When the whole world was watching the World Cup final between Spain and Argentina on Sunday evening, mathematician Levent Alpöge posted a few inconspicuous lines on X that sent ripples through the mathematical community. According to him, the Jacobian conjecture, an open problem since 1939, is false. In the post, he thanked one friend for asking him about the problem and another friend named Fable for working during the match. That second friend was Anthropic’s Claude Fable 5 model.
An 87-year-old unsolved conjecture
The Jacobian conjecture poses a seemingly simple question. Take a polynomial mapping that transforms several numbers into new ones using addition and multiplication. If the Jacobian of such a mapping is equal to a nonzero constant, does it always have an inverse function composed of polynomials? Classical calculus shows that a nonzero Jacobian is necessary for local invertibility. The conjecture claimed that it should also be sufficient globally.
The problem was first formulated for two variables by Ludwig Kraus in 1884, while Ott-Heinrich Keller gave it its fully general form in 1939. Over time, it earned a reputation as a trap. At least five published proofs turned out to be incorrect, and there were dozens of unpublished attempts. In 1998, Stephen Smale included the problem on his famous list of the most important mathematical challenges for the new century, alongside the Riemann hypothesis. In 2008, mathematician Tzuong-Tsieng Moh estimated that it might take humans another hundred years to solve it.
Levent Alpöge is a number theorist who now works at Anthropic and previously served as a Junior Fellow at Harvard. The question was put to him by Akhil Mathew, an algebraic geometer at the University of Chicago. Alpöge gave it to the Fable 5 model, and while the final was playing on the screens, he received a verifiable counterexample. The entire verification took one evening.
What the counterexample shows
Alpöge did not publish a proof of the conjecture, but a disproof. He wrote down a specific polynomial mapping from three-dimensional complex space to itself. Its Jacobian is constant and equal to minus two, so it satisfies all the conditions required by the conjecture. Nevertheless, three different points—specifically (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2)—are mapped to the same point, (-1/4, 0, 0). A function that maps multiple inputs to a single output cannot be inverted. A single such counterexample is enough to bring down the conjecture.
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
— levent (@__alpoge__) July 20, 2026
((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,…
The formula was just 216 characters long, and anyone can verify it by hand or using standard mathematical software. Alpöge attached links to Wolfram Alpha to the post. Within a few hours, the Wikipedia entry on the Jacobian conjecture was edited to mention the claimed disproof.
Mathematicians’ reactions
Stanford number theorist Jared Duker Lichtman described the result as remarkable and noted that it concerns one of the central open problems in algebraic geometry. Bartosz Naskręcki pointed out that finding such a counterexample is no joke and resembles searching for a needle in a haystack because it requires genuine understanding, and he is awaiting a full analysis from Alpöge. Vidit Nanda of Oxford suggested that the finding could trigger a wave of similar discoveries. New Scientist wrote that the answer took mathematicians by surprise and called the problem the hardest mathematical challenge ever cracked by artificial intelligence.
Andrew Blumberg of Columbia University, who is involved in a project testing artificial intelligence’s capabilities in research mathematics, told Mashable that it had not changed his previous expectations, because this is exactly what he expects from such models. OpenAI researcher Aaron Lou also said that an internal version of their Codex model had independently found essentially the same counterexample.
Fable 5 and further implications
The Fable 5 model was released in June and was designed primarily for demanding programming projects and several days of independent work. Alpöge, however, used it more as a research collaborator. If the counterexample passes peer review, it should also have implications for the Dixmier and Poisson conjectures. For now, the result is awaiting review on arXiv, where a preprint analyzing the construction is already available.
Incidentally, this story also contains a piece of history. The Jacobian conjecture once helped derail mathematician Yitang Zhang’s PhD after he was told that he had failed to solve it. Zhang later became famous for his work on gaps between prime numbers and became one of the legends of the field. Many mathematicians find it almost poetically just that the problem on which he foundered was ultimately brought down by a formula typed out on X during a soccer final.
Sources: glitchwire.com, coindesk.com and newscientist.com
