{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-07-01-grothendieck-group-scheme-counterexample/","md":"https://postcutoff.com/e/2026-07-01-grothendieck-group-scheme-counterexample/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":"2026-07-01-grothendieck-group-scheme-counterexample","date":"2026-07-01","date_precision":"month","short_title":"AI-assisted counterexample answers Grothendieck's question on finite flat group schemes, merged into Mathlib","deck":null,"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).","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"Benjamin Antieau: Akhil Mathew and AI","url":"https://antieau.github.io/2026/08/10/akhil-mathew-ai.html","type":"discussion","group":"reactions","domain":"antieau.github.io"},{"n":2,"title":"Xena Project: Human mathematicians are being out-counterexampled","url":"https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/","type":"discussion","group":"reactions","domain":"xenaproject.wordpress.com"}],"official":0,"filed":"2026-09-29","updated":"2026-09-29","orgs":["OpenAI","Anthropic"],"title":"AI-assisted counterexample answers Grothendieck's question on finite flat group schemes, merged into Mathlib","summary":"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). This answers Grothendieck's question negatively. The Lean proof was merged into Mathlib on 3 Aug 2026.","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"],"key_numbers":[],"tags":["math","algebraic-geometry","counterexample","lean","mathlib"],"science":{"field":"mathematics","subfield":"algebraic geometry / group schemes","problem":"Grothendieck's question: is every finite locally free group scheme of order n killed by n?","result":"A counterexample of order 4 not killed by 4, over a non-reduced base.","open_since":"","ai_system":["GPT-5.6 Sol","Claude Fable 5"],"human_role":"AI-assisted: Akhil Mathew directed the search and verified","verification":{"key":"lean","label":"Lean-verified","text":"Formal proof in Lean (Mathlib)"},"status":"confirmed","shock":""},"body_md":"## What happened\n\nIn 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.\n\n## Why it matters\n\nKevin 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.","disputed":[],"related":[{"id":"2026-07-20-jacobian-conjecture-counterexample","url":"https://postcutoff.com/e/2026-07-20-jacobian-conjecture-counterexample/","date":"2026-07-20","date_precision":"day","short_title":"Claude Fable 5 finds a counterexample to the Jacobian conjecture in dimension 3","deck":null,"takeaway":"The Jacobian conjecture is one of the most famous open problems in algebra.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":9,"official":6,"filed":"2026-09-29","updated":"2026-11-10","orgs":["Anthropic"]}],"people":[{"id":"kevin-buzzard","name":"Kevin Buzzard","url":"https://postcutoff.com/person/kevin-buzzard/"},{"id":"benjamin-antieau","name":"Benjamin Antieau","url":"https://postcutoff.com/person/benjamin-antieau/"}],"posts":[],"videos":[],"models":[],"changes":[{"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":"2026-09-29","human_review":null,"version":{"date":"2026-10-04"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":null,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":null,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":null,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":null,"in_training_data":false}],"short_url":null}