AI breaks math: Claude Fable 5 disproves the 87-year-old Jacobian Conjecture

Levent Alpöge, a mathematician working at Anthropic, dropped a bombshell on social media by thanking his friend Akhil and the newly minted Claude Fable 5 for cracking the problem. The AI generated a specific mathematical counterexample to the famous Jacobian Conjecture, an algebra puzzle first proposed in 1939. If this holds up, it means generations of human geniuses have spent their lives trying to prove something that was flat-out false all along.

The core of the conjecture is about whether a complex mathematical mapping is globally reversible—meaning you can always trace any output back to its unique starting point. The math community assumed that if a function behaves nicely locally at every single point, it must behave nicely as a whole. Claude Fable 5 proved this assumption wrong by cooking up a 3D formula where three completely different input coordinates lead to the exact same output, creating a mathematical dead-end where reconstruction is impossible.

What makes this discovery a massive headache for traditionalists is how easily it can be verified. Usually, checking a major math breakthrough requires a committee of professors and three years of peer review. In this case, because the AI provided a concrete formula, other researchers can simply plug in the numbers, calculate the Jacobian matrix, and watch it fail in real-time on a basic computer.

However, the AI didn't kill the entire problem. The counterexample only disproves the conjecture for three dimensions and higher. The two-dimensional version of the puzzle remains unsolved, leaving a small, desperate sliver of hope for human mathematicians who want to beat the machine.

This is a major departure from how AI has behaved in mathematics so far. While tools like Google DeepMind's AlphaGeometry or AlphaProof have been praised for solving high-school olympiad geometry, they were merely proving things we already knew were true. Having an LLM actively find a loophole to dismantle an established historical theory is a completely different beast.

The exact level of autonomy behind this discovery remains a mystery. Anthropic has not yet disclosed whether the model generated the formulas entirely on its own, how many millions of failed attempts it went through, or if it was heavily guided by Levent Alpöge using external computer algebra systems.

Watching an AI casually dismantle nearly nine decades of human intellectual struggle during a lunch break is both deeply inspiring and slightly terrifying. The line between AI as a glorified auto-complete tool and AI as an autonomous scientific pioneer has just become dangerously thin.

Source: X