An exact census of prime-detecting MacMahon combinations

A self-contained, certified-computation artifact. The ℚn-module of prime-vanishing MacMahon combinations ∑ Pk(n)Mk(n) has exactly five primitive generators within (K ≤ 8, deg ≤ 10, validation n ≤ 3000) — top levels 2, 3, 4, 5, 7, with none at level 6 — and generates the same module as Craig–van Ittersum–Ono Table 1. Proven rather than computed: the dimension law dim V(K,D) = (K−1)(D−1) for 2 ≤ K ≤ 5 at every degree, via a syzygy module over ℚx; completeness of the census for K ≤ 5; the level-6 gap with no appeal to the classification theorem; and the degree-threshold rule 2(K − K′). The cuspidal components of M6, M7, M8 are computed exactly; M8 alone carries the weight-16 cusp form ΔE4, giving P6(x) = (x−13)P7(x)/24. All arithmetic is exact (integers and rationals, no floating point). Every result file, figure and the PDF regenerate bit-identically from standard-library Python; verify.py checks the SHA-256 manifest and every certified claim in one command.