{ "schemaVersion": "1.6", "slug": "gaussian-radon-one-point-extension", "title": "A one-point extension of Gaussian Radon log-concavity", "shortTitle": "One more point, two useful deletions", "url": "https://evidencepress.org/releases/gaussian-radon-one-point-extension/", "oneLine": "A two-deletion identity extends a known Gaussian convex-hull intersection theorem from d+2 to d+3 points, with an explicit log-concavity margin.", "abstract": "Split independent points from one nonsingular Gaussian law into two fixed labelled groups. For d+3 points in dimension d, the probability that their convex hulls intersect is log-concave in group size. A deterministic identity counts exactly two deletions preserving a feasible split and reduces the result to the known d+2-point Gaussian theorem. The paper gives an explicit margin, exact slack and local equality criteria, and a conditional transfer to exchangeable laws. General arrangement identities and qualitative ultra-log-concavity convolution are inherited; cases with at least d+4 points remain unresolved here.", "datePublished": "2026-11-06", "dateModified": "2026-11-06", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.23997793", "doiUrl": "https://doi.org/10.5281/zenodo.23997793", "conceptDoi": null, "pdfUrl": "https://github.com/ipitchford/gaussian-radon-one-point-extension/releases/download/v0.1.0-candidate/gaussian-radon-one-point-extension-v0.1.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23997793", "repoUrl": "https://github.com/ipitchford/gaussian-radon-one-point-extension", "releaseUrl": "https://github.com/ipitchford/gaussian-radon-one-point-extension/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/gaussian-radon-one-point-extension/index.md", "bibtexUrl": "https://evidencepress.org/releases/gaussian-radon-one-point-extension/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/gaussian-radon-one-point-extension.mp3", "imageUrl": "https://evidencepress.org/assets/og/gaussian-radon-one-point-extension.png?v=959b99060f", "coverArtUrl": "https://evidencepress.org/assets/art/gaussian-radon-one-point-extension.svg?v=4b3ed6d374", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/gaussian-radon-one-point-extension.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/gaussian-radon-one-point-extension.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": false, "detail": "Unrefereed AI-assisted mathematical candidate. Internal checking is not external peer review, unaffiliated reproduction or formal verification; bounded source comparison does not guarantee priority." }, "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/23997793", "note": "Published GitHub and Zenodo archives; all three assets match the frozen local SHA-256 manifest by full public download. This establishes availability and byte parity, not proof correctness." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "not-assessed", "note": "Pure written mathematical proof with an analytic order-counting example. No numerical scientific replay is applicable; passed packaging gates are not proof verification." }, { "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 established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "No unaffiliated independent reimplementation established." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant verification claimed." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "Internal AI-assisted checking 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 established." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "not-assessed", "note": "Manuscript source and source-audit records are supplied. No experimental dataset or numerical replay environment is required by this proof." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical development and revision", "Internal AI-assisted source and proof checking", "Codex publication preparation" ], "humanRole": "Publication authority only; no human mathematical contribution inferred.", "reviewStatus": "Internal AI-assisted review is not unaffiliated 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 manuscript and bibliography on 6 November 2026. All three frozen archive assets verified by full public downloads from GitHub and Zenodo against local SHA-256; availability is not mathematical validation.", "grandfatheredAtSchemaVersion": null, "keywords": [ "Radon partition", "Gaussian random polytope", "log-concavity", "deletion identity", "ultra-log-concavity" ], "keyResults": [ "For d+3 independent points from one nonsingular Gaussian law, fixed labelled convex-hull intersection probabilities are log-concave in group size.", "Exactly two deletions preserve any feasible split in affine general position, yielding 2q_k=k a_(k−1)+(n−k)a_k.", "An explicit margin and exact local slack refine the inherited qualitative convolution principle.", "Transfer to exchangeable laws requires a positive-interior log-concave common deletion marginal.", "The higher-nullity cases n≥d+4 and strictness at every Gaussian interior index remain unresolved here." ], "reviews": [], "evidencePackage": "Conventional written proof and an exact four-point line example derived by order counting. Five packaging gates passed; no numerical scientific replay or formal verification is claimed.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Chan, Kalai, Narayanan, Ter-Saakov and White (2026). Unimodality for Radon Partitions of Random Vectors. Discrete & Computational Geometry. The inherited d+2 Gaussian base.", "url": "https://doi.org/10.1007/s00454-026-00870-6", "doi": "10.1007/s00454-026-00870-6" }, { "citation": "White (2025). Radon Partitions of Random Gaussian Polytopes. arXiv:2507.05449v1.", "url": "https://arxiv.org/abs/2507.05449v1", "doi": "10.48550/arXiv.2507.05449" }, { "citation": "Streltsova and Wagner (2025). Levels in Arrangements: Linear Relations, the g-Matrix, and Applications to Crossing Numbers. SoCG 2025, 75:1–75:18.", "url": "https://doi.org/10.4230/LIPIcs.SoCG.2025.75", "doi": "10.4230/LIPIcs.SoCG.2025.75" }, { "citation": "Liggett (1997). Ultra Logconcave Sequences and Negative Dependence. Journal of Combinatorial Theory, Series A 79, 315–325.", "url": "https://doi.org/10.1006/jcta.1997.2790", "doi": "10.1006/jcta.1997.2790" }, { "citation": "Gurvits (2008). A short, based on the mixed volume, proof of Liggett's theorem on the convolution of ultra-logconcave sequences. arXiv:0804.1181, Theorem 1.1.", "url": "https://arxiv.org/abs/0804.1181", "doi": "10.48550/arXiv.0804.1181" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:gaussian-radon-one-point-extension", "attemptIds": [ "ep-attempt:gaussian-radon-one-point-extension-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output" ], "lineageId": null, "accelerationPrimitives": [ "structural-compression", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "bound", "description": "A one-point Gaussian Radon log-concavity extension with a quantitative local margin.", "scope": "n=d+3 independent identically distributed nonsingular Gaussian points; conditional exchangeable transfer retains its deletion-marginal premise." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "A two-dimensional affine-dependence sign cone yields a two-deletion count, whose expectation transfers the known smaller-sample inequality.", "remainingRisks": [ "The known Gaussian base theorem remains load-bearing.", "The normalized random-side law differs from fixed labelled split probabilities.", "Bounded novelty assessment is not exhaustive priority; larger nullity remains open." ] }, "humanJudgmentGates": [ "Publication authority is not scientific endorsement.", "External specialist validation remains open." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "specialistReview", "independentRerun", "priorityAssessment" ], "nextAction": "External source-led review of the deletion identity, normalization, quantitative slack and inherited Gaussian base.", "claimCeiling": "Written source-dependent proof; no numerical scientific replay, external validation, formal verification or absolute originality guarantee." }, "impactClaims": [ { "id": "gaussian-radon-one-point-extension-no-impact", "aim": "science", "outcome": "Potential reuse of a deletion identity in random convex geometry.", "setting": "Random polytopes and probabilistic combinatorics", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "No matched research-workflow comparison.", "estimand": "No acceleration or downstream impact effect estimated.", "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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