Registry — Book II: Categorical Holomorphy

II.R01 Boundary-First Principle remark II.R02 Split-Complex Necessity remark II.D01 Split-Complex Codomain H_tau definition II.R03 Inverted Dependency Chain remark II.D02 Tau-Admissible Point definition II.D03 Constraint Lattice definition II.T01 Point Set Well-Defined theorem II.D04 Omega Readout definition II.T02 Fiber Degeneration at Omega theorem II.P01 Lemniscate as Coordinate Limit proposition II.D05 Base tau^1 definition II.D06 Fiber T^2 definition II.D07 Fibered Product tau^3 definition II.T03 Fibration Structure theorem II.D08 Interior Bipolar Decomposition definition II.P02 Sector Inheritance proposition II.R04 ABCD vs Quaternions remark II.P03 Four-Ray Rigidity proposition II.D09 Cylinder Domain definition II.D10 Stage-k Cylinder definition II.D11 Clopen Basis definition II.T04 Cylinder Basis Theorem theorem II.D12 First Disagreement Depth definition II.D13 Ultrametric Distance definition II.T05 Ultrametric Triangle Inequality theorem II.P04 Cylinders Are Balls proposition II.T06 Holomorphic Implies Continuous theorem II.L01 Naturality Forces Cylinder Compatibility lemma II.R05 Topological Skeleton remark II.D14 Stone Space definition II.T07 Compactness theorem II.T08 Hausdorff Property theorem II.T09 Total Disconnectedness theorem II.T10 Topology Uniqueness theorem II.R06 Topology as Invariant remark II.D15 Tau-Dimension definition II.T11 Dimension Four theorem II.D16 Radial-Solenoidal Split definition II.T12 Boundary Minimality theorem II.D17 Angular Sectors definition II.P05 Lobes as Clopen Sets proposition II.T13 Torus Degeneration Theorem theorem II.D18 Pinch Map definition II.T14 Fundamental Group Degeneration theorem II.D18a Two-Readout Principle definition II.D18b Spine Address Path definition II.R06a Refinement Rays remark II.R06b Compatibility Axiom Schema remark II.D19 Betweenness Relation definition II.T15 Betweenness Axioms theorem II.D20 Congruence Relation definition II.T16 Congruence Axioms theorem II.T17 Pasch Axiom theorem II.T18 Parallel Postulate theorem II.D21 Wave-Type PDE definition II.D22 Causal Structure definition II.T19 Euclidean as Static Limit theorem II.R07 Orthodox Denotation Bridge remark II.D23 Approximation Sequence definition II.D24 Alpha-Ray Line definition II.D25 Level Circle definition II.T20 R as Inverse Limit theorem II.D26 Solenoidal Circle definition II.T21 S^1 as Profinite Limit theorem II.D27 Geometric-Topological Unification definition II.D28 Geometric Pi definition II.D29 Archimedes Polygon Sequence definition II.T22 Three Perspectives on Pi theorem II.D30 e as Iterator Eigenvalue definition II.D31 Growth Base definition II.T23 e from Index Arithmetic theorem II.D32 Interior j-Unit definition II.D33 Bipolar Idempotents Interior definition II.T24 j Replaces i theorem II.R08 Polarity vs Rotation remark II.T25 Master Constant Confirmed theorem II.D34 Archimedean Bridge definition II.R09 Refinement Equals Resolution remark II.P06 Refinement Resolution Bound proposition II.D35 Calibrated Split-Complex Codomain definition II.R10 Geometric Meaning of Bipolar Idempotents remark II.D36 BndLift Construction definition II.T26 BndLift Existence theorem II.P07 Bipolar Channel Independence proposition II.L02 Refinement-Spectral Equivalence lemma II.L03 Spectral-Germ Equivalence lemma II.L04 Germ-Character Equivalence lemma II.L05 Character-Hartogs Equivalence lemma II.T27 Mutual Determination (5-Way Equivalence) theorem II.D37 Evolution Operator definition II.D38 Causal Arrow definition II.T28 B/C Asymmetry Implies Time Arrow theorem II.D39 Composition of Holomorphic Maps definition II.D40 Identity Map definition II.T29 Associativity of Holomorphic Composition theorem II.D41 Holomorphic Endomorphism Category definition II.D42 Laurent Expansion definition II.D43 Residue definition II.D44 Meromorphic Function definition II.T30 Residue Theorem theorem II.D45 Canonical Holomorphic Basis definition II.D46 Cylinder Generator definition II.T31 Finite Spectral Support theorem II.P08 Projection Formula proposition II.D47 Holomorphic Presheaf definition II.L06 Gluing Lemma lemma II.T32 Sheaf Axioms theorem II.D48 Canonical Decomposition definition II.L07 Idempotent Decomposition Lemma lemma II.P09 Decomposition Functoriality proposition II.L08 Branch Factorization lemma II.L09 Prime-Split Support lemma II.L10 Polarity Symmetry lemma II.T33 Holomorphic iff Idempotent-Supported theorem II.D49 Tau-Regularity definition II.T34 Regularity Criterion theorem II.R11 Positive vs Negative Regularity remark II.D50 Pre-Yoneda Embedding definition II.P10 Functions as Tau-Objects proposition II.R12 Probe Naturality Equals Holomorphy remark II.D51 Code Map definition II.D52 Decode Map definition II.T35 Code/Decode Bijection theorem II.R13 Diagonal Protection via K5 remark II.D53 Self-Enrichment Structure definition II.D54 Hom Object definition II.P11 Hom Bipolar Decomposition proposition II.L11 Probe Naturality iff Yoneda lemma II.T36 Tau-Yoneda Embedding theorem II.D55 2-Category Structure definition II.D56 2-Morphism definition II.P12 Enrichment Iteration proposition II.R14 E1 Preview remark II.D57 E1 Enrichment Layer definition II.R15 Self-Description Property remark II.D58 E0/E1 Transition definition II.R16 Enrichment Ladder Preview remark II.D59 Idempotent-Supported Character definition II.P14 Character Algebra Ring Structure proposition II.P13 Character Decomposition proposition II.L12 Extension in Split-Complex Codomain lemma II.T37 Hartogs Extension Uniqueness theorem II.L13 Stagewise Naturality lemma II.T38 Extensions Are Omega-Germ Transformers theorem II.L14 Yoneda Application lemma II.T39 Omega-Germs iff Holomorphic Functions theorem II.D60 Spectral Algebra definition II.T40 Central Theorem theorem II.C01 Holographic Principle corollary II.T41 Liouville Categorical Dodge theorem II.T42 Categoricity theorem II.D61 Moduli Space definition II.C02 Uniqueness of Category Tau corollary II.R17 Complete Dependency Chain Audit remark II.D62 Tau-Manifold definition II.D63 Tau-Analytic Atlas definition II.D64 Tau-Exterior Derivative definition II.P15 Tau3 Is a Tau-Manifold proposition II.R18 BSD Bridge remark II.D65 Proto-Rationality definition II.R19 Spectral Forces Preview remark II.D67 E1 Export Package definition II.R20 Open Questions and Research Agenda remark II.R21 Differential-Geometric Agenda remark II.R22 Enrichment Ladder Forward Declaration remark II.D77 Geometric Bi-Square definition II.T49 Geometric Bi-Square Theorem theorem II.R33 Algebraic-to-Geometric Audit remark II.R34 Scaling Chain Forward remark II.D68 Structural Sign Classification definition II.R23 The Rewiring Thesis remark II.D69 The Infinity Trade-Off definition II.T43 Structural Incompatibility of Unique Omega and Archimedean Density theorem II.R24 What Each Path Buys remark II.D70 PDE Type Classification definition II.T44 Asymmetric Determination theorem II.R25 Why Hartogs Is the Wave Front remark II.D71 Stage-Finite Euclidean Geometry definition II.T45 Parallel Preservation theorem II.R26 The Minkowski Comparison remark II.D72 The Rewiring Table definition II.R27 Honest Accounting remark II.R28 Spectral Forces Inherit Hyperbolicity remark II.D73 The Physics Quadrant Matrix definition II.D74 The Unification Obstruction definition II.T46 Fourth Quadrant Resolution theorem II.R29 Why Every Unification Attempt Fails remark II.R30 Books IV-V Preview remark II.D75 Archimedean-Elliptic Engine definition II.D76 Dimensional Rigidity definition II.T47 Simultaneous Rung Theorem theorem II.T48 Fourth Quadrant Ladder Collapse theorem II.R31 Categoricity Implies No Ladder remark II.R32 Honest Scope: Ladder vs. SCV remark II.D78 τ-Connection definition II.D79 Parallel Transport definition II.T50 Flat Connection Existence theorem II.P16 Holonomy Triviality proposition II.D80 τ-Curvature definition II.D81 τ-Geodesic definition II.T51 Flat Curvature Vanishing theorem II.P17 Geodesic Completeness proposition II.T52 Lemniscate Holonomy theorem II.D82 L² Inner Product definition II.D83 L² Norm definition II.T53 Cauchy-Schwarz Inequality theorem II.P18 L² Completeness proposition II.D84 Chain Complex definition II.D85 Homology via SES definition II.T54 Tower Coherence theorem II.P19 Long Exact Sequence proposition II.D86 Čech Complex definition II.D87 Sheaf Cohomology Groups definition II.T55 H⁰ = Global Sections theorem II.P20 Čech-to-Derived Comparison proposition II.R35 Denotation Map as Shadow Functor remark II.P21 HolEnd Exponentials proposition II.D88 Tau-Exponential definition II.R36 Tau-Exponential Sigma-Equivariance remark II.R37 Shadow Families in SCV remark II.D89 Five Comparison Modes definition II.D90 Six Comparison Axes definition II.D91 Mode C Catalog definition II.D92 Mode E Catalog definition II.D93 Combined Gain-Cost Ledger definition II.T56 Trade-Off Atomicity theorem II.R38 Protection Architecture remark II.R39 Epistemic Upgrade Principle remark II.R40 The Reality Test remark II.R41 Bridge Declaration remark