{
  "schemaVersion": "1.4",
  "slug": "lps-structural-reductions",
  "title": "LPS structural reductions: duality, quartic obstruction, and p-adic certificates",
  "shortTitle": "LPS structural reductions",
  "url": "https://evidencepress.org/releases/lps-structural-reductions/",
  "oneLine": "A complementary-minor duality compresses a growing Polydegree obstruction to a fixed-width flag, while repaired quartic and p-adic certificates expose the remaining uniform radical-membership target.",
  "abstract": "This anonymous, unrefereed three-manuscript candidate develops a structural route through the Lewis--Perry--Straub formulation of the Polydegree programme. A complementary-minor congruence, rigidity duality and inverse-coordinate descent compress the second radical-membership obstruction to an adjacent-flag transversality problem in fixed width. For d=5, an exact companion studies the resulting quartic geometry, separates centered and backward coefficient gauges, and proves only their localized bridge. A p-adic addendum gives a certificate theorem, a degeneration identity, twelve designated-prime instances through n=57 and an explicit n=15 exception to the designated reduced-gcd route. The package passes paired 20-check replay, mutation and fail-closed controls, a five-role internal review after repair and bounded confirmation. The uniform radical membership and slope-four law remain open. No target conjecture, independent reproduction, formal verification, external specialist review, peer review, novelty or priority is claimed.",
  "datePublished": "2026-08-25",
  "dateModified": "2026-08-25",
  "version": "0.2.0-candidate",
  "doi": "10.5281/zenodo.22100350",
  "doiUrl": "https://doi.org/10.5281/zenodo.22100350",
  "conceptDoi": "10.5281/zenodo.22100349",
  "pdfUrl": "https://github.com/ipitchford/lps-structural-reductions/releases/download/v0.2.0-candidate/LPS-Structural-Reductions.pdf",
  "altPdfUrl": "https://zenodo.org/records/22100350/files/LPS-Structural-Reductions.pdf?download=1",
  "altPdfLabel": null,
  "zenodoUrl": "https://zenodo.org/records/22100350",
  "repoUrl": "https://github.com/ipitchford/lps-structural-reductions",
  "releaseUrl": "https://github.com/ipitchford/lps-structural-reductions/releases/tag/v0.2.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/lps-structural-reductions/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/lps-structural-reductions/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/lps-structural-reductions.mp3",
  "imageUrl": "https://evidencepress.org/assets/og/lps-structural-reductions.png",
  "coverArtUrl": "https://evidencepress.org/assets/art/lps-structural-reductions.svg?v=8a54bb88be",
  "media": [
    {
      "type": "audio",
      "url": "https://evidencepress.org/assets/audio/lps-structural-reductions.mp3",
      "name": "Audio briefing — LPS structural reductions",
      "description": "Plain-English synthetic-voice summary of the fixed-width reduction, quartic gauge repair, p-adic signal and open theorem boundary; a communication aid, not additional mathematical evidence.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/lps-structural-reductions.txt"
    },
    {
      "type": "video",
      "url": "https://youtu.be/pakMlru6YEI",
      "name": "Compressing the LPS Obstruction  Duality and Fixed Width Transv",
      "description": "Evidence Press video overview of this release. This is a communication aid; it does not extend the release's claims or assurance status and is not additional research evidence."
    }
  ],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Anonymous, unrefereed structural and exact-computation candidate. Written determinant and reduction arguments are presented for expert checking; selected quartic and p-adic predicates have producer-side exact replay. The first frozen successor received Major Revision from all five internal roles and was repaired. A bounded confirmation found no unresolved P0 or P1 publication blocker. The centered and backward gauges are identified only after localization; transversality is not promoted to emptiness and zero-dimensionality is not promoted to saturation. No unaffiliated rerun, independent implementation, proof-assistant formalization, external specialist review, journal peer review or absolute priority determination has occurred. The package does not prove any of the five target conjectures."
  },
  "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/22100350",
      "note": "The tagged GitHub prerelease and published Zenodo record expose the exact ZIP, three PDFs and manifest; public asset digests 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": "The final GitHub integrity job passed. Frozen ordinary and optimized producer receipts report 20 matching outputs, and the validator rejects five semantic receipt mutations and three fail-open p-adic sentinels.",
      "evidenceUrl": "https://github.com/ipitchford/lps-structural-reductions/actions/runs/32878582712"
    },
    {
      "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": "Implementation-diverse checks remain producer-coordinated; no separately authored checker 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 determinant duality, fixed-width descent, quartic bridge or p-adic theorem."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "The five-role reports and 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/lps-structural-reductions/blob/v0.2.0-candidate/ENVIRONMENT.txt",
      "note": "Dependencies, producer environment, commands, manifests and receipts are public, but no independently recreated container, Nix or Guix environment is reported."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "AI systems under human direction",
      "OpenAI Codex",
      "Python 3.14.6",
      "SymPy 1.14.0",
      "Singular 4.4.1"
    ],
    "humanRole": "Research direction, mediation, claim-boundary decisions 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, code, review, editing, replay and release engineering in one producer-coordinated workflow. Model agreement, exact replay and matching public hashes are not independent reproduction, external specialist review or editorial peer review."
  },
  "attributionBoundary": null,
  "problem": {
    "name": "Lewis--Perry--Straub approach to the Polydegree Conjecture",
    "url": "https://doi.org/10.1016/j.jpaa.2019.04.002"
  },
  "orderedProductAverage": null,
  "corrections": [],
  "publicCorrections": [],
  "publicCorrectionIndexes": null,
  "pageStructureVersion": "2.0",
  "pageStructureVariant": "reader-first-reference",
  "pageStructureWaivers": [],
  "recordMaturity": "current",
  "metadataProvenance": "Authored prospectively for the reader-first schema 2.0 against immutable GitHub and Zenodo identities, the final 20-check receipts and frozen internal review record.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "Polydegree Conjecture",
    "Furter rigidity",
    "polynomial automorphisms",
    "Strong Factorial Conjecture",
    "Jacobian Conjecture",
    "complementary minors",
    "adjacent flags",
    "radical membership",
    "p-adic certificates",
    "computer-assisted mathematics",
    "unrefereed candidate"
  ],
  "keyResults": [
    "A complementary-minor congruence and inverse-coordinate descent compress the second LPS radical-membership obstruction to a fixed-width adjacent-flag problem.",
    "For d=5, the companion separates the centered staircase ideal from the backward p-adic ideal and verifies their bridge only on the chart where the explicit transition determinant is a unit.",
    "The repaired logic distinguishes transversality from emptiness and zero-dimensional initial-ideal evidence from saturation or radical membership.",
    "A p-adic certificate theorem and degeneration identity support twelve designated-prime instances through n=57, while n=15 records a designed exceptional-prime diagnostic.",
    "Ordinary and optimized replay pass the same 20 selected predicates with matching outputs; five receipt mutations and three fail-open p-adic sentinels are rejected."
  ],
  "reviews": [],
  "evidencePackage": "A 10-page structural paper, 9-page computational companion and 4-page p-adic addendum with accessible Markdown; 22 indexed formal statements, 13 catalogue rows, 11 bounded rows, five open obligations and five non-claims; exact structural, quartic and p-adic code and data; paired normal and optimized 20-check receipts with matching output hashes; five semantic receipt mutations and three fail-closed contract controls; a complete 189-payload-file manifest; frozen v0.1.0 lineage; a five-role substantive review, response matrix and bounded confirmation report; a fresh-extraction ten-gate publication preflight; GitHub CI; and byte-matched GitHub and Zenodo assets. These establish availability, producer-side replay, integrity and internal editorial closure, not the truth of the open uniform argument or independent mathematical validation.",
  "openProblems": [
    "Prove the slope-four valuation law from an integral presentation or Smith form of the scaled observation block at p=n+2.",
    "Prove reduced norm coprimality over the relevant finite fields while treating the n=15 designated-prime exception explicitly.",
    "Give a parameter-uniform elimination, initial-ideal or saturation certificate across the quartic residual charts.",
    "Prove adjacent-flag transversality directly without assuming the stronger global pair-quasismoothness statement.",
    "Obtain unaffiliated specialist reconstructions and a separately authored implementation, preserving objections or corrections as versioned successors."
  ],
  "relatedWorks": [
    {
      "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"
    },
    {
      "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"
    },
    {
      "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"
    }
  ],
  "operatingModel": {
    "version": "1.0",
    "workId": "ep-work:lps-structural-reductions",
    "attemptIds": [
      "ep-attempt:lps-structural-reductions"
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    "aims": [
      "science"
    ],
    "artifactRoles": [
      "research-output",
      "evidence-assessment",
      "method-demonstration",
      "communication"
    ],
    "lineageId": null,
    "accelerationPrimitives": [
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      "adversarial-controls",
      "productive-failure",
      "assurance-vector",
      "agent-readable-research-object"
    ],
    "decisionObject": {
      "type": "executable-open-problem",
      "description": "A fixed-width adjacent-flag and localized quartic research target with exact structural and p-adic evidence, explicit failure boundaries and replayable next obligations.",
      "scope": "The determinant duality, fixed-width LPS reduction, selected quartic identities, localized gauge bridge and bounded p-adic certificates; not the missing uniform radical membership or any target conjecture."
    },
    "bottleneckTargeted": [
      "assurance",
      "publication"
    ],
    "semanticBridge": {
      "state": "explicit",
      "description": "The package maps determinant complement duality through rigidity and inverse-coordinate descent to the adjacent-flag target, distinguishes centered and backward coefficient ideals with an explicit localized transition determinant, and indexes every formal statement, finite gate, non-claim and remaining obligation.",
      "remainingRisks": [
        "The written determinant, flag-descent and p-adic arguments have not been independently reconstructed by specialists.",
        "The centered/backward bridge is localized and does not discharge residual charts or global radical membership.",
        "Finite norm data and same-producer implementations can share a translation or specification defect.",
        "The targeted antecedent search cannot establish absolute novelty or priority."
      ]
    },
    "humanJudgmentGates": [
      "Check the complementary-minor, rigidity and inverse-coordinate bridges as mathematics rather than inferring them from successful replay.",
      "Keep transversality separate from emptiness and zero-dimensionality separate from saturation or radical membership.",
      "Keep centered and backward gauges separate outside the explicit localized chart.",
      "Require external specialist assessment before any target-conjecture, first, priority or field-acceptance language.",
      "Confirm Anonymous scholarly attribution, component licences, immutable identifiers and release authority before publication."
    ],
    "parentLinks": [],
    "assuranceTarget": {
      "dimensions": [
        "independentRerun",
        "independentReimplementation",
        "formalVerification",
        "specialistReview",
        "dataEnvironmentReproducibility",
        "semanticValidation",
        "noveltyAssessment",
        "priorityAssessment"
      ],
      "nextAction": "Obtain an unaffiliated specialist reconstruction of the structural and quartic bridges, a separately authored implementation of the finite predicates, and a proof or decisive obstruction for the slope-four and residual-chart obligations.",
      "claimCeiling": "A producer-side anonymous unrefereed structural and exact-computation candidate with fixed-width reductions, repaired gauge and logic boundaries, paired replay, bounded p-adic certificates and internal editorial closure; not a target-conjecture proof, independent reproduction, formal verification, external specialist or editorial peer review, absolute priority or demonstrated workflow impact."
    },
    "impactClaims": [
      {
        "id": "lps-structural-science-no-impact",
        "aim": "science",
        "outcome": "Faster or more reliable resolution and assurance of difficult linked algebraic conjectures",
        "setting": "AI-assisted structural, symbolic and p-adic research 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
      }
    ]
  }
}