{
  "schema_version": 1,
  "problem_number": "OWR-12339-005",
  "title": "Unit-Coefficient Simplex Decompositions under Weak Nondegeneracy",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let P be a finite union of compact convex polytopes in real d-space, and let S contain its algebraic vertices. Suppose that every d+2 points of S affinely span the ambient space. We prove that the unit-density measure on P is a sum of distinct simplex measures with coefficients +1 or -1 and all simplex vertices in S. All simplices may share any prescribed point of S as a vertex. The proof combines the signed-section and coning results of Akopyan, Barany and Robins with an elementary bounded-coefficient lemma for a configuration having at most one affine dependence. This answers the coefficient question in the weakly nondegenerate setting of Pasechnik's 2013 Oberwolfach report; it does not resolve the later conjecture for arbitrary configurations. A more general bound by the largest facet jump holds for integer-valued polyhedral functions. Exact regression scripts accompany the written proof. This AI-assisted preprint is self-audited and unrefereed; independent review, formal certification and exhaustive priority determination are not claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.MG",
    "math.CO"
  ],
  "keywords": [
    "generalized polytope",
    "signed simplex decomposition",
    "weak nondegeneracy",
    "affine circuit",
    "unit coefficients",
    "OWR-12339-005"
  ],
  "manuscript_version_date": "2026-10-18",
  "publication_date": "2026-10-18",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-18",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-12339-005/",
  "pdf_url": "https://eulersolve.org/papers/owr-12339-005/paper.pdf?v=08d14fd0316d",
  "doi": "10.5281/zenodo.23456421",
  "zenodo_record_url": "https://zenodo.org/records/23456421",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The coefficient-existence question is answered under the weak nondegeneracy assumption of the 2013 Oberwolfach report. The later arbitrary-configuration conjecture is not resolved. Known integer decomposition, signed-section and coning theorems are credited. AI-assisted, self-audited and unrefereed; no independent review, formal certification or absolute-priority claim.",
  "files": {
    "paper.pdf": {
      "sha256": "08d14fd0316d098982f7d5b308d3c6934cb8a61966df4171904cabfcd06b55d9"
    },
    "source.zip": {
      "sha256": "6dbb164945fe24f7db7ebbf35bc7a081ec496b7d2adb76ded2d24ba837449c65"
    },
    "verification_report.md": {
      "sha256": "33ff6c6c5226941d2b321a98a159c4d6b3e1a9b48009c9a4cc9a5bd0cac1cf10"
    }
  },
  "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."
}