Claude Fable 5 AI finds a tiny formula that topples an 87-year-old math conjecture

Mathematician Daniel Larson at Anthropic used Claude Fable 5 to discover a counterexample disproving the Jacobian conjecture, one of mathematics' longest-standing open problems. The conjecture, posed in 1939, claims that certain polynomial functions are invertible. Larson's team found a simple formula that violates this claim in three or more dimensions, settling a question that has eluded mathematicians for over eight decades.

The Jacobian conjecture relates to whether a polynomial map with a nonzero Jacobian determinant must be invertible. This connects to fundamental questions in algebraic geometry and dynamical systems. Mathematicians have attempted countless approaches without resolution, making it a benchmark problem in the field. The conjecture remains true in two dimensions, but Larson's counterexample proves it fails once dimensions exceed two.

The discovery highlights AI's expanding role in mathematical research. Claude Fable 5, Anthropic's reasoning-focused model, helped identify a formula simple enough that humans can verify it directly. This represents a shift from AI serving as a computational tool to assisting in conceptual breakthroughs. The model appears to have found patterns or relationships that conventional mathematical techniques overlooked.

The counterexample's simplicity matters. Complex proofs often face skepticism and require extensive peer review. A straightforward formula can be checked rapidly, lending immediate credibility to the result. Larson's work demonstrates that even notoriously difficult problems can sometimes yield to unexpected approaches when combined with advanced computational reasoning.

This result doesn't diminish the conjecture's mathematical importance. The two-dimensional case remains open and challenging. Understanding why the conjecture holds in two dimensions but fails in higher dimensions could reveal deep structural properties of polynomial systems. Researchers now focus on characterizing the exact boundary where the conjecture transitions from true to false.

The collaboration between