The part of Navier-Stokes no one is talking about

Sep 11, 2026 04:22 AM - 3 days ago 5

Yesterday OpenAI announced a impervious that settled a long-standing mobility astir the Navier-Stokes equations from fluid dynamics. The announcement has created a batch of buzz, arsenic 1 would expect. But there’s an facet of OpenAI’s activity that I haven’t seen anyone talk about: they posted a Lean 4 general impervious astatine the aforesaid clip arsenic their accepted human-readable proof.

Quite a fewer different mathematical conjectures person been settled precocious utilizing AI, and these person besides been accompanied pinch general proofs, utilizing Lean 4 successful particular.

Until very recently, generating machine-verifiable general proofs has been excruciatingly tedious. In 2005, Henk Barendregt and Freek Wiedijk wrote

To springiness an denotation of really overmuch activity is needed for formalisation, we estimate that it takes astir 1 work-week (five work-days of 8 work-hours) to formalise 1 page from an undergraduate mathematics textbook.

That was the norm of thumb: forty hours per page. And this successful the discourse of undergraduate textbooks. Research publications are overmuch denser than textbooks. Furthermore, page 100 of a textbook astir apt depends mostly connected worldly connected pages 1 done 99. A condemnation successful a investigation article could mention thing that has been published before.

Say a investigation article takes 20 times much effort to formalize than page successful an undergraduate textbook. Then formalizing the 166-page insubstantial from OpenAI would return 132,800 person-hours. It took OpenAI 17 hours to verify their impervious successful Lean. I hesitate to usage the connection “revolutionary,” but lowering the costs of thing by four orders of magnitude is revolutionary.

I’ve utilized AI to make general proofs to cheque my activity conscionable for a small blog post. I wouldn’t dream of doing that if I had to salary personification a week’s net to cheque my work.

Formal verification doesn’t conscionable use to mathematics. You could, for example, formally verify that a group of information policies are accordant and that, fixed definite assumptions, they execute their purpose. You could formally verify that a smart statement imposes a definite maximum liability. You could verify the correctness of mission-critical algorithms. These problems are easier than formalizing mathematics research, and it is easier to quantify the return connected investment.