{
  "schema_version": 1,
  "problem_number": "OWR-14299288-010",
  "title": "A Sharp Sampling Bound for Primitive Real Toric Arrangements",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let A be a finite spanning collection of primitive integral normals in dimension n, and let E_A be the least common multiple of the largest Smith invariant factors of its nonsingular full-rank submatrices. If E_A > 1, every grid of denominator m with E_A dividing m and m >= n E_A meets every ordinary stratum of the associated real toric arrangement. The multiplier n is sharp in every dimension n >= 2 for every prescribed E_A >= 2. For arbitrary integral normals, the successful denominators are exactly E_A times the integers at least the maximum codegree of the scaled stratum closures. The proof combines the interior-lattice-point framework of Hanlon and Painter with the standard Smith lcm period; primitivity excludes unimodular simplex closures through divisibility of facet constants. A sufficient bound for realizing all Bondal-Thomsen summands follows for smooth toric varieties with no torus factor.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.AG"
  ],
  "keywords": [
    "real toric arrangement",
    "rational sampling",
    "Smith invariant factor",
    "codegree",
    "Bondal-Thomsen collection",
    "OWR-14299288-010"
  ],
  "manuscript_version_date": "2026-10-19",
  "publication_date": "2026-10-19",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-19",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-14299288-010/",
  "pdf_url": "https://eulersolve.org/papers/owr-14299288-010/paper.pdf?v=b75641dd2770",
  "doi": "10.5281/zenodo.23484274",
  "zenodo_record_url": "https://zenodo.org/records/23484274",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "A complete proof of the stated sampling theorem, not a full resolution of the broad Bondal-Thomsen polytope question associated with OWR-14299288-010. The necessary denominator and exact codegree threshold concern ordinary strata, not the coarser line-bundle labels. The toric-Frobenius corollary is restricted to smooth toric varieties without a torus factor. Hanlon-Painter's framework and the established Smith lcm period are credited. AI-assisted, self-audited and unrefereed; no independent peer review, formal certification, dataset-maintainer acceptance or absolute priority is claimed.",
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    "paper.pdf": {
      "sha256": "b75641dd2770f48ba8809ed7345bcac717f3e3d3645f693b9e397b053713dfd5"
    },
    "source.zip": {
      "sha256": "e025caf90e2aca6089741a148da5a672592f8be4ae3f58a1298023d3b3274506"
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    "verification_report.md": {
      "sha256": "5930441cdaa419e030aabe13fa50f59cc36d587fc061eecdb23923c670274ede"
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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."
}