Levent Alpöge, a mathematician affiliated with the artificial intelligence firm Anthropic, announced on July 22, 2026, that he has identified a counterexample to the Jacobian conjecture. This breakthrough, shared via social media, was facilitated by the recently released large language model Claude Fable 5, as reported by the science news outlet Phys.
The Jacobian conjecture concerns the nature of polynomial mappings within coordinate spaces. It posits that if a function possesses a constant, nonzero Jacobian determinant, then an inverse polynomial function must exist to reverse the mapping. The conjecture suggests that such functions preserve the distinctness of points in space without folding or merging them, acting as a fundamental assumption in the study of algebraic geometry.
Ludwig Kraus first proposed the two-dimensional version of this problem in 1884. Ott-Heinrich Keller expanded the scope to higher dimensions in 1939, and the conjecture subsequently became a prominent challenge in algebraic geometry. It was even included in a 1998 list of significant mathematical problems compiled by Fields Medalist Stephen Smale, who viewed it as a critical hurdle for the next century of research.
Previous attempts to resolve the conjecture involved numerous claims of proof, including those by Beniamino Segre and Wolfgang Gröbner. Each of these historical arguments eventually succumbed to subtle errors identified by the broader mathematical community. Computational efforts had previously verified the conjecture for polynomials up to degree 100 in two dimensions, yet the general case remained elusive for decades.
The counterexample discovered by Alpöge is a three-dimensional function with a constant Jacobian determinant of -2. Because the function maps multiple distinct input points to the same output location, it is inherently non-reversible. This finding demonstrates that the conjecture is false for all dimensions greater than two, while the original two-dimensional case remains an open question for researchers to investigate.
The brevity of the counterexample allowed for rapid verification by other mathematicians who could quickly test the function’s properties. Its simplicity stands in stark contrast to the complex, lengthy proofs often associated with such high-level mathematical problems. The result highlights the capacity of large language models to explore vast search spaces for specific mathematical objects that might otherwise remain hidden from human intuition.
This discovery follows a series of recent AI-assisted advancements in mathematics, including the disproof of the unit distance conjecture and the resolution of Erdos’ problem 1196. These successes demonstrate that artificial intelligence models can effectively synthesize concepts from disparate mathematical fields. The ability to navigate enormous sets of potential mappings appears to be a critical strength of these systems, allowing them to identify anomalies that human researchers might overlook.
The primary challenge in this instance was not the construction of a complex proof, but the identification of a specific, elusive object within a massive parameter space. This shift suggests that AI may serve as a powerful tool for empirical discovery in pure mathematics. Researchers are now evaluating whether this approach can be applied to other long-standing conjectures that have defied traditional analytical methods for generations.
The exact methodology and prompting techniques used to elicit this result from Claude Fable 5 remain undisclosed. Future peer-reviewed documentation will likely provide deeper insight into how the model structured its search. The mathematical community now faces the task of determining how these computational tools will alter the standard practices of formal proof and conjecture testing, potentially ushering in a new era of machine-aided discovery. This development underscores the necessity for mathematicians to integrate AI-driven computational search into their traditional theoretical frameworks to keep pace with the evolving nature of mathematical exploration.
