{
  "schema_version": 1,
  "problem_number": "AMR-021-0015",
  "title": "A Reciprocal-Block Counterexample to the Forsgård–Shapiro Coefficient-Parity Bound",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "The coefficient-parity conjecture of Forsgård and Shapiro uses the indices at which a_k^2 - a_{k-1}a_{k+1} is nonnegative to bound the number of real zeros of a polynomial with positive coefficients. We give an explicit degree-77 polynomial with positive rational coefficients for which the selected indices change parity only once, but which has at least three distinct negative real zeros. The construction joins a degree-38 block to its reciprocal. A weighted sum-of-squares identity proves the required root crossing; all coefficients and signs can also be checked by rational arithmetic.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "keywords": [
    "AMR-021-0015"
  ],
  "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-021-0015/",
  "pdf_url": "https://eulersolve.org/papers/amr-021-0015/paper.pdf?v=ed1345279b6e",
  "doi": "10.5281/zenodo.23050917",
  "zenodo_record_url": "https://zenodo.org/records/23050917",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "This is a complete counterexample to Conjecture 10 in Section 7 of Shapiro's 2015 source, the exact original unweighted coefficient-parity inequality. It does not settle neighboring Conjecture 9 or weighted variants. No minimum degree, exact total root count, absolute historical priority, human-referee approval or formal proof-assistant certificate is claimed. Self-audited and unrefereed, with AI assistance; no independent human peer review or absolute priority is claimed.",
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      "sha256": "ed1345279b6edda6bf5fe3f3a8dc1d95828a93d8bec8c4dedaf075c354b44fc7"
    },
    "source.zip": {
      "sha256": "a8a5b22fd8dc57b7ade24cbffd3e10437216d38f5cfdae55b090745c078c92d6"
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    "verification_report.md": {
      "sha256": "472b8a9ca31e7187586d054ff041edb43f68de8d1d8f3b6ae74b6e654a7a2cc1"
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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."
}