OpenAI recently announced a proof addressing a long-standing problem related to the Navier-Stokes equations in fluid dynamics. While the mathematical breakthrough has attracted significant attention, an equally important aspect is the simultaneous release of a formal proof verified with the Lean 4 proof assistant.
Formal proofs that are machine-verifiable have traditionally been extremely labor-intensive. In 2005, experts estimated that formalizing just one page of an undergraduate mathematics textbook could require around 40 hours of work. Given that research papers are far denser and rely on extensive prior work, formalizing a 166-page research article would have been prohibitively time-consuming, potentially demanding over 130,000 person-hours.
OpenAI’s approach changed this dynamic by verifying their entire proof in Lean 4 in just 17 hours, representing a reduction in effort by roughly four orders of magnitude. This advancement could be considered transformative for the field of formal verification.
The impact extends beyond pure mathematics. Formal verification can be applied to ensure the consistency and effectiveness of security policies, validate smart contracts’ constraints, and confirm the correctness of critical algorithms. These applications generally require less effort than formalizing advanced mathematical research, making the new efficiency gains highly relevant for practical and mission-critical use cases.