Fudan researchers reconstruct part of OpenAI Navier–Stokes work, defer key step

Fudan University researchers Zhen Lei and Xiao Ren have reconstructed the profile-building portion of OpenAI’s claimed Navier–Stokes result. They call the work a major advance, but their paper does not validate the full proof.

Lei and Ren’s paper, submitted September 28 and revised the next day, lays out how OpenAI’s construction produces self-similar profiles with specified stress and remainder properties. The next step—cancelling the remaining stress through oscillatory pulses—is deferred to a planned companion paper.

Lei and Ren describe their work as a readable presentation of the profile-construction part of OpenAI’s manuscript. Both are listed with Fudan University’s Center for Applied Mathematics; neither is listed as affiliated with OpenAI. The arXiv record calls the paper expository and says it will not be submitted to a journal.

The profiles are smooth and axisymmetric. Inserted into the Navier–Stokes equations, they leave a residual that splits into the divergence of a stress and a remainder. That remainder vanishes to infinite order as the proposed blowup time approaches within fixed similarity regions. The paper also says the stress and radial shear satisfy an admissible cone condition required by the broader construction.

But that does not amount to an exact unforced self-similar solution. Lei and Ren formulate a profile-and-matching problem and warn that making the remainder outside the stress divergence flat does not make the full residual small; the stress itself need not be small. Their planned Part II is intended to explain how oscillatory pulses cancel the remaining divergence-form stress.

The paper therefore offers outside technical engagement with one part of OpenAI’s manuscript, not confirmation of the complete finite-time blowup claim. The residual-correction argument is not included in the evidence reviewed for this story.

OpenAI announced on September 8 that an internal system had produced an analytical proof and a Lean formalization showing finite-time singularity formation from initially smooth fluid at rest under smooth forcing, with finite energy. OpenAI said the result resolves statements C and D in the official Millennium Prize formulation, while also saying it did not intend to claim the prize.

Separately, OpenAI formed a nine-member mathematics advisory group in September. That group does not provide evidence for the Navier–Stokes claim.

The Clay Mathematics Institute said on September 11 that the problem had apparently been settled. It also emphasized that evaluation would be deliberately unhurried and that the work still required analysis and interrogation. Under Clay’s published rules, a proposed solution is considered only after publication in a qualifying outlet, a wait of at least two years and general acceptance in the global mathematics community.

The available record does not show that those conditions have been met. Nor does it show journal peer review for Lei and Ren’s Part I, which assigns the remaining residual-correction stage to the planned companion paper.