{
  "schemaVersion": "1.5",
  "slug": "o01d0-polar-fibre-structural-reductions",
  "title": "O-01d0 polar-fibre structural reductions",
  "shortTitle": "O-01d0 polar-fibre reductions",
  "url": "https://evidencepress.org/releases/o01d0-polar-fibre-structural-reductions/",
  "oneLine": "Exact polar-fibre and Hermite reductions, four rejected proof architectures, and a four-gate executable handoff sharpen an open all-index transversality problem.",
  "abstract": "This anonymous, unrefereed candidate studies the internal O-01d0 target: whether a four-generator weighted-homogeneous ideal has empty weighted-projective zero locus for every index n at least 6. It proves a fixed-index properness and finite-freeness criterion, reduces the lower balanced obstruction to triple coprimality for an explicit Hermite family, and supplies exact incidence, squarefreeness, confluent-boundary, exceptional-fibre-avoidance, and fifth-derivative connection results. Producer evidence gives fixed-index consequences at n=21 and 25; n=29 is sealed but not rerun. Exact computations reject bounded scalar transfer, a canonical monic remainder tail, degree-at-most-twelve projective transfer, and the claim that every structural window prime works for the upper determinant. Large bounded scans support, but do not prove, a narrower pair-content and four-factor conjecture. O-01d0 remains open and unfalsified. The release does not prove AF5 transversality or any linked headline conjecture, and it has no independent reconstruction, formal verification, specialist review, editorial peer review, or priority determination.",
  "datePublished": "2026-08-28",
  "dateModified": "2026-08-28",
  "version": "0.1.0-candidate",
  "doi": "10.5281/zenodo.22143955",
  "doiUrl": "https://doi.org/10.5281/zenodo.22143955",
  "conceptDoi": "10.5281/zenodo.22143954",
  "pdfUrl": "https://github.com/ipitchford/o01d0-polar-fibre-structural-reductions/releases/download/v0.1.0-candidate/paper.pdf",
  "altPdfUrl": "https://zenodo.org/records/22143955/files/paper.pdf?download=1",
  "altPdfLabel": null,
  "zenodoUrl": "https://zenodo.org/records/22143955",
  "repoUrl": "https://github.com/ipitchford/o01d0-polar-fibre-structural-reductions",
  "releaseUrl": "https://github.com/ipitchford/o01d0-polar-fibre-structural-reductions/releases/tag/v0.1.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/o01d0-polar-fibre-structural-reductions/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/o01d0-polar-fibre-structural-reductions/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/o01d0-polar-fibre-structural-reductions.mp3?v=c9123f4513",
  "imageUrl": "https://evidencepress.org/assets/og/o01d0-polar-fibre-structural-reductions.png?v=146d27c4ba",
  "coverArtUrl": "https://evidencepress.org/assets/art/o01d0-polar-fibre-structural-reductions.svg?v=ccdb2edf5e",
  "media": [
    {
      "type": "audio",
      "url": "https://evidencepress.org/assets/audio/o01d0-polar-fibre-structural-reductions.mp3",
      "name": "Audio briefing — O-01d0 polar-fibre reductions",
      "description": "Plain-English OpenAI API synthetic-voice summary of the exact reductions, rejected architectures, bounded evidence, and four remaining gates; a communication aid, not additional evidence.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/o01d0-polar-fibre-structural-reductions.txt"
    }
  ],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Anonymous, unrefereed structural-reductions candidate with internal PASS_WITH_NOTES. Written proofs and computer-assisted all-index statements are supplied for specialist checking; finite computations retain their exact bounds. The n=21 and n=25 producer transcripts replay, while n=29 is sealed and source-bound but not rerun. O-01d0 remains open and unfalsified. No unaffiliated rerun, independent implementation, proof-assistant formalization, external specialist review, journal peer review, absolute novelty or priority determination, or impact evaluation has occurred."
  },
  "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/22143955",
      "note": "The tagged GitHub prerelease and published Zenodo record expose the PDF, deterministic ZIP, claims and experiment registries, citation metadata, and manifest; all six public asset bytes match the local release files."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Ordinary and optimized producer replay agree on all 18 portable and PassageMath/Sage output hashes locally; hosted portable replay and hostile controls pass on Python 3.11, 3.12, and 3.13.",
      "evidenceUrl": "https://github.com/ipitchford/o01d0-polar-fibre-structural-reductions/actions/runs/33170023609"
    },
    {
      "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 immutable package in a separately provisioned environment."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "The exact programs and cross-model reviews remain producer-coordinated; no separately authored implementation has reconstructed the mathematical predicates."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "No proof assistant formalizes the fixed-index criterion, Hermite lemmas, all-index computer-assisted theorem, or projective assembly."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "The five-role review and bounded confirmation are internal AI editorial quality control, not external specialist review."
    },
    {
      "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/o01d0-polar-fibre-structural-reductions/blob/v0.1.0-candidate/ENVIRONMENT.md",
      "note": "Pinned dependencies, environment notes, commands, manifests, and paired receipts are public, but no unaffiliated clean-environment reconstruction or declarative system image is reported."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "AI systems under human direction",
      "OpenAI Codex",
      "Claude",
      "Python 3.14.6",
      "SymPy 1.14.0",
      "PassageMath 10.8.9",
      "Singular 4.4.1"
    ],
    "humanRole": "Research direction, falsification priorities, claim-boundary decisions, mediation, and publication authorization by the human creator; repository maintenance and Evidence Press publication are not scholarly authorship, and scholarly attribution remains Anonymous.",
    "disclosure": "AI systems contributed proof exploration, exact computation, code, review, editing, replay, and release engineering in one producer-coordinated workflow. Cross-model agreement, exact replay, and matching public hashes are not independent reconstruction, specialist review, or editorial peer review."
  },
  "attributionBoundary": null,
  "problem": {
    "name": "O-01d0 uniform polar-fibre primaryness",
    "url": "https://github.com/ipitchford/o01d0-polar-fibre-structural-reductions/blob/v0.1.0-candidate/CLAIMS.json"
  },
  "orderedProductAverage": null,
  "corrections": [],
  "publicCorrections": [],
  "publicCorrectionIndexes": null,
  "pageStructureVersion": "2.0",
  "pageStructureVariant": "reader-first-reference",
  "pageStructureWaivers": [],
  "recordMaturity": "current",
  "metadataProvenance": "Authored prospectively for reader-first schema 2.0 against immutable GitHub and Zenodo identities, the final paired replay record, and the frozen internal review record.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "Polydegree Conjecture",
    "Furter rigidity",
    "weighted complete intersections",
    "polar fibres",
    "Hermite sections",
    "resultants",
    "p-adic degeneration",
    "computer-assisted algebra",
    "negative results",
    "unrefereed candidate"
  ],
  "keyResults": [
    "A finite-dimensional fixed-index polar fibre implies a regular sequence and finite freeness over the deformation parameter, with explicit rank (n-1)n(n+1)(n+2)/24.",
    "The lower balanced obstruction is reduced to triple coprimality for J_A, J_(A+5), and J_(A+10); structural triple gcds are proved squarefree.",
    "Uniform confluent initial forms, joint avoidance of the critical and first infinity-collision fibres, and an exact fifth-derivative A+20-to-A connection constrain the remaining Hermite boundary problem.",
    "Four tempting proof architectures are rejected exactly, including the universal all-window-prime upper determinant claim.",
    "Bounded pair-content scans through A=3000 and primes below 12000 support a narrower four-factor conjecture but do not establish it."
  ],
  "reviews": [],
  "evidencePackage": "A 14-page manuscript and accessible Markdown; 20 indexed claims and non-claims; 13 experiment records with stopping rules and exclusions; 12 portable and 6 PassageMath/Sage checks replayed in ordinary and optimized modes with all 18 paired output hashes identical; fixed-index transcripts at n=21 and n=25 plus a sealed, not-rerun n=29 record; exact counterexamples and hostile controls; a 147-entry manifest; pinned environment files; a five-role producer-coordinated internal review, ten-item repair record, and bounded confirmation; public CI on Python 3.11, 3.12, and 3.13; and byte-matched GitHub and Zenodo assets. These establish availability, internal replay, integrity, exact finite statements in their declared domains, and internal editorial closure—not the open uniform theorem or independent mathematical validation.",
  "openProblems": [
    "Prove or falsify the pair-content and four-factor theorem controlling denominator-valid adjacent-pair content and the third-row same-root event.",
    "Prove that every index admits at least one structural prime that simultaneously avoids the lower exceptions and makes the upper two-level determinant nonzero.",
    "Construct degree- and (x3,x4)-compatible graded multiplier lifts in both odd residue classes.",
    "Assemble the affine and residual charts into a complete projective-emptiness proof, or isolate a counterexample.",
    "Obtain an unaffiliated specialist reconstruction and a separately authored implementation while preserving corrections as versioned successors."
  ],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "Anonymous. (2026). LPS structural reductions: duality, quartic obstruction, and p-adic certificates (Version 0.2.0-candidate). Evidence Press.",
      "url": "https://doi.org/10.5281/zenodo.22100350",
      "doi": "10.5281/zenodo.22100350"
    },
    {
      "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, 5346--5359.",
      "url": "https://doi.org/10.1016/j.jpaa.2019.04.002",
      "doi": "10.1016/j.jpaa.2019.04.002"
    },
    {
      "citation": "Furter, J.-P. (2015). Polynomial composition rigidity and plane polynomial automorphisms. Journal of the London Mathematical Society, 91, 180--202.",
      "url": "https://doi.org/10.1112/jlms/jdu064",
      "doi": "10.1112/jlms/jdu064"
    },
    {
      "citation": "Edo, E., & van den Essen, A. (2014). The Strong Factorial Conjecture. Journal of Algebra, 397, 443--456.",
      "url": "https://doi.org/10.1016/j.jalgebra.2013.09.011",
      "doi": "10.1016/j.jalgebra.2013.09.011"
    }
  ],
  "operatingModel": {
    "version": "1.0",
    "workId": "ep-work:o01d0-polar-fibre-structural-reductions",
    "attemptIds": [
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    "aims": [
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    "artifactRoles": [
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    "lineageId": null,
    "accelerationPrimitives": [
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    ],
    "decisionObject": {
      "type": "executable-open-problem",
      "description": "A four-gate proof-or-falsification handoff for O-01d0, centred on a one-variable Hermite pair-content theorem and a simultaneous one-good-prime upper determinant theorem.",
      "scope": "The exact polar-fibre reductions, Hermite incidence and boundary lemmas, declared finite scans, rejected transfer architectures, graded multiplier obligations, and projective chart assembly; not uniform O-01d0, AF5 transversality, or any linked conjecture."
    },
    "bottleneckTargeted": [
      "discovery",
      "assurance",
      "publication"
    ],
    "semanticBridge": {
      "state": "explicit",
      "description": "The package maps weighted-projective emptiness through a fixed-index properness criterion and two-level degeneration to triple coprimality of J_A sections, separates lower, upper, multiplier, and chart-cover gates, and indexes the exact scope and evidence of every positive and negative claim.",
      "remainingRisks": [
        "The written reductions and computer-assisted all-index squarefreeness theorem have not been independently reconstructed by specialists.",
        "Bounded pair-content survival can miss a later nonlinear common factor or an exceptional prime outside the scan.",
        "The n=29 fixed-index record is sealed rather than rerun.",
        "The lower and upper prime choices, multiplier lifts, and residual projective charts have not been assembled into uniform emptiness."
      ]
    },
    "humanJudgmentGates": [
      "Check each written polar-fibre and Hermite reduction as mathematics rather than inferring it from successful replay.",
      "Keep squarefree separate from coprime and finite scan survival separate from an all-index content theorem.",
      "Keep one bad structural prime separate from failure of the one-good-prime route.",
      "Require a proved graded multiplier map and all-chart cover before promoting local or residue-class calculations to O-01d0.",
      "Require external specialist assessment before any target-conjecture, first, priority, or field-acceptance language."
    ],
    "parentLinks": [
      {
        "relation": "extends-result",
        "workId": "ep-work:lps-structural-reductions",
        "legacyReleaseSlug": null,
        "externalUrl": null,
        "inheritedClaim": "The parent release compresses the LPS obstruction to fixed-width adjacent-flag transversality and exposes the O-01 uniform radical-membership programme.",
        "inheritedAssuranceCeiling": "The parent is an anonymous unrefereed producer-coordinated candidate; reuse of its reductions is dependency evidence, not independent confirmation."
      }
    ],
    "assuranceTarget": {
      "dimensions": [
        "independentRerun",
        "independentReimplementation",
        "formalVerification",
        "specialistReview",
        "dataEnvironmentReproducibility",
        "semanticValidation",
        "noveltyAssessment",
        "priorityAssessment"
      ],
      "nextAction": "Obtain an unaffiliated reconstruction of the Hermite reductions, then prove or decisively falsify the pair-content/four-factor theorem before attempting the upper determinant, multiplier lifts, and chart assembly.",
      "claimCeiling": "An anonymous unrefereed structural-reductions candidate with written exact reductions, producer-replayed fixed-index and bounded evidence, exact failures of four proof architectures, paired replay, and internal editorial closure; not O-01d0, AF5 transversality, any linked headline conjecture, independent validation, formal verification, external specialist or editorial peer review, absolute priority, or demonstrated workflow impact."
    },
    "impactClaims": [
      {
        "id": "o01d0-structural-science-no-impact",
        "aim": "science",
        "outcome": "Faster or more reliable resolution and assurance of difficult linked algebraic conjectures",
        "setting": "AI-assisted exact algebra, falsification, and structural reduction around the LPS Polydegree programme",
        "status": "NO_IMPACT_EVIDENCE",
        "designClass": "none",
        "comparator": "No matched conventional research, theorem-review, or publication workflow was registered.",
        "estimand": "No effect on discovery time, active human effort, compute, correction rate, proof quality, independent-assurance time, uptake, or citation was estimated.",
        "evidenceRefs": [],
        "registeredDesignRef": null,
        "independentAssessment": null
      }
    ]
  }
}