The Schrijver-Seymour Bound for Four Weight Classes

Let a finite matroid be labeled by an arbitrary abelian group, and let S be the set of its basis-weight sums. We prove the Schrijver-Seymour stabilizer bound whenever the nonloop labels occupy at most four cosets modulo the stabilizer of S. There is no restriction on the rank, ground-set size, or ambient group. The proof uses the classical four-variable submodular cone: its primitive generators have zero-one elemental slacks, permitting an integer decomposition by subtraction inside the minimal face. Rado's theorem transfers this decomposition to an exact sumset identity. Nine atomic types have immediate bounds; the Vamos type reduces to a four-element difference set, and Kneser's theorem completes the argument. The result also extends to qualifying direct sums. The unrestricted Schrijver-Seymour conjecture (OPG-369) remains unresolved. This is a six-page unrefereed preprint; classical classification and theorem inputs are credited, and no absolute priority or independent-review claim is made. The accompanying source package includes exact cone certificates and reproducible finite regressions that supplement, rather than replace, the written proof.