{ "schemaVersion": "1.6", "slug": "three-gaussian-projection-counterexample", "title": "Every projection passes. The joint comparison fails.", "shortTitle": "Three Gaussian components break projection tests", "url": "https://evidencepress.org/releases/three-gaussian-projection-counterexample/", "oneLine": "A centred Gaussian and a three-component Gaussian mixture pass every one-dimensional convex-order test, yet an explicit joint convex function separates them.", "abstract": "An exact planar construction separates projected from joint convex order for a centred Gaussian compared with three centred Gaussian components. The written proof covers every projection direction, gives a strictly positive rational lower bound for a six-plane convex witness, and preserves the counterexample after adding a specified positive covariance to every component. An explicit family explains how directional scale and probability allocation separate. Numerical quadrature is diagnostic only.", "datePublished": "2026-10-10", "dateModified": "2026-10-10", "version": "1.1.0-candidate", "doi": "10.5281/zenodo.23284191", "doiUrl": "https://doi.org/10.5281/zenodo.23284191", "conceptDoi": "10.5281/zenodo.23284190", "pdfUrl": "https://github.com/ipitchford/three-gaussian-projection-counterexample/releases/download/v1.1.0-candidate/three-gaussian-projection-counterexample-v1.1.0-candidate.pdf", "altPdfUrl": "https://zenodo.org/records/23284191/files/three-gaussian-projection-counterexample-v1.1.0-candidate.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23284191", "repoUrl": "https://github.com/ipitchford/three-gaussian-projection-counterexample", "releaseUrl": "https://github.com/ipitchford/three-gaussian-projection-counterexample/releases/tag/v1.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/three-gaussian-projection-counterexample/index.md", "bibtexUrl": "https://evidencepress.org/releases/three-gaussian-projection-counterexample/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/three-gaussian-projection-counterexample.mp3", "imageUrl": "https://evidencepress.org/assets/og/three-gaussian-projection-counterexample.png?v=4478b86f57", "coverArtUrl": "https://evidencepress.org/assets/art/three-gaussian-projection-counterexample.svg?v=1292c57a1c", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/three-gaussian-projection-counterexample.mp3", "name": "Audio briefing", "description": "Synthetic speech; communication, not additional mathematical evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/three-gaussian-projection-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 internal finite replay. No external peer review, unaffiliated reproduction, formal verification or absolute priority is 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/23284191", "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": "Exact identities, two numerical diagnostics and wrong-weight/wrong-gap controls pass normally and under optimized Python. The continuum claim rests on written proof.", "evidenceUrl": "https://github.com/ipitchford/three-gaussian-projection-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/three-gaussian-projection-counterexample/blob/v1.1.0-candidate/ENVIRONMENT.txt", "note": "Python, NumPy and SciPy versions and replay route supplied; external reproduction not established." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted supplied research; 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 internal finite replay. No external peer review, unaffiliated reproduction, formal verification or absolute priority is established." }, "attributionBoundary": null, "problem": { "name": "Does projected convex order suffice for three centred Gaussian components?", "url": "https://arxiv.org/abs/2610.07404" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Supplied candidate and model-assisted review; publication revisions dated 10 October 2026. No historical research duration or independent reviewer status inferred.", "grandfatheredAtSchemaVersion": null, "keywords": [ "convex order", "Gaussian mixtures", "projection tests", "multivariate stochastic orders" ], "keyResults": [ "Every projection direction passes the convex-order comparison; an exact joint witness reverses it.", "Witness expectation gap is strictly greater than 37/18432.", "The failure persists with positive definite covariance matrices, with an explicit positive lower bound." ], "reviews": [], "evidencePackage": "Written all-directions proof; exact rational checks; two numerical implementations and semantic corruption controls; supplied model-assisted review and itemised response.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Guéant (2026), Convex Order Beyond Dimension One: Theorem 4 proves the two-component case; Section 4 leaves larger mixtures open.", "url": "https://arxiv.org/abs/2610.07404", "doi": "10.48550/arXiv.2610.07404" }, { "citation": "Jourdain and Pagès, Convex comparison of Gaussian mixtures: inspected arXiv v1, Theorem 1.1; final journal body not inspected.", "url": "https://arxiv.org/abs/2410.07958v1", "doi": "10.48550/arXiv.2410.07958" }, { "citation": "Koshevoy and Mosler (1998), Lift zonoids, random convex hulls and the variability of random vectors: classical weaker projection order.", "url": "https://wisostat.uni-koeln.de/sites/dep_econometrics/pdf_publikationen/1998_KoshevoyM98.pdf", "doi": null }, { "citation": "Pinelis, explicit finite-support failure of projection sufficiency; inspected arXiv v3.", "url": "https://arxiv.org/abs/2510.04269v3", "doi": "10.48550/arXiv.2510.04269" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:three-gaussian-projection-counterexample", "attemptIds": [ "ep-attempt:three-gaussian-projection-counterexample-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "evidence-assessment", "communication" ], "lineageId": null, "parentLinks": [], "accelerationPrimitives": [ "counterexample-proxy-first", "adversarial-controls", "agent-readable-research-object" ], "decisionObject": { "type": "other", "description": "Failure of projection sufficiency for three centred Gaussian components.", "scope": "One centred Gaussian and one three-component centred Gaussian mixture in two dimensions." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "An exhaustive directional inequality establishes all projected comparisons; a max-of-affine convex witness has a strictly reversed expectation.", "remainingRisks": [ "The analytic argument is not formally verified.", "Priority is bounded by inspected literature; one final journal body remains uninspected." ] }, "humanJudgmentGates": [ "External specialist assessment remains open; no human research judgement claimed." ], "assuranceTarget": { "dimensions": [ "specialistReview", "formalVerification" ], "nextAction": "Audit the projection criterion, gap estimate and common-noise extension.", "claimCeiling": "Unrefereed written-proof candidate with internal finite replay. 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