# Brief Catalog Skeletons — Area G (Analysis)

Phase B bulk skeletons for Real Analysis / Measure Theory (14), Complex Analysis (15), Functional Analysis (13), Harmonic Analysis (5), Differential Equations (7). Total: 54 items.

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## Real Analysis / Measure Theory

### Monotone convergence theorem
**Terminal:** `s_monotone_convergence_theorem`
**Axioms:** `s_real_analysis`
**Steps:**
1. input: `⟨s_real_analysis⟩` --[t_exhaustion_squeeze]--> output: `s_monotone_convergence_theorem`

### Fatou's lemma
**Terminal:** `s_fatou_lemma`
**Axioms:** `s_real_analysis`
**Steps:**
1. input: `⟨s_real_analysis⟩` --[t_exhaustion_squeeze]--> output: `s_fatou_lemma`

### Dominated convergence theorem
**Terminal:** `s_dominated_convergence_theorem`
**Axioms:** `s_real_analysis`
**Steps:**
1. input: `⟨s_real_analysis⟩` --[t_exhaustion_squeeze]--> output: `s_dominated_convergence_theorem`

### Fubini's theorem
**Terminal:** `s_fubini_theorem`
**Axioms:** `s_real_analysis`
**Steps:**
1. input: `⟨s_real_analysis⟩` --[t_exhaustion_squeeze]--> output: `s_fubini_theorem`

### Tonelli's theorem
**Terminal:** `s_tonelli_theorem`
**Axioms:** `s_real_analysis`
**Steps:**
1. input: `⟨s_real_analysis⟩` --[t_exhaustion_squeeze]--> output: `s_tonelli_theorem`

### Radon–Nikodym theorem
**Terminal:** `s_radon_nikodym_theorem`
**Axioms:** `s_real_analysis`
**Steps:**
1. input: `⟨s_real_analysis⟩` --[t_duality]--> output: `s_radon_nikodym_theorem`

### Lebesgue differentiation theorem
**Terminal:** `s_lebesgue_differentiation_theorem`
**Axioms:** `s_real_analysis`
**Steps:**
1. input: `⟨s_real_analysis⟩` --[t_exhaustion_squeeze]--> output: `s_lebesgue_differentiation_theorem`

### Vitali covering lemma
**Terminal:** `s_vitali_covering_lemma`
**Axioms:** `s_real_analysis`
**Steps:**
1. input: `⟨s_real_analysis⟩` --[t_compactness_argument]--> output: `s_vitali_covering_lemma`

### Egorov's theorem
**Terminal:** `s_egorov_theorem`
**Axioms:** `s_real_analysis`
**Steps:**
1. input: `⟨s_real_analysis⟩` --[t_exhaustion_squeeze]--> output: `s_egorov_theorem`

### Lusin's theorem
**Terminal:** `s_lusin_theorem`
**Axioms:** `s_real_analysis`
**Steps:**
1. input: `⟨s_real_analysis⟩` --[t_axiomatize_from_instances]--> output: `s_lusin_theorem`

### Rademacher's theorem
**Terminal:** `s_rademacher_theorem`
**Axioms:** `s_real_analysis`
**Steps:**
1. input: `⟨s_real_analysis⟩` --[t_exhaustion_squeeze]--> output: `s_rademacher_theorem`

### Sard's theorem
**Terminal:** `s_sard_theorem`
**Axioms:** `s_smooth_function`
**Steps:**
1. input: `⟨s_smooth_function⟩` --[t_exhaustion_squeeze]--> output: `s_sard_theorem`

### Baire category theorem
**Terminal:** `s_baire_category_theorem`
**Axioms:** `s_complete_metric_space`
**Steps:**
1. input: `⟨s_complete_metric_space⟩` --[t_compactness_argument]--> output: `s_baire_category_theorem`

### Arzelà–Ascoli theorem
**Terminal:** `s_arzela_ascoli_theorem`
**Axioms:** `s_continuous_function_on_closed_interval`
**Steps:**
1. input: `⟨s_continuous_function_on_closed_interval⟩` --[t_compactness_argument]--> output: `s_arzela_ascoli_theorem`

---

## Complex Analysis

### Picard's little theorem
**Terminal:** `s_picard_little_theorem`
**Axioms:** `s_holomorphic_function_on_domain`
**Steps:**
1. input: `⟨s_holomorphic_function_on_domain⟩` --[t_compactness_argument]--> output: `s_picard_little_theorem`

### Picard's great theorem
**Terminal:** `s_picard_great_theorem`
**Axioms:** `s_holomorphic_function_on_domain`
**Steps:**
1. input: `⟨s_holomorphic_function_on_domain⟩` --[t_compactness_argument]--> output: `s_picard_great_theorem`

### Casorati–Weierstrass theorem
**Terminal:** `s_casorati_weierstrass_theorem`
**Axioms:** `s_holomorphic_function_on_domain`
**Steps:**
1. input: `⟨s_holomorphic_function_on_domain⟩` --[t_reductio_ad_absurdum]--> output: `s_casorati_weierstrass_theorem`

### Residue theorem
**Terminal:** `s_residue_theorem`
**Axioms:** `s_holomorphic_function_on_domain`
**Steps:**
1. input: `⟨s_holomorphic_function_on_domain⟩` --[t_conserved_quantity]--> output: `s_residue_theorem`

### Argument principle
**Terminal:** `s_argument_principle`
**Axioms:** `s_holomorphic_function_on_domain`
**Steps:**
1. input: `⟨s_holomorphic_function_on_domain⟩` --[t_conserved_quantity]--> output: `s_argument_principle`

### Rouché's theorem
**Terminal:** `s_rouche_theorem`
**Axioms:** `s_holomorphic_function_on_domain`
**Steps:**
1. input: `⟨s_holomorphic_function_on_domain⟩` --[t_conserved_quantity]--> output: `s_rouche_theorem`

### Morera's theorem
**Terminal:** `s_morera_theorem`
**Axioms:** `s_holomorphic_function_on_domain`
**Steps:**
1. input: `⟨s_holomorphic_function_on_domain⟩` --[t_axiomatize_from_instances]--> output: `s_morera_theorem`

### Schwarz lemma
**Terminal:** `s_schwarz_lemma`
**Axioms:** `s_holomorphic_function_on_domain`
**Steps:**
1. input: `⟨s_holomorphic_function_on_domain⟩` --[t_conserved_quantity]--> output: `s_schwarz_lemma`

### Phragmén–Lindelöf principle
**Terminal:** `s_phragmen_lindelof_principle`
**Axioms:** `s_holomorphic_function_on_domain`
**Steps:**
1. input: `⟨s_holomorphic_function_on_domain⟩` --[t_conserved_quantity]--> output: `s_phragmen_lindelof_principle`

### Mittag-Leffler theorem
**Terminal:** `s_mittag_leffler_theorem`
**Axioms:** `s_holomorphic_function_on_domain`
**Steps:**
1. input: `⟨s_holomorphic_function_on_domain⟩` --[t_interpolate_and_continue]--> output: `s_mittag_leffler_theorem`

### Weierstrass factorization theorem
**Terminal:** `s_weierstrass_factorization_theorem`
**Axioms:** `s_holomorphic_function_on_domain`
**Steps:**
1. input: `⟨s_holomorphic_function_on_domain⟩` --[t_interpolate_and_continue]--> output: `s_weierstrass_factorization_theorem`

### Montel's theorem
**Terminal:** `s_montel_theorem`
**Axioms:** `s_holomorphic_function_on_domain`
**Steps:**
1. input: `⟨s_holomorphic_function_on_domain⟩` --[t_compactness_argument]--> output: `s_montel_theorem`

### Hurwitz's theorem
**Terminal:** `s_hurwitz_theorem`
**Axioms:** `s_holomorphic_function_on_domain`
**Steps:**
1. input: `⟨s_holomorphic_function_on_domain⟩` --[t_compactness_argument]--> output: `s_hurwitz_theorem`

### Bloch's theorem
**Terminal:** `s_bloch_theorem`
**Axioms:** `s_holomorphic_function_on_domain`
**Steps:**
1. input: `⟨s_holomorphic_function_on_domain⟩` --[t_compactness_argument]--> output: `s_bloch_theorem`

### Gauss–Lucas theorem
**Terminal:** `s_gauss_lucas_theorem`
**Axioms:** `s_complex_polynomial_p_z`
**Steps:**
1. input: `⟨s_complex_polynomial_p_z⟩` --[t_conserved_quantity]--> output: `s_gauss_lucas_theorem`

---

## Functional Analysis

### Banach–Steinhaus theorem
**Terminal:** `s_banach_steinhaus_theorem`
**Axioms:** `s_complete_metric_space`
**Steps:**
1. input: `⟨s_complete_metric_space⟩` --[t_compactness_argument]--> output: `s_banach_steinhaus_theorem`

### Open mapping theorem
**Terminal:** `s_open_mapping_theorem`
**Axioms:** `s_complete_metric_space`
**Steps:**
1. input: `⟨s_complete_metric_space⟩` --[t_compactness_argument]--> output: `s_open_mapping_theorem`

### Closed graph theorem
**Terminal:** `s_closed_graph_theorem`
**Axioms:** `s_complete_metric_space`
**Steps:**
1. input: `⟨s_complete_metric_space⟩` --[t_structural_isomorphism]--> output: `s_closed_graph_theorem`

### Banach–Alaoglu theorem
**Terminal:** `s_banach_alaoglu_theorem`
**Axioms:** `s_real_vector_space`
**Steps:**
1. input: `⟨s_real_vector_space⟩` --[t_compactness_argument]--> output: `s_banach_alaoglu_theorem`

### Kreĭn–Milman theorem
**Terminal:** `s_krein_milman_theorem`
**Axioms:** `s_real_vector_space`
**Steps:**
1. input: `⟨s_real_vector_space⟩` --[t_compactness_argument]--> output: `s_krein_milman_theorem`

### Riesz representation theorems
**Terminal:** `s_riesz_representation_theorems`
**Axioms:** `s_L2_function_space`
**Steps:**
1. input: `⟨s_L2_function_space⟩` --[t_duality]--> output: `s_riesz_representation_theorems`

### Spectral theorem for self-adjoint operators
**Terminal:** `s_spectral_theorem_self_adjoint`
**Axioms:** `s_L2_function_space`
**Steps:**
1. input: `⟨s_L2_function_space⟩` --[t_svd_and_spectral_decomposition]--> output: `s_spectral_theorem_self_adjoint`

### Gelfand–Naimark theorem
**Terminal:** `s_gelfand_naimark_theorem`
**Axioms:** `s_L2_function_space`
**Steps:**
1. input: `⟨s_L2_function_space⟩` --[t_structural_isomorphism]--> output: `s_gelfand_naimark_theorem`

### Lax–Milgram lemma
**Terminal:** `s_lax_milgram_lemma`
**Axioms:** `s_L2_function_space`
**Steps:**
1. input: `⟨s_L2_function_space⟩` --[t_duality]--> output: `s_lax_milgram_lemma`

### Sobolev embedding theorem
**Terminal:** `s_sobolev_embedding_theorem`
**Axioms:** `s_L2_function_space`
**Steps:**
1. input: `⟨s_L2_function_space⟩` --[t_exhaustion_squeeze]--> output: `s_sobolev_embedding_theorem`

### Rellich–Kondrachov theorem
**Terminal:** `s_rellich_kondrachov_theorem`
**Axioms:** `s_L2_function_space`
**Steps:**
1. input: `⟨s_L2_function_space⟩` --[t_compactness_argument]--> output: `s_rellich_kondrachov_theorem`

### Stone's theorem on one-parameter unitary groups
**Terminal:** `s_stone_one_parameter_unitary_theorem`
**Axioms:** `s_L2_function_space`
**Steps:**
1. input: `⟨s_L2_function_space⟩` --[t_svd_and_spectral_decomposition]--> output: `s_stone_one_parameter_unitary_theorem`

### Riesz–Fischer theorem
**Terminal:** `s_riesz_fischer_theorem`
**Axioms:** `s_L2_function_space`
**Steps:**
1. input: `⟨s_L2_function_space⟩` --[t_fourier_transform]--> output: `s_riesz_fischer_theorem`

---

## Harmonic Analysis

### Plancherel theorem
**Terminal:** `s_plancherel_theorem`
**Axioms:** `s_L2_function_space`
**Steps:**
1. input: `⟨s_L2_function_space⟩` --[t_fourier_transform]--> output: `s_plancherel_theorem`

### Riesz–Thorin interpolation theorem
**Terminal:** `s_riesz_thorin_interpolation_theorem`
**Axioms:** `s_L2_function_space`
**Steps:**
1. input: `⟨s_L2_function_space⟩` --[t_interpolate_and_continue]--> output: `s_riesz_thorin_interpolation_theorem`

### Marcinkiewicz interpolation theorem
**Terminal:** `s_marcinkiewicz_interpolation_theorem`
**Axioms:** `s_L2_function_space`
**Steps:**
1. input: `⟨s_L2_function_space⟩` --[t_interpolate_and_continue]--> output: `s_marcinkiewicz_interpolation_theorem`

### Carleson's theorem
**Terminal:** `s_carleson_theorem`
**Axioms:** `s_L2_function_space`
**Steps:**
1. input: `⟨s_L2_function_space⟩` --[t_frequency_decomposition]--> output: `s_carleson_theorem`

### Hardy–Littlewood maximal inequality
**Terminal:** `s_hardy_littlewood_maximal_inequality`
**Axioms:** `s_L2_function_space`
**Steps:**
1. input: `⟨s_L2_function_space⟩` --[t_compactness_argument]--> output: `s_hardy_littlewood_maximal_inequality`

---

## Differential Equations

### Picard–Lindelöf theorem
**Terminal:** `s_picard_lindelof_theorem`
**Axioms:** `s_complete_metric_space`
**Steps:**
1. input: `⟨s_complete_metric_space⟩` --[t_contraction_fixed_point]--> output: `s_picard_lindelof_theorem`

### Peano existence theorem
**Terminal:** `s_peano_existence_theorem`
**Axioms:** `s_continuous_function_on_closed_interval`
**Steps:**
1. input: `⟨s_continuous_function_on_closed_interval⟩` --[t_compactness_argument]--> output: `s_peano_existence_theorem`

### Cauchy–Kovalevskaya theorem
**Terminal:** `s_cauchy_kovalevskaya_theorem`
**Axioms:** `s_holomorphic_function_on_domain`
**Steps:**
1. input: `⟨s_holomorphic_function_on_domain⟩` --[t_interpolate_and_continue]--> output: `s_cauchy_kovalevskaya_theorem`

### Frobenius theorem (DE / distributions)
**Terminal:** `s_frobenius_theorem_distributions`
**Axioms:** `s_smooth_manifold_with_boundary`
**Steps:**
1. input: `⟨s_smooth_manifold_with_boundary⟩` --[t_reduce_to_canonical_form]--> output: `s_frobenius_theorem_distributions`

### Poincaré–Bendixson theorem
**Terminal:** `s_poincare_bendixson_theorem`
**Axioms:** `s_real_analysis`
**Steps:**
1. input: `⟨s_real_analysis⟩` --[t_compactness_argument]--> output: `s_poincare_bendixson_theorem`

### Hartman–Grobman theorem
**Terminal:** `s_hartman_grobman_theorem`
**Axioms:** `s_smooth_function`
**Steps:**
1. input: `⟨s_smooth_function⟩` --[t_reduce_to_canonical_form]--> output: `s_hartman_grobman_theorem`

### KAM theorem
**Terminal:** `s_kam_theorem`
**Axioms:** `s_smooth_function`
**Steps:**
1. input: `⟨s_smooth_function⟩` --[t_contraction_fixed_point]--> output: `s_kam_theorem`
