# Phase A Integration Log

Automated merge of three mathematician Phase-A derivation-chain files
(`mathematician_relationships_r1.md`, `_r2.md`, `_r3.md`) into
`knowledge_graph.json`.

## Counts

- Baseline nodes: **358**
- Baseline edges: **371**
- Chains parsed:  **42**
- New theorem nodes added: **42**
- New state   nodes added: **99**
- New axiom   nodes added: **2**
- New edges added:         **325**
- Final nodes: **501**
- Final edges: **696**

## New theorems

- `s_heron_formula` — Heron's Formula
- `s_menelaus_theorem` — Menelaus's Theorem
- `s_brahmagupta_formula` — Brahmagupta's Formula
- `s_brahmagupta_midpoint_theorem` — Brahmagupta's Theorem
- `s_aryabhata_sine_pi_kuttaka` — Aryabhata's Theorems
- `s_alkhwarizmi_six_quadratic_types` — Al-Khwārizmī's Quadratic Framework
- `s_khayyam_cubic_geometric_solution` — Omar Khayyām's Geometric Solution of Cubics
- `s_viete_formulas` — Viète's Formulas
- `s_flt_n_equals_4` — Fermat's Last Theorem, Origin — n = 4 descent
- `s_descartes_rule_of_signs` — Descartes' Rule of Signs
- `s_descartes_angular_defect` — Descartes' Theorem on Total Angular Defect
- `s_pascal_mystic_hexagram` — Pascal's Theorem on Hexagons in a Conic / Mystic Hexagram
- `s_pascal_triangle_identity` — Pascal's Triangle Identity
- `s_newton_binomial_theorem` — Newton's Generalized Binomial Theorem
- `s_newton_identities` — Newton's Identities
- `s_rolle_theorem` — Rolle's Theorem
- `s_torricelli_law` — Torricelli's Law
- `s_wallis_product` — Wallis Product for π
- `s_bernoulli_lln` — Bernoulli's Law of Large Numbers
- `s_de_moivre_laplace` — De Moivre–Laplace Theorem
- `s_euler_totient_theorem` — Euler's Totient Theorem
- `s_euler_homogeneous_functions` — Euler's Theorem on Homogeneous Functions
- `s_lagrange_mvt` — Lagrange's Mean Value Theorem
- `s_wilson_theorem` — Wilson's Theorem
- `s_bayes_theorem` — Bayes's Theorem
- `s_cauchy_schwarz` — Cauchy–Schwarz Inequality
- `s_bolzano_ivt` — Bolzano's Intermediate Value Theorem
- `s_bolzano_weierstrass` — Bolzano–Weierstrass Theorem
- `s_greens_theorem` — Green's Theorem
- `s_divergence_theorem` — Divergence Theorem / Gauss–Ostrogradsky
- `s_liouville_bounded_entire` — Liouville's Theorem (complex analysis)
- `s_cayley_hamilton` — Cayley–Hamilton Theorem
- `s_cantor_bernstein_schroeder` — Cantor–Bernstein–Schroeder Theorem
- `s_poincare_recurrence` — Poincaré Recurrence Theorem
- `s_zermelo_well_ordering` — Zermelo's Well-Ordering Theorem
- `s_noether_isomorphism_theorems` — Noether's Isomorphism Theorems
- `s_godel_completeness_theorem` — Gödel's Completeness Theorem
- `s_banach_tarski_paradox` — Banach–Tarski Paradox
- `s_stone_weierstrass_theorem` — Stone–Weierstrass Theorem
- `s_stone_representation_theorem` — Stone Representation Theorem
- `s_full_modularity_theorem_BCDT` — Modularity Theorem — full, Breuil–Conrad–Diamond–Taylor 2001
- `s_abc_conjecture_mochizuki_claimed` — abc Conjecture — Mochizuki's disputed proof

## New axioms

- `s_first_order_logic_language` — first order logic language
- `s_compact_hausdorff_space` — compact hausdorff space

## New states

- `s_incircle_tangent_decomposition` — incircle tangent decomposition
- `s_area_squared_equals_s_times_product` — area squared equals s times product
- `s_perpendiculars_from_vertices_to_transversal` — perpendiculars from vertices to transversal
- `s_ratio_product_cancels_heights` — ratio product cancels heights
- `s_two_triangles_with_supplementary_angles` — two triangles with supplementary angles
- `s_area_squared_symmetric_in_four_sides` — area squared symmetric in four sides
- `s_orthodiagonal_cyclic_quadrilateral` — orthodiagonal cyclic quadrilateral
- `s_perpendicular_from_diagonal_intersection_to_side` — perpendicular from diagonal intersection to side
- `s_sine_second_difference_recurrence` — sine second difference recurrence
- `s_aryabhata_sine_table` — aryabhata sine table
- `s_aryabhata_pi_3_1416` — aryabhata pi 3 1416
- `s_kuttaka_extended_euclid` — kuttaka extended euclid
- `s_six_canonical_quadratic_forms` — six canonical quadratic forms
- `s_geometric_completion_of_square` — geometric completion of square
- `s_fourteen_irreducible_cubic_types` — fourteen irreducible cubic types
- `s_cubic_as_intersection_of_two_conics` — cubic as intersection of two conics
- `s_polynomial_as_product_of_root_factors` — polynomial as product of root factors
- `s_elementary_symmetric_polynomials` — elementary symmetric polynomials
- `s_flt_general_conjecture` — flt general conjecture
- `s_flt_n_4_strengthened_claim` — flt n 4 strengthened claim
- `s_sign_change_parity_conjecture` — sign change parity conjecture
- `s_sign_change_increment_under_positive_factor` — sign change increment under positive factor
- `s_vertex_angular_defect` — vertex angular defect
- `s_total_defect_equals_2pi_chi` — total defect equals 2pi chi
- `s_inscribed_hexagon_in_circle` — inscribed hexagon in circle
- `s_three_opposite_side_intersection_points` — three opposite side intersection points
- `s_pascal_line` — pascal line
- `s_binomial_recurrence_conjecture` — binomial recurrence conjecture
- `s_distinguished_element_bijection` — distinguished element bijection
- `s_generalized_binomial_coefficient` — generalized binomial coefficient
- `s_vandermonde_convolution_extended` — vandermonde convolution extended
- `s_log_derivative_generating_function` — log derivative generating function
- `s_power_sum_series` — power sum series
- `s_attained_extrema_on_closed_interval` — attained extrema on closed interval
- `s_interior_extremum_or_constant` — interior extremum or constant
- `s_galilean_free_fall_speed` — galilean free fall speed
- `s_fluid_column_energy_balance` — fluid column energy balance
- `s_wallis_integer_quadrature_table` — wallis integer quadrature table
- `s_wallis_half_integer_interpolation` — wallis half integer interpolation
- `s_binomial_tail_ratios` — binomial tail ratios
- `s_binomial_tail_bound` — binomial tail bound
- `s_stirling_central_term_asymptotic` — stirling central term asymptotic
- `s_gaussian_density_as_limit` — gaussian density as limit
- `s_unit_group_mod_n` — unit group mod n
- `s_orbit_of_a_mod_n` — orbit of a mod n
- `s_homogeneous_function_definition` — homogeneous function definition
- `s_scaling_vector_field_identity` — scaling vector field identity
- `s_auxiliary_function_with_equal_endpoints` — auxiliary function with equal endpoints
- `s_rolle_interior_critical_point` — rolle interior critical point
- `s_polynomial_x_p_minus_1_factored_mod_p` — polynomial x p minus 1 factored mod p
- `s_factored_form_of_cyclic_unit_polynomial` — factored form of cyclic unit polynomial
- `s_conditional_probability_definition` — conditional probability definition
- `s_symmetric_chain_rule_for_intersection` — symmetric chain rule for intersection
- `s_inner_product_space` — inner product space
- `s_non_negative_quadratic_in_t` — non negative quadratic in t
- `s_supremum_set_S` — supremum set S
- `s_candidate_root_c_as_sup` — candidate root c as sup
- `s_bounded_sequence_in_Rn` — bounded sequence in Rn
- `s_bounded_sequence_in_closed_box` — bounded sequence in closed box
- `s_nested_intervals_with_infinitely_many_terms` — nested intervals with infinitely many terms
- `s_simple_region_decomposition_in_R2` — simple region decomposition in R2
- `s_per_slice_ftoc_identity` — per slice ftoc identity
- `s_z_simple_region_decomposition_in_R3` — z simple region decomposition in R3
- `s_per_axis_flux_identity` — per axis flux identity
- `s_cauchy_derivative_bound` — cauchy derivative bound
- `s_vanishing_derivative_on_all_of_C` — vanishing derivative on all of C
- `s_square_matrix_with_char_poly` — square matrix with char poly
- `s_adjugate_matrix_polynomial_identity` — adjugate matrix polynomial identity
- `s_pair_of_injections_with_orphans` — pair of injections with orphans
- `s_trajectory_partition_of_A` — trajectory partition of A
- `s_measurable_set_of_positive_measure` — measurable set of positive measure
- `s_non_returning_subset_B` — non returning subset B
- `s_disjoint_preimages_of_B_sum_bounded` — disjoint preimages of B sum bounded
- `s_axiom_of_choice_function_phi` — axiom of choice function phi
- `s_maximal_gamma_set_on_M` — maximal gamma set on M
- `s_normal_subgroup_N_in_G` — normal subgroup N in G
- `s_group_homomorphism_f` — group homomorphism f
- `s_abstract_subobject_quotient_morphism_pattern` — abstract subobject quotient morphism pattern
- `s_canonical_quotient_by_kernel_map` — canonical quotient by kernel map
- `s_consistent_first_order_theory_T` — consistent first order theory T
- `s_consistent_theory_with_witnesses` — consistent theory with witnesses
- `s_maximal_consistent_henkin_theory_T_star` — maximal consistent henkin theory T star
- `s_free_group_F2_on_two_generators` — free group F2 on two generators
- `s_paradoxical_decomposition_of_F2` — paradoxical decomposition of F2
- `s_F2_as_subgroup_of_SO3` — F2 as subgroup of SO3
- `s_paradoxical_sphere_decomposition` — paradoxical sphere decomposition
- `s_point_separating_unital_subalgebra_A_of_C_X_R` — point separating unital subalgebra A of C X R
- `s_closure_of_A_is_a_lattice` — closure of A is a lattice
- `s_pointwise_matching_family_f_xy` — pointwise matching family f xy
- `s_boolean_algebra_B` — boolean algebra B
- `s_ultrafilter_spectrum_Spec_B` — ultrafilter spectrum Spec B
- `s_compact_totally_disconnected_stone_space` — compact totally disconnected stone space
- `s_ramification_stratification_of_elliptic_curves` — ramification stratification of elliptic curves
- `s_non_semistable_modularity_cases_handled` — non semistable modularity cases handled
- `s_coprime_triple_a_b_c_with_a_plus_b_equals_c` — coprime triple a b c with a plus b equals c
- `s_mochizuki_IUT_framework` — mochizuki IUT framework
- `s_anabelian_reconstruction_of_E` — anabelian reconstruction of E
- `s_theta_link_between_hodge_theaters` — theta link between hodge theaters
- `s_log_volume_inequality` — log volume inequality

## Unknown technique ids referenced (skipped steps)

- chain **Cantor–Bernstein–Schroeder Theorem**, step 2: `t_iteration_of_trajectories`

## Near-duplicate candidates flagged for philosopher review

- `s_orthodiagonal_cyclic_quadrilateral` "orthodiagonal cyclic quadrilateral"  ⇔  `s_cyclic_quadrilateral` "Cyclic quadrilateral"  (chain: Brahmagupta's Theorem)
- `s_flt_n_equals_4` "Fermat's Last Theorem, Origin — n = 4 descent"  ⇔  `s_flt` "Fermat's Last Theorem"  (chain: Fermat's Last Theorem, Origin — n = 4 descent)

## Integrity checks

- New theorems lacking an incoming edge: **0**
- New state / axiom nodes with zero incident edges: **0**
- New theorems NOT reachable from any axiom by forward BFS: **0**
