{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:the-proper-hat-guessing-number-of-k-7-e", "title": "The proper hat-guessing number of $K_7-e$", "canonical_statement": "Determine the exact proper hat-guessing number of the complete graph on seven vertices with one edge removed. The general bounds leave $\\mathrm{HG}_P(K_7-e)\\in\\{11,12\\}$.", "plain_summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.", "current_status": "candidate", "solution_events": [ { "id": "vibemathed:the-proper-hat-guessing-number-of-k-7-e:event", "problem_id": "vibemathed:the-proper-hat-guessing-number-of-k-7-e", "type": "proved", "status": "candidate", "occurred_at": "2026-09-03T00:00:00.000Z", "title": "Reviewed K7-e public research disclosure v0.1 (GitHub)", "summary": "We prove $\\mathrm{HG}_P(K_7-e)=12$. One lower-bound proof uses two explicit block-disjoint $S(5,6,12)$ Witt designs. A second uses orbit maps from an explicitly regenerated sharply five-transitive twelve-point permutation group, combining one set-symmetric and one order-sensitive rule. Both satisfy a general coordinate-line twin-completion criterion, and Hall's theorem completes the clique strategy. The release also proves a disjoint completion-design theorem, an even-$n$ obstruction scoped to set-symmetric line-permutation twins in this sufficient framework, and a prime-admissibility theorem for the design parameters. It does not solve the general $K_n-e$ problem or $K_8-e$.", "ai_contribution": "ai_co_developed", "attempt_ids": [ "vibemathed:the-proper-hat-guessing-number-of-k-7-e:attempt" ], "method_family_ids": [ "vibemathed:the-proper-hat-guessing-number-of-k-7-e:method" ], "source_assertion_ids": [ "vibemathed:the-proper-hat-guessing-number-of-k-7-e:assertion" ] } ], "attempts": [ { "id": "vibemathed:the-proper-hat-guessing-number-of-k-7-e:attempt", "problem_id": "vibemathed:the-proper-hat-guessing-number-of-k-7-e", "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. The Witt-design and orbit-map proof architecture was developed in a human-directed model-assisted research process. Matthew Protti selected and framed the target, directed the research programme, evaluated candidate arguments, commissioned independent adversarial review, required exact checks, approved the public scope, and accepts responsibility.", "failed_routes": [], "artifacts": [ { "label": "Four-page K7-e preprint source", "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/blob/39f7716c2d8f4e9a84963c45fd7b8b256bf7ac08/research/proper_hat_guessing_k7_minus_e/preprint/proper_hat_guessing_K7_minus_e_v0.1.tex", "kind": "paper", "license": null }, { "label": "Verifier, certificates and public disclosure", "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/tree/39f7716c2d8f4e9a84963c45fd7b8b256bf7ac08/research/proper_hat_guessing_k7_minus_e", "kind": "code", "license": null }, { "label": "Independent adversarial review acceptance", "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/blob/39f7716c2d8f4e9a84963c45fd7b8b256bf7ac08/research/proper_hat_guessing_k7_minus_e/REVIEW/INDEPENDENT_ADVERSARIAL_REVIEW_ACCEPTANCE_20260904.md", "kind": "other", "license": null }, { "label": "Public draft PR #2 and release record", "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/pull/2", "kind": "announcement", "license": null }, { "label": "Hat guessing with proper colorings (Adriaensen et al.)", "url": "https://arxiv.org/abs/2603.04909", "kind": "problem-record", "license": null } ], "sources": [ { "label": "Reviewed K7-e public research disclosure v0.1 (GitHub)", "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/commit/39f7716c2d8f4e9a84963c45fd7b8b256bf7ac08", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: The proper hat-guessing number of $K_7-e$", "url": "https://vibemathed.com/problem/the-proper-hat-guessing-number-of-k-7-e", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:the-proper-hat-guessing-number-of-k-7-e:verification", "problem_id": "vibemathed:the-proper-hat-guessing-number-of-k-7-e", "solution_event_id": "vibemathed:the-proper-hat-guessing-number-of-k-7-e:event", "level": "unreviewed", "mathematical_correctness": "unknown", "statement_fidelity": "unaudited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "Unreviewed on this site's ladder, but better evidenced than either sibling. An independent adversarial review rebuilt both 132-block S(5,6,12) completion designs, all 792 pentad completions, design disjointness, the 95,040-element group, sharp five-transitivity, all 495 set lines, all 59,400 ordered lines and both Hall-degree censuses, returning ACCEPT_K7E_THEOREM with no mathematical repairs required; STATUS.json pins the review package by SHA-256. That is a real check by someone other than the author, but it is an adversarial review of an artefact rather than a named expert endorsing the theorem or a proof assistant checking it, and this site has not rebuilt it. Conventional peer review is not claimed.", "sources": [ { "label": "VibeMathed: The proper hat-guessing number of $K_7-e$", "url": "https://vibemathed.com/problem/the-proper-hat-guessing-number-of-k-7-e", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "Reviewed K7-e public research disclosure v0.1 (GitHub)", "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/commit/39f7716c2d8f4e9a84963c45fd7b8b256bf7ac08", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "Reviewed K7-e public research disclosure v0.1 (GitHub)", "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/commit/39f7716c2d8f4e9a84963c45fd7b8b256bf7ac08", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: The proper hat-guessing number of $K_7-e$", "url": "https://vibemathed.com/problem/the-proper-hat-guessing-number-of-k-7-e", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "verify", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:the-proper-hat-guessing-number-of-k-7-e:signal:verify_now", "problem_id": "vibemathed:the-proper-hat-guessing-number-of-k-7-e", "kind": "verify_now", "reason": "The source labels this result as a candidate; verification should precede reuse.", "generated_at": "2026-09-13T16:28:40.178Z" } ], "uncertainties": [ "The source status is candidate and must not be represented as solved.", "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." ], "generated_at": "2026-09-13T16:28:40.178Z" }