Failure of unique factorization in ℤ[√(−5)]		The ring ℤ[√(−5)] is not a UFD: 6 = 2·3 = (1+√(−5))(1−√(−5)) gives two essentially distinct factorizations into irreducibles.
s_hurwitz_sharpness_sqrt5	state	Sharpness of √5 in Hurwitz's theorem		The constant √5 in Hurwitz's approximation theorem is best possible: for α = (1+√5)/2 the inequality |α − p/q| < 1/(cq²) has infinitely many solutions iff c ≤ √
s_waring_g_k_lower_bound	state	Lower bound for g(k) in Waring's problem		g(k) ≥ 2ᵏ + ⌊(3/2)ᵏ⌋ − 2 for all k ≥ 2, attained by integers just below 3ᵏ that require many k-th powers.
s_waring_G_k_lower_bounds	state	Lower bounds for G(k) in Waring's problem		G(k) ≥ k+1 for all k ≥ 2 by counting arguments; G(2ᶿ) ≥ 2ᶿ⁺² for θ ≥ 2 and G(4) ≥ 16 by congruence obstructions.
s_farey_neighbor_theorem	theorem	Farey neighbor property		If h/k and h'/k' are consecutive fractions in the Farey sequence F_n, then k·h' − h·k' = 1.
s_farey_mediant_theorem	theorem	Farey mediant property		If h/k, h″/k″, h'/k' are three consecutive terms of F_n, then h″ = h + h' and k″ = k + k', i.e., the middle term is the mediant of its neighbors.
s_borel_normal_number_theorem	theorem	Borel's normal number theorem	Borel's theorem on normal numbers	Almost all real numbers in the sense of Lebesgue measure are absolutely normal.
s_kempner_series_convergence	theorem	Kempner series convergence	Kempner's series	The sum of 1/n over all positive integers n whose base-10 representation omits a given digit converges.
s_lagrange_periodic_cf_theorem	theorem	Lagrange's periodic continued fraction theorem	Lagrange's theorem on continued fractions	A real number has an eventually periodic continued fraction expansion if and only if it is a quadratic irrational.
s_galois_purely_periodic_cf_theorem	theorem	Galois's theorem on purely periodic continued fractions		A continued fraction [a₀; a₁, a₂, …] is purely periodic if and only if it represents a reduced quadratic surd α > 1 whose conjugate α' satisfies −1 < α' < 0.
s_leudesdorf_theorem	theorem	Leudesdorf's theorem		A generalization of Wolstenholme's theorem giving divisibility conditions on sums of reciprocals of integers coprime to a composite modulus.
s_bauer_identical_congruence	theorem	Bauer's identical congruence		For prime p, the product ∏_{a=1}^{p−1}(x − a) ≡ x^{p−1} − 1 (mod p) holds as an identity in x.
s_kronecker_approximation_theorem	theorem	Kronecker's approximation theorem	Kronecker's theorem	If θ is irrational, then the set {nθ mod 1 : n ∈ ℕ} is dense in [0, 1); for any α and ε > 0 there exist integers n, p with |nθ − p − α| < ε.
s_kronecker_simultaneous_approximation	theorem	Kronecker's simultaneous approximation theorem	Kronecker's theorem in k dimensions	If θ₁, …, θₖ, 1 are linearly independent over ℚ, then the vectors (nθ₁, …, nθₖ) mod 1 are dense in [0,1)ᵏ.
s_rogers_ramanujan_continued_fraction	theorem	Rogers–Ramanujan continued fraction	Ramanujan's continued fraction	The continued fraction R(q) = q^{1/5}/(1 + q/(1 + q²/(1 + ⋯))) equals q^{1/5}·∏(1−q^{5n−1})(1−q^{5n−4})/∏(1−q^{5n−2})(1−q^{5n−3}) for |q| < 1.
s_euclidean_quadratic_fields_classification	theorem	Classification of Euclidean quadratic fields		The ring of integers of ℚ(√d) is Euclidean under the norm for d ∈ {−11, −7, −3, −2, −1, 2, 3, 5, 6, 7, 11, 13, …}; a finite list for imaginary quadratic fields.
s_minkowski_theorem_linear_forms	theorem	Minkowski's theorem on linear forms		If ξ₁, …, ξₙ are homogeneous linear forms with determinant Δ and λ₁⋯λₙ ≥ |Δ|, then there exist integers x₁, …, xₙ not all zero with |ξᵢ| ≤ λᵢ.
s_tchebotaref_inhomogeneous_form_theorem	theorem	Tchebotaref's theorem on inhomogeneous forms	Chebotarev's theorem on forms	For homogeneous linear forms ξᵢ with determinant Δ and arbitrary real shifts ρᵢ: inf |(ξ₁−ρ₁)⋯(ξₙ−ρₙ)| ≤ 2⁻ⁿ|Δ| over all integer points.
t_quaternion_norm_descent	technique	Quaternion norm descent		Use Hurwitz quaternion arithmetic with multiplicative norm to express primes as sums of four squares via descent.
t_durfee_square_decomposition	technique	Durfee square decomposition		Decompose a partition at its Durfee square to derive generating function identities relating partitions to q-series.
t_franklin_involution	technique	Franklin's involution		A sign-reversing involution on partitions into distinct parts that bijects even-part and odd-part partitions except at pentagonal numbers, proving Euler's penta
t_lettenmeyer_kronecker_proof	technique	Lettenmeyer's proof of Kronecker's theorem	Estermann's proof	An elementary geometric proof of Kronecker's approximation theorem by pigeonhole on ε-neighborhoods in the unit torus, extended to k dimensions by induction.
t_elliptic_curve_chord_tangent_addition	technique	Chord-and-tangent addition on elliptic curves	elliptic curve group law	Given points P, Q on an elliptic curve, draw line PQ (or tangent if P=Q), find the third intersection R with the curve, and reflect to obtain P+Q = −R, defining
s_polygonal_number	axiom	Polygonal number p_m(k)		The k-th m-gonal number p_m(k) = (m−2)·k(k−1)/2 + k, generalizing triangular, square, and pentagonal numbers to arbitrary polygon order m ≥ 3.
s_cauchy_polygonal_number_theorem	theorem	Cauchy’s polygonal number theorem		Every positive integer is the sum of at most m m-gonal numbers, of which all but four equal 0 or 1, completing the proof of Fermat’s polygonal number conjecture
s_cauchy_lemma_polygonal	theorem	Cauchy’s lemma (polygonal numbers)		If b ≥ 2a and b ≡ a (mod 2), then every integer in the interval [2a + b, 4a + 2b − 2] is a sum a·x₁² + b·x₂² with x₁, x₂ ∈ {0,1}, used in proving the polygonal 
s_pepin_five_pentagonal_theorem	theorem	Pépin’s theorem on five pentagonal numbers		Every sufficiently large positive integer is the sum of at most five pentagonal numbers, sharpening the general polygonal number theorem for m = 5.
s_waring_g_function	axiom	Waring’s function g(k)		g(k) is the smallest s such that every positive integer is representable as a sum of at most s perfect k-th powers.
s_waring_G_function	axiom	Asymptotic Waring function G(k)		G(k) is the smallest s such that every sufficiently large integer is representable as a sum of s perfect k-th powers.
s_wieferich_kempner_theorem	theorem	Wieferich–Kempner theorem (g(3) = 9)		Every positive integer is the sum of at most nine positive cubes, and nine are necessary (for 23 and 239), establishing g(3) = 9.
s_linnik_seven_cubes	theorem	Linnik’s theorem on sums of seven cubes (G(3) ≤ 7)		Every sufficiently large integer is the sum of at most seven positive cubes, i.e., G(3) ≤ 7, proved by Linnik using an elementary additive method.
s_sums_two_cubes_impossibility	state	Impossibility of representing all integers as sums of few cubes		Integers ≡ ±4 (mod 9) cannot be sums of three cubes, since cubes are congruent to 0 or ±1 modulo 9, giving G(3) ≥ 4.
s_ideal_waring_conjecture	state	Ideal Waring conjecture		g(k) = 2^k + ⌊(3/2)^k⌋ − 2 for all k ≥ 1, asserting that the lower bound from numbers just below 2^k is always tight.
t_hilbert_multilinear_identity_proof	technique	Hilbert’s proof via multilinear identities		Hilbert’s original proof of the Waring–Hilbert theorem using multilinear algebraic identities to reduce higher powers to sums of 2k-th powers, combined with ind
t_schur_trick	technique	Schur’s trick		Replaces a polynomial identity with possibly negative coefficients by a sum-of-squares identity with non-negative integer coefficients, used to simplify Hilbert
s_stridsberg_improvement	theorem	Stridsberg’s improvement of Hilbert’s bound		Improved the explicit upper bound on g(k) obtained from Hilbert’s algebraic identity method by optimizing the number of variables in the multilinear identity ap
t_dress_simplification	technique	Dress’s simplification of Hilbert’s proof		Streamlined Hilbert’s proof of the Waring theorem by replacing complicated algebraic identity constructions with a cleaner inductive argument.
s_hurwitz_identity_sums_of_squares	state	Hurwitz identity for sums of squares		A composition identity (x₁² + ··· + xₙ²)(y₁² + ··· + yₙ²) = z₁² + ··· + zₙ² with each zᵢ bilinear in x and y, existing only for n = 1, 2, 4, 8.
s_weyl_inequality	theorem	Weyl’s inequality		If α has a rational approximation a/q with |α − a/q| ≤ q⁻², then |∑_{n≤N} e(αnᵏ)| ≪ N^{1+ε}(q⁻¹ + N⁻¹ + qN⁻ᵏ)^{2^{1−k}}, bounding exponential sums via Diophanti
t_hua_iterative_squaring	technique	Hua’s lemma proof by iterative squaring		Proves Hua’s inequality by iteratively squaring the exponential sum and applying Weyl differencing at each step, reducing the mean value estimate to counting so
s_singular_series_waring	state	Singular series S(n) for Waring’s problem		S(n) = ∑_{q≥1} ∑_{(a,q)=1} q⁻ˢ S(q,a)ˢ e(−an/q), the arithmetic factor in the Hardy–Littlewood asymptotic formula for representations of n as a sum of s k-th po
s_singular_integral_waring	axiom	Singular integral J(n) for Waring’s problem		J(n) = ∫(∫₀¹ e(βxᵏ)dx)ˢ e(−βn)dβ, the real-analytic density factor in the Hardy–Littlewood asymptotic formula measuring the volume of the real solution set.
s_positivity_singular_series	theorem	Positivity of the singular series for Waring’s problem		For s sufficiently large relative to k, the singular series S(n) satisfies S(n) ≥ c > 0 for all n, proved by showing local solubility at every prime via Hensel’
s_complete_exponential_sum_mod_q	axiom	Complete exponential sum mod q		S(q,a) = ∑_{x=1}^{q} e(axᵏ/q), the complete exponential sum of a k-th power phase over a full residue system modulo q, the local factor in the singular series.
t_pruning	technique	Pruning (circle method)		Restricts the major arcs to a narrower set where singular series and singular integral approximations are more precise, showing the contribution from the pruned
s_local_solubility_condition	state	Local solubility condition for Waring’s problem		For every prime p and every n, the congruence x₁ᵏ + ··· + xₛᵏ ≡ n (mod pʲ) has a solution for all j ≥ 1, ensuring the p-adic singular series factor is positive.
s_three_prime_asymptotic_formula	theorem	Asymptotic formula for three-prime representations		The number of representations of a large odd N as p₁ + p₂ + p₃ is asymptotic to ½·𝔖(N)·N²/(log N)³, where 𝔖(N) is the singular series for the ternary Goldbach p
t_major_arc_approximation_prime_sums	technique	Major arc approximation for prime exponential sums		On major arcs, approximates the exponential sum over primes by a product of a Ramanujan-type sum and a smooth integral, using the Siegel–Walfisz theorem for pri
s_goldbach_exceptional_set	state	Goldbach exceptional set		E(X) = |{n ≤ X : n even, n ≠ p₁ + p₂}| is the count of even numbers up to X failing binary Goldbach; the best known bound is E(X) ≪ X^{0.879}.
s_vinogradov_bound_G_k	theorem	Vinogradov’s bound on G(k)		G(k) ≤ 2k(log k + log log k + O(1)), proved using the circle method with Vinogradov’s mean value theorem, giving the best known general upper bound on the asymp
s_easy_waring_problem	axiom	Easy Waring problem		The variant of Waring’s problem asking for the minimal s such that almost all positive integers are sums of s k-th powers, where the threshold is lower than G(k
s_segre_mahler_wright_theorem	theorem	Segre–Mahler–Wright theorem		Almost all positive integers are sums of O(k log k) perfect k-th powers, giving a density-one result with fewer summands than G(k) requires for all sufficiently
t_efficient_differencing	technique	Efficient differencing (Vinogradov–Wooley)		Refines Vinogradov’s mean value method by reusing variables across differencing steps, achieving near-optimal bounds on the mean value integral for exponential 
s_davenport_G4_equals_15	theorem	Davenport’s theorem G(4) = 15		Every sufficiently large integer is the sum of at most 15 fourth powers, and 15 is best possible since numbers of the form 16ᵐ·31 require 15 fourth powers.
s_G2_equals_4	theorem	G(2) = 4		Every sufficiently large integer is the sum of at most four squares, and four is necessary since integers of the form 4ᵃ(8b+7) require four squares.
s_wooley_refinement	theorem	Wooley’s refinement of Vinogradov’s mean value theorem		Establishes the main conjecture for Vinogradov’s mean value theorem: Jₛ,ₖ(X) ≪ X^{s+ε} + X^{2s−k(k+1)/2+ε} for all s, k, via efficient congruencing or decouplin
s_asymptotic_basis	axiom	Asymptotic basis of order h		A set A ⊆ ℕ₀ is an asymptotic basis of order h if every sufficiently large positive integer is representable as a sum of at most h elements of A.
s_schnirelmann_theorem_primes	theorem	Schnirelmann’s theorem (primes form an additive basis)		The set of primes together with {0, 1} is an additive basis of finite order: there exists h such that every integer ≥ 2 is a sum of at most h primes.
s_schnirelmann_inequality	theorem	Schnirelmann’s inequality		σ(A + B) ≥ σ(A) + σ(B) − σ(A)σ(B) for sets A, B ⊆ ℕ₀ with 0 ∈ A ∩ B, where σ denotes Schnirelmann density.
s_mann_theorem	theorem	Mann’s theorem		σ(A + B) ≥ min(1, σ(A) + σ(B)) for sets A, B ⊆ ℕ₀ with 0 ∈ A ∩ B, strengthening Schnirelmann’s inequality to a sharp lower bound on sumset density.
s_khinchin_addition_theorem	theorem	Khinchin’s addition theorem		If A is an asymptotic basis of order h and B has positive Schnirelmann density, then A + B is an asymptotic basis of order at most h.
s_essential_component	axiom	Essential component		A set B ⊆ ℕ₀ is an essential component if σ(A + B) > σ(A) for every set A with 0 < σ(A) < 1, meaning B always increases Schnirelmann density.
s_primes_essential_component	theorem	Primes are an essential component		The set of primes (with 0 and 1 adjoined) is an essential component: for any A ⊆ ℕ₀ with 0 < σ(A) < 1, we have σ(A + P) > σ(A).
s_khinchin_essential_components_theorem	theorem	Khinchin’s theorem on essential components		Every asymptotic basis of finite order is an essential component, providing a sufficient condition for a set to increase the Schnirelmann density of any incompl
s_erdos_essential_components_theorem	theorem	Erdős’s theorem on essential components		If ∑_{n=1}^∞ B(n)/n² = ∞ where B(n) = |B ∩ [1,n]|, then B is an essential component, giving a counting-function criterion weaker than being an asymptotic basis.
t_mann_sequential_addition	technique	Mann’s proof via sequential addition		Proves Mann’s theorem by constructing the sumset incrementally, adding one element of B at a time and tracking how each addition increases the counting function
s_selberg_sieve_goldbach	state	Selberg sieve applied to Goldbach		Applying the Selberg upper-bound sieve to binary Goldbach yields that representations of even n as p + p’ number ≪ n·∏_{p|n}(1+1/p)/(log n)².
s_romanov_theorem	theorem	Romanov’s theorem		A positive proportion of positive integers can be represented as p + 2ᵏ for some prime p and non-negative integer k.
s_dickson_pillai_result	theorem	Dickson–Pillai theorem on g(k)		g(k) = 2ᵏ + ⌊(3/2)ᵏ⌋ − 2 provided 2ᵏ{(3/2)ᵏ} + ⌊(3/2)ᵏ⌋ ≤ 2ᵏ, reducing the ideal Waring conjecture to a condition on fractional parts of powers of 3/2.
s_balasubramanian_deshouillers_dress_theorem	theorem	Balasubramanian–Deshouillers–Dress theorem (g(4) = 19)		Every positive integer is the sum of at most 19 fourth powers, establishing g(4) = 19 via a combination of computational verification and analytic methods.
s_multiplicativity_singular_series	state	Euler product for the singular series		The singular series factors as S(n) = ∏_p σ_p(n), expressing the global arithmetic factor as a product of local p-adic densities for Waring representations.
s_hua_theorem_cubes_of_primes	theorem	Hua’s theorem on sums of cubes of primes		Every sufficiently large odd integer is the sum of nine cubes of primes, establishing a Waring–Goldbach result for cubes using the circle method.
s_waring_problem_for_primes	state	Waring’s problem for primes (Waring–Goldbach problem)		The problem of determining the smallest s such that every sufficiently large integer satisfying necessary congruence conditions is the sum of s k-th powers of p
t_davenport_heilbronn_method	technique	Davenport–Heilbronn method		Adapts the circle method to Diophantine inequalities rather than equations by exploiting irrationality of coefficients to avoid a singular series.
s_representation_function_polygonal	axiom	Representation function for polygonal numbers		r_{m,s}(n) counts the number of ways to write n as a sum of s m-gonal numbers, the fundamental counting function in the additive theory of figurate numbers.
s_representation_function_waring	axiom	Representation function r_{s,k}(n)		r_{s,k}(n) = |{(x₁,…,xₛ) ∈ ℕ₀ˢ : x₁ᵏ + ··· + xₛᵏ = n}| counts representations of n as a sum of s non-negative k-th powers.
s_liouville_identity_four_squares	state	Liouville’s identity (sums of squares)		Liouville’s identity expressing r₄(n) = 8∑_{d|n, 4∤d} d, giving an exact divisor-sum formula for the number of representations as a sum of four squares.
s_quadratic_forms_and_squares	state	Connection between quadratic forms and sums of squares		The representability of integers as sums of k squares is equivalent to representation by the standard positive-definite quadratic form of rank k, linking Waring
s_h_fold_sumset	axiom	h-fold sumset hA		For A ⊆ ℕ₀ and h ≥ 1, the h-fold sumset hA = {a₁ + ··· + aₕ : aᵢ ∈ A} is the set of all sums of exactly h elements of A with repetition.
s_h_fold_representation_function	axiom	h-fold representation function r_{A,h}(n)		r_{A,h}(n) = |{(a₁,…,aₕ) ∈ Aʰ : a₁ + ··· + aₕ = n}| counts ordered representations of n as a sum of h elements from A.
s_counting_function_A_n	axiom	Counting function A(n)		A(n) = |A ∩ [0,n]| for A ⊆ ℕ₀, the counting function measuring elements of A up to n, fundamental to Schnirelmann and asymptotic density.
t_schnirelmann_original_density_argument	technique	Schnirelmann’s original density argument		Schnirelmann’s proof that positive Schnirelmann density implies basis property, using iterated sumset density inequalities to show σ(hA) → 1 as h increases.
s_generating_function_waring	axiom	Generating function for Waring’s problem		f(α) = ∑_{m=0}^{N^{1/k}} e(αmᵏ), the exponential generating function whose s-th power’s Fourier coefficients equal the Waring representation function r_{s,k}(n)
s_p_adic_solubility	state	p-adic solubility for Waring’s problem		The equation x₁ᵏ + ··· + xₛᵏ = n has a non-trivial solution in ℤ_p for every prime p, equivalent to positivity of the local factor σ_p(n) in the singular series
s_minor_arc_estimate_waring	state	Minor arc estimate for Waring’s problem		∫_{𝔪} |f(α)|ˢ e(−nα)dα = o(n^{s/k−1}) for s sufficiently large, showing the minor arc contribution is negligible compared to the main term.
s_minor_arc_estimate_three_primes	state	Minor arc estimate for the three-primes problem		sup_{α ∈ 𝔪} |S(α)| ≪ N/(log N)^A for any A > 0 on the minor arcs, where S(α) = ∑_{n≤N} Λ(n)e(nα), by Vinogradov’s method.
s_major_arc_contribution_waring	state	Major arc contribution for Waring’s problem		∫_{𝔐} f(α)ˢ e(−nα)dα = S(n)·J(n)·n^{s/k−1} + o(n^{s/k−1}), expressing the major arc integral as the product of singular series and singular integral.
s_four_cubes_impossibility	state	Four cubes impossibility modulo 9		Integers ≡ ±4 (mod 9) cannot be represented as sums of three cubes, since cubes are ≡ 0, ±1 (mod 9), constraining local solubility of sums of few cubes.
s_23_and_239_require_9_cubes	state	23 and 239 require nine cubes		The integers 23 and 239 are the only positive integers requiring exactly nine positive cubes for their representation, establishing that g(3) = 9 is tight.
t_descent_four_squares	technique	Descent method for four squares		Proves Lagrange’s four-square theorem by showing that if mp is a sum of four squares with m > 1, then m can be reduced, iterating until m = 1.
t_linnik_elementary_method	technique	Linnik’s elementary method for sums of cubes		An elementary approach avoiding the full circle method, using combinatorial arguments and local density estimates to prove G(3) ≤ 7.
s_subgroup	axiom	Subgroup		A subset of a group that is itself a group under the inherited operation.
s_normal_subgroup	axiom	Normal subgroup		A subgroup N of G such that gNg⁻¹ = N for all g in G.
s_quotient_group	axiom	Quotient group		The group G/N of cosets of a normal subgroup N in G, with multiplication induced from G.
s_index_of_a_subgroup	axiom	Index of a subgroup		The number of left (or right) cosets of a subgroup H in G, denoted [G:H].
s_center_of_a_group	axiom	Center of a group		The set Z(G) = {z ∈ G : zg = gz for all g ∈ G} of elements that commute with every element of G.
s_centralizer	axiom	Centralizer		The subgroup C_G(S) of elements of G that commute with every element of a subset S.
s_normalizer	axiom	Normalizer		The subgroup N_G(H) = {g ∈ G : gHg⁻¹ = H} of elements that normalize a subgroup H.
s_commutator_subgroup	axiom	Commutator subgroup		The subgroup [G,G] generated by all commutators [x,y] = xyx⁻¹y⁻¹, also called the derived subgroup.
s_pgroup	axiom	p-group		A group in which every element has order a power of the prime p.
s_sylow_psubgroup	axiom	Sylow p-subgroup		A maximal p-subgroup of a finite group G, having order p^a where p^a divides |G| but p^(a+1) does not.
s_group_presentation	axiom	Group presentation		A description of a group as a quotient of a free group by a normal subgroup generated by specified relations.
s_orbit_of_a_group_action	axiom	Orbit of a group action		The set Orb(x) = {g·x : g ∈ G} of all images of an element x under a group action.
s_stabilizer_of_a_group_action	axiom	Stabilizer of a group action		The subgroup Stab(x) = {g ∈ G : g·x = x} of elements fixing a point x under a group action.
s_conjugacy_class	axiom	Conjugacy class		The orbit of an element under the conjugation action, i.e., the set {gxg⁻¹ : g ∈ G}.
s_exponent_of_a_group	axiom	Exponent of a group		The least common multiple of the orders of all elements of a group, or infinity if no such finite number exists.
s_lower_central_series	state	Lower central series		The descending series G = γ₁(G) ⊇ γ₂(G) ⊇ ··· where γᵢ₊₁(G) = [γᵢ(G), G].
s_upper_central_series	state	Upper central series		The ascending series 1 = Z₀(G) ≤ Z₁(G) ≤ ··· where Zᵢ₊₁(G)/Zᵢ(G) = Z(G/Zᵢ(G)).
s_fitting_subgroup	state	Fitting subgroup		The unique largest nilpotent normal subgroup F(G) of a finite group G, generated by all nilpotent normal subgroups.
s_socle_of_a_group	state	Socle of a group		The subgroup generated by all minimal normal subgroups of a group G.
s_automorphism_group_of_a_group	state	Automorphism group of a group		The group Aut(G) of all group automorphisms of G under composition.
s_inner_automorphism_group	state	Inner automorphism group		The normal subgroup Inn(G) of Aut(G) consisting of conjugation maps x ↦ gxg⁻¹ for g ∈ G.
s_outer_automorphism_group	state	Outer automorphism group		The quotient group Out(G) = Aut(G)/Inn(G) measuring automorphisms not arising from conjugation.
s_lagranges_theorem_group_theory	theorem	Lagrange's theorem (group theory)		The order of a subgroup H of a finite group G divides the order of G, with |G| = [G:H]·|H|.
s_second_isomorphism_theorem_for_groups	theorem	Second isomorphism theorem for groups		If H ≤ G and N ⊴ G then HN/N ≅ H/(H ∩ N).
s_third_isomorphism_theorem_for_groups	theorem	Third isomorphism theorem for groups		If N ⊆ M are normal subgroups of G then (G/N)/(M/N) ≅ G/M.
s_lattice_isomorphism_theorem_for_groups	theorem	Lattice isomorphism theorem for groups		Subgroups of G/N correspond bijectively with subgroups of G containing N, preserving normality and index.
s_sylows_first_theorem	theorem	Sylow's first theorem		If p^a divides the order of a finite group G, then G has a subgroup of order p^a; in particular, Sylow p-subgroups exist.
s_sylows_second_theorem	theorem	Sylow's second theorem		All Sylow p-subgroups of a finite group G are conjugate to each other.
s_sylows_third_theorem	theorem	Sylow's third theorem		The number n_p of Sylow p-subgroups of a finite group G satisfies n_p ≡ 1 (mod p) and n_p divides |G|.
s_fittings_theorem_on_fitting_subgroup	theorem	Fitting's theorem on Fitting subgroup		The Fitting subgroup of a finite group G is nilpotent and contains every nilpotent normal subgroup of G.
t_sylow_theory_argument	technique	Sylow theory argument		Using the Sylow theorems to constrain the structure of finite groups by analyzing numbers and conjugacy of p-subgroups.
t_counting_via_group_actions	technique	Counting via group actions		Applying the orbit-stabilizer theorem or Burnside's lemma to enumeration problems by identifying the relevant group action.
t_composition_series_analysis	technique	Composition series analysis		Analyzing a group by examining its composition series and composition factors, using the Jordan-Hölder theorem to extract structural invariants.
s_ideal_of_a_ring	axiom	Ideal (of a ring)		A two-sided ideal I of a ring R is an additive subgroup such that ra ∈ I and ar ∈ I for all r ∈ R, a ∈ I.
s_quotient_ring	axiom	Quotient ring		The ring R/I of cosets of a two-sided ideal I, with operations induced from R.
s_nilpotent_element	axiom	Nilpotent element		An element a of a ring R such that a^n = 0 for some positive integer n.
s_nilradical_of_a_ring	state	Nilradical of a ring		The ideal nil(R) of all nilpotent elements in a commutative ring R, equal to the intersection of all prime ideals.
s_jacobson_radical_of_a_ring	state	Jacobson radical of a ring		The ideal J(R) equal to the intersection of all maximal ideals (or all maximal left ideals) of a ring R.
s_fraction_field	state	Fraction field		The field of fractions Frac(R) of an integral domain R, obtained by localizing at the multiplicative set R \ {0}.
s_group_of_units	state	Group of units		The group R× = R* of invertible elements of a ring R under multiplication.
s_first_isomorphism_theorem_for_rings	theorem	First isomorphism theorem for rings		If f: R → S is a ring homomorphism then R/ker(f) ≅ im(f) as rings.
s_every_maximal_ideal_is_prime	theorem	Every maximal ideal is prime		In a commutative ring with unity, every maximal ideal is a prime ideal.
s_every_pid_is_a_ufd	theorem	Every PID is a UFD		Every principal ideal domain is a unique factorization domain.
s_euclidean_domain_is_a_pid	theorem	Euclidean domain is a PID		Every Euclidean domain is a principal ideal domain.
s_zorns_lemma_implies_maximal_ideals_exist	theorem	Zorn's lemma implies maximal ideals exist		Every nonzero ring with unity contains at least one maximal ideal, proved using Zorn's lemma.
s_nilradical_equals_intersection_of_primes	theorem	Nilradical equals intersection of primes		The nilradical of a commutative ring equals the intersection of all its prime ideals.
s_gausss_lemma_content_of_polynomials	theorem	Gauss's lemma (content of polynomials)		The product of two primitive polynomials over a UFD is primitive; equivalently, the content is multiplicative.
s_eisensteins_irreducibility_criterion	theorem	Eisenstein's irreducibility criterion		A polynomial f(x) = aₙxⁿ + ··· + a₀ over a UFD is irreducible if a prime p divides a₀,...,aₙ₋₁ but not aₙ, and p² does not divide a₀.
s_polynomial_ring_over_ufd_is_ufd	theorem	Polynomial ring over UFD is UFD		If R is a unique factorization domain, then the polynomial ring R[x] is also a unique factorization domain.
s_module_over_a_ring	axiom	Module over a ring		An abelian group M with a scalar multiplication R × M → M satisfying associativity, distributivity, and unital conditions with respect to a ring R.
s_submodule	axiom	Submodule		A subset N of an R-module M that is itself an R-module under the inherited operations.
s_quotient_module	axiom	Quotient module		The R-module M/N of cosets of a submodule N, with operations induced from M.
s_projective_module	axiom	Projective module		An R-module P such that every surjection onto P splits, equivalently a direct summand of a free module.
s_exact_sequence	axiom	Exact sequence		A sequence of module homomorphisms ··· → Mᵢ₋₁ → Mᵢ → Mᵢ₊₁ → ··· where the image of each map equals the kernel of the next.
s_short_exact_sequence	axiom	Short exact sequence		An exact sequence 0 → A → B → C → 0, equivalently A embeds in B with quotient C.
s_length_of_a_module	state	Length of a module		The length ℓ(M) of a composition series of a module M, well-defined by the Jordan-Hölder theorem; finite iff M is both Noetherian and Artinian.
s_bimodule	axiom	Bimodule		An abelian group M that is simultaneously a left R-module and a right S-module with compatible actions: (rm)s = r(ms).
s_cokernel_of_a_module_homomorphism	state	Cokernel of a module homomorphism		The quotient module coker(f) = N/im(f) for a homomorphism f: M → N.
s_structure_theorem_for_modules_over_a_pid	theorem	Structure theorem for modules over a PID		Every finitely generated module over a PID is isomorphic to a direct sum of cyclic modules R/(d₁) ⊕ ··· ⊕ R/(dₖ) ⊕ Rʳ with d₁|d₂|···|dₖ.
s_characterization_of_projective_modules	theorem	Characterization of projective modules		An R-module P is projective if and only if P is a direct summand of a free module, if and only if every short exact sequence ending in P splits.
s_baers_criterion_for_injectivity	theorem	Baer's criterion for injectivity		An R-module Q is injective if and only if every homomorphism from an ideal of R to Q extends to a homomorphism from R to Q.
s_every_module_embeds_in_an_injective_module	theorem	Every module embeds in an injective module		For any ring R and any R-module M, there exists an injective R-module containing M as a submodule.
s_schanuels_lemma	theorem	Schanuel's lemma		If 0 → K₁ → P₁ → M → 0 and 0 → K₂ → P₂ → M → 0 with P₁, P₂ projective, then K₁ ⊕ P₂ ≅ K₂ ⊕ P₁.
s_eilenberg_swindle	theorem	Eilenberg swindle		Over a ring with infinite direct sums, every projective module is a direct summand of a free module by the trick P ⊕ P^∞ ≅ P^∞.
s_jordanholder_theorem_for_modules	theorem	Jordan-Hölder theorem for modules		Any two composition series of a module have the same length and isomorphic composition factors up to permutation.
t_localization_of_modules	technique	Localization of modules		Extending scalars of a module M to S⁻¹R to study local properties, using the exact functor S⁻¹(−).
t_diagram_chasing	technique	Diagram chasing		Proving exactness or isomorphism properties in commutative diagrams by systematically tracking elements through the maps.
t_nakayamas_lemma_argument	technique	Nakayama's lemma argument		Applying Nakayama's lemma to conclude that a module or ideal vanishes, or that generators lift from a quotient.
s_content_of_a_polynomial	axiom	Content of a polynomial		The greatest common divisor c(f) of the coefficients of a polynomial f over a UFD.
s_primitive_polynomial	axiom	Primitive polynomial		A polynomial over a UFD whose content is a unit.
s_symmetric_polynomial	axiom	Symmetric polynomial		A polynomial f(x₁,...,xₙ) invariant under all permutations of the variables.
s_power_sum_symmetric_polynomials	state	Power sum symmetric polynomials		The symmetric polynomials pₖ(x₁,...,xₙ) = x₁ᵏ + ··· + xₙᵏ for positive integers k.
s_resultant_of_two_polynomials	state	Resultant of two polynomials		The determinant Res(f,g) of the Sylvester matrix of polynomials f and g, vanishing iff f and g share a common root.
s_discriminant_of_a_polynomial	state	Discriminant of a polynomial		The quantity Δ(f) = (−1)^{n(n−1)/2} Res(f,f')/aₙ for a polynomial f of degree n, measuring separation of roots.
s_newtons_identities	theorem	Newton's identities		Recursive relations connecting power sum symmetric polynomials pₖ to elementary symmetric polynomials eₖ: kEₖ = Σᵢ₌₁ᵏ (−1)ⁱ⁻¹pᵢeₖ₋ᵢ.
s_minimal_polynomial	axiom	Minimal polynomial		The unique monic polynomial of smallest degree in K[x] having α as a root, generating the kernel of the evaluation map K[x] → K(α).
s_algebraic_extension	axiom	Algebraic extension		A field extension L/K in which every element of L is algebraic over K.
s_separable_extension	axiom	Separable extension		An algebraic extension L/K in which the minimal polynomial of every element of L over K is separable.
s_purely_inseparable_extension	axiom	Purely inseparable extension		An algebraic extension L/K in characteristic p where every element α satisfies α^{p^n} ∈ K for some n.
s_separable_degree	state	Separable degree		The number [L:K]_s of K-embeddings of L into an algebraic closure, equals [L:K] when the extension is separable.
s_inseparable_degree	state	Inseparable degree		The quotient [L:K]_i = [L:K]/[L:K]_s, always a power of the characteristic.
s_finite_field	axiom	Finite field		A field with finitely many elements; for each prime power q = pⁿ there exists a unique finite field 𝔽_q up to isomorphism.
s_existence_and_uniqueness_of_finite_fields	theorem	Existence and uniqueness of finite fields		For each prime power q = pⁿ there exists a unique (up to isomorphism) field with q elements, namely the splitting field of x^q − x over 𝔽_p.
s_frobenius_generates_galois_group_of_finite_field	theorem	Frobenius generates Galois group of finite field		The Galois group Gal(𝔽_{pⁿ}/𝔽_p) is cyclic of order n, generated by the Frobenius automorphism x ↦ xᵖ.
s_galois_extension	axiom	Galois extension		A finite field extension L/K that is both normal and separable, equivalently |Aut(L/K)| = [L:K].
s_fixed_field	axiom	Fixed field		The subfield L^H = {x ∈ L : σ(x) = x for all σ ∈ H} of elements fixed by a subgroup H of Aut(L/K).
s_radical_extension	axiom	Radical extension		A field extension obtained by successive adjunction of n-th roots: K = K₀ ⊂ K₁ ⊂ ··· ⊂ Kₘ with Kᵢ₊₁ = Kᵢ(αᵢ^{1/nᵢ}).
s_krull_topology	axiom	Krull topology		The profinite topology on the Galois group of an infinite Galois extension, making it a compact totally disconnected topological group.
s_artins_lemma_on_fixed_fields	theorem	Artin's lemma on fixed fields		If G is a finite group of automorphisms of a field L, then [L:L^G] = |G| and L/L^G is a Galois extension with Galois group G.
s_dedekinds_independence_of_characters	theorem	Dedekind's independence of characters		Distinct characters (group homomorphisms to a field's multiplicative group) are linearly independent over that field.
s_hilberts_theorem_90_additive_form	theorem	Hilbert's Theorem 90 (additive form)		For a cyclic Galois extension L/K with generator σ, an element a ∈ L has trace 0 if and only if a = b − σ(b) for some b ∈ L.
s_kummer_theory_classification	theorem	Kummer theory classification		If K contains a primitive n-th root of unity, abelian extensions of K with exponent dividing n correspond bijectively to subgroups of K×/(K×)ⁿ via L ↦ L× ∩ K×/(
s_solvability_by_radicals_criterion	theorem	Solvability by radicals criterion		A polynomial f ∈ K[x] is solvable by radicals if and only if its Galois group over K is a solvable group (assuming char(K) = 0).
s_infinite_galois_correspondence	theorem	Infinite Galois correspondence		For an infinite Galois extension L/K, closed subgroups of Gal(L/K) in the Krull topology correspond bijectively with intermediate fields.
t_galois_correspondence_method	technique	Galois correspondence method		Using the bijection between subgroups of the Galois group and intermediate fields to translate group-theoretic properties into field-theoretic ones and vice ver
t_kummer_theory_technique	technique	Kummer theory technique		Classifying abelian extensions by adjoining n-th roots when n-th roots of unity are available, reducing field extension problems to multiplicative group computa
t_radical_extension_analysis	technique	Radical extension analysis		Determining solvability of polynomial equations by analyzing whether the Galois group admits a composition series with cyclic factors.
s_integral_element_over_a_ring	axiom	Integral element over a ring		An element α in a ring extension S/R that satisfies a monic polynomial equation αⁿ + rₙ₋₁αⁿ⁻¹ + ··· + r₀ = 0 with coefficients in R.
s_integrally_closed_domain	axiom	Integrally closed domain		An integral domain R that equals its own integral closure in its fraction field.
s_noetherian_ring	axiom	Noetherian ring		A ring satisfying the ascending chain condition on ideals, equivalently every ideal is finitely generated.
s_jacobson_ring	axiom	Jacobson ring		A ring in which every prime ideal is an intersection of maximal ideals.
s_integrality_is_transitive	theorem	Integrality is transitive		If S is integral over R and T is integral over S, then T is integral over R.
s_incomparability_theorem	theorem	Incomparability theorem		If R ⊆ S is integral and 𝔮₁ ⊊ 𝔮₂ are primes of S lying over the same prime of R, this is impossible: distinct primes over the same prime are incomparable.
s_hilberts_nullstellensatz_weak_form	theorem	Hilbert's Nullstellensatz (weak form)		Every maximal ideal of k[x₁,...,xₙ] over an algebraically closed field k has the form (x₁ − a₁,...,xₙ − aₙ) for some aᵢ ∈ k.
s_hilberts_nullstellensatz_strong_form	theorem	Hilbert's Nullstellensatz (strong form)		For an ideal I in k[x₁,...,xₙ] over an algebraically closed field k, the ideal of functions vanishing on V(I) equals the radical: I(V(I)) = √I.
s_fractional_ideals_of_dedekind_domains_are_invertible	theorem	Fractional ideals of Dedekind domains are invertible		Every nonzero fractional ideal of a Dedekind domain is invertible: I · I⁻¹ = R.
s_krulls_intersection_theorem	theorem	Krull's intersection theorem		In a Noetherian local ring (R, 𝔪), the intersection ∩ₙ 𝔪ⁿ = 0.
t_goingupgoingdown_method	technique	Going-up/going-down method		Using the going-up and going-down theorems to lift or descend chains of prime ideals through integral extensions.
t_noether_normalization_technique	technique	Noether normalization technique		Applying Noether normalization to reduce questions about finitely generated algebras to questions about polynomial rings.
s_algebraic_independence	axiom	Algebraic independence		Elements α₁,...,αₙ in an extension L/K are algebraically independent if no nonzero polynomial in K[x₁,...,xₙ] vanishes at (α₁,...,αₙ).
s_purely_transcendental_extension	axiom	Purely transcendental extension		A field extension L/K isomorphic to K(x₁,...,xₙ) or K({xᵢ}_{i∈I}) for a set of algebraically independent indeterminates.
s_linearly_disjoint_extensions	axiom	Linearly disjoint extensions		Field extensions L₁/K and L₂/K inside a common overfield are linearly disjoint if the natural map L₁ ⊗_K L₂ → L₁L₂ is injective.
s_separably_generated_extension	axiom	Separably generated extension		A finitely generated field extension L/K admitting a separating transcendence basis {t₁,...,tᵣ} such that L is separable algebraic over K(t₁,...,tᵣ).
s_module_of_kahler_differentials	axiom	Module of Kähler differentials		The A-module Ω_{A/R} representing derivations from A to A-modules, universal among R-derivations of A.
s_derivation_on_an_algebra	axiom	Derivation on an algebra		An R-linear map D: A → M satisfying the Leibniz rule D(ab) = aD(b) + bD(a).
s_transcendence_degree_is_welldefined	theorem	Transcendence degree is well-defined		Any two transcendence bases of a field extension L/K have the same cardinality.
s_luroths_theorem	theorem	Lüroth's theorem		Every intermediate field K ⊊ E ⊆ K(t) of a purely transcendental extension of transcendence degree 1 is itself purely transcendental: E = K(u) for some u.
s_mac_lanes_separability_criterion	theorem	Mac Lane's separability criterion		A finitely generated extension L/K in characteristic p is separably generated if and only if L and K^{1/p} are linearly disjoint over K.
s_first_exact_sequence_of_kahler_differentials	theorem	First exact sequence of Kähler differentials		For ring maps R → A → B, there is an exact sequence B ⊗_A Ω_{A/R} → Ω_{B/R} → Ω_{B/A} → 0.
s_second_exact_sequence_of_kahler_differentials	theorem	Second exact sequence of Kähler differentials		For an ideal I with A = R/I, there is an exact sequence I/I² → A ⊗_R Ω_{R/K} → Ω_{A/K} → 0 (conormal sequence).
s_affine_algebraic_set	axiom	Affine algebraic set		The set V(I) = {a ∈ kⁿ : f(a) = 0 for all f ∈ I} of common zeros of an ideal I ⊆ k[x₁,...,xₙ].
s_coordinate_ring_of_an_algebraic_set	state	Coordinate ring of an algebraic set		The quotient ring k[V] = k[x₁,...,xₙ]/I(V) of polynomial functions on an algebraic set V.
s_irreducible_algebraic_set	axiom	Irreducible algebraic set		An algebraic set that cannot be expressed as a union of two proper algebraic subsets.
s_function_field_of_a_variety	state	Function field of a variety		The field of fractions of the coordinate ring of an irreducible variety, representing rational functions on the variety.
s_nullstellensatz_correspondence	theorem	Nullstellensatz correspondence		Over an algebraically closed field k, there is an inclusion-reversing bijection between radical ideals of k[x₁,...,xₙ] and algebraic subsets of kⁿ.
s_dimension_equals_transcendence_degree	theorem	Dimension equals transcendence degree		The Krull dimension of the coordinate ring of an irreducible variety equals the transcendence degree of its function field over the base field.
s_reduced_trace	state	Reduced trace		For a central simple algebra A over K, the trace Trd: A → K induced from the matrix trace after splitting.
s_reduced_norm	state	Reduced norm		For a central simple algebra A over K, the norm Nrd: A → K induced from the matrix determinant after splitting.
s_index_of_a_central_simple_algebra	state	Index of a central simple algebra		The degree of the unique division algebra D Brauer-equivalent to a central simple algebra A, denoted ind(A) = √(dim_K D).
s_tensor_product_of_central_simple_algebras	theorem	Tensor product of central simple algebras		If A and B are central simple algebras over K, then A ⊗_K B is central simple over K.
s_brauer_group_is_torsion	theorem	Brauer group is torsion		Every element of the Brauer group Br(K) has finite order, and the order equals the index.
s_representation_of_a_group	axiom	Representation of a group		A group homomorphism ρ: G → GL(V) from a group G to the general linear group of a vector space V over a field k.
s_mackey_decomposition_formula	theorem	Mackey decomposition formula		For subgroups H, K ≤ G and an H-module V: Res_K Ind_H^G V ≅ ⊕_{g∈K\G/H} Ind_{K∩gHg⁻¹}^K (gV), where the sum is over double coset representatives.
s_mackey_irreducibility_criterion	theorem	Mackey irreducibility criterion		An induced representation Ind_H^G V is irreducible if and only if V is irreducible and for every g ∉ H, the representations V and gV have no common irreducible 
t_character_table_computation	technique	Character table computation		Systematically computing the character table of a finite group using orthogonality relations, induction, and restriction.
t_frobenius_reciprocity_method	technique	Frobenius reciprocity method		Using Frobenius reciprocity to decompose induced representations or compute intertwining numbers between representations.
s_general_linear_group	axiom	General linear group		The group GL_n(K) of n×n invertible matrices over a field K, or equivalently the automorphism group of K^n.
s_special_linear_group	axiom	Special linear group		The normal subgroup SL_n(K) = ker(det: GL_n(K) → K×) of matrices with determinant 1.
s_orthogonal_group	axiom	Orthogonal group		The group O(V,q) = {g ∈ GL(V) : q(gv) = q(v) for all v} of isometries of a quadratic form.
s_unitary_group	axiom	Unitary group		The group U(V,h) of linear transformations preserving a non-degenerate hermitian form h.
s_parabolic_subgroup	axiom	Parabolic subgroup		A subgroup of a linear algebraic group containing a Borel subgroup.
s_elementary_matrix	axiom	Elementary matrix		A matrix differing from the identity by a single off-diagonal entry: Eᵢⱼ(a) = I + aeᵢⱼ for i ≠ j.
s_sl_n_generated_by_elementary_matrices	theorem	SL_n generated by elementary matrices		For any field K and n ≥ 2, the special linear group SL_n(K) is generated by elementary matrices.
s_psl_2f_q_is_simple	theorem	PSL_2(F_q) is simple		The projective special linear group PSL_2(𝔽_q) is simple for all prime powers q ≥ 4.
s_bilinear_form	axiom	Bilinear form		A map B: V × V → K that is linear in each variable, which may be symmetric, alternating, or hermitian.
s_quadratic_form	axiom	Quadratic form		A map Q: V → K with Q(av) = a²Q(v) such that the associated bilinear form B(x,y) = Q(x+y) − Q(x) − Q(y) is bilinear.
s_nondegenerate_bilinear_form	axiom	Non-degenerate bilinear form		A bilinear form B on V such that the map V → V* given by v ↦ B(v,−) is an isomorphism.
s_hyperbolic_plane	axiom	Hyperbolic plane		A two-dimensional vector space with a non-degenerate symmetric bilinear form represented by the matrix (0 1; 1 0).
s_anisotropic_form	axiom	Anisotropic form		A quadratic form Q such that Q(v) = 0 implies v = 0.
s_witt_index	state	Witt index		The maximum dimension of a totally isotropic subspace of a quadratic space (V, Q).
s_witt_ring	state	Witt ring		The ring structure on W(K) with tensor product of quadratic forms as multiplication.
s_even_clifford_algebra	state	Even Clifford algebra		The subalgebra Cl⁰(V,Q) of elements of even degree in the Clifford algebra, a central simple or product of central simple algebras.
s_pfaffian	state	Pfaffian		For an alternating matrix A of even size 2n, the polynomial Pf(A) satisfying Pf(A)² = det(A).
s_witts_cancellation_theorem	theorem	Witt's cancellation theorem		If (V₁, Q₁) ⊥ (W, Q) ≅ (V₂, Q₂) ⊥ (W, Q) as quadratic spaces, then (V₁, Q₁) ≅ (V₂, Q₂).
s_witt_decomposition_theorem	theorem	Witt decomposition theorem		Every quadratic space decomposes as an orthogonal sum of an anisotropic space and a number of hyperbolic planes: (V,Q) ≅ V_an ⊥ ℍʳ.
s_sylvesters_law_of_inertia	theorem	Sylvester's law of inertia		A real symmetric bilinear form is classified up to congruence by its signature (p, q), the numbers of positive and negative diagonal entries after diagonalizati
s_structure_of_clifford_algebras	theorem	Structure of Clifford algebras		The Clifford algebra Cl(V,Q) is central simple when dim V is odd, and a product of two central simple algebras when dim V is even.
s_every_alternating_form_has_even_rank	theorem	Every alternating form has even rank		A non-degenerate alternating bilinear form exists only on even-dimensional spaces; every alternating form has even rank.
s_minimal_polynomial_of_a_linear_map	state	Minimal polynomial of a linear map		The unique monic polynomial m_A(t) of lowest degree satisfying m_A(A) = 0.
s_adjoint_functor_pair	axiom	Adjoint functor pair		Functors F: C → D and G: D → C with a natural bijection Hom_D(FX, Y) ≅ Hom_C(X, GY).
s_representable_functor	axiom	Representable functor		A functor naturally isomorphic to Hom(A, −) (covariant) or Hom(−, A) (contravariant) for some object A.
s_additive_category	axiom	Additive category		A category enriched over abelian groups: Hom sets are abelian groups, composition is bilinear, and finite products and coproducts exist and coincide.
s_product_of_objects_in_a_category	axiom	Product of objects in a category		An object ∏Xᵢ with projection morphisms satisfying the universal property: morphisms to ∏Xᵢ correspond to families of morphisms to each Xᵢ.
s_coproduct_of_objects_in_a_category	axiom	Coproduct of objects in a category		An object ∐Xᵢ with injection morphisms satisfying the universal property: morphisms from ∐Xᵢ correspond to families from each Xᵢ.
s_direct_limit_filtered_colimit	axiom	Direct limit (filtered colimit)		The colimit of a directed system of objects and morphisms in a category.
s_inverse_limit	axiom	Inverse limit		The limit of an inverse system of objects and morphisms in a category.
s_left_exact_functor	axiom	Left exact functor		An additive functor preserving left-exact sequences: if 0 → A → B → C is exact then 0 → F(A) → F(B) → F(C) is exact.
s_right_exact_functor	axiom	Right exact functor		An additive functor preserving right-exact sequences: if A → B → C → 0 is exact then F(A) → F(B) → F(C) → 0 is exact.
s_exact_functor	axiom	Exact functor		A functor that is both left exact and right exact, preserving all short exact sequences.
s_projective_resolution	axiom	Projective resolution		An exact sequence ··· → P₁ → P₀ → M → 0 with each Pᵢ a projective module.
s_injective_resolution	axiom	Injective resolution		An exact sequence 0 → M → I⁰ → I¹ → ··· with each Iⁿ an injective module.
s_enough_projectives	axiom	Enough projectives		A category has enough projectives if for every object M there exists an epimorphism P → M with P projective.
s_enough_injectives	axiom	Enough injectives		A category has enough injectives if for every object M there exists a monomorphism M → I with I injective.
s_hom_is_left_exact	theorem	Hom is left exact		The functor Hom_R(M, −) is left exact for any module M, and Hom_R(−, N) is left exact contravariant.
s_tensor_is_right_exact	theorem	Tensor is right exact		The functor − ⊗_R M is right exact for any R-module M.
s_comparison_theorem_for_resolutions	theorem	Comparison theorem for resolutions		Any morphism M → N lifts to a chain map between projective (or injective) resolutions, unique up to chain homotopy.
s_horseshoe_lemma	theorem	Horseshoe lemma		Given a short exact sequence 0 → A → B → C → 0 and projective resolutions of A and C, one can construct a projective resolution of B fitting into a short exact 
s_long_exact_sequence_from_short_exact_sequence_of_complexes	theorem	Long exact sequence from short exact sequence of complexes		A short exact sequence of chain complexes 0 → A• → B• → C• → 0 induces a long exact sequence ··· → Hₙ(A) → Hₙ(B) → Hₙ(C) → Hₙ₋₁(A) → ···.
s_nine_lemma_33_lemma	theorem	Nine lemma (3×3 lemma)		In a 3×3 commutative diagram where all columns and two of the three rows are exact, the third row is also exact.
s_group_extension	axiom	Group extension		A short exact sequence 1 → N → G → Q → 1 of groups, expressing G as an extension of Q by N.
s_factor_set_2cocycle	state	Factor set (2-cocycle)		A function f: Q × Q → N satisfying the cocycle condition, determining a group extension of Q by N up to equivalence.
s_corestriction_transfer_map	state	Corestriction (transfer) map		The map Cor: Hⁿ(H, M) → Hⁿ(G, M) for a subgroup H of finite index in G, a transfer homomorphism on cohomology.
s_cup_product_in_group_cohomology	state	Cup product in group cohomology		A bilinear pairing Hᵖ(G, M) × Hᵍ(G, N) → Hᵖ⁺ᵍ(G, M ⊗ N) making H*(G, −) a graded ring.
s_crossed_homomorphism	axiom	Crossed homomorphism		A map f: G → M satisfying f(στ) = σ·f(τ) + f(σ), a 1-cocycle in group cohomology with H¹(G,M) classifying these modulo principal ones.
s_h_classifies_group_extensions	theorem	H² classifies group extensions		The second cohomology group H²(G, A) is in bijection with equivalence classes of extensions of G by the abelian group A with the given G-action.
s_herbrand_quotient_is_multiplicative	theorem	Herbrand quotient is multiplicative		For a short exact sequence 0 → A → B → C → 0 of modules for a cyclic group, h(B) = h(A)·h(C) when all quotients are defined.
s_cohomology_of_cyclic_groups_is_periodic	theorem	Cohomology of cyclic groups is periodic		For a cyclic group G, Tate cohomology is 2-periodic: Ĥⁿ(G, M) ≅ Ĥⁿ⁺²(G, M) for all n.
s_left_derived_functor	axiom	Left derived functor		The functors LₙF obtained by applying a right exact functor F to a projective resolution and taking homology.
s_right_derived_functor	axiom	Right derived functor		The functors RⁿF obtained by applying a left exact functor F to an injective resolution and taking cohomology.
s_universal_functor	axiom	Universal δ-functor		A connected sequence of functors {Tⁿ} with connecting homomorphisms such that any natural transformation from T⁰ extends uniquely to the whole sequence.
s_effaceable_functor	axiom	Effaceable functor		An additive functor T such that for every object M there exists a monomorphism M → I with T(I) = 0.
s_projective_dimension	state	Projective dimension		The minimum length pd(M) of a projective resolution of a module M, or ∞ if no finite resolution exists.
s_injective_dimension	state	Injective dimension		The minimum length id(M) of an injective resolution of a module M.
s_global_dimension_of_a_ring	state	Global dimension of a ring		The supremum gl.dim(R) = sup{pd(M)} over all R-modules M.
s_weak_flat_dimension	state	Weak (flat) dimension		The supremum w.dim(R) of flat dimensions of all R-modules, always ≤ gl.dim(R).
s_ext_is_independent_of_resolution	theorem	Ext is independent of resolution		The groups Extⁿ_R(M, N) computed via a projective resolution of M or an injective resolution of N are naturally isomorphic.
s_tor_is_independent_of_resolution	theorem	Tor is independent of resolution		The groups Torₙ^R(M, N) computed via projective resolutions of either variable are naturally isomorphic.
s_long_exact_sequence_for_ext	theorem	Long exact sequence for Ext		A short exact sequence 0 → A → B → C → 0 induces long exact sequences ··· → Extⁿ(M,A) → Extⁿ(M,B) → Extⁿ(M,C) → Extⁿ⁺¹(M,A) → ···.
s_long_exact_sequence_for_tor	theorem	Long exact sequence for Tor		A short exact sequence 0 → A → B → C → 0 induces ··· → Torₙ(M,A) → Torₙ(M,B) → Torₙ(M,C) → Torₙ₋₁(M,A) → ···.
s_ext_classifies_extensions	theorem	Ext¹ classifies extensions		The group Ext¹_R(C, A) is in bijection with equivalence classes of short exact sequences 0 → A → B → C → 0.
s_tor_characterizes_flatness	theorem	Tor₁ characterizes flatness		An R-module M is flat if and only if Tor₁^R(M, N) = 0 for all R-modules N.
s_grothendiecks_universality_theorem	theorem	Grothendieck's universality theorem		An effaceable δ-functor is universal, so derived functors are characterized by their universal property.
s_dimension_shifting	theorem	Dimension shifting		If 0 → K → P → M → 0 with P projective, then Extⁿ⁺¹(M, N) ≅ Extⁿ(K, N) for n ≥ 1, reducing higher Ext to lower Ext.
s_auslanderbuchsbaumserre_theorem	theorem	Auslander-Buchsbaum-Serre theorem		A Noetherian local ring is regular if and only if it has finite global dimension, in which case gl.dim equals the Krull dimension.
t_dimension_shifting_technique	technique	Dimension shifting technique		Reducing computation of higher Ext or Tor groups to lower ones by splicing short exact sequences from projective or injective resolutions.
t_resolution_comparison_technique	technique	Resolution comparison technique		Using the comparison theorem to relate different resolutions of the same module, showing derived functor independence of resolution choice.
t_derived_functor_computation_via_acyclic_resolutions	technique	Derived functor computation via acyclic resolutions		Computing derived functors using acyclic resolutions (F-acyclic objects) instead of projective or injective resolutions.
s_filtered_complex	axiom	Filtered complex		A chain or cochain complex C with a decreasing filtration F^p C ⊇ F^{p+1} C compatible with the differential.
s_double_complex	axiom	Double complex		A bigraded module C^{p,q} with horizontal and vertical differentials d_h and d_v satisfying d_h² = d_v² = 0 and d_h d_v + d_v d_h = 0.
s_total_complex_of_a_double_complex	state	Total complex of a double complex		The single complex Tot(C)^n = ⊕_{p+q=n} C^{p,q} with differential d = d_h + d_v.
s_edge_homomorphism	state	Edge homomorphism		Natural maps from the abutment to E₂^{0,n} or from E₂^{n,0} to the abutment, coming from the filtration on convergence of a spectral sequence.
s_balancing_tor_via_double_complexes	theorem	Balancing Tor via double complexes		Torₙ^R(M, N) can be computed from either variable's resolution using a double complex, establishing the symmetry of Tor.
t_spectral_sequence_argument	technique	Spectral sequence argument		Using spectral sequence convergence to compute or constrain (co)homology groups, especially via degeneration at E₂.
t_double_complex_filtration_technique	technique	Double complex filtration technique		Choosing row or column filtration of a double complex to obtain spectral sequences computing the total complex's cohomology.
s_depth_of_a_module	state	Depth of a module		The length of a maximal regular sequence in an ideal I on a module M, denoted depth_I(M).
s_cohenmacaulay_module	axiom	Cohen-Macaulay module		A finitely generated module M over a Noetherian local ring such that depth(M) = dim(M).
s_koszul_complex_acyclicity	theorem	Koszul complex acyclicity		If x₁,...,xₙ is a regular sequence on a module M, then the Koszul complex K(x₁,...,xₙ; M) is acyclic (exact except in degree 0).
s_regular_local_ring_is_ufd	theorem	Regular local ring is UFD		Every regular local ring is a unique factorization domain.
s_regular_local_ring_has_finite_global_dimension	theorem	Regular local ring has finite global dimension		A regular local ring of dimension d has global dimension d, proved by using the Koszul complex on a regular system of parameters.
t_koszul_complex_technique	technique	Koszul complex technique		Using the Koszul complex K(x₁,...,xₙ) to compute Tor, detect regular sequences, and study depth and projective dimension.
t_long_exact_sequence_construction	technique	Long exact sequence construction		Constructing long exact sequences in (co)homology from short exact sequences of complexes via the connecting homomorphism.
s_ordered_field	axiom	Ordered field		A field K with a total order compatible with addition and multiplication: a > 0 and b > 0 imply a + b > 0 and ab > 0.
s_valuation_ring_of_an_absolute_value	state	Valuation ring of an absolute value		The subring O_v = {a ∈ K : |a| ≤ 1} of a field with a non-archimedean absolute value.
s_sturms_theorem	theorem	Sturm's theorem		The number of distinct real roots of a polynomial in an interval (a,b) equals the difference in the number of sign changes of the Sturm sequence at a and b.
s_ostrowskis_theorem	theorem	Ostrowski's theorem		Every nontrivial absolute value on ℚ is equivalent to either the usual real absolute value or a p-adic absolute value for some prime p.
s_extension_of_absolute_values	theorem	Extension of absolute values		Every absolute value on a field K extends to any algebraic extension L/K, and the extension is unique if K is complete.
s_dihedral_group	state	Dihedral Group D_{2n}		The group of symmetries of a regular n-gon, of order 2n, generated by a rotation r of order n and a reflection s with srs = r^{-1}.
s_quaternion_group_q8	state	Quaternion Group Q_8		The non-abelian group of order 8 consisting of {+/-1, +/-i, +/-j, +/-k} with i^2 = j^2 = k^2 = ijk = -1, the smallest non-abelian group in which every subgroup 
s_klein_four_group	state	Klein Four-Group V_4		The abelian group Z/2Z x Z/2Z of order 4, the smallest non-cyclic group, isomorphic to the normal subgroup {e, (12)(34), (13)(24), (14)(23)} in S_4.
s_lattice_of_subgroups	state	Lattice of Subgroups		The partially ordered set of all subgroups of a group G ordered by inclusion, forming a complete lattice with meet given by intersection and join by the subgrou
s_abelianization	state	Abelianization		The quotient G/[G,G] of a group G by its commutator subgroup, the largest abelian quotient of G satisfying the universal property that every homomorphism from G
t_classification_groups_small_order	technique	Classification of Groups of Small Order		Systematic enumeration of all groups of a given small order using Sylow theory, semidirect product constructions, and extensions to determine all isomorphism ty
t_holder_program	technique	Holder Program		The program to classify all finite groups by first classifying all finite simple groups (composition factors via Jordan-Holder) and then solving the extension p
s_chinese_remainder_theorem_rings	theorem	Chinese Remainder Theorem for Commutative Rings		If I_1, ..., I_n are pairwise coprime ideals (I_j + I_k = R for j != k) in a commutative ring R, then R/(I_1 cap ... cap I_n) is isomorphic to R/I_1 x ... x R/I
s_quotient_by_prime_is_domain	theorem	Quotient by Prime Ideal is Integral Domain		An ideal P of a commutative ring R is prime if and only if the quotient ring R/P is an integral domain.
s_quotient_by_maximal_is_field	theorem	Quotient by Maximal Ideal is Field		An ideal M of a commutative ring R with identity is maximal if and only if the quotient ring R/M is a field.
s_associate_elements	state	Associate Elements in a Ring		Two elements a and b of an integral domain are associates if a = ub for some unit u, forming an equivalence relation that identifies elements differing only by 
s_irreducible_element	state	Irreducible Element		A nonzero non-unit element a of an integral domain such that if a = bc then b or c is a unit; in a UFD, irreducibility is equivalent to generating a maximal pri
s_prime_element	state	Prime Element in a Ring		A nonzero non-unit element p of an integral domain such that if p divides ab then p divides a or p divides b; equivalently (p) is a prime ideal, and every prime
s_polynomial_ring_over_field_is_ed	theorem	Polynomial Ring over a Field is a Euclidean Domain		If F is a field, then F[x] is a Euclidean domain with the degree function as Euclidean norm, and hence is also a principal ideal domain and a unique factorizati
s_rational_root_theorem	theorem	Rational Root Theorem		If p/q in lowest terms is a rational root of a polynomial a_n x^n + ... + a_0 with integer coefficients, then p divides a_0 and q divides a_n.
t_mod_p_irreducibility_test	technique	Mod p Irreducibility Test		If a monic polynomial f in Z[x] reduces modulo a prime p to an irreducible polynomial in F_p[x] of the same degree, then f is irreducible over Q.
s_evaluation_homomorphism	state	Evaluation Homomorphism		The ring homomorphism phi_a: R[x] -> R defined by phi_a(f) = f(a) for a fixed element a, whose kernel when R is a field is the principal ideal (x - a) if a is a
s_root_factor_theorem	theorem	Root-Factor Theorem		An element a is a root of a polynomial f(x) over a field F if and only if (x - a) divides f(x) in F[x], so a polynomial of degree n has at most n roots in F.
s_splitting_lemma_modules	theorem	Splitting Lemma for Modules		A short exact sequence 0 -> A -> B -> C -> 0 of R-modules splits (B is isomorphic to A direct sum C) if and only if there exists a section C -> B, equivalently 
s_companion_matrix	state	Companion Matrix		The n x n matrix C(p) associated to a monic polynomial p(x) = x^n + a_{n-1}x^{n-1} + ... + a_0 with last column (-a_0, -a_1, ..., -a_{n-1})^T and 1's on the sub
s_jordan_block	state	Jordan Block		The n x n matrix J_n(lambda) with eigenvalue lambda on the diagonal, 1's on the superdiagonal, and 0's elsewhere; the building block of Jordan canonical form co
s_fx_module_structure_vector_space	state	F[x]-Module Structure on a Vector Space		A finite-dimensional vector space V over F equipped with an endomorphism T becomes an F[x]-module via f(x).v = f(T)(v), establishing an equivalence between stud
t_galois_groups_small_degree	technique	Computing Galois Groups of Small Degree Polynomials		Determination of the Galois group of a low-degree polynomial using the discriminant, resolvent cubic/sextic, and transitive subgroup classification to identify 
s_galois_closure	state	Galois Closure		The smallest Galois extension of the base field containing a given finite extension L/K, obtained as the splitting field of the minimal polynomial of a primitiv
s_composite_field_extension	state	Composite Field Extension (Compositum)		For field extensions E/K and F/K inside a common overfield, their compositum EF is the smallest subfield containing both E and F; if E/K and F/K are Galois with
s_inner_product_of_class_functions	state	Inner Product of Class Functions		The Hermitian inner product <phi, psi> = (1/|G|) sum_{g in G} phi(g) psi(g)* on the space of class functions of a finite group G, under which the irreducible ch
s_permutation_representation	state	Permutation Representation		A representation rho: G -> GL(V) arising from a group action of G on a finite set X by permuting basis vectors {e_x}_{x in X} of the free vector space V = k^X, 
s_symmetric_exterior_power_representations	state	Symmetric and Exterior Powers of Representations		For a representation V of G, the symmetric power Sym^n(V) and exterior power Lambda^n(V) are representations whose characters are expressible via Newton's ident
s_frobenius_schur_indicator	state	Frobenius-Schur Indicator		The quantity nu(chi) = (1/|G|) sum_{g in G} chi(g^2) for an irreducible character chi, taking values +1, -1, or 0 according to whether the representation is rea
t_representations_as_modules	technique	Representations as k[G]-Modules		The category of finite-dimensional k-representations of a group G is equivalent to the category of finitely generated left k[G]-modules, translating representat
t_character_theory_simplicity_proofs	technique	Character Theory for Simplicity Proofs		Using the fact that normal subgroups are exactly unions of conjugacy classes and kernels of characters to prove simplicity by showing no proper nontrivial subse
s_von_dycks_theorem	theorem	Von Dyck's theorem		If G = ⟨S | R⟩ is a group presentation and H is any group whose generators satisfy the relations R, then there exists a surjective homomorphism G → H.
s_chief_series	state	Chief series		A normal series G = G₀ ▷ G₁ ▷ ··· ▷ Gₙ = {e} where each Gᵢ is normal in G and each factor Gᵢ/Gᵢ₊₁ is a minimal normal subgroup of G/Gᵢ₊₁.
s_chief_factors	state	Chief factors		The factor groups Gᵢ/Gᵢ₊₁ of a chief series, each of which is characteristically simple.
s_indecomposable_group	axiom	Indecomposable group		A nontrivial group that cannot be expressed as a direct product of two nontrivial subgroups.
s_krull_schmidt_theorem_groups	theorem	Krull-Schmidt theorem for groups		If a group G satisfies both ACC and DCC on normal subgroups, then any two decompositions of G into direct products of indecomposable factors have the same numbe
s_free_product_with_amalgamation	axiom	Free product with amalgamation		For groups G₁, G₂ and monomorphisms φᵢ: H → Gᵢ, the quotient of the free product G₁ * G₂ by the normal closure of {φ₁(h)φ₂(h)⁻¹ : h ∈ H}.
s_subnormal_subgroup	axiom	Subnormal subgroup		A subgroup H of G for which there exists a finite chain H = Hₖ ◁ Hₖ₋₁ ◁ ··· ◁ H₀ = G where each term is normal in its predecessor.
s_zero_divisor	axiom	Zero divisor		A nonzero element a of a ring R such that ab = 0 or ba = 0 for some nonzero b ∈ R.
s_nil_ideal	axiom	Nil ideal		An ideal of a ring in which every element is nilpotent.
s_nilpotent_ideal	axiom	Nilpotent ideal		An ideal I of a ring R such that Iⁿ = 0 for some positive integer n.
s_primitive_ring	axiom	Primitive ring		A ring possessing a faithful simple left module.
s_semiprimitive_ring	axiom	Semiprimitive ring		A ring R whose Jacobson radical is zero: J(R) = 0.
s_dense_ring_of_linear_transformations	state	Dense ring of linear transformations		A subring D of End_k(V) such that for any finite linearly independent set v₁,...,vₙ and arbitrary w₁,...,wₙ, there exists d ∈ D with d(vᵢ) = wᵢ.
s_group_ring	axiom	Group ring		For a ring R and group G, the ring R[G] of formal finite sums Σ aᵍg with convolution multiplication.
s_semiprime_ring	axiom	Semiprime ring		A ring with no nonzero nilpotent two-sided ideals.
s_essential_submodule	axiom	Essential submodule		A submodule N of M such that N ∩ K ≠ 0 for every nonzero submodule K of M.
s_divisible_abelian_group	axiom	Divisible abelian group		An abelian group D such that for every d ∈ D and nonzero integer n, there exists x ∈ D with nx = d.
s_invariant_basis_number	state	Invariant basis number		A ring R has IBN if any two bases of a free R-module have the same cardinality; all commutative rings and all left-Noetherian rings have IBN.
s_comparison_theorem_resolutions	theorem	Comparison theorem for resolutions		Given a module map f: M → N and projective resolutions P• → M and Q• → N, there exists a chain map lifting f, unique up to chain homotopy.
s_torsion_module	state	Torsion module		An R-module M over a domain such that every element is torsion: for each m ∈ M there exists nonzero r ∈ R with rm = 0.
s_cyclic_module	axiom	Cyclic module		An R-module generated by a single element, isomorphic to R/I for some left ideal I.
s_progenerator	state	Progenerator		A finitely generated projective generator module; rings R and S are Morita equivalent iff S ≅ End_R(P) for a progenerator P.
s_endomorphism_ring_module	axiom	Endomorphism ring of a module		The ring End_R(M) = Hom_R(M, M) with composition as multiplication, for an R-module M.
s_maclane_separability_criterion	theorem	MacLane separability criterion		A field extension F/K in characteristic p is separable if and only if F and K^{1/p} are linearly disjoint over K.
s_separating_transcendence_basis	state	Separating transcendence basis		A transcendence basis B of F/K such that F is separable algebraic over K(B).
s_separable_closure_in_extension	state	Separable closure in an extension		For F/K, the subfield Kₛ of elements separable over K; F/Kₛ is purely inseparable and Kₛ/K is separable.
s_compositum_of_fields	axiom	Compositum of fields		For subfields E, F of a common overfield containing K, the smallest subfield containing both E and F.
s_artin_schreier_map	state	Artin-Schreier map		The additive map ℘: K → K defined by ℘(x) = xᵖ − x in characteristic p, whose cokernel K/℘(K) classifies cyclic extensions of degree p.
s_natural_transformation	axiom	Natural transformation		Given functors F, G: C → D, a family of morphisms ηₐ: F(A) → G(A) indexed by objects of C such that G(f)∘ηₐ = ηᵦ∘F(f) for every morphism f: A → B.
s_functor_category	axiom	Functor category		The category [C, D] whose objects are functors C → D and whose morphisms are natural transformations.
s_equalizer	axiom	Equalizer		For parallel morphisms f, g: A → B in a category, an object E with e: E → A satisfying fe = ge, universal among such.
s_coequalizer	axiom	Coequalizer		For parallel morphisms f, g: A → B, an object Q with q: B → Q satisfying qf = qg, universal among such; the categorical dual of equalizer.
s_pullback_categorical	axiom	Pullback (categorical)		For morphisms f: A → C and g: B → C, a limit over the diagram A → C ← B; the categorical fiber product.
s_pushout_categorical	axiom	Pushout (categorical)		For morphisms f: C → A and g: C → B, a colimit over the diagram A ← C → B; the categorical dual of pullback.
s_complete_category	axiom	Complete category		A category in which all small limits exist.
s_cocomplete_category	axiom	Cocomplete category		A category in which all small colimits exist.
s_initial_object	axiom	Initial object		An object I in a category such that for every object A there exists exactly one morphism I → A.
s_terminal_object	axiom	Terminal object		An object T in a category such that for every object A there exists exactly one morphism A → T.
s_zero_object_category	axiom	Zero object (category)		An object that is both initial and terminal in a category.
s_cone_over_diagram	state	Cone over a diagram		A family of morphisms from a fixed object to each object in a diagram, compatible with the diagram morphisms.
s_cocone_under_diagram	state	Cocone under a diagram		A family of morphisms from each diagram object to a fixed object, compatible with the diagram morphisms.
s_unit_of_adjunction	state	Unit of adjunction		The natural transformation η: Id_C → GF defined by η_A corresponding to id_{F(A)} under the adjunction bijection Hom(FA,FA) ≅ Hom(A,GFA).
s_counit_of_adjunction	state	Counit of adjunction		The natural transformation ε: FG → Id_D defined by ε_B corresponding to id_{G(B)} under the adjunction bijection.
s_additive_functor	axiom	Additive functor		A functor F between additive categories such that F: Hom(A,B) → Hom(FA,FB) is a group homomorphism for all A, B.
s_image_in_abelian_category	state	Image in abelian category		For a morphism f in an abelian category, im(f) = ker(coker(f)), the smallest subobject through which f factors.
s_triangle_identities_adjunction	theorem	Triangle identities for adjunction		For an adjunction (F ⊣ G, η, ε), the composites εF∘Fη = id_F and Gε∘ηG = id_G hold.
s_limits_from_products_and_equalizers	theorem	Limits from products and equalizers		A category has all small limits if and only if it has all small products and all equalizers.
s_colimits_from_coproducts_and_coequalizers	theorem	Colimits from coproducts and coequalizers		A category has all small colimits if and only if it has all small coproducts and all coequalizers.
s_right_adjoints_preserve_limits	theorem	Right adjoints preserve limits		If a functor G has a left adjoint, then G preserves all small limits.
s_left_adjoints_preserve_colimits	theorem	Left adjoints preserve colimits		If a functor F has a right adjoint, then F preserves all small colimits.
s_characterization_equivalence_categories	theorem	Characterization of equivalence of categories		A functor F: C → D is an equivalence of categories iff F is full, faithful, and essentially surjective.
s_p_primary_component	state	p-primary component		For a prime p in a PID R and torsion module M, the submodule M_p = {m ∈ M : p^k m = 0 for some k ≥ 1}.
s_matrix_equivalence	axiom	Matrix equivalence		Matrices A and B over a ring R are equivalent if B = PAQ for invertible matrices P, Q.
s_matrix_similarity	axiom	Matrix similarity		Matrices A and B over a field F are similar if B = P⁻¹AP for some invertible P.
s_embedded_associated_prime	state	Embedded associated prime		An associated prime of an ideal that properly contains another associated prime.
s_isolated_associated_prime	state	Isolated associated prime		An associated prime of an ideal that is minimal among all associated primes.
s_minimal_primary_decomposition	state	Minimal primary decomposition		A primary decomposition I = Q₁ ∩ ··· ∩ Qₙ where no Qᵢ is redundant and all associated primes rad(Qᵢ) are distinct.
t_von_dyck_presentation_argument	technique	Von Dyck presentation argument		Construct an epimorphism from a presented group to any group whose generators satisfy the given relations.
t_horseshoe_lemma_construction	technique	Horseshoe lemma construction		Construct a projective resolution of the middle term from resolutions of the outer terms using direct sums.
t_comparison_theorem_lifting	technique	Comparison theorem lifting		Lift a module homomorphism to a chain map between projective resolutions, unique up to chain homotopy.
t_universal_property_argument	technique	Universal property argument		Verify existence and uniqueness of morphisms by checking the universal property diagram commutes.
t_yoneda_argument	technique	Yoneda argument		Use the bijection Nat(Hom(A,−), F) ≅ F(A) to reduce categorical statements to set-theoretic ones.
t_adjunction_construction_universal_arrows	technique	Adjunction construction via universal arrows		Assemble universal arrows into an adjoint functor pair.
t_diagram_chasing_abelian_categories	technique	Diagram chasing in abelian categories		Chase elements through diagrams using the embedding theorem to prove exactness or isomorphism results.
t_cycle_decomposition_of_permutations	technique	Cycle decomposition of permutations		Writing a permutation as a product of disjoint cycles to determine its order, cycle type, and conjugacy class.
s_simplicity_of_alternating_group	theorem	Simplicity of A_n for n ≥ 5	A_n is simple	The alternating group A_n is simple (has no proper nontrivial normal subgroups) for all n ≥ 5.
s_ufd_polynomial_ring_theorem	theorem	R UFD implies R[x] UFD		If R is a unique factorization domain, then the polynomial ring R[x] is also a unique factorization domain.
s_fundamental_theorem_of_algebra_via_real_closed_fields	theorem	Fundamental theorem of algebra via real closed fields	FTA via real closed fields	If R is a real closed field, then R[i] = R[x]/(x² + 1) is algebraically closed; in particular ℂ = ℝ[i] is algebraically closed.
s_budan_fourier_theorem	theorem	Budan–Fourier theorem		The number of real roots of a polynomial f in an interval (a, b] is at most the difference in the number of sign changes of f, f', f'', ... evaluated at a and b
s_distributive_lattice	axiom	Distributive lattice		A lattice satisfying the distributive law: a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) for all elements a, b, c.
s_complemented_lattice	axiom	Complemented lattice		A bounded lattice in which every element a has a complement b satisfying a ∧ b = 0 and a ∨ b = 1.
s_jordan_dedekind_chain_condition	theorem	Jordan–Dedekind chain condition		In a modular lattice of finite length, all maximal chains between any two comparable elements have the same length.
s_universal_algebra	axiom	Universal algebra (Ω-algebra)	Ω-algebra | general algebra	A set A equipped with a family Ω of finitary operations ω: A^{n_ω} → A, providing a general framework encompassing groups, rings, lattices, and other algebraic 
s_congruence_relation	axiom	Congruence relation		An equivalence relation on an algebra compatible with all operations: if a_i ≡ b_i then ω(a_1,...,a_n) ≡ ω(b_1,...,b_n) for every operation ω.
s_free_algebra_universal	state	Free algebra (universal algebra)		The Ω-algebra F(X) on a generating set X satisfying the universal property: every map X → A to an Ω-algebra A extends uniquely to a homomorphism F(X) → A.
s_variety_equational_class	axiom	Variety (equational class)	equational class | variety of algebras	A class of Ω-algebras defined by a set of universally quantified identities; equivalently, a class closed under homomorphic images, subalgebras, and direct prod
s_birkhoff_hsp_theorem	theorem	Birkhoff's HSP theorem	Birkhoff's variety theorem	A class of algebras is a variety (definable by identities) if and only if it is closed under homomorphic images (H), subalgebras (S), and direct products (P).
s_subdirect_product	state	Subdirect product		A subalgebra of a direct product ∏ A_i whose projection onto each factor A_i is surjective.
s_birkhoff_subdirect_representation_theorem	theorem	Birkhoff's subdirect representation theorem		Every algebra is isomorphic to a subdirect product of subdirectly irreducible algebras.
s_completely_reducible_module	axiom	Completely reducible module	semisimple module	A module in which every submodule is a direct summand; equivalent to being a direct sum of simple submodules (semisimple).
s_composition_length	state	Composition length		The length of a composition series of a module, when one exists; well-defined by the Jordan–Hölder theorem.
s_krull_dimension_of_ring	state	Krull dimension	Krull dimension of a ring	The supremum of lengths of all chains of prime ideals p_0 ⊊ p_1 ⊊ ⋯ ⊊ p_d in a commutative ring R.
s_prufer_domain	axiom	Prüfer domain	Prufer domain	An integral domain in which every finitely generated nonzero ideal is invertible; equivalently, every localization at a maximal ideal is a valuation ring.
s_prufer_domain_characterizations	theorem	Characterizations of Prüfer domains		An integral domain R is a Prüfer domain if and only if R_M is a valuation ring for every maximal ideal M, if and only if every finitely generated torsion-free R
s_homology_of_chain_complex	state	Homology of a chain complex		The graded module H_n(C) = ker(d_n)/im(d_{n+1}) measuring the failure of exactness of a chain complex C at position n.
s_abelian_p_extensions_via_witt_vectors	theorem	Abelian p-extensions via Witt vectors		Over a field of characteristic p, abelian extensions of exponent p^n correspond bijectively to quotients of the additive group of Witt vectors W_n(K) modulo the
s_integrality_of_characters	theorem	Integrality of characters		The character values χ(g) of a finite group representation over ℂ are algebraic integers, and |G|/dim(V) times the character value at any element is an algebrai
t_modular_representation_theory	technique	Modular representation theory		Study of representations of finite groups over fields whose characteristic divides the group order, where Maschke's theorem fails and indecomposable modules nee
s_division_algorithm_for_integers	theorem	Division algorithm for integers		For integers a, b with b > 0, there exist unique integers q, r with a = bq + r and 0 <= r < b.
t_euclidean_algorithm_for_integers	technique	Euclidean algorithm for integers		Iterative procedure computing gcd(a, b) by repeated division: gcd(a, b) = gcd(b, a mod b), terminating when the remainder is 0.
s_division_algorithm_for_polynomials	theorem	Division algorithm for polynomials		For polynomials f, g in F[X] with g nonzero, there exist unique q, r in F[X] with f = gq + r and either r = 0 or deg(r) < deg(g).
t_euclidean_algorithm_for_polynomials	technique	Euclidean algorithm for polynomials		Iterative procedure computing gcd(f, g) in F[X] by repeated polynomial division, analogous to the integer Euclidean algorithm.
s_irreducible_polynomials_over_r	theorem	Irreducible polynomials over R		The irreducible polynomials over the reals are precisely the linear polynomials and the quadratic polynomials with negative discriminant.
s_sign_of_a_permutation	axiom	Sign of a permutation		The sign sgn(sigma) = (-1)^(number of transpositions) defines a group homomorphism from S_n to {+/-1}, with even permutations forming the kernel.
t_row_reduction_gaussian_elimination	technique	Row reduction (Gaussian elimination)		Systematic algorithm transforming a matrix to row echelon form via elementary row operations; the reduced row echelon form is unique.
s_matrix_algebra	axiom	Matrix algebra		The set of n x n matrices over a field F with addition, scalar multiplication, and matrix multiplication, forming a generally non-commutative ring with identity
s_matrix_inverse	axiom	Matrix inverse		A square matrix A is invertible if there exists A^{-1} with AA^{-1} = A^{-1}A = I, computable via row reduction of the augmented matrix [A|I].
s_one_sided_inverse_implies_two_sided	theorem	One-sided inverse implies two-sided inverse		For a square matrix A, if AB = I for some matrix B then BA = I, so B = A^{-1}.
s_vector_space	axiom	Vector space		A set V over a field F with vector addition and scalar multiplication satisfying eight axioms: associativity, commutativity, additive identity/inverses, scalar 
s_subspace_of_vector_space	axiom	Subspace of a vector space		A nonempty subset W of a vector space V that is closed under addition and scalar multiplication, forming a vector space in its own right.
s_linear_independence	axiom	Linear independence		Vectors v_1, ..., v_n are linearly independent if the equation c_1 v_1 + ... + c_n v_n = 0 implies c_1 = ... = c_n = 0.
s_basis_extension_theorem	theorem	Basis extension theorem		Any linearly independent set in a finite-dimensional vector space can be extended to a basis, and any spanning set contains a basis.
s_linear_map	axiom	Linear map		A function T: V -> W between vector spaces preserving addition T(u+v) = T(u)+T(v) and scalar multiplication T(cv) = cT(v).
s_kernel_and_image_of_linear_map	axiom	Kernel and image of a linear map		For T: V -> W, ker(T) = {v in V : T(v) = 0} and im(T) = {T(v) : v in V} are subspaces; T is injective iff ker(T) = {0}.
s_matrix_of_a_linear_map	axiom	Matrix of a linear map		Given ordered bases for V and W, each linear map T: V -> W is uniquely represented by the matrix whose columns are the coordinate vectors of T applied to the ba
s_quotient_vector_space	axiom	Quotient vector space V/W		For a subspace W of V, the space of cosets v + W with induced vector space operations, satisfying dim(V/W) = dim(V) - dim(W).
s_internal_direct_sum_of_subspaces	axiom	Internal direct sum of subspaces		V = W_1 + ... + W_k is an internal direct sum if every v in V has a unique expression w_1 + ... + w_k with w_i in W_i.
s_external_direct_sum_of_vector_spaces	axiom	External direct sum of vector spaces		The space of tuples with componentwise operations, satisfying dim(V_1 + ... + V_k) = sum of dim(V_i), with a universal mapping property.
s_dimension_formula_for_subspaces	theorem	Dimension formula for subspaces		For subspaces U, W of V: dim(U + W) = dim(U) + dim(W) - dim(U intersection W).
s_multiplicativity_of_determinants	theorem	Multiplicativity of determinants		For square matrices A, B: det(AB) = det(A) det(B); also det(A^T) = det(A).
s_determinant_and_invertibility	theorem	Determinant and invertibility		A square matrix A is invertible if and only if det(A) is nonzero; when invertible, A^{-1} = adj(A)/det(A).
s_inner_product	axiom	Inner product		A positive-definite bilinear (real) or sesquilinear Hermitian (complex) form on V, inducing the norm ||v|| = sqrt(<v,v>).
s_self_adjoint_linear_map	axiom	Self-adjoint (Hermitian) linear map		A linear map T on an inner-product space with T* = T, satisfying <Tv, w> = <v, Tw>; represented by symmetric (real) or Hermitian (complex) matrices.
s_unitary_orthogonal_linear_map	axiom	Unitary (orthogonal) linear map		A linear map U preserving inner products: <Uv, Uw> = <v, w>; real version is orthogonal (Q^T Q = I), complex version is unitary (U*U = I).
s_positive_semidefinite_operator	axiom	Positive semidefinite operator		A self-adjoint operator T with <Tv, v> >= 0 for all v, equivalently all eigenvalues >= 0; has a unique positive semidefinite square root.
s_polar_decomposition	theorem	Polar decomposition		Every linear map T on a finite-dimensional inner-product space factors as T = UP with U unitary/isometric and P = sqrt(T*T) positive semidefinite.
s_free_abelian_group	axiom	Free abelian group		An abelian group isomorphic to Z^n with a basis of well-defined rank; every subgroup of a free abelian group of finite rank is itself free abelian.
s_classification_of_groups_of_order_pq	theorem	Classification of groups of order pq		For primes p < q: if p does not divide q-1 then every group of order pq is cyclic; if p divides q-1, there also exists a nonabelian semidirect product.
s_minimal_polynomial_divides_characteristic	theorem	Minimal polynomial divides characteristic polynomial		The minimal polynomial m_A(t) divides the characteristic polynomial chi_A(t), and they share the same irreducible factors over the base field.
s_diagonalizability_criterion	theorem	Diagonalizability criterion		A matrix is diagonalizable iff its minimal polynomial is a product of distinct linear factors, equivalently iff algebraic and geometric multiplicities coincide 
s_projection_operator_idempotent	axiom	Projection operator (idempotent)		A linear map E with E^2 = E, yielding the decomposition V = im(E) + ker(E); complementary projections E_1 + ... + E_k = I with E_i E_j = 0 correspond to direct 
s_t_invariant_subspace	axiom	T-invariant subspace		A subspace W of V with T(W) contained in W, so T restricts to a well-defined linear map T|_W: W -> W.
s_primary_decomposition_theorem_linear_algebra	theorem	Primary decomposition theorem (linear algebra)		If the minimal polynomial factors as m(t) = p_1(t)^{e_1} ... p_k(t)^{e_k} with distinct irreducible factors, then V decomposes as a direct sum of the kernels ke
s_generalized_eigenspace	state	Generalized eigenspace		The subspace V_lambda = ker((T - lambda I)^k) for sufficiently large k, forming the primary component when the irreducible factor is (t - lambda).
s_determinant_over_commutative_ring	axiom	Determinant over a commutative ring		Extension of the determinant to matrices over any commutative ring with 1 via the Leibniz formula, used for characteristic polynomials over polynomial rings.
s_gram_matrix	state	Gram matrix		For vectors v_1, ..., v_n in an inner-product space, the matrix G with G_{ij} = <v_i, v_j>; the vectors are linearly independent iff det(G) is nonzero.
s_equivalence_class_partition_theorem	theorem	Equivalence class partition theorem		An equivalence relation on a set S induces a partition of S into disjoint equivalence classes, and conversely every partition defines an equivalence relation.
s_well_ordering_principle	theorem	Well-ordering principle		Every nonempty set can be well-ordered, meaning equipped with a total order in which every nonempty subset has a least element; equivalent to the Axiom of Choic
s_semisimple_module_decomposition_theorem	theorem	Semisimple module decomposition theorem		Every semisimple module is isomorphic to a direct sum of simple modules, and every submodule of a semisimple module is a direct summand.
s_ascending_chain_condition_modules	axiom	Ascending chain condition (Noetherian condition for modules)		A module M satisfies the ACC if every ascending chain of submodules eventually stabilizes; equivalently, every nonempty set of submodules has a maximal element.
s_descending_chain_condition_modules	axiom	Descending chain condition (Artinian condition for modules)		A module M satisfies the DCC if every descending chain of submodules eventually stabilizes; equivalently, every nonempty set of submodules has a minimal element
s_composition_series_existence_criterion	theorem	Composition series existence criterion		A module M has a composition series if and only if it satisfies both the ascending chain condition and the descending chain condition on submodules.
s_tensor_product_distributes_over_direct_sum	theorem	Tensor product distributes over direct sum		The tensor product commutes with direct sums: M tensor (direct sum N_i) is naturally isomorphic to the direct sum of (M tensor N_i), and dually for Hom.
s_cochain_complex	axiom	Cochain complex		A sequence of module homomorphisms with d^{n+1} composed with d^n = 0 for all n, obtained by reversing the grading of a chain complex.
s_cohomology_of_cochain_complex	state	Cohomology of a cochain complex		The n-th cohomology module H^n(C) = ker(d^n)/im(d^{n-1}), measuring the failure of the cochain complex to be exact at position n.
s_chain_map	axiom	Chain map		A morphism between chain complexes consisting of module maps f_n: C_n -> D_n commuting with the differentials, inducing well-defined maps on homology.
t_left_derived_functor_construction	technique	Derived functor construction (left derived functors)		Given a right exact additive functor F, the left derived functors L_n F(M) are obtained by choosing a projective resolution, deleting M, applying F, and taking 
t_right_derived_functor_construction	technique	Derived functor construction (right derived functors)		Given a left exact additive functor F, the right derived functors R^n F(M) are obtained by choosing an injective resolution, deleting M, applying F, and taking 
s_long_exact_sequence_of_derived_functors	theorem	Long exact sequence of derived functors		A short exact sequence 0 -> A -> B -> C -> 0 induces a long exact sequence connecting the derived functors L_n F or R^n F with connecting homomorphisms.
t_hom_tensor_change_of_rings	technique	Hom-tensor change of rings		Given a ring homomorphism R -> S, extension of scalars M -> S tensor_R M is left adjoint to restriction of scalars, mediating change of base ring via bimodule s
s_hom_exactness_failure	state	Hom functor exactness failure		The functor Hom_R(M, -) is left exact but not exact in general; the failure of full exactness is measured by the Ext functors.
s_tensor_exactness_failure	state	Tensor product exactness failure		The functor M tensor_R - is right exact but not exact in general; the failure of full exactness is measured by the Tor functors, and M is flat precisely when th
s_diagonalization_of_symmetric_bilinear_forms	theorem	Diagonalization of symmetric bilinear forms		Every symmetric bilinear form on a finite-dimensional vector space over a field of characteristic not 2 admits a basis in which the form is diagonal.
s_normal_form_of_alternating_bilinear_forms	theorem	Normal form of alternating bilinear forms		Every alternating bilinear form on a finite-dimensional vector space has a symplectic basis yielding a block-diagonal matrix of 2x2 blocks [[0,1],[-1,0]], with 
s_tensor_product_of_vector_spaces	axiom	Tensor product of two vector spaces		For vector spaces V, W over F, the tensor product V tensor W is a vector space with a bilinear map V x W -> V tensor W through which every bilinear map from V x
s_existence_and_uniqueness_of_tensor_product	theorem	Existence and uniqueness of tensor product		The tensor product V tensor W exists for any vector spaces V, W and is unique up to canonical isomorphism, with dim(V tensor W) = dim V * dim W.
s_tensor_product_of_linear_maps	state	Tensor product of linear maps		Given linear maps f: V -> V' and g: W -> W', there is a unique linear map f tensor g: V tensor W -> V' tensor W' satisfying (f tensor g)(v tensor w) = f(v) tens
s_tensor_product_of_multiple_vector_spaces	axiom	Tensor product of multiple vector spaces		For vector spaces V_1, ..., V_k, the tensor product V_1 tensor ... tensor V_k is defined by the universal property for k-multilinear maps and is associative up 
s_associativity_of_tensor_product	theorem	Associativity of tensor product		(V tensor W) tensor U is canonically isomorphic to V tensor (W tensor U), so the tensor product of multiple spaces is well-defined without parentheses.
s_universal_property_of_the_tensor_algebra	theorem	Universal property of the tensor algebra		Any linear map from V into an associative algebra A with identity extends uniquely to a unital algebra homomorphism T(V) -> A.
s_universal_property_of_the_symmetric_algebra	theorem	Universal property of the symmetric algebra		Any linear map from V into a commutative associative algebra A with identity extends uniquely to an algebra homomorphism S(V) -> A, and S(V) is isomorphic to th
s_universal_property_of_the_exterior_algebra	theorem	Universal property of the exterior algebra		Any linear map f: V -> A into an associative algebra with identity satisfying f(v)^2 = 0 for all v extends uniquely to an algebra homomorphism Lambda(V) -> A.
s_determinant_via_exterior_algebra	state	Determinant via exterior algebra		For a linear endomorphism T of an n-dimensional space V, the induced map on Lambda^n(V) is multiplication by det(T), giving a coordinate-free definition of the 
s_basis_for_exterior_powers	theorem	Basis for exterior powers		If {e_1,...,e_n} is a basis for V, then the set of all e_{i_1} wedge ... wedge e_{i_k} with i_1 < ... < i_k is a basis for Lambda^k(V), so dim Lambda^k(V) = C(n
s_dual_space_and_tensor_products	state	Dual space and tensor products		For finite-dimensional vector spaces V and W, the space Hom(V, W) is naturally isomorphic to V* tensor W, and multilinear forms on V correspond to elements of t
s_universal_property_of_free_groups	theorem	Universal property of free groups		Any set map from the generating set S to a group G extends uniquely to a group homomorphism F(S) -> G, making F(S) the coproduct of |S| copies of Z in the categ
s_schreier_index_formula	theorem	Schreier index formula		If H is a subgroup of index j in a free group of rank n, then H is free of rank j(n-1) + 1.
s_simple_extension	axiom	Simple extension		A field extension K/F of the form K = F(alpha) for a single element alpha; if alpha is algebraic with minimal polynomial p(x) of degree n, then F(alpha) is isom
t_lagrange_resolvent	technique	Lagrange resolvent		For a cyclic Galois extension with generator sigma, the Lagrange resolvent (a, zeta) = sum_{i=0}^{n-1} sigma^i(a) * zeta^{-i} is used to express roots in terms 
s_field_polynomial	state	Field polynomial		For alpha in a finite extension K/F, the field polynomial is the characteristic polynomial of the F-linear map x -> alpha*x on K; it is a power of the minimal p
s_discriminant_determines_alternating_galois_group	theorem	Discriminant determines alternating Galois group		The Galois group of a separable polynomial f over F is contained in the alternating group A_n if and only if the discriminant disc(f) is a square in F.
t_reduction_modulo_p_for_galois_groups	technique	Reduction modulo p for Galois groups		If f mod p is separable over F_p with factorization type (d_1,...,d_k), then the Galois group of f over Q contains a permutation of that cycle type.
s_characterization_of_semisimple_modules	theorem	Characterization of semisimple modules		For a module M, the following are equivalent: M is a sum of simple submodules, M is a direct sum of simple submodules, and every submodule of M is a direct summ
s_acc_and_dcc_imply_composition_series	theorem	ACC and DCC imply composition series		A module has a composition series if and only if it satisfies both the ascending and descending chain conditions on submodules.
s_indecomposable_module	axiom	Indecomposable module		A nonzero module M that cannot be written as a direct sum M = A + B with both A and B nonzero; every simple module is indecomposable but not conversely.
s_direct_sum_and_direct_product_of_modules	axiom	Direct sum and direct product of modules		The direct sum of a family of modules consists of tuples with finitely many nonzero components, serving as the coproduct in the category of R-modules, while the
s_symmetric_group_as_galois_group	theorem	Symmetric group S_n as Galois group		For each n, there exists a polynomial of degree n over Q whose Galois group is the full symmetric group S_n; for n >= 5 this yields explicit polynomials not sol
s_multiplicativity_of_norm_and_trace	theorem	Multiplicativity of norm and trace in towers		For field extensions F subset K subset L, the norm and trace satisfy N_{L/F} = N_{K/F} o N_{L/K} and Tr_{L/F} = Tr_{K/F} o Tr_{L/K}.
s_primary_decomposition_for_modules_over_pids	theorem	Primary decomposition for modules over PIDs		A finitely generated torsion module over a PID decomposes uniquely as a direct sum of primary cyclic modules R/(p^k), giving the elementary divisor form of the 
s_localization_is_exact	theorem	Localization is exact		The localization functor S^{-1}(-) is exact, preserving short exact sequences of modules; equivalently, S^{-1}R is a flat R-module.
s_archimedean_and_nonarchimedean_places	axiom	Archimedean and non-archimedean places		A place of a number field K is an equivalence class of absolute values; archimedean places correspond to embeddings K → ℝ or ℂ, non-archimedean places to prime 
s_artins_theorem_on_simple_rings	theorem	Artin's theorem on simple rings		A left Artinian simple ring is isomorphic to M_n(D) for some division ring D and positive integer n; equivalently, a simple ring is left semisimple if and only 
t_buchbergers_algorithm	technique	Buchberger's algorithm		An algorithm that computes a Gröbner basis from any finite generating set of a polynomial ideal by iteratively computing S-polynomials, reducing, and adjoining 
s_buchbergers_criterion	theorem	Buchberger's criterion		A finite set G is a Gröbner basis for the ideal it generates iff for every pair g_i, g_j in G, the remainder of S(g_i,g_j) on division by G is zero.
s_burnsides_theorem_on_matrix_algebras	theorem	Burnside's theorem on matrix algebras		If V is a finite-dimensional vector space over an algebraically closed field F, then any subalgebra of End_F(V) acting irreducibly on V equals all of End_F(V) ≅
s_change_of_rings_for_ext	theorem	Change of rings for Ext		For a ring homomorphism φ: R → S, there are natural maps and spectral sequences relating Ext over R and Ext over S; in particular, if S is flat over R, then Ext
s_change_of_rings_for_tor	theorem	Change of rings for Tor		For a ring homomorphism R → S and modules M_R, _SN, there is a natural isomorphism Tor_n^R(M, N) ≅ Tor_n^S(M⊗_R S, N) when S is flat as a right R-module.
s_cocycle_and_coboundary_group_cohomology	axiom	Cocycle and coboundary (group cohomology)		An n-cocycle is an n-cochain f: G^n → M satisfying δf = 0; an n-coboundary is f = δg for an (n-1)-cochain g; H^n(G,M) = Z^n/B^n where Z^n = ker δ, B^n = im δ.
s_comparison_theorem_projective_resolutions	theorem	Comparison theorem (projective resolutions)		Any module homomorphism M → N lifts to a chain map between projective resolutions of M and N, and this lift is unique up to chain homotopy.
t_composition_of_binary_quadratic_forms_gauss_composition	technique	Composition of binary quadratic forms (Gauss composition)		Gauss's law of composition defines a group operation on proper equivalence classes of primitive binary quadratic forms of fixed discriminant D, making them into
s_correspondence_between_forms_and_ideals	theorem	Correspondence between forms and ideals		There is a canonical bijection between proper equivalence classes of primitive binary quadratic forms of discriminant D and strict equivalence classes of ideals
s_crossed_product_algebra	state	Crossed product algebra		For a Galois extension K/F with group G and factor set f, the crossed product (K/F, G, f) = ⊕_{σ∈G} Ku_σ with multiplication u_σ u_τ = f(σ,τ)u_{στ} and u_σ a = 
s_cyclic_algebra	state	Cyclic algebra		For a cyclic extension K/F of degree n with generator σ and a ∈ F×, the cyclic algebra (K/F, σ, a) = ⊕_{i=0}^{n-1} Ku^i with u^n = a and ua = σ(a)u; a central s
s_universal_delta_functor	axiom	Delta-functor (universal)		A sequence of functors {T^n}_{n≥0} with connecting homomorphisms δ^n: T^n(C) → T^{n+1}(A) for each short exact sequence, forming long exact sequences; a univers
s_discriminant_of_a_number_field	state	Discriminant of a number field		For a number field K with [K:ℚ] = n and integral basis {ω_1,...,ω_n}, the discriminant d_K = det(σ_i(ω_j))^2 where σ_i are the n embeddings K → ℂ; an invariant 
s_real_complex_embeddings_of_number_field	axiom	Embedding of K into ℝ^n (real and complex places)		A number field K of degree n has r_1 real embeddings σ_i: K → ℝ and r_2 pairs of complex conjugate embeddings τ_j, τ̄_j: K → ℂ, with r_1 + 2r_2 = n; these defin
s_euler_product	state	Euler product		A product representation ∏_p f(p,s) over primes p for a Dirichlet series, expressing multiplicativity of the coefficients; the Riemann zeta function has ζ(s) = 
s_exponent_of_a_central_simple_algebra	state	Exponent of a central simple algebra		The order of [A] in the Brauer group Br(F); always divides the index and shares the same prime factors: if ind(A) = p_1^{a_1}...p_k^{a_k}, then exp(A) = p_1^{b_
s_ext1_classifies_extensions	theorem	Ext^1 and extensions		There is a natural bijection between Ext^1_R(C,A) and equivalence classes of short exact sequences 0 → A → B → C → 0, with the Baer sum making this corresponden
s_finiteness_of_form_classes	theorem	Finiteness of form classes		For any discriminant D < 0, the number of proper equivalence classes of primitive positive definite binary quadratic forms of discriminant D is finite.
s_form_class_group	state	Form class group		The finite abelian group of proper equivalence classes of primitive binary quadratic forms of discriminant D under Gauss composition; its order h(D) is the clas
s_frobenius_theorem_real_division_algebras	theorem	Frobenius theorem (real division algebras)		The only finite-dimensional associative division algebras over ℝ are ℝ, ℂ, and the quaternions ℍ.
s_fundamental_identity_efg_equals_n	theorem	Fundamental identity e·f·g = n		For a prime p in ℤ and K/ℚ of degree n, if pO_K = P_1^{e_1}...P_g^{e_g} with f_i = f(P_i|p), then ∑_{i=1}^g e_i f_i = n; in the Galois case all e_i and f_i are 
s_gausss_lemma_quadratic_residues	theorem	Gauss's lemma (quadratic residues)		If p is an odd prime and gcd(a,p)=1, then (a/p) = (-1)^n where n is the number of least positive residues of a, 2a, ..., ((p-1)/2)a modulo p that exceed p/2.
s_genus_of_a_binary_quadratic_form	axiom	Genus of a binary quadratic form		A genus is a set of proper equivalence classes of binary quadratic forms of discriminant D that represent the same values modulo D (or 4D); genera partition the
s_hilberts_theorem_on_hilbert_polynomial	theorem	Hilbert's theorem on Hilbert polynomial		For a finitely generated graded module over k[X_0,...,X_n], the Hilbert function agrees with a polynomial (the Hilbert polynomial) for all sufficiently large ar
s_hom_functor	axiom	Hom functor		The bifunctor Hom_R(-,-): R-Mod^op × R-Mod → Ab; contravariant and left exact in the first variable, covariant and left exact in the second; its right derived f
s_hurwitz_formula	theorem	Hurwitz formula		For a nonconstant morphism f: C -> C' of smooth projective curves of genera g and g', 2g-2 = deg(f)*(2g'-2) + sum_{P in C} (e_P - 1), where e_P is the ramificat
s_induced_module	state	Induced module		For a subgroup H ≤ G and an H-module M, the induced module Ind_H^G(M) = ℤ[G] ⊗_{ℤ[H]} M; satisfies Shapiro's lemma: H^n(G, Ind_H^G M) ≅ H^n(H, M).
s_inner_automorphism	axiom	Inner automorphism		An automorphism of a ring or algebra A of the form x ↦ uxu^{-1} for some invertible element u ∈ A; the Skolem-Noether theorem shows all automorphisms of a centr
s_brauer_group_h2_isomorphism	theorem	Isomorphism Br(K/F) ≅ H²(Gal(K/F), K×)		For a finite Galois extension K/F, the map sending a factor set to its crossed product algebra induces a group isomorphism from H²(Gal(K/F), K×) to the relative
s_jacobian_criterion_for_nonsingularity	theorem	Jacobian criterion for nonsingularity		A point P on the zero set V(I) in k^n is nonsingular iff the Jacobian matrix (partial f_i/partial x_j) evaluated at P has rank n - dim(V), where the f_i generat
s_left_noetherian_ring	axiom	Left Noetherian ring		A ring R with identity whose left ideals satisfy the ascending chain condition (ACC): every ascending chain of left ideals stabilizes.
s_macaulays_theorem	theorem	Macaulay's theorem		The Hilbert function of an ideal I equals the Hilbert function of its leading term ideal LT(I), so computing with Gröbner bases reduces Hilbert function computa
s_minkowskis_discriminant_bound	theorem	Minkowski's discriminant bound		For any number field K ≠ ℚ, |d_K| > 1; combined with the Dedekind discriminant theorem, this proves every number field other than ℚ has a ramified prime.
s_minkowskis_lattice_point_theorem	theorem	Minkowski's lattice point theorem		If Λ is a lattice in ℝ^n and S is a convex symmetric body with vol(S) > 2^n vol(ℝ^n/Λ), then S contains a nonzero lattice point; used to prove finiteness of cla
s_norm_map_nkf	state	Norm map N_{K/F}		For a finite extension K/F, N_{K/F}(α) = ∏_{σ} σ(α) over all F-embeddings σ of K; for Galois K/F, N_{K/F}: K× → F× is surjective onto its image and its kernel a
s_nullstellensatz_fieldtheoretic_form	theorem	Nullstellensatz (field-theoretic form)		If K is a field and L is a finitely generated K-algebra that is a field, then L is a finite algebraic extension of K.
s_periodicity_of_cohomology_for_cyclic_groups	theorem	Periodicity of cohomology for cyclic groups		If G is a finite cyclic group and M is a G-module, then Ĥ^n(G,M) ≅ Ĥ^{n+2}(G,M) for all n ∈ ℤ in Tate cohomology; in particular H²(G,M) ≅ Ĥ^0(G,M) = M^G/N(M).
s_principal_genus	axiom	Principal genus		The genus containing the principal form x^2 - (D/4)y^2 (D even) or x^2 + xy + (1-D)/4 y^2 (D odd); it equals the subgroup of squares in the form class group.
s_proj_construction	state	Proj construction		For a graded ring S = direct sum S_d, Proj(S) is the scheme whose points are homogeneous prime ideals not containing the irrelevant ideal S_+, with structure sh
s_proper_equivalence_of_binary_quadratic_forms	axiom	Proper equivalence of binary quadratic forms		Two binary quadratic forms f and g are properly equivalent if g is obtained from f by a linear substitution with matrix in SL_2(ℤ); this preserves the discrimin
s_quadratic_number_field	axiom	Quadratic number field		A number field of the form ℚ(√d) where d is a squarefree integer; its ring of integers is ℤ[(1+√d)/2] if d ≡ 1 mod 4, and ℤ[√d] otherwise.
s_quaternion_algebra	axiom	Quaternion algebra		A 4-dimensional central simple algebra over a field F generated by i,j with i^2=a, j^2=b, ij=-ji for a,b ∈ F×; denoted (a,b)_F; splits (≅ M_2(F)) iff the norm f
s_ramification_of_primes	axiom	Ramification of primes		A prime p ∈ ℤ ramifies in O_K if pO_K = ∏ P_i^{e_i} with some e_i > 1; p is unramified if all e_i = 1; p is totally ramified if there is a single prime P with e
s_reduced_binary_quadratic_form	axiom	Reduced binary quadratic form		A positive definite form ax^2+bxy+cy^2 with D<0 is reduced if |b| ≤ a ≤ c, and b ≥ 0 when |b|=a or a=c; each proper equivalence class contains a unique reduced 
s_relative_brauer_group	state	Relative Brauer group		For a field extension K/F, the relative Brauer group Br(K/F) = ker(Br(F) → Br(K)) consists of classes of CSAs over F split by K; Br(F) = ∪_K Br(K/F) over all fi
s_relative_brauer_group_for_cyclic_extensions	theorem	Relative Brauer group for cyclic extensions		If K/F is a cyclic Galois extension of degree n with generator σ, then Br(K/F) ≅ F×/N_{K/F}(K×); every element is represented by a cyclic algebra (K/F, σ, a) fo
s_restriction_and_corestriction_maps	state	Restriction and corestriction maps		For a subgroup H ≤ G: restriction res: H^n(G,M) → H^n(H,M) and corestriction cor: H^n(H,M) → H^n(G,M) satisfy cor ∘ res = [G:H]; these are fundamental maps in g
s_riemann_inequality_for_curves	theorem	Riemann's inequality		For any divisor D on a smooth projective curve C of genus g, l(D) >= deg(D) + 1 - g, with equality for divisors of sufficiently large degree.
s_riemann_roch_space	state	Riemann-Roch space		For a divisor D on C, L(D) = {f in k(C)* : div(f) + D >= 0} union {0}, a finite-dimensional k-vector space whose dimension l(D) measures sections of the associa
s_riemann_roch_theorem_for_curves	theorem	Riemann-Roch theorem for algebraic curves		For a smooth projective curve C of genus g and divisor D, l(D) - l(K_C - D) = deg(D) - g + 1, where K_C is a canonical divisor.
s_second_cohomology_group_h2	state	Second cohomology group H²(G,M)		H²(G,M) classifies equivalence classes of factor sets (2-cocycles) f: G×G → M modulo 2-coboundaries; equivalently classifies group extensions 1 → M → E → G → 1 
s_segre_embedding	state	Segre embedding		The map P^m x P^n -> P^{mn+m+n} sending ([x_i],[y_j]) to [x_i*y_j]; realizes the product of projective spaces as a projective variety.
t_semisimple_decomposition_via_idempotents	technique	Semisimple decomposition via idempotents		In a semisimple ring R ≅ ∏ M_{n_i}(D_i), the identity 1 = e_1 + ... + e_k decomposes into central primitive idempotents corresponding to the simple factors; eac
s_semisimplicity_and_tensor_products	theorem	Semisimplicity and tensor products		If A is a central simple algebra over F and B is a semisimple F-algebra, then A ⊗_F B is semisimple; the tensor product of two central simple algebras over F is
s_splitting_field_of_a_central_simple_algebra	axiom	Splitting field of a central simple algebra		A field extension K/F such that A ⊗_F K ≅ M_n(K); every CSA is split by some finite separable extension, and K splits A iff A ⊗_F K is trivial in Br(K).
t_splitting_field_of_a_polynomial_over_a_finite_field	technique	Splitting field of a polynomial over a finite field		The technique of factoring a polynomial f(x) ∈ ℤ[x] modulo a prime p to determine the splitting of (p) in O_K = ℤ[α]/f(α) via Kummer's criterion; the degrees of
s_splitting_of_primes	axiom	Splitting of primes		A prime p in ℤ splits in O_K as pO_K = P_1^{e_1}...P_g^{e_g} with ∑ e_i f_i = [K:ℚ] where f_i = [O_K/P_i : ℤ/p]; p splits completely if g = [K:ℚ] (all e_i = f_i
s_tensor_product_of_algebras	axiom	Tensor product of algebras		For F-algebras A and B, the tensor product A ⊗_F B is an F-algebra with multiplication (a₁⊗b₁)(a₂⊗b₂) = a₁a₂ ⊗ b₁b₂; preserves simplicity and centrality when bo
s_tor_and_flatness_criterion	theorem	Tor and flatness criterion		An R-module M is flat if and only if Tor_1^R(M,R/I) = 0 for every finitely generated ideal I; equivalently Tor_1^R(M,N) = 0 for all R-modules N.
s_trace_map_trkf	state	Trace map Tr_{K/F}		For a finite extension K/F, Tr_{K/F}(α) = ∑_σ σ(α) summed over all F-embeddings σ: K → F̄; an F-linear map K → F used to define the discriminant and different.
s_units_of_quadratic_number_fields	state	Units of quadratic number fields		The group of units of the ring of integers of ℚ(√d): for d < 0, it is finite ({±1} or {±1, ±i} or the 6th roots of unity); for d > 0, it is {±ε^n : n ∈ ℤ} for a
s_universal_coefficients_theorem_algebraic	theorem	Universal coefficients theorem (algebraic)		For a chain complex C• of free abelian groups, there are natural short exact sequences 0 → H_n(C)⊗G → H_n(C;G) → Tor_1(H_{n-1}(C),G) → 0 and 0 → Ext^1(H_{n-1}(C
s_veronese_embedding	state	Veronese embedding		The map v_d: P^n -> P^N sending [x_0:...:x_n] to all monomials of degree d, where N = binom(n+d,d)-1; an embedding whose image is a projective variety.
s_wedderburns_main_theorem	theorem	Wedderburn's main theorem		If R is a finite-dimensional associative algebra over a field of characteristic 0, then R contains a semisimple subalgebra S such that R = S ⊕ rad(R) as vector 
s_wedderburns_theorem_semisimple_rings	theorem	Wedderburn's theorem (semisimple rings)		A left semisimple ring R with identity is isomorphic to a finite product M_{n_1}(D_1) × ... × M_{n_k}(D_k) of full matrix rings over division rings D_i; the fac
s_wedderburnartin_radical	state	Wedderburn-Artin radical		For a left Artinian ring R, rad(R) is the sum of all nilpotent left ideals; it is the unique maximal nilpotent two-sided ideal, and R/rad(R) is semisimple.
s_weierstrass_equation	state	Weierstrass equation		A model y^2 = x^3 + ax + b (char != 2,3) or y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6 for an elliptic curve, obtained via Riemann-Roch; nonsingular iff
s_weil_differential_repartition	axiom	Weil differential (repartition)		A k-linear functional on the adele space of k(C) that vanishes on the adeles of bounded denominators at each place, used to define the canonical class and prove
s_zariski_topology_on_specr	axiom	Zariski topology on Spec(R)		The topology on Spec(R) with closed sets V(I) = {p : p supset I} and basis of open sets D(f) = {p : f not in p}; Spec(R) is always quasi-compact.
s_zariski_topology_on_affine_space	axiom	Zariski topology on affine space		The topology on k^n (or an affine variety V) where closed sets are affine algebraic sets V(I); has very few open sets (not Hausdorff except for finite sets) but
s_zariski_topology_on_projective_space	axiom	Zariski topology on projective space		The topology on P^n where closed sets are V(I) for homogeneous ideals I of k[X_0,...,X_n]; the projective analog of the affine Zariski topology.
s_zariskis_theorem_on_nonsingular_points	theorem	Zariski's theorem on nonsingular points		A point P on an irreducible variety V is nonsingular iff the local ring O_{V,P} is a regular local ring; equivalently, dim(m_P/m_P^2) = dim(V), and nonsingular 
s_adele_of_a_function_field	axiom	adele of a function field		An element of the restricted direct product of completions of the function field k(C) at all places (discrete valuations), used in the proof of Riemann-Roch.
s_adelic_dirichlet_unit_theorem	theorem	adelic Dirichlet unit theorem		The compactness of J_K^1/K* implies that the unit group O_K* modulo torsion is a free abelian group of rank r_1+r_2-1, where r_1 and r_2 count real and complex 
s_adelic_finiteness_of_class_number	theorem	adelic finiteness of class number		The compactness of J_K^1/K* implies the class group Cl(K) = J_K / (K* * prod O_v*) is finite, giving the adelic proof of finiteness of the class number.
s_affine_variety	axiom	affine variety		An irreducible affine algebraic set, i.e., a set V(p) for a prime ideal p in k[x_1,...,x_n]; equivalently, one that cannot be written as a union of two proper c
s_algebraic_curve	axiom	algebraic curve		An irreducible algebraic variety of dimension 1; equivalently, defined by a function field of transcendence degree 1 over the base field, studied via its discre
s_birational_equivalence_of_smooth_projective_curves	theorem	birational equivalence of smooth projective curves		Two smooth projective curves are birationally equivalent if and only if they are isomorphic; a rational map from a smooth curve to a projective variety is alway
s_canonical_embedding	state	canonical embedding		For a smooth curve C of genus g >= 2 that is not hyperelliptic, the complete linear system |K_C| defines an embedding C -> P^{g-1} called the canonical embeddin
s_classification_of_irreducible_curves	theorem	classification of irreducible curves		Every irreducible curve is birationally equivalent to a smooth projective curve, unique up to isomorphism; the category of smooth projective curves is equivalen
s_compactness_of_adele_quotient	theorem	compactness of adele quotient		The quotient A_K / K (adeles modulo the diagonally embedded global field) is compact, encoding both the strong approximation theorem and finiteness results.
s_compactness_of_idele_class_group_normone	theorem	compactness of idele class group (norm-one)		The quotient J_K^1 / K* of norm-one ideles modulo K* is compact, simultaneously encoding the finiteness of the class number and the Dirichlet unit theorem.
s_complete_linear_system_determines_morphism	theorem	complete linear system determines morphism		A base-point-free complete linear system |D| on a smooth curve C defines a morphism phi_D: C -> P^{l(D)-1}; the morphism is an embedding iff |D| separates point
s_completion_of_a_valued_field	state	completion of a valued field		The unique (up to isomorphism) complete valued field K-hat containing K as a dense subfield with the absolute value extending that of K, constructed as the quot
s_decomposition_field	state	decomposition field		The fixed field L^{D(P|p)} of the decomposition group, the largest intermediate field where p does not split further below P.
s_degree_of_a_divisor	axiom	degree of a divisor		For a divisor D = sum n_P * P on a curve, deg(D) = sum n_P; the degree map is a group homomorphism Div(C) -> Z.
s_degree_of_a_morphism_of_curves	axiom	degree of a morphism of curves		For a nonconstant morphism f: C -> C' of smooth projective curves, deg(f) = [k(C):f*k(C')], the degree of the induced function field extension.
s_degree_of_a_principal_divisor_is_zero	theorem	degree of a principal divisor is zero		For any nonzero rational function f on a smooth projective curve C, deg(div(f)) = 0; equivalently, f has as many zeros as poles counted with multiplicity.
s_degree_of_a_projective_variety	state	degree of a projective variety		For a projective variety V of dimension d, deg(V) = d! times the leading coefficient of its Hilbert polynomial; equals the number of intersection points with a 
s_different_of_an_extension	state	different of an extension		The ideal D_{L/K} in O_L that is the inverse of the fractional ideal {x in L : Tr_{L/K}(x*O_L) subset O_K}, measuring ramification of L/K.
s_dimension_of_an_affine_variety	axiom	dimension of an affine variety		The Krull dimension of the coordinate ring k[V], equivalently the transcendence degree of the function field k(V) over k for an irreducible variety V.
s_discriminant_of_an_extension	state	discriminant of an extension		The ideal d_{L/K} = N_{L/K}(D_{L/K}) in O_K, the norm of the different; a prime p ramifies in L/K if and only if p divides d_{L/K}.
s_divisor_class_group_of_a_curve	state	divisor class group of a curve		The quotient group Cl(C) = Div(C) / Prin(C) of divisors modulo principal divisors on C; its degree-zero part Cl^0(C) is the Jacobian.
s_effective_divisor	axiom	effective divisor		A divisor D = sum n_P * P where all coefficients n_P >= 0; written D >= 0.
s_elimination_ideal	state	elimination ideal		The l-th elimination ideal of I subset k[x_1,...,x_n] is I_l = I intersect k[x_{l+1},...,x_n], obtained by eliminating the first l variables.
s_elimination_theorem	theorem	elimination theorem		If G is a Gröbner basis for I with respect to a lexicographic order with x_1 > ... > x_n, then G intersect k[x_{l+1},...,x_n] is a Gröbner basis for the l-th el
s_elliptic_curve_genusone_curve_with_point	axiom	elliptic curve (genus-one curve with point)		A smooth projective curve of genus 1 together with a specified rational point O, admitting a group law making it an abelian variety of dimension 1.
s_equivalence_of_absolute_values	axiom	equivalence of absolute values		Two absolute values |·|_1 and |·|_2 on K are equivalent if they define the same topology, which holds iff |·|_2 = |·|_1^s for some s > 0.
s_extension_theorem	theorem	extension theorem		Under suitable conditions on the leading coefficients, a solution of the elimination ideal I_l can be extended to a solution of I by finding values for the elim
s_function_field_of_an_algebraic_curve	axiom	function field of an algebraic curve		The field k(C) of rational functions on an irreducible curve C, a finitely generated extension of k of transcendence degree 1; determines the curve up to birati
s_fundamental_identity_for_valuations	theorem	fundamental identity for valuations		For a finite extension L/K of number fields and a prime p of K, the sum of e(P_i|p)*f(P_i|p) over all primes P_i of L above p equals [L:K].
s_fundamental_theorem_for_infinite_galois_extensions	theorem	fundamental theorem for infinite Galois extensions		For a Galois extension L/K, there is an inclusion-reversing bijection between intermediate fields K subset F subset L and closed subgroups H of Gal(L/K) in the 
s_generic_point_of_a_scheme	state	generic point of a scheme		For an irreducible scheme X = Spec(R/p), the point eta = p in Spec(R) whose closure is all of X; the stalk O_{X,eta} is the function field of X.
s_genus_of_an_algebraic_curve	axiom	genus of an algebraic curve		The nonneg integer g such that Riemann's inequality l(D) >= deg(D) + 1 - g holds for all divisors D, and g is the smallest such integer; equals the geometric ge
s_geometric_dimension_noetherian_topological_space	axiom	geometric dimension (Noetherian topological space)		The supremum of lengths of chains of irreducible closed subsets Z_0 subset Z_1 subset ... subset Z_d in a Noetherian topological space; agrees with Krull dimens
s_homogeneous_coordinate_ring	state	homogeneous coordinate ring		The graded ring S(V) = k[X_0,...,X_n]/I(V) for a projective variety V, where I(V) is the homogeneous ideal of V; a graded integral domain.
s_homogeneous_ideal	axiom	homogeneous ideal		An ideal I of k[X_0,...,X_n] generated by homogeneous polynomials, equivalently I = direct sum (I intersect k[X_0,...,X_n]_d); needed to define projective algeb
s_homogeneous_polynomial	axiom	homogeneous polynomial		A polynomial F in k[X_0,...,X_n] where every monomial has the same total degree d; satisfies F(lambda*x) = lambda^d * F(x).
s_hyperelliptic_curve	axiom	hyperelliptic curve		A smooth projective curve C of genus g >= 2 admitting a degree-2 morphism to P^1; equivalently, C has a divisor D of degree 2 with l(D) >= 2.
s_hypersurface	axiom	hypersurface		The zero locus V(f) of a single irreducible polynomial f in k[x_1,...,x_n] or a single homogeneous polynomial in projective space; has codimension 1.
s_ideal_of_an_algebraic_set	state	ideal of an algebraic set		I(V) = {f in k[x_1,...,x_n] : f(P)=0 for all P in V}, a radical ideal; by the Nullstellensatz, I(V(J)) = sqrt(J).
s_inertia_field	state	inertia field		The fixed field L^{I(P|p)} of the inertia group, the largest intermediate field where p is unramified below P.
s_infinite_galois_group	state	infinite Galois group		The Galois group Gal(L/K) of an infinite Galois extension L/K, equipped with the Krull topology making it a profinite group isomorphic to lim_F Gal(F/K) over fi
s_intersection_multiplicity_two_curves	axiom	intersection multiplicity (two curves)		For curves F and G in P^2 meeting at P, I(P; F,G) = dim_k O_{P^2,P}/(F,G), the dimension of the local ring quotient; it is symmetric and additive.
s_intersections_in_projective_space	theorem	intersections in projective space		Two projective varieties in P^n of dimensions r and s with r+s >= n have nonempty intersection; more precisely, every component of the intersection has dimensio
s_irreducible_decomposition	theorem	irreducible decomposition		Every algebraic set V can be written uniquely as a finite union V = V_1 union ... union V_r of irreducible components (maximal irreducible closed subsets), with
s_linear_system_of_divisors	state	linear system of divisors		The projective space |D| = P(L(D)) of effective divisors linearly equivalent to D on a curve C; has dimension l(D)-1 and determines maps to projective space.
s_local_ring_at_a_point_of_a_variety	state	local ring at a point of a variety		The ring O_{V,P} of rational functions regular at P, a local ring with maximal ideal m_P = {f in O_{V,P} : f(P)=0}; isomorphic to the localization of k[V] at th
s_localization_of_a_ring_at_a_prime	state	localization of a ring at a prime		The local ring R_p obtained by inverting all elements not in a prime ideal p; its spectrum is the set of primes contained in p, modeling a neighborhood of p in 
s_locally_ringed_space	axiom	locally ringed space		A topological space X equipped with a sheaf of rings O_X such that every stalk O_{X,x} is a local ring; the category in which schemes live.
s_monomial_order	axiom	monomial order		A total order on the monomials of k[x_1,...,x_n] that is compatible with multiplication and is a well-ordering; examples include lexicographic, graded lexicogra
s_morphism_of_affine_schemes	theorem	morphism of affine schemes		Morphisms Spec(S) -> Spec(R) of affine schemes correspond bijectively to ring homomorphisms R -> S; the functor Spec is a fully faithful contravariant functor f
s_morphism_of_affine_varieties	axiom	morphism of affine varieties		A map phi: V -> W between affine varieties given by phi(P) = (f_1(P),...,f_m(P)) for polynomials f_i; equivalently, a map inducing a k-algebra homomorphism phi*
s_morphism_of_projective_varieties	axiom	morphism of projective varieties		A map phi: V -> W between projective varieties that is locally given by homogeneous polynomials of the same degree; a continuous map in the Zariski topology.
t_multivariate_division_algorithm	technique	multivariate division algorithm		Given a polynomial f and an ordered tuple (g_1,...,g_s) with a fixed monomial order, produces f = sum a_i*g_i + r where no term of r is divisible by any LT(g_i)
s_nonsingular_point_of_a_variety	axiom	nonsingular point of a variety		A point P on a variety V where the local ring O_{V,P} is a regular local ring, equivalently dim(m_P/m_P^2) = dim(V).
s_normalization_of_a_curve	state	normalization of a curve		For an irreducible curve C, its normalization C~ is the unique smooth curve with a birational morphism C~ -> C that is an isomorphism away from the singular poi
s_padic_absolute_value	axiom	p-adic absolute value		The absolute value |x|_p = p^{-v_p(x)} on Q, where v_p(x) is the p-adic valuation measuring the exact power of p dividing x.
s_padic_integers	axiom	p-adic integers		The ring Z_p of p-adic integers, defined as the inverse limit of Z/p^n Z, forming a complete local ring with maximal ideal (p).
s_padic_numbers	axiom	p-adic numbers		The field Q_p obtained as the fraction field of Z_p, or equivalently as the completion of Q with respect to the p-adic absolute value |·|_p.
s_place_of_a_field	axiom	place of a field		An equivalence class of nontrivial absolute values on a field K; for a number field, places are either finite (nonarchimedean, from primes) or infinite (archime
s_presheaf_on_a_topological_space	axiom	presheaf on a topological space		A contravariant functor from the category of open sets of X (with inclusions) to a target category (e.g., abelian groups, rings), assigning sections F(U) and re
s_prime_spectrum_specr	axiom	prime spectrum Spec(R)		The set of all prime ideals of a commutative ring R, equipped with the Zariski topology where closed sets are V(I) = {p in Spec(R) : I subset p} for ideals I; t
s_projective_algebraic_set	axiom	projective algebraic set		The set V(S) in P^n of common zeros of a set S of homogeneous polynomials; closed in the Zariski topology on P^n.
s_projective_closure	state	projective closure		The Zariski closure of an affine variety V subset A^n in projective space P^n under the standard embedding A^n -> P^n; obtained by homogenizing the defining pol
s_projective_plane_curve	axiom	projective plane curve		The zero locus V(F) in P^2 of a nonconstant homogeneous polynomial F(X,Y,Z), considered up to scalar multiples; its degree is the degree of F.
s_projective_space_pn	axiom	projective space P^n		The set of equivalence classes of nonzero points in k^{n+1} under scalar multiplication, i.e., P^n(k) = (k^{n+1} - {0}) / k*; the natural domain for homogeneous
s_projective_variety	axiom	projective variety		An irreducible projective algebraic set, i.e., V(p) for a homogeneous prime ideal p not equal to the irrelevant ideal; a complete algebraic variety.
s_ramification_divisor	state	ramification divisor		For a morphism f: C -> C' of curves, the divisor R = sum (e_P-1)*P on C measuring the ramification; appears in the Hurwitz formula as deg(R) = 2g-2 - deg(f)*(2g
s_rational_curve	axiom	rational curve		A smooth projective curve of genus 0; by Riemann-Roch, it is isomorphic to P^1.
s_rational_function_on_a_variety	axiom	rational function on a variety		An equivalence class of pairs (U, f) where U is a nonempty open set of V and f is a regular function on U, with (U,f) ~ (U',f') if f=f' on U intersect U'; eleme
s_rational_map_of_varieties	axiom	rational map of varieties		An equivalence class of morphisms defined on dense open subsets of a variety V to W; a rational map V --> W may not be defined everywhere.
s_regular_function_at_a_point	axiom	regular function at a point		A rational function f on V that is defined (representable as g/h with h(P)!=0) at a point P; the set of regular functions at P forms the local ring O_{V,P}.
s_residue_class_degree	axiom	residue class degree		For an extension L/K with primes P|p, the residue class degree f(P|p) = [O_P/P : O_p/p] is the degree of the residue field extension.
s_separable_closure	state	separable closure		The maximal separable subextension K^{sep} of an algebraic closure K-bar of K; K-bar/K^{sep} is purely inseparable.
s_sheaf_on_a_topological_space	axiom	sheaf on a topological space		A presheaf F on X satisfying: (1) if s|_{U_i}=0 for all U_i in an open cover of U, then s=0 (locality); (2) if s_i in F(U_i) agree on overlaps, they glue to s i
s_singular_point_of_a_variety	axiom	singular point of a variety		A point P on a variety V where dim(m_P/m_P^2) > dim(V), equivalently where the Jacobian matrix does not have maximal rank; the locus of singular points is a pro
s_splitting_of_primes_in_extensions	state	splitting of primes in extensions		For a prime p of O_K in an extension L/K, the factorization p*O_L = P_1^{e_1} ... P_g^{e_g}; primes can split, remain inert, or ramify depending on the e_i and 
s_structure_sheaf_on_specr	state	structure sheaf on Spec(R)		The sheaf O on Spec(R) with O(D(f)) = R_f (localization at f) and stalks O_p = R_p (localization at prime p); makes (Spec(R), O) into a locally ringed space.
s_tangent_space_zariski	state	tangent space (Zariski)		The Zariski tangent space at a point P of a variety V is (m_P/m_P^2)*, the dual of the cotangent space m_P/m_P^2 of the local ring O_{V,P}; its dimension equals
s_tensor_product_decomposition_of_completions	theorem	tensor product decomposition of completions		For a finite separable extension L/K and a place v of K, K_v tensor_K L is isomorphic to the product of completions L_w over all places w of L above v.
s_totally_ramified_extension	axiom	totally ramified extension		An extension L/K of valued fields where e(P|p) = [L:K] and f(P|p) = 1, so the entire degree is accounted for by ramification.
s_unramified_extension	axiom	unramified extension		An extension L/K of valued fields where e(P|p)=1 and the residue field extension is separable, so ramification index is trivial.
s_zeta_function_of_a_variety_over_a_finite_field	state	zeta function of a variety over a finite field		Z(V,t) = exp(sum_{m>=1} |V(F_{q^m})| * t^m / m), a formal power series encoding point counts of V over all finite extensions of F_q; rational by the Weil conjec
s_ideal_sum	axiom	Sum of ideals		The sum 𝔞 + 𝔟 of two ideals is the smallest ideal containing both, equal to {a + b : a ∈ 𝔞, b ∈ 𝔟}.
s_ideal_product	axiom	Product of ideals		The product 𝔞𝔟 of two ideals is the ideal generated by all products ab with a ∈ 𝔞, b ∈ 𝔟.
s_ideal_quotient	axiom	Ideal quotient (colon ideal)		The ideal quotient (𝔞 : 𝔟) is the set of all ring elements x such that x𝔟 ⊆ 𝔞, forming an ideal of A.
s_coprime_ideals	axiom	Coprime ideals		Two ideals 𝔞, 𝔟 in a commutative ring are coprime (comaximal) if 𝔞 + 𝔟 = (1).
s_prime_avoidance_lemma	theorem	Prime avoidance lemma		If an ideal 𝔞 is contained in a finite union of prime ideals 𝔭₁ ∪ ··· ∪ 𝔭ₙ, then 𝔞 ⊆ 𝔭ᵢ for some i.
s_extension_and_contraction_of_ideals	axiom	Extension and contraction of ideals		For a ring homomorphism f: A → B, the extension 𝔞ᵉ of an ideal 𝔞 ⊆ A is the ideal 𝔞B generated by f(𝔞), and the contraction 𝔟ᶜ of an ideal 𝔟 ⊆ B is f⁻¹(𝔟).
s_jacobson_radical_characterization	theorem	Jacobson radical characterization		An element x belongs to the Jacobson radical J(A) if and only if 1 − xy is a unit in A for every y ∈ A.
s_characterization_of_local_rings	theorem	Characterization of local rings		A ring is local if and only if its set of non-units forms an ideal, equivalently if it has a unique maximal ideal.
s_radical_equals_intersection_of_primes_containing_ideal	theorem	Radical equals intersection of primes containing ideal		The radical √𝔞 of an ideal 𝔞 equals the intersection of all prime ideals containing 𝔞.
s_restriction_of_scalars	axiom	Restriction of scalars		Given a ring homomorphism f: A → B and a B-module N, restriction of scalars gives N the structure of an A-module via a·n = f(a)·n.
s_algebra_over_a_ring	axiom	Algebra over a ring		An A-algebra is a ring B together with a ring homomorphism f: A → B, making B simultaneously an A-module and a ring compatibly.
s_flatness_ideal_criterion	theorem	Ideal-theoretic characterization of flatness		An A-module M is flat if and only if for every ideal 𝔞 of A, the natural map 𝔞 ⊗_A M → M is injective.
s_ring_of_fractions	axiom	Ring of fractions		For a multiplicative subset S of a ring A, the ring of fractions S⁻¹A consists of equivalence classes of pairs (a, s) with a ∈ A, s ∈ S, under the relation (a,s
s_module_of_fractions	axiom	Module of fractions		For a multiplicative subset S and an A-module M, the module of fractions S⁻¹M consists of equivalence classes m/s with the natural S⁻¹A-module structure, isomor
s_local_property	axiom	Local property of modules		A property P of modules or ring elements is local if it holds globally precisely when it holds after localization at every prime (or every maximal) ideal.
s_flatness_is_a_local_property	theorem	Flatness is a local property		An A-module M is flat if and only if M_𝔭 is a flat A_𝔭-module for every prime ideal 𝔭 of A.
s_prime_correspondence_under_localization	theorem	Prime correspondence under localization		The prime ideals of S⁻¹A are in inclusion-preserving bijection with the prime ideals of A that do not meet S, via 𝔭 ↦ S⁻¹𝔭.
s_integral_closure_is_a_subring	theorem	Integral closure is a subring		The set of elements in a ring extension B/A that are integral over A forms a subring of B (the integral closure of A in B).
s_integral_element_equivalences	theorem	Equivalences for integral elements		An element x ∈ B is integral over A iff A[x] is a finitely generated A-module, iff A[x] is contained in a finitely generated A-submodule of B, iff there exists 
s_valuation_ring_characterization	theorem	Characterization of valuation rings		An integral domain V with fraction field K is a valuation ring if and only if for every x ∈ K×, either x ∈ V or x⁻¹ ∈ V, equivalently if its ideals are totally 
s_integral_closure_via_valuation_rings	theorem	Integral closure via valuation rings		The integral closure of an integral domain A in its field of fractions equals the intersection of all valuation rings of K containing A.
s_noetherian_characterization_exact_sequences	theorem	Noetherian characterization via exact sequences		In a short exact sequence 0 → M' → M → M'' → 0 of A-modules, M is Noetherian if and only if both M' and M'' are Noetherian.
s_hilbert_basis_theorem_power_series	theorem	Hilbert basis theorem for power series rings		If A is a Noetherian ring, then the formal power series ring A[[x]] is also Noetherian.
s_irreducible_ideal	axiom	Irreducible ideal		An ideal 𝔞 is irreducible if 𝔞 = 𝔟 ∩ 𝔠 implies 𝔞 = 𝔟 or 𝔞 = 𝔠; it cannot be written as a non-trivial intersection of two strictly larger ideals.
s_irreducible_ideals_are_primary_in_noetherian_rings	theorem	Irreducible ideals are primary in Noetherian rings		In a Noetherian ring, every irreducible ideal is primary.
s_every_ideal_is_finite_intersection_of_irreducibles	theorem	Every ideal is a finite intersection of irreducible ideals		In a Noetherian ring, every ideal can be expressed as a finite intersection of irreducible ideals.
s_symbolic_power_of_a_prime	axiom	Symbolic power of a prime ideal		The n-th symbolic power 𝔭^(n) of a prime ideal 𝔭 is the 𝔭-primary component of 𝔭ⁿ, equivalently 𝔭^(n) = 𝔭ⁿA_𝔭 ∩ A.
s_artinian_ring_has_dimension_zero	theorem	Artinian rings have Krull dimension zero		A commutative ring is Artinian if and only if it is Noetherian and has Krull dimension zero (every prime ideal is maximal).
s_structure_theorem_for_artinian_rings	theorem	Structure theorem for Artinian rings		Every commutative Artinian ring is uniquely (up to isomorphism) a finite direct product of Artinian local rings.
s_artinian_ring_finitely_many_maximal_ideals	theorem	Artinian ring has finitely many maximal ideals		A commutative Artinian ring has only finitely many maximal ideals.
s_dvr_characterizations	theorem	Characterizations of discrete valuation rings		A Noetherian local domain of dimension 1 is a DVR iff it is integrally closed, iff its maximal ideal is principal, iff it is a regular local ring of dimension 1
s_dedekind_domain_characterizations	theorem	Characterizations of Dedekind domains		A Noetherian domain of dimension ≤ 1 is Dedekind iff it is integrally closed, iff every primary ideal is a prime power, iff every localization at a nonzero prim
s_filtered_module	axiom	Filtered module		A module M over a filtered ring A equipped with a decreasing sequence of submodules M = M₀ ⊇ M₁ ⊇ ··· compatible with the filtration on A.
s_completion_is_exact_on_fg_modules	theorem	Completion is exact on finitely generated modules		For a Noetherian ring A with 𝔞-adic topology, the completion functor M ↦ M̂ is exact on the category of finitely generated A-modules.
s_flatness_of_completion	theorem	Flatness of completion		For a Noetherian ring A, the 𝔞-adic completion Â is a flat A-algebra.
s_completion_of_noetherian_ring_is_noetherian	theorem	Completion of a Noetherian ring is Noetherian		The 𝔞-adic completion of a Noetherian ring is again Noetherian.
s_height_of_a_prime_ideal	axiom	Height of a prime ideal		The height of a prime ideal 𝔭 is the supremum of lengths of chains of prime ideals 𝔭₀ ⊊ 𝔭₁ ⊊ ··· ⊊ 𝔭ₙ = 𝔭 descending from 𝔭.
s_coheight_of_a_prime_ideal	axiom	Coheight of a prime ideal		The coheight of a prime ideal 𝔭 in a ring A is the Krull dimension of A/𝔭, the supremum of lengths of chains of primes ascending from 𝔭.
s_regular_local_ring_is_integral_domain	theorem	Regular local ring is an integral domain		Every regular local ring is an integral domain.
s_dimension_inequality_for_extensions	theorem	Dimension inequality for ring extensions		If A ⊆ B are Noetherian domains and 𝔮 is a prime of B lying over 𝔭 in A, then ht(𝔮) + tr.deg(κ(𝔮)/κ(𝔭)) ≤ ht(𝔭) + tr.deg(B/A).
t_primary_decomposition_technique	technique	Primary decomposition technique		Decomposing an ideal or submodule into primary components to study its associated primes and reduce problems to the primary case.
t_integral_closure_technique	technique	Integral closure technique		Passing to the integral closure to improve ring-theoretic properties while preserving dimension and prime structure.
t_noetherian_induction	technique	Noetherian induction		Proving a statement for all ideals of a Noetherian ring by assuming it fails and taking a maximal counterexample via the ACC.
t_completion_technique	technique	Completion technique		Passing to the 𝔞-adic completion of a local ring to exploit completeness, Hensel's lemma, and Cohen structure theorems.
t_graded_ring_technique	technique	Graded/associated-graded technique		Passing from a filtered ring or module to its associated graded object to reduce questions about a local ring to questions about a polynomial ring.
t_hilbert_function_method	technique	Hilbert function method		Using the Hilbert function or Hilbert-Samuel polynomial to extract numerical invariants from graded or filtered algebraic structures.
t_determinant_trick	technique	Determinant trick		Applying the Cayley-Hamilton theorem to an endomorphism of a finitely generated module to produce integral dependence or annihilation relations.
s_residue_field	state	Residue field		The residue field of a prime ideal 𝔭 in a commutative ring R, defined as the quotient of the localization R_𝔭 by its maximal ideal 𝔭R_𝔭.
s_idempotents_and_connected_components_of_spec	theorem	Idempotents and connected components of Spec		The connected components of Spec(R) correspond bijectively to primitive idempotents of R; R decomposes as a product of rings iff Spec(R) is disconnected.
s_prime_filtration	state	Prime filtration of a module		A chain 0 = M₀ ⊂ M₁ ⊂ ··· ⊂ Mₙ = M with each successive quotient Mᵢ/Mᵢ₋₁ ≅ R/𝔭ᵢ for prime ideals 𝔭ᵢ; exists for every finitely generated module over a Noetheria
s_finiteness_of_associated_primes	theorem	Finiteness of associated primes		For a finitely generated module M over a Noetherian ring R, the set Ass(M) of associated primes is finite.
s_zero_divisors_equal_union_of_associated_primes	theorem	Zero divisors equal union of associated primes		For a finitely generated module M over a Noetherian ring, the set of zero divisors on M equals the union of the associated primes of M.
s_ufd_implies_integrally_closed	theorem	UFD implies integrally closed		Every unique factorization domain is integrally closed in its field of fractions.
s_zariskis_lemma	theorem	Zariski's lemma		If k is a field and K is a finitely generated k-algebra that is also a field, then K is a finite (algebraic) extension of k.
s_filtration_of_a_module	axiom	Filtration of a module		A descending chain of submodules M = M₀ ⊇ M₁ ⊇ M₂ ⊇ ··· compatible with the ring filtration; the I-adic filtration Mₙ = IⁿM is the primary example.
s_stable_filtration	state	Stable filtration (I-stable)		An I-filtration {Mₙ} is I-stable if Mₙ₊₁ = I·Mₙ for all sufficiently large n, equivalently the Rees module ⊕Mₙ is finitely generated over R[It].
s_tangent_cone	state	Tangent cone		The associated graded ring of a local ring (R, 𝔪) with respect to the maximal ideal, whose Proj is the projectivized tangent cone at the corresponding point of 
s_localization_is_flat	theorem	Localization is flat		For any multiplicative set S in a commutative ring R, the localization S⁻¹R is a flat R-algebra.
s_flat_base_change_for_tor	theorem	Flat base change for Tor		If R → S is a flat ring homomorphism, then Torᵢᴿ(M, N) ⊗_R S ≅ Torᵢˢ(M ⊗_R S, N ⊗_R S) for all i.
s_lazards_theorem	theorem	Lazard's theorem		An R-module is flat if and only if it is a filtered colimit of finitely generated free modules.
s_flat_equals_locally_free_for_fp	theorem	Flat equals locally free for finitely presented modules		A finitely presented module over a commutative ring is flat if and only if it is locally free (free after localization at every prime ideal).
s_weierstrass_preparation_theorem_formal	theorem	Weierstrass preparation theorem (formal power series)		In k[[x₁,...,xₙ]], a power series distinguished of order d in xₙ factors uniquely as a unit times a Weierstrass polynomial of degree d in xₙ.
t_lifting_idempotents	technique	Lifting idempotents		Lifting an idempotent from R/I to the I-adically complete ring R by iterative Newton-type approximation, used for decomposing modules over complete local rings.
s_embedding_dimension	state	Embedding dimension		The minimal number of generators of the maximal ideal 𝔪 of a local ring (R, 𝔪, k), equal to dim_k(𝔪/𝔪²); a regular local ring has edim(R) = dim(R).
s_fiber_ring	state	Fiber ring of a ring map		The fiber of a ring map φ: R → S over a prime 𝔭 of R, defined as κ(𝔭) ⊗_R S; its Spec is the scheme-theoretic fiber of Spec S → Spec R over 𝔭.
s_total_ring_of_fractions	state	Total ring of fractions		The localization of R at the multiplicative set of all non-zero-divisors; generalizes the fraction field to rings that are not necessarily domains.
s_i_adic_valuation	state	I-adic valuation		For a Noetherian ring R and ideal I with ∩Iⁿ = 0, the function assigning to each element the largest power of I containing it, inducing the I-adic topology.
s_reduced_ring	axiom	Reduced ring		A commutative ring with no nonzero nilpotent elements, equivalently one whose nilradical is zero.
s_homogeneous_nullstellensatz	theorem	Homogeneous Nullstellensatz (projective Nullstellensatz)		A homogeneous ideal I in k[x₀,...,xₙ] over an algebraically closed field defines the empty projective variety iff rad(I) contains the irrelevant ideal (x₀,...,x
t_rees_algebra_technique	technique	Rees algebra technique		Encoding an I-adic filtration as a graded module over the Rees algebra R[It] to reduce filtration questions to graded algebra; key to proving the Artin-Rees lem
s_geometric_primary_decomposition	state	Geometric interpretation of primary decomposition		Primary decomposition I = ∩Qᵢ decomposes V(I) into irreducible components (minimal primes) with embedded structure (embedded primes encoding nilpotent or higher
s_module_of_finite_length	state	Module of finite length		A module admitting a composition series of finite length, equivalently both Noetherian and Artinian; the length ℓ(M) is additive on short exact sequences.
s_completion_is_faithfully_flat	theorem	Completion is faithfully flat		For a Noetherian local ring (R, 𝔪), the 𝔪-adic completion R̂ is faithfully flat over R; in particular R → R̂ is injective and detects exactness.
s_affine_rings_universally_catenary	theorem	Affine rings are universally catenary		Every finitely generated algebra over a field or over ℤ is universally catenary.
s_krull_dimension_of_a_module	state	Krull dimension of a module		The Krull dimension of an R-module M, defined as the Krull dimension of the quotient ring R/Ann(M).
s_codimension_of_an_ideal	state	Codimension (height) of an ideal		The codimension of an ideal I in a ring R is the infimum of heights of prime ideals containing I.
s_equidimensional_ring	axiom	Equidimensional ring		A Noetherian ring in which all minimal primes have the same coheight, equivalently all irreducible components of Spec(R) have the same dimension.
s_dimension_via_m_primary_ideals	theorem	Dimension via 𝔪-primary ideals		The Krull dimension of a Noetherian local ring (R, 𝔪) equals the minimum number of generators of an 𝔪-primary ideal.
s_existence_of_systems_of_parameters	theorem	Existence of systems of parameters		Every Noetherian local ring (R, 𝔪) of dimension d admits a system of parameters: d elements x₁,...,x_d generating an 𝔪-primary ideal.
s_dimension_drop_by_one_element	theorem	Dimension drop by one element		If (R, 𝔪) is a Noetherian local ring and x ∈ 𝔪, then dim(R/(x)) ≥ dim(R) − 1, with equality when x avoids all minimal primes of maximal dimension.
s_parameter_ideal	state	Parameter ideal		An ideal in a Noetherian local ring of dimension d generated by a system of parameters; it is 𝔪-primary with exactly d generators.
s_regular_system_of_parameters	state	Regular system of parameters		In a regular local ring (R, 𝔪), a system of parameters x₁,...,x_d that generates the maximal ideal 𝔪, equivalently whose images form a k-basis of 𝔪/𝔪².
s_dimension_of_fiber	theorem	Dimension of fibers of a ring map		For a local homomorphism φ: (R, 𝔪) → (S, 𝔫) of Noetherian local rings, dim(S) ≤ dim(R) + dim(S/𝔪S), with equality when φ is flat.
s_serre_criterion_for_normality	theorem	Serre's criterion for normality (R₁ + S₂)		A Noetherian ring is normal (integrally closed in its total ring of fractions) iff it satisfies R₁ (regular in codimension ≤ 1) and S₂ (depth ≥ min(2, ht 𝔭) at 
s_serre_condition_rk	axiom	Serre's condition Rₖ		A Noetherian ring satisfies (Rₖ) if the localization R_𝔭 is a regular local ring for every prime 𝔭 of height at most k.
s_serre_condition_sk	axiom	Serre's condition Sₖ		A Noetherian ring satisfies (Sₖ) if depth(R_𝔭) ≥ min(k, ht(𝔭)) for every prime ideal 𝔭; Cohen-Macaulay rings satisfy Sₖ for all k.
s_serre_criterion_for_reducedness	theorem	Serre's criterion for reducedness (R₀ + S₁)		A Noetherian ring is reduced if and only if it satisfies R₀ (regular at all minimal primes) and S₁ (no embedded associated primes of the zero ideal).
s_divisor_class_group_ring	state	Divisor class group of a normal domain		The group of Weil divisor classes on a Noetherian normal domain R: the free abelian group on height-1 primes modulo principal divisors div(f).
s_reflexive_module	axiom	Reflexive module		A finitely generated module M such that the canonical map M → M** = Hom(Hom(M, R), R) is an isomorphism; rank-1 reflexives over a normal domain correspond to di
s_ufd_iff_trivial_class_group	theorem	UFD iff trivial divisor class group		A Noetherian normal domain is a UFD if and only if its divisor class group Cl(R) is trivial, equivalently every height-1 prime ideal is principal.
s_conductor_ideal	state	Conductor ideal		For a reduced ring R with integral closure R̄, the conductor 𝔠 = {x ∈ R̄ : xR̄ ⊆ R} is the largest ideal of R̄ that is also an ideal of R.
s_integral_closure_1dim_is_dedekind	theorem	Integral closure of a 1-dimensional Noetherian domain is Dedekind		The integral closure of a one-dimensional Noetherian domain in its fraction field is a Noetherian ring, hence a Dedekind domain.
s_multiplicity_one_implies_regularity	theorem	Multiplicity one implies regularity		A Noetherian local ring has Hilbert-Samuel multiplicity e(R) = 1 if and only if it is a regular local ring.
s_associativity_formula_for_multiplicities	theorem	Associativity formula for multiplicities		For a Noetherian local ring (R, 𝔪) of dimension d and 𝔪-primary ideal q, e(q; M) = Σ_𝔭 e(q; R/𝔭)·ℓ_{R_𝔭}(M_𝔭), summed over minimal primes 𝔭 with dim(R/𝔭) = dim(
s_reduction_of_an_ideal	state	Reduction of an ideal		An ideal J ⊆ I is a reduction of I if JIⁿ = Iⁿ⁺¹ for some n ≥ 0; minimal reductions have exactly dim(R) generators and share the same multiplicity as I.
s_noether_normalization_nagata	theorem	Noether normalization (Nagata version)		A version of Noether normalization for finitely generated algebras over any Noetherian ring, where the polynomial subring may require localization; due to Nagat
s_noether_normalization_graded	theorem	Noether normalization for graded rings		For a finitely generated graded k-algebra S, there exist algebraically independent homogeneous elements y₁,...,y_d such that S is a finite module over k[y₁,...,
s_finiteness_of_integral_closure_affine	theorem	Finiteness of integral closure of affine domains		The integral closure of a finitely generated domain over a field (or over ℤ) in a finite extension of its fraction field is a finitely generated module.
s_generic_freeness_lemma	theorem	Generic freeness lemma (Grothendieck)		If R is a Noetherian domain and M is a finitely generated R-algebra or R-module, then there exists nonzero f ∈ R such that M_f is free over R_f.
s_dimension_formula_affine_domains	theorem	Dimension formula for affine domains		For a prime 𝔭 in a finitely generated domain R over a field k: ht(𝔭) + dim(R/𝔭) = dim(R).
s_semicontinuity_of_fiber_dimension	theorem	Semicontinuity of fiber dimension		For a morphism of finite type between Noetherian schemes, the function y ↦ dim(f⁻¹(y)) is upper semicontinuous on the target.
s_going_down_for_flat_extensions	theorem	Going-down for flat extensions		If R → S is a flat ring homomorphism, then the going-down property holds: primes in R that are contained in the contraction of a prime of S lift to primes of S.
s_hilbert_polynomial_of_hypersurface	theorem	Hilbert polynomial of a hypersurface		The Hilbert polynomial of a hypersurface V(f) ⊆ Pⁿ of degree d is P(t) = C(t+n,n) − C(t+n−d,n), so deg(V(f)) = d.
t_dimension_via_parameter_chains	technique	Dimension computation via parameter chains		Computing Krull dimension by iteratively choosing elements avoiding minimal primes to reduce dimension by one at each step, building a system of parameters.
s_extension_theorem_elimination	theorem	Extension theorem (elimination theory)		Conditions under which a partial solution in the elimination ideal extends to a full solution of the original polynomial system over an algebraically closed fie
s_fiber_dimension_theorem	theorem	Fiber dimension theorem		For a dominant morphism of irreducible varieties f: X → Y, every irreducible component of a fiber f⁻¹(y) has dimension at least dim(X) − dim(Y), with equality o
s_lexicographic_order	axiom	Lexicographic monomial order		A monomial order on k[x₁,...,xₙ] where x^α > x^β iff the leftmost nonzero entry of α − β is positive; favored for elimination but computationally expensive.
s_grevlex_order	axiom	Graded reverse lexicographic order (grevlex)		A monomial order comparing first by total degree, then breaking ties by the rightmost nonzero entry of α − β being negative; the most efficient order for Gröbne
s_initial_term	state	Initial term (leading term)		The largest monomial term of a polynomial f with respect to a fixed monomial order >, written in(f) or LT(f).
s_initial_ideal	state	Initial ideal		The monomial ideal generated by the initial terms of all nonzero elements of an ideal I; determines the Hilbert function and leading behavior of I.
s_s_polynomial	state	S-polynomial		For polynomials f, g, the combination S(f,g) = (lcm(LM(f),LM(g))/LT(f))·f − (lcm(LM(f),LM(g))/LT(g))·g designed to cancel leading terms; central to Buchberger's
s_reduced_groebner_basis	state	Reduced Gröbner basis		A Gröbner basis where each element is monic and no monomial of any element lies in the initial ideal of the others; unique for a given ideal and monomial order.
s_flat_degeneration_to_initial_ideal	theorem	Flat degeneration to initial ideal		There exists a flat one-parameter family over A¹ whose general fiber is R/I and special fiber is R/in(I), realizing the initial ideal as a flat degeneration pre
s_dicksons_lemma	theorem	Dickson's lemma		Every monomial ideal in k[x₁,...,xₙ] is finitely generated, equivalently every antichain in (ℕⁿ, ≤) is finite.
t_groebner_basis_syzygies	technique	Gröbner basis and syzygies technique		Computing syzygies from a Gröbner basis by lifting S-polynomial reductions to syzygy relations, yielding free resolutions via Schreyer's theorem.
s_universal_derivation	state	Universal derivation		The canonical R-derivation d: S → Ω_{S/R} such that every R-derivation from S to an S-module M factors uniquely through d, characterizing the module of Kähler d
s_conormal_module	state	Conormal module I/I²		For a surjection S ↠ S/I, the S/I-module I/I² measuring first-order deformation of the ideal I; appears in the second exact sequence of Kähler differentials.
s_kahler_differentials_polynomial_ring	theorem	Kähler differentials of a polynomial ring		The module Ω_{R[x₁,...,xₙ]/R} is the free R[x₁,...,xₙ]-module of rank n with basis dx₁,...,dxₙ.
s_kahler_differentials_and_separability	theorem	Kähler differentials and separability		A finite field extension L/K is separable if and only if Ω_{L/K} = 0; more generally dim_L Ω_{L/K} = tr.deg(L/K) for finitely generated field extensions.
s_smoothness_via_differentials	theorem	Smoothness via Kähler differentials		A finitely presented k-algebra A (k perfect) is smooth over k iff Ω_{A/k} is projective of rank equal to dim(A).
s_generic_smoothness	theorem	Generic smoothness		A dominant morphism of varieties over a perfect field is smooth on a dense open subset of the source; equivalently, the generic fiber is geometrically regular.
s_jacobian_criterion_smoothness	theorem	Jacobian criterion for smoothness		For A = k[x₁,...,xₙ]/(f₁,...,fₘ) with k perfect, A is smooth at a prime 𝔭 iff the Jacobian matrix (∂fᵢ/∂xⱼ) has rank n − dim(A_𝔭) modulo 𝔭.
s_koszul_complex_as_tensor_product	theorem	Koszul complex as tensor product		The Koszul complex K(x₁,...,xₙ; R) is isomorphic to the tensor product K(x₁;R) ⊗_R ··· ⊗_R K(xₙ;R) of one-element Koszul complexes.
s_koszul_homology	state	Koszul homology		The homology groups Hᵢ(K(x₁,...,xₙ) ⊗ M) of the Koszul complex; H₀ = M/(x₁,...,xₙ)M and higher Hᵢ detect failure of the sequence to be regular.
s_depth_via_koszul_homology	theorem	Depth via Koszul homology		For a Noetherian local ring (R, 𝔪) and ideal I = (x₁,...,xₙ), depth_I(M) = n − max{i : Hᵢ(x₁,...,xₙ; M) ≠ 0}.
s_self_duality_of_koszul_complex	theorem	Self-duality of the Koszul complex		The Koszul complex K(x₁,...,xₙ; R) is self-dual: Hom_R(Kᵢ, R) ≅ K_{n−i}, inducing isomorphisms Hᵢ(K) ≅ Hⁿ⁻ⁱ(K).
s_koszul_and_tor	theorem	Koszul homology computes Tor		If x₁,...,xₙ is a regular sequence on R, then Hᵢ(x₁,...,xₙ; M) ≅ Torᵢᴿ(R/(x₁,...,xₙ), M) for all i.
s_regular_sequence_permutation	theorem	Regular sequences are permutable in local rings		In a Noetherian local ring, any permutation of a regular sequence on a finitely generated module is again a regular sequence.
s_grade_of_an_ideal	state	Grade of an ideal		The length of a maximal regular sequence in I on M, equal to min{i : Extⁱ_R(R/I, M) ≠ 0}; agrees with depth_I(M).
s_depth_inequality	theorem	Depth inequality (depth ≤ dim)		For a finitely generated module M ≠ 0 over a Noetherian local ring (R, 𝔪), depth(M) ≤ dim(M); equality defines the Cohen-Macaulay condition.
s_cm_polynomial_ring	theorem	Polynomial rings over Cohen-Macaulay rings are Cohen-Macaulay		If R is Cohen-Macaulay, then R[x₁,...,xₙ] and R[[x₁,...,xₙ]] are also Cohen-Macaulay; in particular k[x₁,...,xₙ] is Cohen-Macaulay.
s_flatness_and_depth	theorem	Flatness and depth formula		If φ: (R, 𝔪) → (S, 𝔫) is a flat local homomorphism and M is a finitely generated S-module, then depth_S(M) = depth_R(M) + depth_{S/𝔪S}(M/𝔪M).
s_cm_implies_catenary	theorem	Cohen-Macaulay rings are universally catenary		Every Cohen-Macaulay ring is universally catenary: all maximal chains of primes between any two comparable primes have the same length.
s_cm_equidimensional	theorem	Cohen-Macaulay local rings are equidimensional		A Cohen-Macaulay local ring is equidimensional: all minimal primes have the same coheight, so all irreducible components have the same dimension.
s_minimal_free_resolution	state	Minimal free resolution		A free resolution of a module M over a local ring (R, 𝔪) where all differentials have entries in 𝔪; unique up to isomorphism with ranks given by Betti numbers.
s_uniqueness_minimal_free_resolution	theorem	Uniqueness of minimal free resolutions		Over a Noetherian local ring, the minimal free resolution of a finitely generated module is unique up to isomorphism of chain complexes.
s_betti_numbers	state	Betti numbers (algebraic)		The ranks βᵢ(M) of the free modules in a minimal free resolution of M over a local ring (R, 𝔪, k); invariants measuring the complexity of M.
s_regularity_preserved_by_localization	theorem	Regularity is preserved by localization		If R is a regular local ring and 𝔭 is a prime ideal, then R_𝔭 is also regular; follows from the Auslander-Buchsbaum-Serre characterization.
s_stably_free_is_free_over_local	theorem	Stably free modules are free over local rings		Over a local ring, every stably free module (M with M ⊕ Rᵐ ≅ Rⁿ) is free; a consequence of Nakayama's lemma.
s_ferrand_vasconcelos	theorem	Ferrand-Vasconcelos theorem		If I is an ideal in a Noetherian local ring with I/I² free over R/I and pd_R(R/I) < ∞, then I is generated by a regular sequence.
s_nagatas_lemma	theorem	Nagata's lemma		If R is a Noetherian domain, x a nonzero nonunit with (x) prime, and R[1/x] is a UFD, then R is a UFD; used to prove regular local rings are UFDs.
s_fitting_ideal	state	Fitting ideal		For a finitely presented module M with presentation matrix A, the i-th Fitting ideal Fittᵢ(M) is generated by the (n−i)-minors of A; independent of presentation
s_fitting_ideals_and_annihilator	theorem	Fitting ideals and annihilator		For a finitely presented module with n generators, Fitt₀(M) ⊆ Ann(M)ⁿ ⊆ Fitt₀(M); in particular V(Fitt₀(M)) = V(Ann(M)) = Supp(M).
s_buchsbaum_eisenbud_acyclicity	theorem	Buchsbaum-Eisenbud acyclicity criterion		A complex of free modules is acyclic iff at each step the rank condition holds and the ideal of appropriately sized minors of each differential has sufficient d
s_eagon_northcott_complex	state	Eagon-Northcott complex		A canonical free complex resolving the ideal of maximal minors of a matrix (or cokernel of a generic map), built from symmetric and exterior powers.
s_determinantal_ideal	axiom	Determinantal ideal		The ideal I_t(A) generated by all t×t minors of a matrix A with entries in a ring R; a fundamental class of ideals in commutative algebra and algebraic geometry
s_expected_codimension_determinantal	theorem	Expected codimension of determinantal ideals		For a generic m×n matrix, the ideal of t×t minors has codimension (m−t+1)(n−t+1); this is the maximum possible codimension for any determinantal ideal.
s_perfect_ideal	axiom	Perfect ideal		An ideal I such that pd_R(R/I) = grade(I); equivalently R/I has a finite free resolution of length equal to the grade of I.
s_perfect_ideals_unmixed	theorem	Perfect ideals are unmixed		A perfect ideal in a Cohen-Macaulay ring is unmixed: all its associated primes have height equal to the grade of the ideal.
s_syzygy_module	state	Syzygy module		The k-th syzygy module of M is the image of the k-th differential in a free resolution, measuring higher-order relations among generators.
s_existence_of_canonical_module	theorem	Existence of canonical module		A Cohen-Macaulay local ring that is a quotient of a Gorenstein local ring possesses a canonical module ω_R, unique up to isomorphism.
s_gorenstein_iff_canonical_is_ring	theorem	Gorenstein iff canonical module is the ring		A Cohen-Macaulay local ring possessing a canonical module is Gorenstein if and only if ω_R ≅ R, equivalently the type r(R) = 1.
s_type_of_cm_ring	state	Type of a Cohen-Macaulay ring		The type of a Cohen-Macaulay local ring (R, 𝔪, k) of dimension d, equal to dim_k Ext^d_R(k, R); Gorenstein rings are precisely the CM rings of type 1.
s_complete_intersection_ring	axiom	Complete intersection ring		A Noetherian local ring whose completion is isomorphic to a regular local ring modulo an ideal generated by a regular sequence; sits between regular and Gorenst
s_complete_intersection_is_gorenstein	theorem	Complete intersections are Gorenstein		Every complete intersection local ring is Gorenstein, giving the hierarchy: regular ⊂ complete intersection ⊂ Gorenstein ⊂ Cohen-Macaulay.
s_duality_maximal_cm_modules	theorem	Duality for maximal Cohen-Macaulay modules		Over a Gorenstein local ring R, the functor Hom_R(−, R) gives a duality (contravariant self-equivalence) on the category of maximal Cohen-Macaulay modules.
s_maximal_cohen_macaulay_module	axiom	Maximal Cohen-Macaulay module		A finitely generated module M over a Cohen-Macaulay local ring (R, 𝔪) with depth(M) = dim(R), the maximum possible value.
s_bass_numbers	state	Bass numbers		Invariants measuring the multiplicity of the indecomposable injective E(R/𝔭) in the i-th term of a minimal injective resolution of M.
s_structure_of_injective_modules_noetherian	theorem	Structure of injective modules over Noetherian rings		Every injective module over a Noetherian ring decomposes uniquely as a direct sum of indecomposable injectives E(R/𝔭) for various primes 𝔭, with multiplicities 
s_poincare_duality_gorenstein	theorem	Poincaré duality for Gorenstein rings		For a Gorenstein local ring of dimension d, H^i_𝔪(R) = 0 for i ≠ d and H^d_𝔪(R) ≅ E(k), giving Poincaré-type duality via local cohomology.
s_local_cohomology_via_cech	theorem	Local cohomology via Čech complex		For an ideal I = (f₁,...,fᵣ), local cohomology H^i_I(M) is computed as the cohomology of the Čech complex built from localizations M_{f_{j₁}···f_{jₖ}}.
s_vanishing_local_cohomology	theorem	Vanishing of local cohomology		H^i_I(M) = 0 for i < depth_I(M) and for i > dim(M); the first nonvanishing degree equals the depth and the last equals the dimension.
s_grothendieck_vanishing	theorem	Grothendieck vanishing theorem		For a Noetherian local ring (R, 𝔪) and finitely generated M, H^i_𝔪(M) = 0 for all i > dim(M).
s_depth_via_local_cohomology	theorem	Depth via local cohomology		For a finitely generated module M over a Noetherian ring, depth_I(M) = min{i : H^i_I(M) ≠ 0}, characterizing depth cohomologically.
s_local_cohomology_sheaf_cohomology	theorem	Local cohomology and sheaf cohomology comparison		For X = Proj(S) and a graded S-module M, there is a long exact sequence relating H^i_𝔪(M), the module M, and sheaf cohomology H^i(X, M̃(n)).
s_divided_power_algebra	axiom	Divided power algebra		The algebra with divided power operations γₙ satisfying γₙ(x)γₘ(x) = C(n+m,n)γ_{n+m}(x); used in constructing Eagon-Northcott and Buchsbaum-Rim complexes.
s_schur_functor	axiom	Schur functor		The functor L_λ associating to a free module F the GL(F)-representation indexed by partition λ, generalizing symmetric and exterior powers.
s_buchsbaum_rim_complex	state	Buchsbaum-Rim complex		A generalization of the Eagon-Northcott complex providing a free resolution of the cokernel of a map of free modules when the Fitting ideal has expected depth.
s_mapping_cone	state	Mapping cone		For a chain map f: C → D, the complex cone(f) with cone(f)ₙ = Cₙ₋₁ ⊕ Dₙ fitting into the exact triangle C → D → cone(f) → C[−1].
s_spectral_sequence_filtered_complex	state	Spectral sequence of a filtered complex		A filtered chain complex gives rise to a spectral sequence converging to the homology of the total complex, with E₀ page determined by the associated graded.
s_spectral_sequence_double_complex	theorem	Spectral sequences of a double complex		A double complex has two spectral sequences from row and column filtrations, both converging to H*(Tot(C)); the E₂ pages are iterated homology in the two direct
s_mittag_leffler_condition	axiom	Mittag-Leffler condition		An inverse system {Aₙ} satisfies the Mittag-Leffler condition if for each n the images im(Aₘ → Aₙ) stabilize for large m; ensures lim¹ = 0 and exactness of inve
s_p_basis	state	p-basis of a field		A set {bᵢ} in a field K of characteristic p such that {bᵢ^{eᵢ} : 0 ≤ eᵢ < p} form a basis of K over K^p; measures the degree of imperfection of K.
s_auslander_depth_formula	theorem	Auslander depth formula		For finitely generated modules M, N over a regular local ring R with Torᵢ(M,N) = 0 for i > 0: depth(M) + depth(N) = depth(R) + depth(M ⊗ N).
s_formally_smooth_algebra	axiom	Formally smooth algebra	smooth algebra | formally smooth morphism	An R-algebra S such that for every R-algebra A and nilpotent ideal N of A, every R-homomorphism S → A/N lifts to S → A.
s_formally_unramified_algebra	axiom	Formally unramified algebra	unramified algebra	An R-algebra S with Ω_{S/R} = 0, equivalently liftings in the formal smoothness definition are unique when they exist.
s_formally_etale_algebra	axiom	Formally étale algebra	étale algebra | formally étale morphism	An R-algebra S that is both formally smooth and formally unramified, so liftings of homomorphisms past nilpotent ideals exist and are unique.
s_g_ring	axiom	G-ring (Grothendieck ring)	Grothendieck ring | G-ring	A Noetherian ring R such that for every prime p, the completion map R_p → R̂_p is a regular homomorphism, i.e., has geometrically regular formal fibres.
s_quasi_excellent_ring	axiom	Quasi-excellent ring	quasi-excellent	A Noetherian G-ring R such that for every finitely generated R-algebra S, the singular locus of Spec(S) is closed.
s_n_1_ring	axiom	N-1 ring	N-1 domain	A Noetherian integral domain whose integral closure in its own fraction field is a finite module over itself.
s_buchsbaum_ring	axiom	Buchsbaum ring	Buchsbaum local ring	A Noetherian local ring (R, m) such that the difference ℓ(R/(x₁,…,x_d)) − e(x₁,…,x_d; R) is independent of the choice of system of parameters.
s_geometrically_regular_formal_fibres	axiom	Geometrically regular formal fibres	regular formal fibres	The condition on a Noetherian local ring (R, m) that all formal fibres R̂ ⊗_R κ(p) are geometrically regular algebras over κ(p) for every prime p.
s_attached_primes	axiom	Attached primes	attached prime ideals | Att(M)	For an Artinian module M over a Noetherian ring, Att(M) = {p ∈ Spec(R) : p = Ann(M/N) for some submodule N}, the dual notion to associated primes.
s_dualizing_complex	axiom	Dualizing complex	residual complex	A bounded complex D• of injective modules with finitely generated cohomology such that the natural map R → RHom_R(D•, D•) is a quasi-isomorphism; exists for quo
s_singular_locus	state	Singular locus	non-regular locus | Sing(R)	Sing(R) = {p ∈ Spec(R) : R_p is not a regular local ring}, the subset of primes where a Noetherian ring fails to be regular.
s_formal_fibre	state	Formal fibre	formal fiber	For a Noetherian local ring (R, m) and prime p, the formal fibre at p is Spec(R̂ ⊗_R κ(p)), the fibre of the completion morphism Spec(R̂) → Spec(R) over p.
s_henselization	state	Henselization	Henselization of a local ring | R^h	The Henselization R^h of a local ring (R, m) is the smallest Henselian local ring through which R → R̂ factors, constructed as a filtered colimit of étale neigh
s_strict_henselization	state	Strict Henselization	strict Henselization | R^{sh}	The strict Henselization R^{sh} of a local ring (R, m) is the Henselization with respect to the separable closure of the residue field, the maximal unramified e
s_local_criterion_for_flatness	theorem	Local criterion for flatness	local flatness criterion	For a local homomorphism (R, m) → (S, n) of Noetherian local rings, S is R-flat iff Tor₁^R(R/m, S) = 0, iff the natural map m ⊗_R S → mS is an isomorphism.
s_equational_criterion_for_flatness	theorem	Equational criterion for flatness	equational flatness criterion | Lazard criterion	An R-module M is flat iff every linear relation ∑aᵢmᵢ = 0 can be trivialized: there exist nⱼ ∈ M and bᵢⱼ ∈ R with mᵢ = ∑bᵢⱼnⱼ and ∑aᵢbᵢⱼ = 0 for all j.
s_generic_flatness_theorem	theorem	Generic flatness theorem	generic freeness | generic flatness lemma	If R is a Noetherian domain and M a finitely generated R-module (or R-algebra), there exists nonzero f ∈ R such that M_f is free over R_f.
s_grothendieck_non_vanishing_theorem	theorem	Grothendieck's non-vanishing theorem	non-vanishing of local cohomology at dimension	For a Noetherian local ring (R, m) and finitely generated module M, H^d_m(M) ≠ 0 where d = dim(M), so local cohomology does not vanish at the dimension.
s_hochster_big_cm_modules	theorem	Hochster's theorem on big Cohen-Macaulay modules	existence of big CM modules | Hochster's big CM theorem	Every Noetherian local ring containing a field admits a (possibly non-finitely-generated) balanced big Cohen-Macaulay module on which every system of parameters
s_openness_of_regular_locus	theorem	Openness of the regular locus	regular locus is open	In an excellent ring (or a ring with geometrically regular formal fibres), the set {p ∈ Spec(R) : R_p is regular} is Zariski-open.
s_fibre_dimension_formula_flat	theorem	Fibre dimension formula for flat local homomorphisms	dimension formula for flat extensions	For a flat local homomorphism φ: (R, m) → (S, n) of Noetherian local rings, dim(S) = dim(R) + dim(S/mS), relating source, target, and fibre dimensions.
s_finitistic_dimension_theorem	theorem	Finitistic dimension theorem	finitistic projective dimension equals depth	For a commutative Noetherian local ring (R, m), the finitistic projective dimension (supremum of pd(M) over modules M with pd(M) < ∞) equals depth(R).
s_cohen_structure_theorem_mixed_char	theorem	Cohen structure theorem (mixed characteristic)	Cohen's structure theorem for mixed characteristic	A complete Noetherian local ring (R, m) of mixed characteristic p with residue field k is a quotient of V[[x₁, …, xₙ]] where V is a complete DVR with residue fi
s_power_series_ring_is_local	theorem	Power series ring over a local ring is local		If (R, m) is a local ring, then R[[x₁, …, xₙ]] is a local ring with maximal ideal (m, x₁, …, xₙ).
s_power_series_ring_faithfully_flat	theorem	Power series ring is faithfully flat		The natural inclusion R → R[[x]] is a faithfully flat ring homomorphism.
s_grothendieck_formal_functions	theorem	Grothendieck's theorem on formal functions	theorem on formal functions	Relates the completion of higher direct image sheaves to the cohomology of formal completions along fibres, connecting local and global cohomology in algebraic 
t_cohen_macaulay_approximation	technique	Cohen-Macaulay approximation	Auslander-Buchweitz approximation	Approximating modules over Cohen-Macaulay rings by maximal Cohen-Macaulay modules via short exact sequences, introduced by Auslander-Buchweitz.
t_secondary_representation	technique	Secondary representation	secondary decomposition	Decomposing an Artinian module as a sum of secondary submodules, dual to primary decomposition for Noetherian modules.
s_quasi_isomorphism	axiom	Quasi-isomorphism	quasi-iso | quasi isomorphism | homology isomorphism	A chain map f: C• → D• that induces isomorphisms H_n(f): H_n(C) → H_n(D) on all homology groups.
s_mapping_cylinder_chain	state	Mapping cylinder (chain complexes)	mapping cylinder of a chain map	For a chain map f: C• → D•, the chain complex cyl(f)_n = C_n ⊕ C_{n-1} ⊕ D_n with differential encoding f and the identity, fitting in a short exact sequence wi
s_split_exact_sequence	axiom	Split exact sequence	split short exact sequence | splitting	A short exact sequence 0 → A → B → C → 0 that admits a section s: C → B (right split) or retraction r: B → A (left split), equivalently B ≅ A ⊕ C.
s_homotopy_equivalence_complexes	state	Homotopy equivalence of chain complexes	chain homotopy equivalence	A chain map f: C• → D• such that there exists g: D• → C• with gf and fg both chain homotopic to the respective identities.
s_erasable_functor	axiom	Erasable (effaceable/coeffaceable) functor	erasable functor | effaceable functor definition	A functor F is effaceable if for every object A there is a monomorphism u: A → M with F(u) = 0; coeffaceable if dually there is an epimorphism with the same pro
s_flat_resolution	state	Flat resolution	flat resolution of a module	An exact sequence ··· → F_1 → F_0 → M → 0 where each F_i is a flat module, used to compute Tor.
s_weak_global_dimension	axiom	Weak (global) dimension	weak dimension | Tor-dimension | w.gl.dim	The supremum of the flat dimensions of all modules over a ring R, equivalently the supremum of n such that Tor_n^R ≠ 0.
s_yoneda_ext	state	Yoneda Ext	Yoneda extension group | Yoneda product	The group Ext^n(A, B) defined as equivalence classes of n-fold extensions of A by B under the Baer sum, isomorphic to the derived functor Ext^n_R(A, B).
s_baer_sum	state	Baer sum of extensions	Baer addition	The group operation on Ext^1(A, B) defined by pulling back the direct sum extension along the diagonal and pushing out along the fold map, making Ext^1(A, B) an
s_lim1_derived_inverse_limit	state	lim¹ (first derived functor of inverse limit)	lim^1 | R^1 lim | derived inverse limit	The first right derived functor R¹lim of the inverse limit functor on inverse systems of abelian groups, measuring the obstruction to surjectivity of the invers
s_auslander_buchsbaum_serre_theorem	theorem	Auslander–Buchsbaum–Serre theorem	regularity and global dimension theorem	A Noetherian local ring (R, m) is regular if and only if it has finite global dimension, in which case gl.dim(R) = dim(R) = the minimal number of generators of 
s_grade_of_module	axiom	Grade of a module	j-number | grade	For a finitely generated module M over a Noetherian ring R, grade(M) = inf{i : Ext^i_R(M, R) ≠ 0}, the smallest degree in which Ext into R is nonzero.
s_derived_couple	state	Derived couple	derived exact couple	Given an exact couple (A, E, i, j, k), the derived couple (A', E', i', j', k') where A' = im(i), E' = H(E, d) with d = jk, producing the next page of the associ
s_transgression_spectral_sequence	state	Transgression	transgression map | transgressive element	In a first-quadrant spectral sequence, the partially defined map d_n: E_n^{0,n-1} → E_n^{n,0} connecting the fiber and base terms, arising as the last possible 
s_cartan_eilenberg_resolution	state	Cartan–Eilenberg resolution	CE resolution	An injective (or projective) resolution of a chain complex C• that is compatible with homology: a double complex P•• whose columns are resolutions of C_n and wh
s_hyperhomology	state	Hyperhomology	hyper-homology | left hyper-derived functor | hypertor	The left hyper-derived functors 𝕃_nF(C•) of an additive functor F applied to a chain complex C•, computed via a Cartan–Eilenberg resolution as the homology of t
s_hypercohomology	state	Hypercohomology	hyper-cohomology | right hyper-derived functor | hyperext	The right hyper-derived functors ℝⁿF(C•) of a left exact functor F applied to a cochain complex, generalizing sheaf cohomology to complexes of sheaves.
s_acyclic_assembly_lemma	theorem	Acyclic assembly lemma		If a first-quadrant double complex has exact columns (resp. rows) except in degree zero, then the natural map from the zeroth column (resp. row) to the total co
s_group_invariants	axiom	G-invariants M^G	invariants | fixed points of G-module | M^G	For a G-module M, the subgroup M^G = {m ∈ M : gm = m for all g ∈ G} of elements fixed by all group elements, a left exact functor whose right derived functors a
s_coinvariants	axiom	G-coinvariants M_G	coinvariants | module of coinvariants | M_G	For a G-module M, the quotient M_G = M/I_G·M where I_G is the augmentation ideal, a right exact functor whose left derived functors are group homology.
s_normalized_bar_resolution	state	Normalized bar resolution	normalized standard resolution | reduced bar resolution	The quotient of the bar resolution B•(G) by the subcomplex of degenerate simplices, a smaller free ℤG-resolution of ℤ with B̄_n generated by [g₁|…|gₙ] with all 
s_five_term_exact_sequence_group	theorem	Five-term exact sequence (group cohomology)	inflation-restriction exact sequence | five-term exact sequence	For a group extension 1 → N → G → Q → 1, the exact sequence 0 → H¹(Q, Mᴺ) → H¹(G, M) → H¹(N, M)^Q → H²(Q, Mᴺ) → H²(G, M) obtained from the Lyndon–Hochschild–Ser
s_principal_crossed_homomorphism	state	Principal crossed homomorphism	principal derivation | inner derivation (group cohomology)	A crossed homomorphism f: G → M of the form f(g) = gm − m for some fixed m ∈ M; the group of principal crossed homomorphisms is B¹(G, M), so H¹(G, M) = Z¹/B¹.
s_factor_set	axiom	Factor set (group extension)	2-cocycle for group extension | factor system	A function f: G × G → A satisfying the 2-cocycle condition f(g,h) + f(gh,k) = g·f(h,k) + f(g,hk), classifying extensions of G by A up to equivalence via H²(G, A
s_tate_cohomology	axiom	Tate cohomology	Tate cohomology groups | Ĥ*(G,M) | complete cohomology	For a finite group G and G-module M, the cohomology theory Ĥⁿ(G, M) defined for all n ∈ ℤ using a complete resolution, with Ĥ⁰ = M^G/N_G·M and Ĥ⁻¹ = ker(N_G)/I_
s_complete_resolution	state	Complete (Tate) resolution	Tate resolution | complete projective resolution	An acyclic complex T• of projective ℤG-modules extending the ordinary projective resolution in both directions, with Hom_G(T•, M) computing Tate cohomology.
s_norm_map_group	state	Norm map (group cohomology)	norm element | norm homomorphism	For a finite group G and G-module M, the map N_G: M → M defined by N_G(m) = Σ_{g∈G} gm, whose kernel and cokernel appear in Tate cohomology.
s_cup_product_group_cohomology	state	Cup product in group cohomology	cup product on H*(G,M)	The associative graded-commutative product ∪: Hⁿ(G, M) ⊗ Hᵐ(G, N) → Hⁿ⁺ᵐ(G, M ⊗ N) induced by the diagonal approximation on the bar resolution.
s_cohomological_dimension_group	axiom	Cohomological dimension of a group	cd(G) | cohomological dimension	The supremum cd(G) of all n such that Hⁿ(G, M) ≠ 0 for some G-module M, equivalently the projective dimension of ℤ as a ℤG-module.
s_group_type_fp	axiom	Group of type FP	type FL | type FP_n | finiteness property of groups	A group G is of type FP_n if ℤ admits a projective ℤG-resolution with all modules finitely generated in degrees ≤ n; type FP if finitely generated in all degree
s_lie_algebra_homology	axiom	Lie algebra homology H_n(𝔤, M)	Lie algebra homology | H_n(g,M)	The left derived functors of the coinvariants functor M ↦ M_𝔤 = M/𝔤·M for a Lie algebra 𝔤 over k with coefficients in a 𝔤-module M, computed via the Chevalley–E
s_lie_algebra_cohomology	axiom	Lie algebra cohomology H^n(𝔤, M)	Lie algebra cohomology | H^n(g,M)	The right derived functors of the invariants functor M ↦ M^𝔤 = {m ∈ M : x·m = 0 ∀x ∈ 𝔤} for a Lie algebra 𝔤 with coefficients in a 𝔤-module M.
s_chevalley_eilenberg_complex	state	Chevalley–Eilenberg complex	CE complex | Koszul–Chevalley–Eilenberg complex	The chain complex C_n(𝔤, M) = M ⊗ Λⁿ(𝔤) with differential involving the Lie bracket and the module action, whose homology is the Lie algebra homology H_n(𝔤, M).
s_casimir_element	state	Casimir element	Casimir operator | Casimir invariant	For a semisimple Lie algebra 𝔤 with Killing form κ, the element c = Σ xᵢyᵢ ∈ U(𝔤) where {xᵢ} and {yᵢ} are dual bases; c lies in the center of U(𝔤) and acts on i
s_relative_lie_algebra_cohomology	state	Relative Lie algebra cohomology H^n(𝔤, 𝔨; M)	relative cohomology of Lie pair	For a Lie subalgebra 𝔨 ⊂ 𝔤, the cohomology of the subcomplex of 𝔨-basic elements in the Chevalley–Eilenberg complex, used in the theory of (𝔤, K)-modules.
s_simplicial_object	axiom	Simplicial object	simplicial object in a category	A contravariant functor X: Δᵒᵖ → C from the simplex category Δ to a category C, specified by objects X_n and face maps dᵢ, degeneracy maps sⱼ satisfying the sim
s_cosimplicial_object	axiom	Cosimplicial object	cosimplicial object in a category	A covariant functor X: Δ → C from the simplex category to a category C, specified by objects X^n with coface and codegeneracy maps satisfying the cosimplicial i
s_simplicial_set	axiom	Simplicial set	simplicial set | semi-simplicial set	A simplicial object in the category of sets, i.e., a contravariant functor Δᵒᵖ → Set, providing a combinatorial model for topological spaces up to weak homotopy
s_face_map_simplicial	axiom	Face map (simplicial)	face operator | face morphism	In a simplicial object X•, the map dᵢ: X_n → X_{n-1} for 0 ≤ i ≤ n, corresponding to the injection [n-1] → [n] that skips i, satisfying dᵢdⱼ = dⱼ₋₁dᵢ for i < j.
s_degeneracy_map_simplicial	axiom	Degeneracy map (simplicial)	degeneracy operator	In a simplicial object X•, the map sⱼ: X_n → X_{n+1} for 0 ≤ j ≤ n, corresponding to the surjection [n+1] → [n] that repeats j, satisfying simplicial identities
s_moore_complex	state	Moore complex	Moore complex of a simplicial group	For a simplicial abelian group A•, the chain complex (A_n, ∂) with ∂ = Σᵢ(-1)ⁱdᵢ, whose homology equals the homotopy groups π_n(A•); the unnormalized version of
s_normalized_chain_complex_simplicial	state	Normalized chain complex (simplicial)	normalized Moore complex | N(A)	For a simplicial abelian group A•, the subcomplex N_n = ∩ᵢ₌₁ⁿ ker(dᵢ) ⊂ A_n with differential d₀, quasi-isomorphic to the Moore complex and corresponding to A• 
s_dold_kan_correspondence	theorem	Dold–Kan correspondence	Dold–Kan theorem | Dold–Kan equivalence | Dold–Puppe theorem	The normalized chain complex functor N: sAb → Ch≥0 is an equivalence of categories between simplicial abelian groups and non-negative chain complexes, with inve
s_alexander_whitney_map	state	Alexander–Whitney map	AW map | Alexander–Whitney diagonal	The natural chain map AW: C(X × Y) → C(X) ⊗ C(Y) defined by AW(σ)= Σ (front-p-face of σ) ⊗ (back-q-face of σ), a chain homotopy inverse to the Eilenberg–Zilber 
s_shuffle_map	state	Shuffle map (Eilenberg–Zilber)	Eilenberg–Zilber map | shuffle product | ∇	The natural chain map ∇: C(X) ⊗ C(Y) → C(X × Y) defined using (p,q)-shuffles, providing a chain homotopy inverse to the Alexander–Whitney map.
s_eilenberg_maclane_space	axiom	Eilenberg–MacLane space K(π, n)	K(π,n) | Eilenberg–MacLane space	A CW complex K(π, n) with π_n(K(π,n)) ≅ π and π_k(K(π,n)) = 0 for k ≠ n, unique up to homotopy equivalence, representing singular cohomology H^n(−; π) as a func
s_comonad	axiom	Comonad (cotriple)	cotriple | comonad	A comonoid in the category of endofunctors: a functor G: C → C with natural transformations δ: G → G² (comultiplication) and ε: G → Id (counit) satisfying coass
s_cotriple_resolution	state	Cotriple resolution	comonadic resolution | comonad resolution	For a comonad G on C and object X, the augmented simplicial object ··· ⇉ G²X ⇉ GX → X with face maps using ε and degeneracies using δ, providing canonical resol
s_andre_quillen_homology	state	André–Quillen homology	AQ homology | cotangent homology	The homology theory D_n(A/k, M) for commutative k-algebras A with coefficients in an A-module M, defined as the left derived functors of derivations and compute
s_cotangent_complex	state	Cotangent complex	L_{A/k} | Illusie cotangent complex	The complex L_{A/k} = Ω_{P•/k} ⊗_{P•} A in the derived category, where P• → A is a simplicial resolution; its homology gives André–Quillen homology D_n(A/k, A) 
s_bar_construction	state	Bar construction	bar construction B(T,A) | simplicial bar construction	For a monad or triple T on C and a T-algebra A, the simplicial object B•(T, A) with B_n = T^{n+1}A, providing a canonical free/projective simplicial resolution 
s_kan_complex	axiom	Kan complex	Kan fibrant simplicial set | fibrant simplicial set	A simplicial set satisfying the Kan extension condition: every horn Λⁿₖ → X has a filler Δⁿ → X, providing the combinatorial analogue of a topological space wit
s_hochschild_homology	axiom	Hochschild homology HH_n(A, M)	Hochschild homology | HH_*(A,M)	For an associative k-algebra A and A-bimodule M, the homology HH_n(A, M) = Tor_n^{A^e}(A, M) of the Hochschild complex, where A^e = A ⊗ Aᵒᵖ is the enveloping al
s_gerstenhaber_bracket	state	Gerstenhaber bracket	Gerstenhaber algebra structure | Gerstenhaber Lie bracket	The degree −1 Lie bracket [−,−] on Hochschild cohomology HH*(A,A) making it a Gerstenhaber algebra: a graded commutative ring under cup product with a compatibl
s_formal_deformation_algebra	axiom	Formal deformation of an algebra	infinitesimal deformation | deformation quantization	A k[[t]]-algebra A_t = A[[t]] with product a ∗ b = ab + μ₁(a,b)t + μ₂(a,b)t² + ··· deforming A, where associativity of ∗ is controlled by Hochschild cohomology 
s_cyclic_homology	axiom	Cyclic homology HC_n(A)	cyclic homology | HC_*(A)	For a k-algebra A, the homology of the total complex of Connes' cyclic bicomplex (or equivalently of the mixed complex (C•(A), b, B)), fitting in the SBI exact 
s_mixed_complex	axiom	Mixed complex	mixed complex (b, B)	A graded module C• with two differentials b: C_n → C_{n-1} of degree −1 and B: C_n → C_{n+1} of degree +1 satisfying b² = B² = bB + Bb = 0, encoding the data fo
s_connes_b_operator	state	Connes' B operator	Connes operator | Connes boundary B	The degree +1 map B: C_n(A) → C_{n+1}(A) on the Hochschild complex defined by B = (1 − t)sN where s is the extra degeneracy, t the cyclic operator, and N = 1 + 
s_sbi_exact_sequence	theorem	SBI exact sequence (Connes)	Connes' exact sequence | Connes' SBI sequence | periodicity exact sequence	The long exact sequence ··· → HH_n(A) →^I HC_n(A) →^S HC_{n-2}(A) →^B HH_{n-1}(A) → ··· relating Hochschild and cyclic homology via the periodicity operator S a
s_cyclic_bicomplex	state	Cyclic bicomplex	Connes' bicomplex | CC(A)	Connes' bicomplex CC•• with columns alternating between the Hochschild complex and its b'-complex, connected by the norm N and extra degeneracy maps, whose tota
s_periodic_cyclic_homology	state	Periodic cyclic homology HP_n(A)	periodic cyclic homology | HP_*(A)	The ℤ/2-graded theory HP_*(A) = lim_S HC_{*+2k}(A) obtained as the inverse limit over the periodicity operator S, satisfying HP_n(A) ≅ HP_{n+2}(A) and related t
s_negative_cyclic_homology	state	Negative cyclic homology HN_n(A)	negative cyclic homology | HN_*(A)	The homology HN_*(A) of the product total complex of the cyclic bicomplex, fitting in an exact sequence with HC_* and HP_* and receiving the Dennis trace from a
s_hkr_theorem	theorem	Hochschild–Kostant–Rosenberg theorem	HKR theorem | HKR isomorphism	For a smooth commutative k-algebra A, HH_n(A) ≅ Ω^n_{A/k}, identifying Hochschild homology with differential forms; under this isomorphism, Connes' B operator c
s_loday_quillen_tsygan_theorem	theorem	Loday–Quillen–Tsygan theorem	LQT theorem	The Lie algebra homology of gl(A) = lim gl_n(A) satisfies H_*(gl(A); k) ≅ Λ*(HC_{*-1}(A)) as a graded Hopf algebra, relating cyclic homology to the homology of 
s_dennis_trace_map	state	Dennis trace map	Dennis trace	The natural map K_n(A) → HH_n(A) from algebraic K-theory to Hochschild homology, factoring through negative cyclic homology: K_n(A) → HN_n(A) → HH_n(A).
s_morita_invariance_hochschild	theorem	Morita invariance of Hochschild and cyclic homology	Morita invariance of HH and HC	If A and B are Morita equivalent rings, then HH_*(A) ≅ HH_*(B) and HC_*(A) ≅ HC_*(B); in particular HH_*(M_r(A)) ≅ HH_*(A) for matrix algebras.
s_homotopy_category_chain	axiom	Homotopy category K(A)	homotopy category of chain complexes | K(A)	The category K(A) with the same objects as Ch(A) but morphisms modulo chain homotopy equivalence, carrying a natural triangulated structure with shift [1] and d
s_distinguished_triangle	axiom	Distinguished triangle	exact triangle | distinguished triangle in triangulated category	In a triangulated category, a triangle X → Y → Z → X[1] isomorphic to a standard triangle arising from a morphism f as X →^f Y → cone(f) → X[1].
s_octahedral_axiom	axiom	Octahedral axiom (TR4)	TR4 | octahedral axiom	Axiom TR4 for triangulated categories: given composable morphisms f: X → Y and g: Y → Z with gf: X → Z, the cones of f, g, and gf fit into a distinguished trian
s_verdier_quotient	state	Verdier quotient	Verdier localization | localization of triangulated category	For a triangulated category T and thick subcategory N, the localization T/N obtained by formally inverting all morphisms whose cone lies in N, yielding a triang
s_ore_conditions	axiom	Ore conditions (calculus of fractions)	calculus of fractions | left Ore condition | right Ore condition	Conditions on a class S of morphisms in a category C ensuring that the localization C[S⁻¹] can be computed by left or right fractions s⁻¹f, without requiring a 
s_derived_category	axiom	Derived category D(A)	derived category | D(A) | D^b(A) | bounded derived category	The localization D(A) = K(A)[quasi-iso⁻¹] of the homotopy category of chain complexes over an abelian category A at the class of quasi-isomorphisms, the natural
s_bounded_derived_category	axiom	Bounded derived category D^b(A)	D^b | D^+(A) | D^-(A) | bounded above/below derived category	The full triangulated subcategory D^b(A) ⊂ D(A) of complexes with bounded cohomology, equivalently the localization of K^b(A) at quasi-isomorphisms; similarly D
s_total_left_derived_functor	state	Total left derived functor LF	LF | left hyper-derived functor | 𝕃F	For a right exact functor F: A → B, the total left derived functor LF: D^-(A) → D^-(B) defined by LF(C•) = F(P•) where P• → C• is a projective resolution in the
s_total_right_derived_functor	state	Total right derived functor RF	RF | right hyper-derived functor | ℝF	For a left exact functor F: A → B, the total right derived functor RF: D^+(A) → D^+(B) defined by RF(C•) = F(I•) where C• → I• is an injective resolution in the
s_rhom_complex	state	RHom complex	RHom | derived Hom | ℝHom	The total right derived functor RHom(A, B) of Hom in the derived category, with H^n(RHom(A,B)) = Ext^n(A,B), providing a derived-category refinement of Ext grou
s_derived_tensor_product	state	Derived tensor product ⊗^L	⊗^L | derived tensor | left derived tensor product	The total left derived functor A ⊗^L_R B of the tensor product in the derived category, with H_n(A ⊗^L B) = Tor_n(A,B), computed by tensoring with a projective/
s_way_out_functor	axiom	Way-out functor	way-out right | way-out left	A triangulated functor F: D^+(A) → D^+(B) is way-out right if for every n there exists p such that H^i(F(C•)) = 0 for i > n whenever H^j(C•) = 0 for j > p; way-
s_way_out_lemma	theorem	Way-out lemma (Hartshorne)	Hartshorne's way-out lemma	If η: F → G is a natural transformation of way-out right (resp. left) functors that is an isomorphism on D^b, then η is an isomorphism on D^+ (resp. D^-); used 
s_t_structure	axiom	t-structure on a triangulated category	t-structure | truncation structure	A pair (D^≤0, D^≥0) of full subcategories of a triangulated category D satisfying: D^≤0[1] ⊂ D^≤0, D^≥0[-1] ⊂ D^≥0, Hom(D^≤0, D^≥1) = 0, and every object fits i
s_heart_of_t_structure	state	Heart of a t-structure	core of a t-structure	The full subcategory C = D^≤0 ∩ D^≥0 of a t-structure (D^≤0, D^≥0), which is always an abelian category; for the standard t-structure on D(A), the heart is equi
s_truncation_functors	state	Truncation functors τ≤n and τ≥n	truncation | τ≤n | τ≥n | canonical truncation	The functors τ≤n: D → D^≤n and τ≥n: D → D^≥n associated to a t-structure, fitting in distinguished triangles τ≤n(X) → X → τ≥n+1(X) → (τ≤n(X))[1] for every objec
s_grothendieck_ab_axioms	axiom	Grothendieck's AB axioms	AB3 | AB4 | AB5 | AB3* | AB4* | AB5* | Grothendieck category axioms	A hierarchy of axioms for abelian categories: AB3 (coproducts exist), AB4 (coproducts are exact), AB5 (filtered colimits are exact), and their duals AB3*, AB4*,
t_chevalley_eilenberg_complex_computation	technique	Chevalley–Eilenberg complex computation		Computes Lie algebra homology and cohomology by constructing the Chevalley–Eilenberg complex C_n = M ⊗ Λⁿ𝔤 and taking homology/cohomology.
t_cartan_eilenberg_resolution	technique	Cartan–Eilenberg resolution construction		Constructs a double complex of projective/injective objects resolving each term and homology of a chain complex, enabling computation of hyper-derived functors 
t_dold_kan_correspondence	technique	Dold–Kan correspondence		Establishes an equivalence between simplicial abelian groups and non-negative chain complexes via the normalized chain complex functor, translating simplicial h
t_spectral_sequence_of_double_complex	technique	Spectral sequence of a double complex		Produces two spectral sequences from the row and column filtrations of a double complex, both converging to the homology of the total complex under suitable bou
t_connes_sbi_computation	technique	SBI exact sequence computation		Uses the long exact sequence ··· → HH_n →^I HC_n →^S HC_{n-2} →^B HH_{n-1} → ··· to compute cyclic homology from Hochschild homology iteratively.
t_derived_category_localization	technique	Derived category construction (localization)		Constructs the derived category D(A) by localizing the homotopy category K(A) at quasi-isomorphisms using calculus of fractions, enabling derived functors as ex
t_cotriple_resolution_technique	technique	Cotriple (comonadic) resolution		Constructs canonical simplicial resolutions using the iterates of a comonad, yielding acyclic resolutions suitable for computing derived functors in non-abelian
t_tate_cohomology_computation	technique	Tate cohomology computation		Computes Tate cohomology by splicing a projective resolution with its dual to form a complete resolution, extending group cohomology to negative degrees.
t_hyperhomology_computation	technique	Hyperhomology/hypercohomology computation		Applies a derived functor to a chain complex via Cartan–Eilenberg resolutions or using the two spectral sequences of the resulting double complex.
s_differential_in_chain_complex	axiom	Differential	boundary map | boundary operator	The differential d_n: C_n -> C_{n-1} in a chain complex is the morphism satisfying d_{n-1} composed with d_n = 0.
s_cycles_of_chain_complex	state	Cycles	n-cycles | cycle group	The module of n-cycles of a chain complex C is Z_n(C) = ker(d_n: C_n -> C_{n-1}), the kernel of the n-th differential.
s_boundaries_of_chain_complex	state	Boundaries	n-boundaries | boundary group	The module of n-boundaries of a chain complex C is B_n(C) = im(d_{n+1}: C_{n+1} -> C_n), the image of the (n+1)-th differential.
s_cocycles_of_cochain_complex	state	Cocycles	n-cocycles	The module of n-cocycles of a cochain complex C is Z^n(C) = ker(d^n: C^n -> C^{n+1}).
s_coboundaries_of_cochain_complex	state	Coboundaries	n-coboundaries	The module of n-coboundaries of a cochain complex C is B^n(C) = im(d^{n-1}: C^{n-1} -> C^n).
s_bounded_chain_complex	axiom	Bounded chain complex	bounded complex | bounded below complex | bounded above complex	A chain complex is bounded below if C_n = 0 for n sufficiently negative, bounded above if C_n = 0 for n sufficiently positive, and bounded if both conditions ho
s_chain_homotopy	axiom	Chain homotopy	homotopy of chain maps	A chain homotopy between chain maps f, g: C -> D is a family of morphisms s_n: C_n -> D_{n+1} such that f_n - g_n = d_{n+1}^D s_n + s_{n-1} d_n^C for all n.
s_chain_homotopy_implies_quasi_isomorphism	theorem	Chain homotopy equivalence implies quasi-isomorphism		Every chain homotopy equivalence is a quasi-isomorphism; chain homotopic maps induce identical maps on homology.
s_naturality_of_long_exact_sequence	theorem	Naturality of the long exact sequence in homology		The long exact sequence in homology is natural: a morphism of short exact sequences of chain complexes induces a morphism of the corresponding long exact sequen
s_mapping_cylinder	axiom	Mapping cylinder		The mapping cylinder cyl(f) of a chain map f: C -> D is the chain complex with cyl(f)_n = C_{n-1} + C_n + D_n, fitting into a short exact sequence 0 -> D -> cyl
s_mapping_cone_exact_triangle	theorem	Mapping cone exact triangle		For a chain map f: C -> D, there is a short exact sequence 0 -> D -> cone(f) -> C[-1] -> 0, and the resulting long exact sequence recovers the connecting homomo
s_translation_shift_functor	axiom	Translation (shift) functor	shift functor | suspension functor | degree shift	The shift functor [p] sends a chain complex C to C[p] defined by C[p]_n = C_{n+p} with differential (-1)^p d_{n+p}.
s_kernel_in_abelian_category	axiom	Kernel (abelian category)		The kernel of a morphism f: A -> B in an abelian category is the equalizer of f and the zero map, i.e., a universal morphism i: K -> A such that f i = 0.
s_cokernel_in_abelian_category	axiom	Cokernel (abelian category)		The cokernel of a morphism f: A -> B in an abelian category is the coequalizer of f and the zero map, i.e., a universal morphism p: B -> Q such that p f = 0.
s_coimage	state	Coimage		The coimage of a morphism f: A -> B in an abelian category is coim(f) = coker(ker(f)), and in an abelian category the canonical map coim(f) -> im(f) is an isomo
s_category_of_chain_complexes	state	Category Ch(A) of chain complexes	Ch(A)	The category Ch(A) of chain complexes over an abelian category A, with chain maps as morphisms, is itself an abelian category.
s_homotopy_category_K_A	state	Homotopy category K(A)	K(A)	The homotopy category K(A) has chain complexes over A as objects and chain homotopy classes of chain maps as morphisms, forming a triangulated category.
s_homology_is_functor	theorem	Homology is a functor		For each n, H_n: Ch(A) -> A is an additive functor from the category of chain complexes to the abelian category A.
s_hom_complex	state	Hom complex	internal Hom complex	For chain complexes C and D, the Hom complex Hom(C, D) is the cochain complex with Hom(C, D)^n = prod_i Hom(C_i, D_{i+n}) and differential (df)(c) = d^D(f(c)) -
s_existence_of_projective_resolutions	theorem	Existence of projective resolutions		In an abelian category with enough projectives, every object admits a projective resolution, constructed by iteratively taking projective covers of successive k
s_existence_of_injective_resolutions	theorem	Existence of injective resolutions		In an abelian category with enough injectives, every object admits an injective resolution, constructed by iteratively embedding successive cokernels into injec
s_comparison_theorem_injective_resolutions	theorem	Comparison theorem for injective resolutions		Any morphism f: A -> B lifts to a chain map between injective resolutions, unique up to chain homotopy; thus injective resolutions are unique up to chain homoto
s_cohomological_delta_functor	axiom	Cohomological delta-functor		A cohomological delta-functor is a sequence of additive functors {T^n}_{n >= 0} with connecting morphisms delta^n: T^n(C) -> T^{n+1}(A) for each short exact seq
s_coeffaceable_functor	axiom	Coeffaceable functor		An additive functor T is coeffaceable if for every object A there exists an epimorphism u: M -> A such that T(u) = 0; a homological delta-functor is coeffaceabl
s_effaceable_delta_functors_are_universal	theorem	Effaceable delta-functors are universal		An effaceable cohomological delta-functor is universal, and dually a coeffaceable homological delta-functor is universal.
s_L0_naturally_isomorphic_to_F	theorem	L_0F is naturally isomorphic to F		For a right exact functor F and its left derived functors, there is a natural isomorphism L_0F = F.
s_R0_naturally_isomorphic_to_G	theorem	R^0G is naturally isomorphic to G		For a left exact functor G and its right derived functors, there is a natural isomorphism R^0G = G.
s_left_derived_functors_form_universal_delta_functor	theorem	Left derived functors form a universal delta-functor		The left derived functors {L_nF}_{n >= 0} of a right exact functor F form a universal homological delta-functor.
s_right_derived_functors_form_universal_delta_functor	theorem	Right derived functors form a universal delta-functor		The right derived functors {R^nG}_{n >= 0} of a left exact functor G form a universal cohomological delta-functor.
s_derived_functors_vanish_on_adapted_objects	theorem	Derived functors vanish on adapted objects		For a right exact functor F, L_nF(P) = 0 for all n >= 1 when P is projective; for a left exact functor G, R^nG(I) = 0 for all n >= 1 when I is injective.
s_left_adjoints_are_right_exact	theorem	Left adjoints are right exact		A functor that is a left adjoint in an adjunction between abelian categories is right exact.
s_right_adjoints_are_left_exact	theorem	Right adjoints are left exact		A functor that is a right adjoint in an adjunction between abelian categories is left exact.
s_balancing_ext	theorem	Balancing Ext		Ext^n_R(A, B) can be computed using either a projective resolution of A or an injective resolution of B: H^n(Hom_R(P, B)) = H^n(Hom_R(A, I)) = Ext^n_R(A, B).
s_independence_of_resolution_derived_functors	theorem	Independence of resolution for derived functors		The left (resp. right) derived functors L_nF (resp. R^nG) are independent of the choice of projective (resp. injective) resolution up to canonical natural isomo
s_tor_0_is_tensor_product	theorem	Tor_0 is the tensor product		For R-modules A and B, Tor_0^R(A, B) is naturally isomorphic to A tensor_R B.
s_ext_0_is_hom	theorem	Ext^0 is Hom		For R-modules A and B, Ext^0_R(A, B) is naturally isomorphic to Hom_R(A, B).
t_construction_connecting_homomorphism_via_snake	technique	Construction of connecting homomorphism via Snake Lemma		Constructs the connecting homomorphism in the long exact sequence of homology by applying the Snake Lemma to the commutative diagram formed by cycles, boundarie
s_free_module_is_projective	theorem	Free modules are projective		Every free R-module is projective; in particular, the category of R-modules has enough projectives since every module is a quotient of a free module.
s_baer_criterion_for_injectivity	theorem	Baer criterion for injectivity	Baer's criterion	An R-module I is injective if and only if for every ideal J of R, every homomorphism J -> I extends to R.
t_balancing_via_double_complexes	technique	Balancing via double complexes		Proves balancing of Ext and Tor by forming the double complex from projective resolutions of both arguments and computing the homology of the total complex in t
s_flat_dimension	axiom	Flat dimension	weak dimension of a module	The flat dimension fd_R(M) of an R-module M is the infimum of the lengths of flat resolutions of M, equivalently the largest n such that Tor_n^R(M, N) is nonzer
s_ext_characterization_projective_dimension	theorem	Ext characterization of projective dimension		For an R-module M, pd_R(M) <= n if and only if Ext^{n+1}_R(M, N) = 0 for all R-modules N, equivalently if and only if the n-th syzygy in any projective resoluti
s_ext_characterization_injective_dimension	theorem	Ext characterization of injective dimension		For an R-module M, id_R(M) <= n if and only if Ext^{n+1}_R(N, M) = 0 for all R-modules N.
s_rings_of_global_dimension_zero	theorem	Rings of global dimension zero are semisimple		A ring R has gl.dim(R) = 0 if and only if R is semisimple, which by the Artin-Wedderburn theorem means R is a finite product of matrix rings over division rings
s_hereditary_ring	axiom	Hereditary ring		A ring R is (left) hereditary if every (left) ideal of R is projective, equivalently if gl.dim(R) <= 1.
s_rings_of_global_dimension_at_most_one	theorem	Rings of global dimension at most one		A ring R has gl.dim(R) <= 1 if and only if every submodule of a projective R-module is projective; for commutative domains this is equivalent to R being a Dedek
s_change_of_rings_dimension_quotient	theorem	Change of rings theorem for dimension (quotient ring)		If x is a central non-zero-divisor in R and M is a nonzero R/xR-module, then pd_R(M) = pd_{R/xR}(M) + 1.
s_global_dimension_regular_local_rings	theorem	Global dimension of regular local rings		A Noetherian local ring (R, m, k) is regular if and only if gl.dim(R) is finite, in which case gl.dim(R) equals the Krull dimension of R.
s_koszul_homology_definition	axiom	Koszul homology (with coefficients)		The Koszul homology H_*(x_1, ..., x_n; M) of an R-module M with respect to elements x_1, ..., x_n is H_*(K(x_1, ..., x_n) tensor_R M).
s_koszul_acyclicity_for_regular_sequences	theorem	Koszul acyclicity for regular sequences		If x_1, ..., x_n is a regular sequence on an R-module M, then H_i(x_1, ..., x_n; M) = 0 for all i >= 1 and H_0 = M/(x_1, ..., x_n)M.
s_global_dimension_equals_supremum_over_cyclic_modules	theorem	Global dimension equals supremum over cyclic modules		For a left Noetherian ring R, gl.dim(R) = sup{pd_R(R/I) : I a left ideal of R}.
s_weak_dimension_leq_global_dimension	theorem	Weak dimension is at most global dimension		For any ring R, w.gl.dim(R) <= gl.dim(R), with equality when R is Noetherian.
s_auslander_theorem_global_dimension	theorem	Auslander's theorem on global dimension		For a commutative Noetherian local ring R with residue field k, gl.dim(R) = pd_R(k).
s_koszul_provides_resolution_of_residue_field	state	Koszul complex provides resolution of residue field		If (R, m, k) is a regular local ring with m = (x_1, ..., x_d) where d = dim R, then the Koszul complex K(x_1, ..., x_d) is a free resolution of k of length d.
s_koszul_H1_detects_regular_sequences	theorem	Koszul H_1 detects regular sequences		For a Noetherian local ring (R, m, k) and x_1, ..., x_n in m, H_1(x_1, ..., x_n; R) = 0 if and only if x_1, ..., x_n is a regular sequence.
t_mapping_cone_and_koszul	technique	Mapping cone and Koszul complex		Constructs K(x_1, ..., x_n) inductively as the mapping cone of multiplication by x_n on K(x_1, ..., x_{n-1}), providing a recursive method for computing Koszul 
t_horseshoe_lemma_application_dimension_shifting	technique	Horseshoe lemma application in dimension shifting		Uses the horseshoe lemma to construct a projective resolution of the middle term in a short exact sequence from resolutions of the outer terms, enabling systema
t_restriction_induction_of_scalars	technique	Restriction and induction of scalars	base change | extension and restriction of scalars	Given a ring map f: R -> S, restriction of scalars views S-modules as R-modules while induction S tensor_R - forms an adjoint pair, used to transfer homological
s_spectral_sequence	axiom	Spectral sequence		A sequence of bigraded modules {E_r^{p,q}} with differentials d_r satisfying E_{r+1} = H(E_r, d_r), providing successive approximations to a limit term E_infini
s_associated_graded_complex	state	Associated graded complex		For a filtered chain complex with filtration {F_p C}, the bigraded complex gr_p C = F_p C / F_{p-1} C whose homology forms the E_0 page of the associated spectr
s_bounded_filtration	axiom	Bounded filtration		A filtration {F_p C} on a chain complex is bounded if for each n there exist s < t such that F_s C_n = 0 and F_t C_n = C_n, ensuring the associated spectral seq
s_weak_convergence_spectral_sequence	axiom	Weak convergence of a spectral sequence		A spectral sequence weakly converges to a graded object H_* if E_infinity^{p,q} is isomorphic to F_p H_{p+q} / F_{p-1} H_{p+q} for a filtration F on H_*.
s_bounded_convergence_spectral_sequence	axiom	Bounded convergence of a spectral sequence		A spectral sequence converges boundedly to H_* if it weakly converges and the filtration on each H_n is finite, which holds when the filtration on the complex i
s_first_quadrant_spectral_sequence	axiom	First quadrant spectral sequence		A spectral sequence with E_r^{p,q} = 0 whenever p < 0 or q < 0, which always converges boundedly since for fixed (p,q) only finitely many differentials can be n
s_transgression	state	Transgression		In a first quadrant spectral sequence, the differential d_n: E_n^{0,n-1} -> E_n^{n,0} connecting the base and fiber terms, a partially-defined map appearing in 
s_five_term_exact_sequence_spectral	theorem	Five-term exact sequence of a spectral sequence		For a first quadrant spectral sequence converging to H_*, the sequence H_2 -> E_2^{2,0} -> E_2^{0,1} -> H_1 -> E_2^{1,0} -> 0 is exact.
s_hyper_derived_functors_left	axiom	Left hyper-derived functors	hyper-Tor	For a right exact functor F, the left hyper-derived functors L_n F(C) are H_n(F(P)) where P -> C is a quasi-isomorphism from a complex of projectives, generaliz
s_hyper_derived_functors_right	axiom	Right hyper-derived functors	hyper-Ext	For a left exact functor F, the right hyper-derived functors R^n F(C) are H^n(F(I)) where C -> I is a quasi-isomorphism to a complex of injectives, generalizing
s_hyperhomology_spectral_sequences	theorem	Hyperhomology spectral sequences		For a right exact functor F and a chain complex C, there are two spectral sequences with E_2 = (L_p F)(H_q(C)) and E_1 = (L_q F)(C_p) both converging to L_{p+q}
s_existence_cartan_eilenberg_resolutions	theorem	Existence of Cartan-Eilenberg resolutions		Every bounded-below chain complex over an abelian category with enough projectives admits a Cartan-Eilenberg resolution.
s_universal_coefficients_spectral_sequence	theorem	Universal coefficients spectral sequence		For a chain complex C of flat R-modules and an R-module M, there is a spectral sequence E_2^{p,q} = Tor_p^R(H_q(C), M) converging to H_{p+q}(C tensor_R M).
s_kunneth_spectral_sequence	theorem	Kunneth spectral sequence		For chain complexes C and D over R with C flat, there is a spectral sequence E_2 = Tor_p^R(H_i(C), H_j(D)) converging to H_{p+q}(C tensor_R D).
s_mapping_lemma_spectral_sequences	theorem	Mapping lemma for spectral sequences	comparison theorem for spectral sequences	A morphism of spectral sequences that is an isomorphism at the E_r page for some r is an isomorphism at all subsequent pages and at E_infinity.
s_degeneration_at_E2	state	Degeneration at E_2	collapse at E_2	A spectral sequence degenerates at E_2 if all differentials d_r = 0 for r >= 2, so E_2 = E_infinity.
t_exact_couple_construction_spectral_sequences	technique	Exact couple construction of spectral sequences		Constructs a spectral sequence from an exact couple by iterating the derived couple operation, producing pages E_r with differentials d_r = j_r k_r.
t_spectral_sequence_comparison	technique	Spectral sequence comparison technique		Computes the abutment of a spectral sequence by using a second spectral sequence for the same double complex whose E_2 page is simpler, exploiting the fact that
t_grothendieck_spectral_sequence_construction	technique	Grothendieck spectral sequence construction technique		Applies the Grothendieck spectral sequence to a composite of left exact functors by analyzing the resulting double complex from Cartan-Eilenberg or injective re
s_augmentation_map_epsilon	axiom	Augmentation map epsilon: ZG -> Z	augmentation homomorphism	The ring homomorphism epsilon: ZG -> Z defined by epsilon(sum n_g g) = sum n_g, making Z into a trivial G-module.
s_coinvariants_M_G	axiom	Coinvariants M_G	module of coinvariants	The module of coinvariants M_G = M / I_G M = Z tensor_{ZG} M is the largest quotient of a G-module M on which G acts trivially.
s_invariants_M_G	axiom	Invariants M^G	fixed points | module of invariants	The module of G-invariants M^G = {m in M : gm = m for all g in G} = Hom_{ZG}(Z, M) is the largest submodule on which G acts trivially.
s_H0_G_M_is_coinvariants	theorem	H_0(G, M) = M_G		The zeroth group homology H_0(G, M) is naturally isomorphic to the coinvariants M_G.
s_H0_cohomology_is_invariants	theorem	H^0(G, M) = M^G		The zeroth group cohomology H^0(G, M) is naturally isomorphic to the invariants M^G.
s_H1_classifies_crossed_homomorphisms	theorem	H^1(G, M) classifies crossed homomorphisms		The first cohomology group H^1(G, M) is isomorphic to the group of crossed homomorphisms from G to M modulo the principal crossed homomorphisms.
s_H1_G_Z_is_abelianization	theorem	H_1(G, Z) = G^{ab}		The first homology group H_1(G, Z) with trivial integer coefficients is naturally isomorphic to the abelianization G/[G, G].
s_hopf_formula_H2	theorem	Hopf formula for H_2	Hopf's formula	If G = F/R with F free, then H_2(G, Z) = (R intersect [F,F]) / [F,R].
s_inhomogeneous_bar_resolution	axiom	Inhomogeneous bar resolution	unnormalized bar resolution	A free ZG-resolution of Z with n-th term the free ZG-module on n-tuples [g_1|...|g_n] of elements of G, with differential using the inhomogeneous bar formulas.
s_homogeneous_bar_resolution	axiom	Homogeneous bar resolution		A free ZG-resolution of Z with n-th term the free abelian group on (n+1)-tuples [g_0, ..., g_n] with diagonal G-action.
s_coinduced_module	axiom	Coinduced module Coind_H^G N	co-induced module	The coinduced module from a subgroup H to G is Coind_H^G N = Hom_{ZH}(ZG, N), the right adjoint to the restriction functor from G-modules to H-modules.
s_ind_equals_coind_finite_index	theorem	Ind = Coind for finite index subgroups		If H is a subgroup of finite index in G, then ZG tensor_{ZH} N and Hom_{ZH}(ZG, N) are naturally isomorphic as G-modules.
s_cor_res_is_index	theorem	cor composed with res equals multiplication by [G:H]		For H a subgroup of finite index [G:H] in G, the composite cor res: H^n(G, M) -> H^n(G, M) is multiplication by the index [G:H].
s_lyndon_hochschild_serre_spectral_sequence	theorem	Lyndon-Hochschild-Serre spectral sequence	LHS spectral sequence	For a group extension 1 -> N -> G -> Q -> 1, there is a first-quadrant spectral sequence E_2^{p,q} = H^p(Q, H^q(N, M)) converging to H^{p+q}(G, M), and a homolo
s_five_term_exact_sequence_group_cohomology	theorem	Five-term exact sequence in group cohomology	inflation-restriction exact sequence (extended)	For 1 -> N -> G -> Q -> 1, the exact sequence 0 -> H^1(Q, M^N) -> H^1(G, M) -> H^1(N, M)^Q -> H^2(Q, M^N) -> H^2(G, M) arises from the LHS spectral sequence.
s_five_term_exact_sequence_group_homology	theorem	Five-term exact sequence in group homology		For 1 -> N -> G -> Q -> 1, the exact sequence H_2(G, Z) -> H_2(Q, Z) -> N/[G,N] -> H_1(G, Z) -> H_1(Q, Z) -> 0.
s_cohomological_dimension_cd_G	theorem	cd(G) = proj.dim_{ZG}(Z)		The cohomological dimension cd(G) equals the projective dimension of Z as a ZG-module.
s_group_of_type_FL	axiom	Group of type FL		A group G is of type FL if the trivial G-module Z admits a finite free resolution over ZG.
s_group_of_type_FP	axiom	Group of type FP		A group G is of type FP if the trivial G-module Z admits a finite projective resolution over ZG.
s_free_groups_have_cd_1	theorem	Free groups have cd = 1	Stallings-Swan theorem	A group G is free if and only if cd(G) <= 1, equivalently if and only if the augmentation ideal I_G is a free ZG-module.
s_complete_tate_resolution	axiom	Complete (Tate) resolution	Tate resolution	An acyclic complex of finitely generated free ZG-modules agreeing with a projective resolution of Z in positive degrees and with the dual in negative degrees, s
s_tate_H0_formula	theorem	Tate cohomology H-hat^0 = M^G / N_G(M)		For a finite group G, the zeroth Tate cohomology is isomorphic to M^G / im(N_G).
s_tate_H_minus1_formula	theorem	Tate cohomology H-hat^{-1} = ker(N_G) / I_G M		For a finite group G, the Tate cohomology group H-hat^{-1}(G, M) is ker(N_G) / I_G M.
s_tate_long_exact_sequence	theorem	Tate cohomology long exact sequence		For a finite group G and a short exact sequence of G-modules, there is a long exact sequence in Tate cohomology extending over all integers n.
s_tate_induced_vanishes	theorem	Tate cohomology of induced modules vanishes		For a finite group G, if M is an induced G-module then H-hat^n(G, M) = 0 for all integers n.
s_cup_product_tate_cohomology	axiom	Cup product in Tate cohomology		The cup product extends to Tate cohomology as a graded pairing H-hat^p(G, M) tensor H-hat^q(G, N) -> H-hat^{p+q}(G, M tensor N) for all integers p, q.
s_periodicity_via_cup_product	theorem	Periodicity in group cohomology via cup product		A finite group G has periodic cohomology of period d if and only if there exists u in H^d(G, Z) such that cup product with u induces isomorphisms on all Tate co
s_maschke_theorem_group_cohomology	theorem	Maschke's theorem for group cohomology		If G is a finite group of order n and multiplication by n is an isomorphism on M, then H^k(G, M) = 0 for all k >= 1.
s_universal_coefficient_theorem_group_cohomology	theorem	Universal coefficient theorem for group cohomology		There is a natural split short exact sequence 0 -> Ext^1_Z(H_{n-1}(G, Z), A) -> H^n(G, A) -> Hom_Z(H_n(G, Z), A) -> 0 for any trivial G-module A.
s_cohomology_of_finite_cyclic_groups	state	Cohomology of finite cyclic groups		For Z/nZ with trivial coefficients: H^0 = Z, H^{2k-1} = Z/nZ for k >= 1, H^{2k} = 0 for k >= 1.
s_cohomology_of_free_abelian_groups	state	Cohomology of free abelian groups		For G = Z^r, the cohomology ring H^*(G, Z) is isomorphic to the exterior algebra Lambda^*(Z^r).
t_diagonal_approximation_bar_resolution	technique	Diagonal approximation on bar resolution		Constructs the cup product in group cohomology by defining a chain map from the bar resolution to its tensor square compatible with the augmentation.
t_bar_resolution_technique_computing	technique	Bar resolution technique for computing group (co)homology		Computes group (co)homology by applying Z tensor_{ZG} - or Hom_{ZG}(-, M) to the bar resolution, producing explicit complexes whose elements are functions on tu
t_dimension_shifting_group_cohomology	technique	Dimension shifting in group cohomology		Embeds a G-module M into a coinduced module to shift cohomological dimension, reducing H^n statements to H^{n-1} via the connecting homomorphism.
s_lie_algebra_module	axiom	Lie algebra module	g-module	A k-module M with a Lie algebra action g x M -> M satisfying [x,y].m = x.(y.m) - y.(x.m), equivalently a left module over U(g).
s_augmentation_of_U_g	axiom	Augmentation of U(g)		The algebra homomorphism epsilon: U(g) -> k defined by epsilon(x) = 0 for all x in g, making k into a trivial g-module.
s_H0_lie_cohomology_invariants	state	H^0(g, M) as invariants		The zeroth Lie algebra cohomology H^0(g, M) = M^g = {m in M : x.m = 0 for all x in g}.
s_H0_lie_homology_coinvariants	state	H_0(g, M) as coinvariants		The zeroth Lie algebra homology H_0(g, M) = M_g = M / g.M, the module of g-coinvariants.
s_H1_lie_cohomology_derivations	state	H^1(g, M) as derivations modulo inner derivations		The first Lie algebra cohomology H^1(g, M) = Der(g, M)/Inn(g, M).
s_H2_lie_classifies_extensions	theorem	H^2(g, M) classifies abelian Lie algebra extensions		The second cohomology H^2(g, M) is in natural bijection with equivalence classes of abelian extensions 0 -> M -> e -> g -> 0.
s_H1_lie_homology_abelianization	state	H_1(g, k) as abelianization		The first homology H_1(g, k) with trivial coefficients is isomorphic to g/[g,g].
s_abelian_lie_algebra_extension	axiom	Abelian Lie algebra extension		A short exact sequence 0 -> M -> e -> g -> 0 of Lie algebras where M is an abelian ideal in e, inheriting a g-module structure.
s_chevalley_eilenberg_as_free_resolution	state	Chevalley-Eilenberg complex as U(g)-free resolution		The complex V_*(g) = U(g) tensor_k Lambda^*(g) with appropriate differential is a free U(g)-resolution of the trivial module k when g is a free k-module.
s_invariant_forms_and_lie_cohomology	state	Invariant forms and H^*(g, k)		For a finite-dimensional Lie algebra g over characteristic zero, H^*(g, k) is computed by g-invariant forms in Lambda^*(g^*); for semisimple g this is the exter
s_cohomology_of_gl_n	state	Cohomology of gl_n		Over characteristic zero, H^*(gl_n(k), k) = Lambda(c_1, c_3, ..., c_{2n-1}) on generators of odd degrees.
s_cohomology_of_sl_n	state	Cohomology of sl_n		Over characteristic zero, H^*(sl_n(k), k) = Lambda(c_3, c_5, ..., c_{2n-1}) on generators of odd degrees.
s_inner_derivation_of_lie_algebra	axiom	Inner derivation of a Lie algebra		For m in M, the derivation d_m: g -> M defined by d_m(x) = x.m, forming the subspace Inn(g, M) of 1-coboundaries in Lie algebra cohomology.
s_Hn_vanishes_semisimple_nontrivial	theorem	H^n vanishes for semisimple g and nontrivial irreducible M		If g is finite-dimensional semisimple over characteristic zero and M is a nontrivial finite-dimensional irreducible g-module, then H^n(g, M) = 0 for all n >= 0.
s_H2_lie_homology_central_extensions	state	H_2(g, k) and central extensions		The second homology H_2(g, k) with trivial coefficients classifies central extensions of the Lie algebra g.
s_hopf_algebra_structure_U_g	state	Hopf algebra structure on U(g)		U(g) carries a cocommutative Hopf algebra structure with comultiplication Delta(x) = x tensor 1 + 1 tensor x for x in g, antipode S(x) = -x, and counit epsilon.
s_primitive_elements_of_U_g	state	Primitive elements of U(g)		The set of primitive elements {x in U(g) : Delta(x) = x tensor 1 + 1 tensor x} is exactly g, recovering the Lie algebra from its universal enveloping algebra.
s_hochschild_serre_spectral_sequence_lie	theorem	Hochschild-Serre spectral sequence for Lie algebras		For an ideal h of a Lie algebra g and a g-module M, there is a convergent spectral sequence E_2^{p,q} = H^p(g/h, H^q(h, M)) => H^{p+q}(g, M).
t_chevalley_eilenberg_coboundary_formula	technique	Chevalley-Eilenberg coboundary formula		Computes the differential in Lie algebra cohomology by the explicit formula involving the g-action and Lie bracket terms on Hom_k(Lambda^n(g), M).
t_casimir_operator_vanishing	technique	Casimir operator technique for vanishing theorems		Uses the Casimir element in the center of U(g) to show it acts as zero on trivial modules but invertibly on nontrivial modules, forcing Ext groups to vanish.
s_simplex_category	axiom	Simplex category	Delta | simplicial category	The category Delta whose objects are the finite totally ordered sets [n] = {0, 1, ..., n} and whose morphisms are the nondecreasing maps, indexing simplicial ob
s_face_maps	axiom	Face maps		The morphisms d_i: X_n -> X_{n-1} for 0 <= i <= n in a simplicial object, induced by the injective order-preserving maps that skip the value i.
s_degeneracy_maps	axiom	Degeneracy maps		The morphisms s_i: X_n -> X_{n+1} for 0 <= i <= n in a simplicial object, induced by the surjective order-preserving maps that repeat the value i.
s_simplicial_identities	axiom	Simplicial identities		The relations d_i d_j = d_{j-1} d_i for i < j, s_i s_j = s_{j+1} s_i for i <= j, and mixed face-degeneracy relations characterizing the simplex category.
s_simplicial_abelian_group	axiom	Simplicial abelian group		A simplicial object in the category of abelian groups, i.e., a contravariant functor Delta^op -> Ab.
s_standard_n_simplex	axiom	Standard n-simplex	representable simplicial set	The simplicial set Delta[n] = Hom_{Delta}(-, [n]) represented by [n], whose k-simplices are order-preserving maps [k] -> [n].
s_singular_simplicial_set	state	Singular simplicial set	singular complex	The simplicial set S(X) of a topological space X whose n-simplices are continuous maps from the standard topological n-simplex to X.
s_geometric_realization	axiom	Geometric realization		The functor |.|: sSet -> Top defined as the coend |X| = coprod_n (X_n x Delta^n) / ~, left adjoint to the singular simplicial set functor.
s_simplicial_map	axiom	Simplicial map	simplicial morphism	A natural transformation f: X -> Y between simplicial objects, consisting of morphisms f_n: X_n -> Y_n commuting with all face and degeneracy maps.
s_simplicial_homotopy	axiom	Simplicial homotopy		A simplicial homotopy between simplicial maps f, g: X -> Y is a simplicial map H: X x Delta[1] -> Y restricting to f and g on the endpoints.
s_simplicial_homotopy_groups	state	Simplicial homotopy groups		For a Kan complex X with basepoint, the groups pi_n(X) defined as equivalence classes of n-simplices with boundary at the basepoint, generalizing homotopy group
s_normalized_chain_complex	state	Normalized chain complex		The subcomplex NA of the Moore complex with (NA)_n = intersection of ker(d_i) for i = 1, ..., n, naturally chain homotopy equivalent to the Moore complex.
s_degeneracy_subcomplex	state	Degeneracy subcomplex		The subcomplex DA of the Moore complex generated by images of all degeneracy maps, satisfying CA = NA direct sum DA as graded abelian groups.
s_dold_kan_inverse_functor_K	state	Dold-Kan inverse functor K		The functor K: Ch_{>=0}(Ab) -> sAb, quasi-inverse to the normalized chain complex functor, reconstructing a simplicial abelian group from a chain complex.
t_alexander_whitney_map	technique	Alexander-Whitney map	AW map	The natural chain map AW: C(A x B) -> C(A) tensor C(B) providing one direction of the Eilenberg-Zilber equivalence.
t_shuffle_map	technique	Shuffle map (Eilenberg-MacLane map)	EML map | Eilenberg-MacLane map	The natural chain map nabla: C(A) tensor C(B) -> C(A x B) defined using (p,q)-shuffles, providing the inverse direction of the Eilenberg-Zilber equivalence.
s_shuffle	axiom	Shuffle	(p,q)-shuffle	A (p,q)-shuffle is a permutation of {0, ..., p+q-1} preserving the relative order within the first p and last q elements, used in defining products on chain com
s_nerve_of_a_category	state	Nerve of a category		The simplicial set N(C) whose n-simplices are composable chains of n morphisms, with face maps by composition and degeneracy maps by identity insertion.
s_classifying_space_via_nerve	state	Classifying space of a group (via nerve)	BG	The geometric realization BG = |N(G)| of the nerve of the one-object category of G, with pi_1(BG) = G and contractible universal cover EG.
s_cotriple_comonad	axiom	Cotriple (comonad)	comonad	An endofunctor G: C -> C with natural transformations epsilon: G -> id (counit) and delta: G -> G^2 (comultiplication) satisfying coassociativity and counitalit
s_cotriple_homology	state	Cotriple homology		For a cotriple G and coefficient functor E: C -> Ab, the groups H_n^G(A, E) = H_n(E(G_*(A))) obtained by applying E to the cotriple resolution.
s_andre_quillen_cohomology	state	Andre-Quillen cohomology	AQ cohomology	For a commutative k-algebra A and A-module M, the groups D^n(A/k, M) defined as derived functors of derivations Der_k(A, M), classifying algebra extensions.
s_bisimplicial_object	axiom	Bisimplicial object	bisimplicial set	A functor X: Delta^op x Delta^op -> C, equivalently a simplicial object in simplicial objects, with commuting horizontal and vertical face and degeneracy maps.
s_diagonal_of_bisimplicial_object	state	Diagonal of a bisimplicial object		The simplicial object diag(X) with diag(X)_n = X_{n,n}, whose geometric realization is naturally homeomorphic to that of the total object.
s_normalization_theorem	theorem	Normalization theorem		The inclusion of the normalized chain complex into the Moore complex of a simplicial abelian group is a chain homotopy equivalence, with the decomposition CA_n 
s_simplicial_homotopy_invariance_homology	theorem	Simplicial homotopy invariance of homology		Simplicially homotopic maps of simplicial abelian groups induce chain homotopic maps on Moore complexes, so simplicial homotopy equivalences induce homology iso
s_acyclicity_of_cotriple_resolutions	theorem	Acyclicity of cotriple resolutions		The augmented simplicial object from a cotriple resolution is aspherical when the cotriple arises from an adjunction.
s_transitivity_exact_sequence_AQ	theorem	Transitivity exact sequence (Andre-Quillen)	Jacobi-Zariski exact sequence	For ring maps k -> A -> B and a B-module M, there is a long exact sequence relating D_n(A/k, M), D_n(B/k, M), and D_n(B/A, M).
s_smooth_algebra_vanishing_AQ	theorem	Smooth algebra vanishing (Andre-Quillen)		A commutative k-algebra A is smooth if and only if D_n(A/k, M) = 0 for all n >= 1 and all A-modules M.
t_extra_degeneracy_contracting_homotopy	technique	Extra degeneracy (contracting homotopy)	extra degeneracy argument	An extra degeneracy s_{-1} on an augmented simplicial object provides a contracting simplicial homotopy proving the augmented complex is aspherical.
s_hochschild_complex	axiom	Hochschild complex (bar complex for algebras)	Hochschild bar complex	The chain complex with C_n(A, M) = M tensor A^{tensor n} and face maps by multiplication of adjacent factors, computing Hochschild homology.
s_formal_deformation_of_algebra	axiom	Formal deformation of an algebra		A k[[t]]-algebra structure on A[[t]] of the form a * b = ab + m_1(a,b)t + m_2(a,b)t^2 + ..., reducing to A modulo t.
s_H2_classifies_infinitesimal_deformations	theorem	HH^2 classifies infinitesimal deformations		HH^2(A, A) classifies equivalence classes of infinitesimal deformations of A, while HH^3(A, A) contains obstructions to extending deformations.
s_connes_B_operator	axiom	Connes' B operator		The operator B: C_n(A) -> C_{n+1}(A) defined by B = (1 - t)sN satisfying B^2 = 0 and bB + Bb = 0, connecting Hochschild and cyclic homology.
s_SBI_exact_sequence	theorem	SBI exact sequence	Connes' exact sequence | periodicity exact sequence	The long exact sequence ... -> HH_n(A) -> HC_n(A) -> HC_{n-2}(A) -> HH_{n-1}(A) -> ... relating Hochschild and cyclic homology via the periodicity operator S an
s_connes_bicomplex	axiom	Connes' bicomplex		The first-quadrant bicomplex with columns C_n^{lambda}(A) = C_n(A)/im(1-t) and horizontal differential induced by B, quasi-isomorphic to the cyclic bicomplex in
s_morita_invariance_hochschild_cyclic	theorem	Morita invariance of Hochschild and cyclic homology		Morita equivalent k-algebras have isomorphic Hochschild and cyclic homology in all degrees.
s_hochschild_kostant_rosenberg	theorem	Hochschild-Kostant-Rosenberg theorem	HKR theorem	For a smooth commutative k-algebra A, the antisymmetrization map gives HH_n(A) = Omega^n_{A/k}.
s_loday_quillen_tsygan	theorem	Loday-Quillen-Tsygan theorem		There is a natural isomorphism HC_{n-1}(A) = prim H_n(gl(A), k) between cyclic homology and the primitive part of the Lie algebra homology of gl(A).
s_excision_hochschild_cyclic	theorem	Excision in Hochschild and cyclic homology		For a surjection A -> B with kernel I, there is a long exact sequence relating HH_*(A), HH_*(B), and HH_*(A, I), with an analogous sequence for cyclic homology 
s_triangulated_category_axioms	axiom	Triangulated category axioms (TR1-TR4)	TR1-TR4	An additive category with shift autoequivalence [1] and distinguished triangles X -> Y -> Z -> X[1] satisfying four axioms: existence, rotation, morphism extens
s_RHom_complex	axiom	RHom complex	derived Hom	The total right derived functor of Hom, yielding RHom(A, B) in D(Ab) with H^n(RHom(A, B)) = Ext^n(A, B).
t_calculus_of_fractions	technique	Calculus of fractions		Describes morphisms in a localized category C[S^{-1}] as equivalence classes of roofs or coroofs when S satisfies Ore conditions.
t_replacing_spectral_sequences_with_triangles	technique	Replacing spectral sequences with triangles		Uses distinguished triangles in the derived category to encode information equivalent to spectral sequence E_2 pages, avoiding convergence issues.
s_abelian_category_axioms_AB	axiom	Abelian category axioms (AB1-AB5)	Grothendieck's axioms	Grothendieck's hierarchy: AB1-AB2 define abelian categories; AB3 requires coproducts; AB4 requires exact coproducts; AB5 requires exact filtered colimits; dual 
s_grothendieck_category	axiom	Grothendieck category	AB5 category with generator	An abelian category satisfying AB5 with a generator, guaranteeing enough injectives by the Grothendieck theorem.
s_inverse_limit_functor_lim	axiom	Inverse limit functor lim	projective limit	For an inverse system {A_i, f_{ij}} indexed by a directed poset, lim A_i is the submodule of prod A_i of compatible elements with f_{ij}(a_j) = a_i for all i <=
s_mittag_leffler_implies_lim1_vanishes	theorem	Mittag-Leffler implies lim^1 vanishes		If a tower of modules satisfies the Mittag-Leffler condition, then lim^1 = 0.
s_lim1_short_exact_sequences_towers	theorem	lim^1 and short exact sequences of towers		A short exact sequence of towers induces a six-term exact sequence 0 -> lim A_n -> lim B_n -> lim C_n -> lim^1 A_n -> lim^1 B_n -> lim^1 C_n -> 0.
s_universal_coefficient_theorem_homology	theorem	Universal coefficient theorem for homology		For a chain complex C of flat R-modules with flat boundaries, there is a natural split short exact sequence 0 -> H_n(C) tensor M -> H_n(C tensor M) -> Tor_1(H_{
s_universal_coefficient_theorem_cohomology	theorem	Universal coefficient theorem for cohomology		For a chain complex C of free abelian groups, there is a natural split short exact sequence 0 -> Ext^1(H_{n-1}(C), G) -> H^n(Hom(C, G)) -> Hom(H_n(C), G) -> 0.
s_symmetry_of_tor	theorem	Symmetry of Tor		For a commutative ring R, there is a natural isomorphism Tor_n^R(A, B) = Tor_n^R(B, A) for all n >= 0.
s_tor_for_abelian_groups	state	Tor for abelian groups		For abelian groups, Tor_n^Z(A, B) = 0 for all n >= 2 since Z has global dimension 1.
s_ext_for_abelian_groups	state	Ext for abelian groups		For abelian groups, Ext^n_Z(A, B) = 0 for n >= 2, and Ext^1_Z(Z/n, B) = B/nB.
s_derivation_grothendieck_LHS_spectral	theorem	Derivation of LHS via Grothendieck spectral sequence		The Lyndon-Hochschild-Serre spectral sequence arises as a special case of the Grothendieck spectral sequence for the composite of the fixed-point functors (-)^N
s_metacategory	axiom	Metacategory		A metagraph equipped with composition and identity arrows satisfying associativity and unit laws, described without requiring a set-theoretic foundation.
s_category	axiom	Category	abstract category	A metacategory whose objects and morphisms each form a collection, with an associative composition law and identity morphisms for every object.
s_small_category	axiom	Small category		A category in which both the collection of objects and the collection of morphisms are sets, not proper classes.
s_locally_small_category	axiom	Locally small category		A category in which for every pair of objects A, B the hom-collection Hom(A, B) is a set.
s_functor	axiom	Functor	covariant functor	A morphism of categories sending objects to objects and arrows to arrows, preserving composition and identities.
s_contravariant_functor	axiom	Contravariant functor		A functor from the opposite category C^op to D, equivalently an assignment reversing the direction of morphisms.
s_natural_isomorphism	axiom	Natural isomorphism		A natural transformation whose every component τ_c is an isomorphism in the target category.
s_isomorphism_of_categories	axiom	Isomorphism of categories		A functor F: C → D for which there exists G: D → C with G∘F = Id_C and F∘G = Id_D as strict equality of functors.
s_monomorphism	axiom	Monomorphism	monic | mono	A morphism m: A → B such that m∘f = m∘g implies f = g for all parallel morphisms into A; the categorical generalization of injectivity.
s_epimorphism	axiom	Epimorphism	epic | epi	A morphism e: A → B such that f∘e = g∘e implies f = g for all parallel morphisms out of B; the categorical generalization of surjectivity.
s_section_categorical	axiom	Section (right inverse)		A morphism s: B → A such that r∘s = id_B for some r: A → B; equivalently, a split monomorphism.
s_retraction_categorical	axiom	Retraction (left inverse)		A morphism r: A → B such that r∘s = id_B for some s: B → A; equivalently, a split epimorphism.
s_universal_element	axiom	Universal element		For a functor F: C → Set, an element u ∈ F(r) such that for every object c and x ∈ F(c) there is a unique f: r → c with F(f)(u) = x.
s_universal_arrow	axiom	Universal arrow		For a functor U: D → C and object c of C, a pair (r, u: c → U(r)) initial in the comma category (c ↓ U) through which every morphism c → U(d) factors uniquely.
s_vertical_composition	axiom	Vertical composition of natural transformations		Given natural transformations α: F ⇒ G and β: G ⇒ H, their vertical composite (β·α)_c = β_c ∘ α_c is a natural transformation F ⇒ H.
s_horizontal_composition	axiom	Horizontal composition (Godement product)	Godement product	Given α: F ⇒ G (C → D) and β: H ⇒ K (D → E), the horizontal composite β∗α: HF ⇒ KG defined by (β∗α)_c = β_{G(c)} ∘ H(α_c) = K(α_c) ∘ β_{F(c)}.
s_interchange_law	theorem	Interchange law		Horizontal and vertical composition of natural transformations satisfy (β'·β) ∗ (α'·α) = (β'∗α')·(β∗α), expressing compatibility of the two composition operatio
s_whiskering	axiom	Whiskering		Pre- or post-composing a natural transformation α: F ⇒ G with a functor H, yielding Hα: HF ⇒ HG or αH: FH ⇒ GH; a special case of horizontal composition.
t_yoneda_embedding_technique	technique	Yoneda embedding technique		Embedding a category fully and faithfully into its presheaf category to reduce categorical statements to set-theoretic verifications.
s_opposite_category	axiom	Opposite category	dual category | C^op	The category with the same objects as C but with every morphism reversed: Hom_{C^op}(A,B) = Hom_C(B,A), with composition reversed.
t_duality_principle	technique	Duality principle (categorical)		Any valid categorical statement remains valid when all arrows are reversed, yielding an automatic dual theorem about opposite categories.
s_product_category	axiom	Product category		The category whose objects are pairs (c,d) and whose morphisms are pairs (f,g) composed componentwise.
s_comma_category	axiom	Comma category	comma category (F ↓ G)	Given functors F: A → C and G: B → C, the category whose objects are triples (a, b, h: F(a) → G(b)) and morphisms are commuting pairs.
s_slice_category	axiom	Slice category	over category	The comma category (Id_C ↓ c) whose objects are morphisms f: a → c and morphisms are commuting triangles over c.
s_coslice_category	axiom	Coslice category	under category	The comma category (c ↓ Id_C) whose objects are morphisms f: c → a; the dual of the slice category.
s_arrow_category	axiom	Arrow category	morphism category	The functor category [2, C] whose objects are morphisms of C and whose morphisms are commutative squares.
s_free_category_on_graph	state	Free category on a graph		Given a directed graph, the category whose objects are vertices and morphisms are finite composable paths, with concatenation as composition.
s_congruence_on_category	axiom	Congruence on a category		An equivalence relation on each hom-set compatible with composition: if f ~ f' and g ~ g' then g∘f ~ g'∘f'.
s_quotient_category	state	Quotient category		Given a congruence ~ on a category C, the category C/~ with the same objects and morphisms the equivalence classes [f].
s_presheaf_categorical	axiom	Presheaf (categorical)	presheaf | set-valued presheaf	A contravariant functor F: C^op → Set; the presheaf category [C^op, Set] serves as the free cocompletion of C.
s_limit_of_functor	axiom	Limit of a functor	projective limit | inverse limit	For a diagram F: J → C, a terminal object in the category of cones over F, with a universal natural transformation from the constant functor to F.
s_colimit_of_functor	axiom	Colimit of a functor	inductive limit | direct limit	For a diagram F: J → C, an initial object in the category of cocones under F, through which every cocone factors uniquely.
t_limit_via_products_equalizers	technique	Limit computation via products and equalizers		Constructing any limit as an equalizer of a pair of morphisms between products, reducing arbitrary limits to products and equalizers.
t_colimit_via_coproducts_coequalizers	technique	Colimit computation via coproducts and coequalizers		Constructing any colimit as a coequalizer of a pair of morphisms between coproducts, dual to the product-equalizer construction.
s_preservation_of_limits	axiom	Preservation of limits		A functor F preserves limits of type J if whenever (L, π) is a limit cone in the domain, then (F(L), F(π)) is a limit cone in the codomain.
s_reflection_of_limits	axiom	Reflection of limits		A functor F reflects limits of type J if whenever (F(L), F(π)) is a limit cone in the codomain, then (L, π) was already a limit cone in the domain.
s_creation_of_limits	axiom	Creation of limits		A functor U creates limits of type J if for every diagram whose image has a limit, there is a unique lift to a limit that U preserves and reflects.
s_subobject	axiom	Subobject		An equivalence class of monomorphisms into a given object, where two monics are equivalent if they factor through each other by an isomorphism.
s_quotient_object	axiom	Quotient object		An equivalence class of epimorphisms out of a given object, dual to subobject; two epics are equivalent if they factor through each other.
s_adjunction	axiom	Adjunction	adjoint pair | adjoint functors	A pair of functors F: C → D and U: D → C with a natural bijection Hom_D(Fc, d) ≅ Hom_C(c, Ud); F is left adjoint to U.
s_special_adjoint_functor_theorem	theorem	Special adjoint functor theorem (SAFT)	SAFT	A limit-preserving functor from a complete, locally small, well-powered category with a cogenerating set has a left adjoint; the solution set condition is autom
s_solution_set_condition	axiom	Solution set condition		For a functor U: D → C and object c, the condition that there exists a set of morphisms {c → U(d_i)} through which every c → U(d) factors.
s_free_forgetful_adjunction	state	Free-forgetful adjunction		The prototypical adjunction where F constructs free algebraic structures and U forgets the structure; the unit is the inclusion of generators.
s_forgetful_functor	axiom	Forgetful functor		A functor that discards algebraic structure, sending an object to its underlying set; often faithful and often has a left adjoint.
s_free_functor	axiom	Free functor		A left adjoint to a forgetful functor, constructing the free algebraic structure on a given set.
s_faithful_functor	axiom	Faithful functor		A functor F: C → D such that for all objects A, B the map F: Hom(A,B) → Hom(FA, FB) is injective.
s_full_functor	axiom	Full functor		A functor F: C → D such that for all objects A, B the map F: Hom(A,B) → Hom(FA, FB) is surjective.
s_fully_faithful_functor	axiom	Fully faithful functor	full and faithful functor	A functor that is both full and faithful, i.e., bijective on all hom-sets; an embedding of one category as a full subcategory.
s_essentially_surjective_functor	axiom	Essentially surjective functor		A functor F: C → D such that for every object d in D there exists c in C with F(c) ≅ d.
s_equivalence_iff_ff_eso	theorem	Equivalence iff fully faithful and essentially surjective		A functor is an equivalence of categories if and only if it is full, faithful, and essentially surjective on objects.
s_adjunctions_give_monads	theorem	Adjunctions give rise to monads		Every adjunction F ⊣ U gives rise to a monad (T = UF, η, μ = UεF) on the base category, where μ: T² ⇒ T is obtained from the counit.
s_filtered_category	axiom	Filtered category		A nonempty small category J where any two objects have a common upper bound and any two parallel morphisms are coequalized by some morphism.
s_filtered_colimits_commute_finite_limits	theorem	Filtered colimits commute with finite limits in Set		In Set, the canonical map colim_J lim_K F(j,k) → lim_K colim_J F(j,k) is a bijection when J is filtered and K is finite.
s_generator_separator	axiom	Generator (separator)	separator	An object G such that for distinct parallel morphisms f, g: A ⇉ B there exists h: G → A with f∘h ≠ g∘h; equivalently, Hom(G,−) is faithful.
s_cogenerator	axiom	Cogenerator (coseparator)	coseparator	An object Q such that for distinct f, g: A ⇉ B there exists h: B → Q with h∘f ≠ h∘g; dual of generator.
s_well_powered_category	axiom	Well-powered category		A category in which for every object the collection of subobjects forms a set rather than a proper class.
s_ends	axiom	End (category theory)	end | wedge	For a bifunctor S: C^op × C → D, the universal wedge: an object with morphisms e_c: E → S(c,c) satisfying the wedge condition for all f: c → c'.
s_coends	axiom	Coend (category theory)	coend | cowedge	For a bifunctor S: C^op × C → D, the universal cowedge: the colimit dual to the end, with morphisms S(c,c) → E satisfying the cowedge condition.
s_nat_trans_as_end	theorem	Natural transformations as an end		The set Nat(F,G) of natural transformations between functors F, G: C → D equals the end ∫_c Hom(F(c), G(c)).
s_limit_as_end	theorem	Limit as an end		For any functor F: J → C, its limit can be expressed as the end ∫_j F(j) with appropriate wedge conditions, unifying limits and ends.
s_co_yoneda_lemma	theorem	Co-Yoneda lemma (density formula)	density formula | Yoneda density	Every presheaf F: C^op → Set is a canonical colimit of representables: F ≅ colim_{(c,x) ∈ el(F)} Hom(−,c).
s_category_of_elements	axiom	Category of elements	Grothendieck construction for sets	For F: C → Set, the category whose objects are pairs (c, x ∈ F(c)) and morphisms (c,x) → (c',x') are f: c → c' with F(f)(x) = x'.
s_repr_functors_preserve_limits	theorem	Representable functors preserve limits		For any object A, the functor Hom(A,−): C → Set preserves all limits; dually, Hom(−,A) sends colimits to limits.
s_algebra_over_monad	axiom	Algebra over a monad (T-algebra)	T-algebra | Eilenberg-Moore algebra	A pair (A, h: TA → A) satisfying h∘η_A = id_A and h∘μ_A = h∘T(h); an action of the monad T on the object A.
s_eilenberg_moore_category	axiom	Eilenberg-Moore category	category of algebras over T	The category of all T-algebras and T-algebra morphisms for a monad T; the terminal resolution of the monad.
s_kleisli_category	axiom	Kleisli category		The category with objects of C and morphisms A → B given by C-morphisms A → T(B), with Kleisli composition using the monad multiplication; the initial resolutio
s_comparison_functor	state	Comparison functor (Eilenberg-Moore)		For an adjunction F ⊣ U inducing monad T, the unique functor K: D → C^T sending d to (U(d), U(ε_d)) comparing D with the category of algebras.
s_every_monad_from_adjunction	theorem	Every monad arises from an adjunction		Every monad (T, η, μ) arises from at least one adjunction: the Kleisli adjunction (initial) and the Eilenberg-Moore adjunction (terminal).
s_beck_monadicity_precise	theorem	Beck's monadicity theorem (precise form)	Beck's precise monadicity theorem | precise tripleability theorem	A functor U is monadic iff U has a left adjoint, reflects isomorphisms, and its domain has coequalizers of all U-split pairs which U preserves.
s_beck_monadicity_crude	theorem	Beck's monadicity theorem (crude form)	crude monadicity theorem	A functor U with left adjoint is monadic if U reflects isomorphisms and the domain has and U preserves all coequalizers.
s_split_coequalizer	axiom	Split coequalizer		A coequalizer with a section and a contracting homotopy making it absolute, i.e., preserved by every functor.
s_reflexive_coequalizer	axiom	Reflexive coequalizer		A coequalizer of a reflexive pair f, g: A ⇉ B where there exists s: B → A with f∘s = g∘s = id_B.
s_absolute_coequalizer	axiom	Absolute coequalizer		A coequalizer preserved by every functor; split coequalizers are the paradigmatic example.
s_monadic_functor	axiom	Monadic functor	tripleable functor	A functor U: D → C such that the comparison functor K: D → C^T is an equivalence, where T is the induced monad.
s_creation_limits_monadic	theorem	Creation of limits by monadic functors		If U: D → C is monadic and C has limits of type J, then D has limits of type J and U creates and preserves them.
s_free_algebra_monad	state	Free algebra over a monad		For a monad T, the free T-algebra on c is (Tc, μ_c: T²c → Tc); the image of the left adjoint in the Eilenberg-Moore adjunction.
t_monad_resolution	technique	Monad resolution		The simplicial resolution of a T-algebra by free algebras, used to compute derived functors and homological invariants of monadic structures.
s_completeness_eilenberg_moore	theorem	Completeness of Eilenberg-Moore category		If C is complete, then C^T is also complete, with limits created by the forgetful functor U^T: C^T → C.
t_beck_conditions_verification	technique	Beck's conditions verification		Proving a functor is monadic by checking existence of left adjoint, reflection of isomorphisms, and preservation/existence of relevant coequalizers.
s_monoid_object	axiom	Monoid object (in a monoidal category)		An object M in a monoidal category with multiplication μ and unit η satisfying associativity and unit axioms as commutative diagrams.
s_strict_monoidal_category	axiom	Strict monoidal category		A monoidal category in which the associator and unitors are identity natural transformations, so tensoring is strictly associative and unital.
s_pentagon_axiom	axiom	Pentagon axiom		The coherence condition requiring that two paths of associators from ((A⊗B)⊗C)⊗D to A⊗(B⊗(C⊗D)) coincide, forming a commutative pentagon.
s_triangle_axiom	axiom	Triangle axiom (unit coherence)		The coherence condition that the diagram involving the associator α_{A,I,B} and the unitors ρ_A⊗id_B and id_A⊗λ_B commutes.
s_monoidal_functor	axiom	Monoidal functor (lax)	lax monoidal functor	A functor F between monoidal categories with φ₀: I_D → F(I_C) and φ_{A,B}: FA⊗FB → F(A⊗B) satisfying coherence conditions.
s_strong_monoidal_functor	axiom	Strong monoidal functor		A monoidal functor whose structure morphisms φ₀ and φ_{A,B} are all isomorphisms.
s_strict_monoidal_functor	axiom	Strict monoidal functor		A monoidal functor whose structure morphisms are identities: F(A⊗B) = FA⊗FB and F(I) = I strictly.
s_closed_monoidal_category	axiom	Closed monoidal category		A monoidal category where −⊗B has a right adjoint [B,−] (internal hom) for every B, giving Hom(A⊗B, C) ≅ Hom(A, [B,C]).
s_cartesian_closed_category	axiom	Cartesian closed category	CCC	A category with finite products that is closed: −×B has a right adjoint B^(−) (exponential) for every B, with Hom(A×B, C) ≅ Hom(A, C^B).
s_internal_hom	axiom	Internal hom (exponential object)	exponential object | function object	In a closed monoidal category, the object [B,C] representing Hom(−⊗B, C), with evaluation ev: [B,C]⊗B → C.
s_eckmann_hilton_argument	theorem	Eckmann-Hilton argument		If a set carries two monoid structures with interchange, the two multiplications coincide and are commutative; a monoid in monoids is a commutative monoid.
s_category_of_monoids	state	Category of monoids Mon(C)		For a monoidal category C, the category whose objects are monoid objects and morphisms preserve multiplication and unit.
s_preadditive_category	axiom	Preadditive category	Ab-enriched category | Ab-category	A category enriched over abelian groups: each hom-set carries an abelian group structure and composition is bilinear.
s_biproduct	axiom	Biproduct		In a preadditive category, an object simultaneously a product and coproduct of A and B, with projections and injections satisfying π_i∘ι_j = δ_{ij}.
s_kernel_categorical	axiom	Kernel (categorical)		The equalizer of a morphism f: A → B and the zero morphism, universal among morphisms k with f∘k = 0.
s_cokernel_categorical	axiom	Cokernel (categorical)		The coequalizer of f: A → B and the zero morphism, universal among morphisms q with q∘f = 0.
s_image_categorical	axiom	Image (categorical)		The image of f: A → B is the kernel of the cokernel of f; the smallest subobject of B through which f factors.
s_freyd_mitchell_embedding	theorem	Freyd-Mitchell embedding theorem	Mitchell embedding theorem | Freyd embedding theorem	Every small abelian category admits an exact, fully faithful embedding into the category of modules over some ring.
s_regular_epimorphism	axiom	Regular epimorphism		A morphism that is the coequalizer of some pair of morphisms; in an abelian category, every epimorphism is regular.
s_extremal_epimorphism	axiom	Extremal epimorphism		An epimorphism e such that if e = m∘g with m mono, then m is an isomorphism; in abelian categories, coincides with regular and ordinary epimorphisms.
s_abelian_group_object	axiom	Abelian group object		A group object in a category with finite products: an object G with multiplication, unit, and inverse morphisms satisfying group axioms as commutative diagrams,
s_filtered_colimit_algebraic	theorem	Filtered colimits in algebraic categories		In categories of algebras, filtered colimits exist and are computed on underlying sets; the forgetful functor to Set preserves and creates them.
s_interchange_limits_colimits	theorem	Interchange of limits and colimits		Under appropriate conditions a canonical morphism colim lim → lim colim exists; precise conditions for it to be an isomorphism involve finality and filteredness
s_final_functor	axiom	Final functor (cofinal)	cofinal functor	A functor φ: J → K such that for every k in K the comma category (k ↓ φ) is nonempty and connected; colimits over K equal colimits over J.
s_final_functor_theorem	theorem	Final functor theorem		If φ: J → K is final and F: K → C is a diagram, then colim_K F ≅ colim_J F∘φ canonically whenever either exists.
s_reflexive_coeq_existence	theorem	Reflexive coequalizer existence		In many algebraic categories and categories of algebras over a monad, reflexive coequalizers exist even when general coequalizers may not.
s_limits_in_functor_categories	theorem	Limits in functor categories		If D has limits of type J, then [C,D] has limits of type J computed pointwise: (lim F_i)(c) = lim(F_i(c)).
s_creating_limits_theorem	theorem	Creating limits theorem		If U: D → C creates limits of type J and C has such limits, then D has them; combined with monadicity, establishes completeness of algebraic categories.
s_left_kan_extension	axiom	Left Kan extension		For K: A → B and F: A → C, the functor Lan_K F with universal η: F ⇒ (Lan_K F)∘K, initial among all factorizations of F through K.
s_right_kan_extension	axiom	Right Kan extension		For K: A → B and F: A → C, the functor Ran_K F with universal ε: (Ran_K F)∘K ⇒ F, terminal among all factorizations of F through K.
s_pointwise_kan_extension	axiom	Pointwise Kan extension		A Kan extension computable objectwise as a (co)limit: (Lan_K F)(b) = colim_{(K↓b)} F∘π or (Ran_K F)(b) = lim_{(b↓K)} F∘π.
s_left_kan_colimit_formula	theorem	Left Kan extension as colimit formula		When C has sufficient colimits, (Lan_K F)(b) = colim_{(K↓b)} F∘π where π: (K↓b) → A is the projection; the fundamental computational formula.
s_right_kan_limit_formula	theorem	Right Kan extension as limit formula		When C has sufficient limits, (Ran_K F)(b) = lim_{(b↓K)} F∘π, dual to the left Kan extension colimit formula.
s_kan_along_yoneda	theorem	Kan extension along Yoneda embedding		Left Kan extension along y: C → [C^op, Set] gives the free cocompletion: Lan_y F is the unique cocontinuous extension of F to the presheaf category.
s_every_functor_kan_extension	theorem	Every functor is a Kan extension of itself		For any functor F, F is both Lan_{Id} F and Ran_{Id} F: the Kan extension of F along the identity is F itself.
s_adjunctions_as_kan_extensions	theorem	Adjunctions as Kan extensions		F has right adjoint G iff Ran_F(Id) exists and is preserved by F, giving G = Ran_F(Id); dually, left adjoints are left Kan extensions of identities.
s_all_concepts_kan_extensions	theorem	All concepts are Kan extensions (Mac Lane)		Mac Lane's thesis that limits, colimits, adjunctions, ends, coends, and all fundamental constructions of category theory are expressible as Kan extensions.
s_dense_functor	axiom	Dense functor	adequate functor	A functor K: A → C such that every object of C is a canonical colimit of objects in K's image; equivalently, Lan_K(K) ≅ Id_C.
s_density_theorem_categorical	theorem	Density theorem (categorical)		A functor K: A → C is dense iff every object c is a canonical colimit c ≅ colim_{(K↓c)} K∘π; equivalently, Lan_K(K) ≅ Id_C.
t_kan_extension_computation	technique	Kan extension computation		Computing Kan extensions via the pointwise formula by identifying the relevant comma category and evaluating the (co)limit.
s_symmetric_monoidal_category	axiom	Symmetric monoidal category	SMC	A monoidal category with a symmetry σ_{A,B}: A⊗B → B⊗A satisfying σ_{B,A}∘σ_{A,B} = id and coherence with associator and unitors.
s_braided_monoidal_category	axiom	Braided monoidal category	BMC	A monoidal category with a braiding σ_{A,B}: A⊗B → B⊗A satisfying two hexagon axioms, but not necessarily σ² = id.
s_hexagon_axioms	axiom	Hexagon axioms		The two coherence conditions for a braiding: hexagonal diagrams involving the braiding σ and associator α must commute.
s_coherence_braided	theorem	Coherence for braided monoidal categories		Every diagram of canonical isomorphisms built from associator, unitors, and braiding commutes; the free braided monoidal category is described by braids.
s_coherence_symmetric	theorem	Coherence for symmetric monoidal categories		In a symmetric monoidal category, every diagram of canonical isomorphisms built from associators, unitors, and symmetries commutes.
s_braid_group_categorical	state	Braid group from braided monoidal categories		The endomorphisms of the n-fold tensor product in the free braided monoidal category on one generator form the braid group B_n.
s_yang_baxter_categorical	theorem	Yang-Baxter equation (categorical)		The braiding satisfies (σ⊗id)(id⊗σ)(σ⊗id) = (id⊗σ)(σ⊗id)(id⊗σ), a consequence of the hexagon axioms and naturality.
s_enriched_category	axiom	Enriched category	V-category | V-enriched category	A category whose hom-sets are replaced by objects of a monoidal category V, with composition and identity V-morphisms satisfying associativity and unit axioms i
s_2_category	axiom	2-category	strict 2-category	A category enriched over Cat: objects, 1-morphisms, and 2-morphisms with vertical and horizontal composition satisfying the interchange law.
s_elementary_topos	axiom	Elementary topos	topos | elementary topos	A cartesian closed category with a subobject classifier Ω: an object with true: 1 → Ω such that every subobject is uniquely classified by a characteristic morph
s_subobject_classifier	axiom	Subobject classifier		An object Ω with true: 1 → Ω such that every monomorphism m: S ↣ A is the pullback of true along a unique characteristic morphism χ_m: A → Ω.
s_grothendieck_topos	axiom	Grothendieck topos	sheaf topos	A category of sheaves on a site; equivalently, a left exact reflective subcategory of a presheaf category; a Grothendieck topos is an elementary topos.
s_grothendieck_topology	axiom	Grothendieck topology		An assignment to each object c of covering sieves stable under pullback, transitive, and containing the maximal sieve.
s_sieve	axiom	Sieve		A subfunctor of Hom(−,c); equivalently, a collection S of morphisms with codomain c closed under precomposition.
s_covering_family	axiom	Covering family		A family of morphisms generating a covering sieve in a Grothendieck topology; the site-level data from which the topology is specified.
t_naturality_argument	technique	Naturality argument		Exploiting the naturality square by applying it to strategically chosen morphisms (often identities) to deduce equalities.
t_comma_category_construction	technique	Comma category construction technique		Encoding relationships by forming the comma category and identifying limits, initial objects, or universal properties in it.
t_representability_criterion	technique	Representability criterion		Checking representability via the category of elements: F is representable iff el(F) has a terminal object.
t_comparison_functor_construction	technique	Comparison functor construction		Constructing the comparison functor K: D → C^T and analyzing when it is an equivalence to recognize algebraic categories.
t_coherence_proof	technique	Coherence proof technique		Mac Lane's method: define the free monoidal category, show every diagram of canonical isomorphisms commutes, transfer via universal property.
t_transfinite_composition	technique	Transfinite composition		Constructing objects by iterating through transfinite ordinals with colimits at limit ordinals; used for free algebras and solution set verification.
t_adjoint_functor_construction	technique	Adjoint functor construction technique		Constructing a left adjoint by verifying the solution set condition and limit preservation, directly building universal arrows, or exhibiting a Kan extension.
t_godement_calculus	technique	Godement calculus		The systematic calculus using horizontal and vertical composition with the interchange law for algebraic manipulation of functors and natural transformations.
s_dual_contragredient_representation	state	Dual (contragredient) representation	contragredient representation | dual representation	The representation on the dual space V* given by (g.f)(v) = f(g^{-1}.v), also called the contragredient representation.
s_hom_representation	state	Hom representation	Hom(V,W) representation	The G-representation on Hom(V,W) defined by (g.phi)(v) = g.phi(g^{-1}.v), naturally isomorphic to V* tensor W.
s_space_of_invariants	state	Space of invariants V^G	G-invariants | fixed subspace	The subspace V^G = {v in V : g.v = v for all g in G} of vectors fixed by every group element, equal to Hom_G(trivial, V).
s_second_orthogonality_relation	theorem	Second orthogonality relation for characters	column orthogonality	For a finite group G, sum_chi chi(g)conj(chi(h)) = |C_G(g)| if g and h are conjugate and 0 otherwise, where the sum runs over all irreducible characters.
s_character_determines_representation	theorem	Character determines representation		Two finite-dimensional complex representations of a finite group are isomorphic if and only if they have the same character.
s_projection_formula_representation	state	Projection formula for representations	idempotent projection formula	The projection onto the isotypic component of type V_i in a representation V is p_i = (dim V_i / |G|) sum_{g in G} conj(chi_i(g)) rho(g).
s_representations_of_abelian_groups	theorem	Representations of abelian groups		Every irreducible complex representation of a finite abelian group is one-dimensional; the number of such equals |G|, and they form the character group G-hat.
s_standard_representation_of_S_n	state	Standard representation of S_n	standard representation of symmetric group	The (n-1)-dimensional irreducible representation of S_n obtained as the complement of the trivial subrepresentation in the permutation representation on C^n.
s_sign_representation	state	Sign (alternating) representation	alternating representation | signature representation | sgn representation	The one-dimensional representation sgn: S_n -> {+1,-1} sending each permutation to its sign; the unique nontrivial one-dimensional representation of S_n for n >
s_character_of_induced_representation	theorem	Character of induced representation	induced character formula	The character of Ind_H^G(sigma) at g in G is chi^G(g) = (1/|H|) sum_{t in G, t^{-1}gt in H} chi_sigma(t^{-1}gt).
s_conjugate_partition	state	Conjugate (transpose) partition	transpose partition | dual partition	The partition lambda' obtained by transposing the Young diagram of lambda, interchanging rows and columns.
s_row_group_column_group	state	Row group and column group of a Young tableau	row stabilizer | column stabilizer	For a Young tableau T, the row group R_T is the subgroup of permutations preserving rows, and the column group C_T preserves columns; they generate the Young sy
s_young_symmetrizer	state	Young symmetrizer	Young idempotent	The element c_T = a_T b_T in C[S_d] where a_T = sum_{sigma in R_T} sigma and b_T = sum_{tau in C_T} sgn(tau) tau; the image c_T C[S_d] is an irreducible S_d-mod
s_specht_module	state	Specht module V_lambda	Specht representation	The irreducible representation of S_d corresponding to a partition lambda, constructed as the image of the Young symmetrizer c_T in C[S_d].
s_branching_rule_S_d	theorem	Branching rule for S_d	restriction rule for symmetric groups	The restriction of the Specht module V_lambda from S_d to S_{d-1} decomposes as Res V_lambda = direct_sum V_mu where mu ranges over all partitions obtained by r
s_sign_twist_specht	theorem	Sign twist of Specht modules		Tensoring the Specht module V_lambda with the sign representation gives V_{lambda'}, where lambda' is the conjugate partition.
s_frobenius_character_formula	theorem	Frobenius character formula		The character value chi^lambda(mu) of the Specht module V_lambda on the conjugacy class of cycle type mu equals the coefficient connecting representation theory
s_murnaghan_nakayama_rule	theorem	Murnaghan--Nakayama rule	Murnaghan-Nakayama formula	A combinatorial formula for the character chi^lambda(mu) of S_d as an alternating sum over rim-hook (border-strip) tableaux of shape lambda and type mu.
s_youngs_rule	theorem	Young's rule		The permutation module M^mu = Ind_{S_mu}^{S_d}(trivial) decomposes as M^mu = direct_sum K_{lambda,mu} V_lambda where K_{lambda,mu} are the Kostka numbers.
s_semistandard_young_tableau	axiom	Semistandard Young tableau	SSYT | column-strict tableau	A filling of a Young diagram with positive integers that is weakly increasing along rows and strictly increasing down columns.
s_young_subgroup	state	Young subgroup	parabolic subgroup of S_d	For a composition mu of d, the subgroup S_mu = S_{mu_1} x ... x S_{mu_k} embedded in S_d as permutations preserving the blocks of sizes mu_i.
s_representation_ring_of_symmetric_groups	state	Representation ring of symmetric groups		The graded ring R = direct_sum_d R(S_d) with multiplication given by induction of external tensor products, isomorphic to the ring of symmetric functions Lambda
s_center_of_group_algebra	state	Center of the group algebra	class sums	The center Z(C[G]) consists of class sums for each conjugacy class C; it has dimension equal to the number of conjugacy classes and acts on irreducible represen
s_idempotents_in_group_algebra	state	Primitive idempotents in the group algebra	primitive central idempotents	The central primitive idempotents e_i = (dim V_i / |G|) sum_{g} chi_i(g^{-1}) g in C[G] satisfy e_i e_j = delta_{ij} e_i and sum e_i = 1, giving the Artin-Wedde
s_schur_weyl_duality	theorem	Schur--Weyl duality	Schur-Weyl duality	The actions of GL(V) and S_d on V^{tensor d} generate each other's commutant, yielding V^{tensor d} = direct_sum S^lambda(V) tensor V_lambda as a (GL(V) x S_d)-
s_plethysm	state	Plethysm		The operation on symmetric functions corresponding to composition of polynomial functors: S^lambda(S^mu(V)), whose expansion in the Schur basis is combinatorial
s_matrix_lie_group	axiom	Matrix Lie group	linear Lie group	A closed subgroup of GL(n,C) or GL(n,R); equivalently a Lie group realized as a group of invertible matrices.
s_one_parameter_subgroup	state	One-parameter subgroup of a Lie group		A smooth homomorphism gamma: R -> G from the additive reals to a Lie group, necessarily of the form gamma(t) = exp(tX) for a unique X in the Lie algebra.
s_representation_of_lie_group	axiom	Representation of a Lie group	Lie group representation	A smooth homomorphism rho: G -> GL(V) from a Lie group to the general linear group of a finite-dimensional vector space.
s_lie_bracket_of_matrices	state	Lie bracket of matrices	matrix commutator bracket	The commutator bracket [A,B] = AB - BA on gl(n), defining the Lie algebra structure on the tangent space at the identity of a matrix Lie group.
s_semisimple_direct_sum_of_simples	theorem	Semisimple Lie algebra is a direct sum of simple ideals		A semisimple Lie algebra decomposes uniquely as a direct sum of simple ideals; conversely any direct sum of simple Lie algebras is semisimple.
s_preservation_of_jordan_decomposition	theorem	Preservation of Jordan decomposition under representations		Every representation of a semisimple Lie algebra preserves the abstract Jordan decomposition: phi(x_s) = phi(x)_s and phi(x_n) = phi(x)_n.
s_lie_group_lie_algebra_rep_correspondence	theorem	Correspondence between Lie group and Lie algebra representations	Lie's third theorem for representations	For a simply connected Lie group G with Lie algebra g, differentiation gives an equivalence between finite-dimensional representations of G and of g.
s_sl2_C	state	The Lie algebra sl_2(C)	sl(2,C) | sl_2	The 3-dimensional simple complex Lie algebra of 2x2 traceless matrices, with standard basis H, E, F satisfying [H,E]=2E, [H,F]=-2F, [E,F]=H.
s_standard_basis_sl2	state	Standard basis {H,E,F} of sl_2		The basis H = diag(1,-1), E = e_{12}, F = e_{21} of sl_2(C) with relations [H,E]=2E, [H,F]=-2F, [E,F]=H.
s_heisenberg_algebra	state	Heisenberg algebra	Heisenberg Lie algebra	The nilpotent Lie algebra with basis {p,q,z} and sole nontrivial bracket [p,q]=z; the Lie algebra of the Heisenberg group.
s_irreducible_representations_of_sl2	theorem	Classification of irreducible representations of sl_2	irreps of sl(2)	For each non-negative integer n, there is a unique (n+1)-dimensional irreducible representation V_n of sl_2(C) with weights n, n-2, ..., -n; these exhaust all f
s_highest_weight_vector	axiom	Highest weight vector	maximal vector | primitive vector	A nonzero vector v in a representation annihilated by all positive root vectors (e_alpha.v = 0 for all alpha > 0), generating a highest weight submodule.
s_sl3_C	state	The Lie algebra sl_3(C)	sl(3,C) | sl_3	The 8-dimensional simple complex Lie algebra of 3x3 traceless matrices, of type A_2 with rank 2; the prototypical example for developing semisimple Lie algebra 
s_root_and_root_space	axiom	Root and root space	root of a Lie algebra	A root alpha is a nonzero weight of the adjoint representation; its root space g_alpha = {x in g : [h,x] = alpha(h)x for all h in h} is one-dimensional for semi
s_weight_diagram	state	Weight diagram	weight lattice diagram	The graphical depiction of the weights of a representation as lattice points in h*_R with multiplicities, displaying representation structure.
s_partial_order_on_weights	state	Partial order on weights	dominance order on weights	The partial order mu <= lambda defined by lambda - mu being a non-negative integer combination of simple roots; every irreducible highest weight module has a un
s_weight_lattice	state	Weight lattice	integral weight lattice	The lattice P = {lambda in h* : <lambda, alpha^v> in Z for all coroots} of integral weights, containing the root lattice Q as a sublattice.
s_weyl_group_action_on_weights	theorem	Weyl group action on weights		The Weyl group W preserves the set of weights of any representation, with equal multiplicities on each W-orbit.
s_weyl_vector_rho	state	Weyl vector rho	half-sum of positive roots	The half-sum of positive roots rho = (1/2) sum_{alpha > 0} alpha = omega_1 + ... + omega_n; satisfies <rho, alpha_i^v> = 1 and appears in Weyl character and dim
s_rank_of_lie_algebra	state	Rank of a Lie algebra	Lie algebra rank	The dimension of a Cartan subalgebra of a semisimple Lie algebra, equivalently the number of simple roots or Dynkin diagram nodes.
s_fundamental_representations_sl_n	state	Fundamental representations of sl_n		The n-1 fundamental representations of sl_n(C) are the exterior powers Lambda^k(C^n) for k = 1,...,n-1, with highest weights omega_k.
s_irreps_sl_n_via_young_diagrams	state	Irreducible representations of sl_n via Young diagrams		The irreducible representations of sl_n are indexed by partitions with at most n-1 parts, realized by the Schur functor S^lambda(C^n).
s_weyl_dimension_formula_sl_n	theorem	Weyl dimension formula for sl_n		For sl_n(C) with highest weight lambda, dim V_lambda = prod_{1<=i<j<=n} (l_i - l_j)/(j - i) where l_i = lambda_i + n - i.
s_fundamental_representations_so_n	state	Fundamental representations of so_n		The fundamental representations of so_n(C): exterior powers and spin/half-spin representations, varying by type B_m or D_m.
s_fundamental_representations_sp_2n	state	Fundamental representations of sp_{2n}		The n fundamental representations of sp_{2n}(C) obtained as kernels of contraction maps on exterior powers of the standard representation.
s_exceptional_isomorphisms_low_rank	theorem	Exceptional isomorphisms of low-rank Lie algebras	accidental isomorphisms	The coincidences A_1=B_1=C_1, B_2=C_2, D_2=A_1xA_1, A_3=D_3 arising from Dynkin diagram coincidences in small rank.
s_spin_representation_odd	state	Spin representation of so_{2m+1}	spinor representation of type B	The unique 2^m-dimensional irreducible representation of so_{2m+1}(C) with highest weight omega_m, constructed via the Clifford algebra.
s_half_spin_representations	state	Half-spin representations of so_{2m}	chiral spinor representations | Weyl spinors	The two 2^{m-1}-dimensional irreducible representations S+ and S- of so_{2m}(C) corresponding to the two chiralities of the even Clifford algebra.
s_triality_so_8	theorem	Triality for so_8	D_4 triality	The Dynkin diagram D_4 has an order-3 automorphism that cyclically permutes the vector and two half-spin representations of so_8, giving an outer automorphism o
s_weyl_denominator_formula	theorem	Weyl denominator formula	Weyl denominator identity	The identity sum_{w in W} (-1)^{l(w)} e^{w(rho)} = e^rho prod_{alpha > 0} (1 - e^{-alpha}), the lambda=0 case of the Weyl character formula.
s_steinbergs_formula	theorem	Steinberg's formula for tensor product decomposition	Steinberg tensor product multiplicity formula	The multiplicity of L(nu) in L(lambda) tensor L(mu) expressed via an alternating sum over pairs of Weyl group elements using the Kostant partition function.
s_klimyks_formula	theorem	Klimyk's formula	Klimyk's tensor product formula	The tensor product L(lambda) tensor L(mu) decomposed by summing over weights of L(mu) with multiplicities, giving a quick but redundant formula.
s_cartan_involution	axiom	Cartan involution		An involutive automorphism theta of a real semisimple Lie algebra such that B_theta(X,Y) = -B(X,theta(Y)) is positive definite; gives the Cartan decomposition g
s_maximal_compact_subgroup	state	Maximal compact subgroup		A compact subgroup K of a connected semisimple Lie group G not contained in any larger compact subgroup; K = G^theta is unique up to conjugacy and G/K is contra
s_real_forms_classification	state	Classification of real forms of a complex semisimple Lie algebra	real forms of semisimple Lie algebras	Real forms are classified by conjugacy classes of involutions; they include the compact real form and one or more non-compact forms including the split form.
s_split_real_form	state	Split real form	maximally split real form	The real form of a complex semisimple Lie algebra in which a Cartan subalgebra is entirely real (maximally split); e.g., sl_n(R) for sl_n(C).
s_satake_diagram	axiom	Satake diagram	Tits-Satake diagram	A decorated Dynkin diagram encoding a real form: painted/unpainted nodes and arrows classifying real forms of complex semisimple Lie algebras up to isomorphism.
s_longest_element_w0	state	Longest element w_0 of the Weyl group	w_0 | longest Weyl group element	The unique element w_0 in W of maximal length with w_0(Phi^+) = Phi^-; its length equals |Phi^+| and it sends highest weights to negatives of dual highest weigh
s_dual_representation_via_w0	theorem	Dual representation via the longest element		The dual of L(lambda) is L(-w_0(lambda)); L(lambda) is self-dual iff -w_0(lambda) = lambda.
s_prv_theorem	theorem	Parthasarathy--Ranga Rao--Varadarajan (PRV) theorem	PRV theorem | PRV conjecture	For dominant weights lambda, mu, the irreducible L(lambda + w(mu)) occurs in L(lambda) tensor L(mu) for each w in W, with L(lambda + w_0(mu)) appearing with mul
s_exceptional_lie_algebra_G2	state	Exceptional Lie algebra G_2	G2	The 14-dimensional exceptional simple complex Lie algebra of rank 2, the automorphism algebra of the octonions.
s_exceptional_lie_algebra_F4	state	Exceptional Lie algebra F_4	F4	The 52-dimensional exceptional simple complex Lie algebra of rank 4, the automorphism algebra of the exceptional Jordan algebra.
s_exceptional_lie_algebra_E6	state	Exceptional Lie algebra E_6	E6	The 78-dimensional exceptional simple complex Lie algebra of rank 6, with a 27-dimensional fundamental representation.
s_exceptional_lie_algebra_E7	state	Exceptional Lie algebra E_7	E7	The 133-dimensional exceptional simple complex Lie algebra of rank 7, with a 56-dimensional fundamental representation.
s_exceptional_lie_algebra_E8	state	Exceptional Lie algebra E_8	E8	The 248-dimensional exceptional simple complex Lie algebra of rank 8; its smallest representation is the adjoint (248-dimensional).
t_decomposition_via_inner_products	technique	Decomposition via character inner products		Compute multiplicities m_i = <chi_V, chi_i> to decompose a representation V into irreducible summands.
t_weight_string_analysis	technique	Weight string analysis	alpha-string through mu	Analyze the string of weights through mu in the direction of root alpha using the sl_2-subalgebra, constraining multiplicities and proving integrality.
t_clifford_construction_of_spin_reps	technique	Clifford algebra construction of spin representations		Construct spin representations via the Clifford algebra Cl(V,Q), identifying so(V) with Cl^2(V) and letting it act on a minimal left ideal.
s_character_of_tensor_product	theorem	Character of tensor product		The character of a tensor product of representations satisfies chi_{V \otimes W}(g) = chi_V(g) \cdot chi_W(g) for all g in G.
s_character_of_dual_representation	theorem	Character of dual representation		The character of the contragredient representation satisfies chi_{V^*}(g) = \overline{chi_V(g)} = chi_V(g^{-1}) for all g in G.
s_decomposition_multiplicity_formula	theorem	Decomposition multiplicity formula		The multiplicity of an irreducible representation V_i in a representation V equals the inner product n_i = (chi_V, chi_{V_i}) of their characters.
s_regular_representation_decomposition	theorem	Regular representation decomposition		The regular representation of a finite group G decomposes as the direct sum of all irreducible representations, each appearing with multiplicity equal to its di
s_dimension_divides_group_order	theorem	Dimension divides group order		The dimension of every irreducible complex representation of a finite group G divides the order |G|.
s_character_irreducibility_criterion	theorem	Character criterion for irreducibility		A representation V of a finite group is irreducible if and only if (chi_V, chi_V) = 1.
s_brauer_character_orthonormality	theorem	Orthonormality of Brauer characters		The Brauer characters of the simple k[G]-modules form an orthonormal set with respect to the inner product restricted to p-regular elements.
s_brauer_nesbitt_theorem	theorem	Brauer-Nesbitt theorem		Two finite-dimensional k[G]-modules have the same composition factors (with multiplicities) if and only if they have the same Brauer character.
s_number_simples_equals_p_regular_classes	theorem	Number of simple modules equals p-regular classes		The number of isomorphism classes of simple k[G]-modules equals the number of p-regular conjugacy classes of G.
s_cartan_matrix_symmetric_positive_definite	theorem	Cartan matrix is symmetric positive definite		The Cartan matrix C = D^T D of a p-modular system for a finite group is symmetric and positive definite, with determinant equal to a power of p.
s_p_regular_element	axiom	p-regular element		An element g of a finite group G whose order is coprime to the prime p.
s_p_singular_element	axiom	p-singular element		An element g of a finite group G whose order is divisible by the prime p.
s_brauer_character	axiom	Brauer character		For a representation of G over a field k of characteristic p, the class function on p-regular elements defined by lifting eigenvalues to characteristic zero and
s_block_of_group_algebra	axiom	Block of group algebra		An indecomposable two-sided ideal B_i of k[G] as a direct summand algebra, so that k[G] = B_1 x ... x B_r and each indecomposable module belongs to exactly one 
s_defect_group	axiom	Defect group		A p-subgroup D of G associated to a block B of k[G], maximal with the property that e_B lies in the image of the relative trace map from C_G(D), unique up to G-
s_p_modular_system	axiom	p-modular system		A triple (K, A, k) where A is a complete discrete valuation ring with field of fractions K of characteristic 0 and residue field k of characteristic p, used to 
s_decomposition_map	state	Decomposition map		The group homomorphism d: R_K(G) -> R_k(G) sending a characteristic-zero representation to its reduction modulo p via an integral form.
s_decomposition_matrix	state	Decomposition matrix		The matrix D = (d_{ij}) where d_{ij} is the multiplicity of the j-th simple k[G]-module as a composition factor in the reduction modulo p of the i-th ordinary i
s_cartan_matrix_modular_representations	state	Cartan matrix for modular representations		The matrix C = (c_{ij}) where c_{ij} is the multiplicity of the j-th simple k[G]-module as a composition factor of the projective cover of the i-th simple k[G]-
s_projective_indecomposable_module	state	Projective indecomposable module		An indecomposable projective k[G]-module, which is the projective cover of a simple module and whose isomorphism classes biject with simple k[G]-modules.
s_cde_triangle	state	CDE triangle		The commutative triangle of group homomorphisms R_K(G) --d--> R_k(G) --c--> R_k(G) --e--> R_K(G) relating the decomposition map d, the Cartan map c, and the lif
s_cartan_map	state	Cartan map		The group homomorphism c: P_k(G) -> R_k(G) from the Grothendieck group of projective k[G]-modules to R_k(G), sending a projective module to its class in the Gro
s_lifting_map_brauer	state	Lifting map (Brauer)		The group homomorphism e: R_k(G) -> R_K(G) sending the class of a simple k[G]-module to the virtual character obtained by lifting its Brauer character.
s_grothendieck_group_modular	state	Grothendieck group of modular representations		The Grothendieck group R_k(G) of finite-dimensional k[G]-modules modulo short exact sequences, where k is a field of characteristic p dividing |G|.
s_generalized_decomposition_numbers	state	Generalized decomposition numbers		The algebraic integers d^u_{chi,phi} expressing the value of an ordinary character chi at a p-singular element us as chi(us) = sum_phi d^u_{chi,phi} phi(u), whe
s_defect_zero_block	state	Defect zero block		A block whose defect group is trivial, containing a single ordinary irreducible character whose degree is divisible by the full p-part of |G| and a single simpl
s_block_idempotent	state	Block idempotent		A primitive central idempotent e_i in k[G] corresponding to a block B_i, satisfying k[G]e_i = B_i and sum e_i = 1.
s_kernel_of_representation	state	Kernel of representation		The normal subgroup ker(rho) = {g in G : rho(g) = id_V} consisting of group elements acting trivially in a representation rho: G -> GL(V).
s_inflation_of_representation	state	Inflation of representation		For a normal subgroup N of G and a representation rho of G/N, the representation inf(rho) of G obtained by composing rho with the quotient map G -> G/N.
s_projective_cover_modular	state	Projective cover		For a finitely generated k[G]-module M, the projective module P(M) with an essential epimorphism P(M) -> M, unique up to isomorphism, whose existence is guarant
t_semidirect_product_representations	technique	Semi-direct product representation construction		Classifies irreducible representations of a semi-direct product by choosing an orbit of H on irreducible representations of N, picking a representative with its
t_averaging_projection	technique	Averaging projection technique		Constructs a G-equivariant projection or intertwiner by averaging an arbitrary linear map over the group: pi = (1/|G|) sum_{g in G} rho(g).
t_reduction_mod_p_representations	technique	Reduction modulo p for representations		Obtains a modular representation by choosing a G-stable A-lattice L in a K[G]-module V and reducing modulo the maximal ideal m to get the k[G]-module L/mL.
s_general_linear_lie_algebra	axiom	General linear Lie algebra gl(V)		The Lie algebra End(V) with bracket [x,y] = xy − yx, the associative algebra of all endomorphisms of a vector space V viewed as a Lie algebra.
s_special_linear_lie_algebra_sl_n	axiom	Special linear Lie algebra sl(n,F)	sl_n | type A_n Lie algebra	The Lie subalgebra of gl(n,F) consisting of traceless matrices, simple for n ≥ 2 when char F does not divide n.
s_classical_lie_algebras_bcd	axiom	Classical Lie algebras (types B, C, D)	orthogonal Lie algebra | symplectic Lie algebra | so(n) | sp(2n)	The orthogonal Lie algebras so(2l+1,F) (type B_l), symplectic sp(2l,F) (type C_l), and orthogonal so(2l,F) (type D_l), defined as matrices preserving a nondegen
s_toral_subalgebra	axiom	Toral subalgebra		A subalgebra of a semisimple Lie algebra consisting entirely of ad-semisimple (ad-diagonalizable) elements; always abelian.
s_maximal_toral_subalgebra	axiom	Maximal toral subalgebra		A toral subalgebra not properly contained in any other toral subalgebra; in a split semisimple Lie algebra over an algebraically closed field, equals the Cartan
s_regular_element_lie_algebra	axiom	Regular element of a Lie algebra		An element x of a Lie algebra L whose centralizer has the minimal possible dimension among all elements; equivalently, α(h) ≠ 0 for all roots α when x = h ∈ H.
s_saturated_set_of_weights	axiom	Saturated set of weights		A subset S of the weight lattice such that if λ ∈ S and α ∈ Φ with ⟨λ,α⟩ > 0, then λ − α ∈ S; the set of weights of any finite-dimensional module is saturated.
s_admissible_lattice_lie	axiom	Admissible lattice		A ℤ-lattice M in a finite-dimensional L-module V stable under all divided powers e_α^(n) = e_α^n/n! for all roots α and n ≥ 0, enabling reduction modulo a prime
s_derived_series_lie	state	Derived series of a Lie algebra		The descending chain L^(0) = L, L^(i+1) = [L^(i), L^(i)] of iterated commutator ideals; its termination at zero defines solvability.
s_lower_central_series_lie	state	Lower central series of a Lie algebra		The descending chain L^0 = L, L^i = [L, L^(i−1)] used to define nilpotency; L is nilpotent iff L^n = 0 for some n.
s_nilradical_of_lie_algebra	state	Nilradical of a Lie algebra		The largest nilpotent ideal of a Lie algebra L, containing [L, Rad(L)] and contained in the radical.
s_reflection_s_alpha	state	Reflection s_α in a root system	simple reflection | Weyl reflection	The orthogonal reflection s_α(β) = β − ⟨β,α⟩α in the hyperplane perpendicular to the root α; generates the Weyl group and preserves the root system.
s_root_string_through_beta	state	Root string	α-string through β	The α-string through β: the set {β + iα : −r ≤ i ≤ q} for maximal r,q ≥ 0 with β−rα, β+qα ∈ Φ, satisfying r − q = ⟨β,α⟩.
s_height_of_root	state	Height of a root		For a root β = Σ k_i α_i expressed in simple roots, ht(β) = Σ k_i; simple roots have height 1 and the highest root has maximal height.
s_fundamental_weyl_chamber	state	Fundamental Weyl chamber	dominant chamber	The open cone C(Δ) = {v ∈ E : (v,α_i) > 0 for all simple roots α_i ∈ Δ}, a fundamental domain for the Weyl group action on the Euclidean space.
s_length_function_weyl_group	state	Length function on the Weyl group		The function l(w) giving the minimum number of simple reflections needed to express w ∈ W; equivalently, l(w) equals the number of positive roots sent to negati
s_automorphism_group_of_root_system	state	Automorphism group of a root system	diagram automorphisms	Aut(Φ) = W ⋊ Γ where W is the Weyl group and Γ is the group of Dynkin diagram automorphisms (graph automorphisms of the Dynkin diagram).
s_triangular_decomposition	state	Triangular decomposition		The direct sum decomposition L = N⁻ ⊕ H ⊕ N⁺ of a semisimple Lie algebra, where N⁺ = ⊕_{α>0} L_α, N⁻ = ⊕_{α<0} L_α, and H is the Cartan subalgebra.
s_filtration_of_enveloping_algebra	state	Filtration of the universal enveloping algebra		The canonical filtration U_0 ⊆ U_1 ⊆ ··· of U(L) by degree, whose associated graded algebra gr(U(L)) is isomorphic to the symmetric algebra S(L) by the PBW theo
s_center_of_enveloping_algebra	state	Center of the universal enveloping algebra		Z(U(L)) for semisimple L is a polynomial algebra in l = rank(L) generators, isomorphic to S(H*)^W via the Harish-Chandra homomorphism.
s_simple_highest_weight_module	state	Simple highest weight module L(λ)	irreducible highest weight module	The unique irreducible quotient L(λ) = M(λ)/N(λ) of the Verma module M(λ); finite-dimensional if and only if λ is a dominant integral weight.
s_dot_action_shifted_weyl	state	Dot action (shifted Weyl group action)	shifted action | ρ-shifted action	The shifted action w · λ = w(λ + ρ) − ρ of the Weyl group W on H*, under which the Harish-Chandra central character is constant on W-orbits.
s_character_of_verma_module	state	Character of a Verma module		ch(M(λ)) = e(λ) / ∏_{α∈Φ⁺}(1 − e(−α)) = e(λ) · P where P is the Kostant partition function generating series.
s_formal_characters_ring	state	Ring of formal characters		The subring ℤ[Λ]^W of Weyl-group-invariant elements of the group ring ℤ[Λ], where ch(V⊗W) = ch(V)·ch(W) and ch(V⊕W) = ch(V)+ch(W).
s_integral_structure_constants	state	Integral structure constants of a Chevalley basis		The structure constants N_{α,β} satisfying [e_α, e_β] = N_{α,β} e_{α+β} with N_{α,β} = ±(r+1) where r is the largest integer with β−rα ∈ Φ; always integers.
s_kostant_z_form	state	Kostant's ℤ-form	Kostant integral form	The ℤ-subalgebra U_ℤ of U(L) generated by the divided powers e_α^(n) = e_α^n/n!, providing an integral form of the universal enveloping algebra compatible with 
s_root_subgroup	state	Root subgroup of a Chevalley group		The one-parameter subgroup X_α = {x_α(t) : t ∈ k} ⊂ G(Φ,k) corresponding to each root α, where x_α(t) = exp(t·e_α); the root subgroups generate the Chevalley gr
s_chevalley_group_steinberg_relations	state	Steinberg relations in Chevalley groups	Chevalley commutator formula	The commutator relations [x_α(t), x_β(u)] = ∏ x_{iα+jβ}(c_{ij} t^i u^j) for α+β ≠ 0 in a Chevalley group, with integer constants c_{ij} determined by the root s
s_dual_coxeter_number	state	Dual Coxeter number		The integer h∨ = 1 + ⟨ρ, θ∨⟩ where θ is the highest root; equals (dim L)/rank − 1 and governs the normalization of the Killing form.
s_highest_root_maximal	state	Highest root	maximal root | longest root	The unique root θ of maximum height in a root system Φ, satisfying ⟨θ, α_i⟩ ≥ 0 for all simple roots α_i; its height equals h − 1 where h is the Coxeter number.
s_invariant_bilinear_form_on_module	state	Invariant bilinear form on a module		A nondegenerate L-invariant symmetric bilinear form on the irreducible module L(λ) exists if and only if λ = −w_0(λ), where w_0 is the longest element of the We
s_minuscule_weight	state	Minuscule weight		A dominant weight ω such that the Weyl group orbit W·ω equals the full set of weights of L(ω), i.e., all weights have multiplicity 1 and form a single orbit.
s_simultaneous_triangularization_theorem	theorem	Simultaneous triangularization (corollary of Lie's theorem)		Every solvable Lie subalgebra of gl(V) over an algebraically closed field of characteristic 0 can be simultaneously upper-triangularized with respect to some ba
s_semisimple_decomposition_into_simples	theorem	Decomposition of semisimple Lie algebras into simple ideals		A finite-dimensional semisimple Lie algebra decomposes uniquely as a direct sum L = L_1 ⊕ ··· ⊕ L_k of simple ideals, and every ideal of L is a sum of some subs
s_all_derivations_inner_semisimple	theorem	All derivations are inner (semisimple case)		For a semisimple Lie algebra L, every derivation is inner: Der(L) = ad(L), so Aut(L)₀ = Int(L).
s_root_decomposition_properties	theorem	Properties of the root space decomposition		For a semisimple Lie algebra: [L_α, L_β] ⊆ L_{α+β}; L_α and L_β are Killing-form-orthogonal unless α+β=0; each root space L_α is 1-dimensional; and α ∈ Φ implie
s_existence_conjugacy_of_bases	theorem	Existence and conjugacy of bases of root systems		Every root system possesses a base (simple system), and any two bases are conjugate under the action of the Weyl group; bases correspond bijectively to Weyl cha
s_simple_reflections_generate_weyl_group	theorem	Simple reflections generate the Weyl group		The Weyl group W is generated by the simple reflections s_{α_1}, ..., s_{α_l}; together with the relations (s_i s_j)^{m_{ij}} = 1 they form a finite Coxeter gro
s_cartan_matrix_determines_root_system	theorem	Cartan matrix determines the root system		The Cartan matrix A = (⟨α_i,α_j⟩) of a base determines the root system up to isomorphism, and equivalently determines the Weyl group and the Lie algebra up to i
s_isomorphism_theorem_semisimple_lie_algebras	theorem	Isomorphism theorem for semisimple Lie algebras		Two semisimple Lie algebras over an algebraically closed field of characteristic 0 are isomorphic if and only if their root systems are isomorphic.
s_conjugacy_of_borel_subalgebras	theorem	Conjugacy of Borel subalgebras		Any two Borel subalgebras (maximal solvable subalgebras) of a semisimple Lie algebra are conjugate under the group Int(L) of inner automorphisms.
s_chevalley_invariant_polynomial_theorem	theorem	Chevalley's theorem on invariant polynomials		The algebra S(H*)^W of W-invariant polynomial functions on the Cartan subalgebra H is a polynomial algebra in l = rank(L) algebraically independent homogeneous 
s_universal_property_enveloping_algebra	theorem	Universal property of the enveloping algebra		For any associative algebra A and Lie algebra homomorphism φ: L → A^{Lie}, there exists a unique algebra homomorphism Φ: U(L) → A extending φ; L-modules corresp
s_chevalley_integral_basis_theorem	theorem	Chevalley's theorem on integral bases		Every complex semisimple Lie algebra admits a Chevalley basis with integer structure constants, obtained by careful normalization of root vectors using sl(2)-re
s_automorphisms_of_semisimple_lie_algebra	theorem	Structure of Aut(L) for semisimple L		For a semisimple Lie algebra L, Aut(L)/Int(L) is isomorphic to the automorphism group Aut(Δ) of the Dynkin diagram; the full automorphism group is a semidirect 
s_simplicity_of_chevalley_groups	theorem	Simplicity of Chevalley groups		For most fields k and irreducible root systems Φ of rank ≥ 1, the quotient G(Φ,k)/Z(G(Φ,k)) is a simple group, with explicit small exceptions for |k| ≤ 3 and ra
t_dynkin_diagram_classification_algorithm	technique	Dynkin diagram classification by elimination		Classifies connected Dynkin diagrams by exploiting the constraint a_{ij}a_{ji} ∈ {0,1,2,3} and cos²θ = a_{ij}a_{ji}/4 to systematically eliminate impossible sub
t_root_system_explicit_construction	technique	Explicit construction of root systems		Constructs each irreducible root system A_l through G_2 explicitly as a set of vectors in ℝ^n using standard basis vectors e_i, verifying root system axioms.
t_pbw_basis_construction	technique	PBW basis construction		Constructs the Poincaré–Birkhoff–Witt monomial basis {x_1^{a_1}···x_n^{a_n}} of U(L) from an ordered basis of L, establishing that gr(U(L)) ≅ S(L).
t_verma_module_induction	technique	Verma module construction by parabolic induction		Induces from the 1-dimensional B-module F_λ to construct M(λ) = U(L) ⊗_{U(B)} F_λ ≅ U(N⁻) as a vector space; the unique simple quotient L(λ) is finite-dimension
t_complete_reducibility_via_casimir	technique	Complete reducibility via Casimir element		Proves Weyl's complete reducibility theorem using the Casimir element c_κ ∈ Z(U(L)): since c_κ acts by the scalar (λ,λ+2ρ) on L(λ), distinct irreducible summand
t_freudenthal_recursive_algorithm	technique	Freudenthal's recursive multiplicity algorithm		Computes weight multiplicities top-down: starting from dim V_λ = 1 and descending through the partial order, using Freudenthal's recursive formula involving inn
t_killing_form_root_computation	technique	Killing form computation via roots		Computes the Killing form explicitly using κ(h,h') = Σ_{α∈Φ} α(h)α(h') for h,h' ∈ H, reducing the trace computation to a sum over roots.
t_cartan_subalgebra_construction	technique	Cartan subalgebra construction from regular elements		Constructs a Cartan subalgebra as the Fitting null-component (generalized 0-eigenspace) of ad x for a regular element x ∈ L.
t_chevalley_group_construction	technique	Chevalley group construction		Exponentiates nilpotent basis elements x_α(t) = exp(t·e_α) in GL(V_k) for an admissible lattice V reduced to a field k, generating the Chevalley group G(Φ,k) of
t_sl2_triple_technique	technique	sl(2)-triple technique	sl(2)-copy method	For each root α, constructs e_α ∈ L_α, f_α ∈ L_{-α}, h_α = [e_α, f_α] spanning a subalgebra ≅ sl(2,F), reducing structural questions to sl(2) representation the
s_ordered_set	axiom	Ordered set (partially ordered set)		A set S with a relation < satisfying trichotomy and transitivity (total order) or reflexivity, antisymmetry, and transitivity (partial order).
s_field_axioms	axiom	Field axioms		A set F with two operations +, * satisfying commutativity, associativity, distributivity, existence of identity and inverse elements for both operations.
s_upper_bound_and_supremum	axiom	Upper bound and supremum		An element b in an ordered set S is an upper bound for E subset S if x <= b for all x in E; the supremum is the smallest such b.
s_lower_bound_and_infimum	axiom	Lower bound and infimum		An element b in an ordered set S is a lower bound for E subset S if b <= x for all x in E; the infimum is the largest such b.
s_dedekind_cut_construction	state	Dedekind cuts construction of R		Construction of the real numbers as the set of Dedekind cuts in Q, where a cut is a pair (A,B) partitioning Q with every element of A less than every element of
s_uniqueness_of_lub	theorem	Uniqueness of least upper bound		If a nonempty subset of an ordered set has a least upper bound, then that least upper bound is unique.
s_existence_of_nth_roots	theorem	Existence of nth roots in R		For every real x > 0 and positive integer n, there exists a unique positive real y such that y^n = x.
s_limit_point_definition	axiom	Limit point of a set		A point p in a metric space X is a limit point of E subset X if every neighborhood of p contains a point q in E with q != p.
s_isolated_point	axiom	Isolated point		A point p in E that is not a limit point of E; equivalently, there exists a neighborhood of p containing no other point of E.
s_perfect_set	axiom	Perfect set		A closed set in which every point is a limit point; equivalently, a closed set with no isolated points.
s_bounded_set_metric	axiom	Bounded set in a metric space		A subset E of a metric space (X,d) is bounded if there exists M > 0 and p in X such that d(x,p) < M for all x in E.
s_dense_set	axiom	Dense set		A subset E of a metric space X is dense if every point of X is a limit point of E or a point of E; equivalently, the closure of E is X.
s_open_cover	axiom	Open cover		A collection {G_alpha} of open sets whose union contains a given set K; K is compact if every open cover has a finite subcover.
s_compact_set_definition	axiom	Compact set		A subset K of a topological space such that every open cover of K has a finite subcover.
s_separated_sets	axiom	Separated sets		Two subsets A, B of a metric space are separated if both A intersect closure(B) and closure(A) intersect B are empty.
s_countability_of_Q	theorem	Countability of the rationals		The set Q of rational numbers is countable, as demonstrated by a bijection with N via diagonal enumeration.
s_countable_union_countable	theorem	Countable union of countable sets is countable		If {E_n} is a sequence of countable sets, then their union is countable.
s_every_neighborhood_is_open	theorem	Every neighborhood is an open set		In a metric space, every open ball B(p,r) = {x : d(x,p) < r} is an open set.
s_de_morgan_open_closed	theorem	De Morgan laws for open and closed sets		The complement of a union of sets is the intersection of complements, and vice versa; a set is open iff its complement is closed.
s_compact_subset_metric_is_closed	theorem	Compact subsets of metric spaces are closed		Every compact subset of a metric space is closed and bounded.
s_infinite_subset_compact_has_limit_point	theorem	Infinite subset of compact set has a limit point		If E is an infinite subset of a compact set K, then E has a limit point in K.
s_k_cell_compactness	theorem	Every k-cell is compact		Every closed k-cell [a1,b1] x ... x [ak,bk] in R^k is compact.
s_compact_metric_is_separable	theorem	Compact metric space is separable		Every compact metric space has a countable dense subset and hence a countable base.
s_connected_subsets_of_R_are_intervals	theorem	Connected subsets of R are intervals		A subset E of R is connected if and only if for any x,y in E with x < z < y, z is in E.
s_cantor_set	state	Cantor set		The set obtained by iteratively removing open middle thirds from [0,1]; a perfect, compact, uncountable, nowhere dense set of measure zero.
t_cantor_diagonalization	technique	Cantor diagonalization technique		Constructs an element differing from each listed element in its corresponding position, proving uncountability or non-surjectivity.
t_nested_intervals	technique	Nested intervals technique		Uses the completeness of R to extract a limit point from a sequence of nested closed intervals whose lengths shrink to zero.
s_convergent_sequence	axiom	Convergent sequence in a metric space		A sequence {p_n} in a metric space (X,d) converges to p if for every epsilon > 0 there exists N such that d(p_n,p) < epsilon for all n >= N.
s_subsequence	axiom	Subsequence		A sequence {p_{n_k}} obtained from {p_n} by selecting terms along a strictly increasing sequence of indices n_1 < n_2 < ... .
s_subsequential_limit	state	Subsequential limit		A point p such that some subsequence of a given sequence converges to p; the set of all subsequential limits is denoted E*.
s_series_convergence	axiom	Series convergence		A series sum a_n converges if the sequence of partial sums s_N = sum_{n=1}^{N} a_n converges; otherwise it diverges.
s_absolute_convergence	state	Absolute convergence		A series sum a_n converges absolutely if sum |a_n| converges; absolute convergence implies convergence.
s_conditional_convergence	state	Conditional convergence		A series converges conditionally if it converges but does not converge absolutely.
s_power_series	axiom	Power series		A series of the form sum c_n (x - a)^n, converging in a disk of radius R (the radius of convergence) centered at a.
s_radius_of_convergence	state	Radius of convergence		The nonneg extended real R such that a power series sum c_n z^n converges absolutely for |z| < R and diverges for |z| > R.
s_rearrangement_of_series	axiom	Rearrangement of a series		A series sum a_{sigma(n)} obtained from sum a_n by composing with a bijection sigma: N -> N.
s_uniqueness_of_limits	theorem	Uniqueness of limits in metric spaces		A convergent sequence in a metric space has exactly one limit.
s_convergent_sequences_bounded	theorem	Convergent sequences are bounded		If {p_n} converges in a metric space, then {p_n} is bounded.
s_algebraic_limit_theorem	theorem	Algebraic limit theorem for sequences		If {s_n} -> s and {t_n} -> t, then s_n + t_n -> s+t, s_n * t_n -> s*t, and s_n/t_n -> s/t (when t != 0).
s_subsequences_converge_to_same_limit	theorem	Subsequences converge to the same limit		If {p_n} converges to p, then every subsequence of {p_n} also converges to p.
s_completeness_of_Rk	theorem	Completeness of R^k		The Euclidean space R^k is a complete metric space: every Cauchy sequence in R^k converges.
s_limsup_liminf_always_exist	theorem	Limsup and liminf always exist in the extended reals		For any sequence {s_n} of real numbers, lim sup s_n and lim inf s_n always exist in the extended real number system.
s_cauchy_criterion_for_series	theorem	Cauchy criterion for series convergence		The series sum a_n converges iff for every epsilon > 0 there exists N such that |sum_{n=m}^{p} a_n| < epsilon for all p >= m >= N.
s_comparison_test	theorem	Comparison test for series		If |a_n| <= c_n for all n >= N_0 and sum c_n converges, then sum a_n converges absolutely.
s_cauchy_condensation_test	theorem	Cauchy condensation test		If a_1 >= a_2 >= ... >= 0, then sum a_n converges if and only if sum 2^k a_{2^k} converges.
s_p_series_convergence	theorem	p-series convergence		The series sum 1/n^p converges if and only if p > 1.
s_convergence_of_e_series	theorem	Convergence of e = sum 1/n!		The series sum 1/n! converges, and its sum defines the number e; moreover, e is irrational.
s_root_test	theorem	Root test (Cauchy)		Let alpha = lim sup |a_n|^{1/n}. If alpha < 1, sum a_n converges absolutely; if alpha > 1, it diverges.
s_ratio_test	theorem	Ratio test (d'Alembert)		If lim sup |a_{n+1}/a_n| < 1, sum a_n converges; if |a_{n+1}/a_n| >= 1 for all n >= N, it diverges.
s_root_test_stronger_than_ratio	theorem	Root test is at least as strong as ratio test		lim inf |a_{n+1}/a_n| <= lim inf |a_n|^{1/n} <= lim sup |a_n|^{1/n} <= lim sup |a_{n+1}/a_n|.
s_absolute_convergence_implies_convergence	theorem	Absolute convergence implies convergence		If a series sum a_n converges absolutely, then it converges.
s_abel_theorem_power_series	theorem	Abel's theorem on power series		If sum c_n converges to s, then lim_{x->1^-} sum c_n x^n = s.
s_cauchy_hadamard_formula	theorem	Cauchy-Hadamard formula for radius of convergence		The radius of convergence of sum c_n z^n is R = 1 / lim sup |c_n|^{1/n}.
s_riemann_rearrangement_theorem	theorem	Riemann rearrangement theorem		If sum a_n is conditionally convergent, then for any alpha in [-inf, +inf] there exists a rearrangement converging to alpha.
s_dirichlet_test_series	theorem	Dirichlet's test for series convergence		If the partial sums of sum b_n are bounded and {a_n} is monotonically decreasing to 0, then sum a_n b_n converges.
t_comparison_test_technique	technique	Comparison test technique		Bounds the terms of a given series by those of a known convergent series to establish convergence.
t_root_ratio_test_technique	technique	Root and ratio test technique		Applies the root or ratio test to determine convergence by comparing the growth rate of terms to a geometric series.
t_abel_summation_technique	technique	Abel summation technique		Rewrites sum a_n b_n using partial summation (discrete integration by parts) to transfer convergence.
s_limit_of_function_at_point	axiom	Limit of a function at a point		f(x) -> q as x -> p means: for every epsilon > 0 there exists delta > 0 such that 0 < d(x,p) < delta implies d(f(x),q) < epsilon.
s_uniformly_continuous_function	axiom	Uniformly continuous function		A function f: X -> Y between metric spaces is uniformly continuous if for every epsilon > 0 there exists delta > 0 such that d(x,y) < delta implies d(f(x),f(y))
s_discontinuities_classification	state	Classification of discontinuities		A function has a discontinuity of the first kind (jump) at x if one-sided limits exist but differ; of the second kind if at least one one-sided limit fails to e
s_composition_continuous_is_continuous	theorem	Composition of continuous functions is continuous		If f: X -> Y and g: Y -> Z are continuous, then g compose f: X -> Z is continuous.
s_continuous_image_compact_is_compact	theorem	Continuous image of compact set is compact		If f: X -> Y is continuous and K subset X is compact, then f(K) is compact in Y.
s_continuous_preserves_connectedness	theorem	Continuous image of connected set is connected		If f: X -> Y is continuous and E subset X is connected, then f(E) is connected.
s_monotonic_jump_discontinuities	theorem	Monotonic functions have countably many discontinuities		A monotonic function on an interval has at most countably many points of discontinuity, all of the first kind (jumps).
s_uniform_continuity_on_compact	theorem	Uniform continuity theorem		A continuous function on a compact metric space is uniformly continuous.
s_continuous_bijection_compact_hausdorff_homeomorphism	theorem	Continuous bijection from compact to Hausdorff is homeomorphism		A continuous bijection from a compact space to a Hausdorff space is a homeomorphism.
t_epsilon_delta_technique	technique	Epsilon-delta technique		The standard technique for proving limits and continuity by choosing delta as a function of epsilon.
s_continuously_differentiable	axiom	Continuously differentiable function (C^1)		A function f is continuously differentiable on an interval if f' exists and is continuous on that interval.
s_differentiability_implies_continuity	theorem	Differentiability implies continuity		If f is differentiable at a point x, then f is continuous at x.
s_local_extremum_theorem	theorem	Local extremum theorem		If f has a local maximum or minimum at an interior point x and f'(x) exists, then f'(x) = 0.
s_differentiation_vector_valued	theorem	Differentiation of vector-valued functions		The derivative of f: [a,b] -> R^k exists iff each component has a derivative, and f' = (f_1', ..., f_k').
t_mvt_technique	technique	Mean value theorem technique		Applies the mean value theorem to connect the global change of a function to its derivative at an intermediate point.
t_lhopital_technique	technique	L'Hopital technique		Evaluates limits of indeterminate forms by replacing f(x)/g(x) with f'(x)/g'(x).
t_taylor_expansion_technique	technique	Taylor expansion technique		Approximates a function near a point by its Taylor polynomial and estimates the remainder.
s_partition_of_interval	axiom	Partition of an interval		A finite set of points P = {x_0, x_1, ..., x_n} with a = x_0 < x_1 < ... < x_n = b, dividing [a,b] into subintervals.
s_upper_lower_rs_sums	state	Upper and lower Riemann-Stieltjes sums		For bounded f on [a,b] with increasing integrator alpha and partition P: U(P,f,alpha) = sum sup f * Delta alpha and L(P,f,alpha) = sum inf f * Delta alpha.
s_riemann_stieltjes_integral_definition	axiom	Riemann-Stieltjes integral		The integral of f with respect to alpha on [a,b], defined as the common value of inf U(P,f,alpha) and sup L(P,f,alpha) when they agree.
s_riemann_integral	axiom	Riemann integral		The special case of the Riemann-Stieltjes integral when alpha(x) = x, giving the classical integral of f over [a,b].
s_bounded_variation	axiom	Function of bounded variation		A function f on [a,b] has bounded variation if sup sum |f(x_i) - f(x_{i-1})| over all partitions is finite; this supremum is the total variation.
s_rectifiable_curve	state	Rectifiable curve and arc length		A continuous curve gamma: [a,b] -> R^k is rectifiable if its arc length Lambda(gamma) = sup sum |gamma(t_i) - gamma(t_{i-1})| is finite.
s_rs_integrability_criterion	theorem	Riemann-Stieltjes integrability criterion		f is R-S integrable with respect to alpha iff for every epsilon > 0 there exists a partition P with U(P,f,alpha) - L(P,f,alpha) < epsilon.
s_continuous_functions_rs_integrable	theorem	Continuous functions are R-S integrable		If f is continuous on [a,b] and alpha is monotonically increasing, then f is R-S integrable with respect to alpha.
s_monotonic_functions_rs_integrable	theorem	Monotonic functions are R-S integrable		If f is monotonic on [a,b] and alpha is continuous and monotonically increasing, then f is R-S integrable with respect to alpha.
s_rs_step_integrator	theorem	R-S integral with step function integrator		If alpha has a jump at s and f is continuous at s, then integral f dalpha = a * f(s).
s_rs_reduces_to_riemann	theorem	R-S integral reduces to Riemann integral		If alpha is differentiable with alpha' Riemann integrable, then integral f dalpha = integral f(x) alpha'(x) dx.
s_linearity_of_rs_integral	theorem	Linearity of the Riemann-Stieltjes integral		The R-S integral is linear in both the integrand and the integrator.
s_additivity_over_intervals	theorem	Additivity of the integral over intervals		If f is integrable on [a,b] and c in (a,b), then integral_a^b f dalpha = integral_a^c f dalpha + integral_c^b f dalpha.
s_ftc_first_form	theorem	Fundamental theorem of calculus (first form)		If f is Riemann integrable and F(x) = integral_a^x f(t) dt, then F is continuous; if f is continuous at x_0, then F'(x_0) = f(x_0).
s_ftc_second_form	theorem	Fundamental theorem of calculus (second form)		If F' = f is Riemann integrable on [a,b], then integral_a^b f(x) dx = F(b) - F(a).
s_rectifiable_curves_finite_arc_length	theorem	Rectifiable curves characterized via bounded variation		A curve gamma is rectifiable iff each component is of bounded variation; if gamma' is continuous, Lambda(gamma) = integral |gamma'(t)| dt.
t_rs_integration_technique	technique	Riemann-Stieltjes integration technique		Computes or estimates integrals by refining partitions and bounding the difference of upper and lower R-S sums.
s_pointwise_convergence_definition	axiom	Pointwise convergence of function sequence		A sequence {f_n} converges pointwise to f on E if for each x in E, lim f_n(x) = f(x).
s_uniform_convergence_definition	axiom	Uniform convergence of function sequence		{f_n} converges uniformly to f on E if for every epsilon > 0 there exists N such that |f_n(x) - f(x)| < epsilon for all n >= N and all x in E.
s_pointwise_bounded_family	axiom	Pointwise bounded family of functions		A family {f_n} on E is pointwise bounded if for each x in E there exists M(x) such that |f_n(x)| <= M(x) for all n.
s_uniformly_bounded_family	axiom	Uniformly bounded family of functions		A family {f_n} on E is uniformly bounded if there exists M such that |f_n(x)| <= M for all n and all x in E.
s_algebra_of_functions	axiom	Algebra of functions and subalgebra of C(X)		An algebra A of continuous functions on a compact set X, closed under pointwise addition, multiplication, and scalar multiplication.
s_separating_points	axiom	Algebra separating points		An algebra A of functions on X separates points if for any distinct x,y in X there exists f in A with f(x) != f(y).
s_self_adjoint_algebra	axiom	Self-adjoint algebra of functions		A complex algebra A of functions is self-adjoint if for every f in A, its complex conjugate f-bar is also in A.
s_cauchy_criterion_uniform_convergence	theorem	Cauchy criterion for uniform convergence		{f_n} converges uniformly on E iff for every epsilon > 0 there exists N such that |f_n(x) - f_m(x)| < epsilon for all m,n >= N and all x in E.
s_weierstrass_m_test	theorem	Weierstrass M-test		If |f_n(x)| <= M_n for all x in E and sum M_n converges, then sum f_n converges uniformly and absolutely on E.
s_uniform_limit_continuous	theorem	Uniform limit of continuous functions is continuous		If {f_n} are continuous and f_n -> f uniformly on E, then f is continuous on E.
s_term_by_term_integration	theorem	Term-by-term integration of uniformly convergent series		If f_n -> f uniformly on [a,b] and each f_n is Riemann integrable, then f is integrable and integral f = lim integral f_n.
s_uniform_limit_rs_integrable	theorem	Uniform limit of R-S integrable functions is R-S integrable		If {f_n} converges uniformly and each f_n is R-S integrable, then the limit is R-S integrable with integral f dalpha = lim integral f_n dalpha.
s_term_by_term_differentiation	theorem	Conditions for term-by-term differentiation		If {f_n} converges pointwise at one point and {f_n'} converges uniformly on [a,b], then {f_n} converges uniformly to a differentiable f with f' = lim f_n'.
s_equicontinuity_implies_uniform_convergence	theorem	Equicontinuity with pointwise convergence implies uniform convergence		If {f_n} is equicontinuous on compact K and converges pointwise, then f_n -> f uniformly on K.
t_m_test_technique	technique	Weierstrass M-test technique		Establishes uniform convergence of a function series by bounding each term and verifying convergence of the bound series.
t_equicontinuous_approximation_technique	technique	Equicontinuous approximation technique		Uses equicontinuity and pointwise boundedness with Arzela-Ascoli to extract uniformly convergent subsequences.
t_stone_weierstrass_technique	technique	Stone-Weierstrass approximation technique		Approximates continuous functions uniformly by members of a separating subalgebra.
s_power_series_continuity	theorem	Power series are continuous on their disk of convergence		A power series sum c_n (x-a)^n defines a continuous function on its open disk of convergence.
s_uniqueness_of_power_series	theorem	Uniqueness theorem for power series		If two power series agree on a set with a limit point, then their coefficients are identical.
s_term_by_term_differentiation_power_series	theorem	Term-by-term differentiation of power series		A power series can be differentiated term by term within its radius of convergence, and the derived series has the same radius.
s_composition_product_power_series	theorem	Composition and product of power series		Power series can be multiplied and composed, with the resulting series converging in an appropriate disk.
s_exponential_functional_equation	theorem	Exponential functional equation		The exponential function satisfies exp(z+w) = exp(z) exp(w) for all complex z,w.
s_gamma_recurrence	theorem	Gamma function recurrence relation		Gamma(s+1) = s * Gamma(s) for all s > 0, and Gamma(n+1) = n! for positive integers n.
s_gamma_analyticity	theorem	Gamma function is analytic on the right half-plane		The integral Gamma(s) = integral_0^infty t^{s-1} e^{-t} dt defines an analytic function for Re(s) > 0.
t_power_series_manipulation	technique	Power series manipulation technique		Manipulates power series by differentiating, integrating, multiplying, or composing term by term within the radius of convergence.
t_fourier_analysis_technique	technique	Fourier analysis technique		Expands functions in Fourier series, computes Fourier coefficients, and applies convergence theorems.
t_gamma_function_technique	technique	Gamma function technique		Uses the Gamma function's recurrence, reflection formula, and analytic properties to evaluate integrals.
s_total_derivative	axiom	Total (Frechet) derivative		A function f: R^n -> R^m is differentiable at x if there exists a linear map A with |f(x+h) - f(x) - Ah| / |h| -> 0 as h -> 0.
s_partial_derivative	axiom	Partial derivative		The derivative of f(x_1,...,x_n) with respect to x_j, holding all other variables fixed.
s_jacobian_matrix	state	Jacobian matrix and determinant		For f: R^n -> R^m differentiable at x, the m x n matrix of partial derivatives; when m=n, its determinant is the Jacobian determinant.
s_c1_mapping	axiom	C^1 mapping		A mapping f: E -> R^m (E open in R^n) is C^1 if all partial derivatives exist and are continuous on E.
s_chain_rule_several_variables	theorem	Chain rule for functions of several variables		If f is differentiable at x and g is differentiable at f(x), then (g compose f)'(x) = g'(f(x)) f'(x), the matrix product of Jacobians.
s_c1_implies_differentiable	theorem	C^1 implies differentiable		If all partial derivatives of f exist and are continuous, then f is differentiable.
s_clairaut_theorem	theorem	Clairaut's theorem (symmetry of mixed partials)		If the mixed partial derivatives are continuous, then D_i D_j f = D_j D_i f.
s_mvt_vector_valued	theorem	Mean value estimate for vector-valued functions		If f: [a,b] -> R^k is continuous and differentiable on (a,b), then |f(b) - f(a)| <= (b-a) sup |f'(t)|.
s_inverse_function_theorem	theorem	Inverse function theorem		If f: R^n -> R^n is C^1 and f'(a) is invertible, then f is a C^1 diffeomorphism on a neighborhood of a.
s_implicit_function_theorem	theorem	Implicit function theorem		If F: R^{n+m} -> R^n is C^1 and D_x F is invertible at (a,b) with F(a,b) = 0, then near (a,b) the equation F(x,y) = 0 defines x as a C^1 function of y.
s_rank_theorem	theorem	Rank theorem (constant rank)		If f: R^n -> R^m is C^1 with constant rank r near a, then there exist C^1 diffeomorphisms putting f in canonical form locally.
s_determinant_multilinear_alternating	theorem	Determinant is the unique multilinear alternating form		The determinant is the unique function on n x n matrices that is multilinear and alternating in rows and equals 1 on the identity.
t_contraction_iteration_technique	technique	Contraction iteration technique		Produces a fixed point by iterating a strict contraction on a complete metric space.
t_inverse_implicit_technique	technique	Inverse and implicit function technique		Uses the contraction mapping theorem to produce local inverses or solve systems of equations.
t_multilinear_algebra_technique	technique	Multilinear algebra technique		Exploits multilinearity and alternating property to compute determinants and volumes.
s_k_form	axiom	k-form on R^n		A k-linear alternating function on R^n; in local coordinates, sum f_I dx_{i_1} wedge ... wedge dx_{i_k}.
s_closed_differential_form	state	Closed differential form		A differential form omega is closed if d(omega) = 0.
s_exact_differential_form	state	Exact differential form		A differential form omega is exact if omega = d(lambda) for some form lambda; every exact form is closed.
s_pullback_of_differential_form	state	Pullback of a differential form		For a C^1 map phi: R^m -> R^n, the pullback phi^* omega of a k-form omega on R^n is a k-form on R^m.
s_k_surface	axiom	k-surface in R^n		A C^1 mapping Phi from a compact subset of R^k into R^n, serving as the domain of integration for k-forms.
s_boundary_operator_surface	state	Boundary operator for k-surfaces		The boundary of a k-surface Phi is a (k-1)-chain defined by oriented faces, satisfying del^2 = 0.
s_orientation_of_surface	axiom	Orientation of a k-surface		A consistent choice of sign for the Jacobian determinant of a k-surface parametrization.
s_d_squared_zero	theorem	d^2 = 0 (exterior derivative is nilpotent)		The exterior derivative d satisfies d(d(omega)) = 0 for every differential form omega.
s_pullback_commutes_with_d	theorem	Pullback commutes with exterior derivative		For a C^1 map phi, phi^*(d omega) = d(phi^* omega).
s_exact_forms_zero_integral	theorem	Exact forms have zero integral over cycles		If omega = d lambda and Phi is a cycle, then integral_Phi omega = 0.
s_closed_forms_exact_simply_connected	theorem	Closed forms are exact in simply connected domains		Every closed k-form on a simply connected open subset of R^n is exact.
t_exterior_calculus_technique	technique	Exterior calculus technique		Converts integrals of forms over boundaries to integrals of exterior derivatives over interiors.
t_pullback_technique	technique	Pullback technique for differential forms		Pulls back differential forms along smooth maps to transfer integration between domains.
t_partition_of_unity_technique	technique	Partition of unity technique		Constructs smooth functions subordinate to an open cover to localize global problems.
t_stokes_theorem_technique	technique	Stokes' theorem technique		Applies generalized Stokes theorem to relate boundary integrals to interior integrals.
s_lebesgue_integral_definition	axiom	Lebesgue integral		For nonneg measurable f, integral f dmu = sup{integral phi dmu : phi simple, 0 <= phi <= f}; for general f, integral f+ dmu - integral f- dmu.
s_l1_space	axiom	L^1 space		The space of measurable functions f with integral |f| dmu < infty, modulo equality a.e., with norm ||f||_1 = integral |f| dmu.
s_measure_zero_set	axiom	Set of measure zero		A measurable set E with mu(E) = 0; a property holds almost everywhere if the set where it fails has measure zero.
s_l2_space	axiom	L^2 space		The space of measurable functions f with integral |f|^2 dmu < infty, forming a Hilbert space with inner product <f,g> = integral f g-bar dmu.
s_regularity_of_lebesgue_measure	theorem	Regularity of Lebesgue measure		Lebesgue measure is regular: every measurable set can be approximated from outside by open sets and inside by compact sets.
s_countable_additivity_measure	theorem	Countable additivity of measures		If {E_n} is a sequence of pairwise disjoint measurable sets, then mu(union E_n) = sum mu(E_n).
s_measurable_functions_form_algebra	theorem	Measurable functions form an algebra		Sums, products, scalar multiples, sup, inf, lim sup, lim inf of measurable functions are measurable.
s_pointwise_limit_measurable	theorem	Pointwise limit of measurable functions is measurable		If {f_n} is a sequence of measurable functions converging pointwise, then the limit is measurable.
s_every_measurable_limit_of_simple	theorem	Every measurable function is a limit of simple functions		Every nonneg measurable function is the pointwise limit of an increasing sequence of simple functions.
s_riemann_vs_lebesgue_comparison	theorem	Comparison of Riemann and Lebesgue integrals		If f is Riemann integrable on [a,b], then f is Lebesgue measurable and the two integrals agree.
s_characterization_riemann_integrability	theorem	Characterization of Riemann integrability		A bounded function on [a,b] is Riemann integrable iff the set of its discontinuities has Lebesgue measure zero.
s_l2_is_hilbert	theorem	L^2 is a Hilbert space		L^2(mu) with inner product <f,g> = integral f g-bar dmu is complete, hence a Hilbert space.
t_lebesgue_integration_technique	technique	Lebesgue integration technique		Computes integrals by approximating measurable functions with simple functions and using monotone convergence.
t_dominated_convergence_technique	technique	Dominated convergence technique		Justifies passing limits under the integral when functions are dominated by an integrable function.
t_monotone_convergence_technique	technique	Monotone convergence technique		Passes limits under the integral for increasing sequences of nonnegative measurable functions.
t_fatou_lemma_technique	technique	Fatou's lemma technique		Provides a one-sided inequality for limits of integrals when domination is unavailable.
s_separable_implies_countable_base	theorem	Separable metric space has a countable base		Every separable metric space is second-countable.
s_measurable_function	state	Measurable function		A function f: X -> Y between measurable spaces such that f^{-1}(E) is in M for every E in N.
s_positive_measure	axiom	Positive measure		A countably additive set function mu: M -> [0, infinity] on a sigma-algebra M with mu(emptyset) = 0.
s_complex_measure	axiom	Complex measure		A countably additive set function nu: M -> C on a sigma-algebra, necessarily of bounded total variation.
s_vitali_caratheodory_theorem	theorem	Vitali-Carathéodory theorem		Every integrable function on a locally compact Hausdorff space can be approximated in L1 norm from above by lower semicontinuous and from below by upper semicon
s_completion_of_measure_space	state	Completion of a measure space		The smallest extension of (X, M, mu) in which every subset of a null set is measurable, obtained by adding all such subsets.
s_hardy_littlewood_maximal_function	state	Hardy-Littlewood maximal function		For f in L^1_loc(R^n), the maximal function Mf(x) = sup_{r>0} (1/|B(x,r)|) integral_{B(x,r)} |f(y)| dy, a measurable function controlling local averages.
s_lebesgue_point	state	Lebesgue point		A point x is a Lebesgue point of f in L^1_loc if lim_{r->0} (1/|B(x,r)|) integral_{B(x,r)} |f(y) - f(x)| dy = 0; almost every point is a Lebesgue point.
s_absolutely_continuous_function	axiom	Absolutely continuous function		A function F on [a,b] such that for every epsilon > 0 there exists delta > 0 with sum |F(b_i) - F(a_i)| < epsilon whenever the intervals (a_i, b_i) are disjoint
s_ftc_lebesgue	theorem	Fundamental theorem of calculus for Lebesgue integration		A function F on [a,b] is absolutely continuous if and only if F' exists a.e., F' is in L^1, and F(x) - F(a) = integral_a^x F'(t) dt for all x in [a,b].
s_rising_sun_lemma	theorem	Rising sun lemma		If f is a continuous real function on R, the open set {x : there exists y > x with f(y) > f(x)} is a disjoint union of intervals (a_n, b_n) with f(a_n) <= f(b_n
s_cantor_function	state	Cantor function (devil's staircase)		A continuous nondecreasing surjection from [0,1] to [0,1] that is constant on each complementary interval of the Cantor set, with derivative zero a.e. but not a
s_fourier_inversion_formula_Rn	theorem	Fourier inversion formula on R^n		If f and its Fourier transform f_hat are both in L^1(R^n), then f(x) = integral_{R^n} f_hat(xi) e^{2 pi i x . xi} d xi for a.e. x.
s_fourier_transform_on_schwartz_space	theorem	Fourier transform is an automorphism of Schwartz space		The Fourier transform is a continuous linear bijection from the Schwartz space S(R^n) onto itself, with continuous inverse given by the inverse Fourier transfor
s_cauchy_integral_formula_derivatives	theorem	Cauchy integral formula for derivatives		If f is holomorphic inside and on a simple closed curve gamma, then f^{(n)}(a) = (n!/(2 pi i)) integral_gamma f(z)/(z-a)^{n+1} dz for all a inside gamma.
s_power_series_representation_holomorphic	theorem	Power series representation of holomorphic functions		Every holomorphic function on a domain has a local power series expansion f(z) = sum a_n (z-z_0)^n converging in the largest disk centered at z_0 contained in t
s_isolated_zeros_theorem	theorem	Isolated zeros of holomorphic functions		The zeros of a nonconstant holomorphic function on a connected domain are isolated, and each zero has a finite positive order.
s_classification_of_isolated_singularities	state	Classification of isolated singularities		An isolated singularity of a holomorphic function is exactly one of: removable (bounded near the point), a pole (|f(z)| -> infinity), or essential (Casorati-Wei
t_calculus_of_residues	technique	Calculus of residues		Evaluation of contour integrals and real definite integrals by identifying poles, computing residues, and applying the residue theorem.
s_cauchy_estimates	theorem	Cauchy estimates for derivatives		If f is holomorphic on |z - z_0| < R and |f| <= M there, then |f^{(n)}(z_0)| <= n! M / R^n.
s_mobius_transformation	state	Möbius transformation		A bijective holomorphic map of the Riemann sphere of the form z -> (az+b)/(cz+d) with ad-bc != 0, forming the group PSL(2,C); the automorphism group of the Riem
s_automorphisms_of_unit_disk	theorem	Automorphisms of the unit disk		Every biholomorphic self-map of the unit disk D has the form z -> e^{i theta} (z - a)/(1 - bar{a} z) for some |a| < 1 and theta in R.
s_normal_family_holomorphic	state	Normal family of holomorphic functions		A family F of holomorphic functions on a domain Omega such that every sequence in F has a subsequence converging uniformly on compact subsets of Omega.
s_blaschke_product	state	Blaschke product		A product B(z) = z^m prod_n (|a_n|/a_n)(a_n - z)/(1 - bar{a_n} z) with sum (1 - |a_n|) < infinity, forming a bounded holomorphic inner function on the unit disk
s_blaschke_condition	theorem	Blaschke condition		A sequence {a_n} in the unit disk is the zero set of some bounded holomorphic function if and only if sum (1 - |a_n|) < infinity.
s_muntz_szasz_theorem	theorem	Müntz-Szász theorem		The span of {x^{lambda_n}} is dense in C([0,1]) if and only if sum 1/lambda_n = infinity, where 0 < lambda_1 < lambda_2 < ... .
s_hardy_space_Hp_disk	axiom	Hardy space H^p on the disk		The space of holomorphic functions f on the unit disk with sup_{0<r<1} (1/(2 pi) integral_0^{2 pi} |f(r e^{i theta})|^p d theta)^{1/p} < infinity, for 0 < p < i
s_fatou_theorem_radial_limits	theorem	Fatou's theorem on radial limits		Every bounded holomorphic function on the unit disk has nontangential limits at almost every point of the boundary circle.
s_poisson_kernel_disk	state	Poisson kernel for the unit disk		The kernel P_r(theta) = (1 - r^2)/(1 - 2r cos theta + r^2) such that the Poisson integral of an L^p boundary function is harmonic in the disk and converges nont
s_poisson_integral_representation	theorem	Poisson integral representation		A harmonic function u in the unit disk with sup_r integral |u(r e^{i theta})| d theta < infinity is the Poisson integral of a unique finite Borel measure on the
s_f_and_m_riesz_theorem	theorem	F. and M. Riesz theorem		If mu is a finite complex Borel measure on the unit circle whose negative Fourier coefficients all vanish, then mu is absolutely continuous with respect to Lebe
s_inner_function	state	Inner function		A bounded holomorphic function on the unit disk whose nontangential boundary values have modulus 1 almost everywhere; decomposes as a Blaschke product times a s
s_outer_function	state	Outer function		A function of the form F(z) = c exp((1/(2 pi)) integral (e^{i theta} + z)/(e^{i theta} - z) log psi(theta) d theta) for a positive function psi with log psi in 
s_canonical_factorization_Hp	theorem	Canonical factorization of H^p functions		Every nonzero H^p function (0 < p <= infinity) factors uniquely as f = B S F where B is a Blaschke product, S a singular inner function, and F an outer function
s_beurling_theorem_invariant_subspaces	theorem	Beurling's theorem on invariant subspaces of H^2		Every closed subspace of H^2 that is invariant under multiplication by z has the form phi H^2 for some inner function phi, unique up to a scalar of modulus 1.
s_subharmonic_function	axiom	Subharmonic function		An upper semicontinuous function u on a domain Omega in R^n (or C) satisfying the sub-mean-value property: u(x) <= (1/|partial B|) integral_{partial B} u d sigm
s_poisson_kernel	state	Poisson kernel		The kernel P(x, y) for the unit ball in R^n reconstructing harmonic functions from boundary values: u(x) = integral_{S^{n-1}} P(x, y) f(y) d sigma(y).
s_m_riesz_conjugate_function_theorem	theorem	M. Riesz conjugate function theorem		The Hilbert transform (conjugate function operator) is bounded on L^p(T) for 1 < p < infinity, with norm depending only on p.
t_vitali_caratheodory_approximation	technique	Vitali-Carathéodory approximation technique		Approximates an integrable function from above by lower semicontinuous and from below by upper semicontinuous functions with controlled L^1 error.
s_nevanlinna_class	axiom	Nevanlinna class		The class N of holomorphic functions f on the unit disk satisfying sup_{0<r<1} integral_0^{2 pi} log^+ |f(r e^{i theta})| d theta < infinity, the largest class 
t_duality_technique_Lp	technique	Duality technique for L^p bounds		Establishes operator bounds on L^p by proving the dual bound on L^q via the duality ||T||_{p->p} = sup {|integral Tf g| : ||f||_p = ||g||_q = 1} with 1/p + 1/q 
t_maximal_function_technique	technique	Maximal function technique		Reduces almost-everywhere convergence of operators to a weak-type (1,1) or strong-type bound on the associated maximal operator.
s_szego_theorem	theorem	Szegő theorem		For a positive Borel measure mu on the unit circle with absolutely continuous part w d theta, the distance from 1 to the closed span of {e^{i n theta} : n >= 1}
s_logarithmic_integral_condition_Hp	theorem	Log-integrability condition for H^p		A nonnegative function psi on the circle is the modulus of the boundary values of some H^p function (0 < p <= infinity) if and only if log psi is integrable wit
t_conformal_mapping_technique	technique	Conformal mapping technique		Transfers function-theoretic problems to a canonical domain (disk or half-plane) via the Riemann mapping theorem, solves there, and maps back.
s_schwarz_christoffel_formula	theorem	Schwarz-Christoffel formula		The conformal map from the upper half-plane to a polygon with interior angles alpha_k pi is f'(z) = C prod (z - x_k)^{alpha_k - 1}, where x_k are the prevertice
s_sigma_algebra	axiom	σ-algebra		A nonempty collection M of subsets of X closed under complementation and countable unions, with X in M.
s_semifinite_measure	axiom	Semifinite measure		A measure μ such that for every measurable set E with μ(E) = ∞, there exists a measurable subset F ⊂ E with 0 < μ(F) < ∞.
s_counting_measure	state	Counting measure		The measure on the power set of X defined by μ(E) = |E| if E is finite and μ(E) = ∞ if E is infinite.
s_monotonicity_of_measures	theorem	Monotonicity of measures		If E ⊂ F are measurable sets, then μ(E) ≤ μ(F).
s_subadditivity_of_measures	theorem	Subadditivity of measures		For any countable collection {E_j} of measurable sets, μ(⋃ E_j) ≤ Σ μ(E_j).
s_continuity_from_below_measures	theorem	Continuity from below for measures		If E_1 ⊂ E_2 ⊂ ··· is an increasing sequence of measurable sets, then μ(⋃ E_j) = lim μ(E_j).
s_continuity_from_above_measures	theorem	Continuity from above for measures		If E_1 ⊃ E_2 ⊃ ··· is a decreasing sequence of measurable sets with μ(E_1) < ∞, then μ(⋂ E_j) = lim μ(E_j).
s_complete_measure	axiom	Complete measure		A measure space (X, M, μ) is complete if every subset of a μ-null set belongs to M.
s_caratheodory_measurability	axiom	Carathéodory measurability (μ*-measurable set)		A set E is μ*-measurable if μ*(A) = μ*(A ∩ E) + μ*(A ∩ E^c) for all A ⊂ X, where μ* is an outer measure.
s_caratheodory_theorem_outer_measure	theorem	Carathéodory's theorem (outer measure restriction)		The collection of μ*-measurable sets forms a σ-algebra, and the restriction of μ* to this σ-algebra is a complete measure.
s_algebra_of_sets	axiom	Algebra of sets		A nonempty collection A of subsets of X closed under finite unions and complementation.
s_outer_measure_induced_by_premeasure	state	Outer measure induced by a premeasure		The outer measure μ*(E) = inf{Σ μ₀(A_j) : E ⊂ ⋃ A_j, each A_j ∈ A} defined from a premeasure μ₀ on an algebra A.
s_translation_invariance_lebesgue	theorem	Translation invariance of Lebesgue measure		For any Lebesgue measurable set E ⊂ ℝⁿ and any x ∈ ℝⁿ, the translate E + x is measurable and m(E + x) = m(E).
s_dilation_property_lebesgue	theorem	Dilation property of Lebesgue measure		For any Lebesgue measurable set E ⊂ ℝⁿ and r > 0, m(rE) = rⁿ m(E).
s_vitali_nonmeasurable_set	state	Non-Lebesgue-measurable set (Vitali set)		A subset of [0,1] constructed via the axiom of choice by selecting one representative from each coset of ℚ in ℝ, which is not Lebesgue measurable.
s_distribution_function	axiom	Distribution function (Lebesgue-Stieltjes)		A right-continuous nondecreasing function F: ℝ → ℝ defining a Borel measure via μ_F((a,b]) = F(b) - F(a).
s_borel_measure_distribution_function_correspondence	theorem	Correspondence between Borel measures and distribution functions		There is a bijection between finite Borel measures on ℝ and right-continuous nondecreasing bounded functions F (modulo additive constants), given by F(x) = μ((-
s_cantor_measure	state	Cantor measure (Cantor-Lebesgue measure)		The Lebesgue-Stieltjes measure associated to the Cantor function, a probability measure supported on the Cantor set that assigns measure zero to every singleton
s_borel_measurable_function	state	Borel measurable function		A function f: X → Y between topological spaces such that the preimage of every open set is a Borel set in X.
s_lebesgue_measurable_function	state	Lebesgue measurable function		A function f: ℝⁿ → ℝ such that f⁻¹((a, ∞)) is Lebesgue measurable for every a ∈ ℝ.
s_continuous_functions_are_measurable	theorem	Continuous functions are Borel measurable		Every continuous function f: X → Y between topological spaces is Borel measurable.
s_measurability_criterion_preimages	theorem	Measurability criterion via generating sets		A function f: X → Y is measurable if and only if f⁻¹(E) ∈ M for every E in a collection that generates the σ-algebra on Y.
s_closure_measurable_functions_under_limits	theorem	Closure of measurable functions under limits		If {f_n} is a sequence of measurable functions, then sup f_n, inf f_n, lim sup f_n, and lim inf f_n are all measurable.
s_simple_function	axiom	Simple function		A measurable function φ: X → ℂ taking only finitely many values, expressible as φ = Σ aᵢ χ_{Eᵢ} for measurable sets Eᵢ.
s_integral_nonneg_simple_function	axiom	Integral of a nonnegative simple function		For a nonnegative simple function φ = Σ aᵢ χ_{Eᵢ}, the integral is defined as ∫φ dμ = Σ aᵢ μ(Eᵢ).
s_integral_nonneg_measurable_function	axiom	Integral of a nonnegative measurable function		For a nonnegative measurable function f, ∫f dμ = sup{∫φ dμ : 0 ≤ φ ≤ f, φ simple}.
s_linearity_integration_nonneg	theorem	Linearity of integration for nonnegative functions		If f, g are nonnegative measurable functions and c ≥ 0, then ∫(f + g) = ∫f + ∫g and ∫(cf) = c∫f.
s_almost_everywhere	axiom	Almost everywhere (a.e.)		A property P holds almost everywhere (a.e.) with respect to μ if the set {x : P(x) fails} has μ-measure zero.
s_convergence_in_L1	axiom	Convergence in L¹		A sequence f_n converges to f in L¹ if ∫|f_n - f| dμ → 0.
s_convergence_in_measure_subsequence	theorem	Convergence in measure subsequence theorem		If f_n → f in measure, then there exists a subsequence f_{n_k} that converges to f almost everywhere.
s_product_sigma_algebra	state	Product σ-algebra		The σ-algebra M ⊗ N on X × Y generated by all measurable rectangles {A × B : A ∈ M, B ∈ N}.
s_measurability_of_sections	theorem	Measurability of sections		If E ∈ M ⊗ N, then for each x ∈ X the section E_x = {y : (x,y) ∈ E} is in N, and for each y the section E^y is in M.
s_lebesgue_measure_Rn	state	Lebesgue measure on ℝⁿ		The completion of the n-fold product of Lebesgue measure on ℝ, equivalently the measure assigning to each box its volume.
s_polar_coordinates_integration	theorem	Polar coordinates integration formula		For integrable f on ℝⁿ, ∫_{ℝⁿ} f dx = ∫_0^∞ (∫_{S^{n-1}} f(rω) dσ(ω)) r^{n-1} dr, where σ is surface measure on S^{n-1}.
s_volume_unit_ball_Rn	state	Volume of the unit ball in ℝⁿ		The Lebesgue measure of the unit ball B_n = {x ∈ ℝⁿ : |x| ≤ 1} is V_n = πⁿ/² / Γ(n/2 + 1).
s_L1_convergence_implies_measure_convergence	theorem	Convergence in L¹ implies convergence in measure		If f_n → f in L¹(μ), then f_n → f in measure.
s_positive_set_signed_measure	axiom	Positive set for a signed measure		A measurable set P is positive for a signed measure ν if ν(E) ≥ 0 for every measurable subset E ⊂ P.
s_negative_set_signed_measure	axiom	Negative set for a signed measure		A measurable set N is negative for a signed measure ν if ν(E) ≤ 0 for every measurable subset E ⊂ N.
s_positive_variation_measure	state	Positive variation of a signed measure		The positive part ν⁺ in the Jordan decomposition ν = ν⁺ - ν⁻, defined by ν⁺(E) = ν(E ∩ P) where P is the positive set in a Hahn decomposition.
s_negative_variation_measure	state	Negative variation of a signed measure		The negative part ν⁻ in the Jordan decomposition ν = ν⁺ - ν⁻, defined by ν⁻(E) = -ν(E ∩ N) where N is the negative set in a Hahn decomposition.
s_absolutely_continuous_measure	axiom	Absolutely continuous measure		A measure ν is absolutely continuous with respect to μ (written ν ≪ μ) if μ(E) = 0 implies ν(E) = 0 for every measurable E.
s_chain_rule_radon_nikodym	theorem	Chain rule for Radon-Nikodym derivatives		If λ ≪ ν ≪ μ, then dλ/dμ = (dλ/dν)(dν/dμ) μ-a.e.
s_total_variation_complex_measure	state	Total variation of a complex measure		For a complex measure ν, the total variation |ν|(E) = sup Σ|ν(E_j)| over all partitions of E into countably many measurable sets.
s_total_variation_complex_is_finite	theorem	Total variation of a complex measure is finite		For any complex measure ν on (X, M), the total variation |ν|(X) < ∞.
s_polar_decomposition_complex_measure	theorem	Polar decomposition of a complex measure		Every complex measure ν can be written as dν = h d|ν| where h is a measurable function with |h| = 1 a.e. with respect to |ν|.
s_radon_nikodym_complex_measures	theorem	Radon-Nikodym theorem for complex measures		If ν is a complex measure and μ is a σ-finite positive measure with ν ≪ μ, then dν = f dμ for some f ∈ L¹(μ).
s_lebesgue_set	state	Lebesgue set		The set of points x where lim_{r→0} (1/m(B_r(x))) ∫_{B_r(x)} |f(y) - f(x)| dy = 0, which has full measure for any locally integrable f.
s_differentiation_borel_measures_Rn	theorem	Differentiation of Borel measures on ℝⁿ		If ν and μ are Borel measures on ℝⁿ with μ the Lebesgue measure, then dν/dμ(x) = lim_{r→0} ν(B_r(x))/μ(B_r(x)) exists a.e. and equals the Radon-Nikodym derivati
s_total_variation_function	state	Total variation function		For f ∈ BV([a,b]), the function T_f(x) = sup{Σ|f(x_i) - f(x_{i-1})| : partitions of [a,x]}, which is nondecreasing.
s_bv_difference_of_monotone	theorem	BV functions as differences of monotone functions		A function f is of bounded variation on [a,b] if and only if f = g - h where g and h are nondecreasing.
s_ac_function_measure_correspondence	theorem	AC functions and absolute continuity of measures		F is absolutely continuous on [a,b] if and only if the associated signed measure μ_F is absolutely continuous with respect to Lebesgue measure.
s_bv_functions_differentiable_ae	theorem	BV functions are differentiable a.e.		Every function of bounded variation on [a,b] is differentiable Lebesgue-almost everywhere.
s_singular_function	axiom	Singular function		A nondecreasing continuous function F on [a,b] whose derivative F'(x) = 0 for Lebesgue-a.e. x, yet F is not constant.
s_decomposition_bv_functions	theorem	Decomposition of BV functions		Every BV function on [a,b] can be uniquely decomposed as f = f_ac + f_s + f_j where f_ac is absolutely continuous, f_s is singular continuous, and f_j is a jump
s_integration_by_parts_ac	theorem	Integration by parts for absolutely continuous functions		If F and G are absolutely continuous on [a,b], then ∫_a^b F'G dx = F(b)G(b) - F(a)G(a) - ∫_a^b FG' dx.
s_lebesgue_radon_nikodym_on_R	theorem	Lebesgue-Radon-Nikodym theorem on ℝ		Every BV function F decomposes as F = F_ac + F_s where F_ac is absolutely continuous and F_s' = 0 a.e., corresponding to the Lebesgue decomposition of μ_F.
s_net_convergence	axiom	Net (generalized sequence)		A function from a directed set (I, ≤) to a topological space X; a net (x_α) converges to x if for every open U containing x there exists α₀ with x_α ∈ U for all
s_subnet	axiom	Subnet		A subnet of a net (x_α)_{α∈A} is a net (x_{φ(β)})_{β∈B} where φ: B → A is an order-preserving cofinal map between directed sets.
s_topology_characterized_by_nets	theorem	Characterization of topology by nets		A set F is closed if and only if it contains all limits of convergent nets in F; a function is continuous if and only if it preserves net convergence.
s_compactness_via_nets	theorem	Compactness via nets		A topological space is compact if and only if every net has a convergent subnet.
s_topological_vector_space	axiom	Topological vector space (TVS)		A vector space X over ℝ or ℂ equipped with a topology making addition and scalar multiplication continuous.
s_seminorm	axiom	Seminorm		A function p: X → [0,∞) on a vector space satisfying p(αx) = |α|p(x) and p(x+y) ≤ p(x) + p(y), but possibly p(x) = 0 for x ≠ 0.
s_bounded_linear_operator	axiom	Bounded linear operator		A linear map T: X → Y between normed spaces such that ||T|| = sup{||Tx|| : ||x|| ≤ 1} < ∞, equivalently a continuous linear map.
s_operator_norm	state	Operator norm		The norm ||T|| = sup{||Tx||/||x|| : x ≠ 0} on the space B(X,Y) of bounded linear operators from X to Y.
s_operator_space_banach	theorem	B(X,Y) is Banach when Y is Banach		If Y is a Banach space, then B(X,Y) with the operator norm is a Banach space.
s_separation_of_points_by_functionals	theorem	Separation of points by continuous linear functionals		For any x ≠ 0 in a normed space X, there exists f ∈ X* with f(x) ≠ 0 and ||f|| = 1, ||f(x)|| = ||x||.
s_second_category_set	axiom	Second category set (non-meager set)		A subset of a topological space that is not a countable union of nowhere dense sets.
s_banach_steinhaus_pointwise_convergence	theorem	Banach-Steinhaus theorem (pointwise convergence corollary)		If T_n ∈ B(X,Y) with X Banach and T_n(x) converges for each x ∈ X, then sup_n ||T_n|| < ∞ and the pointwise limit T is in B(X,Y) with ||T|| ≤ lim inf ||T_n||.
s_bounded_inverse_theorem	theorem	Bounded inverse theorem		If T: X → Y is a bounded bijective linear operator between Banach spaces, then T⁻¹ is also bounded.
s_weak_topology_on_normed_space	axiom	Weak topology on a normed space		The coarsest topology on a normed space X making every f ∈ X* continuous; x_α → x weakly iff f(x_α) → f(x) for all f ∈ X*.
s_closest_point_convex_set	theorem	Closest point to a closed convex set in Hilbert space		For every nonempty closed convex subset K of a Hilbert space H and x ∈ H, there exists a unique y ∈ K minimizing ||x - y||.
s_separable_hilbert_isomorphic_l2	theorem	Separable Hilbert spaces are isomorphic to ℓ²		Every infinite-dimensional separable Hilbert space is unitarily isomorphic to ℓ²(ℕ).
s_quotient_normed_space	state	Quotient normed space		For a closed subspace M of a normed space X, the quotient X/M with norm ||x + M|| = inf{||x + m|| : m ∈ M} is a normed space; it is Banach when X is.
s_chebyshev_markov_Lp	theorem	Chebyshev-Markov inequality for Lp		For f ∈ Lp and α > 0, μ({x : |f(x)| > α}) ≤ α⁻ᵖ ||f||_p^p.
s_lp_inclusions_finite_measure	theorem	Lp inclusions under finite measure		If μ(X) < ∞ and 1 ≤ p ≤ q ≤ ∞, then Lq(μ) ⊂ Lp(μ) with ||f||_p ≤ μ(X)^{1/p - 1/q} ||f||_q.
s_dual_of_Lp	theorem	Dual of Lp		For 1 ≤ p < ∞ and σ-finite μ, the dual (Lp)* is isometrically isomorphic to Lq where 1/p + 1/q = 1, via the pairing ⟨f, g⟩ = ∫fg dμ.
s_minkowski_integral_inequality	theorem	Minkowski's inequality for integrals		For 1 ≤ p < ∞ and nonneg measurable F on X × Y, (∫_X (∫_Y F(x,y) dν(y))^p dμ(x))^{1/p} ≤ ∫_Y (∫_X F(x,y)^p dμ(x))^{1/p} dν(y).
s_weak_Lp_space	axiom	Weak Lp space		The space Lp,∞(μ) of measurable functions f with ||f||_{p,∞} = sup_{α>0} α·μ({|f| > α})^{1/p} < ∞, a quasi-normed space containing Lp.
s_Cc_space	axiom	Cc(X) — continuous functions with compact support		The vector space of continuous real- or complex-valued functions on a locally compact Hausdorff space X whose support is compact.
s_C0_space	axiom	C₀(X) — continuous functions vanishing at infinity		The Banach space of continuous functions f on a locally compact Hausdorff space X such that {x : |f(x)| ≥ ε} is compact for every ε > 0, with the sup norm.
s_lusin_theorem_radon	theorem	Lusin's theorem for Radon measures		If μ is a Radon measure on a LCH space X and f is measurable, then for every ε > 0 there exists a compact K with μ(K^c) < ε such that f|_K is continuous.
s_dual_of_C0	theorem	Dual of C₀(X)		The dual space C₀(X)* is isometrically isomorphic to the space M(X) of complex Radon measures on X, via the pairing I(f) = ∫f dμ.
s_approximate_identity_Rn	state	Approximate identity on ℝⁿ		A family {φ_ε} ⊂ L¹(ℝⁿ) with ∫φ_ε = 1, supp(φ_ε) shrinking to {0}, and φ_ε * f → f in Lp and pointwise a.e. for f ∈ Lp.
s_fourier_transform_L2_extension	state	Fourier transform on L² (extension from L¹ ∩ L²)		The Fourier transform, initially defined on L¹ ∩ L², extends uniquely to a unitary operator on L²(ℝⁿ) by the Plancherel theorem.
s_test_function_space	axiom	Test function space D(Ω) = C_c^∞(Ω)		The space of smooth functions with compact support in an open set Ω ⊂ ℝⁿ, equipped with the inductive limit topology.
s_distribution_space	axiom	Distribution space D'(Ω)		The space of continuous linear functionals on D(Ω) = C_c^∞(Ω), whose elements are called distributions on Ω.
s_support_of_distribution	state	Support of a distribution		The support of a distribution u is the complement of the largest open set on which u vanishes.
s_distributions_compact_support	state	Distributions with compact support E'(Ω)		The subspace of distributions u ∈ D'(Ω) with compact support, isomorphic to the dual of C^∞(Ω).
s_tempered_distribution	axiom	Tempered distribution S'(ℝⁿ)		A continuous linear functional on the Schwartz space S(ℝⁿ); the space S'(ℝⁿ) of tempered distributions includes all polynomially growing functions and all Lp fu
s_convolution_of_distributions	state	Convolution of distributions		If u has compact support and v is any distribution, their convolution extends the classical convolution to distributions.
s_sobolev_space_Wkp	axiom	Sobolev space W^{k,p}(Ω)		The space of functions f ∈ Lp(Ω) whose weak derivatives D^α f exist in Lp for all |α| ≤ k, with norm ||f||_{k,p} = (Σ_{|α|≤k} ||D^α f||_p^p)^{1/p}.
s_order_of_distribution	state	Order of a distribution		The smallest integer k such that |u(φ)| ≤ C sup_{|α|≤k} ||D^α φ||_∞ for all φ ∈ D(K) for each compact K ⊂ Ω.
s_singular_support	state	Singular support of a distribution		The complement of the largest open set on which the distribution u agrees with a smooth function.
s_fourier_transform_tempered_distributions	state	Fourier transform of tempered distributions		The extension of the Fourier transform to S'(ℝⁿ) defined by ⟨û, φ⟩ = ⟨u, φ̂⟩ for φ ∈ S(ℝⁿ), yielding an automorphism of S'.
s_borel_sigma_algebra_on_Rn	axiom	Borel σ-algebra on ℝⁿ		The σ-algebra on ℝⁿ generated by the open sets in the product topology, equivalently by the open rectangles or the coordinate projections.
s_random_vector	axiom	Random vector		A measurable function X: (Ω, F) → (ℝⁿ, B(ℝⁿ)), equivalently an n-tuple of real-valued random variables.
s_pi_system	axiom	π-system		A nonempty collection of subsets of Ω closed under finite intersections.
s_lambda_system	axiom	λ-system (Dynkin system)		A collection of subsets containing Ω, closed under proper differences and countable increasing unions.
s_independence_of_sigma_algebras	axiom	Independence of σ-algebras		Sub-σ-algebras (F_i)_{i∈I} of F are independent if P(A₁ ∩ ··· ∩ Aₖ) = P(A₁)···P(Aₖ) for all finite selections Aⱼ ∈ F_{iⱼ} with distinct indices.
s_convergence_in_Lp	axiom	Convergence in Lᵖ		A sequence fₙ converges to f in Lᵖ if ∫|fₙ − f|ᵖ dμ → 0, for 1 ≤ p < ∞.
s_stopping_time	axiom	Stopping time		A random variable τ: Ω → {0,1,2,…} ∪ {∞} such that {τ ≤ n} ∈ Fₙ for every n.
s_tail_sigma_algebra	axiom	Tail σ-algebra		The σ-algebra T = ∩ₙ σ(Xₙ, Xₙ₊₁, …) capturing events whose occurrence is unaffected by any finite number of initial random variables.
s_backward_martingale	axiom	Backward (reversed) martingale		An integrable adapted process (Xₙ)_{n≤0} with respect to a decreasing filtration (Fₙ)_{n≤0} satisfying E[Xₙ | Fₘ] = Xₘ for m ≤ n ≤ 0.
s_regular_conditional_distribution	axiom	Regular conditional distribution		A function P(·|·): Ω × B(E) → [0,1] that is a probability measure in the second argument for each ω and a measurable function in ω for each Borel set.
s_sufficient_statistic	axiom	Sufficient statistic		A statistic T(X) such that the conditional distribution of the sample X given T(X) = t does not depend on the unknown parameter θ.
s_moment_generating_function	axiom	Moment generating function		The function M_X(t) = E[e^{tX}] defined for real t in the domain where the expectation is finite.
s_martingale_transform	axiom	Martingale transform		The discrete stochastic integral (H·M)ₙ = Σₖ₌₁ⁿ Hₖ(Mₖ − Mₖ₋₁) where H is a predictable process and M is a martingale.
s_convergence_determining_class	axiom	Convergence-determining class		A class C of bounded continuous functions on a metric space S such that ∫f dμₙ → ∫f dμ for all f ∈ C implies μₙ →_w μ.
s_separating_class	axiom	Separating class		A class C of bounded measurable functions such that ∫f dμ = ∫f dν for all f ∈ C implies μ = ν.
s_properties_of_conditional_expectation	state	Properties of conditional expectation		Conditional expectation is linear, monotone, satisfies the tower property E[E[X|G]|H] = E[X|H] for H ⊂ G, and the taking-out-what-is-known rule E[YX|G] = YE[X|G
s_relations_between_convergence_modes	state	Relations between modes of convergence		Almost sure convergence implies convergence in probability which implies convergence in distribution; Lᵖ convergence implies convergence in probability; none of
s_properties_of_characteristic_functions	state	Properties of characteristic functions		A characteristic function φ_X(t) = E[e^{itX}] is uniformly continuous, bounded by 1 with φ(0) = 1, Hermitian (φ(−t) = φ̄(t)), and positive definite.
s_dynkin_pi_lambda_theorem	theorem	Dynkin π-λ theorem		If a π-system P is contained in a λ-system L, then σ(P) ⊆ L; equivalently, two measures agreeing on a π-system agree on the generated σ-algebra.
s_monotone_class_theorem	theorem	Monotone class theorem		The smallest monotone class containing an algebra of sets coincides with the σ-algebra generated by that algebra.
s_kolmogorov_maximal_inequality	theorem	Kolmogorov's maximal inequality		For independent zero-mean random variables with finite variances, P(max_{1≤k≤n} |Sₖ| ≥ λ) ≤ Var(Sₙ)/λ² where Sₖ = X₁ + ··· + Xₖ.
s_levy_upward_theorem	theorem	Lévy's upward theorem		If Fₙ ↑ F∞ = σ(∪ₙ Fₙ) and X ∈ L¹, then E[X | Fₙ] → E[X | F∞] almost surely and in L¹.
s_levy_downward_theorem	theorem	Lévy's downward theorem		If Fₙ ↓ F∞ = ∩ₙ Fₙ and X ∈ L¹, then E[X | Fₙ] → E[X | F∞] almost surely and in L¹.
s_levy_inversion_formula	theorem	Lévy inversion formula		The distribution function F of a random variable X is recovered from its characteristic function φ by F(b) − F(a) = lim_{T→∞} (2π)⁻¹ ∫_{−T}^{T} (e^{−ita} − e^{−
s_uniqueness_theorem_characteristic_functions	theorem	Uniqueness theorem for characteristic functions		Two probability distributions on ℝ are equal if and only if their characteristic functions are equal.
s_cramer_wold_device	theorem	Cramér-Wold device		Random vectors Xₙ converge in distribution to X in ℝᵈ if and only if t·Xₙ →_d t·X for every t ∈ ℝᵈ.
s_method_of_moments	theorem	Method of moments (convergence)		If the moment sequence E[Xₙᵏ] → E[Xᵏ] for all k ≥ 1 and the distribution of X is determined by its moments, then Xₙ →_d X.
s_conditional_jensen_inequality	theorem	Conditional Jensen's inequality		If φ is convex and X, φ(X) ∈ L¹, then φ(E[X | G]) ≤ E[φ(X) | G] almost surely.
s_neyman_fisher_factorization_theorem	theorem	Neyman-Fisher factorization theorem		A statistic T(X) is sufficient for θ if and only if the density f(x; θ) factors as g(T(x), θ) · h(x) for some nonnegative functions g and h.
s_disintegration_of_measures	theorem	Disintegration of measures		A probability measure μ on a product of Polish spaces admits a disintegration into a family of regular conditional probability measures μ_y indexed by the margi
s_scheffe_lemma	theorem	Scheffé's lemma		If nonneg integrable functions fₙ → f a.e. and ∫fₙ → ∫f, then ∫|fₙ − f| → 0.
s_de_la_vallee_poussin_criterion	theorem	de la Vallée-Poussin criterion		A family of random variables is uniformly integrable if and only if there exists a nonneg convex function φ with φ(x)/x → ∞ such that sup E[φ(|Xᵢ|)] < ∞.
t_truncation_technique	technique	Truncation technique		Replaces a random variable by its truncation X·1_{|X|≤a} to reduce to bounded variables, then controls the remainder separately.
t_symmetrization_technique	technique	Symmetrization technique		Introduces independent copies X_i' and studies X_i − X_i' to exploit symmetry, converting one-sided tail bounds to two-sided via Lévy's symmetrization inequalit
t_characteristic_function_method	technique	Characteristic function method		Proves convergence in distribution by showing pointwise convergence of characteristic functions and applying the Lévy continuity theorem.
t_tightness_prokhorov_method	technique	Tightness-Prokhorov method		Establishes weak convergence by verifying tightness of the sequence of measures (via Prokhorov's theorem) and identifying the limit through finite-dimensional d
t_monotone_class_argument	technique	Monotone class argument		Extends a property verified on a π-system or an algebra to the generated σ-algebra by showing the collection of sets satisfying the property is a λ-system or mo
t_dyadic_approximation_technique	technique	Dyadic approximation technique (measure theory)		Approximates a nonneg measurable function from below by dyadic simple functions fₙ = Σₖ (k/2ⁿ)·1_{k/2ⁿ ≤ f < (k+1)/2ⁿ}, yielding a pointwise increasing sequence
t_upcrossing_argument	technique	Upcrossing argument (Doob)		Proves almost sure convergence of a submartingale by bounding the expected number of upcrossings of an interval [a,b] and applying the upcrossing inequality.
t_martingale_convergence_technique	technique	Martingale technique for proving convergence		Constructs a martingale or submartingale from the sequence of interest and applies Doob's convergence theorems to deduce almost sure or Lᵖ convergence.
s_g_delta_set	axiom	G_δ set		A subset of a topological space that is a countable intersection of open sets.
s_f_sigma_set	axiom	F_σ set		A subset of a topological space that is a countable union of closed sets.
s_positive_negative_parts	axiom	Positive and negative parts of a function		For a measurable function f, the decomposition f = f⁺ − f⁻ where f⁺ = max(f,0) and f⁻ = max(−f,0), with |f| = f⁺ + f⁻.
s_cauchy_in_measure	axiom	Cauchy in measure		A sequence {fₙ} of measurable functions is Cauchy in measure if for every ε, η > 0, there exists N such that μ({|fₙ − fₘ| ≥ η}) < ε for all n, m ≥ N.
s_non_atomic_measure	axiom	Non-atomic (diffuse) measure		A measure with no atoms: for every measurable set A with μ(A) > 0, there exists a measurable B ⊆ A with 0 < μ(B) < μ(A).
s_atom_of_measure	axiom	Atom of a measure		A measurable set A with μ(A) > 0 such that for every measurable B ⊆ A, either μ(B) = 0 or μ(B) = μ(A).
s_baire_sigma_algebra	axiom	Baire σ-algebra		On a compact Hausdorff space, the σ-algebra generated by the compact G_δ sets, equivalently the smallest σ-algebra making all continuous real-valued functions m
s_uniform_integrability_definition	axiom	Uniform integrability		A family {fₐ} of integrable functions is uniformly integrable if supₐ ∫_{|fₐ|>M} |fₐ| dμ → 0 as M → ∞ and for every ε > 0 there exists δ > 0 with μ(A) < δ imply
s_weak_convergence_in_lp	axiom	Weak convergence in Lp		A sequence {fₙ} in Lp converges weakly to f (fₙ ⇀ f) if ∫ fₙg dμ → ∫ fg dμ for every g ∈ Lq where 1/p + 1/q = 1.
s_modular_function_group	state	Modular function of a locally compact group		The continuous homomorphism Δ: G → (0, ∞) defined by μ(Eg) = Δ(g)μ(E), relating left Haar measure to right translation.
s_density_of_rationals_in_r	theorem	Density of rationals in ℝ		Between any two distinct real numbers there exists a rational number.
s_simple_approximation_theorem	theorem	Simple approximation theorem		Every nonnegative measurable function is the pointwise limit of an increasing sequence of nonnegative simple functions; if bounded, the convergence is uniform.
s_continuity_of_measure_below	theorem	Continuity of measure from below		If {Eₙ} is an ascending sequence of measurable sets, then μ(⋃ Eₙ) = lim μ(Eₙ).
s_continuity_of_measure_above	theorem	Continuity of measure from above		If {Eₙ} is a descending sequence of measurable sets with μ(E₁) < ∞, then μ(⋂ Eₙ) = lim μ(Eₙ).
s_regularity_lebesgue_measure	theorem	Regularity of Lebesgue measure		Lebesgue measure is regular: every measurable set can be approximated from outside by open sets and from inside by closed sets to arbitrary precision in measure
s_clarkson_inequalities	theorem	Clarkson's inequalities		For 2 ≤ p < ∞: ‖(f+g)/2‖ₚᵖ + ‖(f−g)/2‖ₚᵖ ≤ (‖f‖ₚᵖ + ‖g‖ₚᵖ)/2, with an analogous inequality for 1 < p < 2 using the conjugate exponent.
s_uniform_convexity_lp	theorem	Uniform convexity of Lp (1 < p < ∞)		For 1 < p < ∞, Lp is uniformly convex: if ‖fₙ‖ₚ ≤ 1, ‖gₙ‖ₚ ≤ 1, and ‖fₙ + gₙ‖ₚ → 2, then ‖fₙ − gₙ‖ₚ → 0.
s_radon_riesz_theorem	theorem	Radon–Riesz theorem		In Lp for 1 < p < ∞, if fₙ ⇀ f weakly and ‖fₙ‖ₚ → ‖f‖ₚ, then fₙ → f strongly in Lp.
s_banach_saks_theorem	theorem	Banach–Saks theorem		If {fₙ} converges weakly to f in Lp for 1 < p < ∞, then there exists a subsequence whose Cesàro means converge strongly to f.
s_weak_lower_semicontinuity_norm	theorem	Weak lower semicontinuity of the norm		If xₙ ⇀ x weakly in a normed space, then ‖x‖ ≤ lim inf ‖xₙ‖.
s_weak_sequential_compactness_lp	theorem	Weak sequential compactness in Lp (1 < p < ∞)		Every bounded sequence in Lp(E) for 1 < p < ∞ has a weakly convergent subsequence.
s_separability_of_lp	theorem	Separability of Lp (1 ≤ p < ∞)		The space Lp(ℝ) is separable for 1 ≤ p < ∞, possessing a countable dense subset.
s_l_infinity_not_separable	theorem	L∞ is not separable		The Banach space L∞ does not have a countable dense subset.
s_inclusion_lp_finite_measure	theorem	Inclusion of Lp spaces on finite measure sets		If μ(E) < ∞ and 1 ≤ p < q ≤ ∞, then Lq(E) ⊆ Lp(E) with ‖f‖ₚ ≤ μ(E)^{1/p−1/q} ‖f‖_q.
s_interpolation_lp_norms	theorem	Interpolation of Lp norms		If f ∈ Lp ∩ Lr for p < r, then f ∈ Ls for all p ≤ s ≤ r with ‖f‖_s ≤ ‖f‖ₚᵅ ‖f‖ᵣᵝ for appropriate α + β = 1.
s_schur_property_l1	theorem	Schur property of ℓ¹		In ℓ¹, weak convergence implies strong convergence: if xₙ ⇀ x weakly then ‖xₙ − x‖₁ → 0.
s_schauder_compact_adjoint	theorem	Schauder's theorem (compact adjoint)		A bounded linear operator T between Banach spaces is compact if and only if its adjoint T* is compact.
s_compact_operators_closed_ideal	theorem	Compact operators form a closed ideal		The set K(X,Y) of compact operators between Banach spaces X, Y is a closed subspace of B(X,Y) and a two-sided ideal under composition.
s_ulam_tightness_theorem	theorem	Ulam's theorem (tightness on Polish spaces)		Every finite Borel measure on a complete separable metric space is tight: for every ε > 0, there exists a compact K with μ(X ∖ K) < ε.
s_intermediate_value_nonatomic	theorem	Intermediate value property for non-atomic measures		If μ is a non-atomic measure and μ(E) > 0, then for every 0 ≤ t ≤ μ(E) there exists a measurable F ⊆ E with μ(F) = t.
s_completeness_convergence_measure	theorem	Completeness of convergence in measure		If {fₙ} is Cauchy in measure, then there exists a measurable function f such that fₙ → f in measure.
s_inversion_formula_haar	theorem	Inversion formula for Haar measure		For left Haar measure μ with modular function Δ: ∫ f(x⁻¹) Δ(x⁻¹) dμ(x) = ∫ f(x) dμ(x).
s_condensation_of_singularities	theorem	Condensation of singularities		If {Tₙ} are bounded linear operators on a Banach space with sup ‖Tₙ‖ = ∞, then {x : sup ‖Tₙx‖ = ∞} is residual (comeager).
s_lp_approximation_by_simple	theorem	Density of simple functions in Lp		Simple functions with support of finite measure are dense in Lp(X, μ) for 1 ≤ p < ∞.
s_lp_approximation_by_continuous	theorem	Density of continuous functions in Lp(ℝ)		The continuous functions with compact support Cc(ℝ) are dense in Lp(ℝ) for 1 ≤ p < ∞.
s_cc_dense_in_lp	theorem	Density of Cc in Lp for Radon measures		If μ is a Radon measure on a locally compact Hausdorff space, then Cc(X) is dense in Lp(X, μ) for 1 ≤ p < ∞.
s_ac_maps_null_to_null	theorem	Absolutely continuous functions map null sets to null sets		If f is absolutely continuous on [a,b] and m(E) = 0, then m(f(E)) = 0.
s_lebesgue_decomposition_bv	theorem	Lebesgue decomposition of BV functions		Every function of bounded variation on [a,b] decomposes as f = f_ac + f_s where f_ac is absolutely continuous and f_s is singular (f_s' = 0 a.e.).
s_daniell_stone_representation	theorem	Daniell–Stone representation theorem		A positive linear functional on a vector lattice of functions satisfying a monotone convergence condition can be represented as integration with respect to a me
s_reflexivity_of_lp	theorem	Reflexivity of Lp (1 < p < ∞)		For 1 < p < ∞, the canonical embedding J: Lp → (Lp)** is surjective, so Lp is reflexive.
s_l1_not_reflexive	theorem	L¹ is not reflexive		L¹(X, μ) is not reflexive when the measure space is not purely atomic with finitely many atoms, since (L¹)* = L∞ but (L∞)* ⊋ L¹.
s_haar_measure_uniqueness	theorem	Uniqueness of Haar measure		The left Haar measure on a locally compact group is unique up to a positive scalar multiple.
s_l2_is_hilbert_space	state	L² is a Hilbert space		L²(X, μ) is a Hilbert space with inner product ⟨f, g⟩ = ∫ fg dμ.
t_exhaustion_sigma_finite	technique	Exhaustion technique (σ-finite decomposition)		Decomposes a σ-finite space into sets of finite measure, proves the result on each piece, then combines via monotone limits or countable additivity.
t_density_argument_lp	technique	Density argument in Lp		Proves an inequality or identity for a dense subclass (simple, continuous, or Cc functions), then extends to all of Lp by continuity and approximation.
t_duality_method_functional_analysis	technique	Duality method (functional analysis)		Proves properties of a Banach space or Lp by studying the action of dual-space elements, e.g., showing f = 0 by verifying ∫fg = 0 for all g in the dual.
t_uniform_integrability_technique	technique	Uniform integrability technique		Establishes Lp convergence by verifying uniform integrability of {|fₙ|ᵖ} and convergence in measure, bypassing dominated convergence when no single dominator ex
t_weak_compactness_argument	technique	Weak compactness argument		Extracts weakly convergent subsequences from bounded sequences in reflexive spaces via Eberlein–Šmulian, then uses weak lower semicontinuity to pass to limits.
t_egoroff_lusin_chain	technique	Egoroff–Lusin technique chain		Combines Egoroff's theorem (a.e. convergence → near-uniform convergence) with Lusin's theorem (measurability → near-continuity) to reduce measurable-function ar
t_baire_category_method	technique	Baire category method in analysis		Uses completeness and the Baire category theorem to show that a countable intersection of dense open sets is dense, proving existence or generic properties.
t_radon_nikodym_technique	technique	Radon–Nikodym technique		Represents absolutely continuous measures or bounded linear functionals as integrals against an Lp function via the Radon–Nikodym theorem, central to Lp duality
t_completeness_via_rapidly_cauchy	technique	Completeness via rapidly Cauchy subsequences		Proves completeness by showing every rapidly Cauchy sequence (with summable consecutive differences) converges, which implies every Cauchy sequence converges.
t_tonelli_fubini_criterion	technique	Tonelli–Fubini criterion		First applies Tonelli's theorem to |f| to verify integrability via the iterated integral, then applies Fubini's theorem to compute ∫∫ f d(μ×ν).
s_chordal_metric	axiom	Chordal metric on the Riemann sphere		The metric d(z,w) = |z-w|/sqrt((1+|z|^2)(1+|w|^2)) on the extended complex plane, induced by stereographic projection onto the unit sphere.
s_stereographic_projection	axiom	Stereographic projection		The conformal bijection from the unit sphere minus the north pole to the complex plane, extending to a homeomorphism S^2 -> C_hat.
s_argument_of_complex_number	axiom	Argument of a complex number		The angle arg(z) = theta such that z = |z|e^{i theta}, defined modulo 2pi; the principal value Arg(z) is the unique theta in (-pi, pi].
s_polar_representation_complex	axiom	Polar representation of complex numbers		Every nonzero complex number z has a unique representation z = r e^{i theta} with r = |z| > 0 and theta = arg(z).
s_complex_conjugate	axiom	Complex conjugate		The involution z-bar = a - bi for z = a + bi, satisfying |z|^2 = z z-bar and fixing R pointwise.
s_modulus_complex_number	axiom	Modulus of a complex number		The absolute value |z| = sqrt(a^2 + b^2) for z = a + bi, defining the Euclidean metric on C and satisfying |zw| = |z||w|.
s_complex_power	axiom	Complex power z^alpha		For z != 0 and alpha in C, z^alpha = e^{alpha log z}, a multi-valued function single-valued only when alpha is an integer.
s_complex_trigonometric_functions	axiom	Complex trigonometric functions		The extensions sin z = (e^{iz} - e^{-iz})/(2i) and cos z = (e^{iz} + e^{-iz})/2, entire functions satisfying cos^2 z + sin^2 z = 1 on all of C.
s_cross_ratio	axiom	Cross-ratio		The projective invariant (z_1,z_2,z_3,z_4) = ((z_1-z_3)(z_2-z_4))/((z_1-z_4)(z_2-z_3)), preserved by Mobius transformations and real iff the four points are con
s_riemann_surface_of_sqrt_z	axiom	Riemann surface of sqrt(z)		The two-sheeted branched covering of C on which w^2 = z becomes single-valued; topologically a sphere with two branch points at 0 and infinity.
s_riemann_surface_of_log_z	axiom	Riemann surface of log z		The infinitely-sheeted unbranched covering of C \ {0} on which the logarithm is single-valued, realized as the universal cover with the map z -> e^z.
s_simply_periodic_function	axiom	Simply periodic function		A meromorphic function f satisfying f(z + omega) = f(z) for a single fundamental period omega != 0.
s_period_lattice	axiom	Period lattice		For an elliptic function with periods omega_1, omega_2 (linearly independent over R), the lattice Lambda = Z*omega_1 + Z*omega_2 of all periods.
s_fundamental_parallelogram	axiom	Fundamental parallelogram		A fundamental domain {s*omega_1 + t*omega_2 : 0 <= s,t < 1} for the lattice Lambda, on which an elliptic function's zeros and poles are counted.
s_order_of_entire_function	axiom	Order of an entire function		The quantity rho = lim sup log log M(r) / log r where M(r) = max_{|z|=r} |f(z)|, measuring the growth rate of an entire function.
s_genus_of_entire_function	axiom	Genus of an entire function		In the Hadamard factorization, the genus is max(deg Q, p) where p is the smallest integer with sum 1/|a_n|^{p+1} convergent and Q is the polynomial in the expon
s_schwarz_triangle_function	axiom	Schwarz triangle function		The conformal map from the upper half-plane onto a circular-arc triangle with angles alpha*pi, beta*pi, gamma*pi, expressible as a ratio of two hypergeometric s
s_analytic_continuation_along_path	axiom	Analytic continuation along a path		The process of extending a germ of an analytic function along a path via overlapping disk neighborhoods, producing a (possibly different) germ at the endpoint.
s_complete_analytic_function	axiom	Complete analytic function (Weierstrass)		The totality of all function elements obtainable by analytic continuation from a given germ, naturally organized as a sheaf of germs forming a Riemann surface.
s_algebraic_function	axiom	Algebraic function		A multi-valued function w(z) defined implicitly by an irreducible polynomial P(z,w) = 0, locally holomorphic away from finitely many branch points.
s_riemann_surface_of_algebraic_function	axiom	Riemann surface of an algebraic function		The compact Riemann surface associated to an irreducible polynomial P(z,w) = 0 of degree n in w, an n-sheeted branched covering of the Riemann sphere.
s_monodromy_group_linear_ode	axiom	Monodromy group of a linear ODE		The representation of pi_1 of the punctured domain in GL(n,C) obtained by analytically continuing a fundamental system of solutions around singular points.
s_harmonic_conjugate	axiom	Harmonic conjugate		For a harmonic u on a simply connected domain, the unique (up to constant) harmonic v such that u + iv is holomorphic, obtained by integrating the Cauchy-Rieman
s_residue_of_f_at_z0	axiom	Residue of f at an isolated singularity		The coefficient c_{-1} in the Laurent expansion about an isolated singularity z_0; equivalently Res(f, z_0) = (1/2pi i) oint f(z) dz around z_0.
s_complex_line_integral	axiom	Complex line integral		The integral of f along a piecewise smooth curve gamma, defined as integral_gamma f(z) dz = integral_a^b f(gamma(t)) gamma'(t) dt.
s_pole_of_order_m	axiom	Pole of order m		An isolated singularity z_0 where the Laurent series has c_{-m} != 0 and c_n = 0 for n < -m; equivalently (z-z_0)^m f(z) extends to a nonzero holomorphic value.
s_symmetry_principle_mobius	theorem	Symmetry principle for Möbius transformations		A Möbius transformation mapping a circle C to a circle C' sends points symmetric with respect to C to points symmetric with respect to C'.
s_conformality_of_analytic_functions	theorem	Conformality of analytic functions		A holomorphic function f is conformal (angle- and orientation-preserving) at every point z_0 where f'(z_0) != 0.
s_automorphisms_upper_half_plane	theorem	Automorphisms of the upper half-plane		Every biholomorphic self-map of the upper half-plane H has the form z -> (az+b)/(cz+d) with a,b,c,d real and ad-bc > 0, giving Aut(H) = PSL(2,R).
s_cauchy_theorem_multiply_connected	theorem	Cauchy's theorem for multiply connected regions		If f is holomorphic in a region bounded by outer contour Gamma_0 and inner contours Gamma_1,...,Gamma_n, then the integral over Gamma_0 equals the sum of integr
s_minimum_modulus_principle	theorem	Minimum modulus principle		If f is holomorphic and non-vanishing on a connected open set, then |f| has no local minimum in the interior.
s_schwarz_theorem_harmonic	theorem	Schwarz's theorem (Poisson integral solves Dirichlet problem for the disk)		The Poisson integral of a continuous function on the unit circle is harmonic in the disk and extends continuously to the given boundary values.
s_harnack_principle	theorem	Harnack's principle		A monotone increasing sequence of harmonic functions on a connected domain either converges uniformly on compacta to a harmonic function or diverges to +infinit
s_taylor_theorem_complex	theorem	Taylor's theorem (complex analysis)		A holomorphic function on a domain has a convergent Taylor series f(z) = sum f^{(n)}(a)/n! (z-a)^n in every disk centered at a contained in the domain.
s_laurent_theorem	theorem	Laurent's theorem		A holomorphic function on an annulus r < |z-a| < R has a unique Laurent series sum_{n=-infty}^{infty} c_n (z-a)^n converging uniformly on compact subannuli.
s_conformal_mapping_of_rectangles	theorem	Conformal mapping of rectangles to the upper half-plane		The Schwarz-Christoffel map for a rectangle yields an elliptic integral, and the conformal modulus (side ratio) is a complete conformal invariant.
s_canonical_mapping_multiply_connected	theorem	Canonical conformal mapping of multiply connected regions		Every finitely connected region with nondegenerate boundary components is conformally equivalent to a canonical slit domain (parallel, circular, or radial slits
s_liouville_theorems_elliptic	theorem	Liouville's theorems on elliptic functions		An elliptic function with no poles is constant; the sum of residues in a period parallelogram is zero; an order-n elliptic function assumes every value exactly 
s_differential_equation_weierstrass_p	theorem	Differential equation for the Weierstrass p-function		(p'(z))^2 = 4p(z)^3 - g_2 p(z) - g_3 where g_2 = 60 G_4, g_3 = 140 G_6, identifying C/Lambda with the elliptic curve y^2 = 4x^3 - g_2 x - g_3.
s_monodromy_theorem	theorem	Monodromy theorem		If a function element can be analytically continued along every path in a simply connected domain, then the continuation defines a single-valued holomorphic fun
s_cauchy_theorem_homological	theorem	General Cauchy theorem (homological version)		If f is holomorphic on U and Gamma is a cycle null-homologous in U, then integral_Gamma f(z) dz = 0 and the Cauchy integral formula holds for points with nonzer
t_normal_families_method	technique	Normal families method		Proves existence of extremal or limit functions by extracting convergent subsequences from a normal family via Montel's theorem.
s_wirtinger_derivatives	axiom	Wirtinger derivatives		The operators ∂/∂z = (1/2)(∂/∂x − i ∂/∂y) and ∂/∂z̄ = (1/2)(∂/∂x + i ∂/∂y), so that f is holomorphic iff ∂f/∂z̄ = 0.
s_primitive_of_holomorphic_function	axiom	Primitive (antiderivative) of a holomorphic function		A holomorphic function F on a domain Ω such that F'(z) = f(z); exists on simply connected domains for every holomorphic f, and ∫_γ f dz = F(γ(1)) − F(γ(0)).
s_ml_inequality	theorem	ML inequality (estimation lemma)		|∫_γ f(z) dz| ≤ M · L, where M = sup_{z ∈ γ} |f(z)| and L = length(γ), bounding contour integrals by supremum times arc length.
s_uniform_limit_of_holomorphic_functions	theorem	Uniform limit of holomorphic functions is holomorphic		If f_n are holomorphic on a domain Ω and f_n → f uniformly on compact subsets, then f is holomorphic on Ω and f_n' → f' uniformly on compacta.
s_zero_of_order_n	axiom	Zero of order n		A point z_0 where f(z_0) = f'(z_0) = ... = f^{(n-1)}(z_0) = 0 and f^{(n)}(z_0) ≠ 0, equivalently f(z) = (z − z_0)^n g(z) with g(z_0) ≠ 0.
s_principal_part_of_laurent_series	axiom	Principal part of a Laurent series		The sum ∑_{n=-∞}^{-1} c_n (z − z_0)^n of negative-power terms in the Laurent expansion about an isolated singularity, determining the singular behavior of f.
s_meromorphic_function	axiom	Meromorphic function		A function f on a domain Ω that is holomorphic except at a set of isolated points which are poles; equivalently, f is locally a ratio of holomorphic functions.
s_branch_of_logarithm	axiom	Branch of the logarithm		A continuous function log on a domain Ω ⊂ ℂ \ {0} satisfying e^{log z} = z; exists iff the winding number of every closed curve in Ω about 0 is zero.
s_simply_connected_characterization	theorem	Characterization of simply connected domains		For a connected open subset Ω of ℂ, the following are equivalent: Ω is simply connected; every holomorphic f on Ω has a primitive; ∫_γ f = 0 for all closed curv
t_keyhole_contour	technique	Keyhole contour technique		A contour shaped like a keyhole (large circle, slit along a ray, small circle) used to evaluate integrals involving branch cuts by exploiting the discontinuity 
s_riemann_sphere	axiom	Riemann sphere		The one-point compactification ℂ ∪ {∞} ≅ S², equipped with charts z and 1/z making it a compact Riemann surface; the natural domain for meromorphic functions an
s_automorphisms_of_riemann_sphere	theorem	Automorphisms of the Riemann sphere		Every biholomorphic self-map of the Riemann sphere ℂ ∪ {∞} is a Möbius transformation z ↦ (az+b)/(cz+d) with ad − bc ≠ 0, giving Aut(ℂ ∪ {∞}) ≅ PSL(2,ℂ).
s_cayley_transform	state	Cayley transform (complex analysis)		The Möbius transformation z ↦ (z − i)/(z + i) mapping the upper half-plane ℍ biholomorphically onto the unit disk 𝔻, with inverse w ↦ i(1+w)/(1−w).
s_elliptic_function	axiom	Elliptic function		A meromorphic function f on ℂ that is doubly periodic: f(z + ω₁) = f(z + ω₂) = f(z) for two ℝ-linearly independent periods ω₁, ω₂.
s_no_elliptic_function_of_order_one	theorem	No nonconstant elliptic function of order one		There is no elliptic function with exactly one simple pole in a period parallelogram; the minimum order of a nonconstant elliptic function is 2.
s_elliptic_function_rational_in_p_and_p_prime	theorem	Every elliptic function is rational in ℘ and ℘'		Every elliptic function with periods ω₁, ω₂ can be written as R(℘(z), ℘'(z)) for a rational function R in two variables.
s_theta_weierstrass_connection	state	Connection between theta functions and Weierstrass functions		The Weierstrass sigma function is expressible via Jacobi theta: σ(z) = (ω₁/π) exp(η₁z²/ω₁) θ₁(πz/ω₁, q) / θ₁'(0, q), and ℘(z) = −(d²/dz²) log θ₁(πz/ω₁, q) + con
s_r2_representation_number	state	Representation number r₂(n) via theta functions		r₂(n) = #{(a,b) ∈ ℤ² : a² + b² = n} equals the n-th Fourier coefficient of θ(τ)², giving r₂(n) = 4 ∑_{d|n} χ₄(d) where χ₄ is the nontrivial character mod 4.
s_r4_representation_number	state	Representation number r₄(n) via theta functions		r₄(n) = #{(a,b,c,d) ∈ ℤ⁴ : a² + b² + c² + d² = n} equals the n-th Fourier coefficient of θ(τ)⁴, yielding Jacobi's formula r₄(n) = 8 ∑_{4∤d, d|n} d.
s_euler_partition_identity	theorem	Euler's partition identity		The number of partitions of n into distinct parts equals the number of partitions of n into odd parts, proved via the generating function identity ∏(1+x^n) = ∏ 
s_weierstrass_p_via_theta_null	state	Weierstrass ℘-function from theta-null values		℘(z) is expressible in terms of theta functions and their values at z = 0 (theta-nulls), linking lattice invariants to modular quantities.
s_two_square_theorem_via_theta	theorem	Two-square theorem via theta functions		An integer n ≥ 1 is a sum of two squares iff every prime p ≡ 3 (mod 4) dividing n appears to an even power, proved via the factorization of θ(τ)² using Dirichle
t_generating_function_theta_method	technique	Generating function method via theta functions		Encode r_k(n) as Fourier coefficients of θ(τ)^k, then use modular form decomposition (Eisenstein + cusp) to derive exact formulas for representation numbers.
s_triangle_inequality_complex	theorem	Triangle inequality (complex)		For all z, w in C, |z + w| <= |z| + |w|, with equality iff one is a nonnegative real multiple of the other.
s_branch_of_multivalued_function	axiom	Branch of a multi-valued function		A single-valued continuous selection of one value of a multi-valued analytic function (such as log z or z^alpha) on a domain, obtained by introducing branch cut
s_fixed_point_classification_mobius	state	Fixed-point classification of Möbius transformations		A non-identity Möbius transformation is classified as parabolic (one fixed point, trace ±2), elliptic (two fixed points, |multiplier| = 1), loxodromic (two fixe
s_homotopy_of_curves	axiom	Homotopy of curves		Two closed curves gamma_0, gamma_1 in a domain G are homotopic if there exists a continuous deformation H: [0,1] x [0,1] -> G with H(0,·) = gamma_0 and H(1,·) =
s_cycle_complex_analysis	axiom	Cycle (complex analysis)		A formal integer linear combination of closed rectifiable curves in a domain, on which integration and winding numbers extend by linearity.
s_homology_null_homologous_cycle	axiom	Null-homologous cycle		A cycle Gamma in a domain G is null-homologous (homologous to zero) if n(Gamma, a) = 0 for every point a in C \ G, where n denotes the winding number.
s_maximum_modulus_theorem	theorem	Maximum modulus theorem		If f is holomorphic on a bounded domain G and continuous on its closure, then |f| attains its maximum on the boundary of G.
s_space_C_G_Omega	axiom	Space C(G, Ω)		The space of continuous functions from an open set G in C to a metric space Ω, equipped with the topology of uniform convergence on compact subsets of G.
s_germ_of_analytic_function	axiom	Germ of an analytic function		An equivalence class of function elements (a, sum a_n(z-a)^n) at a point a, where two elements are equivalent if they agree on a neighborhood of a.
s_sheaf_of_germs_holomorphic	state	Sheaf of germs of holomorphic functions		The étale space over a domain whose stalk at each point is the set of germs of holomorphic functions at that point, forming a Hausdorff topological space with a
s_riemann_surface_abstract	axiom	Riemann surface (abstract)		A connected one-dimensional complex manifold: a Hausdorff topological space with an atlas of charts to open subsets of C whose transition functions are biholomo
s_converse_mean_value_property	theorem	Converse of the mean value property		A continuous function on a domain satisfying the mean value property over all sufficiently small circles centered at each point is harmonic.
s_maximum_principle_harmonic	theorem	Maximum principle for harmonic functions		A nonconstant harmonic function on a connected open set attains neither its maximum nor its minimum in the interior.
s_dirichlet_problem_disk	state	Dirichlet problem for the disk		The problem of finding a function harmonic in the open unit disk and continuous on the closed disk that agrees with a given continuous function on the boundary 
s_superharmonic_function	axiom	Superharmonic function		A lower semicontinuous function u on a domain satisfying u(z_0) >= (1/2pi) integral_0^{2pi} u(z_0 + re^{i theta}) d theta for all small r, equivalently -u is su
s_maximum_principle_subharmonic	theorem	Maximum principle for subharmonic functions		A nonconstant subharmonic function on a connected open set cannot attain its maximum in the interior.
s_barrier_boundary_point	axiom	Barrier at a boundary point		A superharmonic function beta on G ∩ B(zeta, r) that is positive, tends to zero as z -> zeta, and is bounded below by a positive constant on G ∩ partial B(zeta,
s_regular_boundary_point	state	Regular boundary point		A boundary point zeta of a domain G at which every Perron solution to the Dirichlet problem attains the prescribed boundary value, equivalently a point admittin
s_poisson_jensen_formula	theorem	Poisson–Jensen formula		For f meromorphic in |z| <= R with zeros a_k and poles b_j, log |f(z)| equals the Poisson integral of log|f| on the boundary plus logarithmic terms from the zer
s_type_of_entire_function	state	Type of an entire function		For an entire function of finite order rho, the type sigma = lim sup (log M(r)) / r^rho, classifying growth as minimal (sigma=0), normal (0 < sigma < infty), or
s_canonical_product	state	Canonical product		The Weierstrass product P(z) = prod E_p(z/a_n) formed from the zeros {a_n} using elementary factors of genus p, representing an entire function with exactly tho
s_exponent_of_convergence	state	Exponent of convergence		For a sequence {a_n} with |a_n| -> infty, the infimum lambda of all alpha > 0 such that sum 1/|a_n|^alpha converges, measuring the density of the sequence.
s_landau_theorem	theorem	Landau's theorem (complex analysis)		There exists a universal constant L such that if f is holomorphic on the unit disk with f(0) = 0 and f'(0) = 1, then the image f(D) contains a disk of radius L.
s_schottky_theorem	theorem	Schottky's theorem		If f is holomorphic on the unit disk and omits the values 0 and 1, then |f(z)| is bounded by a constant depending only on |f(0)| and |z| < r < 1.
s_montel_three_omitted	theorem	Montel's theorem (omitting two values)		A family of holomorphic functions on a domain, each omitting two fixed values in C, is a normal family.
s_simple_connectivity_characterization	theorem	Characterization of simple connectivity		For a domain G in C, the following are equivalent: G is simply connected; every holomorphic function on G has a primitive; C_hat \ G is connected; every nonvani
s_conformal_equivalence	axiom	Conformal equivalence		Two domains are conformally equivalent if there exists a biholomorphic (bijective holomorphic with holomorphic inverse) map between them.
s_hyperbolic_distance	state	Hyperbolic distance on the disk		The distance d_D(z,w) = arctanh(|z-w|/|1-bar{w}z|) induced by the Poincaré metric ds = 2|dz|/(1-|z|^2), invariant under automorphisms of D.
s_caratheodory_boundary_correspondence	theorem	Carathéodory boundary correspondence theorem		A conformal map from the unit disk onto a Jordan domain extends to a homeomorphism from the closed disk onto the closed Jordan domain.
s_covering_space	axiom	Covering space		A surjective continuous map p: E -> X such that every point of X has a neighborhood U whose preimage is a disjoint union of open sets each mapped homeomorphical
s_analytic_covering_map	axiom	Analytic covering map		A holomorphic covering map between Riemann surfaces: a holomorphic surjection that is a covering space map, so that each fiber is discrete and the map is a loca
s_deck_transformation	state	Deck transformation		A homeomorphism phi: E -> E of a covering space satisfying p ∘ phi = p; the group of all deck transformations acts freely and properly discontinuously on E.
s_universal_covering_surface	state	Universal covering surface		The simply connected covering space of a Riemann surface, unique up to isomorphism; by the uniformization theorem, biholomorphic to D, C, or C_hat.
s_schlicht_class_S	axiom	Schlicht class S		The class of all univalent (injective holomorphic) functions f on the unit disk normalized by f(0) = 0 and f'(0) = 1, a compact family in the compact-open topol
s_koebe_function	state	Koebe function		The function k(z) = z/(1-z)^2 mapping the unit disk onto C minus the ray (-infty, -1/4], extremal for the Koebe 1/4 theorem and the coefficient bound |a_n| <= n
s_area_theorem_gronwall	theorem	Area theorem (Gronwall)		If g(z) = z + sum_{n=0}^{infty} b_n z^{-n} is univalent in |z| > 1, then sum n|b_n|^2 <= 1, with equality iff g maps onto a complement of a set of area zero.
s_bieberbach_inequality	theorem	Bieberbach's inequality		For f(z) = z + a_2 z^2 + ... in class S, |a_2| <= 2, with equality only for rotations of the Koebe function.
s_distortion_theorem	theorem	Distortion theorem (Koebe)		For f in class S and |z| < 1, (1-|z|)/(1+|z|)^3 <= |f'(z)| <= (1+|z|)/(1-|z|)^3, with equality for rotations of the Koebe function.
s_growth_theorem	theorem	Growth theorem (univalent functions)		For f in class S and |z| < 1, |z|/(1+|z|)^2 <= |f(z)| <= |z|/(1-|z|)^2, with equality for rotations of the Koebe function.
t_loewner_method	technique	Loewner's method		Embeds a univalent function in a continuously parametrized family f_t satisfying the Loewner ODE, reducing coefficient estimates to the analysis of a driving fu
s_loewner_differential_equation	state	Loewner's differential equation (radial)		The ODE df_t/dt = -f_t * (kappa(t) + f_t)/(kappa(t) - f_t) where |kappa(t)| = 1 is the driving function, whose solutions give a subordination chain of univalent
s_loewner_chain	state	Loewner chain		A family {f_t}_{t >= 0} of univalent functions on the disk with f_s(D) subset f_t(D) for s < t, satisfying the Loewner differential equation.
s_robertson_conjecture	state	Robertson conjecture		If g(z) = z + c_3 z^3 + c_5 z^5 + ... is an odd univalent function on the disk, then sum_{k=1}^n |c_{2k-1}|^2 <= n for all n, implied by the Milin conjecture.
s_milin_conjecture	state	Milin conjecture		For f in class S with log(f(z)/z) = 2 sum gamma_k z^k, the inequality sum_{k=1}^n (n-k+1)(k|gamma_k|^2 - 1/k) <= 0 holds for all n; proved by de Branges (1984),
s_boundary_values_Hp	theorem	Boundary values of H^p functions		Every function in H^p(D) for 0 < p <= infty has nontangential boundary values f*(e^{i theta}) existing almost everywhere on the unit circle, with f* in L^p.
s_Hp_norm_completeness	theorem	H^p norm and completeness		For 1 <= p <= infty, the Hardy space H^p(D) is a Banach space under the norm ||f||_p = sup_r (integral |f(re^{i theta})|^p d theta / 2pi)^{1/p}; for 0 < p < 1 i
s_singular_inner_function	state	Singular inner function		A function of the form S_mu(z) = exp(-integral (e^{i theta} + z)/(e^{i theta} - z) d mu(theta)) where mu is a positive singular measure on the circle, inner wit
s_Hp_duality	theorem	H^p duality		For 1 < p < infty, (H^p)* is isometrically isomorphic to H^q (1/p + 1/q = 1); (H^1)* is isometrically isomorphic to BMO via the pairing (f, g) -> integral f g* 
s_green_function_domain	state	Green's function of a domain		The function G(z, a) = -log|z - a| + h_a(z) where h_a is harmonic, with G vanishing on the boundary, serving as the fundamental solution for the Laplacian with 
s_symmetry_green_function	theorem	Symmetry of Green's function		The Green's function of a domain satisfies G(z, w) = G(w, z) for all distinct points z, w in the domain.
s_logarithmic_potential	state	Logarithmic potential		The function U^mu(z) = -integral log|z - w| d mu(w) associated to a compactly supported Borel measure mu, superharmonic on C and harmonic outside the support of
s_energy_of_measure	state	Energy of a measure		The quantity I(mu) = integral integral -log|z - w| d mu(z) d mu(w), whose infimum over probability measures supported on a compact set K equals the Robin consta
s_equilibrium_measure	state	Equilibrium measure		The unique probability measure mu_K supported on a compact set K that minimizes the energy I(mu), whose potential is constant quasi-everywhere on K.
s_logarithmic_capacity	state	Logarithmic capacity		The quantity cap(K) = e^{-V(K)} where V(K) = inf I(mu) is the Robin constant, equivalently the transfinite diameter; K is polar iff cap(K) = 0.
s_robin_constant	state	Robin constant		The value V(K) = inf{I(mu) : mu probability measure on K}, equal to -log cap(K); the equilibrium potential equals V(K) quasi-everywhere on K.
s_polar_set	axiom	Polar set (potential theory)		A set E such that there exists a subharmonic function u with E subset {z : u(z) = -infty}, equivalently a set of logarithmic capacity zero.
s_riesz_decomposition_subharmonic	theorem	Riesz decomposition theorem (subharmonic functions)		Every subharmonic function u on a domain equals h - U^mu where h is harmonic and U^mu is the logarithmic potential of the Riesz measure mu = (1/2pi) Delta u.
s_evans_theorem	theorem	Evans's theorem		Every compact polar set K is the minus-infinity set of some subharmonic function continuous on C \ K, so polar sets carry no positive mass for capacity-related 
s_weierstrass_product_domains	theorem	Weierstrass product theorem for domains		On any domain G in C, given a discrete set of prescribed zeros with multiplicities, there exists a holomorphic function on G with exactly those zeros.
s_lindelof_boundary_theorem	theorem	Lindelöf's theorem (boundary behavior)		If f is bounded and holomorphic on the unit disk and has a limit along a curve ending at e^{i theta}, then f has the same nontangential limit at e^{i theta}.
s_corona_theorem	theorem	Corona theorem		If f_1, ..., f_n in H^infty(D) satisfy inf_{z in D} sum |f_k(z)| > 0, then there exist g_1, ..., g_n in H^infty(D) with sum f_k g_k = 1, equivalently the open d
s_separation_properties_tvs	theorem	Separation properties of topological vector spaces		A Hausdorff topological vector space is automatically completely regular; in particular every TVS with T₁ separation is T₃½, since translations of balanced neig
s_continuity_boundedness_equivalence_linear_maps	theorem	Continuity–boundedness equivalence for linear maps		A linear map between topological vector spaces is continuous if and only if it is bounded (maps bounded sets to bounded sets), which is equivalent to continuity
s_seminorm_characterization_local_convexity	theorem	Seminorm characterization of local convexity		A topological vector space is locally convex if and only if its topology is generated by a family of seminorms; when the family is countable and separating, the
s_bilinear_map_continuity_theorem	theorem	Bilinear map continuity theorem (separate implies joint)		A separately continuous bilinear map from a product of Fréchet spaces into a TVS is jointly continuous; follows from Banach–Steinhaus applied to the family of p
s_balanced_convex_hull	axiom	Balanced convex hull		The balanced convex hull of a set A in a TVS is the smallest balanced convex set containing A; equivalently the set of all finite sums Σ λᵢxᵢ with xᵢ ∈ A, Σ|λᵢ|
s_extreme_point	axiom	Extreme point		A point x of a convex set K is an extreme point if x = ty + (1−t)z with 0 < t < 1 and y,z ∈ K implies y = z = x; equivalently, x is not an interior point of any
s_milman_converse_theorem	theorem	Milman's converse theorem		If K is a compact convex subset of a locally convex space and the closed convex hull of a subset S equals K, then every extreme point of K lies in the closure o
s_weak_star_convergence	axiom	Weak* convergence		A net (f_α) in the dual X* converges weak* to f if f_α(x) → f(x) for every x ∈ X; the weak* topology σ(X*,X) is the coarsest topology making the evaluation maps
s_schauder_basis	axiom	Schauder basis		A sequence (eₙ) in a Banach space X such that every x ∈ X has a unique expansion x = Σ aₙeₙ converging in the norm topology; the coordinate functionals aₙ: X → 
s_day_lp_trivial_dual	theorem	Trivial dual of Lp for 0 < p < 1		For 0 < p < 1 and a non-atomic measure μ, the only continuous linear functional on Lp(μ) is the zero functional; consequently Lp(μ) has no nonzero closed convex
s_lyapunov_vector_measure_theorem	theorem	Lyapunov's vector measure theorem		The range of a non-atomic vector-valued measure μ = (μ₁,…,μₙ) with each μᵢ a finite non-atomic scalar measure is a compact convex subset of ℝⁿ.
s_space_E_omega	axiom	Space E(Ω) = C^∞(Ω)		The space of all infinitely differentiable functions on an open set Ω ⊂ ℝⁿ, equipped with the topology of uniform convergence of all derivatives on compact subs
t_regularization_by_convolution	technique	Regularization of distributions by convolution		Convolves a distribution with a smooth compactly supported mollifier to produce a smooth function that converges to the original distribution in D'; used to app
s_completeness_of_distribution_space	theorem	Completeness of distribution spaces		If a sequence of distributions (uₙ) in D'(Ω) satisfies uₙ(φ) → L(φ) for every test function φ, then L is a distribution; D'(Ω) is complete as a locally convex s
s_sobolev_lemma	theorem	Sobolev's lemma (regularity from L² derivatives)		If u ∈ L²(ℝⁿ) has all distributional derivatives up to order s in L², and s > n/2 + k, then u agrees a.e. with a Cᵏ function; proved via the Fourier transform a
s_local_structure_theorem_distributions	theorem	Local structure theorem for distributions		Every distribution on an open set Ω ⊂ ℝⁿ is locally a finite-order distributional derivative of a continuous function; equivalently, for each compact K ⊂ Ω ther
s_elliptic_regularity_theorem	theorem	Elliptic regularity theorem		If P is an elliptic linear partial differential operator with smooth coefficients and u is a distribution satisfying Pu = f with f ∈ C^∞, then u ∈ C^∞; every el
s_hypoelliptic_operator	axiom	Hypoelliptic operator		A linear partial differential operator P on Ω is hypoelliptic if every distribution solution u of Pu = f is smooth wherever f is smooth; equivalently, sing supp
s_characterization_hypoelliptic_constant_coefficients	theorem	Characterization of hypoelliptic operators with constant coefficients		A constant-coefficient linear partial differential operator P(D) is hypoelliptic if and only if for every multi-index α ≠ 0, |D^α P(ξ)/P(ξ)| → 0 as |ξ| → ∞ alon
s_spanning_translates_L1	theorem	Spanning of translates in L¹(ℝⁿ)		The closed translation-invariant subspace of L¹(ℝⁿ) generated by f equals L¹(ℝⁿ) if and only if the Fourier transform f̂ has no real zeros; a key ingredient in 
s_spectral_synthesis_problem	state	Spectral synthesis problem		The question of whether every closed translation-invariant subspace of L¹(G) (or equivalently every closed ideal in L¹(G)) is the intersection of the maximal id
s_failure_of_spectral_synthesis	theorem	Failure of spectral synthesis (Malliavin)		Malliavin proved that spectral synthesis fails for every non-compact locally compact abelian group; there exist closed ideals in L¹(G) that are not intersection
s_wiener_general_tauberian_theorem	theorem	Wiener's general Tauberian theorem		If K ∈ L¹(ℝ) has Fourier transform that never vanishes, and if lim_{x→∞}∫K(x−t)φ(t)dt = A∫K for a bounded measurable φ that is slowly oscillating, then lim_{x→∞
s_ideals_quotient_banach_algebra	axiom	Ideals and quotient algebras in Banach algebras		A closed two-sided ideal I of a Banach algebra A makes the quotient A/I into a Banach algebra under the quotient norm; maximal ideals are automatically closed i
s_shilov_boundary	axiom	Shilov boundary		The smallest closed subset S of the maximal ideal space Δ(A) of a commutative unital Banach algebra A such that max_{φ∈Δ(A)} |â(φ)| = max_{φ∈S} |â(φ)| for all a
s_wiener_lemma	theorem	Wiener's lemma		If f is a nowhere-vanishing continuous function on the circle with absolutely convergent Fourier series (f ∈ A(T)), then 1/f also has absolutely convergent Four
t_symbolic_calculus_banach_algebra	technique	Symbolic calculus in commutative Banach algebras		Applies a function analytic in a neighborhood of the spectrum σ(a) to an element a of a commutative Banach algebra via the contour integral f(a) = (2πi)⁻¹∮f(λ)R
s_toeplitz_operator	axiom	Toeplitz operator		For φ ∈ L^∞(T), the Toeplitz operator T_φ on the Hardy space H² is defined by T_φf = P₊(φf), where P₊ is the orthogonal projection onto H²; the map φ ↦ T_φ is a
s_shift_operator	axiom	Shift operator (unilateral and bilateral)		The unilateral shift S on ℓ²(ℕ) maps (x₁,x₂,…) to (0,x₁,x₂,…); it is an isometry but not unitary, with S*S = I ≠ SS*, and its spectrum is the closed unit disk; 
s_positive_square_root	theorem	Positive square root of a positive operator		Every bounded positive operator T on a Hilbert space has a unique positive square root S = T^{1/2}, which commutes with every operator commuting with T; constru
s_unbounded_operator	axiom	Unbounded operator		A linear operator T defined on a dense linear subspace D(T) of a Hilbert space H, not required to be bounded; the domain D(T) is part of the data defining T.
s_closable_operator	axiom	Closable operator		An operator T is closable if the closure of its graph in H × H is the graph of an operator (called the closure of T); equivalently, if xₙ → 0 and Txₙ → y implie
s_symmetric_operator	axiom	Symmetric operator		A densely defined operator T on a Hilbert space is symmetric if ⟨Tx,y⟩ = ⟨x,Ty⟩ for all x,y ∈ D(T); equivalently T ⊂ T* (the domain of T is contained in the dom
s_adjoint_unbounded_operator	axiom	Adjoint of an unbounded operator		For a densely defined operator T on a Hilbert space, the adjoint T* has domain D(T*) = {y ∈ H : ∃z with ⟨Tx,y⟩ = ⟨x,z⟩ ∀x ∈ D(T)}, and T*y = z; T* is always clo
s_essentially_self_adjoint_operator	axiom	Essentially self-adjoint operator		A symmetric operator T is essentially self-adjoint if its closure T̄ is self-adjoint; equivalently, T has a unique self-adjoint extension, equivalently the defi
s_deficiency_indices	axiom	Deficiency indices		For a closed symmetric operator T, the deficiency indices n₊ = dim ker(T* − iI) and n₋ = dim ker(T* + iI) measure the 'distance' from self-adjointness; T has se
s_von_neumann_extension_theory	theorem	Von Neumann's extension theory for symmetric operators		A closed symmetric operator T has self-adjoint extensions if and only if its deficiency indices are equal (n₊ = n₋); when they are equal, the self-adjoint exten
s_nelson_analytic_vectors_theorem	theorem	Nelson's theorem on analytic vectors		If T is a symmetric operator on a Hilbert space and the set of analytic vectors for T (vectors v with Σ ||T^n v|| tⁿ/n! < ∞ for some t > 0) is dense, then T is 
s_semigroup_generation_by_laplacian	theorem	Semigroup generation by the Laplacian		The Laplacian Δ (or −Δ) on L²(ℝⁿ), or on L²(Ω) with Dirichlet/Neumann boundary conditions, is essentially self-adjoint on C_c^∞ and generates a strongly continu
t_neumann_series	technique	Neumann series		The geometric series Σₙ₌₀^∞ Tⁿ converging in operator norm to (I − T)⁻¹ whenever ‖T‖ < 1, used to construct resolvents and prove openness of the resolvent set.
s_trace_class_operator	axiom	Trace class operator		A compact operator T on a Hilbert space whose singular values {sₙ} satisfy Σ sₙ < ∞, forming the Banach ideal L¹(H) with trace norm ‖T‖₁ = Σ sₙ.
s_fredholm_index	state	Fredholm index		The integer ind(T) = dim ker T − dim coker T for a Fredholm operator T, invariant under compact perturbations and continuous in the operator norm topology.
s_spectral_multiplicity_theory	state	Spectral multiplicity theory		The classification of normal operators up to unitary equivalence by their spectral measure class and multiplicity function, generalizing the Jordan form to infi
s_tensor_product_of_hilbert_spaces	axiom	Tensor product of Hilbert spaces		The completion of the algebraic tensor product H₁ ⊗ H₂ under the inner product ⟨φ₁⊗φ₂, ψ₁⊗ψ₂⟩ = ⟨φ₁,ψ₁⟩⟨φ₂,ψ₂⟩, fundamental for multiparticle quantum mechanics.
s_relative_bound	axiom	Relative bound (Kato bound)		The infimum of constants a such that ‖Bφ‖ ≤ a‖Aφ‖ + b‖φ‖ for all φ ∈ D(A), measuring the subordination of operator B to operator A.
s_klmn_theorem	theorem	KLMN theorem		If a symmetric quadratic form b is relatively form-bounded with respect to a closed semibounded form a with relative bound less than 1, then a + b is closed and
s_form_representation_theorem	theorem	Form representation theorem (Kato)		Every closed, densely defined, semibounded quadratic form on a Hilbert space is the form of a unique self-adjoint operator, establishing a bijection between suc
t_friedrichs_extension	technique	Friedrichs extension		Construction of the canonical self-adjoint extension of a semibounded symmetric operator by closing its associated quadratic form and applying the form represen
s_analytic_vector	axiom	Analytic vector		A vector φ in the domain of all powers of an operator A such that Σ ‖Aⁿφ‖ tⁿ/n! converges for some t > 0, used in Nelson's theorem to establish essential self-a
s_kato_inequality	theorem	Kato's inequality		The distributional inequality Δ|u| ≥ Re(sgn(ū)·Δu) for u ∈ L¹_loc with Δu ∈ L¹_loc, a key tool for proving self-adjointness and semigroup domination.
s_hypercontractive_semigroup	state	Hypercontractive semigroup		A semigroup e^{−tH} that maps Lᵖ to Lᵍ for q > p after sufficient time, with the optimal bound given by Nelson's best hypercontractive estimate for the Ornstein
s_nelson_best_hypercontractive_estimate	theorem	Nelson's best hypercontractive estimate		The Ornstein–Uhlenbeck semigroup e^{−tN} is a contraction from Lᵖ to Lᵍ if and only if e^{−2t} ≤ (p−1)/(q−1), with sharpness attained on Gaussian functions.
s_diamagnetic_inequality	theorem	Diamagnetic inequality		The pointwise bound |e^{−t(−i∇−A)²}f(x)| ≤ e^{tΔ}|f|(x) showing that magnetic fields raise the ground state energy and reduce the semigroup kernel.
s_nelson_commutator_theorem	theorem	Nelson's commutator theorem		If H is symmetric and N is a self-adjoint operator with D(N) ⊂ D(H) satisfying ±i[H,N] ≤ cN on D(N), then H is essentially self-adjoint on D(N).
s_kato_class	axiom	Kato class		The class K_d of potentials V for which lim_{r→0} sup_x ∫_{|x−y|<r} |V(y)|·g_d(x−y) dy = 0 where g_d is the Green's function, ensuring −Δ+V is essentially self-
t_cook_method	technique	Cook's method		Proof of existence of wave operators by showing ∫₀^∞ ‖V e^{−iH₀t} φ‖ dt < ∞ for a dense set of φ, reducing existence to an integrability condition on the pertur
t_enss_method	technique	Enss method		A geometric, time-dependent proof of asymptotic completeness using phase space decomposition into incoming and outgoing states via propagation observables.
s_kato_smoothness	axiom	Kato smoothness (H-smoothness)		An operator A is H-smooth if ∫_{−∞}^{∞} ‖A e^{−iHt} φ‖² dt ≤ C‖φ‖² for all φ, equivalently sup_{ε>0,λ} ‖A(H−λ−iε)⁻¹A*‖ < ∞, implying absolute continuity of H on
s_kato_smoothness_absolute_continuity	theorem	Kato smoothness implies absolute continuity		If A is H-smooth, then the spectral measure of H restricted to Ran(A*) is absolutely continuous, providing a criterion for absence of singular spectrum.
s_lippmann_schwinger_equation	state	Lippmann–Schwinger equation		The integral equation ψ± = φ + (H₀ − E ∓ i0)⁻¹ V ψ± for scattering states, connecting the resolvent formalism to the time-independent scattering amplitudes.
t_stationary_scattering_theory	technique	Stationary scattering theory		Construction of the scattering matrix from boundary values of the resolvent at the real axis, relating the time-independent and time-dependent approaches via th
s_multichannel_scattering	state	Multichannel scattering		Scattering theory for N-body systems with multiple asymptotic cluster decompositions, each channel labeled by a partition of particles into bound clusters with 
s_asymptotic_completeness_n_body	theorem	Asymptotic completeness for N-body systems		The Hilbert space decomposes as H = H_pp ⊕ (⊕_α Ran Ω±_α) where the sum runs over all cluster decompositions α, proved by Sigal–Soffer and Derezinski for short-
s_dollard_modified_wave_operators	state	Dollard modified wave operators		Modified wave operators Ω±_D = s-lim_{t→±∞} e^{iHt} e^{−iH₀t} e^{−iΦ(t)} incorporating a long-range phase correction Φ(t) to handle Coulomb-type potentials wher
s_hvz_theorem	theorem	HVZ theorem (Hunziker–van Winter–Zhislin)		The essential spectrum of an N-body Hamiltonian H equals [Σ, ∞) where Σ = min_α inf σ(H_α) is the minimum over all two-cluster thresholds, identifying the botto
s_zhislin_theorem	theorem	Zhislin's theorem		A neutral or negatively charged atom (N ≥ Z) has infinitely many bound states below the essential spectrum, proved using the HVZ theorem and variational methods
s_analytic_family_type_A	axiom	Analytic family of type (A)		A family T(κ) of closed operators with common domain D independent of κ such that T(κ)φ is analytic in κ for each φ ∈ D, the Kato framework for non-degenerate p
s_analytic_family_type_B	axiom	Analytic family of type (B)		A family of operators defined via analytic quadratic forms t(κ)[φ] with form domain independent of κ, generalizing type (A) to handle perturbations defined only
t_asymptotic_perturbation_theory	technique	Asymptotic perturbation theory		Analysis of divergent perturbation series that are nonetheless asymptotic to the true eigenvalues, with error bounds of order (n!)κⁿ requiring summability metho
t_semiclassical_approximation	technique	Semiclassical approximation		Systematic expansion of quantum quantities in powers of ℏ recovering classical mechanics at leading order, including WKB approximation for eigenfunctions and We
s_perturbation_determinant	state	Perturbation determinant		The regularized determinant det_p(I + A(z)) for trace-class-type perturbations, whose zeros locate the eigenvalues of H₀ + V and whose logarithmic derivative gi
s_resonance_schrodinger	state	Resonance (Schrödinger operator)		A pole of the meromorphic continuation of the resolvent (H − z)⁻¹ across the continuous spectrum into the unphysical sheet, with imaginary part determining the 
s_absence_of_singular_continuous_spectrum	theorem	Absence of singular continuous spectrum		The singular continuous part σ_sc(H) = ∅ for Schrödinger operators under suitable conditions, typically proved via the Mourre estimate or Kato smoothness theory
s_virial_theorem	theorem	Virial theorem		For a Schrödinger operator H = −Δ + V with Hψ = Eψ, the identity 2⟨ψ, Tψ⟩ = ⟨ψ, x·∇V ψ⟩ holds, constraining eigenvalues and ruling out positive eigenvalues for 
s_temple_inequality	theorem	Temple's inequality		A lower bound for the lowest eigenvalue: if ψ is a trial vector and η₁ = inf σ_ess(H), then E₁ ≥ ⟨ψ,Hψ⟩ − (⟨ψ,H²ψ⟩ − ⟨ψ,Hψ⟩²)/(η₁ − ⟨ψ,Hψ⟩), improving on the Ra
s_kato_birman_invariance_principle	theorem	Kato–Birman invariance principle		If ψ(λ) is a monotone function and the wave operators Ω±(H, H₀) exist and are complete, then Ω±(ψ(H), ψ(H₀)) also exist and are complete, allowing passage betwe
t_ims_localization_formula	technique	IMS localization formula		The identity H = Σₐ Jₐ H Jₐ − Σₐ |∇Jₐ|² for a smooth partition of unity, decomposing the Hamiltonian into localized pieces for N-body analysis and proof of the 
s_rollnik_class	axiom	Rollnik class		The class of potentials V on ℝ³ satisfying ∫∫ |V(x)||V(y)|/|x−y|² dx dy < ∞, ensuring that the Birman–Schwinger kernel is Hilbert–Schmidt and enabling spectral 
s_simon_absence_positive_eigenvalues	theorem	Simon's theorem on absence of positive eigenvalues		For Schrödinger operators −Δ + V with V satisfying suitable decay conditions, there are no positive eigenvalues embedded in the continuous spectrum, extending t
t_decoupling_inequalities	technique	Decoupling inequalities		Inequalities bounding the interaction energy between particle clusters from below, used in proving stability of matter and analyzing N-body quantum systems.
s_hahn_banach_first_geometric_form	theorem	Hahn-Banach First Geometric Form		If A and B are disjoint nonempty convex subsets of a normed space with A open, then there exists a closed hyperplane separating A and B.
s_hahn_banach_second_geometric_form	theorem	Hahn-Banach Second Geometric Form		If A and B are disjoint nonempty convex subsets of a normed space with A closed and B compact, then there exists a closed hyperplane strictly separating A and B
s_conjugate_convex_function	axiom	Conjugate Convex Function (Legendre-Fenchel Transform)		For a function f: E -> (-inf, +inf], the conjugate f*(x*) = sup_{x in E} (<x*, x> - f(x)) is a convex lower semicontinuous function on E*.
s_fenchel_moreau_theorem	theorem	Fenchel-Moreau Theorem		A proper lower semicontinuous convex function on a locally convex space equals its biconjugate, f** = f.
s_young_fenchel_inequality	theorem	Young-Fenchel Inequality		For any function f and its conjugate f*, the inequality f(x) + f*(x*) >= <x*, x> holds for all x in E and x* in E*.
s_subdifferential	axiom	Subdifferential		The subdifferential of a convex function f at x is the set of all x* in E* satisfying f(y) >= f(x) + <x*, y - x> for all y, generalizing the derivative to nonsm
s_complemented_subspace	axiom	Complemented Subspace		A closed subspace M of a Banach space X is complemented if there exists a closed subspace N such that X = M + N and M ∩ N = {0}, equivalently if there exists a 
s_weak_star_topology	axiom	Weak* Topology σ(E*, E)		The coarsest topology on the dual E* making all evaluation maps x* -> <x*, x> continuous for each x in E, always coarser than the weak topology on E*.
s_finite_rank_operator	axiom	Finite-Rank Operator		A bounded linear operator whose range is finite-dimensional; finite-rank operators are compact and their norm closure equals the space of compact operators when
s_courant_fischer_min_max	theorem	Courant-Fischer Min-Max Principle		The k-th eigenvalue of a compact self-adjoint operator equals the min over k-dimensional subspaces of the max of the Rayleigh quotient, providing a variational 
s_maximal_monotone_operator	axiom	Maximal Monotone Operator		A multivalued operator A: D(A) subset H -> 2^H is maximal monotone if <Au - Av, u - v> >= 0 for all u, v in D(A) and the graph of A is not properly contained in
s_yosida_approximation	axiom	Yosida Approximation		For a maximal monotone operator A, the Yosida approximation A_lambda = (1/lambda)(I - J_lambda) is a bounded Lipschitz monotone operator converging to the minim
s_resolvent_of_monotone_operator	axiom	Resolvent of a Monotone Operator		For a maximal monotone operator A and lambda > 0, the resolvent J_lambda = (I + lambda A)^{-1} is a contraction defined on all of H, mapping each point to the u
s_m_dissipative_operator	axiom	m-Dissipative Operator		An operator A on a Banach space is m-dissipative if it is dissipative and R(I - lambda A) = X for some (equivalently all) lambda > 0, ensuring generation of a c
s_c0_semigroup	axiom	Strongly Continuous (C₀) Semigroup		A family {S(t)}_{t >= 0} of bounded linear operators on a Banach space satisfying S(0) = I, S(t+s) = S(t)S(s), and strong continuity lim_{t -> 0+} S(t)u = u for
s_infinitesimal_generator	axiom	Infinitesimal Generator of a Semigroup		The unbounded operator A defined by Au = lim_{t -> 0+} (S(t)u - u)/t on the domain of all u where this limit exists, characterizing the semigroup through the ab
s_exponential_formula_semigroups	theorem	Exponential Formula for Semigroups		For a C₀ contraction semigroup with generator A, the semigroup is recovered by S(t)u = lim_{n -> inf} (I - (t/n)A)^{-n} u for all u and t >= 0.
s_crandall_liggett_theorem	theorem	Crandall-Liggett Theorem		For an m-accretive operator A on a Banach space, the limit S(t)u = lim_{n -> inf} (I + (t/n)A)^{-n} u defines a nonlinear contraction semigroup on the closure o
s_sobolev_W1p_1d_characterization	theorem	Characterization of W^{1,p} in One Dimension		A function u belongs to W^{1,p}(I) for an interval I if and only if u has a representative that is absolutely continuous and whose classical derivative belongs 
s_compact_embedding_W1p_C_1d	theorem	Compact Embedding W^{1,p} into C in One Dimension		The injection W^{1,p}(I) into C(I-bar) is compact for any bounded interval I and 1 <= p <= inf, a one-dimensional precursor to Rellich-Kondrachov.
s_eigenvalues_second_derivative_1d	state	Eigenvalues of -d²/dx² on (0,L)		The eigenvalues of -u'' = lambda u on (0,L) with Dirichlet boundary conditions are lambda_n = (n pi / L)^2 for n >= 1, with eigenfunctions sin(n pi x / L).
s_sobolev_extension_operator	theorem	Sobolev Extension Operator		For a bounded domain Omega with Lipschitz boundary, there exists a bounded linear operator P: W^{1,p}(Omega) -> W^{1,p}(R^N) such that Pu|_Omega = u for 1 <= p 
s_morrey_inequality	theorem	Morrey's Inequality		If p > N, then W^{1,p}(R^N) embeds continuously into C^{0,alpha}(R^N) with alpha = 1 - N/p, so Sobolev functions with high integrability are Holder continuous.
s_poincare_wirtinger_inequality	theorem	Poincaré-Wirtinger Inequality		For u in W^{1,p}(Omega) on a bounded connected Lipschitz domain, ||u - u_avg||_{Lp} <= C ||grad u||_{Lp}, where u_avg is the mean value of u over Omega.
s_green_formula_sobolev	theorem	Green's Formula in Sobolev Spaces		For u in H^2(Omega) and v in H^1(Omega), the identity integral(-Delta u)v dx = integral grad u . grad v dx - integral_{dOmega} (du/dn)v dsigma holds, extending 
s_negative_sobolev_space_H_minus_1	axiom	Negative Sobolev Space H^{-1}(Ω)		The dual space of H^1_0(Omega), consisting of distributions that can be written as f_0 + sum d f_i / d x_i with each f_i in L^2(Omega).
s_sobolev_space_W0_1_p	axiom	Sobolev Space W₀^{1,p}(Ω)		The closure of C_c^inf(Omega) in W^{1,p}(Omega), consisting of W^{1,p} functions with zero trace on the boundary, generalizing H^1_0 = W_0^{1,2}.
s_trace_space_fractional_sobolev	axiom	Trace Space W^{1-1/p,p}(∂Ω)		The fractional Sobolev space on the boundary that is the image of the trace operator gamma_0: W^{1,p}(Omega) -> W^{1-1/p,p}(partial Omega), characterizing bound
s_eigenvalues_dirichlet_laplacian	theorem	Eigenvalues of -Δ on H¹₀(Ω)		For a bounded open set Omega, -Delta with Dirichlet conditions has eigenvalues 0 < lambda_1 < lambda_2 <= ... -> inf with an orthonormal basis of L^2(Omega) con
s_variational_characterization_lambda_1	theorem	Variational Characterization of First Eigenvalue		The first eigenvalue lambda_1 of -Delta on H^1_0(Omega) equals inf{integral |grad u|^2 / integral |u|^2 : u in H^1_0, u != 0}, achieved by a positive eigenfunct
s_domain_monotonicity_eigenvalues	theorem	Domain Monotonicity of Dirichlet Eigenvalues		If Omega_1 subset Omega_2 are bounded open sets, then lambda_k(Omega_1) >= lambda_k(Omega_2) for each k, so enlarging the domain decreases eigenvalues.
s_faber_krahn_inequality	theorem	Faber-Krahn Inequality		Among all bounded domains of fixed volume, the ball minimizes the first Dirichlet eigenvalue: lambda_1(Omega) >= lambda_1(Omega*) where Omega* is the ball with 
s_bv_space	axiom	Space of Functions of Bounded Variation BV(Ω)		The space of L^1 functions whose distributional gradient is a finite Radon measure, strictly containing W^{1,1} and fundamental for free boundary and image proc
s_polarization_identity	theorem	Polarization Identity		In a real inner product space, <x,y> = (1/4)(||x+y||^2 - ||x-y||^2), recovering the inner product from the norm; in the complex case, an analogous formula with 
s_orthogonal_decomposition_hilbert	theorem	Orthogonal Decomposition Theorem (Hilbert Space)		If M is a closed subspace of a Hilbert space H, then H = M + M^perp as an orthogonal direct sum, so every element decomposes uniquely into its projection onto M
s_duality_lp_spaces	theorem	Duality of Lp Spaces		For 1 <= p < inf with conjugate exponent q, the dual (L^p)* is isometrically isomorphic to L^q via the pairing integral fg; for p = 1, (L^1)* = L^inf.
s_lp_reflexivity	theorem	Reflexivity of Lp for 1 < p < ∞		For 1 < p < inf, the space L^p is reflexive, meaning the canonical embedding J: L^p -> (L^p)** is surjective.
s_density_Cc_infinity_in_Lp	theorem	Density of C_c^∞ in Lp		For 1 <= p < inf, the space C_c^inf(R^N) is dense in L^p(R^N), enabling approximation of any Lp function by smooth compactly supported functions.
s_continuity_of_translation_in_lp	theorem	Continuity of Translation in Lp		For 1 <= p < inf and f in L^p(R^N), the map h -> tau_h f is continuous from R^N to L^p, that is ||tau_h f - f||_p -> 0 as h -> 0.
s_interpolation_inequality_lp	theorem	Interpolation Inequality for Lp Norms		If f in L^p ∩ L^q with 1 <= p <= r <= q <= inf, then f in L^r and ||f||_r <= ||f||_p^a ||f||_q^{1-a} where 1/r = a/p + (1-a)/q.
s_obstacle_problem_variational	state	Obstacle Problem (Variational Inequality Formulation)		Find u in K = {v in H^1_0 : v >= psi a.e.} minimizing integral |grad v|^2 - 2 integral fv, equivalently satisfying a(u, v-u) >= (f, v-u) for all v in K, a proto
s_gagliardo_nirenberg_interpolation_general	theorem	Gagliardo-Nirenberg Interpolation Inequality (General)		||D^j u||_{Lr} <= C ||D^m u||_{Lp}^a ||u||_{Lq}^{1-a} for suitable exponents satisfying 1/r = j/N + a(1/p - m/N) + (1-a)/q, interpolating between different orde
s_necas_inequality	theorem	Nečas Inequality		A distribution f on a bounded Lipschitz domain belongs to L^p if and only if f and all its first-order distributional derivatives lie in W^{-1,p}, with equivale
t_minkowski_gauge	technique	Minkowski Gauge Technique		Constructs a sublinear functional p_C(x) = inf{alpha > 0 : x/alpha in C} from a convex set C, used to prove geometric forms of the Hahn-Banach theorem from the 
t_difference_quotient_regularity	technique	Difference Quotient Technique for Regularity		Approximates partial derivatives by difference quotients D_h u = (u(x+he)-u(x))/h, derives uniform H^1 bounds, then passes to the limit to gain one order of Sob
t_symmetric_rearrangement_technique	technique	Symmetric Decreasing Rearrangement Technique		Replaces a function by its equimeasurable radially symmetric decreasing rearrangement, used together with Polya-Szego inequality to prove symmetry of extremals 
t_weak_convergence_methods_sobolev	technique	Weak Convergence Methods in Sobolev Spaces		Combines weak sequential compactness in reflexive spaces, compact Sobolev embeddings, and weak lower semicontinuity of convex integrals to prove existence of mi
s_hardy_littlewood_maximal_theorem	theorem	Hardy–Littlewood maximal theorem		The Hardy–Littlewood maximal operator M is of weak type (1,1) with bound |{Mf > λ}| ≤ C_n‖f‖₁/λ, and bounded on L^p for 1 < p ≤ ∞.
s_dyadic_cubes	axiom	Dyadic cubes in ℝⁿ		The collection 𝒟 = {2^{−k}([0,1)ⁿ + m) : k ∈ ℤ, m ∈ ℤⁿ} of half-open cubes forming a nested filtration of ℝⁿ with each cube split into 2ⁿ children.
s_weak_type_p_p_estimate	state	Weak-type (p,p) estimate		A sublinear operator T is of weak type (p,p) if |{|Tf| > λ}| ≤ (C‖f‖_p / λ)^p for all f ∈ L^p and λ > 0.
s_maximal_function_characterization_of_Lp	theorem	Maximal function characterization of L^p		A function f belongs to L^p(ℝⁿ) for 1 < p ≤ ∞ if and only if its Hardy–Littlewood maximal function Mf belongs to L^p, with equivalent norms.
s_hilbert_transform	axiom	Hilbert transform		The singular integral operator Hf(x) = (1/π) p.v. ∫ f(y)/(x − y) dy, equivalently the Fourier multiplier with symbol −i sgn(ξ).
s_truncated_singular_integral	state	Truncated singular integral		The ε-truncated singular integral T_ε f(x) = ∫_{|x−y|>ε} K(x−y)f(y) dy, a well-defined absolutely convergent integral for f ∈ L^p.
s_pointwise_convergence_truncated_singular_integrals	theorem	Pointwise convergence of truncated singular integrals		For f ∈ L^p(ℝⁿ) with 1 ≤ p < ∞ and T a Calderón–Zygmund operator, the truncated integrals T_ε f(x) converge to Tf(x) as ε → 0 for almost every x.
s_maximal_singular_integral_operator	state	Maximal singular integral operator		The operator T*f(x) = sup_{ε>0} |T_ε f(x)| controlling the pointwise convergence of truncated Calderón–Zygmund singular integrals.
s_cotlar_inequality	theorem	Cotlar's inequality		For a Calderón–Zygmund operator T, the maximal singular integral satisfies T*f(x) ≤ C(M(Tf)(x) + Mf(x)) pointwise, reducing L^p bounds for T* to those for T.
t_calderon_zygmund_rotation_method	technique	Calderón–Zygmund rotation method		Reduces L^p bounds for singular integrals with odd homogeneous kernels on ℝⁿ to one-dimensional Hilbert transform bounds by averaging over rotations on S^{n−1}.
s_hormander_kernel_condition	axiom	Hörmander kernel condition		The integral condition sup_{y≠0} ∫_{|x|>2|y|} |K(x−y) − K(x)| dx ≤ C, sufficient together with L² boundedness to imply L^p boundedness of the associated operato
s_fourier_multiplier_representation_singular_integrals	theorem	Fourier multiplier representation of singular integrals		Every translation-invariant Calderón–Zygmund singular integral operator T with kernel K has Fourier multiplier representation (Tf)^∧ = m f̂ where m(ξ) = K̂(ξ) i
s_Lp_boundedness_of_riesz_transforms	theorem	L^p boundedness of Riesz transforms		Each Riesz transform R_j is bounded on L^p(ℝⁿ) for 1 < p < ∞ and of weak type (1,1), with ‖R_j f‖_p ≤ C_p‖f‖_p.
s_riesz_transforms_and_laplacian	theorem	Riesz transforms and the Laplacian		The Riesz transforms satisfy R_j = ∂_j(−Δ)^{−1/2} and Σ_j R_j² = −I, providing a singular-integral factorization of the Laplacian: ∂_i∂_j(−Δ)^{−1} = −R_iR_j.
s_conjugate_poisson_integral	state	Conjugate Poisson integral		The vector-valued function Q_y ∗ f(x) where Q_y is the conjugate Poisson kernel, forming the system of conjugate harmonic functions in the upper half-space.
s_spherical_harmonic_decomposition	theorem	Spherical harmonic decomposition of L²(S^{n−1})		L²(S^{n−1}) = ⊕_{k=0}^∞ H_k where H_k is the finite-dimensional space of spherical harmonics of degree k, mutually orthogonal eigenspaces of the Laplace–Beltram
s_hecke_identity_spherical_harmonics	theorem	Hecke's identity for spherical harmonics		The Fourier transform of a function of the form P_k(x)f(|x|), where P_k is a solid harmonic of degree k, factors as i^{−k}P_k(ξ)g(|ξ|) for an explicit radial fu
s_boundary_behavior_poisson_integrals	theorem	Boundary behavior of Poisson integrals in ℝ^{n+1}_+		The Poisson integral u(x,y) = P_y ∗ f(x) in the upper half-space ℝ^{n+1}_+ converges nontangentially to f(x) at almost every x ∈ ℝⁿ for f ∈ L^p.
s_multiplier_formula_spherical_harmonics	theorem	Multiplier formula via spherical harmonics		A homogeneous-of-degree-0 Fourier multiplier of a Calderón–Zygmund operator can be expanded in spherical harmonics, and L^p boundedness is determined by the dec
s_littlewood_paley_g_function	axiom	Littlewood–Paley g-function		The square function g(f)(x) = (∫₀^∞ |∇u(x,t)|² t dt)^{1/2} where u = P_t ∗ f is the Poisson integral, measuring oscillation of f via its harmonic extension.
s_littlewood_paley_g_lambda_star_function	axiom	Littlewood–Paley g*_λ function		The nontangential variant g*_λ(f)(x) = (∫∫_{ℝ^{n+1}_+} (t/(t+|x−y|))^{nλ} |∇u(y,t)|² t^{1−n} dy dt)^{1/2}, controlling the area integral with parameter λ.
s_lusin_area_integral	axiom	Lusin area integral (S-function)		The area integral S(f)(x) = (∫∫_{Γ(x)} |∇u(y,t)|² t^{1−n} dy dt)^{1/2} over the cone Γ(x) = {(y,t) : |x−y| < t}, measuring nontangential variation of the Poisso
s_S_function_Lp_equivalence	theorem	S-function L^p norm equivalence		For 1 < p < ∞, the Lusin area integral satisfies ‖S(f)‖_p ≈ ‖f‖_p, providing an equivalent characterization of L^p norms via the area function.
s_marcinkiewicz_multiplier_theorem	theorem	Marcinkiewicz multiplier theorem		A bounded function m on ℝⁿ satisfying coordinate-wise bounded variation conditions in each variable separately defines an L^p-bounded Fourier multiplier for 1 <
s_partial_sums_operator_fourier	state	Partial sums operator for Fourier integrals		The operator S_R f = (χ_{B(0,R)} f̂)^∨ projecting f onto frequencies in the ball of radius R, whose L^p boundedness in dimension n ≥ 2 fails for p ≠ 2.
s_Lp_boundedness_partial_sums_1d	theorem	L^p boundedness of Fourier partial sums in one dimension		In one dimension the partial sums operator S_R is bounded on L^p(ℝ) for 1 < p < ∞, reducible to compositions of the Hilbert transform.
s_bessel_potential_space	axiom	Bessel potential space L^p_α		The space L^p_α(ℝⁿ) = {f ∈ 𝒮′ : (1+|ξ|²)^{α/2} f̂ ∈ L^p} = J_{−α}(L^p), where J_α = (I − Δ)^{−α/2} is the Bessel potential operator.
s_equivalence_sobolev_bessel_potential_spaces	theorem	Equivalence of Sobolev and Bessel potential spaces		For integer k ≥ 0 and 1 < p < ∞, W^{k,p}(ℝⁿ) = L^p_k(ℝⁿ) with equivalent norms, identifying classical Sobolev spaces with Bessel potential spaces.
s_lipschitz_space_Lambda_alpha	axiom	Lipschitz space Λ_α		The space Λ_α(ℝⁿ) defined for 0 < α < 1 by the Hölder condition |f(x+h) − f(x)| ≤ C|h|^α, and for general α > 0 by iterated differences of order k > α.
s_zygmund_class	axiom	Zygmund class Λ_*		The space Λ_*(ℝⁿ) of continuous functions satisfying |f(x+h) + f(x−h) − 2f(x)| ≤ C|h|, strictly larger than Lip₁ and the correct substitute for Lipschitz order 
s_characterization_Lambda_alpha_via_poisson	theorem	Characterization of Λ_α via Poisson integral		A tempered distribution f belongs to Λ_α if and only if its Poisson integral u(x,t) satisfies |∂_t^k u(x,t)| ≤ Ct^{α−k} for all integers k > α.
s_characterization_Lp_alpha_via_littlewood_paley	theorem	Characterization of L^p_α via Littlewood–Paley theory		A tempered distribution f belongs to L^p_α(ℝⁿ) if and only if (Σ_j 2^{2jα}|Δ_j f|²)^{1/2} is in L^p, where Δ_j is the j-th Littlewood–Paley projection.
s_homogeneous_sobolev_space	axiom	Homogeneous Sobolev space Ẇ^{s,p}		The homogeneous Sobolev space Ẇ^{s,p}(ℝⁿ) defined by the seminorm ‖(−Δ)^{s/2}f‖_{L^p}, lacking lower-order terms present in the inhomogeneous space W^{s,p}.
s_riesz_potential_characterization_of_sobolev_spaces	theorem	Riesz potential characterization of Sobolev spaces		For 1 < p < n/k, the homogeneous Sobolev space Ẇ^{k,p}(ℝⁿ) consists precisely of functions representable as I_k(g) for g ∈ L^p, where I_k is the Riesz potential
s_difference_characterization_bessel_potential_spaces	theorem	Difference characterization of Bessel potential spaces		For α > 0 and 1 < p < ∞, f ∈ L^p_α(ℝⁿ) iff f ∈ L^p and ∫(|Δ_h^M f(x)|^p / |h|^{n+αp}) dh ∈ L¹ for M > α, characterizing Bessel potential spaces via integrated d
s_extension_theorem_sobolev_lipschitz_domains	theorem	Extension theorem for Sobolev spaces on Lipschitz domains		There exists a bounded linear extension operator E : W^{k,p}(Ω) → W^{k,p}(ℝⁿ) with Ef|_Ω = f, valid simultaneously for all k and p when Ω has Lipschitz boundary
s_total_extension_operator	state	Stein's total extension operator		A single linear operator extending functions from a half-space to all of ℝⁿ that is bounded on every W^{k,p} simultaneously, constructed via regularized reflect
s_restriction_theorem_riesz_potentials	theorem	Restriction theorem for Riesz potentials		Riesz potentials I_α f of L^p functions restrict to lower-dimensional submanifolds with bounds determined by the codimension and the relation of α, p, and n.
t_reflection_method	technique	Reflection method for Sobolev extension		Extends a Sobolev function from the half-space to all of ℝⁿ by defining the extension for x_n < 0 as a linear combination of f evaluated at reflected points, pr
s_nontangential_maximal_function	axiom	Nontangential maximal function		The function N(u)(x) = sup_{(y,t)∈Γ(x)} |u(y,t)| where Γ(x) = {(y,t) : |x−y| < t} is the cone at x, controlling the size of a function in the upper half-space.
s_nontangential_convergence	state	Nontangential convergence		A function u(y,t) in the upper half-space converges nontangentially to L at x₀ if u(y,t) → L as (y,t) → (x₀,0) within every cone Γ_α(x₀) = {(y,t) : |y−x₀| < αt}
s_fatou_theorem_upper_half_space	theorem	Fatou's theorem for the upper half-space		A harmonic function u in ℝ^{n+1}_+ with bounded L^p means for some 1 ≤ p ≤ ∞ has nontangential boundary limits at almost every point of ℝⁿ.
s_harmonic_function_characterization_area_integral	theorem	Area integral characterization of harmonic functions		For a harmonic function u in the upper half-space, the L^p norm of the area integral S(u) is comparable to the L^p norm of the nontangential maximal function N(
s_Hp_Lp_equivalence	theorem	H^p–L^p equivalence for p > 1		For 1 < p < ∞ the real Hardy space H^p(ℝⁿ) coincides with L^p(ℝⁿ) with equivalent norms; the two spaces differ only when p ≤ 1.
s_singular_integrals_Linfty_to_BMO	theorem	Singular integrals map L^∞ to BMO		A Calderón–Zygmund singular integral operator T maps L^∞(ℝⁿ) boundedly into BMO(ℝⁿ) with ‖Tf‖_{BMO} ≤ C‖f‖_∞, the endpoint substitute for the failing L^∞ → L^∞ 
s_area_integral_characterization_Hp	theorem	Area integral characterization of H^p		A tempered distribution f belongs to H^p(ℝⁿ) iff the area integral S(u) of its Poisson extension u = P ∗ f belongs to L^p, with ‖S(u)‖_p ≈ ‖f‖_{H^p}.
s_nontangential_maximal_characterization_Hp	theorem	Nontangential maximal function characterization of H^p		A tempered distribution f belongs to H^p(ℝⁿ) iff the nontangential maximal function N(P∗f) belongs to L^p, and ‖f‖_{H^p} ≈ ‖N(P∗f)‖_p.
s_conjugate_function_characterization_Hp	theorem	Conjugate function characterization of H^p		For n/(n+1) < p ≤ 1, f ∈ H^p(ℝⁿ) iff f and all its Riesz transforms R_jf belong to L^p, with ‖f‖_{H^p} ≈ ‖f‖_p + Σ_j‖R_jf‖_p.
s_sharp_function	axiom	Fefferman–Stein sharp function		The sharp maximal function f^#(x) = sup_Q (1/|Q|)∫_Q |f(y) − f_Q| dy, measuring local oscillation of f and controlling the BMO seminorm.
s_fefferman_stein_sharp_function_inequality	theorem	Fefferman–Stein inequality for the sharp function		For 1 < p < ∞, ‖f‖_p ≤ C_p‖f^#‖_p, so that L^p norms are controlled by the sharp maximal function, bridging pointwise oscillation and global integrability.
s_Lp_differentiability	state	L^p differentiability		A function f is L^p-differentiable at x if there exists a linear map L such that (r^{−n}∫_{B(x,r)} |f(y) − f(x) − L(y−x)|^p dy)^{1/p} = o(r) as r → 0.
s_calderon_theorem_Lp_differentiability	theorem	Calderón's theorem on L^p differentiability		Every function in the Sobolev space W^{1,p}(ℝⁿ) for 1 < p ≤ ∞ is L^p-differentiable at almost every point.
s_CZ_theorem_ae_differentiability	theorem	Calderón–Zygmund theorem on a.e. differentiability		If f ∈ W^{k,p}(ℝⁿ) with kp > n, then f is classically differentiable of order k−1 a.e.; for kp ≤ n, f is L^p-differentiable of order k at almost every point.
s_maximal_operator_sobolev_functions	state	Maximal operator for Sobolev functions		The Hardy–Littlewood maximal function Mf of f ∈ W^{1,p}(ℝⁿ), 1 < p ≤ n, belongs to W^{1,p} with ‖∇Mf‖_p ≤ C‖∇f‖_p.
s_lebesgue_point_theorem_sobolev_functions	theorem	Lebesgue point theorem for Sobolev functions		Every f ∈ W^{1,p}(ℝⁿ) has Lebesgue points at all x outside a set of p-capacity zero, a finer statement than the measure-zero exceptional set of classical Lebesg
s_p_capacity	axiom	p-capacity		Cap_p(E) = inf{‖u‖_{W^{1,p}}^p : u ≥ 1 on E, u ∈ W^{1,p}(ℝⁿ)}, measuring the size of E from the viewpoint of Sobolev functions; the correct notion of exceptiona
t_real_interpolation_method	technique	Real interpolation method (K-functional)		Constructs the interpolation space (A₀,A₁)_{θ,q} using the Peetre K-functional K(t,a) = inf_{a=a₀+a₁}(‖a₀‖_{A₀} + t‖a₁‖_{A₁}) with norm (∫(t^{−θ}K(t,a))^q dt/t)
t_complex_interpolation_method	technique	Complex interpolation method		Constructs [A₀,A₁]_θ as the values f(θ) where f is bounded analytic on the strip {0 < Re z < 1} with continuous boundary values in A₀ and A₁.
t_good_lambda_inequality	technique	Good-λ inequality technique		Proves ‖Sf‖_p ≤ C‖Tf‖_p by showing |{Sf > 2λ, Tf ≤ γλ}| ≤ c(γ)|{Sf > λ}| for small γ, then integrating over λ to transfer L^p bounds.
t_CZ_L2_boundedness_via_fourier	technique	L² boundedness of CZ operators via Fourier transform		Establishes L² boundedness of a CZ operator by showing its Fourier multiplier m = K̂ is bounded, using homogeneity and smoothness of K; combined with CZ theorem
t_duality_argument_weak_type	technique	Duality argument for weak-type estimates		Proves weak-type (1,1) bounds by dualizing and using the adjoint T* together with the Calderón–Zygmund decomposition to handle the singular part.
t_subordination_formula	technique	Subordination formula for Poisson semigroup		Recovers the Poisson semigroup from the heat semigroup via e^{−t√{−Δ}}f = (t/(2√π))∫₀^∞ s^{−3/2}e^{−t²/(4s)}e^{sΔ}f ds, transferring heat kernel estimates to Po
s_multiplier_problem_for_the_ball	state	Multiplier problem for the ball		The characteristic function χ_{B(0,1)} is not an L^p Fourier multiplier for p ≠ 2 when n ≥ 2 (Fefferman's theorem), in contrast to the one-dimensional case.
s_bochner_riesz_conjecture	state	Bochner–Riesz conjecture		The conjecture that Bochner–Riesz means S^δ_R are bounded on L^p(ℝⁿ) iff δ > max(n|1/p − 1/2| − 1/2, 0); proved for n = 2, open in higher dimensions.
s_CZ_lemma_dyadic	theorem	Calderón–Zygmund lemma (dyadic version)		For f ∈ L¹(ℝⁿ) and λ > 0, there exist disjoint dyadic cubes {Q_j} with averages exceeding λ and summing to at most ‖f‖₁/λ, yielding the canonical good/bad decom
s_fractional_laplacian	axiom	Fractional Laplacian (−Δ)^s		The operator (−Δ)^s defined via ((−Δ)^s f)^∧(ξ) = |ξ|^{2s}f̂(ξ), equivalently c_{n,s} p.v. ∫(f(x) − f(y))/|x−y|^{n+2s} dy, interpolating between identity and La
s_singular_integral_on_domain	state	Singular integral operator restricted to a domain		The operator T_Ω f(x) = ∫_Ω K(x,y)f(y) dy for f supported in Ω, whose L^p boundedness follows from that of T on ℝⁿ by extension and restriction.
s_elliptic_regularity_in_Lp	theorem	L^p elliptic regularity via singular integrals		For a uniformly elliptic operator L of order 2k with 1 < p < ∞, Lu = f with f ∈ L^p implies u ∈ W^{2k,p}_{loc}, proved via representation through Calderón–Zygmu
s_newtonian_potential	axiom	Newtonian potential		The convolution Nf(x) = c_n ∫ f(y)/|x−y|^{n−2} dy solving −Δ(Nf) = f, whose second derivatives are Calderón–Zygmund singular integrals yielding W^{2,p} regulari
s_calderon_commutator_theorem	theorem	Calderón commutator theorem		The first Calderón commutator C₁f(x) = p.v. ∫(A(x)−A(y))/(x−y)² f(y) dy is bounded on L² when A is Lipschitz, with ‖C₁‖ ≤ C‖A′‖_∞.
s_homogeneous_distribution	axiom	Homogeneous distribution		A tempered distribution u satisfying u(tx) = t^λ u(x) in the distributional sense for all t > 0; Calderón–Zygmund kernels are homogeneous distributions of degre
s_laplace_beltrami_operator_sphere	axiom	Laplace–Beltrami operator on S^{n−1}		The intrinsic Laplacian Δ_{S^{n−1}} on the unit sphere, with eigenvalues −k(k+n−2) for k = 0,1,2,… and eigenspaces the spaces of spherical harmonics of degree k
s_stein_maximal_principle	theorem	Stein's maximal principle		If {T_t} converges to the identity a.e. on a dense subclass of L^p and the maximal operator T*f = sup_t|T_tf| is of weak type, then T_tf → f a.e. for all f ∈ L^
s_semigroup_characterization_bessel_potentials	theorem	Semigroup characterization of Bessel potentials		The Bessel potential J_α f = (I−Δ)^{−α/2}f equals (1/Γ(α/2))∫₀^∞ e^{−t}e^{tΔ}f t^{α/2−1} dt, expressing Bessel potentials via the heat semigroup.
s_heat_kernel_Rn	axiom	Heat kernel on ℝⁿ		The Gaussian kernel W_t(x) = (4πt)^{−n/2}exp(−|x|²/(4t)), the fundamental solution of the heat equation ∂_tu = Δu, whose convolution W_t∗f defines the heat semi
s_dual_characterization_H1	theorem	Dual characterization of H¹		H¹(ℝⁿ) is the subspace of L¹ consisting of distributions f with sup{|⟨f,φ⟩| : φ ∈ BMO, ‖φ‖_{BMO} ≤ 1} finite, equivalently the predual of BMO.
s_interpolation_H1_Linfty	theorem	Interpolation between H¹ and L^∞		The complex interpolation space [H¹, L^∞]_θ = L^p for 1/p = 1−θ, extending the classical L^p interpolation scale through the Hardy space endpoint.
s_calderon_formula_sobolev_norms	theorem	Calderón's formula for Sobolev norms		The L^p_α norm of f equals ‖(∫₀^∞ |t^{−α}ψ_t∗f|² dt/t)^{1/2}‖_p for suitable ψ, connecting the Calderón reproducing formula to Sobolev-space norms.
s_circle_group_T	axiom	Circle group 𝕋		The quotient group ℝ/ℤ (equivalently the unit circle {z ∈ ℂ : |z| = 1} under multiplication), the fundamental domain for Fourier series.
s_fourier_coefficients_on_T	axiom	Fourier coefficients on 𝕋		The map f ↦ f̂(n) = ∫_𝕋 f(t)e^{−2πint} dt assigning to each integrable function on 𝕋 its sequence of Fourier coefficients.
s_trigonometric_polynomial	axiom	Trigonometric polynomial		A finite linear combination p(t) = ∑_{|n|≤N} c_n e^{2πint} of exponentials, forming a dense subspace of C(𝕋) and all L^p(𝕋).
s_convolution_on_T	state	Convolution on 𝕋		The operation (f ∗ g)(x) = ∫_𝕋 f(x − t)g(t) dt turning L¹(𝕋) into a commutative Banach algebra with (f̂∗g)(n) = f̂(n)ĝ(n).
s_approximate_identity_on_T	axiom	Approximate identity on 𝕋		A family {K_n} in L¹(𝕋) satisfying ∫K_n = 1, sup_n ‖K_n‖₁ < ∞, and ∫_{|t|>δ} |K_n(t)| dt → 0 for each δ > 0, so that K_n ∗ f → f in L^p and uniformly for contin
s_dirichlet_kernel_katznelson	state	Dirichlet kernel on 𝕋		D_N(t) = ∑_{n=−N}^{N} e^{2πint} = sin(π(2N+1)t)/sin(πt), the convolution kernel for the N-th partial sum of the Fourier series; ‖D_N‖₁ ~ (4/π²) log N.
s_fejer_kernel_katznelson	state	Fejér kernel on 𝕋		F_N(t) = (1/N)∑_{k=0}^{N−1} D_k(t) = (1/N)(sin(Nπt)/sin(πt))², a non-negative approximate identity giving the Cesàro means of the Fourier series.
s_fejer_theorem_katznelson	theorem	Fejér's theorem (uniform Cesàro convergence)		The Cesàro means σ_N(f) = F_N ∗ f converge uniformly to f for every continuous function f on 𝕋, and converge in L^p norm for every f ∈ L^p(𝕋) with 1 ≤ p < ∞.
s_poisson_kernel_on_T	state	Poisson kernel on 𝕋		P_r(t) = ∑_{n∈ℤ} r^{|n|} e^{2πint} = (1 − r²)/(1 − 2r cos 2πt + r²), the non-negative approximate identity for Abel summability of Fourier series.
s_abel_summability_fourier	theorem	Abel summability of Fourier series		For f ∈ L¹(𝕋), the Abel means P_r ∗ f(x) = ∑ f̂(n)r^{|n|}e^{2πinx} converge to f(x) at every Lebesgue point of f as r → 1⁻, and uniformly for continuous f.
s_uniqueness_theorem_fourier_series	theorem	Uniqueness theorem for Fourier series		If f̂(n) = 0 for all n ∈ ℤ then f = 0 a.e.; equivalently, the Fourier coefficient map L¹(𝕋) → c₀(ℤ) is injective.
s_riemann_lebesgue_on_T	theorem	Riemann–Lebesgue lemma on 𝕋		For f ∈ L¹(𝕋), the Fourier coefficients satisfy f̂(n) → 0 as |n| → ∞, i.e., the Fourier transform maps L¹(𝕋) into c₀(ℤ).
s_parseval_theorem_on_T	theorem	Parseval's theorem on 𝕋		The Fourier coefficient map f ↦ (f̂(n))_{n∈ℤ} is a unitary isomorphism from L²(𝕋) onto ℓ²(ℤ), with ‖f‖₂² = ∑|f̂(n)|².
s_bessel_inequality_on_T	theorem	Bessel's inequality on 𝕋		For f ∈ L²(𝕋) and any orthonormal system, the sum of squared Fourier coefficients is bounded by ‖f‖₂², with equality (Parseval) for complete systems.
s_weierstrass_trigonometric_approximation	theorem	Weierstrass trigonometric approximation theorem		The trigonometric polynomials are uniformly dense in C(𝕋); proved as a corollary of Fejér's theorem by showing σ_N(f) → f uniformly.
s_formal_multiplication_fourier_series	state	Formal multiplication of Fourier series		The Fourier series of f · g has coefficients (f̂ · ĝ)^∧(n) obtained via discrete convolution of Fourier coefficient sequences: ∑_k f̂(k)ĝ(n − k).
s_partial_sum_operator_on_T	state	Partial sum operator on 𝕋		The N-th partial sum S_N(f)(x) = ∑_{|n|≤N} f̂(n)e^{2πinx} = (D_N ∗ f)(x), a bounded operator on L^p for 1 < p < ∞ but unbounded on L¹ and C(𝕋).
s_dini_test	theorem	Dini's test for pointwise convergence		If ∫₀^δ |f(x₀ + t) + f(x₀ − t) − 2s|/t dt < ∞ for some s and δ > 0, then S_N(f)(x₀) → s as N → ∞.
s_jordan_test	theorem	Jordan's test for Fourier convergence		If f is of bounded variation in a neighborhood of x₀, then S_N(f)(x₀) → ½(f(x₀⁺) + f(x₀⁻)) as N → ∞.
s_dirichlet_jordan_theorem	theorem	Dirichlet–Jordan theorem		If f is of bounded variation on 𝕋, the Fourier series of f converges at every point to ½(f(x⁺) + f(x⁻)), and converges uniformly on every closed interval of con
s_du_bois_reymond_theorem	theorem	du Bois-Reymond theorem (divergent Fourier series)		There exists a continuous function on 𝕋 whose Fourier series diverges at a given point, proved via the unboundedness of the Dirichlet kernel norms ‖D_N‖₁ → ∞ an
s_localization_principle	theorem	Localization principle for Fourier series		If f = g on an open interval containing x₀, then S_N(f)(x₀) − S_N(g)(x₀) → 0, so convergence of Fourier series at a point depends only on the local behavior of 
s_kolmogorov_divergence_example	theorem	Kolmogorov's divergence example		There exists an integrable function f ∈ L¹(𝕋) whose Fourier series diverges at every point, showing Carleson's theorem cannot extend to L¹.
s_uniform_boundedness_and_divergence	theorem	Banach–Steinhaus theorem applied to Fourier divergence		Since ‖S_N‖ = ‖D_N‖₁ → ∞, the Banach–Steinhaus theorem implies that the set of continuous functions whose Fourier series diverges at a fixed point is a dense G_
s_lebesgue_constants	state	Lebesgue constants for Fourier partial sums		The operator norms L_N = ‖S_N‖ = ‖D_N‖₁ = (4/π²) log N + O(1), measuring the worst-case amplification of the N-th partial sum operator on C(𝕋).
s_best_approximation_fourier	theorem	Best approximation by trigonometric polynomials		The partial sum S_N(f) satisfies ‖f − S_N(f)‖_∞ ≤ (1 + L_N)E_N(f) where E_N(f) is the best uniform approximation by degree-N trigonometric polynomials.
s_conjugate_fourier_series	state	Conjugate Fourier series		The formal series f̃(x) ~ ∑_{n≠0} (−i sgn n) f̂(n) e^{2πinx}, obtained by replacing cos by sin in the Fourier expansion; the boundary value of the conjugate har
s_conjugate_function_on_T	state	Conjugate function on 𝕋		The function f̃(x) = limₑ→₀ ∫_{ε<|t|<½} f(x − t) cot(πt)/2 dt, the periodic analogue of the Hilbert transform, existing a.e. for f ∈ L¹(𝕋).
s_hilbert_transform_on_T	axiom	Hilbert transform on 𝕋		The bounded linear operator H : f ↦ f̃ on L^p(𝕋) for 1 < p < ∞, acting on Fourier coefficients by f̃̂(n) = −i sgn(n) f̂(n) and serving as the periodic Hilbert t
s_m_riesz_theorem_conjugate_function	theorem	M. Riesz theorem (L^p boundedness of conjugate function)		The conjugate function operator f ↦ f̃ is bounded on L^p(𝕋) for 1 < p < ∞, with norm growing as C_p ~ 1/(p−1) as p → 1⁺ and as p for p → ∞.
s_kolmogorov_weak_type_conjugate	theorem	Kolmogorov's weak-type (1,1) inequality for conjugate function		The conjugate function operator satisfies |{x : |f̃(x)| > λ}| ≤ C‖f‖₁/λ for all λ > 0 and f ∈ L¹(𝕋), the best possible substitute for L¹ boundedness which fails
s_zygmund_L_log_L_theorem	theorem	Zygmund's L log L theorem for conjugate function		If f ∈ L log L(𝕋) (i.e., ∫|f| log⁺|f| < ∞), then the conjugate function f̃ belongs to L¹(𝕋), identifying the precise Orlicz class below which L¹ boundedness of 
s_hardy_space_Hp_T	axiom	Hardy space H^p on the disk (Katznelson)		The space H^p(𝔻) of analytic functions F on the open unit disk with sup_{0<r<1} (∫_𝕋 |F(re^{2πit})|^p dt)^{1/p} < ∞; boundary values lie in L^p(𝕋) with non-nega
s_conjugate_poisson_kernel	state	Conjugate Poisson kernel on 𝕋		Q_r(t) = 2r sin(2πt)/(1 − 2r cos(2πt) + r²), the imaginary part of the Cauchy kernel (1 + re^{2πit})/(1 − re^{2πit}), whose convolution with f gives the conjuga
s_analytic_function_factorization_Hp	theorem	Inner-outer factorization in H^p		Every non-zero H^p function factors as F = BSO where B is a Blaschke product, S is a singular inner function, and O is an outer function, each determined up to 
s_helson_lowdenslager_prediction	theorem	Helson–Lowdenslager prediction theorem		The subspace ∨{e^{2πint} : n ≥ 1} applied to L²(w dt) equals L²(w dt) if and only if ∫ log w = −∞ (Szegő's condition); otherwise the complement is one-dimension
s_riesz_thorin_katznelson	theorem	Riesz–Thorin interpolation theorem (Katznelson)		If a linear operator T is bounded from L^{p₀} to L^{q₀} and from L^{p₁} to L^{q₁}, then T is bounded from L^{p_θ} to L^{q_θ} with 1/p_θ = (1−θ)/p₀ + θ/p₁, and l
s_marcinkiewicz_interpolation_katznelson	theorem	Marcinkiewicz interpolation theorem (Katznelson)		If a sublinear operator T is of weak type (p₀,p₀) and weak type (p₁,p₁) with p₀ < p₁, then T is bounded on L^p for all p₀ < p < p₁.
s_hausdorff_young_katznelson	theorem	Hausdorff–Young inequality (Katznelson)		For 1 ≤ p ≤ 2 and p' = p/(p−1), the Fourier coefficients satisfy ‖f̂‖_{ℓ^{p'}} ≤ ‖f‖_{L^p}, proved by interpolating between L¹ → ℓ^∞ (trivial) and L² → ℓ² (Pars
s_young_convolution_katznelson	theorem	Young's convolution inequality on 𝕋 (Katznelson)		If f ∈ L^p(𝕋), g ∈ L^q(𝕋) with 1/p + 1/q ≥ 1, then ‖f ∗ g‖_r ≤ ‖f‖_p ‖g‖_q where 1/r = 1/p + 1/q − 1.
t_three_lines_lemma	technique	Three-lines lemma (Hadamard)		If F is analytic and bounded on the strip 0 ≤ Re z ≤ 1 with |F(it)| ≤ M₀ and |F(1+it)| ≤ M₁, then |F(θ+it)| ≤ M₀^{1−θ} M₁^θ; the key analytic tool in the proof 
s_lacunary_series_hadamard	axiom	Lacunary series (Hadamard gaps)		A trigonometric series ∑ c_k e^{2πin_k t} whose frequencies {n_k} satisfy the Hadamard gap condition n_{k+1}/n_k ≥ q > 1, exhibiting pseudo-random behavior.
s_lacunary_convergence_theorem	theorem	Lacunary convergence theorem		If {n_k} is a lacunary sequence and f ∈ L¹(𝕋), then the partial sums ∑_{k≤K} f̂(n_k)e^{2πin_k x} converge a.e., in contrast to the general L¹ case where diverge
s_sidon_set	axiom	Sidon set		A subset Λ ⊂ ℤ such that every bounded function on Λ extends to the Fourier–Stieltjes transform of a measure on 𝕋; equivalently, every continuous function with 
s_sidon_set_characterization	theorem	Characterization of Sidon sets		A set Λ ⊂ ℤ is Sidon if and only if for every f ∈ C(𝕋) with supp f̂ ⊂ Λ, the Fourier series converges absolutely, if and only if ‖f‖_A ≤ C‖f‖_∞ for such f.
s_riesz_product	state	Riesz product		The weak-∗ limit μ = ∏_{k=1}^∞ (1 + a_k cos 2πn_k t) defines a positive measure on 𝕋 with Fourier coefficients supported on sums of ±n_k, used to construct sing
s_quasi_analytic_class	axiom	Quasi-analytic class C{M_n}		The class C{M_n} of C^∞ functions f on 𝕋 with ‖f^{(n)}‖_∞ ≤ A^n M_n for all n; the class is quasi-analytic if the only f ∈ C{M_n} vanishing to infinite order at
s_denjoy_carleman_theorem	theorem	Denjoy–Carleman theorem		The class C{M_n} is quasi-analytic if and only if ∑_{n=1}^∞ M_{n−1}/M_n = ∞ (equivalently ∑ (inf_{n} M_n^{1/n}/n) = ∞), characterizing when growth conditions on
s_rudin_shapiro_polynomials	state	Rudin–Shapiro polynomials		Trigonometric polynomials P_n of degree 2ⁿ − 1 with coefficients ±1 satisfying ‖P_n‖_∞ ≤ C√(2ⁿ), showing that flat polynomials with ±1 coefficients achieve opti
s_lacunary_Lp_equivalence	theorem	L^p equivalence for lacunary series		If f ∈ L¹(𝕋) has Fourier support on a lacunary set, then f ∈ L^p for all 1 ≤ p < ∞ and the L^p norms are mutually equivalent: c_p ‖f‖₂ ≤ ‖f‖_p ≤ C_p ‖f‖₂.
s_fourier_transform_L1_R	axiom	Fourier transform on L¹(ℝ)		The map f ↦ f̂(ξ) = ∫_ℝ f(x)e^{−2πixξ} dx, a bounded linear map from L¹(ℝ) to C₀(ℝ) (by Riemann–Lebesgue) that is injective but not surjective.
s_fourier_inversion_formula_R	theorem	Fourier inversion formula on ℝ		If both f and f̂ belong to L¹(ℝ), then f(x) = ∫_ℝ f̂(ξ)e^{2πixξ} dξ a.e., and f agrees a.e. with a continuous function.
s_plancherel_theorem_R	theorem	Plancherel theorem on ℝ		The Fourier transform extends uniquely from L¹ ∩ L² to a unitary operator on L²(ℝ) with ‖f̂‖₂ = ‖f‖₂, the L² isometry for the real line.
s_schwartz_space_katznelson	axiom	Schwartz space 𝒮(ℝ) (Katznelson)		The space 𝒮(ℝ) of C^∞ functions on ℝ with x^α ∂^β f → 0 as |x| → ∞ for all α, β; the Fourier transform is an automorphism of 𝒮(ℝ).
s_tempered_distributions_katznelson	axiom	Tempered distributions 𝒮′(ℝ) (Katznelson)		The topological dual 𝒮′(ℝ) of the Schwartz space, on which the Fourier transform extends by duality to a continuous automorphism; includes L^p functions, measur
s_poisson_summation_formula_katznelson	theorem	Poisson summation formula (Katznelson)		For suitable f on ℝ (e.g., f ∈ 𝒮(ℝ)), the identity ∑_{n∈ℤ} f(n) = ∑_{n∈ℤ} f̂(n) holds, relating the periodization of f to the sampling of f̂.
s_heisenberg_uncertainty_katznelson	theorem	Heisenberg uncertainty principle (Fourier analysis formulation)		A non-zero L² function and its Fourier transform cannot both be concentrated near the origin: ‖xf‖₂ · ‖ξf̂‖₂ ≥ ‖f‖₂²/(4π), with equality only for Gaussians.
s_wiener_tauberian_theorem_katznelson	theorem	Wiener's Tauberian theorem (Katznelson)		If f ∈ L¹(ℝ) and f̂(ξ) ≠ 0 for all ξ ∈ ℝ, then the closed ideal generated by f in L¹(ℝ) is all of L¹(ℝ); equivalently, the translates of f span a dense subspace
s_fourier_transform_of_measures_R	state	Fourier–Stieltjes transform on ℝ		The map μ ↦ μ̂(ξ) = ∫_ℝ e^{−2πixξ} dμ(x) extending the Fourier transform to finite Borel measures, with μ̂ bounded, uniformly continuous, and ‖μ̂‖_∞ ≤ ‖μ‖.
s_wiener_algebra_A_R	state	Wiener algebra A(ℝ)		The space A(ℝ) = {f̂ : f ∈ L¹(ℝ)} with norm ‖g‖_A = ‖f‖₁ where g = f̂, a regular commutative Banach algebra of continuous functions vanishing at infinity.
s_bernstein_inequality_fourier	theorem	Bernstein's inequality for trigonometric polynomials		For a trigonometric polynomial p of degree at most N, the derivative satisfies ‖p′‖_∞ ≤ 2πN‖p‖_∞, with equality for p(t) = e^{2πiNt}; extends to L^p norms.
s_lca_group_katznelson	axiom	Locally compact abelian group (Katznelson)		A topological group G that is locally compact, Hausdorff, and abelian; the natural domain for abstract Fourier analysis, encompassing ℝ, ℤ, 𝕋, ℝⁿ, ℤ_p, finite a
s_haar_measure_katznelson	theorem	Existence and uniqueness of Haar measure (Katznelson)		Every LCA group G admits a nonzero, translation-invariant, regular Borel measure (Haar measure), unique up to a positive multiplicative constant.
s_dual_group	axiom	Dual group (Pontryagin dual)		The group Ĝ = Hom(G, 𝕋) of continuous characters χ : G → 𝕋 under pointwise multiplication and the compact-open topology; Ĝ is itself an LCA group.
s_pontryagin_duality_katznelson	theorem	Pontryagin duality theorem (Katznelson)		The natural evaluation map G → Ĝ̂ sending x to the character χ ↦ χ(x) is a topological group isomorphism, so every LCA group is canonically isomorphic to its do
s_fourier_transform_lca	state	Fourier transform on LCA groups		The map f ↦ f̂(χ) = ∫_G f(x) χ(−x) dx from L¹(G) to C₀(Ĝ), generalizing both Fourier series (G = 𝕋) and the Fourier transform (G = ℝ).
s_plancherel_theorem_lca	theorem	Plancherel theorem for LCA groups		The Fourier transform extends to a unitary isomorphism from L²(G) onto L²(Ĝ) (with appropriately normalized Haar measures), generalizing Parseval/Plancherel for
s_fourier_inversion_lca	theorem	Fourier inversion formula on LCA groups		For f ∈ L¹(G) with f̂ ∈ L¹(Ĝ), one has f(x) = ∫_Ĝ f̂(χ) χ(x) dχ for appropriately dual-normalized Haar measures, recovering f from its Fourier transform.
s_bochner_theorem_katznelson	theorem	Bochner's theorem on LCA groups (Katznelson)		A continuous function φ on an LCA group G is positive definite if and only if it is the Fourier–Stieltjes transform of a unique non-negative finite Borel measur
s_structure_theorem_lca	theorem	Structure theorem for LCA groups		Every LCA group G is topologically isomorphic to ℝⁿ × H where n ≥ 0 and H contains a compact open subgroup; in particular, every compactly generated LCA group i
s_dual_group_examples	state	Standard dual group identifications		The canonical dual group identifications: ℝ̂ ≅ ℝ, 𝕋̂ ≅ ℤ, ℤ̂ ≅ 𝕋, (ℤ/nℤ)̂ ≅ ℤ/nℤ, and (G₁ × G₂)̂ ≅ Ĝ₁ × Ĝ₂.
s_annihilator_subgroup	state	Annihilator of a subgroup		For a closed subgroup H of LCA group G, the annihilator H⊥ = {χ ∈ Ĝ : χ|_H = 1} is a closed subgroup of Ĝ, with (G/H)̂ ≅ H⊥ and Ĥ ≅ Ĝ/H⊥.
s_positive_definite_function_lca	axiom	Positive definite function on LCA group		A continuous function φ on G such that ∑_{i,j} c_i c̄_j φ(x_i − x_j) ≥ 0 for all finite sets {x_i} and scalars {c_i}; the class of functions representable as Fo
s_commutative_banach_algebra_katznelson	axiom	Commutative Banach algebra (Katznelson)		A commutative algebra A over ℂ that is a Banach space with ‖ab‖ ≤ ‖a‖‖b‖, equipped with the Gelfand theory framework for spectral analysis of convolution operat
s_maximal_ideal_space	state	Maximal ideal space (spectrum) of a Banach algebra		The space Δ(A) (or M_A) of all non-zero multiplicative linear functionals on A, equipped with the weak-∗ topology, a compact Hausdorff space (compact if A is un
s_gelfand_transform_katznelson	state	Gelfand transform (Katznelson)		The map â(φ) = φ(a) sending each element a ∈ A to the continuous function â on the maximal ideal space, a contractive algebra homomorphism A → C₀(Δ(A)).
s_gelfand_representation_theorem_katznelson	theorem	Gelfand representation theorem (Katznelson)		The Gelfand transform is a continuous algebra homomorphism from A into C(Δ(A)) with ‖â‖_∞ ≤ ‖a‖ and r(a) = ‖â‖_∞ = lim ‖aⁿ‖^{1/n} (spectral radius formula).
s_wiener_lemma_katznelson	theorem	Wiener's lemma (absolutely convergent Fourier series)		If f has an absolutely convergent Fourier series and never vanishes on 𝕋, then 1/f also has an absolutely convergent Fourier series; proved elegantly via the Ge
s_wiener_algebra_A_T	axiom	Wiener algebra A(𝕋)		The Banach algebra A(𝕋) = {f ∈ C(𝕋) : ‖f‖_A = ∑|f̂(n)| < ∞} of functions with absolutely convergent Fourier series, with pointwise multiplication and norm ‖f‖_A
s_gelfand_mazur_katznelson	theorem	Gelfand–Mazur theorem (Katznelson)		Every commutative Banach algebra in which every non-zero element is invertible is isometrically isomorphic to ℂ.
s_spectral_synthesis	axiom	Spectral synthesis		The question of whether every closed ideal I in a group algebra (or regular Banach algebra) is the intersection of the maximal ideals containing it, i.e., wheth
s_spectral_analysis_katznelson	theorem	Spectral analysis holds in L¹(G)		Every proper closed ideal in L¹(G) for an LCA group G has non-empty hull (i.e., is contained in a maximal ideal), proved via the regularity of L¹(G) as a Banach
s_malliavin_theorem	theorem	Malliavin's theorem (failure of spectral synthesis)		For every non-compact locally compact abelian group G, spectral synthesis fails in L¹(G): there exist closed ideals not determined by their hulls, answering a q
s_regular_banach_algebra	axiom	Regular Banach algebra		A commutative Banach algebra A is regular if for every closed set E ⊂ Δ(A) and φ₀ ∉ E, there exists a ∈ A with â(φ₀) = 1 and â|_E = 0; L¹(G) is regular for ever
s_hull_kernel_topology	state	Hull-kernel topology		The topology on Δ(A) defined by closure operation E ↦ h(k(E)) where k(E) = ∩{ker φ : φ ∈ E} and h(I) = {φ : I ⊂ ker φ}; coincides with Gelfand topology iff A is
s_ideal_theory_group_algebra	state	Ideal theory of L¹(G)		The correspondence between closed ideals of the group algebra L¹(G), closed subsets of the dual group Ĝ, and translation-invariant closed subspaces of L¹(G), go
t_gelfand_theory_technique	technique	Gelfand theory technique		Reducing algebraic and analytic questions about a commutative Banach algebra to function-theoretic questions on its maximal ideal space via the Gelfand transfor
t_approximate_identity_method	technique	Approximate identity method in harmonic analysis		Using families of kernels with mass 1, bounded L¹ norm, and vanishing tails to approximate functions via convolution; the common mechanism behind Fejér, Poisson
t_summability_method	technique	Summability method for Fourier series		Replacing partial sums S_N(f) by weighted averages (Cesàro/Abel/de la Vallée-Poussin) to obtain convergence; each method corresponds to convolution with a speci
s_de_la_vallee_poussin_kernel	state	de la Vallée-Poussin kernel		V_N = 2F_{2N} − F_N, a trigonometric polynomial with V̂_N(n) = 1 for |n| ≤ N, providing a smooth approximate identity that also reproduces trigonometric polynom
s_L1_algebra_characterization_via_gelfand	theorem	Gelfand space of L¹(G) is Ĝ		The maximal ideal space of the group algebra L¹(G) is canonically homeomorphic to the dual group Ĝ, with the Gelfand transform reducing to the Fourier transform
s_vibrating_string_equation	axiom	Vibrating string equation		The PDE ∂²u/∂t² = c²∂²u/∂x² with boundary conditions u(0,t) = u(L,t) = 0 modeling the transverse displacement of a vibrating string fixed at both ends.
s_dalembert_solution_wave	theorem	D’Alembert’s solution to the wave equation		The general solution u(x,t) = ½[f(x+ct) + f(x−ct)] + (1/(2c))∫_{x−ct}^{x+ct} g(s) ds of the one-dimensional wave equation with initial displacement f and veloci
s_standing_wave_solution	state	Standing wave solution		A separable solution u_n(x,t) = sin(nπx/L)(A_n cos(nπct/L) + B_n sin(nπct/L)) of the vibrating string equation, representing a mode with n half-wavelengths.
t_superposition_principle_linear_PDE	technique	Superposition principle for linear PDEs		For a linear homogeneous PDE, any convergent linear combination of solutions is again a solution, enabling construction of general solutions from elementary mod
t_fourier_program	technique	Fourier’s program for solving PDEs		The strategy of expanding a solution as a Fourier series, substituting into the PDE to determine coefficients, then verifying convergence and boundary behavior.
s_formal_solution_dirichlet_problem_disk	state	Formal solution to Dirichlet problem on the disk		The formal series u(r,θ) = ∑_{n∈ℤ} f̂(n) r^{|n|} e^{inθ} for the Laplace equation on the disk with boundary data f, whose rigorous justification involves the Po
s_riemann_integrable_on_circle	axiom	Riemann integrable function on the circle		A function f on the circle 𝕋 that is Riemann integrable, the natural domain for classical Fourier series before Lebesgue theory.
s_convolution_identity_fourier_coefficients	theorem	Convolution identity for Fourier coefficients		The nth Fourier coefficient of the convolution f ∗ g equals the product f̂(n)ĝ(n), converting convolution on the circle to pointwise multiplication of Fourier c
s_approximation_theorem_good_kernels	theorem	Approximation theorem for good kernels		If {K_n} is a family of good kernels and f is continuous on 𝕋, then K_n ∗ f converges to f uniformly; for integrable f, convergence holds at points of continuit
s_cesaro_mean_definition	axiom	Cesàro mean (Nth)		The Nth Cesàro mean of a sequence of partial sums, σ_N = (s_0 + s_1 + ... + s_{N−1})/N, the basic building block of Cesàro summability.
s_dirichlet_kernel_not_good	theorem	Dirichlet kernel is not a good kernel		The L¹ norms of the Dirichlet kernels satisfy ‖D_N‖₁ ∼ (4/π²) log N → ∞, so {D_N} does not form a family of good kernels.
s_fejer_kernel_is_good	theorem	Fejér kernel is a good kernel		The Fejér kernel F_N is non-negative with ∫F_N = 1 and ∫_{|t|>δ} F_N(t) dt → 0 for every δ > 0, making it a family of good kernels.
s_poisson_kernel_is_good	theorem	Poisson kernel is a good kernel		The Poisson kernel P_r is a non-negative approximate identity as r → 1⁻, satisfying all three conditions of a good kernel family.
s_L2_inner_product_circle	axiom	L² inner product on the circle		The Hermitian inner product ⟨f,g⟩ = ∫₀¹ f(t)g̅(t) dt on L²(𝕋) making it a separable Hilbert space with orthonormal basis {e^{2πint}}_{n∈ℤ}.
s_orthonormality_of_exponentials	theorem	Orthonormality of exponentials on 𝕋		The complex exponentials {e^{2πint}}_{n∈ℤ} form an orthonormal system in L²(𝕋).
s_mean_square_convergence_fourier	theorem	Mean-square convergence of Fourier series		For f ∈ L²(𝕋), the partial sums S_N(f) converge to f in the L² norm, equivalent to Parseval’s identity.
s_completeness_trigonometric_system	theorem	Completeness of the trigonometric system		The exponentials {e^{2πint}}_{n∈ℤ} form a complete orthonormal system in L²(𝕋): if all Fourier coefficients vanish, then f = 0 a.e.
s_holder_pointwise_convergence_fourier	theorem	Pointwise convergence of Fourier series under Hölder condition		If f satisfies |f(x₀+t) − f(x₀)| ≤ C|t|^α for some α > 0 near x₀, then the Fourier series converges pointwise to f(x₀).
t_fourier_proof_isoperimetric	technique	Fourier analytic proof of the isoperimetric inequality		Proving the isoperimetric inequality by expanding the curve’s coordinates as Fourier series, applying Parseval’s identity and the arc-length constraint.
s_weyl_criterion_equidistribution	theorem	Weyl’s criterion for equidistribution		A sequence {x_n} is equidistributed mod 1 if and only if (1/N)∑_{n=1}^{N} e^{2πikx_n} → 0 for every nonzero integer k.
s_continuous_nowhere_differentiable	state	Continuous nowhere-differentiable function		A continuous function on 𝕋 differentiable at no point, constructed via a lacunary Fourier series f(x) = ∑ b^{−nα} e^{2πib^n x}.
t_lacunary_fourier_construction	technique	Construction via lacunary Fourier series		Using Fourier series with rapidly growing frequencies to construct continuous nowhere-differentiable functions or other pathological examples.
s_heat_equation_circle	axiom	Heat equation on the circle		The parabolic PDE ∂u/∂t = ∂²u/∂x² for u(x,t) periodic in x with initial condition u(x,0) = f(x), modeling heat diffusion on a circular ring.
s_heat_equation_circle_fourier_solution	theorem	Solution of heat equation on circle via Fourier series		The solution to the heat equation on 𝕋 with initial data f is u(x,t) = ∑ f̂(n) e^{−4π²n²t} e^{2πinx}, converging for all t > 0.
s_heat_kernel_circle	state	Heat kernel on the circle		The periodized Gaussian H_t(x) = ∑_{n∈ℤ} e^{−4π²n²t} e^{2πinx}, the fundamental solution of the heat equation on 𝕋.
s_heat_kernel_circle_good_kernel	theorem	Heat kernel on circle is a good kernel		As t → 0⁺, the heat kernel H_t on 𝕋 forms a family of good kernels: H_t ≥ 0, ∫H_t = 1, and ∫_{|x|>δ} H_t → 0.
s_smoothing_property_heat	theorem	Smoothing property of the heat equation		For any integrable initial data f, the solution u(x,t) = H_t ∗ f of the heat equation is C^∞ for all t > 0.
s_uniqueness_heat_equation_circle	theorem	Uniqueness of solution to heat equation on circle		A continuous solution of the heat equation on 𝕋 that is C² for t > 0 and satisfies u(x,0) = f(x) is uniquely determined.
s_steady_state_heat_solution	state	Steady-state solution of the heat equation		A time-invariant solution of the heat equation satisfying Laplace’s equation; on the circle, steady states are constants; on the disk, they are harmonic functio
s_basic_properties_fourier_transform	theorem	Basic properties of the Fourier transform on ℝ		The Fourier transform satisfies linearity, translation-modulation duality, dilation, and conjugation symmetry on L¹(ℝ).
s_fourier_transform_of_gaussian	theorem	Fourier transform of the Gaussian		The Gaussian e^{−πx²} is an eigenfunction of the Fourier transform with eigenvalue 1.
s_fourier_maps_schwartz_to_schwartz	theorem	Fourier transform maps 𝒮(ℝ) to 𝒮(ℝ)		The Fourier transform is a continuous linear bijection from the Schwartz space onto itself, with inverse given by the inverse Fourier transform.
s_multiplication_formula_fourier	theorem	Multiplication formula for the Fourier transform		For f, g ∈ L¹(ℝ), ∫f̂(x)g(x) dx = ∫f(x)ĝ(x) dx, a self-adjointness relation used in extending the Fourier transform to L².
s_convolution_theorem_R	theorem	Convolution theorem on ℝ		The Fourier transform converts convolution to pointwise multiplication: (f ∗ g)^(ξ) = f̂(ξ)ĝ(ξ).
s_equality_case_uncertainty	theorem	Equality case in the Heisenberg uncertainty principle		Equality in the Heisenberg uncertainty principle holds if and only if f(x) = Ce^{−αx²} for constants C ∈ ℂ and α > 0.
s_heat_equation_solution_R	theorem	Solution of the heat equation on ℝ via Fourier transform		The solution of ∂u/∂t = ∂²u/∂x² on ℝ with u(x,0) = f(x) is u(x,t) = W_t ∗ f where W_t is the heat kernel.
s_heat_kernel_R_good_kernel	theorem	Heat kernel on ℝ is a good kernel		The Gaussian heat kernel W_t(x) = (4πt)^{−1/2}e^{−x²/(4t)} forms a family of good kernels as t → 0⁺.
s_fourier_differentiation_correspondence	theorem	Fourier transform and differentiation correspondence		Differentiation corresponds to multiplication by 2πiξ in frequency domain, and multiplication by −2πix corresponds to differentiation of the transform.
s_moderate_decrease_functions	axiom	Functions of moderate decrease		A continuous function f on ℝ satisfying |f(x)| ≤ A/(1+|x|)^{1+ε} for some A, ε > 0, sufficient for the Poisson summation formula.
s_schwartz_space_Rd	axiom	Schwartz space 𝒮(ℝ^d)		The space 𝒮(ℝ^d) of C^∞ functions with sup |x^α ∂^β f(x)| < ∞ for all multi-indices α, β, on which the Fourier transform is an automorphism.
t_wave_equation_solution_via_FT	technique	Solution of wave equation via Fourier transform		Taking the spatial Fourier transform of the wave equation reduces it to an ODE in t for each frequency, yielding the solution via inverse transform.
s_spherical_means	axiom	Spherical means		The average M_f(x,r) of f over the sphere of radius r centered at x, satisfying the Darboux equation and fundamental to wave equation solutions in higher dimens
s_kirchhoff_formula_wave_R3	theorem	Kirchhoff’s formula for the wave equation in ℝ³		The solution of u_tt = Δu in ℝ³ is u(x,t) = ∂/∂t[tM_f(x,t)] + tM_g(x,t), depending only on data on the sphere |y−x| = t.
s_huygens_principle	theorem	Huygens’ principle		In odd spatial dimensions d ≥ 3, wave propagation is sharp: the solution at (x,t) depends only on initial data on the sphere |y−x| = ct.
s_failure_huygens_even_dim	theorem	Failure of Huygens’ principle in even dimensions		In even spatial dimensions, the wave equation solution at (x,t) depends on initial data in the entire ball |y−x| ≤ ct, causing residual signals.
t_method_of_descent	technique	Hadamard’s method of descent		Deriving the wave equation solution in ℝ^d from the known solution in ℝ^{d+1} by treating initial data as independent of the extra variable.
s_wave_solution_R2_poisson	theorem	Wave equation solution in ℝ² (Poisson’s formula)		The solution of u_tt = Δu in ℝ², obtained from Kirchhoff’s formula via the method of descent, involving integration over the disk |y−x| ≤ t.
s_radon_transform	axiom	Radon transform		The integral transform Rf(ω,s) = ∫_{x·ω=s} f(x) dx integrating f over hyperplanes, fundamental to CT imaging and related to the Fourier transform via the Fourie
s_fourier_slice_theorem	theorem	Fourier slice theorem (projection-slice theorem)		The 1D Fourier transform of the Radon transform Rf(ω,·) equals the restriction of the d-dimensional Fourier transform f̂ to the line through the origin in direc
s_radon_inversion_formula	theorem	Radon inversion formula		The function f can be recovered from its Radon transform via filtered backprojection, using the Fourier slice theorem and inverse Fourier transform.
s_radial_functions_FT_bessel	theorem	Fourier transform of radial functions via Bessel functions		If f is radial on ℝ^d, then f̂ is also radial and expressible via Bessel functions: f̂(ξ) = 2π|ξ|^{−(d/2−1)} ∫ f₀(r) J_{d/2−1}(2π|ξ|r) r^{d/2} dr.
s_FT_indicator_ball	state	Fourier transform of the indicator of a ball		The Fourier transform of the characteristic function of the ball B(0,R) in ℝ^d, expressed via the Bessel function J_{d/2}.
s_discrete_fourier_transform	axiom	Discrete Fourier transform (DFT)		The DFT of f on ℤ/Nℤ is f̂(k) = ∑_{n=0}^{N−1} f(n) e^{−2πink/N}, a linear isomorphism of ℂ^N.
s_inverse_DFT	theorem	Inverse discrete Fourier transform		The DFT is inverted by f(n) = (1/N)∑ f̂(k) e^{2πink/N}, proved using orthogonality of characters of ℤ/Nℤ.
s_parseval_identity_DFT	theorem	Parseval’s identity for the DFT		The DFT preserves ℓ² norms up to scaling: ∑|f(n)|² = (1/N)∑|f̂(k)|², the finite analogue of Parseval’s theorem.
s_convolution_on_ZNZ	axiom	Convolution on ℤ/Nℤ		The circular convolution (f ∗ g)(n) = ∑ f(m)g(n−m mod N) on ℤ/Nℤ, satisfying (f∗g)^ = f̂ĝ under the DFT.
s_convolution_theorem_DFT	theorem	Convolution theorem for the DFT		The DFT converts circular convolution on ℤ/Nℤ to pointwise multiplication.
t_fast_fourier_transform	technique	Fast Fourier transform (FFT)		The Cooley-Tukey algorithm computing the DFT in O(N log N) operations by recursively splitting into even- and odd-indexed sub-problems.
s_FFT_complexity	theorem	FFT complexity		The FFT computes the DFT of N = 2^m points using O(N log N) multiplications and additions, compared to O(N²) naively.
t_polynomial_multiplication_FFT	technique	Polynomial multiplication via FFT		Multiplying two polynomials by evaluating at roots of unity via FFT, pointwise multiplying, and interpolating via inverse FFT.
t_fast_integer_multiplication	technique	Fast integer multiplication via FFT		The Schönhage-Strassen algorithm multiplying large integers by representing them as polynomials and using FFT-based polynomial multiplication.
s_non_vanishing_L1_complex_characters	theorem	Non-vanishing of L(1,χ) for complex Dirichlet characters		For a complex (non-real) Dirichlet character χ, L(1,χ) ≠ 0, proved by showing ∏_χ L(s,χ) has nonnegative coefficients and a pole at s = 1.
s_non_vanishing_L1_real_characters	theorem	Non-vanishing of L(1,χ) for real Dirichlet characters		For a real non-principal Dirichlet character, L(1,χ) ≠ 0, proved by a more delicate argument involving the theta function or Dirichlet’s class number formula.
s_analytic_continuation_L_function_s_gt_0	theorem	Analytic continuation of L(s,χ) to Re(s) > 0		For non-principal χ, L(s,χ) extends analytically to Re(s) > 0 via Abel summation, since ∑_{n≤x} χ(n) is bounded.
s_natural_density_primes_AP	theorem	Natural density of primes in arithmetic progressions		Primes p ≡ a (mod d) have natural density 1/φ(d) among all primes, a quantitative refinement of Dirichlet’s theorem.
s_gibbs_phenomenon	theorem	Gibbs phenomenon		At a jump discontinuity, the Fourier partial sums S_N(f) overshoot by approximately 9% of the jump size, and this overshoot persists as N → ∞.
s_absolute_convergence_fourier_smoothness	theorem	Absolute convergence of Fourier series under smoothness		If f is twice continuously differentiable on 𝕋, then f̂(n) = O(1/n²), hence the Fourier series converges absolutely and uniformly.
s_decay_fourier_coefficients_smoothness	theorem	Decay of Fourier coefficients and smoothness		If f ∈ C^k(𝕋), then f̂(n) = o(|n|^{−k}); conversely, rapid decay implies smoothness, establishing a precise smoothness-decay duality.
s_wirtinger_inequality	theorem	Wirtinger’s inequality		For a mean-zero C¹ function on 𝕋, ‖f‖₂ ≤ (1/(2π))‖f’‖₂, with equality iff f is a first-order trigonometric polynomial.
s_gronwall_inequality_integral_form	theorem	Gronwall inequality (integral form)		If u(t) ≤ α + ∫₀ᵗ β(s)u(s) ds with β ≥ 0, then u(t) ≤ α exp(∫₀ᵗ β(s) ds); the integral version of Gronwall’s inequality used for stability and continuous depend
s_continuous_dependence_on_initial_conditions	theorem	Continuous dependence on initial conditions		Solutions of y' = f(t, y) depend continuously on the initial data (t₀, y₀) in the topology of uniform convergence on compact subintervals of the maximal existen
s_continuous_dependence_on_parameters	theorem	Continuous dependence on parameters		If f(t, y, λ) is continuous in all variables and Lipschitz in y, then the solution y(t; λ) of y' = f(t, y, λ) is continuous in the parameter λ uniformly on comp
s_differentiability_wrt_initial_conditions	theorem	Differentiability with respect to initial conditions		If f(t, y) is Cᵏ, then the solution y(t; t₀, y₀) is Cᵏ in the initial data (t₀, y₀), and the derivative ∂y/∂y₀ satisfies the variational equation.
s_variational_equation	axiom	Variational equation		The linearization Z' = (∂f/∂y)(t, y(t))Z of an ODE y' = f(t, y) along a solution y(t), governing the first-order sensitivity of solutions to perturbations in in
s_characteristic_multipliers	state	Characteristic multipliers (Floquet multipliers)		The eigenvalues ρ₁, …, ρₙ of the monodromy matrix M = Φ(T) of a T-periodic linear system; their moduli determine stability: |ρᵢ| < 1 for all i implies asymptoti
s_floquet_normal_form	state	Floquet normal form		The representation Φ(t) = P(t)e^{tB} of the fundamental matrix of a T-periodic system, where P(t) is T-periodic and B is a constant matrix (the Floquet exponent
s_stability_zones_periodic_ode	state	Stability zones for periodic equations		Regions in parameter space where all solutions of a periodic ODE (e.g., Hill’s or Mathieu’s equation) remain bounded, separated by instability zones where solut
s_self_adjointness_sturm_liouville	theorem	Self-adjointness of Sturm–Liouville operators		The operator L[y] = −(py’)’ + qy with separated boundary conditions on [a,b] is self-adjoint in L²([a,b], w), ensuring real eigenvalues and orthogonal eigenfunc
s_sturm_separation_theorem	theorem	Sturm separation theorem		Between any two consecutive zeros of one non-trivial solution of a second-order linear ODE, there lies exactly one zero of every linearly independent solution.
s_sturm_oscillation_theorem	theorem	Sturm oscillation theorem		The n-th eigenfunction of a regular Sturm–Liouville problem has exactly n−1 zeros in the open interval (a, b), and the eigenvalues form an increasing sequence t
s_greens_function_bvp	state	Green’s function for ODE boundary value problem		The kernel G(t, s) that inverts a linear differential operator with boundary conditions, so that the unique solution of L[y] = f is y(t) = ∫ G(t, s)f(s) ds.
s_eigenfunction_expansion_theorem	theorem	Eigenfunction expansion theorem (Sturm–Liouville)		Every function in L²([a,b], w) can be expanded as a convergent series in the eigenfunctions of a regular Sturm–Liouville problem, with convergence in L² and poi
s_min_max_principle_eigenvalues	theorem	Min-max principle for eigenvalues (Sturm–Liouville)		The n-th eigenvalue λₙ of a self-adjoint Sturm–Liouville operator equals the min over n-dimensional subspaces V of the max Rayleigh quotient over V, providing v
s_asymptotic_stability	axiom	Asymptotic stability		An equilibrium y* is asymptotically stable if it is Lyapunov stable and additionally all solutions starting sufficiently close converge to y* as t → ∞.
s_lyapunov_instability_theorem	theorem	Lyapunov instability theorem		If there exists a function V with V̇ > 0 in a punctured neighborhood of an equilibrium and V takes positive values arbitrarily close to the equilibrium, then th
s_lyapunov_equation_linear_systems	theorem	Lyapunov equation for linear systems		The linear matrix equation AᵀP + PA = −Q; a Hurwitz matrix A admits a unique positive-definite solution P for every positive-definite Q, providing a Lyapunov fu
s_exponential_stability	axiom	Exponential stability		An equilibrium is exponentially stable if nearby solutions satisfy ‖y(t) − y*‖ ≤ Ce^{−αt}‖y(0) − y*‖ for constants C, α > 0; stronger than asymptotic stability.
s_uniform_stability	axiom	Uniform stability		The equilibrium of a non-autonomous system y' = f(t, y) is uniformly stable if the δ in Lyapunov stability can be chosen independent of the initial time t₀.
s_phase_plane	axiom	Phase plane		The (y₁, y₂)-plane for a two-dimensional autonomous system, in which solution trajectories (phase curves) provide a qualitative picture of the global dynamics.
s_omega_limit_set	axiom	ω-limit set		The set ω(γ) of all accumulation points of a forward trajectory γ(t) as t → +∞; a non-empty compact connected invariant set when the trajectory is bounded.
s_alpha_limit_set	axiom	α-limit set		The set α(γ) of all accumulation points of a trajectory γ(t) as t → −∞; the time-reversed analogue of the ω-limit set.
s_limit_cycle	state	Limit cycle		An isolated periodic orbit in the phase plane, i.e., a closed trajectory with no other periodic orbit in a neighborhood; the simplest non-equilibrium attractor 
s_bendixson_criterion	theorem	Bendixson criterion (negative criterion)		If div(f) = ∂f₁/∂y₁ + ∂f₂/∂y₂ does not change sign in a simply connected region, then the planar system y' = f(y) has no periodic orbit in that region.
s_orbital_stability	axiom	Orbital stability		A periodic orbit γ is orbitally stable if every trajectory starting near γ remains close to the orbit (not to a specific point on it) for all future time.
s_index_of_closed_curve	state	Index of a closed curve (Poincaré index)		The winding number of the vector field f(y) around a simple closed curve not passing through any equilibrium; equals the sum of indices of equilibria enclosed.
s_index_theorem_equilibria	theorem	Index theorem for equilibria		The index of an isolated equilibrium of a planar vector field is +1 for a node, focus, or center, and −1 for a simple saddle; the index of any closed curve equa
s_poincare_expansion_theorem	theorem	Poincaré expansion theorem		If f(t, y, ε) is analytic in ε, the solution y(t, ε) of the initial value problem is analytic in ε for |ε| sufficiently small, justifying regular perturbation s
s_singular_perturbation	axiom	Singular perturbation		A perturbation problem εy'' + a(t)y' + b(t)y = 0 where setting ε = 0 reduces the order of the equation, causing non-uniform convergence and boundary layers.
s_boundary_layer	state	Boundary layer		A narrow region (of width O(ε) or O(ε^α)) near the boundary where the solution of a singularly perturbed ODE transitions rapidly between the outer solution and 
s_classification_equilibria_r2	state	Classification of equilibria in ℝ²		For a linear system ẏ = Ay in ℝ², the eigenvalues of A classify the equilibrium as a stable/unstable node, saddle, stable/unstable focus, center, or degenerate 
s_separatrices	state	Separatrices		Distinguished trajectories (typically stable/unstable manifolds of saddle points) that divide the phase plane into regions of qualitatively different behavior.
s_topological_equivalence_phase_portraits	axiom	Topological equivalence of phase portraits		Two autonomous systems are topologically equivalent if there is a homeomorphism mapping orbits of one to orbits of the other preserving orientation; the natural
s_weyl_limit_point_limit_circle	theorem	Weyl limit-point/limit-circle classification		At a singular endpoint of a Sturm–Liouville problem, either all solutions are L² (limit-circle case) or exactly one solution is L² (limit-point case); determine
s_weyl_m_function	state	Weyl m-function		The Herglotz–Nevanlinna function m(z) selecting the unique L² solution at a singular endpoint of a Sturm–Liouville problem; its boundary values on ℝ encode the 
s_titchmarsh_weyl_theory	theorem	Titchmarsh–Weyl theory		The spectral theory of singular Sturm–Liouville operators via the Weyl m-function, establishing the eigenfunction expansion and spectral measure for operators o
s_singular_sturm_liouville_problem	axiom	Singular Sturm–Liouville problem		A Sturm–Liouville problem −(py’)’ + qy = λwy on an interval where at least one endpoint is singular (infinite, or p, w vanish/q is unbounded), requiring Weyl th
s_eigenfunction_expansion_half_line	theorem	Eigenfunction expansion on half-line		For a singular Sturm–Liouville problem on [0, ∞), every L²-function admits a generalized Fourier transform f̂(λ) = ∫ f(t)φ(t,λ) dt with Parseval equality via th
s_regular_singular_point	axiom	Regular singular point of an ODE		A point z₀ where the coefficient functions of a linear ODE have poles, but (z−z₀)p(z) and (z−z₀)²q(z) are analytic; solutions have at most algebraic-logarithmic
s_irregular_singular_point	axiom	Irregular singular point of an ODE		A singular point that is not regular; solutions generically have essential singularities with exponential growth/decay, and formal series solutions are typicall
s_logarithmic_solutions_regular_singular	state	Logarithmic solutions at regular singular points		When indicial exponents differ by an integer at a regular singular point, a second linearly independent solution may require a log(z−z₀) term: y₂ = cy₁ log(z−z₀
s_fuchs_theorem	theorem	Fuchs’s theorem		A linear ODE has all its singular points regular (is Fuchsian) if and only if the sum of all indicial exponents at all singular points (including ∞) satisfies t
s_asymptotic_solutions_irregular_singular	theorem	Asymptotic solutions at irregular singular points		Near an irregular singular point of rank r, there exist formal solutions of the form e^{Q(z)} z^ρ Σ aₙ z⁻ⁿ where Q is a polynomial of degree r; these are asympt
s_ordinary_point_ode_complex	axiom	Ordinary point of an ODE in ℂ		A point z₀ where all coefficient functions of a linear ODE are analytic; solutions are analytic in a disk centered at z₀ with radius equal to the distance to th
s_monodromy_representation	state	Monodromy representation		The group homomorphism ρ: π₁(ℂ \ {singularities}, z₀) → GL(n, ℂ) sending each loop to the linear transformation on the solution space obtained by analytic conti
s_riemann_hilbert_problem_ode	theorem	Riemann–Hilbert problem (for linear ODE)		Given points a₁, …, aₘ in ℂP¹ and a representation ρ: π₁(ℂP¹ \ {aᵢ}) → GL(n, ℂ), find a Fuchsian system with those singularities and that monodromy; solvable wi
s_perron_asymptotic_integration	theorem	Perron’s theorem on asymptotic integration		For the system y' = (A + R(t))y where A is a constant matrix with distinct eigenvalues and R(t) → 0 as t → ∞, there exist solutions asymptotic to e^{λᵢt}vᵢ wher
s_levinson_asymptotic_integration	theorem	Levinson’s theorem (asymptotic integration)		Refines Perron’s theorem: if A has distinct real parts and ∫₀^∞ ‖R(t)‖ dt < ∞, then y' = (A + R(t))y has a fundamental system of solutions yₖ(t) ~ e^{λₖt}vₖ as 
s_global_existence_linear_ode	theorem	Global existence for linear ODE systems		If A(t) and b(t) are continuous on [a, b], then the solution of y' = A(t)y + b(t), y(t₀) = y₀ exists on the entire interval [a, b]; linear systems cannot blow u
t_euler_polygonal_approximation	technique	Euler polygonal approximation method		Constructs piecewise-linear approximations yₙ₊₁ = yₙ + hf(tₙ, yₙ) used in Peano’s existence proof and as the simplest numerical ODE integrator.
t_greens_function_construction_bvp	technique	Green’s function construction for BVP		Constructs G(t, s) by patching two solutions satisfying opposite boundary conditions at t = s with a jump in the derivative determined by the Wronskian.
t_prufer_transformation	technique	Prüfer transformation		Transforms a second-order ODE into a system for amplitude and angle, reducing oscillation and comparison questions to the monotonicity analysis of the angle var
t_regular_perturbation	technique	Regular perturbation method		Expands the solution in powers of a small parameter ε and solves order-by-order; valid when the solution depends analytically on ε.
t_matched_asymptotic_expansions	technique	Matched asymptotic expansions		Constructs separate inner (boundary-layer) and outer expansions and determines free constants by matching in an overlap region where both are valid.
t_averaging_method	technique	Averaging method		Replaces a rapidly oscillating non-autonomous system by its time-averaged autonomous approximation, with error O(ε) on time intervals of length O(1/ε).
t_analytic_continuation_of_solutions	technique	Analytic continuation of ODE solutions		Extends a local analytic solution of a linear ODE along a path in the complex plane; the result depends on the homotopy class of the path, generating the monodr
t_reduction_of_order	technique	Reduction of order		Given one solution y₁ of a homogeneous linear ODE, substitutes y = y₁v and reduces to a lower-order equation in v’, lowering the problem by one degree.
t_differential_inequality	technique	Differential inequality technique		Bounds solutions of an ODE above or below by comparing with solutions of a simpler inequality via the comparison principle, yielding a priori estimates.
t_higher_order_ode_reduction	technique	Higher-order linear ODE reduction to first-order system		Converts a scalar n-th order linear ODE to an equivalent first-order system by introducing y₁ = y, y₂ = y', …, yₙ = y⁽ⁿ⁻¹⁾, reducing all higher-order theory to 
s_phase_space_ode	axiom	Phase space of a dynamical system		The space M of all possible states of a system, in which the evolution of the system is described as motion of a point along a phase curve.
s_phase_curve	axiom	Phase curve (integral curve of an ODE)		The image {g^t(x₀) : t ∈ I} of a solution trajectory in phase space; the orbit of a point under the phase flow.
s_phase_portrait	axiom	Phase portrait		The partition of the phase space into the union of all phase curves, providing a complete qualitative picture of the dynamics.
s_phase_flow	axiom	Phase flow		The one-parameter group of diffeomorphisms g^t: M → M defined by g^t(x₀) = φ(t; x₀), where φ is the solution of ẋ = v(x) with φ(0) = x₀.
s_phase_velocity_vector_field	axiom	Phase velocity vector field		The vector field v(x) = (d/dt)g^t(x)|_{t=0} whose integral curves are the phase curves of the flow, equivalently the right-hand side of ẋ = v(x).
s_non_autonomous_ode	axiom	Non-autonomous ODE		An ordinary differential equation ẋ = f(t, x) in which the right-hand side depends explicitly on time, reducible to an autonomous system on the extended phase s
s_direction_field	axiom	Direction field (line element field)		The field assigning to each point (x, y) the line through (x, y) with slope f(x, y), whose integral curves are the solutions of y' = f(x, y).
t_quadrature_barrow	technique	Solution by quadrature (Barrow's formula)		Solving a one-dimensional autonomous ODE ẋ = v(x) by direct integration: t − t₀ = ∫_{x₀}^{x} ds/v(s), the fundamental antiderivative method.
t_separation_of_variables_ode	technique	Separation of variables for ODEs		Solving a first-order ODE dy/dx = f(x)g(y) by separating the variables and integrating each side independently.
s_rectification_theorem	theorem	Rectification theorem (flow box theorem)		Near any non-singular point of a smooth vector field, there exists a local coordinate system in which the field equals the constant field ∂/∂x₁ and the flow is 
s_extension_theorem_ode	theorem	Extension theorem for ODE solutions		A solution of ẋ = f(t, x) on a maximal interval (α, β) with β < ∞ must eventually leave every compact subset of the domain; equivalently, bounded solutions on o
s_differentiable_dependence_on_initial_conditions	theorem	Differentiable dependence on initial conditions		If the right-hand side f of ẋ = f(t, x) is Cᵏ in (t, x), then the solution φ(t; t₀, x₀) is Cᵏ in the initial condition x₀, with the derivative satisfying the va
s_superposition_principle_linear_ode	theorem	Superposition principle for linear ODE systems		Any linear combination of solutions of a homogeneous linear system ẋ = A(t)x is again a solution; the solution set forms a vector space of dimension n.
s_wronskian	state	Wronskian		The determinant of the matrix whose columns are n solutions of an n-dimensional linear system; nonzero at one point if and only if the solutions are linearly in
s_quasi_polynomial	axiom	Quasi-polynomial		A function of the form p(t)e^{λt} where p is a polynomial, arising as a solution component of a constant-coefficient linear ODE when λ is an eigenvalue of multi
t_complexification_linear_operator	technique	Complexification of a real linear operator		Extending a real linear operator A to the complexified space V ⊗ ℂ to exploit the algebraic closure of ℂ, enabling diagonalization and spectral decomposition ev
t_decomplexification	technique	Decomplexification (realification)		Converting complex eigendata of a real operator back to real form: a conjugate pair λ, λ̄ with eigenvectors z, z̄ yields a real invariant 2-plane on which the o
s_stable_node	axiom	Stable node		An equilibrium of a planar linear system where both eigenvalues are real and negative, so all trajectories approach the origin tangent to the eigenvector of the
s_unstable_node	axiom	Unstable node		An equilibrium of a planar linear system where both eigenvalues are real and positive, so all trajectories recede from the origin; the time-reversal of a stable
s_saddle_point_ode	axiom	Saddle point (planar linear system)		An equilibrium of a planar linear system with one positive and one negative real eigenvalue; the stable and unstable eigenspaces are invariant lines, and all ot
s_stable_focus	axiom	Stable focus (spiral sink)		An equilibrium of a planar linear system with complex conjugate eigenvalues having negative real part, so trajectories spiral inward toward the origin.
s_unstable_focus	axiom	Unstable focus (spiral source)		An equilibrium of a planar linear system with complex conjugate eigenvalues having positive real part, so trajectories spiral outward from the origin.
s_center_equilibrium	axiom	Center (elliptic equilibrium)		An equilibrium of a planar linear system with purely imaginary eigenvalues, surrounded by a family of closed elliptical orbits; Lyapunov stable but not asymptot
s_topological_classification_planar_linear_systems	theorem	Topological classification of planar linear systems		Two non-degenerate planar linear systems are topologically equivalent (related by a homeomorphism sending orbits to orbits preserving orientation) if and only i
s_stability_equilibrium_linear_system	theorem	Stability of equilibrium of a linear system		The equilibrium x = 0 of ẋ = Ax is Lyapunov stable if and only if all eigenvalues have non-positive real part and those on the imaginary axis have equal algebra
t_proof_differentiable_dependence_gronwall	technique	Proof of differentiable dependence via Gronwall		Establishing that the difference quotient of the flow in the initial condition converges to the solution of the variational equation, using Gronwall's inequalit
s_tangent_vector	axiom	Tangent vector		An equivalence class of curves through p in a smooth manifold, or equivalently a derivation on the algebra of germs of smooth functions at p, representing an in
s_tangent_space	axiom	Tangent space		The vector space TₚM of all tangent vectors at a point p of an n-dimensional smooth manifold M, isomorphic to ℝⁿ via any chart.
s_one_parameter_group_of_diffeomorphisms	axiom	One-parameter group of diffeomorphisms		A smooth action g : ℝ × M → M such that each g^t is a diffeomorphism, g⁰ = id, and g^{s+t} = g^s ∘ g^t, equivalently a global flow on M.
s_completeness_of_vector_field	axiom	Completeness of a vector field		A vector field is complete if every maximal integral curve is defined for all t ∈ ℝ, equivalently if it generates a global one-parameter group of diffeomorphism
s_compactness_implies_completeness_vector_field	theorem	Compactness implies completeness of vector fields		Every smooth vector field on a compact manifold is complete: since solutions cannot escape a compact set, the extension theorem guarantees they exist for all ti
s_lie_bracket_vector_fields	axiom	Lie bracket of vector fields		The Lie bracket [X, Y] = XY − YX as derivations on C^∞(M), measuring the failure of the flows of X and Y to commute to second order.
s_lie_algebra_of_vector_fields	state	Lie algebra of vector fields		The space 𝔛(M) of all smooth vector fields on M equipped with the Lie bracket, forming an infinite-dimensional Lie algebra over ℝ satisfying skew-symmetry and t
s_commutativity_of_flows_vanishing_bracket	theorem	Commutativity of flows iff vanishing Lie bracket		Two complete vector fields on a manifold have commuting flows (g^s_X ∘ g^t_Y = g^t_Y ∘ g^s_X for all s, t) if and only if their Lie bracket vanishes identically
s_lie_derivative_vector_field	state	Lie derivative of a vector field		The derivative of Y along the flow of X, defined as L_X Y = lim_{t→0} ((g^{-t}_*)Y − Y)/t, which equals the Lie bracket [X, Y].
s_pushforward_of_vector_field	axiom	Pushforward of a vector field by a diffeomorphism		The image of a vector field X under a diffeomorphism φ, defined by (φ_*X)_{φ(p)} = dφ_p(X_p), giving the natural action of Diff(M) on 𝔛(M).
t_linearization_at_equilibrium	technique	Linearization at an equilibrium		Replacing a nonlinear system by its first-order Taylor approximation at an equilibrium point to study local qualitative behavior, justified by the Hartman-Grobm
s_lyapunov_indirect_method	theorem	Lyapunov's indirect method (stability by linearization)		If all eigenvalues of the Jacobian Df(0) have strictly negative real part, the equilibrium of the nonlinear system is asymptotically stable; if any eigenvalue h
s_periodic_orbit	axiom	Periodic orbit		A phase curve γ for which there exists a minimal T > 0 such that φ(t + T; x₀) = φ(t; x₀) for all t, forming a closed loop in phase space distinct from an equili
s_poincare_index	axiom	Poincaré index of a singular point		The winding number of the vector field v around a simple closed curve enclosing an isolated singular point p, counting the number of full turns of v(x) as x tra
s_sturm_liouville_equation	axiom	Sturm-Liouville equation		A second-order linear ODE eigenvalue problem of the form −(p(x)y')' + q(x)y = λw(x)y with p > 0, w > 0, and prescribed boundary conditions, whose spectral theor
s_boundary_value_problem_ode	axiom	Boundary value problem for second-order ODEs		The problem of finding a solution of a second-order ODE on an interval [a, b] subject to conditions imposed at both endpoints, in contrast to an initial value p
s_homoclinic_orbit	axiom	Homoclinic orbit		A non-equilibrium trajectory that is biasymptotic to the same equilibrium point, approaching it as t → +∞ and as t → −∞, lying in the intersection of the stable
s_heteroclinic_orbit	axiom	Heteroclinic orbit (saddle connection)		A trajectory connecting two distinct equilibrium points, lying in the intersection of the unstable manifold of p and the stable manifold of q.
s_first_integral	axiom	First integral (constant of motion)		A smooth function F that is constant along every solution trajectory, satisfying ∇F · v = 0, whose level sets are invariant under the flow and reduce the dimens
s_invariance_ode_under_coordinate_changes	theorem	Invariance of ODE systems under coordinate changes		Under a smooth change of coordinates y = φ(x), the vector field v transforms by the pushforward φ_*v, showing that the qualitative theory of ODEs is intrinsic t
s_transport_equation_PDE	axiom	Transport equation		The linear first-order PDE u_t + b · Du = 0 with velocity field b, whose solutions are constant along characteristic lines x(t) = x_0 + bt.
s_classification_second_order_PDE	state	Classification of second-order linear PDEs		A second-order PDE Σ a_{ij} u_{x_ix_j} + ⋯ = 0 is elliptic if A is definite, parabolic if det A = 0 with rank n−1, and hyperbolic if A has signature (n−1,1).
s_fundamental_solution_laplace	state	Fundamental solution of the Laplace equation		The function Φ(x) = −(1/(n(n−2)α(n)))|x|^{2−n} for n ≥ 3 (Φ(x) = (1/(2π)) log|x| for n = 2) satisfying −ΔΦ = δ_0 in the distributional sense.
s_poisson_formula_ball	theorem	Poisson's formula for the ball		The unique harmonic function in B(0,r) with boundary data g is u(x) = (r²−|x|²)/(nα(n)r) ∫_{∂B} g(y)/|x−y|^n dS(y), the convolution of g with the Poisson kernel
s_smoothness_of_harmonic_functions	theorem	Smoothness of harmonic functions		Every harmonic function u on an open set Ω is infinitely differentiable (in fact real-analytic) in Ω, as follows from the Poisson integral representation.
s_derivative_estimates_harmonic	theorem	Derivative estimates for harmonic functions		If u is harmonic in B(x_0,r), then |D^α u(x_0)| ≤ C(n,|α|) r^{−n−|α|} ||u||_{L^1(B(x_0,r))} for each multi-index α.
s_analyticity_of_harmonic_functions	theorem	Analyticity of harmonic functions		Every harmonic function u on an open set Ω is real-analytic: at each x_0 ∈ Ω, u equals its Taylor series in a neighborhood, as a consequence of the derivative e
s_infinite_speed_of_propagation_heat	state	Infinite speed of propagation for the heat equation		If u solves u_t − Δu = 0 with compactly supported initial data u(·,0) = g ≥ 0, g ≢ 0, then u(x,t) > 0 for all x ∈ ℝ^n and t > 0.
s_mean_value_property_heat	theorem	Mean value property for the heat equation		A solution of the heat equation satisfies u(x,t) = (1/(4r^n)) ∫∫_{E(x,t;r)} u(y,s) |x−y|²/(t−s)² dy ds over the heat ball E(x,t;r).
s_strong_maximum_principle_heat	theorem	Strong maximum principle for the heat equation		If u attains its maximum at an interior point of the parabolic cylinder Ω_T, then u is constant on the connected component reachable by paths going backward in 
s_euler_poisson_darboux_equation	state	Euler-Poisson-Darboux equation		The PDE U_{tt} + ((n−1)/t) U_t = U_{rr} satisfied by the spherical mean of a wave equation solution, reducing wave propagation to a singular 1D problem.
s_finite_speed_of_propagation_wave	theorem	Finite speed of propagation for the wave equation		The solution of the wave equation at point (x,t) depends only on initial data in the backward cone {y : |x−y| ≤ ct}, so disturbances travel at most at speed c.
s_domain_of_dependence_wave	state	Domain of dependence for the wave equation		The domain of dependence of a point (x_0, t_0) for the wave equation u_tt = c²Δu is the closed ball B(x_0, ct_0) in the initial hyperplane t = 0.
s_greens_identities	state	Green's identities		First: ∫_Ω (Δu)v dx = −∫_Ω Du·Dv dx + ∫_{∂Ω} (∂u/∂ν)v dS. Second: ∫_Ω (uΔv − vΔu) dx = ∫_{∂Ω} (u ∂v/∂ν − v ∂u/∂ν) dS.
s_characteristic_equations	state	Characteristic equations for first-order PDEs		The ODE system ẋ = D_p F, ṗ = −D_x F − D_z F · p, ż = p · D_p F that defines the characteristic strips along which a first-order PDE reduces to ODEs.
s_hamilton_jacobi_equation_PDE	axiom	Hamilton-Jacobi equation (PDE)		The PDE u_t + H(Du, x) = 0 for t > 0 with initial data u(x,0) = g(x), where H is the Hamiltonian; a fundamental equation linking optimal control, calculus of va
s_hopf_lax_formula	theorem	Hopf-Lax formula		The function u(x,t) = min_y [tL((x−y)/t) + g(y)] where L = H* is the Legendre transform of H, gives the unique viscosity solution of u_t + H(Du) = 0.
s_scalar_conservation_law	axiom	Scalar conservation law		The PDE u_t + F(u)_x = 0 for a scalar unknown u and flux function F, modeling transport with nonlinear wave speeds F'(u).
s_weak_solution_conservation_law	state	Weak solution of a conservation law		A bounded measurable function u satisfying ∫∫ [u φ_t + F(u) φ_x] dx dt + ∫ g φ(x,0) dx = 0 for all test functions φ ∈ C^1_c, allowing discontinuous (shock) solu
s_lax_entropy_condition	state	Lax entropy condition		A shock is admissible if the characteristics impinge on it: F'(u_l) > s > F'(u_r), selecting the physically relevant weak solution among multiple possibilities.
s_entropy_entropy_flux_pair	axiom	Entropy/entropy-flux pair		A pair (η, q) with η convex and q' = η' F' such that entropy solutions of u_t + F(u)_x = 0 satisfy η(u)_t + q(u)_x ≤ 0 in the distributional sense.
s_lax_oleinik_formula	theorem	Lax-Oleinik formula		An explicit minimization formula giving the entropy solution of the scalar conservation law u_t + F(u)_x = 0 with convex flux, analogous to the Hopf-Lax formula
s_rarefaction_wave	state	Rarefaction wave		A continuous self-similar weak solution u(x,t) = v(x/t) of a conservation law connecting two constant states, existing when the characteristic speed is monotone
s_formation_of_singularities_first_order	theorem	Formation of singularities for first-order PDEs		For the conservation law u_t + F(u)_x = 0 with F'' > 0, if g'(x_0) < 0 at some point, then Du blows up at time T* = −1/min(F''(g) g'), forming a shock.
t_hopf_cole_transformation	technique	Hopf-Cole transformation		The substitution u = −2ε (w_x/w) transforms the viscous Burgers equation into the linear heat equation w_t = ε w_xx, yielding an explicit solution formula.
t_similarity_solutions	technique	Similarity solutions (self-similar reduction)		Seeking solutions of the form u(x,t) = t^α f(x/t^β) reduces a PDE to an ODE for the profile f, exploiting the scaling invariance of the equation.
s_sobolev_conjugate_exponent	state	Sobolev conjugate exponent		The critical exponent p* = np/(n−p) such that W^{1,p}(ℝ^n) embeds into L^{p*}(ℝ^n), the largest Lebesgue exponent achievable by the Gagliardo-Nirenberg-Sobolev 
s_trace_zero_characterization	theorem	Trace zero characterization of W^{1,p}_0		For Ω with C^1 boundary and 1 ≤ p < ∞, a function u ∈ W^{1,p}(Ω) belongs to W^{1,p}_0(Ω) if and only if its trace Tu vanishes on ∂Ω.
s_difference_quotient_characterization_sobolev	theorem	Difference quotient characterization of Sobolev spaces		A function u ∈ L^p(Ω) belongs to W^{1,p}(Ω) (1 < p ≤ ∞) if and only if its difference quotients D_h^i u are uniformly bounded in L^p.
s_general_sobolev_inequalities	theorem	General Sobolev inequalities (higher order)		For u ∈ W^{k,p}(ℝ^n): if kp < n then W^{k,p} ↪ L^{np/(n−kp)}; if kp > n then W^{k,p} ↪ C^{j,α}; with appropriate j, α depending on n, k, p.
s_negative_sobolev_space_W_minus_k_p	axiom	Negative Sobolev space W^{−k,p'}(Ω)		The dual space (W^{k,p}_0(Ω))*, consisting of distributions f representable as f = Σ_{|α|≤k} D^α f_α with f_α ∈ L^{p'}, the natural space for right-hand sides o
s_interpolation_inequalities_sobolev	theorem	Interpolation inequalities for Sobolev spaces		Intermediate derivatives are controlled by interpolation: ||D^j u||_q ≤ ε||D^k u||_p + C(ε)||u||_r for suitable exponents, relating intermediate to highest and 
s_campanato_space	axiom	Campanato space L^{p,λ}(Ω)		The space of L^p functions u with sup_{x,r} r^{−λ} ∫_{B(x,r)∩Ω} |u − u_{x,r}|^p dy < ∞; for λ ∈ (n, n+p], Campanato's theorem identifies L^{p,λ} with C^{0,(λ−n)
s_campanato_characterization_holder	theorem	Campanato's characterization of Hölder spaces		The Campanato space L^{p,λ}(Ω) is isomorphic to C^{0,α}(Ω) with α = (λ−n)/p when n < λ ≤ n+p, providing an integral characterization of Hölder continuity.
s_uniform_ellipticity	axiom	Uniform ellipticity condition		A second-order operator Lu = −Σ(a^{ij} u_{x_i})_{x_j} + lower order is uniformly elliptic if θ|ξ|² ≤ Σ a^{ij}(x)ξ_iξ_j ≤ Θ|ξ|² for all x ∈ Ω, ξ ∈ ℝ^n.
s_bilinear_form_elliptic_operator	state	Bilinear form B[u,v] for elliptic operator		The bilinear form B[u,v] = ∫_Ω [a^{ij} u_{x_i} v_{x_j} + b^i u_{x_i} v + cuv] dx associated to a divergence-form elliptic operator, encoding the weak formulatio
s_interior_H2_regularity_elliptic	theorem	Interior H^2 regularity for elliptic equations		If u ∈ H^1(Ω) weakly solves Lu = f with f ∈ L^2 and smooth coefficients, then u ∈ H^2_loc(Ω) with ||u||_{H^2(V)} ≤ C(||f||_{L^2} + ||u||_{L^2}) for V ⋐ Ω.
s_boundary_H2_regularity_elliptic	theorem	Boundary H^2 regularity for elliptic equations		If Ω has C^2 boundary and u ∈ H^1_0(Ω) weakly solves Lu = f ∈ L^2, then u ∈ H^2(Ω) with ||u||_{H^2} ≤ C(||f||_{L^2} + ||u||_{L^2}).
t_elliptic_regularity_bootstrap	technique	Elliptic regularity bootstrapping		Iteratively applying regularity estimates: if f ∈ H^m and coefficients are C^∞, then u ∈ H^{m+2}_loc, and inducting gives u ∈ C^∞ when f ∈ C^∞.
s_weak_maximum_principle_elliptic_divergence	theorem	Weak maximum principle for elliptic equations in divergence form		If u ∈ H^1(Ω) satisfies B[u,v] ≤ 0 for all v ≥ 0 in H^1_0(Ω), then sup_Ω u ≤ sup_{∂Ω} u^+, where boundary values are in the trace sense.
s_strong_maximum_principle_elliptic_divergence	theorem	Strong maximum principle for elliptic equations in divergence form		If u ∈ H^1(Ω) ∩ C(Ω̄) is a subsolution of Lu = 0 in a connected domain with c = 0, and u attains its maximum at an interior point, then u is constant.
s_eigenvalues_symmetric_elliptic_operator	theorem	Eigenvalue theory for symmetric elliptic operators		The eigenvalue problem Lu = λu in Ω with u = 0 on ∂Ω has a countable sequence of eigenvalues 0 < λ_1 ≤ λ_2 ≤ ⋯ → ∞ with eigenfunctions forming an orthonormal ba
s_principal_eigenvalue_elliptic	state	Principal eigenvalue of an elliptic operator		The smallest eigenvalue λ_1 of −L on H^1_0(Ω), characterized by λ_1 = min{B[u,u]/||u||²_{L^2} : u ∈ H^1_0, u ≠ 0}, with a corresponding eigenfunction of one sig
s_energy_estimates_elliptic	theorem	Energy estimates for elliptic PDEs		From coercivity B[u,u] ≥ α||u||²_{H^1_0} and the Poincaré inequality, the weak solution satisfies ||u||_{H^1_0} ≤ C||f||_{H^{-1}}, the fundamental a priori esti
s_parabolic_cylinder	axiom	Parabolic cylinder		The space-time domain Ω_T = Ω × (0,T] with parabolic boundary ∂_p Ω_T = (∂Ω × [0,T]) ∪ (Ω × {0}), the natural domain for parabolic initial-boundary value proble
s_weak_solution_parabolic_PDE	state	Weak solution of a parabolic PDE		A function u satisfying ⟨u',v⟩ + B[u,v;t] = ⟨f,v⟩ for a.e. t and all v ∈ H^1_0(Ω), with u(0) = g, the distributional formulation of u_t + Lu = f.
s_existence_weak_solutions_parabolic	theorem	Existence of weak solutions for parabolic equations		For u_t + Lu = f with L uniformly elliptic, there exists a unique weak solution u ∈ L^2(0,T; H^1_0) with u' ∈ L^2(0,T; H^{-1}), proved by Galerkin approximation
s_energy_estimates_parabolic	theorem	Energy estimates for parabolic equations		Testing u_t + Lu = f against u and using coercivity and Gronwall gives sup_{t} ||u||²_{L^2} + ∫ ||Du||²_{L^2} ≤ C(||g||²_{L^2} + ||f||²_{L^2(H^{-1})}).
s_parabolic_regularity	theorem	Parabolic regularity		If f and coefficients are smooth and ∂Ω is smooth, the weak solution of u_t + Lu = f satisfies u ∈ C^∞(Ω × (0,T]); proved by difference quotients, energy estima
s_weak_maximum_principle_parabolic	theorem	Weak maximum principle for parabolic equations		A subsolution of u_t + Lu = 0 in the parabolic cylinder Ω_T attains its maximum on the parabolic boundary ∂_p Ω_T.
s_strong_maximum_principle_parabolic	theorem	Strong maximum principle for parabolic equations		If a subsolution of u_t + Lu = 0 attains its maximum at an interior space-time point (x_0,t_0), then u is constant on Ω × (0,t_0].
s_lions_magenes_lemma	theorem	Lions-Magenes lemma		If V ↪ H ↪ V* is an evolution triple and u ∈ L^2(0,T; V) with u' ∈ L^2(0,T; V*), then u ∈ C([0,T]; H) with the energy equality d/dt ||u||²_H = 2⟨u',u⟩.
s_weak_solution_hyperbolic_PDE	state	Weak solution of a second-order hyperbolic PDE		A function u satisfying ⟨u'',v⟩ + B[u,v;t] = ⟨f,v⟩ for a.e. t and all v ∈ H^1_0(Ω), with u(0) = g ∈ H^1_0 and u'(0) = h ∈ L^2.
s_existence_weak_solutions_hyperbolic	theorem	Existence of weak solutions for hyperbolic equations		For u_tt + Lu = f with L symmetric uniformly elliptic, there exists a unique weak solution, proved by Galerkin approximation and hyperbolic energy estimates.
s_energy_estimates_hyperbolic	theorem	Energy estimates for hyperbolic equations		Testing u_tt + Lu = f against u_t and applying Gronwall gives ||u'||²_{L^2} + B[u,u] ≤ C(||h||² + B[g,g] + ||f||²), controlling the solution.
s_finite_speed_propagation_hyperbolic_general	theorem	Finite speed of propagation for general hyperbolic equations		For general second-order hyperbolic equations, the domain of influence of initial data on a set S is contained in a cone, proved by energy estimates in truncate
s_quasiconvexity_morrey	axiom	Quasiconvexity (Morrey)		L is quasiconvex if ∫_D L(A + Dφ) dx ≥ |D| L(A) for all bounded D, matrix A, and φ ∈ W^{1,∞}_0(D; ℝ^m); the natural convexity condition for weak lower semiconti
s_polyconvexity_ball	axiom	Polyconvexity (Ball)		L(A) = g(A, adj₂A, …, det A) for some convex g of all minors; implies quasiconvexity and is verifiable, applicable in nonlinear elasticity.
s_weak_lower_semicontinuity_integral_functionals	theorem	Weak lower semicontinuity of integral functionals		If L(x,z,p) is convex in p and satisfies growth conditions, then I[u] = ∫ L(x,u,Du) dx is sequentially weakly lower semicontinuous on W^{1,p}.
s_null_lagrangian	state	Null Lagrangian		A Lagrangian L for which ∫ L(Du) dx depends only on boundary values of u; equivalently, L is both quasiconvex and quasiconcave, consisting of linear combination
s_regularity_of_minimizers	theorem	Regularity of minimizers in the calculus of variations		If L is smooth and uniformly convex (D²_{pp} L ≥ θI), any minimizer u ∈ H^1(Ω) of I[u] = ∫ L(Du) dx is smooth in the interior.
t_minimax_methods_critical_points	technique	Minimax methods for critical points		Finding critical points by characterizing critical values as c = inf_{A∈Γ} max_{u∈A} I(u) over a suitable class Γ, combining deformation lemmas with topological
s_deformation_lemma	theorem	Deformation lemma		If I ∈ C^1(X,ℝ) has no critical values in [c−ε, c+ε], the sublevel set {I ≤ c+ε} can be continuously deformed into {I ≤ c−ε} via the negative gradient flow.
s_convexity_conditions_lagrangian	state	Convexity conditions on the Lagrangian		The hierarchy convex ⟹ polyconvex ⟹ quasiconvex ⟹ rank-one convex for integrands L(x,z,p) in the calculus of variations; convexity in p ensures weak lower semic
t_direct_method_calculus_of_variations	technique	Direct method of the calculus of variations		Taking a minimizing sequence, extracting a weakly convergent subsequence by coercivity and reflexivity, then showing the limit is a minimizer by weak lower semi
t_method_of_sub_supersolutions	technique	Method of sub- and supersolutions		If u_− ≤ u_+ satisfy −Δu_− ≤ f(u_−) and −Δu_+ ≥ f(u_+), then there exists a solution u with u_− ≤ u ≤ u_+, via monotone iteration or fixed-point arguments.
t_moser_iteration	technique	Moser iteration technique		Testing the equation against u^{p−1} for increasing p and using Sobolev embedding to bootstrap integrability iteratively: ||u||_{L^∞} ≤ C||u||_{L^2}.
t_stampacchia_truncation	technique	Stampacchia truncation method		Testing against (u−k)_+ to bound the measure of superlevel sets {u > k}, iterating De Giorgi-style to obtain L^∞ bounds from L^2 data.
s_leray_schauder_fixed_point	theorem	Leray-Schauder fixed point theorem		If T : X → X is compact and {u : u = σTu for σ ∈ [0,1]} is bounded, then T has a fixed point; a nonlinear generalization of Schauder's theorem via topological d
s_comparison_principle_viscosity	theorem	Comparison principle for viscosity solutions		If u is a viscosity subsolution and v a supersolution of F(D²u,Du,u,x) = 0 with u ≤ v on ∂Ω, then u ≤ v in Ω; proved by doubling of variables.
t_doubling_of_variables	technique	Doubling of variables technique		Introducing Φ(x,y) = u(x) − v(y) − |x−y|²/(2ε) and analyzing its maximum as ε → 0 to transfer viscosity conditions to the same test point.
s_dynamic_programming_principle	theorem	Dynamic programming principle (Bellman)		The value function V(x,t) = inf_α ∫_t^T L(y,α) ds + g(y(T)) satisfies the Hamilton-Jacobi-Bellman equation V_t + H(DV,x) = 0, connecting optimal control to PDE 
s_stability_viscosity_solutions	theorem	Stability of viscosity solutions		If u_ε → u locally uniformly and H_ε → H, then u is a viscosity solution of the limiting equation; the viscosity framework is designed for this stability proper
s_hopf_lax_as_viscosity_solution	theorem	Hopf-Lax formula gives the viscosity solution		The Hopf-Lax formula gives the unique viscosity solution of u_t + H(Du) = 0, u(·,0) = g, establishing the connection between the explicit formula and the viscos
t_vanishing_viscosity_method	technique	Vanishing viscosity method		Approximating a first-order PDE by adding εΔu, solving the parabolic problem, and passing ε → 0 to obtain a viscosity/entropy solution of the original equation.
s_system_of_conservation_laws	axiom	System of conservation laws		The system U_t + F(U)_x = 0 for U : ℝ × [0,∞) → ℝ^m with flux F : ℝ^m → ℝ^m, generalizing scalar conservation laws to vector-valued unknowns.
s_strict_hyperbolicity	axiom	Strict hyperbolicity		A system of conservation laws is strictly hyperbolic if the Jacobian DF(U) has m distinct real eigenvalues λ_1(U) < ⋯ < λ_m(U) for all U.
s_genuine_nonlinearity	axiom	Genuine nonlinearity		The k-th characteristic field (λ_k, r_k) is genuinely nonlinear if Dλ_k · r_k ≠ 0 for all U, so wave speed varies monotonically across rarefactions.
s_linear_degeneracy	axiom	Linear degeneracy		The k-th characteristic field (λ_k, r_k) is linearly degenerate if Dλ_k · r_k = 0 for all U; waves propagate as contact discontinuities rather than shocks or ra
s_contact_discontinuity	state	Contact discontinuity		A discontinuity across which the characteristic speed equals the propagation speed (λ_k = s), occurring in linearly degenerate fields of a conservation law syst
s_lax_theorem_riemann_problem	theorem	Lax's theorem on the Riemann problem		For a strictly hyperbolic system with each field genuinely nonlinear or linearly degenerate, the Riemann problem with U_L close to U_R has a unique solution of 
t_glimm_scheme	technique	Glimm's random choice method		Solving local Riemann problems on a staggered grid and randomly sampling the solution, converging to a global weak solution when the total variation of initial 
s_glimm_functional	state	Glimm interaction functional		The functional Q(t) = Σ |σ_α||σ_β| over approaching wave pairs, whose decrease at interactions compensates total variation increase, ensuring convergence of Gli
s_existence_global_entropy_solutions_glimm	theorem	Existence of global entropy solutions (Glimm's theorem)		For strictly hyperbolic conservation systems with small total variation initial data, Glimm's scheme converges to a global entropy solution in BV.
s_rankine_hugoniot_systems	state	Rankine-Hugoniot condition for systems		For U_t + F(U)_x = 0, a discontinuity at speed s satisfies s(U_R − U_L) = F(U_R) − F(U_L), the conservation condition across the shock.
s_lax_entropy_condition_systems	state	Lax entropy condition for systems		A k-shock is admissible if λ_k(U_L) > s > λ_k(U_R) and λ_{k−1}(U_L) < s < λ_{k+1}(U_R), ensuring exactly one family of characteristics enters from each side.
s_simple_waves_systems	state	Simple waves (rarefaction waves) for systems		A continuous self-similar solution U(ξ) with U' parallel to r_k(U) and ξ = λ_k(U(ξ)); exists for genuinely nonlinear fields in strictly hyperbolic systems.
s_young_measures	axiom	Young measures		A family of probability measures {ν_x} such that for continuous f, f(u_j) ⇀ ∫ f(λ) dν_x(λ) weakly-*, encoding oscillation behavior of weakly convergent sequence
t_tartar_compensated_compactness_framework	technique	Tartar's compensated compactness framework		Combining the div-curl lemma with entropy conditions to show Young measures are Dirac masses, proving strong convergence of vanishing viscosity approximations.
s_semilinear_wave_local_existence	theorem	Local existence for semilinear wave equations		For u_tt − Δu = f(u) with smooth f and Sobolev data with s > n/2, there exists a unique local solution, proved by fixed-point iteration in Sobolev spaces.
s_global_existence_small_data_wave	theorem	Global existence for small data (semilinear wave)		For u_tt − Δu = |u|^p with p above the Strauss exponent p_c(n) and sufficiently small initial data, the solution exists globally in time.
s_blow_up_finite_time_semilinear_wave	theorem	Blow-up in finite time for semilinear wave equations		For u_tt − Δu = |u|^{p−1}u with suitable data and subcritical exponent, the solution blows up in finite time; proved by energy/virial arguments or Levine's conc
t_levine_concavity_method	technique	Levine's concavity method for blow-up		Setting F(t) = ||u(t)||² and showing F'' ≥ c(F')^{1+δ} or that 1/F is concave, proving F → ∞ in finite time.
s_strauss_conjecture	state	Strauss conjecture (critical exponent for semilinear wave)		The critical power p_c(n) is the positive root of (n−1)p² − (n+1)p − 2 = 0; for p > p_c small-data global existence holds, for 1 < p ≤ p_c blow-up occurs.
s_bochner_space	axiom	Bochner space L^p(0,T; X)		The space of strongly measurable functions u : (0,T) → X with ∫_0^T ||u(t)||_X^p dt < ∞, the natural setting for time-dependent PDE solutions valued in Sobolev 
s_evolution_triple	axiom	Evolution triple (Gelfand triple)		A triple V ⊂ H ⊂ V* with V reflexive Banach densely embedded in Hilbert H; the standard framework for weak solutions of parabolic PDEs (e.g., H^1_0 ⊂ L^2 ⊂ H^{-
s_holder_space_Ck_gamma	axiom	Hölder space C^{k,γ}(Ω̄)		The Banach space of functions with continuous derivatives up to order k whose k-th derivatives are γ-Hölder continuous, with norm ||u||_{C^{k,γ}} = Σ_{|α|≤k} ||
t_duhamel_principle	technique	Duhamel's principle		Representing the solution of an inhomogeneous linear evolution equation as the homogeneous solution plus a source convolution, the continuous analogue of variat
s_poisson_equation_newtonian_potential	theorem	Solution of Poisson's equation via Newtonian potential		The convolution u(x) = ∫ Φ(x−y) f(y) dy with the fundamental solution Φ of Laplace's equation solves −Δu = f in ℝ^n.
t_energy_wave_equation	technique	Energy methods for the wave equation		Multiplying u_tt − Δu = f by u_t and integrating yields dE/dt = ∫ f u_t dx; E is conserved when f = 0, giving uniqueness and continuous dependence.
s_open_subset_Rn	axiom	Open subset Ω ⊂ ℝⁿ		An open subset of ℝⁿ serving as the domain for test functions and distributions.
s_multi_index_notation	axiom	Multi-index notation		An n-tuple α = (α₁,…,αₙ) of non-negative integers with |α| = α₁+⋯+αₙ, Dᵅ = ∂^|α|/(∂x₁^α₁⋯∂xₙ^αₙ), and α! = α₁!⋯αₙ!, the notational framework for PDE theory.
s_smooth_functions_C_infinity	axiom	Smooth functions C^∞(Ω)		The space of infinitely differentiable functions on Ω with the topology of uniform convergence of all derivatives on compact subsets.
s_test_function_space_Cc_infinity	axiom	Test function space C^∞_c(Ω) = 𝒟(Ω)		The space of smooth functions with compact support in Ω, topologized as the strict inductive limit of Fréchet spaces C^∞_K(Ω) over compact exhaustions K ⋐ Ω.
s_LF_space_topology_on_D	axiom	Strict inductive limit topology on 𝒟(Ω)		φⱼ → φ in 𝒟(Ω) iff there exists K ⋐ Ω with supp φⱼ ⊂ K for all j and Dᵅφⱼ → Dᵅφ uniformly for every α; this makes 𝒟(Ω) a non-metrizable locally convex space.
s_partition_of_unity_smooth	state	Smooth partition of unity		A locally finite family of non-negative smooth functions {ψᵢ} subordinate to an open cover, with ∑ψᵢ ≡ 1 on Ω; exists because ℝⁿ is paracompact and admits smoot
s_cutoff_function	state	Smooth cutoff function		A smooth function identically 1 on a neighborhood of a compact set K and compactly supported in U, constructed by mollifying characteristic functions.
s_mollifier_standard	state	Standard mollifier (Friedrich's mollifier)		The function ρ_ε(x) = ε⁻ⁿρ(x/ε) where ρ(x) = c·exp(−1/(1−|x|²)) for |x| < 1, ρ = 0 for |x| ≥ 1, with ∫ρ = 1; convolution with ρ_ε provides smooth approximation.
t_mollification	technique	Mollification (Friedrich's smoothing)		Convolving a locally integrable function with a smooth compactly supported approximate identity ρ_ε to produce a smooth approximation converging to f in L^p_{lo
s_distribution_on_omega	axiom	Distribution u ∈ 𝒟'(Ω)		A continuous linear functional on the test function space 𝒟(Ω); equivalently, a linear form u such that for every K ⋐ Ω there exist C, N with |⟨u,φ⟩| ≤ C ∑_{|α|
s_locally_integrable_as_distribution	state	Regular distribution (L¹_loc ↪ 𝒟')		The embedding f ↦ u_f defined by ⟨u_f, φ⟩ = ∫f φ dx, which is injective and identifies L¹_loc(Ω) with a subspace of 𝒟'(Ω); regular distributions have order 0.
s_distributions_compact_support_E_prime	axiom	Distributions of compact support ℰ'(Ω)		The space ℰ'(Ω) = {u ∈ 𝒟'(Ω) : supp u compact}, canonically identified with the topological dual of C^∞(Ω); every u ∈ ℰ' has finite order.
s_sheaf_property_distributions	theorem	Sheaf property of distributions		If u_i ∈ 𝒟'(Ω_i) agree on overlaps Ω_i ∩ Ω_j, then there exists a unique u ∈ 𝒟'(⋃Ω_i) restricting to each u_i; proved using partitions of unity.
s_fundamental_solution	axiom	Fundamental solution of a differential operator		A distribution E satisfying P(D)E = δ, so that P(D)(E ∗ f) = f for f ∈ ℰ'(ℝⁿ), providing a right inverse for P(D) via convolution.
s_leibniz_rule_distributions	theorem	Leibniz rule for distributions		Dᵅ(au) = ∑_{β≤α} (α choose β)(Dᵝa)(D^{α−β}u) for smooth a and distributional u, extending the classical product rule.
s_heaviside_derivative_is_delta	state	Derivative of the Heaviside function		The distributional derivative of the Heaviside step function H(x) = 1_{x>0} is the Dirac delta: ⟨H', φ⟩ = −⟨H, φ'⟩ = −∫₀^∞ φ' dx = φ(0).
s_division_of_distributions	state	Division of distributions by smooth functions		The equation a·v = u for v ∈ 𝒟'(Ω) given smooth a; solvable when a has only zeros of finite order, yielding solutions modulo distributions supported on {a = 0}.
s_principal_value_distribution	state	Principal value distribution p.v.(1/x)		The distribution ⟨p.v.(1/x), φ⟩ = lim_{ε→0} ∫_{|x|>ε} φ(x)/x dx, an order-1 distribution that extends 1/x across the origin; satisfies x · p.v.(1/x) = 1.
s_structure_theorem_distributions	theorem	Structure theorem for distributions		Every distribution is locally a finite sum of derivatives of continuous functions: for each K ⋐ Ω, u|_K = ∑_{|α|≤N} Dᵅf_α with f_α ∈ C⁰(K), where N is the local
s_structure_theorem_compact_support	theorem	Structure theorem for ℰ'(ℝⁿ)		Every distribution of compact support is a finite sum of derivatives of compactly supported continuous functions: u = ∑_{|α|≤N} Dᵅf_α with f_α ∈ C_c⁰(ℝⁿ).
s_distributions_supported_at_point	theorem	Distributions supported at a point		A distribution supported at a single point a is a finite linear combination of the Dirac delta and its derivatives at a: u = ∑_{|α|≤N} c_α Dᵅδ_a.
s_convolution_algebra_E_prime	state	Convolution algebra ℰ'(ℝⁿ)		ℰ'(ℝⁿ) equipped with convolution is an associative algebra with unit δ; satisfies supp(u∗v) ⊂ supp u + supp v and sing supp(u∗v) ⊂ sing supp u + sing supp v.
s_convolution_support_rule	theorem	Convolution support theorem		The support of a convolution is contained in the Minkowski sum of the supports; more refined: ch(supp(u∗v)) ⊂ ch(supp u) + ch(supp v) where ch denotes convex hu
s_singular_support_convolution	theorem	Singular support of a convolution		The singular support of u ∗ v is contained in the Minkowski sum of the singular supports, provided the convolution is defined.
s_regularization_by_convolution	theorem	Regularization of distributions by convolution		Convolution of any distribution with a test function yields a smooth function: u ∗ φ ∈ C^∞ with (u ∗ φ)(x) = ⟨u, φ(x − ·)⟩, and D^α(u ∗ φ) = (D^α u) ∗ φ = u ∗ (
s_approximation_by_smooth_functions	theorem	Density of C^∞_c in 𝒟'		Every distribution is the limit of smooth functions: u ∗ ρ_ε → u in 𝒟'(Ω) as ε → 0, where ρ_ε is a mollifier; more precisely, C^∞_c(Ω) is sequentially dense in 
t_regularization_convolution	technique	Regularization by convolution with mollifier		Approximating distributions by smooth functions via convolution with a mollifier; fundamental tool for transferring pointwise arguments to distributional settin
s_tensor_product_distributions	state	Tensor product of distributions		The unique distribution u ⊗ v on the product satisfying ⟨u ⊗ v, φ ⊗ ψ⟩ = ⟨u,φ⟩⟨v,ψ⟩, with supp(u ⊗ v) = supp u × supp v.
s_schwartz_kernel_theorem_hormander	theorem	Schwartz kernel theorem		Every continuous linear map T: C^∞_c(Ω₂) → 𝒟'(Ω₁) has a unique distributional kernel K ∈ 𝒟'(Ω₁ × Ω₂) with ⟨Tφ, ψ⟩ = ⟨K, ψ ⊗ φ⟩; establishes the canonical isomor
t_kernel_method	technique	Schwartz kernel method		Representing a continuous linear operator between distribution spaces as a distributional kernel on the product, reducing operator questions to distribution-the
s_pullback_distribution	state	Pullback of a distribution by a diffeomorphism		For a diffeomorphism κ, the pullback κ*u is defined by ⟨κ*u, φ⟩ = ⟨u, (φ ∘ κ⁻¹)|det Dκ⁻¹|⟩, extending the change-of-variables formula to distributions.
s_pullback_submersion	state	Pullback of a distribution by a submersion		For a smooth submersion f (surjective differential), pullback f*u is defined by fibered integration; extends u ∘ f from smooth functions to distributions.
s_distribution_on_submanifold	state	Distribution on a submanifold (restriction/trace)		The restriction of a distribution to a smooth submanifold, well-defined when the wavefront set of u has no conormal directions to M.
s_push_forward_distribution	state	Push-forward of a distribution		For proper f, the push-forward f_*u defined by ⟨f_*u, φ⟩ = ⟨u, f*φ⟩; dual operation to pullback.
t_change_of_variables_distributions	technique	Change of variables for distributions		Pulling back distributions through diffeomorphisms or submersions to transfer distributional problems between coordinate systems.
s_schwartz_space_S	axiom	Schwartz space 𝒮(ℝⁿ)		The space of rapidly decreasing smooth functions: φ ∈ C^∞(ℝⁿ) with sup_x |x^α Dᵝφ(x)| < ∞ for all α, β, topologized by these seminorms; dense in L^p for 1 ≤ p <
s_tempered_distributions_S_prime	axiom	Tempered distributions 𝒮'(ℝⁿ)		The dual of the Schwartz space: continuous linear functionals on 𝒮(ℝⁿ); includes all distributions of polynomial growth, all L^p functions, all polynomials, and
s_fourier_transform_on_S	state	Fourier transform on 𝒮(ℝⁿ)		The map φ̂(ξ) = ∫ e^{−ix·ξ} φ(x) dx is a topological automorphism of 𝒮(ℝⁿ), with inverse φ(x) = (2π)⁻ⁿ ∫ e^{ix·ξ} φ̂(ξ) dξ.
s_fourier_transform_on_S_prime	state	Fourier transform on 𝒮'(ℝⁿ)		Extended by duality: ⟨û, φ⟩ = ⟨u, φ̂⟩ for u ∈ 𝒮', φ ∈ 𝒮; an automorphism of 𝒮' interchanging differentiation with polynomial multiplication and support with smo
s_fourier_transform_exchange_rules	theorem	Fourier exchange rules (derivative ↔ multiplication)		The Fourier transform satisfies (D^α u)^ = (iξ)^α û and (x^α u)^ = (iD_ξ)^α û, exchanging differentiation with polynomial multiplication and converting PDEs int
s_plancherel_theorem_hormander	theorem	Plancherel theorem (L² isometry)		The Fourier transform extends from 𝒮(ℝⁿ) to a unitary isomorphism of L²(ℝⁿ): ‖φ̂‖₂ = (2π)^{n/2} ‖φ‖₂, and ∫φ̂ψ̄ dξ = (2π)ⁿ ∫φψ̄ dx (Parseval).
s_fourier_transform_of_E_prime	theorem	Fourier transform of ℰ'(ℝⁿ) (entire analytic functions)		The Fourier transform of u ∈ ℰ' is the entire analytic function û(ζ) = ⟨u, e^{−i⟨·,ζ⟩}⟩ satisfying |û(ζ)| ≤ C(1+|ζ|)^N e^{A|Im ζ|} where supp u ⊂ B̄(0,A).
s_paley_wiener_schwartz_theorem	theorem	Paley–Wiener–Schwartz theorem		A distribution u has compact support (u ∈ ℰ') iff its Fourier transform û extends to an entire function of exponential type: û ∈ O(ℂⁿ) with |û(ζ)| ≤ C(1+|ζ|)^N 
s_paley_wiener_smooth	theorem	Paley–Wiener theorem for C^∞_c		φ ∈ C^∞_c(ℝⁿ) with supp φ ⊂ B̄(0,A) iff φ̂ extends to an entire function with |φ̂(ζ)| ≤ C_N(1+|ζ|)^{−N} exp(A|Im ζ|) for all N.
s_sobolev_space_Hs	axiom	Sobolev space H^s(ℝⁿ) via Fourier transform		H^s(ℝⁿ) = {u ∈ 𝒮'(ℝⁿ) : (1+|ξ|²)^{s/2} û ∈ L²(ℝⁿ)}, with norm ‖u‖_s = ‖(1+|ξ|²)^{s/2} û‖₂; for s ∈ ℕ₀, equivalent to requiring D^α u ∈ L² for |α| ≤ s.
s_sobolev_embedding_via_fourier	theorem	Sobolev embedding via Fourier transform		If s > n/2 + k then H^s(ℝⁿ) embeds continuously into C^k(ℝⁿ) with bounded derivatives: proved by showing (1+|ξ|²)^{−s/2} ∈ L² when s > n/2, hence û ∈ L¹ by Cauc
s_fourier_inversion_formula	theorem	Fourier inversion formula		The inverse Fourier transform recovers φ from φ̂: φ(x) = (2π)⁻ⁿ ∫ e^{ix·ξ} φ̂(ξ) dξ, valid pointwise for φ ∈ 𝒮 and in 𝒮' for tempered distributions.
s_convolution_fourier_exchange	theorem	Convolution theorem (Fourier)		The Fourier transform converts convolution to pointwise multiplication: (u ∗ v)^ = û · v̂ when u ∈ 𝒮' and v ∈ ℰ', or when both are in L¹ or L².
t_fourier_analysis_of_PDE	technique	Fourier transform method for constant-coefficient PDE		Applying the Fourier transform to reduce a constant-coefficient PDE to an algebraic equation, then solving by division; requires analyzing the zero set of the s
s_wavefront_set_definition	axiom	Wavefront set WF(u)		WF(u) = {(x,ξ) ∈ Ω × (ℝⁿ\0) : ξ ∉ Σ_x(u)} where Σ_x(u) is the set of directions in which the localized Fourier transform φu decays rapidly; a refinement of sing
s_wavefront_set_projection_is_sing_supp	theorem	Projection of WF(u) is sing supp u		The projection of the wavefront set onto the base space equals the singular support: x ∈ sing supp u iff there exists ξ with (x,ξ) ∈ WF(u).
s_wavefront_set_of_derivative	theorem	Wavefront set under differentiation		Differentiation does not create new singularities: WF(Dᵅu) ⊂ WF(u) for all multi-indices α, and more generally WF(Pu) ⊂ WF(u) for any differential operator P wi
s_wavefront_set_of_product	theorem	Wavefront set of a product of distributions		When the product uv is well-defined, WF(uv) ⊂ WF(u) ∪ WF(v) ∪ {(x,ξ+η) : (x,ξ) ∈ WF(u), (x,η) ∈ WF(v)}, providing microlocal control of the product's singularit
s_hormander_condition_product	theorem	Hörmander's condition for multiplication of distributions		The product uv of two distributions is well-defined as a distribution if there is no (x,ξ) ∈ WF(u) with (x,−ξ) ∈ WF(v); i.e., the fiber-wise sum WF(u)_x + WF(v)
s_wavefront_set_pullback	theorem	Wavefront set under pullback		For a smooth map f: Ω₁ → Ω₂, if f*u is well-defined then WF(f*u) ⊂ {(x, ᵗf'(x)η) : (f(x),η) ∈ WF(u)}, the canonical pullback of the wavefront set via the transp
s_microlocal_regularity_elliptic	theorem	Elliptic regularity (microlocal version)		If the principal symbol p_m(x₀,ξ₀) ≠ 0 (ellipticity at a covector), then (x₀,ξ₀) ∈ WF(u) implies (x₀,ξ₀) ∈ WF(f); thus WF(u) \ WF(f) ⊂ Char(P), the characterist
s_characteristic_set	state	Characteristic set of a differential operator		The zero set of the principal symbol p_m(x,ξ) in the cotangent bundle (minus zero section); WF(u) \ WF(Pu) ⊂ Char(P) by microlocal elliptic regularity.
s_cone_and_conic_set	axiom	Conic set in T*Ω \ 0		A subset Γ of the cotangent bundle (with zero section removed) that is invariant under positive fiber dilations; wavefront sets are always closed conic sets.
s_microlocal_smoothness	axiom	Microlocal smoothness (absence from WF)		A distribution u is microlocally smooth at (x₀,ξ₀) if there exists φ ∈ C^∞_c with φ(x₀) ≠ 0 and a conic neighborhood Γ of ξ₀ such that |φ̂u(ξ)| ≤ C_N(1+|ξ|)^{−N
s_wavefront_set_convolution	theorem	Wavefront set of a convolution		The wavefront set of u ∗ v (when defined) satisfies WF(u∗v) ⊂ {(x+y,ξ) : (x,ξ) ∈ WF(u) and (y,ξ) ∈ WF(v)}, refining the singular support addition rule with dire
t_microlocal_analysis	technique	Microlocal analysis (wavefront set calculus)		Tracking the wavefront set through operations (differentiation, multiplication, pullback, push-forward, convolution) to determine well-definedness and regularit
s_oscillatory_integral	state	Oscillatory integral		An integral I(x) = ∫ e^{iΦ(x,θ)} a(x,θ) dθ with phase Φ and amplitude a in a symbol class, defined by regularization; the wavefront set is contained in {(x, Φ'_
t_stationary_phase	technique	Stationary phase method		Asymptotic expansion of oscillatory integrals: contributions concentrate at critical points of the phase Φ, with leading term involving (det Φ'')^{−1/2}; fundam
s_distribution_of_finite_order_global	state	Distribution of finite order		A distribution u such that |⟨u,φ⟩| ≤ C ∑_{|α|≤N} sup|Dᵅφ| for all φ ∈ 𝒟(Ω) with supp φ ⊂ K and a single N valid for all K; extends to Cᴺ_c(Ω) by continuity.
s_positive_distribution	theorem	Positive distributions are measures		If ⟨u,φ⟩ ≥ 0 for all non-negative test functions φ, then u is a positive Radon measure on Ω; this is order 0 and follows from the Riesz representation theorem.
s_fourier_laplace_transform	state	Fourier–Laplace transform		The extension of the Fourier transform of u ∈ ℰ' to complex arguments: û(ζ) = ⟨u, e^{−i⟨·,ζ⟩}⟩ for ζ ∈ ℂⁿ, an entire function of exponential type characterized 
s_hormander_propagation_of_singularities_distribution	theorem	Propagation of singularities (distribution-theoretic version)		For a principal-type operator P with real principal symbol, WF(u) \ WF(f) is a union of maximally extended bicharacteristic strips of P inside Char(P); singular
s_fbi_transform	axiom	FBI transform		The Fourier–Bros–Iagolnitzer transform Tu(x,ξ,λ) = ∫ e^{iλ⟨y−x,ξ⟩ − λ|y−x|²/2} u(y) dy, a Gaussian-windowed Fourier transform that characterizes microlocal regu
s_analytic_wave_front_set	axiom	Analytic wave front set WF_A(u)		The analytic wave front set WF_A(u) ⊂ T*Ω \ 0 refines WF(u) by measuring failure of analytic (rather than C^∞) regularity at each codirection, defined via expon
s_fbi_characterization_analytic_regularity	theorem	FBI characterization of analytic regularity		A distribution u is real-analytic near x₀ if and only if WF_A(u) ∩ ({x₀} × (ℝⁿ\0)) = ∅, characterized by exponential decay of the FBI transform Tu(x,ξ,λ) as λ →
t_schwartz_kernel_method	technique	Schwartz kernel method		Representing continuous linear maps between distribution spaces via their Schwartz kernel to reduce operator-theoretic questions to distributional properties of
s_p_convexity_for_supports	axiom	P-convexity for supports		An open set Ω is P-convex for supports if for every compact K ⊂ Ω, the P-support hull of K relative to Ω is compact; equivalent to surjectivity of P(D): 𝒟'(Ω) →
s_p_convexity_for_singular_supports	axiom	P-convexity for singular supports		An open set Ω is P-convex for singular supports if for every compact K ⊂ Ω, the set of points whose singular support hull meets K is relatively compact; require
s_malgrange_exponential_polynomial_approximation	theorem	Exponential-polynomial solution approximation (Malgrange)		For a convex open set Ω, exponential-polynomial solutions x^α e^{⟨ζ,x⟩} with P(ζ) = 0 are dense in the space of all C^∞ solutions of P(D)u = 0 on Ω in the C^∞ t
s_geometric_characterization_p_convexity	theorem	Geometric characterization of P-convexity		P-convexity for supports is characterized by the condition that for each compact K ⊂ Ω, the convex hull of K with respect to the propagation cone of P remains c
s_existence_theorem_p_convex_domains	theorem	Existence theorem for PDE on P-convex domains		If Ω is P-convex for supports, then P(D): 𝒟'(Ω) → 𝒟'(Ω) is surjective; if also P-convex for singular supports, then P(D): C^∞(Ω) → C^∞(Ω) is surjective.
s_convexity_implies_p_convexity	theorem	Convexity implies P-convexity		Every convex open set Ω ⊂ ℝⁿ is P-convex for supports and for singular supports for every constant-coefficient operator P(D).
s_regularity_index_hypoelliptic	state	Regularity index of a hypoelliptic operator		The supremum δ such that |ξ|^δ = O(|P(ξ)|) as |ξ| → ∞; measures Sobolev regularity gain: if P(D)u ∈ H^s then u ∈ H^{s+mδ}. For elliptic operators δ = 1; for the
s_analytically_hypoelliptic_operator	axiom	Analytically hypoelliptic operator		A linear PDE operator P is analytically hypoelliptic if whenever Pu is real-analytic on an open set U, the solution u is also real-analytic on U; stronger than 
s_characterization_analytically_hypoelliptic	theorem	Characterization of analytically hypoelliptic operators		A constant-coefficient operator P(D) is analytically hypoelliptic if and only if the variety V(P) = {ζ ∈ ℂⁿ : P(ζ) = 0} satisfies |Im ζ| → ∞ when |ζ| → ∞ on V(P
s_well_posedness_garding_petrowsky	theorem	Well-posedness of Cauchy problem (Gårding–Petrowsky)		The Cauchy problem for P(D) with data on t = 0 is well-posed in C^∞ if and only if P is hyperbolic with respect to the t-direction.
s_hyperbolic_polynomial	axiom	Hyperbolic polynomial		A homogeneous polynomial P(ξ) of degree m is hyperbolic with respect to N if P(N) ≠ 0 and for every ξ the equation P(ξ + τN) = 0 has only real roots τ; the hype
s_petrowsky_lacuna	axiom	Petrowsky lacuna		A connected component of the complement of the wave front surface of a hyperbolic fundamental solution where it vanishes identically; the sharp Huygens principl
s_herglotz_petrovsky_leray_formula	theorem	Herglotz–Petrovsky–Leray formula		An explicit integral representation of the fundamental solution of a hyperbolic operator as a period integral over a cycle in projective space determined by the
s_atiyah_bott_garding_lacuna_criterion	theorem	Atiyah–Bott–Gårding lacuna criterion		A Petrowsky lacuna is characterized by the vanishing of the Petrovsky cycle in the homology of the complement of the complexified characteristic surface; gives 
s_operator_of_constant_strength	axiom	Operator of constant strength		A variable-coefficient operator P(x,D) has constant strength if for every pair x₀, x₁, the frozen operators P(x₀,D) and P(x₁,D) are equally strong: their symbol
s_existence_theorem_constant_strength	theorem	Existence theorem for operators of constant strength		If P(x,D) has constant strength, the equation P(x,D)u = f is locally solvable, proved by reducing to the constant-coefficient case via parametrix construction a
s_hypoellipticity_constant_strength	theorem	Hypoellipticity for operators of constant strength		If P(x,D) has constant strength and each frozen operator P(x₀,D) is hypoelliptic, then P(x,D) itself is hypoelliptic with the same regularity index.
s_abstract_cauchy_kovalevsky_theorem	theorem	Abstract Cauchy–Kovalevsky theorem (Nirenberg–Nishida)		In a scale of Banach spaces {E_s}, if F: E_s → E_{s'} satisfies a Lipschitz condition with norm ≤ C/(s−s'), then du/dt = F(u), u(0) = u₀ has a unique solution o
s_holmgren_john_global_uniqueness	theorem	Holmgren–John global uniqueness		Extension of Holmgren's theorem: uniqueness of solutions vanishing on one side of a non-characteristic surface propagates along bicharacteristics, sweeping the 
s_ehrenpreis_fundamental_principle	theorem	Ehrenpreis fundamental principle		Every solution of a constant-coefficient PDE system P(D)u = 0 on a convex open set can be represented as an integral of exponential-polynomial solutions over th
s_mean_periodic_function	axiom	Mean-periodic function		A continuous function f is mean-periodic if there exists a nonzero μ ∈ ℰ'(ℝ) with μ ∗ f = 0; generalizes periodic functions and solutions of constant-coefficien
s_spectral_synthesis_mean_periodic	theorem	Spectral synthesis for mean-periodic functions		Every mean-periodic function on ℝ can be approximated by finite sums of x^k e^{λx} where λ ranges over zeros of μ̂(ζ); the Delsarte–Schwartz theorem on spectral
s_noetherian_operator_pde	state	Noetherian operator (PDE)		In the Ehrenpreis–Palamodov theory, differential operators with polynomial coefficients that, applied to e^{i⟨x,ζ⟩} and integrated over V(P), generate all solut
s_division_theorem_ehrenpreis_malgrange	theorem	Division theorem for distributions (Ehrenpreis–Malgrange)		For any nonzero polynomial P and tempered distribution u, there exists v ∈ 𝒮'(ℝⁿ) with Pv = u; multiplication by P is surjective on 𝒮'(ℝⁿ). Key ingredient in th
s_slowly_decreasing_condition	axiom	Slowly decreasing condition		The Fourier–Laplace transform μ̂(ζ) of μ ∈ ℰ'(ℝⁿ) is slowly decreasing if for every ζ there exists ζ' nearby with |μ̂(ζ')| polynomially bounded below; necessary
s_surjectivity_convolution_operators_smooth	theorem	Surjectivity of convolution operators on C^∞		The convolution operator f ↦ μ ∗ f is surjective on C^∞(ℝⁿ) if and only if μ̂ is slowly decreasing; generalizes Malgrange–Ehrenpreis from polynomial operators t
s_supports_theorem_convolution	theorem	Supports theorem for convolution (Titchmarsh generalization)		For u, v ∈ ℰ'(ℝⁿ), ch(supp(u ∗ v)) = ch(supp u) + ch(supp v) where ch is convex hull and + is Minkowski sum; generalizes Titchmarsh's convolution theorem.
s_subelliptic_estimate	state	Subelliptic estimate		An a priori estimate ‖u‖_{H^{s+ε}} ≤ C(‖Pu‖_{H^s} + ‖u‖_{H^{s₀}}) with 0 < ε < 1, weaker than elliptic estimates; satisfied by Hörmander sum-of-squares operator
t_kohn_subelliptic_multiplier_method	technique	Kohn's subelliptic multiplier method		Establishing subelliptic estimates by iteratively constructing an ideal of multipliers using Lie brackets and algebraic operations until the ideal contains a no
s_rothschild_stein_lifting_theorem	theorem	Rothschild–Stein lifting theorem		Vector fields satisfying Hörmander's bracket condition of step r can be locally lifted to left-invariant vector fields on a free nilpotent Lie group of step r, 
s_classical_symbol_class	axiom	Classical symbol class S^m_cl		The subclass of S^m_{1,0} admitting an asymptotic expansion a ~ ∑_{j≥0} a_{m-j} with a_{m-j} homogeneous of degree m−j in ξ for |ξ| ≥ 1; the leading term a_m is
s_pseudodifferential_operator_definition	state	Pseudodifferential operator Op(a)		The operator Op(a)u(x) = (2π)^{−n} ∫ e^{i⟨x,ξ⟩} a(x,ξ) û(ξ) dξ defined by Kohn–Nirenberg quantization; differential operators of order m correspond to polynomia
s_asymptotic_expansion_of_symbols	state	Asymptotic expansion of symbols		Given a_j ∈ S^{m_j}_{ρ,δ} with m_j ↘ −∞, there exists a ∈ S^{m_0} such that a − ∑_{j<N} a_j ∈ S^{m_N} for all N, unique modulo S^{−∞}; fundamental for parametri
s_composition_formula_pseudodifferential	theorem	Composition formula for pseudodifferential operators		For a ∈ S^{m₁}_{ρ,δ} and b ∈ S^{m₂}_{ρ,δ} with ρ > δ, the composition Op(a) ∘ Op(b) = Op(c) with c ~ ∑_α (1/α!) ∂^α_ξ a · D^α_x b; the principal symbol of the c
s_adjoint_formula_pseudodifferential	theorem	Adjoint formula for pseudodifferential operators		The L² adjoint of Op(a) with a ∈ S^m_{ρ,δ} (ρ > δ) is Op(a*) where a* ~ ∑_α (1/α!) ∂^α_ξ D^α_x ā; the principal symbol of the adjoint is the complex conjugate o
s_kernel_characterization_pseudodifferential	theorem	Kernel characterization of pseudodifferential operators		A continuous linear operator belongs to Ψ^m_{ρ,δ} if and only if its Schwartz kernel is C^∞ off the diagonal and satisfies specific singular estimates near the 
s_elliptic_symbol	axiom	Elliptic symbol		A symbol a ∈ S^m is elliptic if |a(x,ξ)| ≥ c|ξ|^m for large |ξ|; ellipticity guarantees existence of a parametrix and full elliptic regularity.
s_elliptic_parametrix_construction	theorem	Elliptic parametrix construction		For an elliptic symbol a, one constructs b₀ = 1/a for large ξ, then iteratively corrects via the composition formula to get Op(a)Op(b) = I + R with R smoothing.
s_elliptic_regularity_pseudodifferential	theorem	Elliptic regularity for pseudodifferential operators		If P ∈ Ψ^m is elliptic and Pu = f ∈ H^s_{loc}, then u ∈ H^{s+m}_{loc}; microlocally, WF(u) = WF(Pu), giving microlocal elliptic regularity.
s_calderon_vaillancourt_theorem	theorem	Calderón–Vaillancourt theorem		If a ∈ S⁰_{ρ,ρ} with 0 ≤ ρ < 1, then Op(a) is bounded on L²(ℝⁿ), with norm controlled by finitely many symbol seminorms; fails for ρ = 1.
s_beals_characterization_pseudodifferential	theorem	Beals characterization of pseudodifferential operators		A bounded operator A on L² belongs to Ψ⁰ if and only if iterated commutators ad(x_j)^α ad(D_j)^β A are bounded on L² for all multi-indices; gives a quantization
s_weyl_quantization	state	Weyl quantization		The symmetric quantization Op^w(a)u(x) = (2π)^{−n} ∫∫ e^{i⟨x−y,ξ⟩} a((x+y)/2, ξ) u(y) dy dξ; real symbols give formally self-adjoint operators, and composition 
s_moyal_product	state	Moyal product		The composition law for Weyl symbols: (a #_w b) ~ ab + (i/2){a,b} + ... where {a,b} is the Poisson bracket; makes the symbol space a noncommutative associative 
s_sharp_garding_inequality	theorem	Sharp Gårding inequality		For a(x,ξ) ≥ 0 with a ∈ S^m_{1,0}, Re⟨Op(a)u, u⟩ ≥ −C‖u‖²_{H^{(m−1)/2}}, improving the classical Gårding inequality by losing only (m−1)/2 derivatives; proved v
s_fefferman_phong_inequality	theorem	Fefferman–Phong inequality		If a ∈ S²_{1,0} satisfies a ≥ 0, then Op^w(a) ≥ −C on L², gaining a full derivative over the sharp Gårding inequality; proved via decomposition into wave packet
t_symbol_smoothing	technique	Symbol smoothing (Friedrichs symmetrization)		Regularizing a symbol by Gaussian convolution in phase space at scale |ξ|^{−1/2} to produce a non-negative symbol differing from the original by a lower-order t
s_pseudodifferential_operators_on_manifolds	axiom	Pseudodifferential operators on manifolds Ψ^m(M)		The algebra Ψ^m(M) of operators on a manifold M that are pseudodifferential with symbol in S^m_{1,0} in every local chart modulo smoothing; the principal symbol
s_principal_symbol_on_manifold	state	Principal symbol on a manifold		The equivalence class of the full symbol modulo S^{m−1}, transforming as a function on T*M \ 0 homogeneous of degree m; invariantly defined and multiplicative u
s_fredholm_property_elliptic_compact_manifold	theorem	Fredholm property of elliptic operators on compact manifolds		An elliptic P ∈ Ψ^m(M) on a compact manifold without boundary is Fredholm: H^s(M) → H^{s−m}(M) with finite-dimensional kernel and cokernel in C^∞(M); index comp
s_seeley_complex_powers	state	Seeley's complex powers		For an elliptic positive self-adjoint P on a compact manifold, the complex powers P^z form a holomorphic family of pseudodifferential operators of order mRe(z);
s_weyl_law_sharp_remainder	theorem	Weyl's law with sharp remainder (Hörmander)		The eigenvalue counting function N(λ) = #{λ_j ≤ λ} for a positive elliptic self-adjoint operator of order m on compact n-manifold satisfies N(λ) = (2π)^{−n} vol
s_calderon_projector	state	Calderón projector		A pseudodifferential projector on ∂M projecting the space of all boundary Cauchy data onto the subspace of Cauchy data of solutions of Pu = 0; its principal sym
s_shapiro_lopatinskii_condition	axiom	Shapiro–Lopatinskii condition		The compatibility condition ensuring an elliptic BVP is Fredholm: at each boundary point, the frozen ODE on the half-line must have a unique decaying solution f
s_boutet_de_monvel_calculus	axiom	Boutet de Monvel calculus		A calculus of 2×2 matrix operators [P₊+G, K; T, S] on manifolds with boundary combining interior pseudodifferential, singular Green, Poisson, and trace operator
s_poisson_operator_boutet	state	Poisson operator (Boutet de Monvel)		An operator in the Boutet de Monvel calculus mapping boundary data on ∂M to solutions in the interior, the off-diagonal entry converting boundary conditions to 
s_singular_green_operator	state	Singular Green operator		An operator in the Boutet de Monvel calculus capturing boundary effects invisible to the interior pseudodifferential calculus; singular only at the boundary dia
s_fredholm_property_elliptic_bvp	theorem	Fredholm property for elliptic BVPs		An elliptic BVP satisfying the Shapiro–Lopatinskii condition on a compact manifold with boundary is Fredholm between Sobolev spaces; index computed by the Atiya
s_hodge_decomposition_elliptic_complexes	theorem	Hodge decomposition for elliptic complexes		For an elliptic complex on a compact Riemannian manifold, each section space decomposes as ker Δ_k ⊕ im d_{k-1} ⊕ im d_k*, with harmonic space ker Δ_k isomorphi
s_symplectic_classification_submanifolds	state	Symplectic classification of submanifolds		Submanifolds of a symplectic manifold classified by ω|_Σ: Lagrangian (ω|_Σ = 0, dim = n), symplectic (non-degenerate), isotropic (ω|_Σ = 0, dim < n), coisotropi
s_bicharacteristic_strip	state	Bicharacteristic strip		An integral curve of the Hamilton vector field H_{p_m} lying on the characteristic set {p_m = 0}; its projection is a bicharacteristic curve (ray). Singularitie
s_operator_of_principal_type	axiom	Operator of principal type		A pseudodifferential operator P is of principal type if dp_m ≠ 0 on Char(P); equivalently, H_{p_m} is nonvanishing on the characteristic set, ensuring bicharact
s_condition_psi_nirenberg_treves	axiom	Condition (Ψ) (Nirenberg–Treves)		Im(p_m) does not change sign from − to + along oriented bicharacteristics of Re(p_m); necessary and sufficient for local solvability (necessity: Hörmander; suff
s_subelliptic_operators_loss_delta	state	Subelliptic operators (loss of δ derivatives)		P is subelliptic with loss δ if ‖u‖_{H^{s+m−δ}} ≤ C(‖Pu‖_{H^s} + ‖u‖_{H^{s₀}}); the loss δ = k/(k+1) corresponds to bracket depth k in the Hörmander condition.
s_microlocal_hypoellipticity	state	Microlocal hypoellipticity		P is microlocally hypoelliptic at (x₀,ξ₀) if (x₀,ξ₀) ∉ WF(u) whenever (x₀,ξ₀) ∉ WF(Pu); the microlocal refinement of hypoellipticity.
s_strictly_hyperbolic_operator	axiom	Strictly hyperbolic operator (PDE)		P is strictly hyperbolic if its principal symbol p_m(t,x,τ,ξ) = 0 has m distinct real roots τ = τ_j(t,x,ξ) for ξ ≠ 0; implies well-posedness of the Cauchy probl
s_well_posedness_strictly_hyperbolic_cauchy	theorem	Well-posedness of strictly hyperbolic Cauchy problem		The Cauchy problem for a strictly hyperbolic operator of order m has a unique solution u ∈ C([0,T]; H^s) with energy estimates controlling ‖u‖ by data and sourc
s_parametrix_strictly_hyperbolic_cauchy	state	Parametrix for strictly hyperbolic Cauchy problem		The solution operator for the strictly hyperbolic Cauchy problem is a Fourier integral operator whose canonical relation is the graph of the bicharacteristic fl
s_propagation_singularities_hyperbolic	theorem	Propagation of singularities for hyperbolic equations		For a strictly hyperbolic P, WF(u) \ WF(f) is a union of bicharacteristic strips, giving the sharp description of singular support propagation along rays.
t_lax_parametrix_construction	technique	Lax parametrix construction		Constructing the parametrix by solving the eikonal equation for the phase and transport equations for the amplitude, producing an FIO approximation to the solut
t_hadamard_fundamental_solution_construction	technique	Hadamard's construction of fundamental solutions		Writing E = ∑ U_j Γ^{j−(n−2)/2} + (log Γ) ∑ V_j Γ^j where Γ is the squared Lorentzian distance; coefficients determined recursively by transport equations along
s_glancing_rays	state	Glancing rays		Bicharacteristics tangent to the boundary at a point, neither entering nor leaving transversally; propagation near glancing rays requires the Melrose–Sjöstrand 
s_diffractive_rays	state	Diffractive rays		Bicharacteristics tangent to the boundary that then leave into the interior; singularities can split between gliding along the boundary and diffracting into the
s_melrose_sjostrand_propagation_theorem	theorem	Melrose–Sjöstrand propagation theorem		Singularities of solutions to the wave equation in a domain with boundary propagate along generalized bicharacteristic rays: reflecting at transversal points, g
s_gliding_rays	state	Gliding rays		Bicharacteristic rays trapped on the boundary ∂Ω carrying singularities in the shadow region; on convex obstacles, these are the geodesics on the boundary (cree
s_non_degenerate_phase_function	axiom	Non-degenerate phase function		A phase function φ(x,θ) homogeneous of degree 1 in θ with dφ ≠ 0 and d(∂φ/∂θ_j) linearly independent on C_φ = {∂φ/∂θ = 0}; the critical set C_φ parameterizes a 
s_canonical_relation	axiom	Canonical relation		A Lagrangian submanifold of T*X × T*Y with the twisted symplectic form, serving as the geometric invariant of a Fourier integral operator; composition of FIOs c
s_lagrangian_distribution	axiom	Lagrangian distribution I^μ(X, Λ)		The space of distributions on X with WF contained in Λ and singularity of order μ, locally expressible as oscillatory integrals ∫ e^{iφ(x,θ)} a(x,θ) dθ with app
s_principal_symbol_lagrangian_distribution	state	Principal symbol of a Lagrangian distribution		The invariantly defined leading term in the half-density bundle tensored with the Maslov bundle; determines u modulo I^{μ−1} and is multiplicative under FIO com
s_maslov_index	axiom	Maslov index		An integer-valued topological invariant measuring the rotation of a Lagrangian plane relative to a reference; gives the phase shift e^{−iπ/2} at each caustic cr
s_maslov_bundle	state	Maslov bundle		A flat complex line bundle on Λ with holonomy e^{−iπμ(γ)/2} around loops; sections of Ω^{1/2}(Λ) ⊗ M_Λ are the natural home for principal symbols of Lagrangian 
s_clean_intersection_composition_fio	theorem	Clean intersection composition of FIOs		When canonical relations compose cleanly with excess e, the FIO composition belongs to I^{μ+ν+e/2} associated to the composed canonical relation; the excess shi
s_transversal_composition_fio	theorem	Transversal composition of FIOs		When canonical relations compose transversally (excess e = 0), the composition is an FIO of order μ + ν with principal symbol equal to the product of principal 
s_l2_continuity_fio	theorem	L² continuity of FIOs		A Fourier integral operator of order 0 whose canonical relation is a local canonical graph is bounded on L²; the order 0 is sharp.
s_hormander_sato_lemma	theorem	Hörmander–Sato lemma on WF of FIOs		For an FIO with canonical relation C, the wavefront set satisfies WF(Au) ⊂ C ∘ WF(u), giving a precise microlocal mapping property showing singularities propaga
t_global_calculus_oscillatory_integrals	technique	Global calculus of oscillatory integrals		Patching local oscillatory integral representations to define FIOs globally via partition of unity; the class I^μ(X,Λ) is independent of the choice of non-degen
t_refined_symbol_calculus_type_1_0	technique	Refined symbol calculus for type (1,0)		Sharper estimates for composition and commutators of type (1,0) operators, with precise remainder dependence on seminorms; needed for Fefferman–Phong, sharp Går
s_carleman_estimate	theorem	Carleman estimate (general form)		An estimate ∑ τ^{m−1−|α|} ‖e^{τφ} D^α u‖_{L²} ≤ C ‖e^{τφ} Pu‖_{L²} for τ ≥ τ₀ and u ∈ C_c^∞, where φ satisfies a pseudo-convexity condition; the main tool for u
s_strong_pseudo_convexity_condition	axiom	Strong pseudo-convexity condition		The condition {p_m, p̄_m}/(2i) > 0 on the complexified characteristic set with the weight φ; ensures Carleman estimates hold and is the natural condition for un
s_holmgren_uniqueness_microlocal	theorem	Holmgren's uniqueness (microlocal version)		If P has analytic coefficients and Pu = 0, the analytic wavefront set WF_A(u) propagates along bicharacteristic strips; u vanishing on one side of a non-charact
s_calderon_uniqueness_theorem	theorem	Calderón uniqueness theorem		If P is a second-order operator with Lipschitz coefficients and u satisfies Pu = 0, vanishing on one side of a non-characteristic surface S, then u = 0 near S; 
s_strong_unique_continuation_property	theorem	Strong unique continuation property		If P is second-order elliptic and |Pu| ≤ C(|u| + |∇u|), and u vanishes to infinite order at x₀, then u ≡ 0 in a connected neighborhood; proved by Carleman estim
s_unique_continuation_across_non_characteristic	theorem	Unique continuation across non-characteristic surfaces		For P of any order with smooth coefficients, if S is strongly pseudo-convex and non-characteristic and u satisfies Pu = 0 with u = 0 on one side, then u = 0 nea
t_tauberian_theorem_spectral_function	technique	Tauberian theorem for spectral function		Converting wave trace distributional information into pointwise asymptotics of N(λ) via Tauberian arguments; the remainder O(λ^{(n−1)/m}) follows from propagati
s_spectral_function_asymptotic	theorem	Spectral function asymptotic		The on-diagonal spectral function e(x,x,λ) = ∑_{λ_j ≤ λ} |φ_j(x)|² satisfies e(x,x,λ) = (2π)^{−n} ∫_{p_m(x,ξ)≤λ} dξ + O(λ^{(n−1)/m}); integrating gives Weyl's l
s_wave_trace_formula	theorem	Wave trace formula		The distributional trace Tr(cos(t√Δ)) has singular support in {0} ∪ {±L : L length of closed geodesic}; singularities at t = ±L encode geometric data about clos
s_sharpness_weyl_remainder	theorem	Sharpness of remainder in Weyl's law		The O(λ^{(n−1)/m}) remainder in Weyl's law is optimal: on the round sphere, eigenvalue clustering produces oscillations of this order; improved remainders hold 
s_local_weyl_law	theorem	Local Weyl law		For P elliptic on compact M and A ∈ Ψ^0, ∑_{λ_j ≤ λ} ⟨Aφ_j, φ_j⟩ = (2π)^{−n} ∫_{p_m ≤ λ} σ_0(A) dω + O(λ^{(n−1)/m}); implies quantum ergodicity when geodesic fl
s_asymptotic_completeness_long_range	theorem	Asymptotic completeness for long-range potentials		For potentials with |∂^α V| ≤ C⟨x⟩^{−ρ−|α|} (ρ > 0), Dollard modified wave operators exist and are complete: Ran(Ω±_D) = H_{ac}(H); combines Mourre estimates, p
s_limiting_absorption_principle_long_range	theorem	Limiting absorption principle (long-range)		Resolvent boundary values (H − λ ± i0)^{−1} exist as bounded operators ⟨x⟩^{−s}L² → ⟨x⟩^s L² (s > 1/2) for λ in the a.c. spectrum; proved via Mourre theory.
s_scattering_matrix_long_range	state	Scattering matrix for long-range potentials		The scattering matrix S(λ) defined via Dollard modified wave operators is a pseudodifferential operator on S^{n−1} whose symbol encodes the long-range scatterin
t_phase_space_scattering_long_range	technique	Phase space characterization of long-range scattering		Characterizing scattering states via the asymptotic velocity operator P± = s-lim x(t)/t and propagation estimates; completeness follows from showing P± exists a
s_sommerfeld_radiation_condition	axiom	Sommerfeld radiation condition (generalized)		The condition ∂u/∂r − iku = o(r^{−(n−1)/2}) selecting the unique outgoing solution of the Helmholtz equation; generalized for variable-coefficient operators via
s_differentiable_manifold	axiom	Differentiable manifold		A topological manifold equipped with a maximal atlas of charts whose transition maps are smooth (C^infty) diffeomorphisms.
s_submanifold	axiom	Submanifold		A subset S of a manifold M that is itself a manifold and whose inclusion map is an injective immersion.
s_orientation_of_manifold	axiom	Orientation of a manifold		A consistent choice of orientation for all tangent spaces of a smooth manifold, equivalently a nowhere-vanishing top-degree differential form.
s_vector_field	axiom	Vector field		A smooth section of the tangent bundle TM, assigning to each point p of a manifold M a tangent vector in T_pM.
s_riemannian_manifold	axiom	Riemannian manifold		A smooth manifold equipped with a Riemannian metric, i.e., a smoothly varying positive-definite inner product on each tangent space.
s_length_of_curve	state	Length of a curve		The integral L(gamma) = integral_a^b |gamma'(t)|_g dt of the speed of a piecewise smooth curve in a Riemannian manifold.
s_riemannian_distance	state	Riemannian distance		The metric d(p, q) = inf { L(gamma) : gamma joins p to q } on a connected Riemannian manifold, making it a metric space compatible with its topology.
s_isometry	axiom	Isometry (Riemannian)		A diffeomorphism f : (M, g) -> (N, h) between Riemannian manifolds satisfying f*h = g, preserving the metric tensor.
s_local_isometry	axiom	Local isometry		A smooth map f : M -> N such that the differential df_p is a linear isometry between the tangent spaces at every point.
s_pullback_metric	state	Pullback metric (induced metric)		The Riemannian metric f*g on M induced by an immersion f : M -> (N, g), defined by (f*g)_p(u, v) = g_{f(p)}(df_p u, df_p v).
s_product_metric	state	Product metric		The Riemannian metric on M x N defined as the direct sum g_M + g_N on the product tangent spaces.
s_affine_connection	axiom	Affine connection		An operator nabla : X(M) x X(M) -> X(M) satisfying C^infty(M)-linearity in the first argument and the Leibniz rule in the second, specifying differentiation of 
s_covariant_derivative_along_curve	axiom	Covariant derivative along a curve		The operator D/dt on vector fields along a curve gamma, induced by a connection, satisfying the Leibniz rule and agreeing with nabla on extendible fields.
s_geodesic_equation_local_coords	state	Geodesic equation in local coordinates		The system of ODEs d^2 x^k/dt^2 + Gamma^k_{ij} (dx^i/dt)(dx^j/dt) = 0 characterizing geodesics in a coordinate chart.
s_normal_coordinates	state	Normal coordinates		Coordinates on a normal neighborhood of p obtained via exp_p, in which Christoffel symbols vanish and the metric is Euclidean to first order at p.
s_normal_neighborhood	state	Normal neighborhood		An open set U containing p such that exp_p maps a star-shaped open subset of T_pM diffeomorphically onto U.
s_geodesic_ball_and_sphere	state	Geodesic ball and geodesic sphere		The geodesic ball B(p, r) = { q in M : d(p,q) < r } and geodesic sphere S(p, r) = { q in M : d(p,q) = r } centered at p.
s_gauss_lemma_riemannian	theorem	Gauss lemma (Riemannian geometry)		Radial geodesics from p are orthogonal to geodesic spheres centered at p; equivalently (d exp_p)_v preserves inner products with radial directions.
s_minimizing_property_of_geodesics	theorem	Minimizing property of geodesics		Within a normal neighborhood, the radial geodesic from p to q is the unique length-minimizing curve, and any locally minimizing curve is a geodesic.
s_convex_neighborhood	axiom	Convex neighborhood		An open set in a Riemannian manifold such that any two points are joined by a unique minimizing geodesic lying entirely within the set.
s_whitehead_convex_neighborhood_theorem	theorem	Whitehead's theorem on convex neighborhoods		Every point of a Riemannian manifold has a fundamental system of convex neighborhoods.
s_symmetries_of_curvature_tensor	state	Symmetries of the curvature tensor		The algebraic identities of the Riemann tensor: skew-symmetry in the first and second pairs, pair symmetry, and the first Bianchi identity.
s_first_bianchi_identity	theorem	First Bianchi identity		For the Riemann curvature tensor of a torsion-free connection: R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0.
s_sectional_curvature	axiom	Sectional curvature		For a 2-plane sigma in T_pM spanned by X, Y, the number K(sigma) = <R(X,Y)Y, X> / (|X|^2|Y|^2 - <X,Y>^2), generalizing Gaussian curvature to higher dimensions.
s_ricci_curvature	axiom	Ricci curvature		The symmetric (0,2)-tensor Ric(X,Y) = trace(Z -> R(Z,X)Y), obtained by contracting the Riemann tensor, measuring average sectional curvature in a direction.
s_scalar_curvature	axiom	Scalar curvature		The smooth function S = trace_g(Ric) obtained by tracing the Ricci tensor with the metric, giving the total average of sectional curvatures at a point.
s_curvature_determined_by_sectional	theorem	Curvature tensor determined by sectional curvature		The Riemann curvature tensor at a point is completely determined by the sectional curvatures of all 2-planes at that point.
s_ricci_determines_curvature_dim_3	state	Ricci determines curvature in dimension 3		In dimension 3, the Riemann curvature tensor is completely determined by the Ricci tensor because the Weyl tensor vanishes identically.
s_jacobi_fields_as_geodesic_variations	state	Jacobi fields as geodesic variations		A Jacobi field along a geodesic gamma arises as the variational vector field J(t) = (d/ds)|_{s=0} gamma_s(t) of a one-parameter family of geodesics.
s_conjugate_point	axiom	Conjugate point		A point gamma(t_0) along a geodesic gamma is conjugate to gamma(0) if there exists a nonzero Jacobi field vanishing at both t = 0 and t = t_0.
s_multiplicity_of_conjugate_point	state	Multiplicity of a conjugate point		The dimension of the space of Jacobi fields along a geodesic that vanish at both endpoints, equivalently the dimension of the kernel of (d exp_p)_{t_0 v}.
s_conjugate_points_exp_singularity	theorem	Conjugate points as singularities of exp		The point gamma(t_0) is conjugate to gamma(0) = p along gamma if and only if t_0 gamma'(0) is a critical point of the exponential map exp_p.
s_geodesics_cease_minimizing_past_conjugate	theorem	Geodesics cease to minimize past conjugate points		A geodesic segment containing an interior conjugate point is not length-minimizing; the first conjugate point bounds the minimizing interval.
s_isometric_immersion	axiom	Isometric immersion		An immersion f : (M, g) -> (N, h) between Riemannian manifolds satisfying f*h = g, preserving the metric.
s_second_fundamental_form	axiom	Second fundamental form		The symmetric bilinear form II(X, Y) = (nabla_X Y)^perp on an isometrically immersed submanifold, measuring extrinsic curvature via the normal component of the 
s_normal_connection	state	Normal connection		The connection nabla^perp on the normal bundle of a submanifold, defined as the normal component of the ambient connection restricted to normal vector fields.
s_shape_operator	state	Shape operator (Weingarten map)		The self-adjoint linear map A_eta : T_pM -> T_pM defined by <A_eta X, Y> = <II(X,Y), eta>, encoding directional extrinsic curvature.
s_gauss_equation	theorem	Gauss equation		Relates intrinsic and extrinsic curvature of a submanifold: <R^M(X,Y)Z, W> = <R^N(X,Y)Z, W> + <II(X,W), II(Y,Z)> - <II(X,Z), II(Y,W)>.
s_codazzi_equation	theorem	Codazzi equation (Codazzi-Mainardi)		The normal component of the ambient curvature restricted to a submanifold equals the difference of covariant derivatives of the second fundamental form: (R^N(X,
s_ricci_equation_submanifold	theorem	Ricci equation (normal curvature equation)		Relates normal bundle curvature to ambient curvature and shape operators: <R^perp(X,Y)eta, xi> = <R^N(X,Y)eta, xi> + <[A_eta, A_xi]X, Y>.
s_totally_geodesic_submanifold	state	Totally geodesic submanifold		A submanifold whose second fundamental form vanishes identically, equivalently every geodesic of the submanifold is a geodesic of the ambient manifold.
s_minimal_submanifold	state	Minimal submanifold		A submanifold whose mean curvature vector vanishes identically, equivalently a critical point of the volume functional for compactly supported variations.
s_geodesic_completeness	axiom	Geodesic completeness		A Riemannian manifold is geodesically complete if every geodesic extends to all of R, equivalently exp_p is defined on all of T_pM for every p.
s_cut_point	axiom	Cut point		The point gamma(t_0) along a geodesic from p beyond which gamma ceases to minimize distance from p, either due to a shorter path or a conjugate point.
s_model_spaces	state	Model spaces S^n, R^n, H^n		The simply connected complete Riemannian manifolds of constant sectional curvature: the n-sphere (K > 0), Euclidean space (K = 0), and hyperbolic space (K < 0).
s_hyperbolic_space	state	Hyperbolic space H^n		The unique simply connected complete Riemannian n-manifold of constant sectional curvature -1, realizable via the hyperboloid, Poincare disk, or upper half-spac
s_cartan_theorem_space_forms	theorem	Cartan's theorem on space forms		Two simply connected complete Riemannian manifolds of the same dimension and same constant sectional curvature are isometric.
s_classification_of_space_forms	state	Classification of space forms		Every complete Riemannian manifold of constant sectional curvature k is isometric to M_k^n / Gamma where M_k^n is the simply connected model space and Gamma is 
s_energy_functional	axiom	Energy functional		The functional E(gamma) = (1/2) integral_a^b |gamma'(t)|^2 dt on piecewise smooth curves, whose critical points are geodesics parametrized proportionally to arc
s_first_variation_of_energy	theorem	First variation of energy		For a variation gamma_s of a curve, dE/ds|_{s=0} = -integral <V, nabla_{gamma'} gamma'> dt + boundary terms, showing critical points of energy are geodesics.
s_first_variation_of_arc_length	theorem	First variation of arc length		The derivative dL/ds|_{s=0} for a variation of a curve, showing critical points of arc length among fixed-endpoint curves are geodesics.
s_second_variation_of_energy	theorem	Second variation of energy		For a proper variation of a geodesic, d^2E/ds^2|_{s=0} = integral_0^l (|V'|^2 - <R(gamma', V)gamma', V>) dt, relating stability of geodesics to curvature.
s_index_form	axiom	Index form		The symmetric bilinear form I(V, W) = integral_0^l (<V', W'> - <R(gamma', V)gamma', W>) dt on vector fields along a geodesic, encoding the second variation of e
s_synge_lemma	theorem	Synge's lemma		On a manifold with positive sectional curvature, a closed geodesic admits a length-decreasing variation via parallel transport whenever orientation is preserved
s_berger_lemma	theorem	Berger's lemma		On a complete Riemannian manifold, if a minimizing geodesic from p to q realizes d(p, q) then its tangent at q makes a controlled angle with other minimizing di
s_klingenberg_injectivity_radius_estimate	theorem	Klingenberg's injectivity radius estimate		On a compact simply connected Riemannian manifold with sectional curvature 0 < K <= K_max, the injectivity radius satisfies inj(M) >= pi / sqrt(K_max).
s_injectivity_radius	state	Injectivity radius		The largest radius r such that exp_p is a diffeomorphism on B(0, r) in T_pM; globally inj(M) = inf_p inj(p) equals the distance from p to its cut locus.
s_toponogov_hinge_version	state	Toponogov comparison (hinge version)		If K >= k and a geodesic hinge in M has sides a, b with angle alpha, then the opposite side length is at most the corresponding length in the model space of cur
s_index_of_geodesic	axiom	Index of a geodesic		The maximum dimension of a subspace on which the index form I is negative definite, counting independent directions that decrease energy along the geodesic.
s_nullity_of_geodesic	state	Nullity of a geodesic		The dimension of the null space of the index form I, equal to the multiplicity of the terminal point as a conjugate point.
s_morse_index_theorem	theorem	Morse index theorem		The index of a geodesic equals the total number of interior conjugate points counted with multiplicity.
s_byers_theorem	theorem	Byers' theorem		Every solvable subgroup of the fundamental group of a compact Riemannian manifold with strictly negative sectional curvature is infinite cyclic.
s_closed_geodesic_in_free_homotopy_class	theorem	Closed geodesic in free homotopy class		On a compact Riemannian manifold, every nontrivial free homotopy class of closed curves contains a closed geodesic of minimal length.
s_geodesic_variation	state	Geodesic variation		A smooth map f : (-epsilon, epsilon) x [a,b] -> M such that each curve t -> f(s,t) is a geodesic, used to derive the Jacobi equation.
s_energy_vs_length_minimizers	state	Energy vs length minimizers		By Cauchy-Schwarz, E(gamma) >= L(gamma)^2/(b-a) with equality iff gamma has constant speed, so among constant-speed curves, energy and length minimizers coincid
s_laplacian_comparison	state	Laplacian comparison theorem		Under Ric >= (n-1)k the Laplacian of the distance function satisfies Delta r <= Delta_k r where Delta_k r is the Laplacian in the model space of curvature k.
s_geodesic_convexity	state	Geodesic convexity		A subset C of a Riemannian manifold is geodesically convex if any two points in C are joined by a unique minimizing geodesic lying entirely in C.
t_normal_coordinate_computation	technique	Normal coordinate computation		Evaluating geometric expressions in normal coordinates where Christoffel symbols vanish and the metric is Euclidean to first order, simplifying tensor computati
t_sturm_liouville_comparison	technique	Sturm-Liouville comparison for Jacobi fields		Applying Sturm comparison theorems to the scalar ODE satisfied by Jacobi field norms to obtain bounds via curvature bounds, the key analytic tool behind Rauch c
t_covering_space_negative_curvature	technique	Covering space argument for negative curvature		Lifting geodesics to the universal cover (diffeomorphic to R^n by Cartan-Hadamard) to transfer geometric rigidity to algebraic constraints on the fundamental gr
t_comparison_of_jacobi_fields	technique	Comparison of Jacobi fields		Comparing Jacobi field magnitudes on a manifold with curvature bounds to those on a constant-curvature model space to deduce distance, volume, and topology esti
s_smoothly_compatible_charts	axiom	Smoothly compatible charts		Two charts (U,φ) and (V,ψ) such that the transition map ψ ∘ φ⁻¹: φ(U ∩ V) → ψ(U ∩ V) is a C^∞ diffeomorphism.
s_smooth_atlas	axiom	Smooth atlas		An atlas in which every pair of charts is smoothly compatible.
s_maximal_smooth_atlas	axiom	Maximal smooth atlas (smooth structure)		A smooth atlas not properly contained in any larger smooth atlas; the unique maximal atlas containing a given smooth atlas.
s_interior_and_boundary_of_manifold_with_boundary	state	Interior and boundary of a manifold with boundary		Int M consists of points with neighborhoods homeomorphic to open subsets of R^n; ∂M consists of points mapped to ∂H^n under some chart.
s_product_manifold	state	Product manifold		The Cartesian product M × N of smooth manifolds carries a unique smooth structure making the product charts smooth.
s_smooth_manifold_chart_lemma	theorem	Smooth manifold chart lemma		If a set M has a countable collection of injections into R^n whose transition maps are smooth and images are open, then M has a unique smooth manifold structure
s_invariance_of_the_boundary	theorem	Invariance of the boundary		For a topological manifold with boundary, ∂M and Int M are disjoint with M = Int M ∪ ∂M; no point is simultaneously interior and boundary.
s_smooth_function_ring	state	Smooth function ring C^∞(M)		The commutative R-algebra of all smooth real-valued functions on M with pointwise operations.
s_smooth_extension_lemma	theorem	Smooth extension lemma		A smooth function on a closed subset A of a smooth manifold (in the sense of local smooth extensions) extends to a smooth function on all of M.
s_exhaustion_function	state	Exhaustion function		A smooth function f: M → R whose sublevel sets {f ≤ c} are compact for every c; every smooth manifold admits one.
s_derivation_at_a_point	axiom	Derivation at a point (tangent vector)		A tangent vector at p ∈ M defined as a linear map v: C^∞(M) → R satisfying the Leibniz rule v(fg) = f(p)v(g) + g(p)v(f).
s_coordinate_basis_for_tangent_space	state	Coordinate basis for tangent space		Given a chart (U,φ) with coordinates (x^1,...,x^n), the partial derivative operators ∂/∂x^i|_p form a basis for T_pM.
s_differential_of_a_smooth_map	axiom	Differential of a smooth map		The differential dF_p: T_pM → T_{F(p)}N defined by (dF_p(v))(f) = v(f ∘ F) for v ∈ T_pM and f ∈ C^∞(N).
s_chain_rule_for_differentials	theorem	Chain rule for differentials		For smooth maps F: M → N and G: N → P, the differential satisfies d(G ∘ F)_p = dG_{F(p)} ∘ dF_p.
s_natural_coordinates_on_TM	state	Natural coordinates on TM		A chart (U,φ) for M induces coordinates (x^1,...,x^n, v^1,...,v^n) on π⁻¹(U) ⊆ TM via v = v^i ∂/∂x^i|_p.
s_velocity_of_a_curve	state	Velocity of a curve		For a smooth curve γ: (a,b) → M, the velocity vector γ'(t) ∈ T_{γ(t)}M is defined as dγ_t(d/dt).
s_smooth_map_of_constant_rank	axiom	Smooth map of constant rank		A smooth map F: M → N has constant rank r if rank(dF_p) = r for all p ∈ M.
s_inverse_function_theorem_manifolds	theorem	Inverse function theorem for manifolds		A smooth map F: M → N is a local diffeomorphism if and only if dF_p is a linear isomorphism for every p ∈ M.
s_global_rank_theorem	theorem	Global rank theorem		A smooth map of constant rank that is surjective is a smooth submersion; one that is injective is a smooth immersion.
s_local_immersion_theorem	theorem	Local immersion theorem		If dF_p is injective at p, then F is a smooth immersion on some neighborhood of p, and in suitable coordinates has the canonical inclusion form.
s_local_submersion_theorem	theorem	Local submersion theorem		If dF_p is surjective at p, then F is a smooth submersion on some neighborhood of p, and in suitable coordinates has the canonical projection form.
s_local_section_of_submersion	theorem	Local section of a submersion		Every smooth submersion admits smooth local sections through every point in its image.
s_smooth_covering_map	axiom	Smooth covering map		A smooth surjective map π: E → M such that every point of M has a neighborhood U evenly covered by π (π⁻¹(U) is a disjoint union of open sets each mapped diffeo
s_embedded_submanifold	axiom	Embedded submanifold (regular submanifold)		A subset S ⊆ M that is a smooth manifold in the subspace topology with the inclusion S ↪ M a smooth embedding.
s_slice_chart	axiom	Slice chart (adapted chart)		A chart (U,φ) for M such that S ∩ U is mapped by φ to the intersection of φ(U) with a coordinate subspace R^k × {0} ⊆ R^n.
s_local_slice_criterion	theorem	Local slice criterion for embedded submanifolds		A subset S ⊆ M is an embedded k-submanifold if and only if around each point of S there exists a slice chart.
s_immersed_submanifold	axiom	Immersed submanifold		A subset S ⊆ M with a smooth manifold topology (possibly finer than subspace topology) such that the inclusion is a smooth immersion.
s_level_set	state	Level set		For F: M → N smooth and c ∈ N, the level set F⁻¹(c) = {p ∈ M : F(p) = c}.
s_regular_value	axiom	Regular value		A point c ∈ N is a regular value of F: M → N if dF_p is surjective for every p ∈ F⁻¹(c).
s_tangent_space_to_level_set	theorem	Tangent space to a level set		If S = F⁻¹(c) for a regular value c, then T_pS = ker(dF_p) for every p ∈ S.
s_set_of_measure_zero_in_manifold	axiom	Set of measure zero in a manifold		A subset A ⊆ M has measure zero if φ(A ∩ U) has measure zero in R^n for every smooth chart (U,φ).
s_critical_point_manifold	axiom	Critical point (manifold context)		A point p ∈ M where dF_p: T_pM → T_{F(p)}N is not surjective; the image F(p) is a critical value.
s_parametric_transversality_theorem	theorem	Parametric transversality theorem		If F: M × S → N is transverse to a submanifold Z ⊆ N, then for almost every s ∈ S the restricted map F(·,s) is transverse to Z.
s_lie_group_isomorphism	axiom	Lie group isomorphism		A Lie group homomorphism that is also a diffeomorphism; equivalently a bijective Lie group homomorphism (the inverse is automatically smooth).
s_embedded_lie_subgroup	state	Embedded Lie subgroup		A Lie subgroup that is an embedded submanifold, equivalently a closed subgroup of a Lie group.
s_left_right_translation	state	Left and right translation		For g ∈ G, the diffeomorphism L_g: G → G defined by L_g(h) = gh (resp. R_g(h) = hg).
s_smooth_group_action	axiom	Smooth group action		A smooth map θ: G × M → M satisfying θ(e,p) = p and θ(g,θ(h,p)) = θ(gh,p) for all g,h ∈ G, p ∈ M.
s_orbit_and_stabilizer	state	Orbit and stabilizer (isotropy group)		The orbit O_p = {g · p : g ∈ G}; the stabilizer G_p = {g ∈ G : g · p = p}, a closed Lie subgroup of G.
s_free_action	axiom	Free action		A group action such that g · p = p implies g = e, equivalently all stabilizers are trivial.
s_proper_action	axiom	Proper action		A continuous group action such that the map (g,p) ↦ (g·p, p) is proper (preimages of compact sets are compact).
s_quotient_manifold_theorem	theorem	Quotient manifold theorem (for group actions)		If a Lie group acts smoothly, freely, and properly on a smooth manifold M, then M/G has a unique smooth manifold structure making the quotient map a smooth subm
s_equivariant_map	axiom	Equivariant map		A map F: M → N between G-spaces satisfying F(g · p) = g · F(p) for all g ∈ G and p ∈ M.
s_covering_group_theorem	theorem	Covering group theorem		If G is a connected Lie group, its universal covering space admits a unique Lie group structure making the covering map a Lie group homomorphism.
s_f_related_vector_fields	axiom	F-related vector fields		Vector fields X on M and Y on N are F-related if dF_p(X_p) = Y_{F(p)} for all p ∈ M.
s_naturality_of_lie_bracket	theorem	Naturality of the Lie bracket		If X₁, X₂ are F-related to Y₁, Y₂ respectively, then [X₁,X₂] is F-related to [Y₁,Y₂].
s_lie_algebra_homomorphism_from_lie_group	theorem	Lie algebra homomorphism induced by Lie group homomorphism		A Lie group homomorphism F: G → H induces a Lie algebra homomorphism dF_e: g → h that preserves brackets.
s_integral_curve	axiom	Integral curve		A smooth curve γ: J → M such that γ'(t) = X_{γ(t)} for all t ∈ J.
s_existence_uniqueness_integral_curves	theorem	Existence and uniqueness of integral curves		For every smooth vector field X on M and every p ∈ M, there exists a unique maximal integral curve of X starting at p.
s_flow_domain	state	Flow domain		The maximal flow domain D ⊆ R × M of a vector field X, an open subset containing {0} × M on which the flow is defined.
s_complete_vector_field	axiom	Complete vector field		A vector field whose flow is defined for all t ∈ R, i.e., the flow domain is all of R × M.
s_flowout_theorem	theorem	Flowout theorem		If S is an embedded submanifold and X a vector field nowhere tangent to S, the flow starting from S maps a neighborhood of {0} × S in R × S diffeomorphically on
s_lie_derivative_equals_lie_bracket	theorem	Lie derivative equals Lie bracket		For smooth vector fields X and Y, the Lie derivative L_X Y equals the Lie bracket [X,Y].
s_commuting_vector_fields_and_flows	theorem	Commuting vector fields and flows		Two smooth vector fields X,Y commute ([X,Y] = 0) if and only if their flows commute wherever both sides are defined.
s_canonical_form_nonvanishing_vector_field	theorem	Canonical form for nonvanishing vector fields		If X_p ≠ 0, there exist smooth coordinates near p in which X = ∂/∂x^1.
s_local_trivialization	axiom	Local trivialization		A fiber-preserving diffeomorphism Φ: π⁻¹(U) → U × R^k restricting to a linear isomorphism on each fiber.
s_transition_function_vector_bundle	state	Transition function of a vector bundle		The map τ_{αβ}: U_α ∩ U_β → GL(k,R) describing the change of fiber coordinates between overlapping trivializations, satisfying the cocycle condition.
s_section_of_vector_bundle	axiom	Section of a vector bundle		A smooth map σ: M → E such that π ∘ σ = id_M; a local section is defined on an open subset.
s_local_frame_vector_bundle	state	Local frame for a vector bundle		An ordered k-tuple (σ_1,...,σ_k) of local sections over U such that (σ_1(p),...,σ_k(p)) is a basis for E_p at every p ∈ U.
s_vector_bundle_construction_theorem	theorem	Vector bundle construction theorem		Given an open cover and smooth transition functions satisfying the cocycle condition, there is a smooth vector bundle realizing these transitions, unique up to 
s_bundle_map	axiom	Bundle map (bundle homomorphism)		A pair of smooth maps (F,f) with F: E → E' and f: M → M' such that π' ∘ F = f ∘ π and F is linear on each fiber.
s_subbundle	axiom	Subbundle		A smooth vector bundle D ⊆ E whose total space is an embedded submanifold of E and whose inclusion is a bundle map with injective fiber maps.
s_covector	axiom	Covector (cotangent vector)		An element of the dual space T_p*M = (T_pM)*; a linear functional ω: T_pM → R.
s_coordinate_covector_fields	state	Coordinate covector fields (dx^i)		Given local coordinates (x^i), the covector fields dx^i defined by dx^i(∂/∂x^j) = δ^i_j, forming the dual basis to coordinate vector fields.
s_covector_field	axiom	Covector field (1-form)		A smooth section ω: M → T*M of the cotangent bundle; a smooth assignment p ↦ ω_p ∈ T_p*M.
s_line_integral_of_covector_field	state	Line integral of a covector field		For a covector field ω and a piecewise smooth curve γ: [a,b] → M, the integral ∫_γ ω = ∫_a^b ω_{γ(t)}(γ'(t)) dt.
s_conservative_covector_field_characterization	theorem	Conservative covector field characterization		A smooth covector field on a smooth manifold is exact if and only if its line integral around every piecewise smooth closed curve is zero.
s_poincare_lemma_covector_fields	theorem	Poincare lemma for covector fields		On a simply connected smooth manifold, every closed covector field is exact.
s_multilinear_map_k_tensor	axiom	Multilinear map (k-tensor)		A multilinear map F: V^k → R; the space of covariant k-tensors on V is T^k(V*) = (V*)^{⊗k}.
s_mixed_tensor	axiom	Mixed tensor		An element of V^{⊗k} ⊗ (V*)^{⊗l}, a multilinear map (V*)^k × V^l → R with k contravariant and l covariant indices.
s_symmetric_tensor	axiom	Symmetric tensor		A covariant k-tensor α invariant under all permutations of its arguments: α(v_{σ(1)},...,v_{σ(k)}) = α(v_1,...,v_k) for every σ ∈ S_k.
s_alternating_tensor	axiom	Alternating tensor		A covariant k-tensor α such that α(v_{σ(1)},...,v_{σ(k)}) = (sgn σ) α(v_1,...,v_k) for every permutation σ; the space is Λ^k(V*).
s_symmetrization_alternation_operators	state	Symmetrization and alternation operators		Sym(α) = (1/k!) Σ_σ α ∘ σ averages over all permutations; Alt(α) = (1/k!) Σ_σ (sgn σ) α ∘ σ antisymmetrizes.
s_pullback_of_tensor_fields	state	Pullback of tensor fields		The pullback F*A defined by (F*A)_p(v_1,...,v_k) = A_{F(p)}(dF_p v_1,...,dF_p v_k).
s_tensor_characterization_lemma	theorem	Tensor characterization lemma		A map F: X(M)^k × Ω¹(M)^l → C^∞(M) is induced by a tensor field if and only if it is C^∞(M)-multilinear.
s_existence_of_riemannian_metrics	theorem	Existence of Riemannian metrics		Every smooth manifold admits a Riemannian metric, constructed via partitions of unity.
s_musical_isomorphisms	state	Musical isomorphisms (flat and sharp)		The Riemannian metric induces ♭: TM → T*M by v ↦ g(v,·) with inverse ♯: T*M → TM.
s_pseudo_riemannian_metric	axiom	Pseudo-Riemannian metric		A smooth symmetric covariant 2-tensor field that is nondegenerate but not necessarily positive definite at each point; the signature (r,s) counts positive and n
s_pseudo_riemannian_manifold	state	Pseudo-Riemannian manifold		A smooth manifold equipped with a pseudo-Riemannian metric; Lorentzian manifolds are the special case of signature (n−1,1).
s_interior_multiplication	axiom	Interior multiplication (contraction)		The interior product ι_X: Ω^k(M) → Ω^{k−1}(M) defined by (ι_X ω)(Y_1,...,Y_{k−1}) = ω(X,Y_1,...,Y_{k−1}), an antiderivation of degree −1.
s_invariant_formula_exterior_derivative	theorem	Invariant formula for the exterior derivative		For a k-form ω and vector fields X_0,...,X_k: dω(X_0,...,X_k) = Σ_i (−1)^i X_i(ω(...X̂_i...)) + Σ_{i<j} (−1)^{i+j} ω([X_i,X_j],...X̂_i...X̂_j...).
s_orientation_of_a_vector_space	axiom	Orientation of a vector space		An equivalence class of ordered bases, where two bases are equivalent if the change-of-basis matrix has positive determinant.
s_oriented_double_cover	state	Oriented double cover		Every connected nonorientable smooth manifold has a connected orientable double cover M̃ with a 2-sheeted smooth covering map π: M̃ → M.
s_riemannian_volume_form	state	Riemannian volume form		The unique n-form dV_g such that dV_g(E_1,...,E_n) = 1 for every oriented orthonormal frame; in coordinates dV_g = √(det g_{ij}) dx^1 ∧ ··· ∧ dx^n.
s_integration_of_forms_on_manifolds	axiom	Integration of differential forms on oriented manifolds		The integral ∫_M ω of a compactly supported n-form on an oriented n-manifold, defined via oriented atlas and partition of unity, independent of choices.
s_induced_boundary_orientation	state	Induced boundary orientation		The boundary ∂M inherits an orientation by the convention that an outward-pointing vector followed by an oriented basis for T_p(∂M) gives an oriented basis for 
s_manifold_with_corners	axiom	Manifold with corners		A second-countable Hausdorff space locally modeled on the corner spaces R^n_k = {x ∈ R^n : x^{n−k+1} ≥ 0,...,x^n ≥ 0}, with smooth transition maps.
s_stokes_for_manifolds_with_corners	theorem	Stokes's theorem for manifolds with corners		Stokes's theorem extends to oriented manifolds with corners: ∫_M dω = ∫_{∂M} ω, where boundary faces are smooth manifolds with corners of one lower dimension.
s_induced_map_on_cohomology	state	Induced map on de Rham cohomology		A smooth map F induces a pullback F*: H^k_{dR}(N) → H^k_{dR}(M), functorial: (G∘F)* = F*∘G* and id* = id.
s_homotopy_invariance_de_rham	theorem	Homotopy invariance of de Rham cohomology		If F,G: M → N are smoothly homotopic, then F* = G* on H^k_{dR}; homotopy equivalent manifolds have isomorphic de Rham cohomology.
t_homotopy_operator	technique	Homotopy operator (cochain homotopy)		Construction of a linear map h satisfying h ∘ d + d ∘ h = i₁* − i₀*, proving homotopy invariance of de Rham cohomology.
s_poincare_lemma	theorem	Poincare lemma		Every closed differential form on a contractible manifold is exact: H^k_{dR}(R^n) = 0 for k ≥ 1.
s_cohomology_of_spheres	theorem	Cohomology of spheres		H^k_{dR}(S^n) ≅ R if k = 0 or k = n, and H^k_{dR}(S^n) = 0 otherwise.
s_smooth_singular_homology	state	Smooth singular homology		The smooth singular homology groups H_k^∞(M; R) defined using smooth singular simplices; naturally isomorphic to ordinary singular homology H_k(M; R).
s_de_rham_homomorphism	state	De Rham homomorphism		The map I: H^k_{dR}(M) → H^k(M; R) defined by [ω] ↦ [σ ↦ ∫_σ ω], integrating closed k-forms over smooth singular k-cycles.
s_distribution_tangent_subbundle	axiom	Distribution (tangent subbundle)		A smooth subbundle D ⊆ TM of rank k, assigning a k-dimensional subspace D_p ⊆ T_pM to each point p smoothly.
s_integral_manifold	axiom	Integral manifold of a distribution		A connected immersed submanifold N ⊆ M such that T_pN = D_p for every p ∈ N.
s_integrable_distribution	axiom	Integrable distribution		A distribution that admits an integral manifold through every point of M.
s_differential_form_criterion_involutivity	theorem	Differential form criterion for involutivity		A rank-k distribution D = ker{ω^1,...,ω^{n-k}} is involutive if and only if each dω^i ∧ ω^1 ∧ ··· ∧ ω^{n-k} = 0.
s_lie_subalgebra_subgroup_correspondence	theorem	Lie subalgebra-Lie subgroup correspondence		Every Lie subalgebra h ⊆ g = Lie(G) is the Lie algebra of a unique connected Lie subgroup H of G, obtained as a maximal integral manifold of the left-invariant 
s_exponential_map_lie_group	state	Exponential map of a Lie group		The map exp: g → G defined by exp(X) = γ_X(1) where γ_X is the one-parameter subgroup tangent to X at e; a smooth map with d(exp)_0 = id_g.
s_continuous_homomorphisms_are_smooth	theorem	Continuous homomorphisms between Lie groups are smooth		Every continuous group homomorphism between Lie groups is automatically smooth.
s_infinitesimal_generator_group_action	state	Infinitesimal generator of a group action		The vector field X̂ on M defined by X̂_p = d/dt|_{t=0} θ(exp(tX), p), generating the flow t ↦ θ(exp(tX), ·).
s_adjoint_representation_lie_algebra	state	Adjoint representation of the Lie algebra		The representation ad: g → gl(g) defined by ad(X)(Y) = [X,Y]; the differential of Ad at the identity.
s_homogeneous_space	state	Homogeneous space		A smooth manifold G/H where G is a Lie group and H a closed subgroup, with natural transitive G-action and smooth structure from the quotient manifold theorem.
s_covering_manifold	state	Covering manifold		A smooth covering map π: M̃ → M where M̃ inherits a unique smooth structure from M making π a local diffeomorphism.
s_symplectic_form_on_vector_space	axiom	Symplectic form on a vector space		A nondegenerate skew-symmetric bilinear form ω: V × V → R; the dimension of V must be even.
s_symplectomorphism	axiom	Symplectomorphism		A diffeomorphism F: (M₁,ω₁) → (M₂,ω₂) satisfying F*ω₂ = ω₁.
s_contact_structure	axiom	Contact structure		A smooth hyperplane distribution ξ = ker α on a (2n+1)-dimensional manifold, where α ∧ (dα)^n ≠ 0 everywhere (maximal non-integrability).
s_contact_manifold	axiom	Contact manifold		An odd-dimensional smooth manifold equipped with a contact structure.
s_cotangent_bundle_symplectic	state	Cotangent bundle as symplectic manifold		T*M carries a canonical symplectic structure ω = −dλ where λ is the tautological 1-form defined by λ_{(p,ξ)}(v) = ξ(dπ(v)).
t_chart_computation	technique	Chart computation technique		Verifying global smooth-manifold properties by working in local coordinate charts and checking expressions transform correctly under transition maps.
t_rank_theorem_technique	technique	Rank theorem technique		Using the constant rank theorem to put smooth maps into canonical form in suitable coordinates, simplifying local analysis of submersions and immersions.
t_flow_box_straightening	technique	Flow box / straightening technique		Using the flow of a nonvanishing vector field to construct coordinates in which the field takes the standard form ∂/∂x^1.
t_mayer_vietoris_induction	technique	Mayer-Vietoris induction		Computing de Rham cohomology by decomposing M = U ∪ V and using the Mayer-Vietoris long exact sequence recursively.
t_exhaustion_by_compact_sets	technique	Exhaustion by compact sets technique		Constructing global objects on noncompact manifolds by working on an increasing sequence of compact subsets covering M.
t_gluing_via_partition_of_unity	technique	Smooth maps into Euclidean space (gluing lemma)		Using partitions of unity to construct global smooth maps and functions by patching together locally defined smooth data.
s_space_form	axiom	Space form		A complete simply connected Riemannian manifold of constant sectional curvature k: Euclidean space (k=0), sphere (k>0), or hyperbolic space (k<0).
s_principal_curvatures	state	Principal curvatures		The eigenvalues κ₁, κ₂ of the shape operator (Weingarten map) at a point of a surface in ℝ³, giving the extremal normal curvatures.
s_mean_curvature_surface	state	Mean curvature of a surface		The arithmetic mean H = (κ₁ + κ₂)/2 of the principal curvatures at a point, equivalently half the trace of the shape operator.
s_riemannian_submanifold	axiom	Riemannian submanifold		An embedded or immersed submanifold of a Riemannian manifold equipped with the pullback (induced) metric from the ambient space.
s_riemannian_distance_induces_topology	theorem	Riemannian distance induces the manifold topology		The metric topology on a connected Riemannian manifold induced by the Riemannian distance function d_g coincides with the original manifold topology.
s_riemannian_density	state	Riemannian density		The canonical density |dV_g| associated to a Riemannian metric, allowing integration on possibly non-orientable manifolds without choosing an orientation.
s_euclidean_space_riemannian	axiom	Euclidean space (Riemannian)		The manifold ℝⁿ equipped with the standard flat Riemannian metric g_E = δᵢⱼ dxⁱ dxʲ, the model space of zero curvature.
s_round_metric_sphere	axiom	Round metric on Sⁿ		The standard Riemannian metric on the n-sphere induced by the embedding Sⁿ ⊂ ℝⁿ⁺¹, having constant sectional curvature +1.
s_poincare_ball_model	state	Poincaré ball model		The open unit ball Bⁿ ⊂ ℝⁿ with the conformally flat metric g_P = 4(1 - |x|²)⁻² g_E, a model of hyperbolic space Hⁿ.
s_poincare_half_space_model	state	Poincaré half-space model		The upper half-space {x ∈ ℝⁿ : xₙ > 0} with metric g_H = xₙ⁻² (dx₁² + ⋯ + dxₙ²), a model of hyperbolic space Hⁿ.
s_homogeneous_riemannian_manifold	axiom	Homogeneous Riemannian manifold		A Riemannian manifold whose isometry group acts transitively, so the geometry is the same at every point.
s_isotropic_riemannian_manifold	axiom	Isotropic Riemannian manifold		A Riemannian manifold such that for every point p and every unit tangent vector at p, there exists an isometry fixing p and mapping the vector to any other unit
s_frame_homogeneous_riemannian_manifold	axiom	Frame-homogeneous Riemannian manifold		A Riemannian manifold whose isometry group acts transitively on the orthonormal frame bundle, the strongest homogeneity condition.
s_classification_frame_homogeneous	theorem	Classification of frame-homogeneous spaces		A connected frame-homogeneous Riemannian manifold is isometric to a Euclidean space, a sphere, a real projective space, or a hyperbolic space.
s_riemannian_covering	state	Riemannian covering		A covering space of a Riemannian manifold equipped with the unique pullback metric making the covering map a local isometry.
s_connection_in_local_frame	state	Connection in a local frame		The connection 1-forms ωⁱⱼ defined by ∇_X Eⱼ = ωⁱⱼ(X) Eᵢ, representing the connection relative to a chosen local frame of the vector bundle.
s_parallel_transport_determines_connection	theorem	Parallel transport characterizes connection		A connection on a vector bundle is uniquely determined by its parallel transport maps along curves; conversely any smooth system of parallel transport maps defi
s_metric_connection	axiom	Metric connection		A connection ∇ on a Riemannian manifold satisfying ∇g = 0, equivalently parallel transport preserves inner products.
s_symmetric_connection	axiom	Symmetric (torsion-free) connection		A connection with vanishing torsion tensor: ∇_X Y - ∇_Y X = [X, Y] for all vector fields X, Y.
s_christoffel_symbols_levi_civita	state	Christoffel symbols of the Levi-Civita connection		The explicit formula Γᵏᵢⱼ = ½gᵏˡ(∂ᵢgⱼˡ + ∂ⱼgᵢˡ - ∂ˡgᵢⱼ) for the Christoffel symbols of the Levi-Civita connection in local coordinates.
s_geodesics_constant_speed	theorem	Geodesics have constant speed		A geodesic γ satisfying ∇_{γ'} γ' = 0 has constant speed |γ'(t)|_g, since d/dt <γ', γ'> = 2<∇_{γ'} γ', γ'> = 0.
s_exp_local_diffeomorphism	theorem	Exponential map is a local diffeomorphism		For each p ∈ M, the exponential map exp_p is a diffeomorphism from a neighborhood of 0 in T_pM onto a neighborhood of p in M, since d(exp_p)_0 = id.
s_fermi_coordinates	state	Fermi coordinates		Coordinates adapted to a geodesic γ constructed via the exponential map of the normal bundle, in which Christoffel symbols vanish along γ and the metric is Eucl
s_riemannian_geodesics_minimize_length	theorem	Riemannian geodesics as locally length-minimizing curves		A curve in a Riemannian manifold is locally length-minimizing if and only if it is a geodesic (up to reparametrization), and sufficiently short geodesic segment
s_variation_of_curve	axiom	Variation of a curve		A smooth one-parameter family of curves F(s, t) with F(0, t) = γ(t), used to derive the first and second variation formulas for length and energy.
s_uniformly_normal_neighborhood	state	Uniformly normal neighborhood		An open set U such that U ⊂ B(p, r) for some normal ball B(p, r) for every p ∈ U, ensuring uniform control of the exponential map.
s_distance_function_from_point	state	Distance function from a point		The function r_p(q) = d(p, q) giving the Riemannian distance from a fixed point p; smooth on M \ ({p} ∪ Cut(p)) with |grad r_p| = 1.
s_cut_locus_measure_zero	theorem	Cut locus has measure zero		The cut locus Cut(p) of any point p in a complete Riemannian manifold has measure zero with respect to the Riemannian volume measure.
s_riemann_curvature_endomorphism	state	Riemann curvature endomorphism		The (1,3)-tensor R(X,Y)Z = ∇_X∇_Y Z - ∇_Y∇_X Z - ∇_{[X,Y]} Z measuring the failure of second covariant derivatives to commute.
s_second_bianchi_identity	theorem	Second Bianchi identity		The differential Bianchi identity: (∇_X R)(Y,Z) + (∇_Y R)(Z,X) + (∇_Z R)(X,Y) = 0 for the Riemann curvature tensor of the Levi-Civita connection.
s_schouten_tensor	state	Schouten tensor		The tensor A = (1/(n-2))(Ric - S/(2(n-1)) g) appearing in the decomposition of the Riemann tensor into Weyl, Ricci, and scalar parts.
s_curvature_tensor_decomposition	theorem	Decomposition of the Riemann curvature tensor		The Riemann curvature tensor decomposes orthogonally as Rm = W + A ∧ g where W is the Weyl tensor and A is the Schouten tensor, separating conformal, Ricci, and
s_curvature_of_constant_curvature_spaces	theorem	Curvature of constant-curvature spaces		A Riemannian manifold has constant sectional curvature k if and only if Rm = k/2 (g ∧ g), i.e., R(X,Y)Z = k(<Y,Z>X - <X,Z>Y).
s_contracted_second_bianchi_identity	theorem	Contracted second Bianchi identity		Contracting the second Bianchi identity yields div(Ric) = ½ dS, equivalently div(Ric - ½Sg) = 0, showing the Einstein tensor is divergence-free.
s_curvature_of_product_manifolds	theorem	Curvature of product manifolds		For a Riemannian product (M₁ × M₂, g₁ ⊕ g₂), the curvature tensor splits: mixed sectional curvatures vanish and R_{M₁×M₂} = R_{M₁} + R_{M₂} in the appropriate s
s_gauss_formula_submanifold	theorem	Gauss formula		For an isometric immersion M ↪ N, the ambient covariant derivative decomposes as ∇̃_X Y = ∇_X Y + II(X, Y) into tangential and normal components.
s_weingarten_equation	theorem	Weingarten equation		For a normal vector field η along a submanifold, ∇̃_X η decomposes as -A_η X + ∇⊥_X η, relating the shape operator A_η to the normal connection.
s_curvature_of_hypersurfaces	theorem	Curvature of hypersurfaces		For a hypersurface with unit normal N and shape operator A, the Gauss equation gives K_M(X,Y) = K_N(X,Y) + <AX,X><AY,Y> - <AX,Y>², relating intrinsic curvature 
s_fundamental_theorem_hypersurface_theory	theorem	Fundamental theorem of hypersurface theory		A simply connected domain in ℝⁿ with a Riemannian metric and symmetric (0,2)-tensor satisfying the Gauss and Codazzi-Mainardi equations can be isometrically imm
s_gauss_bonnet_local	theorem	Local Gauss-Bonnet theorem		For a geodesic polygon P on an oriented Riemannian surface, ∫∫_P K dA + ∫_{∂P} κ_g ds + Σ θᵢ = 2π, relating total Gaussian curvature, geodesic curvature of the 
s_gauss_bonnet_with_boundary	theorem	Gauss-Bonnet theorem with boundary		For a compact oriented Riemannian 2-manifold M with boundary, ∫_M K dA + ∫_{∂M} κ_g ds = 2π χ(M), extending Gauss-Bonnet to manifolds with boundary.
s_euler_characteristic	axiom	Euler characteristic		The alternating sum χ(X) = Σ (-1)^k rank H_k(X) of Betti numbers, a topological invariant equal to V - E + F for polyhedra.
s_jacobi_fields_existence_uniqueness	theorem	Existence and uniqueness of Jacobi fields		Given initial conditions J(0) and J'(0) along a geodesic, there exists a unique Jacobi field satisfying the Jacobi equation; the space of Jacobi fields along a 
s_jacobi_fields_constant_curvature	theorem	Jacobi fields on constant-curvature spaces		On a space of constant sectional curvature K, normal Jacobi fields with J(0) = 0 have the form J(t) = sn_K(t) E(t) where sn_K is sin(√K t)/√K, t, or sinh(√|K| t
s_no_conjugate_before_cut	theorem	No conjugate points before cut point		Along a geodesic from p, the first conjugate point (if it exists) occurs at or after the cut point; there are no conjugate points along a minimizing geodesic se
s_second_variation_of_arc_length	theorem	Second variation of arc length		For a normal variation of a unit-speed geodesic γ with variation field V, d²L/ds²|_{s=0} = ∫₀ˡ (|V'⊥|² - <R(γ',V)γ',V>) dt, relating curvature to length stabili
s_index_form_conjugate_points	theorem	Index form characterization of conjugate points		A geodesic segment has no interior conjugate points if and only if the index form I is positive definite on the space of proper normal vector fields along it.
s_hessian_comparison_theorem	theorem	Hessian comparison theorem		Under a sectional curvature bound K ≤ k (or K ≥ k), the Hessian of the distance function r from a point satisfies Hess(r) ≥ (≤) the corresponding Hessian in the
s_relative_volume_comparison	theorem	Relative volume comparison		Under Ric ≥ (n-1)k, the ratio Vol(B(p,r))/Vol_k(r) is non-increasing in r, where Vol_k(r) is the volume of a ball of radius r in the space form of curvature k.
s_milnor_theorem_fundamental_group	theorem	Milnor's theorem on fundamental groups		The fundamental group of a compact Riemannian manifold with Ric ≥ 0 has polynomial growth; if Ric > 0 then π₁ is finite.
s_weinstein_theorem	theorem	Weinstein's theorem (Riemannian)		An orientation-preserving isometry of a compact even-dimensional Riemannian manifold with positive sectional curvature has a fixed point.
s_berger_quarter_pinched_sphere_theorem	theorem	Berger's quarter-pinched sphere theorem		A compact simply connected Riemannian n-manifold with ¼ < K ≤ 1 is homeomorphic to Sⁿ; the sharp version (Brendle-Schoen) gives diffeomorphism for ¼-pinching.
s_killing_vector_field	axiom	Killing vector field		A vector field X whose flow consists of isometries, equivalently satisfying the Killing equation ∇_i X_j + ∇_j X_i = 0.
s_killing_fields_lie_algebra	theorem	Killing fields form a Lie algebra		The space of Killing vector fields on a Riemannian manifold is a finite-dimensional Lie algebra under the Lie bracket, isomorphic to the Lie algebra of the isom
s_max_dimension_isometry_group	theorem	Maximum dimension of the isometry group		The isometry group of an n-dimensional Riemannian manifold has dimension at most n(n+1)/2, with equality if and only if the manifold has constant sectional curv
s_cartan_ambrose_hicks_theorem	theorem	Cartan-Ambrose-Hicks theorem		A connection-preserving map between simply connected complete Riemannian manifolds that preserves the curvature tensor along all broken geodesics from a base po
s_local_to_global_isometry_extension	theorem	Local-to-global isometry extension		A local isometry from a complete connected Riemannian manifold into a simply connected Riemannian manifold is a covering map; if both are simply connected, it i
s_flat_riemannian_manifold	axiom	Flat Riemannian manifold		A Riemannian manifold with vanishing Riemann curvature tensor, locally isometric to Euclidean space.
s_geodesic_curvature	state	Geodesic curvature		The tangential component of the acceleration of a unit-speed curve on a surface, measuring deviation from being a geodesic; κ_g = <∇_{γ'}γ', n> where n is the u
s_normal_jacobi_field	state	Normal Jacobi field		The component of a Jacobi field perpendicular to the geodesic, satisfying the Jacobi equation independently; governs infinitesimal distance changes between near
s_tangential_jacobi_field	state	Tangential Jacobi field		The component of a Jacobi field tangent to the geodesic, always of the form (a + bt)γ'(t) and geometrically trivial, corresponding to reparametrization.
s_parallel_transport_preserves_inner_products	theorem	Parallel transport preserves inner products		For a metric connection (in particular the Levi-Civita connection), parallel transport along any curve is a linear isometry between the tangent spaces at the en
t_variational_methods_geodesics	technique	Variational methods for geodesics		Using first and second variation formulas of length and energy functionals to characterize geodesics as critical points and study their stability and minimizing
t_symmetry_group_action_methods	technique	Symmetry and group action methods		Exploiting transitive or cohomogeneity-one isometric group actions to reduce geometric problems to algebraic ones on homogeneous or cohomogeneity-one spaces.
t_index_form_methods	technique	Index form methods		Using the index form and Morse index theorem to determine whether geodesics minimize, count conjugate points, and extract topological information from curvature
t_geodesic_deviation_analysis	technique	Geodesic deviation analysis		Analyzing the Jacobi equation to determine how nearby geodesics converge or diverge, detecting conjugate points and estimating distances.
s_busemann_function	state	Busemann function		The function b_γ(q) = lim_{t→∞} (t - d(q, γ(t))), a convex exhaustion function on non-negatively curved manifolds, used in the splitting theorem and soul theore
t_busemann_function_technique	technique	Busemann function technique		Constructing and analyzing Busemann functions to detect lines, produce splittings, and find totally convex compact submanifolds (souls) in nonnegative curvature
t_maximum_principle_riemannian	technique	Maximum principle (Riemannian)		Applying the strong maximum principle or Laplacian comparison to subharmonic or superharmonic functions on complete Riemannian manifolds to deduce rigidity.
s_product_riemannian_manifold	state	Product Riemannian manifold		The Cartesian product of Riemannian manifolds with the direct sum metric g₁ ⊕ g₂, having curvature decomposing as the direct sum of the factors' curvatures.
s_warped_product_metric	state	Warped product metric		The metric g_B + f² g_F on the product B × F where f is a positive smooth function on B, generalizing surfaces of revolution and model spaces.
s_curvature_warped_products	theorem	Curvature of warped products		For a warped product B ×_f F, the sectional curvatures are expressible in terms of curvatures of B and F and derivatives of f: e.g., K(X, V) = -(Hess f(X,X))/(f
s_riemannian_submersion	axiom	Riemannian submersion		A smooth surjective submersion π : (M, g) → (B, h) whose differential restricted to horizontal spaces is a linear isometry, relating the geometry of total space
s_oneill_formula	theorem	O'Neill's formula		For a Riemannian submersion π : M → B, the sectional curvature of B satisfies K_B(X,Y) = K_M(X̃,Ỹ) + ¾|[X̃,Ỹ]^V|² where X̃, Ỹ are horizontal lifts, showing the 
s_ambrose_singer_holonomy_theorem	theorem	Ambrose-Singer holonomy theorem		The Lie algebra of the holonomy group of a connection equals the subspace of End(T_pM) generated by all parallel translates of curvature endomorphisms R(X,Y) al
s_volume_of_geodesic_balls	state	Volume of geodesic balls		Vol(B(p,r)) = ω_n r^n (1 - S(p)/(6(n+2)) r² + O(r⁴)) where ω_n is the Euclidean ball volume and S(p) is scalar curvature, quantifying how curvature affects volu
s_riemannian_laplacian	state	Laplacian (Riemannian)		The Laplace-Beltrami operator Δf = div(grad f) = g^{ij}(∂²f/∂xⁱ∂xʲ - Γᵏᵢⱼ ∂f/∂xᵏ) on a Riemannian manifold, generalizing the Euclidean Laplacian.
s_riemannian_hessian	state	Hessian (Riemannian)		The covariant Hessian Hess f(X, Y) = <∇_X (grad f), Y> = (∇²f)(X,Y), the symmetric (0,2)-tensor of second covariant derivatives of f.
s_riemannian_gradient	state	Gradient (Riemannian)		The vector field grad f = (df)♯ defined by g(grad f, X) = df(X) = X(f) for all X, the metric dual of the differential.
s_riemannian_divergence	state	Divergence (Riemannian)		The divergence div X = trace(Y ↦ ∇_Y X) = (1/√det g) ∂ᵢ(√det g Xⁱ), measuring the infinitesimal volume change of the flow of X.
s_conformally_flat_manifold	axiom	Conformally flat manifold		A Riemannian manifold locally conformal to flat space: every point has a neighborhood where g = e^{2u} g_E for some smooth function u.
s_weyl_vanishing_conformal_flatness	theorem	Weyl tensor vanishing and conformal flatness		In dimension n ≥ 4, a Riemannian manifold is conformally flat if and only if its Weyl tensor vanishes identically.
s_cotton_tensor	state	Cotton tensor		The tensor C_{ijk} = ∇_i R_{jk} - ∇_j R_{ik} - (1/(2(n-1)))(∇_i S g_{jk} - ∇_j S g_{ik}), the obstruction to conformal flatness in dimension 3.
s_conformal_flatness_dim_3	theorem	Conformal flatness in dimension 3		In dimension 3, the Weyl tensor vanishes identically and a Riemannian manifold is conformally flat if and only if the Cotton tensor vanishes.
s_normal_bundle	state	Normal bundle		The vector bundle NM over a Riemannian submanifold whose fiber at p is the orthogonal complement of T_pM in T_pN.
s_comparison_triangle	state	Comparison triangle		A geodesic triangle Δ̄ in the model space M_k² of constant curvature k with the same side lengths as a given triangle Δ in M, used in Toponogov and Alexandrov c
s_alexandrov_angle_comparison	theorem	Alexandrov angle comparison		On a Riemannian manifold with sectional curvature ≥ k, angles in a geodesic triangle are ≥ the corresponding angles in the comparison triangle in the space form
s_ricci_identity	theorem	Ricci identity (commutation of covariant derivatives)		For a tensor field T, (∇²_{X,Y} - ∇²_{Y,X}) T = R(X,Y) · T, expressing the commutator of second covariant derivatives in terms of the curvature tensor acting on
t_parallel_transport_along_geodesics	technique	Parallel transport along geodesics		Constructing parallel frames along geodesics to reduce geometric computations to ODE analysis, used in Synge's lemma, holonomy arguments, and comparison theorem
t_curvature_estimation_via_comparison	technique	Curvature estimation via comparison		Using Rauch, Toponogov, Bishop-Gromov, and Laplacian comparison to convert curvature bounds into quantitative estimates for distances, volumes, eigenvalues, and
s_principal_fibre_bundle	axiom	Principal fibre bundle		A fibre bundle P(M, G) with fibre G and a free right G-action such that M = P/G and the projection π: P → M is the orbit map.
s_frame_bundle	state	Frame bundle L(M)		The principal GL(n,ℝ)-bundle over M whose fibre at p consists of all ordered bases (linear frames) of T_pM.
s_orthonormal_frame_bundle	state	Orthonormal frame bundle O(M)		The principal O(n)-subbundle of L(M) consisting of orthonormal frames with respect to a Riemannian metric.
s_connection_on_principal_bundle	axiom	Connection on a principal bundle		A smooth G-invariant distribution of horizontal subspaces H_u ⊂ T_uP complementary to the vertical subspace at each point of a principal bundle.
s_horizontal_lift_of_curve	state	Horizontal lift of a curve		The unique curve in a principal bundle P projecting to a given base curve γ, with tangent vectors lying in the horizontal distribution of the connection.
s_restricted_holonomy_group	state	Restricted holonomy group		The identity component Φ⁰(u) of the holonomy group, consisting of parallel transport around null-homotopic loops based at u.
s_exterior_covariant_derivative	state	Exterior covariant derivative		The operator D on tensorial forms on a principal bundle defined by Dφ(X₁,…,X_{k+1}) = dφ(hX₁,…,hX_{k+1}), projecting arguments to horizontal components.
s_cartan_structure_equation	theorem	Cartan structure equation		Ω = dω + ½[ω,ω], relating the curvature form Ω to the connection form ω on a principal bundle.
s_bianchi_identity_principal_bundle	theorem	Bianchi identity on principal bundle		DΩ = 0, the vanishing of the exterior covariant derivative of the curvature form on a principal bundle.
s_reduction_theorem_connections	theorem	Reduction theorem for connections		A connection on P(M,G) reduces to a connection on a subbundle Q(M,H) if and only if the holonomy group is contained in H.
s_existence_of_connections	theorem	Existence of connections on principal bundles		Every principal bundle over a paracompact base manifold admits a connection.
s_induced_connection_associated_bundle	state	Induced connection on associated bundle		A connection on a principal bundle P induces a covariant derivative on every associated vector bundle P ×_G V.
s_gauge_transformation	axiom	Gauge transformation		A G-equivariant diffeomorphism f: P → P of a principal bundle covering the identity on the base manifold M.
s_linear_connection	axiom	Linear connection		A connection on the frame bundle L(M), equivalently a covariant derivative operator ∇ on TM satisfying C^∞-linearity in the first argument and the Leibniz rule.
s_torsion_tensor	state	Torsion tensor		The (1,2)-tensor T(X,Y) = ∇_X Y − ∇_Y X − [X,Y] of a linear connection, measuring the antisymmetric failure of covariant derivatives to commute with Lie bracket
s_canonical_form_solder_form	state	Canonical form (solder form) on L(M)		The ℝⁿ-valued 1-form θ on L(M) defined by θ_u(X) = u⁻¹(π_*X), providing the canonical identification of horizontal vectors with ℝⁿ.
s_torsion_form_on_frame_bundle	state	Torsion form on L(M)		The ℝⁿ-valued 2-form Θ = Dθ on L(M), the exterior covariant derivative of the canonical form, encoding the torsion of the connection.
s_structure_equations_linear_connection	theorem	Structure equations of a linear connection		Θ = dθ + ω∧θ and Ω = dω + ω∧ω on L(M), expressing torsion and curvature in terms of the connection form and canonical form.
t_development_of_curves	technique	Development of curves		Rolling a manifold with affine connection onto its tangent affine space along a curve, producing a curve in ℝⁿ that encodes parallel transport and torsion.
s_parallel_section	state	Parallel section		A section s of a vector bundle satisfying ∇s = 0, constant along horizontal lifts of all curves.
s_fundamental_theorem_riemannian_geometry	theorem	Fundamental theorem of Riemannian geometry		Every Riemannian manifold admits a unique torsion-free metric-compatible linear connection, the Levi-Civita connection.
s_koszul_formula	theorem	Koszul formula		2g(∇_X Y, Z) = Xg(Y,Z) + Yg(Z,X) − Zg(X,Y) − g(X,[Y,Z]) + g(Z,[X,Y]) + g(Y,[Z,X]), determining the Levi-Civita connection explicitly from the metric.
s_interior_product	state	Interior product		The operator ι_X: Ω^k(M) → Ω^{k−1}(M) defined by contraction of a differential form with a vector field X.
s_cartans_magic_formula	theorem	Cartan's magic formula		L_X = ι_X ∘ d + d ∘ ι_X, expressing the Lie derivative of differential forms as the sum of interior product composed with exterior derivative and vice versa.
s_affine_transformation	axiom	Affine transformation		A diffeomorphism of a manifold with linear connection that preserves the connection, mapping geodesics to geodesics with affine parameter.
s_infinitesimal_affine_transformation	state	Infinitesimal affine transformation		A vector field X with L_X∇ = 0, generating a local one-parameter group of affine transformations.
s_killing_equation	theorem	Killing equation		∇_iX_j + ∇_jX_i = 0, the system of first-order PDEs characterizing Killing vector fields as infinitesimal isometries.
s_maximum_dimension_isometry_group	theorem	Maximum dimension of isometry group		dim Isom(Mⁿ, g) ≤ n(n+1)/2, with equality if and only if (M,g) has constant sectional curvature.
s_de_rham_decomposition_theorem	theorem	De Rham decomposition theorem		A complete simply-connected Riemannian manifold with reducible holonomy decomposes as a Riemannian product of irreducible factors, uniquely up to order.
s_holonomy_of_flat_connections	theorem	Holonomy of flat connections		A connection is flat iff its restricted holonomy group is trivial; the full holonomy is then a quotient of the fundamental group π₁(M).
s_symmetric_space	axiom	Symmetric space		A connected Riemannian manifold admitting at each point an involutive isometry (geodesic symmetry) with that point as isolated fixed point.
s_locally_symmetric_space	axiom	Locally symmetric space		A Riemannian manifold with parallel curvature tensor ∇R = 0, equivalently admitting local geodesic symmetries at every point.
s_locally_symmetric_iff_parallel_curvature	theorem	Locally symmetric iff parallel curvature		A Riemannian manifold is locally symmetric if and only if the curvature tensor satisfies ∇R = 0.
s_cartan_local_to_global_symmetric	theorem	Cartan's local-to-global symmetric space theorem		Every complete simply-connected locally symmetric Riemannian manifold is a globally symmetric space G/H.
s_totally_umbilic_submanifold	axiom	Totally umbilic submanifold		A submanifold whose second fundamental form is proportional to the metric: h(X,Y) = g(X,Y)H for all tangent vectors X,Y.
s_gauss_map	state	Gauss map		For a hypersurface in Euclidean space, the map sending each point to its unit outward normal vector in the unit sphere Sⁿ⁻¹.
s_flat_normal_connection	axiom	Flat normal connection		A normal connection with vanishing curvature R^⊥ = 0, equivalent to all shape operators commuting.
s_reduction_of_codimension_theorem	theorem	Reduction of codimension theorem		If the first normal space of a submanifold has constant dimension and the normal connection is flat on its orthogonal complement, then the submanifold lies in a
s_hermitian_metric	axiom	Hermitian metric		A Riemannian metric g on a manifold with almost complex structure J satisfying g(JX, JY) = g(X,Y) for all tangent vectors.
s_kahler_form_fundamental_2_form	state	Kähler form (fundamental 2-form)		The 2-form Ω(X,Y) = g(JX,Y) associated to a Hermitian metric, whose closure dΩ = 0 characterizes Kähler manifolds.
s_kahler_potential	state	Kähler potential		A local smooth real-valued function Φ on a Kähler manifold such that the Kähler metric is g_{iȷ̄} = ∂²Φ/∂zⁱ∂z̄ʲ.
s_holomorphic_sectional_curvature	state	Holomorphic sectional curvature		The sectional curvature K(X, JX) restricted to J-invariant 2-planes (complex lines) in the tangent space of a Kähler manifold.
s_constant_holomorphic_sectional_curvature	theorem	Constant holomorphic sectional curvature characterization		A Kähler manifold has constant holomorphic sectional curvature c iff its curvature tensor equals (c/4) times the standard expression involving g, J, and the sym
s_complex_space_form	axiom	Complex space form		A simply connected complete Kähler manifold of constant holomorphic sectional curvature: ℂPⁿ (c>0), ℂⁿ (c=0), or ℂHⁿ (c<0).
s_complex_projective_space	state	Complex projective space ℂPⁿ		The quotient (ℂⁿ⁺¹ \ {0})/ℂ* with the Fubini–Study metric, the unique (up to scale) complete Kähler manifold of constant positive holomorphic sectional curvatur
s_fubini_study_metric	axiom	Fubini–Study metric		The canonical Kähler metric on ℂPⁿ induced from the round metric on S²ⁿ⁺¹ via the Hopf fibration, having constant holomorphic sectional curvature.
s_type_decomposition_differential_forms	state	Type decomposition of differential forms		On a complex manifold, the decomposition Ωᵏ = ⊕_{p+q=k} Ω^{p,q} into (p,q)-type forms with p holomorphic and q antiholomorphic indices.
s_holomorphic_vector_bundle	axiom	Holomorphic vector bundle		A complex vector bundle over a complex manifold whose transition functions are holomorphic maps.
s_ricci_form_kahler	state	Ricci form of a Kähler manifold		The closed (1,1)-form ρ = −i∂∂̄ log det(g_{jk̄}), representing 2π times the first Chern class c₁(M).
s_bochner_curvature_tensor	state	Bochner curvature tensor		The Kähler analogue of the Weyl conformal curvature tensor, the conformally invariant component of the curvature respecting the complex structure.
s_kahler_submanifold	state	Kähler submanifold		A complex submanifold of a Kähler manifold that inherits a Kähler structure and is necessarily minimal.
s_reductive_homogeneous_space	axiom	Reductive homogeneous space		A homogeneous space G/H with an Ad(H)-invariant decomposition 𝔤 = 𝔥 ⊕ 𝔪, identifying 𝔪 with the tangent space at the origin.
s_naturally_reductive_homogeneous_space	axiom	Naturally reductive homogeneous space		A reductive homogeneous space satisfying ⟨[X,Y]_𝔪, Z⟩ + ⟨Y, [X,Z]_𝔪⟩ = 0 for all X,Y,Z ∈ 𝔪, ensuring geodesics are orbits of one-parameter subgroups.
s_invariant_connection_homogeneous	state	Invariant connection on G/H		A G-invariant connection on a homogeneous space, determined by a linear map Λ: 𝔪 → 𝔤𝔩(𝔪) compatible with the Ad(H)-action.
s_canonical_connection_reductive	state	Canonical connection on reductive homogeneous space		The unique G-invariant connection with T(X,Y) = −[X,Y]_𝔪 and ∇R = ∇T = 0, where parallel transport corresponds to the reductive decomposition.
s_curvature_naturally_reductive_spaces	theorem	Curvature of naturally reductive spaces		R(X,Y)Z = −[[X,Y]_𝔥, Z] for X,Y,Z ∈ 𝔪, reducing curvature computation to Lie bracket operations in a naturally reductive homogeneous space.
s_isotropy_representation	state	Isotropy representation		The representation of the isotropy group H on T_o(G/H) ≅ 𝔪 via h ↦ (dL_h)_o, determining invariant geometric structures on the homogeneous space.
s_invariant_metric_homogeneous	state	Invariant metric on G/H		A G-invariant Riemannian metric on G/H, corresponding to an Ad(H)-invariant inner product on 𝔪.
s_invariant_almost_complex_structure_homogeneous	state	G-invariant almost complex structure on G/H		An Ad(H)-invariant endomorphism J: 𝔪 → 𝔪 with J² = −Id satisfying an integrability condition for complex structure on the homogeneous space.
s_geodesic_symmetry	state	Geodesic symmetry		The involutive diffeomorphism s_p defined by s_p(exp_p v) = exp_p(−v), which is an isometry in a symmetric space.
s_riemannian_symmetric_pair	axiom	Riemannian symmetric pair		A pair (G, H) with an involutive automorphism σ of G such that (G^σ)₀ ⊆ H ⊆ G^σ, inducing a symmetric space structure on G/H.
s_rank_of_symmetric_space	state	Rank of a symmetric space		The dimension of a maximal flat totally geodesic submanifold, equivalently the dimension of a maximal abelian subspace of 𝔭 in the Cartan decomposition.
s_symmetric_space_compact_type	state	Symmetric space of compact type		A symmetric space G/H with G compact semisimple, having non-negative sectional curvature and negative definite Killing form on 𝔭.
s_symmetric_space_noncompact_type	state	Symmetric space of noncompact type		A symmetric space G/H with G noncompact semisimple of finite center, having non-positive sectional curvature, diffeomorphic to ℝⁿ.
s_duality_of_symmetric_spaces	theorem	Duality of symmetric spaces		A one-to-one correspondence between simply connected symmetric spaces of compact type and noncompact type, obtained by 𝔭 ↦ i𝔭 in the complexification, reversing
s_curvature_of_symmetric_spaces	theorem	Curvature of symmetric spaces		The curvature tensor of a symmetric space G/H is R(X,Y)Z = −[[X,Y],Z] for X,Y,Z ∈ 𝔭 ≅ T_oM, determined entirely by the Lie bracket.
s_de_rham_decomposition_symmetric_spaces	theorem	De Rham decomposition for symmetric spaces		A simply connected symmetric space decomposes uniquely as M₀ × M₁ × ··· × M_k where M₀ is Euclidean and each Mᵢ is irreducible symmetric.
s_maximal_flat_submanifold	state	Maximal flat submanifold		A maximal totally geodesic flat submanifold through a point in a symmetric space, isometric to ℝʳ or a flat torus Tʳ where r is the rank.
s_compact_symmetric_nonneg_curvature	theorem	Compact symmetric space has non-negative curvature		K(X,Y) = |[X,Y]|²/(|X|²|Y|² − ⟨X,Y⟩²) ≥ 0 for a symmetric space of compact type.
s_symmetric_space_is_complete	theorem	Symmetric space is complete		Every Riemannian symmetric space is geodesically complete.
s_isotropy_irreducible_symmetric_space	state	Isotropy-irreducible symmetric space		A symmetric space G/H where the isotropy representation of H on 𝔭 is irreducible, forcing the invariant metric to be unique up to scaling.
t_weil_homomorphism	technique	Weil homomorphism		The ring homomorphism from Ad-invariant polynomials I^k(G) to de Rham cohomology H^{2k}(M;ℝ), mapping P to the closed form P(Ω,…,Ω).
t_chern_weil_construction	technique	Chern–Weil construction		Constructing characteristic classes from curvature: invariant polynomials applied to the curvature form produce closed forms whose cohomology classes are connec
s_total_chern_class	state	Total Chern class		c(E) = 1 + c₁(E) + c₂(E) + ··· ∈ H*(M;ℤ), the generating series of Chern classes satisfying c(E⊕F) = c(E)∪c(F).
s_whitney_product_formula_chern	theorem	Whitney product formula for Chern classes		c(E ⊕ F) = c(E) ∪ c(F), the multiplicativity of the total Chern class under direct sum of complex vector bundles.
s_first_chern_class	state	First Chern class		c₁(E) ∈ H²(M;ℤ), classifying complex line bundles up to isomorphism; for a Kähler manifold c₁(M) = [ρ/2π] with ρ the Ricci form.
s_total_pontryagin_class	state	Total Pontryagin class		p(E) = 1 + p₁(E) + p₂(E) + ··· ∈ H*(M;ℤ), the generating series of Pontryagin classes of a real vector bundle.
s_euler_class	axiom	Euler class		The characteristic class e(E) ∈ Hⁿ(M;ℤ) of an oriented real rank-n vector bundle, satisfying e(E)² = p_{n/2}(E) when n is even.
s_chern_number	state	Chern number		An integer topological invariant obtained by evaluating a monomial c₁^{a₁}c₂^{a₂}··· on the fundamental class [M] of a compact complex manifold.
s_pontryagin_number	state	Pontryagin number		An integer topological invariant obtained by evaluating a monomial in Pontryagin classes on the fundamental class [M] of a compact oriented manifold.
t_transgression	technique	Transgression		Constructing a (2k−1)-form TP whose exterior derivative equals the difference P(Ω₁) − P(Ω₀) of Chern–Weil forms for two connections, yielding secondary characte
t_splitting_principle	technique	Splitting principle technique		Reducing characteristic class computations for general vector bundles to line bundles by pulling back to a space where the bundle splits as a direct sum of line
s_chern_class_of_line_bundle	theorem	Chern class of a line bundle		For a complex line bundle L, c₁(L) = [(i/2π)Ω] classifies L up to isomorphism via its curvature 2-form.
s_characteristic_classes_flat_bundles_vanish	theorem	Characteristic classes of flat bundles vanish		A flat connection on a principal bundle gives vanishing real characteristic classes in positive degrees via Chern–Weil theory.
s_morse_function	axiom	Morse function		A smooth real-valued function f: M → ℝ on a smooth manifold all of whose critical points are non-degenerate (the Hessian at each critical point is non-singular)
s_critical_point_of_smooth_function	axiom	Critical point of a smooth function		A point p ∈ M at which df_p = 0, i.e., all partial derivatives of f vanish in any local coordinate system.
s_index_of_critical_point	axiom	Index of a critical point		The number of negative eigenvalues of the Hessian matrix of f at a non-degenerate critical point p, denoted λ or ind(p); equals the dimension of the maximal sub
s_hessian_at_critical_point	state	Hessian of a smooth function at a critical point		The symmetric bilinear form H_p(f)(v,w) = v(w̃(f)) on T_pM at a critical point p, represented in local coordinates by the matrix (∂²f/∂xᵢ∂xⱼ)(p); well-defined i
s_sublevel_set	axiom	Sublevel set		For f: M → ℝ and a ∈ ℝ, the sublevel set Mᵃ = f⁻¹(−∞, a] = {x ∈ M : f(x) ≤ a}.
s_handle_attachment_theorem	theorem	Handle attachment theorem		If p is a non-degenerate critical point of f with index λ and f(p) = c, then for sufficiently small ε > 0, M^{c+ε} has the homotopy type of M^{c−ε} with a λ-cel
s_handle_of_index_lambda	axiom	Handle of index λ		A thickened cell D^λ × D^{n−λ} attached to the boundary of a manifold along S^{λ−1} × D^{n−λ}, modeling the topological change at a critical point of index λ.
s_cw_structure_from_morse_function	theorem	CW complex structure from a Morse function		If M is a compact manifold admitting a Morse function f with critical points p₁,…,pₖ of indices λ₁,…,λₖ, then M has the homotopy type of a CW complex with one c
s_morse_number	axiom	Morse number Cₖ		The number of critical points of index k of a Morse function f on a manifold M, denoted Cₖ(f) or simply Cₖ.
s_weak_morse_inequalities	theorem	Weak Morse inequalities		For a Morse function f on a compact manifold M: Cₖ ≥ βₖ for each k, where βₖ = rank Hₖ(M; F) is the k-th Betti number with coefficients in a field F.
s_strong_morse_inequalities	theorem	Strong Morse inequalities		For a Morse function on a compact manifold: ∑_{k=0}^{t} (−1)^{t−k} Cₖ ≥ ∑_{k=0}^{t} (−1)^{t−k} βₖ for all t, with equality when t = dim M (recovering the Euler 
s_poincare_polynomial_morse	state	Poincaré polynomial (Morse context)		P_t(M) = ∑ βₖ tᵏ, the generating polynomial for the Betti numbers of M; the Morse inequalities state M_t(f) = ∑ Cₖ tᵏ satisfies M_t(f) = P_t(M) + (1+t)Q(t) for 
s_lacunary_principle_morse	theorem	Lacunary principle		If a Morse function has critical points only in even dimensions (Cₖ = 0 for k odd), then the Morse inequalities are equalities: βₖ = Cₖ for all k, so M has no t
s_euler_characteristic_via_morse	theorem	Euler characteristic via Morse theory		χ(M) = ∑ₖ (−1)ᵏ Cₖ for any Morse function on a compact manifold M, equating the Euler characteristic with the alternating sum of critical point counts.
s_gradient_flow_morse	state	Gradient flow of a Morse function		The flow φ_t of the negative gradient vector field −grad f on a Riemannian manifold (M,g), satisfying dφ_t(x)/dt = −grad f(φ_t(x)), used to deform sublevel sets
s_gradient_like_vector_field	axiom	Gradient-like vector field		A vector field ξ on M such that ξ(f) > 0 away from critical points of f, and near each critical point p the Morse lemma coordinates give ξ = −2∑_{i≤λ} xᵢ ∂/∂xᵢ 
s_stable_manifold_critical_point	state	Stable manifold of a critical point		W^s(p) = {x ∈ M : lim_{t→∞} φ_t(x) = p}, the set of points flowing into the critical point p under the negative gradient flow; has dimension n − λ where λ = ind
s_unstable_manifold_critical_point	state	Unstable manifold of a critical point		W^u(p) = {x ∈ M : lim_{t→−∞} φ_t(x) = p}, the set of points flowing out of the critical point p under the negative gradient flow; has dimension λ = ind(p).
s_cobordism_via_morse	state	Cobordism via Morse theory		A Morse function on a cobordism W with ∂W = M₀ ⊔ M₁ decomposes W into a sequence of elementary cobordisms, each corresponding to a single handle attachment at a
s_path_space	axiom	Path space Ω(M; p, q)		The space of piecewise smooth paths ω: [0,1] → M with ω(0) = p and ω(1) = q, topologized with the compact-open topology; the domain for Morse theory of geodesic
s_energy_sublevel_set	state	Energy sublevel set Ωᶜ		Ωᶜ = {ω ∈ Ω(M;p,q) : E(ω) ≤ c}, the sublevel set of the energy functional; has the homotopy type of a finite-dimensional manifold with corners when M is complet
s_finite_dimensional_approximation_path_space	state	Finite-dimensional approximation of path space		The space Ω(t₀,…,tₖ) of piecewise geodesic paths with breakpoints at fixed parameter values t₀ < t₁ < ⋯ < tₖ; a finite-dimensional manifold that approximates Ω(
s_morse_index_of_geodesic	state	Morse index of a geodesic		The index of a geodesic γ as a critical point of the energy functional E: the maximal dimension of a subspace on which I is negative definite; equals the number
s_focal_point	axiom	Focal point		Given a submanifold N ⊂ M and a geodesic γ perpendicular to N at γ(0), a point γ(t₀) is a focal point of N along γ if there exists a non-zero Jacobi field J alo
s_index_theorem_with_focal_points	theorem	Index theorem with focal points		For the energy functional on paths from a submanifold N to a point q, the index of a geodesic γ perpendicular to N equals the number of focal points of N along 
s_homotopy_groups_classical_lie_groups	state	Homotopy groups of classical Lie groups		The stable homotopy groups πₖ(O), πₖ(U), πₖ(Sp) for the infinite orthogonal, unitary, and symplectic groups, computed via Bott periodicity: πₖ(U) cycles with pe
s_loop_space_of_symmetric_space	state	Loop space of a symmetric space		The space Ω(G/K; p, q) of paths in a symmetric space G/K between two points; Milnor shows the energy functional on this space is a perfect Morse function with c
s_bott_samelson_theorem	theorem	Bott–Samelson theorem		On a compact symmetric space, every geodesic is periodic, the set of geodesics from p to q of a given energy forms a smooth manifold, and the energy functional 
s_bott_iteration_formula	theorem	Bott's iteration formula		The index of an iterated geodesic γᵏ (traversing γ exactly k times) is computed from the index and nullity of γ by a formula involving the eigenvalues of the ho
s_perfect_morse_function	axiom	Perfect Morse function		A Morse function for which the Morse inequalities are equalities over ℤ: Cₖ = βₖ for all k; equivalently M_t(f) = P_t(M), the Morse polynomial equals the Poinca
s_geodesic_as_critical_point_of_energy	state	Geodesic as critical point of energy		A path ω ∈ Ω(M;p,q) is a critical point of the energy functional E iff ω is a geodesic from p to q; follows from the first variation formula.
s_existence_of_morse_functions	theorem	Existence of Morse functions		Every smooth manifold admits a Morse function; moreover Morse functions form an open and dense subset of C^∞(M,ℝ) in the C² topology (genericity of Morse functi
s_compact_lie_group_as_symmetric_space	state	Compact Lie group as symmetric space		A compact Lie group G with a bi-invariant metric is a symmetric space of type (G×G)/Δ(G), where Δ(G) is the diagonal; geodesics are one-parameter subgroups.
s_bi_invariant_metric	axiom	Bi-invariant metric on a Lie group		A Riemannian metric on a Lie group G that is invariant under both left and right multiplication; exists on every compact Lie group, and geodesics through e are 
s_geodesics_in_symmetric_space	state	Geodesics in a symmetric space		In a symmetric space G/K, the geodesic through eK with initial velocity X ∈ 𝔭 is t ↦ exp(tX) · eK; all geodesics arise this way, and they are orbits of one-para
s_periodicity_of_geodesics_compact_symmetric	theorem	Periodicity of geodesics in compact symmetric spaces		In a compact symmetric space of rank r, every geodesic is periodic (closed); the minimal period depends on the direction and the root system of the symmetric sp
s_morse_theory_on_loop_space_symmetric	state	Morse theory on the loop space of a symmetric space		Applying Morse theory to the energy functional E on Ω(G/K; p, q): the critical manifolds are totally geodesic submanifolds, the indices are computed from the ro
s_non_degenerate_critical_submanifold	state	Non-degenerate critical submanifold		A connected submanifold C of critical points of f such that the null space of the Hessian at each point p ∈ C equals T_pC; generalizes non-degenerate critical p
s_morse_bott_theory	theorem	Morse–Bott theory		Extension of Morse theory where critical points form non-degenerate critical submanifolds C_i: M has the homotopy type of a space built by attaching disk bundle
s_minimax_principle_morse	theorem	Minimax principle (Morse theory)		If Hₖ(Mᵇ, Mᵃ) ≠ 0 for a < b, then f has a critical value in [a,b]; more generally c = inf_{[α] ∈ H_k \{0}} sup_{x ∈ supp α} f(x) is a critical value, yielding e
s_normal_bundle_of_critical_submanifold	state	Normal bundle of a critical submanifold (negative bundle)		The subbundle ν⁻ of the normal bundle of a critical submanifold C on which the Hessian of f is negative definite; its fibre dimension is the index λ, and the di
s_palais_smale_condition	axiom	Condition (C) (Palais–Smale condition)		A function f on a complete Riemannian manifold (possibly infinite-dimensional) satisfies Condition (C) if every sequence {xₙ} with f(xₙ) bounded and |grad f(xₙ)
s_energy_satisfies_condition_c	theorem	Energy functional satisfies Condition (C)		On a complete Riemannian manifold M, the energy functional E on the path space Ω(M;p,q) (with the H¹ Sobolev metric) satisfies the Palais–Smale condition (C), e
s_homotopy_type_path_space_sublevel	theorem	Homotopy type of path space sublevel sets		For a complete M, the sublevel set Ω^c = {ω : E(ω) ≤ c} has the homotopy type of a CW complex with one cell of dimension λ for each geodesic from p to q with en
s_existence_of_geodesics_via_morse	theorem	Existence of geodesics via Morse theory		If M is complete and Ω(M;p,q) is not contractible, the energy functional must have critical points at arbitrarily high energy levels; hence there exist infinite
s_theorem_of_serre	theorem	Theorem of Serre on closed geodesics		On any compact simply connected Riemannian manifold, there exist infinitely many geometrically distinct geodesics connecting any two points; proved using the no
s_curvature_bounds_and_conjugate_points	theorem	Curvature bounds and conjugate points		If K ≥ δ > 0, then every geodesic has a conjugate point within distance π/√δ from the starting point (Bonnet's theorem); if K ≤ 0 there are no conjugate points 
s_homology_of_loop_space_via_morse	state	Homology of the loop space via Morse theory		The Morse theory of the energy functional on Ω(M;p,p) computes the homology of the loop space ΩM; for symmetric spaces, the energy is a perfect Morse function y
t_gradient_flow_deformation	technique	Gradient flow deformation		Using the flow of −grad f to deform the manifold, retracting M^b onto M^a when [a,b] contains no critical values; the fundamental technique underlying sublevel 
t_handle_attachment_technique	technique	Handle attachment technique		Attaching a λ-cell to the sublevel set when passing a critical value c of a Morse function, using the gradient flow to identify the attaching map from the stabl
t_second_variation_analysis	technique	Second variation analysis		Computing the second variation of energy to determine the index of a geodesic: constructing variation fields that make d²E/ds² < 0 detects negative eigenvalues 
t_jacobi_field_index_computation	technique	Jacobi field method for index computation		Computing the Morse index of a geodesic by solving the Jacobi equation along γ and counting zeros of Jacobi fields: each zero contributes its multiplicity to th
t_finite_dimensional_approximation_paths	technique	Finite-dimensional approximation of path spaces		Replacing the infinite-dimensional path space by the finite-dimensional manifold of broken geodesics with fixed breakpoints; the energy restricts to a genuine M
t_symmetric_space_morse_theory	technique	Symmetric space Morse theory		Exploiting the group action on a symmetric space to show the energy functional is equivariantly perfect Morse–Bott: critical manifolds are explicitly computable
t_morse_sublevel_deformation	technique	Morse theory technique (deforming sublevel sets)		The master technique of Morse theory: sweeping through critical values of f, using gradient flow to deform sublevel sets across regular intervals, and attaching
s_normal_bundle_along_geodesic	state	Normal bundle along a geodesic		The vector bundle ν(γ) over [0,1] whose fibre at t is {v ∈ T_{γ(t)}M : g(v, γ̇(t)) = 0}; vector fields along γ in this bundle are the relevant variations for th
s_broken_geodesic	state	Broken geodesic (piecewise geodesic path)		A path in Ω(M;p,q) that is geodesic on each subinterval [tᵢ, t_{i+1}] but may have corners (discontinuities of the tangent vector) at the breakpoints; forms the
s_topology_of_spheres_via_morse	state	Topology of Sⁿ via Morse theory		The height function on Sⁿ ⊂ ℝⁿ⁺¹ is a Morse function with two critical points (index 0 and index n); by Reeb's theorem Sⁿ is homeomorphic to Sⁿ, and by CW struc
s_topology_of_projective_spaces_via_morse	state	Topology of projective spaces via Morse theory		A natural Morse function on ℂPⁿ (or ℝPⁿ) obtained from a generic linear function on the ambient space has n+1 critical points of indices 0, 2, 4, …, 2n (resp. 0
s_topology_of_grassmannians_via_morse	state	Topology of Grassmannians via Morse theory		The function f(V) = ∑ᵢ λᵢ dim(V ∩ Eᵢ) on the Grassmannian G(k,n) is a Morse function whose critical points are the coordinate k-planes; the Schubert cell decomp
s_stable_unitary_group	axiom	Stable unitary group U		U = lim_{n→∞} U(n), the colimit of unitary groups under the standard inclusions U(n) ↪ U(n+1); the classifying space BU classifies complex vector bundles.
s_stable_orthogonal_group	axiom	Stable orthogonal group O		O = lim_{n→∞} O(n), the colimit of orthogonal groups; πₖ(O) is periodic with period 8 by Bott periodicity.
s_stable_symplectic_group	axiom	Stable symplectic group Sp		Sp = lim_{n→∞} Sp(n), the colimit of compact symplectic groups; πₖ(Sp) is periodic with period 8, shifted by 4 relative to πₖ(O).
s_bott_periodicity_chain	state	Bott periodicity chain of symmetric spaces		The sequence of symmetric spaces O → O/U → U/Sp → Sp → Sp/U → U/O → O (period 8), where each space is the loop space of the next; encodes the 8-fold periodicity
s_index_form_positive_definite	theorem	Index form is positive definite before first conjugate point		If a geodesic γ: [0,1] → M has no conjugate points in (0,1], then the index form I(W,W) > 0 for all non-zero W vanishing at endpoints; equivalently the Morse in
s_isotropy_representation_symmetric_space	state	Isotropy representation of a symmetric space		The representation Ad|_K: K → GL(𝔭) of the isotropy group K on the tangent space 𝔭 ≅ T_{eK}(G/K) of a symmetric space G/K; its orbits classify geodesic directio
s_quasi_projective_variety	axiom	Quasi-projective variety		A locally closed subset of projective space, i.e. an open subset of a projective variety; the most general class of classical varieties.
s_ringed_space	axiom	Ringed space		A topological space X equipped with a sheaf of rings O_X; the general framework before requiring stalks to be local rings.
s_separated_morphism	axiom	Separated morphism		A morphism f: X → Y of schemes such that the diagonal morphism X → X ×_Y X is a closed immersion; the scheme-theoretic analogue of the Hausdorff property.
s_proper_morphism	axiom	Proper morphism		A morphism that is separated, of finite type, and universally closed; the algebraic geometry analogue of a proper continuous map between compact spaces.
s_projective_morphism	axiom	Projective morphism		A morphism f: X → Y that factors as a closed immersion X → P^n_Y followed by the projection P^n_Y → Y for some n.
s_quasi_coherent_sheaf	axiom	Quasi-coherent sheaf		An O_X-module that is locally a cokernel of free modules, equivalently one that restricts to the sheaf associated to a module on each affine open.
s_coherent_sheaf	axiom	Coherent sheaf		A quasi-coherent O_X-module that is locally of finite presentation: locally the cokernel of a map between finite-rank free sheaves.
s_invertible_sheaf	axiom	Invertible sheaf		A locally free O_X-module of rank one; equivalently a line bundle, the sheaf-theoretic incarnation of a Cartier divisor class.
s_ideal_sheaf	state	Ideal sheaf		A sheaf of ideals I in O_X defining a closed subscheme Z = V(I), with O_Z = O_X/I as the structure sheaf of the subscheme.
s_very_ample_divisor	axiom	Very ample divisor		A divisor D on X such that the associated invertible sheaf O(D) defines a closed immersion of X into some projective space.
s_flat_morphism	axiom	Flat morphism		A morphism f: X → Y such that for every x in X, the local ring O_{X,x} is a flat O_{Y,f(x)}-module; the algebraic notion of a continuously varying family.
s_regular_scheme	axiom	Regular scheme		A locally Noetherian scheme all of whose local rings are regular local rings; the scheme-theoretic generalization of nonsingularity.
s_normal_scheme	axiom	Normal scheme		A scheme whose local rings are integrally closed domains; equivalently a scheme satisfying Serre conditions R_1 and S_2.
s_arithmetic_genus	axiom	Arithmetic genus		For a projective variety X of dimension n, p_a(X) = (-1)^n(P(0) − 1) where P is the Hilbert polynomial; equals h^1(O_X) for curves.
s_geometric_genus	axiom	Geometric genus		For a smooth projective variety X, p_g(X) = h^0(ω_X) = dim H^0(X, ω_X), the number of linearly independent holomorphic n-forms.
s_ruled_surface	axiom	Ruled surface		A smooth projective surface that is birational to C × P^1 for some curve C; equivalently a P^1-bundle over a curve.
s_rational_surface	axiom	Rational surface		A smooth projective surface birational to P^2; equivalently a surface with p_g = q = 0 and P_2 = 0.
s_minimal_model_surface	axiom	Minimal model (surface)		A smooth projective surface containing no exceptional curves of the first kind (−1 curves); every surface has a minimal model obtained by contracting such curve
s_exceptional_curve	axiom	Exceptional curve of the first kind		A smooth rational curve E on a surface with self-intersection E² = −1; can be blown down by Castelnuovo's criterion.
s_closed_immersion	axiom	Closed immersion		A morphism f: X → Y inducing a homeomorphism of X onto a closed subset of Y, with O_Y → f_*O_X surjective; corresponds to an ideal sheaf in O_Y.
s_open_immersion	axiom	Open immersion		A morphism of schemes that is an isomorphism of X onto an open subscheme of Y.
s_finite_morphism	axiom	Finite morphism		A morphism f: X → Y such that f^{−1}(U) = Spec(B) for each affine open Spec(A) = U in Y, where B is a finite A-algebra.
s_unramified_morphism	axiom	Unramified morphism		A morphism of finite type such that the sheaf of relative differentials Ω_{X/Y} vanishes; together with flatness gives an étale morphism.
s_formal_scheme	axiom	Formal scheme		A topological ringed space locally isomorphic to Spf(A) for an I-adically complete ring A; captures formal neighborhoods and completions in algebraic geometry.
s_elliptic_curve	axiom	Elliptic curve		A smooth projective curve of genus 1 with a specified rational point O; carries a natural group law and is its own Jacobian.
s_canonical_sheaf	state	Canonical sheaf		For a smooth variety X of dimension n, the invertible sheaf ω_X = det(Ω^1_X) = ∧^n(Ω^1_{X/k}), the sheaf of regular n-forms.
s_dualizing_sheaf	state	Dualizing sheaf		For a projective scheme X over k, the sheaf ω_X° representing the functor F → H^n(X, F)' via Serre duality; agrees with the canonical sheaf when X is smooth.
s_conormal_sheaf	state	Conormal sheaf		For a closed immersion Y → X defined by ideal sheaf I, the sheaf I/I² on Y; its dual is the normal sheaf N_{Y/X}.
s_euler_sequence	state	Euler sequence		The fundamental exact sequence 0 → Ω^1_{P^n} → O(−1)^{n+1} → O → 0 on projective space, relating the cotangent sheaf to twisting sheaves.
s_twisted_sheaf	state	Twisting sheaf O(n)		The invertible sheaf O_{P^r}(n) on projective space, where O(1) is the tautological quotient line bundle; tensoring with O(n) is Serre's twist.
s_sheaf_of_differentials	state	Sheaf of relative differentials		For a morphism f: X → Y, the quasi-coherent sheaf Ω^1_{X/Y} on X generated by symbols df subject to the Leibniz rule and f*-linearity.
s_strict_transform	state	Strict transform		For a subvariety Y of X and the blowup π: Bl_Z(X) → X, the closure of π^{−1}(Y \ Z) in the blowup; the proper transform of Y.
s_higher_direct_image	state	Higher direct image sheaf		For a morphism f: X → Y and a sheaf F on X, the sheaves R^i f_*(F) on Y obtained as the right derived functors of the direct image functor f_*.
s_irregularity	state	Irregularity		For a smooth projective surface X, q(X) = h^1(O_X) = dim H^0(Ω^1_X); equals the dimension of the Albanese variety.
s_albanese_variety	state	Albanese variety		The universal abelian variety Alb(X) receiving a morphism from X; dual to the Picard variety Pic°(X), with dimension equal to the irregularity q(X).
s_cubic_surface	state	Cubic surface		A smooth surface of degree 3 in P^3; isomorphic to P^2 blown up at 6 points in general position and contains exactly 27 lines.
s_serre_vanishing_theorem	theorem	Serre vanishing theorem		For a coherent sheaf F on a projective scheme X over a Noetherian ring, H^i(X, F(n)) = 0 for all i > 0 and n sufficiently large.
s_adjunction_formula	theorem	Adjunction formula		For a smooth divisor Y on a smooth variety X, ω_Y ≅ (ω_X ⊗ O(Y))|_Y, relating the canonical sheaves via the normal bundle.
s_noether_formula	theorem	Noether's formula		For a smooth projective surface X, χ(O_X) = (1/12)(K_X² + e(X)) where e(X) is the topological Euler characteristic.
s_semicontinuity_theorem	theorem	Semicontinuity theorem (sheaf cohomology)		For a proper morphism f: X → Y and a coherent sheaf F on X flat over Y, the function y → dim_k(y) H^i(X_y, F_y) is upper semicontinuous on Y.
s_flat_base_change_theorem	theorem	Flat base change theorem		For a flat base change u: Y' → Y and a proper morphism f: X → Y with coherent sheaf F, there is a natural isomorphism u* R^i f_*(F) ≅ R^i f'_*(v* F).
s_projection_formula_sheaves	theorem	Projection formula (sheaves)		For a morphism f: X → Y, a quasi-coherent sheaf F on X and a locally free sheaf E on Y, R^i f_*(F ⊗ f*E) ≅ R^i f_*(F) ⊗ E.
s_enriques_classification	theorem	Enriques classification of surfaces		Every minimal smooth projective surface over an algebraically closed field of characteristic 0 falls into exactly one class: rational, ruled, abelian, K3, Enriq
s_castelnuovo_de_franchis_theorem	theorem	Castelnuovo–de Franchis theorem		If ω_1, ω_2 are linearly independent 1-forms on a smooth projective surface X with ω_1 ∧ ω_2 = 0, there exists a fibration f: X → C to a curve of genus ≥ 2.
s_degree_genus_formula	theorem	Degree-genus formula		A smooth plane curve of degree d in P^2 has genus g = (d−1)(d−2)/2.
s_valuative_criterion_properness	theorem	Valuative criterion of properness		A morphism f: X → Y of finite type is proper iff for every valuation ring R with fraction field K, every K-valued point of X over Y extends uniquely to an R-val
s_valuative_criterion_separatedness	theorem	Valuative criterion of separatedness		A morphism f: X → Y of finite type is separated iff for every valuation ring R with fraction field K, every K-valued point of X over Y extends to at most one R-
s_serre_criterion_affineness	theorem	Serre's criterion for affineness		A Noetherian scheme X is affine if and only if H^i(X, F) = 0 for all quasi-coherent sheaves F and all i > 0.
s_cohomology_of_projective_space	theorem	Cohomology of projective space		H^0(P^n, O(d)) = k[x_0,...,x_n]_d for d ≥ 0, H^n(P^n, O(d)) is dual to H^0(P^n, O(−d−n−1)), and all other cohomology groups vanish.
s_cartier_theorem_smoothness	theorem	Cartier's theorem on smoothness		Every group scheme of finite type over a field of characteristic zero is smooth; equivalently every reduced group scheme over a perfect field is smooth.
s_stein_factorization	theorem	Stein factorization		Every proper morphism f: X → Y factors as X → Z → Y where X → Z has connected fibers and Z → Y is finite, with Z = Spec(f_* O_X).
t_spec_construction	technique	Spec construction		Constructs the affine scheme Spec(A) from a ring A: points are prime ideals, topology is Zariski, structure sheaf assigns localizations A_f to basic opens D(f).
t_sheaf_hom	technique	Sheaf Hom construction		Constructs the sheaf Hom(F, G) assigning to each open U the module Hom_{O_X|U}(F|_U, G|_U); the internal hom in the category of O_X-modules.
t_tensor_product_of_sheaves	technique	Tensor product of sheaves		Forms the tensor product F ⊗_{O_X} G by sheafifying the presheaf U → F(U) ⊗_{O_X(U)} G(U); preserves quasi-coherence and coherence.
t_veronese_embedding_technique	technique	Veronese embedding technique		Re-embeds P^n via all degree-d monomials to convert degree-d hypersurfaces into hyperplane sections, reducing problems to the linear case.
t_segre_embedding_technique	technique	Segre embedding technique		Embeds a product of projective spaces into a single projective space via the bilinear map [x_i] × [y_j] → [x_i y_j], realizing products as projective varieties.
t_linear_system_embedding	technique	Linear system embedding		Uses a base-point-free linear system |D| to define a morphism to projective space by sending x to [s_0(x):...:s_n(x)] for a basis of sections.
t_adjunction_technique_ag	technique	Adjunction technique (algebraic geometry)		Computes the canonical class of a subvariety Y from that of the ambient variety X via K_Y = (K_X + Y)|_Y, yielding genus formulas and classification tools.
t_flattening_stratification	technique	Flattening stratification		Decomposes the base Y of a morphism into locally closed strata over which a coherent sheaf has constant Hilbert polynomial, reducing to the flat case.
t_semicontinuity_method	technique	Semicontinuity method		Uses upper semicontinuity of h^i(X_y, F_y) and constancy of the Euler characteristic to deduce cohomological information across fibers of a flat family.
t_formal_completion	technique	Formal completion		Replaces a scheme X with the formal scheme along a closed subscheme Y, capturing infinitesimal neighborhoods; connects local and global properties via the theor
t_castelnuovo_contraction	technique	Castelnuovo contraction		Contracts an exceptional curve E on a surface S to a smooth point, producing a smooth surface S' with K_{S'}² = K_S² + 1; used iteratively to reach the minimal 
t_minimal_model_program_step	technique	Minimal model program step		Iteratively contracts (−1)-curves on a surface until none remain, reaching a minimal model that is either ruled (if κ = −∞) or has nef canonical class.
t_taking_stalks	technique	Taking stalks		Extracts the stalk F_x = colim_{U ∋ x} F(U) at a point x, reducing sheaf-theoretic statements to module-theoretic ones by exploiting that sheaf properties can b
t_cech_cohomology_computation	technique	Čech cohomology computation		Computes sheaf cohomology via an open affine cover using the Čech complex with terms ∏ F(U_{i_0} ∩ ... ∩ U_{i_p}); agrees with derived functor cohomology for se
t_canonical_embedding_technique	technique	Canonical embedding technique		Embeds a non-hyperelliptic curve of genus g into P^{g−1} via the complete canonical linear system |K_C|, producing a curve of degree 2g−2 whose geometry encodes
s_holomorphic_map	axiom	Holomorphic map		A map between complex manifolds that is locally given by holomorphic functions in local coordinates.
s_dolbeault_complex	axiom	Dolbeault complex		The complex of smooth (p,q)-forms with the ∂̄-operator as differential, whose cohomology yields Dolbeault cohomology.
s_theta_divisor	axiom	Theta divisor		The canonical effective divisor on the Jacobian variety defined as the image of the (g−1)-fold symmetric product of C under the Abel–Jacobi map.
s_schubert_cycle	axiom	Schubert cycle		A cohomology class on the Grassmannian defined by a Schubert variety, indexed by a partition λ, forming an additive basis for the cohomology ring.
s_grothendieck_residue	axiom	Grothendieck residue		A multivariate generalization of the classical residue, assigning to a meromorphic n-form on an n-dimensional complex manifold a complex number at an isolated p
s_hodge_filtration	axiom	Hodge filtration		The decreasing filtration on the complexified cohomology of a compact Kähler manifold defined by the Hodge decomposition, varying holomorphically in families.
s_variation_of_hodge_structure	axiom	Variation of Hodge structure		A family of Hodge structures on a local system over a base S, with Hodge filtration varying holomorphically and satisfying Griffiths transversality.
s_dolbeault_cohomology	state	Dolbeault cohomology		The cohomology of the Dolbeault complex, computing sheaf cohomology H^q(X, Ω^p_X) on compact complex manifolds via the ∂̄-operator.
s_abel_jacobi_map	state	Abel–Jacobi map		The map sending a degree-zero divisor on a curve C to a point in the Jacobian via integration of holomorphic 1-forms, inducing an isomorphism Pic^0(C) ≅ J(C).
s_lefschetz_pencil	state	Lefschetz pencil		A one-parameter family of hyperplane sections of a projective variety with only isolated ordinary double point singularities in the total space.
s_pluricanonical_system	state	Pluricanonical system		The complete linear system of the m-th tensor power of the canonical bundle, whose behavior for large m determines the Kodaira dimension and the Iitaka fibratio
s_period_mapping	state	Period mapping		The holomorphic map from a parameter space of algebraic varieties to the period domain, sending each variety to its Hodge structure modulo monodromy.
s_dolbeault_theorem	theorem	Dolbeault theorem		The isomorphism H^{p,q}_{∂̄}(X) ≅ H^q(X, Ω^p_X) identifying Dolbeault cohomology with sheaf cohomology of holomorphic p-forms.
s_hard_lefschetz_theorem	theorem	Hard Lefschetz theorem		On a compact Kähler manifold of dimension n, the iterated cup product with the Kähler class gives isomorphisms L^k: H^{n−k}(X) → H^{n+k}(X) for all k ≥ 0.
s_lefschetz_1_1_theorem	theorem	Lefschetz (1,1) theorem		A class in H^2(X, ℤ) on a compact Kähler manifold is the first Chern class of a holomorphic line bundle if and only if its image in H^2(X, ℂ) lies in H^{1,1}(X)
s_riemann_roch_for_surfaces	theorem	Riemann–Roch for surfaces		For a divisor D on a smooth projective surface X: χ(𝒪_X(D)) = (1/2)D·(D − K_X) + χ(𝒪_X), combining intersection theory with Euler characteristic.
s_abel_theorem	theorem	Abel's theorem		A divisor of degree zero on a compact Riemann surface is principal if and only if its image under the Abel–Jacobi map is zero in the Jacobian.
s_jacobi_inversion_theorem	theorem	Jacobi inversion theorem		The Abel–Jacobi map from the g-fold symmetric product of a genus-g curve to the Jacobian is surjective, so every point of J(C) is represented by an effective di
s_torelli_theorem	theorem	Torelli theorem		A compact Riemann surface is determined up to isomorphism by its polarized Hodge structure, i.e., by its period matrix with the intersection pairing.
s_bertini_theorem	theorem	Bertini's theorem		A general member of a basepoint-free linear system on a smooth projective variety is smooth; more generally, the singular locus has the expected codimension.
s_gaga_principle	theorem	GAGA principle		Serre's theorem that on a projective algebraic variety over ℂ, the categories of algebraic and analytic coherent sheaves are equivalent, and their cohomologies 
s_riemann_bilinear_relations	theorem	Riemann bilinear relations		The conditions on the period matrix Ω of a compact Riemann surface: ΩJΩ^T = 0 and iΩJΩ̄^T > 0, characterizing which complex tori are Jacobians.
s_wirtinger_theorem	theorem	Wirtinger's theorem		The volume of a k-dimensional complex submanifold of a Kähler manifold equals (1/k!) times the integral of the k-th power of the Kähler form, and complex subman
s_newlander_nirenberg_theorem	theorem	Newlander–Nirenberg theorem		An almost complex structure J on a smooth manifold is integrable (comes from a complex structure) if and only if its Nijenhuis tensor vanishes.
s_castelnuovo_rationality_criterion	theorem	Castelnuovo's rationality criterion		A smooth projective surface over ℂ is rational if and only if its irregularity q = 0 and its second plurigenus P_2 = 0.
s_enriques_kodaira_classification	theorem	Enriques–Kodaira classification		The classification of compact complex surfaces into ten classes by Kodaira dimension (−∞, 0, 1, 2), refining Enriques's classification of algebraic surfaces.
s_exponential_sheaf_sequence	theorem	Exponential sheaf sequence		The short exact sequence 0 → ℤ → 𝒪_X → 𝒪_X^* → 0, whose long exact cohomology sequence relates Pic(X) to H^1(X, 𝒪_X) and H^2(X, ℤ).
s_grothendieck_residue_theorem	theorem	Grothendieck residue theorem		The global residue theorem for meromorphic n-forms on a compact complex n-manifold: the sum of all Grothendieck residues at the poles equals zero.
s_kodaira_spencer_deformation_theory	theorem	Kodaira–Spencer deformation theory		The theory describing infinitesimal deformations of compact complex manifolds via the Kodaira–Spencer map ρ: T_0 S → H^1(X, T_X), with obstructions in H^2(X, T_
t_dolbeault_cohomology_computation	technique	Dolbeault cohomology computation		Computing Dolbeault cohomology groups via ∂̄-closed forms modulo ∂̄-exact forms, using symmetry, vanishing theorems, or explicit representatives.
t_hodge_theoretic_methods	technique	Hodge-theoretic methods		Using Hodge decomposition, Lefschetz operators, and Hodge-Riemann bilinear relations to constrain cohomology, prove vanishing, or detect algebraic classes.
t_residue_calculus_on_manifolds	technique	Residue calculus on manifolds		Computing integrals and cohomology classes via Grothendieck residues and the residue exact sequence on complex manifolds.
t_schubert_calculus	technique	Schubert calculus		Computing intersection numbers on Grassmannians by multiplying Schubert classes using the Pieri and Giambelli formulas.
t_blow_up_technique	technique	Blow-up technique		Resolving singularities or separating infinitely near points by replacing a subvariety with its projectivized normal cone.
t_adjunction_formula_technique	technique	Adjunction formula technique		Computing the canonical class and genus of a subvariety by restricting the canonical and normal bundles from the ambient space.
t_lefschetz_pencil_method	technique	Lefschetz pencil method		Studying the topology of a projective variety by fibering it into hyperplane sections and analyzing vanishing cycles and monodromy.
t_period_mapping_torelli_arguments	technique	Period mapping and Torelli-type arguments		Using the period map to moduli of Hodge structures to prove uniqueness results or study moduli of algebraic varieties.
t_deformation_theory_methods	technique	Deformation theory methods		Studying infinitesimal and local deformations of complex structures via Kodaira–Spencer classes and obstruction theory in H^2(X, T_X).
s_smooth_variety	axiom	Smooth variety		A variety all of whose points are nonsingular, i.e., the local ring at every point is a regular local ring.
s_complete_variety	axiom	Complete variety		A variety X such that for every variety Y the projection X×Y → Y is a closed map; the algebraic analogue of compactness.
s_unirational_variety	axiom	Unirational variety		A variety X admitting a dominant rational map from projective space Pⁿ ⇢ X; equivalently, k(X) embeds into a purely transcendental extension.
s_rational_variety	axiom	Rational variety		A variety birational to projective space Pⁿ; equivalently, its function field is a purely transcendental extension k(t₁,…,tₙ).
s_algebraic_cycle	axiom	Algebraic cycle		A formal Z-linear combination of irreducible closed subvarieties of X; cycles of codimension p form Zᵖ(X), modulo rational equivalence yielding the Chow group C
s_pure_hodge_structure	axiom	Pure Hodge structure		A finite-dimensional Z-module H_Z with a decomposition H_C = ⊕_{p+q=n} H^{p,q} of its complexification satisfying H^{p,q} = conjugate(H^{q,p}).
s_plurigenera	state	Plurigenera		The birational invariants P_m = h⁰(X, mK_X) for m ≥ 1, measuring sections of pluricanonical bundles, fundamental in the classification of algebraic surfaces.
s_formal_neighborhood	state	Formal neighborhood		The formal scheme Spf(Ô_{X,p}) capturing the infinitesimal structure of a variety X at a point p up to all orders via the m-adic completion of the local ring.
s_multiplicity_of_a_singularity	state	Multiplicity of a singularity		The leading coefficient mult_p(X) of the Hilbert-Samuel polynomial of O_{X,p}, equal to the degree of the tangent cone; equals 1 if and only if p is smooth.
s_intersection_number_global	state	Intersection number (global)		For divisors D₁,…,Dₙ on a complete n-dimensional variety, the integer (D₁·…·Dₙ) defined by multilinearity and the moving lemma, counting intersections with mult
s_degree_of_a_finite_morphism	state	Degree of a finite morphism		For a dominant morphism f: X → Y of varieties of equal dimension, deg(f) = [k(X):k(Y)], the degree of the induced function field extension.
s_genus_formula_plane_curves	theorem	Genus formula for plane curves		A smooth projective plane curve of degree d has arithmetic genus g = (d−1)(d−2)/2, computed via the adjunction formula K_C = (K_{P²} + C)|_C.
s_chevalley_theorem_constructible_sets	theorem	Chevalley's theorem on constructible sets		The image of a morphism of finite type between Noetherian schemes is a constructible set (a finite Boolean combination of locally closed sets).
s_moving_lemma	theorem	Moving lemma		On a smooth quasi-projective variety, any two cycles of complementary dimension can be moved by rational equivalence to intersect properly.
s_noether_theorem_rational_curves_surfaces	theorem	Noether's theorem on rational curves on surfaces		On a smooth projective surface with ample canonical class, rational curves form a bounded family and do not cover the surface.
t_adjunction_formula	technique	Adjunction formula		Relating the canonical class of a smooth hypersurface to that of the ambient space via K_Y = (K_X + Y)|_Y.
t_linear_system_technique	technique	Linear system technique		Using a basis of the Riemann-Roch space L(D) to define a morphism to projective space for embeddings, contractions, and fibrations.
t_projection_from_point	technique	Projection from a point		The rational map sending x to the line ⟨p,x⟩ ∩ P^{n−1}; restricts to a finite morphism on varieties not containing p, reducing dimension.
t_normalization_technique	technique	Normalization of a variety		Taking the integral closure of the structure sheaf in the function field to produce a normal variety resolving non-normal singularities.
t_base_change_technique	technique	Base change technique		Forming the fiber product to study fibers, specialize, or extend scalars; with flatness yields cohomology base change isomorphisms.
t_descent_technique	technique	Descent technique		Recovering geometric objects over a base from data over a cover with gluing data satisfying the cocycle condition; includes faithfully flat and Galois descent.
t_specialization_argument	technique	Specialization argument		Deducing properties of a general fiber from a special fiber using semicontinuity and openness of geometric conditions in flat families.
t_generic_point_argument	technique	Generic point argument		Reducing geometric statements to the generic fiber, then spreading out to an open subset of the base using finite presentation arguments.
s_smooth_homotopy	axiom	Smooth homotopy		A smooth map F: M × [0,1] → N with F(·,0) = f and F(·,1) = g; the differentiable-topology analogue of continuous homotopy.
s_degree_mod_2	state	Degree modulo 2		For a smooth map f between compact manifolds of equal dimension, deg₂(f) = #f⁻¹(y) mod 2 for any regular value y; independent of the choice of regular value.
s_sign_of_preimage_point	axiom	Sign of a regular preimage point		The sign +1 or −1 assigned to x ∈ f⁻¹(y) according to whether df_x: T_xM → T_yN preserves or reverses orientation.
s_homotopy_invariance_of_degree	theorem	Homotopy invariance of Brouwer degree		If f, g: M → N are smoothly homotopic maps between compact oriented manifolds without boundary, then deg(f) = deg(g).
s_boundary_theorem_degree	theorem	Boundary theorem for degree		If f: ∂W → N extends to a smooth map F: W → N from a compact oriented manifold with boundary, then deg(f) = 0.
s_degree_of_composition	theorem	Degree of a composition		For smooth maps f: M → N and g: N → P between compact oriented manifolds of equal dimension, deg(g ∘ f) = deg(g) · deg(f).
s_degree_determines_surjectivity	theorem	Nonzero degree implies surjectivity		A smooth map f: M → N between compact connected oriented manifolds of equal dimension is surjective if deg(f) ≠ 0.
s_classification_of_smooth_compact_1_manifolds	theorem	Classification of compact 1-manifolds		Every compact connected smooth 1-manifold is diffeomorphic to [0,1] or to S¹.
s_smooth_approximation_theorem	theorem	Smooth approximation theorem		Every continuous map between smooth manifolds is homotopic to a smooth map; if already smooth on a closed subset, the homotopy can be taken relative to that sub
s_index_of_isolated_zero_of_vector_field	state	Index of an isolated zero of a vector field		The degree of the map x ↦ v(x)/|v(x)| from a small sphere around the isolated zero p to Sⁿ⁻¹, measuring the local winding of the vector field.
s_linking_number	state	Linking number		The integer lk(K,L) counting signed crossings of K over L in a diagram, equivalently the degree of the map (x,y) ↦ (x−y)/|x−y| from K × L to S².
s_hopf_fibration	state	Hopf fibration		The fiber bundle η: S³ → S² defined by η(z₁,z₂) = z₁/z₂ (in complex coordinates), with fiber S¹; the generator of π₃(S²) ≅ ℤ.
s_smooth_retraction	axiom	Smooth retraction		A smooth map r: W → A where A ⊆ W is a submanifold and r|_A = id_A; a smooth left inverse of the inclusion.
s_framed_submanifold	axiom	Framed submanifold		A submanifold N ⊆ M together with a trivialization of its normal bundle ν(N,M), i.e., a smooth choice of basis for each normal fiber.
s_framed_cobordism	axiom	Framed cobordism		An equivalence relation: N₀ and N₁ in M are framed cobordant if there exists a framed submanifold W ⊆ M × [0,1] whose boundary components are N₀ × {0} and N₁ × 
s_pontryagin_manifold	state	Pontryagin manifold		The framed submanifold f⁻¹(y) ⊆ Sⁿ⁺ᵏ for a regular value y ∈ Sⁿ, with framing induced by df; represents the homotopy class of f via the Pontryagin-Thom correspo
s_smooth_isotopy	axiom	Smooth isotopy		A smooth homotopy F: N × [0,1] → M such that F(·,t) is an embedding for every t ∈ [0,1]; a continuous path through the space of embeddings.
s_density_of_regular_values	theorem	Density of regular values		For a smooth map f: M → N, the set of regular values is dense in N; an immediate corollary of Sard's theorem since the set of critical values has measure zero.
s_well_definedness_of_mod_2_degree	theorem	Well-definedness of mod 2 degree		For a smooth map f: M → N between compact manifolds of equal dimension (M without boundary), the parity #f⁻¹(y) mod 2 is the same for every regular value y.
s_boundary_theorem_mod_2	theorem	Boundary theorem (mod 2)		If M = ∂W for a compact manifold W and f: M → N extends smoothly to F: W → N, then deg₂(f) = 0.
s_index_nondegenerate_zero_equals_sign_det	theorem	Index of a non-degenerate zero		If p is a non-degenerate zero of a vector field v (i.e., the Jacobian dv_p is nonsingular), then the index of v at p equals sign(det dv_p) = ±1.
s_nonvanishing_vector_fields_on_odd_spheres	theorem	Nonvanishing vector fields on odd-dimensional spheres		For every odd n, the sphere Sⁿ admits a nowhere-vanishing smooth tangent vector field, since χ(Sⁿ) = 0 when n is odd.
s_linking_number_homotopy_invariance	theorem	Linking number is a homotopy invariant		The linking number lk(K,L) is invariant under smooth isotopies of K and L that keep K and L disjoint throughout.
s_existence_of_maps_with_arbitrary_degree	theorem	Existence of maps of any given degree		For every integer d and every n ≥ 1, there exists a smooth map f: Sⁿ → Sⁿ of degree d.
s_browns_refinement_of_sard	theorem	Brown's refinement of Sard's theorem		The set of critical values of a Cᵏ map f: ℝⁿ → ℝᵐ has measure zero provided k ≥ max(n−m+1, 1), extending Sard's result to finite differentiability.
t_induction_on_dimension_sard	technique	Induction on dimension (Sard proof)		Prove Sard's theorem by inducting on the source dimension, stratifying the critical set by vanishing order of derivatives and applying Fubini's theorem on each 
t_proof_of_brouwer_via_sard	technique	Proof of Brouwer fixed point theorem via Sard		Derive Brouwer's fixed point theorem from the no-retraction theorem, which itself follows from Sard's theorem applied to a hypothetical retraction Dⁿ → Sⁿ⁻¹.
t_one_manifold_classification_technique	technique	One-manifold classification argument		Classify compact 1-manifolds by following flow lines or using local charts to show each component is diffeomorphic to S¹ or [0,1]; used to establish boundary an
t_diagonal_self_intersection	technique	Diagonal map and self-intersection number		Compute the Euler characteristic χ(M) as the self-intersection number of the diagonal Δ ⊆ M × M, connecting the Lefschetz number of the identity to the Poincaré
t_degree_based_surjectivity	technique	Degree-based surjectivity argument		Show a map is surjective by computing its degree as nonzero; every point in N has a nonempty preimage since the signed count of preimages is constant over regul
s_local_parametrization	axiom	Local parametrization		A diffeomorphism phi from an open subset of R^k onto an open neighborhood of a point in a k-dimensional manifold, providing local coordinates.
s_derivative_of_smooth_map_between_manifolds	axiom	Derivative of a smooth map between manifolds		The linear map df_x: T_xX -> T_{f(x)}Y induced by a smooth map f between manifolds, defined via local parametrizations as df_x = d(psi^{-1} circ f circ phi)_0.
s_tangent_space_of_preimage	state	Tangent space of a preimage		At a regular value y, the tangent space to the preimage f^{-1}(y) at a point x equals the kernel of the derivative df_x: T_xX -> T_yY.
s_transversal_preimage_theorem	theorem	Transversal preimage theorem		If f: X -> Y is smooth and transverse to a submanifold Z subset Y, then f^{-1}(Z) is a submanifold of X with codim f^{-1}(Z) = codim Z.
s_transverse_intersection_dimension_formula	theorem	Transverse intersection dimension formula		If submanifolds X and Z intersect transversally in Y, then X cap Z is a submanifold of dimension dim X + dim Z - dim Y.
s_stability_of_immersions	theorem	Stability of immersions		If f: X -> Y is an immersion and X is compact, then every map sufficiently close to f in the C^1 topology is also an immersion.
s_stability_of_submersions	theorem	Stability of submersions		If f: X -> Y is a submersion and X is compact, then every map sufficiently close to f in the C^1 topology is also a submersion.
s_stability_of_embeddings	theorem	Stability of embeddings		If f: X -> Y is an embedding and X is compact, then every map sufficiently close to f in the C^1 topology is also an embedding.
s_stability_of_diffeomorphisms	theorem	Stability of diffeomorphisms		If f: X -> Y is a diffeomorphism between compact manifolds, then every map sufficiently close to f in the C^1 topology is also a diffeomorphism.
s_stability_of_transversality	theorem	Stability of transversality		If f: X -> Y is transverse to a submanifold Z subset Y and X is compact, then every map sufficiently close to f in the C^1 topology is also transverse to Z.
s_nondegenerate_critical_points_are_isolated	theorem	Nondegenerate critical points are isolated		A nondegenerate critical point of a smooth function (one with nonsingular Hessian) is an isolated critical point.
s_compact_morse_finite_critical_points	theorem	Compact manifolds have finitely many Morse critical points		A Morse function on a compact manifold has only finitely many critical points, since nondegenerate critical points are isolated.
s_boundary_is_manifold	theorem	Boundary of a manifold is a manifold		If X is a k-dimensional manifold with boundary, then the boundary dX is a smooth (k-1)-dimensional manifold without boundary.
s_epsilon_neighborhood_theorem	theorem	Epsilon-neighborhood theorem		Every compact manifold X embedded in R^N possesses an epsilon-neighborhood X^epsilon that is a tubular neighborhood diffeomorphic to the normal bundle of X.
s_transversality_homotopy_theorem	theorem	Transversality homotopy theorem		For any smooth map f: X -> Y and submanifold Z subset Y, there exists a smooth map g homotopic to f such that g is transverse to Z.
s_extension_theorem_transversality	theorem	Extension theorem for transversality		If f: X -> Y is already transverse to Z on a closed subset C subset X, then f can be deformed rel C to a map transverse to Z on all of X.
s_intersection_number_mod_2	axiom	Intersection number mod 2		The mod 2 count I_2(f, Z) = #f^{-1}(Z) mod 2 for f transverse to Z, where X is compact without boundary and dim X = codim Z; a homotopy invariant.
s_intersection_number_mod_2_homotopy_invariance	theorem	Homotopy invariance of intersection number mod 2		If f_0 and f_1 are homotopic smooth maps from a compact boundaryless manifold X to Y, then I_2(f_0, Z) = I_2(f_1, Z) for any closed submanifold Z.
s_mod_2_intersection_number_of_submanifolds	state	Mod 2 intersection number of submanifolds		For complementary-dimensional compact boundaryless submanifolds X, Z of Y, the mod 2 intersection number I_2(X, Z) defined via the inclusion map.
s_degree_mod_2_homotopy_invariance	theorem	Homotopy invariance of degree mod 2		If f, g: X -> Y are homotopic smooth maps between compact connected boundaryless manifolds of equal dimension, then deg_2(f) = deg_2(g).
s_no_retraction_theorem	theorem	No retraction theorem		There is no smooth retraction from the closed unit ball B^n to its boundary sphere S^{n-1}, proved via degree theory.
s_brouwer_fixed_point_mod_2	theorem	Brouwer fixed-point theorem (mod 2 proof)		Every smooth map f: B^n -> B^n has a fixed point, proved via the no-retraction theorem using mod 2 degree theory.
s_oriented_intersection_number	state	Oriented intersection number		The integer I(f, Z) = sum_{x in f^{-1}(Z)} sign(x) for f transverse to Z, where X, Y, Z are oriented and X is compact without boundary.
s_oriented_intersection_number_homotopy_invariance	theorem	Homotopy invariance of oriented intersection number		If f_0 and f_1 are smoothly homotopic maps from a compact oriented boundaryless manifold to an oriented manifold Y, then I(f_0, Z) = I(f_1, Z).
s_oriented_intersection_number_of_submanifolds	state	Oriented intersection number of submanifolds		For complementary-dimensional compact oriented boundaryless submanifolds X, Z of an oriented manifold Y, the integer I(X, Z) defined via the inclusion.
s_oriented_boundary_theorem	theorem	Oriented boundary theorem		If f: X -> Y extends to a smooth map F: W -> Y where dW = X with induced orientation, then I(f, Z) = 0 for every closed oriented submanifold Z of complementary 
s_brouwer_degree	state	Brouwer degree		The integer deg(f) = sum_{x in f^{-1}(y)} sign(df_x) for any regular value y, well-defined and independent of the choice of regular value.
s_hopf_degree_theorem	theorem	Hopf degree theorem		Two smooth maps f, g: S^n -> S^n are smoothly homotopic if and only if deg(f) = deg(g), establishing [S^n, S^n] = Z.
s_local_lefschetz_number	state	Local Lefschetz number		The local contribution of an isolated fixed point x to the Lefschetz number, equal to sign(det(I - df_x)) when x is a nondegenerate fixed point.
s_euler_characteristic_via_diagonal_intersection	theorem	Euler characteristic as self-intersection of diagonal		For a compact oriented manifold X, the Euler characteristic equals chi(X) = I(Delta, Delta), the self-intersection number of the diagonal Delta subset X x X.
s_euler_equals_lefschetz_of_identity	theorem	Euler characteristic equals Lefschetz number of identity		For a compact oriented manifold X, chi(X) = L(id_X), since the graph of the identity is the diagonal in X x X.
s_self_intersection_number	state	Self-intersection number		The oriented intersection number I(X, X) computed by perturbing X to a transverse copy via a section of the normal bundle, then counting signed intersections.
s_euler_characteristic_of_spheres	theorem	Euler characteristic of spheres		The Euler characteristic of the n-sphere is chi(S^n) = 1 + (-1)^n, which equals 2 for even n and 0 for odd n.
s_brouwer_fixed_point_oriented	theorem	Brouwer fixed-point theorem (oriented proof)		Every smooth map f: B^n -> B^n has a fixed point, proved using the Lefschetz fixed-point theorem: L(f) = 1 since B^n is contractible.
s_intersection_number_symmetry	theorem	Intersection number symmetry		For compact oriented boundaryless submanifolds X, Z of complementary dimension in Y, I(X, Z) = (-1)^{dim X * dim Z} I(Z, X).
s_pullback_commutes_with_wedge	theorem	Pullback commutes with wedge product		For a smooth map f: X -> Y and differential forms omega, eta on Y, the pullback satisfies f*(omega wedge eta) = f*omega wedge f*eta.
s_change_of_variables_for_forms	theorem	Change of variables for differential forms		If phi: X -> Y is an orientation-preserving diffeomorphism of compact oriented manifolds, then integral_Y omega = integral_X phi*omega for every n-form omega on
s_degree_formula_via_integration	theorem	Degree formula via integration		For a smooth map f: M -> N between compact oriented manifolds of equal dimension, deg(f) * integral_N omega = integral_M f*omega for every n-form omega on N.
t_dimension_counting_transversality	technique	Dimension counting for transversality		When dim X < codim Z in Y, a generic smooth map f: X -> Y has empty preimage f^{-1}(Z), since transversality forces f^{-1}(Z) to have negative dimension.
t_parametric_transversality_method	technique	Parametric transversality method (Thom's method)		Achieve transversality for a specific map by embedding it in a parametric family and applying Sard's theorem to the parameter space.
t_cobordism_argument	technique	Cobordism argument		Prove a topological invariant is unchanged by exhibiting a cobordism W between two manifolds and extending the map over W, using the boundary theorem.
t_isotopy_deformation_argument	technique	Isotopy and deformation argument		Deform a smooth map through a one-parameter family to achieve a desired geometric property such as transversality or general position.
t_diagonal_intersection_method	technique	Diagonal intersection method		Detect and count fixed points of a self-map f by computing the intersection number of the diagonal Delta with the graph of f inside X x X.
t_degree_counting_preimages	technique	Degree computation by counting preimages		Compute the Brouwer degree of a smooth map between compact oriented manifolds by choosing a regular value and summing the orientation signs at all preimage poin
t_integration_pairing_topological_invariants	technique	Integration pairing for topological invariants		Pair a de Rham cohomology class with a homology cycle by integration, yielding topological invariants such as the degree, Euler characteristic, or linking numbe
s_compactly_supported_de_rham_cohomology	axiom	Compactly supported de Rham cohomology		The cohomology H^k_c(M) of the complex Ω^*_c(M) of differential forms with compact support, covariant for proper maps and contravariant (with reversed arrows) f
s_compact_vertical_support_cohomology	axiom	Cohomology with compact vertical support		The cohomology H^*_cv(E) of forms on a vector bundle E that have compact support along the fibers, used to define the Thom class and prove the Thom isomorphism 
s_bump_form	axiom	Bump form		A smooth compactly supported n-form on ℝⁿ with total integral 1, serving as the generator of H^n_c(ℝⁿ) ≅ ℝ and the basic building block for Thom forms and Poinc
s_relative_de_rham_cohomology	axiom	Relative de Rham cohomology		The cohomology H*(Ω*(M, A)) of the complex of forms ω on M with i*ω = 0 on A, fitting into a long exact sequence with H*(M) and H*(A).
s_acyclic_cover	axiom	Acyclic cover (Leray cover)		An open cover U = {U_α} such that every finite intersection U_{α₀} ∩ ··· ∩ U_{αₚ} has vanishing higher cohomology with respect to the sheaf in question; for suc
s_tautological_bundle_grassmannian	axiom	Tautological bundle over the Grassmannian		The rank-k vector bundle γ^k → G_k(V) whose fiber over a k-plane W ∈ G_k(V) is W itself; the universal bundle for rank-k vector bundles, pulling back to any ran
s_integration_along_fiber	state	Integration along the fiber		The pushforward map π_*: Ω^{n+k}(E) → Ω^n(M) defined by fiberwise integration of (n+k)-forms over the k-dimensional fiber, commuting with d and inducing a map o
s_global_angular_form	state	Global angular form		A (k−1)-form ψ on the total space of an oriented sphere bundle S^{k-1} → E → M whose restriction to each fiber is the volume form of S^{k-1}, with dψ = −π*e(E) 
s_diagonal_class	state	Diagonal class		The cohomology class δ_M ∈ H^n(M × M) that is Poincaré dual to the diagonal Δ ⊂ M × M; under the Künneth decomposition δ_M = Σ_a η_a × η_a^∨ where {η_a} and {η_
s_spectral_sequence_collapse	state	Collapsing of a spectral sequence		A spectral sequence collapses at the E_r page if d_k = 0 for all k ≥ r, so E_r ≅ E_{r+1} ≅ ··· ≅ E_∞; this occurs when E_r^{p,q} = 0 unless p or q lies in a sin
s_cohomology_of_SO_n	state	Cohomology of SO(n)		The de Rham cohomology ring of SO(n), computed inductively using the Wang sequence for the fibration SO(n−1) → SO(n) → S^{n−1}; for odd n, H*(SO(2k+1)) is an ex
s_flag_bundle	state	Flag bundle		The associated fiber bundle Fl(E) → M whose fiber over x is the complete flag manifold of E_x; the pullback of E to Fl(E) splits as a direct sum of line bundles
s_poincare_lemma_compact_support	theorem	Poincaré lemma for compactly supported cohomology		H^k_c(ℝⁿ) ≅ ℝ if k = n and H^k_c(ℝⁿ) = 0 otherwise, the compactly supported analogue of the Poincaré lemma with cohomology concentrated in the top degree.
s_mayer_vietoris_compact_support	theorem	Mayer-Vietoris sequence for compact supports		For M = U ∪ V open, the long exact sequence ··· → H^k_c(U∩V) → H^k_c(U) ⊕ H^k_c(V) → H^k_c(M) → H^{k+1}_c(U∩V) → ···, with arrows reversed relative to the ordin
s_finite_dimensionality_de_rham	theorem	Finite-dimensionality of de Rham cohomology		For a manifold admitting a finite good cover, each H^k_{dR}(M) is a finite-dimensional vector space, proved by Mayer-Vietoris induction on the number of open se
s_generalized_mayer_vietoris_principle	theorem	Generalized Mayer-Vietoris principle		For a good cover U of M, the natural maps Ω*(M) → Č*(U, Ω*) ← Č*(U, ℝ) are quasi-isomorphisms of the de Rham complex into and out of the Čech-de Rham double com
s_partition_of_unity_presheaf_acyclicity	theorem	Partition of unity argument for presheaves of forms		The presheaf Ω^q of differential q-forms on a manifold has vanishing higher Čech cohomology Ȟ^p(U, Ω^q) = 0 for p ≥ 1 with respect to any open cover, because sm
s_long_exact_sequence_compact_support	theorem	Long exact sequence of a pair for compact supports		For U ⊂ M open, the long exact sequence ··· → H^k_c(U) → H^k_c(M) → H^k_c(M \ U) → H^{k+1}_c(U ∩ V) → ··· induced by extension by zero of compactly supported fo
s_infinite_grassmannian_classifying_space	theorem	Infinite Grassmannian as classifying space		G_k(ℝ^∞) ≅ BO(k) and G_k(ℂ^∞) ≅ BU(k); every rank-k real (resp. complex) vector bundle over a paracompact base is the pullback of the tautological bundle via a 
s_cohomology_of_BU_k	theorem	Cohomology of BU(k)		H*(BU(k); ℝ) ≅ ℝ[c₁, c₂, …, cₖ], a polynomial ring generated by the universal Chern classes cᵢ in degree 2i, computed via the Leray-Hirsch theorem and induction
s_cohomology_of_BO_k	theorem	Cohomology of BO(k)		H*(BO(k); ℝ) ≅ ℝ[p₁, p₂, …, p_{⌊k/2⌋}] (times an additional Euler class generator e in degree k when k is even), a polynomial ring in the universal Pontryagin c
s_naturality_of_characteristic_classes	theorem	Naturality of characteristic classes		For a smooth map f: N → M and vector bundle E → M, the characteristic classes satisfy c_k(f*E) = f*(c_k(E)), and similarly for Pontryagin, Euler, and Stiefel-Wh
s_homotopy_axiom_cech_cohomology	theorem	Homotopy axiom for Čech cohomology		If f: X → Y is a homotopy equivalence between spaces admitting good covers, then f*: Ȟ*(Y; ℝ) → Ȟ*(X; ℝ) is an isomorphism; proved via the generalized Mayer-Vie
t_zigzag_double_complex	technique	Zigzag argument in a double complex		Constructing a global cocycle from compatible local data by alternately applying horizontal and vertical coboundary operators in a double complex, descending th
t_cochain_homotopy	technique	Cochain homotopy construction		Explicitly constructing a degree −1 operator K via integration along the fiber of the projection M × [0,1] → M, proving that homotopic maps induce identical map
t_compact_exhaustion_direct_limit	technique	Compact exhaustion and direct limit for H*_c		Computing compactly supported cohomology of a non-compact manifold as the direct limit over an exhaustion by precompact open subsets, using extension by zero to
s_symplectic_subspace	axiom	Symplectic subspace		A linear subspace W of a symplectic vector space (V,ω) such that ω|_W is nondegenerate; equivalently W ∩ W^ω = {0}.
s_isotropic_subspace	axiom	Isotropic subspace		A linear subspace W of (V,ω) with ω|_W = 0, equivalently W ⊂ W^ω; necessarily dim W ≤ n.
s_coisotropic_subspace	axiom	Coisotropic subspace		A linear subspace W of (V,ω) satisfying W^ω ⊂ W; necessarily dim W ≥ n.
s_lagrangian_subspace	axiom	Lagrangian subspace		A linear subspace L of (V,ω) with L = L^ω, equivalently isotropic of maximal dimension n; simultaneously isotropic and coisotropic.
s_symplectic_complement	state	Symplectic complement		The symplectic complement W^ω = {v ∈ V : ω(v,w) = 0 for all w ∈ W}; satisfies dim W + dim W^ω = dim V and (W^ω)^ω = W.
s_lagrangian_grassmannian	state	Lagrangian Grassmannian Λ(n)		The manifold of all Lagrangian subspaces of (ℝ²ⁿ, ω₀), diffeomorphic to U(n)/O(n); has dimension n(n+1)/2 and fundamental group π₁(Λ(n)) ≅ ℤ.
s_compatible_complex_structure_symplectic	axiom	Compatible complex structure on symplectic vector space		A linear map J with J² = -Id such that g(u,v) = ω(u,Jv) defines a positive definite inner product; equivalently ω(Ju,Jv) = ω(u,v) and ω(v,Jv) > 0.
s_positive_compatible_triple	state	Positive compatible triple (ω, J, g)		A triple (ω, J, g) on a vector space V where any two determine the third via g(u,v) = ω(u,Jv); the space of such triples is contractible.
s_space_of_compatible_complex_structures	state	Space of compatible complex structures J(V,ω)		The space of all ω-compatible complex structures on V, diffeomorphic to Sp(2n)/U(n); contractible, hence all compatible choices are homotopy equivalent.
s_polar_decomposition_sp2n	theorem	Polar decomposition in Sp(2n)		Every A ∈ Sp(2n) decomposes uniquely as A = PO with P symmetric positive definite symplectic and O ∈ Sp(2n) ∩ O(2n) = U(n); proves Sp(2n) deformation retracts o
s_maslov_index_for_paths	state	Maslov index for paths of Lagrangian subspaces		An integer index for a path of Lagrangian subspaces relative to a fixed Lagrangian L, defined by counting signed crossings of the Maslov cycle Σ_L; extends the 
s_maslov_cycle	state	Maslov cycle		The algebraic variety Σ_L = {L' ∈ Λ(n) : L' ∩ L ≠ {0}} of codimension 1 in the Lagrangian Grassmannian; its complement has two connected components and generate
s_conley_zehnder_index	state	Conley-Zehnder index		An integer-valued index for nondegenerate paths in Sp(2n) starting at the identity; assigns a winding number measuring the rotation of eigenvalues; fundamental 
s_moser_isotopy_lemma_relative	theorem	Moser isotopy lemma (relative version)		If two symplectic forms on a compact manifold agree on a neighborhood of a submanifold and are connected by a path with constant cohomology class, there exists 
s_weinstein_isotropic_neighbourhood	theorem	Weinstein isotropic neighborhood theorem		A neighborhood of a compact isotropic submanifold N in (M,ω) is symplectomorphic to a standard neighborhood determined by the symplectic normal bundle; generali
s_isotropic_submanifold	axiom	Isotropic submanifold		A submanifold N of (M,ω) with ω|_N = 0, equivalently TₚN is isotropic in (TₚM, ωₚ) for all p; dim N ≤ n.
s_coisotropic_submanifold	axiom	Coisotropic submanifold		A submanifold C of (M,ω) with (TₚC)^ω ⊂ TₚC for all p; the null distribution (TC)^ω is integrable, giving the characteristic foliation.
s_symplectic_fiber_bundle	state	Symplectic fiber bundle		A locally trivial fiber bundle whose fiber is a symplectic manifold (F,σ) and whose structure group is the symplectomorphism group Symp(F,σ).
s_symplectic_connection_on_fibration	state	Symplectic connection on a fibration		A horizontal distribution on a symplectic fibration whose parallel transport maps are symplectomorphisms of fibers; equivalently an Ehresmann connection preserv
s_hamiltonian_isotopy	axiom	Hamiltonian isotopy		A smooth family of symplectomorphisms {φ_t}_{t∈[0,1]} with φ₀ = id generated by a time-dependent Hamiltonian H_t via dφ_t/dt = X_{H_t} ∘ φ_t; two Hamiltonian is
s_poisson_bracket	state	Poisson bracket		The bracket {F,G} = ω(X_F, X_G) = X_F(G) making C^∞(M) a Lie algebra; satisfies Leibniz rule, antisymmetry, Jacobi identity; {F,G} = 0 iff F is constant along X
s_existence_uniqueness_moment_maps	theorem	Existence and uniqueness of moment maps		A symplectic G-action admits a moment map iff the lifted map g → C^∞(M) is a Lie algebra homomorphism; obstructions lie in H¹(g, H¹(M;ℝ)) and uniqueness fails b
s_symplectic_reduction_at_general_level	theorem	Symplectic reduction at a general level		For coadjoint orbit value ξ, if G_ξ acts freely on μ⁻¹(ξ), the quotient μ⁻¹(ξ)/G_ξ inherits a natural symplectic structure; generalizes Marsden-Weinstein reduct
s_coadjoint_orbit	state	Coadjoint orbit		The orbit O_ξ = {Ad*_g(ξ) : g ∈ G} ⊂ g* under the coadjoint action; carries the Kirillov-Kostant-Souriau symplectic form ω_ξ(ad*_X ξ, ad*_Y ξ) = ξ([X,Y]).
s_coadjoint_orbits_symplectic	theorem	Coadjoint orbits as symplectic manifolds		Every coadjoint orbit of a Lie group carries a natural symplectic structure via the Kirillov-Kostant-Souriau form; the inclusion O_ξ ↪ g* is a moment map for th
s_flux_homomorphism	state	Flux homomorphism		The homomorphism Flux({φ_t}) = ∫₀¹ [ι_{X_t}ω] dt from the universal cover of Symp₀(M,ω) to H¹(M;ℝ); its kernel is the Hamiltonian group Ham(M,ω).
s_flux_conjecture_ono	theorem	Flux conjecture (Ono)		The flux group Γ_ω = Image(Flux: π₁(Symp₀) → H¹(M;ℝ)) is discrete; equivalently Ham(M,ω) is C¹-closed in Symp₀(M,ω). Proved by Ono (2006).
s_calabi_homomorphism	state	Calabi homomorphism		A surjective homomorphism Cal: Ham_c(M,ω) → ℝ defined by Cal(φ_H) = ∫₀¹ ∫_M H_t ωⁿ dt for compactly supported Hamiltonian diffeomorphisms; its kernel is a simpl
s_hofer_metric	axiom	Hofer metric on Ham(M,ω)		The bi-invariant Finsler metric d(φ,ψ) = inf_H ∫₀¹ (max H_t - min H_t) dt on Ham(M,ω); nondegeneracy proved by Hofer using holomorphic curves or capacities.
s_hofer_nondegeneracy_theorem	theorem	Hofer's nondegeneracy theorem		The Hofer metric on Ham(M,ω) is nondegenerate: d(id, φ) > 0 for φ ≠ id; equivalently the Hofer norm ||φ|| = inf_H ∫₀¹ osc(H_t) dt is a genuine norm.
s_hofer_zehnder_capacity	state	Hofer-Zehnder capacity		The symplectic capacity c_{HZ}(M,ω) = sup{max H : H admissible, flow of H has no nonconstant periodic orbit of period ≤ 1}; satisfies monotonicity, conformality
s_displacement_energy	state	Displacement energy		e(A) = inf{||φ||_Hofer : φ ∈ Ham(M,ω), φ(A) ∩ A = ∅}; satisfies e(A) ≥ c(A) for any capacity c, providing a lower bound on energy needed to displace A.
s_tame_almost_complex_structure	axiom	Tame almost complex structure		An almost complex structure J on (M,ω) such that ω(v, Jv) > 0 for all nonzero v; weaker than compatibility but sufficient for Gromov's theory of J-holomorphic c
s_compatible_almost_complex_structure_manifold	axiom	Compatible almost complex structure on symplectic manifold		An almost complex structure J satisfying ω(v,Jv) > 0 and ω(Jv,Jw) = ω(v,w); the triple (ω,J,g) with g = ω(·,J·) is a compatible Hermitian structure.
s_contractibility_of_J_M_omega	theorem	Contractibility of J(M,ω)		The space J(M,ω) of ω-compatible (or ω-tame) almost complex structures on a symplectic manifold is nonempty and contractible; proved using the pointwise contrac
s_symplectic_capacity_axiomatic	axiom	Symplectic capacity (axiomatic)		A functor c satisfying monotonicity under symplectic embeddings, conformality c(M,λw) = |λ|c(M,ω), and normalization c(B²ⁿ(1)) = π = c(Z²ⁿ(1)).
s_gromov_width	state	Gromov width		w_G(M,ω) = sup{πr² : B²ⁿ(r) symplectically embeds into (M,ω)}; the smallest symplectic capacity, measuring the radius of the largest embeddable ball.
s_ekeland_hofer_capacities	state	Ekeland-Hofer capacities		An infinite sequence of symplectic capacities c_k defined via variational methods on the loop space; c₁ equals the Gromov width on ellipsoids and c_k(B²ⁿ(1)) = 
s_j_holomorphic_curve	axiom	J-holomorphic curve		A smooth map u: (Σ,j) → (M,J) satisfying the Cauchy-Riemann equation du ∘ j = J ∘ du, equivalently ∂̄_J(u) = 0; the fundamental object of Gromov's symplectic to
s_energy_j_holomorphic_curve	state	Energy of a J-holomorphic curve		E(u) = ½∫_Σ |du|² vol_Σ = ∫_Σ u*ω for ω-tame J; the energy identity E(u) = [ω]·[u] shows energy equals the symplectic area of the homology class.
s_energy_identity_j_holomorphic	theorem	Energy identity for J-holomorphic curves		For an ω-compatible J, the energy E(u) = ½∫|du|² equals the symplectic area ∫u*ω; implies J-holomorphic curves are absolute minima of energy in their homology c
s_moduli_space_j_holomorphic	state	Moduli space of J-holomorphic curves		The space of equivalence classes of J-holomorphic maps u: Σ → M representing class A with k marked points, modulo reparametrization; for generic J, a smooth man
s_fredholm_theory_dbar_J	state	Fredholm theory for the ∂̄_J operator		The linearization of the Cauchy-Riemann operator at a J-holomorphic curve u is a real Cauchy-Riemann operator, hence Fredholm; its index gives the virtual dimen
s_virtual_dimension_moduli	state	Virtual dimension of moduli space of J-holomorphic curves		virdim M_{g,k}(M,A,J) = n(2-2g) + 2c₁(A) + 2k - 6; determined by the Fredholm index of D_u via Riemann-Roch; equals actual dimension when transversality holds.
s_simple_j_holomorphic_curve	axiom	Simple (somewhere injective) J-holomorphic curve		A J-holomorphic curve u: Σ → M not factoring as u = v ∘ φ for a branched cover φ of degree > 1; equivalently ∃z with u⁻¹(u(z)) = {z} and du(z) ≠ 0. Transversali
s_transversality_simple_curves	theorem	Transversality for simple J-holomorphic curves		For a generic (Baire second category) ω-tame J, every simple J-holomorphic curve is regular (D_u surjective) and the moduli space is a smooth manifold of the ex
s_bubbling_phenomenon	state	Bubbling phenomenon for J-holomorphic curves		When a sequence of J-holomorphic curves has bounded energy but gradient blow-up, rescaling produces a nonconstant J-holomorphic sphere; the energy splits: E(u_∞
s_stable_map_gw	axiom	Stable map		A J-holomorphic map from a nodal marked Riemann surface with finite automorphism group on each unstable component; the objects compactifying the moduli space M̄
s_evaluation_map_gw	state	Evaluation map in Gromov-Witten theory		The map ev = (ev₁,...,ev_k) sending a stable map to the images of its marked points; used to define GW invariants as ∫_{[M̄]^{vir}} ev₁*α₁ ∧ ⋯ ∧ ev_k*α_k.
s_removal_of_singularities_j_holo	theorem	Removal of singularities for J-holomorphic curves		A J-holomorphic map u: D\{0} → (M,J) with finite energy extends smoothly across the puncture to ū: D → M; the symplectic analogue of Riemann's removable singula
s_monotonicity_lemma_j_holo	theorem	Monotonicity lemma for J-holomorphic curves		If u: Σ → (M,J,g) is J-holomorphic passing through p, the area of u(Σ) ∩ B(p,r) is at least πr² for small r; provides uniform local area bounds preventing energ
s_symplectic_packing	state	Symplectic packing problem		The problem of determining the maximal volume fraction fillable by k disjoint symplectically embedded balls; Gromov showed the answer is not always 1, revealing
s_symplectic_form_on_surfaces	state	Symplectic form on surfaces		On a closed oriented surface, any area form is symplectic; Moser's theorem implies the symplectic structure is determined up to symplectomorphism by total area 
s_eliashberg_rigidity	theorem	Eliashberg rigidity theorem		The group Symp(M,ω) is C⁰-closed in Diff(M); a C⁰-limit of symplectomorphisms that is smooth is itself a symplectomorphism. Shows symplectic topology is not pur
s_mcduff_rational_ruled	theorem	McDuff's theorem on rational and ruled symplectic 4-manifolds		A closed symplectic 4-manifold containing a symplectically embedded sphere of nonnegative self-intersection is rational or ruled; proved using J-holomorphic cur
s_taubes_sw_equals_gr	theorem	Taubes' theorem (SW = Gr)		On a closed symplectic 4-manifold, Seiberg-Witten invariants equal the Gromov invariants counting embedded J-holomorphic curves; establishes SW(M, s_can) = 1 br
s_existence_j_holo_curves_taubes	theorem	Existence of J-holomorphic curves via Taubes		On any closed symplectic 4-manifold with b₂⁺ > 1, the canonical class K has nonvanishing SW invariant; by SW = Gr there exist J-holomorphic curves representing 
s_symplectic_isotopy_problem	state	Symplectic isotopy problem		The question of whether symplectomorphic submanifolds in the same homology class are symplectically isotopic; related to the topology of Symp(M,ω) and braid mon
s_positivity_of_intersections	theorem	Positivity of intersections of J-holomorphic curves		Two distinct J-holomorphic curves in an almost complex 4-manifold intersect with positive local intersection number at each point; algebraic and geometric inter
s_adjunction_formula_j_holomorphic	theorem	Adjunction formula for J-holomorphic curves		For a simple closed J-holomorphic curve u: Σ → M⁴ representing A: 2g(Σ) - 2 = A·A - c₁(A) + 2δ(u) where δ(u) ≥ 0 counts singularities; embedded iff δ(u) = 0.
s_recurrence_hamiltonian	theorem	Recurrence theorem for Hamiltonian diffeomorphisms		On a closed symplectic manifold, every Hamiltonian diffeomorphism is recurrent: for every open set U, there exists n > 0 with φⁿ(U) ∩ U ≠ ∅; follows from Liouvi
s_liouville_measure_symplectic	state	Liouville measure on a symplectic manifold		The canonical volume measure μ_L = ωⁿ/n! on a 2n-dimensional symplectic manifold; invariant under all symplectomorphisms and Hamiltonian flows (Liouville's theo
s_contact_hypersurface	axiom	Contact-type hypersurface in a symplectic manifold		A hypersurface Σ ⊂ (M,ω) admitting a Liouville vector field Y (L_Yω = ω) defined near Σ and transverse to Σ; then α = ι_Yω|_Σ is a contact form on Σ.
s_liouville_vector_field	state	Liouville vector field		A vector field Y satisfying L_Yω = ω, equivalently d(ι_Yω) = ω; its flow dilates the symplectic form by e^t. Exists globally on exact symplectic manifolds where
s_symplectization	state	Symplectization of a contact manifold		The symplectic manifold (ℝ × Σ, d(e^t α)) associated to a contact manifold (Σ, α); equivalently the symplectic cone (ℝ₊ × Σ, d(rα)).
s_symplectic_vector_bundle	axiom	Symplectic vector bundle		A vector bundle E → M with a smoothly varying nondegenerate skew-symmetric bilinear form on each fiber; the structure group reduces from GL(2n) to Sp(2n).
s_characteristic_foliation_coisotropic	state	Characteristic foliation on a coisotropic submanifold		The integrable distribution (TC)^ω on C whose leaves are isotropic; the leaf space C/(TC)^ω inherits a symplectic structure when smooth (symplectic reduction).
s_exact_lagrangian_submanifold	axiom	Exact Lagrangian submanifold		A Lagrangian L in (M, dλ) with λ|_L exact (λ|_L = df for f: L → ℝ); in T*Q the zero section is the canonical example.
s_exact_symplectic_manifold	axiom	Exact symplectic manifold		A symplectic manifold (M,ω) with ω = dλ for a global 1-form λ; necessarily noncompact since ∫_M ωⁿ ≠ 0. Examples: T*Q, Stein manifolds.
s_spectral_invariants_floer	state	Spectral invariants in Hamiltonian Floer theory		Real numbers c(a, H) defined as minimax values of the action functional over Floer cycles representing a ∈ QH*(M); satisfy Hofer-Lipschitz continuity, spectrali
s_floer_homology_hamiltonian	state	Hamiltonian Floer homology		Homology of a chain complex generated by 1-periodic orbits of X_H with differential counting Floer cylinders; isomorphic to quantum homology QH*(M) by the PSS i
s_energy_capacity_inequality	theorem	Energy-capacity inequality		For any compact A ⊂ (M,ω), the displacement energy e(A) ≥ c(A) for any symplectic capacity c; in particular e(B²ⁿ(r)) ≥ πr². Links Hofer geometry to symplectic 
t_moser_method	technique	Moser's method		Constructs a diffeomorphism isotopy by solving for the vector field X_t via ι_{X_t}ω_t = -σ_t; the key technique behind Darboux, Moser stability, and Weinstein 
t_thurston_symplectic_fibration	technique	Thurston's construction for symplectic fiber bundles		Constructs a symplectic form on the total space of a symplectic fibration by adding a large multiple of the pullback of the base form to a closed extension of t
t_symplectic_cutting	technique	Symplectic cutting		Lerman's construction: forms M × ℂ with diagonal S¹-action and reduces at level c to obtain a symplectic manifold that cuts M along μ⁻¹(c).
t_generating_function_lagrangian	technique	Generating function for Lagrangians		Represents a Lagrangian as the critical set of a generating function S(x,ξ); reduces symplectic intersection questions to Morse-theoretic ones via the Chaperon-
t_neck_stretching	technique	Neck-stretching technique		Deforms the almost complex structure near a contact-type hypersurface by stretching the neck to infinity; in the limit, J-holomorphic curves decompose into holo
t_gromov_compactification	technique	Gromov compactification of moduli spaces		Compactifies the moduli space of J-holomorphic curves by adding nodal stable maps as boundary strata; the key tool for defining Gromov-Witten invariants.
t_symplectic_sum_construction	technique	Symplectic sum construction		Gompf's construction: removes tubular neighborhoods of codimension-2 symplectic submanifolds with opposite normal Euler classes and glues boundaries to produce 
t_symplectic_blowup	technique	Symplectic blowup		Replaces a symplectic ball B²ⁿ(ε) at p by its projectivization, producing (M̃, ω̃) with exceptional divisor E ≅ ℂPⁿ⁻¹ of area ε; depends on the embedding and ε.
t_hamiltonian_perturbation	technique	Hamiltonian perturbation technique		Displaces a Lagrangian by a Hamiltonian diffeomorphism to achieve transversality; intersection points L ∩ φ_H(L) correspond to time-1 Hamiltonian chords, connec
s_uncountable_set	axiom	Uncountable set		A set that is not countable; it cannot be put into bijection with any subset of the natural numbers.
s_subbasis_for_a_topology	axiom	Subbasis for a topology		A collection S of subsets of X whose union equals X; the topology generated by S has as basis all finite intersections of members of S.
s_box_topology	axiom	Box topology		The topology on a product of spaces whose basis consists of all products of open sets, with no restriction to finitely many non-trivial factors.
s_metric_topology	axiom	Metric topology		The topology on a metric space (X, d) generated by the collection of all open balls B(x, epsilon).
s_metrizable_space	axiom	Metrizable space		A topological space whose topology is induced by some metric.
s_hausdorff_unique_limits	theorem	Hausdorff spaces have unique limits		In a Hausdorff space, every convergent sequence (or net) has a unique limit.
s_separation_of_a_topological_space	axiom	Separation of a topological space		A pair (U, V) of disjoint nonempty open sets whose union is X, witnessing that X is disconnected.
s_connected_subspace	axiom	Connected subspace		A subspace A of a topological space X that is connected in the subspace topology; equivalently, A cannot be written as a union of two nonempty sets each open in
s_path_connected_implies_connected	theorem	Path connectedness implies connectedness		Every path-connected space is connected; the converse does not hold in general.
s_path_component	axiom	Path component		A maximal path-connected subset of a topological space; the path components partition the space.
s_locally_path_connected_space	axiom	Locally path-connected space		A topological space in which every point has a neighborhood base of path-connected open sets.
s_compact_space	axiom	Compact space		A topological space in which every open cover has a finite subcover.
s_compact_subspace	axiom	Compact subspace		A subspace Y of a topological space X that is compact in the subspace topology.
s_tube_lemma	theorem	Tube lemma		If x_0 times Y is contained in an open set N of the product X times Y with Y compact, then there is an open W containing x_0 such that W times Y is contained in
s_limit_point_compactness	axiom	Limit point compactness		A topological space in which every infinite subset has a limit point; implied by compactness, equivalent to it in metrizable spaces.
s_metrizable_implies_normal	theorem	Every metrizable space is normal		If X is metrizable, then X is normal: any two disjoint closed sets can be separated by disjoint open sets.
s_regular_second_countable_implies_normal	theorem	Every regular second-countable space is normal		A regular Hausdorff space with a countable basis is normal.
s_paracompact_hausdorff_implies_normal	theorem	Every paracompact Hausdorff space is normal		A paracompact Hausdorff space is normal; more generally, any two disjoint closed sets can be separated by a continuous function.
s_smirnov_metrization_theorem	theorem	Smirnov metrization theorem		A topological space is metrizable if and only if it is paracompact Hausdorff and locally metrizable.
s_baire_space	axiom	Baire space		A topological space in which every countable intersection of dense open sets is dense; equivalently, the complement of any meager set is dense.
s_baire_category_theorem_lch	theorem	Baire category theorem (locally compact Hausdorff version)		Every locally compact Hausdorff space is a Baire space: countable intersections of dense open sets are dense.
s_homotopy_of_paths	axiom	Homotopy of paths		A continuous deformation of one path to another through paths sharing the same endpoints; formally, a continuous map F: [0,1] x [0,1] to X with F(s,0) = f(s), F
s_homotopy_equivalence	axiom	Homotopy equivalence		A continuous map f: X to Y such that there exists g: Y to X with g composed with f homotopic to id_X and f composed with g homotopic to id_Y; X and Y are then s
s_path_homotopy_class	axiom	Path homotopy class		The equivalence class [f] of a path f under the relation of path homotopy (homotopy relative to endpoints).
s_simply_connected_space	axiom	Simply connected space		A path-connected topological space whose fundamental group is trivial; equivalently, every loop is null-homotopic.
s_fundamental_group_of_circle	theorem	Fundamental group of the circle		The fundamental group pi_1(S^1) is isomorphic to Z, generated by the homotopy class of the loop that winds once around the circle.
s_path_lifting_property	theorem	Path lifting property		Given a covering map p: E to B, every path in B starting at b and every lift e_0 of b, there exists a unique path in E starting at e_0 that lifts the given path
s_homotopy_lifting_property	theorem	Homotopy lifting property		Given a covering map p: E to B, every homotopy F: [0,1]^2 to B and a lift of F(-, 0) can be uniquely lifted to a homotopy in E.
s_lifting_criterion	theorem	Lifting criterion (general lifting lemma)		A continuous map f: (Y, y_0) to (B, b_0) with Y path-connected and locally path-connected lifts to a map to the covering space E iff f_*(pi_1(Y)) is contained i
s_deformation_retract	axiom	Deformation retract		A subspace A of X such that there exists a continuous map H: X x [0,1] to X with H(x,0) = x, H(x,1) in A for all x, and H(a,t) = a for all a in A and t; A is th
s_deformation_retract_preserves_fundamental_group	theorem	Deformation retract preserves fundamental group		If A is a deformation retract of X, then the inclusion-induced homomorphism pi_1(A) to pi_1(X) is an isomorphism.
s_induced_homomorphism_fundamental_group	state	Induced homomorphism on fundamental groups		A continuous map f: (X, x_0) to (Y, y_0) induces a group homomorphism f_*: pi_1(X, x_0) to pi_1(Y, y_0) defined by f_*([alpha]) = [f composed with alpha]; this 
s_fundamental_group_of_product	theorem	Fundamental group of a product space		For path-connected spaces X and Y, pi_1(X x Y, (x_0, y_0)) is isomorphic to pi_1(X, x_0) x pi_1(Y, y_0).
s_jordan_separation_theorem	theorem	Jordan separation theorem		A topological embedding of S^(n-1) into S^n separates S^n into exactly two components.
s_nonseparation_theorem_for_arcs	theorem	Nonseparation theorem for arcs		A topological embedding of [0,1] into S^2 does not separate S^2; the complement of any arc in S^2 is connected.
s_fundamental_group_of_wedge_sum	theorem	Fundamental group of a wedge sum		If X and Y are path-connected with well-behaved base points, then pi_1(X v Y) is isomorphic to the free product pi_1(X) * pi_1(Y), by the Seifert-van Kampen the
t_van_kampen_surface_computation	technique	Computing pi_1 of surfaces via van Kampen		Applies the Seifert-van Kampen theorem to the CW-structure of a surface given by a fundamental polygon to compute its fundamental group.
s_fundamental_group_of_torus	theorem	Fundamental group of the torus		The fundamental group pi_1(T^2) is isomorphic to Z x Z, the free abelian group on two generators.
s_fundamental_group_of_rp2	theorem	Fundamental group of the real projective plane		The fundamental group pi_1(RP^2) is isomorphic to Z/2Z.
s_surface_2_manifold	axiom	Surface (2-manifold)		A second-countable Hausdorff topological space in which every point has a neighborhood homeomorphic to an open subset of R^2.
s_fundamental_polygon	axiom	Fundamental polygon		A convex polygon with edge identifications that yields a compact surface as its quotient space; encodes the surface via a word in edge labels.
s_orientable_surface	axiom	Orientable surface		A compact surface that can be presented by a fundamental polygon whose boundary word has each edge label appearing once with each sign; equivalently, a surface 
s_equivalence_of_covering_spaces	axiom	Equivalence of covering spaces		Two covering spaces p_1: E_1 to B and p_2: E_2 to B are equivalent if there exists a homeomorphism h: E_1 to E_2 with p_2 composed with h = p_1.
s_lifting_correspondence	state	Lifting correspondence (monodromy action)		For a covering map p: E to B and a path class [gamma] in pi_1(B, b_0), the lifting of gamma defines a bijection from the fiber p^{-1}(b_0) to itself, giving a g
s_classification_of_covering_spaces	theorem	Classification of covering spaces		For a path-connected, locally path-connected, semilocally simply connected space B, there is a bijection between equivalence classes of connected coverings of B
s_semilocally_simply_connected	axiom	Semilocally simply connected space		A space X such that every point x has a neighborhood U with the inclusion-induced map pi_1(U, x) to pi_1(X, x) trivial; the necessary condition for the existenc
s_existence_of_universal_covering	theorem	Existence of universal covering space		A connected, locally path-connected, semilocally simply connected space admits a simply connected covering space (universal cover), unique up to covering equiva
s_normal_covering_space	axiom	Normal (regular) covering space		A covering space p: E to B for which the group of deck transformations acts transitively on each fiber; equivalently, p_*(pi_1(E)) is a normal subgroup of pi_1(
s_covering_index_formula	theorem	Index formula for coverings		For a connected covering space p: E to B, the number of sheets equals the index [pi_1(B, b_0) : p_*(pi_1(E, e_0))].
t_urysohn_function_construction	technique	Urysohn function construction (dyadic rational technique)		Constructs a continuous function via inductive assignment of open sets indexed by dyadic rationals, using normality at each step to interpolate between existing
t_lebesgue_number_argument	technique	Lebesgue number argument		Uses sequential compactness of a compact metric space to obtain a Lebesgue number delta > 0 such that every set of diameter less than delta lies in some member 
t_tube_lemma_argument	technique	Tube lemma argument		Extracts a tube neighborhood around a compact slice {x_0} x Y inside a product open set by first covering the slice with finitely many product-basis sets.
s_sequence_lemma	theorem	Sequence lemma		In a first-countable space, a point x lies in the closure of A iff there is a sequence of points in A converging to x; a map is continuous iff it preserves conv
s_metric_second_countable_iff_separable	theorem	Metric space is second-countable iff separable		A metric space is second-countable if and only if it is separable; equivalently, it is Lindelof if and only if it is separable.
s_long_line	axiom	Long line		The order topology on S_Omega x [0,1) in the dictionary order, minus the minimal element; a path-connected, locally homeomorphic to R space that is not second-c
s_topologists_sine_curve	axiom	Topologist's sine curve		The closure in R^2 of the graph of sin(1/x) for x > 0; a connected space that is not path-connected, providing a standard counterexample.
s_t0_space	axiom	T0-space (Kolmogorov)		A topological space in which for any two distinct points, at least one has a neighborhood not containing the other.
s_pseudometric_space	axiom	Pseudometric space		A set with a function d satisfying d(x,x)=0, symmetry, and triangle inequality, but not necessarily d(x,y)=0 implies x=y.
s_quotient_map	axiom	Quotient map (identification map)		A surjection p: X → Y such that U is open in Y if and only if p⁻¹(U) is open in X.
s_locally_compact_space	axiom	Locally compact space		A topological space in which every point has a neighborhood base consisting of compact sets.
s_equicontinuous_family	axiom	Equicontinuous family		A family F of functions from X to a uniform space Y such that for each x and each entourage V, a single neighborhood U of x satisfies (f(y),f(x)) ∈ V for all f 
s_totally_bounded_uniform_space	axiom	Totally bounded (precompact) uniform space		A uniform space in which for every entourage V there is a finite cover by V-small sets.
s_cauchy_net	axiom	Cauchy net in a uniform space		A net (x_α) in a uniform space such that for every entourage V there exists α₀ with (x_α, x_β) ∈ V for all α, β ≥ α₀.
s_base_for_uniformity	axiom	Base for a uniformity		A collection B of entourages such that every entourage contains a member of B, determining the uniformity as a filter base.
s_even_continuity	axiom	Even continuity		A family F of maps from X to Y is evenly continuous if for each (x,y) and each neighborhood W of y, there exist neighborhoods U of x and V of y such that f(x) ∈
s_convergence_class	axiom	Convergence class		A class of pairs (net, point) satisfying Kelley's axioms (constant nets converge, subnets of convergent nets converge, Urysohn property, iterated limits) that u
s_realcompact_space	axiom	Realcompact space (Hewitt-Nachbin space)		A completely regular Hausdorff space homeomorphic to a closed subspace of a product of copies of ℝ; equivalently, every real maximal ideal of C(X) is fixed.
s_morse_kelley_set_theory	axiom	Morse-Kelley set theory		An axiomatic set theory with proper classes as first-class objects, stronger than NBG due to impredicative class comprehension.
s_mk_classification_axiom_scheme	axiom	Classification axiom scheme (MK comprehension)		For every formula φ (possibly quantifying over classes), {x : x is a set and φ(x)} is a class; the impredicative comprehension scheme of Morse-Kelley set theory
s_axiom_of_choice	axiom	Axiom of choice		For every family of nonempty sets, there exists a function choosing one element from each set.
s_topology_generated_by_subbase	state	Topology generated by a subbase		The coarsest topology containing a given subbase S, whose open sets are unions of finite intersections of members of S.
s_eventually_in	state	Eventually in		A net (x_α) is eventually in a set A if there exists α₀ such that x_α ∈ A for all α ≥ α₀.
s_frequently_in	state	Frequently in		A net (x_α) is frequently in a set A if for every α₀ there exists α ≥ α₀ with x_α ∈ A.
s_cluster_point_of_net	state	Cluster point of a net		A point x such that the net is frequently in every neighborhood of x.
s_universal_net	state	Universal net (ultranet)		A net that is eventually in A or eventually in the complement of A for every subset A of the space.
s_eventuality_filter	state	Eventuality filter (filter generated by a net)		The filter of all sets that a given net is eventually in; establishes the correspondence between nets and filters.
s_evaluation_map_diagonal	state	Evaluation map (diagonal map)		The map e: X → ∏Y_f sending x to (f(x))_f, used to embed a Tychonoff space into a product of intervals.
s_partition_of_unity	state	Partition of unity		A locally finite family of continuous functions φ_i: X → [0,1] subordinate to an open cover with Σφ_i ≡ 1 at every point.
s_hewitt_realcompactification	state	Hewitt realcompactification		The realcompactification υX of a Tychonoff space X, the smallest realcompact space containing X densely, to which every continuous real-valued function extends.
s_gauge_family_pseudometrics	state	Gauge (family of pseudometrics)		A family of pseudometrics generating a uniformity; every uniformity arises from a gauge and the topology is the coarsest making all pseudometrics continuous.
s_metrizable_uniformity	state	Metrizable uniformity		A uniformity induced by a single metric; a uniformity is metrizable if and only if it has a countable base.
s_existence_of_universal_subnets	theorem	Existence of universal subnets		Every net has a universal (ultra) subnet; proved using Zorn's lemma or the ultrafilter lemma.
s_kelleys_convergence_class_theorem	theorem	Kelley's convergence class theorem		A convergence class satisfying Kelley's four axioms uniquely determines a topology on the underlying set.
s_existence_of_ultrafilters	theorem	Existence of ultrafilters (ultrafilter lemma)		Every proper filter can be extended to an ultrafilter; equivalent to the Boolean prime ideal theorem.
s_equivalence_of_nets_and_filters	theorem	Equivalence of nets and filters		Nets and filters are equivalent descriptions of convergence: every net determines an eventuality filter and every filter arises from a net, preserving convergen
s_universal_property_product_topology	theorem	Universal property of product topology		A map into a product space is continuous if and only if each component is continuous; the product topology is initial for projections.
s_universal_property_quotient_topology	theorem	Universal property of quotient topology		A function out of a quotient space is continuous if and only if its composition with the quotient map is continuous.
s_product_of_continuous_maps_continuous	theorem	Product of continuous functions is continuous		If f_i: X → Y_i is continuous for each i, then the product map x ↦ (f_i(x)) into ∏Y_i is continuous.
s_closed_surjection_is_quotient	theorem	Closed continuous surjection is a quotient map		A continuous surjection that is a closed map is automatically a quotient map.
s_open_surjection_is_quotient	theorem	Open continuous surjection is a quotient map		A continuous surjection that is an open map is automatically a quotient map.
s_completely_regular_separates_points_closed	theorem	Completely regular spaces separate points from closed sets		A space is completely regular iff for each point x and closed set F not containing x, there is a continuous f: X → [0,1] with f(x)=0 and f|_F=1.
s_tychonoff_embedding_theorem	theorem	Tychonoff embedding theorem		Every Tychonoff space embeds homeomorphically into a product of unit intervals [0,1]^κ via the evaluation map.
s_normal_implies_completely_regular	theorem	Normal spaces are completely regular		Every normal T1 space is completely regular, as Urysohn's lemma provides the separating functions.
s_filter_characterization_compactness	theorem	Filter characterization of compactness		A space is compact if and only if every ultrafilter converges, equivalently if every filter has a cluster point.
s_tychonoff_equivalent_to_ac	theorem	Tychonoff theorem equivalent to axiom of choice		Tychonoff's theorem for arbitrary products of compact spaces is equivalent to the axiom of choice in ZF (Kelley, 1950).
s_alexandroff_hausdorff_iff_locally_compact	theorem	Alexandroff compactification Hausdorff iff locally compact Hausdorff		The one-point compactification of X is Hausdorff if and only if X is locally compact and Hausdorff.
s_stone_cech_universal_property	theorem	Universal property of Stone-Čech compactification		Every continuous map from a Tychonoff space X to a compact Hausdorff space K extends uniquely through βX.
s_compact_implies_paracompact	theorem	Compact implies paracompact		Every compact space is paracompact, since every open cover has a finite (hence locally finite) refinement.
s_compact_hausdorff_unique_uniformity	theorem	Unique uniformity on compact Hausdorff space		A compact Hausdorff space admits exactly one uniformity compatible with its topology, consisting of all neighborhoods of the diagonal.
s_baire_category_complete_metric	theorem	Baire category theorem (complete metric space)		Every complete metric space is a Baire space: countable intersections of dense open sets are dense.
s_uniform_space_completely_regular	theorem	Hausdorff uniform space is completely regular		Every Hausdorff uniform space is completely regular (Tychonoff).
s_uniform_continuity_implies_continuity	theorem	Uniform continuity implies continuity		Every uniformly continuous map between uniform spaces is continuous with respect to the induced uniform topologies.
s_existence_uniqueness_completion	theorem	Existence and uniqueness of completion		Every uniform space has a Hausdorff completion, unique up to uniform isomorphism, constructed via equivalence classes of Cauchy filters.
s_extension_uniformly_continuous	theorem	Extension of uniformly continuous functions		A uniformly continuous function from a dense subspace to a complete Hausdorff uniform space extends uniquely to the whole space.
s_compact_open_refines_pointwise	theorem	Compact-open topology refines pointwise convergence		The compact-open topology on C(X,Y) is finer than the topology of pointwise convergence.
s_evaluation_map_continuity_theorem	theorem	Evaluation map continuity		The evaluation map ev: C(X,Y) × X → Y is jointly continuous when C(X,Y) has the compact-open topology and X is locally compact Hausdorff.
s_compact_open_equals_uniform_convergence	theorem	Compact-open equals uniform convergence for compact domain		When X is compact, the compact-open topology on C(X,Y) coincides with the topology of uniform convergence.
s_equicontinuity_topology_agreement	theorem	Equicontinuity and topology agreement		On an equicontinuous family, the topologies of pointwise convergence, compact convergence, and compact-open all coincide.
s_even_continuity_equicontinuity_relation	theorem	Even continuity and equicontinuity relationship		Equicontinuity implies even continuity; the converse holds under compactness hypotheses on the codomain.
s_ascoli_via_even_continuity	theorem	Ascoli theorem via even continuity (Kelley)		A subset of C(X,Y) is compact in the compact-open topology iff it is closed, evenly continuous, and pointwise precompact; Kelley's generalization beyond metric 
t_net_argument_compactness	technique	Net argument for compactness		Proves compactness properties by extracting convergent or universal subnets from arbitrary nets using Zorn's lemma.
t_filter_argument_compactness	technique	Filter argument for compactness		Proves compactness by showing every ultrafilter converges, using the finite intersection property and ultrafilter extension.
t_diagonal_argument_metrization	technique	Diagonal argument for metrization		Constructs a compatible metric from a countable base by summing weighted pseudometrics via a diagonal enumeration.
t_zorn_in_topology	technique	Zorn's lemma in topology		Applies Zorn's lemma to produce maximal filters (ultrafilters), maximal chains, or universal nets in topological arguments.
t_cofinal_subnet_extraction	technique	Cofinal subnet extraction		Extracts a subnet with a desired property by constructing a cofinal map from a refined directed set.
t_cauchy_filter_completion_construction	technique	Cauchy filter completion construction		Constructs the completion of a uniform space as equivalence classes of minimal Cauchy filters with extended uniformity.
t_cofinite_topology_trick	technique	Cofinite topology trick (Kelley's AC proof)		Derives the axiom of choice from Tychonoff's theorem by endowing each set with the cofinite topology and extracting a choice function from the product.
t_evaluation_map_embedding	technique	Evaluation map embedding		Embeds a Tychonoff space into [0,1]^κ via the evaluation map x ↦ (f(x))_f for a separating family of continuous functions.
s_cw_complex	axiom	CW complex		A topological space X built inductively by attaching n-cells via maps φ_α: S^{n-1} → X^{n-1}, equipped with the weak topology where A ⊆ X is closed iff A ∩ clos
s_n_skeleton	state	n-skeleton		The subspace X^n of a CW complex X consisting of all cells of dimension at most n, constructed inductively from X^{n-1} by attaching n-cells.
t_cell_attachment	technique	Cell attachment		Constructs a new space by gluing an n-disk to A along its boundary via the attaching map φ, forming the adjunction space A ∪_φ D^n.
s_contractible_space	axiom	Contractible space		A topological space X that is homotopy equivalent to a point, equivalently the identity map id_X is homotopic to a constant map.
s_homotopy_extension_property	axiom	Homotopy extension property		A pair (X, A) has the HEP if every homotopy of A and every map of X extending the initial map of A can be extended to a homotopy of X; equivalently the inclusio
s_change_of_basepoint_isomorphism	theorem	Change of basepoint isomorphism		For a path γ from x₀ to x₁ in X, the map [f] ↦ [γ̄·f·γ] is a group isomorphism between the fundamental groups at the two basepoints.
s_universal_covering_space	axiom	Universal covering space		A covering space p: X̃ → X where X̃ is simply connected; it is unique up to isomorphism for connected, locally path-connected, semi-locally simply connected X a
s_fundamental_group_of_graph	theorem	Fundamental group of a graph		The fundamental group of a connected graph is a free group, with rank equal to 1 minus the Euler characteristic of the graph (i.e., the number of edges not in a
s_fundamental_group_of_surfaces	theorem	Fundamental group of closed surfaces		The fundamental group of the closed orientable surface of genus g has presentation ⟨a₁,b₁,…,a_g,b_g | [a₁,b₁]⋯[a_g,b_g]⟩, and similarly for non-orientable surfa
s_singular_n_simplex	axiom	Singular n-simplex		A continuous map from the standard n-simplex Δⁿ to a topological space X, serving as the generating element for singular chain groups.
s_singular_chain_group	axiom	Singular chain group Cₙ(X)		The free abelian group generated by all singular n-simplices in X, forming the degree-n component of the singular chain complex.
s_boundary_operator	axiom	Boundary operator ∂ₙ		The homomorphism defined by ∂ₙ(σ) = Σᵢ(-1)ⁱ σ|[v₀,…,v̂ᵢ,…,vₙ], satisfying ∂ₙ₋₁∘∂ₙ = 0 and making C_*(X) a chain complex.
s_singular_homology_group	axiom	Singular homology group Hₙ(X)		The n-th homology group of X, defined as the quotient of n-cycles by n-boundaries in the singular chain complex, measuring n-dimensional holes in X.
s_reduced_homology	state	Reduced homology		A modified version of singular homology using the augmented chain complex ε: C₀(X) → ℤ, satisfying H̃ₙ(X) = Hₙ(X) for n ≥ 1 and H̃₀(X) ⊕ ℤ ≅ H₀(X), with H̃ₙ(pt)
s_homotopy_invariance_of_homology	theorem	Homotopy invariance of homology		If f, g: X → Y are homotopic maps then they induce identical homomorphisms on all singular homology groups, so homotopy equivalent spaces have isomorphic homolo
t_chain_homotopy_technique	technique	Chain homotopy		Constructs a sequence of homomorphisms Pₙ: Cₙ(X) → Cₙ₊₁(Y) satisfying ∂P + P∂ = g_# - f_#, proving f_# and g_# induce the same maps on homology.
s_relative_homology_group	axiom	Relative homology group Hₙ(X, A)		The homology of the quotient chain complex C_*(X)/C_*(A), measuring n-dimensional holes in X modulo those in A, fitting into the long exact sequence of the pair
t_barycentric_subdivision	technique	Barycentric subdivision		Iteratively subdivides singular simplices into smaller ones whose diameters approach zero, used to prove excision by making chains small relative to a cover.
s_good_pair	axiom	Good pair (X, A)		A pair (X, A) where A is a nonempty closed subspace that is a deformation retract of some open neighborhood in X, ensuring that the quotient map X → X/A induces
s_homology_of_spheres	theorem	Homology of spheres		The singular homology groups of the n-sphere are ℤ in dimensions 0 and n and zero elsewhere, computed via the Mayer-Vietoris sequence or cellular homology.
s_local_homology_group	state	Local homology group		The relative homology group at a point x ∈ X, detecting the local topological structure; for an n-manifold it equals ℤ in dimension n and zero otherwise.
s_homology_with_coefficients	axiom	Homology with coefficients		Homology computed from the chain complex C_*(X) tensored with an abelian group G, allowing detection of torsion phenomena invisible to integer homology.
s_exact_sequence_of_triple	theorem	Long exact sequence of a triple		For A ⊆ B ⊆ X, the relative homology groups fit into a long exact sequence relating the three pairs (B,A), (X,A), and (X,B).
s_homology_of_real_projective_space	state	Homology of ℝPⁿ		H₀(ℝPⁿ) ≅ ℤ, Hₖ(ℝPⁿ) ≅ ℤ/2 for odd k with 0 < k < n, Hₙ(ℝPⁿ) ≅ ℤ if n is odd (0 if n even), and all other groups vanish.
s_homology_of_complex_projective_space	state	Homology of ℂPⁿ		H₂ₖ(ℂPⁿ) ≅ ℤ for 0 ≤ k ≤ n and all odd-dimensional homology groups vanish, following from the CW structure with one cell in each even dimension.
t_transfer_homomorphism_covering	technique	Transfer homomorphism (covering spaces)		Lifts chains from the base to the total space of a finite-sheeted covering by summing over all lifts, yielding a chain map whose composition with p_* is multipl
s_coboundary_operator	axiom	Coboundary operator δⁿ		The dual of the boundary operator, defined by (δφ)(σ) = φ(∂σ) for cochains φ, satisfying δⁿ⁺¹∘δⁿ = 0 and making C*(X; G) a cochain complex.
s_cross_product_cohomology	state	Cross product (cohomology)		The external product defined by a × b = p₁*(a) ∪ p₂*(b) where p₁, p₂ are projections, relating the cohomology of a product to those of its factors via the Künne
s_cap_product	axiom	Cap product		A bilinear pairing defined by σ ∩ φ = φ(σ|[v₀,…,vₚ])·σ|[vₚ,…,vₙ], making homology a module over the cohomology ring and used to state Poincaré duality as [M] ∩ 
s_fundamental_class_manifold	axiom	Fundamental class of a manifold		For a closed connected oriented n-manifold M, the unique generator [M] of Hₙ(M; ℤ) ≅ ℤ that restricts to the local orientation at every point.
s_cohomology_with_compact_support	axiom	Cohomology with compact support		The direct limit of cohomology groups Hⁿ(X, X∖K; R) over compact subsets K ⊆ X, used to formulate Poincaré duality for non-compact manifolds.
t_bockstein_homomorphism	technique	Bockstein homomorphism		The connecting homomorphism in the long exact cohomology sequence induced by a short exact sequence of coefficient groups, detecting elements not liftable to G-
s_orientation_double_cover	state	Orientation double cover		The unique connected double covering of a connected manifold M whose total space is orientable; M is orientable iff this cover is disconnected (trivial).
s_higher_homotopy_group	axiom	Higher homotopy group πₙ(X)		The set of basepoint-preserving homotopy classes of maps (Sⁿ, s₀) → (X, x₀), forming an abelian group for n ≥ 2 under the composition inherited from the group s
s_pi_n_abelian_for_n_geq_2	theorem	πₙ is abelian for n ≥ 2		For n ≥ 2 the homotopy group πₙ(X, x₀) is abelian, following from the Eckmann-Hilton argument since elements can be composed in two independent directions.
s_relative_homotopy_group	axiom	Relative homotopy group πₙ(X, A)		The set of homotopy classes of maps (Dⁿ, S^{n-1}, s₀) → (X, A, x₀), forming a group for n ≥ 2 (abelian for n ≥ 3), fitting into the long exact sequence of the p
s_long_exact_sequence_of_pair_homotopy	theorem	Long exact sequence of a pair (homotopy)		For a pair (X, A) with basepoint in A, the inclusion and boundary maps form an exact sequence of homotopy groups extending down to π₀.
s_long_exact_sequence_of_fibration	theorem	Long exact sequence of a fibration		For a fibration F → E → B, the homotopy groups fit into a long exact sequence relating the fiber, total space, and base, induced by the boundary map ∂: πₙ(B) → 
s_cw_approximation_theorem	theorem	CW approximation theorem		Every topological space X admits a CW complex Z and a weak homotopy equivalence Z → X, and the construction is functorial up to homotopy.
s_relative_hurewicz_theorem	theorem	Relative Hurewicz theorem		If (X, A) is (n-1)-connected with n ≥ 2, A simply connected and nonempty, then Hᵢ(X, A) = 0 for i < n and the Hurewicz map πₙ(X, A) → Hₙ(X, A) is an isomorphism
s_n_connected_space	axiom	n-connected space		A topological space X such that πᵢ(X) = 0 for all i ≤ n, meaning every map Sⁱ → X for i ≤ n extends to a map Dⁱ⁺¹ → X.
s_n_connected_pair	axiom	n-connected pair		A pair (X, A) such that πᵢ(X, A) = 0 for all i ≤ n, equivalently every map (Dⁱ, S^{i-1}) → (X, A) for i ≤ n is homotopic rel S^{i-1} to a map into A.
s_postnikov_tower	state	Postnikov tower		A sequence of fibrations Xₙ → Xₙ₋₁ with fibers K(πₙ(X), n) such that X → Xₙ induces isomorphisms on πᵢ for i ≤ n and πᵢ(Xₙ) = 0 for i > n, decomposing the homot
s_k_invariant	state	k-invariants		The cohomology class that classifies the fibration Xₙ → Xₙ₋₁ in a Postnikov tower, determining how the n-th homotopy group is attached to the lower stages.
t_obstruction_theory	technique	Obstruction theory		Determines whether a map defined on a subcomplex extends over the next skeleton by computing an obstruction cocycle; the map extends over all of X iff all obstr
s_loopspace	state	Loop space ΩX		The space of basepoint-preserving loops in X with the compact-open topology, satisfying πₙ(ΩX) ≅ πₙ₊₁(X) and serving as the fiber of the path-loop fibration PX 
s_path_loop_fibration	state	Path-loop fibration		The fibration where PX is the contractible space of paths starting at the basepoint, the projection sends a path to its endpoint, and the fiber over x₀ is the l
s_pi_3_S2_isomorphic_Z	theorem	π₃(S²) ≅ ℤ		The third homotopy group of the 2-sphere is infinite cyclic, generated by the Hopf fibration η: S³ → S², proved using the long exact sequence of the Hopf bundle
s_fiber_sequence_puppe	state	Fiber sequence (Puppe sequence)		The iterated extension of a fibration F → E → B to the left by taking loop spaces, yielding an infinite sequence where each three consecutive terms form a fiber
s_cofiber_sequence_puppe	state	Cofiber sequence (Puppe sequence)		The iterated extension of a cofibration A → X to the right by taking cones and suspensions, yielding an exact sequence on homotopy classes [−, Y] for any Y.
s_pi_1_action_on_pi_n	state	π₁ action on πₙ		The action of the fundamental group on higher homotopy groups defined by transporting spheroids along loops, making πₙ(X) a module over ℤ[π₁(X)]; X is n-simple 
s_compression_lemma	theorem	Compression lemma		If (X, A) is a CW pair and (Y, B) is a pair with πₙ(Y, B) = 0 for all n such that X∖A has cells of dimension n, then any map (X, A) → (Y, B) is homotopic rel A 
s_delta_complex	axiom	Δ-complex		A space built from a collection of simplices identified along faces by face maps (without requiring distinct vertices), intermediate between simplicial complexe
t_acyclic_models	technique	Acyclic models		Proves that two chain complex-valued functors are naturally chain homotopy equivalent by checking they agree on a set of acyclic model objects, used to prove co
s_naturality_of_les	theorem	Naturality of long exact sequences		A map of pairs f: (X, A) → (Y, B) induces a commutative ladder between the long exact homology sequences of the two pairs, so connecting homomorphisms are natur
s_suspension_space	state	Suspension ΣX		The quotient of X × [0,1] obtained by collapsing X×{0} to one point and X×{1} to another point; satisfies H̃ₙ(ΣX) ≅ H̃ₙ₋₁(X) and Σ: πₙ(X) → πₙ₊₁(ΣX).
s_suspension_isomorphism_cohomology	theorem	Suspension isomorphism (cohomology)		The natural isomorphism between the reduced cohomology of X and the shifted reduced cohomology of its suspension, characterizing cohomology operations via stabi
s_homology_of_wedge	state	Homology of a wedge sum		The reduced homology of a wedge of spaces is the direct sum of the reduced homologies of the summands, provided the wedge has the topology of a CW complex.
s_attaching_cells_and_homology	theorem	Effect of cell attachment on homology		If Y = X ∪_φ eⁿ is obtained by attaching an n-cell, then Hₖ(Y, X) ≅ ℤ for k = n and 0 otherwise, and the long exact sequence determines how H_*(Y) differs from 
s_cellular_equals_singular	theorem	Cellular homology equals singular homology		For a CW complex X, the homology of the cellular chain complex (with Cₙ = Hₙ(Xⁿ, Xⁿ⁻¹) free abelian on n-cells) is naturally isomorphic to singular homology.
s_degree_and_local_degree	theorem	Degree as sum of local degrees		For f: Sⁿ → Sⁿ and a regular value y, the degree of f equals the sum of local degrees at preimage points, where each local degree is ±1 for a local homeomorphis
s_antipodal_map_degree	theorem	Degree of the antipodal map		The antipodal map a: Sⁿ → Sⁿ has degree (-1)^{n+1}, since it is the composition of (n+1) reflections each of degree -1.
s_mapping_degree_formula	theorem	Mapping degree formula		For a map f: M → N between closed connected oriented n-manifolds, the induced map on top homology is multiplication by the integer deg(f), which equals the sign
s_cohomology_ring_of_real_projective_space	theorem	Cohomology ring of ℝPⁿ		The mod-2 cohomology ring of ℝPⁿ is a truncated polynomial ring on one generator α in degree 1, proved using the cup product structure.
s_cohomology_ring_of_complex_projective_space	theorem	Cohomology ring of ℂPⁿ		The integral cohomology ring of ℂPⁿ is a truncated polynomial ring on one generator α in degree 2, distinguishing ℂPⁿ from a product of spheres.
s_graded_commutativity_of_cup_product	theorem	Graded commutativity of cup product		For α ∈ Hᵖ(X; R) and β ∈ Hᵍ(X; R), the cup product satisfies α ∪ β = (-1)^{pq} β ∪ α, making H*(X; R) a graded-commutative ring.
s_cohomological_mayer_vietoris	theorem	Cohomological Mayer-Vietoris sequence		For X = A ∪ B with A, B open, the long exact sequence in cohomology relating H*(X), H*(A) ⊕ H*(B), and H*(A∩B), dual to the homological Mayer-Vietoris sequence.
s_cup_product_distinguishes_spaces	theorem	Cup product distinguishes spaces		Spaces with isomorphic cohomology groups but different ring structures (e.g., ℂP² vs S² ∨ S⁴) are not homotopy equivalent, demonstrating that the cup product ca
s_fiber_bundle_is_serre_fibration	theorem	Fiber bundle is a Serre fibration		Every fiber bundle F → E → B with paracompact base is a Serre fibration (has the homotopy lifting property for CW complexes), so its long exact homotopy sequenc
s_intersection_form	state	Intersection form		The bilinear form on middle homology of a closed oriented 2n-manifold defined by algebraic intersection number, equivalently by the cup product pairing via Poin
s_homology_manifold	axiom	Homology manifold		A topological space X whose local homology groups H_*(X, X∖{x}; ℤ) at every point x are isomorphic to those of ℝⁿ, allowing Poincaré duality to hold without a m
t_smith_theory	technique	Smith theory		Uses transfer and localization in mod-p homology to bound the total Betti numbers of the fixed-point set of a ℤ/p-action by those of the ambient space.
s_homotopy_type	axiom	Homotopy type		The equivalence class of a topological space under homotopy equivalence; two spaces have the same homotopy type iff there exist continuous maps f: X → Y and g: 
s_retraction	axiom	Retraction (topological)		A continuous map r: X → A where A ⊆ X such that r restricts to the identity on A; A is then called a retract of X.
s_fundamental_groupoid	axiom	Fundamental groupoid		The category whose objects are points of X and whose morphisms are path-homotopy classes of paths, with composition given by concatenation; it generalizes the f
s_h_space	axiom	H-space		A topological space X with a continuous multiplication μ: X × X → X and a two-sided identity e up to homotopy (μ(e,x) ≃ x ≃ μ(x,e)); examples include loop space
s_co_h_space	axiom	Co-H-space (H-cogroup)		A pointed space X with a comultiplication ν: X → X ∨ X that is coassociative and counital up to homotopy; the suspension ΣA is the prototypical co-H-space, and 
s_hurewicz_fibration	axiom	Hurewicz fibration		A continuous map p: E → B satisfying the homotopy lifting property with respect to all topological spaces, strictly stronger than a Serre fibration which only r
s_mapping_path_space	state	Mapping path space (path-space fibration)		For a map f: X → Y, the space of pairs (x, γ) where γ is a path in Y starting at f(x); the endpoint projection Ef → Y is a Hurewicz fibration factoring f, used 
s_abstract_simplicial_complex	axiom	Abstract simplicial complex		A collection Σ of finite subsets (simplices) of a vertex set V, closed under taking subsets; its geometric realization |K| is a topological space obtained by as
s_star_and_link	axiom	Star and link of a simplex		For a simplex σ in a simplicial complex K, the star St(σ) is the union of interiors of all simplices containing σ, and the link Lk(σ) is the subcomplex of simpl
s_cellular_map	axiom	Cellular map		A continuous map between CW complexes that sends the n-skeleton of the domain into the n-skeleton of the codomain for every n; the cellular approximation theore
s_relative_cw_complex	axiom	Relative CW complex		A pair (X, A) where X is obtained from A by successively attaching cells of increasing dimension; generalizes CW complexes (where A = ∅) and provides the natura
s_relative_singular_homology	state	Relative singular homology		The homology of the quotient chain complex C_*(X)/C_*(A), fitting into the long exact sequence ⋯ → Hₙ(A) → Hₙ(X) → Hₙ(X,A) → Hₙ₋₁(A) → ⋯ and satisfying excision
s_short_five_lemma	theorem	Short five lemma		If 0 → A → B → C → 0 and 0 → A' → B' → C' → 0 are short exact sequences with vertical maps α, β, γ forming a commutative diagram, then if α and γ are isomorphis
s_slant_product	state	Slant product		A bilinear pairing that takes a cohomology class on a product and a homology class on one factor and produces a cohomology class on the other factor; used in th
s_kronecker_pairing	state	Kronecker pairing		The evaluation pairing between cohomology and homology defined by ⟨[φ],[c]⟩ = φ(c), inducing the homomorphism Hⁿ(X; R) → Hom(Hₙ(X; R), R) that appears in the un
s_n_connected_map	axiom	n-connected map		A continuous map f: X → Y such that all homotopy fibers are (n-1)-connected, equivalently f_*: πₖ(X) → πₖ(Y) is an isomorphism for k < n and a surjection for k 
s_suspension_loop_adjunction	theorem	Suspension-loop space adjunction (Σ ⊣ Ω)		The natural bijection between pointed homotopy classes of maps from the suspension ΣX to Y and pointed maps from X to the loop space ΩY, making Σ and Ω an adjoi
s_representability_of_cohomology	theorem	Representability of cohomology		For CW complexes X, the n-th cohomology group with coefficients in G is naturally isomorphic to the set of homotopy classes of maps from X to the Eilenberg-MacL
s_obstruction_cocycle	state	Obstruction cocycle		The (n+1)-cochain that measures the failure of extending a map f defined on the n-skeleton to the (n+1)-skeleton; it is a cocycle whose cohomology class is the 
s_difference_cochain	state	Difference cochain		For two maps f, g: Xⁿ → Y that agree on Xⁿ⁻¹, the n-cochain measuring their difference on each n-cell; satisfies δd(f,g) = cⁿ⁺¹(g) - cⁿ⁺¹(f), providing the key 
s_moore_postnikov_decomposition	state	Moore-Postnikov decomposition		A factorization of a map f: X → Y into a tower of fibrations where each stage X[n] → X[n-1] is a principal fibration with fiber K(πₙ(f), n), refining the Postni
s_cohomology_operation	axiom	Cohomology operation		A natural transformation between cohomology functors; by the representability theorem, classified by [K(G,n), K(G',m)] and including Steenrod squares, Steenrod 
s_steenrod_squares	state	Steenrod squares		The unique stable cohomology operations Sqⁱ in mod-2 cohomology satisfying Sq⁰ = id, Sqⁿ(x) = x² for |x| = n, Sqⁱ(x) = 0 for i > |x|, the Cartan formula, and th
s_steenrod_reduced_powers	state	Steenrod reduced powers		The stable cohomology operations Pⁱ in mod-p cohomology for odd primes p, satisfying P⁰ = id, the Cartan formula, and Adem relations; together with the Bockstei
s_steenrod_algebra	state	Steenrod algebra		The graded associative algebra over 𝔽_p generated by the Steenrod squares (p=2) or the Bockstein and reduced powers (p odd), subject to the Adem relations; it i
s_thom_space	state	Thom space		For a vector bundle ξ over B, the quotient of the disk bundle by the sphere bundle, equivalently the one-point compactification of the total space; carries the 
s_cone_topological	state	Cone CX		The quotient of X × [0,1] by collapsing X × {1} to a point; CX is always contractible, and the mapping cone Cf = CX ∪_f Y of a map f: X → Y is the basic buildin
s_join_of_spaces	state	Join of spaces		The quotient of X × Y × [0,1] identifying (x,y₁,0) ~ (x,y₂,0) and (x₁,y,1) ~ (x₂,y,1); satisfies Sⁿ * Sᵐ ≅ Sⁿ⁺ᵐ⁺¹ and X * S⁰ ≅ ΣX, used in constructing the Miln
s_homotopy_pullback	state	Homotopy pullback		The homotopy-invariant version of the categorical pullback, defined as the space of triples (x, γ, y) where γ is a path in B from f(x) to g(y); it is invariant 
s_homotopy_fiber	state	Homotopy fiber		The homotopy fiber of f: X → B over b₀, defined as the pullback of f along the path-space fibration; it fits into the fiber sequence hofiber(f) → X → B and its 
s_induced_fibration	state	Induced fibration (pullback bundle)		For a fibration p: E → B and a map f: X → B, the pullback f*E → X is again a fibration with the same fiber; this construction is fundamental in classifying bund
s_hopf_algebra_h_space_homology	state	Hopf algebra structure on H-space homology		For a connected H-space X with field coefficients, the homology H_*(X; k) is a graded connected Hopf algebra with product from the Pontryagin product and coprod
s_tensor_product_chain_complexes	axiom	Tensor product of chain complexes		The chain complex with degree-n component ⊕_{p+q=n} Cₚ ⊗ Dq and differential ∂(c ⊗ d) = ∂c ⊗ d + (-1)^|c| c ⊗ ∂d; the Eilenberg-Zilber theorem identifies C_*(X 
s_smash_product	state	Smash product		The quotient of the product X × Y by the wedge X ∨ Y for pointed spaces; satisfies Sⁿ ∧ Sᵐ ≅ Sⁿ⁺ᵐ and ΣX ≅ S¹ ∧ X, and serves as the symmetric monoidal product 
s_wedge_sum	axiom	Wedge sum		The coproduct in the category of pointed spaces, formed by identifying the basepoints of X and Y; satisfies π₁(X ∨ Y) ≅ π₁(X) * π₁(Y) for well-pointed spaces an
s_homotopy_pushout	state	Homotopy pushout		The homotopy-invariant version of the categorical pushout for a span X ← A → Y, constructed as X ⊔ (A × [0,1]) ⊔ Y with appropriate identifications; the double 
t_spectral_sequence_computation	technique	Spectral sequence computation		Computes homology or cohomology of a total space, filtered complex, or composition of functors by identifying the E₂ page, propagating differentials, and readin
s_ndr_pair	axiom	NDR pair (neighborhood deformation retract pair)		A pair (X, A) together with maps u: X → [0,1] and h: X × I → X such that A = u⁻¹(0), h₀ = id, h_t|_A = id_A, and h₁(u⁻¹([0,1))) ⊂ A; equivalently, A ↪ X is a co
s_ndr_pair_characterization_of_cofibrations	theorem	NDR pair characterization of cofibrations		An inclusion A ↪ X of compactly generated spaces is a cofibration if and only if (X, A) is an NDR pair.
s_loop_eilenberg_maclane_delooping	theorem	Loop space of Eilenberg-MacLane space delooping		The loop space of an Eilenberg-MacLane space K(G, n+1) is homotopy equivalent to K(G, n) for n ≥ 1 and any abelian group G.
s_spectrum_stable_homotopy	axiom	Spectrum (stable homotopy theory)		A sequence of pointed spaces {Eₙ} with structure maps σₙ: ΣEₙ → Eₙ₊₁ representing generalized cohomology theories in the stable homotopy category.
s_omega_spectrum	axiom	Omega spectrum (Ω-spectrum)		A spectrum in which the adjoint structure maps Eₙ → ΩEₙ₊₁ are weak homotopy equivalences; every generalized cohomology theory is represented by an Ω-spectrum.
s_stiefel_whitney_numbers	state	Stiefel-Whitney numbers		For a closed n-manifold M, the mod-2 integers ⟨w_{i₁}(TM) ⌣ ··· ⌣ w_{iₖ}(TM), [M]⟩ with i₁ + ··· + iₖ = n; two manifolds are unoriented-cobordant iff all their 
s_wu_formula	theorem	Wu formula		The Stiefel-Whitney classes of a closed manifold M are determined by the Wu classes vᵢ via Sq(v) = w, where vᵢ is characterized by ⟨vᵢ ⌣ x, [M]⟩ = ⟨Sqⁱ(x), [M]⟩
s_localization_of_spaces	state	Localization of spaces at a set of primes		For a set of primes T, the T-localization X_T of a nilpotent space X with πₙ(X_T) ≅ πₙ(X) ⊗ ℤ_T, initial among maps inducing T-local isomorphisms on homotopy gr
s_rationalization_of_space	state	Rationalization of a space		The localization of a nilpotent space X at the empty set of primes, yielding X_ℚ with πₙ(X_ℚ) ≅ πₙ(X) ⊗ ℚ; captures the rational homotopy type.
s_p_localization_of_space	state	p-localization of a space		The localization of a nilpotent space X at the prime p, yielding X_(p) with πₙ(X_(p)) ≅ πₙ(X) ⊗ ℤ_(p); retains only p-primary torsion and rational information.
s_p_completion_of_space	state	p-completion of a space		The Bousfield-Kan p-completion of a nilpotent space X, with πₙ(X_p^∧) ≅ πₙ(X) ⊗ ℤ_p^∧ for finitely generated πₙ(X); captures p-adic homotopy information.
s_sullivan_arithmetic_square	theorem	Sullivan arithmetic square (fracture theorem)		A nilpotent space X of finite type is the homotopy pullback of its rationalization X_ℚ and profinite completion ∏_p X_p^∧ over their common rationalization, fra
s_formal_space_rational_homotopy	axiom	Formal space (rational homotopy theory)		A simply connected space X whose rational homotopy type is a formal consequence of its rational cohomology ring; compact Kähler manifolds are formal by Deligne-
s_hopf_algebra_structure_on_loop_space_homology	state	Hopf algebra structure on loop space homology		The homology H*(ΩX; k) carries a graded Hopf algebra structure with product from the Pontryagin product, coproduct from the diagonal, and antipode from loop inv
s_cellular_chain_complex	state	Cellular chain complex		For a CW complex X, the chain complex with Cₙ(X) = Hₙ(Xⁿ, Xⁿ⁻¹) ≅ ℤ^{number of n-cells} and boundary maps given by degrees of attaching maps; its homology equal
s_graph_as_1_dim_cw_complex	axiom	Graph as 1-dimensional CW complex		A graph is a 1-dimensional CW complex with 0-cells as vertices and 1-cells as edges; every connected graph deformation-retracts onto a maximal spanning tree wit
t_based_vs_free_homotopy	technique	Based versus free homotopy comparison		Relates pointed homotopy classes [X, Y]_* to free homotopy classes [X, Y] via the action of π₁(Y); the free classes are the orbit set of this action.
t_help_lemma	technique	HELP lemma (homotopy extension and lifting property)		Combines HEP for cofibrations and HLP for fibrations to solve simultaneous extension-lifting problems for n-connected maps and low-dimensional CW pairs.
t_zabrodsky_mixing	technique	Zabrodsky mixing		Constructs new homotopy types by assembling different p-localizations via the fracture theorem; produces exotic H-spaces and spaces with prescribed local behavi
t_serre_ss_computation_eilenberg_maclane	technique	Computing H*(K(ℤ,n)) via Serre spectral sequence		Uses the Serre spectral sequence of the path-loop fibration with contractible total space to inductively determine the cohomology of Eilenberg-MacLane spaces.
s_boolean_lattice	axiom	Boolean lattice B_n		The poset of all subsets of an n-element set ordered by inclusion, isomorphic to the product of n copies of the 2-element chain.
s_partition_lattice_pi_n	state	Partition lattice Pi_n		The lattice of all set partitions of {1,...,n} ordered by refinement, with Möbius function μ(0̂,1̂) = (-1)^{n-1}(n-1)!.
s_youngs_lattice	state	Young's lattice		The poset of all integer partitions ordered by inclusion of Young diagrams, forming a distributive lattice and a differential poset of rank 1.
s_zeta_function_of_a_poset	state	Zeta function of a poset		The element ζ in the incidence algebra of a locally finite poset P defined by ζ(x,y) = 1 for all x ≤ y, whose inverse is the Möbius function μ.
s_philip_halls_theorem_posets	theorem	Philip Hall's theorem (Möbius function)		For a finite poset P, μ(x,y) = ∑_{k≥0} (-1)^k c_k where c_k is the number of chains x = z_0 < z_1 < ··· < z_k = y of length k from x to y.
s_crosscut_theorem_rota	theorem	Crosscut theorem (Rota)		For a finite lattice L with crosscut C (an antichain meeting every maximal chain), μ(0̂,1̂) = ∑_{k≥0} (-1)^k N_k where N_k counts k-element subsets of C with jo
s_euler_characteristic_mobius_function	theorem	Euler characteristic and Möbius function		For a finite poset P, μ_P(x,y) = χ̃(Δ(x,y)) where χ̃ is the reduced Euler characteristic of the order complex of the open interval (x,y).
s_order_complex	state	Order complex Δ(P)		The simplicial complex whose k-simplices are the chains of length k in the poset P.
s_order_polynomial	state	Order polynomial Ω(P, m)		The number of order-preserving maps σ: P → {1,...,m} from a finite poset P, which is a polynomial in m of degree |P|.
s_strict_order_polynomial	state	Strict order polynomial Ω̄(P, m)		The number of strictly order-preserving maps σ: P → {1,...,m}, which is a polynomial in m of degree |P|.
s_reciprocity_theorem_order_polynomials	theorem	Reciprocity theorem for order polynomials (Stanley)		Ω̄(P, m) = (-1)^{|P|} Ω(P, -m), relating the strict and weak order polynomials of a poset P via evaluation at negative integers.
s_exponential_generating_function	axiom	Exponential generating function		The formal power series EGF(a_n; x) = ∑_{n≥0} a_n x^n/n!, encoding sequences with labeled combinatorial interpretations.
s_rational_generating_function	state	Rational generating function		A generating function f(x) = P(x)/Q(x) with P, Q polynomials, equivalent to the sequence satisfying a linear recurrence with constant coefficients.
s_d_finite_generating_function	state	D-finite (holonomic) generating function		A formal power series satisfying a linear ODE with polynomial coefficients, equivalently whose coefficient sequence satisfies a polynomial recurrence (P-recursi
t_transfer_matrix_method	technique	Transfer matrix method		Encodes walks in a directed graph via a matrix whose (i,j)-entry is the weight of edge i→j, yielding GF as entries of (I - xT)^{-1}.
t_sieve_inclusion_exclusion	technique	Sieve methods (inclusion-exclusion)		Counts elements avoiding specified properties by alternating over-/under-counting via subsets of conditions.
s_principle_of_inclusion_exclusion	theorem	Principle of inclusion-exclusion		|A_1 ∪ ··· ∪ A_n| = ∑_i |A_i| - ∑_{i<j} |A_i ∩ A_j| + ··· + (-1)^{n-1}|A_1 ∩ ··· ∩ A_n|, expressing the union's cardinality as an alternating sum over intersect
s_composition_of_an_integer	axiom	Composition of an integer		An ordered tuple (a_1,...,a_k) of positive integers with a_1 + ··· + a_k = n; the number of compositions of n is 2^{n-1}.
s_descent_of_a_permutation	state	Descent of a permutation		A position i such that σ(i) > σ(i+1) in a permutation σ ∈ S_n; the descent set Des(σ) ⊆ {1,...,n-1} encodes the permutation's shape.
s_descent_number_and_major_index	state	Descent number and major index		For σ ∈ S_n, des(σ) = |Des(σ)| and maj(σ) = ∑_{i ∈ Des(σ)} i; both are equidistributed with inv(σ) via the Carlitz q-Eulerian polynomial.
s_inversion_of_a_permutation	state	Inversion of a permutation		A pair (i,j) with i < j and σ(i) > σ(j); the inversion number inv(σ) equals the minimum number of adjacent transpositions to sort σ.
s_eulerian_polynomial	state	Eulerian polynomial A_n(t)		A_n(t) = ∑_{σ ∈ S_n} t^{des(σ)+1} = ∑_{k=1}^n A(n,k) t^k, satisfying ∑_{m≥0} m^n t^m = A_n(t)/(1-t)^{n+1}.
s_q_factorial	state	q-analog of n!		[n]_q! = [1]_q [2]_q ··· [n]_q = (1-q)(1-q^2)···(1-q^n)/(1-q)^n, the generating function for inversions: ∑_{σ ∈ S_n} q^{inv(σ)} = [n]_q!.
s_exponential_formula_combinatorics	theorem	Exponential formula (combinatorics)		If C(x) = ∑ c_n x^n/n! is the EGF for connected structures, then exp(C(x)) is the EGF for structures decomposable into connected components.
s_compositional_formula	theorem	Compositional formula (exponential)		If F(x) = ∑ f_n x^n/n! counts structures on [n] with components counted by G(x), then F(x) = exp(G(x)) or more generally F(x) = H(G(x)) for assembly operations 
s_p_partition_generating_function	state	P-partition generating function		The formal power series F_P(x) = ∑_{σ: P-partition} x_1^{σ(1)} x_2^{σ(2)} ···, a quasisymmetric function encoding P-partitions of a labeled poset P.
s_fundamental_theorem_of_p_partitions	theorem	Fundamental theorem of P-partitions (Stanley)		The generating function for P-partitions of a naturally labeled poset P decomposes as a sum over linear extensions: F_P(q) = ∑_{w ∈ L(P)} q^{maj(w)} / (1-q)(1-q
s_linear_extension_of_a_poset	state	Linear extension of a poset		A total order on the elements of P that is compatible with the partial order; the number e(P) of linear extensions is #P-hard to compute in general.
s_graded_poset	axiom	Rank function and graded poset		A poset with a rank function ρ: P → ℕ satisfying ρ(y) = ρ(x) + 1 whenever y covers x; equivalently, all maximal chains between two elements have the same length
s_rank_generating_function	state	Rank generating function		For a graded poset P with rank function ρ, the polynomial F(P,q) = ∑_{x ∈ P} q^{ρ(x)} encoding the Whitney numbers of the second kind.
s_characteristic_polynomial_poset	state	Characteristic polynomial of a lattice		For a finite graded lattice L, χ_L(t) = ∑_{x ∈ L} μ(0̂,x) t^{ρ(1̂)-ρ(x)}, generalizing the chromatic polynomial of a graph to geometric lattices.
s_chromatic_polynomial	state	Chromatic polynomial χ_G(k)		The polynomial giving the number of proper k-colorings of a graph G; satisfies the deletion-contraction recurrence χ_G = χ_{G-e} - χ_{G/e}.
s_stirling_numbers_second_kind	state	Stirling numbers of the second kind S(n,k)		The number of partitions of an n-element set into exactly k nonempty blocks, satisfying S(n,k) = kS(n-1,k) + S(n-1,k-1) with x^n = ∑_k S(n,k) x(x-1)···(x-k+1).
s_stirling_numbers_first_kind	state	Stirling numbers of the first kind c(n,k)		The unsigned number |s(n,k)| counting permutations of n elements with exactly k cycles, satisfying x(x+1)···(x+n-1) = ∑_k c(n,k) x^k.
s_binomial_poset	axiom	Binomial poset		A locally finite graded poset with a unique minimal element in which every interval [x,y] of rank k has the same number B(k) of maximal chains.
s_eulerian_poset	axiom	Eulerian poset		A graded poset in which every nontrivial interval [x,y] satisfies μ(x,y) = (-1)^{ρ(y)-ρ(x)}, equivalently the h-vector of every interval is symmetric.
s_differential_poset	axiom	Differential poset		A graded poset satisfying DU - UD = rI for the up and down operators on its linear span, where r is a positive integer; Young's lattice is the unique r=1 exampl
s_monomial_symmetric_function	state	Monomial symmetric function m_λ		The sum of all distinct monomials x_1^{a_1} x_2^{a_2}··· obtainable by permuting the parts of the partition λ = (a_1,a_2,...); the {m_λ} form a Z-basis of Λ.
s_homogeneous_symmetric_function	state	Homogeneous symmetric function h_k		h_k = ∑_{i_1 ≤ ··· ≤ i_k} x_{i_1}···x_{i_k} = m_{(1^k)} + ··· + m_{(k)}, algebraically independent generators of the ring of symmetric functions over Z.
s_hall_inner_product	axiom	Hall inner product		The bilinear form ⟨·,·⟩ on the ring of symmetric functions defined by ⟨s_λ, s_μ⟩ = δ_{λμ}, equivalently ⟨h_λ, m_μ⟩ = δ_{λμ}.
s_knuth_equivalence	state	Knuth equivalence		The equivalence relation on words generated by xzy ~ zxy (for x ≤ y < z) and yxz ~ yzx (for x < y ≤ z); two permutations are Knuth-equivalent iff they have the 
t_jeu_de_taquin	technique	Jeu de taquin (Schützenberger)		Slides entries into inner corners by a sequence of local moves, rectifying a skew tableau to a straight-shape tableau; defines Schützenberger's theory of slides
t_promotion_and_evacuation	technique	Promotion and evacuation (Schützenberger)		Promotion rotates the entries via jeu de taquin; evacuation applies promotion n times giving an involution whose fixed points count self-complementary tableaux.
s_littlewood_richardson_tableau	state	Littlewood-Richardson tableau		A semistandard skew tableau of shape λ/μ whose reverse reading word is a lattice word (Yamanouchi word); the number of such tableaux of content ν is c^λ_{μν}.
s_border_strip	state	Border strip (rim hook)		A connected skew shape containing no 2×2 square; its height ht(β) = (number of rows) - 1 determines the sign (-1)^{ht(β)} in the Murnaghan-Nakayama rule.
s_dual_cauchy_identity	theorem	Dual Cauchy identity		∏_{i,j}(1 + x_i y_j) = ∑_λ s_λ(x) s_{λ'}(y), where λ' is the conjugate partition, the companion identity to the Cauchy identity.
t_omega_involution	technique	Omega involution ω		The ring involution on Λ defined by ω(e_k) = h_k, equivalently ω(s_λ) = s_{λ'}, interchanging elementary and homogeneous symmetric functions.
s_skew_schur_function	state	Skew Schur function s_{λ/μ}		Defined by ⟨s_{λ/μ}, s_ν⟩ = c^λ_{μν} (the LR coefficient), equivalently the generating function for semistandard tableaux of skew shape λ/μ.
s_hall_littlewood_polynomial	state	Hall-Littlewood polynomial P_λ(x;t)		A one-parameter family of symmetric functions interpolating between Schur functions (t=0) and monomial symmetric functions (t=1), with connections to p-adic gro
s_macdonald_polynomial	state	Macdonald polynomial P_λ(x;q,t)		A two-parameter family of symmetric functions that are joint eigenfunctions of Macdonald's q-difference operators, specializing to Hall-Littlewood (q=0), Jack (
s_macmahon_plane_partition_gf	theorem	MacMahon's plane partition generating function		The generating function for all plane partitions is ∏_{k≥1} 1/(1-q^k)^k, proved by MacMahon using the theory of partitions.
s_lattice_path	axiom	Lattice path		A sequence of steps in ℤ^d from a fixed step-set, typically unit steps East and North in ℤ^2; lattice paths model binomial coefficients, Catalan numbers, and ba
t_reflection_principle_andre	technique	Reflection principle (André)		Counts paths staying above a line by reflecting the initial segment at the first crossing, establishing a bijection with paths from a reflected origin.
s_chromatic_symmetric_function	state	Chromatic symmetric function X_G (Stanley)		X_G(x) = ∑_{κ proper} x_{κ(1)} x_{κ(2)}···x_{κ(n)}, a symmetric function generalizing the chromatic polynomial; conjectured to distinguish non-isomorphic trees.
s_stanley_symmetric_function	state	Stanley symmetric function F_w		A symmetric function indexed by a permutation w ∈ S_n, defined as a sum over reduced decompositions of w; its Schur expansion gives the number of reduced words 
s_fundamental_quasisymmetric_function	state	Fundamental quasisymmetric function F_{n,S}		For S ⊆ {1,...,n-1}, F_{n,S} = ∑_{i_1 ≤ ··· ≤ i_n, i_j < i_{j+1} if j ∈ S} x_{i_1}···x_{i_n}; the {F_{n,S}} form a basis of the degree-n quasisymmetric function
s_quasisymmetric_function	axiom	Quasisymmetric function		A formal power series f(x_1,x_2,...) of bounded degree invariant under the operation x_{i_1}^{a_1}···x_{i_k}^{a_k} ↦ x_{j_1}^{a_1}···x_{j_k}^{a_k} whenever i_1<
s_gessels_formula_p_partitions	theorem	Gessel's formula for P-partitions		The P-partition generating function of a labeled poset P equals ∑_{w ∈ L(P)} F_{n,Des(w)}, expressing it as a sum of fundamental quasisymmetric functions over l
s_schur_positivity	state	Schur positivity		A symmetric function f is Schur-positive if all coefficients in its expansion f = ∑_λ c_λ s_λ satisfy c_λ ≥ 0, typically implying a representation-theoretic int
s_hook_content_formula	theorem	Hook content formula		The number of semistandard Young tableaux of shape λ with entries in {1,...,n} is s_λ(1^n) = ∏_{u ∈ λ} (n + c(u))/h(u) where c(u) is the content and h(u) the ho
t_involution_principle	technique	Involution principle (Garsia-Milne)		Constructs bijective proofs of combinatorial identities by composing two sign-reversing involutions, with the non-cancelled terms providing the desired bijectio
s_schensted_theorem	theorem	Schensted's theorem		Under the RSK correspondence, the length of the longest increasing subsequence of a permutation σ equals the length of the first row of its insertion tableau P(
s_greenes_theorem	theorem	Greene's theorem		For a permutation σ with RSK insertion tableau P(σ) of shape λ, the maximum length of a union of k increasing subsequences equals λ_1 + ··· + λ_k for all k.
t_hillman_grassl_correspondence	technique	Hillman-Grassl correspondence		A weight-preserving bijection between reverse plane partitions and hook-indexed arrays, providing a combinatorial proof of the generating function for reverse p
s_stanley_hook_content_formula_plane_partitions	theorem	Stanley's hook content formula for column-strict plane partitions		The generating function for column-strict plane partitions of shape λ with entries ≤ n is ∏_{u ∈ λ} (1-q^{n+c(u)})/(1-q^{h(u)}).
t_combinatorial_species	technique	Theory of combinatorial species (Joyal)		Joyal's framework treating combinatorial structures as functors from finite sets with bijections, enabling systematic passage to generating functions and compos
t_gaussian_elimination_q_counting	technique	Gaussian elimination and q-counting		Uses row reduction over finite fields to count subspaces, yielding q-binomial coefficients and q-analogs of combinatorial identities via the lattice of subspace
s_f_vector_h_vector_simplicial_complex	state	f-vector and h-vector of a simplicial complex		For a (d-1)-dimensional simplicial complex Δ, f_i = number of i-dimensional faces; the h-vector (h_0,...,h_d) is defined by ∑ h_i t^{d-i} = ∑ f_{i-1}(t-1)^{d-i}
s_shellability	axiom	Shellability		A pure simplicial complex is shellable if its maximal faces can be ordered F_1,...,F_t so that F_k ∩ (F_1 ∪ ··· ∪ F_{k-1}) is pure of codimension 1 for all k; i
t_el_shellability	technique	EL-shellability and CL-shellability		An edge labeling of a bounded poset's Hasse diagram is an EL-labeling if every interval has a unique increasing maximal chain which is lexicographically first; 
t_polyhedral_geometry_of_partitions	technique	Polyhedral geometry of partitions		Realizes partition-counting problems as lattice point enumeration in rational polytopes, connecting to Ehrhart polynomials, Barvinok's algorithm, and inside-out
s_multigraph_definition	axiom	Multigraph		A graph allowing multiple edges between the same pair of vertices.
s_subgraph	axiom	Subgraph		A graph H is a subgraph of G if V(H) ⊆ V(G) and E(H) ⊆ E(G).
s_induced_subgraph	axiom	Induced subgraph		The subgraph G[S] induced by S ⊆ V(G) contains all edges of G between vertices in S.
s_spanning_subgraph	axiom	Spanning subgraph		A subgraph H of G with V(H) = V(G).
s_graph_path	axiom	Path (graph theory)		A non-empty graph P with vertices x₀, …, xₖ and edges xᵢxᵢ₊₁, all vertices distinct.
s_graph_cycle	axiom	Cycle (graph theory)		A closed path of length ≥ 3 with all vertices distinct except first and last.
s_connected_graph	axiom	Connected graph		A graph in which every two vertices are linked by a path.
s_tree_graph	axiom	Tree (graph theory)		A connected graph containing no cycle; equivalently, a connected graph on n vertices with exactly n−1 edges.
s_forest_graph	axiom	Forest		An acyclic graph; equivalently, a disjoint union of trees.
s_spanning_tree	axiom	Spanning tree		A spanning subgraph that is a tree.
s_vertex_degree	axiom	Vertex degree		The degree d(v) of a vertex v is the number of edges incident with v.
s_minimum_degree_delta	axiom	Minimum degree δ(G)		The minimum vertex degree δ(G) = min{d(v) : v ∈ V(G)}.
s_euler_tour	axiom	Euler tour		A closed walk in a graph that traverses every edge exactly once.
s_euler_tour_theorem	theorem	Euler's theorem on Euler tours		A connected graph has an Euler tour if and only if every vertex has even degree.
s_bipartite_characterization	theorem	Bipartite characterization		A graph is bipartite if and only if it contains no odd cycle.
s_handshaking_lemma	theorem	Handshaking lemma		The sum of all vertex degrees equals twice the number of edges: Σ d(v) = 2|E|.
s_graph_complement	axiom	Complement graph		The complement Ḡ has edge set consisting of all pairs not in E(G).
s_complete_graph_kn	axiom	Complete graph K_n		The graph on n vertices where every pair of distinct vertices is joined by an edge.
s_complete_bipartite_graph	axiom	Complete bipartite graph K_{r,s}		The bipartite graph with parts of sizes r and s where every cross-pair is adjacent.
s_graph_contraction	axiom	Edge contraction		The operation of contracting edge e = xy, identifying x and y and removing loops.
s_graph_minor_definition	axiom	Graph minor		H is a minor of G if H can be obtained from a subgraph of G by contracting edges.
s_topological_minor_subdivision	axiom	Topological minor (subdivision)		H is a topological minor of G if G contains a subdivision of H (edges replaced by internally disjoint paths).
s_vertex_cover	axiom	Vertex cover		A set of vertices S such that every edge has at least one endpoint in S.
s_independent_set_graph	axiom	Independent set		A set of pairwise non-adjacent vertices in a graph.
s_edge_cover	axiom	Edge cover		A set of edges F such that every vertex is incident to at least one edge in F.
s_gallai_identity	theorem	Gallai's identities		For a graph without isolated vertices: α(G) + β(G) = |V(G)| and α'(G) + β'(G) = |V(G)|.
s_hall_theorem_diestel	theorem	Hall's marriage theorem		A bipartite graph G = (A ∪ B, E) has a matching saturating A iff |N(S)| ≥ |S| for every S ⊆ A.
s_konig_theorem_diestel	theorem	König's theorem (matching-cover duality)		In a bipartite graph, the maximum matching size equals the minimum vertex cover size.
s_tutte_theorem_diestel	theorem	Tutte's 1-factor theorem		G has a perfect matching iff for every S ⊆ V(G), the number of odd components of G − S is at most |S|.
s_deficiency_version_hall	theorem	Deficiency version of Hall's theorem		The maximum matching in bipartite G = (A ∪ B, E) has size |A| − max_{S ⊆ A}(|S| − |N(S)|).
s_berge_theorem_matching	theorem	Berge's theorem (maximum matching)		A matching M is maximum if and only if G contains no M-augmenting path.
s_k_connected_graph	axiom	k-connected graph		G is k-connected if |V(G)| > k and G − S is connected for every S with |S| < k.
s_vertex_connectivity	axiom	Vertex connectivity κ(G)		The maximum k such that G is k-connected.
s_edge_connectivity	axiom	Edge connectivity λ(G)		The minimum number of edges whose removal disconnects G.
s_whitney_inequality	theorem	Whitney's inequality		For every graph G: κ(G) ≤ λ(G) ≤ δ(G).
s_menger_theorem_diestel	theorem	Menger's theorem		The maximum number of internally vertex-disjoint s–t paths equals the minimum s–t separating set size.
s_menger_edge_version	theorem	Menger's theorem (edge version)		The maximum number of edge-disjoint s–t paths equals the minimum edge cut size separating s from t.
s_menger_global_version	theorem	Menger's theorem (global version)		G is k-connected iff every pair of vertices can be joined by k internally vertex-disjoint paths.
s_block_graph	axiom	Block (2-connected component)		A maximal 2-connected subgraph, or a bridge with its endpoints.
s_block_cut_tree	state	Block-cut tree		The tree whose vertices are blocks and cut-vertices, with edges joining each cut vertex to its blocks.
s_ear_decomposition	state	Ear decomposition		A sequence of paths (ears) decomposing a 2-connected graph, each ear sharing only its endpoints with earlier structure.
s_ear_decomposition_theorem	theorem	Ear decomposition theorem (Whitney)		A graph is 2-connected if and only if it has an ear decomposition starting from a cycle.
s_tutte_3connected_structure	theorem	Tutte's wheel theorem		Every 3-connected graph can be obtained from a wheel by edge additions and vertex splits.
t_ear_decomposition_technique	technique	Ear decomposition technique		Constructing 2-connected graphs by adding ears to prove structural results.
s_face_planar_graph	axiom	Face of a plane graph		A connected region of ℝ² \ G in a plane embedding, including the unbounded outer face.
s_euler_formula_planar	theorem	Euler's formula for planar graphs		For a connected plane graph: |V| − |E| + |F| = 2.
s_planar_edge_bound	theorem	Planar graph edge bound		Every planar graph on n ≥ 3 vertices has at most 3n − 6 edges; if triangle-free, at most 2n − 4.
s_k5_k33_nonplanar	theorem	Non-planarity of K₅ and K₃,₃		Neither K₅ nor K₃,₃ is planar, as verified by the edge bound from Euler's formula.
s_kuratowski_theorem_diestel	theorem	Kuratowski's theorem		A graph is planar iff it contains no subdivision of K₅ or K₃,₃.
s_wagner_theorem_diestel	theorem	Wagner's theorem		A graph is planar iff it has no K₅ or K₃,₃ minor.
s_dual_graph	axiom	Dual graph (planar)		The graph G* whose vertices are faces of G, with edges for shared boundaries.
s_mac_lane_planarity	theorem	Mac Lane's planarity criterion		A graph is planar iff its cycle space has a basis where every edge belongs to at most two basis cycles.
s_planar_graph_six_color	theorem	Six-color theorem for planar graphs		Every planar graph is 6-colorable, corollary of the edge bound δ(G) ≤ 5.
t_kempe_chain_argument	technique	Kempe chain argument		Recoloring along a maximal bichromatic path to free a color at a vertex.
s_outerplanar_graph	axiom	Outerplanar graph		A planar graph admitting an embedding with all vertices on the outer face.
s_graph_genus	axiom	Genus of a graph		The minimum genus of a surface in which the graph can be embedded without crossings.
s_vertex_coloring_proper	axiom	Proper vertex coloring		An assignment of colors to vertices such that no two adjacent vertices share the same color.
s_chromatic_number	axiom	Chromatic number χ(G)		The minimum number of colors needed for a proper vertex coloring of G.
s_clique_number	axiom	Clique number ω(G)		The maximum size of a complete subgraph in G.
s_brooks_theorem_diestel	theorem	Brooks' theorem		Every connected graph with Δ(G) ≥ 3 satisfies χ(G) ≤ Δ(G), unless G is complete or an odd cycle.
s_greedy_coloring_bound	theorem	Greedy coloring bound		The greedy algorithm with any vertex ordering produces a proper coloring using at most Δ(G) + 1 colors.
s_chromatic_polynomial_diestel	state	Chromatic polynomial P(G, k)		The polynomial P(G, k) counting proper k-colorings of G.
s_edge_coloring	axiom	Proper edge coloring		An assignment of colors to edges so that no two edges sharing an endpoint share a color.
s_chromatic_index	axiom	Chromatic index χ'(G)		The minimum number of colors needed for a proper edge coloring.
s_vizing_theorem_diestel	theorem	Vizing's theorem		For every simple graph: Δ(G) ≤ χ'(G) ≤ Δ(G) + 1.
s_list_coloring	axiom	List coloring		A proper coloring where each vertex must receive a color from its prescribed list.
s_choosability	axiom	Choosability (list chromatic number)		The minimum k such that G is L-colorable for every list assignment with |L(v)| ≥ k.
s_perfect_graph_theorem_lovasz	theorem	Perfect graph theorem (Lovász)		A graph is perfect iff its complement is perfect.
s_strong_perfect_graph_diestel	theorem	Strong perfect graph theorem (Chudnovsky–Robertson–Seymour–Thomas)		A graph is perfect iff neither it nor its complement contains an induced odd cycle of length ≥ 5.
s_mycielski_construction	state	Mycielski construction		A construction producing triangle-free graphs with arbitrarily large chromatic number.
t_deletion_contraction	technique	Deletion-contraction		Computing graph invariants recursively via P(G,k) = P(G−e,k) − P(G/e,k).
s_degeneracy_coloring	axiom	Degeneracy (k-degenerate graph)		A graph is k-degenerate if every subgraph has a vertex of degree at most k.
s_nowhere_zero_flow	axiom	Nowhere-zero flow		An assignment of non-zero group elements to edges satisfying Kirchhoff's law at every vertex.
s_nowhere_zero_k_flow	axiom	Nowhere-zero k-flow		A ℤ-flow with all values in {±1, …, ±(k−1)} satisfying flow conservation.
s_flow_coloring_duality	theorem	Flow-coloring duality (planar graphs)		A planar graph has a proper k-coloring iff its dual has a nowhere-zero k-flow.
s_tutte_flow_conjectures	state	Tutte's flow conjectures		Every bridgeless graph has a nowhere-zero 5-flow (5-flow conjecture); without Petersen minor a 4-flow.
s_six_flow_theorem	theorem	Six-flow theorem (Seymour)		Every bridgeless graph has a nowhere-zero 6-flow.
s_eight_flow_theorem	theorem	Eight-flow theorem (Jaeger)		Every bridgeless graph has a nowhere-zero 8-flow.
s_four_color_flow_equivalence	theorem	Four-color theorem via flows		The four-color theorem is equivalent to every bridgeless planar graph having a nowhere-zero 4-flow.
s_group_valued_flow	axiom	Group-valued flow		A flow with edge values in a finite abelian group Γ satisfying flow conservation.
s_tutte_group_flow_theorem	theorem	Tutte's theorem on group-valued flows		Existence of a nowhere-zero Γ-flow depends only on |Γ|, not on the group structure.
s_turan_theorem_diestel	theorem	Turán's theorem		The max edges in a K_{r+1}-free n-vertex graph is (1 − 1/r)n²/2, achieved uniquely by T(n,r).
s_turan_graph_definition	axiom	Turán graph T(n,r)		The complete r-partite graph on n vertices with parts as equal as possible.
s_extremal_number	axiom	Extremal number ex(n, H)		The max edges in an n-vertex graph containing no subgraph isomorphic to H.
s_kovari_sos_turan_diestel	theorem	Kővári–Sós–Turán theorem		ex(n, K_{s,t}) ≤ ½((t−1)^{1/s} n^{2−1/s} + (s−1)n) for s ≤ t.
s_szemeredi_regularity_diestel	theorem	Szemerédi regularity lemma		For every ε > 0 there exists M such that V(G) can be partitioned into ≤ M classes with all but εk² pairs ε-regular.
s_epsilon_regular_pair	axiom	ε-regular pair		A pair (A, B) is ε-regular if for all large subsets the edge density deviates by less than ε from d(A,B).
s_graph_removal_lemma_diestel	theorem	Graph removal lemma		For H and ε > 0 there exists δ > 0 so that an n-vertex graph with < δn^{v(H)} copies of H can be made H-free by removing ≤ εn² edges.
s_ramsey_theorem_diestel	theorem	Ramsey's theorem (finite, graph version)		For all r, s there exists n such that every 2-coloring of E(K_n) has a red K_r or blue K_s.
s_ramsey_number	axiom	Ramsey number R(r, s)		The minimum n for which every 2-coloring of E(K_n) contains monochromatic K_r or K_s.
s_erdos_ramsey_lower	theorem	Erdős probabilistic lower bound for Ramsey numbers		R(k, k) ≥ 2^{k/2} by the probabilistic method.
s_zarankiewicz_problem	axiom	Zarankiewicz problem		Determining the max 1s in a 0-1 matrix of given size with no all-1 submatrix of specified dimensions.
t_regularity_method_diestel	technique	Regularity method		Three-step method: apply regularity lemma, clean up, find structure in reduced graph.
t_double_counting_technique	technique	Double counting (combinatorics)		Counting the same quantity two ways to establish an identity or inequality.
t_probabilistic_method_graphs	technique	Probabilistic method (graph theory)		Proving existence by showing a random construction has positive probability of satisfying the desired property.
s_konig_infinity_lemma	theorem	König's infinity lemma		Every infinite finitely-branching rooted tree contains an infinite path.
s_ray_graph	axiom	Ray (one-way infinite path)		A one-way infinite path x₀x₁x₂… in a graph.
s_end_of_graph	axiom	End of a graph		An equivalence class of rays, where two rays are equivalent if no finite vertex set separates them.
s_locally_finite_graph	axiom	Locally finite graph		A graph in which every vertex has finite degree.
s_halin_theorem_ends	theorem	Halin's theorem (grid minor from thick ends)		A graph has a thick end iff it contains a subdivision of the hexagonal half-grid.
s_star_comb_lemma	theorem	Star-comb lemma		For infinite U in connected G, G contains either a comb with teeth in U or a subdivided star with leaves in U.
s_compactness_graph_coloring	theorem	De Bruijn–Erdős compactness theorem for coloring		An infinite graph is k-colorable iff every finite subgraph is k-colorable.
s_ramsey_theorem_infinite_diestel	theorem	Infinite Ramsey theorem		For every finite coloring of r-element subsets of an infinite set, there exists an infinite monochromatic subset.
s_canonical_ramsey_theorem	theorem	Canonical Ramsey theorem (Erdős–Rado)		For every coloring of pairs of ℕ, there exists an infinite set on which the coloring is canonical.
s_graph_ramsey_number	axiom	Graph Ramsey number R(G, H)		The minimum n such that every 2-coloring of E(K_n) contains red G or blue H.
s_ramsey_multiplicity	state	Ramsey multiplicity		The minimum number of monochromatic copies of H in any 2-coloring of E(K_n).
s_hamilton_cycle	axiom	Hamilton cycle		A cycle visiting every vertex exactly once.
s_hamilton_path	axiom	Hamilton path		A path visiting every vertex exactly once.
s_dirac_theorem_hamilton	theorem	Dirac's theorem (Hamilton cycles)		Every graph on n ≥ 3 vertices with δ(G) ≥ n/2 contains a Hamilton cycle.
s_chvatal_theorem_hamilton	theorem	Chvátal's theorem (degree sequence condition)		G on n vertices with degree sequence d₁ ≤ … ≤ dₙ is Hamiltonian if for all i < n/2: dᵢ ≤ i implies d_{n−i} ≥ n − i.
s_chvatal_erdos_theorem_diestel	theorem	Chvátal–Erdős theorem		If κ(G) ≥ α(G) for G with |V| ≥ 3, then G is Hamiltonian.
s_tutte_hamilton_planar	theorem	Tutte's theorem (4-connected planar graphs are Hamiltonian)		Every 4-connected planar graph has a Hamilton cycle.
s_toughness	axiom	Toughness of a graph		t(G) = min |S|/c(G−S) over all cutsets S, where c counts components.
s_chvatal_toughness_conjecture	state	Chvátal's toughness conjecture		There exists t₀ such that every t₀-tough graph is Hamiltonian.
s_bondy_chvatal_closure	theorem	Bondy–Chvátal closure theorem		G is Hamiltonian iff its closure cl(G) is Hamiltonian, where cl(G) adds edges between non-adjacent vertices with degree sum ≥ n.
s_erdos_renyi_model	axiom	Erdős–Rényi random graph model G(n, p)		The random graph on n vertices where each edge is included independently with probability p.
s_threshold_function	axiom	Threshold function		A function p*(n) such that P[G(n,p) has Q] → 0 if p/p* → 0, and → 1 if p/p* → ∞.
s_connectivity_threshold	theorem	Connectivity threshold for G(n, p)		The threshold for G(n, p) to be connected a.a.s. is p = (log n)/n.
s_giant_component_threshold	theorem	Giant component threshold (Erdős–Rényi)		In G(n, c/n): if c < 1 all components are O(log n); if c > 1 there is a unique giant component of Θ(n) vertices.
s_evolution_random_graphs	state	Evolution of random graphs		As p increases in G(n,p), the graph undergoes phase transitions: trees → unicyclic → giant component → connected → Hamiltonian.
s_zero_one_law	theorem	Zero-one law for random graphs		For constant p ∈ (0,1), every first-order graph property has probability tending to 0 or 1 in G(n, p).
s_chromatic_number_random	theorem	Chromatic number of random graphs		χ(G(n, 1/2)) = Θ(n/log n) almost surely.
s_hamilton_threshold	theorem	Hamilton cycle threshold		The threshold for G(n, p) to contain a Hamilton cycle a.a.s. is p = (log n + log log n)/n.
t_first_moment_method	technique	First moment method (Markov bound)		Proving P[X ≥ 1] ≤ E[X] for non-negative X, to show a property a.a.s. does not hold.
t_second_moment_method	technique	Second moment method (Chebyshev)		Proving P[X > 0] ≥ (E[X])²/E[X²] to show a property a.a.s. does hold.
s_tree_width	axiom	Tree-width		The minimum width of a tree-decomposition of G, where width is max bag size minus one.
s_tree_decomposition	axiom	Tree-decomposition		A pair (T, {Bₜ}) covering all vertices and edges, with bags of any vertex inducing a connected subtree of T.
s_bramble	axiom	Bramble (screen)		A family of pairwise touching connected subgraphs; its order is the minimum hitting set size.
s_tree_width_duality	theorem	Tree-width duality theorem (Seymour–Thomas)		The tree-width of G equals the maximum bramble order minus one.
s_robertson_seymour_theorem	theorem	Robertson–Seymour theorem (graph minor theorem)		The minor relation on finite graphs is a well-quasi-ordering.
s_minor_closed_characterization	theorem	Finite forbidden minor characterization		Every minor-closed class has a finite set of excluded minors (corollary of Robertson–Seymour).
s_well_quasi_ordering	axiom	Well-quasi-ordering (WQO)		A quasi-order with no infinite antichains and no infinite strictly descending sequences.
s_graph_structure_theorem	theorem	Graph structure theorem (Robertson–Seymour)		Graphs with no H-minor can be built by clique-sums from graphs almost-embeddable in surfaces excluding H.
s_grid_minor_theorem	theorem	Grid minor theorem (Robertson–Seymour)		For every k there exists f(k) such that every graph of tree-width ≥ f(k) contains a k × k grid minor.
s_excluded_minor_planar	theorem	Excluded minor characterization of planar graphs		Planar graphs are exactly those with no K₅ minor and no K₃,₃ minor.
s_path_width	axiom	Path-width		The minimum width of a path-decomposition (tree-decomposition where the tree is a path).
s_clique_sum	axiom	Clique-sum		Gluing two graphs along a common clique and possibly deleting some clique edges.
s_haven	axiom	Haven		A strategy for a robber to escape fewer than k cops; equivalent to tree-width ≥ k.
s_cops_and_robber	state	Cops and robbers characterization of tree-width		tw(G) + 1 equals the minimum number of cops needed to catch a robber on G.
t_well_quasi_ordering_technique	technique	Well-quasi-ordering technique		Using WQO theory (Higman's lemma, Kruskal's theorem) to prove finiteness of obstruction sets.
t_graph_minor_algorithm	technique	Graph minor testing algorithm		Robertson–Seymour's O(n³) algorithm for testing whether a fixed H is a minor of input G.
s_petersen_graph	axiom	Petersen graph		The 3-regular graph on 10 vertices; a fundamental counterexample in graph theory.
s_line_graph	axiom	Line graph		L(G) has edges of G as vertices, adjacent when the edges share an endpoint.
s_tutte_polynomial_diestel	state	Tutte polynomial T(G; x, y)		A bivariate polynomial encoding extensive combinatorial information, generalizing chromatic and flow polynomials.
s_girth	axiom	Girth of a graph		The length of the shortest cycle; ∞ if acyclic.
s_erdos_girth_chromatic	theorem	Erdős theorem on girth and chromatic number		For all g, k there exists a graph with girth > g and chromatic number > k.
s_kirchhoff_matrix_tree	theorem	Kirchhoff's matrix tree theorem		The number of spanning trees equals any cofactor of the Laplacian matrix.
s_kruskal_tree_theorem	theorem	Kruskal's tree theorem		Finite rooted trees are WQO by the homeomorphic embedding relation.
s_higman_lemma	theorem	Higman's lemma		If (Q, ≤) is a WQO, then finite sequences over Q are WQO under subsequence embedding.
s_gallai_milgram_theorem	theorem	Gallai–Milgram theorem		The minimum directed path cover size of a digraph equals the maximum independent set in the underlying graph.
s_alon_tarsi_theorem	theorem	Alon–Tarsi theorem		G is L-colorable from lists of size k if the numbers of even and odd Eulerian subgraphs differ.
s_hadwiger_conjecture	state	Hadwiger's conjecture		Every graph with χ(G) ≥ k has a K_k minor.
t_augmenting_path_method	technique	Augmenting path method		Finding alternating paths between exposed vertices to increase matching size by one.
t_discharging_method_diestel	technique	Discharging method		Assigning charges via Euler's formula and redistributing by local rules to derive contradictions.
t_extremal_graph_counting	technique	Extremal counting (Zykov symmetrization)		Showing the extremal graph is complete multipartite via successive vertex identification.
s_network_flow_theory	axiom	Network flow		An assignment of flow values to directed edges satisfying conservation at internal vertices.
s_max_flow_min_cut_diestel	theorem	Max-flow min-cut theorem (Ford–Fulkerson)		In a network, the max flow value equals the min s–t cut capacity.
s_mader_theorem_average_degree	theorem	Mader's theorem (average degree forces topological minors)		For every H there exists c(H) such that average degree ≥ c(H) forces a topological minor of H.
s_cycle_space	axiom	Cycle space		The GF(2)-vector space generated by edge sets of cycles; dimension |E| − |V| + c(G).
s_cut_space	axiom	Cut space		The GF(2)-vector space generated by minimal edge cuts; orthogonal complement of cycle space.
t_tree_decomposition_technique	technique	Tree-decomposition dynamic programming		Solving NP-hard problems in polynomial time on bounded tree-width graphs via DP along tree-decomposition.
s_graph_product_types	axiom	Graph products (Cartesian, tensor, strong)		Three standard products defining adjacency in V(G) × V(H) differently.
s_heawood_map_coloring	theorem	Heawood's map coloring theorem		The chromatic number of graphs on a surface of genus g ≥ 1 is at most ⌊(7 + √(1+48g))/2⌋.
s_edge_connectivity_submodularity	theorem	Submodularity of the edge boundary		|∂(A)| + |∂(B)| ≥ |∂(A∩B)| + |∂(A∪B)|.
s_degree_of_vertex	axiom	Degree of a vertex		The degree d(v) of a vertex v in a graph G is the number of edges incident to v.
s_walk_graph	axiom	Walk (graph theory)		A walk in a graph is a sequence of vertices v₀, v₁, …, vₖ where each consecutive pair is joined by an edge.
s_trail_graph	axiom	Trail (graph theory)		A trail is a walk in which all edges are distinct.
s_prufer_sequence	axiom	Prüfer sequence		A unique sequence of n−2 integers from {1,…,n} encoding a labeled tree on n vertices, establishing a bijection used to prove Cayley's formula.
s_konig_edge_coloring_theorem	theorem	König's edge-coloring theorem		For every bipartite graph G, the chromatic index equals the maximum degree: χ'(G) = Δ(G).
s_clique	axiom	Clique		A clique in a graph G is a set of pairwise adjacent vertices, equivalently a subset inducing a complete subgraph.
s_electrical_network_on_graph	axiom	Electrical network on a graph		A graph with positive real conductances (or resistances) assigned to each edge, modeling an electrical resistor network.
s_kirchhoff_current_law_graph	axiom	Kirchhoff's current law (graph)		At every non-terminal vertex of an electrical network, the sum of currents flowing in equals the sum flowing out.
s_kirchhoff_voltage_law_graph	axiom	Kirchhoff's voltage law (graph)		Around every cycle in an electrical network, the sum of the voltage drops (current times resistance) is zero.
s_effective_resistance	state	Effective resistance		The effective resistance R_eff(a,b) between vertices a and b is the voltage difference needed to drive a unit current from a to b.
s_rayleigh_monotonicity_law	theorem	Rayleigh's monotonicity law		Increasing any edge resistance in a network does not decrease the effective resistance between any two nodes.
t_series_parallel_reduction	technique	Series-parallel reduction		Simplifying a resistor network by replacing series resistors with their sum and parallel resistors with their harmonic mean.
s_random_walk_on_graph	axiom	Random walk on a graph		A stochastic process on the vertices of a graph where at each step the walker moves to a uniformly random neighbour of the current vertex.
s_hitting_time	state	Hitting time		The expected number of steps for a random walk starting at vertex u to first reach vertex v.
s_commute_time	state	Commute time		The expected number of steps for a random walk to travel from u to v and back, equal to 2|E| · R_eff(u,v) by the commute-time identity.
s_random_walk_electrical_correspondence	theorem	Random walk–electrical network correspondence		Voltages in an electrical network on a graph equal hitting probabilities of the corresponding random walk, and commute times equal 2|E| times effective resistan
s_nash_williams_criterion	theorem	Nash-Williams recurrence criterion		A connected locally finite graph has a recurrent random walk if and only if there exist disjoint finite cutsets with divergent sum of reciprocal capacities.
t_ford_fulkerson_algorithm	technique	Ford–Fulkerson algorithm		Iteratively finding augmenting s–t paths in the residual graph and pushing flow along them until no augmenting path remains.
s_f_factor	axiom	f-factor		For a function f: V(G) → ℤ≥₀, an f-factor is a spanning subgraph H with d_H(v) = f(v) for every vertex v.
s_tutte_f_factor_theorem	theorem	Tutte's f-factor theorem		A graph G has an f-factor if and only if for every disjoint pair S, T ⊆ V, the Tutte deficiency condition q(S,T) ≤ Σ_{v∈S} f(v) holds.
s_erdos_szekeres_ramsey_bound	theorem	Erdős–Szekeres bound on Ramsey numbers		R(s,t) ≤ C(s+t−2, s−1), establishing that the Ramsey number R(s,t) is at most the central binomial coefficient.
s_thomassen_5_list_coloring	theorem	Thomassen's theorem (5-list-coloring of planar graphs)		Every planar graph is 5-choosable, that is, it admits a proper coloring from any assignment of lists of size 5 to vertices.
s_voigt_non_4_choosable_planar	state	Voigt's example (non-4-choosable planar graph)		A planar graph constructed by Voigt (1993) that is not 4-choosable, showing that the list chromatic number of planar graphs can exceed 4.
s_chordal_graph	axiom	Chordal graph		A graph in which every cycle of length four or more has a chord; equivalently, a graph with a perfect elimination ordering.
s_shannon_multigraph_bound	theorem	Shannon's multigraph edge-coloring bound		For any multigraph G, the chromatic index satisfies χ'(G) ≤ ⌊3Δ(G)/2⌋.
s_vizing_adjacency_lemma	theorem	Vizing's adjacency lemma		If G is a simple graph with χ'(G) = Δ(G) + 1, then every vertex of maximum degree is adjacent to at least two other vertices of maximum degree.
s_class_1_class_2_graph	axiom	Class 1 and Class 2 graphs		A simple graph is Class 1 if χ'(G) = Δ(G) and Class 2 if χ'(G) = Δ(G) + 1, the only possibilities by Vizing's theorem.
t_vizing_fan_argument	technique	Vizing fan argument		Constructing a fan of edges at a vertex and recoloring along Kempe-type paths to extend a partial edge coloring, proving Vizing's theorem.
s_ramsey_theorem_multicolor	theorem	Ramsey's theorem (multicolor version)		For any positive integers r, k₁, …, kᵣ, there exists N such that every r-coloring of the edges of K_N contains a monochromatic K_{kᵢ} in color i for some i.
s_diagonal_ramsey_bounds	state	Diagonal Ramsey number bounds		The diagonal Ramsey number satisfies √2^k ≤ R(k,k) ≤ 4^k, with the lower bound from Erdős's probabilistic argument and the upper from the Erdős–Szekeres recurre
s_off_diagonal_ramsey_bounds	state	Off-diagonal Ramsey bounds		For fixed s ≥ 3, the Ramsey number R(s,t) satisfies c₁ t^{(s+1)/2-1}/(log t)^{...} ≤ R(s,t) ≤ c₂ t^{s-1}/log^{s-2} t as t → ∞.
s_schur_number	axiom	Schur number S(r)		The largest integer n such that {1,…,n} can be r-colored with no monochromatic solution to x + y = z.
s_combinatorial_line	axiom	Combinatorial line		A set of k points in [k]^n obtained by fixing some coordinates and letting the remaining coordinates run through all of {1,…,k} in unison.
s_stepping_up_lemma	theorem	Erdős–Rado stepping-up lemma		A recursive tool converting Ramsey-type bounds for r-uniform hypergraphs to bounds for (r+1)-uniform hypergraphs, yielding tower-type lower bounds for hypergrap
s_compactness_principle_ramsey	theorem	Compactness principle for Ramsey theory		The finite Ramsey theorem follows from the infinite version by a compactness (König's infinity lemma) argument: if no finite bound existed, an infinite countere
s_gnm_random_graph_model	axiom	Uniform random graph model G(n, M)		The random graph chosen uniformly at random from all graphs on n labeled vertices with exactly M edges.
s_subgraph_appearance_threshold	theorem	Subgraph appearance threshold		The threshold for the appearance of a fixed subgraph H in G(n,p) is p*(n) = n^{−1/ρ(H)} where ρ(H) = max_{J⊆H} |E(J)|/|V(J)| is the maximum density.
s_double_jump_random_graph	state	Double jump in random graph evolution		The phase transition near p = 1/n in G(n,p) where the largest component jumps from O(log n) to Θ(n^{2/3}) at p = 1/n and then to Θ(n) for p > 1/n.
s_clique_number_random_graph	state	Clique number of random graphs		Almost surely, ω(G(n,1/2)) = (2 + o(1)) log₂ n, concentrating on at most two consecutive values.
s_janson_inequality	theorem	Janson's inequality		For a family of events determined by independent random variables, the probability that none occur is at most exp(−μ + Δ/2), where μ is the sum of probabilities
s_graph_automorphism_group	axiom	Automorphism group of a graph		The group Aut(G) of all adjacency-preserving permutations of V(G), capturing the symmetries of the graph.
s_vertex_transitive_graph	axiom	Vertex-transitive graph		A graph G such that for every pair of vertices u, v there exists an automorphism mapping u to v.
s_frucht_theorem	theorem	Frucht's theorem		Every finite group is isomorphic to the automorphism group of some finite simple graph.
s_adjacency_matrix_graph	axiom	Adjacency matrix of a graph		The n×n matrix A with A_{ij} = 1 if vertices i and j are adjacent and 0 otherwise.
s_graph_spectrum	axiom	Spectrum of a graph		The multiset of eigenvalues of the adjacency matrix A(G), providing algebraic invariants of the graph.
s_spectral_radius_graph	state	Spectral radius of a graph		The largest eigenvalue λ₁ of the adjacency matrix of G, satisfying d_avg ≤ λ₁ ≤ Δ(G) with equality iff G is regular.
s_algebraic_connectivity	state	Algebraic connectivity (Fiedler value)		The second-smallest eigenvalue λ₂ of the Laplacian matrix L(G), positive if and only if G is connected.
s_expander_graph	axiom	Expander graph		A sparse graph with strong connectivity properties: every subset S with |S| ≤ |V|/2 has neighbourhood |N(S)| ≥ c|S| for a constant c > 0.
s_spectral_gap_graph	state	Spectral gap of a graph		The difference d − λ₂ between the degree d and the second-largest eigenvalue of the adjacency matrix; a large spectral gap implies good expansion.
s_expander_mixing_lemma	theorem	Expander mixing lemma		For a d-regular graph with second eigenvalue λ, and any vertex sets S, T: |e(S,T) − d|S||T|/n| ≤ λ√(|S||T|).
s_discrete_cheeger_inequality	theorem	Discrete Cheeger inequality		For a d-regular graph, (d − λ₂)/2 ≤ h(G) ≤ √(2d(d − λ₂)), relating the Cheeger constant h(G) to the spectral gap.
s_cheeger_constant_graph	axiom	Cheeger constant of a graph		h(G) = min_{S: 0<|S|≤|V|/2} |∂S|/|S|, the minimum edge-boundary-to-volume ratio over all small vertex subsets.
s_ramanujan_graph	axiom	Ramanujan graph		A connected d-regular graph whose nontrivial adjacency eigenvalues all satisfy |λ| ≤ 2√(d−1), achieving the Alon–Boppana bound.
s_alon_boppana_bound	theorem	Alon–Boppana bound		For any d-regular graph on n vertices, the second-largest eigenvalue satisfies λ₂ ≥ 2√(d−1) − o(1) as n → ∞.
s_lps_construction	state	Lubotzky–Phillips–Sarnak (LPS) construction		An explicit construction of (p+1)-regular Ramanujan graphs using quaternion algebras and the Ramanujan–Petersson conjecture for GL₂.
s_strongly_regular_graph	axiom	Strongly regular graph srg(n, k, λ, μ)		A regular graph on n vertices of degree k where every adjacent pair has λ common neighbours and every non-adjacent pair has μ common neighbours.
s_eigenvalue_interlacing	theorem	Eigenvalue interlacing		If H is an induced subgraph of G on m vertices, the eigenvalues of A(H) interlace those of A(G): λᵢ(G) ≥ λᵢ(H) ≥ λ_{n−m+i}(G).
s_cover_time	state	Cover time of a graph		The expected time for a random walk to visit every vertex of the graph, starting from the worst-case vertex.
s_blanket_time	state	Blanket time		The expected time until a random walk has visited every vertex at least a fraction of the stationary-measure times, within a constant factor of the cover time b
s_mixing_time	state	Mixing time of a Markov chain		The minimum time t_mix(ε) until the total variation distance between the distribution at time t and the stationary distribution is at most ε for all initial sta
s_total_variation_distance_markov	axiom	Total variation distance (Markov chains)		d_TV(μ, ν) = (1/2)Σ_x |μ(x) − ν(x)| = max_A |μ(A) − ν(A)|, measuring the distance between a chain's distribution and stationarity.
s_cutoff_phenomenon	state	Cutoff phenomenon		A family of Markov chains exhibits cutoff if the total variation distance drops abruptly from near 1 to near 0 over a window of o(t_mix) steps around the mixing
s_rapid_mixing	state	Rapid mixing		A Markov chain mixes rapidly if its mixing time is polynomial in the natural size parameter, often established via spectral gap, coupling, or canonical paths.
s_whitney_rank_polynomial	state	Whitney rank polynomial (rank-nullity generating function)		R(G; x,y) = Σ_{A⊆E} x^{r(E)−r(A)} y^{|A|−r(A)}, encoding the rank and nullity of each edge subset; related to the Tutte polynomial by T(G;x,y) = R(G;x−1,y−1).
s_flow_polynomial	state	Flow polynomial		The polynomial F(G; k) counting the number of nowhere-zero k-flows on G, obtained from the Tutte polynomial as F(G;k) = (−1)^{|E|−|V|+c} T(G; 0, 1−k).
s_reliability_polynomial	state	Reliability polynomial		The probability R(G; p) that a graph remains connected when each edge fails independently with probability 1−p, expressible via the Tutte polynomial.
s_cographic_matroid	axiom	Cographic matroid		The matroid M*(G) on the edges of G whose circuits are the minimal edge cuts (bonds) of G, dual to the graphic matroid M(G).
s_matroid_minor	axiom	Matroid minor		A matroid obtained from M by a sequence of deletions M\e and contractions M/e of elements, generalizing graph minors to matroids.
s_potts_model	axiom	Potts model		A statistical mechanical model on a graph G with q spin states per vertex and interaction energy β, whose partition function Z_Potts(G; q, β) is an evaluation o
s_tutte_polynomial_complexity	state	Computational complexity of the Tutte polynomial		Evaluating the Tutte polynomial T(G; x, y) is #P-hard for all (x,y) except along the hyperbola (x−1)(y−1) = 1 and at finitely many special points, by a result o
s_partition_function_graph	state	Partition function of a graph (statistical mechanics)		Z(G) = Σ_{A⊆E} w(A), a weighted sum over spanning subgraphs whose specializations yield the chromatic, flow, and reliability polynomials via the Tutte polynomia
s_dichromatic_polynomial	state	Dichromatic polynomial (Whitney)		Whitney's two-variable polynomial Q(G; q, v) = Σ_{A⊆E} q^{k(A)} v^{|A|}, the precursor of the Tutte polynomial encoding the number of components and edges of ea
t_uniform_random_coloring	technique	Uniform random coloring		Color each element independently and uniformly at random from a palette, then compute expected bad events to prove existence of a good coloring.
s_property_b_existence_uniform	theorem	Property B existence for uniform hypergraphs		Every n-uniform hypergraph with fewer than 2^{n-1} edges is 2-colorable (has Property B).
s_max_cut_probabilistic_lower_bound	theorem	Max-cut probabilistic lower bound		Every graph on m edges has a bipartite subgraph (cut) with at least m/2 edges.
s_sum_free_set_probabilistic_bound	theorem	Sum-free subset probabilistic bound		Every set of n nonzero integers contains a sum-free subset of size greater than n/3.
a_property_b_hypergraph	axiom	Property B of a hypergraph		A hypergraph has Property B if its vertex set can be 2-colored so that no edge is monochromatic.
t_linearity_of_expectation	technique	Linearity of expectation technique		Decompose a random variable as a sum of indicator random variables, compute E[X] = sum E[X_i] without requiring independence, then apply the first moment method
s_tournament_domination_bound	theorem	Tournament dominating set bound		Every tournament on n players has a dominating set of size at most ceil(log_2 n) + 1.
s_balancing_vectors_bound	theorem	Balancing vectors bound		Given n vectors v_1,...,v_n in R^n with ||v_i||_2 <= 1, there exist signs epsilon_i in {+1,-1} with ||sum epsilon_i v_i||_infinity <= sqrt(2 ln(2n)).
s_unbalancing_lights	theorem	Unbalancing lights		For any n x n matrix A with entries in {+1,-1}, there exist epsilon in {+/-1}^n with ||A epsilon||_infinity >= sqrt(n * 2/pi).
s_independence_number_greedy_expectation	theorem	Independence number greedy-expectation bound		For any graph G, alpha(G) >= sum_{v in V} 1/(d(v)+1) where d(v) is the degree of vertex v.
s_crossing_number_inequality	theorem	Crossing number inequality		For any graph with m >= 4n edges, the crossing number satisfies cr(G) >= m^3/(64n^2).
s_ramsey_lower_bound_alteration	theorem	Ramsey lower bound via alteration		R(k,k) >= (1+o(1))(k/(e*sqrt(2))) * 2^{k/2}, improving the basic probabilistic bound by alteration.
s_shearer_independence_triangle_free	theorem	Shearer's bound for triangle-free graphs		Every triangle-free d-regular graph on n vertices satisfies alpha(G) >= Omega(sqrt(n log n / d)).
s_independent_transversal_alteration	theorem	Independent transversal via alteration		If each partition class has size >= 2*Delta(G), then an independent transversal exists.
s_paley_zygmund_inequality	theorem	Paley-Zygmund inequality		For a nonnegative random variable X, Pr[X > 0] >= (E[X])^2 / E[X^2].
s_paley_graph_properties	state	Paley graph properties		The Paley graph P(q) on F_q has no clique or independent set of size exceeding (1+o(1)) log_2 q.
a_variance	axiom	Variance		Var[X] = E[X^2] - (E[X])^2, measuring the expected squared deviation of X from its mean.
t_lovasz_local_lemma_technique	technique	Lovasz Local Lemma technique		Define bad events, construct a dependency graph, verify the LLL condition ep(d+1) <= 1 (symmetric) or x_i weights (general), then conclude all bad events can be
s_lovasz_local_lemma_symmetric	theorem	Lovasz Local Lemma (symmetric form)		If each event A_i has Pr[A_i] <= p and is dependent on at most d others, and ep(d+1) <= 1, then Pr[intersection of all A_i^c] > 0.
s_lopsided_lovasz_local_lemma	theorem	Lopsided Lovasz Local Lemma		The LLL holds under the weaker condition of a lopsidependency graph, where Pr[A_i | intersection of A_j^c for j in S] <= Pr[A_i] for non-neighbors S.
s_moser_tardos_algorithmic_lll	theorem	Moser-Tardos algorithmic LLL		The Moser-Tardos resampling algorithm finds a satisfying assignment in expected O(sum x_i/(1-x_i)) steps under the LLL conditions.
s_k_sat_lll_bound	theorem	k-SAT LLL bound		A k-CNF formula in which each clause shares variables with at most 2^k/e - 1 other clauses is satisfiable.
s_latin_transversal_lll	theorem	Latin transversal via LLL		An n x n array where each symbol appears at most n/(4e) times has a Latin transversal.
a_lopsidependency_graph	axiom	Lopsidependency graph		A graph on events where Pr[A_i | intersection of A_j^c for j in S] <= Pr[A_i] for every set S of non-neighbors of i.
s_fkg_inequality	theorem	FKG inequality		On a finite distributive lattice with a log-supermodular measure mu, for increasing functions f and g: E[fg] >= E[f]E[g].
s_four_functions_theorem	theorem	Four functions theorem (Ahlswede-Daykin)		If alpha, beta, gamma, delta are nonneg functions on a finite lattice with alpha(a)beta(b) <= gamma(a v b)delta(a ^ b) for all a,b, then (sum alpha)(sum beta) <
s_reimer_inequality	theorem	Reimer inequality (BK inequality)		For events A, B on {0,1}^n, Pr[A box B] <= Pr[A]*Pr[B], where A box B denotes disjoint occurrence. Proved by Reimer, resolving the van den Berg-Kesten conjectur
a_increasing_event	axiom	Increasing event		An event A in {0,1}^n such that omega in A and omega <= omega' coordinatewise implies omega' in A.
a_disjoint_occurrence	axiom	Disjoint occurrence (box product)		A box B is the event that disjoint witness sets of coordinates certify A and B simultaneously.
t_azuma_hoeffding_technique	technique	Azuma-Hoeffding technique		Construct a Doob martingale by sequential exposure, verify bounded-difference conditions, then apply Azuma-Hoeffding to obtain exponential concentration.
t_vertex_exposure_martingale	technique	Vertex exposure martingale		Expose vertices of G(n,p) one by one; the resulting Doob martingale gives concentration for Lipschitz graph parameters.
t_edge_exposure_martingale	technique	Edge exposure martingale		Expose potential edges of G(n,p) one by one; yields tighter concentration than vertex exposure when the parameter is edge-Lipschitz.
s_chromatic_number_concentration	theorem	Chromatic number concentration		The chromatic number chi(G(n,1/2)) is concentrated in an interval of width O(sqrt(n)).
s_kim_vu_polynomial_concentration	theorem	Kim-Vu polynomial concentration		Tight concentration bounds for multilinear polynomials of independent {0,1} random variables.
a_doob_martingale_combinatorial	axiom	Doob martingale (combinatorial)		M_i = E[f | F_i] defines a martingale by sequential exposure of random variables; the fundamental tool for applying martingale concentration in combinatorics.
a_lipschitz_function_product_space	axiom	Lipschitz function on product space		A function f on a product space is c-Lipschitz if changing any single coordinate changes f by at most c.
t_stein_chen_poisson_approximation	technique	Stein-Chen Poisson approximation		Approximate the distribution of a sum of weakly dependent indicator random variables by a Poisson distribution, with explicit total-variation bounds.
s_triangle_threshold_gnp	theorem	Triangle threshold in G(n,p)		The threshold for triangles in G(n,p) is p = n^{-1}; when p = c/n the number of triangles converges in distribution to Poisson(c^3/6).
s_bonferroni_inequalities	theorem	Bonferroni inequalities		Truncated inclusion-exclusion sums yield alternating upper and lower bounds for Pr[union of A_i].
t_expander_mixing_technique	technique	Expander mixing technique		Use the spectral gap of a regular graph to show that edge distribution between vertex subsets approximates that of a random graph.
s_paley_graph_pseudo_random	theorem	Paley graph pseudo-randomness		The Paley graph is an (n, (n-1)/2, O(sqrt(n)))-graph, exhibiting strong pseudo-random properties.
a_ndlambda_graph	axiom	(n,d,lambda)-graph		A d-regular graph on n vertices whose second-largest eigenvalue (in absolute value) of the adjacency matrix is at most lambda.
a_erdos_renyi_gnm	axiom	Erdos-Renyi G(n,m) model		A uniform random graph on n labelled vertices with exactly m edges.
s_katona_cyclic_permutation_proof	state	Katona's cyclic permutation proof of EKR		Katona's elegant proof of the Erdos-Ko-Rado theorem using counting over cyclic permutations and arcs.
s_lovasz_theta_sandwich	theorem	Lovasz theta sandwich theorem		For any graph G, alpha(G) <= theta(G) <= chi_bar(G), sandwiching the Lovasz theta function between the independence number and the fractional chromatic number.
s_shannon_capacity_c5	theorem	Shannon capacity of C_5		The Shannon capacity of the 5-cycle is Theta(C_5) = sqrt(5), determined by Lovasz via the theta function.
s_lovasz_theta_ramsey_bound	theorem	Ramsey bound via Lovasz theta function		R(3,k) >= Omega(k^2/log^2 k), obtained using the Lovasz theta function.
a_shannon_capacity	axiom	Shannon capacity of a graph		Theta(G) = lim_{n->inf} (alpha(G^{box n}))^{1/n}, the zero-error capacity of a noisy channel with confusability graph G.
a_lovasz_theta_function	axiom	Lovasz theta function		An SDP-computable graph parameter theta(G) satisfying alpha(G) <= theta(G) <= chi_bar(G).
t_partial_coloring_method	technique	Partial coloring method		Find a partial coloring of low discrepancy by pigeonhole or entropy arguments, then iterate to color all elements.
t_beck_fiala_technique	technique	Beck-Fiala technique		Fix colors of elements appearing in few active sets by a dimension/linear algebra argument, iterating until all elements are colored.
s_partial_coloring_lemma	theorem	Partial coloring lemma		There exists a coloring with at least n/2 nonzero entries and discrepancy O(sqrt(n)) for any n sets on n elements.
s_discrepancy_arithmetic_progressions	theorem	Discrepancy of arithmetic progressions		The discrepancy of the set system of arithmetic progressions in [n] is Theta(n^{1/4}).
a_combinatorial_discrepancy	axiom	Combinatorial discrepancy		disc(S) = min_{chi: X -> {-1,+1}} max_{S in S} |sum_{x in S} chi(x)|, the minimum worst-case imbalance over all 2-colorings.
t_conditional_expectations_method	technique	Method of conditional expectations		Sequentially set each random variable to the value that keeps the conditional expectation favorable, derandomizing a probabilistic proof.
t_pairwise_independence_derandomization	technique	Pairwise independence derandomization		Replace full independence with pairwise independence, preserving first and second moment arguments while reducing randomness.
t_k_wise_independence_derandomization	technique	k-wise independence derandomization		Use k-wise independent random variables constructible from O(k log n) random bits to derandomize arguments depending on k-th moments.
s_derandomized_ramsey_bound	theorem	Derandomized Ramsey lower bound		Constructive proof that R(k,k) >= 2^{k/2} via the method of conditional expectations.
s_deterministic_max_cut	theorem	Deterministic max-cut algorithm		A polynomial-time greedy algorithm produces a cut with at least m/2 edges, derandomizing the probabilistic max-cut bound.
a_k_wise_independent_rv	axiom	k-wise independent random variables		A collection of random variables where every subset of size at most k is mutually independent.
a_pairwise_independent_rv	axiom	Pairwise independent random variables		Random variables satisfying E[X_i X_j] = E[X_i]E[X_j] for all i != j, i.e., 2-wise independence.
t_graph_property_testing	technique	Graph property testing		Test a graph property by querying a random constant-size induced subgraph and checking the property on it.
s_graph_property_testability	theorem	Testability of hereditary graph properties		Every hereditary graph property is testable with a constant number of queries (depending only on epsilon).
s_triangle_removal_lemma	theorem	Triangle removal lemma		A graph with at most delta*n^3 triangles can be made triangle-free by removing at most epsilon*n^2 edges.
s_ruzsa_szemeredi_theorem	theorem	Ruzsa-Szemeredi (6,3)-theorem		A graph on n vertices in which every edge belongs to exactly one triangle has o(n^2) edges.
t_chernoff_bound_technique	technique	Chernoff bound technique		Bound tail probabilities by optimizing the exponential moment E[exp(lambda X)] over lambda (exponential tilting / Cramér method).
t_janson_inequality_technique	technique	Janson inequality technique		Control pairwise correlations among dependent indicators to obtain lower tail bounds, bounding Pr[X=0] <= exp(-mu + Delta/2).
s_janson_extended_inequality	theorem	Janson extended inequality		Pr[X=0] <= exp(-mu^2/(2 Delta)) when the dependency measure Delta >= mu, giving a stronger bound in the high-dependency regime.
s_subgraph_count_concentration	theorem	Subgraph count concentration		The number of copies of a fixed subgraph H in G(n,p) is concentrated around its expectation.
s_weierstrass_product_inequality	theorem	Weierstrass product inequality		For x_i in [0,1]: prod_{i}(1 - x_i) >= 1 - sum_{i} x_i, a basic inequality used in probabilistic lower bounds.
t_entropy_method_combinatorics	technique	Entropy method in combinatorics		Use subadditivity of Shannon entropy, the chain rule, and the maximum entropy principle to derive combinatorial counting and discrepancy bounds.
s_shearer_entropy_lemma	theorem	Shearer's entropy lemma		If each coordinate is covered at least t times by sets A_1,...,A_m, then t*H(X) <= sum_{i} H(X_{A_i}).
s_bregman_theorem	theorem	Bregman's theorem (permanent bound)		For a 0-1 matrix A with row sums r_1,...,r_n, the permanent satisfies perm(A) <= prod_{i} (r_i!)^{1/r_i}.
s_kahn_zhao_independent_sets	theorem	Kahn-Zhao theorem on independent sets		The number of independent sets in a d-regular bipartite graph on 2n vertices is at most (2^{d+1} - 1)^{n/d}, with equality for a disjoint union of K_{d,d}.
s_shearer_independent_set_bound	theorem	Shearer's independent set bound via entropy		Entropy-based bound on the number of independent sets in a graph, strengthening the naive 2^n bound using local structure.
t_container_method	technique	Hypergraph container method		Construct a small collection of 'containers' (almost-independent vertex sets) such that every independent set of the hypergraph is contained in some container.
s_saxton_thomason_container_theorem	theorem	Saxton-Thomason container theorem		For uniform hypergraphs satisfying codegree conditions, there exists a small family of containers for independent sets with controlled supersaturation.
s_balogh_morris_samotij_container_theorem	theorem	Balogh-Morris-Samotij container theorem		An independent container theorem for hypergraphs using a supersaturation approach, yielding strong bounds on the number of independent sets.
s_sum_free_sets_count_container	theorem	Sum-free subsets count via containers		The number of sum-free subsets of [n] is 2^{n/2 + o(n)}, proved using the container method.
s_triangle_free_graph_count_container	theorem	Triangle-free graph count via containers		The number of triangle-free graphs on n labelled vertices is 2^{n^2/4 + o(n^2)}, proved using the container method.
t_sparse_regularity_method	technique	Sparse regularity method		Adapt Szemeredi's regularity lemma to sparse graphs with edge density p, obtaining meaningful partitions and counting lemmas relative to p.
s_sparse_regularity_lemma	theorem	Kohayakawa-Rodl sparse regularity lemma		Every graph satisfying an upper-uniformity condition admits an epsilon-regular partition relative to its density, extending Szemeredi regularity to sparse graph
s_sparse_counting_lemma	theorem	Sparse counting lemma		A counting lemma for sparse epsilon-regular graphs: the number of copies of a fixed graph in a sparse regular setup matches the expected count up to a small err
s_sparse_ramsey_theorem	theorem	Rodl-Rucinski sparse Ramsey theorem		For any graph H and integer r, there exists a threshold p_0(n) such that for p >> p_0, every r-coloring of edges of G(n,p) a.a.s. contains a monochromatic copy 
s_shamir_spencer_chromatic_concentration	theorem	Shamir-Spencer chromatic concentration		For constant p in (0,1), the chromatic number chi(G(n,p)) is concentrated on an interval of length about sqrt(n).
s_ramsey_upper_bound_lll	state	Off-diagonal Ramsey bound via LLL		R(3,k) >= Omega(k^2/log k) obtained via the Lovasz Local Lemma and Property B arguments.
s_chromatic_lower_bound_theta	theorem	Chromatic number lower bound via theta		chi(G) >= n/theta(G), giving a lower bound on the chromatic number using the Lovasz theta function of the complement.
a_2_density	axiom	2-density of a graph		m_2(H) = max_{J subseteq H, v(J) >= 3} (e(J)-1)/(v(J)-2), the maximum 2-density over subgraphs, governing thresholds for balanced subgraph appearance.
a_upper_uniformity_condition	axiom	Upper uniformity condition		A no-dense-spots condition requiring that no subgraph is significantly denser than the ambient density p, needed for sparse regularity to be meaningful.
s_k_uniform_set_system	axiom	k-uniform set system (k-graph)		A family F ⊆ ([n] choose k) where every member has exactly k elements; the combinatorial analogue of a k-uniform hypergraph.
s_chain_in_poset	axiom	Chain in a poset		A totally ordered subset of a partially ordered set; in 2^[n], a sequence ∅ ⊂ A₁ ⊂ A₂ ⊂ ··· ⊂ [n].
s_shattering_trace	axiom	Shattering		A set system F shatters S ⊆ X if the trace F|_S = {A ∩ S : A ∈ F} equals 2^S, realizing all 2^|S| subsets.
s_trace_of_set_system	axiom	Trace of a set system		For F ⊆ 2^X and S ⊆ X, the trace F|_S = {A ∩ S : A ∈ F}, the projection of F onto the coordinate set S.
s_covering_number_tau	axiom	Covering number (τ)		The minimum size of a set T ⊆ X such that T ∩ A ≠ ∅ for all A ∈ F; the transversal number of the hypergraph.
s_packing_number_matching	axiom	Packing number (matching number)		The maximum number of pairwise disjoint sets in a family F; the matching number ν(F) of the hypergraph.
s_epsilon_net_def	axiom	Epsilon-net		For a set system (X, F) and probability measure μ on X, a subset N ⊆ X is an ε-net if N ∩ A ≠ ∅ for every A ∈ F with μ(A) ≥ ε.
s_boolean_function_def	axiom	Boolean function		A function f: {0,1}^n → {0,1}; the fundamental object of study in combinatorial complexity and analysis of Boolean functions.
s_sensitivity_boolean	axiom	Sensitivity of a Boolean function		The sensitivity s(f) = max_x |{i : f(x) ≠ f(x ⊕ eᵢ)}|, the maximum number of sensitive coordinates over all inputs.
s_block_sensitivity	axiom	Block sensitivity		bs(f) = max_x max over disjoint blocks B₁,...,Bₖ with f(x) ≠ f(x ⊕ Bⱼ) for all j; always satisfies bs(f) ≥ s(f).
s_certificate_complexity	axiom	Certificate complexity		C(f,x) = min{|S| : S ⊆ [n], for all y with y|_S = x|_S we have f(y) = f(x)}, the minimum bits certifying the output on input x.
s_decision_tree_complexity	axiom	Decision tree complexity		D(f) = minimum depth of a decision tree computing f; the worst-case number of variable queries needed to evaluate f.
s_influence_of_variable	axiom	Influence of a variable		Inf_i(f) = Pr[f(x) ≠ f(x ⊕ eᵢ)] where x is uniform on {0,1}^n; the probability that the i-th variable is pivotal.
s_total_influence	axiom	Total influence		I(f) = ∑ᵢ Inf_i(f), the sum of all variable influences; equals the average sensitivity and the edge-boundary measure on the Boolean cube.
s_error_correcting_code	axiom	Error-correcting code		A subset C ⊆ F_q^n with minimum Hamming distance d(C) = min{d(x,y) : x ≠ y ∈ C}; an [n,k,d]-code if |C| = q^k.
s_partition_regularity	axiom	Partition regularity		A system of equations Ax = 0 is partition regular if for every finite coloring of ℕ there exists a monochromatic solution; characterized by Rado's theorem.
s_noise_sensitivity	axiom	Noise sensitivity		A Boolean function f is noise sensitive if Corr(f(x), f(x_ε)) → 0 as n → ∞ where x_ε flips each bit of x independently with probability ε.
s_monotone_boolean_function	axiom	Monotone Boolean function		A Boolean function f with f(x) ≤ f(y) whenever x ≤ y componentwise; computable by a circuit with only AND and OR gates.
s_turan_density	state	Turán density π(F)		For a forbidden k-uniform hypergraph F, π(F) = lim_{n→∞} ex_k(n,F)/(n choose k), the asymptotic extremal density.
s_extremal_number_set_system	state	Extremal number for set systems		The maximum |G| over k-uniform families G on [n] containing no copy of the forbidden configuration F; the hypergraph Turán number ex_k(n,F).
s_hypergraph_ramsey_number	state	Hypergraph Ramsey number		R_k(s,t) = min N such that every 2-coloring of ([N] choose k) contains a monochromatic s-set or t-set; grows as a tower of height k−1.
s_van_der_waerden_number	state	Van der Waerden number W(k,r)		The minimum N such that every r-coloring of [N] contains a monochromatic k-term arithmetic progression.
s_hales_jewett_number	state	Hales-Jewett number HJ(k,r)		The minimum n such that every r-coloring of [k]^n contains a monochromatic combinatorial line.
s_dual_set_system	state	Dual set system		For F ⊆ 2^[n], the dual F* has ground set F and sets {A ∈ F : i ∈ A} for each i ∈ [n]; transposes the incidence matrix.
s_growth_function_shatter	state	Growth function (shatter function)		π_F(m) = max_{|S|=m} |F|_S|, the maximum trace size on m-element sets; bounded by Φ_d(m) = ∑_{i≤d} (m choose i) when VC-dim(F) = d.
s_nisan_wigderson	state	Nisan-Wigderson design		A combinatorial design yielding pseudorandom generators: a family of sets with bounded pairwise intersections used for hardness-based derandomization.
s_symmetric_chain_decomposition	state	Symmetric chain decomposition of 2^[n]		A partition of the Boolean lattice 2^[n] into (n choose ⌊n/2⌋) symmetric chains, each a maximal chain symmetric about level n/2.
s_normalized_matching_property	state	Normalized matching property		A graded poset satisfies the normalized matching property if for consecutive levels, the ratio of up-degree to level size is non-increasing; implies the LYM ine
s_turan_forbidden_trace	state	Turán problem for forbidden traces		The problem of bounding the maximum size of F ⊆ 2^[n] whose trace on every m-set avoids a forbidden configuration; connects VC dimension to forbidden subconfigu
s_hilton_milner	theorem	Hilton-Milner theorem		If n ≥ 2k and F ⊆ ([n] choose k) is intersecting but not a principal family, then |F| ≤ (n-1 choose k-1) − (n-k-1 choose k-1) + 1.
s_frankl_t_intersecting	theorem	Frankl's theorem on t-intersecting families		For n sufficiently large relative to k and t, if F ⊆ ([n] choose k) is t-intersecting, then |F| ≤ (n-t choose k-t).
s_lovasz_kruskal_katona	theorem	Lovász continuous Kruskal-Katona theorem		If |F| = (x choose k) for real x ≥ k, then |∂F| ≥ (x choose k−1); a continuous extension of the Kruskal-Katona shadow bound.
s_ramsey_theorem_hypergraph	theorem	Ramsey's theorem (hypergraph version)		For all k, r, s there exists N such that every r-coloring of ([N] choose k) contains a monochromatic s-element set; numbers grow as towers of height k.
s_rado_partition_regularity	theorem	Rado's theorem (partition regularity)		A system Ax = 0 is partition regular over ℕ if and only if the columns of A satisfy the columns condition, a nested divisibility condition on column sums.
s_pajor_lemma	theorem	Pajor's lemma		For a down-closed family F ⊆ 2^[n], the number of subsets shattered by F equals |F|; relates the trace structure to family size.
s_haussler_packing	theorem	Haussler packing lemma		For a set system of VC dimension d, the maximum number of ε-separated sets (packing number) is O((1/ε)^d).
s_epsilon_net_theorem	theorem	Epsilon-net theorem (Haussler-Welzl)		For a set system of VC dimension d and any probability measure, there exists an ε-net of size O((d/ε) log(1/ε)).
s_fractional_helly	theorem	Fractional Helly theorem		If among all (d+1)-tuples of n convex sets in ℝ^d at least α-fraction intersect, then some point lies in at least β(α,d)·n of the sets.
s_colorful_helly	theorem	Colorful Helly theorem (Bárány)		Given d+1 families of convex sets in ℝ^d each with nonempty common intersection, if every colorful (d+1)-tuple intersects, then one family has a common point.
s_p_q_theorem	theorem	(p,q)-theorem (Alon-Kleitman)		For p ≥ q ≥ d+1, there exists c(p,q,d) such that if among every p convex sets in ℝ^d some q intersect, then c points pierce all sets.
s_ray_chaudhuri_wilson	theorem	Ray-Chaudhuri-Wilson theorem		If F ⊆ 2^[n] and |A ∩ B| takes at most s distinct values for A ≠ B ∈ F, then |F| ≤ (n choose s).
s_oddtown_theorem	theorem	Oddtown theorem		If F ⊆ 2^[n] with |A| odd for all A ∈ F and |A ∩ B| even for all distinct A, B ∈ F, then |F| ≤ n.
s_two_distance_sets	theorem	Two-distance sets theorem		The maximum number of points in ℝ^d with only two distinct pairwise distances is at most (d+2 choose 2).
s_combinatorial_nullstellensatz	theorem	Combinatorial Nullstellensatz (Alon)		If f ∈ F[x₁,...,xₙ] has degree ∑tᵢ and the coefficient of ∏xᵢ^{tᵢ} is nonzero, then for sets Sᵢ with |Sᵢ| > tᵢ there exists a ∈ ∏Sᵢ with f(a) ≠ 0.
s_johnson_bound	theorem	Johnson bound		An upper bound on code size refining the Hamming bound by bounding the list-decodability radius as a function of code parameters.
s_kkl_theorem	theorem	KKL theorem (Kahn-Kalai-Linial)		For any Boolean function f with Pr[f=1] = p ∈ (0,1), there exists a variable i with Inf_i(f) ≥ Ω(p(1−p) log n / n).
s_friedgut_sharp_thresholds	theorem	Friedgut's theorem on sharp thresholds		A monotone Boolean function with bounded total influence and transitive symmetry exhibits a sharp threshold: transition from Pr[f=1] ≈ 0 to ≈ 1 in an interval o
s_bollobas_two_families_general	theorem	Bollobás two-families theorem (generalized)		If (Aᵢ,Bᵢ) are set pairs with |Aᵢ|=aᵢ, |Bᵢ|=bᵢ, and Aᵢ ∩ Bⱼ = ∅ iff i=j, then ∑ 1/(aᵢ+bᵢ choose aᵢ) ≤ 1.
s_katona_intersection	theorem	Katona's intersection theorem		For an intersecting family F ⊆ 2^[n] (not necessarily uniform), |F| ≤ 2^{n−1}; the non-uniform generalization of Erdős-Ko-Rado.
s_bondy_theorem_traces	theorem	Bondy's theorem on traces		If |F| > Φ_{d−1}(n) = ∑_{i<d} (n choose i) then F shatters some d-element set; a converse to the Sauer-Shelah lemma.
s_alon_babai_suzuki	theorem	Alon-Babai-Suzuki theorem		If F ⊆ ([n] choose k) with k ≡ 0 mod p and |Aᵢ ∩ Aⱼ| mod p ∈ L for i ≠ j where 0 ∉ L and |L| = s, then |F| ≤ (n choose s).
s_elias_bassalygo	theorem	Elias-Bassalygo bound		An upper bound on code rate: R ≤ 1 − H_q(J_q(δ)) where J_q is the Johnson radius, improving the Hamming bound for certain distance ranges.
s_threshold_phenomenon	theorem	Threshold phenomenon for monotone properties		Every monotone increasing graph property has a threshold p_c such that Pr_{G(n,p)}[property holds] transitions sharply from 0 to 1 as p crosses p_c.
s_sunflower_conjecture	theorem	Improved sunflower lemma (Alweiss-Lovett-Wu-Zhang)		A family of more than (C·log s)^s sets of size s contains a sunflower with 3 petals, exponentially improving the Erdős-Ko (p−1)^s·s! bound.
t_compression_shifting	technique	Compression / shifting		Replacing A by (A\{j})∪{i} for i<j when the result is absent; pushes families toward extremal configurations while preserving size and intersection properties.
t_sunflower_extraction	technique	Sunflower extraction (greedy peeling)		Iteratively selecting and removing sets to extract a sunflower; the greedy argument underlying the sunflower lemma proof.
t_dimension_argument	technique	Dimension argument (linear algebra bound)		Associating vectors to combinatorial objects and bounding family size by the dimension of the ambient vector space; if vectors are independent in F^n, the famil
t_lexicographic_cascading	technique	Cascading representation		Decomposing integers via binomial coefficients in the cascading (falling factorial) representation; used to state and prove the Kruskal-Katona theorem.
t_weight_function_method	technique	Weight function method		Assigning weights to sets (e.g., 1/(n choose |A|) in LYM) and showing the total weight is at most 1 to derive bounds on family size.
t_product_tensor_argument	technique	Product / tensor argument		Taking Cartesian products or tensor powers of structures to force patterns; used in Hales-Jewett and density arguments where product structure guarantees substr
t_stepping_up_ramsey	technique	Stepping-up technique		A recursive construction lifting Ramsey lower bounds from k-uniform to (k+1)-uniform hypergraphs by encoding colorings via orderings; produces tower-type growth
t_derandomization	technique	Derandomization via conditional expectations		Converting a probabilistic existence proof into a deterministic construction by iteratively fixing random choices to maintain or improve the expected value.
t_fourier_analysis_boolean	technique	Fourier analysis on the Boolean cube		Expanding functions on {0,1}^n in the Walsh-Fourier basis to analyze influence, noise sensitivity, and threshold phenomena via spectral properties.
t_chain_partition	technique	Chain partition technique		Constructing chain decompositions of posets via matching, flow, or greedy algorithms; the constructive dual of antichain bounds.
t_multilinear_extension	technique	Multilinear polynomial representation		Representing sets as multilinear polynomials over a field and using polynomial degree/independence to bound family size; key to the linear algebra method.
t_algebraic_rank_argument	technique	Algebraic rank argument		Constructing a matrix whose rows correspond to sets, showing full rank to bound family size; proves Oddtown, Fisher's inequality, and Frankl-Wilson.
t_turan_extremal_method	technique	Turán-type extremal method		Determining the maximum size of a structure avoiding a forbidden substructure by analyzing the extremal configuration; generalizes from graphs to hypergraphs an
t_amalgamation	technique	Amalgamation technique		Combining Ramsey-type and density-type arguments by gluing smaller certified pieces while maintaining global properties.
t_inclusion_exclusion	technique	Inclusion-exclusion as counting technique		Computing union sizes via alternating sums of intersection sizes; used in sieve methods, derangement counting, and chromatic polynomial computations.
s_difference_set	axiom	Difference set A−B		A−B = {a−b : a ∈ A, b ∈ B}; the set of all pairwise differences between elements of A and B in an abelian group.
s_iterated_sumset	axiom	Iterated sumset kA		kA = A + A + ⋯ + A (k copies), the k-fold sumset; distinct from the dilation k·A = {ka : a ∈ A}.
s_doubling_constant	axiom	Doubling constant σ(A)		σ(A) = |A+A|/|A|, measuring how far A is from being a subgroup or coset; σ(A) = 1 iff A is a coset of a finite subgroup.
s_ruzsa_distance	axiom	Ruzsa distance d(A,B)		d(A,B) = log(|A−B|/√(|A||B|)), a pseudo-metric on finite subsets of an abelian group satisfying the triangle inequality up to additive constants.
s_additive_energy	axiom	Additive energy E(A,B)		E(A,B) = |{(a₁,a₂,b₁,b₂) ∈ A²×B² : a₁+b₁ = a₂+b₂}|; measures the additive structure shared by A and B, with E(A,A) ∈ [|A|², |A|³].
s_freiman_homomorphism	axiom	Freiman homomorphism of order s		A map φ: A → B such that a₁+⋯+aₛ = a'₁+⋯+a'ₛ implies φ(a₁)+⋯+φ(aₛ) = φ(a'₁)+⋯+φ(a'ₛ); preserves additive structure up to s-fold sums.
s_freiman_isomorphism	axiom	Freiman isomorphism of order s		A bijective Freiman s-homomorphism whose inverse is also a Freiman s-homomorphism; the correct notion of structural equivalence in additive combinatorics.
s_generalized_arithmetic_progression	axiom	Generalized arithmetic progression (GAP)		P = {a₀ + x₁v₁ + ⋯ + xₐvₐ : 0 ≤ xᵢ < Nᵢ} for base point a₀, generators v₁,…,vₐ, and lengths N₁,…,Nₐ; a d-dimensional GAP has rank/dimension d.
s_proper_gap	axiom	Proper generalized arithmetic progression		A GAP P = {a₀ + ∑xᵢvᵢ : 0 ≤ xᵢ < Nᵢ} is proper if |P| = ∏Nᵢ, i.e., all sums are distinct and the representation map is injective.
s_bohr_set	axiom	Bohr set B(S,ε)		B(S,ε) = {x ∈ G : |χ(x)−1| ≤ ε for all χ ∈ S} for a finite set S of characters; a structured neighborhood of 0 that generalizes subgroups.
s_dissociated_set	axiom	Dissociated set		A set Λ ⊂ G such that any equation ∑_{λ∈Λ} ε_λ λ = 0 with ε_λ ∈ {−1,0,1} forces all ε_λ = 0; the combinatorial analog of linear independence for sumset problems
s_k_approximate_group	axiom	K-approximate group		A finite symmetric set A containing the identity in a group G such that A·A can be covered by at most K translates of A; generalizes the notion of a finite subg
s_combinatorial_dimension	axiom	Combinatorial dimension of a Bohr set		The cardinality |S| of the frequency set S in a Bohr set B(S,ε); controls the size and structure of the Bohr set via |B(S,ε)| ≈ εᵈ|G| for d = |S|.
s_progression_free_set	axiom	Progression-free set (3-AP-free set)		A subset A of an abelian group containing no non-trivial 3-term arithmetic progression; the maximum size r₃(N) of such a set in {1,…,N} is the subject of Roth's
s_sumset_growth_function	axiom	Sumset growth function		The function K ↦ f(K) bounding |kA−lA| in terms of |A| when |A+A| ≤ K|A|; the key quantitative object in Plünnecke–Ruzsa theory.
s_plunnecke_graph	axiom	Plünnecke graph		A commutative layered bipartite graph (V₀,…,Vₕ) with a matching from Vᵢ to Vᵢ₊₁ modeling the map A → A+B; the combinatorial device underlying Plünnecke's inequa
s_magnification_ratio	axiom	Magnification ratio		For a Plünnecke graph, Dₕ = min_{∅≠X⊆V₀} |im_h(X)|/|X|, the minimal expansion ratio from V₀ to Vₕ; the quantity controlled in Plünnecke's inequality.
s_additive_energy_normalized	axiom	Normalized additive energy		e(A) = E(A,A)/|A|³, the additive energy per unit cube; e(A) = 1 iff A is a coset, and e(A) ≥ 1/|A| always holds.
s_regularity_pair	axiom	ε-regular pair (Szemerédi)		A pair (U,W) in a graph is ε-regular if for all X ⊂ U, Y ⊂ W with |X| ≥ ε|U|, |Y| ≥ ε|W|, |d(X,Y)−d(U,W)| ≤ ε; the quasirandom condition in the regularity lemma
s_popular_difference	axiom	Popular difference		An element d ∈ G is a popular difference for A if |{a ∈ A : a+d ∈ A}| ≥ δ|A| for some threshold δ; appears in density increment and Balog–Szemerédi arguments.
s_additive_quadruple	axiom	Additive quadruple		A tuple (a₁,a₂,b₁,b₂) ∈ A²×B² with a₁+b₁ = a₂+b₂; the additive energy E(A,B) counts the number of such quadruples.
s_coset_progression	axiom	Coset progression		A set of the form P + H where P is a GAP and H is a finite subgroup; the natural generalization of GAP needed for Freiman's theorem in arbitrary abelian groups.
s_convolution_counting_function	axiom	Convolution (representation) function r_{A+B}		r_{A+B}(x) = |{(a,b) ∈ A×B : a+b = x}|, counting representations of x as a sum from A and B; satisfies ∑_x r_{A+B}(x) = |A||B|.
s_spectrum_of_set	axiom	Spectrum (large Fourier coefficients) of a set		Spec_δ(A) = {γ ∈ Ĝ : |Â(γ)| ≥ δ|A|}, the set of characters on which the Fourier transform of 1_A is large; controlled by Chang's theorem.
s_pseudorandom_measure	axiom	Pseudorandom measure (Green–Tao)		A non-negative function ν: ℤ_N → ℝ≥0 with 𝔼ν = 1+o(1) satisfying a linear forms condition and a correlation condition; the majorant for W-tricked primes in the 
s_w_trick	axiom	W-trick (Green–Tao)		Restricting to primes p ≡ b (mod W) for W = ∏_{p≤w} p to remove small-prime biases; converts the primes into a set with better pseudorandomness properties.
s_linear_forms_condition	axiom	Linear forms condition		The condition 𝔼_{x∈ℤ_N^d} ∏ᵢ ν(ψᵢ(x)) = 1+o(1) for a pseudorandom measure ν and a finite system of affine-linear forms ψᵢ with finite complexity; key hypothesis
s_correlation_condition	axiom	Correlation condition (Green–Tao)		The condition 𝔼_x ∏_{j=1}^m ν(x+h_j) ≤ ∑_{1≤i<j≤m} τ(h_i−h_j) for a weight τ with controlled moments; ensures the pseudorandom measure is not too concentrated.
s_gowers_uniformity_norm	axiom	Gowers uniformity norm U^k		‖f‖_{U^k}^{2^k} = 𝔼_{x,h₁,…,hₖ} ∏_{ω∈{0,1}^k} C^{|ω|} f(x+ω·h), where C is conjugation; measures k-step uniformity and controls counts of (k+1)-term APs.
s_gowers_anti_uniformity	state	Gowers anti-uniform function		A dual function F with ‖F‖∞ ≤ 1 maximizing |⟨f,F⟩| subject to ‖F‖_{U^k} being small; used in the density/energy increment strategy for Szemerédi's theorem.
s_plunnecke_inequality	theorem	Plünnecke's inequality		If |A+B| ≤ K|A| then |hB−lB| ≤ K^{h+l}|A| for all h,l ≥ 0; proved via magnification ratios in Plünnecke graphs, controlling iterated sumset growth from a single
s_ruzsa_triangle_inequality	theorem	Ruzsa triangle inequality		|A−C| ≤ |A−B||B−C|/|B| for finite non-empty subsets A,B,C of an abelian group; the key tool making Ruzsa distance into a quasi-metric.
s_ruzsa_covering_lemma	theorem	Ruzsa covering lemma		If |A+B| ≤ K|A| then B ⊆ A−A+T for some set T with |T| ≤ K; used to embed sets of small doubling into translates of a structured set.
s_ruzsa_sum_triangle_inequality	theorem	Ruzsa sum-triangle inequality		|A+C| ≤ |A+B||B+C|/|B| for finite non-empty subsets of an abelian group; the sum-version complementing the difference-version of Ruzsa's triangle inequality.
s_ruzsa_distance_controls_energy	theorem	Ruzsa distance–energy duality		E(A,B) ≥ |A|²|B|²/|A−B| by Cauchy–Schwarz, so small Ruzsa distance implies large additive energy; the bridge between metric and energy viewpoints.
s_freiman_ruzsa_f2n	theorem	Freiman–Ruzsa theorem in 𝔽₂ⁿ		If A ⊆ 𝔽₂ⁿ satisfies |A+A| ≤ K|A|, then A is contained in a subspace of size at most 2^{O(K)}|A|; the model-case inverse sumset theorem with polynomial bounds.
s_balog_szemeredi_gowers	theorem	Balog–Szemerédi–Gowers theorem		If E(A,A) ≥ |A|³/K then there exists A' ⊂ A with |A'| ≥ c|A|/K and |A'+A'| ≤ CK⁴|A'|; converts large additive energy into a dense subset with small doubling, wi
s_erdos_heilbronn_conjecture	theorem	Erdős–Heilbronn conjecture (Dias da Silva–Hamidoune)		|{a+b : a,b ∈ A, a ≠ b}| ≥ min(p, 2|A|−3) for A ⊆ ℤ/pℤ; the restricted-sumset analog of Cauchy–Davenport, proved via the Combinatorial Nullstellensatz.
s_sum_product_estimate_reals	theorem	Elekes sum-product estimate		max(|A+A|, |A·A|) ≥ c|A|^{5/4} for finite A ⊂ ℝ; proved via the Szemerédi–Trotter theorem, confirming a weak form of the Erdős–Szemerédi conjecture.
s_erdos_szemeredi_conjecture	state	Erdős–Szemerédi sum-product conjecture		For finite A ⊂ ℝ, max(|A+A|, |A·A|) ≥ c_ε|A|^{2−ε} for every ε > 0; asserts that a set cannot be simultaneously additively and multiplicatively structured.
s_bergelson_leibman	theorem	Bergelson–Leibman theorem (polynomial Szemerédi)		For any polynomials p₁,…,pₖ ∈ ℤ[x] with pᵢ(0) = 0 and δ > 0, any sufficiently dense subset of {1,…,N} contains a configuration {a, a+p₁(d), …, a+pₖ(d)}; extends
s_inverse_littlewood_offord	theorem	Inverse Littlewood–Offord theorem (Tao–Vu)		If P(v₁ε₁+⋯+vₙεₙ = x) ≥ n^{−C} for random signs εᵢ and non-degenerate vᵢ, then most vᵢ lie in a proper GAP of small rank and size O(n^{1/2+o(1)}); the structura
s_littlewood_offord_problem	state	Littlewood–Offord problem		Given non-zero v₁,…,vₙ ∈ ℝ, maximize P(|v₁ε₁+⋯+vₙεₙ| ≤ 1) over random signs εᵢ ∈ {±1}; the classical answer is C(n,⌊n/2⌋)/2ⁿ ∼ c/√n by the Erdős bound, with the
s_chang_theorem	theorem	Chang's theorem (large Fourier spectrum)		If A ⊆ ℤ/Nℤ with |A| = αN, then Spec_δ(A) is contained in a subspace of dimension O(δ⁻² log(1/α)); sharp control on the structure of the large spectrum using di
s_bogolyubov_lemma	theorem	Bogolyubov's lemma		If A ⊆ ℤ/Nℤ has density α, then 2A−2A contains a Bohr set B(S,1/4) with |S| ≤ O(1/α²); iterated sumsets of dense sets contain highly structured subsets.
s_sanders_quasi_polynomial_freiman_ruzsa	theorem	Sanders' quasi-polynomial Freiman–Ruzsa theorem		If A ⊆ 𝔽₂ⁿ with |A+A| ≤ K|A|, then A is contained in a subspace of size at most 2^{O(K(log K)^C)}|A|; the quasi-polynomial improvement toward the polynomial Fre
s_polynomial_freiman_ruzsa_conjecture	state	Polynomial Freiman–Ruzsa conjecture		If A ⊆ 𝔽₂ⁿ with |A+A| ≤ K|A|, then A can be covered by poly(K) translates of a subspace of size ≤ |A|; equivalently, the covering number in Freiman's theorem sh
s_gowers_inverse_u3	theorem	Gowers inverse theorem for U³		If ‖f‖_{U³} ≥ δ with |f| ≤ 1, then |⟨f, e(qφ)⟩| ≥ c(δ) for some quadratic phase φ; the inverse theorem identifying U³-structured functions with quadratic phases
s_gowers_inverse_uk	theorem	Gowers inverse conjecture for U^k (Green–Tao–Ziegler)		If ‖f‖_{U^{k+1}} ≥ δ then f correlates with a nilsequence of step ≤ k and complexity O_{δ,k}(1); proved by Green–Tao–Ziegler, completing the inverse program cha
s_generalized_von_neumann	theorem	Generalized von Neumann theorem		The count of k-APs 𝔼_{x,d} ∏ᵢ fᵢ(x+id) is controlled by min_i ‖fᵢ‖_{U^{k−1}}; the structure theorem reducing AP-counting to Gowers norm estimates.
s_solymosi_sum_product	theorem	Solymosi's sum-product estimate		max(|A+A|, |A·A|) ≥ c|A|^{4/3}/log^{1/3}|A| for finite A ⊂ ℝ; an improvement over the Elekes 5/4 exponent using the multiplicative energy approach.
s_sum_product_finite_fields	theorem	Sum-product estimate in finite fields (Bourgain–Katz–Tao)		For A ⊆ 𝔽_p with p^δ ≤ |A| ≤ p^{1−δ}: max(|A+A|, |A·A|) ≥ |A|^{1+ε} for some ε = ε(δ) > 0; the finite field sum-product phenomenon.
s_ruzsa_modeling_lemma	theorem	Ruzsa modeling lemma		If A ⊆ ℤ/Nℤ has density α, then A is Freiman 2-isomorphic to a subset of ℤ/Mℤ of density ≥ α^{O(1)} for some M ≤ N; used to transfer Freiman-type results betwee
s_ruzsa_analog_freiman	theorem	Ruzsa's proof of Freiman's theorem (via covering lemma)		Streamlined proof: if |A+A| ≤ K|A|, use the Ruzsa covering lemma to embed A in few translates of A−A, then the Plünnecke–Ruzsa inequality and a geometry-of-numb
s_gowers_bound_szemeredi	theorem	Gowers quantitative bound for Szemerédi's theorem		r_k(N) ≤ N/(log log N)^{c_k} for the maximum size of a k-AP-free subset of {1,…,N}; the first effective bound for arbitrary k, via the Gowers uniformity norms a
s_regularity_counting_lemma	theorem	Regularity counting lemma (embedding lemma)		In an ε-regular partition with appropriately dense pairs, the count of copies of any fixed graph H is (1+o(1)) times the expected count for a random graph with 
s_koopman_von_neumann_structure	theorem	Koopman–von Neumann structure theorem (in additive combinatorics)		Any bounded function f on ℤ_N decomposes as f = f_s + f_u + f_e where f_s is structured (low U^k norm correlation), f_u is uniform (small U^k norm), and f_e is 
s_arithmetic_regularity_lemma	theorem	Arithmetic regularity lemma (Green)		For any f: ℤ_N → [0,1] and ε > 0, there exists a Bohr set B such that f is ε-uniform on cosets of B in the Gowers U² norm; the Fourier-analytic analog of graph 
s_goldston_yildirim_majorant	state	Goldston–Yıldırım pseudorandom majorant for primes		The function ν(n) = (φ(W)/W)(log R)⁻¹(∑_{d|n, d≤R} μ(d)log(R/d))² satisfying the linear forms and correlation conditions; provides the pseudorandom measure domi
s_hypergraph_regularity	theorem	Hypergraph regularity lemma (Gowers, Nagle–Rödl–Schacht–Skokan)		Every k-uniform hypergraph admits a regular partition with respect to a (k−1)-complex of controlled complexity; the higher-order generalization enabling the reg
s_hypergraph_removal	theorem	Hypergraph removal lemma		If a k-uniform hypergraph on n vertices contains o(nᵏ⁺¹) copies of a fixed k-uniform hypergraph H, then all copies can be removed by deleting o(nᵏ) edges; gener
s_multidimensional_szemeredi	theorem	Multidimensional Szemerédi theorem (Furstenberg–Katznelson)		For any finite S ⊂ ℤᵈ and δ > 0, any sufficiently large subset of {1,…,N}ᵈ of density ≥ δ contains a homothetic copy a+tS for some t > 0; the multi-dimensional 
s_long_aps_in_sumsets	theorem	Long arithmetic progressions in sumsets		If A, B ⊆ {1,…,N} with |A| = αN, |B| = βN, then A+B contains an AP of length at least exp(c(αβ log N)^{1/2}); combining probabilistic, Fourier, and combinatoria
s_bourgain_3ap_bound	theorem	Bourgain's bound for 3-AP-free sets		r₃(N) ≤ N(log log N)^{1/2}/(log N)^{1/2}; improved Roth's original log log N bound using iterated density increment with Bohr set techniques.
s_meshulam_roth_f3n	theorem	Meshulam's theorem (Roth in 𝔽₃ⁿ)		The maximum size of a 3-AP-free subset of 𝔽₃ⁿ is O(3ⁿ/n); the clean model-case Roth theorem in the finite field setting.
s_bateman_katz_f3n	theorem	Bateman–Katz bound for cap sets		A 3-AP-free subset of 𝔽₃ⁿ has size O(3ⁿ/n^{1+ε}) for some ε > 0; improved the Meshulam O(3ⁿ/n) bound in the cap set problem.
s_rankin_construction	state	Rankin's construction (k-AP-free sets)		A k-AP-free subset of {1,…,N} of size N·exp(−c_k(log N)^{1/(⌈log₂ k⌉)}); the higher-k generalization of Behrend's sphere-based construction.
s_green_tao_linear_equations_primes	theorem	Green–Tao linear equations in primes		Systems of linear equations of finite complexity have the expected number of prime solutions (up to local factors), asymptotically; generalizes the Green–Tao th
t_plunnecke_graph_method	technique	Plünnecke graph method		Model iterated sumset growth via a commutative layered graph with matchings; derive magnification ratio bounds by Menger's theorem and the submodularity of imag
t_ruzsa_calculus	technique	Ruzsa calculus		The systematic use of Ruzsa's triangle inequality, covering lemma, and distance to chain sumset estimates: derive |kA−lA| bounds from |A+A|/|A|, embed sets into
t_fourier_analysis_finite_abelian_groups	technique	Fourier analysis on finite abelian groups		Expand indicator functions 1_A as ∑_γ Â(γ)γ over the dual group; use Parseval, convolution identities, and large-value estimates to count additive configuration
t_density_increment_argument	technique	Density increment argument		If A has no k-AP, find a long subprogression (or Bohr set) where A has higher density; iterate O(1/δ) times until the density exceeds 1, yielding a contradictio
t_energy_increment_argument	technique	Energy increment argument		Iteratively refine a partition/decomposition so that the 'energy' (L² norm of the structured part) increases by a definite amount at each step; must terminate i
t_regularity_method_arithmetic	technique	Regularity method for arithmetic combinatorics		Apply the Szemerédi regularity lemma to an auxiliary graph encoding the arithmetic structure (e.g., the graph on A with edges {(a,b): a+b ∈ A}), then use the co
t_graph_removal_technique	technique	Graph removal technique		Encode an arithmetic problem (e.g., 3-APs) as a subgraph counting problem in a tripartite graph; apply the removal lemma to deduce density results. The bridge f
t_probabilistic_method_additive_comb	technique	Probabilistic method in additive combinatorics		Use random sampling, random dilations, or random restrictions to show existence of sets/configurations with desired additive properties; includes random subsets
t_polynomial_method_additive	technique	Polynomial method (Combinatorial Nullstellensatz applications)		Encode additive problems as polynomial nonvanishing: choose a polynomial whose nonzero coefficient certifies the configuration, then apply Alon's Nullstellensat
t_sum_product_technique	technique	Sum-product technique		Exploit the tension between additive and multiplicative structure: if A has small sumset then A·A must be large, via incidence geometry (Szemerédi–Trotter) or e
t_incidence_geometry_method	technique	Incidence geometry method		Bound the number of incidences between points and curves (lines, circles) using cell decomposition or crossing number inequalities; apply to sumset and sum-prod
t_tensor_product_trick	technique	Tensor product trick (Ruzsa)		Embed A ⊂ ℤ into A^d ⊂ ℤᵈ and exploit the multiplicativity of sumset growth under Cartesian products to amplify Freiman-type bounds; converts approximate bounds
t_ergodic_theory_approach	technique	Ergodic theory approach (Furstenberg)		Via the Furstenberg correspondence, translate combinatorial density problems to ergodic recurrence; prove the recurrence using weak mixing/compact decomposition
t_hypergraph_regularity_method	technique	Hypergraph regularity method		Extend graph regularity to k-uniform hypergraphs via a hierarchy of regular approximations (simplicial complexes); combined with a hypergraph counting lemma, yi
t_cell_decomposition_technique	technique	Cell decomposition (for incidence bounds)		Partition ℝ² into O(r²) cells using r vertical lines and r polynomial curves so each cell has few points; aggregate local incidence bounds using the Szemerédi–T
t_dyadic_pigeonholing	technique	Dyadic pigeonholing		Partition a range into O(log N) dyadic intervals [2ᵏ, 2ᵏ⁺¹) and select the interval contributing the most to a sum; loses only a logarithmic factor while reduci
t_compression_symmetrization	technique	Compression/symmetrization		Replace a set A by its successive compressions (projections onto coordinate hyperplanes followed by downsets) to reduce to the extremal case without increasing 
t_additive_energy_amplification	technique	Additive energy amplification		Boost additive energy by passing to a popular-difference subset or applying iterated sumset operations; combined with Balog–Szemerédi–Gowers, converts partial s
t_bohr_set_method	technique	Bohr set method		Use Bohr sets as substitutes for subgroups in ℤ/Nℤ: perform density increments on Bohr sets, control their size via dimension, and exploit their near-invariance
t_weyl_differencing	technique	Weyl differencing		Reduce the degree of a phase function by squaring an exponential sum and changing variables: |∑ e(f(n))|² = ∑_h ∑_n e(f(n+h)−f(n)); iterating reduces polynomial
t_van_der_corput_trick	technique	Van der Corput trick/inequality		|𝔼_n f(n)|² ≤ (1/H) ∑_{|h|<H} (1−|h|/H) 𝔼_n f(n+h)f̄(n); converts a single average into a double average over differences, reducing equidistribution to correlat
t_dependent_random_choice	technique	Dependent random choice		Choose a random subset by selecting common neighbors of a random tuple in a graph; the resulting set is small but has the property that any small sub-tuple has 
t_stepanov_method_addcomb	technique	Stepanov method		Construct an auxiliary polynomial vanishing to high order at the points of interest, then bound its degree against its vanishing order to get a point count; an 
t_character_sum_method	technique	Character sum method		Express indicator functions of structured sets (squares, kth powers) via multiplicative characters and bound the resulting character sums using Weil-type estima
t_geometry_of_numbers_in_freiman	technique	Geometry of numbers in Freiman's theorem		Apply Minkowski's lattice point theorem or John's ellipsoid theorem to find a proper GAP containing A: given A−A ⊆ a coset progression from covering, find a GAP
t_structure_randomness_dichotomy	technique	Structure–randomness dichotomy		Decompose a function/set into a structured (low-complexity) component and a pseudorandom (uniform) component; the structured part determines the count of the de
t_epsilon_net_argument	technique	ε-net/covering argument for Bohr sets		Construct a maximal separated subset (ε-net) inside a Bohr set to control its metric entropy; used to estimate |B(S,ε)| and to find large structured subsets via
s_nilsequence	axiom	Nilsequence		A sequence n ↦ F(g(n)Γ) where G/Γ is a nilmanifold, g: ℤ → G is a polynomial sequence, and F is Lipschitz; the higher-order structured objects arising in the in
s_nilmanifold	axiom	Nilmanifold G/Γ		A compact homogeneous space G/Γ where G is a connected nilpotent Lie group and Γ ≤ G is a cocompact discrete subgroup; the phase space for nilsequences in the G
s_stabilizer_of_sumset	axiom	Stabilizer (period) of a sumset		H(A+B) = {g ∈ G : g+A+B = A+B}, the largest subgroup preserving A+B; the critical object in Kneser's theorem controlling how much |A+B| can fall below |A|+|B|.
s_davenport_constant	axiom	Davenport constant D(G)		The smallest d such that any sequence of d elements in G contains a non-empty subsequence summing to 0; D(ℤ/nℤ) = n and the Erdős–Ginzburg–Ziv theorem states D(
s_sum_free_set	axiom	Sum-free set		A set A ⊆ G such that (A+A) ∩ A = ∅; the maximum size of a sum-free subset of {1,…,n} is ⌈n/2⌉, and the study of sum-free sets connects to Schur's theorem and R
s_covering_number_sumset	axiom	Covering number (Ruzsa)		cov(B, A−A) = the minimum number of translates of A−A needed to cover B; the Ruzsa covering lemma bounds this by |A+B|/|A|.
s_szemeredi_regularity_graph_arithmetic	theorem	Szemerédi's theorem via graph regularity		Proof of Szemerédi's theorem using the regularity lemma: encode k-APs as cliques in an auxiliary (k−1)-partite graph, apply regularity + counting lemma; the ori
s_roth_via_fourier	theorem	Roth's theorem via Fourier analysis		Proof: the 3-AP count T₃(A) = ∑_r |Â(r)|²Â(−2r)·N⁻¹; if T₃(A) is small, a large Fourier coefficient gives a density increment on a subprogression. Iterate to co
s_roth_via_triangle_removal	theorem	Roth's theorem via triangle removal		For A ⊆ ℤ/Nℤ, construct a tripartite graph where triangles encode 3-APs in A; the triangle removal lemma gives |{3-APs}| = o(N²) implies |A| = o(N). A clean gra
s_green_tao_structure	state	Structure of the Green–Tao proof		Step 1: W-trick to derandomize primes. Step 2: Construct Goldston–Yıldırım majorant ν. Step 3: Verify linear forms + correlation conditions. Step 4: Apply relat
s_additive_combinatorics_in_fp	axiom	Additive combinatorics in 𝔽_p (finite field model)		The study of sumsets, additive energy, and structure theorems in ℤ/pℤ or 𝔽_pⁿ, where the absence of torsion issues and availability of exact Fourier analysis ma
s_inverse_sumset_problem	state	Inverse sumset problem		Given that |A+A| ≤ K|A|, characterize the structure of A; the central question of additive combinatorics, answered (in various precisions) by Freiman's theorem,
s_cap_set_problem	state	Cap set problem		Determine the maximum size of a 3-AP-free subset of 𝔽₃ⁿ; the Ellenberg–Gijswijt (2016) bound of O(2.756ⁿ) via the polynomial method was a breakthrough (outside 
s_gowers_norm_controls_ap_count	theorem	Gowers norms control AP counts		|Λ_k(f) − Λ_k(1)| ≤ min_i ‖fᵢ − 1‖_{U^{k-1}} where Λ_k(f) = 𝔼_{x,d} ∏ f(x+id); the U^{k-1} norm is the correct notion of uniformity for k-APs, as small U^{k-1} 
t_balog_szemeredi_gowers_graph	technique	Balog–Szemerédi–Gowers graph argument		Build a bipartite graph on A×A with edges where the sumset representation is popular; use graph-theoretic path-counting (or dependent random choice) to find a d
s_ap_count_formula_fourier	theorem	AP count formula via Fourier analysis		The number of k-APs in A ⊆ ℤ/Nℤ equals N ∑_r |Â(r)|^k · (Fourier coefficient of the k-AP kernel); for k=3, this is N⁻¹ ∑_r Â(r)²Â(−2r), making the problem amena
s_sumset_lower_bound_cauchy_davenport	state	Sumset lower bound problem		The direct sumset problem: given |A|, |B| and the ambient group, determine the minimum possible |A+B|; answered by Cauchy–Davenport for ℤ/pℤ and Kneser for gene
s_vosper_theorem	theorem	Vosper's theorem		If A, B ⊆ ℤ/pℤ with |A|, |B| ≥ 2 and |A+B| = |A|+|B|−1 < p, then A and B are arithmetic progressions with the same common difference; the inverse theorem for eq
s_freiman_inverse_2_3k_4	theorem	Freiman's 3k−4 theorem		If A ⊂ ℤ with |A+A| ≤ 3|A|−4, then A is contained in an arithmetic progression of length |A+A|−|A|+1 ≤ 2|A|−3; the tight inverse theorem for very small doubling
s_erdos_littlewood_offord	theorem	Erdős–Littlewood–Offord inequality		For non-zero v₁,…,vₙ and random signs εᵢ ∈ {±1}: P(|∑ εᵢvᵢ − x| ≤ 1) ≤ C(n,⌊n/2⌋)/2ⁿ ∼ c/√n for any x; the optimal anti-concentration bound, proved by Erdős via
s_halasz_inequality	theorem	Halász-type inequality (additive combinatorics)		P(∑ εᵢvᵢ ∈ a+S) ≤ C·sup_t |𝔼 e^{itS}| · ∏ cos(tvᵢ) bounds; anti-concentration via characteristic function methods, underlying the Littlewood–Offord inverse theo
s_small_ball_probability	state	Small ball probability (Lévy concentration function)		Q(X, ε) = sup_x P(|X−x| ≤ ε), the concentration function of a random variable; the Littlewood–Offord problem amounts to bounding Q(∑εᵢvᵢ, 1) and the inverse pro
s_probability_space	axiom	Probability space	probability triple	A triple (Ω, F, P) where Ω is a sample space, F is a σ-algebra of events, and P is a probability measure with P(Ω) = 1.
s_probability_measure	axiom	Probability measure		A countably additive set function P: F → [0,1] on a σ-algebra F with P(Ω) = 1.
s_random_variable	axiom	Random variable	measurable function to ℝ	A measurable function X: (Ω, F) → (ℝ, B(ℝ)), i.e., {ω : X(ω) ≤ x} ∈ F for every x ∈ ℝ.
s_law_of_random_variable	axiom	Distribution (law) of a random variable	law of X | push-forward measure	The probability measure μ_X(B) = P(X ∈ B) on (ℝ, B(ℝ)) induced by a random variable X, also written L(X) or P ∘ X⁻¹.
s_independence_of_events	axiom	Independence of events	mutual independence of events	Events A₁, …, Aₙ are independent if P(Aᵢ₁ ∩ ··· ∩ Aᵢₖ) = P(Aᵢ₁)···P(Aᵢₖ) for every subcollection of distinct indices.
s_independence_of_random_variables	axiom	Independence of random variables		Random variables X₁, …, Xₙ are independent if the σ-algebras σ(X₁), …, σ(Xₙ) are independent, equivalently P(X₁ ∈ B₁, …, Xₙ ∈ Bₙ) = ∏ P(Xᵢ ∈ Bᵢ).
s_almost_sure_convergence	axiom	Almost sure convergence	convergence with probability one | a.s. convergence	Xₙ → X almost surely if P({ω : Xₙ(ω) → X(ω)}) = 1.
s_convergence_in_probability	axiom	Convergence in probability		Xₙ → X in probability if P(|Xₙ − X| > ε) → 0 for every ε > 0.
s_convergence_in_distribution	axiom	Convergence in distribution	weak convergence of random variables | convergence in law	Xₙ →_d X if Fₙ(x) → F(x) at every continuity point x of F, where Fₙ, F are the respective distribution functions.
s_characteristic_function	axiom	Characteristic function	Fourier transform of a distribution	The function φ_X(t) = E[e^{itX}] for t ∈ ℝ, the Fourier transform of the distribution of X, which uniquely determines the distribution.
s_stable_distribution	axiom	Stable distribution	α-stable law | Lévy stable distribution	A distribution with index α ∈ (0,2] such that the sum of n i.i.d. copies, suitably normalized, has the same distribution; the Gaussian (α=2) and Cauchy (α=1) ar
s_domain_of_attraction	axiom	Domain of attraction	domain of normal attraction	A distribution F belongs to the domain of attraction of a stable law G with index α if there exist aₙ > 0, bₙ such that (Sₙ − bₙ)/aₙ →_d G.
s_tightness_of_measures	axiom	Tightness of a family of probability measures	uniform tightness	A family {μ_α} of probability measures on a metric space is tight if for every ε > 0 there exists a compact K with μ_α(K) > 1 − ε for all α.
s_poisson_limit_theorem	theorem	Poisson limit theorem	law of rare events | Poisson approximation theorem	If Xₙ ~ Binomial(n, pₙ) with npₙ → λ, then Xₙ →_d Poisson(λ); the Poisson distribution arises as the limit of rare Bernoulli events.
s_local_central_limit_theorem	theorem	Local central limit theorem	local CLT | local limit theorem	For i.i.d. integer-valued random variables with span 1, the probability P(Sₙ = k) is asymptotically (σ√(2πn))⁻¹ exp(−(k − nμ)²/(2nσ²)) uniformly in k.
s_strong_markov_property_rw	theorem	Strong Markov property for random walks	strong Markov property (discrete)	For a random walk Sₙ and a stopping time τ < ∞ a.s., the post-τ increments (S_{τ+k} − S_τ)_{k≥1} are independent of F_τ and have the same distribution as the or
s_arc_sine_law	theorem	Arc sine law	arc sine law for random walks | Lévy's arc sine law	For a symmetric random walk, the fraction of time spent positive up to step 2n converges in distribution to the arc sine distribution with density 1/(π√(x(1−x))
s_gambler_ruin	theorem	Gambler's ruin	gambler's ruin problem	For a simple random walk with absorbing barriers at 0 and N, the ruin probability starting from k is (q/p)^k − (q/p)^N)/(1 − (q/p)^N) when p ≠ q, and 1 − k/N wh
s_ladder_variables	axiom	Ladder variables	ladder heights | ladder epochs	For a random walk Sₙ, the ascending ladder time T⁺ = inf{n ≥ 1 : Sₙ > 0} and ladder height H⁺ = S_{T⁺}; iterating these defines the ascending ladder process.
s_convergence_of_random_series	state	Convergence of random series		The series Σ Xₙ of independent random variables converges a.s. if and only if it converges in probability, which holds iff the three series of Kolmogorov's thre
s_kronecker_lemma	theorem	Kronecker's lemma		If bₙ ↑ ∞ and Σ aₙ/bₙ converges, then (1/bₙ) Σₖ₌₁ⁿ aₖ → 0; the key tool converting convergence of a series to convergence of Cesàro-type averages in SLLN proofs
s_tower_property	theorem	Tower property of conditional expectation	law of total expectation | iterated expectations	If H ⊂ G are sub-σ-algebras, then E[E[X | G] | H] = E[X | H] a.s.; the law of iterated expectations.
s_ui_and_l1_convergence	theorem	Uniform integrability and L¹ convergence	UI characterization of L¹ convergence	A uniformly integrable martingale (Xₙ, Fₙ) converges a.s. and in L¹ to X∞ = E[X∞ | Fₙ]; conversely, an L¹-bounded martingale converges in L¹ iff it is uniformly
s_doob_decomposition	theorem	Doob's decomposition theorem (discrete)	Doob decomposition	Every adapted integrable process Xₙ admits a unique decomposition Xₙ = Mₙ + Aₙ where M is a martingale and A is a predictable process with A₀ = 0.
s_backward_martingale_convergence	theorem	Backward martingale convergence theorem	reversed martingale convergence	If (Xₙ)_{n≤0} is a backward martingale with respect to (Fₙ)_{n≤0}, then Xₙ → E[X₀ | F₋∞] a.s. and in L¹ as n → −∞.
s_levy_zero_one_law	theorem	Lévy's 0-1 law		If Fₙ ↑ F∞ and A ∈ F∞, then P(A | Fₙ) → 1_A a.s.; conditional probabilities of a tail event converge to its indicator.
s_branching_process	axiom	Branching process (Galton-Watson)	Galton-Watson process	A Markov chain Zₙ where Z₀ = 1 and each individual in generation n independently produces a random number of offspring with a fixed distribution, giving Zₙ₊₁ = 
s_branching_process_extinction	theorem	Extinction criterion for branching processes	Galton-Watson extinction theorem	The extinction probability q = P(Zₙ → 0) equals 1 if the mean offspring μ ≤ 1, and equals the unique fixed point in [0,1) of the offspring generating function i
s_martingale_clt	theorem	Martingale central limit theorem	CLT for martingale differences	For a martingale difference array (Xₙᵢ) with conditional variances Σᵢ E[Xₙᵢ² | Fₙ,ᵢ₋₁] →_P 1 and a Lindeberg-type condition, the partial sums converge in distri
s_exponential_martingale	state	Exponential martingale	Wald's exponential martingale	For i.i.d. Xᵢ with E[e^{θXᵢ}] = M(θ) < ∞, the process exp(θSₙ − n log M(θ)) is a martingale; a key tool for large deviation and stopping time computations.
s_markov_chain	axiom	Markov chain	discrete-time Markov chain | DTMC	A sequence of random variables (Xₙ) in a countable state space S with the Markov property: P(Xₙ₊₁ = j | X₀, …, Xₙ) = P(Xₙ₊₁ = j | Xₙ) = p(Xₙ, j).
s_transition_matrix	axiom	Transition matrix	stochastic matrix | transition probability matrix	A matrix P = (p(i,j))_{i,j ∈ S} with nonneg entries and row sums equal to 1, encoding one-step transition probabilities of a Markov chain.
s_chapman_kolmogorov_equations	theorem	Chapman-Kolmogorov equations		The n-step transition probabilities satisfy p^{(m+n)}(i,j) = Σ_k p^{(m)}(i,k) p^{(n)}(k,j), equivalently P^{m+n} = P^m P^n.
s_markov_property	theorem	Markov property	memoryless property	Conditional on Xₙ = i, the future process (Xₙ₊ₖ)_{k≥0} is independent of the past (X₀, …, Xₙ₋₁) and has the law of a chain started at i.
s_strong_markov_property_mc	theorem	Strong Markov property for Markov chains		For a Markov chain and stopping time τ with P(τ < ∞) > 0, conditional on τ < ∞ and X_τ = i, the post-τ process is independent of F_τ and has the law of a chain 
s_communication_classes	axiom	Communication classes		States i, j communicate (i ↔ j) if i → j and j → i, where i → j means p^{(n)}(i,j) > 0 for some n; communication is an equivalence relation partitioning the sta
s_irreducibility_mc	axiom	Irreducibility of a Markov chain	irreducible chain	A Markov chain is irreducible if every state communicates with every other state, i.e., the state space consists of a single communication class.
s_recurrence_mc	axiom	Recurrence for Markov chains	recurrent state	A state i is recurrent if P_i(Xₙ = i for some n ≥ 1) = 1, equivalently Σₙ p^{(n)}(i,i) = ∞.
s_transience_mc	axiom	Transience for Markov chains	transient state	A state i is transient if P_i(Xₙ = i for some n ≥ 1) < 1, equivalently Σₙ p^{(n)}(i,i) < ∞.
s_recurrence_is_class_property	theorem	Recurrence is a class property		If i ↔ j and i is recurrent, then j is recurrent; thus recurrence and transience are properties of communication classes, not individual states.
s_positive_recurrence	axiom	Positive recurrence	positive recurrent state | ergodic state	A recurrent state i is positive recurrent if the expected return time E_i[T_i] < ∞, where T_i = inf{n ≥ 1 : Xₙ = i}.
s_null_recurrence	axiom	Null recurrence	null recurrent state	A recurrent state i is null recurrent if the expected return time E_i[T_i] = ∞; the state is visited infinitely often but the average return time is infinite.
s_period_of_state	axiom	Period of a state	periodicity of a Markov chain state	The period of state i is d(i) = gcd{n ≥ 1 : p^{(n)}(i,i) > 0}; a state with period 1 is called aperiodic.
s_existence_uniqueness_stationary	theorem	Existence and uniqueness of stationary distribution		An irreducible positive recurrent Markov chain has a unique stationary distribution π with π(i) = 1/E_i[T_i]; an irreducible chain has a stationary distribution
s_convergence_theorem_mc	theorem	Convergence theorem for Markov chains	basic limit theorem for Markov chains	For an irreducible, aperiodic, positive recurrent Markov chain with stationary distribution π, p^{(n)}(i,j) → π(j) as n → ∞ for all i, j.
s_detailed_balance	axiom	Detailed balance	reversibility condition | time-reversibility	A distribution π and transition matrix P satisfy detailed balance if π(i)p(i,j) = π(j)p(j,i) for all i, j; this implies π is stationary for P.
s_ratio_limit_theorem	theorem	Ratio limit theorem		For an irreducible recurrent Markov chain, Σₖ₌₀ⁿ p^{(k)}(i,j) / Σₖ₌₀ⁿ p^{(k)}(i,i) → π(j)/π(i) (or E_i[T_i]/E_j[T_j]) as n → ∞.
s_mean_return_time	state	Mean return time	expected return time	The expected first return time E_i[T_i] where T_i = inf{n ≥ 1 : Xₙ = i}; equals 1/π(i) for positive recurrent state i with stationary distribution π.
s_absorbing_state	axiom	Absorbing state	absorbing barrier	A state i is absorbing if p(i,i) = 1; once the chain enters state i it remains there forever.
s_ergodicity_definition	axiom	Ergodicity	metric transitivity | ergodic transformation	A measure-preserving transformation T on (Ω, F, μ) is ergodic if every T-invariant set A (T⁻¹A = A) satisfies μ(A) = 0 or μ(A) = 1.
s_maximal_ergodic_lemma	theorem	Maximal ergodic lemma	Hopf maximal ergodic lemma	For f ∈ L¹ and T measure-preserving, ∫_{A} f dμ ≥ 0 where A = {ω : sup_{n≥1} (1/n)Σₖ₌₀ⁿ⁻¹ f(Tᵏω) > 0}; the key lemma in proving the pointwise ergodic theorem.
s_strong_mixing	axiom	Strong mixing	mixing	A measure-preserving transformation T is strongly mixing if μ(A ∩ T⁻ⁿB) → μ(A)μ(B) for all measurable A, B as n → ∞.
s_stationary_process	axiom	Stationary process	strictly stationary process	A stochastic process (Xₙ) whose joint distributions are shift-invariant: (X_{n₁+k}, …, X_{nₘ+k}) has the same distribution as (X_{n₁}, …, X_{nₘ}) for all k.
s_shannon_mcmillan_breiman	theorem	Shannon-McMillan-Breiman theorem	asymptotic equipartition property | AEP	For an ergodic stationary process, −(1/n) log μ(partition cell containing (X₁,…,Xₙ)) → h a.s., where h is the entropy rate; typical sequences have probability ≈
s_kac_lemma	theorem	Kac's lemma	Kac's recurrence theorem	For an ergodic measure-preserving transformation T on a probability space and A with μ(A) > 0, the expected return time to A is E_A[τ_A] = 1/μ(A).
s_kolmogorov_centsov_continuity	theorem	Kolmogorov-Čentsov continuity theorem	Kolmogorov continuity criterion	If E[|X_t − X_s|^α] ≤ C|t − s|^{1+β} for some α, β > 0, then X has a continuous modification with locally Hölder continuous paths of any exponent γ < β/α.
s_markov_property_bm	theorem	Markov property of Brownian motion		For Brownian motion B_t and fixed time s, the process (B_{s+t} − B_s)_{t≥0} is a standard Brownian motion independent of F_s.
s_strong_markov_property_bm	theorem	Strong Markov property of Brownian motion		For Brownian motion B_t and an a.s. finite stopping time τ, the process (B_{τ+t} − B_τ)_{t≥0} is a standard Brownian motion independent of F_τ.
s_blumenthal_zero_one_law	theorem	Blumenthal's 0-1 law		Every event in the germ σ-algebra F_{0+} = ∩_{t>0} F_t of a Brownian motion has probability 0 or 1; the immediate future of BM is deterministic given the starti
s_reflection_principle_bm	theorem	Reflection principle for Brownian motion		For Brownian motion and the hitting time τ_a = inf{t : B_t = a}, the reflected process B̃_t = B_t for t ≤ τ_a and B̃_t = 2a − B_t for t > τ_a is also a Brownian
s_distribution_of_max_bm	theorem	Distribution of the maximum of Brownian motion		P(max_{0≤s≤t} B_s ≥ a) = 2P(B_t ≥ a) = 2(1 − Φ(a/√t)) for a ≥ 0, derived from the reflection principle.
s_arc_sine_law_bm	theorem	Arc sine law for Brownian motion	Lévy's arc sine laws	The last zero of BM before time 1, the time at which BM attains its maximum on [0,1], and the Lebesgue measure of {t ∈ [0,1] : B_t > 0} all follow the arc sine 
s_nowhere_differentiability_bm	theorem	Nowhere differentiability of Brownian motion	Paley-Wiener-Zygmund theorem	Almost surely, the paths of Brownian motion are nowhere differentiable: for a.e. ω, there is no t at which lim_{h→0} (B_{t+h}(ω) − B_t(ω))/h exists.
s_quadratic_variation_bm	theorem	Quadratic variation of Brownian motion		The quadratic variation of Brownian motion on [0,t] equals t: Σᵢ (B_{tᵢ} − B_{tᵢ₋₁})² → t in L² (and in probability) as the mesh of the partition goes to 0.
s_bm_harmonic_functions	theorem	Brownian motion and harmonic functions	Kakutani's theorem | probabilistic solution of Dirichlet problem	For a bounded domain D ⊂ ℝⁿ with regular boundary and continuous f on ∂D, the function u(x) = E_x[f(B_{τ_D})] solves the Dirichlet problem Δu = 0 in D with u = 
s_recurrence_transience_bm	theorem	Recurrence and transience of Brownian motion		Brownian motion in ℝᵈ is recurrent (visits every open set infinitely often) for d ≤ 2 and transient (|B_t| → ∞ a.s.) for d ≥ 3.
s_bm_time_inversion	theorem	Time inversion of Brownian motion	BM inversion	If B_t is a standard Brownian motion, then the process X_t = tB_{1/t} for t > 0 and X_0 = 0 is also a standard Brownian motion.
s_bm_scaling_property	theorem	Scaling property of Brownian motion	Brownian scaling | self-similarity of BM	For any c > 0, the process (c⁻¹/² B_{ct})_{t≥0} is a standard Brownian motion; BM is self-similar with exponent 1/2.
s_bm_as_martingale	state	Brownian motion as a martingale		Standard Brownian motion B_t is a continuous martingale w.r.t. its natural filtration; moreover B_t² − t is also a martingale.
s_construction_of_bm	state	Construction of Brownian motion	Lévy-Ciesielski construction | Wiener's construction	Brownian motion can be constructed via Lévy's interpolation (dyadic), the Wiener series (random Fourier series on [0,1]), or the Kolmogorov extension theorem co
s_hitting_times_bm	state	Hitting times of Brownian motion	first passage time for BM	The hitting time τ_a = inf{t ≥ 0 : B_t = a} is a.s. finite for all a ∈ ℝ (d=1), has the distribution of a²/N² where N ~ N(0,1) (related to inverse Gaussian), an
t_subsequence_argument	technique	Subsequence argument		Extracts an a.s. convergent subsequence from a sequence converging in probability, then extends the conclusion to the full sequence by monotonicity or interpola
t_borel_cantelli_technique	technique	Borel-Cantelli technique		Proves almost sure convergence or divergence by bounding the sum of probabilities of bad events and applying the first or second Borel-Cantelli lemma.
t_moment_method	technique	Moment method (probabilistic)		Establishes convergence in distribution by computing all moments of Xₙ, showing convergence to moments of the target, and verifying the target is moment-determi
t_skorokhod_representation_technique	technique	Skorokhod representation technique		Replaces convergence in distribution with almost sure convergence by constructing equivalent random variables on a common probability space via Skorokhod's repr
t_maximal_inequality_technique	technique	Maximal inequality technique		Controls the fluctuations of partial sums or martingales using Kolmogorov's, Doob's, or Lévy's maximal inequality to deduce almost sure convergence or moment bo
t_stopping_time_technique	technique	Stopping time technique		Analyzes a process at a random time by applying optional stopping theorems, the strong Markov property, or Wald's equations to extract information about hitting
t_diagonalization_technique	technique	Diagonalization technique (probability)		Extracts a single subsequence satisfying countably many convergence requirements simultaneously by iterating subsequence extraction and taking the diagonal.
s_sample_space	axiom	Sample space		The set Ω of all possible outcomes of a random experiment, forming the first component of a probability space (Ω, F, P).
s_classical_probability	axiom	Classical probability (Laplace)		For a finite sample space with equally likely outcomes, P(A) = |A|/|Ω|; the combinatorial definition of probability due to Laplace.
s_multinomial_coefficient	axiom	Multinomial coefficient		The number n!/(k₁!k₂!⋯kᵣ!) of ways to partition n objects into r groups of sizes k₁,…,kᵣ with k₁+⋯+kᵣ=n.
s_multinomial_distribution	axiom	Multinomial distribution		The joint distribution of counts (X₁,…,Xᵣ) in n independent trials with r outcomes of probabilities p₁,…,pᵣ, with P(X₁=k₁,…,Xᵣ=kᵣ) = n!/(k₁!⋯kᵣ!) p₁^{k₁}⋯pᵣ^{kᵣ
s_hypergeometric_distribution	axiom	Hypergeometric distribution		The distribution of the number of successes in n draws without replacement from a population of N containing K successes: P(X=k) = C(K,k)C(N−K,n−k)/C(N,n).
s_birthday_problem	state	Birthday problem		The probability that among n persons at least two share a birthday; for n=23 this exceeds 1/2, illustrating the surprising frequency of coincidences in moderate
s_bose_einstein_statistics	axiom	Bose–Einstein statistics		The occupancy model in which n indistinguishable particles are distributed among r cells with all arrangements equally likely; the number of such arrangements i
s_fermi_dirac_statistics	axiom	Fermi–Dirac statistics		The occupancy model in which n indistinguishable particles are placed in r cells with at most one per cell, all C(r,n) arrangements equally likely.
s_first_passage_time	state	First passage time		For a random walk (Sₙ), the first time T_a = inf{n ≥ 1 : Sₙ = a} that the walk reaches level a; a fundamental quantity in fluctuation theory.
s_matching_problem	state	Matching problem (problème des rencontres)		The problem of counting fixed points (matches) in a random permutation of n objects; the probability of at least one match converges to 1 − e⁻¹ as n → ∞.
s_law_of_total_probability	theorem	Law of total probability		If {B₁,…,Bₙ} is a partition of Ω with P(Bᵢ) > 0, then P(A) = Σᵢ P(A|Bᵢ)P(Bᵢ); the marginal probability as a mixture over conditioning events.
s_polya_urn_model	state	Pólya urn model		An urn initially containing a red and b blue balls; at each step a ball is drawn and returned together with c additional balls of the same color, producing exch
s_bernoulli_trial	axiom	Bernoulli trial		A sequence of independent experiments each having exactly two outcomes (success with probability p, failure with probability 1−p); the fundamental building bloc
s_negative_binomial_distribution	axiom	Negative binomial distribution		The distribution of the number of failures before the r-th success in independent Bernoulli trials with success probability p: P(X=k) = C(k+r−1,k) pʳ(1−p)ᵏ.
s_covariance	axiom	Covariance		Cov(X,Y) = E[(X−EX)(Y−EY)] = E[XY] − E[X]E[Y], measuring the linear dependence between two random variables.
s_standard_deviation	axiom	Standard deviation		σ(X) = √Var(X), the square root of the variance, measuring dispersion in the same units as X.
s_probability_generating_function	axiom	Probability generating function		G_X(s) = E[sˣ] = Σₖ P(X=k)sᵏ for |s| ≤ 1; encodes the distribution of a non-negative integer-valued random variable and converts convolution to multiplication.
s_pgf_convolution_theorem	theorem	Convolution theorem for PGFs		If X₁,…,Xₙ are independent ℕ-valued random variables, then G_{X₁+⋯+Xₙ}(s) = G_{X₁}(s)⋯G_{Xₙ}(s); independence turns convolution into multiplication of generatin
s_continuity_theorem_pgf	theorem	Continuity theorem for probability generating functions		If Gₙ(s) → G(s) for all s ∈ [0,1) and G is a PGF, then the corresponding distributions converge; the PGF analogue of Lévy's continuity theorem.
s_random_sum	state	Random sum (compound distribution)		S_N = X₁ + X₂ + ⋯ + X_N where N is a random variable independent of the i.i.d. summands Xᵢ; its PGF satisfies G_{S_N}(s) = G_N(G_X(s)).
s_compound_poisson_distribution	state	Compound Poisson distribution		The distribution of S_N = Σᵢ₌₁ᴺ Xᵢ where N ~ Poisson(λ) is independent of i.i.d. summands Xᵢ with distribution F; characterized by φ(t) = exp(λ(φ_X(t)−1)).
s_branching_process_criticality	axiom	Criticality classification of branching processes		A Galton–Watson process with offspring mean μ is subcritical if μ < 1, critical if μ = 1, or supercritical if μ > 1; the extinction probability equals 1 iff μ ≤
s_recurrent_event_feller	axiom	Recurrent event (Feller)		An event E occurring at epochs n₁ < n₂ < ⋯ such that the inter-occurrence times are i.i.d.; Feller's axiomatization of renewal phenomena via generating function
s_persistent_recurrent_event	axiom	Persistent (certain) recurrent event		A recurrent event is persistent (certain) if the probability of its eventual occurrence equals 1, i.e., Σₙ fₙ = 1 where fₙ is the first-occurrence probability a
s_transient_recurrent_event	axiom	Transient (uncertain) recurrent event		A recurrent event is transient (uncertain) if the probability of its eventual occurrence is strictly less than 1, i.e., Σₙ fₙ < 1.
s_periodic_recurrent_event	axiom	Periodic and aperiodic recurrent event		The period of a recurrent event is d = gcd{n : fₙ > 0}; the event is periodic if d > 1 and aperiodic if d = 1, where fₙ is the first-occurrence probability at t
s_renewal_equation	theorem	Renewal equation		The integral equation Z(t) = z(t) + ∫₀ᵗ Z(t−s) dF(s) where F is the inter-renewal distribution; solved by Z = z * U where U is the renewal measure.
s_elementary_renewal_theorem	theorem	Elementary renewal theorem		For a renewal process with inter-arrival mean μ, the expected number of renewals satisfies E[N(t)]/t → 1/μ as t → ∞.
s_excess_life	state	Excess life (residual waiting time)		γ(t) = S_{N(t)+1} − t, the time from t until the next renewal; its asymptotic distribution is (1/μ)∫₀ˣ (1−F(u)) du when the inter-renewal distribution F has mea
s_delayed_renewal_process	state	Delayed (modified) renewal process		A renewal process in which the first inter-arrival time has a different distribution from the subsequent i.i.d. inter-arrivals; used to construct stationary ren
s_duration_gamblers_ruin	theorem	Duration of the gambler's ruin		The expected duration of the gambler's ruin game starting with capital a against an opponent with capital b; for p ≠ 1/2, E[D] = (a/(q−p)) − ((a+b)/(q−p)) · (1−
t_duality_principle_random_walk	technique	Duality principle for random walks		The path (S₁,…,Sₙ) has the same joint distribution as (Sₙ−Sₙ₋₁, Sₙ−Sₙ₋₂,…,Sₙ−S₀), enabling transfer of results about maxima to results about last-visit times.
s_aperiodic_markov_chain	axiom	Aperiodic Markov chain		A Markov chain (or state) is aperiodic if every state has period 1; equivalently, gcd{n ≥ 1 : p^{(n)}(i,i) > 0} = 1 for all states i.
s_convergence_periodic_markov_chain	theorem	Convergence theorem for periodic Markov chains		For an irreducible positive recurrent chain with period d, the Cesàro average (1/n)Σₖ p^{(k)}(i,j) → π(j) and p^{(nd)}(i,i) → d·π(i) as n → ∞.
s_absorption_probability	state	Absorption probability		The probability hᵢⱼ = Pᵢ(Xₙ = j for some n) of a Markov chain starting in transient state i being eventually absorbed into absorbing state j, satisfying hᵢⱼ = Σ
s_mean_absorption_time	state	Mean absorption time		The expected number of steps gᵢ = Eᵢ[T] until an absorbing Markov chain is absorbed, satisfying gᵢ = 1 + Σⱼ pᵢⱼ gⱼ summed over transient states j.
t_canonical_form_absorbing_chain	technique	Canonical form for absorbing Markov chains		Rearranges the transition matrix of an absorbing chain into P = [[Q, R],[0, I]] with Q for transient states and R for absorption; the fundamental matrix N = (I−
s_doubly_stochastic_matrix	axiom	Doubly stochastic matrix		A non-negative matrix whose rows and columns each sum to 1; the stationary distribution of the corresponding Markov chain is the uniform distribution.
s_exponential_distribution	axiom	Exponential distribution		The continuous distribution with density f(x) = λe^{−λx} for x ≥ 0 and CDF F(x) = 1−e^{−λx}; the unique continuous memoryless distribution with mean 1/λ.
s_memoryless_property	theorem	Memoryless property		P(X > s+t | X > s) = P(X > t) for all s,t ≥ 0; characterizes the exponential distribution among continuous distributions and the geometric among discrete distri
s_superposition_poisson_processes	theorem	Superposition of Poisson processes		The superposition (merger) of independent Poisson processes with rates λ₁,…,λₖ is a Poisson process with rate λ₁+⋯+λₖ.
s_thinning_poisson_process	theorem	Thinning of a Poisson process		If each arrival of a Poisson process with rate λ is independently retained with probability p, the retained arrivals form a Poisson process with rate λp.
s_birth_process	axiom	Pure birth process		A continuous-time Markov chain on ℕ with transitions only from n to n+1 at rate λₙ; the simplest continuous-time chain, generalizing the Poisson process (λₙ = λ
s_yule_process	state	Yule process (pure birth with linear rate)		A pure birth process with birth rate λₙ = nλ, modeling population growth where each individual independently gives birth at rate λ; the population size grows as
s_birth_death_process	axiom	Birth-death process		A continuous-time Markov chain on ℕ with transitions n→n+1 at rate λₙ (birth) and n→n−1 at rate μₙ (death); the canonical model for population dynamics and queu
s_kolmogorov_forward_equation	theorem	Kolmogorov forward equations		P'(t) = P(t)Q, or p'ᵢⱼ(t) = Σₖ pᵢₖ(t)qₖⱼ, governing the evolution of transition probabilities forward in time; also called the Fokker–Planck equations in the di
s_generator_matrix	axiom	Generator matrix (Q-matrix)		The matrix Q with off-diagonal entries qᵢⱼ ≥ 0 (transition rates) and diagonal entries qᵢᵢ = −Σⱼ≠ᵢ qᵢⱼ, satisfying P(t) = e^{tQ}; the infinitesimal description 
s_inspection_paradox	state	Inspection paradox (waiting time paradox)		At a random time t, the inter-renewal interval containing t is stochastically longer than a typical inter-renewal interval, because longer intervals are more li
t_method_of_generating_functions	technique	Method of generating functions		Translates a recurrence or convolution equation into an algebraic or functional equation for the generating function, solves it, then extracts coefficients.
t_first_step_analysis	technique	First-step analysis		Conditions on the first transition of a Markov chain to derive a system of linear equations for hitting probabilities, expected hitting times, or other quantiti
s_inversion_formula_characteristic_function	theorem	Inversion formula for characteristic functions		F(b) − F(a) = lim_{T→∞} (1/2π) ∫_{−T}^{T} (e^{−ita} − e^{−itb})/(it) φ(t) dt at continuity points a,b of F; recovers the distribution from its characteristic fu
s_uniqueness_theorem_distributions	theorem	Uniqueness theorem for characteristic functions		Two probability distributions on ℝ are identical if and only if their characteristic functions are identical; φ_X = φ_Y implies X =ᵈ Y.
s_gamma_distribution	axiom	Gamma distribution		The continuous distribution with density f(x) = β^α x^{α−1} e^{−βx}/Γ(α) for x > 0, generalizing the exponential (α=1) and chi-squared (β=1/2, α=k/2) distributi
s_beta_distribution	axiom	Beta distribution		The continuous distribution on [0,1] with density f(x) = x^{α−1}(1−x)^{β−1}/B(α,β); conjugate prior for the Bernoulli parameter and the distribution of order st
s_cauchy_distribution	axiom	Cauchy distribution		The distribution with density f(x) = 1/(πγ(1+((x−x₀)/γ)²)); a stable distribution of index 1 having no finite moments, and the ratio of two independent standard
s_uniform_distribution_continuous	axiom	Continuous uniform distribution		The distribution on [a,b] with constant density f(x) = 1/(b−a); the maximum-entropy distribution on a bounded interval.
s_exponential_family	axiom	Exponential family of distributions		A parametric family with densities f(x|θ) = h(x) exp(η(θ)·T(x) − A(θ)); encompasses normal, exponential, gamma, Poisson, binomial, and many other standard distr
s_f_distribution	state	F-distribution (Fisher–Snedecor)		The distribution of (U/d₁)/(V/d₂) where U ~ χ²(d₁) and V ~ χ²(d₂) are independent; fundamental in analysis of variance and F-tests.
s_lognormal_distribution	state	Lognormal distribution		The distribution of e^X where X ~ N(μ,σ²); equivalently, Y has a lognormal distribution if log Y is normally distributed.
s_pareto_distribution	state	Pareto distribution		The power-law distribution with survival function P(X > x) = (x_m/x)^α for x ≥ x_m; the prototypical heavy-tailed distribution, arising in the study of income a
s_weibull_distribution	state	Weibull distribution		The distribution with CDF F(x) = 1 − exp(−(x/λ)ᵏ) for x ≥ 0; generalizes the exponential (k=1) and models failure times with increasing (k>1) or decreasing (k<1
s_logistic_distribution	state	Logistic distribution		The distribution with CDF F(x) = 1/(1 + e^{−(x−μ)/s}) and bell-shaped density similar to the normal but with heavier tails.
s_order_statistics	axiom	Order statistics		The values X_{(1)} ≤ X_{(2)} ≤ ⋯ ≤ X_{(n)} obtained by sorting a random sample; the joint density of all order statistics from a continuous distribution with de
s_marginal_distribution	axiom	Marginal distribution		The distribution of a subset of random variables obtained by integrating (or summing) the joint distribution over the remaining variables: f_X(x) = ∫ f_{X,Y}(x,
s_edgeworth_expansion	state	Edgeworth expansion		A refinement of the CLT giving P((Sₙ−nμ)/(σ√n) ≤ x) = Φ(x) + n^{−1/2} p₁(x)φ(x) + n⁻¹ p₂(x)φ(x) + ⋯ where pₖ are polynomials in the cumulants; provides higher-o
t_saddlepoint_approximation	technique	Saddlepoint approximation		Approximates the density of Sₙ/n at x by (2πnK''(ŝ))^{−1/2} exp(n(K(ŝ)−ŝx)) where ŝ solves K'(ŝ)=x; yields relative error O(1/n) uniformly in x, far superior to
s_cramer_decomposition_theorem	theorem	Cramér's decomposition theorem		If X + Y is normally distributed and X, Y are independent, then both X and Y are normally distributed; the normal distribution is indecomposable among non-degen
s_raikov_theorem	theorem	Raikov's theorem		If X + Y has a Poisson distribution and X, Y are independent, then both X and Y have Poisson distributions; the Poisson analogue of Cramér's decomposition theor
s_key_renewal_theorem	theorem	Key renewal theorem		If F is non-lattice with mean μ and z is directly Riemann integrable, then ∫₀^∞ z(t−x)dU(x) → (1/μ)∫₀^∞ z(t)dt as t → ∞; the continuous-time sharpening of the B
s_age_renewal_theory	state	Age (backward recurrence time)		δ(t) = t − S_{N(t)}, the elapsed time since the last renewal before time t; the backward analogue of the excess life γ(t).
s_equilibrium_renewal_distribution	state	Equilibrium (stationary) renewal distribution		F_e(x) = (1/μ)∫₀ˣ (1−F(t))dt, the distribution whose use as the first inter-arrival time makes the renewal process stationary; also the asymptotic excess-life d
s_renewal_reward_theorem	theorem	Renewal reward theorem		If rewards Rₙ with E[|Rₙ|] < ∞ accrue at renewal epochs with mean inter-renewal time μ, then the long-run average reward rate converges to E[R]/μ a.s.
s_abelian_theorem_laplace	theorem	Abelian theorem for Laplace transforms		If f(t) ~ Ct^{ρ} as t → ∞, then its Laplace transform satisfies f̂(s) ~ CΓ(ρ+1)/s^{ρ+1} as s → 0⁺; transfers asymptotic information from the original function t
s_regularly_varying_function	axiom	Regularly varying function		A measurable function f: (0,∞) → (0,∞) satisfying f(tx)/f(x) → t^ρ for every t > 0 as x → ∞, for some ρ ∈ ℝ (the index of regular variation); includes slowly va
s_feller_semigroup	axiom	Feller semigroup		A strongly continuous positive contraction semigroup {Tₜ} on C₀(E) (continuous functions vanishing at infinity) that maps C₀(E) into itself; the semigroup gener
s_diffusion_process	axiom	Diffusion process		A continuous-path strong Markov process whose generator is a second-order differential operator L = (1/2)σ²(x)d²/dx² + b(x)d/dx; the continuous analogue of a ra
t_esscher_transform	technique	Esscher transform (exponential tilting)		Defines a new probability measure dF_θ(x) = e^{θx} dF(x)/M(θ) where M(θ) = E[e^{θX}] is the MGF; shifts the mean of the distribution and is used for large-devia
s_levy_inequality	theorem	Lévy's inequality		For independent symmetric random variables X₁,…,Xₙ with partial sums Sₖ, P(max_{1≤k≤n} |Sₖ| ≥ λ) ≤ 2P(|Sₙ| ≥ λ); a maximal inequality exploiting symmetry.
s_occupation_time	state	Occupation time		The total time a stochastic process spends in a given set: Γ_A(t) = ∫₀ᵗ 1_{X_s ∈ A} ds (continuous time) or Σₖ₌₀ⁿ 1_{Xₖ ∈ A} (discrete time).
s_local_time_random_walk	state	Local time for random walk		L(x,n) = #{0 ≤ k ≤ n : Sₖ = x}, the number of visits to site x by the random walk up to time n; for the simple symmetric walk, the arc-sine law governs the frac
s_resolvent_markov_process	state	Resolvent of a Markov process		R_λ f(x) = E_x[∫₀^∞ e^{−λt} f(X_t) dt], the Laplace transform of the semigroup; satisfies the resolvent equation R_λ − R_μ = (μ−λ)R_λ R_μ and determines the pro
s_levy_khintchine_triplet	state	Lévy–Khintchine triplet		The triple (γ, σ², ν) where γ ∈ ℝ is the drift, σ² ≥ 0 the Gaussian variance, and ν a Lévy measure, uniquely characterizing an infinitely divisible distribution
s_compound_poisson_representation	theorem	Compound Poisson representation of infinitely divisible laws		Every infinitely divisible distribution is a weak limit of compound Poisson distributions; equivalently, its Lévy–Khintchine exponent can be approximated by com
s_maximal_coupling	state	Maximal coupling		A coupling (X,Y) of distributions μ and ν that maximizes P(X=Y), achieving P(X≠Y) = ‖μ−ν‖_{TV}; optimal for bounding total variation distance.
s_symmetrization_inequality	theorem	Symmetrization inequality		P(|Sₙ − ESₙ| > t) ≤ 2P(|S̃ₙ| > t/2) where S̃ₙ = Σ(Xᵢ − X'ᵢ) is the symmetrized sum with independent copies X'ᵢ; reduces tail bounds to the symmetric case.
s_record_values	state	Record values		X_{n} is an upper record if X_{n} > max(X₁,…,X_{n−1}); for i.i.d. continuous observations, record times form a renewal process with E[number of records in n tri
s_cf_inversion_lattice	theorem	Characteristic function inversion formula on a lattice		For a lattice distribution with span h, P(X = a+kh) = (h/2π) ∫_{−π/h}^{π/h} e^{−it(a+kh)} φ(t) dt; the discrete analogue of the Fourier inversion formula.
s_type_convergence_of_types	axiom	Type (convergence of types)		Two distributions F and G are of the same type if G(x) = F(ax+b) for constants a > 0, b; the convergence-of-types theorem states that if (Sₙ−bₙ)/aₙ →ᵈ F and (Sₙ
s_stable_laws_classification	theorem	Classification of stable laws (Lévy)		Every stable distribution is characterized by four parameters: index α ∈ (0,2], skewness β ∈ [−1,1], scale γ > 0, and location δ ∈ ℝ, with characteristic expone
s_operator_stable_law	state	Operator stable distribution		A distribution μ on ℝᵈ such that for each n there exist matrices Aₙ and vectors bₙ with μⁿ = μ ∘ Aₙ⁻¹ * δ_{bₙ}; the multivariate generalization of stable laws w
s_subexponential_distribution	axiom	Subexponential distribution		A distribution F on [0,∞) with P(X₁+X₂ > x)/P(X₁ > x) → 2 as x → ∞ (for i.i.d. X₁,X₂ ~ F); the tail of the sum is dominated by the maximum, implying P(Sₙ > x) ~
s_self_decomposable_distribution	axiom	Self-decomposable distribution		A distribution μ is self-decomposable (class L) if for every c ∈ (0,1) there exists a distribution μ_c with μ = (μ scaled by c) * μ_c; equivalently, it is the l
s_ehrenfest_urn_model	state	Ehrenfest urn model (diffusion model)		A Markov chain on {0,1,…,N} modeling diffusion: at each step, one of N balls is chosen uniformly and moved to the other urn, so the chain transitions from k to 
s_ruin_probability_half_line	theorem	Ruin probability on the half-line		For a random walk with negative drift, the probability P(sup_{n≥0} Sₙ ≥ x) of ever exceeding level x, which decreases exponentially in x when the adjustment coe
s_fourier_analysis_on_groups	axiom	Fourier analysis on locally compact abelian groups		The extension of Fourier analysis to locally compact abelian groups via the dual group Ĝ: the Fourier transform of a measure μ is μ̂(χ) = ∫ χ(x) dμ(x), and conv
s_domain_of_normal_attraction	state	Domain of normal attraction		A distribution F belongs to the domain of normal attraction of a stable law G_α if (S_n − bn)/n^{1/α} →ᵈ G_α with the specific normalization n^{1/α}; a strict s
s_n_step_transition_probability	state	n-step transition probability		The probability p^{(n)}(x,y) = P_x(X_n = y) of reaching state y from state x in exactly n steps; the (x,y)-entry of P^n.
s_initial_distribution_mc	axiom	Initial distribution of a Markov chain		A probability measure μ on the state space E that, together with the transition matrix P, uniquely determines the law of the Markov chain.
t_canonical_construction_mc	technique	Canonical construction of a Markov chain		Constructs the Markov chain on the canonical path space Ω = E^ℕ with coordinate maps X_n(ω) = ω_n, using the Kolmogorov extension theorem to build the unique me
s_accessibility_mc	axiom	Accessibility (leads to) in a Markov chain		State y is accessible from state x (written x → y) if p^{(n)}(x,y) > 0 for some n ≥ 0; the fundamental asymmetric reachability relation on the state space.
s_closed_communicating_set	axiom	Closed (absorbing) set in a Markov chain		A nonempty subset C of the state space is closed if p(x,y) = 0 for all x ∈ C, y ∉ C; equivalently, C is absorbing — once the chain enters C it never leaves.
s_solidarity_of_period	theorem	Solidarity of period		All states in the same communication class have the same period; hence the period is a class property, and one may speak of the period of an irreducible chain.
s_cyclic_decomposition_mc	theorem	Cyclic decomposition of a periodic chain		For an irreducible chain of period d, the state space decomposes into d cyclic classes C_0, ..., C_{d-1} such that from any state in C_r the chain moves to C_{r
s_essential_inessential_states	axiom	Essential and inessential states		A state x is essential if for every state y accessible from x, x is also accessible from y; otherwise x is inessential (transient).
s_first_return_time_decomposition	state	First return time decomposition		The first return probabilities f^{(n)}(x,x) = P_x(T_x = n) satisfy the renewal equation p^{(n)}(x,x) = Σ_{k=1}^n f^{(k)}(x,x) p^{(n-k)}(x,x), linking n-step ret
s_recurrence_criterion_green_function	theorem	Recurrence criterion via Green function		A state x is recurrent if and only if G(x,x) = Σ_{n=0}^∞ p^{(n)}(x,x) = ∞, and transient if and only if G(x,x) < ∞.
s_solidarity_positive_null_recurrence	theorem	Solidarity of positive/null recurrence		Positive recurrence and null recurrence are class properties: if one state in a communication class is positive (resp. null) recurrent, then all states in the c
s_chung_fuchs_theorem	theorem	Chung-Fuchs theorem		A random walk on ℝ^d with mean-zero i.i.d. increments is recurrent if d ≤ 2 and transient if d ≥ 3; recurrence is determined by the integrability of Re(1/(1-φ(t
s_number_of_visits_geometric	state	Number of visits to a transient state		For a transient state x, the total number of visits N_x started from x has a geometric distribution with parameter f(x,x) = P_x(T_x < ∞); hence E_x[N_x] = G(x,x
s_invariant_measure_mc	axiom	Invariant measure for a Markov chain		A σ-finite measure μ on the state space satisfying μP = μ; generalizes the stationary distribution to the σ-finite setting, essential for null recurrent chains.
s_existence_uniqueness_invariant_measure_mc	theorem	Existence and uniqueness of invariant measure for irreducible recurrent chains		An irreducible recurrent Markov chain admits a unique (up to scalar multiples) invariant measure μ, given by μ(y) = E_x[Σ_{k=0}^{T_x-1} 1_{X_k = y}] for any ref
s_ratio_limit_theorem_chung_erdos_orey	theorem	Ratio limit theorem (Chung-Erdős-Orey form)		For an irreducible recurrent chain, p^{(n)}(x,y)/p^{(n)}(x,z) → μ(y)/μ(z) as n → ∞ along multiples of the period, where μ is the invariant measure.
s_cesaro_convergence_mc	theorem	Cesàro convergence for Markov chains		For an irreducible positive recurrent chain with stationary distribution π, (1/n) Σ_{k=0}^{n-1} p^{(k)}(x,y) → π(y) as n → ∞, regardless of periodicity.
s_orey_convergence_theorem	theorem	Orey's convergence theorem		For an irreducible, aperiodic, positive recurrent Markov chain, ‖P^n(x,·) − π(·)‖_{TV} → 0 as n → ∞ for all x; convergence to stationarity holds in total variat
s_doeblin_condition	axiom	Doeblin's condition		There exist n₀ ≥ 1, ε > 0, and a probability measure φ such that P^{n₀}(x,A) ≥ εφ(A) for all states x and measurable sets A; a strong uniform mixing condition e
t_doeblin_decomposition	technique	Doeblin decomposition		Decomposes the state space into at most countably many ergodic classes plus a transient set; every initial distribution eventually concentrates on the ergodic c
s_geometric_ergodicity_mc	state	Geometric ergodicity		An ergodic Markov chain is geometrically ergodic if ‖P^n(x,·) − π(·)‖_{TV} ≤ C(x)ρⁿ for some ρ < 1; exponential convergence to the stationary distribution in to
s_taboo_probability	axiom	Taboo probability		The taboo transition probability ₐp^{(n)}(x,y) = P_x(X_n = y, X_k ∉ A for 1 ≤ k ≤ n-1), the probability of going from x to y in n steps while avoiding set A.
s_taboo_green_function	state	Taboo Green function		The expected number of visits to y starting from x while avoiding set A: ₐG(x,y) = Σ_{n=0}^∞ ₐp^{(n)}(x,y); a restricted potential kernel fundamental to decompo
s_last_exit_decomposition	theorem	Last exit decomposition		The Green function decomposes via the last visit to an intermediate state z: G(x,y) = Σ_z G(x,z)·f(z,y) using taboo probabilities; dually G(x,y) = Σ_z f(x,z)·G(
s_harmonic_function_mc	axiom	Harmonic function for a Markov chain		A function h: E → ℝ satisfying Ph = h, i.e., h(x) = Σ_y p(x,y)h(y) for all x; the discrete analogue of harmonic functions, with h(X_n) forming a martingale.
s_liouville_property_recurrent_mc	theorem	Liouville property for recurrent Markov chains		An irreducible recurrent Markov chain has no non-constant bounded harmonic functions; every bounded function h with Ph = h is constant.
t_discrete_dirichlet_problem	technique	Discrete Dirichlet problem		Solves for the unique function h harmonic on D with h = f on ∂D via h(x) = E_x[f(X_{τ_{∂D}})]; the probabilistic solution of discrete boundary value problems.
s_superharmonic_optional_stopping	theorem	Superharmonic functions and optional stopping		If h is superharmonic (Ph ≤ h) for a Markov chain, then h(X_n) is a supermartingale; optional stopping yields h(x) ≥ E_x[h(X_τ)] for stopping times τ.
s_potential_kernel_recurrent_mc	state	Potential kernel (deviation matrix) for recurrent chains		The deviation matrix A(x,y) = Σ_{n=0}^∞ [p^{(n)}(x,y) − π(y)], measuring cumulative deviation from equilibrium; the analogue of the Green function in the recurr
s_potential_of_charge_mc	state	Potential of a charge (recurrent chain)		For a signed measure ν with ν(E) = 0, the potential Uν(x) = Σ_y A(x,y)ν(y) where A is the deviation matrix; the discrete analogue of the potential of a balanced
s_green_kernel_transient_mc	state	Green kernel for transient Markov chains		G(x,y) = Σ_{n=0}^∞ p^{(n)}(x,y) = E_x[number of visits to y], finite for transient chains; the kernel for the potential operator Gf(x) = Σ_y G(x,y)f(y).
s_riesz_decomposition_discrete_mc	theorem	Riesz decomposition for Markov chains		Every non-negative superharmonic function on a transient Markov chain decomposes uniquely as h = Gν + f where Gν is a potential (ν ≥ 0) and f is harmonic.
s_excessive_function_mc	axiom	Excessive function (α-excessive)		A non-negative function f satisfying αPf ≤ f and α^n P^n f(x) → f(x) for all x; when α = 1, these are the non-negative superharmonic functions with a regularity
s_excessive_function_characterization	theorem	Characterization of excessive functions		A non-negative function f is excessive iff Pf ≤ f and P^n f → f pointwise; equivalently, f is the pointwise limit of an increasing sequence of potentials.
s_minimum_principle_excessive	theorem	Minimum principle for excessive functions		If an excessive function f dominates a potential g outside a finite set, then f ≥ g everywhere; the dual of the maximum principle for subharmonic functions.
s_domination_principle_mc	theorem	Domination principle for potentials		If Gμ ≤ Gν on the support of μ, then Gμ ≤ Gν everywhere; the discrete analogue of the classical domination principle in potential theory.
s_cone_of_excessive_functions	state	Cone of excessive functions		The set of all excessive functions for a Markov chain, forming a convex cone closed under pointwise infima and increasing limits; the fundamental object of prob
s_energy_of_measure_mc	axiom	Energy of a measure (discrete potential theory)		ℰ(μ) = ⟨μ, Gμ⟩ = Σ_{x,y} μ(x)G(x,y)μ(y), the mutual energy of measure μ under the Green kernel G of a transient chain.
s_dirichlet_form_discrete	axiom	Dirichlet form (discrete Markov chain)		The quadratic form ℰ(f,f) = (1/2) Σ_{x,y} π(x)p(x,y)(f(x)−f(y))² measuring energy dissipation; for reversible chains, connected to the spectral gap.
s_capacity_discrete_mc	state	Capacity of a set (discrete Markov chain)		Cap(A) = Σ_{x∈A} e_A(x) where e_A is the equilibrium measure; measures the ease with which the chain hits A and characterizes transience vs. recurrence.
s_equilibrium_measure_mc	state	Equilibrium measure (discrete potential theory)		The unique measure e_A on A such that Ge_A(x) = P_x(T_A < ∞) for transient chains; the charge distribution on A generating the hitting probability as its potent
s_equilibrium_potential_mc	state	Equilibrium potential (discrete Markov chain)		h_A(x) = P_x(T_A < ∞) = Ge_A(x); an excessive function equal to 1 on A, the unique smallest excessive function dominating 1_A.
s_capacity_transience_criterion	theorem	Capacity and transience criterion		A state is transient if and only if it has positive capacity; a Markov chain is transient iff Cap({x}) > 0 for some (hence all, in the irreducible case) states.
s_energy_principle_mc	theorem	Energy principle (discrete potential theory)		The energy ℰ(μ) = ⟨μ, Gμ⟩ is minimized over probability measures on A by the normalized equilibrium measure; the minimum equals 1/Cap(A).
t_balayage_mc	technique	Balayage (sweeping) for Markov chains		Sweeps an excessive function f onto a set A, producing the réduite R_A f(x) = E_x[f(X_{T_A})], the smallest excessive function dominating f on A.
s_reduite_mc	state	Réduite (reduced function) for Markov chains		R_A f(x) = inf{g(x) : g excessive, g ≥ f on A}, the smallest excessive function that dominates f on A; equals E_x[f(X_{T_A})].
s_balayage_of_measure_mc	state	Balayage of a measure (discrete potential theory)		The swept measure μ^A representing the distribution of the chain started with measure μ and stopped upon hitting A.
s_balayage_principle_mc	theorem	Balayage principle for Markov chains		For every excessive function f and set A, the réduite R_A f is excessive, equals f on A (up to a polar set), and is harmonic on A^c.
s_hunt_balayage_theorem	theorem	Hunt's balayage theorem		Every excessive function is the supremum of potentials of measures it dominates; gives a complete representation theory for the cone of excessive functions.
s_ctmc_definition	axiom	Continuous-time Markov chain (CTMC)		A continuous-time stochastic process (X_t)_{t≥0} on a countable state space satisfying the Markov property with time-homogeneous transition rates given by a Q-m
s_transition_semigroup_ctmc	axiom	Transition semigroup of a CTMC		The family {P(t)}_{t≥0} of transition matrices satisfying P(s+t) = P(s)P(t), P(0) = I, encoding transition probabilities of a continuous-time Markov chain.
s_holding_time_ctmc	state	Holding time of a CTMC		The time spent in state i before jumping, exponentially distributed with rate q_i = -q_{ii}; the memoryless property ensures the Markov property in continuous t
s_jump_chain_ctmc	state	Jump chain (embedded chain) of a CTMC		The discrete-time Markov chain recording successive states visited by the CTMC, with transition probabilities p̃(i,j) = q_{ij}/q_i; together with holding times,
s_minimal_solution_ctmc	state	Minimal solution of the Kolmogorov equations		The smallest non-negative solution to both Kolmogorov equations, constructed via successive approximations; honest iff no explosion occurs.
s_explosion_ctmc	state	Explosion of a CTMC		The event that infinitely many jumps occur in finite time (ζ = lim J_n < ∞); when explosion occurs, the minimal chain is dishonest (Σ_y p_t(x,y) < 1).
s_non_explosion_criterion_ctmc	theorem	Non-explosion criterion for CTMCs		A CTMC is non-explosive if Σ_n 1/q_{Y_n} = ∞ a.s.; sufficient conditions include uniform boundedness of rates or existence of a Lyapunov function f with Qf ≤ cf
t_uniformization_ctmc	technique	Uniformization of a CTMC		Represents a CTMC with bounded rates as P(t) = e^{-qt} Σ (qt)^n/n! · P̃^n where P̃ = I + Q/q; reduces analysis to a Poisson-subordinated discrete chain.
s_resolvent_as_lambda_potential	state	Resolvent as λ-potential kernel		R_λ(x,y) = ∫_0^∞ e^{-λt} p(t,x,y) dt; the λ-potential kernel generalizing the Green function, with R_λ f = E_x[∫_0^∞ e^{-λt} f(X_t) dt].
s_lambda_excessive_function	axiom	λ-excessive function		A non-negative function f with e^{-λt} P_t f ≤ f for all t ≥ 0 and e^{-λt} P_t f → f as t ↓ 0; the natural class for resolvent potential theory.
s_resolvent_characterization_excessive	theorem	Resolvent characterization of λ-excessive functions		A non-negative function f is λ-excessive iff αR_{λ+α} f ↑ f as α → ∞; equivalently, f is the pointwise supremum of λ-potentials.
s_potential_operator_continuous	state	Potential operator for continuous-time processes		Gf(x) = E_x[∫_0^∞ f(X_t) dt] = lim_{λ↓0} R_λ f(x); the continuous-time analogue of the discrete Green kernel, defined for transient processes.
s_recovery_semigroup_from_resolvent	theorem	Recovery of semigroup from resolvent		The transition semigroup is uniquely determined by the resolvent family via P_t f = lim (nR_{n/t})^{[nt]} f or Laplace inversion; content of the Hille-Yosida ge
s_recurrence_transience_ctmc	axiom	Recurrence and transience for CTMCs		A CTMC state is recurrent if ∫_0^∞ p_t(x,x) dt = ∞ (returns a.s.) and transient if the integral is finite; recurrence/transience are class properties.
s_invariant_measure_ctmc	axiom	Invariant measure for a CTMC		A σ-finite measure π satisfying πQ = 0, equivalently πP(t) = π for all t ≥ 0; unique up to scalar for irreducible recurrent CTMCs.
s_ergodic_theorem_ctmc	theorem	Ergodic theorem for CTMCs		For an irreducible positive recurrent CTMC with stationary distribution π, (1/t)∫_0^t f(X_s) ds → Σ_x f(x)π(x) a.s. and p_t(x,y) → π(y) as t → ∞.
s_continuous_time_green_function	state	Green function for continuous-time processes		G(x,y) = ∫_0^∞ p_t(x,y) dt = E_x[total time at y]; the continuous-time Green function, finite when the process is transient.
s_riesz_decomposition_continuous_mc	theorem	Riesz decomposition for continuous-time processes		Every non-negative excessive function for a transient process decomposes uniquely as f = Gμ + h where Gμ is a potential and h is harmonic.
s_capacity_continuous_process	state	Capacity for continuous-time processes		Cap(A) = sup{μ(A) : supp(μ) ⊂ A, Gμ ≤ 1}; measures the size of A from the process's perspective, related to hitting probabilities and equilibrium measures.
s_hunt_process	axiom	Hunt process		A strong Markov process with right-continuous paths and quasi-left-continuity; the standard framework for probabilistic potential theory, generalizing Feller pr
s_quasi_left_continuity	axiom	Quasi-left-continuity		For every increasing sequence of stopping times T_n ↑ T, X_{T_n} → X_T a.s. on {T < ∞}; prevents jumps at predictable times and is the key regularity assumption
s_additive_functional	axiom	Additive functional		A non-decreasing right-continuous process A_t with A_0 = 0 satisfying A_{t+s} = A_t + A_s ∘ θ_t; the building block for time changes and Revuz correspondence.
s_revuz_measure	state	Revuz measure		The unique σ-finite measure μ_A associated to a positive continuous additive functional via lim_{t↓0} (1/t) E_m[∫_0^t f(X_s) dA_s] = ∫ f dμ_A.
s_revuz_correspondence	theorem	Revuz correspondence		A bijection between positive continuous additive functionals (up to equivalence) and smooth measures via the Revuz measure; a cornerstone connecting additive fu
s_fine_topology	axiom	Fine topology		The coarsest topology making all excessive functions continuous; the natural topology for potential theory, in which thin sets are precisely those of capacity z
t_time_change_markov	technique	Time change of a Markov process		Defines Y_s = X_{τ_s} where τ_s = inf{t : A_t > s}; the time-changed process is again a strong Markov process with transformed generator and state space.
t_killing_subprocess	technique	Killing and subprocess construction		Constructs a subprocess by killing X_t via a multiplicative functional M_t, yielding semigroup P̃_t f(x) = E_x[M_t f(X_t)]; amounts to modifying the generator b
t_renewal_argument_mc	technique	Renewal argument for Markov chains		Exploits regenerative structure at return times to a recurrent state, reducing long-run averages and limit theorems to renewal theory.
t_cycle_trick_mc	technique	Cycle trick for Markov chains		Decomposes the trajectory into i.i.d. excursions from a reference state, enabling the law of large numbers, CLT, and large deviations for additive functionals.
t_doeblin_coupling_mc	technique	Doeblin's coupling method		Runs two copies of a Markov chain exploiting Doeblin's condition to ensure coalescence in bounded expected time; yields geometric convergence bounds.
t_lyapunov_foster_criterion	technique	Lyapunov function method (Foster's criterion)		Establishes positive recurrence or geometric ergodicity via a Lyapunov function V with PV ≤ V − ε outside a finite set; the drift condition PV ≤ λV + b·1_C with
s_foster_lyapunov_geometric_ergodicity	theorem	Foster-Lyapunov criterion for geometric ergodicity		An irreducible aperiodic chain is geometrically ergodic if PV ≤ λV + b·1_C for V ≥ 1, λ < 1, b < ∞, and finite C; the convergence rate depends on λ.
t_taboo_probability_technique	technique	Taboo probability technique		Analyzes the chain via transition probabilities conditioned on avoiding a set, yielding restricted Green functions and decomposition formulas for passage times 
s_harris_recurrence	axiom	Harris recurrence		An irreducible chain on a general state space is Harris recurrent if every set A with ψ(A) > 0 is visited infinitely often a.s., where ψ is the irreducibility m
t_doob_h_transform	technique	Doob's h-transform		Defines the transformed kernel P^h(x,y) = h(y)p(x,y)/h(x), yielding a new Markov chain; used to condition on events of probability zero such as never hitting a 
s_martin_boundary	state	Martin boundary		A compactification of the state space via the Martin kernel K(x,y) = G(x,y)/G(x₀,y); boundary points represent extremal harmonic functions and classify all posi
s_martin_representation_theorem	theorem	Martin representation theorem		Every positive harmonic function h for a transient chain has a unique representation h(x) = ∫_{∂_m} K(x,ξ) dν_h(ξ) over the minimal Martin boundary; the probabi
s_convergence_parameter	state	Convergence parameter		R = sup{r ≥ 0 : Σ_n r^n p^{(n)}(x,y) < ∞}; the radius of convergence of the Green function series, independent of x,y for irreducible chains, classifying the ch
s_R_recurrence_transience	axiom	R-recurrence and R-transience		An irreducible chain is R-recurrent if Σ_n R^n p^{(n)}(x,y) = ∞ and R-transient if finite, where R is the convergence parameter; generalizes ordinary recurrence
s_solidarity_theorem_general	theorem	Solidarity theorem (general form)		For an irreducible chain, the convergence parameter R and the R-recurrence/R-transience classification are the same for all pairs (x,y); a unifying solidarity r
s_birth_death_chain_discrete	axiom	Birth-death chain (discrete time)		A discrete-time Markov chain on ℕ with transitions only to nearest neighbors: p(n,n+1) = p_n, p(n,n−1) = q_n; the simplest non-trivial class admitting explicit 
t_subadditive_argument_period	technique	Subadditive argument for period		Uses closure under addition of {n : p^{(n)}(x,x) > 0} and supermultiplicativity of transition probabilities to establish periodicity results and solidarity of p
s_wiener_test_discrete	theorem	Wiener test (discrete potential theory)		A point z is regular for the discrete Dirichlet problem iff Σ_n Cap(A_n ∩ D^c)/Cap(A_n) = ∞ for annuli A_n around z; the discrete analogue of Wiener's regularit
s_markov_kernel	axiom	Markov kernel (transition kernel on measurable space)		A function P: X × B(X) → [0,1] such that P(x, ·) is a probability measure for each x and P(·, A) is measurable for each A; the general-state-space analogue of a
s_phi_irreducibility	axiom	φ-irreducibility (general state space)		A Markov chain with kernel P on measurable space (X, B(X)) is φ-irreducible if there exists a non-trivial σ-finite measure φ such that for every x ∈ X and every
s_maximal_irreducibility_measure	theorem	Existence and uniqueness of maximal irreducibility measure		For a φ-irreducible chain, there exists a unique (up to equivalence) maximal irreducibility measure ψ such that every other irreducibility measure is absolutely
s_small_set	axiom	Small set (Meyn-Tweedie)		A set C ∈ B(X) is (ν_m, m)-small for kernel P if P^m(x, A) ≥ ν_m(A) for all x ∈ C and A ∈ B(X), where ν_m is a non-trivial measure; the existence of small sets 
s_petite_set	axiom	Petite set		A set C is petite if there exists a probability distribution a on ℤ₊ and a non-trivial measure ν_a such that the sampled kernel K_a(x, ·) = Σ_n a(n) P^n(x, ·) ≥
s_sampled_chain	axiom	Sampled chain (sampling distribution)		Given kernel P and a probability distribution a on ℤ₊, the sampled kernel K_a = Σ_n a(n) P^n averages transition probabilities over the sampling distribution; u
s_atom_general_state_space	axiom	Atom (general state space Markov chain)		A set α ∈ B⁺(X) is an atom for the kernel P if there exists a non-trivial measure ν such that P(x, ·) = ν(·) for all x ∈ α; the chain regenerates at each visit 
s_minorization_condition	axiom	Minorization condition		There exist a set C, an integer m ≥ 1, a constant δ > 0, and a probability measure ν such that P^m(x, B) ≥ δ ν(B) for all x ∈ C and B ∈ B(X); the fundamental co
s_aperiodicity_general_state_space	axiom	Aperiodicity (φ-irreducible chain)		A φ-irreducible chain is aperiodic if the gcd of {n ≥ 1 : C is (ν_n, n)-small for some ν_n} equals 1 for some small set C; equivalently, the chain has period d 
s_t_chain	axiom	T-chain		A Markov chain whose sampled kernel K_a for some sampling distribution a has a non-trivial continuous component: K_a(x, ·) ≥ T(x, ·) where T(x, ·) is a non-triv
s_e_chain	axiom	e-chain (equicontinuous chain)		A Markov chain on a topological state space such that the family of transition kernels {P^n(x, ·) : n ≥ 0} is equicontinuous at every x in the weak topology; e-
s_feller_property_markov_chain	axiom	Feller property (Markov chain on topological space)		A Markov chain has the (weak) Feller property if Pf is continuous whenever f is bounded and continuous; equivalently, P(x, ·) is weakly continuous in x.
s_strong_feller_property	axiom	Strong Feller property		A Markov kernel P has the strong Feller property if Pf is continuous for every bounded measurable f; implies that the chain has a transition density with respec
s_skeleton_chain	state	Skeleton chain		The discrete-time chain (X_{nδ})_{n≥0} obtained by sampling a continuous-time process at intervals of length δ; used to transfer discrete-time results (φ-irredu
s_occupation_measure_mc	state	Occupation measure (Markov chain)		The random measure η_n = (1/n) Σ_{k=0}^{n-1} δ_{X_k} recording the empirical fraction of time spent in each set; its almost-sure convergence to the invariant pr
s_recurrence_transience_dichotomy_general	theorem	Recurrence-transience dichotomy (general state space)		A φ-irreducible chain is either Harris recurrent or transient; there is no intermediate behavior, paralleling the classical result for countable chains.
s_positive_harris_recurrence	axiom	Positive Harris recurrence		A Harris recurrent chain is positive Harris recurrent if it admits a finite invariant measure (equivalently, an invariant probability measure π); this is the co
s_kac_theorem_general	theorem	Kac's theorem (general state space)		For a positive Harris recurrent chain with invariant probability π and a set A with π(A) > 0, the expected return time satisfies E_π[τ_A | X_0 ∈ A] = 1/π(A).
s_drift_criterion_recurrence	theorem	Foster's drift criterion for recurrence (general state space)		A φ-irreducible chain is Harris recurrent if there exists a petite set C and a function V ≥ 0 with V(x) → ∞ as x leaves compact sets such that PV(x) ≤ V(x) for 
s_drift_criterion_transience	theorem	Drift criterion for transience		A φ-irreducible chain is transient if there exists a non-negative function V, a set A with ψ(A) > 0, and ε > 0 such that PV(x) ≥ V(x) + ε for all x ∉ A and V is
s_drift_criterion_positive_recurrence	theorem	Foster's drift criterion for positive recurrence (general state space)		A φ-irreducible aperiodic chain is positive Harris recurrent if there exists a petite set C, a function V ≥ 1, and ε > 0 such that PV(x) ≤ V(x) − ε for x ∉ C an
s_ergodicity_general_state_space	theorem	Ergodicity (Meyn-Tweedie sense)		A positive Harris recurrent, aperiodic chain is ergodic: ‖P^n(x, ·) − π(·)‖_{TV} → 0 as n → ∞ for all x.
s_uniform_ergodicity	state	Uniform ergodicity		A chain is uniformly ergodic if sup_x ‖P^n(x, ·) − π(·)‖_{TV} → 0 as n → ∞ at geometric rate; equivalently, the entire state space is a small set, or Doeblin's 
s_geometric_ergodicity_general	theorem	Geometric ergodicity (general state space, V-norm)		A φ-irreducible aperiodic chain is V-geometrically ergodic if there exist r > 1 and V ≥ 1 such that ‖P^n(x, ·) − π(·)‖_V ≤ V(x)r^{−n}; characterized by the drif
s_v_norm_ergodicity	axiom	V-norm (f-norm) for measures		For a measurable function V ≥ 1 and a signed measure μ, the V-norm is ‖μ‖_V = sup{|∫ f dμ| : |f| ≤ V}; the natural norm for measuring convergence rates in ergod
s_subgeometric_ergodicity	state	Subgeometric ergodicity		A chain is subgeometrically ergodic if ‖P^n(x, ·) − π(·)‖_V ≤ V(x) r(n) where r(n) → 0 slower than any geometric rate; arises from drift conditions PV ≤ V − φ(V
s_polynomial_ergodicity	state	Polynomial ergodicity		A chain is polynomially ergodic of order β > 0 if ‖P^n(x, ·) − π(·)‖_V = O(n^{−β}) as n → ∞; corresponds to drift conditions with φ(v) = cv^{1−1/α} and arises i
s_drift_criterion_geometric_ergodicity_general	theorem	Drift criterion for geometric ergodicity (general state space)		A φ-irreducible aperiodic chain is geometrically ergodic in the V-norm if and only if there exist V ≥ 1, λ ∈ (0,1), b < ∞, and a petite set C such that PV(x) ≤ 
s_drift_criterion_subgeometric	theorem	Drift criterion for subgeometric ergodicity		If PV ≤ V − φ(V) + b·1_C for a petite set C, V ≥ 1, and a concave non-decreasing function φ with φ(v)/v → 0, then the chain is subgeometrically ergodic at rate 
s_lln_harris_chain	theorem	Law of large numbers for Harris recurrent chains		For a positive Harris recurrent chain with invariant probability π and f ∈ L¹(π), (1/n) Σ_{k=0}^{n-1} f(X_k) → π(f) almost surely for every initial condition.
s_clt_harris_chain	theorem	Central limit theorem for Harris recurrent chains		For a geometrically ergodic chain with invariant probability π and f with π(f²) < ∞, n^{1/2}((1/n) Σ f(X_k) − π(f)) converges in distribution to N(0, σ²_f).
s_lil_harris_chain	theorem	Law of the iterated logarithm for Harris chains		For a geometrically ergodic Harris chain with π(|f|^{2+δ}) < ∞, limsup |Σ(f(X_k) − π(f))| / √(2nσ²_f log log n) = 1 almost surely, provided σ²_f > 0.
s_asymptotic_variance_mc	state	Asymptotic variance (Markov chain CLT)		The quantity σ²_f = lim n · Var_π((1/n) Σ f(X_k)) = π((f − π(f))²) + 2 Σ_{k≥1} π((f − π(f)) P^k(f − π(f))); determines the CLT scaling.
s_kendall_theorem	theorem	Kendall's theorem (uniform ergodicity characterization)		An irreducible aperiodic chain is uniformly ergodic if and only if the whole state space is petite (or small), equivalently if a Doeblin-type minorization condi
s_harris_ergodic_theorem	theorem	Harris' ergodic theorem		If a φ-irreducible, aperiodic chain is positive Harris recurrent with invariant probability π, then ‖P^n(x, ·) − π‖_{TV} → 0 as n → ∞ for every x.
s_comparison_theorem_drift	theorem	Comparison theorem for drift functions		If two kernels P₁, P₂ share the same small set structure and P₁V ≤ P₂V pointwise for a Lyapunov function V, then ergodicity properties established for P₂ transf
s_multiplicative_mean_drift_criterion	theorem	Multiplicative mean drift criterion (geometric case)		For a φ-irreducible chain, if E[V(X_{n+1})/V(X_n) | X_n = x] ≤ λ < 1 for x outside a petite set C with V bounded on C, then the chain is geometrically ergodic.
s_doeblin_minorization_equivalence	theorem	Doeblin-minorization equivalence for uniform ergodicity		A φ-irreducible aperiodic chain is uniformly ergodic if and only if it satisfies a uniform minorization P^m(x, ·) ≥ δν(·) for all x (Doeblin's condition); equiv
s_moment_bound_ergodic_chain	theorem	Moment bounds from drift conditions		If PV ≤ λV + b·1_C for a geometrically ergodic chain, then π(V) < ∞ and sup_n E_x[V(X_n)] ≤ λ^n V(x) + b/(1−λ); drift conditions yield explicit moment bounds.
s_stochastic_monotonicity_mc	axiom	Stochastic monotonicity (Markov chain)		A Markov chain on an ordered state space is stochastically monotone if x ≤ y implies P(x, ·) ≤_{st} P(y, ·) in the stochastic order; enables construction of mon
s_iterated_function_system	axiom	Iterated function system (IFS) as Markov chain		A Markov chain X_{n+1} = f_{θ_{n+1}}(X_n) where (θ_n) are i.i.d.; the transition kernel P(x, A) = ∫ 1_A(f_θ(x)) μ(dθ) arises in random dynamical systems, MCMC, 
s_state_space_collapse_petite	theorem	Compact sets are petite for T-chains		If the chain is a T-chain, then every compact subset of the state space is petite; bridges the topological and measure-theoretic structures in the Meyn-Tweedie 
s_v_uniform_ergodicity	state	V-uniform ergodicity		A chain is V-uniformly ergodic if sup_{x: V(x)<∞} ‖P^n(x, ·) − π(·)‖_V / V(x) → 0 as n → ∞; a refinement of geometric ergodicity where convergence is uniform ov
s_irreducibility_plus_strong_feller_implies_harris	theorem	Irreducibility plus strong Feller implies Harris recurrence		If a Markov chain on a locally compact space is φ-irreducible, has the strong Feller property, and admits a reachable compact set, then it is a T-chain; combine
s_ergodic_decomposition_harris	theorem	Ergodic decomposition for general Markov chains		A φ-irreducible chain that is not Harris recurrent decomposes into a maximal Harris recurrent absorbing set H and a transient complement; the absorbing set carr
s_number_of_visits_general	state	Expected number of visits (general state space)		For a transient chain, E_x[Σ_{n=0}^∞ 1_A(X_n)] = G(x, A) defines the Green kernel; for recurrent chains, G(x, A) = ∞ for all A with ψ(A) > 0.
s_nonlinear_state_space_model	axiom	Nonlinear state space model as Markov chain		The model X_{n+1} = f(X_n, W_{n+1}) with observations Y_n = g(X_n, V_n), where (W_n), (V_n) are i.i.d. noise sequences; the hidden state (X_n) is a Markov chain
s_stability_criterion_nonlinear_sde	theorem	Stability criterion for Markov chains from SDEs		For X_{n+1} = f(X_n, W_{n+1}) where f satisfies E[V(f(x, W))] ≤ λV(x) + b, the chain is geometrically ergodic; applies the Meyn-Tweedie drift framework to discr
s_lyapunov_criterion_continuous_time	theorem	Lyapunov criterion for continuous-time chains (generator form)		For a non-explosive continuous-time Markov process with generator L, if LV(x) ≤ −cV(x) + b·1_C(x) for a petite set C and V ≥ 1, then the process is positive Har
s_basic_adjoint_relationship	theorem	Basic adjoint relationship (Revuz measure)		For a Harris recurrent chain with invariant measure π and a set A, E_A[Σ_{k=0}^{τ_A-1} 1_B(X_k)] = π(B)/π(A) for all B; establishes the Revuz correspondence bet
s_resolvent_characterization_general	theorem	Resolvent characterization of recurrence (general state space)		A φ-irreducible chain is recurrent if and only if R_λ(x, A) → ∞ as λ ↑ 1 for all A with ψ(A) > 0; the potential kernel U = Σ_n P^n is infinite on ψ-positive set
s_two_series_theorem_chain	theorem	Convergence parameter characterization of R-transience		For R-transient chains, the R-resolvent Σ_n R^n P^n(x, A) < ∞ characterizes transient behavior; R-recurrence is equivalent to existence of an R-invariant measur
s_mcmc_ergodicity	theorem	MCMC ergodicity via Meyn-Tweedie theory		A Metropolis-Hastings chain targeting π is Harris ergodic if the proposal is φ-irreducible and aperiodic and a suitable Lyapunov drift condition holds; the stan
s_f_modulated_drift	theorem	f-modulated drift criterion		If PV ≤ V − f + b·1_C for V ≥ 1, f ≥ 1, and a petite set C, then the chain is positive Harris recurrent with π(f) < ∞; simultaneously establishes recurrence and
s_total_variation_convergence_bound	theorem	Explicit total variation convergence bound via drift and minorization		Given a drift condition PV ≤ λV + b·1_C and minorization P^m(x, ·) ≥ δν(·) on C, ‖P^n(x, ·) − π‖_{TV} ≤ (1 − δ)^{⌊n/m⌋} V(x) + residual; provides computable non
s_splitting_construction_continuous_time	state	Splitting construction for continuous-time chains		Extends Nummelin splitting to continuous-time Markov processes by applying the splitting to the skeleton chain at a suitably chosen sampling rate δ > 0.
s_accessible_set_general	axiom	Accessible set (general state space)		A set A ∈ B(X) is accessible from x if P^n(x, A) > 0 for some n ≥ 1; the chain is φ-irreducible if every ψ-positive set is universally accessible.
s_absorbing_set_general	axiom	Absorbing set (general state space)		A set H ∈ B(X) is absorbing if P(x, H) = 1 for all x ∈ H; the maximal Harris component is always absorbing, and the chain restricted to an absorbing set inherit
s_taboo_kernel_general	state	Taboo kernel (general state space)		The kernel _A P^n(x, B) = P_x(X_n ∈ B, X_k ∉ A for 1 ≤ k ≤ n) giving transition probabilities conditional on avoiding set A for n steps; used in passage time an
t_nummelin_splitting	technique	Nummelin splitting technique		Augments the state space by introducing a binary auxiliary variable at visits to a small set C, using the minorization P^m(x, ·) ≥ ν_m(·) to construct an artifi
t_regeneration_technique	technique	Regeneration technique (for general state space chains)		Decomposes the sample path into independent identically distributed excursion blocks between successive regeneration times at an atom; reduces limit theorems to
t_coupling_general_state_space	technique	Coupling construction for general state space chains		At visits to the small set C, attempts to couple two copies using the minorization measure; succeeds with probability δ at each attempt, giving geometric tail b
t_monotone_coupling	technique	Monotone coupling (stochastically ordered chains)		Constructs a coupling (X_n, Y_n) of two copies of a stochastically monotone chain such that if X_0 ≤ Y_0 then X_n ≤ Y_n for all n a.s.; yields bounds on converg
s_conditional_expectation_as_projection	state	Conditional expectation as L² projection		E[X | G] is the orthogonal projection of X onto L²(G) in the Hilbert space L²(Ω, F, P); it minimizes E[(X − Y)²] over all G-measurable Y.
s_stochastic_process_cinlar	axiom	Stochastic process		A family of random variables {X_t : t ∈ T} defined on a probability space (Ω, F, P) taking values in a measurable state space (E, ℰ), indexed by a totally order
s_natural_filtration	state	Natural filtration		The filtration F_t^X = σ(X_s : s ≤ t) generated by the process X, the smallest filtration to which X is adapted.
s_predictable_process	axiom	Predictable process		A process H measurable with respect to the predictable σ-algebra P generated by all left-continuous adapted processes; in discrete time, H_n is F_{n-1}-measurab
s_predictable_sigma_algebra	axiom	Predictable σ-algebra		The σ-algebra P on Ω × [0,∞) generated by all left-continuous adapted processes, or equivalently by sets of the form (s,t] × A with A ∈ F_s and {0} × A with A ∈
s_optional_sigma_algebra	axiom	Optional σ-algebra		The σ-algebra O on Ω × [0,∞) generated by all right-continuous adapted processes (equivalently, all càdlàg adapted processes); contains the predictable σ-algebr
s_stopped_sigma_algebra	state	Stopped σ-algebra		For a stopping time τ, F_τ = {A ∈ F : A ∩ {τ ≤ t} ∈ F_t for all t ≥ 0}, the σ-algebra of events determined by time τ.
s_stopped_process	state	Stopped process		The process X^τ_t = X_{t ∧ τ} obtained by freezing X at the stopping time τ; if X is a martingale (submartingale, supermartingale) then so is X^τ.
s_submartingale	axiom	Submartingale		An adapted integrable process (X_t) satisfying E[X_t | F_s] ≥ X_s for all s ≤ t; equivalently, a process whose conditional expected future value is at least as 
s_right_continuous_martingale	axiom	Right-continuous martingale		A continuous-time martingale with right-continuous sample paths (and left limits, i.e., càdlàg); the standard regularity assumption for continuous-time martinga
s_usual_conditions	axiom	Usual conditions (conditions habituelles)		A filtration (F_t) satisfies the usual conditions if it is right-continuous (F_t = F_{t+} = ∩_{s>t} F_s) and complete (F_0 contains all P-null sets). Required f
s_cadlag_modification	theorem	Existence of càdlàg modification		A right-continuous submartingale (or supermartingale) with respect to a filtration satisfying the usual conditions admits a càdlàg modification; in particular, 
s_class_D_process	axiom	Class (D) process		A process X is of class (D) if the family {X_τ : τ a finite stopping time} is uniformly integrable; the strongest uniform integrability condition on a process, 
s_class_DL_process	axiom	Class (DL) process		A process X is of class (DL) if for each t > 0, the family {X_τ : τ a stopping time with τ ≤ t} is uniformly integrable; weaker than class (D), sufficient for t
s_predictable_quadratic_variation	state	Predictable quadratic variation (angle bracket process)		The unique predictable increasing process ⟨M⟩ such that M² − ⟨M⟩ is a local martingale; for continuous local martingales, ⟨M⟩ = [M]. Determines the law of the m
s_semimartingale	axiom	Semimartingale		A càdlàg adapted process X that decomposes as X = M + A where M is a local martingale and A is an adapted process of locally bounded variation; the most general
s_markov_process_cinlar	axiom	Markov process (general state space)		A stochastic process (X_t)_{t≥0} on a measurable state space (E, ℰ) with transition semigroup (P_t) satisfying E_x[f(X_{t+s}) | F_t] = P_s f(X_t) a.s. for all b
s_feller_process	axiom	Feller process		A strong Markov process whose transition semigroup (P_t) maps C_0(E) into C_0(E) and is strongly continuous; equivalently, a process whose semigroup is a Feller
s_resolvent_operator	state	Resolvent operator (of a semigroup/generator)		R_λ = ∫_0^∞ e^{-λt} T_t dt = (λI − A)^{-1} where A is the infinitesimal generator; the family {R_λ}_{λ>0} satisfies the resolvent equation R_λ − R_μ = (μ − λ)R_
s_resolvent_equation	theorem	Resolvent equation		The resolvent operators satisfy R_λ − R_μ = (μ − λ)R_λ R_μ for all λ, μ > 0; a fundamental algebraic identity characterizing resolvent families and equivalent t
s_positive_maximum_principle	axiom	Positive maximum principle		A linear operator A on C_0(E) satisfies the positive maximum principle if whenever f ∈ D(A) attains its supremum at x₀ with f(x₀) ≥ 0, then Af(x₀) ≤ 0. Characte
s_martingale_problem	axiom	Martingale problem (Stroock–Varadhan)		A probability measure P on path space solves the martingale problem for (A, μ) if the initial distribution is μ and f(X_t) − ∫_0^t Af(X_s) ds is a P-martingale 
s_poisson_process_as_markov	state	Poisson process as a Markov/Lévy process		The Poisson process N_t with rate λ is simultaneously a Markov process (with generator Af(n) = λ[f(n+1) − f(n)]), a Lévy process (stationary independent increme
s_compound_poisson_process	state	Compound Poisson process		The process X_t = Σ_{k=1}^{N_t} Y_k where N_t is Poisson(λt) and Y_k are i.i.d. with distribution F, independent of N; a Lévy process with finitely many jumps i
s_bm_construction_kolmogorov	state	Construction of Brownian motion via Kolmogorov extension		Brownian motion is constructed by specifying consistent finite-dimensional Gaussian distributions (BM has independent N(0, t−s) increments), applying the Kolmog
s_bm_generator	state	Generator of Brownian motion		The infinitesimal generator of Brownian motion on ℝ^d is A = (1/2)Δ (half the Laplacian), with domain containing C²_c(ℝ^d); the Feller semigroup is P_t f(x) = ∫
s_bm_path_properties	state	Path properties of Brownian motion		Sample paths of Brownian motion are a.s. continuous, nowhere differentiable, have unbounded variation on every interval, and are Hölder continuous of order α < 
s_right_continuous_filtration	state	Right-continuous filtration		A filtration (F_t)_{t≥0} is right-continuous if F_t = F_{t+} := ∩_{s>t} F_s for all t ≥ 0; a key component of the usual conditions, ensuring the débuts of right
s_debut_theorem	theorem	Début theorem		If (F_t) satisfies the usual conditions and A is a progressively measurable set in Ω × [0,∞), then the début D_A(ω) = inf{t ≥ 0 : (ω, t) ∈ A} is a stopping time
s_section_theorem	theorem	Section theorem		For a measurable set H in the optional (resp. predictable) σ-algebra on Ω × [0,∞), and for every ε > 0, there exists a stopping time (resp. predictable time) T 
s_optional_projection	state	Optional projection		For a bounded measurable process Y, its optional projection °Y is the unique optional process satisfying °Y_T = E[Y_T | F_T] a.s. on {T < ∞} for every stopping 
s_predictable_projection	state	Predictable projection		For a bounded measurable process Y, its predictable projection ^pY is the unique predictable process satisfying ^pY_T = E[Y_T | F_{T-}] a.s. on {T < ∞} for ever
s_compensator_process	state	Compensator (dual predictable projection)		The unique predictable increasing process Ã such that A − Ã is a local martingale, where A is an adapted increasing process (or the increasing part of a submart
s_levy_ito_decomposition	theorem	Lévy–Itô decomposition		Every Lévy process X_t decomposes as X_t = bt + σW_t + ∫_{|x|<1} x Ñ(t,dx) + ∫_{|x|≥1} x N(t,dx) where W is Brownian motion, N is the Poisson random measure of 
s_poisson_random_measure	axiom	Poisson random measure		A random counting measure N on [0,∞) × E such that N(A) is Poisson(ν(A)) for measurable A, and N(A₁), …, N(Aₖ) are independent for disjoint Aᵢ. The jump measure
s_markov_semigroup_on_banach	state	Markov semigroup on a Banach space		A strongly continuous semigroup (T_t) of positive contractions on a Banach lattice (typically C_0(E) or L^p) satisfying T_t 1 ≤ 1 (sub-Markov) or T_t 1 = 1 (Mar
s_generator_core	axiom	Core of a generator		A subspace D₀ of the domain D(A) of a semigroup generator A such that A is the closure of A|_{D₀}; the generator is uniquely determined by its action on any cor
s_diffusion_generator	state	Generator of a diffusion process		The infinitesimal generator of an Itô diffusion dX_t = b(X_t)dt + σ(X_t)dW_t is the second-order operator Af(x) = (1/2)Σ_{ij} a_{ij}(x) ∂²f/∂x_i∂x_j + Σ_i b_i(x
s_ito_diffusion	state	Itô diffusion		The solution X_t of the SDE dX_t = b(X_t)dt + σ(X_t)dW_t; under Lipschitz conditions on b, σ, the solution exists, is unique, and is a strong Markov process wit
s_stochastic_exponential	state	Stochastic exponential (Doléans-Dade exponential)		For a semimartingale M, the unique solution Z_t = ℰ(M)_t of dZ = Z_{-} dM with Z_0 = 1; for continuous M, ℰ(M)_t = exp(M_t − (1/2)⟨M⟩_t). Used in Girsanov's the
s_novikov_condition	theorem	Novikov's condition		If M is a continuous local martingale with E[exp((1/2)⟨M⟩_∞)] < ∞, then the stochastic exponential ℰ(M) is a uniformly integrable martingale. Ensures that the G
s_multiplicative_functional	axiom	Multiplicative functional		A non-negative adapted process M_t satisfying M_0 = 1 and M_{t+s} = M_t · (M_s ∘ θ_t) a.s. for all s, t ≥ 0, where θ_t is the shift operator. Used to construct 
s_killed_process	state	Killed process		The process obtained by sending the Markov process to a cemetery state ∂ at a rate determined by a multiplicative functional or killing measure κ; the semigroup
t_coupling_technique	technique	Coupling technique		Constructs two random variables (X, Y) on a common probability space with marginals μ and ν to bound distances; the coupling inequality gives ‖μ−ν‖_{TV} ≤ P(X ≠
t_optional_stopping_technique_cinlar	technique	Optional stopping technique		Applies the optional stopping theorem or optional sampling theorem to a (super)martingale at carefully chosen stopping times to derive identities or bounds; a w
t_martingale_method_cinlar	technique	Martingale method		Embeds the problem into martingale theory by constructing a martingale from the Markov process (e.g., via the generator: f(X_t) − ∫_0^t Af(X_s) ds), then applie
t_semigroup_method	technique	Semigroup method		Studies Markov processes or evolution equations through the associated C₀-semigroup {T_t} and its generator A, using Hille–Yosida theory, resolvent analysis, an
t_localization_technique	technique	Localization via stopping times		Reduces problems involving local martingales or locally bounded processes to bounded/integrable ones by stopping at an increasing sequence τₙ ↑ ∞, proving the r
t_generator_characterization	technique	Generator characterization technique		Identifies the Markov process by characterizing its generator: verifies the positive maximum principle, density of the domain, and range condition, then applies
t_doob_meyer_technique	technique	Doob–Meyer technique		Decomposes a submartingale of class (DL) into a local martingale plus a predictable increasing process (the compensator), enabling separate analysis of the mart
t_change_of_measure_technique	technique	Change of measure (Girsanov) technique		Changes the probability measure via a positive martingale density process L_t to simplify the process dynamics; under Q the drift changes while the martingale p
t_lyapunov_function_technique	technique	Lyapunov function technique (for Markov processes)		Establishes recurrence, positive recurrence, or ergodicity of a Markov process by finding a function V ≥ 1 with AV ≤ −cV + b·1_K outside a compact set K, so tha
t_potential_theory_technique	technique	Potential-theoretic technique for Markov processes		Analyzes a Markov process through its potential kernel, excessive functions, and harmonic functions; uses the Riesz decomposition, maximum principles, and balay
s_conditional_dominated_convergence	theorem	Conditional dominated convergence theorem		If X_n → X a.s. and |X_n| ≤ Y with E[Y] < ∞, then E[X_n | G] → E[X | G] a.s. and in L¹; the dominated convergence theorem lifted to conditional expectations.
s_conditional_fatou	theorem	Conditional Fatou's lemma		If X_n ≥ 0 are integrable, then E[lim inf X_n | G] ≤ lim inf E[X_n | G] a.s.; the conditional version of Fatou's lemma.
s_martingale_Lp_convergence	theorem	Martingale Lᵖ convergence theorem		If (M_n) is a martingale bounded in Lᵖ for some p > 1, then M_n → M_∞ a.s. and in Lᵖ. The converse also holds: an Lᵖ-convergent martingale is bounded in Lᵖ.
s_dambis_dubins_schwarz	theorem	Dambis–Dubins–Schwarz theorem		Every continuous local martingale M with ⟨M⟩_∞ = ∞ a.s. can be represented as a time-changed Brownian motion: M_t = B_{⟨M⟩_t} where B is a Brownian motion with 
s_occupation_times_formula	theorem	Occupation times formula		For a continuous semimartingale X with local time L^a_t, ∫_0^t f(X_s) d⟨X⟩_s = ∫_{-∞}^{∞} f(a) L^a_t da for every non-negative Borel function f; converts time i
s_local_time	state	Local time of a continuous semimartingale		A jointly measurable process L^a_t measuring the time density of the semimartingale X at level a up to time t; defined via the occupation times formula or as L^
s_tanaka_formula	theorem	Tanaka's formula		|B_t − a| = |a| + ∫_0^t sgn(B_s − a) dB_s + L^a_t, where L^a_t is the local time of Brownian motion at level a. Shows that |B_t − a| is a submartingale and prov
s_entrance_boundary	axiom	Entrance boundary (Feller classification)		A boundary point of a one-dimensional diffusion that can be reached from the interior in finite time and from which the process can enter the interior, but cann
s_exit_boundary	axiom	Exit boundary (Feller classification)		A boundary point of a one-dimensional diffusion that can be reached from the interior in finite time but from which the process cannot re-enter; characterized b
s_speed_measure	state	Speed measure of a one-dimensional diffusion		The unique (up to scalar) σ-finite measure m on the state interval (l,r) such that the generator of the diffusion acts as Af = (1/2)(d/dm)(df/ds) where s is the
s_scale_function	state	Scale function of a one-dimensional diffusion		A continuous strictly increasing function s on the state interval (l,r) such that s(X_t) is a local martingale; equivalently, s'(x) = exp(−∫^x 2b(y)/σ²(y) dy) f
s_feller_boundary_classification	state	Feller boundary classification for one-dimensional diffusions		Classification of each boundary point of a one-dimensional diffusion into four types — regular, exit, entrance, or natural — based on convergence of integrals i
s_column_space	axiom	Column Space (Range)		The subspace of ℂ^m spanned by the columns of A, equal to the image (range) of the linear map x ↦ Ax.
s_row_space	axiom	Row Space		The subspace of ℂ^n spanned by the rows of A, equivalently the column space of the conjugate transpose A*.
s_null_space	axiom	Null Space (Kernel)		The subspace of ℂ^n consisting of all vectors mapped to zero by A.
s_matrix_rank	axiom	Rank of a Matrix		The dimension of the column space of A, equivalently the number of nonzero singular values.
s_unitary_matrix	axiom	Unitary Matrix		A square complex matrix whose columns form an orthonormal basis of ℂ^m; equivalently Q* = Q⁻¹, preserving inner products and 2-norms.
s_conjugate_transpose	axiom	Conjugate Transpose (Adjoint)		The matrix obtained by transposing A and taking complex conjugates of all entries; satisfies ⟨Ax, y⟩ = ⟨x, A*y⟩.
s_frobenius_norm	axiom	Frobenius Norm		The entrywise 2-norm of a matrix, equal to the square root of the sum of squared singular values: ‖A‖_F = (σ₁² + ⋯ + σ_r²)^{1/2}.
s_matrix_operator_norm	axiom	Matrix Operator Norm (Induced Norm)		The norm on matrices induced by vector norms on domain and codomain; the 2-norm equals the largest singular value σ₁.
s_submultiplicativity_of_matrix_norms	theorem	Submultiplicativity of Matrix Norms		For any operator norm (and the Frobenius norm), the norm of a product is at most the product of the norms.
s_singular_values	axiom	Singular Values		The nonnegative square roots of the eigenvalues of A*A, ordered decreasingly; they measure the semiaxes of the image of the unit ball under A.
s_reduced_svd	state	Reduced (Thin) SVD		The economy-size singular value decomposition retaining only the first n left singular vectors, reducing storage from O(m²) to O(mn).
s_eckart_young_theorem	theorem	Eckart–Young–Mirsky Theorem		The best rank-k approximation to A in the 2-norm or Frobenius norm is the truncated SVD A_k = ∑_{i=1}^{k} σᵢ uᵢ vᵢ*.
s_moore_penrose_pseudoinverse	axiom	Moore–Penrose Pseudoinverse		The unique matrix satisfying the four Penrose conditions (AA⁺A=A, A⁺AA⁺=A⁺, (AA⁺)*=AA⁺, (A⁺A)*=A⁺A); computable via SVD as A⁺ = VΣ⁺U*.
s_orthogonal_projector	axiom	Orthogonal Projector		An idempotent Hermitian matrix projecting onto a subspace S along S⊥; satisfies ‖P‖₂ = 1 (unless P = 0) and range(P) = S.
s_qr_factorization	state	QR Factorization		Decomposition of a matrix into a unitary factor Q and an upper triangular factor R; unique (up to signs) when A has full column rank and diagonal entries of R a
s_upper_triangular_matrix	axiom	Upper Triangular Matrix		A square matrix with all entries below the main diagonal equal to zero; its eigenvalues are the diagonal entries.
s_lower_triangular_matrix	axiom	Lower Triangular Matrix		A square matrix with all entries above the main diagonal equal to zero.
s_householder_reflector	axiom	Householder Reflector		A unitary Hermitian matrix that reflects vectors across the hyperplane orthogonal to v; used to zero out subdiagonal entries column by column.
s_least_squares_problem	axiom	Least Squares Problem		The problem of finding x minimizing the 2-norm of the residual Ax − b, arising when the system Ax = b is overdetermined (m > n).
s_normal_equations	state	Normal Equations		The n×n system whose solution gives the least squares minimizer; equivalent to projecting b onto col(A), but squaring the condition number makes direct solution
s_condition_number_of_linear_system	theorem	Conditioning of Ax = b		The relative error in the solution of Ax = b is bounded by κ(A) = ‖A‖‖A⁻¹‖ times the relative perturbation in data, so κ(A) measures sensitivity of linear syste
s_machine_epsilon	axiom	Machine Epsilon		The smallest positive floating-point number such that fl(1 + ε) > 1; equivalently half the spacing between 1 and the next representable number (ε_mach ≈ 1.11 × 
s_fundamental_axiom_of_floating_point	axiom	Fundamental Axiom of Floating-Point Arithmetic		Every elementary floating-point operation (+ − × ÷) produces a result that is the exact result rounded: the relative error of each operation is at most machine 
s_backward_stability	axiom	Backward Stability		A numerical algorithm is backward stable if for every input x, its computed output f̃(x) is the exact answer to a nearby problem f(x + δx) with ‖δx‖/‖x‖ = O(ε_m
s_backward_stability_implies_accuracy	theorem	Backward Stability Implies Forward Accuracy		If an algorithm is backward stable and the problem has condition number κ, then the relative forward error is at most O(κ · ε_mach).
s_accuracy_equals_condition_times_stability	theorem	Accuracy = Condition × Stability		The central principle of numerical analysis: the accuracy achievable by an algorithm is the product of the problem's condition number and the algorithm's backwa
s_backward_stability_of_householder_qr	theorem	Backward Stability of Householder QR		Householder triangularization is backward stable: the computed factors Q̃, R̃ are the exact QR factors of a matrix A + δA with ‖δA‖/‖A‖ = O(ε_mach).
s_lu_factorization	state	LU Factorization		Decomposition of a square matrix into a product of a lower triangular matrix L and an upper triangular matrix U; exists without pivoting when all leading princi
s_partial_pivoting	axiom	Partial Pivoting		Row interchange strategy in Gaussian elimination that selects the largest magnitude entry in the current column as pivot, ensuring |l_{ij}| ≤ 1 in L.
s_permutation_matrix	axiom	Permutation Matrix		A matrix with exactly one entry of 1 in each row and column and 0s elsewhere, representing a permutation of coordinates; orthogonal and its inverse is its trans
s_growth_factor	state	Growth Factor		The ratio of the largest entry encountered during Gaussian elimination to the largest entry of the original matrix; controls the backward error of LU factorizat
s_backward_stability_of_gepp	theorem	Backward Stability of GEPP		Gaussian elimination with partial pivoting is backward stable provided the growth factor ρ is moderate; in practice ρ = O(1) for virtually all matrices.
s_hermitian_positive_definite_matrix	axiom	Hermitian Positive Definite Matrix		A Hermitian matrix all of whose eigenvalues are strictly positive; equivalently x*Ax > 0 for every nonzero vector x.
s_cholesky_factorization	state	Cholesky Factorization		The unique decomposition of a Hermitian positive definite matrix into the product of an upper triangular matrix and its conjugate transpose; costs (1/3)n³ flops
s_hessenberg_matrix	axiom	Hessenberg Matrix		An upper Hessenberg matrix has zeros below the first subdiagonal; a natural intermediate form for eigenvalue computation since it preserves eigenvalues under un
s_tridiagonal_matrix	axiom	Tridiagonal Matrix		A matrix with nonzero entries only on the main diagonal and the two adjacent diagonals; arises from Hessenberg reduction of Hermitian matrices and enables O(n) 
s_givens_rotation	axiom	Givens Rotation		A rotation in the (i,j)-plane by angle θ that selectively zeros a single matrix entry; used in QR factorization of Hessenberg matrices and in Jacobi eigenvalue 
s_bidiagonal_matrix	axiom	Bidiagonal Matrix		A matrix with nonzero entries only on the main diagonal and the first superdiagonal; the intermediate form in SVD computation.
s_sparse_matrix	axiom	Sparse Matrix		A matrix in which most entries are zero, typically with O(n) nonzeros; enables matrix-vector products in O(n) time, making Krylov methods practical for large sy
s_preconditioner	axiom	Preconditioner		A matrix M approximating A such that M⁻¹A has clustered spectrum or small condition number, transforming the original system into one for which Krylov methods c
s_wilkinson_shift	state	Wilkinson Shift		The shift strategy for QR iteration that selects the eigenvalue of the trailing 2×2 submatrix of H closer to h_{nn}; guarantees at least quadratic convergence f
s_ritz_values	state	Ritz Values and Vectors		The eigenvalues of the projected Hessenberg matrix H_k obtained from Arnoldi iteration; they approximate eigenvalues of A, with extremal eigenvalues typically c
s_ilu_preconditioner	state	Incomplete LU (ILU) Preconditioner		An approximate LU factorization that discards fill-in entries outside a prescribed sparsity pattern, yielding a sparse triangular preconditioner applicable in O
s_incomplete_cholesky_preconditioner	state	Incomplete Cholesky Preconditioner		The symmetric positive definite analogue of ILU: an approximate Cholesky factor with fill limited to a sparsity pattern, preserving positive definiteness under 
s_domain_decomposition_preconditioner	state	Domain Decomposition Preconditioner		Decomposes the computational domain into overlapping or non-overlapping subdomains, solves local subproblems independently or with interface coupling, and assem
s_convergence_via_polynomial_approximation	theorem	Convergence of Krylov Methods via Polynomial Approximation		The error of a Krylov method after k steps is bounded by the best polynomial approximation to zero on the spectrum, connecting iterative solver convergence to a
s_gmres_convergence	theorem	GMRES Convergence Bound		GMRES convergence is governed by how well polynomials of degree k bounded by 1 at the origin can be made small on the spectrum (or field of values) of A.
s_lanczos_gauss_quadrature_connection	theorem	Lanczos–Gauss Quadrature Connection		The k-step Lanczos approximation to bᵀf(A)b equals a k-point Gauss quadrature rule for the Riemann–Stieltjes integral ∫f(λ)dμ(λ) defined by the spectral measure
s_cg_lanczos_equivalence	theorem	CG–Lanczos Equivalence		The conjugate gradient method applied to Ax = b produces iterates equivalent to the Lanczos process applied to A with starting vector b; the CG residuals are sc
s_cg_convergence_bound	theorem	CG Convergence Bound		The A-norm of the error in conjugate gradients after k iterations is bounded by 2((√κ(A) − 1)/(√κ(A) + 1))^k, where κ(A) = λ_max/λ_min is the spectral condition
s_faber_manteuffel_theorem	theorem	Faber–Manteuffel Theorem		A short-recurrence Krylov method minimizing a norm of the residual at each step can exist only for matrices that are normal (up to a shift and scaling), explain
s_bauer_fike_theorem	theorem	Bauer–Fike Theorem		For a diagonalizable matrix A = VDV⁻¹, every eigenvalue μ of A + E satisfies min_λ|μ − λ| ≤ κ(V)‖E‖, quantifying eigenvalue sensitivity via the eigenvector cond
s_cauchy_interlacing_theorem	theorem	Cauchy Interlacing Theorem		The eigenvalues of a principal submatrix of a Hermitian matrix interlace with the eigenvalues of the full matrix.
t_classical_gram_schmidt	technique	Classical Gram–Schmidt (CGS)		Orthogonalizes columns of A by subtracting projections onto all previously computed basis vectors simultaneously; numerically unstable due to cancellation.
t_modified_gram_schmidt	technique	Modified Gram–Schmidt (MGS)		Orthogonalizes columns of A by sequentially projecting out each new basis vector from all remaining columns, yielding improved numerical stability over CGS.
t_householder_triangularization	technique	Householder Triangularization		Computes the QR factorization by applying a sequence of Householder reflectors to introduce zeros below the diagonal; costs 2mn² − (2/3)n³ flops and is backward
t_least_squares_via_qr	technique	Least Squares via QR Factorization		Solves the least squares problem by computing the reduced QR factorization A = Q̂R̂ and back-substituting; backward stable and O(2mn² − (2/3)n³) flops.
t_least_squares_via_svd	technique	Least Squares via SVD		Solves the least squares problem using the pseudoinverse from the SVD; most robust method, especially for rank-deficient problems.
t_back_substitution	technique	Back Substitution		Solves an upper triangular system by computing x_n, x_{n−1}, ..., x_1 sequentially in O(n²) flops; backward stable.
t_forward_substitution	technique	Forward Substitution		Solves a lower triangular system by computing x_1, x_2, ..., x_n sequentially in O(n²) flops.
t_gaussian_elimination_with_partial_pivoting	technique	Gaussian Elimination with Partial Pivoting (GEPP)		Computes PA = LU by choosing the largest-magnitude pivot in each column, then solves via forward and back substitution; costs (2/3)n³ flops and is backward stab
t_cholesky_algorithm	technique	Cholesky Algorithm		Computes the Cholesky factor by a modified form of Gaussian elimination exploiting symmetry; no pivoting needed, always stable, and costs (1/3)n³ flops.
t_hessenberg_reduction	technique	Hessenberg Reduction		Reduces A to upper Hessenberg form via n−2 Householder reflectors applied from left and right; costs (10/3)n³ flops for general A, (4/3)n³ for Hermitian A yield
t_power_iteration	technique	Power Iteration		Iterates v^{(k+1)} = Av^{(k)}/‖Av^{(k)}‖ to converge to the eigenvector of the largest-magnitude eigenvalue; convergence rate is |λ₂/λ₁|.
t_inverse_iteration	technique	Inverse Iteration		Applies power iteration to (A − μI)⁻¹ to find the eigenvalue closest to a given shift μ; converges as |λ_nearest − μ|/|λ_next − μ| per step.
t_rayleigh_quotient_iteration	technique	Rayleigh Quotient Iteration		Inverse iteration with the shift updated to the Rayleigh quotient ρ(v^{(k)}) = (v^{(k)*}Av^{(k)})/(v^{(k)*}v^{(k)}) at each step; achieves cubic convergence for
t_qr_algorithm	technique	QR Algorithm (Unshifted)		Iterates A^{(k+1)} = R^{(k)}Q^{(k)} where A^{(k)} = Q^{(k)}R^{(k)} is a QR factorization; equivalent to simultaneous iteration with identity matrix, converging 
t_shifted_qr_algorithm	technique	Shifted QR Algorithm		QR iteration with shifts μ^{(k)} chosen to accelerate convergence; each step factors H − μI = QR then forms H_new = RQ + μI, achieving quadratic or cubic conver
t_simultaneous_iteration	technique	Simultaneous Iteration (Subspace Iteration)		Block generalization of power iteration: multiply a block of vectors by A and orthogonalize at each step to converge to the dominant invariant subspace.
t_implicit_qr_step_francis	technique	Implicit QR Step (Francis Algorithm)		Performs a QR step implicitly without forming Q and R: introduces a bulge in H via the implicit Q theorem and chases it down the subdiagonal, enabling real arit
t_deflation	technique	Deflation in Eigenvalue Computation		When a subdiagonal entry of a Hessenberg matrix becomes negligible, decouple the problem into two independent smaller eigenvalue problems.
t_jacobi_eigenvalue_algorithm	technique	Jacobi Eigenvalue Algorithm		Diagonalizes a real symmetric matrix by iteratively applying plane (Givens) rotations to zero out the largest off-diagonal entry; converges unconditionally with
t_divide_and_conquer_eigenvalue	technique	Divide-and-Conquer Eigenvalue Algorithm		Splits a tridiagonal matrix into two half-sized tridiagonals plus a rank-one correction, recurses, then merges via the secular equation; O(n² to n³) depending o
t_bisection_for_eigenvalues	technique	Bisection for Symmetric Tridiagonal Eigenvalues		Counts eigenvalues in an interval using the Sturm sequence property of LDLT factorizations of T − μI, then bisects to isolate each eigenvalue to prescribed accu
t_mrrr_algorithm	technique	MRRR Algorithm (Multiple Relatively Robust Representations)		Computes all eigenvalues and eigenvectors of a symmetric tridiagonal matrix in O(n²) time by choosing locally well-conditioned LDLT representations near each ei
t_svd_algorithm	technique	SVD Algorithm (Golub–Kahan)		Computes the SVD by first reducing A to bidiagonal form via Householder reflectors, then applying implicit shifted QR to the bidiagonal matrix to converge to Σ.
t_bidiagonalization	technique	Bidiagonalization (Golub–Kahan)		Reduces A to upper bidiagonal form by alternating Householder reflectors on the left and right; costs 4mn² − (4/3)n³ flops when m ≫ n.
t_arnoldi_iteration	technique	Arnoldi Iteration		Builds an orthonormal basis of the Krylov subspace K_k(A,b) and a (k+1)×k upper Hessenberg matrix H̃_k via modified Gram-Schmidt; foundation of GMRES and non-He
t_implicitly_restarted_arnoldi	technique	Implicitly Restarted Arnoldi (IRAM / ARPACK)		Runs Arnoldi to length k+p, then applies p implicit QR shifts to filter out unwanted Ritz values, effectively restarting with an improved Krylov basis; the algo
t_gmres	technique	GMRES (Generalized Minimal Residual)		At step k, finds the vector in the affine Krylov subspace x_0 + K_k(A, r_0) that minimizes the 2-norm of the residual; implemented via Arnoldi plus a least squa
t_lanczos_iteration	technique	Lanczos Iteration		The specialization of Arnoldi to Hermitian matrices producing a symmetric tridiagonal matrix T_k via a three-term recurrence; costs only O(n) per step but suffe
t_bicg	technique	BiCG (Biconjugate Gradient)		Extends CG to nonsymmetric systems by running coupled Lanczos-type recurrences with A and A*; generates biorthogonal residual sequences but may break down and d
t_cgs_method	technique	CGS (Conjugate Gradient Squared)		Applies the BiCG polynomial twice without requiring A*, squaring the convergence rate but also amplifying irregular convergence behavior.
t_bicgstab	technique	BiCGSTAB (Bi-Conjugate Gradient Stabilized)		Stabilized variant of BiCG due to van der Vorst that avoids the irregular convergence of CGS by combining a BiCG step with a minimization step; does not require
t_qmr	technique	QMR (Quasi-Minimal Residual)		Based on the Lanczos biorthogonalization, applies a quasi-minimization of the residual norm to obtain smooth convergence behavior; more robust against breakdown
t_preconditioned_cg	technique	Preconditioned Conjugate Gradient		Applies CG to the preconditioned system M⁻¹Ax = M⁻¹b (in implicit form preserving symmetry via M = CC*) with convergence governed by κ(M⁻¹A) instead of κ(A).
s_chebyshev_points_second_kind	axiom	Chebyshev points of the second kind	Chebyshev-Lobatto points | Chebyshev extreme points | Chebyshev grid	The set of n+1 points x_j = cos(jπ/n), j=0,...,n in [-1,1], the projections onto the real axis of equispaced points on the upper unit semicircle.
s_chebyshev_three_term_recurrence	theorem	Chebyshev three-term recurrence	Recurrence for Chebyshev polynomials	The Chebyshev polynomials of the first kind satisfy T_0=1, T_1=x, and T_{k+1}(x)=2x T_k(x) − T_{k-1}(x) for k≥1.
s_chebyshev_orthogonality	theorem	Chebyshev orthogonality	Orthogonality of Chebyshev polynomials	The Chebyshev polynomials of the first kind are orthogonal on [-1,1] under the weight (1−x²)^{-1/2}, with ⟨T_j,T_k⟩=0 for j≠k and ⟨T_k,T_k⟩=π (k=0) or π/2 (k≥1)
s_chebyshev_series	state	Chebyshev series	Chebyshev expansion (series object)	The expansion f(x)=Σ a_k T_k(x) of a function on [-1,1] in Chebyshev polynomials of the first kind, with coefficients a_k=(2/π)∫_{-1}^1 f(x)T_k(x)(1−x²)^{-1/2}d
s_equivalence_chebyshev_fourier_cosine	theorem	Equivalence of Chebyshev and Fourier cosine series	Chebyshev-Fourier correspondence | cos-substitution equivalence	Under x=cos θ the Chebyshev series of f on [-1,1] coincides with the Fourier cosine series of the even 2π-periodic function f(cos θ), making the two theories in
s_chebyshev_projection	state	Chebyshev projection (truncated Chebyshev series)	Truncated Chebyshev series | Chebyshev L2 projection	The degree-n polynomial f_n=Σ_{k=0}^n a_k T_k obtained by truncating the Chebyshev series, equal to the orthogonal projection of f onto P_n in the Chebyshev-wei
s_chebyshev_aliasing	theorem	Aliasing of Chebyshev coefficients	Chebyshev aliasing | Aliasing relation interpolant vs projection | Trigonometric aliasing on Chebyshev grid	On the n+1 Chebyshev points T_k is indistinguishable from T_m whenever k≡±m (mod 2n), so the interpolation coefficient equals the sum of the true Chebyshev coef
t_barycentric_interpolation_formula	technique	Barycentric interpolation formula	Second barycentric formula | Barycentric Lagrange interpolation	An algorithm evaluating the degree-n polynomial interpolant as p(x)=(Σ_j w_j f_j/(x−x_j))/(Σ_j w_j/(x−x_j)) using barycentric weights w_j=1/∏_{k≠j}(x_j−x_k).
s_chebyshev_barycentric_weights	state	Barycentric weights for Chebyshev points	Chebyshev barycentric weights	The closed-form barycentric weights w_j=(−1)^j δ_j (δ_j=1/2 at the two endpoints, 1 otherwise) for interpolation at Chebyshev points of the second kind, enablin
s_numerical_stability_barycentric	theorem	Numerical stability of the barycentric formula	Backward stability of barycentric interpolation	The second barycentric interpolation formula with Chebyshev weights is backward stable in floating-point arithmetic, evaluating polynomial interpolants with rel
s_bernstein_ellipse	axiom	Bernstein ellipse	Bernstein regularity ellipse | ρ-ellipse	For ρ>1 the open region E_ρ in ℂ bounded by the image of the circle |z|=ρ under the Joukowski map x=(z+z^{-1})/2, an ellipse with foci ±1 and semiaxis sum ρ.
t_joukowski_map	technique	Joukowski map	Joukowsky map | Zhukovsky transform	The conformal map x=(z+z^{-1})/2 sending circles |z|=ρ to Bernstein ellipses and the unit circle to [-1,1], underlying the transfer between Laurent/Fourier and 
s_chebyshev_coeff_decay_bv	theorem	Chebyshev coefficient decay for bounded-variation functions	BV Chebyshev coefficient bound	If f has bounded variation V on [-1,1] then its Chebyshev coefficients satisfy |a_k|≤2V/(πk) for k≥1.
s_chebyshev_coeff_decay_differentiable	theorem	Chebyshev coefficient decay for differentiable functions	Algebraic Chebyshev coefficient decay	If f,...,f^{(ν-1)} are absolutely continuous and f^{(ν)} has bounded variation V on [-1,1], then |a_k|≤2V/(πk(k-1)...(k-ν))=O(k^{-ν-1}).
s_chebyshev_coeff_decay_analytic	theorem	Chebyshev coefficient decay for analytic functions	Geometric Chebyshev coefficient decay	If f is analytic in the Bernstein ellipse E_ρ with |f|≤M there, then its Chebyshev coefficients satisfy |a_k|≤2M ρ^{-k}, decaying geometrically.
s_geometric_convergence_analytic_chebyshev	theorem	Geometric convergence of Chebyshev approximation for analytic functions	Spectral/geometric convergence of Chebyshev interpolation	If f is analytic in E_ρ with |f|≤M, its degree-n Chebyshev projection and interpolant satisfy ‖f−p_n‖_∞≤2Mρ^{-n}/(ρ−1) (with an extra factor for the interpolant
s_convergence_rate_differentiable_chebyshev	theorem	Algebraic convergence of Chebyshev approximation for differentiable functions	Finite-smoothness Chebyshev convergence rate	If f has a ν-th derivative of bounded variation V on [-1,1], its degree-n Chebyshev projection and interpolant satisfy ‖f−p_n‖_∞≤C_ν V n^{-ν}, giving algebraic 
s_optimal_chebyshev_rate_rho	theorem	Estimate of optimal Chebyshev convergence rate ρ	Root test for Chebyshev convergence rate	The geometric convergence factor ρ equals the parameter of the largest Bernstein ellipse to which f extends analytically; equivalently 1/ρ=limsup|a_k|^{1/k}, re
s_lebesgue_constant_interpolation	state	Lebesgue constant (interpolation operator norm)	Lebesgue constant for interpolation	The norm Λ_n=‖L_n‖ of the polynomial interpolation operator on a given node set, equal to the maximum over the interval of the sum of absolute values of the Lag
s_lebesgue_constant_chebyshev_points	theorem	Lebesgue constant for Chebyshev points	Logarithmic growth of Chebyshev Lebesgue constant	The Lebesgue constant for polynomial interpolation in Chebyshev points grows only logarithmically, Λ_n~(2/π)log(n+1)+O(1), so Chebyshev interpolation is well-co
s_lebesgue_constant_equispaced_points	theorem	Lebesgue constant for equispaced points	Exponential growth of equispaced Lebesgue constant	The Lebesgue constant for polynomial interpolation in equally spaced points grows exponentially, Λ_n~2^n/(e n log n), quantifying the instability behind the Run
s_lebesgue_constant_bound	theorem	Lebesgue constant interpolation error bound	Near-best interpolation bound | Operator-norm inequality for interpolation	Polynomial interpolation error obeys ‖f−L_n f‖_∞≤(1+Λ_n)E_n(f), bounding the interpolant in terms of the best approximation via the interpolation Lebesgue const
s_conditioning_polynomial_interpolation	theorem	Conditioning of polynomial interpolation	Stability of polynomial interpolation	The sensitivity of a polynomial interpolant to perturbations of the data values is governed by the Lebesgue constant, so Chebyshev-point interpolation is stable
s_existence_uniqueness_best_approximation	theorem	Existence and uniqueness of best polynomial approximation	Uniqueness of minimax polynomial	For each f∈C([-1,1]) and n≥0 there exists a unique best (minimax) degree-n polynomial approximant in the supremum norm, via compactness and the Haar property of
s_de_la_vallee_poussin_alternation_lemma	theorem	de la Vallée Poussin lower bound (alternation lemma)	Vallée-Poussin alternation lemma | de la Vallée Poussin lower bound	If a polynomial p∈P_n produces an error f−p that alternates in sign at n+2 points x_i, then min_i|f(x_i)−p(x_i)| is a lower bound for the minimax error E_n(f).
s_haar_condition	axiom	Haar condition (Haar/Chebyshev system)	Chebyshev system | Haar system	A finite-dimensional space of continuous functions on an interval in which every nonzero element has at most n−1 zeros (equivalently any n points admit unique i
s_linear_approximation_scheme	axiom	Linear approximation scheme	Linear approximation	An approximation in which the approximant is drawn from a fixed finite-dimensional linear subspace (polynomials, trigonometric polynomials, fixed-knot splines),
s_nonlinear_approximation	axiom	Nonlinear approximation	Nonlinear approximation theory	An approximation drawn from a set that is not a linear subspace (rational functions of fixed type, free-knot splines, sums of exponentials), so approximants are
s_node_polynomial	state	Node polynomial	Nodal polynomial	The monic polynomial ℓ(x)=∏_{k=0}^n(x−x_k) whose roots are the interpolation nodes, governing interpolation error through the Hermite/Cauchy formula.
s_hermite_integral_formula	theorem	Hermite integral formula (Cauchy interpolation error formula)	Cauchy interpolation formula | Hermite-Cauchy error formula	The polynomial interpolation error equals (1/2πi)∮_Γ [ℓ(x)/ℓ(t)] f(t)/(t−x) dt over a contour Γ enclosing the nodes and x but inside the region of analyticity o
t_potential_theory_of_approximation	technique	Potential theory of polynomial approximation	Potential-theoretic convergence analysis	The framework identifying (1/n)log|ℓ(z)| with the logarithmic potential of the limiting node-density measure, so interpolation/approximation convergence rates a
s_green_function_complement_log_singularity	state	Green's function of the complement with logarithmic pole	Green's function with logarithmic singularity | complement Green's function	The Green's function g_E(z) of the complement of a compact set E with pole at infinity: harmonic off E, vanishing on E, with g_E(z)=log|z|−log cap(E)+o(1) at in
s_convergence_rate_from_green_function	theorem	Convergence rate from Green's function	Geometric convergence region (potential theory)	For nodes distributed according to the equilibrium measure, polynomial interpolants of f converge geometrically at rate exp(−g_E(z)) inside the largest level cu
s_equispaced_equilibrium_potential	state	Equispaced equilibrium potential	Uniform-density logarithmic potential	The logarithmic potential of the uniform (Lebesgue) density on [-1,1], whose level curves pass through ±1 (not the interval itself), explaining via potential th
s_impossibility_stable_equispaced_approximation	theorem	Impossibility of stable high-order equispaced approximation	No stable geometric equispaced reconstruction | Platte-Trefethen-Kuijlaars theorem	Any method reconstructing analytic functions from n+1 equispaced samples that converges geometrically must be exponentially ill-conditioned, and any well-condit
s_interlacing_orthogonal_polynomial_roots	theorem	Interlacing of orthogonal polynomial roots	Separation/interlacing of orthogonal polynomial zeros	All roots of any orthogonal polynomial are real, simple, and lie in the support interval, and the roots of p_n and p_{n+1} strictly interlace.
s_colleague_matrix	state	Colleague matrix	Chebyshev companion matrix	The Chebyshev-basis analogue of the companion matrix: a tridiagonal-plus-rank-one matrix built from the Chebyshev three-term recurrence whose eigenvalues are th
t_rootfinding_via_eigenvalues	technique	Rootfinding via eigenvalues (colleague/companion method)	Eigenvalue rootfinding | companion matrix rootfinding	Computing all roots of a polynomial by forming its colleague (Chebyshev) or companion (monomial) matrix and numerically computing the matrix eigenvalues.
s_max_degree_exactness_gauss_optimality	theorem	Maximal algebraic degree of exactness (Gauss optimality)	Gauss quadrature optimality	No n-point quadrature rule can be exact for all polynomials of degree 2n, and the Gauss rule (nodes at orthogonal-polynomial roots) uniquely attains exactness t
t_golub_welsch_algorithm	technique	Golub-Welsch algorithm	Golub-Welsch | Jacobi matrix quadrature	Computing Gauss quadrature nodes and weights as the eigenvalues and squared first eigenvector components of the symmetric tridiagonal Jacobi matrix formed from 
s_comparable_accuracy_cc_gauss	theorem	Comparable accuracy of Clenshaw-Curtis and Gauss quadrature	Trefethen's observation on Clenshaw-Curtis vs Gauss	For functions of finite smoothness or analyticity, n-point Clenshaw-Curtis quadrature achieves accuracy essentially as good as n-point Gauss quadrature despite 
t_caratheodory_fejer_approximation	technique	Carathéodory-Fejér (CF) approximation	CF approximation | Caratheodory-Fejer method	A near-best polynomial/rational approximation method computing an SVD of a Hankel matrix of Chebyshev coefficients and reading the near-minimax approximant from
s_cf_hankel_singular_value_identity	theorem	CF singular value / Hankel eigenvalue identity	Hankel singular value = best approximation error	For an analytic function the (n+1)st singular value of the Hankel matrix of Chebyshev coefficients essentially equals the best degree-n approximation error E_n(
s_aak_theory	theorem	AAK (Adamyan-Arov-Krein) theory	Adamyan-Arov-Krein theorem | AAK theorem	The distance in sup-norm from a bounded analytic function (Hankel symbol) to the rational functions of type (m,n) equals a singular value of the associated Hank
s_rational_best_approximation_type_mn	state	Rational best approximation of type (m,n)	Best rational minimax approximant | R_{mn} best approximation	The rational function p/q with deg p≤m, deg q≤n minimizing the sup-norm error to a continuous function on an interval, characterized by an equioscillation whose
s_rational_equioscillation_characterization	theorem	Rational equioscillation characterization theorem	Rational Chebyshev characterization	A rational function r∈R_{mn} is the best sup-norm approximation to f iff the error f−r equioscillates between extrema at least m+n+2−d times, where d is the def
s_defect_of_rational_approximant	axiom	Defect of a rational approximant	Rational approximation defect	The nonnegative integer d=min(m−deg p, n−deg q) measuring how far a reduced rational p/q falls short of fully using the available degrees in type (m,n), enterin
s_existence_uniqueness_rational_best_approximation	theorem	Existence and uniqueness of rational best approximation	Rational best approximation existence/uniqueness	On a real interval the best sup-norm rational approximation of given type to a continuous function exists and is unique, despite R_{mn} being non-convex and app
s_root_exponential_convergence_rational	theorem	Root-exponential convergence of rational approximation	exp(−C√n) rational convergence	For functions with branch-point or corner singularities (such as |x|), rational best approximations converge at the root-exponential rate exp(−C√n), strictly fa
t_pole_clustering	technique	Pole clustering near singularities	Exponential pole clustering | lightning/AAA pole placement	The construction principle that good rational approximants of functions with singularities place poles in a geometrically clustered arrangement approaching the 
s_newman_theorem_abs_x	theorem	Newman's theorem (rational approximation of |x|)	Newman rational approximation of |x|	|x| on [-1,1] can be approximated by rationals of degree n with error between (1/2)exp(−√n) and 3exp(−√n), establishing root-exponential convergence far beyond 
s_newman_construction_points	state	Newman's construction (Newman points)	Newman points | Newman rational	The explicit rational r(x)=x(p(x)−p(−x))/(p(x)+p(−x)) with p(x)=∏(x+ξ^k), ξ=exp(−1/√n), giving geometrically clustered approximation of |x|.
s_stahl_theorem_sharp_rates	theorem	Stahl's theorem (sharp |x| and exp rational rates)	Stahl asymptotics | Halphen constant rate	Stahl's sharp asymptotics: best rational approximation error to |x| on [-1,1] satisfies E_{nn}~8exp(−π√n), and best rational approximation of e^x on (−∞,0] deca
s_zolotarev_first_problem	theorem	Zolotarev's first problem (best rational approximation of sign)	Zolotarev sign approximation	The best rational approximation of given type to sign(x) on two intervals symmetric about the origin, solved explicitly by Zolotarev via Jacobi elliptic functio
s_zolotarev_problems	theorem	Zolotarev problems (elliptic extremal rational functions)	Four Zolotarev problems	A family of four classical extremal problems solved by Zolotarev giving rational functions best separating/approximating on disjoint intervals, with explicit Ja
s_chebyshev_pade_approximation	state	Chebyshev-Padé (Clenshaw-Lord) approximation	Clenshaw-Lord approximation | Chebyshev-Pade	The rational approximation analogous to Padé but matching the Chebyshev series of f on [-1,1] rather than its Taylor series, giving rationals well-suited to app
s_convergence_pade_pole_detection	theorem	Convergence of Padé approximants and pole detection	Montessus de Ballore / Stahl convergence | Padé pole detection	Padé approximants of a meromorphic function converge to it in regions avoiding its singularities, with denominator roots converging to the true poles, providing
t_analytic_continuation_via_rational_approximation	technique	Analytic continuation via rational approximation	Rational analytic continuation | Padé continuation	Using Padé or rational approximants computed from local data (Taylor/Chebyshev coefficients or samples) to extend a function analytically beyond its original do
s_froissart_doublets	axiom	Froissart doublets (spurious poles)	Spurious poles | Froissart doublets	Near-coincident pole-zero pairs of a Padé or rational approximant that arise from noise or rounding and do not correspond to true singularities, signaling overf
s_quasimatrix	axiom	Quasimatrix	Continuous-analogue matrix	A matrix-like object continuous in one dimension and discrete in the other — a finite collection of functions of a continuous variable treated as the columns of
s_chebfun	state	Chebfun (numerical function representation)	Chebfun representation	A representation of a function on an interval by an adaptively chosen Chebyshev interpolant of degree sufficient to capture it to machine precision, enabling nu
t_chebfun_adaptive_chopping	technique	Chebfun adaptive degree selection (chopping)	Chopping | standardChop | resolution test	Choosing the polynomial degree for a chebfun by sampling at successively doubled Chebyshev grids and truncating once the tail of the Chebyshev coefficients fall
s_spectral_differentiation_matrix	state	Spectral differentiation matrix	Chebyshev differentiation matrix | spectral derivative matrix	The dense matrix D mapping the vector of function values at (typically Chebyshev) grid points to the values of the interpolating polynomial's derivative at thos
s_chebyshev_differentiation_matrix_entries	theorem	Chebyshev differentiation matrix entries formula	Negative sum trick | explicit Chebyshev D matrix	The explicit entries D_{ij}=(c_i/c_j)(−1)^{i+j}/(x_i−x_j) for i≠j of the Chebyshev spectral differentiation matrix, with diagonal entries determined by the nega
s_spectral_accuracy	theorem	Spectral accuracy (spectral convergence)	Spectral convergence | exponential accuracy of spectral methods	For analytic functions the error of a spectral (Chebyshev) method decays geometrically (root-exponentially or faster) with the number of grid points, faster tha
t_spectral_collocation_method	technique	Spectral collocation method for differential equations	Chebyshev collocation | pseudospectral method	Solving boundary value problems by replacing derivatives with spectral differentiation matrices, imposing boundary conditions by deleting/modifying rows and col
s_affine_set	axiom	Affine set		A set that contains the entire line through any two of its points, i.e. is closed under affine combinations (coefficients summing to one).
s_affine_hull	axiom	Affine hull		The set of all affine combinations of points of a given set, equivalently the smallest affine set containing it.
s_convex_set	axiom	Convex set		A set that contains the line segment between any two of its points, i.e. is closed under convex combinations.
s_convex_hull	state	Convex hull		The set of all convex combinations of points of a given set, equivalently the smallest convex set containing it.
s_convex_cone	axiom	Convex cone		A set closed under all conic (nonnegative) combinations of its points, equivalently closed under nonnegative scaling and addition.
s_conic_hull	state	Conic hull		The set of all conic combinations (nonnegative-coefficient combinations) of points of a given set, the smallest convex cone containing it.
s_hyperplane	axiom	Hyperplane		The codimension-one affine set of points satisfying a single nontrivial linear equation a^T x = b.
s_halfspace	axiom	Halfspace		The convex set of points satisfying a single nontrivial linear inequality a^T x ≤ b, one of the two regions a hyperplane divides space into.
s_euclidean_ball	state	Euclidean ball		The convex set of points within Euclidean distance r of a center x_c.
s_ellipsoid_set	state	Ellipsoid (set)		The convex set obtained by an invertible linear image of a Euclidean ball, parameterized by a center x_c and positive-definite shape matrix P.
s_second_order_cone	state	Second-order (norm) cone		The convex cone of vector–scalar pairs whose vector part has Euclidean norm bounded by the scalar part, also called the ice-cream or Lorentz cone.
s_polyhedron_convex_opt	state	Polyhedron		The solution set of finitely many linear inequalities and equalities, equivalently an intersection of finitely many halfspaces and hyperplanes.
s_simplex_convex	state	Simplex		The convex hull of k+1 affinely independent points, the simplest bounded polyhedron of a given affine dimension.
s_positive_semidefinite_cone	state	Positive semidefinite cone		The convex cone of symmetric n×n positive semidefinite matrices within the space of symmetric matrices.
s_intersection_preserves_convexity	theorem	Intersection preserves convexity		The intersection of any collection of convex sets is convex.
s_affine_map_preserves_convexity	theorem	Affine image and preimage preserve convexity		The image and the inverse image of a convex set under an affine map are convex.
t_perspective_function	technique	Perspective function		The map P(x,t)=x/t on the domain t>0 whose image and inverse image preserve convexity of sets.
t_linear_fractional_function	technique	Linear-fractional function		The composition of an affine map with a perspective function, f(x)=(Ax+b)/(c^T x+d), which preserves convexity of sets on its domain.
s_separating_hyperplane_theorem	theorem	Separating hyperplane theorem		Any two disjoint convex sets can be separated by a hyperplane with one set in each closed halfspace.
s_dual_cone	state	Dual cone		The set of vectors making a nonnegative inner product with every element of a cone K, itself always a closed convex cone.
s_self_dual_cone	theorem	Self-dual cones		The nonnegative orthant, the second-order cone, and the positive semidefinite cone each equal their own dual cone.
s_proper_cone	axiom	Proper cone		A convex cone that is closed, solid (nonempty interior), and pointed (contains no line).
s_generalized_inequality	axiom	Generalized inequality		The partial order x ⪯_K y defined by y − x ∈ K for a proper cone K, generalizing componentwise and matrix inequalities.
s_minimal_element_generalized_inequality	state	Minimum and minimal elements		For a generalized inequality, a minimum element is comparable to and below all others, while a minimal element has no other element strictly below it.
s_dual_characterization_minimal_elements	theorem	Dual characterization of minimal elements		A point is minimal for ⪯_K if it minimizes λ^T x over the set for some strictly dual-feasible λ ≻_{K^*} 0 (with a converse for minimum elements).
s_convex_function	axiom	Convex function		A function whose domain is convex and which satisfies f(θx+(1−θ)y) ≤ θf(x)+(1−θ)f(y) for all θ in [0,1].
s_concave_function	axiom	Concave function		A function whose negative is convex, equivalently satisfying the reversed convexity inequality.
t_extended_value_extension	technique	Extended-value extension		Extending a convex function to all of R^n by assigning value +∞ outside its domain, simplifying handling of domains in convexity statements.
s_first_order_condition_convexity	theorem	First-order condition for convexity		A differentiable function on a convex domain is convex iff f(y) ≥ f(x) + ∇f(x)^T(y−x) for all x, y in the domain.
s_second_order_condition_convexity	theorem	Second-order condition for convexity		A twice-differentiable function on a convex domain is convex iff its Hessian is positive semidefinite (∇²f ⪰ 0) throughout the domain.
s_epigraph	state	Epigraph		The set of points lying on or above the graph of a function, which is convex iff the function is convex.
s_nonneg_weighted_sum_convexity	theorem	Nonnegative weighted sum and integral preserve convexity		A nonnegative weighted sum, or weighted integral, of convex functions is convex.
s_affine_composition_convexity	theorem	Composition with an affine map preserves convexity		If f is convex then g(x)=f(Ax+b) is convex.
s_pointwise_supremum_convexity	theorem	Pointwise maximum and supremum preserve convexity		The pointwise maximum of finitely many, or supremum of an arbitrary family of, convex functions is convex.
s_scalar_composition_convexity	theorem	Scalar composition rules		Conditions on the monotonicity and convexity of h and g under which the composition f(x)=h(g(x)) of scalar functions is convex.
s_vector_composition_convexity	theorem	Vector composition rules		Conditions on h and the component functions g_i under which the composition f(x)=h(g_1(x),...,g_k(x)) is convex.
s_partial_minimization_convexity	theorem	Partial minimization preserves convexity		If f(x,y) is jointly convex and C is convex, then g(x)=inf_{y in C} f(x,y) is convex in x.
t_perspective_of_function	technique	Perspective of a function		Forming g(x,t)=t f(x/t) from a function f, which is convex whenever f is convex.
s_quasiconvex_function	axiom	Quasiconvex function		A function on a convex domain all of whose sublevel sets are convex, equivalently f(θx+(1−θ)y) ≤ max{f(x),f(y)}.
s_quasiconvex_first_order_condition	theorem	First-order condition for quasiconvexity		A differentiable f on a convex domain is quasiconvex iff f(y) ≤ f(x) implies ∇f(x)^T(y−x) ≤ 0.
s_log_concave_log_convex_function	axiom	Log-concave and log-convex functions		A positive function is log-concave (log-convex) if its logarithm is a concave (convex) function.
s_k_convex_function	axiom	K-convex function		A vector-valued function convex with respect to a generalized inequality ⪯_K, satisfying f(θx+(1−θ)y) ⪯_K θf(x)+(1−θ)f(y).
s_optimization_standard_form	axiom	Standard-form optimization problem		The canonical statement of minimizing an objective f_0 subject to inequality constraints f_i ≤ 0 and equality constraints h_j = 0.
s_convex_optimization_standard_form	axiom	Convex optimization problem (standard form)		A standard-form problem with convex objective and convex inequality constraint functions and affine equality constraints.
s_optimality_condition_convex	theorem	Optimality condition for convex problems		For a convex problem with differentiable objective, a feasible x is optimal iff ∇f_0(x)^T(y−x) ≥ 0 for every feasible y.
t_equivalence_transformations	technique	Equivalence transformations		Reformulations such as variable change, constraint substitution, slack variables, or epigraph form that preserve a problem's solution set.
t_quasiconvex_bisection	technique	Quasiconvex optimization via bisection		Solving a quasiconvex problem by bisecting on the objective value and checking convex feasibility of each sublevel set.
s_linear_fractional_program	state	Linear-fractional program		A quasiconvex program minimizing a ratio of affine functions over a polyhedron, reducible to a linear program.
s_quadratic_program	state	Quadratic program (QP)		A convex problem minimizing a convex quadratic objective subject to affine inequality and equality constraints.
s_qcqp	state	Quadratically constrained quadratic program (QCQP)		A convex problem with a convex quadratic objective and convex quadratic inequality constraints.
s_second_order_cone_program	state	Second-order cone program (SOCP)		A convex problem minimizing a linear objective subject to second-order cone constraints.
s_robust_linear_program	state	Robust linear program		A linear program with constraints required to hold for all parameter values in an uncertainty set, reducing (e.g. for ellipsoidal uncertainty) to an SOCP.
s_geometric_program	state	Geometric program (GP)		An optimization problem in posynomial and monomial functions that becomes convex under a logarithmic change of variables and objective.
s_semidefinite_program	state	Semidefinite program (SDP)		A convex problem minimizing a linear objective subject to a linear matrix inequality (matrix-valued affine constraint in the PSD cone).
s_cone_program	state	Conic (cone) program		A convex problem in conic form with a linear objective and a generalized inequality constraint with respect to a proper cone K, generalizing LP, SOCP, and SDP.
s_vector_optimization	axiom	Vector (multicriterion) optimization		An optimization problem whose objective is vector-valued and minimized with respect to a generalized inequality induced by a proper cone.
s_pareto_optimal_point	state	Pareto optimal point		A feasible point of a vector optimization problem whose achieved objective value is minimal, i.e. no feasible point does at least as well in every component and
t_scalarization	technique	Scalarization		Finding Pareto optimal points by minimizing the scalar objective λ^T f_0(x) for dual-feasible weight vectors λ ≻_{K^*} 0.
s_lagrangian	state	Lagrangian		The weighted sum of the objective and constraint functions of an optimization problem, with Lagrange multipliers λ and ν on the constraints.
s_lagrange_dual_function	state	Lagrange dual function		The infimum of the Lagrangian over x, a concave function of the multipliers that lower-bounds the optimal value for λ ⪰ 0.
s_lagrange_dual_problem	state	Lagrange dual problem		The convex problem of maximizing the Lagrange dual function over nonnegative inequality multipliers, yielding the best lower bound on the primal optimal value.
s_weak_duality	theorem	Weak duality		The optimal value of the dual problem is always a lower bound on the optimal value of the primal problem, for any problem.
s_duality_gap	state	Duality gap		The nonnegative difference between the primal and dual optimal values; zero exactly when strong duality holds.
s_strong_duality	theorem	Strong duality		The condition that the dual optimal value equals the primal optimal value (zero duality gap), which holds for convex problems under a constraint qualification.
s_slater_condition	theorem	Slater's condition		A constraint qualification stating that strict feasibility of a convex problem (a point strictly satisfying the nonlinear inequalities) implies strong duality.
t_epigraph_duality_interpretation	technique	Geometric/epigraph interpretation of duality		Interpreting the dual function and duality gap via supporting hyperplanes of the set of achievable constraint–objective value pairs.
s_complementary_slackness	theorem	Complementary slackness		At a primal–dual optimal pair with zero duality gap, each inequality multiplier and its constraint cannot both be nonzero (active).
s_kkt_conditions	theorem	Karush-Kuhn-Tucker (KKT) conditions		The system of primal feasibility, dual feasibility, complementary slackness, and stationarity of the Lagrangian that is necessary, and for convex problems suffi
s_saddle_point_duality	theorem	Max-min (saddle-point) interpretation of duality		Expressing primal and dual optimal values as inf-sup and sup-inf of the Lagrangian, with strong duality corresponding to a saddle point.
s_strong_max_min_sion	theorem	Strong max-min equality (Sion's minimax theorem)		For a function convex in one argument and concave in the other over suitable convex sets, the order of inf and sup can be exchanged.
t_alternatives_via_duality	technique	Theorems of alternatives via duality		Deriving theorems of alternatives by applying Lagrange duality to a feasibility problem so that exactly one of a primal system and a dual system is solvable.
s_sensitivity_shadow_price	theorem	Sensitivity (shadow price) interpretation		Under strong duality and differentiability, optimal dual variables equal the negative partial derivatives of the optimal value with respect to constraint right-
t_global_sensitivity_analysis	technique	Perturbation and global sensitivity analysis		Using optimal dual variables to bound how the optimal value changes under tightening or loosening of the constraints.
s_duality_generalized_inequalities	theorem	Duality for generalized inequalities		The extension of Lagrange duality to problems with generalized-inequality constraints, using dual-cone multipliers λ ⪰_{K^*} 0.
s_sdp_cone_duality	theorem	SDP and cone program duality		The explicit primal–dual pair and strong-duality conditions for semidefinite and conic programs, with matrix or cone-valued dual variables.
s_norm_approximation_problem	state	Norm approximation problem		The problem of minimizing the norm of the residual Ax − b, whose form (LP, QP, etc.) depends on the chosen norm.
s_chebyshev_minimax_approximation	state	Chebyshev (minimax) approximation		Norm approximation in the ℓ∞ norm, minimizing the largest residual magnitude, expressible as a linear program.
s_penalty_function_approximation	state	Penalty function approximation		Minimizing a sum of a convex penalty function applied to each residual (e.g. quadratic, ℓ1, deadzone, or Huber penalty).
s_regularized_approximation	state	Regularized (Tikhonov) approximation		A bi-objective approximation that trades off residual norm against a regularization term penalizing the solution size; the squared-ℓ2 case is Tikhonov regulariz
s_robust_least_squares	state	Robust least squares		Least-squares approximation minimizing the worst-case residual over uncertainty in the data matrix A, reducible to an SOCP or SDP.
s_mle_convex_problem	state	Maximum likelihood estimation as a convex problem		Formulating maximum likelihood estimation as maximizing a concave log-likelihood, hence a convex optimization problem, for log-concave models.
s_map_estimation	state	MAP estimation as a convex problem		Maximum a posteriori estimation cast as maximizing a concave log-posterior (log-likelihood plus log-prior) when the posterior is log-concave.
s_optimal_experiment_design	state	Optimal experiment design		Choosing measurement proportions to minimize a scalar measure (D-, A-, or E-optimality) of the estimation error covariance, formulated as an SDP or logdet progr
s_chebyshev_center	state	Chebyshev center		The center of the largest Euclidean ball contained in a convex set, computable by linear programming for a polyhedron.
s_max_volume_inscribed_ellipsoid	state	Maximum volume inscribed ellipsoid		The largest-volume ellipsoid contained in a convex set, found by a convex logdet maximization program.
s_min_volume_enclosing_ellipsoid	state	Minimum volume enclosing (Löwner-John) ellipsoid		The smallest-volume ellipsoid containing a given set, the Löwner-John ellipsoid, found by a convex logdet program or SDP.
s_linear_discrimination	state	Linear discrimination		Finding a separating hyperplane that strictly classifies two point sets, a linear-programming feasibility problem.
s_support_vector_machine	state	Support vector machine		Robust linear discrimination maximizing the classification margin (with slack penalties for nonseparable data), a convex quadratic program.
s_convex_projection_problem	state	Projection onto a convex set		Finding the point of a convex set nearest to a given point, a convex problem (QP or SOCP depending on the norm and set).
s_floor_planning_convex	state	Floor planning (placement)		Arranging non-overlapping rectangular cells to minimize bounding-box area subject to placement constraints, formulated as a convex (geometric-programming) probl
s_strong_convexity_assumption	axiom	Strong convexity		The assumption that a function's Hessian is bounded below by a positive multiple of the identity, giving a quadratic lower bound used in convergence analysis.
t_line_search	technique	Line search (exact and backtracking)		Choosing the step length along a descent direction, either by exact minimization or by backtracking (Armijo) until sufficient decrease is achieved.
t_gradient_descent_method	technique	Gradient descent method		A descent method that at each step moves along the negative gradient direction Δx = −∇f(x) with a line search.
t_steepest_descent_method	technique	Steepest descent method		A descent method using the normalized direction of steepest decrease with respect to a chosen norm, of which gradient descent is the Euclidean-norm case.
s_newton_step_decrement	state	Newton step and Newton decrement		The Newton search direction obtained from the Hessian and gradient, together with the Newton decrement that measures proximity to optimality and serves as a sto
t_newton_method_optimization	technique	Newton's method (damped and pure)		An affine-invariant descent method using the Newton step with a damped (line-search) phase followed by quadratically convergent pure Newton steps near the optim
s_self_concordant_function	axiom	Self-concordant function		A convex function whose third derivative is bounded by its second derivative to the 3/2 power (and its multidimensional restriction along lines), the Nesterov-N
s_self_concordant_convergence	theorem	Convergence analysis for self-concordant functions		A bound on the number of Newton iterations to minimize a self-concordant function that depends only on the initial gap and the Newton tolerance, free of unknown
t_newton_system_factorization	technique	Newton system via factorization and structure		Computing the Newton step by Cholesky or other factorization of the Hessian, exploiting sparsity or block structure for efficiency.
s_kkt_system_equality	state	KKT system (Newton system for equality constraints)		The block matrix linear system whose solution gives the Newton step for an equality-constrained minimization.
t_newton_method_equality_constrained	technique	Newton's method with equality constraints		Newton's method for minimizing a function subject to affine equality constraints, generating feasible iterates by solving the KKT system at each step.
t_infeasible_start_newton	technique	Infeasible-start Newton method		A Newton method that uses the combined primal–dual residual and need not start at a feasible point, attaining feasibility and optimality together.
t_newton_step_elimination	technique	Newton step via elimination		Solving the equality-constrained Newton/KKT system by eliminating variables to reduce it to a smaller positive-definite system.
s_logarithmic_barrier_function	state	Logarithmic barrier function		A smooth convex function that grows to infinity as x approaches the boundary of the inequality-feasible set, used to approximate indicator constraints.
s_central_path	state	Central path		The curve of minimizers, parameterized by t > 0, of the objective plus a scaled logarithmic barrier, converging to an optimal point as t → ∞.
s_central_path_suboptimality_bound	theorem	Central path suboptimality bound		A central path point is no more than m/t suboptimal, where m is the number of inequality constraints, via an associated dual feasible point.
t_barrier_method_interior_point	technique	Barrier method (SUMT)		An interior-point method that solves a sequence of barrier-augmented problems along the central path with geometrically increasing t, also known as the sequenti
s_barrier_method_convergence	theorem	Barrier method convergence and complexity		A bound on the number of centering steps, and hence Newton iterations, the barrier method needs to reach a given accuracy, polynomial for self-concordant barrie
t_phase_one_feasibility	technique	Phase I (finding a strictly feasible point)		An auxiliary optimization that minimizes the maximum constraint violation to compute a strictly feasible starting point for an interior-point method.
t_primal_dual_interior_point	technique	Primal-dual interior-point method		An interior-point method that updates primal and dual variables simultaneously by solving the perturbed KKT (modified Newton) equations, without an explicit inn
s_surrogate_duality_gap	state	Surrogate duality gap		A computable gap used in primal–dual methods that equals the actual duality gap when the iterates are primal and dual feasible.
t_generalized_inequality_interior_point	technique	Generalized-inequality interior-point methods		Extensions of barrier and primal–dual interior-point methods to cone programs using generalized logarithmic (e.g. −log det) barriers for the second-order and PS
t_schur_complement	technique	Schur complement		The matrix S = C − B^T A^{-1} B obtained by eliminating a block from a partitioned matrix, used in factorization and elimination.
s_schur_complement_psd	theorem	Schur complement characterization of positive semidefiniteness		For A ≻ 0, the block matrix is positive semidefinite iff its Schur complement is, converting nonlinear matrix inequalities into linear matrix inequalities.
s_matrix_fractional_convexity	theorem	Convexity of the matrix-fractional function		The matrix-fractional function x^T Y^{-1} x is jointly convex in x and the positive-definite matrix Y.
s_log_det_concavity	theorem	Concavity of log-determinant		The function log det X is concave on the cone of symmetric positive-definite matrices.
s_entropy	axiom	Entropy H(X)		The expected value −Σ p(x) log p(x) of the self-information of a discrete random variable, measuring its average uncertainty in bits.
s_joint_entropy	axiom	Joint entropy H(X,Y)		The entropy −ΣΣ p(x,y) log p(x,y) of a pair of random variables treated as a single vector-valued variable.
s_conditional_entropy	axiom	Conditional entropy H(Y|X)		The expected entropy −ΣΣ p(x,y) log p(y|x) of Y averaged over the conditioning variable X.
s_mutual_information	axiom	Mutual information I(X;Y)		The relative entropy D(p(x,y)‖p(x)p(y)) between the joint distribution and the product of marginals, equal to H(X)−H(X|Y).
s_conditional_mutual_information	state	Conditional mutual information I(X;Y|Z)		The expected reduction in uncertainty about X from knowing Y, given Z, equal to H(X|Z)−H(X|Y,Z).
s_mi_entropy_relations	theorem	Mutual information–entropy relations		The identities I(X;Y)=H(X)−H(X|Y)=H(Y)−H(Y|X)=H(X)+H(Y)−H(X,Y) and I(X;X)=H(X).
s_chain_rule_entropy	theorem	Chain rule for entropy		The joint entropy decomposes as H(X_1,...,X_n)=Σ_i H(X_i | X_{i−1},...,X_1).
s_chain_rule_mutual_information	theorem	Chain rule for mutual information		The mutual information decomposes as I(X_1,...,X_n; Y)=Σ_i I(X_i; Y | X_{i−1},...,X_1).
s_chain_rule_relative_entropy	theorem	Chain rule for relative entropy		The joint relative entropy decomposes as D(p(x,y)‖q(x,y))=D(p(x)‖q(x))+D(p(y|x)‖q(y|x)).
s_information_inequality	theorem	Information inequality (Gibbs' inequality)		Relative entropy is nonnegative, D(p‖q) ≥ 0, with equality if and only if p=q.
s_nonnegativity_mutual_information	theorem	Nonnegativity of mutual information		I(X;Y) ≥ 0 with equality if and only if X and Y are independent.
s_conditioning_reduces_entropy	theorem	Conditioning reduces entropy		H(X|Y) ≤ H(X), with equality iff X and Y are independent; conditioning never increases entropy on average.
s_uniform_maximizes_entropy	theorem	Uniform distribution maximizes entropy		For a finite alphabet of size m, H(X) ≤ log m with equality iff X is uniformly distributed.
s_independence_bound_entropy	theorem	Independence bound on entropy		H(X_1,...,X_n) ≤ Σ_i H(X_i), with equality iff the X_i are mutually independent.
s_log_sum_inequality	theorem	Log-sum inequality		For nonnegative a_i, b_i, Σ a_i log(a_i/b_i) ≥ (Σ a_i) log((Σ a_i)/(Σ b_i)), with equality iff a_i/b_i is constant.
s_convexity_relative_entropy	theorem	Convexity of relative entropy		D(p‖q) is jointly convex in the pair (p,q).
s_concavity_of_entropy	theorem	Concavity of entropy		H(p) is a concave function of the probability distribution p.
s_mi_concave_convex	theorem	Concavity/convexity of mutual information		For fixed p(y|x), I(X;Y) is concave in p(x); for fixed p(x), I(X;Y) is convex in p(y|x).
s_data_processing_inequality	theorem	Data-processing inequality		If X→Y→Z form a Markov chain then I(X;Y) ≥ I(X;Z); no processing of Y can increase its information about X.
s_fanos_inequality	theorem	Fano's inequality		For an estimate X̂ of X from Y with error probability P_e, H(P_e)+P_e log(|X|−1) ≥ H(X|Y), lower-bounding the error probability by the conditional entropy.
s_le_cam_l1_entropy	theorem	L1 bound on entropy difference		If the L1 distance between two distributions p and q on an alphabet of size m is at most 1/2, then |H(p)−H(q)| ≤ −‖p−q‖_1 log(‖p−q‖_1/m).
s_aep	theorem	Asymptotic Equipartition Property (AEP)		For i.i.d. X_1,...,X_n ~ p, −(1/n) log p(X_1,...,X_n) → H(X) in probability.
s_typical_set	state	Typical set A_ε^{(n)}		The set of sequences whose empirical entropy rate −(1/n) log p(x^n) is within ε of H(X), i.e. whose probability is near 2^{−nH}.
s_typical_set_properties	theorem	Properties of the typical set		The typical set has probability near 1, has at most 2^{n(H+ε)} elements, at least (1−ε)2^{n(H−ε)} elements, and each member has probability near 2^{−nH}.
s_aep_source_coding	theorem	AEP-based source coding bound		i.i.d. sequences can be encoded with about nH(X) bits using roughly n(H+ε) bits per typical sequence and a negligible-probability overflow code.
s_high_probability_set	state	Smallest high-probability set B_δ^{(n)}		A minimum-cardinality set carrying probability at least 1−δ, whose size to first order in the exponent equals that of the typical set (≈ 2^{nH}).
s_entropy_rate	axiom	Entropy rate H(𝒳)		The per-symbol limit H(𝒳)=lim_{n→∞} (1/n) H(X_1,...,X_n) of the entropy of a stationary stochastic process.
s_entropy_rate_conditional_form	theorem	Equivalence of entropy-rate definitions		For a stationary process the limit H'=lim H(X_n | X_{n−1},...,X_1) exists and equals the entropy rate H(𝒳).
s_markov_entropy_rate	theorem	Entropy rate of a Markov chain		For a stationary Markov chain with stationary distribution μ and transition matrix P, the entropy rate is H(𝒳)=−Σ_{ij} μ_i P_{ij} log P_{ij}.
s_hidden_markov_entropy_rate	theorem	Entropy rate of a hidden Markov / functions of Markov chains		For a function Y_i=φ(X_i) of a stationary Markov chain, the entropy rate is squeezed between H(Y_n|Y_{n−1},...,Y_1,X_1) and H(Y_n|Y_{n−1},...,Y_1), both converg
s_second_law_thermodynamics_markov	theorem	Markov chains and the second law		For a Markov chain, relative entropy D(μ_n‖μ'_n) to any other distribution evolving under the same transition decreases with time, and H(X_n) increases toward s
s_prefix_code	axiom	Prefix (instantaneous) code		A code in which no codeword is a prefix of any other, so each symbol is decodable as soon as its codeword ends.
s_entropy_bound_codeword_length	theorem	Entropy bound on expected codeword length		The expected length L of any prefix D-ary code satisfies L ≥ H_D(X), with equality iff the lengths are l_i=−log_D p_i.
s_shannon_code	state	Shannon code lengths		The codeword-length assignment l_i=⌈log(1/p_i)⌉ which satisfies Kraft and achieves expected length within one bit of entropy.
s_shannon_code_optimality_bound	theorem	Optimal code length bound (H ≤ L < H+1)		The minimum expected length L* of a prefix code satisfies H_D(X) ≤ L* < H_D(X)+1, and block coding drives the per-symbol length to the entropy rate.
s_wrong_code_penalty	theorem	Penalty for using the wrong code		Using a code optimal for q to encode a source p costs an extra D(p‖q) bits per symbol over the entropy, i.e. expected length is between H(p)+D(p‖q) and H(p)+D(p
t_huffman_code	technique	Huffman coding		A bottom-up algorithm that builds an optimal prefix code by repeatedly merging the least probable symbols into a combined symbol.
s_huffman_optimality	theorem	Optimality of Huffman codes		The Huffman code achieves the minimum expected codeword length over all prefix (hence all uniquely decodable) codes for a given source.
t_shannon_fano_elias_code	technique	Shannon–Fano–Elias coding		A coding scheme using the rounded cumulative distribution function F̄(x) truncated to ⌈log(1/p(x))⌉+1 bits to assign prefix-free codewords.
t_arithmetic_coding	technique	Arithmetic coding		A coding technique that represents an entire sequence by an interval of the unit line determined by cumulative probabilities, encoded by a single number.
s_competitive_optimality_shannon	theorem	Competitive optimality of Shannon code		For Shannon code lengths l(x), no other uniquely decodable code l'(x) can satisfy l'(x) < l(x)−c with probability exceeding 2^{−c+1}; the Shannon code is essent
s_horse_race	axiom	Horse race / gambling model		A repeated investment model in which a gambler distributes wealth across m outcomes with win probabilities p_i and payoff odds o_i.
s_doubling_rate	state	Doubling rate W(b,p)		The expected logarithmic growth rate W(b,p)=Σ p_i log(b_i o_i) of wealth per race under bet allocation b.
s_kelly_proportional_betting	theorem	Kelly (proportional) gambling optimality		The doubling rate is maximized by proportional betting b_i=p_i, yielding optimal doubling rate W*(p)=Σ p_i log o_i − H(p).
s_conservation_theorem_gambling	theorem	Conservation theorem (gambling)		For uniform fair odds o_i=m, the sum of the doubling rate and the entropy is constant: W*(p)+H(p)=log m.
s_gambling_side_information	theorem	Value of side information in gambling		The increase ΔW in doubling rate due to side information Y equals the mutual information I(X;Y) between the horse race outcome and the side information.
s_dependent_horse_races	theorem	Doubling rate of dependent horse races		For a sequence of races, the growth-rate-optimal wealth grows as 2^{nW*} where the per-race doubling rate relates to the entropy rate via W*=Σ log o − H(𝒳) for 
s_entropy_of_english	state	Entropy of English		The estimated entropy rate of English (about 1.3 bits/letter) obtained via models and Shannon's guessing/gambling estimation games.
s_discrete_channel	axiom	Discrete memoryless channel (DMC)		A channel specified by a conditional distribution p(y|x) that acts independently on each transmitted symbol.
s_channel_capacity	state	Channel capacity C		The maximum mutual information C=max_{p(x)} I(X;Y) over input distributions, the information capacity of the channel.
s_bsc	state	Binary symmetric channel (BSC)		A binary-input binary-output channel that flips each bit independently with probability p, having capacity 1−H(p).
s_bec	state	Binary erasure channel (BEC)		A binary-input channel that erases each symbol with probability α and otherwise transmits it correctly, having capacity 1−α.
s_symmetric_channel	state	Symmetric and weakly symmetric channels		A channel whose transition-matrix rows are permutations of one another (and columns sums equal), for which capacity equals log|Y|−H(row) achieved by the uniform
s_jointly_typical_set	state	Jointly typical set A_ε^{(n)}		The set of pairs (x^n,y^n) that are individually typical and whose joint empirical entropy is near H(X,Y).
s_joint_aep	theorem	Joint AEP		Jointly typical pairs have probability near 1, number about 2^{nH(X,Y)}, and an independently drawn pair is jointly typical with probability about 2^{−nI(X;Y)}.
s_channel_code	axiom	Channel code (M, n)		An encoding of M messages into length-n channel inputs together with a decoding rule, with rate R=(log M)/n bits per channel use.
s_channel_coding_theorem	theorem	Channel coding theorem (achievability)		For any rate R < C there exist codes with maximal error probability tending to 0; all rates below capacity are achievable.
s_channel_coding_converse	theorem	Converse to the channel coding theorem		Any sequence of (2^{nR},n) codes with error probability tending to 0 must have R ≤ C; capacity is the supremum of achievable rates.
t_random_coding	technique	Random coding argument		A proof method that generates a codebook at random from p(x), bounds the expected error probability, and concludes a good code exists.
t_jointly_typical_decoding	technique	Jointly typical decoding		A decoder that declares the unique codeword jointly typical with the received sequence, erring only on atypicality or collision.
s_feedback_capacity	theorem	Feedback capacity theorem		Feedback does not increase the capacity of a discrete memoryless channel: C_FB=C.
s_source_channel_separation	theorem	Source–channel separation theorem		A source of entropy rate H can be transmitted reliably over a channel of capacity C iff H < C, and separate optimal source and channel coding is asymptotically 
s_zero_error_capacity	state	Zero-error capacity		The largest rate at which messages can be sent over a channel with exactly zero probability of error, related to the independence number of the channel's confus
s_differential_entropy	axiom	Differential entropy h(X)		The quantity h(X)=−∫ f(x) log f(x) dx, the continuous analogue of entropy for a random variable with density f.
s_differential_entropy_uniform_gaussian	state	Differential entropy of uniform and Gaussian		The uniform on [0,a] has differential entropy log a, and the N(0,σ²) Gaussian has differential entropy (1/2) log(2πeσ²).
s_aep_differential	theorem	AEP for differential entropy		For i.i.d. continuous X_i with density f, −(1/n) log f(X^n) → h(X), and the typical set has volume about 2^{nh(X)}.
s_relation_discrete_differential	theorem	Relation of differential to discrete entropy		The entropy of an n-bit quantization of a continuous X satisfies H(X^{Δ}) ≈ h(X)+n, so differential entropy is discrete entropy minus the log of the quantizatio
s_joint_conditional_differential_entropy	state	Joint and conditional differential entropy		The differential entropies h(X,Y)=−∫ f log f and h(X|Y)=h(X,Y)−h(Y) defined for jointly continuous variables.
s_relative_entropy_continuous	state	Continuous relative entropy and mutual information		The relative entropy D(f‖g)=∫ f log(f/g) and mutual information I(X;Y)=h(X)−h(X|Y) for continuous variables, both nonnegative.
s_multivariate_gaussian_entropy	theorem	Entropy of the multivariate Gaussian		A multivariate normal with covariance K has differential entropy (1/2) log((2πe)^n |K|).
s_gaussian_maximizes_differential_entropy	theorem	Gaussian maximizes differential entropy		Among all densities with a given covariance K, the multivariate Gaussian uniquely maximizes differential entropy.
s_hadamard_information_inequality	theorem	Hadamard's inequality (via entropy)		For a positive-definite covariance matrix K, |K| ≤ Π K_{ii}, derived from the independence bound on differential entropy.
s_gaussian_channel	axiom	Gaussian channel		A continuous channel adding independent Gaussian noise of power N to the input, typically under an input power constraint E[X²] ≤ P.
s_gaussian_channel_capacity	theorem	Capacity of the Gaussian channel		The power-constrained Gaussian channel has capacity C=(1/2) log(1+P/N) bits per transmission, achieved by Gaussian inputs.
t_sphere_packing_gaussian	technique	Sphere-packing argument		A geometric heuristic counting how many noise spheres of radius √(nN) pack into a signal sphere of radius √(n(P+N)), yielding 2^{nC} codewords.
s_bandlimited_channel_capacity	theorem	Capacity of the band-limited Gaussian channel		A band-limited channel of bandwidth W with power P and noise spectral density N_0/2 has capacity W log(1+P/(N_0 W)) bits per second.
s_parallel_gaussian_channels	theorem	Parallel Gaussian channels capacity		For k independent Gaussian channels under a total power constraint, capacity is Σ (1/2) log(1+P_i/N_i) maximized over a power allocation summing to P.
t_water_filling	technique	Water-filling power allocation		The optimal allocation P_i=(ν−N_i)^+ that pours power into the lowest-noise channels up to a common water level ν.
s_colored_gaussian_noise	theorem	Capacity with colored Gaussian noise		For a Gaussian channel with non-white noise of covariance K_Z, capacity is obtained by water-filling over the eigenvalues of the noise covariance under a power 
s_gaussian_feedback_capacity	theorem	Gaussian channel with feedback		For the colored Gaussian channel feedback increases capacity by at most half a bit and at most a factor of two; for the memoryless (white) Gaussian channel feed
s_rate_distortion_function	state	Rate-distortion function R(D)		The minimum mutual information R(D)=min_{p(x̂|x): E d(X,X̂) ≤ D} I(X;X̂) needed to describe a source within average distortion D.
s_distortion_measure	axiom	Distortion measure d(x,x̂)		A nonnegative function quantifying the cost of representing source symbol x by reconstruction x̂ (e.g. Hamming or squared-error).
s_rate_distortion_theorem	theorem	Rate-distortion theorem		The operational rate-distortion function (minimum rate for distortion ≤ D) equals the information rate-distortion function R(D)=min I(X;X̂).
s_bernoulli_rate_distortion	theorem	Rate-distortion of a Bernoulli source		A Bernoulli(p) source under Hamming distortion has R(D)=H(p)−H(D) for 0 ≤ D ≤ min(p,1−p) and 0 beyond.
s_gaussian_rate_distortion	theorem	Rate-distortion of a Gaussian source		A N(0,σ²) source under squared-error distortion has R(D)=(1/2) log(σ²/D) for D ≤ σ² and 0 for D ≥ σ².
s_parallel_gaussian_rate_distortion	theorem	Rate-distortion for parallel Gaussian sources (reverse water-filling)		For independent Gaussian sources under total squared-error distortion, the optimal distortion allocation D_i=min(λ,σ_i²) gives reverse water-filling, allocating
s_rate_distortion_convexity	theorem	Convexity and monotonicity of R(D)		The rate-distortion function R(D) is a nonincreasing convex function of the distortion level D.
t_blahut_arimoto	technique	Blahut–Arimoto algorithm		An alternating-minimization iteration that computes channel capacity or the rate-distortion function by treating them as a double minimization over two distribu
s_channel_capacity_double_minimization	theorem	Capacity / R(D) as double optimization		Both channel capacity and the rate-distortion function can be written as a minimization of relative entropy over two distributions, justifying the Blahut–Arimot
s_multivariate_aep_general	theorem	Strong typicality / method of types (preview)		Sequences classified by their empirical type (histogram) so that the probability of a type and the number of sequences of a type are governed by entropy and rel
s_binary_entropy_function	state	Binary entropy function H(p)		The entropy −p log p −(1−p) log(1−p) of a Bernoulli(p) variable, a concave function peaking at p=1/2.
s_type_of_sequence	axiom	Type (empirical distribution) of a sequence		The empirical probability distribution P_{x^n} of a length-n sequence, recording the relative frequency of each symbol of the alphabet.
s_number_of_types_bound	theorem	Polynomial bound on the number of types		The number of distinct types of length-n sequences over an alphabet of size |X| is at most (n+1)^{|X|}, polynomial in n.
t_method_of_types	technique	Method of types		A combinatorial proof technique that partitions sequences by their type and replaces probability calculations with counting over the polynomially many type clas
s_probability_of_sequence_by_type	theorem	Probability of a sequence under Q in terms of type		Under i.i.d. distribution Q, the probability of a sequence of type P equals 2^{-n(H(P)+D(P||Q))}, depending on the sequence only through its type.
s_type_class_size_bound	theorem	Size of a type class		The number of sequences of type P satisfies (n+1)^{-|X|} 2^{nH(P)} ≤ |T(P)| ≤ 2^{nH(P)}, so to first order |T(P)| ≈ 2^{nH(P)}.
s_type_class_probability_bound	theorem	Probability of a type class under Q		Under i.i.d. Q the probability of type class T(P) satisfies (n+1)^{-|X|}2^{-nD(P||Q)} ≤ Q^n(T(P)) ≤ 2^{-nD(P||Q)}, decaying like 2^{-nD(P||Q)}.
s_universal_source_code_via_types	state	Universal source code via types		A fixed-rate block code that encodes all sequences whose type has entropy below the rate R, achieving asymptotically optimal compression without knowing the sou
s_i_projection	state	I-projection (relative-entropy projection)		The distribution P* in a set E minimizing the relative entropy D(P||Q), the closest distribution in E to Q in the Kullback–Leibler sense.
s_conditional_limit_theorem	theorem	Conditional limit theorem		Conditioned on the empirical distribution lying in a convex set E, the marginal distribution of a single coordinate converges to the I-projection P* of Q onto E
s_pythagorean_relative_entropy	theorem	Pythagorean theorem for relative entropy		For the I-projection P* of Q onto a convex set E, every P in E satisfies D(P||Q) ≥ D(P||P*) + D(P*||Q).
t_large_deviation_theory	technique	Large deviation theory		The study of exponential decay rates of probabilities of rare events, identifying the rate as a relative-entropy (or rate-function) minimization over the event.
s_hypothesis_testing_problem	axiom	Binary hypothesis testing problem		The decision problem of choosing between two candidate distributions P_1 and P_2 for observed data, with error probabilities of type I (alpha) and type II (beta
s_likelihood_ratio_test	state	Likelihood ratio test		The optimal test deciding between two hypotheses by comparing the ratio of their likelihoods to a threshold T.
s_chernoff_stein_lemma	theorem	Chernoff–Stein lemma		For fixed type I error bounded by epsilon, the best achievable exponent of the type II error is the relative entropy D(P_1||P_2), i.e. beta_n ≈ 2^{-nD(P_1||P_2)
s_chernoff_information	state	Chernoff information		The quantity C(P_1,P_2) = -min_{0≤λ≤1} log Σ P_1^λ(x) P_2^{1-λ}(x), the best achievable exponent of the overall (Bayesian) probability of error in binary hypoth
s_bayesian_error_exponent	theorem	Bayesian probability of error exponent		The minimal Bayesian probability of error in deciding between P_1 and P_2 from n i.i.d. samples decays with exponent equal to the Chernoff information C(P_1,P_2
s_relative_entropy_fisher_information	theorem	Relative entropy and Fisher information (local approximation)		For nearby parameters, the relative entropy between f(·;θ) and f(·;θ+dθ) is to second order (1/2) J(θ) dθ^2, making Fisher information the local curvature of th
s_score_function	state	Score function		The derivative of the log-likelihood with respect to the parameter, whose expectation is zero and whose variance is the Fisher information.
s_maximum_entropy_principle	axiom	Maximum entropy principle		The principle of choosing, among all distributions satisfying given constraints, the one of maximum (differential) entropy as the least-biased model.
s_maximum_entropy_distribution	state	Maximum entropy distribution		The entropy-maximizing distribution under moment constraints, which has exponential-family form f*(x)=exp(λ_0+Σ λ_i r_i(x)) with multipliers fixing the moments.
s_max_entropy_theorem	theorem	Maximum entropy theorem		The distribution of exponential-family form matching given moment constraints uniquely maximizes differential entropy among all distributions meeting those cons
s_burg_maximum_entropy_theorem	theorem	Burg's maximum entropy theorem		Among all stationary processes with prescribed autocorrelation values R_0,...,R_p, the one of maximum entropy rate is the p-th order Gauss–Markov (autoregressiv
s_maximum_entropy_spectral_estimation	state	Maximum entropy spectral estimation		The spectral density estimate obtained by maximizing entropy rate subject to known autocorrelations, yielding an all-pole (autoregressive) spectrum.
s_anomalous_max_entropy_no_solution	theorem	Existence of maximum entropy distribution		A maximum entropy distribution of the exponential form exists matching the constraints precisely when such an exponential-form distribution is normalizable and 
s_universal_source_coding	axiom	Universal source coding		Lossless coding that asymptotically achieves the entropy rate of an unknown source drawn from a known class, without prior knowledge of the source statistics.
s_minimax_redundancy	state	Minimax redundancy		The smallest worst-case excess of expected code length over entropy, minimized over codes and maximized over sources in the class, characterizing the price of u
s_redundancy_of_code	state	Redundancy of a code		The expected per-symbol excess of a code's length over the source entropy, equal to the relative entropy between the true distribution and the coding distributi
t_lz78_coding	technique	Lempel–Ziv (LZ78) coding		A universal compression algorithm that incrementally parses the input into distinct phrases, each the shortest new string, coded as a pointer to a previous phra
t_lz77_sliding_window	technique	Lempel–Ziv (LZ77) sliding-window coding		A universal compression algorithm that encodes the next text segment as a pointer (offset and match length) into a sliding window of recently seen text.
s_tree_structured_lz_parsing	state	LZ78 phrase / dictionary tree		The dictionary of distinct phrases produced by incremental LZ78 parsing, whose count c(n) controls the codeword length.
s_optimality_of_lz78	theorem	Optimality of LZ78		For a stationary ergodic source, the compression rate of LZ78 converges almost surely to the entropy rate, making it asymptotically optimal without knowledge of
s_optimality_of_lz77	theorem	Optimality of sliding-window (LZ77) coding		Sliding-window Lempel–Ziv compression achieves the entropy rate of a stationary ergodic source as the window size grows, using recurrence-time properties of the
s_ziv_lempel_distinct_phrases_bound	theorem	Bound on number of distinct LZ phrases		The number c(n) of distinct phrases in the LZ78 parsing of a length-n string satisfies c(n) ≤ n/((1-ε_n) log n), the key combinatorial bound for proving optimal
s_kolmogorov_complexity	axiom	Kolmogorov complexity		The length of the shortest binary program that makes a universal computer output the string x and halt, the algorithmic measure of descriptive complexity.
s_conditional_kolmogorov_complexity	state	Conditional Kolmogorov complexity		The length of the shortest program that outputs x given the string y (and its length) as additional input to the universal computer.
s_universality_kolmogorov_complexity	theorem	Universality (invariance) of Kolmogorov complexity		For any other computer A there is a constant c_A such that K_U(x) ≤ K_A(x) + c_A, so complexity relative to a universal machine is machine-independent up to an 
s_kolmogorov_complexity_bounds	theorem	Upper bound on Kolmogorov complexity		For any string, K(x | ℓ(x)) ≤ ℓ(x) + c and K(x) ≤ ℓ(x) + 2 log ℓ(x) + c, so complexity is at most the length plus a small description of the length.
s_kolmogorov_complexity_and_entropy	theorem	Kolmogorov complexity and entropy		For an i.i.d. source with entropy H, the expected Kolmogorov complexity per symbol of a length-n sequence converges to H, linking algorithmic and Shannon inform
s_universal_probability	state	Universal probability		The probability that a universal computer fed random program bits outputs x, dominating every computable distribution up to a multiplicative constant.
s_universal_probability_complexity_relation	theorem	Universal probability ≈ 2^{-K(x)}		The universal probability of a string and its Kolmogorov complexity satisfy log(1/P_U(x)) = K(x) + O(1), so the most likely program is essentially the shortest 
s_incompressible_sequence	axiom	Incompressible sequence		A finite string whose Kolmogorov complexity is at least its length (no shorter description exists), of which most strings of each length are examples.
s_algorithmically_random_sequence	axiom	Algorithmically random (infinite) sequence		An infinite sequence all of whose prefixes are essentially incompressible, the algorithmic notion of randomness (Martin-Löf / Kolmogorov).
s_chaitin_omega	state	Chaitin's Omega		The halting probability of a universal prefix machine fed random bits, a well-defined real number that is algorithmically random and noncomputable.
s_undecidability_of_kolmogorov_complexity	theorem	Noncomputability of Kolmogorov complexity		There is no algorithm that computes K(x) for all strings x; the complexity function is uncomputable, related to the undecidability of the halting problem.
s_kolmogorov_sufficient_statistic	state	Kolmogorov sufficient statistic		The smallest finite-complexity description (model) S* containing x relative to which the remaining bits of x are random, separating structure from noise in a si
s_kolmogorov_structure_function	state	Kolmogorov structure function		The function giving, for each model-complexity budget k, the smallest log-cardinality of a set containing x describable in ≤ k bits, tracing the structure/rando
s_occams_razor	theorem	Occam's razor (algorithmic form)		The principle, justified by universal probability, that the shortest computable explanation consistent with the data should be preferred because it has the high
t_universal_gambling	technique	Universal gambling scheme		A betting strategy that distributes wealth across all programs proportional to 2^{-ℓ(p)}, guaranteeing capital growth tied to the Kolmogorov complexity of the r
s_network_information_theory	axiom	Network information theory		The study of information flow in networks with multiple senders and receivers, characterizing achievable rate regions rather than single capacities.
s_multiple_access_channel	axiom	Multiple-access channel (MAC)		A channel with several independent senders and one receiver, modeled by a single conditional distribution of the output given all the inputs.
s_mac_capacity_region	theorem	Multiple-access channel capacity region		The set of achievable rate pairs (R_1,R_2) for the MAC, the closure of the convex hull of all rates satisfying R_1 ≤ I(X_1;Y|X_2), R_2 ≤ I(X_2;Y|X_1), and R_1+R
s_gaussian_mac_capacity	theorem	Gaussian multiple-access channel capacity		The capacity region of the Gaussian MAC with powers P_1,P_2 and noise N, given by R_i ≤ C(P_i/N) per user and R_1+R_2 ≤ C((P_1+P_2)/N) jointly, with C(x)=(1/2)l
t_successive_cancellation_decoding	technique	Successive cancellation (onion-peeling) decoding		A decoding strategy that decodes one user's message, subtracts its effect, and decodes the next, achieving corner points of the multiple-access capacity region.
s_broadcast_channel	axiom	Broadcast channel		A channel with one sender and several receivers, each receiving a possibly different output of a single transmitted signal.
s_degraded_broadcast_channel	axiom	Degraded broadcast channel		A broadcast channel in which the receivers can be ordered so that the worse receiver's output is a degraded (further-processed) version of the better receiver's
s_degraded_broadcast_capacity_region	theorem	Capacity region of the degraded broadcast channel		The achievable rate region for a degraded broadcast channel, R_1 ≤ I(X;Y_1|U), R_2 ≤ I(U;Y_2), achieved by superposition coding over an auxiliary variable U.
t_superposition_coding	technique	Superposition coding		An encoding that superimposes a fine codeword for the strong receiver on a coarse cloud-center codeword for the weak receiver, achieving the degraded broadcast 
s_gaussian_broadcast_channel	theorem	Gaussian broadcast channel capacity		The capacity region of the degraded Gaussian broadcast channel, obtained by splitting power αP and (1−α)P between the two users with superposition coding.
s_slepian_wolf_rate_region	state	Slepian–Wolf rate region		The achievable rate region for separate lossless encoding of correlated sources, the pentagon defined by R_1 ≥ H(X|Y), R_2 ≥ H(Y|X), R_1+R_2 ≥ H(X,Y).
s_correlated_sources_over_mac	theorem	Transmission of correlated sources over a MAC		A sufficient condition for sending correlated sources over a multiple-access channel, in which the source correlation can be exploited and source–channel separa
s_relay_channel	axiom	Relay channel		A channel with a sender, a receiver, and an intermediate relay node that helps convey the message by transmitting based on its own received signal.
s_relay_channel_capacity_degraded	theorem	Capacity of the degraded relay channel		The capacity of a degraded relay channel, C = max min{I(X,X_1;Y), I(X;Y_1|X_1)}, achieved by block-Markov decode-and-forward coding.
s_interference_channel	axiom	Interference channel		A channel with two sender–receiver pairs in which each receiver wants only its own sender's message while suffering interference from the other transmission.
s_gaussian_interference_strong	theorem	Capacity of the strong-interference Gaussian interference channel		For strong interference, the Gaussian interference channel capacity region coincides with that of the corresponding compound multiple-access channel, since each
s_gelfand_pinsker_channel	theorem	Gelfand–Pinsker theorem (channel with side information)		The capacity of a channel with state known noncausally at the encoder is C = max [I(U;Y) − I(U;S)] over auxiliary U and encoding x(u,s).
s_writing_on_dirty_paper	theorem	Writing on dirty paper (Costa)		For a Gaussian channel with additive Gaussian interference known at the transmitter, the capacity equals that of the interference-free channel, so the known int
t_binning_for_side_information	technique	Gelfand–Pinsker binning		An encoding that organizes auxiliary codewords into bins per message and chooses within a bin one jointly typical with the known state, the dual of Slepian–Wolf
s_source_channel_separation_networks	theorem	Failure of source–channel separation in networks		Unlike the point-to-point case, separately optimal source coding and channel coding are not in general optimal for networks, as exhibited by correlated sources 
s_gaussian_multiuser_channels	axiom	Gaussian multiuser channels		The family of additive white Gaussian noise network models (MAC, broadcast, relay, interference) used to obtain explicit multiuser capacity regions.
s_stock_market_model	axiom	Stock market (price relative vector)		A model of m stocks given by a random vector X of price relatives, where X_i is the ratio of end-of-day to start-of-day price of stock i.
s_portfolio	axiom	Portfolio		An allocation of wealth across stocks given by a probability vector b, where b_i is the fraction invested in stock i.
s_doubling_rate_portfolio	state	Doubling rate		The expected logarithm of the wealth relative b^T X of a portfolio, the exponential growth rate of wealth under repeated reinvestment.
s_log_optimal_portfolio	state	Log-optimal portfolio		The portfolio b* maximizing the expected log wealth (doubling rate), which maximizes the long-run growth rate of capital.
s_kuhn_tucker_log_optimal	theorem	Kuhn–Tucker characterization of the log-optimal portfolio		A portfolio b* is log-optimal iff E[X_i / (b*^T X)] ≤ 1 for every stock, with equality for every stock held with positive weight.
s_competitive_optimality_log_optimal	theorem	Competitive optimality of the log-optimal portfolio		For the log-optimal portfolio b* and any other portfolio b, the probability that b does at least as well as b* by a factor t is at most 1/t, so b* is competitiv
s_expected_ratio_bound	theorem	Expected-ratio bound for log-optimal wealth		For the log-optimal portfolio b* and any portfolio b, E[(b^T X)/(b*^T X)] ≤ 1, the key inequality underlying competitive optimality.
s_side_information_growth_rate	theorem	Side information increases growth rate by mutual information		The increase in the optimal doubling rate of a stock market due to side information Y is at most the mutual information I(X;Y) between the market and the side i
s_growth_rate_stationary_market	theorem	Asymptotic optimality of log-optimal investment (AEP for investment)		For a stationary ergodic market, the wealth of the log-optimal (causal) portfolio strategy grows to first order in the exponent like 2^{nW*}, and no causal stra
t_universal_portfolio	technique	Universal portfolio		Cover's causal investment strategy that performs a wealth-weighted average over all constant-rebalanced portfolios, achieving the same exponential growth as the
s_constant_rebalanced_portfolio	state	Constant rebalanced portfolio		An investment strategy that rebalances to the same fixed proportion vector b every trading period regardless of past prices.
s_universal_portfolio_optimality	theorem	Asymptotic optimality of the universal portfolio		The universal portfolio's wealth is within a polynomial factor of the best constant rebalanced portfolio in hindsight, so it achieves the same growth-rate expon
s_entropy_power	state	Entropy power		The variance of a Gaussian having the same differential entropy per dimension as X, the quantity through which entropy and second-moment inequalities connect.
s_entropy_power_inequality	theorem	Entropy power inequality (EPI)		For independent random vectors X and Y, the entropy power of the sum dominates the sum of entropy powers: 2^{(2/n)h(X+Y)} ≥ 2^{(2/n)h(X)} + 2^{(2/n)h(Y)}.
s_fisher_information_matrix_vector	state	Fisher information (of a random vector under translation)		The translation Fisher information of a random vector, the expected squared gradient of its log density, central to the Fisher information and entropy-power cha
s_fisher_information_inequality	theorem	Fisher information inequality		For independent random vectors X and Y, the reciprocal Fisher information is superadditive: 1/J(X+Y) ≥ 1/J(X) + 1/J(Y).
s_cramer_rao_fisher_form	theorem	Cramér–Rao (estimation form) via Fisher information		The Fisher information of a random variable is at least the reciprocal of its variance, J(X) ≥ 1/var(X), with equality for the Gaussian.
s_de_bruijn_identity	theorem	de Bruijn's identity		The rate of change of the differential entropy of X + √t Z (Z standard Gaussian) with respect to t equals half the Fisher information: (d/dt) h(X+√t Z) = (1/2) 
s_minkowski_determinant_inequality	theorem	Minkowski determinant inequality		For positive-definite K_1 and K_2, |K_1+K_2|^{1/n} ≥ |K_1|^{1/n} + |K_2|^{1/n}, the determinant analogue of the entropy power inequality.
s_szasz_inequality	theorem	Szász's inequality		A chain of inequalities relating the determinant of a positive-definite matrix to the geometric means of the determinants of its principal submatrices of each o
s_han_inequality	theorem	Han's inequality (subset entropy inequality)		The average of the entropies of all size-k subsets of a set of random variables, normalized by k, is nonincreasing in k, so larger subsets give smaller per-vari
s_han_inequality_relative_entropy	theorem	Han's inequality for relative entropies		The analogue of Han's inequality for relative entropy, in which averaged subset relative entropies are monotone, used to derive determinant inequalities for rat
s_subset_inequalities_determinant	theorem	Subset inequalities for determinants		A family of inequalities, generalizing Hadamard and Szász, bounding the determinant of a covariance matrix by symmetric functions of the determinants of its pri
s_cauchy_schwarz_inequality_it	theorem	Cauchy–Schwarz inequality (covariance form)		For random variables with finite second moments, (E[XY])^2 ≤ E[X^2]E[Y^2], used to bound covariances and underlie the Cramér–Rao bound.
s_invariant_sigma_algebra	state	Invariant σ-algebra of a measure-preserving transformation		For a measure-preserving map T on (Ω,F,P), the sub-σ-field 𝓘 = {A ∈ F : T⁻¹A = A} of T-invariant events, whose triviality (mod P) characterizes ergodicity of T.
s_sigma_algebra_generated_by_rv	state	σ-algebra generated by a random variable		For a measurable map X : (Ω,F) → (S,𝓢), the smallest sub-σ-field σ(X) = X⁻¹(𝓢) = {X⁻¹(B) : B ∈ 𝓢} making X measurable, encoding exactly the information carried 
s_stieltjes_measure_function	axiom	Stieltjes measure function		A nondecreasing right-continuous function F : ℝ → ℝ that induces a unique Lebesgue–Stieltjes measure μ on (ℝ,𝓑) via μ((a,b]) = F(b) − F(a).
s_feller_clt_converse	theorem	Feller's converse to the central limit theorem		For a uniformly asymptotically negligible triangular array whose row sums converge to a standard normal, the Lindeberg condition is necessary, i.e. asymptotic n
s_esseen_smoothing_inequality	theorem	Esseen's smoothing inequality		Bounds the sup-norm distance between two distribution functions (one with bounded density) by an integral of the difference of their characteristic functions ov
t_lindeberg_replacement	technique	Lindeberg replacement (swapping) method		Proves a central limit theorem by replacing the summands one at a time with matching-moment Gaussians and Taylor-expanding a smooth test function, telescoping t
t_standard_machine	technique	Standard machine (measure-theoretic bootstrap)		Establishes a linear/monotone identity or inequality for general measurable functions by proving it first for indicator functions, then extending by linearity t
s_mixing_implies_ergodicity	theorem	Mixing implies ergodicity		Any (strongly or weakly) mixing measure-preserving transformation is ergodic, since asymptotic independence P(A ∩ T⁻ⁿB) → P(A)P(B) forces every invariant set to
s_kolmogorov_cycle_condition	theorem	Kolmogorov cycle condition		An irreducible positive-recurrent Markov chain with stationary distribution is reversible if and only if for every finite cycle x₀ → x₁ → … → xₙ = x₀ the produc
t_multiplication_counting_principle	technique	Multiplication (fundamental counting) principle		The number of ways to perform a sequence of independent choices equals the product of the number of options available at each stage.
s_ordered_samples_with_replacement	state	Ordered samples with replacement		The set of ordered r-tuples drawn with repetition allowed from n distinct objects, of which there are exactly n^r.
s_ordered_samples_without_replacement	state	Ordered samples without replacement (permutations of r from n)		The set of ordered r-tuples of distinct elements chosen from n objects, of which there are exactly the falling factorial (n)_r = n!/(n−r)!.
t_combinations_with_repetition	technique	Combinations with repetition (stars and bars)		A bijective counting technique placing r identical items into n categories by arranging r stars and n−1 bars, giving C(n+r−1, r) multisets.
s_occupancy_model	axiom	Occupancy (balls in cells) model		The combinatorial probability model in which r balls are distributed among n cells, parametrizing classical distinguishability and exclusion assumptions.
s_maxwell_boltzmann_statistics	state	Maxwell–Boltzmann statistics		The occupancy scheme in which r distinguishable balls are placed independently and uniformly into n cells, yielding n^r equally likely arrangements.
s_boole_inequality	theorem	Boole's inequality (subadditivity)		The probability of a finite or countable union of events is at most the sum of their individual probabilities.
s_multinomial_theorem	theorem	Multinomial theorem		The expansion of (x_1+…+x_m)^n as a sum over compositions of n of multinomial-coefficient-weighted monomials.
s_probability_exactly_m_events	theorem	Probability of realizing exactly m of N events		The inclusion-exclusion-type formula expressing P(exactly m of N events occur) as an alternating sum of the binomial-weighted symmetric sums S_m, S_{m+1}, ….
s_returns_to_origin_random_walk	state	Returns to the origin (u_{2n})		The probability u_{2n} that a simple symmetric random walk is back at the origin after 2n steps, equal to C(2n,n)2^{-2n}.
s_iterated_generating_functions_branching	theorem	Iterated generating functions for branching processes		The generating function of the n-th generation size in a Galton–Watson process is the n-fold functional composition of the offspring generating function.
s_sparre_andersen_fluctuation	theorem	Sparre Andersen fluctuation theorem		A distribution-free combinatorial identity stating that for exchangeable increments the law of the number of positive partial sums equals the law of the index o
s_independent_increments_process	axiom	Independent increments process		A stochastic process whose increments over disjoint time intervals are mutually independent random variables.
s_stationary_increments_process	axiom	Stationary increments		A stochastic process for which the distribution of an increment X_{t+h}−X_t depends only on the length h and not on the time t.
s_bessel_function_density	state	Bessel function densities		Probability densities expressed through modified Bessel functions, arising e.g. as first-passage densities of random walks and as laws of sums of independent ex
s_kolmogorov_canonical_representation	theorem	Kolmogorov canonical representation		The canonical representation of the characteristic function of an infinitely divisible law with finite variance via a drift and a finite spectral measure K(dx),
s_index_of_stability	axiom	Index of stability α		The exponent α ∈ (0,2] characterizing a stable distribution through the scaling relation of sums of i.i.d. copies and governing its tail behavior.
s_types_of_distributions	axiom	Types of distributions (affine equivalence)		The equivalence classes of distribution functions under affine transformations X ↦ aX+b (a>0), within which location and scale vary but shape is fixed.
t_duality_lemma_random_walk	technique	Duality lemma for random walks		The time-reversal identity that the partial sums (S_1,…,S_n) have the same joint law as the reversed walk, relating first-passage and ladder events to maxima.
s_transitive_closure	state	Transitive closure TC(x)	TC(x)	The smallest transitive set containing x as a subset, obtained by iterating union over ω.
s_burali_forti_paradox	theorem	Burali-Forti paradox	Burali-Forti	The class Ord of all ordinals is not a set; assuming it were yields an ordinal strictly greater than every ordinal.
s_ordinal_arithmetic	state	Ordinal arithmetic	ordinal addition | ordinal multiplication	Addition, multiplication, and exponentiation of ordinals defined by transfinite recursion; associative but non-commutative.
s_cantor_normal_form	theorem	Cantor normal form	CNF	Every ordinal α has a unique representation α=ω^{β₁}k₁+···+ω^{βₙ}kₙ with β₁>···>βₙ and positive integers kᵢ.
s_epsilon_numbers	state	Epsilon numbers (ε-numbers)	ε-numbers | ε₀	Fixed points of ordinal exponentiation, i.e. ordinals ε satisfying ω^ε=ε, with ε₀ the least such ordinal.
s_order_type	state	Order type	otp	The unique ordinal order-isomorphic to a given well-ordered set, given by the Mostowski collapse of its ∈-relation.
s_rank_function	state	Rank function	set-theoretic rank	The function assigning to each set x the least ordinal α with x∈V_{α+1}, measuring its position in the cumulative hierarchy.
t_epsilon_induction	technique	∈-induction (Foundation induction)		Proving a property holds of all sets by showing it holds of x whenever it holds of all elements of x, justified by Foundation.
s_well_founded_relation	axiom	Well-founded relation	well-founded	A binary relation R on a class such that every nonempty subset has an R-minimal element, equivalently admits no infinite R-descending chain.
t_well_founded_recursion	technique	Well-founded recursion		Defining a function uniquely along a set-like well-founded relation by specifying its value at x in terms of its values at R-predecessors of x.
s_hereditarily_small_sets_H_kappa	state	Hereditarily small sets H(κ)	H(κ) | H_κ	The set of all sets whose transitive closure has cardinality strictly less than κ; H(κ) models ZFC minus Power Set for regular κ.
s_aleph_function	state	Aleph function ℵ_α	ℵ_α	The order-isomorphism enumerating the infinite cardinals (initial ordinals) in increasing order, with ℵ₀ the least.
s_initial_ordinal	state	Initial ordinal		An ordinal not equinumerous with any smaller ordinal; the initial ordinals are exactly the cardinal numbers.
s_cardinal_arithmetic	state	Cardinal arithmetic	κ+λ | κ·λ | κ^λ	Sum, product, and exponentiation of cardinals defined via cardinality of disjoint union, Cartesian product, and function space.
s_hessenberg_kappa_squared	theorem	Hessenberg's theorem (κ²=κ)	κ²=κ	Every infinite cardinal κ satisfies κ·κ=κ, proved via the Gödel pairing well-ordering of κ×κ.
s_godel_pairing_function	state	Gödel pairing function	canonical well-ordering of pairs	A definable well-ordering of Ord×Ord whose restriction to κ×κ is a bijection onto κ for every infinite cardinal κ.
s_cofinality	state	Cofinality cf(α)	cf(α)	The least order type (equivalently least cardinality) of an unbounded subset of a limit ordinal α.
s_regular_cardinal	axiom	Regular cardinal	regular	A cardinal κ with cf(κ)=κ, i.e. not the union of fewer than κ sets each of size less than κ.
s_singular_cardinal	axiom	Singular cardinal	singular	A cardinal κ with cf(κ)<κ, i.e. expressible as a union of fewer than κ smaller sets.
s_successor_cardinals_regular	theorem	Successor cardinals are regular		Under AC, every successor cardinal ℵ_{α+1} is regular.
s_continuum_function	state	Continuum function	2^κ	The cardinal exponential κ↦2^κ giving the size of the power set; 2^{ℵ₀}=𝔠 is the continuum.
s_hausdorff_formula	theorem	Hausdorff formula		For all α,β: ℵ_{α+1}^{ℵ_β}=ℵ_α^{ℵ_β}·ℵ_{α+1}, reducing successor-cardinal exponentiation.
s_gimel_function	state	Gimel function ℷ(κ)	ℷ | gimel	The function ℷ(κ)=κ^{cf(κ)}; all cardinal exponentiation reduces to gimel values and the continuum function.
s_beth_function	state	Beth function ℶ_α	ℶ_α | beth numbers	The hierarchy ℶ₀=ℵ₀, ℶ_{α+1}=2^{ℶ_α}, ℶ_λ=sup_{α<λ}ℶ_α of iterated power-set cardinals.
s_generalized_continuum_hypothesis	state	Generalized Continuum Hypothesis (GCH)	GCH	The assertion 2^{ℵ_α}=ℵ_{α+1} for every ordinal α, equivalently ℶ_α=ℵ_α for all α.
s_tukey_lemma	theorem	Tukey's lemma		Every nonempty family of finite character has a maximal element under inclusion; equivalent to the Axiom of Choice.
s_trichotomy_of_cardinals	theorem	Trichotomy of cardinals	cardinal comparability	For any sets X,Y either |X|≤|Y| or |Y|≤|X|; this comparability of all cardinals is equivalent to AC.
s_ideal_set_theory	axiom	Ideal (set-theoretic)	dual ideal	A family of subsets closed downward and under finite unions and not containing the whole set; the dual notion to a filter.
s_kappa_complete_filter	axiom	κ-complete filter	κ-complete	A filter closed under intersections of fewer than κ of its members.
s_principal_filter	state	Principal vs non-principal ultrafilter	principal ultrafilter | free ultrafilter	A principal ultrafilter is the set of all supersets of a fixed point; a non-principal one contains the cofinite filter (Fréchet filter).
s_boolean_prime_ideal_theorem	theorem	Boolean Prime Ideal Theorem	BPIT | ultrafilter theorem	Every proper filter on a set (or Boolean algebra) extends to an ultrafilter; weaker than full AC.
s_complete_boolean_algebra	axiom	Complete Boolean algebra	cBa	A Boolean algebra in which every subset has a least upper bound (and dually a greatest lower bound).
s_regular_open_algebra	state	Regular open algebra RO(P)	RO(P)	The complete Boolean algebra of regular open subsets of a topological space (or separative poset), serving as its canonical Boolean completion.
s_club_set	state	Closed unbounded (club) set	club | closed unbounded set	A subset of a regular uncountable cardinal κ that is unbounded and closed under suprema of its bounded subsets.
s_club_filter	state	Club filter		The κ-complete normal filter on a regular uncountable κ generated by the closed unbounded subsets of κ.
s_normal_filter	state	Normal filter	normal	A filter on a cardinal closed under diagonal intersection △_{α<κ}C_α; for the club filter this is equivalent to closure under regressive-function fibers.
s_club_intersection_theorem	theorem	Intersection of clubs is club		The intersection of fewer than κ club subsets of a regular uncountable κ is again club, so the club sets generate a κ-complete filter.
s_tree_set_theoretic	axiom	Tree (set-theoretic)	set-theoretic tree	A partially ordered set (T,<) in which the predecessors of every node are well-ordered by <.
s_tree_branch_level_height	state	Branch, level, and height of a tree	branch | level | height	A branch is a maximal chain, the α-th level is the set of nodes whose predecessors have order type α, and the height is the supremum of the levels.
s_aronszajn_tree	state	Aronszajn tree	κ-Aronszajn tree	A κ-tree of height κ all of whose levels have size <κ and which has no branch of length κ.
s_aronszajn_tree_existence	theorem	Existence of an Aronszajn tree		An ℵ₁-Aronszajn tree exists in ZFC.
s_suslin_tree	state	Suslin tree		An ℵ₁-tree with no uncountable chain and no uncountable antichain, equivalently a ccc Aronszajn tree.
s_suslin_problem	state	Suslin's problem (Suslin Hypothesis)	Suslin's Hypothesis | SH | Suslin line	The question whether every ccc dense complete linear order without endpoints is isomorphic to ℝ; SH asserts that it is.
s_suslin_tree_line_equivalence	theorem	Equivalence of Suslin tree and Suslin line		A Suslin line exists if and only if a Suslin tree exists.
s_kurepa_tree	state	Kurepa tree / Kurepa's hypothesis	Kurepa's hypothesis | KH	An ℵ₁-tree with countable levels and at least ℵ₂ uncountable branches; Kurepa's Hypothesis asserts one exists.
s_special_aronszajn_tree	state	Special Aronszajn tree		An Aronszajn tree that is the union of countably many antichains, equivalently admitting an order-preserving map into ℚ.
s_partition_calculus	state	Partition calculus (arrow notation)	arrow notation | κ→(λ)^n_m	The notation κ→(λ)^n_m meaning every m-coloring of the n-element subsets of κ has a homogeneous subset of order type λ.
s_square_principle	axiom	Square principle □_κ (Jensen)	□_κ | Jensen square	The existence of a coherent sequence ⟨C_α:α<κ⁺⟩ of clubs C_α⊆α with otp(C_α)≤κ and coherence under limit points, holding in L.
s_club_principle	state	Club principle ♣	♣ | clubsuit	A guessing principle giving a sequence ⟨A_α⟩ of cofinal subsets A_α⊆α such that every uncountable X⊆ω₁ contains some A_α; a weakening of ◊.
s_diamond_implies_ch	theorem	◊ implies CH		The diamond principle ◊ on ω₁ implies the Continuum Hypothesis.
s_singular_cardinal_hypothesis	state	Singular Cardinal Hypothesis (SCH)	SCH	The assertion that for every singular κ with 2^{cf(κ)}<κ one has κ^{cf(κ)}=κ⁺.
s_levy_hierarchy	state	Lévy hierarchy (Σ_n/Π_n)	Σ_n | Π_n	The classification of set-theoretic formulas by the number of alternations of bounded-then-unbounded quantifiers into Σ_n, Π_n, Δ_n classes.
t_absoluteness	technique	Absoluteness		The method of showing a formula's truth is preserved between transitive models; Δ₀ formulas are absolute and Σ₁ formulas are upward absolute.
t_elementary_submodel_method	technique	Elementary submodel method		Taking a countable elementary submodel M≺H(θ) and applying its Mostowski collapse as a core tool for combinatorial set theory.
t_skolem_hull	technique	Skolem functions / Skolem hull		Closing a set of parameters under definable Skolem witnessing functions to obtain an elementary substructure.
t_godel_operations	technique	Gödel operations 𝒢₁–𝒢₁₀		A finite list of ten basic finitary set operations whose closure yields exactly the Σ₀-definable (rudimentary) functions used to build L.
s_constructible_wellordering	state	Constructible well-ordering <_L	<_L	The canonical Σ₁-definable well-ordering of the constructible universe L obtained from the stagewise construction of the L_α.
s_ac_in_L	theorem	AC holds in L		The constructible universe L satisfies the Axiom of Choice via the definable global well-ordering <_L.
s_relative_constructibility_L_A	state	Relative constructibility L[A]	L[A]	The smallest inner model of ZF closed under the definable-power-set operation relativized to a predicate A.
s_L_of_A	state	Constructible closure L(A)	L(A)	The smallest inner model of ZF containing the set A (and its transitive closure) as an element.
s_diamond_holds_in_L	theorem	◊ holds in L (Jensen)		V=L implies the diamond principle ◊_κ on every regular uncountable κ, proved via condensation.
s_square_holds_in_L	theorem	□_κ holds in L (Jensen)		V=L implies Jensen's square principle □_κ for every uncountable cardinal κ, proved by fine-structure methods.
s_kurepa_tree_in_L	theorem	Kurepa tree exists in L		V=L (via ◊⁺) implies the existence of a Kurepa tree.
t_rudimentary_functions	technique	Rudimentary functions		The class of functions generated by the Gödel operations and closed under composition and bounded quantification, used to define the J-hierarchy.
s_jensen_J_hierarchy	state	Jensen J-hierarchy J_α	J_α | J-hierarchy	A refinement of the constructible hierarchy built by rudimentary closure, satisfying smooth condensation at every level.
s_sigma_n_projectum	state	Σ_n-projectum ρ_n	ρ_n | projectum	For a fine-structural level, the least ordinal ρ such that some Σ_n-definable (with parameters) subset of ρ is not a member of the level.
t_fine_structural_condensation	technique	Fine-structural condensation		The principle that Σ_n-elementary submodels of J-levels collapse to J-levels, refining Gödel's condensation lemma.
s_silver_indiscernibles	state	Silver indiscernibles		A closed unbounded class of ordinal order-indiscernibles for L that generate L via Skolem functions; they exist iff 0# exists.
s_zero_sharp_implies_V_not_L	theorem	0# exists implies V≠L		If 0# exists then every uncountable cardinal is inaccessible (indeed indiscernible) in L, so V≠L.
s_covering_implies_SCH	theorem	Covering lemma implies SCH		If 0# does not exist, Jensen's covering lemma holds and implies the Singular Cardinal Hypothesis.
s_notion_of_forcing	state	Notion of forcing	forcing poset | forcing notion	A separative partial order (P,≤,𝟙) of conditions used to extend a ground model by a generic filter.
s_dense_predense	state	Dense and predense sets	dense set | predense	A subset D of a forcing poset is dense if every condition has an extension in D, and predense if every condition is compatible with a member of D.
s_compatible_conditions	state	Compatible and incompatible conditions	compatibility | forcing antichain	Two conditions are compatible if they have a common extension and incompatible otherwise; an antichain is a set of pairwise incompatible conditions.
s_rasiowa_sikorski_lemma	theorem	Rasiowa–Sikorski lemma		For any countable family of dense subsets of a poset there is a filter meeting every member of the family.
t_p_names	technique	P-names		Hereditarily P-labeled sets that name potential elements of a forcing extension and are interpreted via a generic filter G.
t_canonical_names	technique	Canonical names (x̌, Ġ)		The check-name x̌ denoting a ground-model set x and the name Ġ denoting the generic filter in the forcing extension.
s_forcing_relation	state	Forcing relation p⊩φ	p⊩φ	The relation holding when p∈P forces φ, i.e. φ is true in M[G] for every generic G containing p.
s_definability_lemma_forcing	theorem	Definability lemma (forcing)		For each formula φ the forcing relation p⊩φ is definable in the ground model.
s_truth_lemma	theorem	Truth lemma		For a generic G, M[G]⊨φ if and only if some condition p∈G forces φ.
s_forcing_theorem	theorem	Forcing theorem (fundamental)		The Definability and Truth lemmas together reduce truth in M[G] to a ground-model-definable forcing relation.
s_ground_model_definability	theorem	Ground model definability		The ground model M is definable, with a parameter, inside any of its set-generic extensions (Laver–Woodin).
t_separative_quotient	technique	Separativity / separative quotient		Replacing a forcing poset by its separative quotient, densely embeddable into the regular open completion / a complete Boolean algebra.
t_cohen_forcing	technique	Cohen forcing		Forcing with finite partial functions Fn(I,2) to adjoin a new real (or I-many mutually generic Cohen reals).
s_ccc_forcing	state	Countable chain condition (ccc)	ccc	A forcing poset has the ccc if every antichain in it is countable.
s_ccc_preserves_cardinals	theorem	ccc forcing preserves cardinals		A ccc forcing notion preserves all cardinals and cofinalities of the ground model.
s_cohen_forcing_is_ccc	theorem	Cohen forcing is ccc		The poset Fn(I,2) satisfies the countable chain condition, proved via the Δ-system lemma.
t_nice_names	technique	Nice names		Antichain-indexed names for subsets of a ground-model set, used to count the subsets appearing in a ccc extension.
s_con_zfc_not_ch_cohen	theorem	Con(ZFC)⟹Con(ZFC+¬CH) (Cohen)		Forcing with Fn(ω₂×ω,2) over a model of ZFC yields a ccc extension in which 2^{ℵ₀}=ℵ₂, so ¬CH is consistent.
s_boolean_valued_universe	state	Boolean-valued universe V^B	V^B	The hierarchy of B-valued names with each formula assigned a truth value ‖φ‖∈B, an alternative formulation of forcing.
t_boolean_truth_value	technique	Boolean truth value ‖φ‖		The recursively defined element of a complete Boolean algebra B giving the degree of truth of a formula in V^B.
s_mixing_lemma_forcing	theorem	Mixing lemma		Given a maximal antichain {b_i} and names {ẋ_i}, there is a single name ẋ with b_i≤‖ẋ=ẋ_i‖ for all i.
s_maximum_principle_forcing	theorem	Maximum principle (forcing)		If ‖∃x φ(x)‖=b in V^B then there is a name ẋ with ‖φ(ẋ)‖=b.
t_levy_collapse	technique	Lévy collapse Coll(ω,κ)		Forcing with finite (or appropriate) conditions that collapses a cardinal κ, or all cardinals below λ, to ω.
s_kappa_closed_forcing	state	<κ-closed forcing	κ-closed | <κ-closed	A forcing notion in which every descending sequence of length <κ has a lower bound.
s_closed_forcing_adds_no_sequences	theorem	<κ-closed forcing adds no new <κ-sequences		A <κ-closed forcing adds no new sequences of length <κ, hence preserves H_κ and all cardinals up to κ.
s_kappa_cc_knaster	state	κ-cc / κ-Knaster	κ-cc | Knaster	A poset is κ-cc if all its antichains have size <κ, and κ-Knaster if every κ-sized subset has a κ-sized pairwise-compatible subset.
s_kappa_cc_preserves_cardinals	theorem	κ-cc forcing preserves cardinals ≥κ		A κ-cc forcing notion preserves all cardinals and cofinalities greater than or equal to κ.
s_product_forcing_lemma	theorem	Product forcing lemma		G×H is P×Q-generic over M iff G is P-generic over M and H is Q-generic over M[G], and then M[G×H]=M[G][H].
t_two_step_iteration	technique	Two-step iteration P∗Q̇		The forcing that first forces with P and then with a P-name Q̇ for a poset, realized by ordered pairs of conditions.
s_martins_axiom_MA_kappa	axiom	Martin's Axiom MA_κ	MA_κ	For every ccc poset and every family of at most κ dense sets there is a filter meeting all of them.
s_martins_axiom	axiom	Martin's Axiom (MA)	MA	The assertion that MA_κ holds for every κ<2^{ℵ₀}.
s_MA_aleph0_theorem	theorem	MA_{ℵ₀} is a theorem of ZFC		Martin's Axiom for countably many dense sets is provable in ZFC, being exactly the Rasiowa–Sikorski lemma.
s_con_MA_not_CH	theorem	Con(MA+¬CH) (Solovay–Tennenbaum)		A finite-support ccc iteration of length ω₂ forces MA together with 2^{ℵ₀}=ℵ₂.
t_finite_support_iteration	technique	Finite-support iteration		An iterated forcing construction in which conditions have finite support, used to obtain ccc-preserving limits.
s_fs_iteration_ccc	theorem	FS iteration of ccc is ccc		A finite-support iteration of ccc forcing notions is itself ccc, proved via the Δ-system lemma.
s_MA_aleph1_negates_CH	theorem	MA_{ℵ₁} implies ¬CH		Martin's Axiom for ℵ₁ dense sets implies 2^{ℵ₀}>ℵ₁, refuting the Continuum Hypothesis.
s_con_SH	theorem	Con(SH) (Solovay–Tennenbaum)		Suslin's Hypothesis is consistent with ZFC, obtained from a model of MA+¬CH which kills all Suslin trees.
s_MA_measure_category	theorem	MA and additivity of measure/category		Under MA_κ the union of at most κ Lebesgue-null (resp. meager) sets of reals is null (resp. meager).
t_countable_support_iteration	technique	Countable-support iteration		An iterated forcing construction with countable support, the standard framework for iterating proper forcings.
s_properness_preserved_CS_iteration	theorem	Properness preserved under CS iteration		Countable-support iterations of proper forcing notions are proper (Shelah).
s_mahlo_cardinal	state	Mahlo cardinal	Mahlo	A regular cardinal κ for which the set of inaccessible cardinals below κ is stationary in κ.
s_weakly_compact_cardinal	state	Weakly compact cardinal	weakly compact	An inaccessible cardinal κ with the tree property, equivalently satisfying κ→(κ)²₂ or Π¹₁-indescribability.
s_weakly_compact_tree_property	theorem	Tree property characterization of weak compactness		For an inaccessible κ, weak compactness is equivalent to the tree property and to the partition relation κ→(κ)²₂.
s_normal_measure	state	Normal measure	normal ultrafilter	A κ-complete nonprincipal ultrafilter on a measurable cardinal κ closed under diagonal intersections; equivalently the constant function is least nontrivial in 
t_ultrapower_Ult_V_U	technique	Ultrapower Ult(V,U)		Forming V^κ/U and collapsing the well-founded structure to a transitive inner model M with elementary embedding j:V→M.
s_measurable_embedding_characterization	theorem	Measurable cardinals and elementary embeddings	critical point characterization	κ is measurable iff there is a nontrivial elementary embedding j:V→M into a transitive class with critical point κ.
t_iterated_ultrapowers	technique	Iterated ultrapowers		Repeatedly taking ultrapowers by images of a measure to produce a directed system of elementary embeddings and iterates.
s_ramsey_cardinal	state	Ramsey cardinal	Ramsey	A cardinal κ satisfying the partition relation κ→(κ)^{<ω}_2.
s_erdos_cardinal	state	Erdős cardinal κ→(α)^{<ω}	Erdős cardinal | κ(α)	For an ordinal α, the least cardinal κ(α) satisfying the partition relation κ→(α)^{<ω}_2.
t_ehrenfeucht_mostowski_models	technique	Ehrenfeucht–Mostowski models		Building elementary submodels generated over a set of order-indiscernibles via Skolem functions, the engine producing sharps.
s_erdos_implies_zero_sharp	theorem	κ→(ω₁)^{<ω} implies 0# exists		The existence of an Erdős/Ramsey-type cardinal with κ→(ω₁)^{<ω} yields uncountably many indiscernibles for L, hence 0#.
s_strongly_compact_cardinal	state	Strongly compact cardinal	strongly compact	A cardinal κ such that every κ-complete filter on any set extends to a κ-complete ultrafilter.
s_huge_cardinal	state	Huge cardinal	huge	A cardinal κ admitting an elementary embedding j:V→M with critical point κ and ^{j(κ)}M⊆M.
t_extenders	technique	Extenders		Coherent systems of ultrafilters coding a (possibly strong) elementary embedding, generalizing normal measures.
s_vopenka_principle	axiom	Vopěnka's principle	Vopěnka	Every proper class of first-order structures of the same type contains two members one of which elementarily embeds into the other.
s_proper_forcing	state	Proper forcing	proper	A forcing notion preserving stationary subsets of [λ]^ω for all λ; equivalently generic conditions exist over countable elementary submodels.
s_cantor_space	state	Cantor space 2^ω	2^ω	The space of infinite binary sequences with the product topology, a compact perfect totally disconnected Polish space.
s_baire_space_irrationals	theorem	Baire space homeomorphic to the irrationals		The Baire space ω^ω is homeomorphic to the space of irrational reals, via continued fractions.
s_universality_of_baire_space	theorem	Universality of Baire space		Every nonempty Polish space is a continuous image of the Baire space ω^ω.
s_tree_on_a_set_body	state	Tree on a set and its body [T]	[T] | tree on ω	A set of finite sequences closed under initial segments; its body [T] of infinite branches is the corresponding closed subset of the product space.
s_well_founded_tree_rank	state	Well-founded tree and its rank	tree rank	A tree with no infinite branch, to which well-founded recursion assigns an ordinal rank.
s_borel_hierarchy	state	Borel hierarchy (Σ⁰_α,Π⁰_α,Δ⁰_α)	Σ⁰_α | Π⁰_α	The transfinite stratification of the Borel sets of a Polish space by iterated countable unions and complements.
t_borel_codes	technique	Borel codes		Real-number codes for Borel sets making Borel relations definable and absolute between transitive models.
s_property_of_baire	state	Property of Baire	Baire property | BP	A set has the Baire property if it differs from an open set by a meager set; the BP sets form a σ-algebra closed under the Suslin operation.
t_suslin_operation	technique	Suslin operation 𝒜		The operation 𝒜_s A_s=⋃_f⋂_n A_{f↾n} on a Suslin scheme that generates the analytic sets from closed sets.
s_coanalytic_set	state	Coanalytic set Π¹₁	Π¹₁	The complement of an analytic set; the first level of the projective hierarchy dual to Σ¹₁.
s_cantor_bendixson_rank	technique	Cantor–Bendixson derivative and rank	CB derivative | CB rank	The iterated removal of isolated points from a closed set, stabilizing at a countable ordinal rank and leaving the perfect kernel.
s_perfect_set_property	state	Perfect set property (PSP)	PSP	A set of reals has the PSP if it is either countable or contains a nonempty perfect subset.
s_analytic_psp	theorem	Analytic sets have the perfect set property		Every uncountable analytic set contains a nonempty perfect subset.
s_analytic_measurable	theorem	Analytic sets are Lebesgue measurable		Every Σ¹₁ set is Lebesgue measurable.
s_analytic_baire_property	theorem	Analytic sets have the Baire property		Every Σ¹₁ set has the property of Baire.
s_universal_analytic_set	state	Universal analytic set		A single Σ¹₁ subset of (ω^ω)² that is universal for all Σ¹₁ subsets of ω^ω, witnessing Σ¹₁≠Π¹₁.
t_pi11_norm	technique	Π¹₁ norm / rank		A regular Π¹₁-norm assigning each member the ordinal rank of an associated well-founded tree, all below ω₁.
s_boundedness_theorem	theorem	Σ¹₁-boundedness theorem	Σ¹₁-bounding	A Σ¹₁ subset of the well-founded trees has ranks bounded below ω₁.
s_projective_hierarchy	state	Projective hierarchy (Σ¹_n,Π¹_n,Δ¹_n)	Σ¹_n | Π¹_n | projective sets	The hierarchy of pointclasses with Σ¹₁ the analytic sets, Π¹_n the complements, and Σ¹_{n+1} the projections of Π¹_n sets.
s_projective_hierarchy_proper	theorem	Projective hierarchy is proper		For each n there is a universal Σ¹_n set, and the projective hierarchy is strictly increasing.
s_uniformization	state	Uniformization		Selecting for a relation R⊆X×Y a function with graph in R, with the graph required to lie in a given pointclass.
s_sigma2_uniformization	theorem	Σ¹₂-uniformization (Novikov–Kondô–Addison)		Every Σ¹₂ relation can be uniformized by a Σ¹₂ function.
s_prewellordering_property	state	Prewellordering property	PWO	A pointclass Γ has the prewellordering property if every Γ set admits a Γ-norm; Π¹₁ and Σ¹₂ have it.
s_scale_property	state	Scale property	scale	A set has a scale if there is a sequence of norms with a convergence/lower-semicontinuity property enabling uniformization.
s_periodicity_theorems	theorem	Periodicity theorems (Moschovakis)	First and Second Periodicity	Under determinacy the prewellordering and scale properties propagate up the projective hierarchy in a periodic pattern.
t_shoenfield_tree	technique	Shoenfield tree		A tree on ω×ω₁ whose projection is a given Σ¹₂ set, reducing membership to well-foundedness and giving Shoenfield absoluteness.
s_sigma2_wellordering_in_L	theorem	Σ¹₂ well-ordering under V=L		If V=L there is a Σ¹₂ well-ordering of the reals, yielding a Δ¹₂ non-Lebesgue-measurable set.
s_constructible_reals	state	Constructible reals ℝ∩L	ℝ∩L	The reals belonging to L, a Σ¹₂ set which under V=L equals all reals.
s_infinite_game	state	Infinite game G(A) and strategy	G(A) | winning strategy	A two-player perfect-information game on ω where players alternately play naturals, I winning iff the resulting real lies in A; a strategy is a function dictati
s_determinacy_of_a_set	state	Determinacy of a set Det(A)	Det(A)	The property that one of the two players has a winning strategy in the game G(A).
s_gale_stewart_theorem	theorem	Gale–Stewart theorem	open determinacy | closed determinacy	Every open game and every closed game (on a tree) is determined.
t_unraveling	technique	Unraveling / covering technique		Lifting a game to an auxiliary tree on which the payoff set becomes clopen, the key device in Martin's proof of Borel determinacy.
s_AD_implies_measurable	theorem	AD implies all sets Lebesgue measurable		Under the Axiom of Determinacy every set of reals is Lebesgue measurable.
s_AD_implies_baire_property	theorem	AD implies the Baire property for all sets		Under the Axiom of Determinacy every set of reals has the property of Baire, via the Banach–Mazur game.
s_AD_implies_PSP	theorem	AD implies perfect set property for all sets		Under the Axiom of Determinacy every uncountable set of reals contains a perfect subset.
t_banach_mazur_game	technique	Banach–Mazur game		An infinite game in which players choose decreasing basic open sets, characterizing comeagerness and the Baire property of A.
s_analytic_determinacy	theorem	Analytic determinacy	Σ¹₁-determinacy	Every analytic (Σ¹₁) game is determined; this is equivalent to the existence of sharps for all reals and follows from a measurable cardinal (Martin, Harrington)
s_sharps_analytic_determinacy_equivalence	state	Sharps and analytic determinacy equivalence	x#	The existence of x# for every real x is equivalent to the determinacy of all analytic games (Harrington–Martin).
t_homogeneous_tree	technique	Homogeneously Suslin set / homogeneous tree		A tree carrying a coherent commuting system of countably complete measures whose projection is a homogeneously Suslin (hence determined) set.
s_martin_steel_pd	theorem	Martin–Steel theorem (projective determinacy)	Martin–Steel	n Woodin cardinals with a measurable above imply Π¹_{n+1}-determinacy; infinitely many Woodins imply full projective determinacy.
s_PD_implies_regularity	theorem	PD implies regularity for all projective sets		Projective Determinacy implies every projective set is Lebesgue measurable, has the Baire property and the perfect set property, and projective relations can be
s_AD_equiconsistent_woodins	theorem	AD equiconsistent with infinitely many Woodins		Con(ZF+AD) is equivalent to Con(ZFC+there are infinitely many Woodin cardinals) (Woodin).
s_wadge_reducibility	state	Wadge reducibility / Wadge hierarchy	≤_W | Wadge degrees	A≤_W B iff A=f^{-1}(B) for a continuous f; under determinacy the induced degrees form a nearly well-ordered hierarchy.
t_solovay_model_construction	technique	Solovay's model construction via Lévy collapse		Lévy-collapsing an inaccessible cardinal to ω₁ and passing to the L(ℝ) of the extension to obtain a ZF+DC model in which all sets are Lebesgue measurable, Baire
s_shelah_inaccessible_necessary	theorem	Inaccessible necessary for all sets Lebesgue measurable (Shelah)		If every set of reals is Lebesgue measurable (with DC) then ω₁ is inaccessible in L, so an inaccessible cardinal is needed for Solovay's model.
s_transitive_set_def	state	Transitive set	transitive set	A set X every element of which is also a subset of X, so that membership never escapes X.
s_delta_0_formula	state	Δ₀ (bounded) formula	Σ₀ formula | Π₀ formula | bounded formula | Delta-0 formula	A first-order set-theoretic formula all of whose quantifiers are bounded, hence absolute between any transitive model and the universe.
t_relativization	technique	Relativization φᴹ		The syntactic operation restricting every quantifier of a formula φ to range over a class or set M, expressing truth-in-M inside the ambient universe.
s_diamond_plus	axiom	◊⁺ (diamond-plus)	diamond-plus | ◊+	A strengthening of ◊ assigning to each α<ω₁ a countable family A_α⊆P(α) such that for every X⊆ω₁ there is a club C with X∩α and C∩α both in A_α for all α∈C.
s_diamond_plus_kurepa	theorem	◊⁺ implies a Kurepa tree (Jensen)	diamond-plus implies Kurepa tree	The principle ◊⁺ implies the existence of a Kurepa tree, a tree of height ω₁ with countable levels and at least ℵ₂ cofinal branches.
s_first_order_signature	axiom	First-order signature (language)		A specification of constant symbols, function symbols, and relation symbols together with their arities, fixing the non-logical vocabulary of a first-order lang
s_l_structure	axiom	L-structure		A nonempty set M together with interpretations of each constant, function, and relation symbol of a signature L as actual elements, functions, and relations on 
s_l_term	axiom	L-term		A syntactic expression built from variables and constant symbols by repeated application of function symbols of the signature L.
s_l_formula	axiom	L-formula		A well-formed syntactic expression built from atomic formulas (equalities and relation symbols applied to terms) using Boolean connectives and quantifiers over 
s_tarski_satisfaction	axiom	Tarski satisfaction relation		The recursively defined relation specifying when an L-structure M with a variable assignment makes an L-formula true, giving the semantics of first-order logic.
s_l_embedding	axiom	L-embedding		An injective map between L-structures preserving the interpretation of all constants, functions, and relations (and their negations on atomic formulas).
s_substructure	axiom	Substructure / extension		An L-structure M whose domain is a subset of N closed under N's functions and constants, with all symbols interpreted as restrictions of N's interpretations.
s_l_theory	axiom	L-theory		A set of L-sentences (closed formulas), the models of which are the L-structures satisfying every sentence in the set.
s_logical_consequence	axiom	Logical consequence (semantic entailment)		The relation holding when every model of a theory T also satisfies the sentence φ.
s_axiomatizable_theory	axiom	Axiomatizable theory		A theory closed under logical consequence that is the set of consequences of some (often finite or recursive) set of axioms.
s_definable_function	axiom	Definable function and definable closure		A function whose graph is a definable set, and the definable closure dcl(A) consisting of all elements that are the unique realization of some formula over A.
t_henkin_construction	technique	Henkin construction		Expand T to a maximal consistent theory with witnessing constants for every existential sentence, then build a model out of equivalence classes of closed terms.
s_tarski_vaught_test	theorem	Tarski–Vaught test		A substructure M of N is an elementary substructure iff for every formula φ(x,b) with parameters in M that has a solution in N, it already has a solution in M.
s_elementary_chain_theorem	theorem	Tarski's elementary chain theorem		The union of an elementary chain of L-structures is an elementary extension of each member of the chain.
s_diagram_lemma	theorem	Diagram lemma		Models of the atomic (elementary) diagram of M correspond exactly to L-structures admitting an embedding (elementary embedding) of M.
s_elementary_diagram	axiom	Elementary diagram		The set of all L_M-sentences true in M after adjoining a new constant for each element of M; its models are the elementary extensions of M.
s_upward_lowenheim_skolem	theorem	Upward Löwenheim–Skolem theorem		Any infinite structure has elementary extensions of every cardinality κ ≥ max(|M|,|L|).
s_lowenheim_skolem_tarski	theorem	Löwenheim–Skolem–Tarski theorem		A first-order theory with an infinite model has a model of every cardinality κ ≥ |L|+ℵ₀.
s_dlo	axiom	Dense linear order without endpoints (DLO)		The complete ℵ₀-categorical theory of a dense linear order having no least and no greatest element.
s_robinson_model_completeness_test	theorem	Robinson's test for model completeness		A theory is model complete iff every embedding between its models preserves existential formulas, equivalently every formula is equivalent to an existential one
s_qe_test	theorem	Quantifier elimination test		A theory admits quantifier elimination iff for any two models with a common substructure A, a primitive existential formula realized in one over A is realized i
s_completeness_decidability_acf	theorem	Completeness and decidability of ACF_p		For each fixed characteristic, the theory of algebraically closed fields is complete and decidable, a consequence of its quantifier elimination.
s_lefschetz_principle	theorem	Lefschetz principle		A first-order sentence holds in the complex field iff it holds in every algebraically closed field of characteristic 0 (and in characteristic p for all large p)
s_ax_injective_surjective	theorem	Ax's theorem (injective endomorphisms are surjective)		Every injective polynomial map of an algebraic variety over an algebraically closed field to itself is surjective, proved by transfer from finite fields.
s_completeness_decidability_rcf	theorem	Completeness and decidability of RCF		The theory of real closed ordered fields is complete and decidable, following from Tarski's quantifier elimination for RCF.
s_hilbert_17th_model_theoretic	theorem	Hilbert's 17th problem (model-theoretic form)		Every positive semidefinite rational function over a real closed field is a sum of squares of rational functions, derived via the model completeness of RCF.
s_presburger_qe	theorem	Quantifier elimination for Presburger arithmetic		The first-order theory of the additive integers admits quantifier elimination in the language expanded by congruence-mod-n predicates.
s_omitting_types_theorem_real	theorem	Omitting Types Theorem		A countable theory has a countable model omitting any given non-isolated (non-principal) partial type, generalizable to countably many such types.
s_isolated_principal_type	axiom	Isolated (principal) type		A complete type generated by a single formula φ, i.e. an isolated point of the Stone space of types, realized in every model containing A.
s_prime_model	axiom	Prime model		A model of a complete theory that elementarily embeds into every other model of the theory.
s_atomic_model	axiom	Atomic model		A model in which every tuple realizes an isolated (principal) complete type over the empty set.
s_prime_iff_atomic	theorem	Prime ⟺ countable atomic		For a complete countable theory, a model is prime iff it is countable and atomic, and such a model is unique up to isomorphism when it exists.
s_existence_uniqueness_prime_model	theorem	Existence and uniqueness of prime/atomic models		A complete countable theory has a (unique up to isomorphism) prime model iff the isolated types are dense in every Stone space Sₙ(∅).
s_uniqueness_saturated_models	theorem	Uniqueness of saturated models		Any two saturated models of a complete theory having the same cardinality are isomorphic, proved by a back-and-forth argument.
s_uniqueness_homogeneous_models	theorem	Uniqueness of homogeneous models		Two homogeneous models of the same cardinality realizing exactly the same types are isomorphic.
s_universal_model	axiom	Universal model		A model into which every model of the theory of cardinality at most |M| elementarily embeds.
s_monster_model	axiom	Monster model		A very large saturated and strongly homogeneous model serving as a universal domain inside which all small models and all types under study are realized.
s_em_type	axiom	Ehrenfeucht–Mostowski type (EM-type)		The complete type, indexed by increasing tuples, shared by an order-indiscernible sequence, determining a functor from linear orders to models.
s_existence_of_indiscernibles	theorem	Existence of indiscernibles		For any theory with infinite models and any infinite linear order, there is a model containing a sequence of order-indiscernibles, obtained via Ramsey's theorem
s_few_types_over_em_models	theorem	Few types over EM models		A model generated as the Skolem hull of order-indiscernibles realizes only few types over itself, used to control the number of types and models.
s_vaughtian_pair	axiom	Vaughtian pair		A pair of models M ≺ N with M ≠ N but having the same infinite solution set for some formula φ, witnessing a two-cardinal phenomenon.
s_no_vaughtian_pairs_categorical	theorem	Uncountably categorical theories have no Vaughtian pairs		An uncountably categorical complete theory admits no Vaughtian pairs, a key step in the proof of Morley's theorem.
s_morley_degree	state	Morley degree		For a definable set of finite Morley rank, the maximum number of disjoint definable subsets of the same rank, measuring multiplicity at that rank.
s_totally_transcendental	axiom	Totally transcendental theory		A theory in which every definable set has an ordinal-valued Morley rank, equivalent to ω-stability for countable languages.
s_omega_stable_implies_tt	theorem	ω-stable ⟹ totally transcendental		An ω-stable theory is totally transcendental, so every definable set carries a well-defined Morley rank and degree.
s_definability_of_types	theorem	Definability of types (and heirs)		In a stable (in particular ω-stable) theory, every type over a model is definable, giving canonical extensions (heirs and coheirs) to larger sets.
s_uncountably_categorical_implies_omega_stable	theorem	Uncountably categorical ⟹ ω-stable		Every uncountably categorical countable complete theory is ω-stable, a central lemma in Morley's categoricity theorem.
s_omega_stable_group	axiom	ω-stable group / group of finite Morley rank		A group definable in an ω-stable theory, whose definable subsets carry Morley rank and degree, modeling algebraic-group-like behavior.
s_dcc_definable_subgroups	theorem	Descending chain condition on definable subgroups		In an ω-stable (finite Morley rank) group there is no infinite strictly descending chain of definable subgroups, yielding a connected component.
s_connected_component_group	state	Connected component G⁰		The smallest definable subgroup of finite index in a group of finite Morley rank, normal in G, called the connected component.
s_generic_type_group	axiom	Generic type of a group		A type of an element of a finite-Morley-rank group whose Morley rank equals that of the whole group, invariant under translation in the connected case.
s_stabilizer_finite_morley	state	Stabilizer of a type		The definable subgroup of group elements fixing the (translated) type p, central to analyzing generic data in finite-Morley-rank groups.
s_zilber_indecomposability	theorem	Zilber indecomposability theorem		In a group of finite Morley rank, the subgroup generated by a family of definable indecomposable subsets each containing the identity is definable and connected
s_macintyre_theorem	theorem	Macintyre's theorem		Every infinite ω-stable field is algebraically closed.
s_omega_stable_division_ring_commutative	theorem	ω-stable division rings are commutative		An infinite division ring whose theory is ω-stable is commutative, hence an algebraically closed field.
s_acl_pregeometry	theorem	Algebraic closure gives a pregeometry		On a strongly minimal set, the model-theoretic algebraic closure operator satisfies finite character, monotonicity, idempotence, and exchange, forming a pregeom
s_exchange_property_acl	theorem	Exchange property for acl (Steinitz exchange)		In a strongly minimal theory, if a is algebraic over B together with c but not over B alone, then c is algebraic over B and a, the abstract Steinitz exchange ax
s_pregeometry	axiom	Pregeometry (combinatorial geometry)		A set with a closure operator satisfying reflexivity, monotonicity, finite character, idempotence, and the exchange property, abstracting linear/algebraic depen
s_dimension_in_pregeometry	state	Dimension and independence in a pregeometry		The common cardinality of any maximal independent (closure-free) subset of a set, well-defined by the exchange property and additive over closures.
s_dimension_determines_model	theorem	Dimension determines the model		Two models of a strongly minimal theory are isomorphic iff their associated pregeometries have the same dimension, the mechanism behind uncountable categoricity
s_trivial_geometry	axiom	Trivial (degenerate) geometry		A pregeometry in which the closure of any set is the union of the closures of its points, exhibiting no nontrivial dependence.
s_locally_modular_geometry	axiom	Locally modular geometry		A pregeometry that becomes modular after localizing at a single point, equivalently dim(A∪B)+dim(A∩B)=dim(A)+dim(B) for closed sets with nonempty intersection.
s_locally_modular_iff_no_field	theorem	Locally modular ⟺ no definable field		A non-trivial strongly minimal set is non-locally-modular iff it interprets an infinite field (an algebraically closed field), linking geometry to definable alg
s_zilber_trichotomy_conjecture	state	Zilber's trichotomy conjecture		The conjecture that every strongly minimal set is either trivial, the geometry of a modular group (vector-space-like), or interprets an algebraically closed fie
s_locally_modular_implies_group_based	theorem	Locally modular ⟹ group-based geometry		A non-trivial locally modular strongly minimal set interprets an abelian group whose geometry is essentially that of a vector space over a division ring (group 
s_robinson_model_completeness_acf	theorem	Model completeness of ACF		The theory of algebraically closed fields is model complete: every embedding between algebraically closed fields is elementary, equivalently a Nullstellensatz s
s_completeness_robinson_test_application	theorem	Robinson's test application to ACF/RCF		A model-complete theory with a model embedding into every model (e.g. the prime field) is complete, used to derive completeness of ACF_p and RCF.
s_signature	axiom	Signature	language | vocabulary	A signature (language) is a set of constant symbols, function symbols, and relation symbols each equipped with an arity.
s_domain_universe	axiom	Domain (universe)	universe | carrier set	The domain or universe of a structure is the underlying set on which its functions and relations are defined.
s_cardinality_of_structure	state	Cardinality of a structure		The cardinality of a structure is the cardinality of its domain.
s_term	axiom	Term		A term is a syntactic expression built from variables and constants by repeated application of function symbols.
s_term_function	state	Term function (value of a term)	term value	The term function is the map sending an assignment of domain elements to variables to the value the term takes under that assignment in the structure.
s_homomorphism	axiom	Homomorphism		A homomorphism is a map between structures of the same signature that commutes with all function symbols and preserves all relation symbols.
s_isomorphism	axiom	Isomorphism		An isomorphism is a bijective homomorphism whose inverse is also a homomorphism, so it preserves and reflects all functions and relations.
s_substructure_generated	state	Substructure generated by a set		The substructure generated by a subset X is the smallest substructure containing X, consisting of all values of terms with parameters in X.
s_reduct_expansion	state	Reduct and expansion		A reduct is a structure obtained by forgetting some symbols of the signature, and its expansion is a structure on the same domain interpreting additional symbol
s_atomic_diagram	state	Diagram (atomic/literal)	literal diagram	The diagram of a structure is the set of all atomic and negated atomic sentences with parameters that hold in the structure expanded by names for its elements.
s_canonical_term_structure	state	Canonical (term) structure	Herbrand structure	A canonical or term structure is a structure whose elements are equivalence classes of closed terms, used in Henkin-style constructions.
s_first_order_formula	axiom	First-order formula		A first-order formula is a syntactic expression built from atomic formulas using boolean connectives and quantification over individual variables.
s_atomic_formula	axiom	Atomic formula		An atomic formula is a relation symbol or equality applied to terms, with no connectives or quantifiers.
s_free_bound_variables	axiom	Free and bound variables		A variable occurrence is bound if it lies within the scope of a quantifier binding it and free otherwise.
s_sentence	axiom	Sentence	closed formula	A sentence is a first-order formula with no free variables.
s_parametrically_definable_set	state	Parametrically definable set	definable with parameters	A parametrically definable set is the solution set in a structure of a formula allowing parameters from the domain.
s_theory_of_structure	state	Theory of a structure Th(A)		The theory of a structure is the set of all first-order sentences true in it.
s_elementary_equivalence	state	Elementary equivalence		Two structures are elementarily equivalent when they satisfy exactly the same first-order sentences.
s_elementary_embedding	axiom	Elementary embedding		An elementary embedding is a map between structures that preserves the truth of all first-order formulas with parameters.
s_elementary_diagram_lemma	theorem	Elementary Diagram Lemma		The elementary diagram lemma states that models of the elementary diagram of a structure correspond exactly to its elementary extensions (up to elementary embed
s_back_and_forth_equivalence	state	Back-and-forth equivalence (potential isomorphism)	potential isomorphism	Back-and-forth equivalence (potential isomorphism) holds between two structures when there is a nonempty family of partial isomorphisms closed under extending a
s_cantor_dlo_theorem	theorem	Cantor's theorem on DLO		Cantor's theorem states that any two countable dense linear orders without endpoints are isomorphic.
s_ehrenfeucht_fraisse_game	state	Ehrenfeucht–Fraïssé game	EF game	The Ehrenfeucht–Fraïssé game is a two-player game of n rounds in which Duplicator must maintain a partial isomorphism against Spoiler's element choices in two s
s_ef_quantifier_rank_theorem	theorem	Ehrenfeucht–Fraïssé theorem	Fraïssé's theorem on games	The Ehrenfeucht–Fraïssé theorem states that Duplicator has a winning strategy in the n-round game exactly when the two structures agree on all sentences of quan
s_quantifier_rank	state	Quantifier rank	quantifier depth	The quantifier rank of a formula is the maximal depth of nesting of quantifiers occurring in it.
s_hintikka_formulas	state	Hintikka formulas	Hintikka normal form	Hintikka formulas are the finitely many formulas of a given quantifier rank that completely describe a tuple up to back-and-forth equivalence of that rank.
s_age_of_structure	state	Age of a structure		The age of a structure is the class of all finitely generated structures that embed into it.
s_hereditary_property	axiom	Hereditary property (HP)	HP	A class has the hereditary property when it is closed under taking finitely generated substructures.
s_joint_embedding_property	axiom	Joint embedding property (JEP)	JEP	A class has the joint embedding property when any two of its members embed into a common member.
s_amalgamation_property	axiom	Amalgamation property (AP)	AP	A class has the amalgamation property when any two extensions of a common member can be embedded into a single member compatibly over that member.
s_fraisse_limit	state	Fraïssé limit		The Fraïssé limit of a suitable class is the unique countable ultrahomogeneous structure whose age is that class.
s_ultrahomogeneous_structure	axiom	Ultrahomogeneous structure	homogeneous structure	A structure is ultrahomogeneous when every isomorphism between finitely generated substructures extends to an automorphism of the whole structure.
s_fraisse_theorem	theorem	Fraïssé's theorem		Fraïssé's theorem states that a countable class closed under isomorphism with HP, JEP, and AP has a unique countable ultrahomogeneous structure as its limit.
s_weak_homogeneity	axiom	Weak homogeneity		Weak homogeneity is the property that any isomorphism between finitely generated substructures can be extended to include any one further element.
s_random_graph	state	The random graph (Rado graph)	Rado graph	The random graph is the unique countable graph that is the Fraïssé limit of all finite graphs and satisfies the extension property.
s_extension_property_graphs	axiom	Extension property (graphs)	one-point extension property	The extension property states that for any two disjoint finite sets of vertices there is another vertex joined to all of the first set and none of the second.
s_random_graph_aleph0_categorical	theorem	ℵ₀-categoricity of the random graph		The theory of the random graph is ℵ₀-categorical, having a unique countable model up to isomorphism.
s_orbit_stabiliser	state	Orbit and stabiliser		For a group acting on a structure, the orbit of an element is its image set under the group and the stabiliser is the subgroup fixing it.
s_permutation_group_topology	axiom	Permutation group topology	topology of pointwise convergence	The topology of pointwise convergence on a permutation group has as basic open sets the cosets of pointwise stabilisers of finite tuples.
s_closed_subgroup_aut_theorem	theorem	Closed-subgroup characterization of automorphism groups		A subgroup of the symmetric group on a set is the automorphism group of some structure exactly when it is closed in the topology of pointwise convergence.
s_k_transitivity	state	k-homogeneity / k-transitivity	k-homogeneity	A permutation group is k-transitive when it acts transitively on ordered k-tuples of distinct elements, and k-homogeneous when transitive on k-element subsets.
s_imaginary_elements	state	Imaginary elements	definable quotients	Imaginary elements are equivalence classes of definable equivalence relations on tuples, adjoined to a structure to name quotients.
s_small_index_property	axiom	Small index property		A countable structure has the small index property when every subgroup of its automorphism group of index less than the continuum is open.
s_interpretation	state	Interpretation Γ		An interpretation of one structure in another is given by a definable subset, definable relations on it, and a definable equivalence relation whose quotient rec
s_coordinate_map	state	Coordinate map		The coordinate map of an interpretation is the definable surjection from the interpreting definable set onto the domain of the interpreted structure.
t_interpretation_reduction	technique	Reduction of theories via interpretation		Interpretation reduces questions about the interpreted structure (such as decidability or definability) to corresponding questions about the interpreting struct
s_aeq_construction	state	A^eq / M^eq construction	Meq	The A^eq construction expands a structure by adding a sort for every ∅-definable equivalence relation together with the canonical projection maps.
s_elimination_of_imaginaries	axiom	Elimination of imaginaries		A theory eliminates imaginaries when every definable equivalence class has a definable canonical representative as a tuple of real elements.
s_pseudo_elementary_class	axiom	Pseudo-elementary class (PC)	PC class	A pseudo-elementary class is the class of reducts to a signature of the models of a theory in a larger signature.
t_word_construction_interpretation	technique	Word construction interpreting groups/fields		The word construction interprets a group or field inside a structure by encoding its elements and operations as definable words or coordinate tuples.
s_completeness_theorem	theorem	Gödel completeness theorem	Gödel's completeness theorem	The completeness theorem states that a first-order sentence is a semantic consequence of a theory if and only if it is provable from it.
s_direct_reduced_product	state	Direct and reduced product	reduced product	The direct product of structures is their coordinatewise product, and the reduced product modulo a filter identifies tuples agreeing on a set in the filter.
t_compactness_via_ultraproducts	technique	Compactness via ultraproducts		This technique proves the compactness theorem by building a model of a finitely satisfiable theory as an ultraproduct of models of its finite subsets.
s_diagonal_embedding_ultrapower	state	Diagonal embedding into an ultrapower		The diagonal embedding sends each element of a structure to the class of the constant sequence, giving an elementary embedding into any ultrapower.
s_los_tarski_preservation	theorem	Łoś–Tarski preservation theorem	Tarski–Łoś theorem	The Łoś–Tarski theorem states that a first-order sentence is preserved under substructures if and only if it is logically equivalent to a universal sentence.
s_chang_los_suszko_theorem	theorem	Chang–Łoś–Suszko theorem		The Chang–Łoś–Suszko theorem states that a sentence is preserved under unions of chains if and only if it is equivalent to a universal-existential (∀∃) sentence
s_lyndon_preservation_theorem	theorem	Lyndon's preservation theorem		Lyndon's theorem states that a sentence is preserved under surjective homomorphisms if and only if it is logically equivalent to a positive sentence.
s_craig_interpolation_theorem	theorem	Craig interpolation theorem		The Craig interpolation theorem states that whenever one sentence implies another there is an interpolant in their common language implied by the first and impl
s_beth_definability_theorem	theorem	Beth definability theorem		The Beth definability theorem states that a relation implicitly definable in a theory is explicitly definable by a formula in the remaining language.
s_realizing_omitting_types	state	Realizing and omitting types		A structure realizes a type when some tuple satisfies every formula in it and omits the type when no tuple does.
s_countably_saturated_model	state	Countably saturated model	ℵ₀-saturated model	A countably (ℵ₀-) saturated model is one realizing every complete type over each of its finite parameter subsets.
s_aleph0_categorical_theory	axiom	ℵ₀-categorical theory	omega-categorical theory	A theory is ℵ₀-categorical when all of its countable models are isomorphic.
s_oligomorphic_permutation_group	axiom	Oligomorphic permutation group		A permutation group is oligomorphic when it has only finitely many orbits on n-tuples for every natural number n.
s_existentially_closed_structure	axiom	Existentially closed structure	e.c. structure	A structure is existentially closed in a class when every existential formula with parameters that has a solution in some extension already has a solution in it
s_inductive_theory	axiom	Inductive theory	inductive class	A theory is inductive when the union of any chain of its models is again a model, equivalently when it is axiomatizable by universal-existential sentences.
s_existence_ec_models	theorem	Existence of existentially closed models		Every model of an inductive theory embeds into an existentially closed model of the same theory.
s_model_companion	state	Model companion / model completion	model completion	The model companion of a theory is a model-complete theory with the same universal consequences, and is a model completion when it additionally has amalgamation
s_existence_uniqueness_model_companion	theorem	Existence and uniqueness of the model companion		A theory has at most one model companion, and when it exists it is exactly the theory of the existentially closed models.
s_quantifier_elimination	axiom	Quantifier elimination	QE	A theory has quantifier elimination when every formula is equivalent modulo the theory to a quantifier-free formula in the same free variables.
s_model_complete_prime_implies_qe	theorem	Model-complete with prime model implies QE		A model-complete theory that has a model embeddable in every model (a prime model) admits quantifier elimination.
t_nullstellensatz_via_model_completeness	technique	Nullstellensatz via model completeness		This technique derives Hilbert's Nullstellensatz from the model completeness of algebraically closed fields by transferring solvability up to an extension field
s_kaiser_hull	state	Kaiser hull		The Kaiser hull of a theory is the set of all ∀∃ sentences holding in every existentially closed model, i.e. the largest inductive theory with the same existent
s_stable_theory_type_counting	axiom	Stability via type-counting	stable theory	A theory is λ-stable when over every parameter set of size at most λ there are at most λ complete types.
s_stable_superstable_omega_stable	state	Stable / superstable / ω-stable hierarchy	superstable theory	A theory is stable if it is λ-stable for some λ, superstable if λ-stable for all large λ, and ω-stable if ℵ₀-stable, forming a strict hierarchy.
s_order_property	axiom	Order property	no order property	A theory has the order property when some formula defines an infinite linear order on a sequence of tuples in some model.
s_indiscernible_sequence	axiom	Indiscernible sequence	order-indiscernibles	An indiscernible sequence is an order-indexed family of tuples such that any two increasing tuples of the same length have the same type.
s_set_indiscernibles	axiom	Set indiscernibles		A set of indiscernibles is a family of elements such that every n-element subset realizes the same type irrespective of ordering.
s_ramsey_erdos_rado	theorem	Ramsey / Erdős–Rado theorem	Ramsey's theorem	The Ramsey and Erdős–Rado theorems guarantee that any finite colouring of the finite subsets of a sufficiently large set has an infinite homogeneous subset.
s_em_model_functor	state	EM model / functor	EM functor	The Ehrenfeucht–Mostowski functor sends a linear order to the model generated by an indiscernible sequence of that order type realizing a fixed EM-type.
t_em_construction	technique	EM construction technique		The Ehrenfeucht–Mostowski construction builds models with many automorphisms and few types by generating them from indiscernible sequences indexed by a chosen l
s_forking	state	Forking	non-forking	Forking is a notion of dependence under which an extension of a type adds essential new information relative to a smaller parameter set.
s_heir_coheir	state	Heir and coheir (non-forking extensions)		An heir of a type is a coheir-dual extension that is finitely satisfiable in (resp. definable over) the original parameter set, giving canonical non-forking ext
s_prime_model_over_set	state	Prime model over a set (t.t.)		For a totally transcendental theory, the prime model over a parameter set is the least elementary extension containing that set, embedding elementarily into eve
s_partial_computable_function	axiom	Partial computable function	partial recursive function | φ_e	A partial function on the naturals whose graph is the set of input-output pairs realized by some Turing machine (defined exactly where the machine halts).
s_total_computable_function	axiom	Total computable function	total recursive function | computable function	A total function on the naturals computed by a Turing machine that halts on every input.
s_primitive_recursion_schema	axiom	Primitive recursion schema	primitive recursion	The function-building rule defining f(x,0)=g(x) and f(x,n+1)=h(x,n,f(x,n)), one of the closure operations generating the primitive recursive functions.
s_mu_operator_minimization	axiom	μ-operator (unbounded minimization)	minimization | unbounded search | μ	The search operator producing g(x)=the least y with f(x,y)=0 (undefined if none exists), whose addition to the primitive recursive base yields exactly the parti
s_partial_recursive_functions	axiom	Partial recursive functions	μ-recursive functions	The smallest class of partial functions containing the basic functions (zero, successor, projections) and closed under composition, primitive recursion, and the
s_register_machine	axiom	Register machine	unlimited register machine | URM | counter machine	An abstract computing device with finitely many unbounded-integer registers manipulated by increment/decrement-and-jump instructions, an alternative model defin
s_church_turing_thesis	axiom	Church–Turing Thesis	Church's thesis | Turing's thesis	The informal claim that every function effectively calculable by an algorithm is computable by a Turing machine (equivalently, is partial recursive).
s_computability_thesis_soare	axiom	Computability Thesis (Soare)	Soare's thesis	Soare's reformulation asserting that the intuitively computable functions coincide with the Turing-computable functions, motivating the terminology 'computable/
s_oracle_turing_machine	axiom	Oracle Turing machine	o-machine | relativized Turing machine	A Turing machine equipped with a query tape that may consult an oracle set A for membership answers during its computation, the device underlying relative compu
s_use_function	axiom	Use function	use | u(A;e,x)	The function u(e,x,s) giving the largest oracle argument queried during the halting computation of Φ_e^A(x) (by stage s), bounding which oracle bits affect the 
s_godel_index	axiom	Gödel number / index	index | program number | e	A natural number coding a Turing machine (or partial recursive program) under a fixed effective enumeration, so that e is an index for the e-th partial computab
s_turing_jump_A_prime	axiom	Turing jump A′	jump | A'	The halting set relativized to oracle A, A′={e : Φ_e^A(e)↓}, the canonical operator strictly raising Turing degree.
s_arithmetical_hierarchy	axiom	Arithmetical hierarchy	Kleene arithmetical hierarchy | Σ0n/Π0n/Δ0n	The classification of subsets of ℕ by the number of alternating number quantifiers prefixing a computable matrix, with Σ0_n existential-leading, Π0_n universal-
s_many_one_reducibility	axiom	Many-one (m) reducibility	m-reducibility | ≤m	A≤_m B iff there is a total computable f with x∈A ⟺ f(x)∈B for all x.
s_one_one_reducibility	axiom	One-one (1) reducibility	1-reducibility | ≤1	A≤_1 B iff there is a total computable injection f with x∈A ⟺ f(x)∈B for all x.
s_truth_table_reducibility	axiom	Truth-table (tt) reducibility	tt-reducibility | ≤tt	A≤_tt B iff there is a total computable procedure assigning to each x a finite Boolean truth-table condition on B-membership of finitely many points that decide
s_weak_truth_table_reducibility	axiom	Weak truth-table (wtt) reducibility	wtt-reducibility | bounded Turing reducibility | ≤wtt	A≤_wtt B iff A≤_T B via an oracle reduction whose use is bounded by a computable function of the input.
s_simple_set	axiom	Simple set		A c.e. set whose complement is infinite yet contains no infinite c.e. subset (Post's first weakening of completeness).
s_immune_set	axiom	Immune set		An infinite set containing no infinite computably enumerable subset.
s_productive_set	axiom	Productive set		A set A with a total computable productive function p such that whenever W_e⊆A then p(e)∈A∖W_e, witnessing that A is not c.e. in an effective way.
s_creative_set	axiom	Creative set		A computably enumerable set whose complement is productive; equivalently a many-one complete c.e. set such as K.
s_diagonal_halting_set_K0	state	Diagonal halting set K0	K_0 | general halting set	The set K0={⟨e,x⟩ : Φ_e(x)↓} of pairs encoding a machine and an input on which it halts, a c.e. many-one complete set.
s_hypersimple_set	state	Hypersimple set		A c.e. set whose complement is hyperimmune, i.e. its principal (enumeration-ordered) function is not dominated by any computable function.
s_hyperimmune_set	state	Hyperimmune set	h-immune set	An infinite set whose principal function (listing its elements in increasing order) is not majorized by any computable function.
s_hyperhypersimple_set	state	Hyperhypersimple set	hhsimple set	A c.e. set whose complement cannot be covered by any uniformly computable array of finite sets meeting it in exactly one block each (strengthening hypersimplici
s_maximal_set	state	Maximal set		A c.e. set whose complement is infinite and cannot be split into two infinite pieces by any c.e. set, i.e. it is a coatom in the lattice of c.e. sets modulo fin
s_stage_approximation_Phi	state	Stage approximation Φ_{e,s}^A(x)	finite-stage approximation | Φ_{e,s}	The s-step truncation of the oracle computation Φ_e^A(x), which converges iff the full computation halts within s stages, providing the computable approximation
s_enumeration_theorem	theorem	Enumeration Theorem	Universal function theorem	There is a single partial computable function (universal function) that, given an index e and input x, simulates φ_e(x).
s_kleene_normal_form_theorem	theorem	Kleene Normal Form Theorem	Normal form theorem | Kleene T-predicate	Every partial computable function can be written as U(μs.T(e,x,s)) for a fixed primitive recursive Kleene T-predicate and primitive recursive output U, using a 
s_ce_equals_sigma01	theorem	c.e. = Σ0_1 characterization	projection characterization of c.e. sets | domain/range characterization | Σ01 normal form for c.e. sets	A set is computably enumerable iff it is the domain of a partial computable function iff it is the projection of a computable relation iff it is definable by a 
s_post_complementation_theorem	theorem	Post's Complementation Theorem	Complementation theorem | Post's theorem (computable iff c.e. and co-c.e.)	A set is computable (decidable) if and only if both it and its complement are computably enumerable.
s_recursion_theorem_with_parameters	theorem	Recursion Theorem with parameters	uniform recursion theorem | parametrized fixed-point theorem	A uniform (parametrized) version of Kleene's recursion theorem producing a computable function of the parameters that returns fixed-point indices uniformly.
s_myhill_isomorphism_theorem	theorem	Myhill Isomorphism Theorem	Myhill's theorem	Two sets that are one-one reducible to each other are computably isomorphic, i.e. there is a computable permutation of ℕ carrying one onto the other.
s_myhill_creative_characterization	theorem	Myhill characterization of creative sets	creative = m-complete	The creative sets are exactly the many-one complete c.e. sets, and all are computably isomorphic to the halting set K.
s_m_completeness_of_K	theorem	m-completeness of K	K is m-complete	Every computably enumerable set many-one reduces to the halting set K, so K is m-complete (indeed 1-complete) among c.e. sets.
s_jump_theorem	theorem	Jump Theorem	properties of the jump | jump monotonicity	The Turing jump satisfies A<_T A′ strictly, is monotone (A≤_T B implies A′≤_m B′ hence A′≤_T B′), and A′ is c.e. in and complete for A.
s_posts_theorem	theorem	Post's Theorem	Post's theorem (hierarchy/jump)	Connects the arithmetical hierarchy to iterated jumps: a set is Σ0_{n+1} iff it is c.e. in the n-th jump 0^{(n)}, and Δ0_{n+1} comprises exactly the sets Turing
s_arithmetical_hierarchy_theorem	theorem	Hierarchy Theorem (arithmetical hierarchy is proper)	arithmetical hierarchy proper | strictness of the arithmetical hierarchy	For every n the arithmetical hierarchy is strict, with Σ0_n, Π0_n, and Δ0_n all distinct, witnessed by the n-th jump.
s_limit_lemma	theorem	Shoenfield Limit Lemma	Limit lemma | Shoenfield's lemma | Δ02 = computable in 0'	A set (or function) is Turing reducible to 0′ if and only if it is the pointwise limit of a computable sequence of approximations; equivalently A is Δ0_2.
s_modulus_lemma	theorem	Modulus Lemma	modulus of convergence lemma	If A has a computable approximation converging to a Δ0_2 set, then any function dominating its modulus of convergence computes A; in particular every c.e. set h
s_post_simple_set_existence	theorem	Existence of a simple set (Post)	Post's simple set	Post constructed a computably enumerable set that is simple, hence c.e., noncomputable, and not many-one complete, advancing toward Post's problem.
s_productive_not_ce	theorem	Productive sets are not c.e.		A productive set cannot be computably enumerable, since its productive function effectively escapes every candidate enumeration W_e⊆A.
s_relativization_principle	theorem	Relativization principle	relativization	The meta-principle that results proved about computable functions and c.e. sets remain valid uniformly when all notions are relativized to an arbitrary oracle A
t_dovetailing	technique	Dovetailing	interleaving computations	A scheduling method that runs infinitely many partial computations together by advancing each one finitely at successive stages, ensuring every computation that
t_pairing_function	technique	Pairing function / coding of tuples	Cantor pairing | sequence coding | tupling	A computable bijection ⟨·,·⟩:ℕ²→ℕ (e.g. Cantor's polynomial) with computable inverses, extended to code finite tuples and sequences as single numbers.
t_construction_by_stages_markers	technique	Construction by stages with movable markers	movable markers | stage construction	A method that builds a c.e. set or function in stages, tracking the current candidate witness for each requirement by a marker that is moved (released) when its
t_permitting	technique	Permitting method	permitting argument | C-permitting	A construction technique that makes a built set computable in a given c.e. set C by only acting on a requirement at a stage when C newly enumerates a small elem
t_reduction_to_K	technique	Reduction to K (proving undecidability)	reduction from the halting problem	A proof method establishing undecidability of a set P by exhibiting a computable many-one reduction from the halting set K (or another known undecidable set) to
t_computable_approximation_delta02	technique	Computable approximation (Δ0_2 method)	finite approximation method | stage approximation method	The technique of representing a Δ0_2 set or function as the limit of a computable stagewise approximation, the workhorse behind Limit-Lemma and injury construct
s_turing_equivalence	axiom	Turing equivalence (≡_T)	≡T	The equivalence relation A≡_T B holding when A≤_T B and B≤_T A, i.e. each set computes the other.
s_turing_degree	axiom	Turing degree	degree of unsolvability | T-degree	An equivalence class of subsets of ℕ under Turing equivalence, representing a level of relative computability.
s_degree_structure_D	axiom	Degree structure D	the degrees | D | global degree structure	The upper semilattice of all Turing degrees partially ordered by relative computability, with least element 0, join operation, and the jump operator.
s_join_of_degrees	state	Join of degrees (a∨b)	least upper bound of degrees | a∨b | A⊕B	The least upper bound of two Turing degrees, computed by the degree of the disjoint sum A⊕B={2x:x∈A}∪{2x+1:x∈B}.
s_degree_0	state	Degree 0	zero degree | the computable degree	The least Turing degree, consisting precisely of the computable (decidable) sets.
s_degree_0_prime	state	Degree 0′ (zero jump)	0' | the halting degree | zero jump	The Turing degree of the halting problem, the jump of 0 and the greatest c.e. degree.
s_jump_hierarchy	state	Jump hierarchy (0,0′,0″,…)	iterated jumps | 0,0',0''	The strictly increasing sequence of iterated jumps of 0, giving canonical representatives 0^{(n)} of degrees high in the arithmetical hierarchy.
s_ce_degrees_R	state	c.e. degrees R	R | the c.e. degrees | r.e. degrees	The substructure of Turing degrees containing a computably enumerable set, an upper semilattice with least element 0 and greatest element 0′.
s_low_set	state	Low set / degree	low degree | low	A set (or degree) whose jump is as low as possible, A′≡_T 0′, so it computes nothing extra at the jump level.
s_lown_degree	state	Low_n degree	low_n | Low_n	A degree whose n-th jump equals 0^{(n)}, generalizing lowness to higher jumps.
s_high_set	state	High set / degree	high degree | high	A degree below 0′ whose jump is as high as possible, A′≡_T 0″, so it computes a function dominating all computable functions.
s_highn_degree	state	High_n degree	high_n | High_n	A degree below 0′ whose n-th jump is maximal, A^{(n)}≡_T 0^{(n+1)}.
s_high_low_hierarchy	state	High/Low hierarchy	high/low hierarchy	The classification of c.e. (or Δ0_2) degrees by how their iterated jumps compare to the corresponding jumps of 0, forming nested Low_n and High_n classes.
s_superlow_degree	state	Superlow degree	superlow	A degree whose jump is truth-table (not merely Turing) equivalent to 0′, a uniform strengthening of lowness.
s_hyperimmune_free_degree	state	Hyperimmune-free degree	computably dominated degree | hyperimmune-free	A nonzero degree every function of which is majorized by a computable function (equivalently, every f≤_T a is computably dominated).
s_hyperimmune_degree	state	Hyperimmune degree	hyperimmune degree	A degree that computes a function not majorized by any computable function (i.e. computes a hyperimmune set's principal function).
s_GL1_degree	state	GL1 degree (generalized low 1)	generalized low | GL_1 | GL1	A degree satisfying the generalized lowness condition a′≡_T a∨0′, the relativized analogue of lowness not requiring a≤_T 0′.
s_GL2_degree	state	GL2 degree (generalized low 2)	generalized low 2 | GL_2 | GL2	A degree satisfying a″≡_T (a∨0′)′; the complement GL2̄ (a not GL2) marks degrees of strong domination/escape properties.
s_minimal_degree	state	Minimal degree	degree minimal over 0	A nonzero Turing degree with no degree strictly between it and 0, an atom of the degree ordering.
s_1_generic_set	state	1-generic set / degree	1-generic | weakly 1-generic	A set G that is Cohen-generic for one-quantifier (Σ0_1) conditions: for each c.e. set of strings, G either meets it or has an initial segment with no extension 
s_n_generic_degree	state	n-generic set / degree	n-generic	A set generic for arithmetical conditions of complexity Σ0_n, refining 1-genericity up the hierarchy.
s_ershov_hierarchy	state	Ershov hierarchy	difference hierarchy | Boolean hierarchy over c.e. sets	The fine classification of Δ0_2 sets by the number (and ordinal type) of mind-changes allowed in a computable approximation, with the finite levels being the n-
s_n_ce_set	state	n-c.e. set	n-r.e. set | n-c.e.	A set admitting a computable approximation A_s that changes its decision on each x at most n times (1-c.e. = c.e.).
s_dce_set	state	d.c.e. set / degree	difference c.e. | 2-c.e. set | d-c.e.	A difference of two computably enumerable sets (the 2-c.e. sets), and the degrees they realize.
s_e_state	state	e-state	e-state of an element	In the maximal/dense-set constructions, the binary vector recording which of the first e c.e. sets a given element currently belongs to, maximized to push the c
s_posts_problem	state	Post's Problem	Post problem	The question, solved affirmatively by Friedberg and Muchnik, of whether there exists a computably enumerable set whose Turing degree is strictly between 0 and 0
s_kleene_post_theorem	theorem	Kleene–Post Theorem	Kleene–Post incomparable degrees below 0'	There exist Turing degrees a and b below 0′ that are incomparable, constructed by the finite extension method (though not necessarily c.e.).
s_incomparable_ce_degrees	theorem	Existence of incomparable c.e. degrees	incomparable c.e. degrees	There are computably enumerable sets of incomparable Turing degree, the content of the Friedberg–Muchnik solution to Post's problem realized within R.
s_low_basis_theorem	theorem	Low Basis Theorem (Jockusch–Soare)	Jockusch–Soare low basis theorem	Every nonempty Π0_1 class (e.g. the paths of an infinite computable binary tree) contains a member of low Turing degree.
s_hyperimmune_free_basis_theorem	theorem	Hyperimmune-free Basis Theorem	Jockusch–Soare hyperimmune-free basis theorem	Every nonempty Π0_1 class contains a member of hyperimmune-free (computably dominated) degree.
s_minimal_pair_theorem	theorem	Minimal Pair Theorem (Lachlan–Yates)	Lachlan–Yates minimal pair	There exist nonzero computably enumerable degrees a and b whose only common lower bound is 0, a minimal pair in R.
s_spector_minimal_degree	theorem	Existence of a minimal degree (Spector)	Spector minimal degree theorem	Spector proved, by forcing with computable perfect trees, that minimal Turing degrees exist.
s_exact_pair_theorem	theorem	Exact Pair Theorem	exact pair (Kleene–Post / Spector)	Every countable ideal of Turing degrees has an exact pair: two degrees whose set of common lower bounds is exactly that ideal.
s_lattice_embedding_into_ce_degrees	theorem	Embedding lattices into the c.e. degrees	distributive lattice embedding theorem	Every finite (indeed countable) distributive lattice embeds, preserving join and meet, into the upper semilattice of computably enumerable degrees.
s_nondistributivity_of_ce_degrees	theorem	Nondistributivity of the c.e. degrees (Lachlan)	Lachlan nondistributivity	Lachlan showed the c.e. degrees fail distributivity by embedding the nonmodular/nondistributive five-element lattices, so R is not a distributive lattice.
s_nondiamond_theorem	theorem	Nondiamond Theorem (Lachlan)	Lachlan nondiamond theorem	Lachlan proved that the diamond lattice cannot be embedded into the c.e. degrees preserving both least and greatest elements (0 and 0′).
s_robinson_low_splitting	theorem	Robinson Low Splitting / interpolation	Robinson splitting theorem | Robinson interpolation	Robinson's theorem providing splitting of c.e. degrees by low c.e. degrees together with an interpolation/density property, refining Sacks splitting with lownes
s_friedberg_jump_inversion	theorem	Friedberg Jump Inversion Theorem	Friedberg completeness criterion | jump inversion (Friedberg)	For every degree c≥0′ there is a degree a whose jump is c (and a∨0′=c), so the jump operator maps onto the cone above 0′.
t_finite_injury_priority_method	technique	Finite-injury priority method	finite injury method | Friedberg–Muchnik method	A stagewise construction satisfying ranked requirements R_0,R_1,… in which lower-priority requirements may injure higher ones only finitely often, so each is ev
t_priority_requirements_and_injury	technique	Priority requirements and injury	injury | priority ordering	The bookkeeping framework assigning a fixed priority order to requirements and declaring a requirement 'injured' when a higher-priority action undoes its curren
t_kleene_post_finite_extension	technique	Kleene–Post finite extension method	finite extension method | forcing the jump	A forcing-style construction building characteristic functions by successive finite extensions, each diagonalizing one requirement, with the whole construction 
t_length_of_agreement_function	technique	Length-of-agreement function	agreement length | ℓ(e,s)	The measure ℓ(e,s) of the longest initial segment on which a candidate reduction Φ_e^A agrees with target B at stage s, used to detect when a requirement appear
t_restraint_function	technique	Restraint function	restraint | r(e,s)	A function r(e,s) imposing a stagewise lower bound below which a strategy protects its computations, preventing lower-priority requirements from enumerating sma
t_hat_trick	technique	Hat trick	hatting | Â_s	A bookkeeping device in infinite-injury arguments that replaces A_s by the 'hatted' approximation reflecting only changes below the current restraint at true st
t_infinite_injury_priority_method	technique	Infinite-injury priority method	infinite injury method | 0''-priority argument	A priority construction in which requirements may be injured infinitely often, organized on a tree of strategies so that along the true path each requirement is
t_tree_of_strategies	technique	Tree of strategies	strategy tree | tree method	A tree whose nodes are strategy versions parametrized by finitely many guesses about the outcomes of higher-priority requirements, with the correct guesses lyin
s_true_path	state	True path	true path of the construction	The leftmost path through the tree of strategies visited infinitely often, along which every requirement's correct guesses are realized.
t_thickness_lemma	technique	Thickness Lemma	thickness requirement	A lemma/technique in infinite-injury constructions guaranteeing a built c.e. set contains 'thick' (cofinite) portions of prescribed columns of a given set, used
t_pinball_machine	technique	Pinball machine	pinball construction	A visualization/technique for 0‴ and density constructions in which elements ('balls') roll past priority-ordered 'gates' that hold or release them, regulating 
t_0_triple_prime_priority_argument	technique	0‴-priority argument	0''' argument | triple-jump priority method	A higher-level priority construction whose verification requires three jumps of bookkeeping (a tree-of-trees), e.g. in Lachlan's nondistributivity and density r
t_minimal_pair_construction	technique	Minimal pair construction	minimal pair method | gap-cogap argument	The technique of building two c.e. sets while preserving, for each potential common reduction, agreement that forces any set computable from both to be computab
t_spector_tree_forcing	technique	Spector's construction (tree forcing)	perfect tree forcing | Spector forcing	A forcing construction with a descending sequence of computable perfect (Spector) trees, each splitting or forcing total/partial behavior, yielding a real of mi
t_cohen_forcing_in_arithmetic	technique	Cohen forcing in arithmetic	arithmetic forcing | finite-condition forcing	The use of finite binary strings as Cohen forcing conditions to build generic reals deciding arithmetical statements, producing n-generic sets and degrees.
t_induction_on_wffs	technique	Induction principle on wffs		
s_truth_table	technique	Truth table method		
t_substitution_of_term	technique	Substitution of a term for a variable		
s_generalization_on_constants	technique	Generalization on constants		
t_alphabetic_variant	technique	Alphabetic variant (bound-variable renaming)		
t_many_sorted_to_one_sorted	technique	Reduction many-sorted to one-sorted		
s_sentence_symbol	axiom	Sentence symbol		Atomic propositional symbols A_n forming the alphabet of sentential logic.
s_sentential_wff	state	Sentential well-formed formula		Strings built from sentence symbols and connectives by the formation rules.
s_unique_readability_sentential	theorem	Unique readability (sentential)		Every sentential wff has a unique parse tree / decomposition.
s_truth_assignment	state	Truth assignment		A function assigning truth values to sentence symbols.
s_extended_truth_assignment	theorem	Extended truth assignment		Unique extension of a truth assignment to all wffs respecting connectives.
s_tautology	state	Tautology		A wff true under every truth assignment.
s_tautological_implication	state	Tautological implication		Sigma tautologically implies tau iff every assignment satisfying Sigma satisfies tau.
s_tautological_equivalence	state	Tautological equivalence		Two wffs with identical truth values under all assignments.
s_functional_completeness	theorem	Functional completeness		Every boolean function is realized by a wff over a complete connective set.
s_normal_forms_sentential	theorem	Normal forms (DNF/CNF)		Every wff is equivalent to one in disjunctive and conjunctive normal form.
s_compactness_sentential	theorem	Compactness (sentential)		A set of wffs is satisfiable iff every finite subset is satisfiable.
s_decidability_of_tautologies	theorem	Decidability of tautologies		There is an effective procedure deciding whether a wff is a tautology.
s_switching_circuits	state	Switching circuits		Application of sentential logic to boolean/switching circuit design.
s_variable_assignment	state	Variable assignment		Function assigning domain elements to first-order variables.
s_logical_implication_validity	state	Logical implication and validity		Gamma logically implies phi iff every model of Gamma models phi; validity is implication from empty set.
s_substitution_lemma	theorem	Substitution lemma		Satisfaction of phi(x/t) relates to satisfaction of phi under modified assignment.
s_homomorphism_theorem_logic	theorem	Homomorphism theorem		Homomorphisms preserve satisfaction of quantifier-free formulas; isomorphisms preserve all.
s_logical_axiom_groups	axiom	Logical axiom groups		The six groups of logical axioms (tautologies, quantifier, equality, etc.).
s_deduction_relation	state	Deducibility relation		Gamma proves phi via logical axioms and modus ponens.
s_generalization_theorem	theorem	Generalization theorem		If Gamma proves phi and x not free in Gamma, then Gamma proves forall x phi.
s_soundness_theorem	theorem	Soundness theorem		Every deducible formula is logically implied (provability implies validity).
s_consistency	state	Consistency		A theory is consistent iff it does not prove a contradiction.
s_lindenbaum_lemma	theorem	Lindenbaum lemma		Every consistent set extends to a maximal consistent set.
s_enumerability_of_theorems	theorem	Enumerability of theorems		The set of theorems of an effectively axiomatized theory is recursively enumerable.
s_prenex_normal_form	theorem	Prenex normal form		Every formula is equivalent to one with all quantifiers in front.
s_skolem_normal_form	state	Skolem normal form		Equisatisfiable universal form obtained by introducing Skolem functions.
s_herbrand_normal_form	state	Herbrand normal form		Dual Skolemization yielding existential/Herbrand form.
s_language_of_number_theory	axiom	Language of number theory		Language with 0, S, +, *, < for arithmetic.
s_standard_structure_N	axiom	Standard structure N		The standard model of arithmetic on the natural numbers.
s_robinson_arithmetic_Q	axiom	Robinson arithmetic Q		Finitely axiomatized weak arithmetic Q.
s_numeral	state	Numeral		Term S...S0 representing a natural number.
s_definable_relation_in_N	state	Definable relation in N		A relation defined by a first-order formula over the standard model.
s_representability_theorem	theorem	Representability theorem		Every recursive function/relation is representable in Q.
s_godel_beta_function	state	Godel beta function		Beta function coding finite sequences by arithmetic means.
s_recursiveness_of_syntax	theorem	Recursiveness of syntax		Syntactic relations (proof, substitution) are primitive recursive after arithmetization.
s_fixed_point_lemma	theorem	Fixed-point (diagonal) lemma		For each phi(x) there is a sentence sigma with proves sigma<->phi(#sigma).
s_undecidability_of_arithmetic	theorem	Undecidability of arithmetic		True arithmetic / Q is undecidable.
s_church_theorem_validity	theorem	Church's theorem		The set of valid first-order sentences is undecidable.
s_essential_undecidability	state	Essential undecidability		A theory all of whose consistent extensions are undecidable.
s_decidable_undecidable_theories	state	Decidable and undecidable theories		Classification of theories by decidability.
s_derivability_conditions	axiom	Hilbert-Bernays derivability conditions		The three derivability conditions on the provability predicate.
s_godel_second_incompleteness	theorem	Godel second incompleteness theorem		No consistent effective theory extending Q proves its own consistency.
s_incompleteness_applications_set_theory	state	Incompleteness applications to set theory		Consequences of incompleteness for ZF/ZFC.
s_second_order_language	axiom	Second-order language		Language permitting quantification over relations and functions.
s_standard_full_semantics_sol	axiom	Standard (full) second-order semantics		Semantics where second-order variables range over the full power set.
s_categoricity_second_order	theorem	Categoricity in second-order logic		Second-order axioms (e.g. PA2) are categorical.
s_failure_compactness_completeness_sol	theorem	Failure of compactness/completeness (SOL)		Full second-order logic lacks compactness and a complete proof system.
s_general_henkin_structures	axiom	General (Henkin) structures		Second-order structures with restricted relation domains.
s_completeness_general_semantics	theorem	Completeness for general semantics		Second-order logic with Henkin semantics is complete.
s_many_sorted_logic	axiom	Many-sorted logic		First-order logic with multiple sorts of variables.
s_topological_dynamical_system	axiom	Topological dynamical system (cascade)		A pair consisting of a compact metric space together with a continuous self-map (discrete time, a cascade) or continuous flow, generating orbits under iteration
s_topological_transitivity	axiom	Topological transitivity		A system is topologically transitive if for every pair of nonempty open sets U,V there exists n with fⁿ(U)∩V≠∅, equivalently it has a dense orbit on a complete 
s_topological_mixing	axiom	Topological mixing		A system is topologically mixing if for every pair of nonempty open sets U,V there exists N such that fⁿ(U)∩V≠∅ for all n≥N.
s_minimality_topological	axiom	Minimality (topological)		A topological system is minimal if every orbit is dense, equivalently the only closed invariant subsets are ∅ and X.
s_nonwandering_set	state	Nonwandering set Ω(f)		The set of points x such that every neighborhood U of x satisfies fⁿ(U)∩U≠∅ for some n>0, a closed invariant set containing all recurrent behavior.
s_chain_recurrent_set	state	Chain recurrent set / ε-chains		The set of points x admitting, for every ε>0, a periodic ε-pseudo-orbit (ε-chain) from x to itself, where each successive jump is within ε of the true image.
s_birkhoff_recurrence_theorem	theorem	Birkhoff recurrence theorem		Every topological dynamical system on a nonempty compact metric space contains a nonempty minimal closed invariant subset, hence a recurrent (almost periodic) p
s_topological_entropy_spanning_separated	axiom	Topological entropy via spanning/separated sets (Bowen–Dinaburg)		The exponential growth rate of the minimal cardinality of (n,ε)-spanning sets (equivalently maximal (n,ε)-separated sets) measuring orbit complexity at finite r
s_topological_entropy_open_covers	axiom	Topological entropy via open covers (Adler–Konheim–McAndrew)		The supremum over open covers U of the exponential growth rate of the minimal cardinality of subcovers of the joined cover ⋁ f^{-i}U, the original definition of
s_periodic_orbit_growth_pn	state	Periodic orbit growth rate p_n(f)		The sequence counting the number of fixed points of fⁿ (periodic points of period dividing n), whose exponential growth rate bounds topological entropy from bel
s_artin_mazur_zeta_function	state	Artin–Mazur zeta function		The formal power series generating function exponentiating the periodic-point counts p_n(f), encoding the dynamical periodic structure of f.
s_zeta_rationality_sft	theorem	Rationality of the zeta function for subshifts of finite type		For a topological Markov chain with transition matrix A, the Artin–Mazur zeta function equals the reciprocal of det(I−zA), hence is rational.
s_li_yorke_theorem	theorem	Li–Yorke theorem (period three implies chaos)		A continuous interval map possessing a periodic point of period three has periodic points of every period and an uncountable scrambled set exhibiting Li–Yorke c
s_denjoy_inequality	theorem	Denjoy inequality		For a C² (or bounded-variation-derivative) circle diffeomorphism with irrational rotation number, the total variation of log of the derivative of fⁿ over a fund
s_denjoy_counterexample	state	Denjoy counterexample		A C¹ (but not C²) circle diffeomorphism with irrational rotation number that is not topologically conjugate to a rotation, possessing a wandering interval and a
s_poincare_classification_circle	theorem	Poincaré classification theorem for circle homeomorphisms		An orientation-preserving circle homeomorphism has a periodic orbit iff its rotation number is rational, and if the rotation number is irrational and the map is
s_rotation_number_rational_iff_periodic	theorem	Rationality of rotation number iff periodic orbit		An orientation-preserving circle homeomorphism has rotation number p/q in lowest terms iff it has a periodic orbit of period q.
s_morse_smale_system	state	Morse–Smale system		A diffeomorphism (or flow) whose nonwandering set consists of finitely many hyperbolic periodic orbits whose stable and unstable manifolds intersect transversal
s_expanding_map_of_circle_Em	state	Expanding map of the circle E_m		The degree-m self-cover of the circle multiplying the angle by an integer m of absolute value at least 2, the canonical uniformly expanding example with topolog
s_doubling_map_dynamics	state	Doubling map		The angle-doubling self-map of the circle, a degree-two expanding map measure-preserving for Lebesgue and conjugate to the one-sided full 2-shift.
s_linear_toral_automorphism_cat_map	state	Linear toral automorphism (cat map)		The automorphism of the torus induced by an integer matrix A of determinant ±1, hyperbolic (Anosov) when A has no eigenvalue on the unit circle, the Arnold cat 
s_circle_rotation_R_alpha	state	Circle rotation R_α		Rigid rotation of the circle by angle α, which is minimal and uniquely ergodic iff α is irrational and periodic iff α is rational.
s_full_shift_sigma_N	state	Full shift σ on Σ_N		The left-shift on the space of bi-infinite (or one-sided) sequences over an N-symbol alphabet, the universal model of symbolic dynamics with topological entropy
s_topological_markov_chain	state	Topological Markov chain / subshift of finite type Σ_A		The shift restricted to the closed invariant set of sequences whose successive symbol transitions are permitted by a 0–1 transition matrix A, the basic building
s_spectral_decomposition_topological_markov_chain	theorem	Spectral decomposition for topological Markov chains		An irreducible topological Markov chain decomposes into finitely many disjoint closed sets cyclically permuted by σ on each of which a power of σ is topological
s_cylinder_set_symbolic	state	Cylinder set (symbolic dynamics)		The set of sequences agreeing with a prescribed finite word on a fixed block of coordinates, the basic clopen sets generating the topology and Borel structure o
s_inclination_lemma	theorem	Inclination lemma (λ-lemma)		If a disk D transversally intersects the stable manifold of a hyperbolic fixed point, its forward iterates accumulate in the C¹ topology on the unstable manifol
s_anosov_flow	state	Anosov flow		A smooth flow whose tangent bundle splits continuously and invariantly into uniformly contracting stable, flow-direction, and uniformly expanding unstable subbu
s_hyperbolic_linear_map	state	Hyperbolic linear map		An invertible linear operator with no eigenvalue of modulus one, so that ℝⁿ splits into complementary contracting and expanding invariant subspaces.
s_unstable_subspace_Eu	state	Unstable subspace E^u		The maximal invariant subspace on which a hyperbolic linear map (or the differential of a hyperbolic system) is uniformly expanding, spanned by generalized eige
s_local_product_structure	state	Local product structure		The property of a locally maximal hyperbolic set that nearby points x,y have a unique intersection of the local stable manifold of x with the local unstable man
s_expansiveness	axiom	Expansiveness		A homeomorphism is expansive if there is a constant c such that any two distinct points are eventually separated by more than c under forward or backward iterat
t_cone_criterion_hyperbolicity	technique	Cone criterion (cone field) for hyperbolicity		A method establishing hyperbolicity of an invariant set by exhibiting continuous fields of stable and unstable cones that are mapped strictly inside themselves 
t_adapted_lyapunov_norm	technique	Adapted (Lyapunov) norm		A reparametrized Riemannian metric, equivalent to the original, in which the contraction on E^s and expansion on E^u of a hyperbolic system become immediate (on
t_graph_transform_method	technique	Graph transform method		The Hadamard contraction-mapping argument that constructs stable and unstable manifolds as fixed points of the operator sending the graph of a Lipschitz functio
s_expanding_map_structural_stability_shub	theorem	Structural stability of expanding maps (Shub)		Every C¹ expanding endomorphism of a compact manifold is structurally stable and topologically conjugate to a fixed expanding linear model on an infranilmanifol
s_anosov_structural_stability	theorem	Anosov structural stability theorem		Every Anosov diffeomorphism is structurally stable: any C¹-small perturbation is topologically conjugate to it via a homeomorphism close to the identity.
s_holder_continuity_hyperbolic_splitting	theorem	Hölder continuity of the hyperbolic splitting		For a hyperbolic set of a C¹⁺ᵅ diffeomorphism the stable and unstable subbundles depend Hölder-continuously on the base point, even when they fail to be C¹.
s_no_cycle_condition	axiom	No-cycle condition		The requirement that the basic sets in Smale's spectral decomposition admit no cyclic chain of heteroclinic connections, equivalently they can be totally ordere
s_omega_stability_theorem	theorem	Ω-stability theorem		An Axiom A diffeomorphism satisfying the no-cycle condition is Ω-stable: any C¹-nearby diffeomorphism is topologically conjugate to it on its nonwandering set.
s_strong_transversality_condition	axiom	Strong transversality condition		The condition that all stable and unstable manifolds of an Axiom A system intersect transversally, characterizing (with Axiom A) C¹ structural stability via the
s_structural_stability_theorem_RRM	theorem	C¹ structural stability theorem (Robbin–Robinson–Mañé)		A C¹ diffeomorphism is C¹ structurally stable if and only if it satisfies Axiom A together with the strong transversality condition.
t_filtration_dynamics	technique	Filtration (dynamical)		A nested sequence of compact submanifolds with boundary, each mapped into its own interior, used to isolate the chain components/basic sets of a system and orde
t_markov_partition_hyperbolic	technique	Markov partition for hyperbolic sets		A finite cover of a hyperbolic set by proper rectangles (products of stable and unstable pieces) whose boundaries map compatibly so that itineraries give a semi
s_markov_partition_existence_sinai_bowen	theorem	Existence of Markov partitions (Sinai–Bowen)		Every locally maximal hyperbolic set admits finite Markov partitions of arbitrarily small diameter, yielding a finite-to-one symbolic coding by a subshift of fi
s_smale_solenoid	state	Smale solenoid attractor		The hyperbolic attractor obtained as the nested intersection of images of a solid torus under an embedding wrapping it twice (or k times) around longitudinally,
s_da_attractor	state	DA (derived from Anosov) attractor		A hyperbolic attractor produced by altering a toral Anosov automorphism near a fixed point so the fixed point becomes a repeller, yielding a non-invertible-on-i
s_plykin_attractor	state	Plykin attractor		A planar (or 2-sphere) hyperbolic attractor with one-dimensional unstable foliation locally homeomorphic to the product of a Cantor set and an interval, the low
s_absolute_continuity_of_foliations	theorem	Absolute continuity of stable/unstable foliations		For a C¹⁺ᵅ (partially) hyperbolic system the stable and unstable foliations are absolutely continuous, i.e. their holonomy maps preserve sets of measure zero, t
t_hopf_argument	technique	Hopf argument		The method proving ergodicity of a volume-preserving hyperbolic system by showing Birkhoff averages are constant along stable and unstable leaves and using abso
s_geodesic_flow_anosov_negative_curvature	theorem	Geodesic flow on negatively curved manifold is Anosov		The geodesic flow on the unit tangent bundle of a compact Riemannian manifold of negative sectional curvature is an Anosov flow, with stable/unstable bundles ta
s_volume_preserving_anosov_ergodic	theorem	Ergodicity of volume-preserving Anosov systems		A C¹⁺ᵅ Anosov diffeomorphism or flow preserving a smooth volume is ergodic, mixing, and in fact Bernoulli with respect to that volume.
s_livsic_theorem	theorem	Livšic theorem		For a transitive hyperbolic system, a Hölder cocycle that sums to zero over every periodic orbit is a Hölder coboundary, i.e. cohomologous to zero.
s_cohomological_equation_coboundary	state	Cohomological equation / coboundary (dynamics)		The equation expressing an observable φ as the coboundary of a transfer (transformation) u under the dynamics f, central to questions of cohomology and equivale
s_zeta_rationality_axiom_a	theorem	Rationality/meromorphy of the zeta function for Axiom A systems		For an Axiom A diffeomorphism the Artin–Mazur zeta function is rational (via Markov partitions), and the associated Ruelle dynamical zeta function for flows is 
t_smale_program_synthesis	technique	Smale's differentiable dynamics program		The overarching synthesis organizing hyperbolic dynamics around horseshoes, Axiom A, the spectral decomposition, structural and Ω-stability, and symbolic coding
s_unique_ergodicity	state	Unique ergodicity		The property that a topological system admits exactly one invariant Borel probability measure, which is then automatically ergodic.
s_unique_ergodicity_criterion	theorem	Unique ergodicity criterion		A continuous system is uniquely ergodic if and only if for every continuous function the Birkhoff averages converge uniformly to a constant (equivalently to the
t_koopman_operator	technique	Koopman operator		The composition operator U_f φ = φ∘f induced on L²(μ) by a measure-preserving transformation, an isometry (unitary if invertible) translating dynamical properti
s_spectral_characterization_mixing	theorem	Spectral characterization of weak and strong mixing		A measure-preserving system is weakly mixing iff its Koopman operator has continuous spectrum on the orthocomplement of constants (no nonconstant L² eigenfuncti
s_discrete_spectrum_system	state	Discrete (pure point) spectrum		The property of a measure-preserving system that the eigenfunctions of its Koopman operator span L²(μ), characterizing systems measurably isomorphic to rotation
s_halmos_von_neumann_theorem	theorem	Halmos–von Neumann theorem		An ergodic system has discrete spectrum iff it is measurably isomorphic to an ergodic rotation on a compact abelian group, and two such systems are isomorphic i
s_rank_one_system	state	Rank-one system		A measure-preserving transformation built by a cutting-and-stacking construction from a single Rokhlin tower at each stage, the simplest class of ergodic system
s_rigid_transformation	state	Rigidity (measurable)		The property of a measure-preserving system that some sequence of iterates of its Koopman operator converges strongly to the identity, the opposite extreme from
s_entropy_of_partition	state	Entropy of a partition H(ξ)		The Shannon entropy of a finite or countable measurable partition ξ, the average information gained by learning which atom of ξ a point lies in.
s_conditional_entropy_partitions	state	Conditional entropy of partitions H(ξ|η)		The average residual information in partition ξ given knowledge of partition η, the basic quantity from which Kolmogorov–Sinai entropy is built.
s_kolmogorov_sinai_entropy	state	Metric (Kolmogorov–Sinai) entropy h_μ(f)		The supremum over finite partitions of the asymptotic per-step entropy of refinements under f, an isomorphism invariant measuring the exponential rate of inform
s_sinai_generator_theorem	theorem	Kolmogorov–Sinai (Sinai) generator theorem		If a finite-entropy partition ξ generates the σ-algebra under f (its iterated refinements separate points mod 0), then the entropy of the system equals the entr
s_abramov_formula	theorem	Abramov formula		The entropy of a suspension (special) flow equals the entropy of its base transformation divided by the integral of the roof (return-time) function, and analogo
s_suspension_flow	state	Suspension (special) flow		The flow obtained from a base transformation T and a positive roof function r by flowing upward at unit speed in the mapping torus and identifying the top of ea
s_kolmogorov_automorphism_k_system	state	K-system (Kolmogorov automorphism)		A measure-preserving transformation admitting an exhaustive and trivial-tail generating sub-σ-algebra, equivalently having completely positive entropy; K-system
s_bernoulli_automorphism	state	Bernoulli automorphism		A measure-preserving system isomorphic to a two-sided Bernoulli shift on an independent identically distributed product space, the most random class of dynamica
s_measure_of_maximal_entropy	state	Measure of maximal entropy		An invariant probability measure whose metric entropy attains the topological entropy, the equilibrium state for the zero potential and unique for many hyperbol
s_variational_principle_pressure	theorem	Variational principle for topological pressure		The topological pressure of a continuous potential φ equals the supremum over invariant measures of metric entropy plus the integral of φ, generalizing the vari
s_equilibrium_state	state	Equilibrium state		An invariant measure realizing the supremum in the variational principle for the pressure of a potential φ, the dynamical analogue of a Gibbs state in statistic
s_gibbs_measure_dynamics	state	Gibbs measure (for a Hölder potential)		An invariant measure whose cylinder masses are comparable, up to a uniform constant, to exp(S_nφ − nP(φ)), the unique equilibrium state of a Hölder potential on
t_measurable_partition_rokhlin	technique	Measurable partition (Rokhlin theory)		Rokhlin's framework treating a partition into possibly uncountably many atoms as a measurable object dual to a sub-σ-algebra, equipped with a canonical system o
t_conditional_measures_disintegration	technique	Conditional measures (disintegration)		The canonical decomposition of a measure on a Lebesgue space into conditional probability measures supported on the atoms of a measurable partition, integrating
s_lebesgue_space_mod_zero	axiom	Lebesgue space (Lebesgue probability space mod 0)		A probability space isomorphic mod null sets to an interval carrying Lebesgue measure together with at most countably many atoms.
s_gauss_measure	axiom	Gauss measure		The probability measure dμ = (1/ln 2)·dx/(1+x) on [0,1) that is invariant under the Gauss (continued-fraction) map.
s_induced_transformation	state	Induced (first-return) transformation T_A		For a measure-preserving T and a positive-measure set A, the map T_A(x)=T^{n_A(x)}(x) sending x to its first return to A, which preserves the normalized restric
s_first_return_time_function	state	First return-time function n_A		The function assigning to x∈A the least integer n≥1 with T^n x ∈ A, finite a.e. by Poincaré recurrence.
t_kakutani_tower_construction	technique	Kakutani–Rokhlin tower (skyscraper) construction		A method that builds a measure-preserving transformation by stacking iterates of a base map according to a return/height function over a base set.
s_bernoulli_shift_object	state	Bernoulli shift		The coordinate shift on a one- or two-sided product probability space equipped with the product measure of a fixed probability vector (p_0,…,p_{k-1}).
s_markov_shift_object	state	Markov shift		The coordinate shift on a sequence space whose measure is determined by a stochastic matrix P and a stationary probability vector p with pP=p.
s_shift_map_sigma	axiom	Two-sided shift map σ		The coordinate-shift homeomorphism (σx)_n = x_{n+1} on a bi-infinite sequence space, preserving any shift-invariant measure.
s_compact_group_endomorphism_mpt	state	Endomorphism/automorphism of a compact group (as m.p.t.)		A continuous surjective group homomorphism of a compact group, which preserves the normalized Haar measure.
s_toral_automorphism_mpt	state	Automorphism/endomorphism of the torus (matrix m.p.t.)		The map T(x)=Ax mod 1 on the n-torus induced by an integer matrix A, preserving Lebesgue measure (an automorphism when det A = ±1).
s_isomorphism_mod_zero	axiom	Isomorphism mod 0 (measure-theoretic isomorphism)		An invertible measure-preserving map between full-measure invariant subsets that conjugates two systems, the basic equivalence relation of ergodic theory.
s_measure_algebra_isomorphism	axiom	Measure-algebra isomorphism (conjugacy of measure algebras)		A measure-preserving Boolean σ-algebra isomorphism between the measure algebras of two systems that intertwines the induced maps.
s_spectral_isomorphism_mpt	axiom	Spectral isomorphism (unitary equivalence of Koopman operators)		Existence of a unitary operator intertwining the Koopman operators of two systems, an equivalence strictly weaker than isomorphism mod 0.
s_conjugacy_implies_spectral_isomorphism	theorem	Conjugacy implies spectral isomorphism		Measure-theoretically isomorphic systems have unitarily equivalent Koopman operators, while the converse fails.
s_point_spectrum_eigenvalues_UT	state	Point spectrum (eigenvalues and eigenfunctions of U_T)		The set of λ for which the Koopman operator has a nonzero eigenfunction f with f∘T = λf a.e.
s_eigenvalues_modulus_one_constant_modulus	theorem	Eigenvalues of modulus one with constant-modulus eigenfunctions		For an ergodic transformation every eigenvalue has modulus one, eigenfunctions have constant modulus a.e., and the eigenvalues form a subgroup of the circle wit
s_weak_mixing_product_characterization	theorem	Weak-mixing product characterization (T×T ergodic)		A transformation is weak-mixing if and only if its Cartesian square is ergodic, equivalently the square is itself weak-mixing.
s_weak_mixing_continuous_spectrum	theorem	Weak-mixing ⇔ continuous spectrum (no nonconstant eigenfunctions)		A transformation is weak-mixing if and only if the constants are the only eigenfunctions of its Koopman operator, i.e. U_T has continuous spectrum on the orthoc
s_weak_mixing_all_powers_ergodic	theorem	Weak-mixing equivalent to all powers ergodic		A transformation is weak-mixing if and only if every power Tⁿ is ergodic, and then every power is itself weak-mixing.
s_koopman_von_neumann_lemma	theorem	Koopman–von Neumann lemma		A bounded nonnegative sequence has Cesàro averages tending to zero if and only if it converges to zero along a subsequence of density one.
s_ergodicity_of_group_rotations	theorem	Ergodicity criterion for group rotations		Rotation by an element a of a compact group G is ergodic with respect to Haar measure if and only if the cyclic subgroup generated by a is dense in G.
s_ergodicity_of_toral_automorphisms	theorem	Ergodicity criterion for toral automorphisms		A toral endomorphism given by integer matrix A is ergodic with respect to Lebesgue measure if and only if no eigenvalue of A is a root of unity.
s_ergodicity_mixing_of_shifts	theorem	Ergodicity and mixing of Bernoulli and Markov shifts		Bernoulli shifts are strong-mixing, and a Markov shift is ergodic if and only if its stochastic matrix is irreducible (mixing if aperiodic).
s_fekete_subadditive_lemma	theorem	Fekete subadditive sequence lemma		For a subadditive real sequence the limit of a_n/n exists and equals the infimum of a_n/n.
s_entropy_of_transformation_wrt_partition	state	Entropy of a transformation with respect to a partition h(T,ξ)		The asymptotic per-step entropy of the iterated joins of a partition under T, whose limit exists by subadditivity.
s_entropy_of_transformation_hT	state	Measure-theoretic entropy of a transformation h(T)		The supremum of h(T,ξ) over all finite measurable partitions, an isomorphism invariant of the system.
s_entropy_powers_inverse_properties	theorem	Entropy behaviour under powers and inverses		Measure-theoretic entropy scales linearly under iteration and is invariant under inversion of an invertible transformation.
s_generator_partition	axiom	Generator (one-sided/two-sided)		A finite measurable partition whose iterated T-preimages (one-sided, or two-sided for invertible T) generate the full σ-algebra mod 0.
s_join_of_partitions	state	Join (refinement) of partitions ξ∨η		The common refinement of two partitions whose atoms are the nonempty intersections of an atom of each.
s_basic_entropy_inequalities	theorem	Basic entropy (in)equalities for partitions		Entropy of partitions satisfies the chain rule H(ξ∨η)=H(η)+H(ξ|η), subadditivity, monotonicity, and concavity in the partition.
s_pinsker_sigma_algebra	state	Pinsker σ-algebra		The largest T-invariant sub-σ-algebra on which the transformation has zero entropy.
s_space_M_X_probability_measures	state	Space M(X) of Borel probability measures		The convex set of Borel probability measures on a compact metric space, compact and metrizable in the weak* topology.
s_space_M_X_T_invariant_measures	state	Space M(X,T) of T-invariant probability measures		The nonempty compact convex subset of M(X) consisting of the T-invariant Borel probability measures of a continuous system.
s_ergodic_measures_extreme_points	theorem	Ergodic measures are the extreme points of M(X,T)		The ergodic invariant measures are exactly the extreme points of M(X,T), which is a Choquet simplex.
s_bowen_metric_dn	state	Bowen (dynamical) metric d_n		The metric measuring the maximal separation of two orbit segments over n iterates, used to define (n,ε)-separated and spanning sets.
s_Lp_mean_ergodic_theorem	theorem	L^p mean ergodic theorem		For 1≤p<∞ and f∈L^p, the Birkhoff averages converge in L^p norm to the conditional expectation onto the invariant σ-algebra.
s_upper_semicontinuity_entropy_map	theorem	Upper semicontinuity of the entropy map and existence of equilibrium states		When the map μ↦h_μ(T) is upper semicontinuous on M(X,T) (e.g. for expansive systems) the supremum in the variational principle is attained, so equilibrium state
s_levy_khinchin_continued_fraction_constants	theorem	Khinchin and Lévy constants for continued fractions		For Lebesgue-almost every x the geometric mean of the continued-fraction partial quotients converges to Khinchin's constant and the denominators grow at the Lév
s_eigenvalue_group_complete_invariant	theorem	Eigenvalue group as complete invariant for discrete spectrum		For ergodic transformations with discrete spectrum the subgroup of the circle formed by the eigenvalues is a complete invariant up to isomorphism.
s_galilean_spacetime	axiom	Galilean space-time structure		A four-dimensional affine space equipped with a surjective time map to ℝ and a Euclidean metric on each simultaneity fiber, providing the kinematic arena of cla
s_galilean_group	axiom	Galilean group		The group of all affine transformations of Galilean space-time preserving the time intervals and the spatial Euclidean distance between simultaneous events.
s_galilean_relativity_principle	axiom	Galileo's principle of relativity		The laws of mechanics are invariant under the Galilean group, so no inertial frame is physically distinguished from another.
s_newton_equation_of_motion	axiom	Newton's equation of motion		The second-order differential equation positing that the acceleration of a point mass is determined by a force function of position, velocity, and time.
s_newton_determinacy_principle	axiom	Newton's principle of determinacy		The initial positions and velocities of all points of a mechanical system uniquely determine its entire future and past motion.
s_galilean_invariance_of_forces	theorem	Galilean invariance constraints on forces		Requiring Newton's equations to be Galilean-invariant forces the interaction to depend only on relative positions and relative velocities of the particles and t
s_conservative_force_field	axiom	Conservative (potential) force field		A force field that is the negative gradient of a scalar potential function, so that work done is path-independent.
s_conservation_of_momentum	theorem	Conservation of momentum		For a closed system the total linear momentum is conserved, following from the translational invariance of the Lagrangian.
s_conservation_of_angular_momentum	theorem	Conservation of angular momentum		For a system invariant under rotations the total angular momentum about the center is conserved, following from rotational invariance of the Lagrangian.
s_period_1d_motion_integral	state	Period of one-dimensional motion (energy integral)		The period of oscillation of a one-degree-of-freedom conservative system expressed as an integral over the classically allowed region between turning points.
s_reduced_mass_two_body	theorem	Reduction of the two-body problem (reduced mass)		The two-body problem with central interaction separates into free motion of the center of mass and motion of a single particle of reduced mass μ in a central fi
t_reduction_central_field_angular_momentum	technique	Reduction of central motion via angular momentum		Uses conservation of angular momentum to confine motion to a plane and eliminate the angular variable, reducing a central-field problem to a single radial degre
s_effective_potential	state	Effective potential		The sum of the true potential and the centrifugal term M²/(2μr²) governing the equivalent one-dimensional radial motion in a central field.
s_central_orbit_quadrature	state	Orbit equation of central motion (quadrature)		The closed-form expression of the orbit shape φ(r) for central motion obtained by integrating the radial and angular equations.
s_kepler_problem	state	Kepler's problem		The central-field problem with an inverse-square attractive force, whose bounded orbits are conics with the center of force at a focus.
s_laplace_runge_lenz_vector	state	Laplace–Runge–Lenz vector		A conserved vector specific to the inverse-square central field, pointing along the major axis and encoding the orientation and eccentricity of the orbit.
t_fall_to_center	technique	Analysis of fall to the center		Determines whether a particle in a central field spirals into the center in finite time by comparing the potential's singularity with the centrifugal barrier.
s_canonical_momentum	state	Generalized (canonical) momentum		The momentum conjugate to a generalized coordinate, defined as the partial derivative of the Lagrangian with respect to the corresponding velocity.
s_routh_reduction	theorem	Cyclic coordinates and Routh reduction		When a coordinate is cyclic (absent from the Lagrangian) its conjugate momentum is conserved, and Routh's procedure eliminates it to obtain a reduced Lagrangian
s_holonomic_constraint	axiom	Holonomic constraint		A constraint expressible as equations relating only the coordinates (and possibly time), confining motion to a submanifold of configuration space.
s_dalembert_lagrange_principle	axiom	D'Alembert–Lagrange principle		The principle that the total virtual work of the constraint forces vanishes for all virtual displacements compatible with the constraints, yielding the equation
s_lagrangian_mechanics_on_manifolds	state	Lagrangian mechanics on a manifold		The formulation of Lagrangian dynamics for a system whose configuration space is a smooth manifold, with the Lagrangian a function on the tangent bundle and mot
t_legendre_transformation_mechanics	technique	Legendre transformation (Lagrangian to Hamiltonian)		Passes from the Lagrangian on the tangent bundle to the Hamiltonian on the cotangent bundle by replacing velocities with conjugate momenta via fiberwise convex 
t_small_oscillations_linearization	technique	Small oscillations / linearization about equilibrium		Approximates motion near a stable equilibrium by expanding the kinetic and potential energies to quadratic order, yielding a linear system of coupled harmonic o
s_normal_modes	theorem	Normal modes and characteristic frequencies		Small oscillations of a system about stable equilibrium decompose into independent harmonic normal modes whose characteristic frequencies are the roots of the g
t_simultaneous_diagonalization_quadratic_forms	technique	Simultaneous diagonalization of two quadratic forms		Finds a single linear change of coordinates that simultaneously diagonalizes a positive-definite quadratic form and another symmetric form, used to decouple the
s_parametric_resonance	theorem	Parametric resonance		A system whose parameters vary periodically in time can be destabilized, with the trivial equilibrium becoming unstable when the driving frequency is near twice
s_rigid_body_configuration_so3	axiom	Rigid-body configuration space SO(3)		The configuration space of a rigid body fixed at a point is the rotation group SO(3), and of a free rigid body the group of orientation-preserving Euclidean mot
s_inertia_tensor	state	Inertia tensor		The symmetric positive operator on the body frame relating angular velocity to angular momentum, encoding the mass distribution of a rigid body.
s_principal_axes_of_inertia	state	Principal axes of inertia		The orthogonal eigenframe of the inertia tensor, in which the inertia operator is diagonal with the principal moments of inertia as eigenvalues.
s_angular_velocity_body_space	state	Angular velocity in body and space frames		The angular velocity vector of a rotating rigid body expressed in the rotating body frame and in the fixed space frame, related by the body's current rotation.
s_euler_rigid_body_equations	theorem	Euler's equations for a rigid body		The equations of motion of a rigid body in the body frame, equating the rate of change of angular momentum plus its gyroscopic term to the applied torque.
s_euler_angles	state	Euler angles		A triple of angles parametrizing the orientation of a rigid body as a composition of three successive rotations, giving local coordinates on SO(3).
s_poinsot_construction	theorem	Poinsot's construction (inertia ellipsoid)		The free rotation of a rigid body is described geometrically as the inertia ellipsoid rolling without slipping on a fixed plane perpendicular to the conserved a
s_lagrange_top	state	Lagrange top (heavy symmetric top)		A heavy symmetric rigid body fixed at a point on its symmetry axis, an integrable case whose motion combines precession, nutation, and spin.
s_stability_rotation_principal_axes	theorem	Stability of rotation about principal axes		Steady rotation of a free rigid body about the axes of largest and smallest moment of inertia is stable, while rotation about the intermediate axis is unstable.
s_poincare_cartan_integral_invariant	theorem	Poincaré–Cartan integral invariant		The 1-form p dq - H dt whose integral around any closed curve in extended phase space is preserved by the Hamiltonian flow, characterizing the canonical structu
s_relative_absolute_integral_invariant	theorem	Relative and absolute integral invariants		An integral over a cycle is an absolute integral invariant of a flow if preserved over arbitrary cycles and relative if preserved only over boundaries, with the
t_separation_of_variables_hamilton_jacobi	technique	Separation of variables in the Hamilton–Jacobi equation		Seeks a complete integral of the Hamilton–Jacobi equation as a sum of functions each depending on one coordinate, reducing the PDE to decoupled quadratures and 
s_euler_lie_algebra_equations	theorem	Euler equations on a Lie algebra		The generalization of the rigid-body and ideal-fluid equations as the flow on the dual of a Lie algebra generated by a left-invariant quadratic energy via the c
s_geodesics_on_lie_groups	theorem	Geodesics on Lie groups (hydrodynamics analogy)		Geodesics of a one-sided invariant Riemannian metric on a Lie group satisfy the Euler equations on its Lie algebra, unifying rigid-body rotation with ideal flui
s_lie_poisson_structure	state	Lie–Poisson structure on the dual of a Lie algebra		The natural Poisson bracket on the dual of a Lie algebra induced by the Lie bracket, whose symplectic leaves are the coadjoint orbits.
s_small_denominators_problem	state	Small denominators problem		The obstruction in perturbation theory of quasi-periodic motion arising from near-resonant frequency combinations k·ω that can make formal series diverge unless
t_adiabatic_invariant	technique	Adiabatic invariant		Identifies the action variable of a one-degree-of-freedom oscillator as approximately conserved under slow variation of the system's parameters.
t_canonical_perturbation_theory	technique	Canonical perturbation theory		Constructs canonical transformations by generating functions to remove the perturbation order by order in action-angle variables for a nearly integrable Hamilto
s_caustic	state	Caustic (Lagrangian singularity)		The singular set of the projection of a Lagrangian submanifold to configuration space, where infinitely many rays or trajectories focus and short-wave amplitude
t_wkb_maslov_asymptotics	technique	Short-wave (WKB) asymptotics with Maslov index		Constructs short-wavelength asymptotic solutions of wave equations from a Lagrangian submanifold, with phase shifts at caustics fixed by the Maslov index to mak
s_extended_phase_space	state	Extended phase space		The phase space enlarged by the time variable and its conjugate momentum -H, on which a time-dependent Hamiltonian system becomes autonomous and the Poincaré–Ca
s_torsion_coefficients_homology	state	Torsion coefficients of homology		The torsion part of a finitely generated homology group, given by the cyclic factors ℤ/k_i in its decomposition into free and torsion subgroups.
s_euler_poincare_theorem	theorem	Euler–Poincaré theorem		The alternating sum of the numbers of n-simplices of a complex equals the alternating sum of its Betti numbers, so the Euler characteristic is a homotopy invari
s_maurer_cartan_form	state	Maurer–Cartan form		The canonical left-invariant Lie-algebra-valued one-form on a Lie group G that maps each tangent vector back to the identity tangent space and satisfies dθ + ½[
s_codifferential_operator	state	Codifferential (adjoint exterior derivative)		The formal adjoint of the exterior derivative defined via the Hodge star, lowering form degree by one and combining with d to give the Hodge Laplacian Δ = dδ + 
s_dolbeault_operators	state	Dolbeault operators ∂ and ∂̄		The holomorphic and antiholomorphic components of the exterior derivative on a complex manifold, splitting d on the (p,q)-bidegree decomposition and each squari
s_chern_connection	state	Chern connection		The unique connection on a holomorphic vector bundle that is compatible with the Hermitian metric and whose (0,1)-part equals the Dolbeault operator ∂̄.
s_structure_group_reduction	state	Reduction of the structure group		A reduction of a principal G-bundle to a subgroup H ⊂ G, given by an H-subbundle of the total space, whose existence encodes geometric structure such as orienta
s_universal_bundle	state	Universal bundle		The principal G-bundle over the classifying space BG with contractible total space EG, from which every principal G-bundle on a space X is obtained by pullback 
s_fundamental_vector_field	state	Fundamental vector field		The vertical vector field on a principal bundle generated by a Lie algebra element through the right action, giving the canonical isomorphism between 𝔤 and the 
s_vertical_subspace_bundle	state	Vertical subspace of a principal bundle		The subspace of the tangent space to a principal bundle tangent to the fibre, namely the kernel of the projection's differential, isomorphic to the Lie algebra 
s_ehresmann_connection	state	Ehresmann connection		A smooth right-invariant choice of horizontal subspace complementary to the vertical subspace at every point of a principal bundle, equivalent to a connection o
s_local_connection_form_gauge_potential	state	Local connection form (gauge potential)		The pullback of the connection one-form by a local section, identified physically with the gauge potential, transforming inhomogeneously across overlapping triv
s_local_field_strength	state	Local field strength		The pullback of the curvature two-form by a local section, equal to dA + A∧A, transforming homogeneously (covariantly) under gauge transformations.
s_topological_instanton_charge	state	Topological instanton charge		The integer-valued second Chern number of a gauge field on a four-manifold, classifying instanton sectors as the integral of tr(F∧F).
s_berry_phase	state	Berry phase		The geometric phase acquired by an adiabatically transported quantum eigenstate around a closed loop in parameter space, equal to the holonomy of the Berry conn
s_berry_connection_curvature	state	Berry connection and curvature		The U(1) connection one-form built from a parameter-dependent quantum eigenstate and its exterior-derivative curvature, whose loop integral gives the Berry phas
s_spin_structure	state	Spin structure		A lift of the orthonormal frame bundle of an oriented Riemannian manifold to a principal Spin(n)-bundle double-covering it fibrewise, existing precisely when th
s_analytical_index	state	Analytical index		The integer difference between the dimensions of the kernel and cokernel of an elliptic operator, the analytic side of the Atiyah–Singer index theorem.
s_topological_index	state	Topological index		The integer computed from characteristic classes and the principal symbol of an elliptic operator, equated to the analytical index by the Atiyah–Singer theorem.
s_chiral_abelian_anomaly	state	Chiral (abelian) anomaly		The quantum non-conservation of the axial current in a gauge background, whose anomaly density equals a characteristic-class form integrating to the Dirac index
s_non_abelian_gauge_anomaly	state	Non-abelian gauge anomaly		The breakdown of gauge invariance of the effective action under infinitesimal non-abelian gauge transformations, measured by the consistent anomaly functional.
s_gravitational_anomaly	state	Gravitational anomaly		The quantum violation of general-coordinate (diffeomorphism) invariance, manifesting as non-conservation of the energy–momentum tensor for chiral fields in cert
s_anomaly_polynomial	state	Anomaly polynomial		The closed characteristic-class form in even dimension whose descent gives the consistent gauge and gravitational anomalies of a chiral theory in two dimensions
s_consistent_vs_covariant_anomaly	state	Consistent versus covariant anomaly		The distinction between the anomaly satisfying the Wess–Zumino consistency condition and the manifestly gauge-covariant anomaly, differing by a local counterter
s_mckean_singer_formula	theorem	McKean–Singer formula		The index of a Dirac-type operator equals the supertrace of the heat kernel for any positive time, the basis of the supersymmetric heat-kernel proof of the inde
s_nambu_goto_action	state	Nambu–Goto action		The relativistic string action proportional to the proper area of the worldsheet, given by the square root of the determinant of the induced metric.
s_polyakov_action	state	Polyakov action		The string action with an independent worldsheet metric γ_{ab} as an auxiliary field, classically equivalent to the Nambu–Goto action and quantizable.
s_worldsheet_diff_weyl_invariance	state	Worldsheet diffeomorphism and Weyl invariance		The local symmetries of the Polyakov action under worldsheet reparametrizations and rescalings of the worldsheet metric, used to fix conformal gauge.
s_conformal_gauge	state	Conformal gauge		The gauge choice fixing the worldsheet metric to be conformally flat, reducing the Polyakov action to a free-field form with residual conformal symmetry.
s_conformal_anomaly_central_charge	state	Conformal anomaly (central charge)		The quantum anomaly in worldsheet Weyl/conformal symmetry encoded by the central charge c appearing in the Virasoro algebra of the energy–momentum modes.
s_critical_dimension	state	Critical dimension		The spacetime dimension in which the total Virasoro central charge cancels (matter plus ghosts), removing the conformal anomaly: 26 for the bosonic string and 1
s_faddeev_popov_ghosts	state	Faddeev–Popov ghosts		The anticommuting ghost fields b and c introduced when gauge-fixing the worldsheet symmetries, contributing central charge −26 that must cancel the matter contr
s_string_mass_spectrum_tachyon	state	String mass spectrum and tachyon		The mass levels of string excitations set by the number operator and normal-ordering constant, whose lowest state is the tachyon with negative mass-squared in t
s_moduli_space_riemann_surfaces	state	Moduli space of Riemann surfaces		The space of inequivalent complex structures on a genus-g surface, the quotient of Teichmüller space by the mapping class group, over which string amplitudes ar
s_topological_defect_classification	theorem	Homotopy classification of topological defects		The classification of stable topological defects of an ordered medium with order-parameter space G/H by the homotopy groups π_n(G/H), with domain walls, strings
t_fujikawa_method	technique	Fujikawa method		A technique deriving the chiral anomaly as the non-invariance (Jacobian) of the fermionic path-integral measure under a chiral transformation, regularized to a 
t_descent_equations_stora_zumino	technique	Stora–Zumino descent equations		A chain of equations dω_{2n+1}=I_{2n+2}, δω_{2n+1}=dω_{2n}^1 that descends an anomaly polynomial through Chern–Simons forms to obtain the consistent gauge anoma
t_susy_proof_index_theorem	technique	Supersymmetric proof of the index theorem		A technique computing the Dirac index via the t→0 limit of the supersymmetric path integral (McKean–Singer supertrace), localizing onto the characteristic-class
t_brst_quantization_string	technique	BRST quantization of the string		A technique quantizing the string by constructing a nilpotent BRST charge from matter and ghost fields, whose cohomology at the right ghost number selects the p
t_heat_kernel_supertrace	technique			Computes the index of a Dirac-type operator by evaluating the t-independent supertrace of the heat kernel and taking its short-time asymptotic expansion to extr
t_gauge_holonomy_transport	technique			Parallel-transports along loops and applies gauge transformations to a connection, producing holonomy phases and the inhomogeneous transformation law of the loc
t_simplicial_euler_computation	technique			Computes the Euler characteristic of a complex as the alternating sum of simplex counts and equates it, via homology, to the alternating sum of Betti numbers.
t_homotopy_classification_order_parameter	technique			Classifies stable topological defects of an ordered medium by computing the homotopy groups of the order-parameter coset space G/H.
t_virasoro_central_charge_computation	technique			Computes the total Virasoro central charge from matter and Faddeev–Popov ghost sectors and imposes its cancellation to fix the critical spacetime dimension and 
s_reduced_root_system	axiom	Reduced root system	reduced root system	A root system in which for every root alpha the only scalar multiples of alpha that are also roots are alpha and -alpha (i.e. 2*alpha is never a root).
s_existence_of_cartan_subalgebras	theorem	Existence of Cartan subalgebras		Every finite-dimensional Lie algebra over an infinite field possesses a Cartan subalgebra, obtained as the generalized zero-eigenspace (nilspace) of ad(X) for a
s_compact_lie_algebra	axiom	Compact Lie algebra	Lie algebra of compact type	A real Lie algebra that admits an Ad-invariant positive-definite inner product, equivalently the Lie algebra of a compact Lie group.
s_killing_form_negative_definite_compact	theorem	Negative-definiteness of the Killing form on a compact semisimple Lie algebra		A real semisimple Lie algebra is the Lie algebra of a compact group if and only if its Killing form is negative definite.
s_weyl_theorem_finite_fundamental_group	theorem	Weyl's theorem on finiteness of the fundamental group	Weyl's finiteness theorem | compactness of the universal cover of a compact semisimple group	The fundamental group of a compact semisimple Lie group is finite, and consequently its universal covering group is again compact.
s_surjectivity_of_exp_compact_connected	theorem	Surjectivity of the exponential map for compact connected Lie groups		For a compact connected Lie group the exponential map exp: g -> G is surjective, so every element lies on a one-parameter subgroup and hence in a maximal torus.
s_conjugation_relative_to_real_form	state	Conjugation relative to a real form	conjugation with respect to a real form	The conjugate-linear involution of a complex Lie algebra g^C that fixes exactly a given real form g_0, satisfying tau([X,Y]) = [tau X, tau Y] and tau(zX) = conj
s_existence_of_compact_real_form	theorem	Existence of a compact real form		Every complex semisimple Lie algebra possesses a compact real form, namely a real form whose Killing form is negative definite.
s_uniqueness_of_compact_real_form	theorem	Uniqueness of the compact real form up to conjugacy		Any two compact real forms of a complex semisimple Lie algebra are conjugate by an inner automorphism of the complex algebra.
s_complexification_of_compact_lie_group	state	Complexification of a compact Lie group	universal complexification of a compact group	The unique connected complex Lie group G^C containing a compact Lie group U as a maximal compact subgroup, with Lie algebra g^C = u (+) iu and U^C-representatio
s_existence_of_cartan_involution	theorem	Existence of a Cartan involution		Every real semisimple Lie algebra admits a Cartan involution, i.e. an involutive automorphism theta for which B_theta(X,Y) = -B(X, theta Y) is positive definite
s_conjugacy_of_cartan_involutions	theorem	Conjugacy of Cartan involutions		Any two Cartan involutions of a real semisimple Lie algebra are conjugate by an inner automorphism, so the maximal compact subalgebra k is unique up to conjugac
s_borel_weil_theorem	theorem	Borel-Weil theorem		For a compact connected Lie group with complexification G^C and Borel subgroup B, the space of holomorphic sections of the line bundle L_lambda over the flag ma
s_reductive_lie_group_harish_chandra_class	axiom	Reductive Lie group (Knapp's class)	Harish-Chandra class | reductive group in the sense of Knapp	A 4-tuple (G,K,theta,B) consisting of a Lie group G with a global Cartan involution theta, maximal compact subgroup K = G^theta, and a nondegenerate Ad(G)-invar
s_cartan_subalgebra_of_real_semisimple	axiom	Cartan subalgebra of a real semisimple Lie algebra		A maximal abelian subalgebra h of a real semisimple Lie algebra g whose adjoint action is everywhere semisimple over the complexification, equivalently a real f
s_theta_stable_cartan_subalgebra	state	Theta-stable Cartan subalgebra	theta-stable CSA	A Cartan subalgebra h of a real semisimple Lie algebra invariant under a fixed Cartan involution theta, hence decomposing as h = (h ∩ k) (+) (h ∩ p) into its co
s_existence_of_theta_stable_cartan_subalgebra	theorem	Existence of a theta-stable Cartan subalgebra		Every Cartan subalgebra of a real semisimple Lie algebra is conjugate by an inner automorphism to a theta-stable one, so theta-stable Cartan subalgebras always 
s_maximally_compact_noncompact_cartan_subalgebras	state	Maximally compact and maximally noncompact Cartan subalgebras	fundamental Cartan subalgebra | maximally split Cartan subalgebra	Among theta-stable Cartan subalgebras of a real semisimple Lie algebra, the maximally compact one has its toral part h ∩ k of maximal dimension and the maximall
s_finiteness_of_conjugacy_classes_of_cartan_subalgebras	theorem	Finiteness of conjugacy classes of Cartan subalgebras		A real semisimple Lie algebra has only finitely many conjugacy classes of Cartan subalgebras, all obtained from a maximally noncompact one by successive Cayley 
s_real_imaginary_complex_roots	state	Real, imaginary, and complex roots		For a theta-stable Cartan subalgebra h = t (+) a of a real semisimple Lie algebra, a root is imaginary if it vanishes on a, real if it vanishes on t, and comple
s_compact_and_noncompact_imaginary_roots	state	Compact and noncompact imaginary roots		An imaginary root (one fixed by theta) is compact if its root space lies in k^C and noncompact if its root space lies in p^C, distinguishing the two types neede
s_vogan_diagram	state	Vogan diagram		The Dynkin diagram of a complex semisimple Lie algebra decorated by an involution (arrows) coming from a maximally compact theta-stable Cartan subalgebra togeth
s_vogan_diagram_theorem	theorem	Vogan diagram theorem		Every Vogan diagram arises from a real form of a complex semisimple Lie algebra, two real forms are isomorphic iff their diagrams are equivalent, and applying t
s_borel_de_siebenthal_theorem	theorem	Borel-de Siebenthal theorem	Borel-de Siebenthal procedure	The maximal-rank reductive (regular) subalgebras of a compact semisimple Lie algebra are classified, up to conjugacy, by deleting one node from the extended (af
s_maximal_abelian_subspace_of_p	state	Maximal abelian subspace a of p	maximal split abelian subalgebra	A maximal abelian subspace a of the (-1)-eigenspace p of a Cartan involution on a real semisimple Lie algebra; all such subspaces are Ad(K)-conjugate and their 
s_real_rank	state	Real rank of G	split rank | R-rank	The dimension of a maximal abelian subspace a of p, equivalently the rank of the restricted root system and the dimension of the vector group A in the Iwasawa d
s_restricted_roots	state	Restricted roots	relative roots | a-roots	The nonzero weights of the adjoint action of a maximal abelian subspace a of p on a real semisimple Lie algebra, i.e. the nonzero linear functionals lambda on a
s_restricted_root_space_decomposition	theorem	Restricted root space decomposition	a-root space decomposition	A real semisimple Lie algebra decomposes as g = g_0 (+) sum over restricted roots lambda of g_lambda, where g_0 = m (+) a is the centralizer of a and g_lambda a
s_restricted_root_multiplicity	state	Restricted root multiplicity		The dimension m_lambda = dim g_lambda of the restricted root space attached to a restricted root lambda, which (unlike for ordinary roots) may exceed one.
s_restricted_root_system	theorem	Restricted root system Sigma	relative root system	The set of restricted roots of a real semisimple Lie algebra forms an abstract root system Sigma in a*, possibly non-reduced (of type BC), with Weyl group W(Sig
s_centralizer_m_of_a_algebra	state	Centralizer m = Z_k(a)		The centralizer of a maximal abelian subspace a in the maximal compact subalgebra k, equal to the compact part of the zero restricted root space g_0 = m (+) a.
s_centralizer_M_of_a_group	state	M = Z_K(a)	M = centralizer of A in K	The centralizer of a in the maximal compact subgroup K, a compact (possibly disconnected) subgroup with Lie algebra m, appearing as the reductive factor of the 
t_ordering_of_restricted_roots	technique	Positivity and ordering of restricted roots		Fixes a lexicographic (or chamber-based) ordering on a* to select a system of positive restricted roots, thereby determining the nilpotent subalgebra n in the I
s_nilpotent_subalgebra_n	state	Nilpotent subalgebra n	Iwasawa nilpotent subalgebra	The nilpotent Lie subalgebra n = sum over positive restricted roots lambda of g_lambda associated to a choice of positivity, which is the n-factor of the Iwasaw
s_iwasawa_decomposition_algebra	theorem	Iwasawa decomposition of the Lie algebra	g = k + a + n	A real semisimple Lie algebra splits as a vector-space direct sum g = k (+) a (+) n, where k is maximal compact, a is a maximal abelian subspace of p, and n is 
s_iwasawa_decomposition_group_kan	theorem	Iwasawa decomposition G = KAN	KAN decomposition	A connected semisimple Lie group factors as G = KAN with K maximal compact, A = exp(a) a simply connected abelian (vector) group, and N = exp(n) simply connecte
s_uniqueness_in_iwasawa_decomposition	theorem	Uniqueness in the Iwasawa decomposition		The multiplication map K × A × N -> G is a diffeomorphism, so every group element g = kan has uniquely determined factors k in K, a in A, n in N.
s_vector_group_A	state	A = exp(a) vector group		The analytic subgroup A = exp(a) of a semisimple Lie group, a simply connected abelian Lie group isomorphic to R^r (r the real rank) on which exp: a -> A is a d
s_nilpotent_group_N	state	N = exp(n) nilpotent group		The analytic subgroup N = exp(n) of a semisimple Lie group, a simply connected nilpotent Lie group on which exp: n -> N is a diffeomorphism, normalized by A.
s_weyl_group_of_restricted_roots	state	Weyl group of the restricted root system W(Sigma)	relative Weyl group | small Weyl group	The Weyl group generated by reflections in the restricted roots, acting on a* and permuting the restricted-root chambers.
s_weyl_group_as_NK_over_ZK	theorem	W(Sigma) = N_K(a)/Z_K(a)		The Weyl group of the restricted root system is realized analytically as the quotient N_K(a)/Z_K(a) of the normalizer of a in K by its centralizer.
s_kak_decomposition	theorem	KAK (Cartan/polar) decomposition	Cartan decomposition of the group | polar decomposition G = KAK	A connected semisimple Lie group satisfies G = K A K, more precisely G = K cl(A+) K with the A+-part of each element unique up to the Weyl group W(Sigma); equiv
s_closed_positive_weyl_chamber_a_plus	state	Closed positive Weyl chamber a+ and A+		The closed chamber cl(a+) = {H in a : lambda(H) >= 0 for all positive restricted roots lambda} and the corresponding A+ = exp(cl(a+)), a fundamental domain for 
s_minimal_parabolic_subgroup	state	Minimal parabolic subgroup P = MAN	minimal parabolic P = MAN	The minimal parabolic subgroup P = MAN of a semisimple Lie group, with M = Z_K(a), A = exp(a), and N the Iwasawa nilpotent group, having Langlands decomposition
s_standard_parabolic_subgroups_real	theorem	Standard parabolic subgroups from subsets of simple restricted roots	standard parabolic subgroups	The parabolic subgroups of a real reductive group containing a fixed minimal parabolic P = MAN are parametrized, up to conjugacy, by subsets of the simple restr
s_langlands_decomposition	theorem	Langlands decomposition of a parabolic	Levi decomposition of a real parabolic	Every parabolic subgroup of a real reductive group factors canonically as P = M_P A_P N_P, where M_P A_P is the Levi factor (M_P reductive, A_P a split central 
s_iwasawa_projection_H	state	Iwasawa projection H(g)	Iwasawa a-projection | H(g) function	The a-valued function H: G -> a defined by writing g = k exp(H(g)) n in the Iwasawa decomposition G = KAN, central to spherical functions and the theory of Eise
s_half_sum_of_positive_restricted_roots	state	Half-sum of positive restricted roots rho	restricted rho	The element rho = (1/2) sum over positive restricted roots lambda of m_lambda * lambda in a*, weighted by restricted-root multiplicities, governing the modular 
s_integral_formula_kan	theorem	Integral formula for the KAN decomposition	Iwasawa integration formula	A Haar integral over G expressed in Iwasawa coordinates g = kan reads ∫_G f = c ∫_K ∫_A ∫_N f(kan) e^{2 rho(log a)} dn da dk, with the exponential factor coming
s_integral_formula_kak	theorem	Integral formula for the KAK decomposition	sinh Jacobian formula | Cartan integration formula	A Haar integral over G in polar coordinates g = k1 (exp H) k2 reads ∫_G f = c ∫_K ∫_{a+} ∫_K f(k1 exp(H) k2) J(H) dk1 dH dk2 with Jacobian J(H) = prod over posi
s_integral_formula_over_G_mod_K	theorem	Integral formula over the symmetric space G/K		Integration over the symmetric space G/K in geodesic polar coordinates about the base point reads ∫_{G/K} f = c ∫_K ∫_{a+} f(k exp(H) o) prod (sinh lambda(H))^{
s_non_unimodularity_of_AN	theorem	Non-unimodularity of AN	modular function of AN	The solvable group AN (and the minimal parabolic MAN) is non-unimodular, with modular function Delta(an) = e^{2 rho(log a)} given by the half-sum of positive re
s_unimodularity_of_reductive_groups	theorem	Unimodularity of semisimple and reductive groups	semisimple groups are unimodular	Every semisimple (and every reductive in Knapp's class) Lie group is unimodular, so its left Haar measure is also right-invariant.
s_hermitian_symmetric_space	axiom	Hermitian symmetric space	Hermitian symmetric space of noncompact/compact type	A Riemannian symmetric space carrying a G-invariant complex structure compatible with the metric (making it a Kahler manifold whose symmetries are holomorphic i
s_criterion_for_hermitian_type	theorem	Criterion for Hermitian type		An irreducible symmetric space G/K of noncompact type is Hermitian if and only if the center Z(k) of the maximal compact subalgebra is nonzero (one-dimensional)
s_bounded_symmetric_domain	state	Bounded symmetric domain	Cartan domain | classical domain	A bounded domain in C^n that is symmetric, i.e. at each point admits a holomorphic involutive isometry fixing that point isolatedly; these are exactly the Hermi
s_harish_chandra_decomposition	theorem	Harish-Chandra decomposition g^C = p- + k^C + p+	g^C = p- + k^C + p+	For a Hermitian symmetric pair, the complexification splits as g^C = p- (+) k^C (+) p+ where p+ and p- are abelian subalgebras (the (+i) and (-i) eigenspaces of
s_harish_chandra_embedding	theorem	Harish-Chandra embedding	Harish-Chandra realization	The map G/K -> p+ given by g K -> the p+-component of log in the decomposition G^C subset P- K^C P+ realizes a Hermitian symmetric space of noncompact type as a
s_strongly_orthogonal_roots	state	Strongly orthogonal roots	strongly orthogonal noncompact roots	A set of roots that are pairwise orthogonal and whose pairwise sums and differences are never roots; a maximal such set of noncompact positive roots yields poly
s_kostant_linear_convexity_theorem	theorem	Kostant's linear convexity theorem	Kostant convexity theorem | linear convexity theorem	For a connected semisimple Lie group, the image under the Iwasawa a-projection H(.) of the orbit Ad(K)X of an element X in a equals the convex hull of the Weyl-
s_opposite_nilpotent_subalgebra	state	Opposite nilpotent subalgebra n-bar = theta(n)	opposite nilradical | n-bar	The nilpotent subalgebra n-bar = theta(n) = sum over negative restricted roots of g_lambda, the image of n under the Cartan involution, giving the opposite (low
s_open_bruhat_cell	theorem	Open Bruhat cell (big cell)	big cell | open dense Bruhat cell	The dense open subset N-bar M A N of a semisimple Lie group (the big Bruhat cell corresponding to the longest Weyl element), on which N-bar -> G/P is an open em
s_invariant_metric_on_G_mod_K_from_B_theta	state	G-invariant metric on G/K from B_theta		The G-invariant Riemannian metric on the symmetric space G/K obtained by transporting the inner product B_theta restricted to p ≅ T_o(G/K), making G/K a symmetr
s_curvature_of_symmetric_space_bracket_formula	theorem	Curvature formula R(X,Y)Z = -[[X,Y],Z]	symmetric space curvature formula	On a symmetric space G/K with tangent space p, the Riemann curvature tensor at the base point is R(X,Y)Z = -[[X,Y],Z] for X,Y,Z in p, expressing curvature purel
s_siegel_domain_realization	state	Siegel domain (tube domain) realization	tube domain | Siegel domain of type I/II	An unbounded realization of a bounded symmetric domain as a Siegel domain of type I (tube domain over a symmetric cone) or type II, biholomorphic to the domain 
t_cayley_transform_lie	technique	Cayley transform (Lie-theoretic)	Cayley transform of a Cartan subalgebra	The inner automorphism c_alpha = Ad(exp((pi/4)(X_alpha - X_{-alpha}))) built from a noncompact root, which interchanges compact and noncompact directions of a C
t_harish_chandra_projection_rho_shift	technique	Harish-Chandra projection (rho-shift)		Projects the center of the universal enveloping algebra onto U(h) ≅ S(h) along the triangular decomposition U(g) = (n- U(g) + U(g) n+) (+) U(h) and then twists 
s_analytic_manifold	axiom	Analytic manifold		A manifold equipped with an atlas whose transition maps are real-analytic (C^omega), so local coordinate changes are given by convergent power series; this is t
s_lie_third_theorem_existence	theorem	Lie's third theorem (existence)		Every finite-dimensional real Lie algebra is the Lie algebra of some Lie group, and there is a unique simply connected Lie group realizing it, so the Lie functo
s_ad_exp_naturality	theorem	Naturality of exp: Ad(exp X) = exp(ad X)		For a Lie group G with Lie algebra g, the adjoint representation satisfies Ad(exp X) = exp(ad X) for all X in g, expressing that exp intertwines ad on g with Ad
s_canonical_coordinates	state	Canonical coordinates of the first and second kind		Local coordinate systems near the identity of a Lie group: first kind sends (x_1,...,x_n) to exp(sum x_i X_i) for a basis {X_i} of g, and second kind to the ord
t_taylor_expansion_product_map	technique	Taylor expansion of the product map		Recovers the Lie bracket and higher structure of a Lie group from the Taylor expansion of the multiplication map mu(x,y) = x + y + (1/2)[x,y] + ... in canonical
s_symmetrization_map_pbw	state	Symmetrization map S(g) -> U(g)		The canonical g-module linear isomorphism of the symmetric algebra S(g) onto the universal enveloping algebra U(g) sending X_1...X_n to (1/n!) sum over Sn of X_
s_infinitesimal_character	state	Infinitesimal (central) character		The algebra homomorphism chi: Z(U(g)) -> C by which the center of the universal enveloping algebra acts as scalars on an irreducible or highest-weight g-module,
t_bost_slope_formalism	technique	Bost slope formalism for Euclidean lattices		An adaptation of the Arakelov-geometric slope and Harder–Narasimhan filtration formalism to Euclidean lattices, assigning a slope and a canonical destabilizing 
s_reverse_minkowski_transference	theorem	Reverse-Minkowski / transference inequalities for lattices		Quantitative inequalities relating the covering radius, smoothing parameter, and theta function of a Euclidean lattice and its dual, proved by Banaszczyk, Regev
s_fekete_points_equidistribution	theorem	Equidistribution of Fekete points via pluripotential theory		Energy-minimizing Fekete configurations on a compact complex manifold equidistribute toward the equilibrium measure of the associated weighted potential as the 
s_coulomb_gas_pluripotential	state	Coulomb gas / determinantal pluripotential point process		A point process weighting configurations by an exponential of a pairwise logarithmic-energy functional whose macroscopic large-deviation limit is an equilibrium
s_eigenvector_normality_random_regular	theorem	Asymptotic normality of eigenvectors of random regular graphs		The entries of eigenvectors of a uniformly random d-regular graph become jointly Gaussian (fully delocalized) in the large-vertex limit, proved by Backhausz and
t_factor_of_iid_regular_tree	technique	Typical processes on the d-regular tree (factor-of-i.i.d. method)		Analyzing eigenvectors of random regular graphs via invariant typical processes (factors of i.i.d.) on the infinite d-regular tree together with entropy inequal
s_coarse_embeddability_nonexact_groups	theorem	Coarse embeddability of non-exact groups into Hilbert space		There exist finitely generated groups that are not exact (lack property A) yet admit a coarse embedding into Hilbert space, separating the two coarse-geometric 
s_property_a_coarse_amenability	axiom	Property A (coarse amenability)		A coarse amenability condition on a bounded-geometry metric space requiring uniformly large controlled-boundary finite subsets, equivalent for groups to exactne
s_crystalline_companions_theorem	theorem	Crystalline companions theorem		For a smooth variety over a finite field, every ℓ-adic lisse sheaf in a compatible system admits ℓ'-adic and crystalline (overconvergent F-isocrystal) companion
s_abe_padic_langlands_function_fields	theorem	Abe's p-adic Langlands correspondence for function fields		A bijection, compatible with L. Lafforgue's ℓ-adic correspondence, between overconvergent F-isocrystals and automorphic forms over a function field, realizing c
s_furstenberg_boundary_cstar_simplicity	theorem	Furstenberg-boundary criterion for C*-simplicity		A discrete group has a simple reduced C*-algebra if and only if its action on its Furstenberg boundary is topologically free, equivalently it has no nontrivial 
s_cstar_simplicity	axiom	C*-simplicity		The property of a discrete group that its reduced group C*-algebra is simple, having no nontrivial closed two-sided ideals.
s_bab_boundedness_fano	theorem	Boundedness of singular Fano varieties (BAB conjecture)		For each dimension d and ε>0 the ε-log-canonical Fano varieties of dimension d form a bounded family, proved by Birkar.
t_theory_of_complements	technique	Theory of complements (boundedness of complements)		A birational-geometry method bounding the index of n-complements of log Fano pairs, used by Birkar to establish boundedness of Fano families.
s_ksba_moduli_stable_varieties	theorem	Moduli of stable (KSBA) varieties of general type		Families of higher-dimensional varieties of general type admit a separated proper moduli space of stable (semi-log-canonical, ample canonical) objects, generali
s_higher_teichmuller_theory	state	Higher Teichmüller theory		The study of connected components of the character variety of a surface group into a higher-rank semisimple Lie group consisting entirely of discrete faithful r
s_anosov_representation	axiom	Anosov representation		A representation of a hyperbolic group into a semisimple Lie group whose flat bundle admits a continuous equivariant transverse limit map with uniform contracti
s_homfly_from_hilbert_schemes	theorem	HOMFLY polynomial from Hilbert schemes of a plane curve singularity		The HOMFLY polynomial of the link of a plane curve singularity equals the generating series of Euler characteristics of Hilbert schemes of points on the curve, 
s_oblomkov_shende_conjecture	state	Oblomkov–Shende / Maulik conjecture		The conjectural identity equating the HOMFLY invariant of a singularity link with a Hilbert-scheme generating function, refined by the perverse filtration.
s_max_zeta_short_intervals	theorem	Maximum of the Riemann zeta function in short intervals		The maximum of |ζ(1/2+it)| over a typical short interval of length ~2π near height T is governed to leading order by log-correlated field / branching random wal
t_log_correlated_field_zeta	technique	Log-correlated field / branching random walk heuristic for zeta		Modeling the local behaviour of log|ζ| on the critical line by a logarithmically correlated Gaussian field to transfer extreme-value results from branching rand
s_sharpness_phase_transition	theorem	Sharpness of the phase transition (Duminil-Copin–Raoufi–Tassion)		For a broad class of percolation and Ising-type models the phase transition is sharp, with exponential decay of correlations throughout the subcritical regime, 
t_osss_sharp_thresholds	technique	OSSS inequality method for sharp thresholds		Using the O'Donnell–Saks–Schramm–Servedio decision-tree inequality to derive differential inequalities on percolation probabilities yielding sharp phase transit
s_disproof_triangulation_conjecture	theorem	Disproof of the Triangulation Conjecture		For every dimension n ≥ 5 there exist closed topological manifolds admitting no triangulation, proved by Manolescu via Pin(2)-equivariant Seiberg–Witten Floer h
t_pin2_sw_floer_homology	technique	Pin(2)-equivariant Seiberg–Witten Floer homology (Manolescu invariants)		A Pin(2)-equivariant Seiberg–Witten Floer homology of homology 3-spheres producing invariants that obstruct order-two elements of the homology cobordism group w
s_homological_stability_hurwitz	theorem	Homological stability for Hurwitz spaces (Cohen–Lenstra over function fields)		Hurwitz moduli spaces of branched covers exhibit homological stability, implying Cohen–Lenstra-type predictions for class groups of quadratic function fields ov
t_homological_stability_scanning	technique	Homological stability method (scanning / group completion)		A topological method establishing that the homology of a sequence of moduli or configuration spaces stabilizes, via arc complexes and scanning/group-completion 
s_infinitely_many_minimal_hypersurfaces	theorem	Infinitely many minimal hypersurfaces (Yau's conjecture, dim 3–7)		Every closed Riemannian manifold of dimension between 3 and 7 contains infinitely many closed embedded minimal hypersurfaces, proved by Marques–Neves and Song.
t_almgren_pitts_min_max	technique	Almgren–Pitts min-max theory / volume spectrum Weyl law		A variational min-max scheme producing minimal hypersurfaces as critical points of the area functional over multiparameter sweepouts, with a Weyl-type asymptoti
s_herman_positive_metric_entropy	state	Herman's positive metric entropy conjecture		The conjecture that arbitrarily close to integrable systems or near an elliptic fixed point there exist smooth area-preserving diffeomorphisms with positive met
s_berger_turaev_theorem	theorem	Berger–Turaev theorem (positive metric entropy)		Any area-preserving C^r diffeomorphism of a surface with an elliptic fixed point can be C^r-perturbed to one with a chaotic island of positive metric entropy, p
s_stability_lamb_oseen_vortex	theorem	Asymptotic stability of the Lamb–Oseen vortex		The two-dimensional Lamb–Oseen vortex is asymptotically stable for the Navier–Stokes equations uniformly in viscosity, with enhanced inviscid damping rates, pro
t_pseudospectral_enhanced_dissipation	technique	Pseudospectral / enhanced-dissipation estimates		Using pseudospectral resolvent bounds on the linearized non-self-adjoint operator together with enhanced dissipation to control transient growth and prove stabi
s_c_infinity_closing_lemma	theorem	C^∞ closing lemma (Asaoka–Irie)		For Hamiltonian and Reeb flows on surfaces and 3-manifolds, periodic orbits are C^∞-dense, proving a smooth closing lemma in these settings, due to Asaoka and I
t_ech_spectral_invariants_weyl	technique	Embedded contact homology spectral invariants (Weyl law)		Spectral invariants from embedded contact homology whose asymptotics obey a Weyl law recovering contact volume, used to force creation and density of periodic R
s_resolvent_estimates_non_self_adjoint	theorem	Resolvent estimates for non-self-adjoint semiclassical operators		Sharp bounds on the resolvent and localization of the pseudospectrum for classes of non-self-adjoint semiclassical pseudodifferential operators, governed by sub
s_definability_of_period_maps	theorem	Definability of period maps		Any period map of an admissible polarized variation of mixed Hodge structure is definable in the o-minimal structure R_an,exp, proved by Bakker–Brunebarbe–Kling
t_definable_chow_ominimal_gaga	technique	Definable Chow / o-minimal GAGA (Peterzil–Starchenko)		A transfer principle of Peterzil–Starchenko stating that a definable complex analytic subset of an algebraic variety is algebraic, used to deduce algebraicity f
s_algebraicity_of_hodge_loci	theorem	Algebraicity of Hodge loci (via o-minimality)		The Hodge locus in a family is an algebraic subvariety, reproved by Bakker–Klingler–Tsimerman using definable Chow / o-minimal GAGA (the theorem of Cattani–Deli
s_maximal_isotopy_transverse_foliation	state	Maximal isotopy & transverse singular foliation		For a surface homeomorphism isotopic to the identity, a maximal identity isotopy carries a transverse singular foliation whose leaves encode the dynamics.
t_le_calvez_tal_forcing	technique	Le Calvez–Tal forcing theory		A forcing method of Le Calvez and Tal using transverse trajectories in a maximal singular foliation to detect periodic orbits and positive topological entropy o
s_nodal_set_volume_estimates	theorem	Nodal-set volume estimates (Yau's nodal conjecture)		The (n−1)-Hausdorff measure of the nodal set of a Laplace eigenfunction with eigenvalue λ on a compact analytic manifold is bounded above and below by polynomia
t_doubling_index_frequency_function	technique	Doubling index / frequency-function method		A combinatorial-analytic technique controlling the local growth (doubling index) of harmonic or eigenfunctions via almost-monotonicity and cube subdivision to b
s_ratner_infinite_volume	theorem	Ratner-type rigidity in infinite-volume hyperbolic manifolds		Classification and equidistribution of orbit closures of unipotent and horospherical flows on geometrically finite infinite-volume hyperbolic manifolds, due to 
s_quantitative_inverse_gowers	theorem	Quantitative inverse theorem for the Gowers norms		A function with non-negligible U^{s+1} Gowers norm correlates with a nilsequence of bounded complexity, with polynomial effective bounds, proved by F. Manners.
t_higher_order_fourier_analysis	technique	Higher-order Fourier analysis (inverse-theorem method)		Decomposing functions via degree-s nilsequences and iterating Cauchy–Schwarz and equidistribution on nilmanifolds to invert the Gowers norms.
s_reconstruction_from_zariski_topology	theorem	Reconstruction of a variety from its Zariski topology		A normal projective variety of dimension ≥ 2 over an algebraically closed field is determined, as a scheme, by the homeomorphism type of its underlying Zariski 
s_zagier_polylogarithm_conjecture	state	Zagier's polylogarithm conjecture		The conjecture that the value ζ_F(n) of a number field is expressible via a regulator determinant of values of the n-th polylogarithm on F-points.
s_goncharov_rudenko_zeta_f4	theorem	Goncharov–Rudenko theorem on ζ_F(4)		Zagier's conjecture holds in weight 4, expressing ζ_F(4) via the tetralogarithm, proved by Goncharov–Rudenko using the weight-4 polylogarithmic motivic complex 
s_goncharov_motivic_polylog_complexes	state	Goncharov motivic polylogarithm complexes		A family of complexes built from polylogarithmic groups conjecturally computing motivic cohomology and underlying Zagier's conjecture.
t_motivic_correlators_cluster_polylog	technique	Motivic correlators / cluster polylogarithm method		Using Goncharov's motivic correlators and the cluster variety structure of M_{0,n} to construct polylogarithm functional equations realizing required cohomology
s_fractal_uncertainty_principle	state	Fractal uncertainty principle		The principle that a function and its Fourier transform cannot both concentrate on a fractal (porous, Ahlfors–David regular) set, quantified by a power-saving o
s_essential_spectral_gap_fractal	theorem	Essential spectral gap via fractal uncertainty		On convex co-compact hyperbolic surfaces the fractal uncertainty principle yields an essential spectral gap (resonance-free strip) depending only on the limit-s
s_finiteness_totally_geodesic_subvarieties	theorem	Finiteness of totally geodesic subvarieties in moduli of curves		There are only finitely many primitive totally geodesic subvarieties of dimension ≥ 2 in strata of translation surfaces / moduli of Riemann surfaces, proved by 
s_asymptotic_counting_minimal_surfaces	theorem	Asymptotic counting of minimal surfaces in hyperbolic 3-manifolds		The number of closed minimal surfaces of bounded genus in a closed hyperbolic 3-manifold grows with an exponential asymptotic analogous to the prime geodesic th
t_equidistribution_minimal_surfaces_frame_flow	technique	Equidistribution of minimal surfaces (frame-flow mixing)		Proving randomly chosen minimal surfaces equidistribute in the Grassmann bundle via exponential mixing of the frame/geodesic flow, to deduce asymptotic counting
s_roelcke_precompact_polish_group	axiom	Roelcke precompact Polish group		A topological group precompact for the lower (Roelcke) uniformity; these arise as automorphism groups of ℵ₀-categorical structures.
s_property_t_roelcke_precompact	theorem	Property (T) for Roelcke-precompact Polish groups		Large classes of Roelcke-precompact Polish groups, such as automorphism groups of homogeneous structures, satisfy a continuous-logic version of Kazhdan's proper
s_classification_nip_fields_johnson	theorem	Classification of NIP fields (Johnson's theorem)		Every infinite NIP field is separably closed, real closed, or admits a nontrivial henselian valuation, confirming Shelah's conjecture in the dp-finite case, pro
s_nip_dependent_theory	axiom	NIP (dependent theory)		A first-order theory has the Non-Independence Property if no formula has the independence property, equivalently every formula has finite VC dimension.
t_dp_rank_valued_fields	technique	dp-rank / dp-finiteness analysis of valued fields		A method classifying NIP fields by bounding dp-rank and reconstructing a canonical henselian valuation from the model-theoretic structure.
s_high_dimensional_expanders	state	High-dimensional expanders		Higher-dimensional analogues of expander graphs: bounded-degree simplicial complexes exhibiting strong spectral, coboundary, or topological expansion in all dim
t_local_spectral_coboundary_expansion	technique	Local spectral expansion / coboundary expansion		A bootstrapping criterion deriving global expansion of a simplicial complex from expansion of all of its vertex links.
s_metric_john_theorem	theorem	Metric John theorem (Naor)		A nonlinear analogue of John's ellipsoid theorem giving average-distortion bounds for embeddings of metric spaces, controlled by nonlinear spectral gaps, due to
s_nonlinear_spectral_gap	state	Nonlinear spectral gap		A generalization of the spectral gap of a graph to maps into nonlinear metric spaces, quantifying the obstruction to low-distortion embeddings.
s_local_marked_length_spectrum_rigidity	theorem	Local marked length spectrum rigidity		On a closed negatively curved (Anosov) manifold the marked length spectrum locally determines the metric up to isometry in all dimensions, proved by Guillarmou–
t_microlocal_geodesic_xray_transform	technique	Microlocal geodesic X-ray transform method		Using the geodesic X-ray transform, its normal operator as an elliptic pseudodifferential operator, and microlocal analysis to prove length-spectrum rigidity.
s_finite_time_implosion	theorem	Finite-time implosion for compressible fluids and NLS		Construction of smooth solutions of the compressible Euler/Navier–Stokes equations and energy-supercritical NLS developing self-similar implosion singularities 
t_self_similar_imploding_profile	technique	Self-similar imploding profile construction		Building finite-time singularities via smooth self-similar profile solutions and proving their nonlinear stability under perturbation.
s_kls_conjecture	state	Kannan–Lovász–Simonovits (KLS) conjecture		The conjecture that the Cheeger isoperimetric constant of an isotropic log-concave probability measure is bounded below by a universal dimension-independent con
t_stochastic_localization_eldan	technique	Stochastic localization (Eldan)		A measure-valued martingale process gradually localizing a log-concave measure into Gaussians, used by Eldan and Y. Chen to obtain near-optimal bounds toward th
s_binary_additive_problems_function_fields	theorem	Binary additive problems for polynomials over finite fields		Asymptotic counts for twin-prime and Goldbach-type additive problems over F_q[t] in the large-q regime, due to Sawin–Shusterman.
t_sheaf_theoretic_counting_function_fields	technique	Sheaf-theoretic counting over function fields		Translating additive prime problems over F_q[t] into computations of compactly-supported étale cohomology and bounding Betti numbers.
s_crystalline_measure	state	Crystalline measure		A tempered measure on Euclidean space whose support and the support of whose Fourier transform are both locally finite (discrete), generalizing Poisson summatio
s_kpi1_affine_artin_groups	theorem	K(π,1) conjecture for affine Artin groups		The Salvetti complex of an affine Coxeter system is aspherical, so its quotient is a classifying space, proving the K(π,1) conjecture for affine Artin groups, d
t_dual_braid_garside_method	technique	Dual braid monoid / Garside structure method		Establishing asphericity via Garside-theoretic structures on the dual braid monoid and the associated noncrossing-partition order complex with discrete Morse th
s_improved_bounds_three_term_aps	theorem	Improved bounds for sets without three-term arithmetic progressions		Improved upper bounds on subsets of {1,...,N} and of F_q^n with no nontrivial 3-term AP, including the cap-set polynomial-method bound and the Bloom–Sisask loga
t_slice_rank_polynomial_method	technique	Slice rank / polynomial method (cap sets)		Bounding extremal set sizes by upper-bounding the slice rank of an associated diagonal multilinear tensor via low-degree polynomials, due to Croot–Lev–Pach and 
s_unbounded_denominators_conjecture	theorem	Unbounded denominators conjecture		A modular form on a finite-index subgroup of SL₂(Z) whose q-expansion has bounded denominators is a modular form for a congruence subgroup, with a vector-valued
t_arithmetic_holonomicity_bound	technique	Arithmetic holonomicity bound		A potential-theoretic and transcendence inequality bounding the dimension of a space of holonomic power series with arithmetic integrality, combined with Nevanl
s_fluctuation_theory_boltzmann_grad	theorem	Fluctuation theory for the Boltzmann–Grad limit		Rigorous derivation, in the Boltzmann–Grad limit, of fluctuation and large-deviation corrections to the Boltzmann equation from a hard-sphere system, extending 
t_bbgky_cumulant_expansion	technique	BBGKY hierarchy / cumulant expansion for fluctuations		Expanding correlations of a hard-sphere system via the BBGKY hierarchy and cluster/cumulant expansions to control fluctuations around the Boltzmann limit.
s_pointwise_ergodic_polynomial_sequences	theorem	Pointwise ergodic theorems along polynomial sequences		Pointwise convergence of ergodic averages along sparse and polynomial sequences, via oscillation and variation estimates, due to J. Bourgain.
s_direct_summand_conjecture	theorem	Direct summand conjecture		For a finite extension of a regular Noetherian ring, the smaller ring is a direct summand as a module, proved in mixed characteristic by André and Bhatt.
t_perfectoid_almost_mathematics	technique	Perfectoid / almost-mathematics method		Reducing problems in mixed-characteristic commutative algebra to perfectoid algebras and tilting, controlling errors via Faltings's almost mathematics.
s_strong_convergence_random_permutation_matrices	theorem	Strong convergence of spectra of random permutation matrices		A sum of independent uniformly random permutation matrices, restricted to the orthogonal complement of the trivial representation, has operator norm converging 
t_strong_freeness_non_backtracking	technique	Strong asymptotic freeness / non-backtracking operator method		Establishing strong (operator-norm) convergence to free limits via the non-backtracking (Ihara–Bass) operator and matched-moment polynomial bounds.
s_mip_star_equals_re	theorem	MIP* = RE		The class of languages decidable by multiprover interactive proofs with entangled (quantum) provers equals the recursively enumerable languages, proved by Ji–Na
s_refutation_connes_embedding	state	Refutation of the Connes embedding conjecture		The conjecture that every separable II₁ factor embeds into an ultrapower of the hyperfinite II₁ factor, shown false as a consequence of MIP* = RE.
s_non_simplicity_area_preserving_homeo_s2	theorem	Non-simplicity of the group of area-preserving homeomorphisms of S²		The group of area- and orientation-preserving homeomorphisms of the 2-sphere is not simple, via unbounded quasimorphisms built from spectral invariants, due to 
s_periodic_floer_homology_spectral_invariants	state	Periodic Floer homology spectral invariants		Sequences of spectral invariants of area-preserving surface maps defined via periodic Floer homology, assembling into unbounded quasimorphisms and a Calabi homo
s_pointwise_convergence_schrodinger_carleson	theorem	Pointwise convergence for the Schrödinger equation (Carleson's problem)		The free Schrödinger evolution of an H^s(ℝⁿ) datum converges pointwise a.e. to the datum iff s ≥ n/(2(n+1)), resolving Carleson's problem, due to Du–Zhang.
t_polynomial_partitioning_schrodinger_maximal	technique	Polynomial partitioning / decoupling for the Schrödinger maximal operator		Bounding the Schrödinger maximal operator via broad–narrow analysis, polynomial partitioning, and ℓ²-decoupling with weighted fractal restriction estimates.
s_well_ordering_growth_rates_hyperbolic	theorem	Well-ordering of exponential growth rates of hyperbolic groups		The set of exponential growth rates of finitely generated subgroups of a hyperbolic group is well-ordered, with growth rates achieved and algebraic, proved by F
s_aspero_schindler_mm_implies_star	theorem	Asperó–Schindler theorem (MM⁺⁺ implies (∗))		Martin's Maximum⁺⁺ implies Woodin's axiom (∗), unifying two strong forcing axioms and settling the continuum at ℵ₂ under them, proved by Asperó–Schindler.
t_pmax_stationary_tower_forcing	technique	Pmax / stationary-tower forcing		Set-theoretic forcing constructions (Pmax, stationary tower) producing canonical models of strong forcing axioms.
s_classification_symmetric_tensor_categories_char_p	theorem	Classification of symmetric tensor categories in positive characteristic		Incompressibility and classification results for symmetric tensor categories of moderate growth over a field of characteristic p, with the Verlinde categories a
s_verlinde_category_ver_p	state	Verlinde category Ver_p		The semisimplification of the category of representations of Z/p in characteristic p, the universal incompressible target for symmetric tensor categories of mod
s_non_uniqueness_leray_hopf_forced_ns	theorem	Non-uniqueness of Leray–Hopf solutions to forced Navier–Stokes		There exist forcing terms for which the 3D Navier–Stokes equations admit two distinct Leray–Hopf weak solutions, exhibiting non-uniqueness in the energy class, 
t_unstable_self_similar_profile_method	technique	Unstable self-similar profile / spectral instability method		Constructing two solutions by exhibiting a linearly unstable self-similar background and bifurcating a second trajectory along an unstable eigenmode.
s_twisted_lang_weil_estimate	theorem	Twisted Lang–Weil estimate		A point-count estimate for definable sets twisted by a Frobenius-type correspondence, quantifying equidistribution, with applications to difference algebra and 
s_rotational_invariance_critical_percolation	theorem	Rotational invariance of critical planar percolation		The scaling limit of critical Bernoulli percolation on a class of planar lattices is rotationally invariant, a step toward full conformal invariance, due to Dum
t_rsw_star_triangle_method	technique	RSW / star-triangle (Yang–Baxter) transformation method		Using Russo–Seymour–Welsh box-crossing estimates with star-triangle (Yang–Baxter) transformations to deform lattices and transport crossing events, yielding rot
t_quadratic_chabauty	technique	Quadratic Chabauty		The depth-two explicit instance of Kim's nonabelian Chabauty method (Balakrishnan-Dogra) that uses p-adic height pairings to cut out an explicit finite locus co
t_kim_nonabelian_chabauty	technique	Kim's Nonabelian Chabauty Method		A generalization of Chabauty's method replacing the Jacobian by the unipotent (de Rham/étale) fundamental group, locating rational points inside a Selmer-type p
s_coleman_effective_chabauty_bound	theorem	Coleman's Effective Chabauty Bound		Under the Chabauty rank hypothesis, a curve of genus g with good reduction at a prime p > 2g has at most #C(𝔽_p) + 2g − 2 rational points.
t_lawrence_venkatesh_method	technique	Lawrence-Venkatesh Method		A proof of the Mordell conjecture using a Kodaira-Parshin family of covers and p-adic Hodge theory to show that the p-adic Galois representations attached to fi
t_skolem_p_adic_method	technique	Skolem's Method (p-adic, integral points)		A p-adic method for integral points expressing solutions as zeros of p-adic analytic functions whose finiteness follows from the Strassmann/Weierstrass preparat
s_croot_lev_pach_lemma	theorem	Croot-Lev-Pach lemma		A low-degree multilinear polynomial that vanishes off the diagonal of a finite-field cube has a number of nonzero diagonal entries bounded by twice the dimensio
s_ellenberg_gijswijt_cap_set	theorem	Ellenberg-Gijswijt cap set theorem		Every cap set in (ℤ/3ℤ)^n has size at most c^n with c ≈ 2.756, i.e. exponentially smaller than 3^n, resolving the cap set problem via the polynomial method.
s_slice_rank_of_a_tensor	axiom	Slice rank of a tensor		The minimum number of terms needed to write a tensor as a sum of products each of which depends in one factor on a single variable group and in the other on the
s_sunflower_free_tricolored_slice_rank	theorem	Sunflower-free / tri-colored sum-free bounds via slice rank		Application of the slice rank method giving exponential upper bounds on sunflower-free set systems and on tri-colored sum-free sets in abelian groups (Naslund-S
s_analytic_l_function_axiomatic	axiom	Analytic L-function (axiomatic)		A Dirichlet series with Euler product, meromorphic continuation, and a functional equation relating s to 1−s, packaged with prescribed analytic data such as con
s_stably_rational_variety	axiom	Stably Rational Variety		A variety X such that X × ℙ^m is rational for some m, an intermediate notion strictly between rational and unirational.
s_luroth_problem	state	Lüroth Problem		The question of whether every unirational variety is rational, answered affirmatively for curves and surfaces but negatively in dimension three and higher.
t_decomposition_of_the_diagonal	technique	Decomposition of the Diagonal Method		An obstruction to (stable) rationality showing that a stably rational variety admits a Chow-theoretic decomposition of its diagonal, whose failure under special
s_non_stable_rationality_hypersurfaces	theorem	Non-Stable-Rationality of Very General Hypersurfaces		A very general hypersurface of sufficiently high degree in projective space (degree at least roughly two-thirds the dimension, after Totaro and Schreieder) is n
s_big_polynomial_ring_uniformity	state	Big Polynomial Ring / Uniformity (Erman-Sam-Snowden)		The phenomenon, formalized via the inverse-limit polynomial ring in infinitely many variables and GL-equivariant commutative algebra, that bounds on invariants 
s_stillmans_conjecture	theorem	Stillman's Conjecture (Theorem)		The projective dimension of an ideal generated by a fixed number of homogeneous polynomials of fixed degrees is bounded independently of the number of variables
s_strength_of_a_polynomial	axiom	Strength of a Polynomial		The least number r such that a homogeneous form is a sum of r products of lower-degree forms, a measure of degeneracy controlling the uniform algebraic behavior
s_tangent_developable_surface	axiom	Tangent Developable Surface		The ruled surface swept out by the tangent lines to a space curve, naturally attached to the curve and whose singular structure encodes the curve's projective g
s_greens_conjecture	theorem	Green's Conjecture		The Clifford index of a smooth projective curve equals the index of the first nonzero linear syzygy in the minimal free resolution of its canonical embedding, p
s_koszul_cohomology_property_np	axiom	Koszul Cohomology / Property N_p		The graded groups computing the syzygies of an embedded variety, with property N_p meaning the resolution is linear up to homological degree p, governing how a 
t_degeneration_to_tangent_developable	technique	Degeneration to Tangent Developable Surfaces		A proof method computing syzygies of canonical curves by specializing to the tangent developable of a rational normal curve and using its explicit, computable r
s_compactified_jacobian_universal_picard	state	Compactified Jacobian / Universal Picard Variety		A compactification of the moduli of line bundles over a family of curves (including degenerate nodal fibers) parametrizing rank-one torsion-free sheaves with a 
t_tropical_dual_graph_boundary	technique	Tropical / Dual-Graph Combinatorics of the Boundary		The encoding of the boundary strata of compactified moduli spaces by the dual graphs of stable curves, linking the geometry to tropical moduli and combinatorial
s_space_of_diagonal_harmonics	axiom	Space of Diagonal Harmonics		The bigraded vector space DH_n of polynomials in two sets of n variables killed by all symmetric-group-invariant differential operators of positive degree, carr
s_nabla_operator	axiom	Nabla Operator		The Garsia-Bergeron linear operator on symmetric functions, diagonal in the modified Macdonald polynomial basis with explicit eigenvalues, whose value on e_n eq
s_parking_function	axiom	Parking Function (combinatorics)		A Dyck path of order n together with a column-increasing labeling of its north steps by 1,...,n, equipped with the statistics area and dinv (diagonal inversions
s_shuffle_conjecture	theorem	Shuffle Conjecture (Theorem)		The symmetric function ∇(e_n) equals the weighted generating function over parking functions of order n with weight q^dinv t^area times the quasisymmetric funct
s_compositional_shuffle_conjecture	theorem	Compositional Shuffle Conjecture (Theorem)		A refinement of the shuffle conjecture indexed by compositions controlling the touch points of the Dyck path, whose proof by Carlsson and Mellit (via the double
s_overconvergent_modular_form	axiom	Overconvergent Modular Form		A p-adic modular form converging on a strict neighborhood of the ordinary locus of the modular curve, organized into Coleman's finite-slope eigenvariety theory.
s_up_operator_slope_decomposition	axiom	U_p Operator and Slope Decomposition		The completely continuous Atkin-Lehner operator on spaces of overconvergent forms whose finite-slope generalized eigenspaces (by p-adic valuation of eigenvalue)
s_coleman_classicality_theorem	theorem	Coleman Classicality Theorem		An overconvergent eigenform of weight k and slope strictly less than k−1 is a classical modular form.
s_rigid_meromorphic_cocycles	state	Rigid Meromorphic Cocycles / p-adic Singular Moduli		The Darmon-Vonk p-adic analogues of singular moduli, constructed from overconvergent forms and real quadratic geodesics, conjecturally algebraic and refining co
s_classical_siegel_weil_formula	theorem	Classical Siegel-Weil Formula		An identity expressing a special value of a Siegel Eisenstein series as the average over a genus of theta series of quadratic forms.
s_special_cycles_shimura	axiom	Special Cycles on Shimura Varieties		Algebraic cycles on orthogonal or unitary Shimura varieties indexed by positive-definite quadratic/Hermitian data, generalizing Heegner points to higher codimen
s_kudla_modularity_conjecture	theorem	Kudla's Modularity Conjecture (Theorem)		The generating series whose coefficients are the classes of special cycles of a fixed codimension in the Chow group of a Shimura variety is a (vector-valued) mo
s_geometric_siegel_weil_formula	theorem	Geometric Siegel-Weil Formula		An identity expressing the special value of a Siegel Eisenstein series as a generating series of degrees/volumes of special cycles on a Shimura variety, lifting
s_arithmetic_siegel_weil_formula	theorem	Arithmetic Siegel-Weil Formula		An identity relating arithmetic intersection numbers of special cycles on integral models of Shimura varieties to derivatives of Siegel Eisenstein series at the
s_kudla_rapoport_conjecture	theorem	Kudla-Rapoport Conjecture (Theorem)		A local identity equating the arithmetic intersection number of special cycles on a Rapoport-Zink space with the derivative of a local representation/Whittaker 
s_rapoport_zink_space	axiom	Rapoport-Zink Space		A formal/rigid moduli space of p-divisible groups with additional structure (polarization, level) serving as the local p-adic uniformization of Shimura varietie
s_arithmetic_inner_product_formula	theorem	Arithmetic Inner Product Formula		An identity relating the Beilinson-Bloch height of an arithmetic theta lift cycle to the central derivative of an L-function, generalizing the Gross-Zagier form
s_beilinson_bloch_conjecture	state	Beilinson-Bloch Conjecture		The conjecture that the rank of the Chow group of homologically trivial cycles on a smooth projective variety equals the order of vanishing of the associated L-
t_macaulay_inverse_system	technique	Macaulay Inverse System		Macaulay's duality between Artinian quotients of a polynomial ring and finitely generated submodules of the dual divided-power (or polynomial) module under cont
s_embedding_q_finitely_presented	theorem	Embedding Q into a Finitely Presented Group (Theorem)		The additive group of rational numbers embeds into a finitely presented group, realized via Belk-Hyde-Matucci using twisted Brin-Thompson groups acting on Canto
s_twisted_brin_thompson_group	axiom	Twisted Brin-Thompson Group		A group of homeomorphisms of a Cantor space built from a base group acting on an alphabet together with prefix-replacement (Thompson-type) moves, simple and fin
s_space_time_white_noise	axiom	Space-time white noise		A Gaussian generalized random field on ℝ₊ × ℝ^d whose values over disjoint space-time regions are independent with covariance equal to the Lebesgue measure of t
s_dynamical_phi4_equation	axiom	Dynamical Phi^4 equation		The renormalized stochastic reaction-diffusion equation ∂_t Φ = ΔΦ − (Φ³ − ∞·Φ) + ξ, the Langevin dynamics of Φ⁴ Euclidean QFT, singular in spatial dimension d 
s_parabolic_anderson_model	axiom	Parabolic Anderson Model		The linear singular SPDE ∂_t u = Δu + u·(ζ − ∞) with spatial white noise potential ζ, requiring renormalization in dimension d ≥ 2.
t_renormalization_counterterm_spde	technique	Renormalization (counter-term subtraction) for singular SPDEs		Mollifying the noise, subtracting divergent counter-terms C_ε (chosen via Gaussian moment / Wick-power computations) from the nonlinearity, and passing to the l
t_da_prato_debussche_trick	technique	Da Prato-Debussche trick		Splitting the solution Φ = u + v into the stochastic linear (Gaussian) part u and a higher-regularity remainder v solving a classically well-posed fixed-point e
s_primary_uncertainty_principle	theorem	Primary uncertainty principle		For operators A, B with ‖A‖_{1→∞} ≤ 1, ‖B‖_{1→∞} ≤ 1 and ‖BAv‖_∞ ≥ k‖v‖_∞, one has ‖v‖₁‖Av‖₁ ≥ k‖v‖_∞‖Av‖_∞, from which classical Fourier uncertainty principles
s_k_hadamard_matrix	axiom	k-Hadamard matrix		A matrix A with entries bounded by 1 satisfying ‖A*Av‖_∞ ≥ k‖v‖_∞ (equivalently A*A invertible with ‖(A*A)⁻¹‖_{∞→∞} ≤ 1/k); precisely the matrices to which the 
s_donoho_stark_uncertainty	theorem	Donoho-Stark support-size uncertainty principle		For a nonzero function f on a finite abelian group G with Fourier transform f̂, the supports satisfy |supp(f)|·|supp(f̂)| ≥ |G|.
s_meshulam_nonabelian_uncertainty	theorem	Meshulam's non-abelian uncertainty principle		An extension of the Donoho-Stark support-size inequality to functions on arbitrary finite groups via rank and min-support notions of support for the Fourier tra
s_hardy_uncertainty_principle	theorem	Hardy's uncertainty principle		If |f(x)| ≤ C e^{−a|x|²} and |f̂(ξ)| ≤ C e^{−b|ξ|²} with ab > 1/4 then f = 0, and with ab = 1/4 then f is a constant multiple of the Gaussian e^{−a|x|²}.
s_dynamical_hardy_reformulation	state	Dynamical (Schrödinger) reformulation of Hardy's uncertainty principle		The reinterpretation of Hardy's theorem as a unique-continuation/rigidity statement for the free Schrödinger equation: a solution Gaussian-bounded at two distin
t_liouville_proof_hardy_uncertainty	technique	Liouville-theorem proof of Hardy's uncertainty principle		A proof deriving the uncertainty principle from the Schrödinger connection together with Liouville's theorem for bounded entire functions rather than Phragmén-L
s_sine_kernel_fredholm_determinant	axiom	Sine-kernel Fredholm determinant		The determinant P_x = det(1 − K_sine) of the trace-class operator with kernel sin(x(z−z'))/(π(z−z')), giving the probability of no GUE eigenvalues in a union of
s_widom_large_gap_asymptotics	theorem	Widom's large-gap asymptotics for the sine-kernel determinant		Widom's rigorous proof that ln P_x = −(x²/2)(1 + o(1)) as x → ∞, the first rigorous large-gap asymptotics for the sine determinant.
s_tracy_widom_painleve_representation	theorem	Tracy-Widom Painlevé representation		The identity F_2(t) = exp(−∫_t^∞ (s−t)u(s)² ds) where u is the Hastings-McLeod solution of the second Painlevé equation u_tt = t u + 2u³, with F_1, F_4 also exp
s_integrable_fredholm_operator	axiom	Integrable Fredholm operator		An integral operator whose kernel has the form (φ(z)ψ(z')−ψ(z)φ(z'))/(z−z'), a structure whose determinant satisfies nonlinear integrable PDEs and admits a Riem
t_nonlinear_steepest_descent_rh	technique	Riemann-Hilbert / nonlinear steepest descent method		Representing Painlevé and integrable-determinant functions as matrix Riemann-Hilbert problems whose asymptotics are computed via the Deift-Zhou nonlinear steepe
s_arnold_geodesic_hydrodynamics	axiom	Arnold's geodesic formulation of ideal hydrodynamics		The principle that the incompressible Euler equations are the geodesic equations of the right-invariant L² (kinetic-energy) metric on the group of volume-preser
s_newton_equations_diffeo_density	state	Newton's equations on diffeomorphism groups and density spaces		The generalization ∇_{q̇} q̇ = −∇U(q) of geodesic flow to infinite dimensions, with a potential U on Diff(M) or Dens(M), unifying compressible/incompressible fl
s_fisher_rao_metric_infinite	axiom	Fisher-Rao information metric (infinite-dimensional)		The diffeomorphism-invariant information-geometric Riemannian metric on probability densities, whose Newton equations yield the μ-Camassa-Holm and Klein-Gordon 
s_madelung_transform_symplectomorphism	theorem	Madelung transform as a symplectomorphism		The map sending a wave function to its density-and-phase (hydrodynamic) variables is a symplectomorphism (and Kähler morphism) intertwining the Schrödinger equa
t_symplectic_poisson_reduction_hydro	technique	Symplectic and Poisson reduction for hydrodynamics		Reduction of Hamiltonian systems with semidirect-product symmetry used to derive Hamiltonian/Poisson structures, Casimir invariants, and equations of motion for
t_frequency_envelopes	technique	Frequency envelopes		A device assigning a slowly varying admissible sequence dominating the Littlewood-Paley pieces of the data, used to track and propagate the dyadic distribution 
t_rough_solutions_as_limits	technique	Rough solutions as limits of smooth solutions		Constructing low-regularity solutions of quasilinear evolutions by regularizing the data, solving smoothly, and passing to the limit using uniform energy and fr
t_quasilinear_energy_estimates	technique	Quasilinear energy estimates		Coercive a priori bounds for symmetric (or symmetrizable) hyperbolic systems ∂_t u = A_j(u)∂_j u obtained by differentiating, symmetrizing and integrating by pa
t_convex_integration_hydrodynamics	technique	Convex integration in hydrodynamics		An iterative construction adding high-frequency oscillatory perturbations (Mikado/intermittent flows) to approximate solutions of the fluid equations to cancel 
s_onsagers_conjecture	state	Onsager's conjecture		The conjecture that weak solutions of the incompressible Euler equations conserve kinetic energy if their Hölder regularity exponent exceeds 1/3 and may dissipa
s_flexible_side_onsager	theorem	Resolution of the flexible side of Onsager's conjecture		The construction (Isett; Buckmaster-De Lellis-Székelyhidi-Vicol) via convex integration of C^{1/3−} weak solutions of the Euler equations that dissipate energy,
s_nonuniqueness_weak_navier_stokes	theorem	Nonuniqueness of weak solutions of Navier-Stokes		Buckmaster-Vicol's theorem that the 3D incompressible Navier-Stokes equations admit nonunique weak solutions with bounded kinetic energy, obtained by intermitte
s_intermittency_convex_integration	state	Intermittency in convex integration		The use of spatially concentrated (high L^∞, low L^p for p<2) building-block flows that mimic turbulent energy concentration and supply the extra integrability 
s_polya_fourier_transform_gaussian	axiom	Polya-Fourier transform with Gaussian factor		The entire function H_{ρ,λ}(z) = ∫ e^{izt} e^{λt²} dρ(t) extending the Riemann ξ-function, whose real-zeros property as λ varies encodes the Riemann Hypothesis.
s_de_bruijn_newman_constant	state	de Bruijn-Newman constant		The finite constant Λ_DN ∈ (−∞, 1/2] such that H_{Φ,λ} has only real zeros iff λ ≥ Λ_DN, so the Riemann Hypothesis is equivalent to Λ_DN ≤ 0.
s_polya_debruijn_newman_bounds	theorem	Polya monotonicity / De Bruijn upper bound / Newman lower bound		The three facts that the real-zeros set P_Φ is an upper half-line (Pólya), contains 1/2 (De Bruijn), and is bounded below (Newman), which together define Λ_DN.
s_rodgers_tao_theorem	theorem	Rodgers-Tao theorem (Λ_DN ≥ 0)		The 2018 proof by Rodgers and Tao of Newman's 1976 conjecture that the de Bruijn-Newman constant is nonnegative, formalizing that the Riemann Hypothesis, if tru
s_newman_wu_weak_convergence	theorem	Newman-Wu weak-convergence theorem for real-zero distributions		A theorem that the purely-real-zeros property of Fourier transforms of a sequence of probability measures controls their tail behavior uniformly, used to classi
s_euler_alignment_equations	axiom	Euler alignment (hydrodynamic alignment) equations		The pressure-augmented continuity-momentum system ∂_t ρ + div(ρu) = 0, ∂_t(ρu) + div(ρu⊗u + P) = A(ρ,u) with nonlocal alignment forcing A weighted by a symmetri
s_p_alignment_hydrodynamics	axiom	p-alignment hydrodynamics		The class of alignment models based on (fractional) weighted 2p-graph Laplacians, generalizing Cucker-Smale alignment to nonlinear p-Laplacian-type interaction 
s_entropic_pressure_tensor	state	Entropic pressure tensor		A pressure law required only to form an energy-dissipative process (no thermodynamic closure assumption), introduced to study long-time alignment dynamics with 
s_flocking_p_alignment_entropic	theorem	Flocking of p-alignment with entropic pressure		Tadmor's result that p-alignment hydrodynamics driven by singular (heavy-tailed) communication kernels flock unconditionally even with general entropic pressure
t_energy_fluctuation_decay_flocking	technique	Energy-fluctuation decay method for flocking		Quantifying flocking by controlling the decay of velocity-energy fluctuations, with enstrophy bounds upgraded to Hölder regularity and uniform dispersion bounds
s_non_euclidean_elasticity	axiom	Non-Euclidean (incompatible) elasticity		The variational model in which a growing thin body carries a prescribed target metric g not realizable by an isometric immersion, and the observed shape minimiz
s_metric_frustration_biological_shape	state	Metric frustration as the origin of biological shape		The principle that morphogenesis of thin tissues results from differential growth inducing an incompatible (e.g. hyperbolic) metric whose frustrated embedding i
s_gamma_convergence_plate_shell_hierarchy	theorem	Gamma-convergence hierarchy of thin elastic plate/shell theories		The derivation, as thickness tends to zero, of a hierarchy of reduced 2D energies (membrane, von Kármán, bending/Kirchhoff) as Γ-limits of 3D non-Euclidean elas
t_convex_integration_prestrained_immersions	technique	Convex-integration construction of low-regularity isometric immersions of prestrained films		The use of Nash-Kuiper-type convex integration to construct C^{1,α} very-weak (non-Euclidean) isometric immersions realizing prescribed growth metrics, with ene
s_monge_ampere_constrained_energy	state	Monge-Ampère constrained energy for weak prestrains		The reduced energy governing weakly prestrained thin films whose admissible out-of-plane deflections are constrained by a Monge-Ampère (Gauss-curvature) equatio
s_boolean_gaussian_noise_stability	axiom	Boolean / Gaussian noise stability		The probability that two ρ-correlated random inputs (on the discrete cube or, in the Gaussian limit, on ℝ^n) yield the same output of a function, the central qu
s_majority_is_stablest	theorem	Majority is Stablest theorem		Among low-influence balanced Boolean functions, weighted majority maximizes noise stability, with optimal value given by Borell's Gaussian (arccos) noise-stabil
s_gaussian_quantitative_arrow	theorem	Quantitative Arrow theorem		A quantitative version of Arrow's impossibility theorem: any aggregation rule ε-far from every dictatorship produces an irrational (non-transitive) outcome with
s_quantitative_gibbard_satterthwaite	theorem	Quantitative Gibbard-Satterthwaite theorem		Any social choice function far from dictatorship has a non-negligible (inverse-polynomial) probability of being manipulable by a random voter, proved via isoper
t_invariance_principle_low_influence	technique	Invariance principle for low-influence functions		A nonlinear Lindeberg-type theorem showing low-degree, low-influence multilinear polynomials have nearly the same distribution under Bernoulli and Gaussian inpu
s_boolean_social_choice_function	axiom	Boolean social choice function		A function assigning a societal outcome to a profile of voter preferences (e.g. a Boolean function aggregating n voters' binary opinions), the basic object of p
s_dolds_theorem_equivariant	theorem	Dold's theorem (equivariant map nonexistence)		For a finite group G of order >1, if X is n-connected with a free G-action and Y is paracompact of dimension ≤ n with a free G-action, then there is no G-equiva
t_test_map_configuration_space	technique	Test map / configuration space scheme		Reducing a combinatorial mass-partition problem to the nonexistence of a G-equivariant map by parametrizing partitions as a configuration space X and encoding t
s_mass_partition_problem	axiom	Mass partition problem		Given a fixed family P of partitions of ℝ^d each into a fixed number of parts and a family H of measures, deciding whether some partition in P splits each measu
s_general_position_point_set	axiom	General position (point set)		A finite set of points in ℝ^d such that no d+1 of them lie on a common hyperplane.
s_necklace_splitting_theorem	theorem	Necklace splitting theorem (Hobby-Rice / Alon)		For m absolutely continuous probability measures on ℝ and r thieves, there is a partition using (r−1)m cuts whose pieces can be distributed among r parties so e
t_moment_curve_lifting	technique	Moment curve lifting (necklace splitting proof)		Embedding measures on ℝ via the moment curve t ↦ (t, t², ..., t^m) into ℝ^m and applying the Ham Sandwich theorem, so a bisecting hyperplane yields the required
s_grunbaum_hadwiger_ramos_problem	state	Grünbaum-Hadwiger-Ramos problem		Determining the smallest dimension d for which a single arrangement of k hyperplanes can equipart any given collection of j measures in ℝ^d.
s_nandakumar_ramana_rao_problem	state	Nandakumar-Ramana-Rao problem		Whether any convex body (or measure) in ℝ^d can be partitioned into k convex pieces simultaneously equal in measure and in a prescribed list of d−1 additional c
s_convex_equipartition_theorem	theorem	Convex equipartition theorem (spicy chicken theorem)		For k a prime power, any absolutely continuous probability measure on ℝ^d admits a partition into k convex pieces of equal measure on which any chosen d−1 conti
s_holmsen_kyncl_valculescu_conjecture	state	Holmsen-Kynčl-Valculescu conjecture		A conjecture asserting bounds on partitions of colored point sets into parts each capturing prescribed numbers of points from each color class while respecting 
s_center_transversal_theorem	theorem	Center Transversal theorem		For any k+1 measures in ℝ^d there exists a k-dimensional affine subspace that is a center transversal: every hyperplane containing it has at least a 1/(d−k+1) f
s_yao_yao_partition_theorem	theorem	Yao-Yao partition theorem		A measure in ℝ^d can be partitioned into 2^d convex cones from a common apex, each of equal measure and with bounded directional spread, enabling efficient geom
s_same_type_lemma	theorem	Same-type lemma		Given finitely many point sets in general position in ℝ^d, large positive-fraction subsets can be chosen so that every transversal (one point per subset) realiz
s_ppa_completeness_ham_sandwich	theorem	PPA-completeness of discrete Ham Sandwich		Computing a discrete ham sandwich cut, with dimension as part of the input, is complete for the complexity class PPA (Polynomial Parity Argument).
t_categorification	technique	Categorification		Lifting an algebraic invariant to a richer categorical object (homology groups or a category) whose decategorification (Euler characteristic or Grothendieck gro
s_reduced_khovanov_homology	state	Reduced Khovanov homology		A variant of Khovanov homology defined using a marked point on the link, splitting unreduced homology and refining the invariant.
s_functoriality_khovanov_cobordism	theorem	Functoriality of Khovanov homology under link cobordisms		A smooth cobordism between links in ℝ³ × [0,1] induces a (well-defined up to sign) map on Khovanov homology, making it a functor from the link cobordism categor
s_lee_spectral_sequence_homology	theorem	Lee spectral sequence and Lee homology		A deformation of the Khovanov differential yields Lee homology of dimension 2^{#components}, with an induced spectral sequence from Khovanov homology converging
s_rasmussen_s_invariant	state	Rasmussen s-invariant		An integer s(K) extracted from the adjacent quantum gradings of the two surviving generators on the E_∞ page of the Lee spectral sequence, defining a concordanc
s_rasmussen_slice_genus_bound	theorem	Rasmussen's slice-genus bound		The s-invariant is a homomorphism from the smooth concordance group and satisfies |s(K)| ≤ 2 g_4(K), giving a combinatorial lower bound on the slice genus.
s_combinatorial_milnor_conjecture	theorem	Combinatorial proof of the Milnor conjecture (torus knots)		Rasmussen's s-invariant gives s(T_{p,q}) = 2 g_4(T_{p,q}) = (p−1)(q−1), yielding the first gauge-theory-free proof that the slice genus of the (p,q)-torus knot 
s_khovanov_stable_homotopy_type	state	Khovanov stable homotopy type (Lipshitz-Sarkar spectrum)		A stable homotopy refinement (a spectrum) whose cohomology recovers Khovanov homology, carrying Steenrod operations that strictly strengthen the homological inv
s_odd_khovanov_homology	state	Odd Khovanov homology		An integral variant of Khovanov homology (Ozsváth-Rasmussen-Szabó) that lifts the Ozsváth-Szabó spectral sequence to ℤ-coefficients and is mutation invariant.
s_kronheimer_mrowka_unknot_detection	theorem	Kronheimer-Mrowka unknot detection		Via a spectral sequence from singular-instanton homology to Khovanov homology, if reduced Khovanov homology of K has rank one then K is the unknot.
s_symplectic_khovanov_homology	state	Symplectic Khovanov homology		A Floer-theoretic (Lagrangian intersection) construction of a link invariant via the geometry of nilpotent slices, conjecturally (and in cases provably) isomorp
t_spectral_sequences_to_khovanov	technique	Spectral sequences from gauge/Floer theories to Khovanov homology		A family of spectral sequences from Khovanov homology to instanton, monopole, or Heegaard Floer invariants used to prove detection results in the spirit of unkn
s_teichmuller_curve	axiom	Teichmüller curve		An algebraic curve V isometrically and holomorphically immersed in the moduli space M_g of genus-g Riemann surfaces with respect to the Teichmüller metric, i.e.
s_veech_group_lattice_surface	axiom	Veech group / lattice surface SL(X, ω)		The group of derivatives of affine automorphisms of a translation surface (X, ω); when it is a lattice in SL_2(ℝ) the surface is a lattice (Veech) surface gener
s_primitive_teichmuller_curve	axiom	Primitive Teichmüller curve		A Teichmüller curve in M_g that does not arise by covering constructions from a Teichmüller curve in a moduli space of lower genus.
s_optimal_dynamics_lattice_billiards	theorem	Optimal dynamics of billiards in lattice polygons		Billiards in a polygon whose unfolding generates a lattice surface have optimal dynamics: every trajectory is either periodic or uniformly distributed; in parti
s_regular_polygons_primitive_teichmuller	theorem	Regular polygons generate primitive Teichmüller curves		The translation surface obtained by unfolding billiards in a regular n-gon has a lattice Veech group and generates a primitive Teichmüller curve.
s_masurs_criterion	theorem	Masur's criterion		If the Teichmüller geodesic ray generated by (X, ω) does not return to a compact set infinitely often, then its horizontal foliation fails to be uniquely ergodi
t_cylinder_decomposition_parabolic	technique	Cylinder decomposition and parabolic affine automorphisms		Decomposing a translation surface into parallel cylinders with commensurable moduli and applying simultaneous Dehn twists to produce parabolic elements of the V
s_weierstrass_curve_wd	state	Weierstrass curve W_D (genus 2 classification)		For each non-square discriminant D, the locus W_D in M_2 of genus-2 surfaces with a double-zero 1-form whose Jacobian admits real multiplication by the order of
s_classification_primitive_teichmuller_genus2	theorem	Classification of primitive Teichmüller curves in genus 2		Every primitive Teichmüller curve in M_2 is one of the Weierstrass curves W_D (with the decagon example a special case), giving a complete classification in gen
s_finiteness_primitive_teichmuller_genus3	theorem	Finiteness of primitive Teichmüller curves in genus 3 outside W_D		In M_3 there are only finitely many primitive Teichmüller curves not belonging to the Weierstrass series W_D.
s_infinitude_primitive_teichmuller_open	state	Infinitude of primitive Teichmüller curves in higher genus (open problem)		The open question of whether there exist infinitely many primitive Teichmüller curves in M_g for g ≥ 5, where no infinite construction is currently known.
s_slice_knot	axiom	Slice knot		A knot K in S³ that bounds a smoothly embedded disk in the 4-ball B⁴.
s_slice_genus	axiom	Slice genus		The minimal genus of a smooth compact oriented connected surface embedded in the 4-ball with boundary K; slice knots are exactly those of slice genus zero.
s_knot_mutation	axiom	Knot mutation		Excising a tangle from a knot diagram and regluing it after a rotation, relating the Conway knot to the Kinoshita-Terasaka knot and preserving many invariants.
s_knot_trace	axiom	Knot trace X(K)		The 4-manifold obtained from the 4-ball by attaching a 0-framed 2-handle along the knot K in S³ = ∂B⁴.
s_conway_knot_not_slice	theorem	The Conway knot is not slice		Piccirillo's theorem that the Conway knot does not bound a smooth disk in the 4-ball, settling the last sub-13-crossing knot whose sliceness was unknown.
s_trace_embedding_lemma	theorem	Trace embedding lemma		A knot K is slice if and only if its trace X(K) embeds smoothly in S⁴, so that sliceness depends only on the diffeomorphism type of the trace.
t_handle_rearrangement_trace_equivalent	technique	Handle-rearrangement construction of trace-equivalent knots		Building the knot trace from a 4-ball plus a 1-handle and two 2-handles and reinterpreting which 2-handle is the attaching knot, producing a different knot K' w
s_trace_invariance_nu	theorem	Trace invariance of nu (knot Floer slice obstruction)		The slice obstruction ν from knot Floer homology is an invariant of the knot trace, so knots sharing a trace have equal ν, allowing K' to obstruct sliceness of 
s_vanishing_classical_slice_obstructions_conway	state	Vanishing of classical slice obstructions for the Conway knot		Every classical sliceness obstruction (Alexander polynomial, signature, Rasmussen s-invariant, etc.) vanishes for the Conway knot, motivating Piccirillo's trace
s_topological_helly_theorem	theorem	Topological Helly theorem		For a good cover (a family of open sets in ℝ^d all of whose nonempty intersections are contractible), if every d+1 sets intersect then all sets have a common po
s_colorful_caratheodory_theorem	theorem	Colorful Caratheodory theorem		Given d+1 color classes in ℝ^d each whose convex hull contains a point x, there is a rainbow selection of one point from each class whose convex hull contains x
s_upper_bound_theorem_convex_sets	theorem	Upper bound theorem for convex sets		Bounds the number of k-fold intersection points of n convex sets in ℝ^d in which no d+r+1 sets have a common point, in terms of the number of empty intersection
s_topological_tverberg_conjecture	state	Topological Tverberg conjecture		Conjecture that for any continuous map from the (d+1)(r−1)-simplex to ℝ^d there exist r pairwise vertex-disjoint faces whose images have a common point; true fo
s_sierksma_conjecture	state	Sierksma's conjecture		Conjecture that the number of Tverberg r-partitions of any (r−1)(d+1)+1 points in general position in ℝ^d is at least ((r−1)!)^d.
s_cascade_conjecture	state	Cascade conjecture		Kalai's conjecture that if Σ_i (1+dim T_i(A)) over i=1..r−1 is less than |A| then the r-fold Tverberg set T_r(A) is nonempty, interpolating between Radon and Tv
t_nerve_complex_method	technique	Nerve complex method		Encoding the intersection pattern of a family of sets as a simplicial complex (the nerve) whose combinatorial-topological properties (d-collapsibility, being d-
s_impagliazzo_wigderson_theorem	theorem	Impagliazzo-Wigderson theorem (hardness vs randomness)		If some problem in E requires Boolean circuits of size 2^{Ω(n)}, then P = BPP, i.e. randomized polynomial time can be fully derandomized.
t_hardness_amplification	technique	Hardness amplification (Yao XOR lemma / direct product)		Converting a function mildly hard on average into one extremely hard on average by taking XORs or direct products of independent instances.
s_zigzag_product_of_graphs	state	Zig-zag product of graphs		A graph product (Reingold-Vadhan-Wigderson) combining a large graph with a small one to produce expanders of controlled degree, yielding an iterative combinator
s_sl_equals_l_theorem	theorem	SL = L theorem (Reingold)		Undirected s-t connectivity can be decided in deterministic logarithmic space (symmetric logspace equals logspace), proved via the zig-zag product.
t_operator_scaling	technique	Operator scaling / noncommutative optimization		A geodesically-convex alternating-minimization algorithm over symmetric (group-orbit) spaces that scales tuples of matrices toward doubly balanced form, solving
s_efficiency_gap	state	Efficiency gap		A scalar measure of partisan advantage in a districting plan equal to the normalized difference between the two parties' total wasted votes (votes for losers pl
s_partisan_symmetry	axiom	Partisan symmetry		The criterion that a districting plan's seats-votes curve be invariant under swapping the two parties' roles, i.e. symmetric under (x,y) ↦ (1−x,1−y).
t_ensemble_outlier_analysis_districting	technique	Ensemble (outlier) analysis of districting plans		Detecting partisan gerrymanders by generating a large random sample of legally valid districting plans and testing whether an enacted plan is a statistical outl
s_recombination_recom_markov_chain	state	Recombination (ReCom) Markov chain		A Markov chain on balanced graph partitions that merges two adjacent districts, samples a uniform spanning tree of the union, and splits it along an edge into t
s_chikina_frieze_pegden_local_outlier_test	theorem	Chikina-Frieze-Pegden local outlier test		For a reversible Markov chain, a rigorous statistical significance bound on a plan being an outlier obtained from short random walks without knowing the mixing 
s_nphardness_balanced_connected_partition	theorem	NP-hardness of balanced connected graph partition		Sampling or counting balanced partitions of a weighted planar graph into connected pieces of equal weight is NP-hard, even into two parts, formalizing the compu
s_sunflower_conjecture_open	state	Sunflower conjecture		Conjecture of Erdős and Rado that for each r there is a constant c(r) such that any family of more than c(r)^w sets of size at most w contains an r-sunflower.
s_spread_robust_sunflower_family	axiom	Spread (robust sunflower) family		A weighted set family is r-spread if for every set T the total weight of members containing T is at most r^{−|T|} of the whole, a pseudorandomness condition for
s_kahn_kalai_expectation_threshold	theorem	Kahn-Kalai expectation-threshold conjecture (Park-Pham theorem)		For any monotone property, the threshold probability is at most a logarithmic factor times the expectation threshold (the fractional lower bound), proved by Par
t_random_restriction_with_spreadness	technique	Random restriction with spreadness		Repeatedly applying random restrictions to a spread set family, shrinking set sizes while preserving approximate spreadness until a sunflower is forced, with Ja
s_chow_ring_of_a_matroid	axiom	Chow ring of a matroid		A graded commutative ring built from the flats of a matroid (via the Bergman fan / order complex) that plays the role of the cohomology of a smooth projective v
s_kahler_package_for_matroids	theorem	Kahler package for matroids		The Chow ring of any matroid satisfies Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations, despite the absence of an underlying varie
s_heron_rota_welsh_conjecture	theorem	Heron-Rota-Welsh conjecture (theorem)		The coefficients of the reduced characteristic polynomial of any matroid form a log-concave sequence, proved via the Hodge-Riemann relations on the Chow ring.
s_dowling_wilson_top_heavy	theorem	Dowling-Wilson top-heavy theorem		For any matroid the number of flats of rank k is at most the number of flats of corank k for k below half the rank (Braden-Huh-Matherne-Proudfoot-Wang).
s_mason_log_concavity_conjecture	theorem	Mason's log-concavity conjecture (theorem)		The numbers of independent sets of each size in a matroid form an (ultra) log-concave sequence, established via the Hodge-theoretic / Lorentzian polynomial fram
t_hodge_riemann_relations_combinatorial	technique	Hodge-Riemann relations (combinatorial use)		Establishing inequalities by proving that a bilinear form on a degree-graded ring is nondegenerate with prescribed signature, transferring algebraic positivity 
s_classical_verification_quantum_computation	axiom	Classical verification of quantum computation		A protocol in which a polynomial-time classical verifier interacts with a single untrusted quantum prover and accepts only if the prover correctly performs a cl
s_trapdoor_claw_free_function_family	axiom	Trapdoor claw-free function family		A family of two-to-one functions for which finding a colliding pair (a claw) is computationally hard while a secret trapdoor allows inversion, instantiated from
s_mahadev_classical_verification_theorem	theorem	Mahadev's classical verification theorem		Assuming the quantum hardness of learning with errors, there is an interactive protocol by which a classical verifier can verify the output of an arbitrary poly
t_cryptographic_measurement_protocol	technique	Cryptographic measurement protocol (qubit commitment)		A cryptographic primitive forcing a quantum prover to commit to a qubit and later reveal a measurement in a verifier-chosen basis, so the prover cannot cheat ac
s_quintic_calabi_yau_threefold	axiom	Quintic Calabi-Yau threefold	quintic threefold	A smooth degree-5 hypersurface in complex projective 4-space, the canonical example of a compact Calabi-Yau threefold with trivial canonical bundle.
s_virtual_fundamental_class	axiom	Virtual fundamental class	virtual class	A homology class of expected (virtual) dimension associated to a moduli space carrying a perfect obstruction theory, enabling enumerative integrals even when th
s_moduli_space_stable_maps	axiom	Moduli space of stable maps	Kontsevich moduli space | moduli of stable maps	The compact moduli stack parametrizing maps from nodal genus-g curves with marked points into a target satisfying Kontsevich's stability condition of finitely m
s_bcov_holomorphic_anomaly_equation	axiom	BCOV holomorphic anomaly equation	holomorphic anomaly equation | BCOV equation	A recursion of Bershadsky-Cecotti-Ooguri-Vafa governing the dependence of higher-genus topological string free energies on the anti-holomorphic moduli, predicti
s_yamaguchi_yau_finiteness_conjecture	state	Yamaguchi-Yau finiteness conjecture	Yamaguchi-Yau polynomial structure conjecture	The conjecture that the genus-g Gromov-Witten potential of the quintic threefold is a polynomial of bounded degree in a finite set of generators built from spec
s_structure_theorem_higher_genus_gw_quintic	theorem	Structure theorem for higher-genus GW invariants of the quintic		The theorem establishing the BCOV holomorphic anomaly equations and the Yamaguchi-Yau polynomial finiteness for the all-genus Gromov-Witten potentials of the qu
t_mixed_spin_p_field	technique	Mixed-Spin-P (MSP) field	MSP field	A moduli-theoretic construction interpolating between the Gromov-Witten theory of the quintic and Fan-Jarvis-Ruan-Witten theory via a master space carrying a to
t_virtual_localization	technique	Virtual (Graber-Pandharipande) localization	Graber-Pandharipande localization | virtual localization formula	An equivariant localization formula expressing virtual integrals over a moduli space with torus action as a sum of contributions from the fixed loci weighted by
s_yau_conjecture_infinitely_many_minimal_hypersurfaces	state	Yau conjecture on infinitely many minimal hypersurfaces		Yau's conjecture that every closed Riemannian manifold of dimension between 3 and 7 contains infinitely many smooth closed embedded minimal hypersurfaces.
s_marques_neves_infinitely_many_minimal_hypersurfaces	theorem	Marques-Neves theorem (infinitely many minimal hypersurfaces)		The theorem of Marques and Neves that any closed Riemannian manifold of dimension 3 to 7 with positive Ricci curvature contains infinitely many smooth embedded 
s_weyl_law_volume_spectrum	theorem	Weyl law for the volume spectrum (Liokumovich-Marques-Neves)	Liokumovich-Marques-Neves Weyl law	An asymptotic law showing the p-th min-max width (volume spectrum) of a closed n-manifold grows like a universal constant times p^{1/n} times the manifold volum
s_volume_spectrum_p_widths	state	Volume spectrum (p-widths)	p-widths | min-max widths	The non-decreasing sequence of min-max critical values of the area functional obtained from p-parameter sweepouts, a nonlinear analogue of Laplacian eigenvalues
s_equidistribution_minimal_hypersurfaces_mns	theorem	Equidistribution of minimal hypersurfaces (Marques-Neves-Song)		The result that for a generic metric on a closed manifold there exists a sequence of closed minimal hypersurfaces that becomes equidistributed, with area-weight
s_generic_density_minimal_hypersurfaces_imn	theorem	Generic density of minimal hypersurfaces (Irie-Marques-Neves)		The theorem that for a C-infinity generic Riemannian metric on a closed manifold of dimension 3 to 7, the union of all closed minimal hypersurfaces is dense.
s_probabilistically_checkable_proof	axiom	Probabilistically checkable proof (PCP)	PCP	A proof system in which a randomized verifier, given oracle access to a proof string, decides membership using bounded randomness and a bounded number of querie
s_unique_games_conjecture	state	Unique Games Conjecture	UGC	Khot's conjecture that for every epsilon there is an alphabet size such that distinguishing unique-constraint label-cover instances that are (1-epsilon)-satisfi
s_two_to_two_games_theorem	theorem	2-to-2 Games Theorem	2-to-2 games conjecture (theorem) | two-to-two games theorem	The theorem of Khot-Minzer-Safra establishing NP-hardness of 2-to-2 games with imperfect completeness, implying the Unique Games Conjecture holds with completen
s_grassmann_graph	axiom	Grassmann graph		The graph whose vertices are the k-dimensional subspaces of an n-dimensional vector space over a finite field, with edges joining subspaces whose intersection h
s_grassmann_graph_expansion_theorem	theorem	Grassmann graph expansion theorem (characterization of non-expanding sets)		The theorem of Khot-Minzer-Safra (with Dinur-Khot-Kindler-Minzer-Safra) that every non-expanding set of vertices in the Grassmann graph is structured, correlate
t_dictatorship_test	technique	Dictatorship test		A probabilistic gadget distinguishing Boolean functions depending on a single coordinate (dictators) from functions with no influential coordinate, used to conv
s_arithmetic_spectral_transition_amo	theorem	Arithmetic spectral (localization) transition for the almost Mathieu operator	arithmetic phase transition | Jitomirskaya arithmetic localization transition	The sharp result (Avila-You-Zhou, Jitomirskaya et al.) that in the localization regime the spectral type of the almost Mathieu operator is determined by an arit
s_universality_critical_almost_mathieu	theorem	Universality of the critical almost Mathieu operator	critical almost Mathieu universality | hidden singularity	Results showing that at critical coupling lambda = 1 the almost Mathieu operator exhibits a hidden singularity with universal continuity of spectral gaps and sh
s_parity_sheaf	axiom	Parity sheaf		An indecomposable complex of sheaves with positive-characteristic coefficients on a stratified variety satisfying parity vanishing on stalks and costalks, provi
s_soergel_bimodule	axiom	Soergel bimodule		A graded bimodule over a polynomial ring built as a direct summand of tensor products of elementary bimodules associated to simple reflections, categorifying th
s_lusztig_modular_representation_conjecture	state	Lusztig modular representation conjecture	Lusztig character formula conjecture	Lusztig's conjecture expressing the characters of simple modules for a reductive algebraic group in characteristic p (above the Coxeter number) via affine Kazhd
s_williamson_counterexamples_modular_bounds	theorem	Williamson counterexamples to expected bounds in modular representation theory	Williamson torsion examples	Williamson's construction of families showing that Lusztig's character formula and the James conjecture fail, with the bound on the prime p growing super-expone
s_p_canonical_basis	state	p-canonical basis		The basis of the Hecke algebra recording graded ranks of indecomposable Soergel bimodules / parity sheaves in characteristic p, controlling modular decompositio
s_symmetry_protected_topological_phase	axiom	Symmetry protected topological (SPT) phase	SPT phase	A gapped phase of a quantum many-body system with unique ground state that is trivial without symmetry but cannot be connected to a product state through symmet
s_gapped_ground_state_phase_equivalence	axiom	Gapped ground state phase equivalence		The equivalence relation on interactions defined by existence of a smooth path of uniformly gapped local Hamiltonians (automorphic equivalence via quasi-local a
s_matrix_product_state	axiom	Matrix product state	MPS	A translation-invariant pure state of a spin chain represented by contracting a fixed tensor at each site, generating all frustration-free gapped ground states 
s_group_cohomology_invariant_spt	state	Group cohomology invariant of SPT phases		The assignment to each symmetric gapped phase of an element of the second group cohomology H^2(G,U(1)) of the symmetry group, extracted from the projective repr
s_ogata_classification_spt_phases	theorem	Ogata classification of SPT phases by H^2(G,U(1))		Ogata's operator-algebraic theorem proving the H^2(G,U(1)) index is a well-defined invariant of symmetric gapped phases of quantum spin chains, giving a rigorou
s_split_property_spin_chains	state	Split property of quantum spin chains		The structural property of a pure gapped ground state that the observable algebras of complementary half-infinite chains are related by a type-I factor, enablin
s_black_hole_stability_problem	state	Black hole stability problem	Kerr stability conjecture	The conjecture that the Kerr family of rotating black hole spacetimes is a nonlinearly asymptotically stable solution of the vacuum Einstein equations under sma
s_kerr_de_sitter_spacetime	axiom	Kerr (Kerr-de Sitter) spacetime	Kerr-de Sitter black hole | Kerr metric	The two-parameter family of stationary axisymmetric asymptotically flat (or de Sitter) vacuum Einstein solutions describing a rotating black hole of given mass 
s_hintz_vasy_kerr_de_sitter_stability	theorem	Hintz-Vasy global stability of Kerr-de Sitter		The theorem of Hintz and Vasy establishing global nonlinear asymptotic stability of slowly rotating Kerr-de Sitter black holes as solutions of the Einstein vacu
t_microlocal_stationary_wave_operators	technique	Microlocal analysis of stationary wave operators (b-calculus / scattering calculus)		A framework using pseudodifferential operators adapted to the geometry near the event horizon and infinity (b- and scattering calculus, radial point estimates) 
s_quasinormal_modes_black_holes	state	Quasinormal modes / resonances of black holes	ringdown frequencies | black hole resonances	The complex frequencies, identified with poles of the meromorphically continued resolvent of the wave operator, that govern the exponentially damped ringdown of
t_redshift_effect_dafermos_rodnianski	technique	Red-shift effect (Dafermos-Rodnianski)		A geometric vector-field multiplier estimate exploiting positive surface gravity at a nondegenerate event horizon to gain a coercive energy estimate (decay) for
s_painleve_conjecture_noncollision_singularities	state	Painleve conjecture (noncollision singularities)	noncollision singularity conjecture	Painleve's conjecture that for N greater than 3 the Newtonian N-body problem admits noncollision singularities, i.e. solutions reaching infinity in finite time 
s_von_zeipel_theorem_singularities	theorem	Von Zeipel theorem on singularities		The theorem that a singularity of the N-body problem is a noncollision singularity if and only if the maximal mutual distance among the bodies becomes unbounded
s_xia_noncollision_singularities_5body	theorem	Xia's example of noncollision singularities (5-body)		Zhihong Xia's construction of a spatial 5-body configuration exhibiting a noncollision singularity, where two binaries oscillate and drive a fifth body to infin
s_xue_painleve_4body	theorem	Xue's solution of the Painleve conjecture for the 4-body problem		Jinxin Xue's theorem constructing noncollision singularities in the planar Newtonian 4-body problem, settling the Painleve conjecture in its lowest open dimensi
t_near_collision_hyperbolic_dynamics	technique	Near-collision (regularization) analysis via hyperbolic dynamics		A method combining McGehee/Levi-Civita regularization of close encounters with hyperbolic invariant sets and shadowing/symbolic dynamics to assemble infinitely 
s_phi4_4_euclidean_field_theory	axiom	Phi^4_4 Euclidean field theory	phi^4 in four dimensions | Phi^4_4 model	The four-dimensional Euclidean scalar field theory with quartic self-interaction, constructed as a continuum scaling limit of lattice models with a single-site 
t_random_current_representation	technique	Random current representation		A graphical/probabilistic reformulation of the Ising model expressing correlation functions as sums over integer-valued currents on the lattice, enabling switch
s_triviality_scaling_limit_d_ge_4	theorem	Triviality of the scaling limit in dimension d >= 4		The result of Aizenman and Frohlich that the continuum scaling limit of the critical Ising and Phi^4 models in dimensions strictly greater than 4 is a free (Gau
s_aizenman_duminil_copin_marginal_triviality_4d	theorem	Aizenman-Duminil-Copin marginal triviality theorem in 4D		The theorem of Aizenman and Duminil-Copin that the scaling limits of the critical nearest-neighbor Ising and Phi^4 models in the marginal dimension 4 are Gaussi
s_aizenman_frohlich_tree_graph_inequality	theorem	Aizenman-Frohlich (tree-graph) inequality	tree-graph inequality	An upper bound on the truncated four-point Ursell function of the Ising / Phi^4 model by a tree-graph sum of two-point functions, the key correlation inequality
s_reductive_p_adic_group	axiom	Reductive p-adic group	p-adic reductive group	A connected reductive linear algebraic group G over a non-archimedean local field F, together with its group of F-points regarded as a locally profinite topolog
s_smooth_representation_p_adic_group	axiom	Smooth representation of a p-adic group		A representation of G(F) on a complex vector space in which every vector has open stabilizer, equivalently the union of fixed spaces of compact open subgroups i
s_admissible_representation_p_adic	axiom	Admissible representation (p-adic group)		A smooth representation of G(F) in which the fixed subspace under every compact open subgroup is finite-dimensional.
s_supercuspidal_representation	axiom	Supercuspidal representation	supercuspidal | cuspidal representation of a p-adic group	An irreducible smooth representation of G(F) all of whose matrix coefficients are compactly supported modulo the center, equivalently one not appearing in any p
t_parabolic_induction_p_adic	technique	Parabolic induction (p-adic group)		The functor producing a smooth representation of a p-adic group from a smooth representation of a Levi subgroup by normalized induction along a parabolic subgro
s_parahoric_subgroup_bruhat_tits_building	axiom	Parahoric subgroup and Bruhat-Tits building	Bruhat-Tits building | parahoric subgroup | Iwahori subgroup	The Bruhat-Tits building is a contractible polysimplicial complex on which G(F) acts strongly transitively; parahoric subgroups are the connected stabilizers of
s_moy_prasad_filtration	axiom	Moy-Prasad filtration		A decreasing filtration of a parahoric subgroup by normal compact open subgroups indexed by non-negative real numbers, defined via the Bruhat-Tits building and 
s_depth_of_a_representation	state	Depth of a representation		The smallest non-negative real number r such that the representation has nonzero vectors fixed by the (r+)-step of the Moy-Prasad filtration of some point, a nu
s_type_bushnell_kutzko	axiom	Type (Bushnell-Kutzko theory)	Bushnell-Kutzko type	A pair (K, rho) of a compact open subgroup K and an irreducible representation rho of K such that the irreducible representations containing rho exactly fill ou
s_tame_elliptic_regular_pair	axiom	Tame elliptic regular pair / generic cuspidal datum	generic cuspidal datum | Yu datum	The input data for Yu's construction: a tame twisted Levi sequence together with generic characters and a depth-zero supercuspidal datum satisfying a genericity
s_twisted_levi_sequence	axiom	Twisted Levi sequence		A chain of reductive subgroups of a p-adic group, each a Levi subgroup over a tame extension, used as the structural backbone of Yu's supercuspidal construction
t_yu_construction_supercuspidals	technique	Yu's construction of supercuspidal representations		A construction producing irreducible smooth representations of G(F) by compact induction from a chain of subgroups built using a tame twisted Levi sequence, Hei
s_supercuspidality_yu_representations_fintzen	theorem	Supercuspidality of Yu-constructed representations (Fintzen)		Fintzen's theorem that the representations output by Yu's construction are irreducible and supercuspidal, with a self-contained proof valid in all residual char
s_fintzen_exhaustion_tame_supercuspidals	theorem	Exhaustion theorem for tame supercuspidals (Fintzen)	Fintzen exhaustion theorem	If p does not divide the order of the absolute Weyl group of G, then every supercuspidal representation of G(F) arises from Yu's construction, sharpening the pr
s_fintzen_kaletha_spice_quadratic_twist	state	Fintzen-Kaletha-Spice quadratic twist		A correction by an explicit sign character of an appropriate compact open subgroup, twisting Yu's construction so the resulting representations match expected c
s_adler_spice_character_formula	theorem	Adler-Spice / DeBacker-Spice character formula		An explicit formula for the Harish-Chandra character of a tame supercuspidal representation on regular semisimple elements, expressed via Gauss sums and the geo
s_depth_zero_supercuspidal_representation	state	Depth-zero supercuspidal representation		A supercuspidal representation of depth zero, obtained by compact induction of an inflation of a cuspidal representation of the reductive quotient of a parahori
s_local_langlands_p_adic_groups	state	Local Langlands correspondence for p-adic groups	local Langlands conjecture for general reductive p-adic groups	The conjectural bijection between L-packets of irreducible smooth representations of a p-adic group and Langlands parameters into its Langlands dual group.
s_mixed_quantum_state_density_matrix	axiom	Mixed quantum state (density matrix)	density matrix | density operator | mixed state	A d-by-d positive semidefinite Hermitian matrix of trace one representing the state of a quantum system of dimension d.
s_quantum_state_tomography_problem	state	Quantum state tomography problem	quantum state tomography	The task of producing, from independent copies of an unknown d-dimensional mixed state, an estimate close to it in trace distance with high probability, using a
s_quantum_spectrum_estimation_problem	state	Quantum spectrum estimation / testing problem		The task of estimating or testing properties of the unordered spectrum (eigenvalue multiset) of an unknown mixed state from independent copies, without estimati
t_weak_schur_sampling	technique	Weak Schur sampling	Keyl-Werner spectrum estimation | Schur sampling	The measurement of rho-tensor-n by the projectors onto the GL(d) x S_n isotypic subspaces indexed by partitions, whose outcome is a random Young diagram.
s_schur_weyl_distribution	state	Schur-Weyl distribution		The probability distribution on partitions of n arising as the outcome of weak Schur sampling on rho-tensor-n, with probabilities given by the Schur polynomial 
s_rsk_realization_schur_weyl_distribution	theorem	RSK realization of the Schur-Weyl distribution	RSK-Schur-Weyl connection	The shape of the RSK tableau produced from a word of n i.i.d. letters drawn from the eigenvalue distribution of rho is distributed exactly as the Schur-Weyl dis
t_empirical_young_diagram_algorithm	technique	Empirical Young Diagram (EYD) algorithm	EYD algorithm	The spectrum-estimation procedure that performs weak Schur sampling on rho-tensor-n and outputs the normalized sorted Young diagram as an estimate of the spectr
s_optimal_sample_complexity_state_tomography	theorem	Optimal sample complexity of state tomography	O'Donnell-Wright / Haah et al. tomography bound	The theorem that Theta(d r / epsilon^2) copies are necessary and sufficient to estimate an unknown rank-r d-dimensional state to trace distance epsilon, with th
s_pure_state_tomography_sample_complexity	theorem	Pure-state tomography sample complexity		The result that Theta(d/epsilon^2) copies suffice and are required to learn an unknown pure state in dimension d to fidelity error epsilon.
s_max_schur_weyl_row_top_eigenvalue	theorem	Convergence of the maximal Schur-Weyl row to the top eigenvalue		The normalized longest row of a Schur-Weyl-distributed diagram concentrates around the largest eigenvalue of rho, generalizing the Vershik-Kerov / Logan-Shepp l
s_quantum_identity_testing_bounds	theorem	Quantum identity / maximal-mixedness testing bounds	quantum state certification	Property-testing results, proved via the Schur-Weyl distribution, giving the optimal number of copies needed to distinguish whether an unknown state equals the 
s_p_adic_galois_representation	axiom	p-adic Galois representation		A continuous representation of the absolute Galois group of a number field on a finite-dimensional vector space over a p-adic field.
s_de_rham_potentially_semistable_representation	axiom	de Rham / potentially semistable representation	de Rham Galois representation | potentially semistable representation | Hodge-Tate representation	A p-adic representation of a local Galois group that is de Rham (equivalently potentially semistable) in Fontaine's p-adic Hodge theory, the local condition exp
s_algebraic_galois_representation	axiom	Algebraic Galois representation		A continuous semisimple p-adic representation of the absolute Galois group of a number field, unramified outside finitely many primes and de Rham at all places 
s_fontaine_mazur_conjecture	state	Fontaine-Mazur conjecture		The conjecture that an irreducible p-adic Galois representation of a number field unramified outside finitely many primes and de Rham at p arises, up to twist, 
s_regular_algebraic_cuspidal_automorphic_representation	axiom	Regular algebraic cuspidal automorphic representation	RACAR | regular algebraic cuspidal form	A cuspidal automorphic representation of GL_n over a number field whose archimedean component has the same infinitesimal character as a finite-dimensional algeb
s_symmetric_power_lifting_sym_m	state	Symmetric power lifting Sym^m	symmetric power functoriality lift | Sym^m lift	The functorial lift along the m-th symmetric power map Sym^m: GL_2 -> GL_{m+1}, sending an automorphic representation of GL_2 to a conjectural automorphic repre
s_thorne_reciprocity_conjecture	state	Reciprocity conjecture (Langlands global reciprocity, Thorne)	global Langlands reciprocity conjecture	The conjecture that every algebraic p-adic Galois representation of a number field corresponds to a regular algebraic automorphic representation of GL_n with ma
s_sato_tate_conjecture	theorem	Sato-Tate conjecture	Sato-Tate equidistribution	The theorem that for a non-CM elliptic curve or holomorphic newform over Q the normalized Frobenius traces are equidistributed with respect to the Sato-Tate (se
s_newton_thorne_symmetric_power_functoriality	theorem	Newton-Thorne symmetric power functoriality theorem	symmetric power functoriality for holomorphic newforms	For every regular algebraic cuspidal automorphic representation of GL_2 over Q without CM and every m >= 1, the symmetric power lift Sym^m exists as a cuspidal 
s_automorphy_lifting_theorem	theorem	Automorphy lifting theorem	modularity lifting theorem	A theorem asserting that a p-adic Galois representation with suitable local conditions whose mod-p reduction is automorphic is itself automorphic, identifying a
t_taylor_wiles_kisin_patching	technique	Taylor-Wiles-Kisin patching	Taylor-Wiles method | Taylor-Wiles-Kisin method	A patching technique proving R = T by adjoining auxiliary Taylor-Wiles primes to enlarge the level and taking an inverse limit to produce a big module over a po
t_brauer_induction_l_functions	technique	Brauer induction reduction of L-functions		Expressing the L-function of a symmetric power representation as a ratio of automorphic L-functions using Brauer's induction theorem on virtual characters, so p
s_coleman_mazur_eigencurve	state	Coleman-Mazur eigencurve	eigencurve | eigenvariety	A rigid analytic curve p-adically interpolating finite-slope overconvergent Hecke eigenforms of fixed tame level, parametrizing systems of Hecke eigenvalues wit
s_overconvergent_finite_slope_eigenform	axiom	Overconvergent finite-slope Hecke eigenform		A p-adic modular form that is overconvergent and an eigenform for the U_p operator with nonzero (finite slope) eigenvalue, the points of the eigencurve.
t_analytic_continuation_functoriality_eigencurve	technique	Analytic continuation of functoriality along the eigencurve		Newton-Thorne's strategy of proving symmetric power functoriality at one accessible point of the eigencurve and propagating it p-adically across the connected e
s_pseudo_representation_pseudocharacter	axiom	Pseudo-representation (pseudocharacter)	pseudocharacter | determinant law	A trace-like function on a group satisfying the formal identities of the trace of an n-dimensional representation, used to construct and interpolate Galois repr
s_future_null_infinity	axiom	Future null infinity (scri-plus)	scri-plus | null infinity | I-plus	The idealized boundary at which outgoing light rays of an asymptotically flat spacetime terminate, foliated by retarded-time cuts on which radiated quantities a
s_bondi_metzner_sachs_group	axiom	Bondi-Metzner-Sachs (BMS) group	BMS group	The asymptotic symmetry group of future null infinity, an infinite-dimensional semidirect product of the Lorentz group with the abelian group of supertranslatio
s_supertranslation	axiom	Supertranslation		An angle-dependent shift of the retarded time coordinate by an arbitrary smooth function on the celestial sphere, generalizing rigid time translation and genera
s_bondi_sachs_energy_momentum	state	Bondi-Sachs energy-momentum	Bondi energy-momentum | Bondi four-momentum	The four-vector of conserved energy and linear momentum at a cut of null infinity, obtained by integrating the mass aspect against the lowest spherical harmonic
s_news_tensor	state	News tensor		The retarded-time derivative of the shear tensor of null infinity, encoding outgoing gravitational radiation and having a simple transformation under supertrans
s_bondi_mass_loss_formula	theorem	Bondi mass loss formula		The identity that the retarded-time derivative of the Bondi energy equals minus the integral of the squared news tensor over the sphere, so radiated gravitation
s_supertranslation_ambiguity_angular_momentum	state	Supertranslation ambiguity of angular momentum		The phenomenon that classical definitions of angular momentum at null infinity change under BMS supertranslations, so different distant observers may measure di
s_wang_yau_quasilocal_mass	state	Wang-Yau quasi-local mass	Wang-Yau quasilocal mass | Wang-Yau quasi-local energy	A quasi-local energy assigned to a spacelike 2-surface defined by comparing its geometry to an optimal isometric embedding into Minkowski space, characterized b
s_optimal_isometric_embedding	state	Optimal isometric embedding		An isometric embedding of a spacelike 2-surface into Minkowski spacetime that is critical for the Wang-Yau quasi-local energy, equivalently making the embedded 
s_chen_wang_yau_angular_momentum	state	Chen-Wang-Yau angular momentum	Chen-Wang-Yau quasilocal angular momentum	A definition of angular momentum at null infinity obtained as the large-radius limit of quasi-local angular momentum, pairing the optimal isometric embedding wi
s_angular_momentum_at_null_infinity	state	Angular momentum at null infinity		A definition of the total angular momentum of an isolated gravitating system extracted from the asymptotic geometry of the spacetime at future null infinity.
s_supertranslation_invariance_cwy_angular_momentum	theorem	Supertranslation invariance of Chen-Wang-Yau angular momentum		Chen-Wang-Yau's theorem that their quasi-local-limit definition of total angular momentum at null infinity is invariant under BMS supertranslations, resolving t
s_supertranslation_invariant_center_of_mass	state	Supertranslation-invariant center of mass at null infinity		A definition of the center-of-mass integral at null infinity shown to be supertranslation invariant, forming with the Bondi energy-momentum a complete set of co
s_basis_generating_polynomial_matroid	state	Basis generating polynomial of a matroid	bases generating polynomial	The multivariate polynomial summing monomials over the bases of a matroid, encoding its combinatorial structure as a homogeneous multiaffine polynomial.
s_completely_log_concave_polynomial	axiom	Completely log-concave polynomial	completely log-concave | strongly log-concave polynomial	A homogeneous polynomial with non-negative coefficients whose every iterated directional derivative is a log-concave function on the positive orthant.
s_lorentzian_polynomial	axiom	Lorentzian polynomial	Lorentzian polynomials (Branden-Huh)	A homogeneous polynomial of degree at least two with non-negative coefficients whose Hessian has at most one positive eigenvalue on the positive orthant, the cl
s_anari_liu_oveisgharan_vinzant_log_concavity	theorem	Anari-Liu-Oveis Gharan-Vinzant log-concavity theorem		The theorem that the basis generating polynomial of every matroid is completely log-concave, equivalently Lorentzian.
s_matroid_expansion_conjecture_mihail_sudan	theorem	Mihail-Sudan / matroid expansion theorem	Mihail-Vazirani matroid expansion conjecture	The result that the basis exchange graph of any matroid has spectral expansion, equivalently the bases-exchange walk is a one-sided spectral expander.
s_local_spectral_expander	axiom	Local spectral expander	high-dimensional expander | local-spectral expander	A simplicial complex whose vertex links are all spectral expanders, a condition propagating up to fast mixing of the top-dimensional random walk.
t_spectral_independence	technique	Spectral independence		A method bounding mixing times of Markov chains by controlling the spectral norm of pairwise influence matrices, derived from local-to-global spectral expansion
s_mihail_vazirani_matroid_sampling	theorem	Mihail-Vazirani matroid sampling consequence		The result that the bases-exchange random walk on any matroid mixes in polynomial time, yielding an efficient algorithm to sample a uniformly random basis.
s_random_lozenge_tiling	axiom	Random lozenge tiling	random rhombus tiling	A uniformly random tiling of a planar region by lozenges, equivalently a uniformly random dimer configuration on the hexagonal lattice or a stepped surface.
s_dimer_model	axiom	Dimer model	perfect matching model	A statistical-mechanical model of perfect matchings on a planar bipartite graph, weighted by edge weights and analyzed through the Kasteleyn matrix.
s_limit_shape_variational_principle_tilings	theorem	Limit shape (variational principle for tilings)	Cohn-Kenyon-Propp variational principle	The law of large numbers asserting that the normalized height function of a random tiling converges to a deterministic surface minimizing a surface-tension func
s_arctic_circle_frozen_boundary	state	Arctic circle / frozen boundary	arctic curve | frozen boundary	The deterministic curve separating the frozen (deterministic) region from the disordered (liquid) region in the limit shape of a random tiling of a region.
s_gaussian_free_field_fluctuations_tilings	theorem	Gaussian free field fluctuations of tilings		The statement that height-function fluctuations of a random tiling in the liquid region converge, after a complex-structure change of variables, to the two-dime
s_airy_line_ensemble_edge_universality	theorem	Airy line ensemble edge universality	Airy process edge universality	The result that local statistics of a random tiling near a smooth point of the arctic boundary converge to the Airy line ensemble independently of the model det
s_sine_kernel_bulk_universality_tilings	theorem	Sine kernel bulk universality for tilings		The result that local correlations of a random tiling in the liquid region converge to the discrete sine-kernel determinantal process determined only by the loc
s_prismatic_cohomology	axiom	Prismatic cohomology	prismatic site cohomology	A cohomology theory for p-adic formal schemes defined via the prismatic site of bounded prisms, specializing to and unifying crystalline, de Rham, and etale p-a
s_prism	axiom	Prism	bounded prism	A delta-ring equipped with an ideal that is a Cartier divisor and over which the Frobenius lift satisfies a compatibility condition with p, the base object of p
s_delta_ring	axiom	Delta-ring	delta-ring | lambda-ring of p-typical type	A commutative ring equipped with a delta-operation whose induced map is a lift of the Frobenius endomorphism modulo p.
s_prismatic_f_crystal	axiom	Prismatic F-crystal		A crystal of finite locally free modules on the prismatic site equipped with a Frobenius structure, conjecturally classifying p-adic motivic data over a base.
s_bhatt_scholze_crystals_equals_lattices	theorem	Bhatt-Scholze crystals equals lattices theorem		The theorem that over a p-adic ring of integers the category of prismatic F-crystals is equivalent to the category of Galois-stable lattices in crystalline p-ad
s_crystalline_cohomology	axiom	Crystalline cohomology		A p-adic Weil cohomology theory for smooth proper schemes over a perfect field of characteristic p, computed via divided-power thickenings and recovered as a sp
s_p_adic_hodge_comparison	theorem	p-adic Hodge theory comparison	p-adic comparison isomorphisms	The collection of comparison isomorphisms relating etale, de Rham, and crystalline cohomologies of p-adic varieties, unified and refined through the prismatic f
s_p_adic_motive	state	p-adic motive		A conjectural universal object capturing the p-adic cohomological invariants of a variety, realized concretely via prismatic F-crystals and their associated Gal
s_circular_unitary_ensemble_characteristic_polynomial	axiom	Circular Unitary Ensemble / CUE characteristic polynomial	CUE | CUE characteristic polynomial	The characteristic polynomial of a Haar-distributed unitary matrix, used as a model for the value distribution of the Riemann zeta function on the critical line
s_log_correlated_gaussian_field	axiom	Log-correlated Gaussian field	log-correlated field	A random field whose covariance decays like the logarithm of the inverse distance, exhibiting branching-random-walk extreme-value behavior.
s_branching_random_walk_maximum	theorem	Branching random walk maximum (BRW)	BRW maximum asymptotics	The theorem giving leading-order and logarithmic-correction asymptotics for the maximum of a branching random walk, the universal template for maxima of log-cor
s_maximum_cue_characteristic_polynomial	theorem	Maximum of CUE characteristic polynomial		The result establishing leading-order and subleading logarithmic asymptotics of the maximum of the logarithm of the CUE characteristic polynomial on the unit ci
t_arithmetic_algebraization	technique	Arithmetic algebraization	arithmetic holonomy method	A method proving that a formal power series with arithmetic integrality and analytic growth/holonomy constraints is algebraic or rational, generalizing the Bore
s_borel_dwork_rationality_criterion	theorem	Borel-Dwork rationality criterion	Borel-Dwork criterion	A power series with integer coefficients that converges p-adically and archimedeanly on domains whose product of radii exceeds one is the expansion of a rationa
s_schur_siegel_smyth_trace_problem	theorem	Schur-Siegel-Smyth trace problem result	Schur-Siegel-Smyth trace problem	Dimitrov's resolution, via arithmetic holonomy bounds, of conjectures on the minimal limit of the absolute trace of totally positive algebraic integers.
s_kronecker_coefficient	state	Kronecker coefficient		The multiplicity of an irreducible representation of the symmetric group in the tensor product of two others, a quantity lacking a known combinatorial positive 
s_complexity_class_sharp_p	axiom	Complexity class #P	#P | sharp-P	The class of counting problems expressible as the number of accepting paths of a nondeterministic polynomial-time Turing machine.
s_combinatorial_interpretation_in_sharp_p	state	Combinatorial interpretation as membership in #P		The formalization that a structure constant has a combinatorial interpretation precisely when its computation lies in the complexity class #P, framing the centr
t_geometric_complexity_theory	technique	Geometric Complexity Theory	GCT	The program of Mulmuley and Sohoni attacking P versus NP and the permanent versus determinant problem through representation theory and the geometry of orbit cl
s_plethysm_coefficient	state	Plethysm coefficient		The multiplicity arising in the decomposition of the composition of two symmetric functions or Schur functors, central to representation-theoretic complexity qu
s_gct_obstruction_vanishing_rectangular_kronecker	theorem	Vanishing of rectangular Kronecker coefficients / GCT obstruction		The Burgisser-Ikenmeyer-Panova result that the conjectured occurrence obstructions of geometric complexity theory cannot separate permanent from determinant, re
s_saturation_theorem_littlewood_richardson	theorem	Saturation theorem for Littlewood-Richardson coefficients	Knutson-Tao saturation theorem	Knutson and Tao's theorem that a Littlewood-Richardson coefficient is nonzero whenever a dilation of its indices is, proven via the honeycomb model.
s_positivity_kronecker_coefficients_np_hard	theorem	Positivity of Kronecker coefficients is NP-hard		The complexity-theoretic result that deciding whether a Kronecker coefficient is positive is NP-hard, contrasting with polynomial-time decidability of Littlewoo
s_clique_width	axiom	Clique-width		A graph parameter equal to the minimum number of labels needed to construct a graph using vertex creation, disjoint union, relabeling, and adding all edges betw
s_hereditary_graph_class	axiom	Hereditary graph class		A class of graphs closed under taking induced subgraphs, equivalently definable by a (possibly infinite) family of forbidden induced subgraphs.
s_clique_width_h_free_dichotomy	theorem	Boundedness of clique-width for H-free graphs		For each fixed graph H, the class of H-free graphs either has bounded clique-width or unbounded clique-width, and the boundary between these cases is characteri
t_cut_distance_convergence	technique	Cut distance and left convergence		A method that metrizes the space of graphs and graphons via the cut metric so that a sequence of dense graphs converges in homomorphism densities to a limit gra
t_flag_algebra_method	technique	Flag algebra method		Razborov's formalism that encodes asymptotic subgraph densities as elements of an algebra of partially labeled flags, reducing extremal inequalities to positive
s_independence_complex_connectedness	axiom	Topological connectedness of the independence complex		The topological connectivity of the simplicial complex whose faces are the independent sets of a graph, a homotopy-theoretic invariant of the graph.
s_aharoni_haxell_topological_hall	theorem	Aharoni–Haxell topological Hall theorem		A system of sets has a system of disjoint representatives whenever, for every subfamily, the topological connectivity of the independence complex of the union e
s_independent_transversal	state	Independent transversal		Given a graph whose vertex set is partitioned into blocks, an independent set containing exactly one vertex from each block.
s_long_paths_cycles_in_expanders	theorem	Long paths and cycles in expanders		Every expander graph with sufficiently good vertex- or spectral-expansion contains a path and a cycle whose length is linear in the number of vertices.
s_supersingular_isogeny_graph	axiom	Supersingular isogeny graph		The Pizer/Ramanujan graph whose vertices are supersingular elliptic curves over a finite field of characteristic p and whose edges are ℓ-isogenies between them.
s_isogeny_graph_hardness_sidh	state	Isogeny-graph hardness / SIDH		The computational assumption that finding an isogeny path between two given supersingular elliptic curves in the isogeny graph is hard, underlying supersingular
s_delta_matroid	axiom	Delta-matroid		A set system (E, F) of feasible subsets satisfying a symmetric-exchange axiom, generalizing matroids by allowing feasible sets of differing cardinality.
s_ribbon_graph_delta_matroid	state	Ribbon graph / embedded graph correspondence		The correspondence assigning to each ribbon graph a delta-matroid whose feasible sets are the spanning quasi-trees, generalizing the cycle matroid of a graph.
s_ordered_ramsey_turan	axiom	Ordered Ramsey/Turán theory		The study of extremal and Ramsey-type problems for graphs with a fixed linear order on the vertices, where forbidden patterns are ordered subgraphs.
s_furedi_hajnal_interval_chromatic	theorem	Füredi–Hajnal theorem and interval chromatic number		The ordered Turán number of an ordered graph H is governed by its interval chromatic number, with ex<(n,H)=n^{2(1−1/χ<(H))+o(1)}.
s_spherical_building	axiom	Spherical building		A simplicial complex equipped with a system of apartments isomorphic to a finite Coxeter complex, axiomatizing the incidence geometry of a semisimple algebraic 
t_combinatorial_construction_buildings	technique	Combinatorial construction of buildings		A method that builds a building from a chamber system or a group with a BN-pair by gluing apartments according to the Weyl group.
s_stanley_reisner_ring	axiom	Stanley–Reisner ring		The quotient of a polynomial ring by the squarefree monomial ideal generated by the non-faces of a simplicial complex, encoding the complex's combinatorics in c
s_cohen_macaulay_shellability	theorem	Shellable complexes are Cohen–Macaulay		Every shellable simplicial complex has a Cohen–Macaulay Stanley–Reisner ring, linking the combinatorial shelling order to the algebraic Cohen–Macaulay property.
s_planar_exceptional_polynomials	state	Planar and exceptional polynomial functions		Polynomials over a finite field that are exceptional (define permutation maps over infinitely many extensions) or planar (whose difference maps are bijective), 
s_graph_edge_decomposition	axiom	Graph edge decomposition		A partition of the edge set of a graph into edge-disjoint subgraphs, each isomorphic to or drawn from a prescribed family.
s_nash_williams_arboricity	theorem	Nash-Williams arboricity theorem		The minimum number of forests into which the edges of a graph can be partitioned equals the maximum over subgraphs H of ⌈|E(H)|/(|V(H)|−1)⌉.
s_linear_star_arboricity_conjectures	state	Linear and star arboricity conjectures		Conjectured tight bounds on the minimum number of linear forests or star forests needed to decompose the edges of a d-regular graph, namely roughly ⌈(d+1)/2⌉ fo
t_switching_method	technique	Switching method		A counting technique that estimates the relative sizes of two classes of structures by bounding the number of local edge-switching operations between them.
s_graph_isomorphism_problem	state	Graph isomorphism problem		The computational decision problem of determining whether two given finite graphs are isomorphic.
s_babai_quasipolynomial_isomorphism	theorem	Babai's quasipolynomial graph isomorphism algorithm		Graph isomorphism can be decided in quasipolynomial time exp((log n)^{O(1)}).
t_weisfeiler_leman_algorithm	technique	Weisfeiler–Leman algorithm and dimension		An iterative color-refinement procedure on k-tuples of vertices whose stable coloring is an isomorphism invariant; the WL dimension is the least k for which a g
s_hypergraph_decomposition_extremal	axiom	Graph and hypergraph decomposition (extremal)		The extremal problem of partitioning the edges of a (hyper)graph into copies of a fixed subgraph subject to divisibility and density conditions.
s_keevash_existence_of_designs	theorem	Existence of designs (Keevash; Glock–Kühn–Lo–Osthus)		For all admissible parameters and sufficiently large n satisfying the divisibility conditions, combinatorial designs (Steiner systems and more generally F-desig
t_absorption_method	technique	Absorption method		A technique that reserves a flexible absorbing substructure in advance so that any small leftover set produced by a near-perfect cover can be absorbed to comple
t_iterative_absorption	technique	Iterative absorption		A refinement of the absorption method that repeatedly absorbs leftovers across a nested sequence of vortices, used to prove exact decomposition and spanning res
s_borel_combinatorics_of_graphs	axiom	Borel combinatorics of graphs		The study of combinatorial structures on Borel graphs defined on standard Borel or measure spaces, requiring the solutions to be Borel or measurable.
s_borel_measurable_chromatic_number	state	Borel/measurable chromatic number		The least number of colors needed to properly color a Borel graph with the color classes required to be Borel or measurable sets.
s_johnson_graph	axiom	Johnson graph		The graph J(n,k) whose vertices are the k-subsets of an n-set, with two subsets adjacent when their intersection has size k−1.
s_codes_designs_johnson_graphs	theorem	Codes and designs in Johnson graphs		Constant-weight codes and combinatorial designs correspond to cliques, cocliques, and perfect codes in Johnson graphs, with bounds derived from the Johnson sche
s_maximal_subgroups_finite_simple_groups	theorem	Maximal subgroups of finite simple groups		The classification and structural description of the maximal subgroups of the finite simple groups, organized via the O'Nan–Scott and Aschbacher reduction theor
s_onan_scott_theorem	theorem	O'Nan–Scott theorem		Every finite primitive permutation group falls into one of a small number of structural types (affine, almost simple, product action, diagonal, twisted wreath) 
s_fair_partition_nrr	state	Fair partition / Nandakumar–Ramana-Rao problem		The problem of partitioning a convex body in the plane into n convex pieces of equal area and equal perimeter.
t_equivariant_topology_test_map	technique	Equivariant topology test-map method		A method that encodes a combinatorial problem as the search for a zero of a continuous equivariant map, then derives existence or nonexistence from equivariant 
s_hypergraph_turan_problem	axiom	Hypergraph Turán problem		The problem of determining the maximum number of edges in an n-vertex r-uniform hypergraph containing no copy of a fixed forbidden r-uniform hypergraph.
s_hypergraph_turan_l2_norm	theorem	Hypergraph Turán in the ℓ₂-norm		Results bounding the codegree-squared (ℓ₂-norm) extremal function of forbidden hypergraphs, a refinement of the hypergraph Turán problem measuring the second mo
s_circuit_imbalance_measure	axiom	Circuit imbalance measure		The maximum absolute ratio of nonzero entries appearing in the support-minimal vectors (circuits) of the kernel of a matrix, a condition-number-like measure of 
s_circuit_imbalance_lp_complexity	theorem	Circuit imbalance and LP complexity		Linear programs can be solved in a number of arithmetic operations depending only on the dimension and the logarithm of the circuit imbalance measure of the con
t_junta_method	technique	Junta method for intersection problems		A technique showing that an extremal or near-extremal family in an intersection problem is essentially determined by a bounded number of coordinates (a junta), 
t_finite_geometry_extremal_constructions	technique	Finite geometry constructions for extremal graphs		A method that builds dense graphs free of a fixed bipartite subgraph from incidence structures of finite projective planes, generalized polygons, and polarities
s_erdos_renyi_polarity_graph	state	Erdős–Rényi C₄-free polarity graph		The C₄-free graph on the points of a projective plane PG(2,q) in which two points are adjacent when one lies on the polar line of the other, attaining the Kővár
s_rainbow_subgraph_transversal	axiom	Rainbow subgraph / transversal		In an edge-colored graph or a collection of graphs, a subgraph all of whose edges have distinct colors, equivalently a transversal selecting one edge per color 
s_aharoni_berger_rainbow_matching	theorem	Aharoni–Berger rainbow matching results		Results and conjectures asserting that n matchings of size n in a bipartite multigraph have a rainbow matching using one edge from each, generalizing Latin-squa
s_explicit_bounds_graph_minors	theorem	Explicit bounds for graph minors		Quantitative bounds making the qualitative Robertson–Seymour structure theory effective, such as the polynomial grid-minor theorem relating treewidth to the lar
s_tropical_geometry_combinatorial	axiom	Tropical geometry (combinatorial)		The geometry over the min-plus (tropical) semiring in which algebraic varieties degenerate to piecewise-linear polyhedral complexes, providing a combinatorial s
t_matroid_chow_intersection_theory	technique	Matroid Chow ring intersection theory		A method that proves combinatorial inequalities by computing degrees of products of divisor classes in the Chow ring of a matroid and invoking the Hodge–Riemann
s_erdos_covering_system	axiom	Erdős covering system		A finite collection of congruences x ≡ aᵢ (mod mᵢ) with distinct moduli whose union covers every integer.
s_minimum_modulus_hough	theorem	Minimum modulus problem (Hough's theorem)		There is an absolute constant bounding the smallest modulus of any covering system with distinct moduli, so the minimum modulus cannot be arbitrarily large.
t_cluster_expansion	technique	Cluster expansion		A technique expressing the logarithm of a polymer-model partition function as an absolutely convergent series over connected clusters of polymers.
s_polymer_model	axiom	Polymer model		A statistical-mechanics abstraction whose partition function sums weighted collections of pairwise-compatible polymers, used to encode combinatorial counting an
s_kotecky_preiss_criterion	theorem	Kotecký–Preiss criterion		A sufficient condition on polymer weights guaranteeing absolute convergence of the cluster expansion of the corresponding polymer-model partition function.
s_sublinear_expander	axiom	Sublinear expander		A graph in which every not-too-large vertex subset expands by a factor that decays only polylogarithmically in its size, a weak expansion notion satisfied by al
s_komlos_szemeredi_sublinear_expander	theorem	Komlós–Szemerédi sublinear expander theorem		Every graph contains a subgraph of comparable average degree that is a sublinear expander, providing a universal expansion structure inside dense-enough graphs.
t_sublinear_expander_embedding	technique	Sublinear-expander embedding method		A technique using the short typical distances and robust connectivity of a sublinear expander to embed subdivisions, minors, and other sparse structures.
s_latin_square_transversal	axiom	Transversal in a Latin square		A set of n cells in an n×n Latin square, one in each row and one in each column, whose entries are all distinct.
s_ryser_brualdi_stein_conjecture	state	Ryser–Brualdi–Stein conjecture		The conjecture that every n×n Latin square has a partial transversal of size n−1, and a full transversal whenever n is odd.
s_montgomery_transversal_theorem	theorem	Montgomery's transversal theorem		Every sufficiently large n×n Latin square contains a full transversal, asymptotically resolving the existence part of the Ryser–Brualdi–Stein conjecture.
s_transshipment_over_time	axiom	Transshipment / flows over time		A network-flow model in which flow on each arc incurs a transit time, and supplies and demands at multiple sources and sinks must be routed within a given time 
s_quickest_transshipment_hoppe_tardos	theorem	Quickest transshipment theorem (Hoppe–Tardos)		The quickest transshipment problem of routing given multi-terminal supplies and demands through a network with transit times in minimum total time is solvable i
s_oriented_trees_paths_digraphs	axiom	Oriented trees and paths in digraphs		The study of when a digraph or tournament of given size or minimum out-degree must contain a prescribed oriented tree or oriented path as a subdigraph.
s_sumner_conjecture	state	Sumner's conjecture		The conjecture that every tournament on 2n−2 vertices contains every oriented tree on n vertices as a spanning subdigraph.
s_gallai_roy_directed_path	theorem	Gallai–Roy and Burr–Erdős directed-path results		The Gallai–Roy theorem that every orientation of a graph with chromatic number k contains a directed path on k vertices, together with related Burr–Erdős bounds
s_sidorenko_conjecture	state	Sidorenko's conjecture		The conjecture that for every bipartite graph H, the homomorphism density of H in any graphon W is at least the edge density of W raised to the number of edges 
t_gcd_graph_method	technique	GCD graph method		A combinatorial-analytic framework encoding overlaps of Diophantine-approximation events as weighted GCD graphs whose iteratively improved 'quality' bounds thei
s_duffin_schaeffer_theorem	theorem	Duffin–Schaeffer theorem (Koukoulopoulos–Maynard)	Koukoulopoulos-Maynard theorem	If the series of phi(q)psi(q)/q diverges, then for almost every real x there are infinitely many coprime fractions a/q with |x-a/q| < psi(q)/q.
t_o_minimal_point_counting	technique	O-minimal point counting (Pila–Wilkie)		Bounds the number of rational/algebraic points of bounded height on the transcendental part of a set definable in an o-minimal structure, polynomially in the he
s_andre_oort_conjecture	theorem	André–Oort conjecture (Pila–Shankar–Tsimerman)		Any subvariety of a Shimura variety containing a Zariski-dense set of special (CM) points is itself a special subvariety.
t_betti_map_height_inequality	technique	Betti-map height inequality		Uses generic submersivity of the real-analytic Betti (period) map to convert geometric non-degeneracy into fibre-uniform lower bounds on Néron–Tate heights.
s_uniform_mordell_lang	theorem	Uniform Mordell–Lang for subvarieties of abelian varieties		The number of cosets needed to cover the intersection of a subvariety with a finite-rank subgroup of an abelian variety is bounded only in terms of the dimensio
s_schinzel_zassenhaus_theorem	theorem	Schinzel–Zassenhaus theorem (Dimitrov)		Every monic non-cyclotomic integer polynomial of degree n has a conjugate of absolute value at least 1 + c/n, giving a uniform repulsion of conjugates from the 
s_picard_rank_jumps_k3	theorem	Exceptional jumps of Picard ranks of K3 reductions		A K3 surface over a number field with everywhere potentially good reduction has infinitely many primes at which the geometric Picard rank of the reduction jumps
s_caraiani_scholze_torsion_vanishing	theorem	Caraiani–Scholze torsion vanishing		The generic part of the mod-p cohomology of unitary Shimura varieties is concentrated in the middle degree.
t_entropy_method_extremal	technique	Entropy (information-theoretic) method		Bounds extremal quantities by analysing the Shannon entropy of suitably chosen random structures and applying submodularity/entropy inequalities.
s_ggmt_polynomial_freiman_ruzsa	theorem	Polynomial Freiman–Ruzsa theorem (Gowers–Green–Manners–Tao)		In a group of bounded torsion, a set with doubling constant K is covered by at most poly(K) cosets of a subgroup no larger than the set, resolving Marton's conj
s_kelley_meka_3ap	theorem	Kelley–Meka bound for three-term progressions		Every subset of {1,...,N} free of nontrivial three-term arithmetic progressions has size at most N·exp(-c(log N)^b) for some absolute b>0, a quasi-polynomial bo
s_sensitivity_conjecture	theorem	Sensitivity theorem (Huang)		For every total Boolean function the degree is at most the square of the sensitivity, so sensitivity is polynomially related to all other standard complexity me
t_signed_adjacency_interlacing	technique	Signed adjacency / eigenvalue interlacing		Assigns a signing to a graph's adjacency matrix so that Cauchy eigenvalue interlacing forces large induced subgraphs to have a large maximum degree.
s_ringel_conjecture	theorem	Ringel's conjecture (Montgomery–Pokrovskiy–Sudakov)		For large n the complete graph K_{2n+1} decomposes into 2n+1 edge-disjoint copies of any fixed tree with n edges.
s_multicolor_ramsey_lower_bound	theorem	Exponential improvement of multicolor Ramsey lower bounds (Conlon–Ferber)		For every fixed number r>=3 of colors the diagonal r-color Ramsey number admits an exponentially improved lower bound via a pseudorandom algebraic-plus-random c
s_erdos_faber_lovasz_theorem	theorem	Erdős–Faber–Lovász theorem (Kang–Kelly–Kühn–Methuku–Osthus)		For all large n, every linear hypergraph on n vertices has chromatic index at most n.
s_random_symmetric_matrix_singularity	theorem	Singularity probability of a random symmetric matrix		A uniformly random n×n symmetric ±1 matrix is singular with probability at most exp(-cn).
s_high_girth_steiner_triple_systems	theorem	High-girth Steiner triple systems		For all admissible orders there exist Steiner triple systems whose girth tends to infinity, proving a 1973 conjecture of Erdős.
s_union_closed_constant_bound	theorem	Union-closed sets constant bound (Gilmer)		In any union-closed family of sets some element belongs to at least a positive constant fraction of the sets, the first constant-fraction bound toward Frankl's 
s_diagonal_ramsey_exponential_improvement	theorem	Exponential improvement for diagonal Ramsey (Campos–Griffiths–Morris–Sahasrabudhe)		The diagonal Ramsey number satisfies R(k,k) <= (4-c)^k for an absolute c>0, the first exponential improvement on the 1935 Erdős–Szekeres upper bound.
t_ramsey_book_algorithm	technique	Book algorithm for Ramsey embeddings		An adaptive embedding process that grows monochromatic 'books' while controlling neighborhood densities to extract larger-than-classical cliques.
s_ramsey_r4t_asymptotics	theorem	Asymptotics of R(4,t) (Mattheus–Verstraete)		The off-diagonal Ramsey number satisfies R(4,t) = Theta-tilde(t^3), confirming an Erdős conjecture up to a polylogarithmic factor.
s_erdos_hajnal_conjecture	state	Erdős–Hajnal conjecture		The conjecture that for every fixed graph H there is c>0 such that every H-free graph on n vertices contains a clique or independent set of size at least n^c.
s_erdos_hajnal_loglog_bound	theorem	First improvement to the Erdős–Hajnal bound (Bucić–Nguyen–Scott–Seymour)		Every H-free graph G contains a clique or stable set of size at least 2^{c·sqrt(log|G|·log log|G|)}, the first asymptotic improvement on the 1977 Erdős–Hajnal b
s_talagrand_selector_process	theorem	Talagrand selector-process theorem (Park–Pham)		The suprema of selector (Bernoulli) processes are governed by a majorizing-measure-type bound, proving Talagrand's selector-process conjecture.
s_hadwiger_improved_bound	theorem	Improved bound for Hadwiger's conjecture (Delcourt–Postle / Norin–Postle–Song)		Every graph with no K_t minor is O(t·log log t)-colorable, the current-best general bound toward Hadwiger's conjecture.
s_cheskidov_luo_sharp_nonuniqueness	theorem	Sharp non-uniqueness for Navier–Stokes (Cheskidov–Luo)		Weak solutions of the Navier–Stokes equations are non-unique in L^p_t L^infty_x for every p<2 in dimension >=2, matching the Ladyzhenskaya–Prodi–Serrin uniquene
s_knotted_3_balls_s4	theorem	Knotted 3-balls in S^4 (Budney–Gabai)		There exist smoothly embedded 3-balls in the 4-sphere with the same boundary that are not smoothly isotopic rel boundary.
s_rectangular_peg_problem	theorem	Rectangular peg problem (Greene–Lobb)		Every smooth Jordan curve in the plane inscribes a rectangle of every prescribed aspect ratio.
s_arithmeticity_geodesic_submanifolds	theorem	Arithmeticity from totally geodesic submanifolds (Bader–Fisher–Miller–Stover)		A finite-volume real hyperbolic manifold containing infinitely many maximal totally geodesic submanifolds of dimension at least 2 must be arithmetic.
s_generalized_smale_conjecture	theorem	Generalized Smale conjecture (Bamler–Kleiner)		For every spherical space form the inclusion of the isometry group into the diffeomorphism group is a homotopy equivalence.
t_pointed_nash_entropy	technique	Pointed Nash entropy method		Uses the pointed Nash entropy and conjugate-heat-kernel bounds to control the geometry and singular set of Ricci flows.
s_noncollapsed_ricci_flow_structure	theorem	Structure of non-collapsed Ricci-flow limits (Bamler)		Non-collapsed limits of Ricci flows are smooth away from a singular set of parabolic codimension at least 4, with an associated compactness and partial-regulari
s_michael_simon_sobolev_brendle	theorem	Sharp Michael–Simon–Sobolev inequality (Brendle)		A sharp Sobolev/isoperimetric inequality holds on submanifolds of Euclidean space of arbitrary dimension and codimension, yielding the sharp isoperimetric inequ
s_generic_mcf_singularities	theorem	Generic singularities of mean curvature flow (Chodosh–Choi–Mantoulidis–Schulze)		For a generic closed embedded surface in R^3 the mean curvature flow encounters only spherical and cylindrical (multiplicity-one) singularities.
s_generic_regularity_minimizers	theorem	Generic regularity of minimizing hypersurfaces in dimensions 9 and 10		For generic boundary data or metrics, area-minimizing hypersurfaces are smooth in ambient dimensions 9 and 10, past the dimension-8 Simons-cone obstruction.
t_convex_hypersurface_theory	technique	Convex hypersurface theory (higher-dimensional)		Extends Giroux's convex surface theory to all dimensions, organizing contact topology via convex hypersurfaces and their dividing sets.
t_floer_homotopy_theory	technique	Floer homotopy theory		Realizes Floer-theoretic moduli as a stable homotopy type, allowing generalized cohomology theories to be applied to symplectic invariants.
s_arnold_conjecture_morava	theorem	Arnold conjecture over arbitrary fields (Abouzaid–Blumberg)		For a closed symplectic manifold the number of nondegenerate periodic Hamiltonian orbits is at least the total Betti number over any field, including positive c
s_telescope_conjecture_disproof	theorem	Disproof of Ravenel's telescope conjecture (Burklund–Hahn–Levy–Schlank)		Telescopic T(n+1)-localization and chromatic K(n+1)-localization of the stable homotopy category differ for every prime and height n+1>=2.
s_chromatic_redshift	theorem	Chromatic redshift for truncated Brown–Peterson spectra (Hahn–Wilson)		The algebraic K-theory of the height-n ring spectrum BP<n> is nontrivial at chromatic height n+1, confirming Rognes' redshift principle in these cases.
t_even_filtration	technique	Even filtration		A canonical filtration right Kan extended from even E-infinity rings, recovering motivic and Adams–Novikov-type filtrations and measuring failure of even concen
s_chromatic_nullstellensatz	theorem	Chromatic Nullstellensatz (Burklund–Schlank–Yuan)		Algebraically closed Lubin–Tate theories are characterized among T(n)-local E-infinity rings by a Nullstellensatz property and jointly detect nilpotence.
s_lqg_metric	theorem	Existence and uniqueness of the Liouville quantum gravity metric (Gwynne–Miller)		For each parameter gamma in (0,2) there is a unique random metric associated with gamma-Liouville quantum gravity, arising as the scaling limit of Liouville fir
s_wave_kinetic_equation_derivation	theorem	Rigorous derivation of the wave kinetic equation (Deng–Hani)		The wave kinetic equation is derived as the effective long-time limit of the cubic nonlinear Schrödinger equation with random data.
s_directed_landscape	state	Directed landscape		The universal scaling-limit space-time random field of the Kardar–Parisi–Zhang universality class, governing last-passage and related growth models.
s_lis_directed_landscape_convergence	theorem	Convergence of the longest increasing subsequence to the directed landscape (Dauvergne–Virág)		The rescaled longest increasing subsequence of a uniform random permutation converges to a geodesic of the directed landscape.
s_kerr_stability_small_a	theorem	Nonlinear stability of slowly rotating Kerr (Klainerman–Szeftel)		The slowly rotating Kerr family with |a|/m small is nonlinearly stable under general perturbations of the Einstein vacuum equations.
s_schwarzschild_nonlinear_stability	theorem	Nonlinear stability of the Schwarzschild family (Dafermos–Holzegel–Rodnianski–Taylor)		The Schwarzschild family of black holes is nonlinearly asymptotically stable under general perturbations of the Einstein vacuum equations in a double-null gauge
t_condensed_mathematics	technique	Condensed mathematics		Replaces topological spaces and topological algebraic structures by sheaves on profinite sets, so that topology-with-algebra forms a well-behaved abelian catego
t_absolute_prismatic_cohomology	technique	Absolute prismatic cohomology (Cartier–Witt stack)		A stacky reformulation of prismatic cohomology over the p-adic integers via the Cartier–Witt stack, organizing the Nygaard filtration and syntomic cohomology.
s_fargues_scholze_geometrization	theorem	Geometrization of the local Langlands correspondence (Fargues–Scholze)		A spectral action of the category of coherent sheaves on the stack of L-parameters on the category of l-adic sheaves on Bun_G over the Fargues–Fontaine curve, a
t_smith_treumann_theory	technique	Smith–Treumann theory		Applies Smith theory for constructible sheaves to relate equivariant cohomology in characteristic p to fixed loci, used to compute modular representation-theore
s_modular_tilting_character_formula	theorem	Character formula for tilting modules (Riche–Williamson)		The characters of indecomposable tilting modules for a reductive group in characteristic p are expressed through the p-canonical basis of the affine Hecke algeb
t_fourier_interpolation	technique	Fourier interpolation (modular-form method)		Constructs radial Schwartz functions interpolating prescribed values of a function and its Fourier transform on a discrete set, via quasimodular-form generating
s_universal_optimality_e8_leech	theorem	Universal optimality of E8 and Leech lattices		The E8 lattice in dimension 8 and the Leech lattice in dimension 24 minimize energy for every completely monotone potential, the strongest possible form of opti
s_nuclear_dimension_one	theorem	Nuclear dimension one for classifiable C*-algebras (Castillejos–Evington–Tikuisis–White–Winter)		Separable, simple, unital, nuclear, Z-stable C*-algebras have nuclear dimension exactly one, so finite nuclear dimension is equivalent to Z-stability.
s_keisler_order_maximal_complexity	theorem	Maximal complexity of Keisler's order (Malliaris–Shelah)		Keisler's order on first-order theories has continuum-many classes, with maximal complexity arising already among simple theories with trivial forking.
s_mip_star_re_connes	theorem	MIP* = RE and refutation of the Connes embedding problem		The class of languages decidable by a classical verifier interacting with entangled provers equals all recursively enumerable languages, which via Tsirelson's p
t_brun_sieve	technique	Brun sieve (combinatorial sieve)	combinatorial sieve | Brun's pure sieve	A truncated inclusion–exclusion sieve using Brun's pure and upper–lower truncation constructions to give the first nontrivial bounds on integers surviving congr
t_beurling_selberg_extremal	technique	Beurling–Selberg extremal functions	Beurling–Selberg majorant | extremal band-limited functions	Explicit entire band-limited functions that majorize or minorize a target (e.g. sgn or an interval indicator) while minimizing the L¹ mass of the error, used to
t_theta_correspondence	technique	Theta correspondence (Howe duality)	Howe duality | theta lift | Weil representation correspondence	A representation-theoretic lift transferring automorphic forms or representations between the two members of a reductive dual pair via the restriction of the We
t_cyclic_base_change	technique	Cyclic base change	Langlands base change | automorphic base change	Transfer of automorphic representations of a reductive group over a field F to the group over a cyclic extension E/F, characterized by matching of trace formula
t_potential_automorphy	technique	Potential automorphy	potential modularity	A method proving that a Galois representation becomes automorphic after restriction to some finite extension, by producing a suitable point via Moret-Bailly and
t_langlands_shahidi	technique	Langlands–Shahidi method	Langlands-Shahidi L-functions	Extraction of automorphic L-functions and their analytic properties from the constant and non-constant Fourier coefficients of Eisenstein series via intertwinin
s_satake_isomorphism	theorem	Satake isomorphism	classical Satake isomorphism | Satake transform	The spherical Hecke algebra of a p-adic reductive group is isomorphic to the Weyl-invariant part of the representation ring (group algebra of the cocharacter la
t_motivic_integration	technique	Motivic integration	arc space integration	Integration over the arc or jet spaces of a variety taking values in a completed Grothendieck ring of varieties, computing invariants such as stringy Hodge numb
t_mordell_weil_sieve	technique	Mordell–Weil sieve	MW sieve	A method combining reduction maps modulo many primes with the structure of the Mordell–Weil group of the Jacobian to rule out the existence of rational points o
s_zilber_pink_conjecture	state	Zilber–Pink conjecture	unlikely intersections conjecture | Zilber–Pink for unlikely intersections	For a subvariety of a mixed Shimura variety (or semiabelian variety), the atypical intersections with special subvarieties are contained in a finite union of pr
s_tate_poitou_duality	theorem	Tate–Poitou duality	Poitou–Tate duality | nine-term exact sequence	For a finite Galois module over a number field there is a perfect nine-term exact sequence relating global Galois cohomology to the restricted product of local 
t_deligne_lusztig_theory	technique	Deligne–Lusztig theory	Deligne–Lusztig induction | Deligne–Lusztig characters	Construction of virtual representations of a finite reductive group as alternating sums of ℓ-adic cohomology of Deligne–Lusztig varieties, classifying its irred
t_affine_sieve	technique	Affine sieve / property (τ) spectral gap	Bourgain–Gamburd–Sarnak affine sieve | property (τ) sieve	Execution of sieve methods in thin or orbit settings using a uniform spectral gap (property (τ) / Bourgain–Gamburd expansion) for the associated arithmetic quot
t_modular_method_frey	technique	Modular method (Frey curve)	Frey curve method | Wiles–Ribet modular method	Attaches a Frey elliptic curve to a hypothetical solution of a Diophantine equation and derives a contradiction by combining modularity with Ribet's level-lower
t_caffarelli_silvestre_extension	technique	Caffarelli–Silvestre extension	Caffarelli–Silvestre harmonic extension | fractional Laplacian extension	Realizes the fractional Laplacian (−Δ)^s as the Dirichlet-to-Neumann map of a degenerate elliptic harmonic extension into one extra dimension with an A₂ weight.
s_tb_theorem	theorem	T(b) theorem	T(b) theorem (McIntosh–Meyer, David–Journé–Semmes)	A singular integral operator is L²-bounded if it satisfies a weak boundedness property and Tb, T*b lie in BMO for suitable para-accretive functions b, generaliz
t_restriction_method	technique	Restriction/extension estimates (Stein program)	Fourier restriction method | extension estimates | Stein restriction program	Bounding the Fourier extension operator from L^q of a curved hypersurface to Lᵖ of physical space, encoding the oscillatory-integral cancellation due to curvatu
t_wave_packet_analysis	technique	Wave packet analysis	wave packet decomposition	Decomposes a function, solution, or operator into space-and-frequency localized wave packets that travel along light rays or bicharacteristics, respecting the u
t_sparse_domination	technique	Sparse domination	sparse bounds | domination by sparse operators	Bounds an operator pointwise (or in a bilinear form) by a positive sum of averages over a sparse family of cubes, immediately yielding sharp weighted (Aₚ) inequ
t_carleson_tile_analysis	technique	Carleson time-frequency / tile analysis	time-frequency analysis | phase-plane analysis | Carleson–Fefferman tile method	Proves bounds for modulation-invariant operators (e.g. the Carleson operator, bilinear Hilbert transform) by organizing phase-plane tiles into trees and forests
t_profile_decomposition	technique	Profile decomposition	bubble decomposition | Gérard–Bahouri–Keraani profile decomposition	Writes a bounded sequence in a scale-invariant space as a sum of asymptotically orthogonal translated and rescaled profiles plus a remainder small in the releva
t_continuity_bootstrap	technique	Bootstrap / continuity argument	continuity argument | continuous induction (PDE)	Assumes a quantitative bound on the maximal connected set where it holds, then strictly improves it there, using openness and closedness to force the set to be 
t_lyapunov_schmidt	technique	Lyapunov–Schmidt reduction	Lyapunov–Schmidt method	Splits a nonlinear equation along the kernel and range of its Fredholm linearization, solving the range part by the implicit function theorem to reduce to a fin
t_nehari_manifold	technique	Nehari manifold method	Nehari manifold technique | fibering method	Minimizes a functional over the Nehari manifold (where the functional's derivative annihilates the solution direction) using fibering maps to obtain ground-stat
t_malliavin_calculus	technique	Malliavin calculus	stochastic calculus of variations | Malliavin–Stein method	A differential calculus on Gaussian (Wiener) space built from the Malliavin derivative and its Skorokhod-integral adjoint, yielding integration-by-parts formula
t_stochastic_quantization	technique	Stochastic quantization	Parisi–Wu stochastic quantization	Constructs a Euclidean quantum field theory measure as the invariant measure of a Langevin-type stochastic PDE driven by space-time white noise, after renormali
t_replica_cavity_method	technique	Replica / cavity method	replica method | cavity method | replica symmetry breaking	Computes free energies and order parameters of disordered spin systems either by the replica trick with replica-symmetry-breaking ansätze or by self-consistent 
t_infrared_bound	technique	Infrared bound / continuous symmetry breaking	Fröhlich–Simon–Spencer infrared bound | Gaussian domination bound	Bounds the Fourier transform of the two-point function by the Gaussian-domination estimate, proving existence of long-range order and spontaneous breaking of a 
t_reflection_positivity	technique	Reflection positivity	Osterwalder–Schrader positivity (statistical mechanics) | chessboard estimate method	Exploits positivity of expectations under a reflection of the lattice to derive chessboard estimates, Gaussian domination, and infrared bounds proving phase tra
t_constructive_qft_phase_cell	technique	Constructive QFT / phase-cell expansion	Glimm–Jaffe phase-cell expansion | multiscale expansion (constructive QFT)	Builds interacting Euclidean quantum field measures through multiscale phase-cell and cluster expansions together with renormalization, controlling the infinite
t_renormalization_group_wilson	technique	Renormalization group (Wilson–Kadanoff)	Wilsonian renormalization group | block-spin renormalization | Kadanoff coarse-graining	Iteratively integrates out short-distance degrees of freedom and rescales, generating a flow of effective actions whose fixed points govern universality and cri
t_mountain_pass_linking	technique	Mountain-pass linking / index theory	linking theorem | Benci–Rabinowitz linking | critical point index theory	A min-max critical point theory for strongly indefinite functionals using linking pairs of sets and topological indices (genus, cohomological index) in place of
t_rodl_nibble	technique	Rödl nibble (semi-random method)	semi-random method | random greedy method | nibble method	Builds a combinatorial structure in many small random increments, tracking key quantities by concentration of measure to achieve asymptotically optimal packings
t_algebraic_shifting	technique	Algebraic shifting / Stanley–Reisner method	exterior algebraic shifting | Stanley–Reisner theory	Associates to a simplicial complex a combinatorially shifted complex (via exterior or symmetric algebraic shifting) and a Stanley–Reisner ideal, transferring f-
t_topological_combinatorics	technique	Topological combinatorics / Borsuk–Ulam method	topological method in combinatorics | Lovász–Kneser method | configuration-space/test-map scheme	Reduces combinatorial existence or lower-bound statements to topological obstructions, typically by applying the Borsuk–Ulam theorem or Z/2-equivariance to an a
t_h_principle	technique	h-principle (Gromov)	homotopy principle | Gromov h-principle | convex integration / holonomic approximation	Reduces solving a partial differential relation to the existence of a formal (jet-level) solution up to homotopy, realized via tools such as convex integration 
t_gromov_hausdorff_collapsing	technique	Gromov–Hausdorff convergence and collapsing	Gromov precompactness | Cheeger–Fukaya–Gromov collapsing	Takes limits of Riemannian manifolds or metric spaces in the Gromov–Hausdorff metric, using precompactness and controlling degeneration and dimension collapse v
t_heegaard_floer	technique	Heegaard Floer homology	Ozsváth–Szabó homology | knot Floer homology	Invariants of 3-manifolds and knots defined as the Lagrangian Floer homology of two tori in the symmetric product of a pointed Heegaard surface.
t_donaldson_gauge_theory	technique	Donaldson gauge theory	instanton gauge theory | Yang–Mills/anti-self-dual moduli | Donaldson invariants	Studies smooth 4-manifolds via the moduli spaces of anti-self-dual (instanton) connections on principal SU(2)/SO(3) bundles, extracting Donaldson polynomial inv
t_persistent_homology	technique	Persistent homology	topological data analysis (persistence) | persistence barcodes	Tracks the birth and death of homology classes across a filtration to produce a stable multiscale barcode or persistence diagram of topological features.
t_normal_surface_theory	technique	Normal surface theory	Haken normal surfaces | normal surfaces	Represents embedded surfaces in a triangulated 3-manifold by their normal intersections with tetrahedra, encoding them as nonnegative integer solutions of linea
t_inner_model_fine_structure	technique	Inner model theory / fine structure	fine structure theory | core model theory | extender models	Constructs canonical minimal inner models (L, extender/core models) and analyzes them level-by-level through Jensen's fine structure and the comparison of itera
t_idempotent_ultrafilter	technique	Idempotent ultrafilter method	Galvin–Glazer method | ultrafilter semigroup method	Uses idempotent ultrafilters in the compact right-topological semigroup βS to prove partition-regularity results such as Hindman's finite-sums theorem.
t_spectral_sparsification	technique	Spectral sparsification	graph sparsification | Spielman–Srivastava sparsifiers	Approximates a graph's Laplacian by that of a sparse reweighted subgraph preserving every quadratic form to within a factor (1±ε), via effective-resistance samp
t_multiplicative_weights	technique	Multiplicative weights update method	multiplicative weights | hedge algorithm | exponentiated gradient	Iteratively multiplies the weights of experts or constraints by factors depending on their observed loss, converging to near-optimal decisions with provable reg
t_sum_of_squares_hierarchy	technique	Sum-of-squares hierarchy (Lasserre)	Lasserre hierarchy | SOS hierarchy | Lasserre–Parrilo relaxations	Approximates polynomial optimization by a hierarchy of semidefinite relaxations that certify nonnegativity as sums of squares, with completeness guaranteed by P
t_smoothed_analysis	technique	Smoothed analysis	Spielman–Teng smoothed analysis	Analyzes an algorithm's performance on worst-case inputs subjected to a small random perturbation, interpolating between worst-case and average-case to explain 
t_interlacing_families_mss	technique	Interlacing families of polynomials (MSS)	method of interlacing families | Marcus–Spielman–Srivastava method | Kadison–Singer method	Proves existence of a structure with small eigenvalues by exhibiting an interlacing family whose average characteristic polynomial is real-rooted, so some membe
t_syracuse_random_variable	technique	Syracuse random variables on 3-adic rings		A construction of explicit random variables ∑ᵢ 3^{i-1} 2^{-(a₁+…+aᵢ)} mod 3ⁿ from i.i.d. geometric exponents that model the value of the accelerated Collatz (Sy
s_syracuse_characteristic_decay	theorem	Characteristic function decay estimate for Syracuse distributions		The characteristic function 𝔼 e^{-2πiξ·Syrac/3ⁿ} of the Syracuse random variable on ℤ/3ⁿℤ decays faster than any fixed negative power of n at every nonzero non-
t_transport_log_density_first_passage	technique	Approximate transport of logarithmic density under Collatz first-passage map		A method showing that the logarithmic-density distribution of n is approximately preserved under the Collatz first-passage random variable, reducing the bounded
s_almost_bounded_collatz_orbits	theorem	Almost-bounded Collatz orbits theorem		For any function f tending to infinity, the minimal value Col_min(N) attained by the Collatz orbit of N is below f(N) for almost all N in the sense of logarithm
s_collatz_conjecture	state	Collatz conjecture		The conjecture that iterating the Collatz map (n↦n/2 if even, 3n+1 if odd) from any positive integer eventually reaches 1.
s_collatz_map	axiom	Collatz map		The function on positive integers sending even n to n/2 and odd n to 3n+1.
s_sendov_conjecture	state	Sendov's conjecture		The conjecture that for a complex polynomial of degree at least 2 with all zeros in the closed unit disk, every zero has a critical point of the polynomial with
s_sendov_high_degree	theorem	Sendov's conjecture for high-degree polynomials		There is an absolute degree threshold n₀ above which Sendov's conjecture holds for all polynomials of degree n, proved by a probabilistic compactness argument o
t_polynomial_zeros_as_point_processes	technique	Probabilistic reformulation of polynomial zeros as point processes		A method recasting statements about all zeros and critical points of a polynomial as statements about random variables drawn uniformly from the multiplicity-cou
t_log_potential_stieltjes_zero_measure	technique	Logarithmic potential and Stieltjes transform of empirical zero measure		The representation of the normalized logarithmic derivative f'/(nf) of a polynomial as the Stieltjes transform of its empirical zero measure, with the associate
t_compactness_limit_high_degree_polynomials	technique	Compactness limiting argument for high-degree polynomials		A strategy extracting weak limits via Prokhorov tightness of the random zero and critical-point variables of a hypothetical sequence of high-degree counterexamp
s_grace_apolarity_theorem	theorem	Grace's apolarity theorem		A classical apolarity criterion stating that the zeros of two apolar polynomials of the same degree cannot be separated by any circle (or more generally any cir
s_local_fourier_uniformity_conjecture	state	Local Fourier uniformity conjecture for multiplicative functions		The conjecture that bounded multiplicative functions such as the Liouville function have negligible averaged Gowers Uᵏ uniformity norm over almost all short int
s_higher_local_fourier_uniformity	theorem	Higher-order local Fourier uniformity of multiplicative functions		The Liouville and Möbius functions have asymptotically vanishing averaged Gowers Uᵏ norm over short intervals of length H≥exp(log^{5/8+ε}X) for every fixed k, e
t_matomaki_radziwill_method	technique	Matomäki–Radziwiłł method		A technique controlling averages of bounded multiplicative functions in short intervals by exploiting their factorization through small prime factors to relate 
t_multiplicativity_small_primes_functional_equation	technique	Multiplicativity at small primes functional equation		A method using the identity λ(pn)=−λ(n) at small primes to force any nilsequence correlating with the Liouville function to be approximately dilation-invariant,
s_chowla_conjecture	state	Chowla conjecture		The conjecture that shifted products ∑_{n≤X} λ(n+h₁)…λ(n+hₖ) of the Liouville (or Möbius) function over distinct shifts exhibit cancellation, i.e. are o(X).
s_infinite_partial_sumsets_primes	theorem	Infinite partial sumsets in the primes		There exist two infinite increasing sequences (aᵢ) and (bⱼ) of natural numbers whose pairwise sums aᵢ+bⱼ for i<j are all prime, resolving an Erdős question on p
t_bergelson_intersectivity_lemma	technique	Bergelson intersectivity lemma		A measure-theoretic selection principle extracting from infinitely many positive-measure sets an infinite subfamily whose every finite intersection still has po
t_maynard_sieve_prime_tuples	technique	Maynard sieve for prime-producing tuples		A multidimensional Selberg-type sieve with smooth weights that, for any admissible tuple of shifts, produces a positive density of n for which many of the shift
s_kmrr_sumset_theorem	theorem	Kra–Moreira–Richter–Robertson sumset theorem		Every set of positive upper density in the natural numbers contains a B+B-type infinite sumset configuration, resolving Erdős's sumset conjecture via topologica
s_polymath8_distribution_estimate	theorem	Polymath8 distribution estimate		A Bombieri–Vinogradov-type equidistribution of primes in arithmetic progressions to smooth moduli at a level of distribution exceeding 1/2, obtained via Weil/De
s_de_bruijn_newman_nonnegativity	theorem	de Bruijn–Newman constant nonnegativity		The de Bruijn–Newman constant Λ, defined so that the heat-deformed Riemann ξ-function H_t has only real zeros iff t≥Λ, satisfies Λ≥0, so the Riemann hypothesis 
t_backward_heat_flow_repulsion	technique	Backward heat flow repulsion of zeros		A method tracking the dynamics of zeros of the heat-deformed ξ-function under reverse time, showing that Λ<0 would force the zeros to be implausibly regularly s
s_erdos_primitive_set_conjecture	state	Erdős primitive set conjecture		The conjecture that among all primitive sets (sets with no element dividing another) the sum ∑_{a∈A} 1/(a log a) is maximized by the set of primes.
s_lichtman_primitive_set_bound	theorem	Lichtman bound for primitive sets		For every primitive set A the reciprocal log-weighted sum ∑_{a∈A} 1/(a log a) is at most the corresponding sum ∑_p 1/(p log p) over the primes, resolving the Er
s_de_grey_chromatic_plane_bound	theorem	de Grey chromatic number of the plane lower bound		The chromatic number of the unit-distance graph of the Euclidean plane is at least five, established via an explicit non-4-colorable finite unit-distance graph.
s_hadwiger_nelson_problem	state	Hadwiger–Nelson problem		The open problem of determining the minimum number of colors needed to color the plane so that no two points at unit distance share a color.
s_bourgain_clozel_kahane_uncertainty	theorem	Bourgain–Clozel–Kahane Fourier uncertainty principle		A real even self-dual function (f=f̂) that is nonpositive at the origin in both physical and frequency space cannot change sign arbitrarily close to the origin,
t_polymath_collaborative_mathematics	technique	Polymath project (massively collaborative mathematics)		A methodology of solving or simplifying mathematical problems through open, public, incremental online collaboration by many contributors.
s_entropic_doubling_constant	axiom	Entropic doubling constant		The Shannon-entropy analogue of the combinatorial doubling constant, defined for a group-valued random variable X as d(X;X)=H(X₁+X₂)−H(X) using two independent 
s_entropic_ruzsa_distance	axiom	Entropic Ruzsa distance		A symmetric, nonnegative, triangle-inequality-obeying pseudo-distance d(X;Y)=H(X'−Y')−H(X)/2−H(Y)/2 between two independent group-valued random variables, the e
t_entropy_method_additive_combinatorics	technique	Entropy method for additive combinatorics		A proof strategy replacing sets and their sumsets with group-valued random variables and analyzing combinatorial doubling via Shannon entropy quantities, workin
s_fibring_inequality_entropy	theorem	Fibring inequality (entropic)		An entropy inequality stating that the entropic doubling of X is at least the doubling of its homomorphic image π(X) plus the conditional doubling along the fib
s_conditional_doubling_constant	axiom	Conditional doubling constant		The entropic doubling constant of a random variable measured conditionally on the value of a homomorphic image, quantifying doubling within the fibers of a homo
s_entropic_balog_szemeredi_gowers	theorem	Entropic Balog–Szemerédi–Gowers lemma		An entropy-formulated analogue of the Balog–Szemerédi–Gowers theorem converting small conditional mutual information into structural control with bounded entrop
t_sumset_intersection_improvement_ops	technique	Sumset/intersection improvement operations		A pair of operations replacing A by A+A or by A∩(A+h) for typical h, each of which strictly reduces the entropic doubling constant unless a rigid conditional-in
t_induction_on_doubling_constant	technique	Induction on the doubling constant		A descent strategy that inductively lowers the entropic doubling constant by repeatedly applying improvement operations until reaching the trivial regime where 
t_characteristic_2_endgame_identity	technique	Characteristic-2 endgame identity		The use of the characteristic-2 cancellation identity (X₁+X₂)+(X₂+X₃)=X₁+X₃ together with conditional independence relative to X₁+X₂+X₃+X₄ to close the PFR argu
s_rao_sunflower_bound	theorem	Rao sunflower bound via Shannon coding		A refinement of the Alweiss–Lovett–Wu–Zhang sunflower bound giving Sun(k,r) ≤ O(r log k)^k, obtained by encoding spread sets through the Shannon noiseless codin
t_entropy_proof_sunflower_lemma	technique	Entropy proof of the sunflower lemma		Tao's information-theoretic reformulation proving the improved sunflower lemma by bounding conditional Shannon entropy of constrained random variables instead o
s_inverse_theorem_entropy	axiom	Inverse theorem entropy		The minimal cardinality of a family of structured functions needed so that every 1-bounded function with seminorm at least η correlates at level ε with some fam
t_random_sampling_entropy_bounds	technique	Random sampling argument for inverse-theorem entropy bounds		A technique conditioning on a randomly chosen subset with tuned probability to construct dual functions with controllable L² bounds, yielding inverse-theorem-en
s_inverse_theorem_u3_finite_abelian	theorem	Inverse theorem for the U³ Gowers norm on arbitrary finite abelian groups		A bounded function on an arbitrary finite abelian group with large U³ norm correlates with a Lipschitz function on an explicit degree-two filtered nilmanifold c
t_manners_nilsequence_encoding	technique	Manners' nilsequence encoding of locally quadratic phases		An algebraic construction encoding a locally quadratic phase function as a nilsequence on a degree-two filtered nilmanifold, adapted to arbitrary finite abelian
t_blueprint_driven_formalization	technique	Blueprint-driven formalization		A formalization methodology using Massot's Blueprint tool to write a human-readable proof outline whose statements link to Lean4 lemmas, with a color-coded depe
s_lean4_formalization_pfr	theorem	Lean4 formalization of PFR		A complete machine-verified proof in the Lean4 proof assistant of the Gowers–Green–Manners–Tao proof of the polynomial Freiman–Ruzsa conjecture over 𝔽₂ⁿ.
