Anthropic Mathematician Uses Claude Fable 5 to Disprove 87-Year-Old Jacobian Conjecture

On Sunday evening, July 19, 2026, Levent Alpöge, a mathematician working at Anthropic who previously worked at Harvard, posted a message on X that spread through the global mathematics community within hours. Alpöge announced that the Jacobian conjecture, formulated and unsuccessfully attacked by mathematicians since 1939, is false. He used the Claude Fable 5 model to find the counterexample.

87 Years Without a Solution

The Jacobian conjecture concerns polynomial functions whose Jacobian determinant is a nonzero constant everywhere. Mathematical intuition suggested that such a function must be globally invertible using polynomials, since it never locally "crumples" space anywhere. For nearly nine decades no one managed to either prove or disprove it, even though the problem drew generation after generation of researchers.

The conjecture's history is littered with failures. Flawed proofs were published that turned out, years later, to have holes, and one special case of the problem became the subject of a rejected doctoral dissertation by Yitang Zhang, the mathematician who later became famous for his breakthrough in prime number theory. The conjecture's inclusion on Stephen Smale's 1998 list cemented its status as one of the key challenges of modern mathematics.

A Counterexample in One Tweet

Alpöge constructed a function in which two different points in three-dimensional space, for example (0, 0, -1/4) and (1, -3/2, 13/2), map to the same resulting point. At the same time, the function satisfies all the conjecture's assumptions: its Jacobian determinant is constant everywhere and equals -2. In other words, the mapping is locally invertible at every point but not globally invertible, exactly what the conjecture ruled out.

The solution's brevity turned out to be its strength. Because the counterexample is a concrete, computable formula rather than a multi-page abstract proof, other mathematicians could check it themselves immediately, including with tools such as SymPy and the Lean proof assistant. Several independent researchers reportedly confirmed the calculation's correctness within hours of publication, though formal peer review is still ongoing.

This is probably the biggest mathematical conjecture disproved so far in which AI played a significant role - Abhishek Saha, Queen Mary University of London

Skeptics Temper the Enthusiasm

Not all mathematicians share the enthusiasm. Chris Bowman-Scargill of the University of York points out that finding counterexamples is different from building new branches of mathematics, which still requires human creativity. Andrew Blumberg of Columbia University went further, pushing back on the narrative of an AI breakthrough in mathematics.

This hasn't changed my prior beliefs - Andrew Blumberg, Columbia University

What's Next for Math and AI

Alpöge himself stressed in his statements that the model was a tool in the mathematician's hands, not an independent creator of the solution. Critics, however, note that the full history of the conversation and prompts used to generate the counterexample has not been disclosed, making it difficult to assess how much of the work was done by the AI itself and how much by the researcher.

The case fits into a recent series of reports about language models tackling open mathematical problems, from Erdős problems to number-theory conjectures. Unlike many of these, the disproof of the Jacobian conjecture concerns a problem with an established, nearly century-long history and a place on Stephen Smale's list of challenges, which explains why the news spread so quickly among professional mathematicians, not just in tech media.