The Jacobian conjecture, a notorious challenge in algebraic geometry formulated by Ott-Heinrich Keller in 1939, has been effectively disproved through the use of artificial intelligence. Levent Alpöge, a Harvard fellow, announced the finding on X on July 19, 2026, crediting Fable 5, the latest frontier model developed by Anthropic, for identifying the specific counterexample.
The conjecture posited that a polynomial map with a non-zero constant Jacobian determinant must possess a neat polynomial inverse. This problem remained one of the most stubborn obstacles in the field for nearly nine decades. Stephen Smale famously included the conjecture on his 1998 list of essential challenges for the 21st century. It has historically attracted numerous flawed proofs, including an attempt that famously impacted the early career of mathematician Yitang Zhang.
Alpöge and Fable 5 did not provide a formal proof of the conjecture but instead identified a single map from three-dimensional space to itself that defies the expected behavior. This counterexample consists of a specific list of polynomials that, while difficult to locate, are straightforward for mathematicians to verify manually. The discovery process required the computational power of Fable 5 to sift through vast search spaces that are impractical for human researchers to process alone.
The model used in this discovery, Fable 5, is the public iteration of a system previously referred to internally as Claude Mythos. Anthropic had initially withheld the release of this model, citing concerns regarding its advanced capabilities. The collaboration between Alpöge and the machine highlights the growing intersection of large-scale computation and pure mathematics. Other researchers, including those at OpenAI, have reported that internal models like Codex arrived at similar conclusions independently.
The mathematical community has responded with a mixture of professional intrigue and cautious skepticism regarding the implications of the discovery. Timothy Gowers, a Fields medallist, described the event as a significant milestone for AI in a field outside his primary area of expertise. Other mathematicians, such as Daniel Litt, expressed surprise at the efficiency with which the machine resolved the long-standing open question. The result serves as a practical demonstration of how AI can act as a high-speed search engine for complex mathematical structures.
The reaction online was particularly loud given the historical weight of the problem. As Stanford number theorist Jared Duker Lichtman noted, the conjecture is notorious for attracting incorrect proofs, making the AI-assisted resolution feel poetic to many observers. The speed at which the machine settled the issue contrasts sharply with the decades of human effort that previously yielded no definitive results.
Not all researchers view the development as a fundamental shift in the discipline. Andrew Blumberg, a mathematician at Columbia University, noted that while the model successfully identified a counterexample, the result does not necessarily advance the broader understanding of mathematical theory. He argued that a counterexample serves only to terminate an argument rather than providing the deep insight typically associated with a formal proof. The distinction remains critical for those who believe the value of mathematics lies in the underlying structure rather than the final result.
Blumberg contrasted this specific achievement with recent work by OpenAI, where a disproof of a geometry conjecture provided a framework that mathematicians could unpack and build upon. He emphasized that the current utility of AI in mathematics is limited to finding needles in a haystack rather than explaining the nature of the haystack itself. This limitation suggests that the current generation of models excels at verification and search but struggles with the conceptual synthesis required for deep theoretical advancement.
The work surrounding the Jacobian conjecture continues to evolve as researchers analyze the implications of the findings. Alpöge has suggested that the structure of the counterexample may contain hints of a positive result, and a full academic write-up is expected to follow. Meanwhile, other developers have begun using different AI models to expand the initial finding into an infinite family of counterexamples. The emergence of these tools confirms that current mathematical research increasingly relies on a hybrid approach where machines generate candidate data and humans provide the necessary theoretical validation.
