What OpenAI’s proposed proof claims, how the model produced it, and what remains to be checked.
A proposed fluid singularity, illustrated.
Yesterday, OpenAI published a proposed solution to the Navier-Stokes existence and smoothness problem.
The equations describe the motion of fluids. We use them to model water around a ship, air over a wing, weather systems, and blood flow. One basic question about them has resisted mathematicians for nearly a century.
Can a fluid that starts perfectly smooth break down in finite time?
OpenAI says its internal model constructed exactly that kind of breakdown.
Before getting into the vortex it produced, it helps to understand the list this problem belongs to.
The seven Millennium Prize problems
In 2000, the Clay Mathematics Institute selected seven problems that had survived generations of work. It assigned one million dollars to each solution.
- P versus NP asks whether every problem whose answer is easy to check is also easy to solve.
- The Riemann Hypothesis concerns the hidden structure behind how prime numbers are distributed.
- The Hodge Conjecture asks whether certain topological features of complex algebraic shapes come from algebraic subspaces.
- The Birch and Swinnerton-Dyer Conjecture connects the behavior of elliptic curves modulo primes to their rational solutions.
- Yang-Mills and the mass gap asks for a rigorous mathematical foundation for a central part of quantum physics.
- Navier-Stokes existence and smoothness asks whether smooth three-dimensional fluid flow can develop a singularity.
- The Poincaré Conjecture asked how to recognize when a three-dimensional space is really a three-dimensional sphere.
Only the last one has been solved. Grigori Perelman released his proof in 2002 and 2003. The other six remain open.
Seven Millennium Prize Problems. Only Poincaré has been solved; Navier-Stokes is highlighted.
There had already been signs that AI systems were becoming useful on open mathematics problems. In May, an internal OpenAI model disproved a conjecture connected to Paul Erdős’s unit-distance problem. It found an infinite family of geometric constructions that beat a bound mathematicians had treated as essentially optimal since 1946. External mathematicians checked the result.
An internal version of Claude later improved a bound related to the Riemann Hypothesis. Claude did not solve 67% of the hypothesis, despite how that number was repeated online. It increased the proven proportion of non-trivial zeta zeros that are both simple and on the critical line from roughly 41.6% to more than 67.25%. Alpöge and Ralph Furman verified the result, and a human-written proof has since reproduced it.
The Navier-Stokes claim is the first of these results aimed directly at one of the six remaining Millennium problems.
What Navier-Stokes asks
For a fluid with constant density, the incompressible Navier-Stokes equation can be written as:
The incompressible Navier-Stokes equations: how fluid velocity changes, with no local accumulation of fluid.
The letter u represents the velocity of the fluid at each position and time. The other quantities describe pressure, density, viscosity, and any external force acting on the fluid.
The first term tracks how velocity changes with time. The next term describes fluid carrying its own velocity as it moves. A fast current entering a slower region changes the motion there, which then changes the current again. This feedback makes the equation nonlinear and difficult to control.
Pressure redirects the flow. If fluid rushes inward from several directions, it cannot pile up at the center. Pressure changes across the fluid and pushes the motion elsewhere. This is part of how the incompressibility condition works. Whatever enters a small region must also leave it.
Viscosity smooths the motion. It spreads sharp differences in velocity into nearby regions and resists shear between fluid layers. Honey has high viscosity and water has less, but viscosity damps sharp velocity differences in both fluids.
The difficulty comes from these effects competing with each other. Fluid motion can fold, stretch, and concentrate. Viscosity tries to spread those concentrations out. In three dimensions, a vortex can also stretch, become thinner, and spin faster, much like a skater pulling their arms inward.
Nobody had proved whether viscosity must always stop that amplification from becoming infinite.
Fluid motion can concentrate while viscosity spreads differences in velocity.
A solution can go in either direction. Someone could prove that every smooth starting flow remains smooth forever. Or they could construct one valid flow that develops a singularity in finite time.
OpenAI’s paper takes the second route.
What OpenAI constructed
The 166-page proof starts the fluid at rest. A smooth external force creates a vortex around a vertical axis.
Fluid spirals toward the axis. Since it cannot accumulate there, it moves upward and downward along the axis. The core shrinks and becomes more slender as the singular time approaches, while the fluid inside spins faster.
Both the height and radius shrink, with the radius decreasing faster. Conceptual illustration, not a simulation.
The velocity eventually becomes unbounded. The total kinetic energy does not.
That is possible because the fastest motion occupies an increasingly tiny volume. The speed inside that region can keep rising while the region shrinks fast enough to keep the energy of the entire flow finite.
The force is the hard part. Anyone can invent a velocity field that blows up and then define the external force as whatever makes the equation balance. That does not resolve the Clay problem if the force itself becomes singular.
The official problem statement allows an external force in the versions OpenAI claims to resolve, but that force must remain smooth.
The collapsing vortex by itself leaves terms in the equation that grow without bound. OpenAI adds rapidly oscillating pulses around it. The pulses produce momentum fluxes chosen to cancel the unbounded part of that leftover error. Further corrections remove smaller errors that remain. The final sum gives the vortex its claimed blowup while the external force and all of its derivatives stay finite.
This is the main technical claim mathematicians now have to check.
How the proof was produced
OpenAI started training the internal model on August 28. On September 1, its researchers heard rumors that two Millennium Prize problems might have been resolved. They sent groups of agents after every open problem and several related questions.
Nearly one hundred agents first worked for about fifty hours on the Euler equations. Euler describes an ideal fluid without viscosity. The agents found an unforced Euler blowup, and OpenAI used that result to focus the larger search on Navier-Stokes.
The final effort used roughly 10,000 concurrent agents. They could read a cached copy of the internet, run code, work in groups, and receive useful results found by other groups. The system reached the Navier-Stokes construction about 88 hours after the first agents launched.
The Navier-Stokes run generated 2.7 million messages and around 130 billion output tokens.
GPT-6 Astra then spent another 17 hours converting the argument into Lean. OpenAI released the Lean formalization with the paper.
An Euler result guided the larger Navier-Stokes search. The proposed proof was then formalized in Lean.
Lean checks whether each encoded step follows from the definitions and previous steps. That makes the proof harder to dismiss as a plausible-looking chain with a hidden algebra mistake. It does not settle whether the encoded theorem matches every requirement of the Clay problem, or whether mathematicians will accept the construction after studying it.
Clay’s rules require publication in a qualifying outlet, at least two years of scrutiny, and general acceptance from the mathematics community. OpenAI has not submitted a prize claim.
The credit dispute
Tristan Buckmaster, a professor at NYU, and Levent Alpöge, a mathematician employed by Anthropic, spent most of the previous year working on related fluid equations. Their collaboration was personal rather than an Anthropic project, and they used Claude and Codex throughout the work.
They built on research by Diego Córdoba and Luis Martínez-Zoroa, who had developed a route to fluid blowups through external forcing. Buckmaster and Alpöge pushed that work to smooth forcing for incompressible porous media, Boussinesq, and three-dimensional Euler. They did not publish a solution to the full Navier-Stokes problem.
In Buckmaster’s statement, he says rumors about their progress reached OpenAI before its agent run began. OpenAI confirms that the rumors triggered the September 1 effort.
Buckmaster says the smooth-force approach raised a red flag because few researchers were pursuing it. He and Alpöge had also stored drafts and arguments in Codex sessions, so he asked whether OpenAI’s model had accessed those sessions or learned from them.
OpenAI says its researchers and agents did not see the pair’s work before publication and did not access specific user data to solve the problem. It says the proofs differ. The company also says it cannot rule out de-identified product usage having improved the model during training.
Buckmaster separately alleges that Sébastien Bubeck proposed a release where Buckmaster would present OpenAI’s result without Alpöge as an author because Alpöge works at Anthropic. Bubeck disputes that account. Buckmaster says that when he threatened to describe the conversations publicly, Bubeck asked, “Why would you ruin your career?” WIRED reported both accounts.
When research passes through shared AI tools, tracing each contribution becomes harder.
The public evidence does not establish that OpenAI used Buckmaster and Alpöge’s proof. Their concern points to a question researchers will have to answer as these tools become part of serious work. How should private drafts be handled when the company operating the research tool can also direct a stronger internal model at the same problem?
What I think happens next
The proposed solution may fail during review. A definition may not match the Clay statement, an assumption may turn out to be too strong, or mathematicians may find a flaw that Lean did not expose because it sits outside the encoded argument.
Even then, the way this proof was produced is difficult to ignore. OpenAI coordinated thousands of agents, used a result from one fluid equation to guide work on another, and generated a formal proof that experts can inspect.
When computing entered mathematics and science, it changed which problems humans could attempt. Numerical weather prediction, large simulations, and computer-assisted proofs grew out of machines doing calculations that no person could finish by hand. Mathematicians did not stop being useful. They learned to formulate problems for machines, verify numerical results, and recognize when the computation revealed a reusable idea.
This feels like another change of that kind, but the machine now participates in the search for the idea itself.
That will change the work around the proof. When ten thousand agents can attack a problem at once, choosing the right question becomes a larger part of the job. Proofs may arrive faster than people can check them, making verification a bottleneck. Mathematicians will also have to turn correct results into explanations whose mechanisms can be understood and reused.
Credit and provenance will become part of that work too. Researchers will need clearer records of what they supplied, what a model produced, and which private material could have influenced the result. The Buckmaster and Alpöge dispute is an early example of a problem that will recur unless research tools make those boundaries visible.
I have no idea what a mathematician trained inside that world will be capable of twenty years from now. I am fairly sure their skill set will look very different from the one we reward today.
That is the part I cannot stop thinking about.
Sources and further reading
- OpenAI, On the Navier-Stokes Millennium Prize Problem
- OpenAI, Finite Time Blowup for Navier-Stokes
- OpenAI, Lean certificates for the Navier-Stokes and Euler results
- Clay Mathematics Institute, The Millennium Prize Problems
- Clay Mathematics Institute, Official Navier-Stokes problem statement
- Clay Mathematics Institute, Rules for the Millennium Prize Problems
- Tristan Buckmaster, Statement on the concurrent fluid-dynamics work and OpenAI
- OpenAI, An OpenAI model has disproved a central conjecture in discrete geometry
- Youness Lamzouri, A new proof that more than two thirds of the zeta zeros are simple and on the critical line
- WIRED, OpenAI Just Claimed a Huge Math Discovery. Some Academics Are Crying Foul