{
  "schema_version": 1,
  "problem_number": "AMR-029-0040",
  "title": "Arithmetic Periodicity of Gonality under Regular Subdivision",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For each fixed finite connected loopless multigraph G and degree d, we give an effective finite union of finitely generated additive submonoids describing all positive k for which the regular k-subdivision has divisorial gonality at most d. Consequently the gonality sequence eventually depends only on gcd(k,M) for a computable M. We also prove that every tricycle in the family of van Dobben de Bruyn, Smit and van der Wegen has gonality five for even k and six for odd k; midpoint necessity yields the all-parameter formula. Their rational rescaling method, tricycle construction, degree-five reduced-divisor restriction and known one-to-two subdivision drop are explicitly credited, together with prior integer-programming certificates. The paper includes complete written proofs and portable exact finite checks. It does not claim an efficient algorithm, small period bound, resolution of odd-fold drops on other graphs, or exhaustive closure of the broad Egres question AMR-029-0040. AI-assisted self-audited preprint; no independent review, formal verification or absolute-priority certification is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO"
  ],
  "keywords": [
    "AMR-029-0040",
    "divisorial gonality",
    "regular subdivision",
    "chip firing",
    "numerical semigroup",
    "eventual periodicity",
    "tricycle"
  ],
  "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/amr-029-0040/",
  "pdf_url": "https://eulersolve.org/papers/amr-029-0040/paper.pdf?v=8db1cb76864c",
  "doi": "10.5281/zenodo.23646761",
  "zenodo_record_url": "https://zenodo.org/records/23646761",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For each fixed finite connected loopless multigraph G and degree d, all positive regular subdivision parameters with divisorial gonality at most d form an effective finite union of finitely generated additive submonoids. Consequently the gonality sequence eventually depends only on gcd(k,M) for a computable M. For every DSW tricycle it is exactly 5 for even k and 6 for odd k, with a necessary midpoint form for degree-five reduced divisors. The theorem establishes effective finite unions of additive monoids and eventual gcd dependence for a fixed graph, and exact parity for the known DSW tricycle family. The DSW rational rescaling argument, tricycle construction and reduced-divisor restriction, and prior integer-programming certificates are credited. It does not settle possible odd-fold drops on other graphs or close the broad source question. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
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    "source.zip": {
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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."
}