Levent Alpoge disproved the Jacobian conjecture using Claude Fable 5, marking a major breakthrough in algebraic geometry.
On July 20, 2026, Anthropic researcher Levent Alpoge made a significant advancement in mathematics by presenting a counterexample to the Jacobian conjecture using the AI model Claude Fable 5. This finding challenges a theory that has perplexed mathematicians since its inception in 1939.
"Hello there, the Jacobian conjecture is false. Thanks to my close friend Akhil for asking about it and my other close friend Fable for working during the World Cup final." — Levent Alpoge
The Jacobian conjecture, initially proposed by mathematician Ott-Heinrich Keller, asserts that a specific mathematical transformation should allow for the complete recovery of original values from transformed results without loss of information. However, Alpoge's formula constructs a scenario where multiple distinct sets of numbers yield the same outcome, thus invalidating the conjecture's core premise.
This breakthrough comes after nearly nine decades of attempts to resolve the conjecture, which had garnered attention for numerous flawed proofs. Mathematicians such as Qiaochu Yuan noted the significance of this development, highlighting that it is the most notable solution to an open problem achieved by a language model. The ease of verification for Alpoge's findings sets it apart from traditional AI claims, as the formula can be tested and recalculated by anyone interested.
As Alpoge elaborated, the question arose from a conversation with friends, and Claude Fable 5 was utilized during a notable time, the World Cup final. He emphasized the uniqueness of their approach which allows for straightforward verification through platforms like Wolfram Alpha.
Mathematician Jared Duker Lichtman remarked on the importance of this result, as the Jacobian conjecture has been a central challenge in the field. Despite the groundbreaking nature of this finding, it has yet to undergo peer review, marking a new chapter in the intersection of artificial intelligence and mathematical inquiry.
AI's Role in Solving Mathematical Problems
This achievement is not the first instance of language models contributing to mathematical problem-solving. Prior claims, such as those made by OpenAI regarding the resolution of Erdős problems, have sparked ongoing discussions regarding the validity of AI in addressing complex mathematical inquiries. While some earlier assertions faced scrutiny, recent developments suggest that AI may be approaching more substantial mathematical challenges.
In January 2026, another significant problem from the Erdős list was solved using AI, garnering confirmation from established mathematicians. This trend indicates a growing capability of AI models in tackling longstanding issues, though experts caution that much of the current progress still involves relatively simpler problems within the field.
