CRT Coverage Maximality: An Analytic Four-Prime Theorem for the All-Orders Bound and an Exact Extremal Certificate at L = 1155

This research memorandum establishes an analytic proof of the all-orders CRT Coverage Maximality conjecture (\(U(L) ≤ Bfull(L)\)) for any odd integer \(L\) containing at most four distinct prime factors (\(k ≤ 4\)), with no squarefreeness constraints. By analyzing the system through prime-power tower complements, the proof establishes a global compensation inequality that successfully bypasses the local rank-four obstruction (\(g(D) = -2\)) present in previous literature. Additionally, this work provides a rigorous, constructive extremal certificate for the squarefree four-prime modulus \(L = 3 · 5 · 7 · 11 = 1155\). It demonstrates via a combinatorial box-overlap graph theorem that the exact maximum coverage is bounded at 855 elements, yielding a precise maximality gap of 4 against the all-orders coprime baseline of \(Bfull(1155) = 859\). Finally, the memorandum demonstrates structural limitations showing why the current weight-region bounding methods cease to guarantee nonnegativity at higher prime ranks (\(k ≥ 5\)).