{ "schemaVersion": "1.4", "slug": "full-e4-polydegree-column", "title": "Effective Fourier certificates for the full e=4 Polydegree column", "shortTitle": "The full e=4 Polydegree column", "url": "https://evidencepress.org/releases/full-e4-polydegree-column/", "oneLine": "An exact residue branch, 14,985 outward-rounded finite certificates and a uniform Fourier envelope yield the e=4 Polydegree containment for every degree d at least 2.", "abstract": "Lewis, Perry and Straub proved the e=4 Polydegree containment for 2 <= d < 20. This anonymous, unrefereed candidate presents an all-degree continuation: for every integer d >= 2, G_(d+4) is contained in the closure of G_(d,5). Writing d=4m+r, the proof uses the published low-degree range, an exact solution for r=1, 14,985 outward-rounded FLINT/Arb certificates for 5 <= m < 5000 in the other residues, and an explicit analytic envelope for every m >= 5000. The package also contains a Lean-checked universal bordered-Jacobian identity over arbitrary commutative rings and a separate conditional finite-pencil boundary-norm transfer theorem. Normal and optimized replay, semantic mutation controls, a pinned Lean build, manifests and internal AI editorial review pass. These are producer-side assurances. No independent implementation, external specialist review, journal peer review or absolute priority determination is claimed. The bundle does not prove Furter's R(3), monotone rigidity, the two-dimensional Jacobian conjecture or the quartic Hessian conjecture.", "datePublished": "2026-08-23", "dateModified": "2026-08-23", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.22072044", "doiUrl": "https://doi.org/10.5281/zenodo.22072044", "conceptDoi": "10.5281/zenodo.22072043", "pdfUrl": "https://github.com/ipitchford/full-e4-polydegree-column/releases/download/v0.1.0-candidate/full-e4-polydegree-column-v0.1.0-candidate.pdf", "altPdfUrl": "https://zenodo.org/records/22072044/files/full-e4-polydegree-column-v0.1.0-candidate.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/22072044", "repoUrl": "https://github.com/ipitchford/full-e4-polydegree-column", "releaseUrl": "https://github.com/ipitchford/full-e4-polydegree-column/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/full-e4-polydegree-column/index.md", "bibtexUrl": "https://evidencepress.org/releases/full-e4-polydegree-column/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/full-e4-polydegree-column.mp3", "imageUrl": "https://evidencepress.org/assets/og/full-e4-polydegree-column.png", "coverArtUrl": "https://evidencepress.org/assets/art/full-e4-polydegree-column.svg?v=164b888ad7", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/full-e4-polydegree-column.mp3", "name": "Audio briefing -- the full e=4 Polydegree column", "description": "Plain-English synthetic-voice summary of the degree stitch, companion papers and assurance boundary; a communication aid, not additional mathematical evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/full-e4-polydegree-column.txt" }, { "type": "video", "url": "https://youtu.be/2u27-f2ocOs", "name": "Stitching Infinity The e=4 Polydegree Certificate", "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 three-paper candidate. The main all-degree e=4 containment is presented as a theoretical and computer-assisted theorem. Its exact, finite-interval and eventual regimes pass producer-side replay, structural audit, cross-version CI and deliberate mutation controls. The arbitrary-ring bordered-Jacobian companion is kernel-checked in the pinned Lean environment; that formal-verification claim does not extend to the analytic or boundary papers. The boundary transfer theorem is conditional on an explicit geometric package. Internal five-role AI review accepted the frozen candidate after repair, but this is not external human review. No unaffiliated rerun, independent implementation, external specialist review, journal peer review or absolute priority determination has occurred. The bounded search supports describing the e=4 result as plausibly highly original and significant in its narrow area, not as first or world-leading." }, "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/22072044", "note": "The tagged GitHub prerelease and published Zenodo record expose the source package, three PDFs and checksum ledger; independent public downloads from both services matched SHA-256 byte for byte." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "Python 3.12 and 3.14 normal and optimized replays, boundary checks, package integrity, three semantic mutation controls, the pinned Lean build and its axiom audit all passed.", "evidenceUrl": "https://github.com/ipitchford/full-e4-polydegree-column/actions/runs/32663897433" }, { "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 finite ledger, eventual verifier, Lean package and boundary checks." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "The structural auditor and optimized paths remain inside the producer workflow; no separately authored checker has reconstructed the coefficient, interval or analytic predicates." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "partial", "evidenceUrl": "https://github.com/ipitchford/full-e4-polydegree-column/tree/v0.1.0-candidate/formal/BorderedJacobianUniversal", "note": "Lean kernel-checks the scoped universal bordered-Jacobian theorems over arbitrary commutative rings. The Polydegree theorem, Fourier persistence argument and boundary geometry are not formalized." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "The archived five-role reports are internal AI editorial quality control; no external polynomial-automorphism, validated-numerics, formalization or boundary-geometry specialist has reviewed the exact package." }, { "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/full-e4-polydegree-column/blob/v0.1.0-candidate/REPRODUCIBILITY.md", "note": "Direct Python dependencies, Lean and Mathlib versions, manifests, receipts and CI are pinned, but no independently recreated container, Nix or Guix environment is reported." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI systems under human direction", "OpenAI Codex", "Lean 4.32.1", "Mathlib v4.32.1", "python-flint 0.8.0" ], "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 source discovery, proof exploration, code, Lean implementation, review, editing, replay and release engineering in one producer-coordinated workflow. Model agreement, exact replay, formal checking of one companion theorem and matching public hashes are not independent reproduction, external specialist review or editorial peer review." }, "attributionBoundary": null, "problem": { "name": "Polydegree Conjecture for plane polynomial automorphisms", "url": "https://doi.org/10.1016/j.jpaa.2019.04.002" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Authored for the reader-first schema 2.0 against immutable GitHub and Zenodo identities and the frozen public assurance record.", "grandfatheredAtSchemaVersion": null, "keywords": [ "plane polynomial automorphisms", "Polydegree Conjecture", "Fourier asymptotics", "Newton--Kantorovich theorem", "interval arithmetic", "FLINT Arb", "Lean formalization", "bordered Jacobian", "computer-assisted mathematics", "unrefereed candidate" ], "keyResults": [ "For every integer d at least 2, G_(d+4) is contained in the closure of G_(d,5), giving the candidate full e=4 Polydegree column.", "The all-degree stitch combines the published range 2 <= d < 20, an exact residue-one solution, 14,985 FLINT/Arb cases for 5 <= m < 5000 in residues 0, 2 and 3, and a uniform analytic certificate for m >= 5000.", "An explicit four-mode Fourier system supplies transverse limiting zeros; exact rational coefficient bounds and 256-bit Arb evaluation yield the eventual Newton--Kantorovich inequalities.", "Lean 4.32.1 and Mathlib v4.32.1 check the signed maximal-minor and bordered-determinant identities for binary-form multiplication over every commutative ring, including zero divisors and degree-zero boundary cases.", "A separate companion proves a cover-degree-free norm lemma, a finite-pencil equivalence and a conditional boundary-norm transfer theorem, with a six-sheet application and a sharp limitation of the method." ], "reviews": [], "evidencePackage": "Three anonymous candidate manuscripts comprising 25 pages and accessible Markdown; exact coefficient normalization; a 14,985-row outward-rounded FLINT/Arb ledger and byte-identical optimized replay ledger; an exact-rational eventual envelope with 256-bit Arb only at cancellation-sensitive Fourier constants; producer and independent structural audit receipts; Python 3.12 and 3.14 CI in normal and optimized modes; three detected semantic byte mutations; a pinned Lean 4.32.1 and Mathlib v4.32.1 package with no sorry, custom axioms or warnings; exact boundary algebra replay; claim, provenance, source, priority, licence and assurance records; two rounds of five-role internal AI editorial review; deterministic manifests; immutable GitHub and Zenodo assets downloaded and matched byte for byte. These establish scoped formal checking, producer-side replay, integrity and availability, not independent reconstruction of the analytic proof or external peer review.", "openProblems": [ "Have unaffiliated specialists reconstruct the Lewis--Perry--Straub bridge, exact normalization, Fourier zeros and Newton--Kantorovich envelope from the definitions.", "Write a separately authored interval implementation and rerun all 14,985 immutable finite cases plus the eventual inequalities without producer intermediates.", "Formalize the Polydegree smooth-zero reduction and analytic persistence theorem, extending the current Lean assurance beyond the bordered-Jacobian companion.", "Obtain external specialist reviews of the complete immutable package and publish objections, corrections or confirmations as versioned successors.", "Investigate e=5 and other fixed columns without inferring Furter's R(3), monotone rigidity or an all-e theorem from the e=3 and e=4 cases.", "Determine where the boundary-norm hypotheses hold beyond the verified six-sheet companion and whether different boundary invariants bypass the finite-pencil limitation." ], "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(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": "Furter, J.-P. (2009). Plane polynomial automorphisms of fixed multidegree. Mathematische Annalen, 343, 901--920.", "url": "https://doi.org/10.1007/s00208-008-0296-2" }, { "citation": "Anonymous. (2026). A Jacobian smooth-point criterion and the full e=3 column of the Polydegree Conjecture (Version 0.1.0-candidate).", "url": "https://doi.org/10.5281/zenodo.21909085" } ], "operatingModel": { "version": "1.0", "workId": "ep-work:full-e4-polydegree-column", "attemptIds": [ "ep-attempt:full-e4-polydegree-column" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "evidence-assessment", "method-demonstration", "communication" ], "lineageId": null, "accelerationPrimitives": [ "certificate-first", "structural-compression", "regime-stitching", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "certificate", "description": "A gap-free all-degree e=4 Polydegree certificate joining published low-degree input, an exact residue branch, 14,985 outward-rounded finite cases and a uniform analytic tail, with scoped Lean and boundary-theorem companions.", "scope": "The containment G_(d+4) contained in the closure of G_(d,5) for every integer d at least 2; not Furter's R(3), monotone rigidity, the full Polydegree Conjecture, JC2 or HC4." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "The manuscript maps the Lewis--Perry--Straub specialization criterion to a smooth-zero problem, gives the exact coefficient normalization and degree partition, and states the interval and eventual predicates checked by the public programs. The Lean companion has a separate theorem-to-declaration concordance, and the boundary companion states every geometric hypothesis before the finite-pencil transfer.", "remainingRisks": [ "The analytic manuscript, finite checker and structural auditor share producer-side mathematical lineage and may share a translation error.", "The Lewis--Perry--Straub implication, Fourier-limit derivation, tail bounds and Newton constants require independent specialist reconstruction.", "Formal verification covers only the bordered-Jacobian companion, not the Polydegree theorem or boundary geometry.", "The bounded literature audit cannot establish absolute priority." ] }, "humanJudgmentGates": [ "Assess the specialization, normalization, Fourier and Newton--Kantorovich bridges as mathematics rather than inferring them from successful replay.", "Keep the scoped Lean theorem separate from the unformalized analytic and geometric papers.", "Keep the conditional boundary transfer separate from any claim about arbitrary Keller maps.", "Require external specialist assessment before first, priority, world-leading, Furter R(3), JC2 or HC4 language.", "Confirm Anonymous scholarly attribution, component licences, immutable identifiers and release authority before publication." ], "parentLinks": [ { "relation": "reuses-method", "workId": "ep-work:polydegree-full-e3-column", "legacyReleaseSlug": null, "externalUrl": null, "inheritedClaim": "The earlier full-e3 candidate supplies a methodological precedent for the smooth-point, Fourier-persistence, exact-eventual-envelope and finite-Arb regime architecture; its theorem is not used as a premise for e=4.", "inheritedAssuranceCeiling": "The earlier release is an anonymous unrefereed producer-side candidate. Method reuse is not independent confirmation of either column and can propagate shared defects." } ], "assuranceTarget": { "dimensions": [ "independentRerun", "independentReimplementation", "formalVerification", "specialistReview", "dataEnvironmentReproducibility", "semanticValidation", "noveltyAssessment", "priorityAssessment" ], "nextAction": "Obtain an unaffiliated specialist reconstruction of the Polydegree and analytic bridges and a separately authored interval implementation, then extend formalization from the bordered-Jacobian companion to the smooth-zero and persistence theorems.", "claimCeiling": "A producer-side theoretical and computer-assisted candidate for the full e=4 Polydegree column, with scoped Lean formalization, exact replay, bounded novelty evidence and immutable public assets; not independent reproduction, complete formal verification, external specialist or editorial peer review, Furter's R(3), monotone rigidity, JC2, HC4, absolute priority or demonstrated workflow impact." }, "impactClaims": [ { "id": "polydegree-e4-science-no-impact", "aim": "science", "outcome": "Faster or more reliable resolution and assurance of difficult all-parameter mathematical problems", "setting": "AI-assisted theoretical, computer-assisted and partially formalized work on the e=4 Polydegree column", "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 } ] } }