{ "schemaVersion": "1.6", "slug": "williamson-order-transfer", "title": "Order transfer through six for Williamson low-storage Runge–Kutta methods", "shortTitle": "Classical accuracy survives the exponential lift", "url": "https://evidencepress.org/releases/williamson-order-transfer/", "oneLine": "A division-free tensor identity transfers classical order through six; three explicit extra tests give a conditional criterion at order seven.", "abstract": "This anonymous mathematical candidate proves that every finite real Williamson low-storage Runge–Kutta coefficient list of classical order at most six retains at least that order under the specified left-multiplication exponential lift. An all-degree repeated-tail identity, polarisation and finite shuffle systems reduce ordered-flow coefficients to classical moments. The result covers zero and negative coefficients and repeated stage nodes. A conditional order-seven criterion reduces the next-order question to three extra tensor identities; automatic seventh-order transfer remains open. Written proofs are accompanied by exact internal replay, separate checker implementations and corruption controls. No external specialist review, independent external reproduction or formal verification is claimed.", "datePublished": "2026-10-20", "dateModified": "2026-10-20", "version": "1.0.0-candidate", "doi": "10.5281/zenodo.23501588", "doiUrl": "https://doi.org/10.5281/zenodo.23501588", "conceptDoi": null, "pdfUrl": "https://github.com/ipitchford/williamson-order-transfer/releases/download/v1.0.0-candidate/williamson-order-transfer-1.0.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23501588", "repoUrl": "https://github.com/ipitchford/williamson-order-transfer", "releaseUrl": "https://github.com/ipitchford/williamson-order-transfer/releases/tag/v1.0.0-candidate", "markdownUrl": "https://evidencepress.org/releases/williamson-order-transfer/index.md", "bibtexUrl": "https://evidencepress.org/releases/williamson-order-transfer/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/williamson-order-transfer.mp3", "imageUrl": "https://evidencepress.org/assets/og/williamson-order-transfer.png?v=4cff819dd4", "coverArtUrl": "https://evidencepress.org/assets/art/williamson-order-transfer.svg?v=3ab18f91c8", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/williamson-order-transfer.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/williamson-order-transfer.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted mathematical candidate. Internal proofs and checks have the stated scope; external specialist review, independent external reproduction and formal verification have not been established." }, "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/23501588", "note": "The versioned research package and manuscript are publicly archived on GitHub and Zenodo. Provider asset parity was checked against frozen local files; archival availability does not establish scientific correctness or external peer review." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "Written universal arguments, 197 ordered-tree coefficient polynomials, exact rational checks, independent internal prefix and rank implementations, optimisation-safe replay, corruption controls and conditional order-seven enumeration." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated rerun has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "partial", "note": "Separate internal checks remain within the coordinated AI-assisted workflow, not external reproduction." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant formalisation." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "AI-assisted review is 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 external editorial peer review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "Code and environment are supplied; empirical checks have the recorded bounded scope." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical research", "OpenAI Codex: review, revision, bounded extensions and publication" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted assessment, not external peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the revised self-contained manuscript and scoped internal verification records. Versioned GitHub and Zenodo research archives were checked against the frozen release assets on 20 October 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "Runge–Kutta methods", "Lie-group integrators", "Williamson low storage", "order conditions", "exact certificates" ], "keyResults": [ "Automatic order transfer for every finite real Williamson list satisfying classical order conditions through any p≤6.", "A division-free repeated-tail tensor identity valid at every tensor length.", "Three explicit additional tensor identities suffice at order seven; automatic transfer at that order remains unproved." ], "reviews": [], "evidencePackage": "Written universal arguments, 197 ordered-tree coefficient polynomials, exact rational checks, independent internal prefix and rank implementations, optimisation-safe replay, corruption controls and conditional order-seven enumeration.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Bazavov (2022), Commutator-free Lie group methods with minimum storage requirements and reuse of exponentials.", "url": "https://doi.org/10.1007/s10543-021-00892-x", "doi": "10.1007/s10543-021-00892-x" }, { "citation": "Owren (2006), Order conditions for commutator-free Lie group methods.", "url": "https://doi.org/10.1088/0305-4470/39/19/S15", "doi": "10.1088/0305-4470/39/19/S15" }, { "citation": "Lystad (2026), All-stage order-three transfer for Williamson 2N-storage Runge–Kutta methods.", "url": "https://doi.org/10.5281/zenodo.21764741", "doi": "10.5281/zenodo.21764741" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:williamson-order-transfer", "attemptIds": [ "ep-attempt:williamson-order-transfer-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "certificate-first", "structural-compression", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "reusable-method", "description": "A division-free tensor identity transfers classical order through six; three explicit extra tests give a conditional criterion at order seven.", "scope": "Finite-stage real Williamson recurrences and the stated left-multiplication exponential lift; no claim for arbitrary Runge–Kutta tableaus or all orders." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Finite-stage real Williamson recurrences and the stated left-multiplication exponential lift; no claim for arbitrary Runge–Kutta tableaus or all orders.", "remainingRisks": [ "Written arguments require external specialist scrutiny.", "The bounded literature check is not exhaustive priority certification." ] }, "humanJudgmentGates": [ "Publication is authorised; scientific endorsement is not inferred.", "External specialist review and independent reproduction remain open." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "independentRerun", "specialistReview", "formalVerification", "priorityAssessment" ], "nextAction": "Independently examine the written reductions and replay the scoped checks.", "claimCeiling": "Finite-stage real Williamson recurrences and the stated left-multiplication exponential lift; no claim for arbitrary Runge–Kutta tableaus or all orders. 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