{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:a-counterexample-to-the-stable-forking-conjecture", "title": "A Counterexample to the Stable Forking Conjecture", "canonical_statement": "Using ChatGPT 5.6, we find a counterexample to the stable forking conjecture. This answers a long-standing open question of Hart, Kim and Pillay (1996).", "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.", "current_status": "resolved", "solution_events": [ { "id": "vibemathed:a-counterexample-to-the-stable-forking-conjecture:event", "problem_id": "vibemathed:a-counterexample-to-the-stable-forking-conjecture", "type": "disproved", "status": "resolved", "occurred_at": "2026-08-31T00:00:00.000Z", "title": "arXiv", "summary": "The paper constructs a simple theory in which forking cannot always be witnessed by a stable formula. Precisely, what fails is: in a simple theory, if $a \\not\\downarrow_C b$ then there is $\\varphi(x,\\bar b) \\in \\mathrm{tp}(a/Cb)$ forking over $C$ whose parameter-free form $\\varphi(x,y)$ is stable.\n\nThe counterexample is an infinite-dimensional vector space over the division ring of fractions of the quantum graph algebra of the random graph. Forking is characterised by an abstract independence relation, and the random graph is encoded into that relation so that it has the order property; stable formulas therefore cannot determine all forking in simple theories.", "ai_contribution": "ai_co_developed", "attempt_ids": [ "vibemathed:a-counterexample-to-the-stable-forking-conjecture:attempt" ], "method_family_ids": [ "vibemathed:a-counterexample-to-the-stable-forking-conjecture:method" ], "source_assertion_ids": [ "vibemathed:a-counterexample-to-the-stable-forking-conjecture:assertion" ] } ], "attempts": [ { "id": "vibemathed:a-counterexample-to-the-stable-forking-conjecture:attempt", "problem_id": "vibemathed:a-counterexample-to-the-stable-forking-conjecture", "model": "GPT 5.6 Sol", "provider": "OpenAI", "attempted_at": "2026-08-31T00:00:00.000Z", "human_collaborators": [ "James Freitag; Scott Mutchnik" ], "exposure": "unknown", "prompt_public": null, "outcome": "GPT-5.6 Sol generated the counterexample through an interactive, human-guided search. Freitag and Mutchnik prompted it with structural restrictions any counterexample would have to satisfy, including using the Kim-Pillay abstract-independence criterion to prove simplicity and characterise forking, and directed the search toward infinite rank - earlier work having placed severe obstructions on a finite-rank example. They independently wrote the proofs and the manuscript.\n\nThe paper says it plainly, in the abstract and again in the text: \"This is an AI-generated result proven with the help of GPT-5.6 Sol. Specifically, we prompted ChatGPT to construct a counterexample to the stable forking conjecture with a detailed series of prompts which took into account the likely restrictions such a counterexample would have to satisfy.\" They add that \"in retrospect, it seems unlikely that this counterexample would have been found in the near term without the use of generative AI\".", "failed_routes": [], "artifacts": [], "sources": [ { "label": "arXiv", "url": "https://arxiv.org/abs/2609.00436", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: A Counterexample to the Stable Forking Conjecture", "url": "https://vibemathed.com/problem/a-counterexample-to-the-stable-forking-conjecture", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:a-counterexample-to-the-stable-forking-conjecture:verification", "problem_id": "vibemathed:a-counterexample-to-the-stable-forking-conjecture", "solution_event_id": "vibemathed:a-counterexample-to-the-stable-forking-conjecture:event", "level": "unreviewed", "mathematical_correctness": "unknown", "statement_fidelity": "unaudited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "Unreviewed. A preprint three days old. The construction is explicit and the authors give the abstract-independence argument in full, so it is checkable by a model theorist, but none has done so on the record.", "sources": [ { "label": "VibeMathed: A Counterexample to the Stable Forking Conjecture", "url": "https://vibemathed.com/problem/a-counterexample-to-the-stable-forking-conjecture", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "arXiv", "url": "https://arxiv.org/abs/2609.00436", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "arXiv", "url": "https://arxiv.org/abs/2609.00436", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: A Counterexample to the Stable Forking Conjecture", "url": "https://vibemathed.com/problem/a-counterexample-to-the-stable-forking-conjecture", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "verify", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:a-counterexample-to-the-stable-forking-conjecture:signal:watch_only", "problem_id": "vibemathed:a-counterexample-to-the-stable-forking-conjecture", "kind": "watch_only", "reason": "Current public evidence does not meet the default replay threshold.", "generated_at": "2026-09-13T16:28:40.178Z" } ], "uncertainties": [ "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" }