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  "slug": "ctln-six-is-minimal",
  "title": "Six is minimal",
  "shortTitle": "Six is minimal",
  "url": "https://evidencepress.org/releases/ctln-six-is-minimal/",
  "oneLine": "An exact all-parameter certificate rules out stable nonclique activity on five or fewer neurons; a known six-neuron graph makes the bound sharp.",
  "abstract": "For nondegenerate legal combinatorial threshold-linear networks with uniform positive drive, the minimum stable nonclique support size is exactly six. Integer-polynomial obstructions cover all 9,846 graph classes through five vertices across the full legal parameter region. The six-neuron graph is due to Geneson. The candidate also supplies an exact phase classification of that seed and an explicit stable family on 5r+6 vertices, with size-dependent parameters and shrinking stability margin. Model checking and exact producer replay are not external human peer review or proof-assistant verification.",
  "datePublished": "2026-10-01",
  "dateModified": "2026-10-01",
  "version": "1.1.0-candidate",
  "doi": "10.5281/zenodo.23089682",
  "doiUrl": "https://doi.org/10.5281/zenodo.23089682",
  "conceptDoi": "10.5281/zenodo.23089681",
  "pdfUrl": "https://github.com/ipitchford/ctln-six-is-minimal/releases/download/v1.1.0-candidate/paper.pdf",
  "altPdfUrl": "https://zenodo.org/records/23089682/files/paper.pdf?download=1",
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  "zenodoUrl": "https://zenodo.org/records/23089682",
  "repoUrl": "https://github.com/ipitchford/ctln-six-is-minimal",
  "releaseUrl": "https://github.com/ipitchford/ctln-six-is-minimal/releases/tag/v1.1.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/ctln-six-is-minimal/index.md",
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      "url": "https://evidencepress.org/assets/audio/ctln-six-is-minimal.mp3",
      "name": "Six is minimal",
      "description": "Synthetic OpenAI fable briefing; explanation, not additional scientific evidence.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/ctln-six-is-minimal.txt"
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    {
      "type": "video",
      "url": "https://youtu.be/EBtwXnxCkKA",
      "name": "Six Is Minimal",
      "description": "Evidence Press video overview of this release. This is a communication aid; it does not extend the release's claims or assurance status and is not additional research evidence."
    }
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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 mathematical candidate. The graph is prior work; the sharp lower bound, family and phase certificates have separate scopes. No formal verification or unaffiliated reproduction is established."
  },
  "assurance": [
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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/23089682",
      "note": "Public GitHub and Zenodo assets verified by complete downloads against the frozen package SHA-256 ledger on 1 October 2026."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Python 3.13.15 hosted clean-extraction exact regeneration, nine semantic corruptions and five wrapper/optimized-mode controls passed. Every frozen archive file matches the tested commit. Internal, not unaffiliated verification."
    },
    {
      "dimension": "independentRerun",
      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed",
      "note": "Supplied model-audit reruns do not establish unaffiliated reproduction."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "Supplied separately written audit code provides implementation diversity, not authenticated institutional independence."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "No end-to-end proof-assistant verification."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "Supplied review is AI generated, not authenticated human specialist review."
    },
    {
      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
      "question": "Has a journal or venue run peer review to a decision?",
      "state": "not-assessed",
      "note": "No editorial peer review established."
    },
    {
      "dimension": "dataEnvironmentReproducibility",
      "label": "Data and environment reproducibility",
      "question": "Are data and computational environment pinned well enough to rebuild?",
      "state": "partial",
      "note": "Python standard-library exact checker and environment instructions supplied; unaffiliated environment reproduction unestablished."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "AI-assisted research; original model identity not established in the supplied package",
      "Codex revision, scoped checks and publication",
      "Supplied AI-generated referee audit; not authenticated human review"
    ],
    "humanRole": "The user authorised publication. No human scientific contribution is inferred.",
    "disclosure": "Anonymous is the scholarly creator. Evidence Press is the publisher. Model review and implementation diversity are distinct from unaffiliated reproduction."
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  "attributionBoundary": null,
  "problem": {
    "name": "Minimum stable nonclique support size in legal CTLNs",
    "url": "https://arxiv.org/abs/2607.21396v1"
  },
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  "corrections": [],
  "publicCorrections": [],
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  "recordMaturity": "current",
  "metadataProvenance": "Prepared from the supplied v1.0 research and model-audit archives, revised as v1.1. Original archive hash and all review dispositions are recorded. No human research role or independent institutional assurance inferred.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "combinatorial threshold-linear networks",
    "CTLN",
    "stable fixed points",
    "target-free clique conjecture",
    "exact certificates",
    "Hurwitz stability",
    "neural network dynamics"
  ],
  "keyResults": [
    "Six is the exact minimum stable nonclique support size under the stated legal and nondegeneracy assumptions.",
    "All 9,846 graph classes through five vertices are covered over the full legal parameter domain.",
    "An exact seed phase region and a family of sizes 5r+6; no fixed-parameter or uniform-margin claim."
  ],
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  "evidencePackage": "Exact integer-polynomial exclusions across 9,846 graph classes; rational six-neuron witness; symbolic all-size and phase certificates; supplied model audit with separately written implementations.",
  "openProblems": [],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "Geneson (2026), A six-neuron counterexample to the target-free clique conjecture. The prior graph and counterexample; not discovered by this release.",
      "url": "https://arxiv.org/abs/2607.21396v1",
      "doi": "10.48550/arXiv.2607.21396"
    },
    {
      "citation": "Curto, Geneson and Morrison (2024), Stable fixed points of combinatorial threshold-linear networks. Small-support results, simply-added splits and composite graphs.",
      "url": "https://doi.org/10.1016/j.aam.2023.102652",
      "doi": "10.1016/j.aam.2023.102652"
    },
    {
      "citation": "Curto, Geneson and Morrison (2019), Fixed points of competitive threshold-linear networks. The fixed-point and graph-rule framework.",
      "url": "https://doi.org/10.1162/neco_a_01151",
      "doi": "10.1162/neco_a_01151"
    }
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        "Prose reductions and checker implementation are not formally verified.",
        "The weighted quotient is not a smaller ordinary CTLN.",
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