On the CRT Coverage Maximality Conjecture and the Erdős–Selfridge Problem Through Rank Seven

This research memorandum establishes the exact algebraic boundaries of the Chinese Remainder Theorem (CRT) Coverage Maximality Conjecture and derives its structural consequences for the Erdős–Selfridge odd distinct covering problem. By mapping complete-divisor residue assignments onto a multivariate support polynomial formulation \(Z_m(t)\), we provide a rigorous, non-computational proof of the complete-divisor maximality inequality \(U(L) \le B_{\text{full}}(L)\) for all finite moduli supported on at most seven distinct odd prime factors (\(k \le 7\)). The proof establishes coordinate-wise monotonicity across the admissible prime-reciprocal box, optimizing independent-set matching criteria via Shearer’s dependency-graph theorem. Conversely, we present an unconditional probabilistic construction demonstrating that this complete-divisor benchmark fails at arbitrarily large ranks. Using an exponential generating function identity for random residue choices, we show that \(U(L_M) > B_{\text{full}}(L_M)\) holds for finite moduli of unbounded prime-support rank, proving that the conjecture cannot serve as an all-rank vehicle for settling the Erdős–Selfridge problem. Finally, the memorandum synthesizes these bounds to establish several explicit, necessary constraints on hypothetical odd covering systems, including a four-support completion construction, deletion inequalities utilizing recent rank-eight and rank-nine density thresholds, multi-block tail bounds, and a fibrewise \(p\)-adic prefix-code partition condition.