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

In the world of mathematics, excitement was brewing as millions of people relaxed after the excitement of the FIFA World Cup Final. Levent Alpöge, a mathematician at Anthropic, an artificial intelligence company, announced on X that he had discovered a counterexample to the Jacobian conjecture, a longstanding problem in algebraic geometry. This groundbreaking discovery came through the use of Anthropic's large language model, Claude Fable 5, which was recently made available to the public.

A conjecture, in the mathematical world, is an idea believed to be true but has yet to be proven or disproven. The Jacobian conjecture, in particular, is a complex mathematical concept involving functions that act like machines taking numbers as input and producing other numbers as output, following a specific rule or equation. These functions typically use polynomials, and their behavior can be analyzed using the Jacobian determinant.

If the Jacobian determinant is always a non-zero constant, the function is considered to never fold or crush space around a particular point. The conjecture posits that when the Jacobian determinant is a non-zero constant, there should be another polynomial function that reverses the original one, returning all points to their original positions.

The Jacobian conjecture has been a subject of intrigue for over a century, with many mathematicians attempting proofs, only to find subtle errors. Despite numerous valid efforts proving the conjecture true under certain restrictions, the general case—both disproving and proving it—remained elusive. Alpöge's breakthrough is significant because he found an example in three dimensions that meets both conditions of the conjecture: a constant Jacobian determinant of -2 and points merging to the same output point, making it non-reversible.

This discovery proves the conjecture false for dimensions larger than two, leaving the two-dimensional version still unproven. Alpöge's concise counterexample simplifies verification for other mathematicians and exemplifies the potential of AI in discovering unexpected mathematical objects and combinations of ideas across various mathematical fields.