What is settled

Claims, checks, and reactions, with dates. Updated 25 September 2026.

The theorem is the forced alternatives (C) and (D) of the Clay statement. The unforced alternatives (A) and (B) are open. Everything else on this page is about how much weight to put on each half of that sentence.

Proved

• For every $\nu>0$ there is a smooth, compactly supported force and a smooth compactly supported flow from rest with bounded energy and unbounded velocity at $t=1$. No smooth bounded-energy solution with the same data exists globally. That is alternative (C), and (D) follows by periodization.
• The singular point is isolated; every derivative of the velocity has a one-sided limit at $t=1$ on compact sets avoiding the origin.
• The total dissipation on $[0,1)$ is finite.
• A Lean 4 formalization of the breakdown statements accompanies the paper, in the repository openai/NavierStokesAndEuler, together with a formalized unforced Euler blow-up. On 24 September 2026 it was built from source here: 11,424 jobs, no sorry, and all four Navier–Stokes theorems depend on the three standard axioms only. The formal statement page has the details and the clause-by-clause comparison with Fefferman.

Not proved, and not claimed

• Nothing about unforced initial data. The force is essential to the method and the paper says so.
• Nothing about turbulence or about any flow one could set up. The force is a construction, not a description of any experiment.
• Nothing about stability. Whether nearby forces or nearby data blow up is not addressed. Cao, Chi and Nie have since shown that forces producing blow-up are dense in a weak topology, $L^1_tH^s_x$ with $s<1/2$; density is not openness.

Constraints found since

• Constantin, Ignatova and Vicol (17 September 2026). A solution with the construction's anisotropic Type II bounds on its angular mean and exact axisymmetry in a collapsing core is regular at the putative singular point if the force is real analytic in space near it, locally uniformly in time, and bounded in $C^2$ up to the singular time. Consequently the force in any such construction can neither vanish near the singular point nor be analytic there. The authors state that they have not verified the construction.
• Duraiswami (15 September 2026). The core geometry is that of an exact steady swirl between rotating porous cylinders, but that boundary value problem does not recast into the similarity variables, and nothing found suggests the mechanism is reachable in a flow one computes or builds.

Verification

The Lean formalization checks that the formalized proposition follows from the axioms. It does not check that the formalized proposition is Fefferman's, and that step is human; the formal statement page does it clause by clause and finds the encoding faithful, with two readings noted. The build reproduced here confirms the axiom claim in the repository's manifest. The Clay Mathematics Institute said on 11 September 2026 that the problem has apparently been settled and that its evaluation will be deliberately unhurried; its rules require publication, two years of scrutiny, and broad acceptance. OpenAI has said it will not claim the prize. No independent verification of the manuscript had been announced as of the date above.

Priority

Córdoba and Martínez-Zoroa established forced blow-up for three-dimensional Euler in 2023 and, with Zheng, for hypodissipative Navier–Stokes in 2024; Fefferman has called them the heroes of the story. Alpöge and Buckmaster obtained smooth-forcing blow-up for Euler on 15 August 2026, verified in Lean a week later, and released it with the porous medium and Boussinesq cases on 7 September, with Coiculescu as coauthor on the porous medium paper. OpenAI's run began on 1 September and its release was 8 September. A dispute over authorship, attribution, and whether product data from the NYU group's Codex sessions could have reached OpenAI's model followed; OpenAI's first paper omitted the Córdoba–Martínez-Zoroa citations and a revision added them. The timeline lists the dates. This site takes no position on the dispute beyond recording it.

What people have said

• Tao, on the human results of the same week: a remarkable achievement, and a strange and unprecedented decoupling between getting answers and getting understanding. He has also written that it should be possible to do without the forcing term.
• Silvestre: the Clay problem is settled, but the main problem for the Navier–Stokes equations is not.
• Córdoba: all fluids we know of are under some kind of external force, so to have the force makes complete sense.
• Palasek: for unforced Navier–Stokes, energy loss from viscosity should outweigh the amplification produced by this construction's instabilities.
• Karniadakis: for air, the singularity appears when the vortex becomes around 70 nanometres wide. The molecular limit page reproduces the arithmetic.

Next: can fluctuations break it?