AI solves a ‘holy grail’ problem from probability theory

Just days following a Fields Medalist's prediction that an artificial intelligence (AI) would solve a complex mathematical problem, Anthropic, an AI company, successfully provided a proof for the conjecture. Percolation theory is a subfield of probability theory that deals with the permeability of networks, specifically focusing on the transition from finite to infinite clusters.

Mathematicians have long sought to find the threshold value (pc) above which a network is likely to have infinite open connections. The discovery of this threshold has been considered a holy grail for the field, with the potential to earn a mathematician a prestigious Fields Medal. Hugo Duminil-Copin, a Fields Medalist, had previously expressed concerns that AI might be the first to solve this conjecture, as no human mathematician had managed to do so.

On August 30, 2026, Anthropic released a proof generated by a large language model, confirming that the transition in percolation theory is continuous in certain dimensions. This discovery generated mixed feelings among mathematicians, as they questioned the future of such prestigious awards and the role of AI in academia.