{ "schemaVersion": "1.6", "slug": "noninteger-sliced-wasserstein", "title": "Sliced Wasserstein distance at the critical moment: the noninteger case", "shortTitle": "When slices control the whole", "url": "https://evidencepress.org/releases/noninteger-sliced-wasserstein/", "oneLine": "For every fixed dimension at least two and noninteger transport power above one, sliced transport uniformly controls full transport on the whole critical moment ball, without a higher-moment or support assumption.", "abstract": "For each fixed d≥2 and noninteger p>1, this candidate proves uniform equivalence of sliced p-Wasserstein and p-Wasserstein distance on all probability measures with pth moment at most one. Quantitatively, Wp≤C(d,p) SWp^[1/(p(p+d+2))], with no higher-moment, common-tail, regularity or support condition. Combined with the integer-power counterexamples of Letrouit and Mérigot, this gives an integer–noninteger classification at the critical moment. The proof lifts pth-moment mass to logarithmic radius and direction, uses classical one-sided cosine multipliers and positive spherical Jackson smoothing, and derives a weighted finite-direction consequence with an explicit angular-coverage error. The exponent is not claimed sharp and its constant is not calibrated.", "datePublished": "2026-10-11", "dateModified": "2026-10-11", "version": "1.1.0-candidate", "doi": "10.5281/zenodo.23295188", "doiUrl": "https://doi.org/10.5281/zenodo.23295188", "conceptDoi": "10.5281/zenodo.23295187", "pdfUrl": "https://github.com/ipitchford/noninteger-sliced-wasserstein/releases/download/v1.1.0-candidate/noninteger-sliced-wasserstein-v1.1.0-candidate.pdf", "altPdfUrl": "https://zenodo.org/records/23295188/files/noninteger-sliced-wasserstein-v1.1.0-candidate.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23295188", "repoUrl": "https://github.com/ipitchford/noninteger-sliced-wasserstein", "releaseUrl": "https://github.com/ipitchford/noninteger-sliced-wasserstein/releases/tag/v1.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/noninteger-sliced-wasserstein/index.md", "bibtexUrl": "https://evidencepress.org/releases/noninteger-sliced-wasserstein/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/noninteger-sliced-wasserstein.mp3", "imageUrl": "https://evidencepress.org/assets/og/noninteger-sliced-wasserstein.png?v=94d06979a5", "coverArtUrl": "https://evidencepress.org/assets/art/noninteger-sliced-wasserstein.svg?v=2cf8fb5f6f", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/noninteger-sliced-wasserstein.mp3", "name": "Audio briefing — When slices control the whole", "description": "Synthetic plain-English explanation; not additional scientific evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/noninteger-sliced-wasserstein.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 AI review and scoped finite diagnostics; no formal verification, external specialist review or unaffiliated reproduction 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/23295188", "note": "Public GitHub and Zenodo assets were verified against the frozen release package; availability and byte parity are not proof, novelty assessment 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": "Producer-side fresh-extraction checks include 320 finite standard-library diagnostics and 20 rejected semantic controls under ordinary and optimized Python. Internal proof audits remain separate; these checks do not verify the unrestricted analytic theorem.", "evidenceUrl": "https://github.com/ipitchford/noninteger-sliced-wasserstein/blob/v1.1.0-candidate/REPLAY_RECEIPT.md" }, { "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 coordinated internal AI checks, public availability or publication." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "Not established by coordinated internal AI checks, public availability or publication." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "Not established by coordinated internal AI checks, public availability or publication." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "Not established by coordinated internal AI checks, public availability or publication." }, { "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 coordinated internal AI checks, public availability or publication." }, { "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/noninteger-sliced-wasserstein/blob/v1.1.0-candidate/ENVIRONMENT.txt", "note": "LaTeX sources, environment information and standard-library diagnostics are supplied; no unaffiliated reproduction is established." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted Codex research workflow", "OpenAI Codex: argument development, source comparison, internal checking and publication preparation" ], "humanRole": "Publication authority only; no human research contribution asserted.", "reviewStatus": "Unrefereed written-proof candidate with internal AI review and scoped finite diagnostics; no formal verification, external specialist review or unaffiliated reproduction established." }, "attributionBoundary": null, "problem": { "name": "Critical-moment equivalence of sliced and full Wasserstein distance for noninteger powers", "url": "https://arxiv.org/abs/2610.12029v1" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Supplied research and review materials, revised with internally audited classical Jackson smoothing and weighted-direction consequences on 11 October 2026. No external review, exhaustive priority or historical research duration inferred.", "grandfatheredAtSchemaVersion": null, "keywords": [ "sliced Wasserstein distance", "optimal transport", "critical moment", "cosine transform", "Mellin transform", "spherical Jackson approximation" ], "keyResults": [ "On the full p-moment ball, sliced p-Wasserstein distance uniformly controls p-Wasserstein distance for every fixed noninteger p>1 and d≥2.", "The explicit non-sharp estimate has exponent 1/[p(p+d+2)]; no higher moment, compact support, common radial law or fixed target is assumed.", "Together with the imported integer counterexamples, the result classifies critical-moment uniform equivalence by whether p is an integer; a weighted finite-direction rule retains an unavoidable angular-coverage error." ], "reviews": [], "evidencePackage": "Analytic written proof, theorem-level source and prior-art audits, internal mathematical review, and scoped finite diagnostics with deliberately corrupted controls.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Letrouit and Mérigot (2026), Embedding large subsets of Wasserstein spaces into Banach spaces: higher-moment comparison, integer critical-moment obstructions and the source question.", "url": "https://arxiv.org/abs/2610.12029v1", "doi": "10.48550/arXiv.2610.12029" }, { "citation": "Haberl (2008), Lp intersection bodies: classical one-sided spherical cosine multipliers and smooth-function inversion.", "url": "https://dmg.tuwien.ac.at/haberl/lp%20intersection%20bodies.pdf", "doi": null }, { "citation": "Cao and Guo (2010), Approximation by Jackson-type operator on the sphere: classical positive finite-degree angular smoothing.", "url": "https://hrcak.srce.hr/en/file/92590", "doi": null }, { "citation": "Han (2025), Sliced Wasserstein distance between probability measures on infinite dimensional Hilbert spaces: directional Lipschitz estimate used by the finite-direction corollary.", "url": "https://arxiv.org/abs/2307.05802v3", "doi": "10.48550/arXiv.2307.05802" }, { "citation": "Boman and Lindskog (2009), Support theorems for the Radon transform and Cramér–Wold theorems: established noninteger recovery antecedent in a different setting.", "url": "https://doi.org/10.1007/s10959-008-0151-0", "doi": "10.1007/s10959-008-0151-0" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:noninteger-sliced-wasserstein", "attemptIds": [ "ep-attempt:noninteger-sliced-wasserstein-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "evidence-assessment", "communication" ], "lineageId": null, "accelerationPrimitives": [ "structural-compression", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "bound", "description": "Uniform critical-moment inverse comparison, integer–noninteger classification and a weighted finite-direction consequence with angular error.", "scope": "All Borel probability pairs on Euclidean R^d with pth moment at most R^p, for each fixed d≥2, R>0 and noninteger p>1; constants may depend on d and p." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "The pth-moment weighted lift records logarithmic radius and direction; a coupling bound converts its bounded-Lipschitz distance to spatial transport, while the projection transform is inverted only after classical angular and radial smoothing.", "remainingRisks": [ "The unrestricted analytic proof is not formally verified or externally reviewed.", "The bounded source search does not establish exhaustive novelty or priority.", "The exponent is not claimed sharp, the constant is unevaluated and conditioning near integer powers is not controlled.", "Finite directional data require a certified coverage error and do not identify every probability measure." ] }, "humanJudgmentGates": [ "External specialist scrutiny of the transport bridge, multiplier inversion and uniform estimates remains open; no human research contribution is asserted." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "specialistReview", "formalVerification" ], "nextAction": "Independently audit the weighted-cylinder transport bridge, classical multiplier normalization, Jackson approximation and uniform reciprocal-multiplier estimates.", "claimCeiling": "Unrefereed written-proof candidate with internal AI review and scoped finite diagnostics; 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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:noninteger-sliced-wasserstein-publication", "status": "result-ready", "resultClass": "positive", "metrics": { "schemaVersion": "1.0", "measurementScope": "assurance-through-publication", "scopeBoundary": "Work after this registration; research, extension, review and package-freezing actions are left-censored. No historical clocks or resource telemetry are reconstructed.", "forecast": { "frozenAt": "2026-10-11T08:47:19.281Z", "procedureClass": "full-candidate", "targetOutcome": "Immutable research publication and canonical site readback", "expectedActiveMinutes": 150, "plausibleLowMinutes": 120, "plausibleHighMinutes": 210, "expectedUnattendedWaitMinutes": 30, "referenceClass": { "label": "Full candidate publication", "sampleSize": 0, "basis": "Route-level procedural estimate recorded before ledger registration; not a measured benchmark" }, "fermiComponents": [ { "component": "Remaining assurance, provider, media and guarded publication stages", "count": 1, "lowMinutesPerUnit": 120, "centralMinutesPerUnit": 150, "highMinutesPerUnit": 210, "basis": "Mandatory remaining publication stages; no assurance gate is waived for time" } ], "probabilityPositiveSignal": 0.9, "probabilityTargetClosure": 0.8, "probabilityHorizonMinutes": 480, "assumptions": [ "The frozen candidate remains unchanged and provider, CI and deployment routes remain accessible." ], "reforecastTriggers": [ "A substantive integrity finding, provider failure or CI failure." ], "stopRule": "No hard cap; complete mandatory gates or report a precise blocker." }, "outcome": null }, "missingnessReason": null } ] } }