{
  "schemaVersion": "1.3",
  "slug": "frankl-concavity-obstruction",
  "title": "A concavity obstruction and repaired support reductions in a Frankl entropy programme",
  "shortTitle": "Frankl entropy concavity obstruction",
  "url": "https://evidencepress.org/releases/frankl-concavity-obstruction/",
  "oneLine": "An exact counterdirection blocks a printed joint-concavity step in Liu's arXiv v1, while independent three-atom reductions, endpoint control and bounded certificates survive; no new Frankl bound is proved.",
  "abstract": "This anonymous, unrefereed candidate audits a conditional entropy programme for Frankl's union-closed sets conjecture. For every protocol parameter 0 < ell <= 1, an explicit feasible perturbation preserves the aggregate constraints used in the proof of Theorem 12 of Liu's arXiv:2306.08824v1 but has positive curvature, so the printed joint-concavity inference cannot justify the shared-weight support conclusion. The claim is source-specific: it does not refute Liu's one-measure lemma, the theorem's conclusion by every possible argument, the paper's analytic non-explicit strict improvement, or the uninspected CISS text. The candidate then reconstructs a binary-latent reduction, proves a conventional positive-semidefinite sublemma through an explicit operator bridge and exact-arithmetic finite-bound certificate, and obtains independently weighted component laws with at most three atoms each. It also proves a sharp endpoint theorem, certifies one one-component face by 32,768 directed-rounding leaves, and eliminates the latent weight exactly. The generic residual problem still has ten variables after elimination, and the global inequality at c = 153/400 remains open. No new lower bound for Frankl's conjecture is established.",
  "datePublished": "2026-08-14",
  "dateModified": "2026-08-14",
  "version": "0.2.0-candidate",
  "doi": "10.5281/zenodo.21938497",
  "doiUrl": "https://doi.org/10.5281/zenodo.21938497",
  "conceptDoi": "10.5281/zenodo.21938496",
  "pdfUrl": "https://github.com/ipitchford/frankl-concavity-obstruction/releases/download/v0.2.0-candidate/frankl-concavity-obstruction-v0.2.0-candidate.pdf",
  "altPdfUrl": "https://zenodo.org/records/21938497/files/frankl-concavity-obstruction-v0.2.0-candidate.pdf?download=1",
  "zenodoUrl": "https://zenodo.org/records/21938497",
  "repoUrl": "https://github.com/ipitchford/frankl-concavity-obstruction",
  "releaseUrl": "https://github.com/ipitchford/frankl-concavity-obstruction/releases/tag/v0.2.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/frankl-concavity-obstruction/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/frankl-concavity-obstruction/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/frankl-concavity-obstruction.mp3",
  "imageUrl": "https://evidencepress.org/assets/og/frankl-concavity-obstruction.png",
  "coverArtUrl": "https://evidencepress.org/assets/art/frankl-concavity-obstruction.svg",
  "media": [
    {
      "type": "audio",
      "url": "https://evidencepress.org/assets/audio/frankl-concavity-obstruction.mp3",
      "name": "Audio briefing — Frankl entropy concavity obstruction",
      "description": "Plain-English synthetic-voice summary of the source-specific obstruction, surviving reductions and open global inequality; a communication aid, not additional mathematical evidence.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/frankl-concavity-obstruction.txt"
    }
  ],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Anonymous unrefereed candidate. The source-specific positive-curvature counterdirection and repaired reductions are presented as mathematical results; the global candidate inequality and any new Frankl bound remain explicitly unproved. Producer-side exact and directed-rounding replays pass within their stated scopes, including same-source runs on macOS arm64 and Linux/amd64. Independent rerun, separately authored reimplementation, proof-assistant formalisation, external specialist review, editorial peer review and priority adjudication are not established. The CISS 2024 full text was not assessed."
  },
  "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://doi.org/10.5281/zenodo.21938497",
      "note": "The GitHub candidate release and Zenodo record expose the byte-bound PDF, source ZIP and SHA-256 ledger. Public availability and identity do not establish theorem truth or independent reproduction."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "The exact PSD suite rejects ten mutations, the directed-rounding face suite rejects seven mutations, and the deterministic package and clean-extraction gates pass within their declared scopes.",
      "evidenceUrl": "https://github.com/ipitchford/frankl-concavity-obstruction/releases/tag/v0.2.0-candidate"
    },
    {
      "dimension": "independentRerun",
      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed",
      "note": "No unaffiliated party has reported rerunning the immutable public package; environment-diverse runs were conducted within the producer-coordinated workflow."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "No separately authored checker has reconstructed the PSD finite bound or directed-rounding face certificate without producer code and formulas."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "The analytic reductions, operator bridge and endpoint theorem have not been formalised in a proof assistant over a declared trusted base."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "The retained developmental and model-assisted audits are not external human specialist review in union-closed sets, entropy inequalities, moment geometry or validated numerics."
    },
    {
      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
      "question": "Has a journal or venue run peer review to a decision?",
      "state": "not-assessed",
      "note": "No journal or comparable venue has conducted editorial peer review to a decision."
    },
    {
      "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/frankl-concavity-obstruction/blob/v0.2.0-candidate/ENVIRONMENT.md",
      "note": "The package pins MPFR 4.2.2 and GMP 6.3.0 for the Tier C certificate and records macOS arm64 and Linux/amd64 same-source receipts, but no unaffiliated environment reconstruction is reported."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "AI systems under human direction",
      "OpenAI Codex",
      "Anthropic Claude in a producer-coordinated review path"
    ],
    "humanRole": "Research direction, mediation, claim-boundary decisions and publication authorisation by the human creator; scholarly attribution remains Anonymous.",
    "disclosure": "AI systems contributed derivations, code, searches, review, drafting, replay and release engineering in one producer-coordinated workflow. Cross-model review, exact replay and matching receipts are not independent reproduction, external specialist review or editorial peer review."
  },
  "problem": null,
  "corrections": [],
  "keywords": [
    "union-closed sets",
    "Frankl's conjecture",
    "entropy method",
    "moment sets",
    "concavity obstruction",
    "positive-semidefinite kernel",
    "computer-assisted mathematics",
    "unrefereed candidate"
  ],
  "keyResults": [
    "For every 0 < ell <= 1, an explicit candidate-feasible perturbation preserves Liu arXiv:2306.08824v1's aggregate constraints but has positive objective curvature, invalidating the printed joint-concavity inference used in its Theorem 12 proof.",
    "The counterdirection does not refute Liu's one-measure concavity lemma, Theorem 12's shared-weight conclusion by every possible argument, the analytic non-explicit strict improvement, or the uninspected CISS text.",
    "A compact extreme-point argument recovers a binary latent variable for the continuous candidate kernel.",
    "An explicit operator bridge and exact-arithmetic finite-bound certificate establish the required positive-semidefinite sublemma within the stated conventional-proof trust boundary.",
    "Sequential two-moment minimisation reduces each component independently to at most three atoms; shared component weights are not recovered.",
    "A sharp endpoint theorem gives the optimal entropy-zero asymptotic ratio, and a 32,768-leaf 256-bit directed-rounding certificate covers one one-component face.",
    "Exact elimination of the latent weight leaves ten generic variables; the six-variable count applies only to the genuine two-by-two subface.",
    "The global inequality at c = 153/400 remains open, and the release establishes no new universal Frankl bound."
  ],
  "reviews": [],
  "evidencePackage": "An 18-page candidate manuscript and accessible Markdown; exact source-to-model and theorem-dependency maps; a frozen arXiv-v1 source identity; analytic repair and endpoint derivations; an exact PSD checker with ten semantic mutations; a 32,768-leaf MPFR 4.2.2 directed-rounding certificate with seven mutations; macOS arm64 and Linux/amd64 same-source replay receipts; exact latent-weight reduction diagnostics; a deterministic manifest, clean-extraction ZIP verifier, licensing records and review responses. These are producer-side and environment-diverse checks, not unaffiliated theorem reconstruction or independent implementation.",
  "openProblems": [
    "Obtain independent specialist review of the arXiv-v1 source map, parameterised counterdirection and exact scope of the challenged inference.",
    "Have an unaffiliated team rederive the operator/entropy bridge and implement the PSD finite-bound check independently.",
    "Prove or refute mean-constraint activity on genuine mixed two-by-two branches without assuming monotonicity of binary entropy above one half.",
    "Find an analytic stratification that reduces every hard residual branch to at most four or five effective dimensions before attempting interval branch-and-bound.",
    "Certify or falsify the full independently weighted three-plus-three-atom inequality at c = 153/400 and C = 1 + 10^-8.",
    "Inspect the CISS 2024 full text lawfully before making any claim about whether the arXiv-v1 concavity wording persisted into the conference version."
  ],
  "relatedWorks": [
    {
      "citation": "Liu, J. (2023). Improving the lower bound for the union-closed sets conjecture via conditionally IID coupling. arXiv:2306.08824v1.",
      "url": "https://arxiv.org/abs/2306.08824v1"
    },
    {
      "citation": "Cambie, S. (2022; revised 2025). Better bounds for the union-closed sets conjecture using the entropy approach. arXiv:2212.12500.",
      "url": "https://arxiv.org/abs/2212.12500"
    },
    {
      "citation": "Sawin, W. (2022). An improved lower bound for the union-closed set conjecture. arXiv:2211.11504.",
      "url": "https://arxiv.org/abs/2211.11504"
    },
    {
      "citation": "Yu, L. (2023). Dimension-Free Bounds for the Union-Closed Sets Conjecture. Entropy, 25(5), 767.",
      "url": "https://doi.org/10.3390/e25050767"
    }
  ],
  "operatingModel": {
    "version": "1.0",
    "workId": "ep-work:frankl-concavity-obstruction",
    "attemptIds": [
      "ep-attempt:frankl-concavity-obstruction"
    ],
    "aims": [
      "science"
    ],
    "artifactRoles": [
      "research-output",
      "evidence-assessment",
      "method-demonstration",
      "communication"
    ],
    "lineageId": null,
    "accelerationPrimitives": [
      "certificate-first",
      "structural-compression",
      "adversarial-controls",
      "productive-failure",
      "assurance-vector",
      "agent-readable-research-object"
    ],
    "decisionObject": {
      "type": "obstruction",
      "description": "A parameterised counterdirection to a printed joint-concavity inference, with repaired independently weighted support reductions and an executable residual problem.",
      "scope": "Liu arXiv:2306.08824v1 only; no claim about the uninspected CISS text, no global inequality and no new Frankl bound."
    },
    "bottleneckTargeted": [
      "assurance",
      "publication"
    ],
    "semanticBridge": {
      "state": "explicit",
      "description": "The source inclusion variable, complement coordinate, polynomial, protocol amplitude, tau = ell squared kernel, objective, aggregate constraints and component-slice failure are mapped explicitly for every 0 < ell <= 1.",
      "remainingRisks": [
        "No independent specialist has validated the source map or analytic PSD bridge.",
        "The CISS 2024 full text has not been assessed.",
        "The global ten-variable inequality is open."
      ]
    },
    "humanJudgmentGates": [
      "Assess the arXiv-v1 source map and exact scope of the counterdirection rather than inferring it from successful arithmetic replay.",
      "Assess the entropy-series and operator bridges as mathematics, including endpoint and compactness arguments.",
      "Require a new release decision before promoting the work beyond an unrefereed candidate or claiming a Frankl bound.",
      "Confirm anonymous scholarly attribution, component licences and public release authority."
    ],
    "parentLinks": [],
    "assuranceTarget": {
      "dimensions": [
        "independentRerun",
        "independentReimplementation",
        "formalVerification",
        "specialistReview",
        "semanticValidation",
        "noveltyAssessment",
        "priorityAssessment"
      ],
      "nextAction": "Obtain focused independent specialist review of the source map, parameterised obstruction, PSD semantic bridge, endpoint theorem and independent-support reduction, preferably with a separately authored arithmetic implementation.",
      "claimCeiling": "A conventional source-specific proof obstruction and repaired reductions with producer-side exact and directed-rounding checks; not a Frankl bound, global certificate, independent reproduction, formal proof, specialist review, editorial peer review or priority claim."
    },
    "impactClaims": [
      {
        "id": "frankl-science-no-impact",
        "aim": "science",
        "outcome": "More reliable reuse of a conditional entropy reduction",
        "setting": "Research on entropy approaches to Frankl's union-closed sets conjecture",
        "status": "NO_IMPACT_EVIDENCE",
        "designClass": "none",
        "comparator": "No matched conventional mathematical-research or review workflow was registered.",
        "estimand": "No effect on discovery time, error rate, review effort, theorem production, uptake or citation was estimated.",
        "evidenceRefs": [],
        "registeredDesignRef": null,
        "independentAssessment": null
      }
    ]
  }
}