On October 25, mathematician Scott Armstrong called OpenAI’s Navier–Stokes paper “unbelievable,” saying it might be beyond what a person could have devised but still within human understanding. His assessment concerns the work’s ambition and comprehensibility, not whether the proof is correct.
OpenAI president Greg Brockman posted the same central wording separately. For context on the wider debate over AI’s role in mathematics, NeoTeo previously covered mathematicians Scott Borcherds and Robin Huang’s differing views.
What the October paper argues about Lean
In an October 6 preprint, A. Bastounis, F. Circelli and A. C. Hansen argue that OpenAI’s Lean formalization does not semantically correspond to its written proof. Lean is a proof assistant: software that checks a proof expressed in a formal, computer-readable language. The authors’ concern is whether that formal version preserves the meaning of the written argument.
One example compares estimates that use different numbers of additional derivatives:
| Estimate discussed in the preprint | Additional derivatives |
| Natural-language proof | 4 |
| Lean formalization | 5 |
The authors also describe a difference between the written proof’s pressure-flux bound and the Lean estimate, including differences in the variables used and the stated time conditions. They explicitly say their paper does not claim that OpenAI’s written proof is incorrect.
A Lean check applies to the formal statement encoded in the system. It does not, by itself, establish that the encoded statement and reasoning preserve the meaning of a separate written proof. The preprint addresses that correspondence, rather than deciding whether the written proof is correct.
OpenAI’s claim and Clay’s dated statement
On September 8, OpenAI said its internal system had produced a finite-time blow-up result for a smooth, initially resting, three-dimensional incompressible fluid acted on by a smooth external force, with finite energy throughout. OpenAI said the result establishes cases C and D in the Clay Mathematics Institute’s problem formulation, and that it did not intend to claim the Millennium Prize for this result. OpenAI’s announcement describes the company’s claim and its written and Lean-formalized proofs.
In a September 11 notice, the Clay Mathematics Institute described the problem as “apparently” settled and said its prize-evaluation process would be deliberately unhurried, with updates to follow.
