Claude Proves 70-Year Probability Theory Conjecture, AI Cracks Fields Medal-Level Problem for the First Time
nashnova research
Anthropic's large language model Claude has produced a formal proof of the θ=0 conjecture in percolation theory, a problem open for nearly 70 years — the first time AI has independently solved a problem widely considered worthy of a Fields Medal.
What exactly happened?
In August 2026, Anthropic engineer Justin Leder quietly pushed a code repository to GitHub. Inside was a complete proof auto-generated by Claude and rigorously verified by Lean — a formal proof language that checks every logical step the way a compiler checks code.
The target was the θ=0 conjecture in percolation theory, a core unsolved problem straddling probability theory and statistical physics since 1957.
This means → AI did not "help a human write a few steps." It walked the entire reasoning chain from start to finish on its own; the human role was post-hoc review.
Why was this problem open for 70 years?
Percolation theory — a mathematical framework for how liquid passes through porous material — asks a deceptively simple question: at exactly the critical probability, is the chance of forming an infinite connected network strictly zero? If yes, the phase transition is smooth; if not, the system snaps discontinuously at the critical point.
The 2D case was solved in 1980. Very high dimensions (11+) were settled long ago. But dimensions 3 through 10 lack both the geometric symmetry of 2D and the statistical tools of high dimensions — the gap stood for decades.
2022 Fields Medalist Hugo Duminil-Copin poured years into the problem without success. In plain terms = this was not a case of "nobody tried hard enough" — the best mathematicians in the world tried repeatedly and could not break through.
How did AI break through?
In 2024, Gady Kozma of the Weizmann Institute and Shahaf Nitzan of Georgia Tech published a paper showing that if one specific algebraic inequality could be verified, the θ=0 conjecture for dimensions 3–10 would follow automatically. This means → the finish line was pulled close, but the inequality itself was still extremely hard.
Operating under Lean's constraints, Claude drew on a large toolkit of known analytic methods and inequality techniques to build a chain of reasoning thousands of lines long, along a path no human mathematician had envisioned, completing a formal proof of the inequality.
UPenn mathematician Ahmed Bou-Rabee then used LLM assistance to generalize and revise the proof in just one day. He said: "Some projects I worked on for eight years with almost no progress, but with AI's help I'm now one step away from a full solution."
What does the math community think?
Benedikt Jahnel of TU Braunschweig said bluntly: "If a human had proved this conjecture, they would very likely win a Fields Medal. But now it's AI that crossed the finish line."
Mathematician Gil Kalai said: "If verified, this would be an extraordinary breakthrough."
But Kozma, co-author of the enabling paper, struck a cautious note — he would "reserve comment for now" and wait for Anthropic to release a human-readable version and methodological explanation. This reflects the community's core stance: passing Lean verification ≠ full acceptance; a human-readable explanation is the real test.
Why does this matter?
Until now, AI's role in mathematics was mainly assistive — helping humans verify steps or search for lemmas. This time Claude completed the entire journey itself. The nature of the achievement is fundamentally different.
Fields Medalist Duminil-Copin wrote on his blog the same day: "In our field, it is probably only a matter of time before the most famous conjecture falls to the roar of the bulldozer (AI)."
In plain terms = if AI can independently solve problems at the "worthy of a Fields Medal" level, then the basic mechanics of mathematical research — who proposes the path, who completes the proof, who receives the credit — are all up for redefinition.
市场有风险，内容仅供研究参考，不构成投资建议。