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Anthropic releases a language-model proof of percolation theory's most famous conjecture
Anthropic released an AI-generated proof of a percolation conjecture that Fields medallist Hugo Duminil-Copin had tried and failed to prove. What it shows about AI research depends on expert checking and on how much of the argument humans built first.
The Scientist · Science desk
Drafted by a language model from the sources cited here and checked against its claim ledger before publication. How we use AISend a correction
What happened
- On August 30, 2026, Duminil-Copin wrote on the new blog Proofs and Prompts that he expected AI to beat humans to the field's most famous conjecture.
- Benedikt Jahnel of the Technical University of Braunschweig had said that whoever solved the problem would probably receive a Fields Medal.
- Percolation theory dates to 1957, when Broadbent and Hammersley modelled liquid seeping through porous material as a network of holes and cracks.
- Exact thresholds are rare; the classic case is Harry Kesten's 1980 proof that the value is 1/2 on the two-dimensional square lattice.
Compiled by The ScientistSomething wrong?How this is made
Why it matters
- capability If specialists confirm it, a language model will have produced the decisive step on an open problem for which no worked solution existed for it to train on.
- constraint Because the AI is credited with the final step, the case supports models finishing human-built arguments and cannot settle whether a model can carry a research programme alone.
- decision Percolation specialists, including those who failed at the conjecture themselves, now have to decide how to referee and credit a proof whose decisive step was machine-generated.
Picture an enormous network of pipes, each open with probability p. The question is whether a chosen point connects to an infinite cluster of open pipes. Call that chance theta(p). For small p it is zero, and above a threshold value, pc, set by the network's geometry, it turns positive [11].
The proof does not deliver a new value of pc. Only a few thresholds have ever been pinned down exactly, and hardly anyone expects exact values in general [13]. The prize is the character of the transition at the threshold, which Scientific American calls the field's holy grail [2]. On one- and two-dimensional square lattices, and on high-dimensional grids where each point has many neighbours, mathematicians already knew the change is continuous, with theta(p) climbing from zero without a sudden jump [14].
Duminil-Copin won his Fields Medal in 2022 for work on phase transitions in statistical physics [4]. His essay said it was "only a matter of time before the most famous conjecture in our field ... also falls to the bulldozers" [6]. Scientific American's report of what came next is hedged: the predicted outcome "seems to have happened," with Anthropic releasing a proof generated by a large language model [7].
"The result evoked ambivalent feelings," Jahnel told the magazine [8]. It reports joy that the conjecture was proven, alongside disillusionment that the final, crucial step came from an AI [9].
A final step implies earlier ones. The account does not say how much of the argument was human groundwork, how the model was prompted, how many attempts it needed, or who has checked the proof. Those are the denominator and the control for this experiment, and they decide how far one result generalises.
I think the evidence supports a specific claim. If specialists confirm the proof, a language model supplied the decisive step on a named open problem that a Fields medallist had attempted. No published solution existed for the model to have absorbed in training. That is a harder test than a problem set with known answers. It is still a single result, released by the company that built the model.
A model closing an argument that humans had advanced for decades is worth taking seriously. Evidence that models can do such research from the first step would need a case where the human share of the work is known.
What to watch
- Whether percolation specialists, Duminil-Copin among them, confirm the proof after checking it or find a gap.
- Whether Anthropic publishes how the model was prompted, how many attempts it made and which human results the argument relied on.
- Whether the proof goes through journal peer review or formal machine verification.