How does a coherent structure emerge from many small random occurrences? For instance, at what point are there enough openings in a sponge for a liquid to flow through? Such questions are critical to a long-standing puzzle of percolation theory, a subfield of probability theory focused on the permeability of networks.
More specifically, for years, mathematicians have sought to better understand a threshold that applies to these networks: above this threshold, the odds are high that a particular network is comprised of infinite open connections, while below it, those odds are low. Better characterizing of this transition has been considered a holy grail for the field. As mathematician Benedikt Jahnel of the Technical University of Braunschweig in Germany has puts it, “If someone manages to solve this problem, they’ll probably receive a Fields Medal.”
Among the many experts who have attempted to find this threshold is French mathematician Hugo Duminil-Copin, who himself received a Fields Medal in 2022 for his work on phase transitions in statistical physics—a topic connected to probability theory. But when it came to the percolation theory conjecture, he, like all the rest, failed. And on August 30, 2026, Duminil-Copin expressed concern in an essay on the new blog Proofs and Prompts that AI would likely beat humans to the punch, writing that it would be “only a matter of time before the most famous conjecture in our field ... also falls to the bulldozers.”
Just days after he posted those words, exactly that seems to have happened: the artificial intelligence company Anthropic released a proof of the conjecture generated by a large language model. “The result evoked ambivalent feelings,” Jahnel tells me. Alongside joy that the conjecture had finally been proven, there was a certain disillusionment that the final, crucial step came from an AI.
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