AI & MATHEMATICS — Formal Methods

Human Mathematicians Are Being Outcounterexampled

In the space of a single summer, AI systems found counterexamples to celebrated problems that had stood unsolved for 60 and 100 years, then translated their reasoning into code a computer could prove correct.

• On 20 May 2026, ChatGPT produced a disproof of the Erdős Unit Distance conjecture, using a 1960s number-theory theorem to construct a concrete counterexample.
• On 11 July, a model called Sol found a counterexample to a 60-year-old question of Grothendieck about group schemes — verified the same day when its reasoning was converted into 1,076 lines of Lean proof.
• Within hours of the World Cup Final, another AI delivered a counterexample to the 100-year-old Jacobian Conjecture in algebraic geometry.

The common thread is a new workflow. An AI proposes an informal answer in plain language; a second system translates it into Lean, a formal proof language where every claim is checked against the axioms of mathematics. If the Lean code compiles, the counterexample is real — no peer review needed. As Fields Medalist Kevin Buzzard put it at Imperial College, the formal file is actually easier to trust than the prose version, because "there is no danger of imagining things."

What makes this moment striking is the leap in what machines can autoformalize. To verify the Erdős disproof, the underlying number theory — a hundred-plus pages of global class-field theory — had to be formalized too. One model, Sol, produced 1.2 million lines of Lean code in three weeks, already half the size of the entire nine-year mathlib library. The same kind of tool recently helped a PhD student finish 250,000 lines of code for a major theorem in roughly two weeks.

The counterexamples are the headline, but the deeper change is procedural. Decades of conjectures are now being formalized by humans into a common language, and a machine can then check or disprove them almost trivially. The next step, mathematicians say, is not more AI output but human insight — understanding exactly what the machines found, so that counterexamples become real knowledge instead of curiosities.