{
  "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. Katkova, Shapiro, and Vishnyakova already disproved this bound in 2024. We give an alternative 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": "1.1",
  "date_modified": "2026-09-30",
  "presentation_revision_only": true,
  "doi_archived_file_version": "1.1",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-021-0015/",
  "pdf_url": "https://eulersolve.org/papers/amr-021-0015/paper.pdf?v=a84131b90aac",
  "doi": "10.5281/zenodo.23056466",
  "zenodo_record_url": "https://zenodo.org/records/23056466",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Alternative valid explicit counterexample to Shapiro's 2015 Section 7 Conjecture 10 (unweighted coefficient parity). KSV2024 already disproved both coefficient-parity conjectures, including their distinct-root interpretations. This manuscript is not a new whole-problem closure or the first disproof. No construction-priority, minimum-degree, exact total root-count, human-review or formal-certificate claim is made. AI-assisted, self-audited and unrefereed; no independent human peer review is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "a84131b90aac04b3937b2a990c44bd3db9f1f8743543b7e80a5582b6c93d416b",
      "bytes": 72247
    },
    "source.zip": {
      "sha256": "6aac09af5adedae28ccdd82c651fd285ebb0a2678fe34ed11d9577cddf1344f1",
      "bytes": 102865
    },
    "verification_report.md": {
      "sha256": "da3ae47ad0eec77281c6d50725c9bea552d3b0e51864fb896c791be4c1f0aaa8",
      "bytes": 1189
    }
  },
  "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.",
  "concept_doi": "10.5281/zenodo.23050916",
  "concept_url": "https://doi.org/10.5281/zenodo.23050916",
  "revision_published_at": "2026-09-30T09:18:37.771366+00:00",
  "revision_note": "The initial version missed direct prior art: Katkova, Shapiro and Vishnyakova, In search of Newton-type inequalities (KSV2024), Section 5, already disproved both the weighted and unweighted coefficient-parity conjectures in 2024. Their counterexamples also imply failure when distinct roots are counted. The present mathematical construction remains valid as an alternative construction; it is not a first resolution of a previously unsolved conjecture.",
  "prior_art_doi": "10.1016/j.jmaa.2024.128349",
  "prior_art_correction": true,
  "new_result": false,
  "new_manuscripts_added": 0,
  "new_problem_closures_added": 0,
  "novelty_confidence": "DIRECT_PRIOR_RESOLUTION_KSV2024",
  "review_disclosure": "Internal AI-assisted checks only; unrefereed preprint, no independent human peer review claimed.",
  "version_history": [
    {
      "version": "1.0",
      "doi": "10.5281/zenodo.23050917",
      "zenodo_record_url": "https://zenodo.org/records/23050917",
      "files": {
        "paper.pdf": {
          "sha256": "ed1345279b6edda6bf5fe3f3a8dc1d95828a93d8bec8c4dedaf075c354b44fc7"
        },
        "source.zip": {
          "sha256": "a8a5b22fd8dc57b7ade24cbffd3e10437216d38f5cfdae55b090745c078c92d6"
        },
        "verification_report.md": {
          "sha256": "472b8a9ca31e7187586d054ff041edb43f68de8d1d8f3b6ae74b6e654a7a2cc1"
        }
      },
      "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.",
      "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.",
      "public_archive_preserved": true,
      "prior_art_assessment_superseded": true
    }
  ]
}