Anthropic Mathematician Uses Claude Fable 5 to Disprove the 87-Year-Old Jacobian Conjecture in a 216-Character Post

Overview

An artificial intelligence model has helped disprove a mathematical conjecture that had gone unsolved for 87 years, and the answer was confirmed within a day because anyone can now check it by hand, according to CoinDesk. Levent Alpöge, a number theorist who works at Anthropic and previously held a fellowship at Harvard, posted the result on X on Sunday night, crediting the company’s Claude Fable 5 model, CoinDesk reported. TheNextWeb similarly reported that Alpöge posted the message on Sunday evening as Spain and Argentina played out the World Cup final.

What We Know

• The Jacobian conjecture was set out by Ott-Heinrich Keller in 1939, according to TheNextWeb. In rough terms, the outlet reported, it says that a certain kind of polynomial map, one whose Jacobian determinant is a non-zero constant, must be reversible with a neat polynomial inverse.
• The problem sits on mathematician Stephen Smale’s influential list of unsolved problems for the century, CoinDesk reported.
• Alpöge posted the counterexample itself on X, writing: “hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0)” — quoted in full by CoinDesk.
• In plain terms, the map passes the Jacobian conjecture’s local test everywhere — its Jacobian determinant is the constant -2 — yet it is not globally reversible, since three distinct input points collide on a single output point rather than the map being one-to-one.
• TheNextWeb reported that the counterexample was 216 characters long, short enough to fit inside the single X post.
• The disproof covers more than the specific three-dimensional case: The Conversation reported that it shows the conjecture is false for every dimension larger than two, with the original conjecture in two dimensions remaining open.
• Reaction from mathematicians has been mixed on what the result demonstrates. Bartósz Naskręcki said the discovery was not trivial, telling TheNextWeb: “It was not a one-line prompt… Searching for a counterexample like this… takes real insight.” Andrew Blumberg was more skeptical, telling Mashable, as reported by TheNextWeb: “This did not cause me to update my priors,” adding, “There are a lot of polynomials, and it is hard for people to check them all, but it is not hard for machines.”
• One developer, Alexis Gallagher, said he used GPT-5.6 to turn the single counterexample into an infinite family, one for every whole number above two, TheNextWeb reported.
• The result follows Anthropic’s public release of Claude Fable 5 weeks earlier, as previously reported.

What We Don’t Know

• The Conversation reported that, at the time of writing, details had not been made public regarding exactly how Alpöge prompted the AI model to produce the counterexample and what its raw output looked like.
• Gallagher’s claimed extension of the counterexample to an infinite family using GPT-5.6 has not been independently verified in the reporting reviewed.
• The result has been announced only through a social-media post rather than a peer-reviewed paper; none of the outlets reviewed reported that it had yet undergone formal journal peer review.

Analysis

The reaction captured by The Conversation frames Alpöge’s discovery as “the latest in a string of high-profile mathematical breakthroughs made by large language models,” citing OpenAI’s disproof of the unit distance conjecture and the proof of Erdős’ problem 1196 by Liam Price, a 23-year-old amateur mathematician, as recent examples. But the outlet also drew a distinction specific to this case: “Unlike many other recent AI-assisted breakthroughs, the counterexample itself is remarkably simple. The difficulty in finding it seems to have lain not in an intricate construction or a lengthy proof, but rather in finding a good way of navigating an enormous search space of possible polynomial mappings to find one with the right properties.” That framing lines up with Blumberg’s characterization of the achievement as a search problem suited to machines rather than a demonstration of deeper mathematical reasoning — even as Naskręcki maintained that finding the right search strategy still required real mathematical insight.