{ "schemaVersion": "1.6", "slug": "sharp-rank-three-chollet-order-five", "title": "Sharp rank-three Chollet inequalities in order five", "shortTitle": "The sharp rank-three constant", "url": "https://evidencepress.org/releases/sharp-rank-three-chollet-order-five/", "oneLine": "An exact rational certificate proves the sharp order-five rank-three permanent inequality, classifies equality, and yields the sharp two-matrix form.", "abstract": "For every real 5 by 5 correlation matrix C of rank at most three, this computer-assisted mathematical candidate proves per(C circle C) at most 13/48 times per(C) squared, with equality exactly at signed simultaneous permutations of J2 direct-sum E3. It also proves the sharp mixed inequality per(A circle B) at most 13/48 per(A)per(B) for real positive-semidefinite rank-at-most-three matrices and classifies all equality cases. The proof combines a written Gram reduction with a symmetry-adapted rational sum-of-squares certificate and exact standard-library replay. The certificate-construction pipeline is unavailable, and no external specialist review or formal verification is claimed.", "datePublished": "2026-10-19", "dateModified": "2026-10-19", "version": "1.0.0", "doi": "10.5281/zenodo.23474553", "doiUrl": "https://doi.org/10.5281/zenodo.23474553", "conceptDoi": "10.5281/zenodo.23474552", "pdfUrl": "https://github.com/ipitchford/sharp-rank-three-chollet-order-five/releases/download/v1.0.0/sharp-rank-three-chollet-order-five-v1.0.0.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23474553", "repoUrl": "https://github.com/ipitchford/sharp-rank-three-chollet-order-five", "releaseUrl": "https://github.com/ipitchford/sharp-rank-three-chollet-order-five/releases/tag/v1.0.0", "markdownUrl": "https://evidencepress.org/releases/sharp-rank-three-chollet-order-five/index.md", "bibtexUrl": "https://evidencepress.org/releases/sharp-rank-three-chollet-order-five/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/sharp-rank-three-chollet-order-five.mp3", "imageUrl": "https://evidencepress.org/assets/og/sharp-rank-three-chollet-order-five.png?v=23394ef8e0", "coverArtUrl": "https://evidencepress.org/assets/art/sharp-rank-three-chollet-order-five.svg?v=c5033afb62", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/sharp-rank-three-chollet-order-five.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/sharp-rank-three-chollet-order-five.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted computer-assisted mathematical candidate. Exact producer replay and separate internal reimplementation are established for the specified finite calculations; certificate generation, external specialist review, formal verification and exhaustive priority assessment are not." }, "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/23474553", "note": "The frozen ZIP, standalone PDF and checksum manifest are publicly archived on GitHub and Zenodo with byte-for-byte parity. Archiving does not establish mathematical correctness." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "Ordinary and optimised exact replay, mutation tests and the mixed regression passed on archive SHA-256 9af25b8aa35c3d4e4341e10147180943fd73b3fed97ad28d2386a924c8b46ae2. This establishes the encoded finite checks, not the full prose proof independently.", "evidenceUrl": "https://github.com/ipitchford/sharp-rank-three-chollet-order-five/blob/v1.0.0/ASSURANCE.md" }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated execution has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "partial", "evidenceUrl": "https://github.com/ipitchford/sharp-rank-three-chollet-order-five/blob/v1.0.0/independent_replay.json", "note": "A separately organised internal implementation reproduced the decisive finite rational calculations. It remained within the same AI-coordinated workflow and is 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 has been established." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "Internal AI-assisted mathematical 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 journal or external editorial peer review has been established." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "not-assessed", "note": "The finite certificate replays without external data, but its construction cannot be regenerated from the released materials." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical research and certificate construction", "OpenAI Codex: review actioning, extension proof, checking and publication" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted mathematical and adversarial 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 final version 1.0.0 manuscript and archive at Git commit d4b3aa01ef6fc33c94f2fc81a5bdbb64f88172f7. GitHub and Zenodo carry the same ZIP, PDF and checksum manifest. Publication metadata distinguishes exact producer replay, separate internal reimplementation, unavailable construction history and absent external assurance.", "grandfatheredAtSchemaVersion": null, "keywords": [ "permanent inequalities", "positive-semidefinite matrices", "rank-three correlation matrices", "sum-of-squares certificate", "computer-assisted proof" ], "keyResults": [ "The sharp coefficient for real order-five correlation matrices of rank at most three is 13/48.", "Self-product equality holds exactly at signed simultaneous permutations of J2 direct-sum E3.", "The same sharp constant holds for two real positive-semidefinite rank-at-most-three matrices, with a complete common-partition equality classification.", "A rational sum-of-squares certificate reduces the remaining polynomial to multiples of 4 by 4 Gram minors, which vanish at rank at most three." ], "reviews": [], "evidencePackage": "Written Gram reduction, explicit rational sum-of-squares certificate, standard-library exact checker, three corruption tests, mixed-equality regression and a separate implementation of the decisive exact calculations. Frozen archive SHA-256 9af25b8aa35c3d4e4341e10147180943fd73b3fed97ad28d2386a924c8b46ae2; GitHub and Zenodo assets were checked byte for byte.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Zeng, Liu and Ratnavelu (2026), Sharp self-Chollet inequalities in order five: Rank two and structured rank three. Candidate coefficient and witness.", "url": "https://doi.org/10.20944/preprints202608.1240.v1", "doi": "10.20944/preprints202608.1240.v1" }, { "citation": "Li, Zhao and Luo (2026), Chollet's permanent conjecture through order six via border induction.", "url": "https://www.preprints.org/manuscript/202608.2196", "doi": null }, { "citation": "Gatermann and Parrilo (2004), Symmetry groups, semidefinite programs, and sums of squares.", "url": "https://doi.org/10.1016/j.jpaa.2003.12.011", "doi": "10.1016/j.jpaa.2003.12.011" }, { "citation": "Peyrl and Parrilo (2008), Computing sum of squares decompositions with rational coefficients.", "url": "https://doi.org/10.1016/j.tcs.2008.09.025", "doi": "10.1016/j.tcs.2008.09.025" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:sharp-rank-three-chollet-order-five", "attemptIds": [ "ep-attempt:sharp-rank-three-chollet-order-five-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "certificate-first", "structural-compression", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "certificate", "description": "Sharp order-five rank-three permanent inequality with complete self-product and mixed equality classifications.", "scope": "Real 5 by 5 positive-semidefinite matrices of rank at most three; not complex matrices, rank four or the unrestricted Chollet conjecture." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "The manuscript maps rank-three correlation matrices to five unit-vector Gram matrices, symmetry-adapted covariants, rational sums of squares and vanishing 4 by 4 Gram minors.", "remainingRisks": [ "The written Gram reduction and equality deductions require specialist scrutiny.", "The certificate construction cannot be regenerated from the released package.", "The bounded literature audit is not exhaustive priority clearance." ] }, "humanJudgmentGates": [ "Publication is authorised; no scientific endorsement is inferred.", "External specialist review of the reduction and equality classification remains open." ], "parentLinks": [ { "relation": "depends-on-claim", "workId": null, "legacyReleaseSlug": null, "externalUrl": "https://doi.org/10.20944/preprints202608.1240.v1", "inheritedClaim": "The candidate sharp coefficient 13/48 and the extremal witness J2 direct-sum E3.", "inheritedAssuranceCeiling": "Preprint proposal and structured-case proofs; the antecedent does not validate the new global certificate or equality classifications." } ], "assuranceTarget": { "dimensions": [ "independentRerun", "specialistReview", "formalVerification", "priorityAssessment" ], "nextAction": "Obtain unaffiliated scrutiny of the Gram reduction and equality argument, then independently replay or formalise the certificate-to-theorem bridge.", "claimCeiling": "Exact internally replayed computer-assisted candidate with a separate internal reimplementation; no external specialist review, formal verification, construction regeneration or exhaustive priority clearance." }, "impactClaims": [ { "id": "sharp-rank-three-chollet-order-five-no-impact", "aim": "science", "outcome": "Inspectable sharp permanent inequality and equality classifications.", "setting": "AI-assisted mathematical research and exact certificate checking", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "None registered.", "estimand": "No causal research-speed or scientific-impact estimate.", "evidenceRefs": [], "registeredDesignRef": null, "independentAssessment": null } ] }, "researchMetrics": { "schemaVersion": "1.0", "policyUrl": "https://evidencepress.org/api/research-metrics-policy.json", "outcomePolicyUrl": "https://evidencepress.org/api/v2/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:sharp-rank-three-chollet-order-five-publication", "status": "published", "resultClass": "positive", "metrics": { "schemaVersion": "1.0", "measurementScope": "assurance-through-publication", "scopeBoundary": "Remaining publication only; prior research, extension work and substantive review are left-censored.", "forecast": { "frozenAt": "2026-10-19T12:22:08.486Z", "procedureClass": "remaining-candidate-delivery", "targetOutcome": "Scoped archived candidate with standard media and canonical Evidence Press readback.", "expectedActiveMinutes": 150, "plausibleLowMinutes": 120, "plausibleHighMinutes": 210, "expectedUnattendedWaitMinutes": 30, "referenceClass": { "label": "Full candidate publication after scientific revision", "sampleSize": 0, "basis": "Procedural judgement, not empirical calibration." }, "fermiComponents": [ { "component": "Freeze, archive, media and guarded delivery", "count": 1, "lowMinutesPerUnit": 120, "centralMinutesPerUnit": 150, "highMinutesPerUnit": 210, "basis": "Final identity, media, site and readback gates remain; no scientific gate is weakened." } ], "probabilityPositiveSignal": 0.9, "probabilityTargetClosure": 0.8, "probabilityHorizonMinutes": 360, "assumptions": [ "No provider, rights or final-byte mismatch blocks publication." ], "reforecastTriggers": [ "Provider failure, public-byte mismatch or a new load-bearing scientific defect." ], "stopRule": "Never weaken assurance or omit canonical readback to meet the forecast." }, "outcome": { "status": "unavailable", "policyVersion": "2.0", "recordedAt": "2026-10-19T13:42:51.447Z", "reason": "Gate and delivery observations do not provide complete effort telemetry. No retrospective reconstruction or calibration is made.", "missingFields": [ { "field": "calendarElapsedMinutes", "reason": "Opening and delivery timestamps are milestones, not a measurement of active effort." }, { "field": "activeAgentMinutes", "reason": "Active agent effort was not instrumented; unknown is not zero." }, { "field": "unattendedWaitMinutes", "reason": "Unattended waiting was not instrumented; unknown is not zero." }, { "field": "blockedMinutes", "reason": "Blocked time was not instrumented; unknown is not zero." }, { "field": "reworkMinutes", "reason": "Rework effort was not instrumented; unknown is not zero." }, { "field": "agentRuns", "reason": "Agent-run counts were not instrumented across the left-censored work." }, { "field": "maxParallelAgents", "reason": "Peak parallelism was not instrumented across the left-censored work." }, { "field": "modelTurns", "reason": "Model turns were not instrumented across the left-censored work." }, { "field": "researchCycles", "reason": "Research-cycle counts were not instrumented across the left-censored work." }, { "field": "falsificationGatesRun", "reason": "Falsification-gate counts were not prospectively aggregated." }, { "field": "candidateArchitecturesTested", "reason": "Candidate-architecture counts do not apply cleanly and were not recorded." }, { "field": "candidateArchitecturesRejected", "reason": "Rejected-architecture counts do not apply cleanly and were not recorded." }, { "field": "substantiveReviewRounds", "reason": "Review evidence exists, but no prospective numeric telemetry series was instrumented." }, { "field": "p0Findings", "reason": "Finding severity counts were not prospectively instrumented." }, { "field": "p1Findings", "reason": "Finding severity counts were not prospectively instrumented." }, { "field": "prepublicationClaimCorrections", "reason": "Claim corrections were not prospectively aggregated as a numeric metric." }, { "field": "resultState", "reason": "The published candidate state is preserved in statusHistory; no calibrated metric is inferred." }, { "field": "positiveSignalObserved", "reason": "Publication evidence is recorded separately; no calibrated signal metric is inferred." }, { "field": "targetReached", "reason": "Delivery is recorded in statusHistory; no calibrated target metric is inferred." }, { "field": "forecastErrorMinutes", "reason": "Active effort was not measured, so forecast error cannot be computed from wall time." }, { "field": "forecastRatio", "reason": "Active effort was not measured, so no forecast ratio can be computed." }, { "field": "withinForecastInterval", "reason": "Active effort was not measured, so interval membership is unknown." }, { "field": "activeHumanMinutes", "reason": "Active human effort was not instrumented; unknown is not zero." }, { "field": "computeMinutes", "reason": "Compute time was not prospectively aggregated; unknown is not zero." }, { "field": "deduplicatedModelTokens", "reason": "Deduplicated token use was not instrumented; unknown is not zero." }, { "field": "uncachedInputTokens", "reason": "Uncached input tokens were not instrumented; unknown is not zero." } ], "calibration": null } }, "missingnessReason": null } ] } }