Anthropic's Claude proves long-standing percolation conjecture

Just days after a Fields Medalist warned that AI would soon bulldoze his field’s biggest unsolved problems, an LLM went ahead and did exactly that.

Anthropic's Claude, guided by researchers, has proved a long-standing core conjecture in percolation theory. Even better, it generated machine-verified Lean code to back up every step, sending quiet shockwaves through pure mathematics without so much as an official press release.

The sponge puzzle that stumped human geniuses

Percolation theory dates back to 1957, when mathematicians started modeling how liquids seep through porous materials like sponges. Imagine a massive grid of pipes where each connection randomly opens or closes with a certain probability.

Below a critical threshold, fluid only forms tiny puddles. Above it, an infinite open pathway suddenly emerges. Figuring out whether this phase transition happens continuously or abruptly has been considered one of the holy grails of probability theory.

As mathematician Benedikt Jahnel once put it, whoever solved it would likely take home a Fields Medal. Even Hugo Duminil-Copin, who won the Fields Medal in 2022 for his work on phase transitions, failed to crack it.

Bridging dimensions 3 through 10

Mathematicians had already solved the extreme cases. In 1 and 2 dimensions, the transition was proven to be continuous decades ago. In 11 dimensions and higher, the math becomes easier because high neighbor counts allow clean statistical averaging, also confirming a continuous shift.

Dimensions 3 through 10, however, remained an impenetrable mathematical dead zone.

On August 30, 2026, Duminil-Copin wrote an essay expressing concern that AI would inevitably beat humans to this answer. Days later, Anthropic showed that Claude had completed the argument, proving that the transition in dimensions 3 to 10 is indeed continuous.

Instead of just spitting out persuasive text that might harbor hallucinations, Claude generated code verified by the Lean proof assistant. Jahnel admitted to feeling ambivalent: immense relief that the problem is finally closed, paired with a distinct disillusionment that an AI took the final leap.

Why it matters

This is a massive milestone for how frontier science gets done. LLMs are graduating from solving high school math olympiad problems to tackling genuinely open conjectures that stumped the greatest living human minds.

In our view, pairing Claude's pattern recognition with Lean's uncompromising formal verification is the real breakthrough here. It completely sidesteps the hallucination debate: if the Lean compiler accepts the formal proof, the math is true, period.

That said, Claude didn't do this in a vacuum. Human mathematicians still had to direct the flow of ideas and break down the proof architecture. The AI acts like an intellectual bulldozer rather than an autonomous visionary, but it turns out a guided bulldozer can clear thirty years of stagnant math in days.

Expect pressure to mount immediately on OpenAI and DeepMind. The race is no longer just about building faster coding assistants, but about who can build the definitive copilot for scientific discovery.