Registry — Book I: Categorical Foundations

I.C01 Constructive Encoding via ABCD definition I.C02 Stages from Limit corollary I.C03 No Identity Decoherence corollary I.C04 NF Deduplication Invariant corollary I.D01 Five Generators definition I.D02 Progression Operator rho definition I.D03 Diagonal Discipline definition I.D04 Static Kernel tau_0 definition I.D05 Orbit Rays definition I.D06 Iterator Ladder definition 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.D103 Polarity Pairing definition I.D104 PolStream definition I.D105 Tau-Weighted Boundary Constants definition I.D106 Tau-Bayesian State 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.D16 NF Address Encoding definition I.D16a Denotational Order definition I.D17 ABCD Coordinate Chart definition I.D18 Algebraic Lemniscate theorem I.D19 Boundary Ring and Scalars definition I.D19a Internal Divisibility definition I.D19b Internal Primes definition I.D19c Tower Atom definition I.D19d Greedy Peel Algorithm definition I.D19e Prime Spectral Signature definition I.D19f Prime Enumeration and Sieve definition I.D20 Split-Complex Scalars definition I.D21 Truth4 Logic definition I.D22 Holomorphic Transformers definition I.D23 Genealogical Decomposition (Spine) definition I.D24 Address DAG definition I.D25 Omega-Tail (Compatible Tower) definition I.D26 Polarized Omega-Germ definition I.D27 Bipolar Spectral Algebra definition I.D28 Boundary Local Ring definition I.D29 CRT Decomposition definition I.D30 Teichmueller Lift definition I.D31 tau-Membership Relation definition I.D32 Set-Theoretic Operations definition I.D33 Bounded Powerset definition 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.D39 Crossing Point definition I.D40 Bipolar Fourier Transform definition I.D41 Subobject Classifier Preview definition I.D42 D-Differentiability definition I.D43 Split-CR Equations definition I.D45 Omega-Germ Transformer definition I.D46 Tower Coherence definition I.D47 Tau-Holomorphic Function definition I.D48 Tau-Holomorphic Map definition I.D49 Hol(L) definition I.D50 Tau-Arrow definition I.D51 Cat_tau definition I.D52 Tau-Functor definition I.D53 Natural Transformation definition I.D54 Yoneda Embedding definition I.D55 Finite Limits in Cat_tau definition I.D56 Tau-Site definition I.D57 Presheaf Topos definition I.D58 Characteristic Morphism definition I.D59 Earned Topos definition I.D60 Categorical Product definition I.D61 Cartesian Monoidal Structure definition I.D62 Categorical Coproduct definition I.D63 Bi-Monoidal Structure definition I.D64 Internal Hom definition I.D65 Spectral Coefficients definition I.D66 Restriction Map definition I.D67 Primorial Thinness definition I.D68 Earned Stage-Determined Point definition I.D75 Generative Counting Principle definition I.D76 Omega-Germ Approach definition I.D77 Meta-Logical Substrate definition I.D78 Diagonal-Linear Correspondence definition I.D79 Program Monoid as Linear Calculus definition I.D80 Self-Hosting Degree Classification definition I.D81 CCC-Linear Dichotomy definition I.D82 Enrichment Frontier Classification definition I.D83 Primorial Presheaf definition I.D84 Constructive Reals definition I.D85 Elliptic Complex Field definition I.D86 Elliptic-Hyperbolic Dichotomy definition I.D87 Elliptic Quaternions definition I.D88 Cyclotomic Fields definition I.D89 Diagonal Resonance definition I.D90 Identity Slippage definition I.D91 Shadow Identity definition I.D92 Identity-Faithful Reception definition I.D93 Structural Instability definition I.D94 Orbit-Set Map definition I.D95 τ-Measure Space definition I.D96 Tower σ-Algebra definition I.D97 Galois Automorphism definition I.D98 Galois Group of Primorial Stage definition I.D99 τ-Integral definition I.K0 Universe Postulate axiom 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.L01 Pentation Non-Injectivity lemma I.L02 NF-Confluence lemma I.L03 No-Tie Determinism lemma I.L04 Strict Remainder Descent lemma I.L05 Growth Escape lemma I.L06 Well-Foundedness of Membership lemma I.L07 Tail Agreement Propagation lemma I.L08 CRT Extension lemma I.P01 Generator Distinctness proposition I.P02 rho Injectivity Per Orbit proposition I.P03 Pairwise Disjointness of Orbits proposition I.P04 Orbit Countability proposition I.P05 Tetration Injectivity proposition I.P06 Arithmetic Laws proposition I.P07 Well-Ordering of Obj(tau) proposition I.P08 Dimension Theorem (dim_tau = 4) proposition I.P09 Metric Inequality proposition I.P10 Membership Decidability proposition I.P11 Distributive Lattice proposition I.P12 Countability of Set(tau) proposition I.P13 Boolean Recovery proposition I.P14 Omega Unique Fixed Seed proposition I.P15 Sum Zero Iff Both Zero proposition I.P16 Positive Core Closure proposition I.P17 Universal Additive Cancellation proposition I.P18 Zero Vacuous proposition I.P19 Integral Domain proposition I.P20 Ultra Dist Self proposition I.P21 Congruent Tails Agree proposition I.P22 Sector Independence proposition I.P23 No Simultaneous Projection proposition I.P24 HolFun Associativity proposition I.P25 Thin Category proposition I.P26 Countable Topos proposition I.P27 Paraconsistent Character proposition I.P28 Self-Enrichment proposition I.P29 Passage to Book II proposition I.P33 Counting as Structural Feature proposition I.P34 No Unearned Decimal Diagonal proposition I.P35 No Unrestricted Comprehension proposition I.P36 No Free Cartesian Diagonal proposition I.P37 Ultrametric Replaces Cardinality proposition I.P38 Classical.em Eliminability proposition I.P39 TauReal Ring Axioms proposition I.P40 Extensionality proposition I.P41 Self-Containment Partition proposition I.P42 Order Bound proposition I.P43 Measure Compatibility proposition I.P44 CRT-Galois Decomposition proposition I.P45 Monotone Convergence proposition I.P46 No Second Linearity proposition I.P47 Bayesian Factorization proposition I.R01 Axiom Independence remark I.R15 Structural Rules Inventory remark I.R16 Linear Logic Glossary remark I.R17 Gap Declaration remark I.R18 Proof-Theoretic Landscape Survey remark I.R19 Barr's Star-Autonomous Categories remark I.R20 E0-E3 Literature Map remark I.R21 Scale Declaration remark I.R22 Hurwitz Classification Preview remark I.R23 Galois Theory Preview remark I.R24 Five Reasons Why The Bug Hides remark I.R25 Orthodox Foundations Under the Lens remark I.R26 Implications for Absolute Meaning remark I.R27 Honest Scope Declaration remark I.R28 Inseparability of N and omega remark I.R29 Finite-Infinite Boundary remark I.R30 Duality and Atoms remark 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.R40 Pairing Density remark I.R41 PolStream Aperiodicity remark I.R42 Lemniscate Lipschitz Forward Reference remark I.R43 Quaternion Open Question remark I.R44 Lean Bayesian Status remark I.R45 Distributivity from Combinators remark I.T01 Ontic Closure theorem I.T02 Iterator Ladder Saturation theorem I.T03 Composition Associativity theorem I.T04 Hyperfactorization Theorem theorem I.T05 Prime Polarity Theorem theorem I.T06 Global Hartogs theorem I.T07 Rigidity of tau theorem I.T08 Categoricity of tau_0 theorem I.T09 FTA on tau-Idx theorem I.T10 Split-Complex Forced theorem 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.T12 Spectral Decomposition theorem I.T13 Explosion Barrier theorem I.T14 No Additive Inverse theorem I.T15 No Ring Negation theorem I.T16 Mul Cancel Fails at Zero theorem I.T17 Multiplicative Cancellation iff Positive theorem I.T18 CRT Coherence Constraint theorem I.T19 Diagonal-Free Protection theorem I.T20 Composition Closure theorem I.T21 Tau-Identity Theorem theorem I.T22 Category Axioms theorem I.T23 Yoneda Lemma theorem I.T24 Grothendieck Topos theorem I.T25 Omega_tau Subobject Classifier theorem I.T26 Product Universal Property theorem I.T27 Distributivity theorem I.T28 Cartesian Closed theorem I.T29 Spectral Determination theorem I.T30 Removable Singularity theorem I.T31 Global Hartogs Extension theorem I.T35 Cantor Diagonal Inapplicability theorem I.T36 Unique Infinity Object theorem I.T37 Diagonal-Linear Correspondence theorem I.T38 Linearity Census theorem I.T39 K5 Structural Exclusion theorem I.T40 Presheaf Characterization theorem I.T41 Bi-Square Characterization theorem I.T42 Archimedean Property theorem I.T43 TauComplex Ring Axioms theorem I.T44 Quaternion Non-Commutativity theorem I.T45 Roots of Unity CRT Decomposition theorem I.T46 Ontic Identity Invariance theorem I.T47 Slippage Breaks Unique Omega theorem I.T48 Structural Instability Theorem theorem I.T49 Countable Additivity theorem I.T50 Fundamental Theorem of Internal Galois Theory theorem I.T51 Linearity of Integration theorem I.T53 Canonical Normalisation (Hinge 7) theorem I.T54 NF Confluence / Church-Rosser for tau-kernel (Hinge 7) theorem I.T55 Genealogical DAG Structure (Hinge 7) theorem I.T56 Cayley Word Metric (Hinge 7) theorem I.T57 Ontic Ultrametric (Hinge 7) theorem I.T58 Address Resolution and Absence of Equations (Hinge 7) theorem I.T59 Hinge 7 Integration Tabulation (Bundle Closure) theorem I.T60 Ontic Identity Invariance (Hinge 8) theorem I.X01 The Generative Act construction I.X02 Earned Category and Topos construction