Did AI Solve One of Math’s Hardest Problems?

StarTalk StarTalk • Oct 01, 2026

Audio Brief

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Did OpenAI's artificial intelligence really solve a million dollar math puzzle that has stumped humans for roughly ninety years, and what happens when a computer proof is too massive for any human to read? On StarTalk, astrophysicist Neil deGrasse Tyson and his American Museum of Natural History colleague Mordecai Mark Mac Low examine OpenAI's announcement that it solved the Navier Stokes existence and smoothness problem, one of seven million dollar Millennium Prize Problems established by the Clay Mathematics Institute in the year two thousand. To see what OpenAI tackled, look at the three nineteenth century equations that track every fluid in the universe. A true fluid is any continuous material that flows without solid forces holding it together, like water, air, or the turbulent astrophysical gas forming stars and planets. Written by Claude Louis Navier and Sir George Stokes, these three partial differential equations track velocity, density, and energy or pressure over time across all fluids simply by changing the parameters. Tracking those fluids reveals a surprising engineering trick: you can fight big, slowing turbulence by creating smaller turbulence on purpose. The dimples on golf balls are designed to generate mini turbulence right at their surfaces, allowing the surrounding air to flow smoothly around the object and helping golf balls travel farther. While engineers use turbulence on the golf course, mathematicians have spent roughly ninety years asking if the equations behind it can mathematically blow up. The Navier Stokes Millennium Prize problem asks whether three dimensional fluid equations are always stable and smooth, or if they can hit a singularity where predicted fluid speeds reach infinity. By running roughly ten thousand AI agents, OpenAI claimed to disprove this smoothness hypothesis, finding a mathematical counterexample strictly within the Navier Stokes equations that runs off to infinity. Before you worry about the ocean hitting infinite speed, remember that a broken equation just exposes the limits of a math shortcut. In the eighteen seventies, Ludwig Boltzmann wrote an equation tracking gases as discrete, colliding Newtonian atoms and molecules rather than a continuous fluid. Forty years later, when Chapman and Enskog derived the Navier Stokes equations from Boltzmann's work, they had to assume regional changes, known as gradients, were small and smooth. When extreme changes occur, Navier Stokes violates its own underlying assumptions and breaks down, whereas the Boltzmann equation handles the physics by tracking the actual particles. So the math can blow up while real fluids stay intact, which brings us to the catch in OpenAI's announcement: a proof that a computer can verify, but no human can understand. OpenAI demonstrated its result using Lean, a computer based mathematical verification language, and while Mac Low hedges that the result is more likely true than not, assuming Lean has no reliability issues, the output is twenty six thousand sub theorems that no human can read or build upon. Meanwhile, mathematicians Tristan Buckmaster at New York University and Levent Alpöge at Anthropic, who Mac Low notes seem to almost be there, or maybe are there, stated they were already translating a solution out of AI slop into a human readable format after using OpenAI tools. When a physics equation predicts an impossibility like infinite speed, do not assume physical reality is broken; check whether extreme conditions violated the simplifying assumptions built into the model. And when using artificial intelligence in science, translate raw computer outputs into steps humans can actually build upon, while verifying literature summaries yourself. As Mac Low notes from his own astrophysics research, even AI tools that guarantee real source papers do not guarantee correct interpretations and will still hallucinate details like who authored the papers.

Episode Overview

  • Astrophysicist Neil deGrasse Tyson and American Museum of Natural History colleague Mordecai-Mark Mac Low examine OpenAI's announcement that its artificial intelligence solved the Navier-Stokes existence and smoothness problem, one of the seven $1,000,000 Millennium Prize Problems established by the Clay Mathematics Institute in 2000.
  • The discussion begins with the fundamentals of fluid dynamics, explaining how the 19th-century Navier-Stokes equations govern continuous flowing matter—from Earth's weather and oceans to turbulent astrophysical gas forming stars, galaxies, and planets.
  • The episode explores the mechanics and controversies of the AI proof, which required $6,000,000 worth of GPU compute across roughly 10,000 AI agents for 88 hours to generate 26,000 sub-theorems in the Lean verification language, alongside claims that OpenAI may have scooped two human mathematicians using its tools.
  • Finally, the hosts contextualize what a mathematical singularity actually means for physical reality, contrasting the continuous approximations of Navier-Stokes with Ludwig Boltzmann's particle-based equations, and evaluating the practical limits of unreadable AI proofs in scientific research. (At 20:40, Tyson also promotes his latest book, Lost in Space, containing 5,000 facts, co-authored with StarTalk Senior Producer Lindsey Nyx Walker in collaboration with StarTalk and National Geographic Books).

Key Concepts

  • Fluids and the Navier-Stokes Equations: A fluid is any continuous material that can flow without solid forces holding it together, such as water, air, or astrophysical gas (whereas poured salt takes the shape of its container but is not continuous). Written in the 19th century by Claude-Louis Navier and Sir George Stokes, the Navier-Stokes equations are three partial differential equations tracking fluid velocity, density, and energy/pressure over time across all fluids simply by changing parameters.
  • Fighting Turbulence with Turbulence: Fluid dynamics reveals counterintuitive engineering solutions where small, intentional turbulence prevents larger, drag-inducing air swirls. For example, aircraft wings and the dimples on golf balls generate mini-turbulence right at the surface so surrounding air flows smoothly around the object, allowing airplanes to fly more efficiently and golf balls to travel farther.
  • The Smoothness Problem and Mathematical Singularities: Unresolved for roughly 90 years, the Navier-Stokes Millennium Prize problem asks whether 3D fluid equations are always stable and "smooth," or if they can "blow up" into a singularity where predicted fluid speeds reach infinity. OpenAI claimed to disprove the smoothness hypothesis by finding a mathematical counterexample strictly within the Navier-Stokes equations that runs off to infinity.
  • Why Physical Fluids Don't Hit Infinite Speed (Boltzmann vs. Navier-Stokes): A mathematical blow-up in the Navier-Stokes equations does not mean real physical fluids can move at infinite speeds; rather, it exposes the limits of an approximation, much like Newton's laws are approximations contained within relativity and quantum mechanics. In the 1870s, Ludwig Boltzmann wrote an equation tracking gases as discrete, colliding Newtonian atoms and molecules rather than a continuous fluid. When Chapman and Enskog derived the Navier-Stokes equations from the Boltzmann equation 40 years later, they had to assume regional changes were small and linearized on the gradients. When extreme gradients occur, Navier-Stokes violates its own underlying assumptions and breaks down, whereas the Boltzmann equation accurately handles the physics by tracking the actual particles.
  • Verification vs. Comprehension in AI Proofs: OpenAI demonstrated its proof using Lean (a computer-based mathematical verification language Mac Low notes is being developed by someone he thinks is at MIT). While Mac Low hedges that the result is "more likely true than not"—assuming Lean has no reliability issues—the output consists of 26,000 sub-theorems that no human can read or build upon. Furthermore, mathematicians Tristan Buckmaster (NYU) and Levent Alpöge (Anthropic), who Mac Low notes "seem to almost be there, maybe are there," stated they were already translating a solution out of "AI slop" into a human-readable format after using OpenAI tools. When they asked if OpenAI's model trained on their inputs, OpenAI hedged, "We don't think so," though Mac Low notes the dual discoveries likely represent independent verification.

Quotes

  • At 7:50 - "So the problem was, are the Navier-Stokes equations always smooth? And if you can find a counterexample where, no, it's not smooth, it goes off to infinity, then you're done. You have shown that that hypothesis is false." - Mac Low clarifying the exact mathematical goal of the $1,000,000 Millennium Prize problem and how finding a single singularity resolves the 90-year-old question.
  • At 13:42 - "And Boltzmann handles that, because it's got the particles, it knows what's going on. But Navier-Stokes, you've put in assumptions that there aren't big, nasty gradients, and you violated them, and sure enough, it's not smooth." - Mac Low explaining why the AI's mathematical singularity reflects a breakdown in linear approximations rather than a breakdown in real-world physics.
  • At 17:21 - "The problem with the OpenAI solution is that wasn't unpacked. So they, as you say, they produced the solution, they provided this immense package of Lean statements, but nobody can read that. Nobody can build on it." - Mac Low contrasting raw, machine-verified AI outputs with human-readable mathematics that advances scientific understanding.

Takeaways

  • Look for broken approximations when mathematical models fail: When a physics equation predicts an impossibility like infinite speed, do not assume physical reality is broken; instead, check whether extreme conditions have violated the simplifying assumptions (such as treating discrete particles as a continuous medium) built into the model.
  • Prioritize human-readable scientific proofs over raw compute outputs: When using AI to solve complex problems, follow the approach of translating results into human-understandable steps rather than leaving them as massive, opaque code packages, ensuring other researchers can actually understand and build upon the discovery.
  • Verify AI literature interpretations and citations manually: When using AI tools to explore scientific literature, do not rely on the model's summaries alone; as Mac Low notes from his own astrophysics research, even tools that guarantee real source papers do not guarantee correct interpretations and will still hallucinate details like who authored the papers.