{
  "schemaVersion": "1.6",
  "slug": "rbm31-exact-kl-radius",
  "title": "The Exact KL Radius of Two Bernoulli Products on Three Bits",
  "shortTitle": "An exact three-bit KL radius",
  "url": "https://evidencepress.org/releases/rbm31-exact-kl-radius/",
  "oneLine": "A computer-assisted proof candidate gives the exact worst-case error of two Bernoulli products on three bits, attained only by the two uniform parity distributions.",
  "abstract": "How badly can a two-component product mixture approximate an arbitrary distribution on three binary variables? This anonymous, AI-assisted, unrefereed candidate proves an exact worst-case forward-KL error of -3/4 log(2 sqrt(3)-3) nats, approximately 0.830615532444 bits. Only the two uniform parity targets attain it. A short log-supermodular multiplier certificate gives the matching lower bound. Reverse local stochastic channels reduce all targets to six support orbits; 16,600 exact witness simplices cover every real weight vector remaining. A standard-library checker verifies 93,792 vertex conditions. The release also supplies a rational target-to-witness program and an elementary conditioning corollary for mixtures of 2^(n-2) products on n bits. The constant and general transfer mechanism have prior antecedents; higher-dimensional sharpness, unaffiliated verification, formal proof and external peer review are not claimed.",
  "datePublished": "2026-09-05",
  "dateModified": "2026-09-05",
  "version": "0.1.0-candidate",
  "doi": "10.5281/zenodo.22339153",
  "doiUrl": "https://doi.org/10.5281/zenodo.22339153",
  "conceptDoi": "10.5281/zenodo.22339152",
  "pdfUrl": "https://github.com/ipitchford/rbm31-exact-kl-radius/releases/download/v0.1.0-candidate/rbm31-exact-kl-radius-0.1.0-candidate.pdf",
  "altPdfUrl": "https://zenodo.org/records/22339153/files/rbm31-exact-kl-radius-0.1.0-candidate.pdf?download=1",
  "altPdfLabel": null,
  "zenodoUrl": "https://zenodo.org/records/22339153",
  "repoUrl": "https://github.com/ipitchford/rbm31-exact-kl-radius",
  "releaseUrl": "https://github.com/ipitchford/rbm31-exact-kl-radius/releases/tag/v0.1.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/rbm31-exact-kl-radius/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/rbm31-exact-kl-radius/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/rbm31-exact-kl-radius.mp3?v=44ce7ca5d1",
  "imageUrl": "https://evidencepress.org/assets/og/rbm31-exact-kl-radius.png?v=c97ce6f101",
  "coverArtUrl": "https://evidencepress.org/assets/art/rbm31-exact-kl-radius.svg?v=93948cdd57",
  "media": [
    {
      "type": "audio",
      "url": "https://evidencepress.org/assets/audio/rbm31-exact-kl-radius.mp3",
      "name": "Audio briefing — the exact three-bit KL radius",
      "description": "Plain-English synthetic-voice account of the parity extremizers, channel reduction, exact convex cover and nonsharp conditioning extension; a communication aid, not additional evidence.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/rbm31-exact-kl-radius.txt"
    }
  ],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Anonymous AI-assisted unrefereed computer-assisted theorem candidate after actioning the supplied minor-revision review. The sharp result is three-bit and two-component; the all-n consequence is an upper bound for the specified product-mixture budget. Four-star potential was considered but is not claimed achieved."
  },
  "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/22339153",
      "note": "The versioned GitHub release and published Zenodo record serve byte-identical PDF, final ZIP, frozen Stage-1 review ZIP and checksum sidecar; all four public downloads were hashed against the final local assets."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Normal and optimized standard-library verification and all 18 tests pass locally; fresh final extraction and hosted Python 3.11/3.13 CI pass. Continuous coverage relies on the explicitly stated analytic bridges. Five differentiated producer-coordinated internal roles recommend Accept, without upgrading unaffiliated assurance.",
      "evidenceUrl": "https://github.com/ipitchford/rbm31-exact-kl-radius/blob/v0.1.0-candidate/verification.json"
    },
    {
      "dimension": "independentRerun",
      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed",
      "note": "No authenticated unaffiliated rerun receipt is supplied."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "Discovery/checker separation and a reviewer-reported alternate check do not establish unaffiliated reconstruction; no separate reviewer implementation was supplied."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "No proof assistant verifies the analytic bridges or checker."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "The supplied review is not authenticated unaffiliated specialist review; the five-role gate is producer-coordinated internal 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 journal or comparable venue has conducted editorial peer review."
    },
    {
      "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/rbm31-exact-kl-radius/blob/v0.1.0-candidate/ENVIRONMENT.md",
      "note": "Exact certificate inputs, source, hashes and standard-library replay route are provided. The environment is not hermetic; optional discovery and cross-platform PDF bytes are separate reproducibility boundaries."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "OpenAI Codex under human direction",
      "Python standard library",
      "XeLaTeX / TeX Live 2026"
    ],
    "humanRole": "Research direction, problem selection, supplied review, scope decisions and full-publication authorization; scholarly attribution remains Anonymous.",
    "disclosure": "AI assistance contributed proof search, mathematical exposition, exact code, revision, producer-coordinated internal editorial review and publishing. Different programs, roles and execution platforms do not establish unaffiliated assurance."
  },
  "attributionBoundary": null,
  "problem": {
    "name": "AIM-PROBABILITY-0002: maximum divergence of RBM(3,1)",
    "url": "https://www.unsolvedmath.com"
  },
  "orderedProductAverage": null,
  "corrections": [],
  "publicCorrections": [],
  "publicCorrectionIndexes": null,
  "pageStructureVersion": "2.0",
  "pageStructureVariant": "canonical-theorem",
  "pageStructureWaivers": [],
  "recordMaturity": "current",
  "metadataProvenance": "Assessed on 2026-09-05T10:54:17.188Z against immutable commit ec82819b13493d12430338b309d81f08d6a4db09, DOI 10.5281/zenodo.22339153, exact four-asset GitHub/Zenodo public readback, hosted Python 3.11/3.13 replay, five-role internal editorial reports and full-audio round-trip. Canonical site readback and terminal workflow metrics are recorded separately; no external assurance dimension is promoted.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "Kullback-Leibler divergence",
    "Bernoulli mixtures",
    "restricted Boltzmann machines",
    "algebraic statistics",
    "convex-cover certificates",
    "unrefereed candidate"
  ],
  "keyResults": [
    "The closed three-bit two-product-mixture model has exact forward-KL radius -3/4 log(2 sqrt(3)-3) nats.",
    "Only the two uniform parity targets maximize the radius; each has four translated nearest distributions.",
    "52 root simplices and 16,600 leaves cover every real target remaining after exact channel reduction; 93,792 vertex conditions include 552 exact parity incidences.",
    "A rational query interface returns certified feasible mixture parameters and a channel, symmetry and leaf trace, not a generally optimal projection.",
    "For n at least three and k at least 2^(n-2), mixtures of k n-bit Bernoulli products have radius at most the same constant; higher-dimensional sharpness is not proved."
  ],
  "reviews": [],
  "evidencePackage": "An 11-page DOI-bearing paper, aligned Markdown, six exact certificate files unchanged from the original reviewed bundle, standalone standard-library checker, 13 verifier controls, five constructive-interface test groups, normal and optimized fresh-extraction replay, a worked channel example, explicit analytic-to-code map, primary-source comparison, original review-response matrix, rights and AI disclosures, and SHA-256 manifests. Internal editorial, immutable availability and hosted-CI receipts are recorded separately before publication. The exact calculation is producer-coordinated, not unaffiliated validation. Hosted replay: https://github.com/ipitchford/rbm31-exact-kl-radius/actions/runs/33961635189. Internal editorial decision: https://github.com/ipitchford/rbm31-exact-kl-radius/blob/v0.1.0-candidate/editorial/DECISION.md. All four public assets match across GitHub, Zenodo and the local release.",
  "openProblems": [
    "Obtain an authenticated unaffiliated reconstruction of the cone, boundary, channel and equality arguments and a separately written exact checker.",
    "Determine whether the conditioning upper bound is sharp for any n greater than three at the stated component budget.",
    "Compress or formally verify the finite convex-cover certificate and its analytic interfaces.",
    "Study other component budgets without conflating product mixtures with larger RBMs.",
    "Assess broader mathematical significance through specialist review and actual downstream reuse, not a producer-assigned four-star label."
  ],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "Montúfar, G. (2018). Restricted Boltzmann Machines: Introduction and Review, Section 9 item 10.",
      "url": "https://arxiv.org/abs/1806.07066",
      "doi": "10.48550/arXiv.1806.07066"
    },
    {
      "citation": "Allman, E. S., Rhodes, J. A., Sturmfels, B., and Zwiernik, P. (2015). Tensors of Nonnegative Rank Two.",
      "url": "https://doi.org/10.1016/j.laa.2013.10.046",
      "doi": "10.1016/j.laa.2013.10.046"
    },
    {
      "citation": "Allman et al. (2019). Maximum Likelihood Estimation of the Latent Class Model through Model Boundary Decomposition.",
      "url": "https://doi.org/10.18409/jas.v10i1.75",
      "doi": "10.18409/jas.v10i1.75"
    },
    {
      "citation": "Seigal, A., and Montúfar, G. (2018). Mixtures and products in two graphical models. A different model.",
      "url": "https://arxiv.org/abs/1709.05276",
      "doi": "10.48550/arXiv.1709.05276"
    },
    {
      "citation": "Alexandr, Y., and Hoşten, S. (2025). Maximum information divergence from linear and toric models, Proposition 2.",
      "url": "https://doi.org/10.1007/s41884-025-00166-3",
      "doi": "10.1007/s41884-025-00166-3"
    },
    {
      "citation": "Montúfar, G., Rauh, J., and Ay, N. (2013). Maximal Information Divergence from Statistical Models Defined by Neural Networks, Lemma 6.",
      "url": "https://arxiv.org/abs/1303.0268",
      "doi": "10.48550/arXiv.1303.0268"
    }
  ],
  "claimAssurance": null,
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    "decisionObject": {
      "type": "certificate",
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      "scope": "Exact three-bit two-product-mixture radius; specified non-sharp all-n conditioning upper bound."
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      "remainingRisks": [
        "Analytic and code interfaces remain unaffiliated-unverified.",
        "The checker is not proof-assistant certified.",
        "Bounded novelty searches may miss differently phrased or unpublished work.",
        "No higher-dimensional sharpness or general-RBM transfer is proved."
      ]
    },
    "humanJudgmentGates": [
      "Check necessary cone inclusion and boundary limits, not just the candidate star fit.",
      "Check that simplex coverage and common-witness convexity justify the continuous quantifiers.",
      "Keep diagnostic floating-point comparisons out of exact proof decisions.",
      "Distinguish feasible approximation from optimal projection and product mixtures from larger RBMs.",
      "Preserve anonymous authorship, rights, prior attribution and separate assurance dimensions."
    ],
    "parentLinks": [
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        "relation": "depends-on-claim",
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        "externalUrl": "https://arxiv.org/abs/1806.07066",
        "inheritedClaim": "Section 9 item 10 records the proposed constant and asks for maximizers.",
        "inheritedAssuranceCeiling": "The recorded conjecture is not a proof or an explicit parity-maximizer identification."
      },
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        "relation": "extends-result",
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        "externalUrl": "https://arxiv.org/abs/1303.0268",
        "inheritedClaim": "Lemma 6 supplies the classical disjoint-support mixture decomposition used in the conditioning transfer.",
        "inheritedAssuranceCeiling": "This mechanism does not establish the candidate sharp three-bit constant or higher-dimensional sharpness."
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    "assuranceTarget": {
      "dimensions": [
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        "independentReimplementation",
        "formalVerification",
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        "semanticValidation",
        "noveltyAssessment",
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      "nextAction": "Obtain unaffiliated specialist reconstruction and a separate checker before stronger assurance claims; investigate higher-dimensional sharpness as new research.",
      "claimCeiling": "Anonymous AI-assisted unrefereed computer-assisted theorem candidate for the exact three-bit two-product-mixture KL radius, with a constructive rational interface and a non-sharp conditioning corollary. Not unaffiliated verification, formal proof, authenticated specialist review, journal peer review, exhaustive novelty, priority, a four-star rating or demonstrated impact."
    },
    "impactClaims": [
      {
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        "aim": "science",
        "outcome": "More reusable exact approximation-error certificates",
        "setting": "Producer-coordinated mathematical research and publication",
        "status": "NO_IMPACT_EVIDENCE",
        "designClass": "none",
        "comparator": "No matched conventional workflow comparator was assigned.",
        "estimand": "No causal discovery-time, effort, reliability, uptake or field-impact effect was estimated.",
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        "registeredDesignRef": null,
        "independentAssessment": null
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    "workLedgerUrl": "https://evidencepress.org/api/work-ledger.json",
    "claimCeiling": "These records describe scoped workflow inputs and outcomes. 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:rbm31-exact-kl-radius-revision-stretch-publication",
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          "schemaVersion": "1.0",
          "measurementScope": "research-through-publication",
          "scopeBoundary": "Prospective review repair, a bounded constructive-oracle and conditioning-lift extension, and full publication. The original three-bit proof and certificate predate this registration; their discovery clocks are excluded and are not reconstructed.",
          "forecast": {
            "frozenAt": "2026-09-05T09:59:53.587Z",
            "procedureClass": "review-repair-bounded-stretch-full-candidate",
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            "expectedActiveMinutes": 120,
            "plausibleLowMinutes": 80,
            "plausibleHighMinutes": 150,
            "expectedUnattendedWaitMinutes": 15,
            "referenceClass": {
              "label": "Established Evidence Press full-candidate route",
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              "basis": "Procedural analogy to the current fixed-d release; not a statistical calibrated sample."
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              },
              {
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                "count": 1,
                "lowMinutesPerUnit": 20,
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                "basis": "One new release using established generators."
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                "centralMinutesPerUnit": 25,
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            "probabilityTargetClosure": 0.8,
            "probabilityHorizonMinutes": 150,
            "assumptions": [
              "Existing certificate replays.",
              "Required authenticated services remain available.",
              "Bounded extension does not alter the sharp three-bit theorem."
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            "reforecastTriggers": [
              "Fatal mathematical or provenance objection.",
              "Unavailable authenticated publishing service.",
              "Overlapping origin-main changes."
            ],
            "stopRule": "Stop the stretch after 30 active minutes if its proof or novelty comparison is unresolved; retain the minor-revised base theorem. Fail closed on scientific, rights, CI, preservation or canonical-integrity blockers."
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            "activeAgentMinutes": 36,
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            "resultSummary": "Review actioned; constructive feasible-witness route and analytic nonsharp conditioning transfer each reached a positive decisive endpoint. Two adversarial gate families covered verifier semantics and query feasibility; repeated runs are not new cycles. Decimal, prior-attribution and independence wording were corrected before publication. Five internal roles accepted; immutable research assets and first canonical publication passed. Four-star significance and external assurance were not promoted. Active-agent minutes are a measured lower bound from explicit runtime generation intervals across six agents, not total elapsed time. Unattended wait records instrumented compaction only; unrecorded waits are not imputed. Rework is an upper attribution bound for the recorded post-CI-failure repair window. Consequently forecast errors and ratios are clock-limited diagnostics, not a complete-effort or acceleration comparison.",
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            "missingFields": [
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                "field": "deduplicatedModelTokens",
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}