Mathematician Levent Alpöge used Anthropic’s Claude Fable 5 language model to discover a counterexample that disproves the general Jacobian conjecture in three dimensions and above, settling an 87-year-old problem that had resisted generations of researchers. Alpöge, who works at the artificial intelligence company Anthropic, announced the finding on X on July 19, 2026, posting a polynomial formula short enough to fit in a single tweet.
The result does not resolve the original two-dimensional version of the conjecture, which remains open. But it closes the general formulation for every dimension larger than two, and it ranks among the most significant mathematical breakthroughs yet tied to collaboration with a large language model.
- Levent Alpöge, a mathematician at Anthropic, announced a counterexample to the general Jacobian conjecture on July 19, 2026.
- The counterexample is a three-variable polynomial map with a constant Jacobian determinant of −2 that is not globally invertible.
- Alpöge credits Anthropic’s Claude Fable 5 AI model with assisting in the discovery.
- The result disproves the conjecture for all dimensions n ≥ 3; the two-dimensional case remains unsolved.
- Mathematicians including Fields Medalist Terry Tao have independently verified the short algebraic calculation.
- No formal peer-reviewed paper has been published, and details of the AI prompting workflow remain undisclosed.
The Jacobian conjecture belongs to the field of algebraic geometry and asks a deceptively simple question about polynomial functions. A polynomial map from n-dimensional space to itself that has a constant, non-zero Jacobian determinant is locally invertible everywhere,it never folds or crushes space in a small neighborhood. The conjecture asserts that such a map must also be globally invertible, with a polynomial inverse that returns every point to its original position.
German mathematician Ott-Heinrich Keller generalized the statement to arbitrary dimensions in 1939, building on a two-dimensional version that Czech mathematician Ludwig Kraus had posed in 1884. The problem proved so intractable that Fields Medalist Stephen Smale included it in his 1998 list of mathematical problems for the next century. Despite numerous claimed proofs over the decades, each was eventually found to contain a subtle error. Computational checks had confirmed the conjecture for two-dimensional polynomials up to degree 100, but the general case remained wide open.
Alpöge’s counterexample is a degree-seven polynomial map from three-dimensional complex space to itself. Its Jacobian determinant is the constant −2 everywhere, satisfying the conjecture’s hypothesis. Yet the map sends three distinct input points to the same output, which means no inverse function,polynomial or otherwise,can separate them. Because a three-dimensional counterexample can be extended to any higher dimension by adding identity coordinates, the conjecture is now known to be false for every n ≥ 3.
The brevity of the formula made verification almost immediate. Within hours of Alpöge’s post, mathematicians checked the determinant and the point collisions by hand and with computer algebra systems. Fields Medalist Terry Tao published a detailed exposition on his blog on July 21, 2026, explaining the algebra and noting that the example has since been understood in more geometric terms. Other researchers reproduced the calculation independently in symbolic computation software such as SymPy.
Unlike lengthy proofs that construct elaborate theoretical machinery, Alpöge’s counterexample is a single, compact mathematical object. The difficulty lay not in verifying it, but in navigating the enormous search space of polynomial mappings to find one with the right properties. As a Math Stack Exchange user observed in 2017, “for all what we know, some smart undergraduate can simply write a formula … that will be a counter-example to this conjecture.” Alpöge’s finding suggests that AI systems may be particularly well suited to exactly this kind of search-driven discovery.
How Claude Fable 5 produced the counterexample remains unclear. Alpöge credited his “close friend fable” in the announcement and thanked the model for working during the 2026 FIFA World Cup final, but he has not released the prompt history, model logs, or a detailed research notebook. Anthropic has not issued an official statement about the discovery. Without that transparency, the exact division of labor between human insight and machine search cannot be independently assessed.
The finding arrives amid a rapid succession of AI-assisted mathematical advances. In recent months, an OpenAI model disproved the unit distance conjecture in combinatorial geometry, and the original report also cites amateur mathematician Liam Price’s proof of Erdős’ problem 1196 as another AI-assisted result. Abhishek Saha, a mathematician at Queen Mary University of London, told New Scientist that the Jacobian conjecture result is “probably the biggest conjecture that AI has played a significant role in so far in mathematics.”
For the mathematics community, the immediate consequence is a shift in research direction. Decades of work had proceeded under the assumption that the general conjecture might be true, with mathematicians seeking proof strategies and partial results. Now that the universal statement is known to be false, researchers are asking what additional hypotheses might rescue a useful theorem and whether the counterexample can be simplified further.
The two-dimensional Jacobian conjecture, the version closest to the historical origin, remains open. Whether it too will fall,to human ingenuity, to AI assistance, or to some combination of both,remains one of the most significant open questions in algebraic geometry.