Chapter 15: The CRT Decomposition Theorem

the relevant chapter established the primorial ladder Prim(k) = p₁ ⋯ p_k as the canonical cofinal filtration of the τ kernel. Every τ-effective property reduces to finitely many primorial-level checks. But a check at level k is a computation in the ring ℤ / Prim(k)ℤ, whose order grows super-exponentially with k. The Chinese Remainder Theorem (CRT) decomposes this single large ring into a product of small prime-level rings,

turning one hard computation into k easy ones. The classical CRT proof relies on the extended Euclidean algorithm, which requires subtraction—a signed operation that the τ kernel does not import (K3 provides divisibility predicates, not signed arithmetic). This chapter proves the CRT constructively using modular B'ezout coefficients obtained from divisibility alone. We then read the CRT as the algebraic Euler product: the endomorphism ring of the primorial presheaf decomposes prime by prime. The Reconstruction Functor assembles local prime-level data into global primorial data via an equivalence of module categories. All Parts IV–VI arguments decompose through this functor.