Mathematics Glossary — 41 entries, 7 categories

The 21-entry mathematics glossary covers the τ-framework’s foundational kernel (Books I–III): postulates K0–K6, canonical definitions (ι_τ, τ-categorical, window-algebra, Yoneda-as-theorem, rank coordinates), Books I–II load-bearing theorems (Hyperfactorization, Rigidity, Categoricity, Central Theorem at rank (3, 15), Yoneda enrichment ladder), the three Book-III conjectural axioms (bridge functor, spectral correspondence O(3), Grand GRH adelic), structures (spectrum functor, holomorphy tower, self-enrichment), and the τ-object class. Each entry carries a mathematical-content section with the orthodox statement, Mathlib bridge, and the categoricity argument fixing the concept inside the kernel. Pilot sealed at v0.4.

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Axiom (3)

• Bridge functor axiom B

  The Bridge functor axiom (III.D71) is the first of three custom axioms in TauLib beyond Mathlib's trusted base. It asserts the existence of a structure-preserving functor betwee…

  MathG-A01-bridge-functor

• Spectral correspondence O(3) axiom SC_{O(3)}

  The Spectral correspondence O(3) axiom (III.T18, surfaced as A02 in this glossary) is the second custom axiom in TauLib. It asserts that the τ-spectral correspondence — the τ-in…

  MathG-A02-spectral-correspondence

• Grand GRH (adelic) axiom GRH_{adelic}

  The Grand GRH (adelic) axiom (III.D31) is the third custom axiom in TauLib. It asserts that the Prime Polarity Scaling Theorem (III.T20) extends to all automorphic L-functions i…

  MathG-A03-grand-grh-adelic

Definition (16)

• Master constant ι_τ ι_τ

  ι_τ = 2/(π + e) is the structural fixed point of the boundary holonomy algebra H_∂[ω] over the categorical kernel τ. It is a theorem about τ, not a parameter — uniquely determin…

  MathG-D01-iota-tau

• τ-categorical structure τ-cat

  The τ-categorical structure is the categorical content of the τ-kernel: τ as a category in its own right, equipped with the τ-site topology that turns it into an earned topos. T…

  MathG-D02-tau-categorical

• Window-algebra integers W_n(k) W_n(k)

  The window-algebra integers W_n(k) are the Book-II numerical invariants of the τ-categorical structure at rank coordinates (n, k). For load-bearing pairs (W₃(4) = 5, W₅(3) = 19,…

  MathG-D03-window-algebra

• Yoneda-as-theorem under self-enrichment Y

  Yoneda-as-theorem (II.D50) is the τ-framework's distinctive treatment of the Yoneda lemma: rather than appearing as an external categorical fact applied to τ, the Yoneda embeddi…

  MathG-D04-yoneda-as-theorem

• Rank coordinates (n, k) (n, k)

  Rank coordinates (n, k) are the indexing scheme used throughout Books I–II to locate τ-categorical content along two structural axes: n is the prime-rank index (which prime-numb…

  MathG-D05-rank-coordinates

• Truth4 Logic T4

  Truth4 Logic (I.D21) is the τ-internal four-valued logic — extending classical bivalence (true/false) with the structural truth values reflecting the kernel's bipolar polarity. …

  MathG-D06-truth4-logic

• 4+1 Sector Decomposition (D, A, B, C, ω)

  The 4+1 Sector Decomposition (III.D13) decomposes τ-categorical content into five canonical sectors: four 'analytic' sectors (D, A, B, C) plus the closed 'ω' sector. The 4+1 str…

  MathG-D07-4-plus-1-sector

• Five Generators (definition) {g_1,…,g_5}

  The Five Generators definition (I.D01) names the five canonical generators of the τ-kernel as a single bundled definition, complementing the K01–K06 axiomatization of MathG-K02.…

  MathG-D08-five-generators-def

• Calibrated Split-Complex Codomain ℝ[j]_cal

  The Calibrated Split-Complex Codomain (II.D35) is the split-complex algebra ℝ[j]/(j²−1) equipped with the τ-categorical calibration — a graded structure on real and j-imaginary …

  MathG-D09-calibrated-split-complex

• Split-Complex Scalars ℝ[j]

  The Split-Complex Scalars (I.D20) are the elements of ℝ[j]/(j²−1) treated as the canonical scalar algebra of the τ-framework. Every τ-categorical scalar — including ι_τ, the ran…

  MathG-D10-split-complex-scalars

• Stage-k Cylinder Cyl_k

  The Stage-k Cylinder (II.D10) is the τ-categorical analogue of the cylinder construction at depth k along the K1 strict-order direction. It packages the rank-(n, k) data into a …

  MathG-D11-stage-k-cylinder

• Progression Operator ρ ρ

  The Progression Operator ρ (I.D02) is the τ-internal advancement operator on kernel atoms — the canonical step that lifts content from one K1 strict-order position to the next. …

  MathG-D12-progression-operator

• Diagonal Discipline Δ-disc

  The Diagonal Discipline (I.D03) is the τ-internal coherence rule: every τ-categorical morphism that factors through the diagonal Δ : A → A × A must commute with K3 composition a…

  MathG-D13-diagonal-discipline

• τ-Stone Space Stone_τ

  The τ-Stone Space (II.D14) is the τ-categorical analogue of the classical Stone space — the labelled topological space underlying the K2 boundary axiom. It is the structural env…

  MathG-D14-stone-space

• Boundary Ring and Scalars B_τ

  The Boundary Ring (I.D19) is the τ-internal commutative ring on the K2 labelled boundary, with the split-complex scalars (D10) as its scalar algebra. The ring structure carries …

  MathG-D15-boundary-ring

• τ-Ultrametric Distance d_τ

  The τ-Ultrametric Distance (II.D13) is the τ-categorical metric on the boundary, satisfying the strong triangle inequality d(x,z) ≤ max(d(x,y), d(y,z)) — the ultrametric propert…

  MathG-D16-ultrametric-distance

Lemma (1)

• Idempotent Decomposition Lemma Idem-decomp

  The Idempotent Decomposition Lemma (II.L07) is the Book-II structural result that every τ-categorical content decomposes orthogonally into idempotent components. With 19 incomin…

  MathG-L01-idempotent-decomposition

Object (2)

• Generic τ-object A ∈ τ

  A generic τ-object is an inhabitant of the τ-categorical kernel — an object of the (∞, 1)-category τ that arises from finite K1–K5 composition (closed under K6). Each τ-object c…

  MathG-O01-tau-object

• Window-algebra object W_n(k)-obj

  A window-algebra object is the categorical presentation of a window in the ABCD coordinate chart at rank coordinate (n, k). It is a τ-object (O01) carrying the K1-iteration dept…

  MathG-O02-window-object

Postulate (3)

• The Universe Postulate (K0) τ

  The Universe Postulate (I.K0) is the foundational axiom of the τ-framework: there exists a small (∞, 1)-category τ — the categorical kernel — that supports the five canonical ge…

  MathG-K01-universe-postulate

• The five canonical generators (K1–K5) K1–K5

  K1–K5 are the five canonical generators of the τ-kernel — strict order, labelled boundary, composition, boundary identification, and generator closure. They are the structural a…

  MathG-K02-five-generators

• The no-ω axiom (K6) K6

  K6 is the closure axiom of the τ-kernel: there is no sixth generator. Any candidate sixth atom must reduce to a finite combination of K1–K5. K6 is what makes the τ-kernel finite…

  MathG-K03-no-omega-axiom

Structure (3)

• Spectrum functor Spec

  The spectrum functor (III.D81) is the τ-internal functor sending each τ-categorical object to its associated spectral data — the analogue of the algebraic-geometric Spec functor…

  MathG-S01-spectrum-functor

• Book I holomorphy tower Hol_τ

  The holomorphy tower (I.D96) is the Book-I structural ladder of holomorphy refinements on the τ-categorical kernel. It exhibits a graded sequence of τ-internal holomorphic objec…

  MathG-S02-holomorphy-tower

• Self-enrichment construction τ-enr

  The self-enrichment construction (II.D53) equips τ with the structure of a category enriched over itself. Hom-sets in τ become τ-internal objects rather than external Set-elemen…

  MathG-S03-self-enrichment

Theorem (13)

• Hyperfactorization theorem Hyperfact

  The Hyperfactorization theorem (I.T04) is the Book-I structural result that every τ-categorical morphism admits a unique hyperfactorization: a canonical decomposition through th…

  MathG-T01-hyperfactorization

• Rigidity of τ (non-ω) Rigid_τ

  The Rigidity theorem (I.T07) states that the τ-kernel admits no non-trivial automorphisms — every endomorphism of τ that fixes K0–K6 is the identity. Combined with the Categoric…

  MathG-T02-rigidity-non-omega

• Categoricity of τ (non-ω) Cat_τ

  The Categoricity theorem (I.T08) states that any two structures satisfying the τ-kernel axioms K0 + K1–K6 are canonically equivalent. Together with the Rigidity theorem (T02 / I…

  MathG-T03-categoricity-non-omega

• Central theorem at rank (3, 15) T_{(3,15)}

  The Central theorem at rank (3, 15) (II.T40) is the Book-II structural categoricity result that pins down the master constant ι_τ. The theorem asserts that the τ-categorical str…

  MathG-T04-central-theorem

• Yoneda enrichment ladder Y_{enrich}

  The Yoneda enrichment ladder (II.T36) is the Book-II theorem that lifts the pre-Yoneda embedding (D04 / II.D50) to a full self-enrichment of τ. The ladder exhibits a sequence of…

  MathG-T05-yoneda-enrichment

• Prime Polarity Theorem PP

  The Prime Polarity Theorem (I.T05) is the most-referenced kernel result in Books I–III: it classifies every prime p with respect to the τ-categorical kernel into a polarity sect…

  MathG-T06-prime-polarity

• Split-Complex Forced SCF

  The Split-Complex Forced theorem (I.T10) proves that the τ-kernel forces the boundary algebra to be the split-complex numbers ℝ[j]/(j²−1) — not the complex numbers ℝ[i]/(i²+1), …

  MathG-T07-split-complex-forced

• CRT Coherence Constraint CRT-coh

  The CRT Coherence Constraint (I.T18) is the τ-internal Chinese Remainder Theorem analogue: it asserts that the τ-categorical kernel decomposes coherently across coprime factor s…

  MathG-T08-crt-coherence

• Algebraic Lemniscate Lem

  The Algebraic Lemniscate (I.D18) is the canonical 1-dimensional boundary structure on the τ-categorical kernel — a real-algebraic curve with two crossing branches (the lemniscat…

  MathG-T09-algebraic-lemniscate

• CRT Decomposition Theorem CRT-decomp

  The CRT Decomposition Theorem (III.T10) generalizes the Book I CRT Coherence Constraint (T08) to spectral data: τ-spectral content decomposes coherently across coprime adelic pl…

  MathG-T10-crt-decomposition

• Mutual Determination (5-Way Equivalence) 5≡

  The Mutual Determination theorem (II.T27) is the Book II equivalence statement: five distinct presentations of τ-categorical content — boundary, interior, spectrum, lemniscate, …

  MathG-T11-mutual-determination

• Global Hartogs Extension Hartogs

  The Global Hartogs Extension theorem (I.T31) is the τ-categorical analogue of the classical Hartogs extension principle, lifted to the holomorphy tower (S02). It asserts that ev…

  MathG-T12-global-hartogs

• Spectral Trichotomy Lemma (B,I,S)

  The Spectral Trichotomy Lemma (III.T14) classifies every τ-spectral datum into one of exactly three sectors: B (boundary-type), I (interior-type), or S (singular-type). The tric…

  MathG-T13-spectral-trichotomy