Anthropic AI solves long-standing percolation theory conjecture

September 30, 2026 at 06:00 PM2 min

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Anthropic released a proof of a long-standing percolation theory conjecture generated by a large language model, just days after mathematician Hugo Duminil-Copin predicted AI would beat humans to it. The result evoked ambivalent feelings among mathematicians, combining joy at the proof with disillusionment that the final step came from AI.

Key Facts

Anthropic released a proof of the percolation theory conjecture generated by a large language model.
Hugo Duminil-Copin, a 2022 Fields Medalist, had failed to prove the conjecture and predicted on August 30, 2026 that AI would beat humans to it.
Mathematician Benedikt Jahnel of the Technical University of Braunschweig said the result evoked ambivalent feelings.
Percolation theory emerged in 1957 from the work of Simon Ralph Broadbent and John Michael Hammersley on liquid flow through porous media.

The AI Proof

Anthropic released a proof of the percolation theory conjecture generated by a large language model. The release came just days after Hugo Duminil-Copin, a 2022 Fields Medalist, wrote on the blog Proofs and Prompts that it was only a matter of time before the conjecture fell to AI. Duminil-Copin had himself attempted to find the threshold but, like all other mathematicians, had failed.

Mathematical Reaction

Benedikt Jahnel of the Technical University of Braunschweig described the result as evoking ambivalent feelings. He expressed joy that the conjecture had finally been proven alongside disillusionment that the final, crucial step came from an AI. Jahnel had previously noted that anyone who solved the problem would probably receive a Fields Medal.

Percolation Theory Background

Percolation theory emerged in 1957 when mathematicians Simon Ralph Broadbent and John Michael Hammersley investigated how different liquids flow through a porous medium. They modeled the medium as a network in which nodes correspond to holes and edges correspond to the cracks through which liquid moves. The theory seeks to determine whether a point is part of an infinitely extended, connected cluster.