A mathematical idea that had resisted resolution since the 19th century was disproved in 2025, not by a team of researchers working over years, but by a mathematician working alongside an AI model during a soccer match. The result has been independently verified. It is being described by mathematicians as the most significant conjecture AI has played a role in resolving so far.
A Conjecture That Survived for Over a Century
The Jacobian conjecture has a long history. Czech mathematician Ludwig Kraus proposed an early version in 1884. German mathematician Ott-Heinrich Keller expanded it into its current form in 1939. The conjecture concerns polynomial functions: mathematical operations that take a set of numbers as input and produce a new set as output, effectively moving points around in space according to fixed rules.
The key question the conjecture poses is whether a specific property of those functions, called the Jacobian determinant, guarantees reversibility. If the determinant is always a nonzero constant, the conjecture holds that there must always exist a way to reverse the function and return every point to its original position. Disproving it requires finding a counterexample: a function where the determinant is a nonzero constant, but points still merge, making reversal impossible.
Mathematicians knew what they were looking for. Finding it was the problem. The search space across all possible polynomial functions is vast, far beyond what human researchers can systematically check. Fields Medal winner Stephen Smale considered the problem significant enough to include it on a 1998 list of 18 major mathematical problems for the 21st century.
How the Disproof Was Found
Levent Alpöge, a mathematician affiliated with both Anthropic and Harvard University, used Anthropic’s Claude Fable 5 to approach the problem. The result was a three-dimensional polynomial function with a constant Jacobian determinant that nonetheless merges multiple input points, making it irreversible. That is precisely the kind of counterexample the conjecture said could not exist.
Alpöge posted the result on X, crediting mathematician Akhil Mathew of the University of Chicago for suggesting the problem. Several independent mathematicians have since verified the result. Abhishek Saha, a mathematician at Queen Mary University of London, described it to New Scientist as “probably the biggest conjecture that AI has played a significant role in proving or disproving so far in mathematics.”
One important boundary: the disproof applies to spaces of three or more dimensions. Whether the conjecture holds in lower-dimensional spaces remains an open question.
What This Result Actually Tells Us About AI in Mathematics
Here is what most coverage of this story underemphasizes. Finding a counterexample is not the same as understanding why the conjecture is false. Columbia University mathematician Andrew Blumberg put it directly: the result mostly shows that there are “a lot of polynomials,” which are hard for humans alone to check. AI, in this case, functioned as an extraordinarily capable search engine across a space of mathematical objects too large for manual exploration.
That is genuinely useful. It is also genuinely limited. A full mathematical proof can run more than 100 pages, documenting not just the answer but the reasoning that leads to it. Current AI models are not yet capable of producing that kind of reliable, step-by-step logical structure. Akhil Mathew, who was credited in Alpöge’s post, described the situation clearly: AI can verify that something is correct, but it does not yet tell a story. It surfaces the “how” without illuminating the “why.”
Blumberg offered a pointed analogy: if Moses descended from a mountain with a tablet reading “cancer can be cured,” the statement alone would not be enough. The value of solving a hard problem lies not just in the answer but in what the path to the answer reveals about the underlying structure of the world. Smale included the Jacobian conjecture on his list precisely because he believed solving it would deepen understanding of how nature is organized.
This tension is shaping how the mathematical community is responding. After a separate AI-assisted breakthrough involving an 80-year-old problem posed by Hungarian mathematician Paul Erdős, 16 mathematicians published the Leiden Declaration on Artificial Intelligence and Mathematics, endorsed by the International Mathematical Union. It calls for disclosure of AI use, proper attribution of prior work, and peer-reviewed publication before public announcements. The declaration reflects a community trying to establish norms quickly, as the pace of AI-assisted discovery accelerates faster than expected.
Mathew described the shift as “very rapid and very unsettling, especially for junior mathematicians.” That concern is worth taking seriously. If AI can locate counterexamples and surface results that previously required years of specialized human effort, the nature of mathematical work itself is changing. The question is not whether AI will contribute to mathematics. It already has. The question is what kind of understanding gets built alongside those contributions, and who develops the capacity to generate it.
In Short
AI did not solve the Jacobian conjecture. It disproved it, by finding a counterexample that mathematicians had been unable to locate for over a century. That is a meaningful milestone, and it is also a partial one. The result shows that AI can search mathematical space at a scale humans cannot match. It does not yet show that AI can reason through that space in the way that produces lasting mathematical understanding. Both things are true at the same time, and keeping them in view is what makes this development genuinely significant rather than simply dramatic.
Based on reporting from Smithsonian Magazine.