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Registry Dashboard — Book III: Categorical Spectrum

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This dashboard provides a complete inventory and formalization status map for Book III: Categorical Spectrum. It covers the enrichment ladder, sector template, spectral algebra, and the Millennium Problem clusters. Each registry object carries a scope label (established, tau-effective, or conjectural) and a formalization status. Use the statistics below to assess coverage, the dependency summary to trace proof chains, and the object list to navigate to individual entries in the registry.

Book III Registry Dashboard

Generated: 2026-03-10

Statistics

Total objects: 289
By type: definition: 113 proposition: 48 remark: 48 theorem: 80
Formalization: formalized: 231 not_applicable: 58
Scope: conjectural: 16 established: 37 metaphorical: 1 tau-effective: 235

Dependency Summary

Max chain depth: 91 (III.R45)
Most depended-on: III.T10 (20 uses), III.T14 (16 uses), III.D23 (13 uses)
No dependencies (foundational): III.R01, III.R02, III.R03, III.R04, III.R10, III.R17, III.D106

Part Coverage

| Part | Objects | Formalized | Planned | N/A | |——|———|————|———|—–| | 0 | 7 | 2 | 0 | 5 | | 1 | 15 | 13 | 0 | 2 | | 2 | 15 | 13 | 0 | 2 | | 3 | 29 | 25 | 0 | 4 | | 4 | 30 | 26 | 0 | 4 | | 5 | 32 | 27 | 0 | 5 | | 6 | 27 | 24 | 0 | 3 | | 7 | 19 | 14 | 0 | 5 | | 8 | 7 | 0 | 0 | 7 | | 9 | 43 | 29 | 0 | 14 | | 10 | 65 | 58 | 0 | 7 |

Object List

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