{ "schemaVersion": "1.6", "slug": "sharp-logconcave-moment-comparison", "title": "Sharp moment bounds for centred log-concave distributions", "shortTitle": "When the extremal distribution changes", "url": "https://evidencepress.org/releases/sharp-logconcave-moment-comparison/", "oneLine": "A computer-assisted mathematical candidate proves the predicted switch between Laplace and centred-exponential extremisers, with equality cases and qualitative stability.", "abstract": "For nondegenerate mean-zero log-concave real random variables, this candidate determines sharp Lp-to-L2 lower bounds for 12. The extremiser changes from symmetric Laplace to centred exponential at a unique p* in (1.682457208, 1.682457209); both families attain equality at the transition, while p=2 is an identity for every admissible law. Eitan's real-exponent reduction is treated self-containedly, and analytic estimates for p>=7 are joined to ball-arithmetic enclosures below 7. A compactness argument gives qualitative total-variation and finite-Wasserstein stability at fixed p other than 2, without an explicit rate. The conjecture and predicted transition are credited to Melbourne, Roysdon, Tang and Tkocz.", "datePublished": "2026-10-15", "dateModified": "2026-10-15", "version": "1.0.0-candidate", "doi": "10.5281/zenodo.23405173", "doiUrl": "https://doi.org/10.5281/zenodo.23405173", "conceptDoi": "10.5281/zenodo.23405172", "pdfUrl": "https://github.com/ipitchford/sharp-logconcave-moment-comparison/releases/download/v1.0.0-candidate/paper.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23405173", "repoUrl": "https://github.com/ipitchford/sharp-logconcave-moment-comparison", "releaseUrl": "https://github.com/ipitchford/sharp-logconcave-moment-comparison/releases/tag/v1.0.0-candidate", "markdownUrl": "https://evidencepress.org/releases/sharp-logconcave-moment-comparison/index.md", "bibtexUrl": "https://evidencepress.org/releases/sharp-logconcave-moment-comparison/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/sharp-logconcave-moment-comparison.mp3", "imageUrl": "https://evidencepress.org/assets/og/sharp-logconcave-moment-comparison.png?v=ac1f1e5f42", "coverArtUrl": "https://evidencepress.org/assets/art/sharp-logconcave-moment-comparison.svg?v=18c2b80e59", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/sharp-logconcave-moment-comparison.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/sharp-logconcave-moment-comparison.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. Research archive published with exact-byte provider parity and producer-side replay. No external peer review, unaffiliated reproduction or formal verification claimed. Site deployment and canonical page readback are separate publication checks." }, "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/23405173", "note": "Research package publicly available at the linked GitHub release and published Zenodo record; asset parity checked against archive SHA-256 670871406b43fc1891b6d2119b6720f78ccb7e5c0110822cb3451519b2e49074. This does not assert canonical Evidence Press deployment or scientific correctness." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "Fresh producer-side normal and optimized certificate replay and package-specific gates passed for archive SHA-256 670871406b43fc1891b6d2119b6720f78ccb7e5c0110822cb3451519b2e49074. Scope is the encoded regional inequalities and retained gate plan, not independent reproduction or formal verification of the written proof." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "Not established by this local authoring draft; producer checks do not confer external independence." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "Not established by this local authoring draft; producer checks do not confer external independence." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "Not established by this local authoring draft; producer checks do not confer external independence." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "Not established by this local authoring draft; producer checks do not confer external independence." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "Not established by this local authoring draft; producer checks do not confer external independence." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "not-assessed", "note": "Not established by this local authoring draft; producer checks do not confer external independence." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "OpenAI Codex coordinated mathematical development, checking and drafting" ], "humanRole": "Publication authority only; no human research contribution inferred from account ownership.", "reviewStatus": "Internal AI-assisted source and analytic checking; supplied reviewer limitations retained. No independent or external peer-review status inferred." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared on 15 October 2026 from the revised manuscript, including qualitative distributional stability, and the dated source audit. Final research identities bind Zenodo record 23405173, GitHub v1.0.0-candidate, and archive SHA-256 670871406b43fc1891b6d2119b6720f78ccb7e5c0110822cb3451519b2e49074. Publication and replay statements are scoped to retained exact-version receipts; no independent assurance is inferred.", "grandfatheredAtSchemaVersion": null, "keywords": [ "log-concavity", "moment inequalities", "sharp constants", "phase transition", "computer-assisted proof", "qualitative stability" ], "keyResults": [ "Sharp variance-normalised moment comparisons for every real p>1, with p=2 explicitly an identity.", "Unique transition in the exact bracket (1.682457208, 1.682457209), with exactly two extremal families up to scaling and reflection.", "Analytic large-p proof and certified regional enclosures address endpoint and p=2 degeneracies.", "Qualitative total-variation and every finite-Wasserstein stability for variance-one near-extremisers at fixed p other than 2; no explicit rate." ], "reviews": [], "evidencePackage": "Written analytic proof, detailed supplements and production Arb enclosure code with certificates. Fresh producer-side normal and optimized replay and package gates are bound to archive SHA-256 670871406b43fc1891b6d2119b6720f78ccb7e5c0110822cb3451519b2e49074. Published GitHub and Zenodo assets have been checked against the frozen package. These checks are not independent reproduction or formal verification.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Melbourne, Roysdon, Tang and Tkocz: variance-normalised conjecture and predicted transition, §3.1.", "url": "https://arxiv.org/abs/2505.00944v2", "doi": "10.48550/arXiv.2505.00944" }, { "citation": "Eitan: real-exponent reduction, existing equality information and even higher-moment case.", "url": "https://arxiv.org/abs/2110.14382", "doi": "10.48550/arXiv.2110.14382" }, { "citation": "Eskenazis, Nayar and Tkocz: symmetric log-concave moment problem and crossing-method background.", "url": "https://arxiv.org/abs/1801.07597", "doi": "10.48550/arXiv.1801.07597" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:sharp-logconcave-moment-comparison", "attemptIds": [ "ep-attempt:sharp-logconcave-moment-comparison-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "certificate-first", "structural-compression", "regime-stitching", "assurance-vector" ], "decisionObject": { "type": "bound", "description": "Sharp moment bounds with complete equality classification and qualitative stability.", "scope": "Nondegenerate centred real log-concave laws; real p>1; p=2 identity; no explicit distributional stability rate." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Equality-sensitive reduction connects arbitrary centred log-concave laws to a one-parameter exponential family; analytic and regional numerical arguments cover the reduced objective.", "remainingRisks": [ "Written reduction and compactness proof require semantic scrutiny.", "Arb checks only encoded inequalities and explicit regions.", "No quantitative distributional stability or general two-exponent solution." ] }, "humanJudgmentGates": [ "The user authorises publication; no scientific endorsement is inferred.", "Development-independent proof, novelty and priority scrutiny remain open." ], "parentLinks": [ { "relation": "reuses-method", "workId": null, "legacyReleaseSlug": null, "externalUrl": "https://arxiv.org/abs/2110.14382", "inheritedClaim": "Established real-exponent reduction to the centred two-sided exponential family and equality-sensitive crossing method.", "inheritedAssuranceCeiling": "Published theorem and inspected arXiv argument; antecedent does not validate the new optimisation and no reduction priority is claimed." } ], "assuranceTarget": { "dimensions": [ "availability", "internalReplay", "semanticValidation", "specialistReview", "noveltyAssessment" ], "nextAction": "Bind final repaired-byte producer replay and archive availability; then seek development-independent scrutiny of the proof and certificate-to-claim bridge.", "claimCeiling": "Unrefereed mathematical candidate with internal producer checks; no formal verification, external peer review, priority clearance or measured impact." }, "impactClaims": [ { "id": "sharp-logconcave-moment-comparison-no-impact", "aim": "science", "outcome": "Inspectable sharp moment comparison and equality classification", "setting": "Coordinated AI-assisted mathematical candidate", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "None registered.", "estimand": "No causal discovery-speed or 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. 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