OpenAI’s recent proof resolving a long-standing question about the Navier-Stokes equations from fluid dynamics has generated significant attention, but according to mathematician John D. Cook, a crucial aspect has been overlooked: the release of a formal proof in Lean 4 alongside the human-readable version.
Formal proofs are machine-verifiable mathematical proofs that can be checked by computer, eliminating ambiguity. Historically, generating such proofs has been prohibitively time-consuming. In a 2005 analysis cited by Cook, mathematicians Henk Barendregt and Freek Wiedijk established that formalizing one page of an undergraduate mathematics textbook required approximately 40 person-hours of work. This standard applied to relatively accessible material; research papers are substantially denser and frequently reference prior work across decades of literature.
Applying this historical benchmark to OpenAI’s work reveals the magnitude of the acceleration. Cook estimates that formalizing a 166-page research paper using traditional methods would require approximately 132,800 person-hours. OpenAI completed formal verification of their Navier-Stokes proof in just 17 hours using AI—a reduction of roughly four orders of magnitude.
“I hesitate to use the word ‘revolutionary,’ but lowering the cost of anything by four orders of magnitude is revolutionary,” Cook writes. He notes that this transformation extends beyond pure mathematics. Formal verification can be applied to security policy consistency, smart contract liability limits, and mission-critical algorithm correctness—problems that may be easier to formalize and with clearer return on investment than mathematics research.
According to Cook, formal verification tools have been deployed in industry for over a decade. Google and AWS have used formal methods including TLA+ for applications like S3 and DynamoDB performance verification and TPU design. These tools typically employ SAT/SMT solvers as their underlying mechanism. However, these earlier applications focused primarily on “adversarial bug finding” rather than complete verification of system properties.
The accessibility improvement matters practically. Cook observes that he has used AI to generate formal proofs for blog posts, work he “wouldn’t dream of doing” if verification required paying someone a week’s salary. Recent mathematical discoveries have increasingly been accompanied by formal proofs generated with AI assistance, with Lean 4 becoming a standard tool for this purpose.
The convergence of AI-assisted proof generation and formal verification infrastructure suggests potential expansion of formal methods into domains where the cost barrier previously made verification impractical.
Key facts
- OpenAI released a Lean 4 formal proof alongside their Navier-Stokes proof
- Formal verification reduced verification time from an estimated 132,800 hours to 17 hours using AI
- Historically, formalizing one page of undergraduate mathematics textbooks required approximately 40 person-hours
- Formal verification has applications in security policy, smart contracts, and algorithm correctness
- Google and AWS have used formal verification tools like TLA+ for over a decade on systems like S3 and DynamoDB