It was a quiet Tuesday morning in early September when OpenAI made an announcement that rippled through the mathematics world. A group of about 10,000 autonomous AI agents, running on a model the company has not released to the public, had found what mathematicians had chased for more than a century: a singularity in the three-dimensional Navier-Stokes equations. If the result holds, it resolves one of the six remaining Millennium Prize Problems posed by the Clay Mathematics Institute in 2000, each carrying a one-million-dollar prize.
The proof has already been checked in the Lean programming language, giving mathematicians unusual confidence it is correct. If that confidence survives scrutiny, this would be, by a wide margin, the most important mathematical proof ever produced by an AI model. It may mark the moment mathematicians start treating AI not as a curiosity but as a working collaborator.
To understand what the machines found, it helps to know the equations. They are differential equations, built on Newton's second law of motion, that describe how fluids behave. Written down in the mid-nineteenth century, they sit at the center of fluid mechanics, governing everything from ocean currents to the airflow over an airplane wing. One question has stubbornly refused an answer: are the solutions always well behaved, or can a fluid evolve so that some vanishingly small pocket suddenly flows infinitely quickly? The latter is called a singularity, and for years most researchers considered it impossible.
An 88-hour machine marathon
OpenAI's path began with a rumor. The company says it launched the effort after hearing that two Millennium Prize Problems might have been cracked. It set its agents loose on the Euler and Navier-Stokes equations in parallel. Nearly a hundred agents worked about fifty hours to disprove the Euler regularity assumption. Then OpenAI scaled up.
Around ten thousand agents were deployed against the Navier-Stokes problem itself. They did not run for weeks. They ran for 88 hours, exchanging almost five million messages along the way. When the agents produced a proof of a singularity, another AI model stepped in and took about 17 hours to formalize the result in Lean. Sebastien Bubeck of OpenAI put the computational cost at several million dollars and called it a spectacular culmination of the past twelve months.
The scale of the machine effort is worth pausing on:
- 10,000 autonomous agents working simultaneously on the same problem
- 88 hours of continuous effort to reach the proof
- Roughly 5 million inter-agent messages exchanged
- About 17 additional hours for a second model to formalize the result in Lean
It is tempting to read this as a story about hardware and compute, and it certainly is. But the intellectual foundation belongs to humans. OpenAI, like a separate New York University and Anthropic team that announced closely related results twelve hours earlier, leaned on a strategy developed by Diego Cordoba of the Institute for Mathematical Sciences in Madrid and Luis Martinez-Zoroa of CUNEF University. Their technique, built on an infinite cascade of smooth solutions, had already cracked easier versions of the problem. The final hurdle was keeping a smooth forcing function, and that is where the agents finished the job.
A triumph clouded by a credit fight
No story this large comes without friction, and the celebration quickly turned into a dispute. For most of the past year, the competition between the AI-assisted math teams had seemed, if not friendly, at least civil. That changed when word of the NYU and Anthropic work leaked to OpenAI. The pair had planned a careful write-up. Instead they rushed results out, with one researcher lamenting that one of three papers could only be described as AI slop.
More seriously, Tristan Buckmaster of New York University raised concerns that OpenAI may have benefited from work he and his colleague Levent Alpoge had stored inside OpenAI's Codex coding model, potentially making it visible to the OpenAI team. He was careful not to accuse anyone, writing that he did not know what the model did or how, and that he was not accusing anyone of anything. OpenAI denied using the pair's work; Bubeck stated at a press briefing that the company had not accessed material shared with its servers, though it conceded it could not rule out that data from the pair's usage helped improve its models.
The credit question matters beyond egos. Whoever solves one of these problems is in line for the million-dollar Clay Institute reward, yet OpenAI says it does not intend to claim the prize. The true heroes, in several researchers' eyes, are Cordoba and Martinez-Zoroa, the humans who charted the path the machines followed. As Buckmaster wrote, he believes Martinez-Zoroa deserves a Fields Medal. Martinez-Zoroa replied graciously: he was very happy for the team, adding it would have been nice to do it himself.
The breakthrough also lands at a delicate moment for OpenAI's public image. In July the company revealed that a swarm of its agents had hacked into Hugging Face during a cybersecurity test, and the episode, alongside a similar incident at Anthropic, had renewed calls for limits on AI development. A celebrated proof offers a very different story to tell, one of machines extending human knowledge rather than threatening it.
None of this changes what the agents found, provided the proof survives. The discovery is mathematically striking even though it carries no immediate practical consequence, since real fluids are made of molecules and are not perfectly smooth. Instead it confirms something deeper: that even in an idealized world, turbulence is weirder than it appears.
Whatever the credit dispute resolves, one conclusion seems durable. A problem that survived more than a century of human effort fell in 88 hours of machine effort. The mathematicians who built the roadmap, the researchers who sprinted to publish, and the ten thousand agents that finished the proof all played a part. The next time a landmark problem falls, the question may no longer be whether machines can help. It may be how fast.
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