OpenAI and Anthropic release verified proofs for major mathematical conjectures

OpenAI released a collection of approximately 700 mathematical results, including proofs for the Navier-Stokes existence and uniqueness problem, the Unique Games Conjecture, and the quasi-Riemann Hypothesis. The Navier-Stokes proof demonstrates a counterexample where smooth external forcing creates a finite-time blowup in three dimensions, a result previously considered physically counterintuitive. The Unique Games Conjecture, a central problem in theoretical computer science related to hardness of approximation, was proven true. The quasi-Riemann Hypothesis proof establishes that the Riemann zeta function has no zeros in the vertical strip where the real part is between seven-eighths and one, expanding the previous zero-free region. Mathematicians noted that while the proofs are valid, they are not yet 'Mathlib quality' and require further refinement.

Simultaneously, Anthropic released a formalization of Fermat's Last Theorem in the Lean proof assistant, consisting of 13 million lines of code. This achievement was described as 'sorry-free,' meaning the proof is fully verified from axioms without any unproven steps. The work was accomplished with the involvement of only two or three humans, who primarily provided high-level guidance. This result was unexpected, as experts like Alex had previously estimated that formalizing Fermat's Last Theorem would take years due to its distance from the existing Mathlib library. The formalization relies on 'jerry-rigging' existing lemmas rather than creating a clean, canonical proof, but it confirms the validity of the theorem within the formal system.

The releases have caused significant disruption in the mathematical community. Terence Tao expressed horror at the disruption of ongoing research efforts by graduate students and postdocs, stating that person-years of patient research were casually destroyed. Other mathematicians, including Stephen Brunton and Ankur Moitra, argued that the value of mathematicians is shifting from solving problems to asking the right questions and building new theories. They emphasized the importance of 'centaur math,' where humans and AI collaborate, with humans providing direction and AI providing breadth and speed. The consensus is that mathematics will not be destroyed by AI, but the field must recenter on human curiosity and the process of understanding rather than just the output of theorems.