AI Model Claude Fable 5 Finds Counterexample to 1939 Math Conjecture

Mathematician Levent Alpöge announced on X that he used Anthropic's Claude Fable 5 AI model to find a counterexample to the Jacobian conjecture, a problem posed by Ott-Heinrich Keller in 1939. The discovery, which awaits peer review, marks a rare instance of a large language model contributing to a breakthrough in pure mathematics.

August 3, 2026 , Inside AI — A mathematician has used Anthropic’s latest AI model to find a counterexample to the Jacobian conjecture, a problem that has stood since 1939 . The discovery, announced by Levent Alpöge on social media platform X, credits Claude Fable 5 as a collaborator, marking a rare instance of a large language model contributing to a breakthrough in pure mathematics. The Jacobian conjecture, first posed by Ott-Heinrich Keller , concerns polynomial mappings in multiple variables. It posits that if the determinant of the Jacobian matrix of a polynomial map is a nonzero constant, then the map has a polynomial inverse. Despite its simple statement, the conjecture has resisted proof or disproof for decades, becoming a central open problem in algebraic geometry. Alpöge’s announcement signals a potential shift in how mathematicians approach long-standing problems. Rather than simply verifying existing proofs, Claude Fable 5 actively participated in the discovery process. The model’s ability to search a high-dimensional space of polynomial mappings and identify a counterexample suggests a new paradigm for mathematical research. The Human-Machine Math Lab Details of the collaboration remain sparse, but the event raises immediate questions about validation. A counterexample to the Jacobian conjecture would require rigorous verification by the mathematical community. The conjecture’s history includes many false proofs and mistaken counterexamples, so skepticism is warranted until peer review confirms the result. " I found a counterexample to the Jacobian conjecture. My collaborator was Claude Fable 5. " Levent Alpöge, Mathematician The use of AI in pure mathematics is not new, but previous successes have been limited. In 2022 , DeepMind’s AlphaTensor discovered a faster matrix multiplication algorithm, a result in discrete mathematics. The Jacobian conjecture, however, lies in continuous algebraic geometry, a domain where AI’s pattern-matching strengths are less obviously applicable. Anthropic’s Claude models are known for their focus on safety and nuanced reasoning, but they are general-purpose language models. That one could contribute to a problem of this caliber underscores the rapid advancement of AI capabilities. It also blurs the line between tool and collaborator. Abhyankar’s Ghost in the Machine The Jacobian conjecture holds particular significance in India due to the work of Prof. Shreeram Abhyankar , a renowned algebraic geometer who founded the Bhaskaracharya Pratishthana in Pune. Abhyankar was deeply interested in the conjecture and popularized it through his lectures, inspiring generations of mathematicians. Abhyankar’s own work on polynomial automorphisms and the conjecture’s two-dimensional case made him a central figure in the field. That an AI model might now resolve what he and others could not is both a testament to human ingenuity and a poignant moment for the community. The broader implications for mathematical research are profound. If AI can reliably generate counterexamples, it could accelerate the falsification of conjectures, allowing mathematicians to focus on refining theories. However, it also raises concerns about the devaluation of human intuition and the potential for AI-generated errors to mislead the field. For now, the mathematical community awaits the full paper. Until then, this episode serves as a powerful reminder that the boundaries of machine intelligence are expanding into the most abstract realms of human thought.