Counterexample to the stable forking conjecture
A simple first-order theory is constructed in which forking cannot always be witnessed by a stable formula, answering a question of Hart, Kim and Pillay open since 1996.
- Lab
- Independent
- Model
- GPT-5.6 Sol
- Field
- Mathematics
- Date
- 2026-08-31
- Human collaborators
- James Freitag, Scott Mutchnik
- Problem posed
- 1996 · open 30 yrs
Sources
Original work
Independent commentary
What was found
The stable forking conjecture asserts that in a simple theory, whenever a does not fork with b over C, there is a formula in the type of a over Cb that forks over C and 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 through an abstract independence relation using the Kim-Pillay criteria, and the random graph is encoded into that relation so that it has the order property, which is what breaks stability.
Novelty check
The conjecture is a named open question in classification theory, attributed in the paper to Hart, Kim and Pillay (1996) and described there as long-standing. The construction runs through the standard route for such results, proving simplicity and then characterising forking independence via the Kim-Pillay criteria, so its relation to the existing literature is explicit rather than assumed. Model-theoretic study of noncommutative rings has precedent, which the authors cite; no prior counterexample appears.
Caveats and known objections
A preprint days old at entry, unrefereed, not formalized, and with no independent check on record. Autonomy is graded collaborative rather than ai-led on the authors' own account of the division of labour: they prompted the model with a detailed series of prompts that encoded the known restrictions such a counterexample must satisfy, and specifically directed it to the standard strategy of proving simplicity and applying the Kim-Pillay criteria, so the framing was substantially theirs. They state that the proofs of the relevant facts and aspects of the setup were written entirely by the authors and that no AI tool was used in writing the manuscript, with AI used for proofreading under manual evaluation. They also record their view that the counterexample would have been unlikely to be found in the near term without generative AI.
Also recorded at
vibematheda-counterexample-to-the-stable-forking-conjecture
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
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Entry history (1 event)
- AddedEntered the registry graded Author verified and Collaborative.
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Graded author verified for verification and collaborative for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). Counterexample to the stable forking conjecture. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-31-stable-forkingBibTeX
@misc{whataifound-independent-2026-forking,
title = {Counterexample to the stable forking conjecture},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Author verified. Autonomy: Collaborative.},
url = {https://whataifound.org/finding/2026-08-31-stable-forking}
}