{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-07-20-jacobian-conjecture-counterexample/","md":"https://postcutoff.com/e/2026-07-20-jacobian-conjecture-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":"2026-07-20-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":"Anthropic mathematician Levent Alpöge posted an explicit polynomial map F: C³→C³ with constant Jacobian determinant −2 that is not injective, found with Claude Fable 5.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"google-deepmind/formal-conjectures #7829: formal proof links for the Dixmier conjecture (9 Nov 2026)","url":"https://github.com/google-deepmind/formal-conjectures/commit/1aab32a062c4fe07048b5c5ce7515529604ec3ea","type":"code","group":"primary","domain":"github.com"},{"n":2,"title":"Palomar registry PALOMAR-2026-11-06-000008 (Lean: Dixmier ⇒ Jacobian, DC_n false for n ≥ 3)","url":"https://palomar-registry.org/entry.html?id=PALOMAR-2026-11-06-000008","type":"code","group":"primary","domain":"palomar-registry.org"},{"n":3,"title":"Shuhong Gao: counterexamples in all dimensions >2 (arXiv 2608.00222)","url":"https://arxiv.org/abs/2608.00222","type":"paper","group":"primary","domain":"arxiv.org"},{"n":4,"title":"de Bondt: Alpöge's counterexample to the JC, Economic Stabilization, and Zhao's Vanishing Conjecture (arXiv 2610.24889)","url":"https://arxiv.org/abs/2610.24889","type":"paper","group":"primary","domain":"arxiv.org"},{"n":5,"title":"Long: An Explicit Counterexample to the Rank-Two Poisson Conjecture (arXiv 2608.23777)","url":"https://arxiv.org/abs/2608.23777","type":"paper","group":"primary","domain":"arxiv.org"},{"n":6,"title":"formal-conjectures #7657: formal proof links for the Poisson conjecture","url":"https://github.com/google-deepmind/formal-conjectures/commit/a64e959c32c270df4dd0d96d2027481f15750534","type":"code","group":"primary","domain":"github.com"},{"n":7,"title":"ScienceDaily: Claude Fable 5 AI finds a tiny formula that topples an 87-year-old math conjecture","url":"https://www.sciencedaily.com/releases/2026/08/260804034634.htm","type":"press","group":"press","domain":"sciencedaily.com"},{"n":8,"title":"Terence Tao: A digestion of the Jacobian conjecture counterexample","url":"https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/","type":"discussion","group":"reactions","domain":"terrytao.wordpress.com"},{"n":9,"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":6,"filed":"2026-09-29","updated":"2026-11-10","orgs":["Anthropic"],"title":"Claude Fable 5 finds a counterexample to the Jacobian conjecture in dimension 3","summary":"Anthropic mathematician Levent Alpöge posted an explicit polynomial map F: C³→C³ with constant Jacobian determinant −2 that is not injective, found with Claude Fable 5. This refutes Keller's 1939 Jacobian conjecture in every dimension n≥3; the two-variable case remains open. Within days mathematicians produced infinite families, a geometric explanation and counterexamples in all dimensions above 2.","key_facts":["Announced on X on 19–20 July 2026 ('hello there the jacobian conjecture is false thanx'); no paper at first","Explicit map with constant Jacobian −2 sending three points to one; checkable by hand or computer algebra","Akhil Mathew suggested the problem; Claude Fable 5 found the map","Follow-ups: infinite family (Gallagher, 20 Jul); 'tangent-sweep' explanation (Speyer, 23 Jul); Tao's 'digestion' (21 Jul); Shuhong Gao, arXiv 2608.00222, including a degree-4 3-D example","The Fields Medallists' September letter criticised announcing it by tweet","17 Oct 2026: Michiel de Bondt (arXiv 2610.24889) applied economical stabilization to a tweaked Alpöge map: counterexamples to HJC(15), the Vanishing Conjecture VC(28) and HVC(30), low-degree ones to JC(12,3), HJC(18,3), VC(34,4), HVC(36,4), and a cubic-linear (Drużkowski) counterexample in dimension 90, all over the (Gaussian) integers. Disclosure: 'This manuscript was written by Google Chrome AI, under supervision of the author … With the exception of section 5, all ideas in the paper are the author's own. The AI-model came with the 7-term formula that we use for xi xj xk and did not mention the 4-term formula.' Unrefereed preprint","Poisson conjecture (math/0608009 setting, Belov-Kanel–Kontsevich): Christopher D. Long (arXiv 2608.23777, 22 Jul 2026) built on the Alpöge core a Poisson endomorphism of C[x,q,p,z] (two canonical pairs) that is not an automorphism, disproving the Poisson conjecture for every n ≥ 2; 'The four-variable construction, including the Hamiltonian correction H, was produced during an interactive research session with ChatGPT 5.6 Sol … Claude Fable 5 subsequently supplied independent algebraic audits'. Lean proof over every characteristic-zero field (zblore/fc-poisson-2, 'I used Claude Code to draft the Lean') linked from google-deepmind/formal-conjectures on 1 Nov 2026 (PR #7657)"],"key_numbers":[],"tags":["math","algebraic-geometry","counterexample","claude","jacobian-conjecture"],"science":{"field":"mathematics","subfield":"algebraic geometry / polynomial automorphisms","problem":"Jacobian conjecture (Keller 1939): a polynomial map with non-zero constant Jacobian determinant is invertible","result":"Explicit non-injective polynomial map C³→C³ with constant Jacobian determinant, disproving the conjecture for all n≥3.","open_since":"1939","ai_system":["Claude Fable 5"],"human_role":"AI-assisted: human chose the problem and verified; Claude found the counterexample","verification":{"key":"none","label":"Unverified claim","text":"Explicit map, checkable by direct computation (constant Jacobian determinant and three points with the same image); digested and generalised by Terence Tao, Gallagher, Speyer and Shuhong Gao; no refereed paper by the discoverers"},"status":"confirmed","shock":"One of Smale's 18 problems for the 21st century, with a long history of false proofs, was refuted by a short formula found by an AI."},"body_md":"## What happened\n\nAlpöge announced the counterexample in a one-line tweet with the explicit map. Because anyone can verify it by expanding a determinant, it was confirmed within hours, and a burst of human follow-up work explained and generalised it.\n\n## Why it matters\n\nThe Jacobian conjecture is one of the most famous open problems in algebra. Its refutation by an AI-found formula is among the most shocking AI results in mathematics so far, and fed the debate about how such results should be announced.","disputed":[],"related":[{"id":"2026-09-11-fields-medalists-letter-ai-mathematics","url":"https://postcutoff.com/e/2026-09-11-fields-medalists-letter-ai-mathematics/","date":"2026-09-11","date_precision":"day","short_title":"Fields Medallists' open letter 'A Severe Misalignment of AI in Mathematics' criticises labs' race for famous problems","deck":null,"takeaway":"It marked open tension between AI labs and the mathematical community at the moment AI began producing major results, and shaped norms for credit and verification.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":8,"official":3,"filed":"2026-09-29","updated":"2026-10-07","orgs":["mathandai.org"]},{"id":"2026-07-24-hessian-conjecture-counterexample","url":"https://postcutoff.com/e/2026-07-24-hessian-conjecture-counterexample/","date":"2026-07-24","date_precision":"day","short_title":"Hessian conjecture refuted in five variables, derived from Claude-found Jacobian counterexample","deck":null,"takeaway":"It shows how AI-found results feed quickly into human follow-up work.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":1,"filed":"2026-09-29","updated":"2026-10-19","orgs":["Independent researchers"]},{"id":"2026-07-01-grothendieck-group-scheme-counterexample","url":"https://postcutoff.com/e/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":"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\".","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":2,"official":0,"filed":"2026-09-29","updated":"2026-09-29","orgs":["OpenAI","Anthropic"]},{"id":"2026-06-09-claude-fable-5-mythos-5","url":"https://postcutoff.com/e/2026-06-09-claude-fable-5-mythos-5/","date":"2026-06-09","date_precision":"day","short_title":"Anthropic releases Claude Fable 5 and Claude Mythos 5","deck":"First generally available Mythos-class model","takeaway":"This was the first time the class of model Anthropic had withheld in April (Mythos Preview) became available to the public.","category":"model-release","category_label":"Model releases","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":7,"official":5,"filed":"2026-09-29","updated":"2026-09-29","orgs":["Anthropic"]}],"people":[{"id":"terence-tao","name":"Terence Tao","url":"https://postcutoff.com/person/terence-tao/"},{"id":"levent-alpoge","name":"Levent Alpöge","url":"https://postcutoff.com/person/levent-alpoge/"}],"posts":[{"id":"2026-07-21-tao-jacobian-conjecture-digestion","title":"A digestion of the Jacobian conjecture counterexample","url":"https://postcutoff.com/p/2026-07-21-tao-jacobian-conjecture-digestion/"}],"videos":[{"id":"fireship-summer-math-fell-to-the-machines","title":"The summer Math fell to the machines...","url":"https://postcutoff.com/v/fireship-summer-math-fell-to-the-machines/"}],"models":[],"changes":[{"date":"2026-11-10","type":"edit","text":"Consequence formalized. A 9 Nov commit to google-deepmind/formal-conjectures (#7829) links Lean proofs (repo zblore/fc-dixmier-jacobian, Palomar entry PALOMAR-2026-11-06-000008) that the Dixmier conjecture implies the Jacobian conjecture in every dimension and that the Dixmier conjecture is therefore false for every n ≥ 3; axioms propext, Classical.choice, Quot.sound only. Contributor: \"I used Claude Code to draft the Lean. I've checked every line myself and can explain all of it.\" The commit credits Pickhardt and the Omniscience Research Agent, and Teller, Naughton and Mayner, for the dimension-3 result."},{"date":"2026-11-01","type":"edit","text":"Added Long's rank-two Poisson conjecture counterexample (arXiv 2608.23777, ChatGPT 5.6 Sol) and its Lean proof linked in formal-conjectures"},{"date":"2026-10-20","type":"edit","text":"Added de Bondt's stabilization paper (arXiv 2610.24889): HJC(15), Zhao's Vanishing Conjecture VC(28), cubic JC(12,3), Drużkowski dimension 90"},{"date":"2026-10-19","type":"edit","text":"Related: back-link to the Hessian counterexample derived from this one"},{"date":"2026-10-19","type":"edit","text":"Verification names who checked it and what is checkable"},{"date":"2026-10-19","type":"edit","text":"Added video(s) fireship-summer-math-fell-to-the-machines"},{"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":"2026-11-10","human_review":null,"version":{"date":"2026-11-10"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":81,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":20,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":111,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":50,"in_training_data":false}],"short_url":"https://postcutoff.com/s/claude-fable-5-finds"}