OpenAI posts Navier-Stokes and Yang-Mills claims

OpenAI reports that a smooth three-dimensional flow can reach unbounded speed in finite time while the energy in the flow stays finite, when a smooth external force is allowed to act.

Clay's official statement C asks for some smooth starting velocity and some smooth force for which no smooth solution lasts forever, and statement D asks the same question inside a periodic box.

The Navier-Stokes equations are Newton's second law for a fluid treated as a continuous medium. The change in motion balances pressure, the transport of momentum, and viscosity. Viscosity damps sharp differences in speed. A singularity means the speed becomes unbounded in a finite time even so, while the fluid is still being treated as a continuum. A real liquid or gas cannot move infinitely fast. Past that point the continuous equations no longer describe the fluid. Aircraft design, weather forecasts, and the study of blood flow use these equations where the speed stays finite. A forced blowup does not replace those calculations. It marks a place where the continuous model stops.

OpenAI's construction starts from rest. A vortex spins, draws inward, and stretches along its axis until the fast region is long and thin. OpenAI compares that stretch to spaghetti. The core shrinks and speeds up, and the energy stays finite because the region that is moving fast is getting smaller. The large terms in the equations cancel closely enough that the force left over is still a smooth function, rather than an infinite push written in by hand. The search started on September 1, 2026. OpenAI dates the resolution to September 5, about 88 hours after the first agents started, with a further 17 hours to check the argument in Lean. The run used on the order of 10,000 agents at once. OpenAI says it will not claim the Millennium Prize.

Statements A and B of the same official problem set the external force to zero. Those are the versions this search did not resolve. Charles Fefferman wrote the four statements for the Clay Mathematics Institute in 2000, and a proof of any one of them meets the written problem. Luis Silvestre, a mathematician at the University of Chicago, told Scientific American that the Clay problem and the main problem for these equations are not the same problem. Scientific American reported a later argument that this construction falls apart if the force is removed, so the method does not extend to the force-free case. Diego Córdoba, also quoted there, noted that real fluids do sit under forces such as gravity, so a force term can be physically reasonable. The force in this construction is one particular smooth function, chosen so the cancellation works. It is not a forecast that an ordinary flow blows up.

The same search produced a blowup for the Euler equations, which drop viscosity and keep the rest of the law. That Euler result has no external force. Levent Alpöge of Anthropic and Tristan Buckmaster, a math professor at New York University, had reached a forced Euler result with an Anthropic model. OpenAI reports that its own Euler argument is the unforced one, that the two proofs differ, and that it did not see their work before the public release. An unforced Euler blowup says a frictionless fluid can reach unbounded speed. Viscosity is still present in the Navier-Stokes equations, and the force-free version of that problem is untouched by the forced construction.

On October 26, 2026, OpenAI released three further manuscripts and called each of them raw research data for other people to explain. The mathematics manuscript takes up the local smoothing conjecture in every dimension. Sogge proposed it in 1991. It says the fiercest focusing of a wave is brief: at a typical moment the wave is much less concentrated than it is at the single worst instant. The model output is dated October 18, and the Lean check the next day. OpenAI posted the writeup and the Lean check publicly and left other researchers to explain them. The argument uses an auxiliary partial differential equation built for the proof. Local smoothing is not one of the seven prize problems.

A Kakeya set, also called a Besicovitch set, contains a unit line segment in every direction. Around 1920, Besicovitch showed that a Kakeya set in the plane can have an area as small as one likes, so the conjecture is not a claim about ordinary area. It says the set must still have Hausdorff dimension equal to the dimension of the space it sits in. That dimension measures geometric size: a curve has dimension one, and a filled region of the plane has dimension two. Davies proved the conjecture in the plane. Wang and Zahl proved it in three dimensions in 2025. From four dimensions upward, only partial bounds were known. OpenAI says a proof of local smoothing in every dimension also proves the Kakeya conjecture in every dimension.

Stein's restriction conjecture is the related question in Fourier analysis. The Fourier transform decomposes a function into frequencies. When the function is smooth enough, that transform can be read along a surface. Stein asked when the transform of a rougher function still defines a function on a curved surface such as a sphere, and how large that surface function can be. A flat surface lines up with plane waves, so the transform can concentrate there. A sphere curves away from those waves, and the conjecture states the full range of the estimate that curvature allows. The two-dimensional case was settled by Fefferman and Stein and by Zygmund. In three and more dimensions the full range had not been proved. The restriction conjecture implies the Kakeya conjecture, because a bound of that strength forces a unit segment in every direction to have the full Hausdorff dimension of the space. OpenAI's account says local smoothing implies both. Guth, Wang, and Zhang had local smoothing in two dimensions in 2019. The manuscript claims that estimate in every dimension.

If the argument holds, the result is a set of theorems in analysis. A Kakeya set in any dimension would have the Hausdorff dimension of the surrounding space. The restriction bound on the sphere would hold across the range Stein asked for. Around 1970, Fefferman used a very thin Kakeya set as the core of a counterexample about Fourier multipliers, the operators that change the size of each frequency in a wave. That example is why the size of a Kakeya set matters for these estimates. A 2019 survey in the Notices of the American Mathematical Society connects the restriction problem with geometric measure theory, dispersive equations, and analytic number theory. For the wave equation, local smoothing limits how long extreme focusing can last. The Lean check of this argument is public. Other mathematicians have not yet published their own account of it.

The physics manuscript argues that certain quantum field theories in four dimensions exist as mathematical objects, including theories with gauge fields and fermions. Quantum chromodynamics is the theory of the strong force between quarks and gluons. Pure Yang-Mills theory keeps the gluons and leaves the quarks out. A natural-language argument for pure Yang-Mills existence is dated September 9, a QCD argument is dated September 18, and a Lean check of the Yang-Mills existence argument is dated October 16. Existence of that pure theory is one piece of the Clay problem. The prize statement also requires a mass gap: every state other than the vacuum carries energy of at least some positive amount, so the force carriers are not massless in the way a classical wave of light is. OpenAI's public account claims the existence piece and does not claim that gap. The institute's statement of the problem asks for both. The Standard Model describes three of the four forces and leaves gravity out. OpenAI says the Standard Model itself cannot be extended to arbitrarily short distances, and that the existence argument applies to the simplest unified theory that still contains the Standard Model's fields. David Gross is leading the group that will write the physics up, with Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov. Asymptotic freedom, the weakening of the strong force at short distances, is the mechanism Gross helped establish, and it is why a rigorous existence proof can be attempted. The group asked to hold the model files and to produce its own account first. Those files are not public, so the steps cannot be checked the way the fluid argument can.

The third manuscript says the exponential time hypothesis is false. That hypothesis has been a working premise in hundreds of papers about how long certain logic problems take in the worst case. A refutation reopens the question of how much faster those tasks could become. It does not hand over the faster algorithms, and whether every quickly checked problem is quickly solvable remains a separate question.

The fluid argument a reader can examine today is a forced blowup, checked in Lean, with the prize left unclaimed. The physics argument is an existence claim whose files are still private.