Results Glossary Entry Canonical mathematics 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 systems, with every τ-internal datum recoverable from its restri…
Results · Mathematics Glossary · Theorem MathG-T08-crt-coherence CRT-coh Canonical

CRT Coherence Constraint

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 systems, with every τ-internal datum recoverable from its restrictions modulo coprime ideals. With 29 incoming edges, it underwrites the framework's modular-arithmetic substrate including Book III's CRT Decomposition Theorem (T10).

Mathematics Glossary Primary: I.T18

τ-Definition

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 systems, with every τ-internal datum recoverable from its restrictions modulo coprime ideals. With 29 incoming edges, it underwrites the framework's modular-arithmetic substrate including Book III's CRT Decomposition Theorem (T10).

Categorical invariant. A coherence isomorphism Tau ≅ ⊕_p Tau_p over coprime prime indexes, with restriction-glue structure compatible with K3 composition.

Primary registry anchor: I.T18

Supporting items: I.T05, I.K1

τ-Derivation Chain

  1. I.K0 — Universe Postulate
  2. I.T05 — Prime Polarity — primes carry polarity structure
  3. I.T18 — CRT Coherence Constraint — τ decomposes coherently across coprime factors

Lean modules referenced: TauLib.BookI.Polarity.ChineseRemainder, TauLib.BookI.Polarity.CRTBasis

Mathematical content

Theorem CRT-coh
Theorem

For every finite collection of pairwise coprime kernel atoms {p_1, …, p_k}, the τ-categorical kernel decomposes as τ ≅ ⊕_i τ_{p_i}, with the gluing structure determined by K3 composition. Equivalently, the canonical restriction map τ → Π_i τ_{p_i} is an isomorphism onto the coherent sub-product.

Proof sketch (expand)

Direct categorical analogue of CRT: Prime Polarity (T06) shows polarity is multiplicative across coprimes; K3 (composition) makes the restriction maps mutual-inverses on coherent inputs; K6 (no-ω) closes off ω-shift extensions that would break the iso. The proof essentially mirrors classical CRT but on τ-categorical kernel atoms rather than ℤ/n.

Consequences:

  • CRT Basis (I.D? — supporting) — modular-coordinate basis on τ.
  • Book III CRT Decomposition Theorem (T10 / III.T10) generalizes T08 to spectral data.
  • Every τ-categorical computation reduces to coherent prime-component computations (the framework's modular substrate).

Lean Coverage

Status: Formalized

Module: TauLib.BookI.Polarity.ChineseRemainder

Lean kind: theorem

Lean symbol: Tau.BookI.Polarity.crtCoherence

Cross-domain bridges

This glossary term sits on the boundary between domains. The τ-framework's cross-domain pivots are the structural junctions where physics, life, and metaphysics readouts meet.

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