This dashboard provides a complete inventory and formalization status map for Book II: Categorical Holomorphy. It covers the tau-3 fibration, split-complex holomorphy, the Central Theorem, and self-enrichment. 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 II Registry Dashboard
Generated: 2026-03-10
Statistics
- Total objects: 219
-
By type: corollary: 2 definition: 89 lemma: 14 proposition: 20 remark: 39 theorem: 55 -
Formalization: formalized: 184 not_applicable: 33 planned: 2 -
Scope: established: 194 tau-effective: 25
Dependency Summary
- Max chain depth: 64 (II.R32)
- Most depended-on: I.D21 (41 uses), I.T10 (31 uses), I.T05 (27 uses)
- No dependencies (foundational): II.R03
Part Coverage
| Part | Objects | Formalized | Planned | N/A | |——|———|————|———|—–| | 0 | 4 | 1 | 0 | 3 | | 1 | 14 | 14 | 0 | 0 | | 2 | 11 | 10 | 0 | 1 | | 3 | 19 | 17 | 0 | 2 | | 4 | 12 | 11 | 0 | 1 | | 5 | 20 | 18 | 0 | 2 | | 6 | 32 | 32 | 0 | 0 | | 7 | 17 | 15 | 0 | 2 | | 8 | 20 | 14 | 2 | 4 | | 9 | 19 | 19 | 0 | 0 | | 10 | 25 | 17 | 0 | 8 | | 11 | 26 | 16 | 0 | 10 |
Object List
| ID | Type | Name | Scope | Lean | |—-|——|——|——-|——| | II.R01 | remark | Boundary-First Principle | estab | — | | II.R02 | remark | Split-Complex Necessity | estab | — | | II.D01 | definition | Split-Complex Codomain H_tau | estab | form | | II.R03 | remark | Inverted Dependency Chain | estab | — | | II.D02 | definition | Tau-Admissible Point | estab | form | | II.D03 | definition | Constraint Lattice | estab | form | | II.T01 | theorem | Point Set Well-Defined | estab | form | | II.D04 | definition | Omega Readout | estab | form | | II.T02 | theorem | Fiber Degeneration at Omega | estab | form | | II.P01 | proposition | Lemniscate as Coordinate Limit | estab | form | | II.D05 | definition | Base tau^1 | estab | form | | II.D06 | definition | Fiber T^2 | estab | form | | II.D07 | definition | Fibered Product tau^3 | estab | form | | II.T03 | theorem | Fibration Structure | estab | form | | II.D08 | definition | Interior Bipolar Decomposition | estab | form | | II.P02 | proposition | Sector Inheritance | estab | form | | II.R04 | remark | ABCD vs Quaternions | estab | form | | II.P03 | proposition | Four-Ray Rigidity | estab | form | | II.D09 | definition | Cylinder Domain | estab | form | | II.D10 | definition | Stage-k Cylinder | estab | form | | II.D11 | definition | Clopen Basis | estab | form | | II.T04 | theorem | Cylinder Basis Theorem | estab | form | | II.D12 | definition | First Disagreement Depth | estab | form | | II.D13 | definition | Ultrametric Distance | estab | form | | II.T05 | theorem | Ultrametric Triangle Inequality | estab | form | | II.P04 | proposition | Cylinders Are Balls | estab | form | | II.T06 | theorem | Holomorphic Implies Continuous | estab | form | | II.L01 | lemma | Naturality Forces Cylinder Compatibility | estab | form | | II.R05 | remark | Topological Skeleton | estab | — | | II.D14 | definition | Stone Space | estab | form | | II.T07 | theorem | Compactness | estab | form | | II.T08 | theorem | Hausdorff Property | estab | form | | II.T09 | theorem | Total Disconnectedness | estab | form | | II.T10 | theorem | Topology Uniqueness | estab | form | | II.R06 | remark | Topology as Invariant | estab | — | | II.D15 | definition | Tau-Dimension | estab | form | | II.T11 | theorem | Dimension Four | estab | form | | II.D16 | definition | Radial-Solenoidal Split | estab | form | | II.T12 | theorem | Boundary Minimality | estab | form | | II.D17 | definition | Angular Sectors | estab | form | | II.P05 | proposition | Lobes as Clopen Sets | estab | form | | II.T13 | theorem | Torus Degeneration Theorem | estab | form | | II.D18 | definition | Pinch Map | estab | form | | II.T14 | theorem | Fundamental Group Degeneration | estab | form | | II.D18a | definition | Two-Readout Principle | estab | form | | II.D18b | definition | Spine Address Path | estab | form | | II.R06a | remark | Refinement Rays | estab | form | | II.R06b | remark | Compatibility Axiom Schema | estab | — | | II.D19 | definition | Betweenness Relation | estab | form | | II.T15 | theorem | Betweenness Axioms | estab | form | | II.D20 | definition | Congruence Relation | estab | form | | II.T16 | theorem | Congruence Axioms | estab | form | | II.T17 | theorem | Pasch Axiom | estab | form | | II.T18 | theorem | Parallel Postulate | estab | form | | II.D21 | definition | Wave-Type PDE | estab | form | | II.D22 | definition | Causal Structure | estab | form | | II.T19 | theorem | Euclidean as Static Limit | estab | form | | II.R07 | remark | Orthodox Denotation Bridge | estab | form | | II.D23 | definition | Approximation Sequence | estab | form | | II.D24 | definition | Alpha-Ray Line | estab | form | | II.D25 | definition | Level Circle | estab | form | | II.T20 | theorem | R as Inverse Limit | estab | form | | II.D26 | definition | Solenoidal Circle | estab | form | | II.T21 | theorem | S^1 as Profinite Limit | estab | form | | II.D27 | definition | Geometric-Topological Unification | estab | form | | II.D28 | definition | Geometric Pi | estab | form | | II.D29 | definition | Archimedes Polygon Sequence | estab | form | | II.T22 | theorem | Three Perspectives on Pi | estab | form | | II.D30 | definition | e as Iterator Eigenvalue | estab | form | | II.D31 | definition | Growth Base | estab | form | | II.T23 | theorem | e from Index Arithmetic | estab | form | | II.D32 | definition | Interior j-Unit | estab | form | | II.D33 | definition | Bipolar Idempotents Interior | estab | form | | II.T24 | theorem | j Replaces i | estab | form | | II.R08 | remark | Polarity vs Rotation | estab | — | | II.T25 | theorem | Master Constant Confirmed | estab | form | | II.D34 | definition | Archimedean Bridge | estab | form | | II.R09 | remark | Refinement Equals Resolution | estab | — | | II.P07 | proposition | Refinement Resolution Bound | estab | form | | II.D35 | definition | Calibrated Split-Complex Codomain | estab | form | | II.R10 | remark | Geometric Meaning of Bipolar Idempotents | estab | form | | II.D36 | definition | BndLift Construction | estab | form | | II.T26 | theorem | BndLift Existence | estab | form | | II.P08 | proposition | Bipolar Channel Independence | estab | form | | II.L02 | lemma | Refinement-Spectral Equivalence | estab | form | | II.L03 | lemma | Spectral-Germ Equivalence | estab | form | | II.L04 | lemma | Germ-Character Equivalence | estab | form | | II.L05 | lemma | Character-Hartogs Equivalence | estab | form | | II.T27 | theorem | Mutual Determination (5-Way Equivalence) | estab | form | | II.D37 | definition | Evolution Operator | estab | form | | II.D38 | definition | Causal Arrow | estab | form | | II.T28 | theorem | B/C Asymmetry Implies Time Arrow | estab | form | | II.D39 | definition | Composition of Holomorphic Maps | estab | form | | II.D40 | definition | Identity Map | estab | form | | II.T29 | theorem | Associativity of Holomorphic Composition | estab | form | | II.D41 | definition | Holomorphic Endomorphism Category | estab | form | | II.D42 | definition | Laurent Expansion | estab | form | | II.D43 | definition | Residue | estab | form | | II.D44 | definition | Meromorphic Function | estab | form | | II.T30 | theorem | Residue Theorem | estab | form | | II.D45 | definition | Canonical Holomorphic Basis | estab | form | | II.D46 | definition | Cylinder Generator | estab | form | | II.T31 | theorem | Finite Spectral Support | estab | form | | II.P09 | proposition | Projection Formula | estab | form | | II.D47 | definition | Holomorphic Presheaf | estab | form | | II.L06 | lemma | Gluing Lemma | estab | form | | II.T32 | theorem | Sheaf Axioms | estab | form | | II.D48 | definition | Canonical Decomposition | estab | form | | II.L07 | lemma | Idempotent Decomposition Lemma | estab | form | | II.P10 | proposition | Decomposition Functoriality | estab | form | | II.L08 | lemma | Branch Factorization | estab | form | | II.L09 | lemma | Prime-Split Support | estab | form | | II.L10 | lemma | Polarity Symmetry | estab | form | | II.T33 | theorem | Holomorphic iff Idempotent-Supported | estab | form | | II.D49 | definition | Tau-Regularity | estab | form | | II.T34 | theorem | Regularity Criterion | estab | form | | II.R11 | remark | Positive vs Negative Regularity | estab | — | | II.D50 | definition | Pre-Yoneda Embedding | estab | form | | II.P11 | proposition | Functions as Tau-Objects | estab | form | | II.R12 | remark | Probe Naturality Equals Holomorphy | estab | form | | II.D51 | definition | Code Map | estab | form | | II.D52 | definition | Decode Map | estab | form | | II.T35 | theorem | Code/Decode Bijection | estab | form | | II.R13 | remark | Diagonal Protection via K5 | estab | — | | II.D53 | definition | Self-Enrichment Structure | estab | form | | II.D54 | definition | Hom Object | estab | form | | II.P12 | proposition | Hom Bipolar Decomposition | estab | form | | II.L11 | lemma | Probe Naturality iff Yoneda | estab | form | | II.T36 | theorem | Tau-Yoneda Embedding | estab | form | | II.D55 | definition | 2-Category Structure | estab | form | | II.D56 | definition | 2-Morphism | estab | form | | II.P13 | proposition | Enrichment Iteration | estab | form | | II.R14 | remark | E1 Preview | estab | — | | II.D57 | definition | E1 Enrichment Layer | estab | form | | II.R15 | remark | Self-Description Property | estab | — | | II.D58 | definition | E0/E1 Transition | estab | form | | II.R16 | remark | Enrichment Ladder Preview | estab | — | | II.D59 | definition | Idempotent-Supported Character | estab | form | | II.P14 | proposition | Character Algebra Ring Structure | estab | form | | II.L12 | lemma | Extension in Split-Complex Codomain | estab | form | | II.T37 | theorem | Hartogs Extension Uniqueness | estab | form | | II.L13 | lemma | Stagewise Naturality | estab | form | | II.T38 | theorem | Extensions Are Omega-Germ Transformers | estab | form | | II.L14 | lemma | Yoneda Application | estab | form | | II.T39 | theorem | Omega-Germs iff Holomorphic Functions | estab | form | | II.D60 | definition | Spectral Algebra | estab | form | | II.T40 | theorem | Central Theorem | estab | form | | II.C01 | corollary | Holographic Principle | estab | form | | II.T41 | theorem | Liouville Categorical Dodge | estab | form | | II.T42 | theorem | Categoricity | estab | form | | II.D61 | definition | Moduli Space | estab | form | | II.C02 | corollary | Uniqueness of Category Tau | estab | form | | II.R17 | remark | Complete Dependency Chain Audit | estab | — | | II.D62 | definition | Tau-Manifold | estab | form | | II.D63 | definition | Tau-Analytic Atlas | estab | form | | II.D64 | definition | Tau-Exterior Derivative | estab | form | | II.P15 | proposition | Tau3 Is a Tau-Manifold | estab | form | | II.R18 | remark | BSD Bridge | estab | — | | II.D65 | definition | Proto-Rationality | estab | form | | II.R19 | remark | Spectral Forces Preview | estab | — | | II.D66 | definition | E1 Export Package | estab | form | | II.R20 | remark | Open Questions and Research Agenda | estab | — | | II.R21 | remark | Differential-Geometric Agenda | estab | — | | II.R22 | remark | Enrichment Ladder Forward Declaration | estab | — | | II.D77 | definition | Geometric Bi-Square | estab | form | | II.T49 | theorem | Geometric Bi-Square Theorem | estab | form | | II.R33 | remark | Algebraic-to-Geometric Audit | estab | — | | II.R34 | remark | Scaling Chain Forward | estab | — | | II.D68 | definition | Structural Sign Classification | estab | form | | II.R23 | remark | The Rewiring Thesis | estab | — | | II.D69 | definition | The Infinity Trade-Off | estab | form | | II.T43 | theorem | Structural Incompatibility of Unique Omega and Archimedean Density | estab | form | | II.R24 | remark | What Each Path Buys | estab | — | | II.D70 | definition | PDE Type Classification | estab | form | | II.T44 | theorem | Asymmetric Determination | estab | form | | II.R25 | remark | Why Hartogs Is the Wave Front | estab | — | | II.D71 | definition | Stage-Finite Euclidean Geometry | estab | form | | II.T45 | theorem | Parallel Preservation | estab | form | | II.R26 | remark | The Minkowski Comparison | estab | — | | II.D72 | definition | The Rewiring Table | estab | form | | II.R27 | remark | Honest Accounting | estab | — | | II.R28 | remark | Spectral Forces Inherit Hyperbolicity | estab | — | | II.D73 | definition | The Physics Quadrant Matrix | estab | form | | II.D74 | definition | The Unification Obstruction | estab | form | | II.T46 | theorem | Fourth Quadrant Resolution | estab | form | | II.R29 | remark | Why Every Unification Attempt Fails | estab | — | | II.R30 | remark | Books IV-V Preview | estab | — | | II.D75 | definition | Archimedean-Elliptic Engine | estab | form | | II.D76 | definition | Dimensional Rigidity | estab | form | | II.T47 | theorem | Simultaneous Rung Theorem | estab | form | | II.T48 | theorem | Fourth Quadrant Ladder Collapse | estab | form | | II.R31 | remark | Categoricity Implies No Ladder | estab | form | | II.R32 | remark | Honest Scope: Ladder vs. SCV | estab | — | | II.D78 | definition | τ-Connection | tau-e | form | | II.D79 | definition | Parallel Transport | tau-e | form | | II.T50 | theorem | Flat Connection Existence | tau-e | form | | II.P16 | proposition | Holonomy Triviality | tau-e | form | | II.D80 | definition | τ-Curvature | tau-e | form | | II.D81 | definition | τ-Geodesic | tau-e | form | | II.T51 | theorem | Flat Curvature Vanishing | tau-e | form | | II.P17 | proposition | Geodesic Completeness | tau-e | form | | II.T52 | theorem | Lemniscate Holonomy | tau-e | form | | II.D82 | definition | L² Inner Product | tau-e | form | | II.D83 | definition | L² Norm | tau-e | form | | II.T53 | theorem | Cauchy-Schwarz Inequality | tau-e | form | | II.P18 | proposition | L² Completeness | tau-e | form | | II.D84 | definition | Chain Complex | tau-e | form | | II.D85 | definition | Homology via SES | tau-e | form | | II.T54 | theorem | Tower Coherence | tau-e | form | | II.P19 | proposition | Long Exact Sequence | tau-e | form | | II.D86 | definition | Čech Complex | tau-e | form | | II.D87 | definition | Sheaf Cohomology Groups | tau-e | form | | II.T55 | theorem | H⁰ = Global Sections | tau-e | form | | II.P20 | proposition | Čech-to-Derived Comparison | tau-e | form | | II.R35 | remark | Denotation Map as Shadow Functor | estab | — | | II.P21 | proposition | HolEnd Exponentials | tau-e | — | | II.D88 | definition | Tau-Exponential | tau-e | — | | II.R36 | remark | Tau-Exponential Sigma-Equivariance | tau-e | — | | II.R37 | remark | Shadow Families in SCV | tau-e | — |