{
  "schemaVersion": "1.3",
  "slug": "full-e3-column-polydegree-conjecture",
  "title": "A Jacobian smooth-point criterion and the full e=3 column of the Polydegree Conjecture",
  "shortTitle": "The full e=3 Polydegree column",
  "url": "https://evidencepress.org/releases/full-e3-column-polydegree-conjecture/",
  "oneLine": "A Jacobian smooth-point reduction, Fourier-limit persistence, an exact eventual envelope and 97,033 finite Arb certificates prove the e=3 containment for every degree d at least 2.",
  "abstract": "Lewis, Perry and Straub reduced a Polydegree Conjecture containment to the existence of a complex specialisation satisfying explicit coefficient and determinant conditions. This anonymous, unrefereed candidate identifies that determinant with the coefficient Jacobian and uses a weighted-Euler syzygy to reformulate the criterion as a smooth codimension-two point away from the next hypersurface. For e=3, factorially normalised coefficients converge in C1 to explicit Fourier-exponential sums with transverse zeros. Quantitative Newton--Kantorovich bounds and an exact-rational defect envelope prove persistence from m=48,550 onward. Exact finite-field checks cover d from 2 through 100, the residue-one class has an exact special point, and 97,033 outward-rounded FLINT/Arb certificates close the remaining finite bridge with no failure. Together these establish G_(d+3) contained in the closure of G_(d,4) for every integer d at least 2. Producer-side replay and cross-model adversarial review are recorded; no unaffiliated full Arb rerun, independent implementation, formal verification, external specialist review or editorial peer review is claimed.",
  "datePublished": "2026-08-12",
  "dateModified": "2026-08-12",
  "version": "0.1.0-candidate",
  "doi": "10.5281/zenodo.21909085",
  "doiUrl": "https://doi.org/10.5281/zenodo.21909085",
  "conceptDoi": "10.5281/zenodo.21909084",
  "pdfUrl": "https://github.com/ipitchford/polydegree-full-e3-column/releases/download/v0.1.0-candidate/polydegree-full-e3-column-v0.1.0-candidate.pdf",
  "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/polydegree-full-e3-column/v0.1.0-candidate/paper/MANUSCRIPT.pdf",
  "zenodoUrl": "https://zenodo.org/records/21909085",
  "repoUrl": "https://github.com/ipitchford/polydegree-full-e3-column",
  "releaseUrl": "https://github.com/ipitchford/polydegree-full-e3-column/releases/tag/v0.1.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/full-e3-column-polydegree-conjecture/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/full-e3-column-polydegree-conjecture/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/full-e3-column-polydegree-conjecture.mp3",
  "imageUrl": "https://evidencepress.org/assets/og/full-e3-column-polydegree-conjecture.png",
  "coverArtUrl": "https://evidencepress.org/assets/art/full-e3-column-polydegree-conjecture.svg",
  "media": [
    {
      "type": "audio",
      "url": "https://evidencepress.org/assets/audio/full-e3-column-polydegree-conjecture.mp3",
      "name": "Audio briefing — the full e=3 Polydegree column",
      "description": "Plain-English synthetic-voice summary of the smooth-point reduction, proof regimes and assurance boundary; a communication aid, not additional mathematical evidence.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/full-e3-column-polydegree-conjecture.txt"
    }
  ],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Anonymous unrefereed candidate. The main all-degree e=3 containment is presented as a theoretical and computer-assisted theorem. Exact algebra, finite-field checks, a rational eventual envelope and the complete finite Arb bridge have passed producer-side replay, and the final review requested only bounded minor revisions. No unaffiliated party has rerun the full immutable Arb package or independently reconstructed the proof; there is no separately authored checker, proof-assistant formalisation, external specialist review, editorial peer review or venue acceptance. The dated public-record search found no earlier published all-degree e=3 proof; it is not a specialist priority adjudication."
  },
  "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/21909085",
      "note": "The immutable GitHub prerelease and published Zenodo record expose the candidate ZIP, PDF and checksum ledger; producer-side public downloads matched. This establishes byte availability and identity, not correctness or independent reproduction."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "A fresh pinned producer environment regenerated all 97,033 finite Arb rows and the terminal receipt byte-for-byte, alongside exact, optimized, envelope, positive-control and semantic-negative-control checks.",
      "evidenceUrl": "https://github.com/ipitchford/polydegree-full-e3-column/releases/tag/v0.1.0-candidate"
    },
    {
      "dimension": "independentRerun",
      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed",
      "note": "No unaffiliated party has reported rerunning the full immutable locator sequence and certificate package."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "The normative specification is public, but no separately authored interval checker has reconstructed the finite and eventual predicates."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "No theorem or checker has been formalised in a proof assistant over a stated trusted base."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "Producer-coordinated cross-model adversarial review is retained; no external human polynomial-automorphism, algebraic-geometry, asymptotic-analysis or interval-certification specialist has reviewed the exact candidate."
    },
    {
      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
      "question": "Has a journal or venue run peer review to a decision?",
      "state": "not-assessed",
      "note": "No journal or comparable venue has conducted editorial peer review to a decision."
    },
    {
      "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/polydegree-full-e3-column/blob/v0.1.0-candidate/verification/ENVIRONMENT.md",
      "note": "The package pins Python-FLINT 0.8.0 by hashes, records Python 3.14.6 and supplies an atomic replay wrapper, but no unaffiliated cross-platform or independently recreated container, Nix or Guix environment is reported."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "AI systems under human direction",
      "OpenAI Codex",
      "Anthropic Claude in a producer-coordinated review path"
    ],
    "humanRole": "Research direction, mediation, claim-boundary decisions and publication authorization by the human creator; scholarly attribution remains Anonymous.",
    "disclosure": "AI systems contributed derivations, code, review, drafting, replay and release engineering in one producer-coordinated workflow. Cross-model review, exact replay, interval certification and matching hashes are not independent reproduction, external specialist review or editorial peer review."
  },
  "problem": {
    "name": "Polydegree Conjecture for plane polynomial automorphisms",
    "url": "https://doi.org/10.1016/j.jpaa.2019.04.002"
  },
  "corrections": [],
  "keywords": [
    "plane polynomial automorphisms",
    "Polydegree Conjecture",
    "Jacobian criterion",
    "Fourier asymptotics",
    "Newton-Kantorovich theorem",
    "interval arithmetic",
    "FLINT Arb",
    "computer-assisted mathematics",
    "unrefereed candidate"
  ],
  "keyResults": [
    "For every integer d at least 2, G_(d+3) is contained in the closure of G_(d,4), proving the full e=3 column of the Polydegree Conjecture.",
    "The Lewis--Perry--Straub determinant is, up to sign, the coefficient Jacobian; an integral weighted-Euler syzygy reduces its criterion to one smooth codimension-two point away from the next coefficient hypersurface.",
    "After explicit factorial normalisation, the e=3 coefficient windows converge in C1 to Fourier-exponential systems with explicit transverse zeros.",
    "An exact-rational envelope gives a defect at most 0.0029969535913992734 at m=48,550 and uniformly thereafter, below the required 0.003 threshold.",
    "The finite bridge contains exactly 97,033 ordered FLINT/Arb certificates with zero failures; a fresh pinned producer replay regenerated byte-identical rows and receipt.",
    "The fixed-e continuation is a conjectural programme and is not proved by this release."
  ],
  "reviews": [],
  "evidencePackage": "A 13-page candidate paper and accessible Markdown; a normative formula-to-checker specification; exact symbolic and finite-field checks; an exact-rational eventual envelope; a pinned Python-FLINT/Arb checker; 97,033 complete result rows with exact Gaussian-rational locators; terminal finite and eventual receipts; fresh producer replay under ordinary and optimized Python; positive and semantic negative controls; claim, provenance, licence, citation, novelty and assurance records; review dispositions; CI; and complete SHA-256 manifests. These establish producer-side replay and artifact identity, not an unaffiliated theorem reconstruction.",
  "openProblems": [
    "Independently reconstruct the Jacobian--Euler, Fourier-limit, Newton--Kantorovich and Lewis--Perry--Straub bridges from the definitions.",
    "Write a separate interval implementation and rerun all 97,033 immutable locators without producer intermediates.",
    "Obtain external specialist reviews in polynomial automorphisms, algebraic geometry, asymptotic analysis and validated numerics, retaining corrections publicly.",
    "Formalise the structural identities and persistence theorem over an explicit trusted base.",
    "Investigate the fixed-e extension only after the e=3 proof survives independent review, publishing changed thresholds and failed routes."
  ],
  "relatedWorks": [
    {
      "citation": "Anonymous (2026). Smooth-Point Certificates for Polydegree Containments: A Jacobian Interpretation of the Lewis--Perry--Straub Determinant. Version 0.4.1-candidate.",
      "url": "https://doi.org/10.5281/zenodo.21864574"
    },
    {
      "citation": "Lewis, D., Perry, K., & Straub, A. (2019). An algorithmic approach to the Polydegree Conjecture for plane polynomial automorphisms. Journal of Pure and Applied Algebra, 223(12), 5346--5359.",
      "url": "https://doi.org/10.1016/j.jpaa.2019.04.002"
    },
    {
      "citation": "Perry, K. A. (2016). Polydegree properties of polynomial automorphisms. Doctoral dissertation, The University of Alabama.",
      "url": "https://ir.ua.edu/items/f71aa1b3-0dbc-42a2-9f59-3f01eb1a5529"
    },
    {
      "citation": "Ortega, J. M., & Rheinboldt, W. C. (1970). Iterative Solution of Nonlinear Equations in Several Variables.",
      "url": "https://doi.org/10.1137/1.9780898719468"
    }
  ],
  "operatingModel": {
    "version": "1.0",
    "workId": "ep-work:polydegree-full-e3-column",
    "attemptIds": [
      "ep-attempt:full-e3-column-polydegree-conjecture"
    ],
    "aims": [
      "science"
    ],
    "artifactRoles": [
      "research-output",
      "method-demonstration",
      "communication"
    ],
    "lineageId": null,
    "accelerationPrimitives": [
      "certificate-first",
      "structural-compression",
      "regime-stitching",
      "adversarial-controls",
      "research-lineage-reuse",
      "assurance-vector",
      "agent-readable-research-object"
    ],
    "decisionObject": {
      "type": "certificate",
      "description": "A gap-free all-degree proof object joining an exact smooth-point reduction, explicit Fourier-limit zeros, a rational eventual envelope and 97,033 ordered finite Arb certificates.",
      "scope": "The containment G_(d+3) contained in the closure of G_(d,4) for every integer d at least 2; not the full Polydegree Conjecture or the conjectural fixed-e extension."
    },
    "bottleneckTargeted": [
      "assurance",
      "publication"
    ],
    "semanticBridge": {
      "state": "explicit",
      "description": "The normative specification maps the exact coefficient normalisation, norms, locators, tails, Newton predicates, third-row condition, Jacobian transport and complete range order to code; the manuscript separately states the Lewis--Perry--Straub implication and analytic derivation.",
      "remainingRisks": [
        "The manuscript, specification and implementation share producer-side lineage and may share a transcription error.",
        "The analytic tail derivation and application of the Lewis--Perry--Straub criterion require independent specialist reconstruction.",
        "No separately authored interval checker has tested the archived locators and range stitch."
      ]
    },
    "humanJudgmentGates": [
      "Assess the Jacobian--Euler and Lewis--Perry--Straub bridges as mathematics rather than inferring them from successful computation.",
      "Assess the Fourier-limit and Newton--Kantorovich derivations, constants and norm translations independently.",
      "Judge novelty and priority against the published record beyond the dated bounded search.",
      "Authorize scholarly attribution, public release and any later claim about the fixed-e programme or workflow impact."
    ],
    "parentLinks": [
      {
        "relation": "extends-result",
        "workId": null,
        "legacyReleaseSlug": "smooth-point-certificates-polydegree-containments",
        "externalUrl": null,
        "inheritedClaim": "The derivative--Jacobian identity, integral weighted-Euler factorisation, smooth-point reformulation and smaller Hensel certificate form the structural layer used by this all-degree successor.",
        "inheritedAssuranceCeiling": "The parent is an anonymous unrefereed producer-side candidate with exact replay and coordinated review, not independent confirmation, external specialist review or formal verification."
      }
    ],
    "assuranceTarget": {
      "dimensions": [
        "independentRerun",
        "independentReimplementation",
        "formalVerification",
        "specialistReview",
        "semanticValidation",
        "noveltyAssessment",
        "priorityAssessment"
      ],
      "nextAction": "An unaffiliated specialist team should reconstruct the analytic and Polydegree bridges and implement the finite and eventual interval predicates independently against the immutable public inputs.",
      "claimCeiling": "A producer-side theoretical and computer-assisted proof of the full e=3 column, with exact and interval replay and bounded public-record novelty evidence; not independent reproduction, formal verification, peer review, the full conjecture, a fixed-e theorem, absolute priority or demonstrated workflow impact."
    },
    "impactClaims": [
      {
        "id": "science-no-impact",
        "aim": "science",
        "outcome": "Faster or more reliable resolution and assurance of difficult all-parameter mathematical problems",
        "setting": "AI-assisted theoretical and computer-assisted work on the e=3 Polydegree Conjecture column",
        "status": "NO_IMPACT_EVIDENCE",
        "designClass": "none",
        "comparator": "No prospective matched conventional research or publication workflow comparator was registered.",
        "estimand": "No effect on discovery time, active human effort, compute, correction rate, proof quality or independent-assurance time was estimated.",
        "evidenceRefs": [],
        "registeredDesignRef": null,
        "independentAssessment": null
      }
    ]
  }
}