{ "schemaVersion": "1.6", "slug": "cgf-majorisation-counterexample", "title": "Positive coefficients, failed majorisation", "shortTitle": "Positive coefficients, failed majorisation", "url": "https://evidencepress.org/releases/cgf-majorisation-counterexample/", "oneLine": "A cyclotomic generating function has strictly positive coefficients, yet its largest parameters break a conjectured ordering—even while every lower-prefix inequality holds.", "abstract": "An explicitly specified primorial cyclotomic polynomial, multiplied by a sufficiently large explicit power of 1+q, refutes the upper-suffix assertion of Billey and Swanson’s Conjecture 33. The largest 5591 numerator parameters sum to less than the corresponding denominator parameters. The construction connects a signed divisor hinge to a known negative smoothed Möbius sum at 18350. A uniform analytic argument establishes all-coefficient positivity; exact finite arithmetic establishes the defect. Cancelling common parameters preserves the obstruction, while the original padded presentation satisfies every lower-prefix inequality. The universal prefix assertion remains unresolved.", "datePublished": "2026-10-09", "dateModified": "2026-10-09", "version": "1.1.0-candidate", "doi": "10.5281/zenodo.23270364", "doiUrl": "https://doi.org/10.5281/zenodo.23270364", "conceptDoi": "10.5281/zenodo.23270363", "pdfUrl": "https://github.com/ipitchford/cgf-majorisation-counterexample/releases/download/v1.1.0-candidate/cgf-majorisation-counterexample-v1.1.0-candidate.pdf", "altPdfUrl": "https://zenodo.org/records/23270364/files/cgf-majorisation-counterexample-v1.1.0-candidate.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23270364", "repoUrl": "https://github.com/ipitchford/cgf-majorisation-counterexample", "releaseUrl": "https://github.com/ipitchford/cgf-majorisation-counterexample/releases/tag/v1.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/cgf-majorisation-counterexample/index.md", "bibtexUrl": "https://evidencepress.org/releases/cgf-majorisation-counterexample/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/cgf-majorisation-counterexample.mp3", "imageUrl": "https://evidencepress.org/assets/og/cgf-majorisation-counterexample.png?v=b2797ed30c", "coverArtUrl": "https://evidencepress.org/assets/art/cgf-majorisation-counterexample.svg?v=71169e8848", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/cgf-majorisation-counterexample.mp3", "name": "Audio briefing", "description": "Synthetic speech; communication, not additional mathematical evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/cgf-majorisation-counterexample.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed written-proof candidate with exact internal arithmetic replay. No external peer review, unaffiliated reproduction, formal verification, minimality or absolute priority is established. The supplied model-assisted review is not authenticated external peer review." }, "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/23270364", "note": "Public GitHub and Zenodo assets verified by full downloads against the frozen archive; availability is not proof or 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": "Clean-extraction certificate regeneration, trial-factorisation check and eight corrupted-certificate controls pass normally and under optimized Python. Analytic positivity is a written proof, not a computationally verified theorem.", "evidenceUrl": "https://github.com/ipitchford/cgf-majorisation-counterexample/blob/v1.1.0-candidate/VERIFICATION.md" }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No qualifying evidence establishing this dimension." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "No qualifying evidence establishing this dimension." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No qualifying evidence establishing this dimension." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "No qualifying evidence establishing this dimension." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No qualifying evidence establishing this dimension." }, { "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/cgf-majorisation-counterexample/blob/v1.1.0-candidate/ENVIRONMENT.txt", "note": "Python standard-library checkers and standalone LaTeX supplied; external reproduction not established." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted supplied research; exact originating model not established", "OpenAI Codex assistant: review actions and publication" ], "humanRole": "Publication authority only; no human research contribution asserted.", "reviewStatus": "Unrefereed written-proof candidate with exact internal arithmetic replay. No external peer review, unaffiliated reproduction, formal verification, minimality or absolute priority is established." }, "attributionBoundary": null, "problem": { "name": "Billey–Swanson Conjecture 33: two-sided majorisation", "url": "https://arxiv.org/html/2305.07620v3" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "User-supplied candidate and model-assisted review, with bounded publication revisions on 9 October 2026. No historical research clock or reviewer independence reconstructed.", "grandfatheredAtSchemaVersion": null, "keywords": [ "cyclotomic generating functions", "majorisation", "Möbius function", "coefficient positivity", "counterexample" ], "keyResults": [ "Strict coefficient positivity coexists with a reversed upper-suffix comparison of length 5591.", "The obstruction survives cancellation to disjoint parameter supports.", "Every lower-prefix inequality holds for this example; the universal prefix question is not settled.", "A general hinge obstruction is preserved by sufficiently large positivity multipliers." ], "reviews": [], "evidencePackage": "Written analytic positivity proof, exact arithmetic certificate, two separately implemented checkers, eight corruption controls, supplied review and itemised response, source comparison and AI index.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Billey and Swanson (2024), Cyclotomic Generating Functions. Conjecture 33 is the exact target; the suffix component is refuted, not the universal prefix component.", "url": "https://doi.org/10.37236/12687", "doi": "10.37236/12687" }, { "citation": "Gatzweiler, Levicán-Santibáñez and Yoshida (2026), Cyclotomic generating functions, empty weighted complete intersections and positivity. Theorem 4.7 and Corollary 4.8 provide inherited positivity multiplication.", "url": "https://arxiv.org/abs/2603.22226v1", "doi": "10.48550/arXiv.2603.22226" }, { "citation": "Daval (2019), doctoral thesis. Remark 21, printed page 81, records the negative smoothed Möbius endpoint used here.", "url": "https://pepite-depot.univ-lille.fr/LIBRE/EDSPI/2019/50376-2019-Daval.pdf", "doi": null }, { "citation": "Powers and Reznick (2001), A new bound for Pólya’s theorem with applications to polynomials positive on polyhedra. Classical quantitative positivity framework.", "url": "https://doi.org/10.1016/S0022-4049(00)00155-9", "doi": "10.1016/S0022-4049(00)00155-9" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:cgf-majorisation-counterexample", "attemptIds": [ "ep-attempt:cgf-majorisation-counterexample-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "evidence-assessment", "communication" ], "lineageId": null, "parentLinks": [], "accelerationPrimitives": [ "certificate-first", "counterexample-proxy-first", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "counterexample", "description": "Strict coefficient positivity without the proposed upper-tail parameter ordering.", "scope": "Billey–Swanson Conjecture 33 suffix assertion; original padded example satisfies lower prefixes." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Möbius divisor identity gives a negative hinge; denominator threshold count converts it into a same-rank suffix failure; low padding enforces positivity without changing the hinge.", "remainingRisks": [ "Written positivity and semantic arguments are not formally verified.", "Priority is supported only by bounded search." ] }, "humanJudgmentGates": [ "External specialist assessment remains open; no human research judgement claimed." ], "assuranceTarget": { "dimensions": [ "specialistReview", "formalVerification" ], "nextAction": "Scrutinise the positivity-to-tail separation and investigate the remaining universal prefix assertion.", "claimCeiling": "Unrefereed written-proof candidate with exact internal arithmetic replay. No external peer review, unaffiliated reproduction, formal verification, minimality or absolute priority is established." }, "impactClaims": [ { "id": "cgf-majorisation-counterexample-no-impact", "aim": "science", "outcome": "Inspectable conjecture counterexample", "setting": "Producer-coordinated mathematical publication", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "None.", "estimand": "No measured research acceleration or downstream benefit.", "evidenceRefs": [], "registeredDesignRef": null, "independentAssessment": null } ] }, "researchMetrics": { "schemaVersion": "1.0", "policyUrl": "https://evidencepress.org/api/research-metrics-policy.json", "workLedgerUrl": "https://evidencepress.org/api/work-ledger.json", "claimCeiling": "These records describe scoped workflow inputs and outcomes. They support calibration and reference-class construction, but do not by themselves establish research acceleration, theorem correctness, novelty, independent assurance or impact.", "attempts": [ { "attemptId": "ep-attempt:cgf-majorisation-counterexample-publication", "status": "result-ready", "resultClass": "positive", "metrics": { "schemaVersion": "1.0", "measurementScope": "assurance-through-publication", "scopeBoundary": "Remaining publication from registration; earlier research and initial revisions are left-censored. No discovery duration reconstructed.", "forecast": { "frozenAt": "2026-10-09T20:32:50.269Z", "procedureClass": "cgf-publication", "targetOutcome": "Immutable research publication and canonical site readback", "expectedActiveMinutes": 60, "plausibleLowMinutes": 45, "plausibleHighMinutes": 75, "expectedUnattendedWaitMinutes": 15, "referenceClass": { "label": "Full candidate publication", "sampleSize": 0, "basis": "Procedure estimate, not observed benchmark" }, "fermiComponents": [ { "component": "Revisions, archive, media, providers and guarded site cycles", "count": 1, "lowMinutesPerUnit": 45, "centralMinutesPerUnit": 60, "highMinutesPerUnit": 75, "basis": "Mandatory remaining work" } ], "probabilityPositiveSignal": 0.95, "probabilityTargetClosure": 0.85, "probabilityHorizonMinutes": 120, "assumptions": [ "No new integrity failure; providers remain accessible." ], "reforecastTriggers": [ "Provider or CI failure." ], "stopRule": "No hard cap; complete mandatory gates or report a precise blocker." }, "outcome": null }, "missingnessReason": null } ] } }