A Counterexample to the Stable Forking Conjecture
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).
- Result
- Disproved(see note)
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Construction
- Field
- Model theory
- Posed by
- Bradd Hart, Byunghan Kim, and Anand Pillay
- Year posed
- 1996
- Years open
- 30y
- Solved
- 2026-08-31
- Model
- GPT 5.6 Sol
- Vendor
- OpenAI
- Collaborators
- James Freitag; Scott Mutchnik
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 26 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
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 then there is forking over whose parameter-free form is stable.
The 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.
What the AI did
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.
The 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".
Verification
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.
Source
- PaperarXiv
Submitted by VibeGene on