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Registry Dashboard — Book II: Categorical Holomorphy

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 II has 230 objects across all types; this dashboard enumerates 219 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 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 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 —

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