{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:conway-s-refinement-conjecture-for-omnific-integers", "title": "Conway's Refinement Conjecture for Omnific Integers", "canonical_statement": "If $a,b,c,d\\in\\mathbf{Oz}$ are omnific integers and\n$$\nab=cd,\n$$\nthen there exist omnific integers $e,f,g,h\\in\\mathbf{Oz}$ such that\n$$\na=ef,\\qquad b=gh,\\qquad c=eg,\\qquad d=fh.\n$$\nEquivalently, every equality of two products in the omnific integers admits a common four-factor refinement.", "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:conway-s-refinement-conjecture-for-omnific-integers:event", "problem_id": "vibemathed:conway-s-refinement-conjecture-for-omnific-integers", "type": "proved", "status": "candidate", "occurred_at": "2026-09-03T00:00:00.000Z", "title": "Github", "summary": "The repository gives a Lean proof of Conway's 1976 refinement conjecture for omnific integers: whenever\n$$\nab=cd\n$$\nwith $a,b,c,d\\in\\mathbf{Oz}$, there exist $e,f,g,h\\in\\mathbf{Oz}$ satisfying\n$$\na=ef,\\quad b=gh,\\quad c=eg,\\quad d=fh.\n$$\nThe proof is formalized twice: once using CombinatorialGames' surreal-number implementation and once with the required surreal definitions inlined over Mathlib. The development also proves stronger structural results about factorization in Hahn-series integer parts and related generalized power-series rings.", "ai_contribution": "ai_co_developed", "attempt_ids": [ "vibemathed:conway-s-refinement-conjecture-for-omnific-integers:attempt" ], "method_family_ids": [ "vibemathed:conway-s-refinement-conjecture-for-omnific-integers:method" ], "source_assertion_ids": [ "vibemathed:conway-s-refinement-conjecture-for-omnific-integers:assertion" ] } ], "attempts": [ { "id": "vibemathed:conway-s-refinement-conjecture-for-omnific-integers:attempt", "problem_id": "vibemathed:conway-s-refinement-conjecture-for-omnific-integers", "model": "ChatGPT; Claude", "provider": "OpenAI; Anthropic", "attempted_at": "2026-09-03T00:00:00.000Z", "human_collaborators": [ "Dan Abramov" ], "exposure": "unknown", "prompt_public": null, "outcome": "Dan Abramov describes the project as an AI-proof experiment. Claude first selected Conway's refinement conjecture as a promising open problem after noting recent progress by L'Innocente and Mantova. Over roughly a month, Abramov repeatedly used ChatGPT and Claude to generate mathematical ideas and then steered the systems into producing Lean proofs, using compilation and formal checking to reject invalid directions. The repository therefore attributes the mathematical proof search and much of the formal proof construction to interactive AI exploration under human steering.", "failed_routes": [], "artifacts": [ { "label": "CombinatorialGames statement of the conjecture", "url": "https://github.com/gaearon/conway-refinement/blob/main/ConwayRefinement/Standalone/CombinatorialGames/ConwayRefinement.lean", "kind": "lean-statement", "license": null }, { "label": "Proof of that statement", "url": "https://github.com/gaearon/conway-refinement/blob/main/ConwayRefinement/Standalone/CombinatorialGames/ConwayRefinementProof.lean", "kind": "lean-proof", "license": null }, { "label": "Mathlib-only statement, no CombinatorialGames dependency", "url": "https://github.com/gaearon/conway-refinement/blob/main/ConwayRefinement/Standalone/Mathlib/InlineConwayRefinement.lean", "kind": "lean-statement", "license": null }, { "label": "Palomar provenance and the comparator statement", "url": "https://github.com/gaearon/conway-refinement/blob/main/PALOMAR-PROVENANCE.md", "kind": "palomar", "license": null }, { "label": "Rebuilt here: lake build, axiom audit and leanchecker replay", "url": "https://github.com/mrconter1/vibemathed/actions/runs/33843993855", "kind": "independent", "license": null } ], "sources": [ { "label": "Github", "url": "https://github.com/gaearon/conway-refinement", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Conway's Refinement Conjecture for Omnific Integers", "url": "https://vibemathed.com/problem/conway-s-refinement-conjecture-for-omnific-integers", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:conway-s-refinement-conjecture-for-omnific-integers:verification", "problem_id": "vibemathed:conway-s-refinement-conjecture-for-omnific-integers", "solution_event_id": "vibemathed:conway-s-refinement-conjecture-for-omnific-integers:event", "level": "unreviewed", "mathematical_correctness": "unknown", "statement_fidelity": "unaudited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "Site-confirmed: rebuilt here on 4 September 2026, not taken from the repository's CI. This site's verify-lean workflow checked out gaearon/conway-refinement at commit 264445c9, the commit the entry cites, installed the toolchain it pins (leanprover/lean4:v4.31.0) with its CombinatorialGames dependency, ran $\\texttt{lake build}$ over every module (3145 jobs), then the project's own scripts/Axioms.lean, then $\\texttt{lake env leanchecker}$ replaying the whole ConwayRefinement environment. 37 minutes, every step green.\n\nBoth formulations of the conjecture - the one over the CombinatorialGames $\\texttt{Surreal}$ type and the Mathlib-only one with the surreal definitions inlined - report $\\texttt{propext}$, $\\texttt{Classical.choice}$ and $\\texttt{Quot.sound}$ and nothing else. No $\\texttt{sorryAx}$, no project axiom.\n\nStill Candidate rather than Resolved, and the reason is the one the author gives himself: a kernel checks the proof against the statement as written, and whether that statement is Conway's conjecture is a reading a surreal-number specialist has to do. The definition used is Conway's own cut $x = \\{x - 1 \\mid x + 1\\}$ and the statement is a few lines, so it is an afternoon's work for the right reader. Nobody without a stake has done it yet.", "sources": [ { "label": "VibeMathed: Conway's Refinement Conjecture for Omnific Integers", "url": "https://vibemathed.com/problem/conway-s-refinement-conjecture-for-omnific-integers", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "Github", "url": "https://github.com/gaearon/conway-refinement", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "Github", "url": "https://github.com/gaearon/conway-refinement", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Conway's Refinement Conjecture for Omnific Integers", "url": "https://vibemathed.com/problem/conway-s-refinement-conjecture-for-omnific-integers", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "verify", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:conway-s-refinement-conjecture-for-omnific-integers:signal:verify_now", "problem_id": "vibemathed:conway-s-refinement-conjecture-for-omnific-integers", "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" }