{
  "schema_version": 1,
  "problem_number": "OWR-2090-008",
  "title": "Sharp Coefficient Ratios and Optimal Constants for Dilated Ehrhart Series",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For every dimension d and integer dilation n >= d+1, the ratios of the coefficients of a dilated Ehrhart h-star polynomial to its last coefficient are maximized simultaneously by the unimodular simplex. The same sharp inequality holds for every nonnegative numerator of degree at most d under the Veronese operator. Combined with known real-rootedness and alternating-inequality results, this gives explicit finite rational-polynomial formulas for every admissible pair of coefficient multiplier and dilation threshold in the Beck-Stapledon theorem and its Ehrhart corollary. The smallest full-chain threshold is d+2 and the smallest eventual integer multiplier is one more than the largest Eulerian number of order d; these two minima need not occur together. Exact polynomial certificates give the least thresholds at that multiplier through dimension eight. The general-numerator root/ratio-only version and the one-dimensional edge case of the original Oberwolfach formulation are treated separately. Previously established root and alternating-inequality bounds, and the prior sufficiency of the Eulerian multiplier, are explicitly credited. The main contributions are the sharp ratio extremality, necessity of the multiplier, and finite optimization of their full tradeoff. AI-assisted and self-audited preprint; no independent review, formal verification, or absolute priority certification is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.AC",
    "math.CA"
  ],
  "keywords": [
    "OWR-2090-008",
    "Ehrhart polynomial",
    "Veronese operator",
    "coefficient ratio",
    "Eulerian number",
    "optimal constants"
  ],
  "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-2090-008/",
  "pdf_url": "https://eulersolve.org/papers/owr-2090-008/paper.pdf?v=6673171f71d3",
  "doi": "10.5281/zenodo.23640629",
  "zenodo_record_url": "https://zenodo.org/records/23640629",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Sharp simultaneous coefficient-ratio maxima for every dimension d and integer dilation n>=d+1 are attained by the unimodular simplex. Explicit finite rational-polynomial formulas determine every admissible multiplier/threshold pair for the complete Beck-Stapledon Ehrhart conclusions, and separately for the general-numerator root/ratio conclusions. The smallest full-chain threshold is d+2 and the smallest eventual integer multiplier is 1+max Eulerian(d,i); their tradeoff is computed exactly through dimension8. The literal one-dimensional strict-hypothesis edge case in the OWR theorem is explicitly separated. The sharp simultaneous coefficient ratios are attained by the unimodular simplex in every dimension. Explicit finite formulas determine all admissible multiplier/threshold pairs, with separate treatment of the root/ratio-only version and the original dimension-one edge case. Previously known real-rootedness and alternating inequalities are credited. The characterization is finite and exact; no efficiency or short-closed-form claim is made. 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": "6673171f71d328c6e3e16043a26b997110d6ae105ceb127d370e0f439e15c96e"
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    "source.zip": {
      "sha256": "b9bfe022f465ad0f7daf20c559a15170b27e48dc8f026137efd267c38637dc3c"
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    "verification_report.md": {
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  },
  "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."
}