A Note on the CRT Coverage Maximality Conjecture: Exact Baseline Formula and 4-Prime Local Obstruction

Let L be an odd integer and choose one residue class Cd (mod d) for every divisor d | L with d > 1. A 2026 preprint proposes that the size of the union of these classes is bounded by an inclusion–exclusion baseline formed from families of pairwise-coprime moduli. The general statement is called CRT Coverage Maximality in that preprint. This memorandum separates what can be proved from what remains open. For squarefree L = p1 · · · pk, we derive an exact all-order formula for the baseline using signed sums of Stirling numbers of the second kind. We then give a self-contained proof of the conjectured inequality for k ≤ 3. For k = 4, an explicit local configuration shows why the same point-by-point argument cannot simply be extended; it is not a counterexample to the global conjecture. We also identify an ambiguity between the source’s third-order Bonferroni expression and its later use of the same symbol for the all-order coprime sum. Finally, a deterministic, reproducible coordinate-ascent search for six prime factors finds no violation, but is emphatically not exhaustive. The general conjecture is left unresolved here. Note: This research note audits, human-verifies, and extends an AI-assisted preprint from April 2026, establishing exact formulations, mapping local structural barriers, and providing reproducible programmatic testbeds for future global proof strategies.