A Shearer-Polynomial Approach to CRT Coverage Maximality for Five Prime Factors

This repository contains the research memorandum and verification scripts establishing an all-orders coprime inclusion-exclusion bound for Chinese Remainder Theorem (CRT) coverage problems with up to five distinct odd prime factors (\(\omega(L) \le 5\)). By mapping divisor intersections onto a dependency graph and utilizing Shearer's Lemma, the methodology bypasses traditional truncation errors through a novel monotonicity identity on a support-indexed matching polynomial. The partial derivatives of the rank-5 matching polynomial collapse into lower-rank domains (\(\le 4\)) that are proven to be strictly positive, guaranteeing coordinate-wise monotonicity across the parameter box and ensuring that the system baseline serves as a valid global lower bound for every induced subgraph. The provided text and exact arithmetic verification scripts confirm a strict bound of \(U(L) \le 12062\) for the squarefree test case \(L = 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13\), providing an airtight, non-constructive combinatorial solution to a classic number-theoretic bottleneck.