{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:the-proper-hat-guessing-number-of-k-8-e", "title": "The proper hat-guessing number of $K_8-e$", "canonical_statement": "Determine the exact proper hat-guessing number of the complete graph on eight vertices with one edge removed. The general bounds leave $\\mathrm{HG}_P(K_8-e)\\in\\{13,14\\}$.", "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-8-e:event", "problem_id": "vibemathed:the-proper-hat-guessing-number-of-k-8-e", "type": "proved", "status": "candidate", "occurred_at": "2026-09-04T00:00:00.000Z", "title": "K8-e public research disclosure v0.1 (GitHub)", "summary": "We prove $\\mathrm{HG}_P(K_8-e)=14$ using an explicit $\\mathrm{PGL}(2,13)$-equivariant strategy. A compact formula fixes 990 normalized twin decisions, and a 48,510-edge residual-orbit matching induces consistent rules on 53,460 labelled clique-view orbits. Independent implementations regenerate the certificate and check all 138,378,240 proper fourteen-colourings with zero failures. Unlike the $n=5,6,7$ constructions, residual right degrees reach eight, so the explicit global matching is load-bearing. A separately labelled companion exhibits one fixed clique completion compatible with $2^{380}$ equivariant twin-rule pairs and proves 380 optimal within the stated independent whole-tail reversal model; that companion's independent review is pending. The release does not solve the general $K_n-e$ family or $K_9-e$.", "ai_contribution": "ai_co_developed", "attempt_ids": [ "vibemathed:the-proper-hat-guessing-number-of-k-8-e:attempt" ], "method_family_ids": [ "vibemathed:the-proper-hat-guessing-number-of-k-8-e:method" ], "source_assertion_ids": [ "vibemathed:the-proper-hat-guessing-number-of-k-8-e:assertion" ] } ], "attempts": [ { "id": "vibemathed:the-proper-hat-guessing-number-of-k-8-e:attempt", "problem_id": "vibemathed:the-proper-hat-guessing-number-of-k-8-e", "model": "GPT-5.6 Pro", "provider": "OpenAI", "attempted_at": "2026-09-04T00:00:00.000Z", "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, certificate design, and manuscript and release preparation. The equivariant construction, residual-matching proof, Hall-obstruction analysis, and shared-completion family were developed in a human-directed model-assisted research process. Matthew Protti selected and framed the target, directed and evaluated the work, commissioned review, required exact checks, approved the public scope, and accepts responsibility.", "failed_routes": [], "artifacts": [ { "label": "K8-e theorem and proof", "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/blob/bacb79e3201dcc59b57339181affda122db771ce/research/proper_hat_guessing_k8_minus_e/THEOREM.md", "kind": "paper", "license": null }, { "label": "Verified construction and full-colouring checker", "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/tree/bacb79e3201dcc59b57339181affda122db771ce/research/proper_hat_guessing_k8_minus_e/accepted_construction", "kind": "code", "license": null }, { "label": "Review-pending shared-completion companion", "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/tree/bacb79e3201dcc59b57339181affda122db771ce/research/proper_hat_guessing_k8_minus_e/shared_completion_companion", "kind": "other", "license": null }, { "label": "Repaired Gate 2/3 Hall-obstruction archive", "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/tree/bacb79e3201dcc59b57339181affda122db771ce/research/proper_hat_guessing_k8_minus_e/supporting_gate23", "kind": "other", "license": null }, { "label": "Public draft PR #3 and review surface", "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/pull/3", "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": "K8-e public research disclosure v0.1 (GitHub)", "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/commit/bacb79e3201dcc59b57339181affda122db771ce", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: The proper hat-guessing number of $K_8-e$", "url": "https://vibemathed.com/problem/the-proper-hat-guessing-number-of-k-8-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-8-e:verification", "problem_id": "vibemathed:the-proper-hat-guessing-number-of-k-8-e", "solution_event_id": "vibemathed:the-proper-hat-guessing-number-of-k-8-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. The frozen commit bacb79e3 carries STATUS.json with headline status PROVED_COMPUTER_ASSISTED and project review ACCEPTED, an AI_USE_AND_PROVENANCE.md naming the model and the division of labour, and a C++ sweep of all 138,378,240 proper fourteen-colourings with zero coverage failures across 48,510 residual orbits. The Gate 2/3 Hall-obstruction archive was reviewed as ACCEPT_WITH_EXPLICIT_REPAIRS and the repairs applied. The 2^380 shared-completion companion is a separate claim whose independent review is still pending and is not part of this entry. As with the siblings: a real adversarial review of an artefact, not a named expert or a proof assistant, and this site has not rebuilt it.", "sources": [ { "label": "VibeMathed: The proper hat-guessing number of $K_8-e$", "url": "https://vibemathed.com/problem/the-proper-hat-guessing-number-of-k-8-e", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "K8-e public research disclosure v0.1 (GitHub)", "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/commit/bacb79e3201dcc59b57339181affda122db771ce", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "K8-e public research disclosure v0.1 (GitHub)", "url": "https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/commit/bacb79e3201dcc59b57339181affda122db771ce", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: The proper hat-guessing number of $K_8-e$", "url": "https://vibemathed.com/problem/the-proper-hat-guessing-number-of-k-8-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-8-e:signal:verify_now", "problem_id": "vibemathed:the-proper-hat-guessing-number-of-k-8-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" }