{
  "schemaVersion": "1.6",
  "slug": "sharp-quartic-hadamard-powers",
  "title": "The sharp exponent for ordinary Hadamard powers of quartics with positive coefficients",
  "shortTitle": "Sharp Hadamard powers of quartics",
  "url": "https://evidencepress.org/releases/sharp-quartic-hadamard-powers/",
  "oneLine": "A written proof candidate gives the exact exponents preserving real roots for every positive-coefficient quartic, with a sharp threshold and an exclusive equality case.",
  "abstract": "Raise each ordinary coefficient of a real-rooted quartic to the same positive exponent. This anonymous, unrefereed candidate classifies the universal preservers as {1} union [p*, infinity), where p*>1 is the unique root above one of 6^p - 2*4^p + 2 = 0, approximately 1.1477203812370143654. Every output above the threshold has four distinct negative roots. At the threshold, a multiple output occurs exactly for inputs c(x+r)^4 with c,r>0. Root perturbations, separated-scale compactness and multiplicity rigidity justify reduction to reciprocal quartics. Exact code supports selected algebra and a rational threshold enclosure; the universal claim rests on the written proof. All-degree preservation, full AIM resolution, historical priority and external validation are not claimed.",
  "datePublished": "2026-09-05",
  "dateModified": "2026-09-05",
  "version": "0.1.0-candidate",
  "doi": "10.5281/zenodo.22346503",
  "doiUrl": "https://doi.org/10.5281/zenodo.22346503",
  "conceptDoi": "10.5281/zenodo.22346502",
  "pdfUrl": "https://github.com/ipitchford/sharp-quartic-hadamard-powers/releases/download/v0.1.0-candidate/sharp-quartic-hadamard-powers-0.1.0-candidate.pdf",
  "altPdfUrl": "https://zenodo.org/records/22346503/files/sharp-quartic-hadamard-powers-0.1.0-candidate.pdf?download=1",
  "altPdfLabel": null,
  "zenodoUrl": "https://zenodo.org/records/22346503",
  "repoUrl": "https://github.com/ipitchford/sharp-quartic-hadamard-powers",
  "releaseUrl": "https://github.com/ipitchford/sharp-quartic-hadamard-powers/releases/tag/v0.1.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/sharp-quartic-hadamard-powers/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/sharp-quartic-hadamard-powers/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/sharp-quartic-hadamard-powers.mp3?v=31a9d7c19e",
  "imageUrl": "https://evidencepress.org/assets/og/sharp-quartic-hadamard-powers.png?v=2216500a83",
  "coverArtUrl": "https://evidencepress.org/assets/art/sharp-quartic-hadamard-powers.svg?v=84acb1b0be",
  "media": [
    {
      "type": "audio",
      "url": "https://evidencepress.org/assets/audio/sharp-quartic-hadamard-powers.mp3",
      "name": "Audio briefing — sharp Hadamard powers of quartics",
      "description": "Plain-English synthetic-voice account of the sharp quartic threshold and its written proof; communication, not additional evidence.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/sharp-quartic-hadamard-powers.txt"
    }
  ],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Anonymous, AI-assisted, unrefereed theorem candidate. Five producer-coordinated internal roles recommend Accept with no P0/P1 scientific findings. This is not external peer review. The higher-degree stretch was considered but no four-star grade or higher-degree result is claimed."
  },
  "assurance": [
    {
      "dimension": "availability",
      "label": "Availability and archiving",
      "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?",
      "state": "passed",
      "evidenceUrl": "https://zenodo.org/records/22346503",
      "note": "All five public assets were downloaded independently from GitHub and Zenodo and matched local SHA-256 values: final ZIP, unchanged reviewed PDF, frozen editorial target, checksum ledger and fresh-extraction receipt."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Fresh local extraction, exact auxiliary algebra, optimized-mode refusals and package negative controls pass. Five-role internal AI review is producer-coordinated. Universal correctness remains a written-proof claim. Final payload Linux Python 3.11/3.13 replay and manuscript-build CI pass.",
      "evidenceUrl": "https://github.com/ipitchford/sharp-quartic-hadamard-powers/actions/runs/33970746672"
    },
    {
      "dimension": "independentRerun",
      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed",
      "note": "No authenticated unaffiliated rerun is supplied."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "Separate coordinated calculations do not establish unaffiliated reconstruction."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "No proof assistant or end-to-end formal certificate verifies the analytical argument."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "AI role reports do not authenticate specialist expertise or unaffiliated review."
    },
    {
      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
      "question": "Has a journal or venue run peer review to a decision?",
      "state": "not-assessed",
      "note": "Internal publication approval is not external journal peer review."
    },
    {
      "dimension": "dataEnvironmentReproducibility",
      "label": "Data and environment reproducibility",
      "question": "Are data and computational environment pinned well enough to rebuild?",
      "state": "partial",
      "evidenceUrl": "https://github.com/ipitchford/sharp-quartic-hadamard-powers/blob/main/ENVIRONMENT.txt",
      "note": "Exact inputs, source and pinned Python requirements are provided. PDF builds are not byte-hermetic; no fully pinned operating system is supplied."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "OpenAI Codex under human direction",
      "Python and SymPy exact auxiliary algebra",
      "Pandoc and pdfLaTeX"
    ],
    "humanRole": "Research direction, candidate selection, supplied review, scope decisions and full-publication authorization; scholarly creator remains Anonymous.",
    "disclosure": "AI assistance contributed mathematical argument, exposition, exact code, review repair, producer-coordinated internal editorial review and publishing. Multiple roles and execution platforms do not establish unaffiliated validation."
  },
  "attributionBoundary": null,
  "problem": {
    "name": "Self-contained ordinary-coefficient quartic subproblem; secondary AIM-LINEAR_ALGEBRA-0005 association is unverified",
    "url": "https://www.unsolvedmath.com"
  },
  "orderedProductAverage": null,
  "corrections": [],
  "publicCorrections": [],
  "publicCorrectionIndexes": null,
  "pageStructureVersion": "2.0",
  "pageStructureVariant": "canonical-theorem",
  "pageStructureWaivers": [],
  "recordMaturity": "current",
  "metadataProvenance": "Research assurance assessed at 2026-09-05T14:10:45.255Z against immutable payload 943c9b6cca8b3dd566ef16011cafb4f57e473ae0, five-role internal Accept synthesis, Linux CI and ten matching downloads. Canonical site readback remains separate and pending; no external assurance is promoted.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "real-rooted polynomials",
    "Hadamard powers",
    "quartic polynomials",
    "coefficientwise preservers",
    "unrefereed candidate"
  ],
  "keyResults": [
    "The universal positive exponent set for strictly positive ordinary-coefficient real-rooted quartics is {1} union [p*, infinity).",
    "The threshold p*>1 is the unique root above one of 6^p - 2*4^p + 2 = 0; exact auxiliary bounds give 1.147720381237014 < p* < 1.147720381237015.",
    "Every output for p>p* has four distinct negative roots; at p*, a multiple output occurs exactly for c(x+r)^4 with c,r>0.",
    "The proof controls noncompact root scales and derives reciprocal boundary inputs without assuming symmetry or monotonicity of the preservation set."
  ],
  "reviews": [],
  "evidencePackage": "A nine-page written proof, Markdown and TeX sources, exact auxiliary algebra, rational endpoint bounds, fail-closed package verifier, four negative-control test groups, pinned dependencies, source and novelty audits, response matrix, five frozen internal editorial reports and separate Accept synthesis. The code does not certify the universal analytical proof. All five final assets match across GitHub and published Zenodo. Final Linux CI: https://github.com/ipitchford/sharp-quartic-hadamard-powers/actions/runs/33970746672. Internal decision: https://github.com/ipitchford/sharp-quartic-hadamard-powers/blob/v0.1.0-candidate/editorial/DECISION.md.",
  "openProblems": [
    "Obtain unaffiliated reconstruction of the scale-escape, last-failure and boundary arguments.",
    "Complete inaccessible-source and citation-graph comparison, and authenticate the original AIM correspondence.",
    "Develop a threshold-relative higher-degree escape theorem and classify new degree-five boundary patterns.",
    "Investigate proof-assistant formalization without conflating auxiliary algebra with the complete proof."
  ],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "Wang, Y., and Zhang, B. (2013). Hadamard powers of polynomials with only real zeros. LAA 439, 3173–3176.",
      "url": "https://doi.org/10.1016/j.laa.2013.08.012",
      "doi": "10.1016/j.laa.2013.08.012"
    },
    {
      "citation": "Białas, S., and Białas-Cież, L. (2017). Comments on 'On Hadamard powers of polynomials'. MCSS 29, article 16.",
      "url": "https://doi.org/10.1007/s00498-017-0202-0",
      "doi": "10.1007/s00498-017-0202-0"
    },
    {
      "citation": "Białas, S., Białas-Cież, L., and Kudra, M. (2024). On the Hurwitz stability of noninteger Hadamard powers of stable polynomials. LAA 683, 111–124.",
      "url": "https://doi.org/10.1016/j.laa.2023.11.022",
      "doi": "10.1016/j.laa.2023.11.022"
    },
    {
      "citation": "Fujie, K., and Ueda, Y. (2023). Law of Large Numbers for Roots of Finite Free Multiplicative Convolution of Polynomials. SIGMA 19, 004.",
      "url": "https://doi.org/10.3842/SIGMA.2023.004",
      "doi": "10.3842/SIGMA.2023.004"
    },
    {
      "citation": "Kudra, M. (2026). The idealizer of the semigroup of stable polynomials, arXiv:2605.07628v2.",
      "url": "https://arxiv.org/html/2605.07628v2",
      "doi": null
    }
  ],
  "claimAssurance": null,
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    "decisionObject": {
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      "description": "Exact universal exponent set supported by a written boundary proof and scoped auxiliary arithmetic.",
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    },
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      "assurance",
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      "remainingRisks": [
        "Written universal argument is not formally verified or externally reconstructed.",
        "Prior-art and original AIM access remain incomplete.",
        "No all-degree extension or numerical conditioning margin is established."
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    "humanJudgmentGates": [
      "Reconstruct the universal quantifier and noncompact boundary argument.",
      "Check ordinary versus normalized powers and real-rootedness versus Hurwitz stability.",
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      "Do not treat replay, a DOI or coordinated model agreement as independent validation."
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        "externalUrl": "https://doi.org/10.1016/j.laa.2013.08.012",
        "inheritedClaim": "Cubic preservation, quartic failure and a degree-dependent sufficient tail are prior work.",
        "inheritedAssuranceCeiling": "The current note proves its own lower-degree and nonsharp-tail inputs; prior work does not establish this candidate's sharp endpoint or historical priority."
      }
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      "dimensions": [
        "independentRerun",
        "independentReimplementation",
        "formalVerification",
        "specialistReview",
        "semanticValidation",
        "noveltyAssessment",
        "priorityAssessment"
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      "nextAction": "Obtain unaffiliated mathematical reconstruction and fuller prior-art reconciliation before stronger assurance claims.",
      "claimCeiling": "Unrefereed quartic theorem candidate with exact auxiliary checks; no external validation, formal proof, full AIM solution, all-degree classification, historical priority, four-star grade or impact established."
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        "setting": "Producer-coordinated mathematical research and publication",
        "status": "NO_IMPACT_EVIDENCE",
        "designClass": "none",
        "comparator": "No matched conventional workflow comparator.",
        "estimand": "No causal discovery-time, reliability, uptake or field-impact effect estimated.",
        "evidenceRefs": [],
        "registeredDesignRef": null,
        "independentAssessment": null
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          "forecast": {
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                "component": "Immutable identities, reader-first page and provenance-bound media",
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            "resultSummary": "The supplied review was actioned and the quartic candidate passed the frozen deterministic package and five-role internal editorial gate. One positive cycle closed the stated quartic publication target; one negative cycle rejected the direct higher-degree stretch as insufficient for a four-star or higher-degree claim. Immutable GitHub and Zenodo assets matched across ten downloads, and the first canonical Evidence Press deployment passed exact protocol and publication-preservation readback. A caught append-only work-log ordering defect required a site-only repair before upload and changed no research or media bytes. The result remains an unrefereed candidate without external reproduction, formal verification, journal peer review, exhaustive priority or demonstrated impact.",
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}