AI-assisted counterexample answers Grothendieck’s question on finite flat group schemes, merged into Mathlib
Result confirmed
The takeaway
Akhil Mathew, using OpenAI’s and Anthropic’s models, found a finite locally free group scheme of order 4 over a non-reduced finite ring with 2⁹ elements that is not killed by 4 (it is killed by 8).
Status
- Claim
Result confirmed
- Our reporting
- Medium confidence
- Verification
- Lean-verified
- Importance
- 3 of 5
- Last verified
- 29 September 2026
Your AI and this story
- GPT-6 AstraAfter its cutoff
- Claude Opus 5.5After its cutoff
- Gemini 4 HydrogenAfter its cutoff
- Grok 4.7After its cutoff
None of these four assistants can know about it: it happened after all their cutoffs.
Key facts
- Known positive cases: commutative (Deligne), reduced base (Grothendieck); pure characteristic-p case still open
- Found by studying deformations of α₂×α₂
- Kevin Buzzard attributes discovery to OpenAI’s Sol and autoformalisation to Claude Fable
- Mathlib PR #41748 (Counterexamples/GrothendieckPower.lean), merged 3 Aug 2026
What happened
In the same weeks as the Jacobian counterexample, Mathew used frontier models to find and formalise a counterexample to a question from the foundations of algebraic geometry.
Why it matters
Kevin Buzzard said this mattered more to him than the Erdős results because it lies in “an area of mathematics that I personally find more interesting”. AI was now reaching core modern algebraic geometry.
Sources
2 sources from 2 sites. Numbers match the chips in the text.
2 sources: 2 reactions
Reactions
- Benjamin Antieau: Akhil Mathew and AIantieau.github.io, discussion
- Xena Project: Human mathematicians are being out-counterexampledxenaproject.wordpress.com, discussion
Changes
- Filed