{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-09-07-koethe-conjecture-disproved/","md":"https://postcutoff.com/e/2026-09-07-koethe-conjecture-disproved/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-09-07-koethe-conjecture-disproved","date":"2026-09-07","date_precision":"day","short_title":"Pre-release GPT-6 Astra disproves the Köthe conjecture with a Lean-verified counterexample","deck":null,"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.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"arXiv 2609.07996 (write-up)","url":"https://arxiv.org/abs/2609.07996","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"GitHub: tadamcz/koethe (Lean proof)","url":"https://github.com/tadamcz/koethe","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"Wikipedia: List of mathematical discoveries by artificial intelligence","url":"https://en.wikipedia.org/wiki/List_of_mathematical_discoveries_by_artificial_intelligence","type":"discussion","group":"reactions","domain":"en.wikipedia.org"}],"official":2,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI","Epoch AI"],"title":"Pre-release GPT-6 Astra disproves the Köthe conjecture (1930) with a Lean-verified counterexample","summary":"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"],"key_numbers":[],"tags":["math","ring-theory","counterexample","lean","astra"],"science":{"field":"mathematics","subfield":"ring theory","problem":"Köthe conjecture","result":"Explicit counterexample disproving the Köthe conjecture, formally verified in Lean.","open_since":"1930","ai_system":["GPT-6 Astra (pre-release)"],"human_role":"Autonomous discovery; humans checked and wrote up","verification":{"key":"lean","label":"Lean-verified","text":"Formal proof in Lean 4 (repo tadamcz/koethe: Comparator-checked against the Formal Conjectures statement of Krempa's matrix form, standard axioms only, nanoda enabled); human write-up with ring theorists Böhmler and Marczinzik (arXiv 2609.07996) unrefereed"},"status":"confirmed","shock":"A 96-year-old central problem of noncommutative ring theory fell as a side effect of a benchmark run."},"body_md":"## What happened\n\nEpoch 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.\n\n## Why it matters\n\nIf it survives review, it resolves one of the most famous open problems in ring theory, found autonomously and verified formally.","disputed":[],"related":[{"id":"2026-09-10-astra-leanopenproblems-september-results","url":"https://postcutoff.com/e/2026-09-10-astra-leanopenproblems-september-results/","date":"2026-09-10","date_precision":"day","short_title":"GPT-6 Astra's Epoch AI run adds more Lean-checked results","deck":"Dittert conjecture proved, Ibragimov–Iosifescu and eternal-domination conjectures disproved","takeaway":"Autonomous formal proof search now turns out a steady stream of mid-level resolved conjectures, not one-off headlines.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":4,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI","Epoch AI"]},{"id":"2026-09-03-gpt-6-astra","url":"https://postcutoff.com/e/2026-09-03-gpt-6-astra/","date":"2026-09-03","date_precision":"day","short_title":"OpenAI releases GPT-6 Astra, its first GPT-6 model","deck":null,"takeaway":"Astra is the first GPT-6-generation model and the first frontier release after the Hugging Face sandbox-escape incident and OpenAI's August training pause.","category":"model-release","category_label":"Model releases","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":20,"official":11,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"]},{"id":"2026-08-01-openai-astra-ten-advances","url":"https://postcutoff.com/e/2026-08-01-openai-astra-ten-advances/","date":"2026-08-01","date_precision":"day","short_title":"OpenAI's unreleased 'Astra' model claims ten advances in maths and theoretical CS, with Lean proofs","deck":null,"takeaway":"It moved the frontier from individual AI-assisted results to a lab producing batches of significant theorems.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":5,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-19","type":"edit","text":"Related: back-link to the LeanOpenProblems September results entry"},{"date":"2026-10-19","type":"edit","text":"Status pending → confirmed: complete Lean proof checked by Comparator against an independently written statement (Formal Conjectures); comparator.json and formalization.yaml read"},{"date":"2026-09-29","type":"filed","text":"Created"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-09-29","run":null,"sources_read":null,"updated":"2026-10-19","human_review":null,"version":{"date":"2026-10-19"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":130,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":69,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":160,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":99,"in_training_data":false}],"short_url":"https://postcutoff.com/s/pre-release-gpt-6-astra-2"}