{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:the-proper-hat-guessing-number-of-k-5-e", "title": "The proper hat-guessing number of $K_5-e$", "canonical_statement": "Determine the exact proper hat-guessing number of the complete graph on five vertices with one edge removed. The existing bounds left $\\mathrm{HG}_P(K_5-e)$ in $\\{7,8\\}$.", "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-5-e:event", "problem_id": "vibemathed:the-proper-hat-guessing-number-of-k-5-e", "type": "proved", "status": "candidate", "occurred_at": "2026-09-02T00:00:00.000Z", "title": "The proper hat-guessing number of K5-e - public research disclosure v0.1", "summary": "The exact value is $\\mathrm{HG}_P(K_5-e)=8$. The lower bound uses explicit legal twin-player rules over $\\mathbb F_2^3$. After those rules cover 3,024 of the 8,400 proper colorings, the residual-coloring/local-view incidence graph has left degree three and right degree at most three; Hall's theorem supplies consistent guesses for the three clique players. The release also proves a general sufficient twin-completion lemma for $K_n-e$. It does not solve the full $K_n-e$ family or determine $\\mathrm{HG}_P(C_5)$.", "ai_contribution": "ai_co_developed", "attempt_ids": [ "vibemathed:the-proper-hat-guessing-number-of-k-5-e:attempt" ], "method_family_ids": [ "vibemathed:the-proper-hat-guessing-number-of-k-5-e:method" ], "source_assertion_ids": [ "vibemathed:the-proper-hat-guessing-number-of-k-5-e:assertion" ] } ], "attempts": [ { "id": "vibemathed:the-proper-hat-guessing-number-of-k-5-e:attempt", "problem_id": "vibemathed:the-proper-hat-guessing-number-of-k-5-e", "model": "GPT-5.6 Pro", "provider": "OpenAI", "attempted_at": "2026-09-02T00:00:00.000Z", "human_collaborators": [], "exposure": "unknown", "prompt_public": null, "outcome": "OpenAI GPT-5.6 Pro contributed substantively to literature search, problem selection, construction search, proof development, code generation, computational verification, adversarial critique, and manuscript preparation. The central $\\mathbb F_2^3$ construction and Hall-completion proof were developed in a model-assisted process. Matthew Protti selected and framed the target, directed and evaluated the work, required exact checks, set the scope, approved disclosure, and accepts responsibility.", "failed_routes": [], "artifacts": [ { "label": "Tagged v0.1 release (ZIP and PDF)", "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/releases/tag/v0.1", "kind": "announcement", "license": null }, { "label": "Source problem: Hat guessing with proper colorings", "url": "https://arxiv.org/abs/2603.04909", "kind": "problem-record", "license": null } ], "sources": [ { "label": "The proper hat-guessing number of K5-e - public research disclosure v0.1", "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: The proper hat-guessing number of $K_5-e$", "url": "https://vibemathed.com/problem/the-proper-hat-guessing-number-of-k-5-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-5-e:verification", "problem_id": "vibemathed:the-proper-hat-guessing-number-of-k-5-e", "solution_event_id": "vibemathed:the-proper-hat-guessing-number-of-k-5-e:event", "level": "unreviewed", "mathematical_correctness": "unknown", "statement_fidelity": "unaudited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "No independent mathematical review yet. The public disclosure contains a self-contained proof, a seven-entry finite certificate, and a dependency-free Python verifier. The verifier has been run successfully from a clean clone and from the v0.1 release archive; it reconstructs the residual incidence graph, a saturating matching, and a complete 6,720-entry strategy, then checks all 8,400 proper colorings. This remains author-controlled verification.", "sources": [ { "label": "VibeMathed: The proper hat-guessing number of $K_5-e$", "url": "https://vibemathed.com/problem/the-proper-hat-guessing-number-of-k-5-e", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "The proper hat-guessing number of K5-e - public research disclosure v0.1", "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "The proper hat-guessing number of K5-e - public research disclosure v0.1", "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: The proper hat-guessing number of $K_5-e$", "url": "https://vibemathed.com/problem/the-proper-hat-guessing-number-of-k-5-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-5-e:signal:verify_now", "problem_id": "vibemathed:the-proper-hat-guessing-number-of-k-5-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" }