Registry — Book I: Categorical Foundations

I.K0 Universe Postulate axiom I.D01 Five Generators definition I.D02 Progression Operator rho definition I.D03 Diagonal Discipline definition I.D04 Static Kernel tau_0 definition I.K1 Strict Order (K1) axiom I.K2 Omega Fixed Point (K2) axiom I.K3 Orbit-Seeded Generation (K3) axiom I.K4 No-Jump / Cover (K4) axiom I.K5 Beacon Non-Successor (K5) axiom I.K6 Object Closure (K6) axiom I.P01 Generator Distinctness proposition I.P02 rho Injectivity Per Orbit proposition I.X01 The Generative Act construction I.D05 Orbit Rays definition I.D06 Iterator Ladder definition I.T01 Ontic Closure theorem I.T02 Iterator Ladder Saturation theorem I.P03 Pairwise Disjointness of Orbits proposition I.P04 Orbit Countability proposition I.D07 tau-Idx (Earned Natural Numbers) definition I.D08 Rank Transfer Maps definition I.D09 Swap Operator sigma definition I.D10 Index Addition definition I.D11 Index Multiplication definition I.D12 Index Exponentiation definition I.D13 Index Tetration definition I.D14 Program Monoid definition I.D15 Three-Level Equality definition I.T03 Composition Associativity theorem I.P05 Tetration Injectivity proposition I.D16 NF Address Encoding definition I.D17 ABCD Coordinate Chart definition I.T04 Hyperfactorization Theorem theorem I.T05 Prime Polarity Theorem theorem I.D18 Algebraic Lemniscate theorem I.D19 Boundary Ring and Scalars definition I.D20 Split-Complex Scalars definition I.D21 Truth4 Logic definition I.D22 Holomorphic Transformers definition I.X02 Earned Category and Topos construction I.T06 Global Hartogs theorem I.R01 Axiom Independence remark I.L01 Pentation Non-Injectivity lemma I.T07 Rigidity of tau theorem I.T08 Categoricity of tau_0 theorem I.P06 Arithmetic Laws proposition I.L02 NF-Confluence lemma I.D16a Denotational Order definition I.P07 Well-Ordering of Obj(tau) proposition I.D19a Internal Divisibility definition I.D19b Internal Primes definition I.T09 FTA on tau-Idx theorem I.D19c Tower Atom definition I.D19d Greedy Peel Algorithm definition I.P08 Dimension Theorem (dim_tau = 4) proposition I.L03 No-Tie Determinism lemma I.L04 Strict Remainder Descent lemma I.C01 Constructive Encoding via ABCD definition I.D19e Prime Spectral Signature definition I.D23 Genealogical Decomposition (Spine) definition I.D24 Address DAG definition I.P09 Metric Inequality proposition I.D25 Omega-Tail (Compatible Tower) definition I.D26 Polarized Omega-Germ definition I.T10 Split-Complex Forced theorem I.D27 Bipolar Spectral Algebra definition I.D28 Boundary Local Ring definition I.D19f Prime Enumeration and Sieve definition I.D29 CRT Decomposition definition I.D30 Teichmueller Lift definition I.T11 Minimal Alphabet Theorem theorem I.T11a Six-Generator Rigidity Failure theorem I.T11b Four-Generator Ladder Incompleteness theorem I.T11c Tetration Algebraic Degradation theorem I.L05 Growth Escape lemma I.D31 tau-Membership Relation definition I.P10 Membership Decidability proposition I.D32 Set-Theoretic Operations definition I.P11 Distributive Lattice proposition I.D33 Bounded Powerset definition I.L06 Well-Foundedness of Membership lemma I.P12 Countability of Set(tau) proposition I.D34 Master Constant iota_tau definition I.D35 Number Tower definition I.D36 Constructive Reals definition I.D37 Lemniscate Characters definition I.D38 Character Group definition I.T12 Spectral Decomposition theorem I.D39 Crossing Point definition I.D40 Bipolar Fourier Transform definition I.T13 Explosion Barrier theorem I.P13 Boolean Recovery proposition I.D41 Subobject Classifier Preview definition I.P14 Omega Unique Fixed Seed proposition I.P15 Sum Zero Iff Both Zero proposition I.T14 No Additive Inverse theorem I.T15 No Ring Negation theorem I.P16 Positive Core Closure proposition I.P17 Universal Additive Cancellation proposition I.T16 Mul Cancel Fails at Zero theorem I.T17 Multiplicative Cancellation iff Positive theorem I.P18 Zero Vacuous proposition I.P19 Integral Domain proposition I.P20 Ultra Dist Self proposition I.P21 Congruent Tails Agree proposition I.D42 D-Differentiability definition I.D43 Split-CR Equations definition I.P22 Sector Independence proposition I.D45 Omega-Germ Transformer definition I.D46 Tower Coherence definition I.D47 Tau-Holomorphic Function definition I.D48 Tau-Holomorphic Map definition I.T18 CRT Coherence Constraint theorem I.P23 No Simultaneous Projection proposition I.T19 Diagonal-Free Protection theorem I.T20 Composition Closure theorem I.P24 HolFun Associativity proposition I.L07 Tail Agreement Propagation lemma I.T21 Tau-Identity Theorem theorem I.D49 Hol(L) definition I.D50 Tau-Arrow definition I.D51 Cat_tau definition I.T22 Category Axioms theorem I.P25 Thin Category proposition I.D52 Tau-Functor definition I.D53 Natural Transformation definition I.D54 Yoneda Embedding definition I.T23 Yoneda Lemma theorem I.D55 Finite Limits in Cat_tau definition I.D56 Tau-Site definition I.D57 Presheaf Topos definition I.T24 Grothendieck Topos theorem I.P26 Countable Topos proposition I.T25 Omega_tau Subobject Classifier theorem I.D58 Characteristic Morphism definition I.D59 Earned Topos definition I.P27 Paraconsistent Character proposition I.D60 Categorical Product definition I.T26 Product Universal Property theorem I.D61 Cartesian Monoidal Structure definition I.D62 Categorical Coproduct definition I.T27 Distributivity theorem I.D63 Bi-Monoidal Structure definition I.D64 Internal Hom definition I.T28 Cartesian Closed theorem I.P28 Self-Enrichment proposition I.D65 Spectral Coefficients definition I.D66 Restriction Map definition I.T29 Spectral Determination theorem I.D67 Primorial Thinness definition I.L08 CRT Extension lemma I.T30 Removable Singularity theorem I.T31 Global Hartogs Extension theorem I.D68 Earned Stage-Determined Point definition I.C02 Stages from Limit corollary I.P29 Passage to Book II proposition I.D75 Generative Counting Principle definition I.P33 Counting as Structural Feature proposition I.T35 Cantor Diagonal Inapplicability theorem I.P34 No Unearned Decimal Diagonal proposition I.P35 No Unrestricted Comprehension proposition I.P36 No Free Cartesian Diagonal proposition I.D76 Omega-Germ Approach definition I.T36 Unique Infinity Object theorem I.P37 Ultrametric Replaces Cardinality proposition I.D77 Meta-Logical Substrate definition I.R15 Structural Rules Inventory remark I.D78 Diagonal-Linear Correspondence definition I.T37 Diagonal-Linear Correspondence theorem I.D79 Program Monoid as Linear Calculus definition I.R16 Linear Logic Glossary remark I.T38 Linearity Census theorem I.P38 Classical.em Eliminability proposition I.R17 Gap Declaration remark I.D80 Self-Hosting Degree Classification definition I.R18 Proof-Theoretic Landscape Survey remark I.D81 CCC-Linear Dichotomy definition I.T39 K5 Structural Exclusion theorem I.R19 Barr's Star-Autonomous Categories remark I.D82 Enrichment Frontier Classification definition I.R20 E0-E3 Literature Map remark I.R21 Scale Declaration remark I.D83 Primorial Presheaf definition I.T40 Presheaf Characterization theorem I.T41 Bi-Square Characterization theorem I.D84 Constructive Reals definition I.T42 Archimedean Property theorem I.P39 TauReal Ring Axioms proposition I.D85 Elliptic Complex Field definition I.D86 Elliptic-Hyperbolic Dichotomy definition I.T43 TauComplex Ring Axioms theorem I.D87 Elliptic Quaternions definition I.T44 Quaternion Non-Commutativity theorem I.R22 Hurwitz Classification Preview remark I.D88 Cyclotomic Fields definition I.T45 Roots of Unity CRT Decomposition theorem I.R23 Galois Theory Preview remark I.D89 Diagonal Resonance definition I.D90 Identity Slippage definition I.D91 Shadow Identity definition I.R24 Five Reasons Why The Bug Hides remark I.R25 Orthodox Foundations Under the Lens remark I.T46 Ontic Identity Invariance theorem I.C03 No Identity Decoherence corollary I.T47 Slippage Breaks Unique Omega theorem I.D92 Identity-Faithful Reception definition I.D93 Structural Instability definition I.T48 Structural Instability Theorem theorem I.R26 Implications for Absolute Meaning remark I.R27 Honest Scope Declaration remark I.D94 Orbit-Set Map definition I.P40 Extensionality proposition I.P41 Self-Containment Partition proposition I.P42 Order Bound proposition I.R28 Inseparability of N and omega remark I.R29 Finite-Infinite Boundary remark I.R30 Duality and Atoms remark I.D95 τ-Measure Space definition I.D96 Tower σ-Algebra definition I.T49 Countable Additivity theorem I.P43 Measure Compatibility proposition I.D97 Galois Automorphism definition I.D98 Galois Group of Primorial Stage definition I.T50 Fundamental Theorem of Internal Galois Theory theorem I.P44 CRT-Galois Decomposition proposition I.D99 τ-Integral definition I.T51 Linearity of Integration theorem I.P45 Monotone Convergence proposition I.D101 Prime Counting in Progressions definition I.D102 Chebyshev Bias Measure definition I.T52 Bias at Primorial Levels theorem I.R31 Wiring Primitives remark I.R32 Product-Free Operator Definitions remark I.R33 Route B Axioms Comparison remark I.R34 Normalization Endofunctors M and E remark I.R35 Address Combinators: Merge and Lift remark I.R36 CGANF to ABCD Design Evolution remark I.R37 Cluster Collapse in ABCD Context remark I.R38 Token-Set vs Tuple Encoding remark I.R39 Dimension Readout Functors remark I.P46 No Second Linearity proposition I.C04 NF Deduplication Invariant corollary I.D103 Polarity Pairing definition I.R40 Pairing Density remark I.D104 PolStream definition I.R41 PolStream Aperiodicity remark I.R42 Lemniscate Lipschitz Forward Reference remark I.R43 Quaternion Open Question remark I.D105 Tau-Weighted Boundary Constants definition I.D106 Tau-Bayesian State definition I.P47 Bayesian Factorization proposition I.R44 Lean Bayesian Status remark I.R45 Distributivity from Combinators remark