AI Model Claude Fable Disproves Long-Standing Mathematical Conjecture
Mathematician Levent Alpoge confirms that the AI model produced a polynomial map with a non-zero Jacobian determinant that is not invertible, settling a decades-old problem in algebraic geometry.
The Jacobian Conjecture, a prominent open problem in algebraic geometry and commutative algebra, has been disproven. The conjecture posited that any polynomial map with a constant non-zero Jacobian determinant must be invertible. This principle has now been overturned by a counterexample generated by the AI model Claude Fable.
The discovery was announced by mathematician Levent Alpoge via a post on X. Alpoge credited the development to a collaboration involving a colleague referred to as Akhil, who raised the question, and Claude Fable, which executed the proof. Alpoge noted that the model worked on the problem during the final of the 2026 FIFA World Cup, a match in which Spain defeated Argentina 1-0 in extra time.
The counterexample involves a specific polynomial map from complex three-space to itself. The map is defined by three components: f1 equals (1 plus xy) cubed z plus y squared times (1 plus xy) times (4 plus 3xy); f2 equals y plus 3x times (1 plus xy) squared z plus 3xy squared times (4 plus 3xy); and f3 equals 2x minus 3x squared y minus x cubed z.
Crucially, this map possesses a Jacobian determinant of negative 2. Under the conditions of the conjecture, a constant non-zero determinant should imply invertibility. However, the map demonstrates that this is not the case, thereby invalidating the conjecture.
To substantiate the claim, Alpoge provided specific coordinate mappings. The map sends the points (0, 0, negative one-quarter), (1, negative three-halves, thirteen-halves), and (negative 1, three-halves, thirteen-halves) all to the single coordinate point of (negative one-quarter, 0, 0). The existence of multiple distinct points mapping to the same image confirms that the function is not injective and therefore not invertible.
This development marks a significant intersection of artificial intelligence and pure mathematics. By utilising computational power to navigate complex algebraic structures, the AI model identified a structural flaw in a hypothesis that had remained unchallenged for considerable time. The announcement highlights the growing role of large language models in solving high-level theoretical problems.
