Pre-release GPT-6 Astra disproves the Köthe conjecture (1930) with a Lean-verified counterexample
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. Mathematicians wrote it up in arXiv 2609.07996.
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
Science result
- Field
- mathematics / ring theory
- Problem
- Köthe conjecture (open since 1930)
- Result
- Explicit counterexample disproving the Köthe conjecture, formally verified in Lean.
- AI system
- GPT-6 Astra (pre-release)
- Human role
- Autonomous discovery; humans checked and wrote up
- Verification
- Formal proof in Lean; pending peer review
- Status
- pending
- Why surprising
- A 96-year-old central problem of noncommutative ring theory fell as a side effect of a benchmark run.
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.
Changelog
- 2026-09-29: created
Related events
- OpenAI releases GPT-6 Astra, its first GPT-6 model ★★★★★
- OpenAI's unreleased 'Astra' model claims ten advances in maths and theoretical CS, with Lean proofs ★★★★★
- GPT-6 Astra's Epoch AI run adds more Lean-checked results: Dittert conjecture proved, Ibragimov–Iosifescu and eternal-domination conjectures disproved ★★★
Sources (3)
- paperarXiv 2609.07996 (write-up)
- codeGitHub: tadamcz/koethe (Lean proof)
- discussionWikipedia: List of mathematical discoveries by artificial intelligence
id: 2026-09-07-koethe-conjecture-disproved · updated 2026-09-29 · open in the interactive timeline