Pre-release GPT-6 Astra disproves the Köthe conjecture with a Lean-verified counterexample
Result confirmed
Importance: major (4 of 5)The takeaway
During an Epoch AI run over the Formal Conjectures collection, pre-release GPT-6 Astra autonomously found an explicit 2×2 matrix counterexample over a nil algebra (Krempa’s matrix form) with a Lean 4 proof, disproving the Köthe conjecture of 1930.
Status
- Claim
Result confirmed
- Our reporting
- High confidence
- Verification
- Lean-verified
- Importance
- Major (4 of 5)
- Last verified
- 19 October 2026
Your AI and this story
- GPT-6 Astra130 days after its cutoff
- Claude Opus 5.569 days after its cutoff
- Gemini 4 Hydrogen160 days after its cutoff
- Grok 4.799 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 69 days before it.
Key facts
- Köthe conjecture (1930): if a ring has no nonzero nil two-sided ideals, it has no nonzero nil one-sided ideals
- Counterexample via Krempa’s equivalent matrix formulation; Lean 4 proof
- Found inside Epoch AI’s LeanOpenProblems evaluation (222 research-open formal problems); repository README: ‘No human saw or steered the proof search’
- Write-up by Adamczewski, Böhmler and Marczinzik; a second counterexample by Greenfeld, King and Vendramin with some Astra help
What happened
Epoch AI ran pre-release Astra against a library of formalised open conjectures. The model returned a Lean-checked counterexample to Köthe’s conjecture, which human algebraists then confirmed and wrote up.
Why it matters
If it survives review, it resolves one of the most famous open problems in ring theory, found autonomously and verified formally.
Sources
3 sources from 3 sites. Numbers match the chips in the text.
3 sources: 2 primary, 1 reaction
Primary
- arXiv 2609.07996 (write-up)arxiv.org, paper
- GitHub: tadamcz/koethe (Lean proof)github.com, code
Reactions
- Wikipedia: List of mathematical discoveries by artificial intelligenceen.wikipedia.org, discussion
Changes
- Related: back-link to the LeanOpenProblems September results entry
- Status pending → confirmed: complete Lean proof checked by Comparator against an independently written statement (Formal Conjectures); comparator.json and formalization.yaml read
- Filed