{
  "schemaVersion": "1.6",
  "slug": "rbm43-eight-point-obstruction",
  "title": "An eight-point obstruction to universality of RBM(4,3)",
  "shortTitle": "Eight states an RBM cannot fit",
  "url": "https://evidencepress.org/releases/rbm43-eight-point-obstruction/",
  "oneLine": "An explicit eight-state distribution cannot be approximated arbitrarily well by a classical binary model with four visible and three hidden units.",
  "abstract": "This unrefereed candidate proves a necessary leakage inequality on the entire real binary RBM(4,3) visible closure. Forty-two integer identities and an exhaustive selector cover exclude a specified eight-point support. The written proof also supplies a strictly positive excluded distribution and non-sharp total-variation separation bounds. A contrasting proposition represents every distribution on a nine-point parity-plus-one support. The result makes at least four hidden units necessary for universality on four visible bits, without showing that four suffice. Exact producer checks and internal AI editorial review are not unaffiliated validation or formal proof.",
  "datePublished": "2026-09-06",
  "dateModified": "2026-09-06",
  "version": "0.2.0-candidate",
  "doi": "10.5281/zenodo.22550044",
  "doiUrl": "https://doi.org/10.5281/zenodo.22550044",
  "conceptDoi": "10.5281/zenodo.22550043",
  "pdfUrl": "https://github.com/ipitchford/rbm43-eight-point-obstruction/releases/download/v0.2.0-candidate/rbm43-eight-point-obstruction-0.2.0-candidate.pdf",
  "altPdfUrl": "https://zenodo.org/records/22550044/files/rbm43-eight-point-obstruction-0.2.0-candidate.pdf?download=1",
  "altPdfLabel": null,
  "zenodoUrl": "https://zenodo.org/records/22550044",
  "repoUrl": "https://github.com/ipitchford/rbm43-eight-point-obstruction",
  "releaseUrl": "https://github.com/ipitchford/rbm43-eight-point-obstruction/releases/tag/v0.2.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/rbm43-eight-point-obstruction/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/rbm43-eight-point-obstruction/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/rbm43-eight-point-obstruction.mp3?v=b8d793b558",
  "imageUrl": "https://evidencepress.org/assets/og/rbm43-eight-point-obstruction.png?v=6e34668eaf",
  "coverArtUrl": "https://evidencepress.org/assets/art/rbm43-eight-point-obstruction.svg?v=4d9b2375a3",
  "media": [
    {
      "type": "audio",
      "url": "https://evidencepress.org/assets/audio/rbm43-eight-point-obstruction.mp3",
      "name": "Audio briefing — eight states an RBM cannot fit",
      "description": "Plain-English synthetic narration; communication, not additional evidence.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/rbm43-eight-point-obstruction.txt"
    }
  ],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Unrefereed candidate for classical real binary RBM(4,3), including the full visible closure. No four-hidden sufficiency, minimal excluded support, complete classification, sharp error radius, practical training penalty, formal proof, unaffiliated reproduction or historical priority is claimed."
  },
  "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/22550044",
      "note": "All four assets downloaded from public GitHub and Zenodo and SHA-256 matched on 2026-09-06 at 19:08 UTC. This is archival availability, not validation of the mathematical result."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Exact integer and rational checks, normal/optimized replay and hostile/read-only controls. Coordinated review is internal only.",
      "evidenceUrl": "https://github.com/ipitchford/rbm43-eight-point-obstruction/blob/v0.2.0-candidate/replay-receipt.json"
    },
    {
      "dimension": "independentRerun",
      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed",
      "note": "Supplied AI review reports additional execution, but its separate audit archive was not retrieved; no authenticated unaffiliated evidence establishes this dimension."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "Supplied AI review reports additional execution, but its separate audit archive was not retrieved; no authenticated unaffiliated evidence establishes this dimension."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "No external specialist/venue decision or proof-assistant theorem establishes this dimension. Internal AI role review does not qualify."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "No external specialist/venue decision or proof-assistant theorem establishes this dimension. Internal AI role review does not qualify."
    },
    {
      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
      "question": "Has a journal or venue run peer review to a decision?",
      "state": "not-assessed",
      "note": "No external specialist/venue decision or proof-assistant theorem establishes this dimension. Internal AI role review does not qualify."
    },
    {
      "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/rbm43-eight-point-obstruction/blob/v0.2.0-candidate/ENVIRONMENT.md",
      "note": "Standard-library replay and pinned recorded runtimes; no hermetic or cross-platform PDF-byte identity claim."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "OpenAI Codex under human direction",
      "Python standard library; Z3 and SymPy for historical discovery",
      "Pandoc 3.9 and pdfLaTeX"
    ],
    "humanRole": "Problem selection, research direction, supplied review and full-publication authorization. Scholarly creator remains Anonymous.",
    "disclosure": "AI assistance contributed proof, code, exposition, repairs and internal editorial review. The supplied AI review reports a separate reconstruction, but its linked audit ZIP was not retrieved here; no unaffiliated assurance promotion."
  },
  "attributionBoundary": null,
  "problem": {
    "name": "RBM(4,3) universality — screening dataset AIM-PROBABILITY-0001",
    "url": "https://aimath.org/pastworkshops/boltzmannrep.pdf"
  },
  "orderedProductAverage": null,
  "corrections": [],
  "publicCorrections": [],
  "publicCorrectionIndexes": null,
  "pageStructureVersion": "2.0",
  "pageStructureVariant": "canonical-theorem",
  "pageStructureWaivers": [],
  "recordMaturity": "current",
  "metadataProvenance": "Research assurance assessed on 2026-09-06 at 19:08 UTC after local replay, hosted Python 3.10/3.12/3.14 CI and byte-identical GitHub/Zenodo readback. Canonical site-publication readback is recorded separately in PUBLISHED.json only after deployment. Historical research and initial publication intake are excluded from the prospectively registered assurance/publication clock.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "restricted Boltzmann machine",
    "universal approximation",
    "algebraic statistics",
    "finite integer certificate",
    "model closure",
    "unrefereed candidate"
  ],
  "keyResults": [
    "A closure-wide necessary inequality excludes the specified eight-point support for real binary RBM(4,3).",
    "A strictly positive distribution is also excluded, with certified non-sharp total-variation lower bounds.",
    "Every law on the specified parity-plus-one support is attainable.",
    "At least four hidden units are necessary; four-hidden sufficiency and smallest excluded support remain undetermined."
  ],
  "reviews": [],
  "evidencePackage": "Nine-page manuscript, printable and JSON 42-row certificate, standalone standard-library verifiers, two exhaustive selector-cover implementations, exact auxiliary constructions, normal/optimized replay, mutation and immutable-receipt controls, manifest, source comparison and scoped review response. Internal editorial reports and exact archive identity are linked with the release.",
  "openProblems": [
    "Obtain an unaffiliated reconstruction with a publicly retrievable receipt and specialist review.",
    "Determine the exact minimum universal hidden-unit count for four visible bits.",
    "Find sharp approximation distances or smaller excluded supports.",
    "Generalize the certificate-to-leakage method while checking the new selector domain separately."
  ],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "American Institute of Mathematics. Boltzmann machines workshop summary, page 2: fixed-size question, parity-plus-one investigation and eight-vertex proof outline.",
      "url": "https://aimath.org/pastworkshops/boltzmannrep.pdf",
      "doi": null
    },
    {
      "citation": "Montúfar (2018). Restricted Boltzmann Machines: Introduction and Review, Sections 6 and 9.",
      "url": "https://arxiv.org/abs/1806.07066",
      "doi": "10.48550/arXiv.1806.07066"
    },
    {
      "citation": "Seigal and Montúfar (2018). Mixtures and products in two graphical models. Real inequalities for RBM(3,2).",
      "url": "https://arxiv.org/abs/1709.05276",
      "doi": "10.48550/arXiv.1709.05276"
    },
    {
      "citation": "Sonthalia, Seigal and Montúfar (2023). Supermodular Rank: Set Function Decomposition and Optimization.",
      "url": "https://arxiv.org/abs/2305.14632",
      "doi": "10.48550/arXiv.2305.14632"
    },
    {
      "citation": "Antolini, Montúfar and Oneto (2025). Hadamard ranks of algebraic varieties. Generic complex rank is a different boundary.",
      "url": "https://arxiv.org/abs/2510.05231",
      "doi": "10.48550/arXiv.2510.05231"
    },
    {
      "citation": "Reinhart and De les Coves (2026). Universal spin models are universal approximators in machine learning. Hidden size may vary.",
      "url": "https://doi.org/10.1103/b3xp-wtwx",
      "doi": "10.1103/b3xp-wtwx"
    }
  ],
  "claimAssurance": null,
  "operatingModel": {
    "version": "1.0",
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    "aims": [
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    "artifactRoles": [
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    "lineageId": null,
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      "agent-readable-research-object"
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    "decisionObject": {
      "type": "obstruction",
      "description": "Finite integer certificate giving a visible leakage inequality and explicit excluded distributions.",
      "scope": "Classical real binary RBM(4,3) including the entire visible closure."
    },
    "bottleneckTargeted": [
      "assurance",
      "publication"
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    "semanticBridge": {
      "state": "explicit",
      "description": "Augmented sufficient-statistic identities imply normalized probability monomials; exhaustive hidden selectors supply an identity at every parameter value; leakage and continuity give the closure-wide obstruction.",
      "remainingRisks": [
        "Written analytic bridge is not formally verified.",
        "Coordinated checkers/reviewers may share errors.",
        "Bounded novelty search does not establish priority.",
        "No practical modelling penalty follows from tiny certified lower bounds."
      ]
    },
    "humanJudgmentGates": [
      "Check the full-model-to-selector and continuity arguments.",
      "Separate real probability laws from generic complex decompositions.",
      "Assess source correspondence and historical priority.",
      "Preserve rights, Anonymous authorship and assurance boundaries."
    ],
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        "relation": "depends-on-claim",
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        "legacyReleaseSlug": null,
        "externalUrl": "https://aimath.org/pastworkshops/boltzmannrep.pdf",
        "inheritedClaim": "Fixed (4,3) universality question and workshop proof outline supply historical problem framing.",
        "inheritedAssuranceCeiling": "Source framing, not a theorem dependency or priority clearance."
      }
    ],
    "assuranceTarget": {
      "dimensions": [
        "independentRerun",
        "independentReimplementation",
        "formalVerification",
        "specialistReview",
        "semanticValidation",
        "noveltyAssessment",
        "priorityAssessment"
      ],
      "nextAction": "Unaffiliated exact reconstruction and specialist assessment; larger architectures remain new research.",
      "claimCeiling": "Unrefereed candidate for classical real binary RBM(4,3), including the full visible closure. No four-hidden sufficiency, minimal excluded support, complete classification, sharp error radius, practical training penalty, formal proof, unaffiliated reproduction or historical priority is claimed."
    },
    "impactClaims": [
      {
        "id": "rbm43-eight-point-obstruction-no-impact",
        "aim": "science",
        "outcome": "Reusable fixed-architecture obstruction",
        "setting": "Producer-coordinated mathematical candidate publication",
        "status": "NO_IMPACT_EVIDENCE",
        "designClass": "none",
        "comparator": "No matched comparator.",
        "estimand": "No research acceleration, reuse or impact effect estimated.",
        "evidenceRefs": [],
        "registeredDesignRef": null,
        "independentAssessment": null
      }
    ]
  },
  "researchMetrics": {
    "schemaVersion": "1.0",
    "policyUrl": "https://evidencepress.org/api/research-metrics-policy.json",
    "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:rbm43-eight-point-obstruction-assurance-publication",
        "status": "published",
        "resultClass": "positive",
        "metrics": {
          "schemaVersion": "1.0",
          "measurementScope": "assurance-through-publication",
          "scopeBoundary": "Prospective assurance and publication after registration only. Scientific discovery, supplied review and the earlier publication intake/token-limit pause predate registration and are excluded, with separate historical receipts retained. No retrospective research forecast.",
          "forecast": {
            "frozenAt": "2026-09-06T18:35:18Z",
            "procedureClass": "minor-revision-assurance-full-candidate",
            "targetOutcome": "Review repairs, exact replay, internal five-role approval, public GitHub/Zenodo and first canonical Evidence Press release. Second ledger-sealing deployment and final administrative closeout measured separately.",
            "expectedActiveMinutes": 150,
            "plausibleLowMinutes": 95,
            "plausibleHighMinutes": 210,
            "expectedUnattendedWaitMinutes": 30,
            "referenceClass": {
              "label": "Evidence Press full-candidate route prior",
              "sampleSize": 0,
              "basis": "Uncalibrated procedural prior, not a measured comparison."
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            "fermiComponents": [
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                "component": "Review repairs and deterministic assurance",
                "count": 1,
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                "centralMinutesPerUnit": 40,
                "highMinutesPerUnit": 60,
                "basis": "Existing finite proof and two bounded checker defects."
              },
              {
                "component": "Five-role frozen editorial review",
                "count": 1,
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                "centralMinutesPerUnit": 30,
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                "basis": "One substantive round; three concurrent reviewer slots."
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              {
                "component": "Archive identity, page and media",
                "count": 1,
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                "centralMinutesPerUnit": 50,
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                "basis": "Established repository generators."
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                "count": 1,
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            "assumptions": [
              "Existing proof certificate replays.",
              "Publishing authentication remains available.",
              "No critical editorial or rights defect."
            ],
            "reforecastTriggers": [
              "Load-bearing defect.",
              "External authentication failure.",
              "Concurrent repository conflict."
            ],
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            "candidateArchitecturesTested": 0,
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            "substantiveReviewRounds": 1,
            "p0Findings": 0,
            "p1Findings": 0,
            "prepublicationClaimCorrections": 0,
            "resultState": "target-closed",
            "resultSummary": "First canonical release, exact public archive bytes, required CI, page/media QA and guarded preservation passed at the unchanged unrefereed claim ceiling. Second ledger-sealing deployment and administrative closeout are outside the frozen first-readback target.",
            "positiveSignalObserved": true,
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            "varianceReason": "The forecast was an uncalibrated full-candidate prior. Existing proof/certificate, concurrent bounded role review and established publication tools made the first canonical release faster despite disclosed tooling repairs. This is descriptive, not causal acceleration evidence.",
            "missingFields": [
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                "field": "activeHumanMinutes",
                "reason": "Human direction and review time not instrumented."
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              {
                "field": "computeMinutes",
                "reason": "Individual exact-check wall/CPU times are retained in receipts; complete nonoverlapping substantive-compute and inference-hardware accounting is unavailable, and ordinary builds/CI/deployment are not substantive research compute."
              }
            ]
          }
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      }
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}