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  "problem_id": "vibemathed:the-ceiling-of-the-single-letter-entropy-method-for-the-union-closed-sets-conjec",
  "title": "The ceiling of the single-letter entropy method for the union-closed sets conjecture",
  "canonical_statement": "Every lower bound on the union-closed constant $c_0$ since Gilmer (2022) — including $(3-\\sqrt5)/2$, Sawin–Yu–Cambie's $0.3823455$ and Liu's $0.382709$ — is certified by the same single-letter inequality: a memoryless protocol coupling two conditional bits, together with a class of admissible joint laws of the two prefix-conditional probabilities. It was not known how far this framework can go: whether it could be pushed to $1/2$ and settle Frankl's conjecture, or whether it has a hard ceiling strictly below it.",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "resolved",
  "solution_events": [
    {
      "id": "vibemathed:the-ceiling-of-the-single-letter-entropy-method-for-the-union-closed-sets-conjec:event",
      "problem_id": "vibemathed:the-ceiling-of-the-single-letter-entropy-method-for-the-union-closed-sets-conjec",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-09-08T00:00:00.000Z",
      "title": "GitHub release v1.2, the stated version of record: paper, code, logs, referee reports and Lean",
      "summary": "Two unconditional ceilings. Every single-letter certificate whose classes contain product laws certifies at most $1-h(1/\\sqrt2)/\\sqrt2 = 0.383099\\ldots$. Every certificate that uses the i.i.d. protocol and whose other classes admit component hiding — a property of every class in the literature — certifies at most $c^{**} = 0.382885260\\ldots$. So the entropy method, in every form used since 2022, cannot reach $1/2$: Frankl's conjecture is out of its range, and only $1.8\\cdot10^{-4}$ of headroom remained above the previous record. The paper also constructs a protocol attaining $0.38284$ (computer-assisted, conditional on two numerically verified hypotheses). Frankl's conjecture itself remains wide open.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:the-ceiling-of-the-single-letter-entropy-method-for-the-union-closed-sets-conjec:attempt"
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  "attempts": [
    {
      "id": "vibemathed:the-ceiling-of-the-single-letter-entropy-method-for-the-union-closed-sets-conjec:attempt",
      "problem_id": "vibemathed:the-ceiling-of-the-single-letter-entropy-method-for-the-union-closed-sets-conjec",
      "model": "Fable 5.1",
      "provider": null,
      "attempted_at": "2026-09-08T00:00:00.000Z",
      "human_collaborators": [
        "Andrew Moffat"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "A team of Claude agents working under the author's direction. Claude Fable 5.1 selected the problem, found both ceiling theorems — including the component-hiding adversary, which is the new idea — and designed the optimal-diagonal protocol and its certification. Claude Opus 5 agents refereed the manuscript over four rounds (catching an error in the lead agent's own lemma), proved the small-entropy lemma, and independently re-implemented the numerical certificate from the written statement alone. Claude Sonnet agents transcribed the cited literature. The author set the goal and the standard, checked the claims, and decided what to publish. Referee reports and logs are in the repository.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "The paper, 31 pages with the Lean appendices, at v1.2",
          "url": "https://github.com/moffatstudio/union-closed-constant/blob/v1.2/paper/paper.pdf",
          "kind": "paper",
          "license": null
        },
        {
          "label": "lean/: Theorems 3.1 and 3.4 machine-checked, with check.sh",
          "url": "https://github.com/moffatstudio/union-closed-constant/tree/v1.2/lean",
          "kind": "lean-proof",
          "license": null
        },
        {
          "label": "Cambie 2022, which studies the entropy approach's boundaries",
          "url": "https://arxiv.org/abs/2212.12500",
          "kind": "problem-record",
          "license": null
        }
      ],
      "sources": [
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        {
          "label": "VibeMathed: The ceiling of the single-letter entropy method for the union-closed sets conjecture",
          "url": "https://vibemathed.com/problem/the-ceiling-of-the-single-letter-entropy-method-for-the-union-closed-sets-conjec",
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      "solution_event_id": "vibemathed:the-ceiling-of-the-single-letter-entropy-method-for-the-union-closed-sets-conjec:event",
      "level": "lean_checked_statement_unaudited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Filed at Lean-checked rather than Lean-verified, and the distinction is the one that tier exists for. The two unconditional ceiling theorems are machine-checked: the lean/ project builds with no sorry and only propext, Classical.choice and Quot.sound, check.sh enforces both, and the lean and verify workflows were green at HEAD on 10 September. What Lean proves, though, is a theorem about Certifies, the author's own finite-model definition of a single-letter certificate. Whether every certificate in the literature is an instance is Lemma 3.3, whose maximal-correlation case is paper-only, and Gilmer's reduction from union-closed families to the certificate is a hypothesis in the development rather than a theorem. So the artifact compiles and proves what the paper's framework says; the framework's fidelity to the informal claim that the entropy method cannot reach $1/2$ was not audited here. The computer-assisted constant $0.38284$ is conditional on two numerically verified hypotheses of the same kind as those behind Liu's record, and is not in the formalisation. Not peer reviewed and not checked by a named expert; four rounds of referee reports were produced by the author's own model agents and are in the repository.",
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          "label": "VibeMathed: The ceiling of the single-letter entropy method for the union-closed sets conjecture",
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    {
      "label": "VibeMathed: The ceiling of the single-letter entropy method for the union-closed sets conjecture",
      "url": "https://vibemathed.com/problem/the-ceiling-of-the-single-letter-entropy-method-for-the-union-closed-sets-conjec",
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      "generated_at": "2026-09-13T16:28:40.178Z"
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  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
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  "generated_at": "2026-09-13T16:28:40.178Z"
}