On 8 September 2026, OpenAI officially announced it had determined the Navier-Stokes differential equations for describing the motion of fluids can develop a "singularity", or "blow up" or "break" to use more expressive terms, while running in a finite period of time.

Here's how they described the open question about the math involved in the Navier-Stokes equations:

The Navier–Stokes equations use Newton’s second law of motion (“F=ma”) to describe how fluids move. Importantly, they treat a fluid as a continuous medium rather than tracking individual molecules. These equations are used for aircraft design, weather forecasting, and the study of blood flow.

A fundamental open question for these dynamical equations has been whether the continuum approximation of the fluid can break down. Specifically, can the Navier–Stokes equations for a three-dimensional incompressible fluid with constant density develop a “singularity,” even when the motion starts smoothly? Here, a singularity means the dynamics lead to speeds in the fluid growing without bound within a finite amount of time. The development of a singularity would have to happen despite the presence of viscosity, which tends to smooth out motion. Because a real fluid cannot move infinitely fast, this would mark a breakdown in how the equations model the fluid. To continue modeling the system, one would then need to track the behaviour of each particle individually.

The equations date to the nineteenth-century work of Claude-Louis Navier and George Gabriel Stokes. In 1934, Jean Leray proved that solutions exist in a generalized sense, but whether they always remain smooth became a central unanswered question. In 2000, the Clay Mathematics Institute named the Navier–Stokes existence and smoothness problem one of seven Millennium Prize Problems.

And here's their result:

Our system produced an analytical proof and a Lean formalization that an initially smooth fluid at rest can develop a singularity in a finite time. The fluid has a smooth force applied to it, and its energy remains finite through the entire dynamics, from rest to the formation of the singularity. This resolves the Navier–Stokes Millennium Prize problem by establishing statement “C” (and also “D”) in the official Millennium Prize formulation⁠.

The solution is a vortex, a spinning swirl of fluid, that spirals inward and gets increasingly elongated, like spaghetti. This central region shrinks while it speeds up in such a way that its energy still stays finite, as required by the laws of physics. The technical challenge is for the equations to develop the breakdown through the motion of the fluid itself, rather than, for example, us putting in an infinite force by hand. More mathematically, the terms in the Navier–Stokes equations that describe the motion—acceleration, pressure gradients, momentum transfer, viscosity—must both become big yet cancel in a precise way. This detailed balance leaves a smooth external force even as the velocity of the fluid grows without bound.

Assuming it holds, OpenAI's solution to the Navier-Stokes Millennium Prize challenge, for which it might win $1 million, will have cost the firm $20 million.

The Clay Mathematics Institute, which established the "Millennium Prize" challenges for solving seven long-standing open problems in mathematics back in 2000, issued a statement indicating the million dollar prize for cracking the Navier-Stokes millennium problem is now pending their official verification.

Controversy soon erupted because of questions about whose mathematical research OpenAI's still-internal AI system used to develop their proof the Navier-Stokes equations will not always work and that they can break down under specific circumstances. The following Code Report from Fireship provides an entertaining rundown of the conflict:

Terry Tao's blog post announcing the accomplishment and assigning credit as best as could be done on 7 September 2026 is here.

The controversy over credit has the potential to arrest the rapid progress in mathematics that is being boosted by AI technologies. That potential exists because of the AI systems' ability to rapidly absorb new developments by mathematicians and apply them to a multitude of other problems without really understanding the work. In doing so, the AI developers are creating a strong, adverse incentive for the mathematicians to hold back their work until it has been fully verified to ensure they retain credit for it.

On 11 September 2026, Tao signed onto a declaration with other Fields Medal-winners (the math equivalent of the Nobel prize for science disciplines) who are alarmed at the major disconnect they're seeing between mathematicians and AI developers, which they argue puts advances in mathematics at risk.

That's a shame because the controversy is obscuring one of the biggest math stories of the year, which points to the massive capabilties that AI technologies are unlocking to advance mathematics. We'll cover that part of the story in Part 2.

References

Tristan Buckmaster. Announcement of three results with Levent Alpöge on finite-time blowup with smooth forcing for incompressible porous media, for Boussinesq, and for 3d incompressible Euler. [Mastodon post]. 7 September 2026.

Terence Tao. Finite time blowup with smooth forcing term for the incompressible porous medium, Boussinesq, and incompressible Euler equations. [Online article]. 7 September 2026.

OpenAI. On the Navier-Stokes Millennium Prize Problem. [Online article]. 8 September 2026.

Complexity. Visualizing the OpenAI solution to the Navier Stokes Equation. [Online video]. 10 September 2026.

Clay Mathematics Institute. Navier-Stokes Announcement. [Online article]. 11 September 2026.

Math and AI. A Severe Misalignment of AI in Mathematics. [Online article]. 11 September 2026.

Previously on Political Calculations

We've been covering developments toward the resolving the open question of when the Navier-Stokes equations describing fluid motion works and when it doesn't for some time. Here's our coverage in chronological order, the articles marked with an asterisk are directly relevant to the resolution of the Clay Mathematical Institute's Navier-Stokes Millennium Problem:

• The Biggest Math Story of 2017*
• Navier-Stokes and the Emergence of Order from Disorder
• The Biggest Math Story of 2019
• A Proof for Batchelor's Law
• The Biggest Math Story of 2022*
• Breakthrough in Unifying Math to Describe How Fluids Behave at Different Scales
• The Biggest Math Story of 2025
• AI Cracks the Navier-Stokes Millennium Problem, Part 1*