AI Solves 87-Year-Old Jacobian Conjecture

Levent Alpöge, a mathematician at Harvard University, announced on 19 July 2026 that he has disproven the 87-year-old Jacobian conjecture. By providing a concise 216-character counterexample, Alpöge demonstrated that the long-standing mathematical assumption is false. This breakthrough is notable for the significant role played by Anthropic's AI model, Claude Fable 5, marking a major milestone in the application of artificial intelligence to complex mathematical problem-solving. The mathematical community views this development as a paradigm shift in the field. While the conjecture was originally proposed by Ott-Heinrich Keller in 1939 and later highlighted by Stephen Smale in 1998, the sudden disproof via AI has surprised experts. The result is considered the most difficult mathematical problem solved by AI to date, sparking intense discussion regarding the future necessity of human mathematicians versus increasingly capable AI systems. The essence of this discovery lies in the efficiency of AI-assisted verification. Although the counterexample is simple to verify, the process used to derive it remains a subject of inquiry. Experts emphasize that while AI can effectively identify counterexamples, the creation of entirely new mathematical theories—a hallmark of human creativity—remains a distinct challenge that differentiates human research from machine-generated results. For the broader scientific community, this event serves as a catalyst for re-evaluating research methodologies. Users should anticipate a rapid acceleration in AI-driven discoveries across various scientific disciplines. The practical takeaway is that AI is transitioning from a supportive tool to a primary driver of fundamental breakthroughs, necessitating a shift in how researchers integrate computational intelligence into their workflows to maintain relevance and innovation.