Registry — Book III: Categorical Spectrum

III.D01 Hartogs Bulk Projection definition III.R01 Naive Coordinate Picture remark III.D02 Millennium-Problem Gluing Table definition III.D03 E₁ as Gluing Principle definition III.R02 Master Schema Preview remark III.R03 Four-Tier Scope Discipline remark III.R04 Enrichment Ladder Preview remark III.D04 Enrichment Functor definition III.R05 Book II Handoff remark III.D05 Layer Template definition III.D06 E₀ Layer (Mathematics) definition III.D07 E₁ Layer (Physics) definition III.D08 E₂ Layer (Computation) definition III.D09 E₃ Layer (Metaphysics) definition III.T01 Non-Emptiness Theorem theorem III.T02 Strictness Theorem theorem III.P01 E₁ Strictness Witness proposition III.T03 Saturation at E₃ theorem III.P02 Functor Category Collapse proposition III.T04 Canonical Ladder Theorem theorem III.D10 Ladder Checker definition III.R06 Part I Export Contracts remark III.D11 Boundary Character Space definition III.D12 Boundary-to-Interior Functor definition III.T05 Sector Preservation Theorem theorem III.D13 4+1 Sector Decomposition definition III.D14 ω-Coupling Sector definition III.P03 E₁ Physical Preview proposition III.D15 Langlands₁ Reflection Bridge definition III.T06 Template Invariance Under Reflection theorem III.D16 Universal Operator definition III.D17 Spectral Polarity definition III.P04 Balanced Sector Uniqueness proposition III.T07 Parity Bridge Theorem theorem III.T08 No Knobs Principle theorem III.D18 Coupling Ledger definition III.R07 Part II Export Contracts remark III.D19 Primorial Ladder definition III.T09 Primorial Cofinality theorem III.R08 Primorial Verification Principle remark III.T10 CRT Decomposition Theorem theorem III.D20 Reconstruction Functor definition III.P05 Independence of Prime-Level Actions proposition III.T11 Constructive Hensel Lifting theorem III.D21 τ-Native Local Field definition III.P06 Completeness Without Topology proposition III.D22 τ-Adele Ring definition III.T12 Adelic Embedding Theorem theorem III.P07 Adelic Euler Product proposition III.D23 Internal Bipolar Classifier definition III.T13 Label Convergence theorem III.P08 Label-Idempotent Compatibility proposition III.T14 Spectral Trichotomy Lemma theorem III.D24 Boundary Normal Form definition III.T15 B/C Non-Collapse Theorem theorem III.R09 Complete Spectral Vocabulary remark III.R10 Coherence Checklist Template remark III.R11 Part III Export Contracts remark III.D25 Mutual Determination Schema definition III.R12 Enrichment Placement Table remark III.D26 Split-Complex Zeta ζ_τ definition III.D27 Functional Equation Involution J definition III.T16 Bipolar Euler Product theorem III.D28 Lemniscate Operator H_L definition III.T17 Self-Adjointness of H_L theorem III.P09 Discrete Spectrum of H_L proposition III.D29 Spectral Parameter Λ(s) definition III.T18 Spectral Correspondence Theorem theorem III.T19 Critical Line Theorem theorem III.P10 K5 Off-Diagonal Exclusion proposition III.D30 τ-Effective RH Statement definition III.P11 Primorial RH Verification Protocol proposition III.R13 Six Proof Obligations Ledger remark III.D31 Grand GRH (τ-effective) definition III.T20 Prime Polarity Scaling Theorem theorem III.D32 L-Function as Spectral Determinant definition III.D33 NP Witness as Canonical Address (v1) definition III.P12 CRT Witness Decomposition (v1) proposition III.T21 τ-Admissibility Collapse Theorem (v1) theorem III.D34 τ-CDCL Architecture definition III.T22 Representation Barrier Theorem theorem III.R14 P vs NP Honest Non-Claim remark III.R15 Perelman's Proof (review) remark III.D35 Simply Connected in Category τ definition III.P13 Poincaré as Gluing Guarantee proposition III.T23 Master Schema Theorem theorem III.R16 Part IV Export Contracts remark III.D36 τ-Admissible Fluid Data definition III.D37 Clopen Cylinder Domain definition III.R17 NS Scope Boundary remark III.D38 Fluid Sector Decomposition definition III.D39 Defect Functional Δ definition III.P14 Defect Contractivity proposition III.D40 Hartogs Flow Operator definition III.T24 Hartogs Flow Theorem theorem III.D41 Operator Polarity Swap definition III.T25 Positive Regularity Theorem theorem III.D42 Stabilized ω-Germ definition III.P15 3-Condition Sufficiency proposition III.R18 NS Export Contracts remark III.D43 Strong Sector at E₁ definition III.D44 τ-Admissible Gauge Data definition III.P16 NF Discreteness Lemma proposition III.T26 τ-Gap Meta-Theorem theorem III.D45 Gap Constant Γ* definition III.P17 Gap Stabilization proposition III.T27 Yang-Mills Gap Theorem theorem III.D46 Strong Defect Functional definition III.R19 YM Export Contracts remark III.D47 σ-Fixed Character definition III.D48 Sector Addressability definition III.P18 Hodge Requires E₁ proposition III.T28 NF-Addressability Theorem theorem III.P19 EM Sector Verification proposition III.P20 Sector-by-Sector Protocol proposition III.T29 Physics Layer Assembly theorem III.R20 Part V Export Contracts remark III.R21 Part V Coherence Summary remark III.D57 Enrichment Functor Enr₀₁ definition III.D58 E₁ Mutual Determination Instance definition III.P24 Three-Reading Equivalence at E₁ proposition III.D59 τ-Rational Point definition III.D60 Rank as Tower Depth definition III.P25 Mordell-Weil Analogue proposition III.D61 Proto-Code definition III.D62 BSD Functional definition III.P26 Bridgehead Proposition proposition III.T35 BSD Coherence Theorem theorem III.P27 BSD Three-Ingredient Proof proposition III.R25 BSD Export Contracts remark III.D63 Automorphic-Galois Duality in τ definition III.D64 Local Langlands Instance definition III.P28 Duality as Mutual Determination on ℤ² proposition III.T36 Functoriality Theorem theorem III.T37 Base Change-Transfer Naturality theorem III.D65 Enriched Bi-Square at E₁⁺ definition III.T38 Finite Factorization Pasting theorem III.T39 Enriched Bi-Square Comparison theorem III.T40 Enrichment Tower Assembly theorem III.R26 Part VI Export Contracts remark III.R27 Part VI Coherence Summary remark III.D49 E₂ Computational Agent definition III.D50 Operational Closure definition III.R22 Computation Cannot Live Below E₂ remark III.D51 τ-Tower Machine definition III.T30 TTM τ-Nativity theorem III.D52 Observable Transition definition III.D53 Interface Width definition III.D54 τ-Admissibility (E₂) definition III.T31 Interface Width Principle theorem III.P21 Earned Admissibility proposition III.D55 NP Witness as Canonical Address definition III.P22 CRT Witness Decomposition proposition III.P23 Polynomial Refinement proposition III.D56 Computational Bi-Square definition III.T32 Product-Meet Collapse theorem III.T33 τ-Admissibility Collapse theorem III.T34 No Barrier Theorem theorem III.R23 1st Edition Error Diagnosis remark III.R24 Part IX Export Contracts remark III.D66 Complete Dependency Chain definition III.P29 Chain Verification Protocol proposition III.T41 Hinge Theorem theorem III.P30 Sector Instantiation Lemma proposition III.R28 Export Contracts: Books IV-VII remark III.R29 Binding Conditions remark III.T42 No Knobs Theorem theorem III.R30 Master Schema Summary remark III.R31 Results Inventory remark III.D67 ZFC as E₂ VM definition III.D68 Gödel Numbering as NF Address definition III.D69 Five Forbidden Moves definition III.T43 Move-Bridge Correspondence theorem III.T44 Incompleteness as VM Boundary theorem III.D70 Host-Level Property definition III.D71 Bridge Axiom definition III.D72 Shadow Diagram definition III.T45 RH Bridge Three-Layer Structure theorem III.T46 Bridge Ledger theorem III.R32 P vs NP Bridge Break remark III.R33 Poincaré Established remark III.T47 Honest Claim Theorem theorem III.R34 Bridge Settlement Criteria remark III.D73 Proof Theory as E₃ definition III.D74 Diagrammatic Sector of E₃ definition III.D75 E₂→E₃ Boundary Crossing definition III.T48 Four Paradox Diagnostic theorem III.T49 Applied Saturation theorem III.P31 Terminal Level Characterization proposition III.R35 Architecture of Reality remark III.R36 Book III Coherence Certificate remark III.R37 Eight Guarantees Ledger (Earned) remark III.T50 Global Cartesian Gluing Theorem theorem III.D76 Minkowski Extension definition III.R38 E₁ Complete remark III.R39 Export Contracts to Books IV-V remark III.R40 Temporal-Spatial Decomposition remark III.R41 Light Cone Consistency remark III.D77 Physical E₂ Agent definition III.T51 Physical Admissibility Theorem theorem III.T52 Physical P=NP theorem III.R42 Coherence Horizon remark III.D78 τ-Native Abstract Turing Machine definition III.T53 Universal Admissibility Theorem theorem III.R43 Actual vs Potential Infinity remark III.D79 Physical Realizability Predicate definition III.R44 The Gödel Analogy remark III.R45 The Complete P vs NP Landscape remark III.R46 The Independence Prediction remark III.D80 q-Expansion Coefficients definition III.T54 Confinement Bridge Identity theorem III.T55 S-Duality Transport theorem III.P32 Bridge Algebraic Reduction proposition III.D81 Spectral Projector definition III.D82 Spectral Measure definition III.T56 Parseval Identity theorem III.P33 Spectral Resolution proposition III.D83 Kleene Fixed Point definition III.D84 E₂ Orbit Structure definition III.T57 Operational Closure Theorem theorem III.P34 E₂ ⊋ E₁ Strict Witness proposition III.D85 Self-Referential Fixed Point definition III.D86 Paradox Absorption Map definition III.T58 E₃ Self-Model Completeness theorem III.P35 Saturation Semantics proposition III.D87 Arithmetic Translation Functor definition III.D88 CRT-Integer Correspondence definition III.T59 Arithmetic Faithfulness theorem III.P36 Arithmetic Preserves Operations proposition III.D89 Topological Translation Functor definition III.D90 Dimension Recovery definition III.T60 Topological Faithfulness theorem III.P37 Boundary Restriction proposition III.D91 Obstruction Cocycle definition III.D92 Forbidden Move Obstruction Classes definition III.T61 Translation Failure Boundary theorem III.P38 P vs NP as Polynomial Translation Obstruction proposition III.D93 RH Spectral Gap Characterization definition III.D94 YM Mass Gap Persistence definition III.T62 NS Flow Causal Arrow theorem III.T63 P vs NP Forbidden Triple theorem III.P39 Bridge Ledger Completeness proposition III.D95 Goldbach Representation definition III.D96 Twin Prime Distribution definition III.T64 Goldbach at Primorial Levels theorem III.P40 Additive-Multiplicative Duality proposition III.D97 Radical Function definition III.D98 ABC Quality definition III.T65 ABC at Primorial Levels theorem III.P41 Radical-Primorial Identity proposition III.D99 Eratosthenes Sieve definition III.D100 Sieve Prime Count definition III.D101 Brun Sieve Count definition III.T66 Sieve Correctness theorem III.T67 Sieve-Tower Compatibility theorem III.P42 Sieve-CRT Compatibility proposition III.D102 Sieve-Accelerated Goldbach definition III.D103 Partition Count at Primorial definition III.D104 Goldbach Obstruction definition III.T68 Goldbach Verified to 500 theorem III.T69 Goldbach at Primorial M₄ theorem III.T70 Partition Growth theorem III.T71 Obstruction Bounded theorem III.P43 CRT-Goldbach Duality proposition III.P44 Goldbach Gap Characterization proposition III.D105 Twin Prime Sieve Count definition III.D106 Hardy-Littlewood Constant definition III.D107 CRT Twin Admissibility definition III.T72 Twin Primes to 500 theorem III.T73 Twin Density Primorial theorem III.T74 HL Constant Convergence theorem III.T75 CRT Admissible Positive theorem III.P45 Twin Admissibility Fraction proposition III.P46 Twin Gap Characterization proposition III.D108 Sieve-Accelerated ABC definition III.D109 High Quality Count definition III.D110 Squarefree ABC Check definition III.T76 ABC Quality 100 theorem III.T77 Squarefree Dominance theorem III.T78 Radical Primorial 5 theorem III.P47 Squarefree Dominance Theorem proposition III.P48 ABC Gap Characterization proposition III.D111 Tower Decidable Check definition III.D112 Gap Type definition III.D113 Forbidden Move Mapping definition III.T79 Tower Finite Decidable theorem III.T80 Bridge Necessary Insufficient theorem III.R47 Classical Comparison remark III.R48 Honest Conclusion remark