A mathematician at Anthropic posted the example, crediting the work to the AI model Fable. The search for a counterexample had assumed any such object would be monstrously complicated. This one is simple enough for a first-year student to check in about ten minutes.
The Jacobian Conjecture, open since 1939, appears to be false. It held that a certain kind of mathematical machine can always be run backwards, and someone has now produced one that cannot be, which is all it takes: a claim about every possible case dies the moment a single case breaks it. Credit for finding that case is going to an AI model.
These machines take a handful of numbers in and hand the same count back, using only the plainest operations, adding and multiplying. The example at issue works on three. The conjecture said that whenever such a machine passes a particular check, there must be a second machine of the same kind that undoes it and returns you to where you began. The new example passes the check and cannot be undone. Two different starting points come out at the same destination, so once you are standing there, no route leads back. Nothing can send one point to two.
The surprise is how modest it is. Hunting for a counterexample had always been organised around how complicated one would have to be, measured the standard way by the highest power in the formula. The guesses ran high. One paper argued that nothing below roughly 100 would do within its own search method, a later paper raised the figure to 108, and an earlier pair of researchers put the real threshold near 200. The example that finally worked scores 7. It had been sitting far beneath the floor everyone was searching above.
It survives well outside the setting the conjecture is usually posed in. Every number involved is an ordinary fraction, so the same example works over the real numbers too, and across a range of other number systems besides. That leaves little room to dismiss it as a quirk of one specialised corner.
The result went up on X from Levent Alpöge, a mathematician who works at Anthropic, as a personal post rather than a company announcement. Alpöge holds a mathematics PhD from Princeton, where he studied under the Fields Medalist Manjul Bhargava. He described the finding as a collaboration: a colleague posed the question, and Fable did the work over the course of a World Cup final.
Almost nobody disputes the mathematics, and that follows directly from how small it is. Commenters saw no realistic chance of an error, precisely because anyone can check it independently. Several put the job at about ten minutes for a first-year undergraduate, working from the Wolfram Alpha link the original post carried.
The argument is about credit. The finding arrived as a post, not a paper, and the conversation with the model was never published alongside it. Some readers argued that Alpöge and his collaborator almost certainly supplied real guiding insight of their own, and that letting the model carry the discovery suits his employer.