{
  "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. Katkova, Shapiro, and Vishnyakova already disproved this bound in 2024. We give an alternative 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": "1.1",
  "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-021-0014/",
  "pdf_url": "https://eulersolve.org/papers/amr-021-0014/paper.pdf?v=574287a6b58c",
  "doi": "10.5281/zenodo.23056319",
  "zenodo_record_url": "https://zenodo.org/records/23056319",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Alternative valid explicit counterexample to Shapiro's 2015 Section 7 Conjecture 9 (weighted 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": "574287a6b58cef23e0131f2d893a12dadaa18e8a77d3fd227128c7792a85d7a8",
      "bytes": 69486
    },
    "source.zip": {
      "sha256": "05b3259eee5790cd0c48c205d2727d3015aad6e7c385a1b5271c28b4e4f2b3e3",
      "bytes": 56374
    },
    "verification_report.md": {
      "sha256": "5216af586e16eecdbaf90b0e0f5200c3633ac4a4d89669e73d8da8ffb24a1f5b",
      "bytes": 1225
    }
  },
  "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.23051792",
  "concept_url": "https://doi.org/10.5281/zenodo.23051792",
  "revision_published_at": "2026-09-30T09:10:17.416436+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.23051793",
      "zenodo_record_url": "https://zenodo.org/records/23051793",
      "files": {
        "paper.pdf": {
          "sha256": "6efdd97d81029adb0bd38b0745020e8358fb08583417fbb92acc32c33fddc34b"
        },
        "source.zip": {
          "sha256": "f720c65f7da453f03f780145a864c0c0ac132653f71148b4bc152db98ab9cfde"
        },
        "verification_report.md": {
          "sha256": "5ec74a77fc36623bd579e6214ad0d29ce557a9096dde61d465aee9873ee4f89a"
        }
      },
      "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.",
      "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.",
      "public_archive_preserved": true,
      "prior_art_assessment_superseded": true
    }
  ]
}