{
  "schema_version": 1,
  "problem_number": "AIM-COMBINATORICS-0209",
  "title": "A Finite Gap Bound for Local Progression-Free Density",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Fix s>=3 and consider increasing integer sequences whose every s consecutive terms contain no nontrivial three-term arithmetic progression. We show that every admissible gap sequence can be decreased coordinatewise to an admissible sequence with all gaps at most B_s=1+2 binomial(s,3). Consequently the unrestricted maximum density is the reciprocal of the minimum cycle mean in an explicitly defined finite graph. In particular, it is rational, is computable for each s, and is attained periodically with integer period at most B_s^(s-1). A nonnegative defect derived from this graph characterizes all extremizers, including those with unbounded gaps, for upper, lower, natural and upper Banach density. A short analytic potential proves the sharp value 4/9 for 5<=s<=8.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "keywords": [
    "local progression-free sequence",
    "extremal density",
    "finite gap bound",
    "minimum cycle mean",
    "periodic optimization",
    "additive combinatorics",
    "AIM-COMBINATORICS-0209",
    "math.CO",
    "math.NT"
  ],
  "manuscript_version_date": "2026-09-29",
  "publication_date": "2026-09-29",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-29",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-combinatorics-0209/",
  "pdf_url": "https://eulersolve.org/papers/aim-combinatorics-0209/paper.pdf?v=9cac4845fa90",
  "doi": "10.5281/zenodo.23032134",
  "zenodo_record_url": "https://zenodo.org/records/23032134",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Effective algorithmic characterization for each s and exact extremizer criteria; not a short closed formula or sharp asymptotic. Self-audited and unrefereed, with AI assistance; no independent peer review or absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "9cac4845fa9052cfe2b62d35f76d3f0aed1c95ec6a503a053ed8365fb27d6b3e"
    },
    "source.zip": {
      "sha256": "ca89c8ddc034e2185116d37aa38ba21592de8b724fec0669ac24ad5bc595405a"
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    "verification_report.md": {
      "sha256": "af90995fda0fa62ca88368e102367231c8931d6243258870fb59ead47fc68aa3"
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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."
}