{
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  "problem_id": "vibemathed:the-proper-hat-guessing-number-of-k-6-e",
  "title": "The proper hat-guessing number of $K_6-e$",
  "canonical_statement": "Determine the exact proper hat-guessing number of the complete graph on six vertices with one edge removed. The known general bounds leave $\\mathrm{HG}_P(K_6-e)\\in\\{9,10\\}$.",
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      "status": "candidate",
      "occurred_at": "2026-09-03T00:00:00.000Z",
      "title": "Reviewed public research disclosure v0.1 (GitHub)",
      "summary": "We prove $\\mathrm{HG}_P(K_6-e)=10$. The lower bound uses two order-sensitive twin-player rules obtained by deleting and repairing one point of an explicit sharply four-transitive eleven-point permutation group. On every coordinate line the repaired rules are derangement permutations, are pointwise unequal, and have fixed-point-free composition. Hall's theorem completes the strategy on the four clique vertices. The release also classifies all repairable orbit labels and proves an even-$n$ obstruction for set-symmetric line-permutation twin rules. It does not solve the general $K_n-e$ family.",
      "ai_contribution": "ai_co_developed",
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        "vibemathed:the-proper-hat-guessing-number-of-k-6-e:attempt"
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      "model": "GPT-5.6 Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-09-03T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "OpenAI GPT-5.6 Pro contributed substantively to literature search, target selection, construction search, proof development, code generation, exact verification, adversarial critique, and manuscript preparation. Matthew Protti selected and framed the target, directed and evaluated the work, required exact checks, determined the claim scope, approved disclosure, and accepts responsibility.",
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          "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/blob/8f36d8c82bea951ae2f0147ebc2e66ea2bd6432e/research/proper_hat_guessing_k6_minus_e/preprint/proper_hat_guessing_K6_minus_e_v0.1.md",
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          "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/tree/8f36d8c82bea951ae2f0147ebc2e66ea2bd6432e/research/proper_hat_guessing_k6_minus_e",
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          "kind": "announcement",
          "license": null
        },
        {
          "label": "Hat guessing with proper colorings (Adriaensen et al.)",
          "url": "https://arxiv.org/abs/2603.04909",
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        }
      ],
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      "level": "unreviewed",
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      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "A separate adversarial technical review independently regenerated the principal finite core and prompted the scope and proof-presentation corrections incorporated into this public version. The dependency-free verifier regenerates the group, repairs, coordinate-line checks, label classification, Witt-design check, and residual right-degree census. Conventional journal peer review and Lean verification are not claimed.",
      "sources": [
        {
          "label": "VibeMathed: The proper hat-guessing number of $K_6-e$",
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      "kind": "verify_now",
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    "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"
}