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Registry Dashboard — Book III: Categorical Spectrum

Browse Objects Registry Overview Filter Rules

Filter rule on this page: dashboard_display — counts the five display types (definition, lemma, proposition, remark, theorem). The full registry root for III has 289 objects across all types; this dashboard enumerates 289 of them. See the Filter Rules manifest for the authoritative definitions, or the Release Manifest for the per-book reconciliation across all surfaces.

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 —

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