{
  "schema_version": 1,
  "problem_number": "OWR-15587-015",
  "title": "Smooth 34-Dimensional IDP Polytopes with Nonunimodal h*-Polynomials",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Ferroni constructed smooth 40-dimensional lattice polytopes with the integer decomposition property whose h-star polynomials are not unimodal. This paper refines the parameters in his Cayley-box construction to obtain infinitely many pairwise lattice-inequivalent examples in dimension 34. For every integer t >= 1, the matrix [100000t*ones_17 | 1310000t*I_17] + J_(17x18) defines a smooth IDP polytope with h-star degree 17 and h15 > h16 < h17. Exact positivity of all coefficients of two gap polynomials in t-1 proves the inequalities uniformly. A standalone verifier independently reconstructs the polynomials by rational interpolation of direct interior counts and checks forward h-vectors. The associated Ehrhart rings are standard graded normal Cohen--Macaulay domains of dimension 35 and are not Gorenstein. This is a dimension refinement of Ferroni's prior counterexamples, not a first disproof of unimodality or a minimal-dimension claim. The general standard graded Gorenstein-domain unimodality question in OWR-15587-015 remains outside the result. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority certification is claimed.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "keywords": [
    "OWR-15587-015",
    "Ehrhart polynomial",
    "IDP polytope",
    "unimodality",
    "smooth lattice polytope",
    "dimension refinement"
  ],
  "manuscript_version_date": "2026-10-25",
  "publication_date": "2026-10-25",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-25",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-15587-015/",
  "pdf_url": "https://eulersolve.org/papers/owr-15587-015/paper.pdf?v=bf1bdcfb044a",
  "doi": "10.5281/zenodo.23638812",
  "zenodo_record_url": "https://zenodo.org/records/23638812",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every integer t>=1, the Cayley-box polytope with matrix [100000t*ones_17 | 1310000t*I_17]+J_(17x18) is smooth, IDP and 34-dimensional, with h-star degree 17 and h15>h16<h17. The fixed-dimensional family is pairwise lattice-inequivalent. Its Ehrhart rings are non-Gorenstein standard graded normal Cohen--Macaulay domains of Krull dimension 35. The five-page paper refines Ferroni's smooth 40-dimensional Cayley-box counterexample to an infinite smooth 34-dimensional IDP family. A strict terminal h-star trough is proved for every integer t>=1 by 36 positive coefficients and independent exact checks. This is not a first disproof, a minimal-dimension theorem, or a solution of general Gorenstein-domain unimodality. Zero new source-subquestion or whole-record closures; the broad frozen source stays PARTIALLY_SOLVED. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "bf1bdcfb044ae425c9d00ca460e8bccd08e5bd91a46d8257272aa32b3169c225"
    },
    "source.zip": {
      "sha256": "afec1bb210fbec80e0faee42ac3ee30b9ea53f3e2f607d54e8acf85a66dcbe75"
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    "verification_report.md": {
      "sha256": "c100cb70a02f5b78e5454dfa87b75ce54e278fcfa5387ede84418219ec4c7276"
    }
  },
  "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."
}