{
  "schema_version": 1,
  "problem_number": "AMR-021-0014",
  "title": "A Positive-Coefficient Counterexample to the Weighted Forsgård–Shapiro Parity Bound",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Forsgård and Shapiro proposed bounding the number of real zeros of a positive-coefficient polynomial by the parity changes among all indices at which (k+1)a_k^2 - k a_(k-1)a_(k+1) is positive. We give a counterexample with positive rational coefficients, parity count one, and at least three distinct negative real zeros. An even-degree shift of a degree-77 reciprocal-block seed makes the undesired weighted local quantities negative. A sufficiently small, strictly log-convex factorial prefix fills every missing coefficient without introducing a parity change or destroying the three roots. The explicit witness has degree 1,000,077; no optimal-degree claim is made. Its coefficients have a compact exact formula.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "keywords": [
    "AMR-021-0014"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-021-0014/",
  "pdf_url": "https://eulersolve.org/papers/amr-021-0014/paper.pdf?v=6efdd97d8102",
  "doi": "10.5281/zenodo.23051793",
  "zenodo_record_url": "https://zenodo.org/records/23051793",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete counterexample to final-journal Shapiro2015 Section7 Conjecture9. The previously public degree77 unweighted seed is credited. No minimum-degree, exact-total-root-count, human-review, formal-certificate or absolute-priority claim is made. Self-audited and unrefereed, with AI assistance; no independent human peer review or absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "6efdd97d81029adb0bd38b0745020e8358fb08583417fbb92acc32c33fddc34b"
    },
    "source.zip": {
      "sha256": "f720c65f7da453f03f780145a864c0c0ac132653f71148b4bc152db98ab9cfde"
    },
    "verification_report.md": {
      "sha256": "5ec74a77fc36623bd579e6214ad0d29ce557a9096dde61d465aee9873ee4f89a"
    }
  },
  "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."
}