{
  "schemaVersion": "1.3",
  "slug": "furter-r3-through-299",
  "title": "Exact finite rigidity for Furter's R(3): all windows through n=299",
  "shortTitle": "Furter's R(3) through n=299",
  "url": "https://evidencepress.org/releases/furter-r3-through-299/",
  "oneLine": "Exact same-prime special-fibre certificates prove Furter's R(3,n) for every 1 <= n <= 299, while a fixed-prime zero-band theorem and an executable challenge isolate what remains open.",
  "abstract": "Let g_(k,3) be the inverse-series coefficient polynomials associated with X(1+x_1X+x_2X^2+x_3X^3). This anonymous, unrefereed candidate proves by exact computer-assisted certificates that the radical of (g_(d,3),g_(d+1,3),g_(d+2,3)) is (x_1,x_2,x_3) for every 2 <= d <= 300. Equivalently, Furter's R(3,n) holds for every 1 <= n <= 299. Each instance is certified over one finite field across all strata of a weighted-projective cover, and properness transfers the empty special fibre to characteristic zero. The release also proves a periodic Kummer-theoretic zero band showing that no fixed prime can certify every degree by this route, records exact Euler-socle and resultant pilots, and publishes a machine-readable Universal R(3) Challenge. Universal R(3), the unrestricted Strong Factorial Conjecture and the general Polydegree Conjecture remain open.",
  "datePublished": "2026-08-14",
  "dateModified": "2026-08-14",
  "version": "0.1.0",
  "doi": "10.5281/zenodo.21939362",
  "doiUrl": "https://doi.org/10.5281/zenodo.21939362",
  "conceptDoi": "10.5281/zenodo.21939361",
  "pdfUrl": "https://github.com/ipitchford/furter-r3-through-299/releases/download/v0.1.0/exact-finite-rigidity-furter-r3-n1-299-v0.1.0.pdf",
  "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/furter-r3-through-299/v0.1.0/manuscript/paper.pdf",
  "zenodoUrl": "https://zenodo.org/records/21939362",
  "repoUrl": "https://github.com/ipitchford/furter-r3-through-299",
  "releaseUrl": "https://github.com/ipitchford/furter-r3-through-299/releases/tag/v0.1.0",
  "markdownUrl": "https://evidencepress.org/releases/furter-r3-through-299/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/furter-r3-through-299/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/furter-r3-through-299.mp3",
  "imageUrl": "https://evidencepress.org/assets/og/furter-r3-through-299.png",
  "coverArtUrl": "https://evidencepress.org/assets/art/furter-r3-through-299.svg",
  "media": [
    {
      "type": "audio",
      "url": "https://evidencepress.org/assets/audio/furter-r3-through-299.mp3",
      "name": "Audio briefing — Furter's R(3) through n=299",
      "description": "Plain-English synthetic-voice summary of the exact finite theorem, special-fibre mechanism and open universal problem; a communication aid, not additional mathematical evidence.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/furter-r3-through-299.txt"
    }
  ],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Anonymous unrefereed candidate. The finite theorem is presented as an exact theoretical and computer-assisted result. The frozen package, ordinary and optimized checks, root receipt and fresh-extraction replay are terminal, and a targeted final review closed the identified claim error. No unaffiliated party has independently reconstructed the proof, rerun it in a separately controlled environment, or reimplemented the large projective lane; no proof assistant or external specialist has verified the theorem; and no journal has peer reviewed it. The dated public-record search found no earlier published extension of the relevant finite R(3) window beyond the LPS range, but it is not an absolute 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/21939362",
      "note": "The published Zenodo record exposes the PDF, tagged source snapshot, complete frozen evidence ZIP, manifest, verification receipt and executable challenge. This establishes availability and byte identity, not theorem correctness."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "The producer-side root and fresh-extraction replay verify all 299 windows, exact inventories, hashes, controls and ordinary/optimized parity with no tolerance-based acceptance path.",
      "evidenceUrl": "https://github.com/ipitchford/furter-r3-through-299/releases/tag/v0.1.0"
    },
    {
      "dimension": "independentRerun",
      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed",
      "note": "The fresh-extraction receipt belongs to the coordinated producer workflow; no unaffiliated party has reported rerunning the immutable package."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "The package contains implementation diversity on bounded lanes, but no unaffiliated implementation has reconstructed the full d=2,...,300 claim."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "Neither the rigidity dictionary, properness bridge nor finite certificates have 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 specialist in polynomial automorphisms, algebraic geometry or computational commutative algebra has reviewed the complete 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/furter-r3-through-299/blob/v0.1.0/evidence/stage0/ENVIRONMENT.md",
      "note": "The package records and hashes the relevant Python and Singular executables and supplies a fail-closed replay, but no independently rebuilt container, Nix or Guix environment or cross-platform reproduction is reported."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "AI systems under human direction",
      "OpenAI Codex",
      "Anthropic Claude and Fable in producer-coordinated research and review paths"
    ],
    "humanRole": "Research direction, mediation, claim-boundary decisions and publication authorization by the human creator; scholarly attribution remains Anonymous.",
    "disclosure": "AI systems contributed mathematical exploration, code, review, drafting, replay and release engineering in one producer-coordinated programme. Cross-model agreement, exact replay, public archiving and matching hashes are not independent reproduction, external specialist review or editorial peer review."
  },
  "problem": {
    "name": "Furter's polynomial-composition Rigidity Conjecture R(3)",
    "url": "https://doi.org/10.1112/jlms/jdu064"
  },
  "corrections": [],
  "keywords": [
    "polynomial composition rigidity",
    "Furter rigidity conjecture",
    "plane polynomial automorphisms",
    "Polydegree Conjecture",
    "Strong Factorial Conjecture",
    "weighted projective space",
    "computer-assisted proof",
    "computational commutative algebra",
    "unrefereed candidate"
  ],
  "keyResults": [
    "For every integer n with 1 <= n <= 299, Furter's R(3,n) holds; equivalently, the three consecutive inverse-coefficient polynomials have only the origin as a common complex zero for every window d=2,...,300.",
    "For every 2 <= k <= 300, Furter's length-two Polydegree closure equality holds for the multidegrees (4,k) and (k,4).",
    "For each d=50,...,300, one exact same-prime certificate proves the unit ideal on all three strata of a weighted-projective cover; properness transfers the empty special fibre to characteristic zero.",
    "The complete finite theorem covers 299 windows. The d<=49 map records 8,064 exact degree-prime cases; the d=50,...,300 lane records 251 selected GOOD primes after 2,854 deterministic attempts.",
    "For every prime p>=7, a periodic Kummer zero band forces infinitely many BAD windows for the one-prime GOOD-fibre architecture, so no fixed prime can certify every degree by that route.",
    "The release verifies exact rational Euler-socle certificates through d=12 and normalized-resultant data through d=25, but neither pilot is promoted to an all-degree theorem."
  ],
  "reviews": [],
  "evidencePackage": "An eight-page candidate paper and accessible Markdown; a 125-entry frozen Stage 0 evidence package; exact modular colength and projective-stratum computations; rational Euler-socle certificates; normalized resultant data; ordinary and optimized Python parity; independent sparse arithmetic and formal-reversion checks on bounded lanes; fail-closed negative controls; exact source, runtime and input hashes; a terminal root receipt covering all 299 windows; a fresh-extraction package replay receipt; a public source snapshot; a formula-first novelty record; and a machine-readable Universal R(3) Challenge. The largest d=50,...,300 lane retains Singular inside its trusted computing boundary and does not export portable Nullstellensatz cofactors for all 251 degrees.",
  "openProblems": [
    "Independently reconstruct the Furter and LPS index dictionary, the same-prime properness bridge and the exact finite coverage partition from the definitions.",
    "Run the immutable package in a separately controlled environment and publish the receipt, then reimplement the d=50,...,300 projective-stratum verifier without producer code.",
    "Prove or refute the two all-degree canonical Euler-socle identities recorded in the Universal R(3) Challenge, including the exceptional initial-state compatibility.",
    "Construct a finite-prime product automaton whose every reachable state has a GOOD component, or produce a reachable all-BAD state that kills that route.",
    "Formalise the algebraic core and the properness transfer in Lean or another proof assistant with an explicit statement-fidelity audit.",
    "Obtain external specialist and editorial review while keeping corrections additive and public."
  ],
  "relatedWorks": [
    {
      "citation": "Furter, J.-P. (2015). Polynomial composition rigidity and plane polynomial automorphisms. Journal of the London Mathematical Society, 91(3), 743-760.",
      "url": "https://doi.org/10.1112/jlms/jdu064"
    },
    {
      "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": "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"
    },
    {
      "citation": "Scheja, G., & Storch, U. (1975). Über Spurfunktionen bei vollständigen Durchschnitten. Journal für die reine und angewandte Mathematik, 278/279, 174-190.",
      "url": "https://eudml.org/doc/151652"
    },
    {
      "citation": "Adamczewski, B., Bostan, A., & Caruso, X. (2023). A sharper multivariate Christol theorem with applications to diagonals and algebraic power series.",
      "url": "https://arxiv.org/abs/2306.02640"
    },
    {
      "citation": "Rowland, E., & Yassawi, R. (2015). Automatic congruences for diagonals of rational functions. Journal de Théorie des Nombres de Bordeaux, 27(1), 245-288.",
      "url": "https://arxiv.org/abs/1310.8635"
    },
    {
      "citation": "Anonymous (2026). A Jacobian smooth-point criterion and the full e=3 column of the Polydegree Conjecture. Unrefereed candidate, Evidence Press.",
      "url": "https://doi.org/10.5281/zenodo.21909085"
    }
  ],
  "operatingModel": {
    "version": "1.0",
    "workId": "ep-work:furter-r3-through-299",
    "attemptIds": [
      "ep-attempt:furter-r3-through-299"
    ],
    "aims": [
      "science"
    ],
    "artifactRoles": [
      "research-output",
      "method-demonstration",
      "communication"
    ],
    "lineageId": null,
    "accelerationPrimitives": [
      "certificate-first",
      "structural-compression",
      "regime-stitching",
      "adversarial-controls",
      "counterexample-proxy-first",
      "assurance-vector",
      "agent-readable-research-object"
    ],
    "decisionObject": {
      "type": "executable-open-problem",
      "description": "A terminal exact certificate for R(3,n), 1<=n<=299, paired with a machine-readable statement of the remaining universal obligations, falsifiers and acceptance criteria.",
      "scope": "Furter's three-variable polynomial-composition rigidity conjecture and its bounded length-two Polydegree and restricted Strong Factorial consequences."
    },
    "bottleneckTargeted": [
      "discovery",
      "assurance",
      "publication"
    ],
    "semanticBridge": {
      "state": "explicit",
      "description": "The paper fixes n=d-1, identifies the radical statement with Furter's one-map and two-map assertions, separates one same-prime projective certificate from mixed-prime nonproofs, and states the exact consequence boundaries for Polydegree and Strong Factorial claims.",
      "remainingRisks": [
        "The Furter and LPS dictionary has not been formally verified or independently reconstructed by an unaffiliated specialist.",
        "The large finite lane keeps Singular inside its trusted computing boundary and lacks portable sparse Nullstellensatz cofactors for all 251 new degrees.",
        "Finite coverage through n=299 supplies no monotonicity or all-degree argument."
      ]
    },
    "humanJudgmentGates": [
      "Decide whether the finite theorem's mathematical exposition is adequate for specialist scrutiny despite the retained CAS boundary.",
      "Decide whether the bounded novelty search warrants the calibrated novelty language without an absolute priority claim.",
      "Keep the universal conjecture open unless an all-degree structural proof passes statement-fidelity and independent-assurance gates."
    ],
    "parentLinks": [
      {
        "relation": "reuses-method",
        "workId": null,
        "legacyReleaseSlug": null,
        "externalUrl": "https://doi.org/10.1016/j.jpaa.2019.04.002",
        "inheritedClaim": "Lewis, Perry and Straub provide the inverse-coefficient window architecture and the earlier finite d<50 computation against which this release is indexed.",
        "inheritedAssuranceCeiling": "Published peer-reviewed antecedent for the cited reduction and finite range; it does not independently verify this release's d=50,...,300 certificates or universal challenge."
      }
    ],
    "assuranceTarget": {
      "dimensions": [
        "independentRerun",
        "independentReimplementation",
        "formalVerification",
        "specialistReview",
        "semanticValidation",
        "noveltyAssessment",
        "priorityAssessment"
      ],
      "nextAction": "An unaffiliated specialist team should reconstruct the Furter/LPS/properness bridges, rerun the immutable package, independently implement the large finite lane and then attack one all-degree challenge route.",
      "claimCeiling": "An exact producer-side computer-assisted theorem for the 299 stated finite instances, with public replay and bounded novelty evidence; not universal R(3), the unrestricted Strong Factorial Conjecture, the general Polydegree Conjecture, independent reconstruction, formal verification, peer review, absolute priority or demonstrated research acceleration."
    },
    "impactClaims": [
      {
        "id": "science-no-impact",
        "aim": "science",
        "outcome": "Faster or more reliable resolution and assurance of open polynomial-rigidity problems",
        "setting": "AI-assisted exact computation, structural route finding and publication for Furter's R(3)",
        "status": "NO_IMPACT_EVIDENCE",
        "designClass": "none",
        "comparator": "No matched conventional research or publication workflow comparator was registered.",
        "estimand": "No effect on discovery time, human effort, compute, error rate, proof quality, independent-assurance time or scientific impact was estimated.",
        "evidenceRefs": [],
        "registeredDesignRef": null,
        "independentAssessment": null
      }
    ]
  }
}