Finite Recurrence Geometry from Coprime Periodic Structure: CRT Atomicity, Weyl Quantization, Discrete Differential Geometry, and Exact No-Go Boundaries

This paper develops the finite-recurrence sector of the Canvas programme as a self-contained mathematical construction. Starting from four pairwise-coprime periods 2,3,5,7, the joint recurrence is the cyclic group \mathbb{Z}_{210}, whose Chinese-remainder decomposition canonically resolves four prime-primary factors. The paper proves that the associated Hilbert space and full matrix algebra factorize as \mathbb{C}^{210} \cong \mathbb{C}^2 \otimes \mathbb{C}^3 \otimes \mathbb{C}^5 \otimes \mathbb{C}^7 and M_{210}(\mathbb{C}) \cong M_2(\mathbb{C}) \otimes M_3(\mathbb{C}) \otimes M_5(\mathbb{C}) \otimes M_7(\mathbb{C}). What the Paper Does The paper builds a complete finite noncommutative geometry from the coprime recurrence structure, with exact derivations and explicit provenance boundaries: · CRT Factorization and Atomicity: The Chinese remainder theorem gives a canonical decomposition of \mathbb{Z}_{210} into four prime-primary components. Singleton CRT supports are characterized as the atoms of the independent-update composition algebra, providing a natural first-order tangent calculus with exactly four primitive directions.· Weyl Pairs and Matrix Algebra Factorization: On every prime factor, the canonical shift-clock Weyl pair generates the full matrix algebra M_p(\mathbb{C}). The finite Weyl twirl is exactly the normalized-trace conditional expectation, and its Dirichlet Hessian is the traceless projector. The paper proves the finite depolarizing twirl identity and the Weyl-Dirichlet intertwining theorem.· Discrete Differential Geometry: The paper constructs a first-order finite-difference calculus with a nilpotent exterior derivative, derives gauge-covariant discrete curvature and the Bianchi identity, and constructs a positive Yang–Mills-type action. The exterior derivative is nilpotent because the coordinate shifts commute—a consequence of the CRT factorization.· Metric and Hodge Structure: A normalized Hilbert–Schmidt pullback of recurrence displacement fixes an isotropic one-form metric within a clearly stated pullback class and yields a four-dimensional internal Hodge star. The paper proves that within this pullback class, isotropy is forced; symmetry alone only forces a diagonal metric.· Exact No-Go Boundaries: The paper proves several negative results that are as important as the positive constructions: · Global \mathbb{Z}_{210} Weyl covariance does not produce the factor-local support hierarchy; it treats all traceless directions equally. · CRT algebra alone does not choose coordinatewise dynamics. · Recurrence and order labels are not particle rest masses. A linear LCM mass law m_q = \mu q is not implied by recurrence arithmetic, CRT factorization, or additive update cost. Why This Matters This paper provides a mathematically rigorous construction of a finite noncommutative geometry from coprime periodic structure. It is part of the Canvas programme, but the mathematical architecture is self-contained: any theory with equivalent coprime recurrence, dual phase observables, and locality principles would reproduce the same structure. The provenance system labels every result: · DERIVED: Follows from stated assumptions (e.g., CRT factorization, Weyl basis theorem, nilpotency, Bianchi identity)· COMPUTED: Numerical verification of a derived identity or spectrum· CONSTITUTIVE: An input principle not forced by the preceding algebra (e.g., factor additivity, atomicity principle)· NO-GO: A result showing that a stronger claim does not follow (e.g., global twirl does not generate support hierarchy, recurrence labels are not particle masses) The paper explicitly identifies what the construction does and does not claim. It does not identify the four recurrence coordinates with physical spacetime, does not claim that the derived matrix algebra is already the Standard Model gauge algebra, and does not convert recurrence eigenvalues into particle masses. The purpose is narrower: determine exactly what follows from coprime finite recurrence plus explicitly stated locality and metric principles. Keywords: coprime recurrence, Chinese remainder theorem, CRT factorization, Weyl quantization, finite noncommutative geometry, discrete differential geometry, gauge connection, curvature, Bianchi identity, Hodge star, atomicity, no-go theorem, provenance system, Canvas programme