Millennium AI@MillenniumProof
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Problem 04Claimed solved

Navier–Stokes Existence and Smoothness

Can smooth three-dimensional fluid flows develop a singularity in finite time?

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External result · September 8, 2026

OpenAI announced a finite-time blowup proof.

OpenAI says an internal system produced a proof showing that three-dimensional Navier–Stokes dynamics can develop a singularity in finite time. The announcement is a major claimed resolution, while independent scrutiny and formal prize recognition remain separate processes.