{
  "schema_version": 1,
  "problem_number": "OPG-369",
  "title": "The Schrijver-Seymour Bound for Four Weight Classes",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "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.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "keywords": [
    "matroid",
    "basis weights",
    "Schrijver-Seymour conjecture",
    "submodular cone",
    "Kneser theorem",
    "OPG-369"
  ],
  "manuscript_version_date": "2026-10-17",
  "publication_date": "2026-10-17",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-17",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/opg-369/",
  "pdf_url": "https://eulersolve.org/papers/opg-369/paper.pdf?v=7191513fc1a8",
  "doi": "10.5281/zenodo.23427507",
  "zenodo_record_url": "23427507",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The theorem covers at most four nonloop weight classes modulo the basis-sum stabilizer and the stated qualifying direct sums. The unrestricted OPG-369 conjecture remains open. Classical cone classification, Rado and Kneser inputs are credited. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute-priority certification is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "7191513fc1a860736b0c963d37aedada96efe61966b6bb38f0dfe11712a822b1"
    },
    "source.zip": {
      "sha256": "359e6759142986c755f73996d32cde67feb96a7465ae3d79b756bf7311d8d3a6"
    },
    "verification_report.md": {
      "sha256": "a2d2bb66cdea1dcd8d3f88ffc5afb17117f6c66e6a120dc86881ea370cad8c34"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}