Claude Fable 5 finds a counterexample to the Jacobian conjecture in dimension 3
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
Science result
- Field
- mathematics / algebraic geometry / polynomial automorphisms
- Problem
- Jacobian conjecture (Keller 1939): a polynomial map with non-zero constant Jacobian determinant is invertible (open since 1939)
- Result
- Explicit non-injective polynomial map C³→C³ with constant Jacobian determinant, disproving the conjecture for all n≥3.
- AI system
- Claude Fable 5
- Human role
- AI-assisted: human chose the problem and verified; Claude found the counterexample
- Verification
- Directly computer-checkable; confirmed independently by multiple mathematicians; formal peer review pending
- Status
- confirmed
- Why surprising
- 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.
What happened
Alpöge 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.
Why it matters
The 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.
Changelog
- 2026-09-29: created
Related posts (1)
- A digestion of the Jacobian conjecture counterexample Terence Tao · blog · 2026-07-21
Tao's expert explanation of the 3D Jacobian conjecture counterexample found with Claude Fable 5, the most-cited human 'digestion' of an AI-found disproof.
Related events
- Anthropic releases Claude Fable 5 and Claude Mythos 5 — first generally available Mythos-class model ★★★★★
- AI-assisted counterexample answers Grothendieck's question on finite flat group schemes, merged into Mathlib ★★★
- Fields Medallists' open letter 'A Severe Misalignment of AI in Mathematics' criticises labs' race for famous problems ★★★
- Hessian conjecture refuted in five variables, derived from Claude-found Jacobian counterexample ★★★
- HRT conjecture (1996) disproved: 12 time-frequency shifts of a Schwartz function are linearly dependent, found with ChatGPT-assisted guesswork ★★★
- Claude-assisted construction claims a complex structure on the 6-sphere, answering Hopf's 1947 problem (pending verification) ★★★★★
Sources (4)
- discussionTerence Tao: A digestion of the Jacobian conjecture counterexample
- paperShuhong Gao: counterexamples in all dimensions >2 (arXiv 2608.00222)
- discussionXena Project: Human mathematicians are being out-counterexampled
- pressScienceDaily: Claude Fable 5 AI finds a tiny formula that topples an 87-year-old math conjecture
id: 2026-07-20-jacobian-conjecture-counterexample · updated 2026-09-29 · open in the interactive timeline