Mathematics & Artificial Intelligence

An AI Helped Find a Counterexample to the Jacobian Conjecture

For 85 years the Jacobian Conjecture was one of algebraic geometry’s best-known open problems. Mathematician Levent Alpòge, working with Anthropic’s Claude Fable 5, has now published an explicit counterexample — and the answer can be verified by hand.

• The construction has a constant Jacobian determinant of −2, but maps three distinct points to the same point, so no inverse function can exist.
• Independent researchers have already reproduced the counterexample, checked it symbolically, and formalised a version in the Lean proof assistant.
• Published only as a public preprint so far — there is no peer-reviewed journal paper yet, but the formula is openly available and reproducible.

Stated in 1939 by the German mathematician Ott-Heinrich Keller, the Jacobian Conjecture asks a simple-looking question. If a polynomial map from n-dimensional space to itself has a Jacobian determinant that is a non-zero constant, must the map have a polynomial inverse? For two variables the answer is known to be yes. But once you get into three variables and beyond, no one has been able to prove it, and for decades most mathematicians suspected the truth was “no.”

What makes this result different from a typical “AI solved hard math” headline is that the answer comes with built-in proof. There is no long, opaque derivation for humans to second-guess. A reader only needs to compute a small determinant and plug in a few points to confirm the map is not one-to-one. Two qualities — symbolic verification and Lean formalisation — mean the counterexample already lives in a form that computers can independently audit.

Alpòge says the candidate structure emerged after a colleague asked Claude Fable 5 the right question, and the model surfaced a construction worth testing. In other words, the AI did not write a proof from scratch; it acted as a search engine over a vast space of possible formulas, pointing a human toward something verifiable. That division of labour — generate a guess, then certify it mechanically — is exactly where large models tend to be most reliable, because the risk of hallucination is removed by a separate, deterministic check.

Assuming the work passes further academic scrutiny, the conjecture would be closed in the negative after more than eight decades of effort. Even before that verdict, the episode is a useful data point for how AI is changing mathematics: not by replacing proof, but by shifting where the human time goes — from guessing the right object, to verifying it.