CRT Decomposition Theorem
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 places, with every τ-spectral datum recoverable from its place-wise restrictions. With 20 incoming edges, it underwrites the adelic structure that Grand GRH (A03) operates on.
τ-Definition
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 places, with every τ-spectral datum recoverable from its place-wise restrictions. With 20 incoming edges, it underwrites the adelic structure that Grand GRH (A03) operates on.
Categorical invariant. An adelic decomposition isomorphism τ-Spec ≅ ⊗_v τ-Spec_v over places v of a number field, restricted to coprime adelic data.
Primary registry anchor:
III.T10
τ-Derivation Chain
Mathematical content
For every τ-spectral datum S and every finite collection of pairwise coprime adelic places {v_1, …, v_k}, S decomposes as S ≅ ⊗_i S_{v_i} with restriction-glue structure compatible with the spectrum functor (S01) and Prime Polarity (T06).
Proof sketch (expand)
Apply the Book I CRT Coherence Constraint (T08) to the restriction τ-Spec → τ over each place; use the spectrum functor's preservation of K3 composition to glue the local pieces back coherently; close under K6 (no-ω) at the adelic level.
Consequences:
- Adelic structure for τ-spectral data (substrate of Grand GRH adelic axiom A03).
- Place-wise computation of τ-spectral invariants.
- Bridge to Mathlib's `Mathlib.NumberTheory.LSeries.Adelic` infrastructure.
Lean Coverage
Status: Formalized
Module: TauLib.BookIII.Spectral.CRT
Lean kind: theorem
Lean symbol: Tau.BookIII.Spectral.crtDecomposition