{
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  "slug": "six-dimensional-zonoid-counterexample",
  "title": "A six-dimensional counterexample to cross-polytope maximality in zonoid approximation",
  "shortTitle": "Beyond the cross-polytope",
  "url": "https://evidencepress.org/releases/six-dimensional-zonoid-counterexample/",
  "oneLine": "A fourteen-vertex body in six dimensions has exact zonoid enclosure factor 122/65, exceeding 15/8 by 1/520.",
  "abstract": "Exact rational lower and upper certificates establish lambda(K)=122/65 for K=conv{±4e_1,...,±4e_6,±(1,1,1,1,2,2)}. Sixty-four signed-sum identities yield an all-zonoid lower bound; a 53-generator zonotope attains it. The cross-polytope value is 15/8. A separate clipped-cube polar enumeration checks the 35 facet pairs.",
  "datePublished": "2026-09-08",
  "dateModified": "2026-09-08",
  "version": "1.0.0-candidate",
  "doi": "10.5281/zenodo.22660175",
  "doiUrl": "https://doi.org/10.5281/zenodo.22660175",
  "conceptDoi": "10.5281/zenodo.22660174",
  "pdfUrl": "https://github.com/ipitchford/six-dimensional-zonoid-counterexample/releases/download/v1.0.0-candidate/six-dimensional-zonoid-counterexample-1.0.0-candidate.pdf",
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  "altPdfLabel": null,
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  "repoUrl": "https://github.com/ipitchford/six-dimensional-zonoid-counterexample",
  "releaseUrl": "https://github.com/ipitchford/six-dimensional-zonoid-counterexample/releases/tag/v1.0.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/six-dimensional-zonoid-counterexample/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/six-dimensional-zonoid-counterexample/cite.bib",
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  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Unrefereed computer-assisted candidate. Exact producer replay and five internal model-assisted editorial reports are not unaffiliated specialist review, independent reproduction or formal verification. Historical priority, least counterexample dimension and global dimension-six extremality are not established."
  },
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      "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/22660175",
      "note": "Public archive byte identity checked before deployment."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Exact lower/upper replay, 16 rejection controls and polar enumeration. Written lemma supplies the universal bridge.",
      "evidenceUrl": "https://github.com/ipitchford/six-dimensional-zonoid-counterexample/blob/v1.0.0-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": "Internal model-assisted review and producer replay do not establish this external dimension."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "Internal model-assisted review and producer replay do not establish this external dimension."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "Internal model-assisted review and producer replay do not establish this external dimension."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "Internal model-assisted review and producer replay do not establish this external dimension."
    },
    {
      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
      "question": "Has a journal or venue run peer review to a decision?",
      "state": "not-assessed",
      "note": "Internal model-assisted review and producer replay do not establish this external 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/six-dimensional-zonoid-counterexample/blob/v1.0.0-candidate/ENVIRONMENT.txt",
      "note": "Standard-library replay on macOS and Linux CI; historical optimization environment not reconstructed."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "OpenAI Codex under human direction",
      "Python exact rational checkers",
      "XeLaTeX"
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    "humanRole": "Research selection, direction, supplied review and publication authorization; scholarly creator Anonymous.",
    "disclosure": "AI-assisted research, manuscript, code, internal role review and publication. Supplied review audit ZIP unavailable; no independent external assurance inferred."
  },
  "attributionBoundary": null,
  "problem": {
    "name": "AIM-CONVEX_GEOMETRY-0032",
    "url": "https://aimath.org/WWN/fourierconvex/fourierconvex.pdf"
  },
  "orderedProductAverage": null,
  "corrections": [],
  "publicCorrections": [],
  "publicCorrectionIndexes": null,
  "pageStructureVersion": "2.0",
  "pageStructureVariant": "canonical-theorem",
  "pageStructureWaivers": [],
  "recordMaturity": "current",
  "metadataProvenance": "Producer-coordinated exact-certificate assurance is separate from canonical publication; availability does not upgrade external validation.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "zonoid",
    "zonotope",
    "convex geometry",
    "cross-polytope",
    "rational certificate",
    "counterexample"
  ],
  "keyResults": [
    "Exact factor 122/65 for a specified fourteen-vertex body in dimension six.",
    "Strict gap 1/520 above the cross-polytope benchmark 15/8.",
    "Universal lower certificate from 64 rational signed-sum identities.",
    "Matching 53-generator enclosure and a separate 70-vertex polar check."
  ],
  "reviews": [],
  "evidencePackage": "Exact certificates, standard-library replay, 16 rejection controls, normal/optimized clean extraction, polar enumeration, six-page proof and five internal role reports. https://github.com/ipitchford/six-dimensional-zonoid-counterexample/blob/v1.0.0-candidate/EDITORIAL_DECISION.md",
  "openProblems": [
    "What is the largest zonoid enclosure factor in dimension six?",
    "What is the smallest dimension admitting a counterexample?",
    "Are optimal enclosures unique, or can fewer than 53 generators suffice?",
    "Assess this particular example and proof against earlier literature independently."
  ],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "AIM Fourier analytic methods in convex geometry, Question 2 (Schneider), page 1.",
      "url": "https://aimath.org/WWN/fourierconvex/fourierconvex.pdf",
      "doi": null
    },
    {
      "citation": "Schneider (2001), On the Busemann area in Minkowski spaces: antecedent support-function methods.",
      "url": "https://eudml.org/doc/121503",
      "doi": null
    },
    {
      "citation": "Henk, Linke and Wills (2010), Minimal zonotopes containing the crosspolytope: fixed-body enclosure questions.",
      "url": "https://doi.org/10.1016/j.laa.2009.12.041",
      "doi": "10.1016/j.laa.2009.12.041"
    },
    {
      "citation": "Siegel (2025), Optimal approximation of zonoids and uniform approximation by shallow neural networks: already-zonoid inputs.",
      "url": "https://doi.org/10.1007/s00365-025-09712-9",
      "doi": "10.1007/s00365-025-09712-9"
    }
  ],
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    "decisionObject": {
      "type": "counterexample",
      "description": "A fourteen-vertex symmetric convex body with exact factor 122/65.",
      "scope": "Specified six-dimensional body, compared with cross-polytope factor 15/8."
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    "bottleneckTargeted": [
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    "semanticBridge": {
      "state": "explicit",
      "description": "Signed-sum identities imply a direction-universal inequality, summed over zonotope generators and extended to zonoids by support-function continuity; exact upper containment closes equality.",
      "remainingRisks": [
        "Written semantic bridge awaits unaffiliated specialist scrutiny.",
        "No formal proof-assistant verification.",
        "Historical priority remains bounded uncertainty."
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    },
    "humanJudgmentGates": [
      "Audit the all-zonoid extension and facet completeness.",
      "Assess contribution-specific priority and significance.",
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        "inheritedClaim": "Definition of lambda and the stated cross-polytope benchmark.",
        "inheritedAssuranceCeiling": "Imported source benchmark, not external validation of this counterexample."
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        "setting": "Producer-coordinated publication",
        "status": "NO_IMPACT_EVIDENCE",
        "designClass": "none",
        "comparator": "No matched comparator.",
        "estimand": "No speed or impact effect estimated.",
        "evidenceRefs": [],
        "registeredDesignRef": null,
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          "scopeBoundary": "Remaining assurance and publication only. Discovery, supplied review and initial intake predate registration and are excluded. No historical research clock reconstructed.",
          "forecast": {
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            "procedureClass": "minor-revision-partial-mathematics-full-candidate",
            "targetOutcome": "Bounded partial-results candidate with fail-closed replay, five-role internal review, public immutable archives and guarded canonical readback.",
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            "plausibleHighMinutes": 230,
            "expectedUnattendedWaitMinutes": 25,
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            "calendarElapsedMinutes": 28,
            "activeAgentMinutes": 28,
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            "modelTurns": 68,
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            "substantiveReviewRounds": 1,
            "p0Findings": 0,
            "p1Findings": 0,
            "prepublicationClaimCorrections": 0,
            "resultState": "target-closed",
            "resultSummary": "Reviewed exact-counterexample candidate published with rational replay, five differentiated internal reports, matching immutable archives and guarded canonical readback. Original numerical forecast retained; copied partial-results wording corrected transparently. Historical research excluded.",
            "positiveSignalObserved": true,
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            "varianceReason": "Existing exact certificates and reusable publisher tools. Active time is the continuous coordinator workflow interval minus explicitly recorded unattended/blocked intervals, including short interleaved waits; not summed agent CPU. Twelve minutes is an approximate operational repair subtotal for packaging, layout, CI schema and intake-descriptor issues, not scientific revision. Model turns count available coordinator reasoning records only, excluding reviewer forks. No comparative acceleration claim.",
            "missingFields": [
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              {
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                "reason": "No complete substantive compute meter across model calls; ordinary publication builds do not stand for research compute."
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              {
                "field": "deduplicatedModelTokens",
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}