AI’s Fable 5 Overnight Refutes Jacobian Conjecture, Dashing Zhang Yitang’s 7‑Year Quest

On a recent evening the mathematics community was shaken when Claude Fable 5, an Anthropic AI model, produced a concise counterexample that disproves the generalized Jacobian conjecture, a problem that had resisted proof for 87 years.

The Jacobian conjecture, posed by Ott‑Heinrich Keller in 1939, asks whether a polynomial map whose Jacobian determinant is a non‑zero constant must have a polynomial inverse. The conjecture’s three‑dimensional version remained open despite many claimed proofs.

Fable 5 examined a polynomial map F : ℂ³→ℂ³ whose Jacobian determinant is the constant ‑2. By evaluating the map at the three distinct points (0,0,‑¼), (1,‑3⁄2,13⁄2) and (‑1,3⁄2,13⁄2) it showed all are sent to the same image (‑¼,0,0), proving the map is not injective and therefore has no polynomial inverse.

This single example refutes the conjecture in three dimensions.

The result was quickly verified by researchers using Wolfram Alpha and sparked widespread reaction: mathematicians such as Jared Duker Lichtman called it “one of the most inspiring stories in modern mathematics,” while Fields‑medalist Timothy Gowers warned that the era of human‑only proofs may be ending.

OpenAI’s Aaron Lou reproduced the counterexample with internal Codex, publishing a full derivation (see https://aaronlou.com/jacobian_counterexample_derivation.pdf). GPT‑5.6 subsequently proposed a revised conjecture stating that a constant‑Jacobian polynomial map that is locally biholomorphic and has no loss at infinity must be an automorphism.

The article also recounts the personal saga of Chinese mathematician Zhang Yitang, who spent seven years on the Jacobian problem under the guidance of his advisor Mo Zongjian, only to discover a faulty lemma that collapsed his work. After years of unrelated jobs, Zhang later gained fame for his breakthrough on the twin‑prime conjecture, while the AI‑generated refutation of the Jacobian conjecture adds a bittersweet chapter to his story.