Fable Language Model “Disproves” the Jacobian Conjecture — but Mathematicians Are Cautious

A claim in a single tweet

A mathematician who posts on X under the handle levent (@alpoge) has claimed that the Jacobian Conjecture turns out to be false. He credited two “close friends” for it: his colleague Ahil, who asked just the right question, and the language model Fable, which — according to the author — was toiling away at the problem right during the World Cup final.

The Jacobian conjecture is one of the famous open problems of algebraic geometry, and it has a bad reputation: the author of the post himself calls it “the canonical mass grave of cranks,” where mistaken “proofs” have been piling up for decades.

What the counterexample is

The counterexample put forward is a polynomial map \(\mathbb{C}^3 \to \mathbb{C}^3\) with a constant Jacobian equal to −2 (that is, nonzero). Yet it sends three distinct points — (0, 0, −¼), (1, −3/2, 13/2) and (−1, 3/2, 13/2) — to one and the same point (−¼, 0, 0). In other words, a map with a constant nonzero Jacobian is not injective, which is something the conjecture is not supposed to allow.

The author attached links to Wolfram Alpha, where you can recompute the determinant of the Jacobian matrix yourself and check the values at the given points. The coefficients and degrees in the formulas are small, which immediately raised a question: was the counterexample found by brute force, by directed search or by “pure reasoning” — and why hadn't it been found earlier?

The reaction: excitement and doubts

The discussion instantly drew hundreds of replies. Some offer congratulations and joke about a “blessed mathematical fairy,” while others demand a full solution and an account of exactly how the result was obtained — this question comes up most often of all. Siddhartha Gadgil proposed writing the example up in formal terms and adding it to the counterexamples folder of the Mathlib library for verification in the Lean system.

There is substantive criticism too. One participant recalls that over the complex numbers a local condition is not enough to disprove the conjecture: the map must be proper, and the points “heading off to infinity” have to be considered separately — that is, an example that works over the reals may not do for the complex case. Another notes that the falsity of the Jacobian conjecture would formally imply the falsity of the related Dixmier and Poisson conjectures as well.

For now this is merely a social-media post, with no accompanying paper and no independent verification. But the episode itself — with its casual tone and an AI model cast in the role of co-author — captures the mood of the moment well: language systems are increasingly being brought in to attack problems that have been considered impregnable for decades.

Source: Hacker News