Claude Fable 5 helped crack the Jacobian Conjecture after 87 years of failure

Levent Alpöge, a mathematician at Anthropic, posted a polynomial map on July 20 that serves as a counterexample to the Jacobian Conjecture, a problem unsolved for 87 years. The map, found with help from Anthropic's Claude Fable after a suggestion from Akhil Mathew, has Jacobian determinant -2 and collapses three distinct inputs to the same output, disproving the conjecture in dimensions three and above. The counterexample has been independently verified with Mathematica code by Oliver Knill and is listed on Wolfram MathWorld, though the two-dimensional case remains open.

Levent Alpöge's Fable-assisted counterexample to the Jacobian Conjecture is short enough to check and big enough to unsettle mathematics. The result looks real, but you should separate the verified algebra from the grand claims about AI doing research alone. The counterexample is the story because it fits in a single social post. Alpöge, a mathematician who works at Anthropic, posted a polynomial map from three complex dimensions to three complex dimensions on July 20 and credited Akhil Mathew for suggesting the problem and Fable for working on it during the World Cup final. According to Wolfram MathWorld's updated entry, the map has Jacobian determinant -2 and sends three different points, 0, 0, -1/4 , 1, -3/2, 13/2 , and -1, 3/2, 13/2 , to the same output, -1/4, 0, 0 . That is the break. The Jacobian Conjecture says a polynomial map with a nonzero constant Jacobian determinant should have a polynomial inverse. This one passes the determinant test and still collapses three separate inputs into one output. A map that does that cannot be globally invertible. The conjecture says it must be. It isn't. Alpöge's X post had the casual tone of someone dropping a joke into a group chat: "hello there the jacobian conjecture is false." He thanked "my close friend akhil" and "my other close friend fable for working during the world cup final." MathOverflow's updated discussion says the counterexample was found by Claude Fable prompted by Alpöge after a suggestion from Mathew, and Kevin Buzzard's Xena Project blog says Mathew had discussed the idea of finding more counterexamples in algebraic geometry with Alpöge before the post went up. That attribution checks out. So does the algebra. The Math Is Short Enough To Survive The Hype What makes this different from the usual failed proof claim is the cost of verification. Past attempts at the Jacobian Conjecture have produced long arguments, and long arguments give errors somewhere to hide. Here, the central claim is a calculation: compute the determinant, plug in three points, and see whether the outputs match. Oliver Knill, a mathematician in Cambridge, posted Mathematica code on July 20 showing exactly that. MathWorld now lists the same map and the same three-point collapse. You don't have to trust Anthropic for that. You don't even have to trust Fable. The object is sitting in the open. The result also has a precise limit, and this is where some of the public framing gets sloppy. The counterexample kills the conjecture in dimension three and, by adding identity coordinates, in every dimension above three. The two-dimensional case remains open. Fortune reported that Mathew, a University of Chicago mathematician, said the calculation can be checked but that mathematicians would still like to be able to tell a story about why it works. That's the right caution. A counterexample can settle a question without explaining the landscape around it. Fields medalist Timothy Gowers put the point cleanly on X, calling it the first example he knew of where an LLM solved a famous problem outside his own area. He also said it was "not in end of mathematics territory" because it was a counterexample rather than a broad theory. Still, he called it pretty amazing. Coming from Gowers, that isn't filler. What AI Actually Did Here Here's the thing: the phrase "AI solved the Jacobian Conjecture" is both defensible and too loose. Alpöge is a serious mathematician, not a prompt tourist. The published trail does not yet show the exact prompt, the intermediate attempts, or how much steering happened before the final map appeared. Explainx.ai's careful write-up makes the same point: the autonomous-discovery claim is not established, but the explicit counterexample has been independently checked and analyzed. That distinction matters for you if you care about AI in research. The impressive part is not that a chatbot produced a confident answer. Models do that all the time, and often badly. The impressive part is that a frontier model appears to have helped find a concrete object in a huge symbolic search space, and that humans could verify the object almost immediately. That is a much stronger story than another benchmark claim. It is also why the crypto angle should be handled carefully. CoinDesk connected the result to Bitcoin because AI capability jumps now move speculative capital, and because cryptography depends on hard mathematical assumptions. Fine. But the Jacobian Conjecture does not break Bitcoin's elliptic curve cryptography, and pretending otherwise would be nonsense. The serious point is narrower: if AI systems keep finding short paths through hard mathematical search spaces, anyone building long-term security on assumed difficulty has to watch that curve closely. No journal paper has finished the ordinary refereeing process. There is no Anthropic release explaining the workflow. There is no public transcript showing exactly what Fable did. But there is a 216-character polynomial map, independent algebra checks, a MathWorld update, and public discussion by mathematicians who know the problem. Peer review will catch up. The calculation is already doing its work. 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