On September 10, 2026, OpenAI announced a formal proof that addresses a long-standing question regarding the Navier-Stokes equations in fluid dynamics. Alongside their conventional proof, OpenAI also released a Lean 4 formal proof. This development has garnered significant attention in the mathematical community.
Recent advancements in AI have led to the resolution of several mathematical conjectures, often accompanied by formal proofs, particularly utilizing Lean 4. Historically, generating machine-verifiable formal proofs has been a labor-intensive process. In 2005, researchers Henk Barendregt and Freek Wiedijk estimated that it required approximately one work-week (40 hours) to formalize a single page from an undergraduate mathematics textbook. Given that research publications are denser than textbooks, they suggested that formalizing a research article could take up to 20 times more effort.
Applying this estimate to OpenAI's 166-page paper implies a potential requirement of 132,800 person-hours for formalization. However, OpenAI managed to verify their proof in Lean in just 17 hours. While the term "revolutionary" is used cautiously, the reduction in effort by four orders of magnitude is noteworthy.
The implications of formal verification extend beyond mathematics; they can also be applied to ensure the consistency of security policies, verify smart contracts, and confirm the correctness of critical algorithms. These applications are generally easier to quantify in terms of return on investment than formalizing mathematical research.