Navier-Stokes Proof: A Year Versus a Sprint

A Hundred-Page Proof Claim (0:00)

Hey, chibis! I'm Aiwee, and today we're talking about the AI proof a mathematician says stole his life's work. If you enjoy stories like this, hit the like button and subscribe if you haven't already — let's go! In the first week of September, an artificial intelligence system reportedly produced a roughly one hundred page proof of a fluid dynamics question that had stumped the world's best mathematicians for nearly a century. The equations had been written down in the eighteen hundreds.

In the year two thousand, a one million dollar prize had been attached to settling them. The mathematician who had spent a year quietly pushing the same rare approach says he did not learn of the run from a press release. He learned it from a rumor. And he says the company that ran the agents then offered him a deal he could not accept. To understand why this story matters, you have to understand the prize it is tangled up in.

In the year two thousand, the Clay Mathematics Institute named seven problems it considered the grandest unsolved questions in mathematics, and placed a one million dollar bounty on each. Twenty six years later, exactly one had fallen. A Russian mathematician named Grigori Perelman proved the Poincaré conjecture, and then refused both the money and the Fields Medal, mathematics' highest honor. The other six remained open. One of them is called the Navier-Stokes problem.

The Navier-Stokes equations were first written down in the eighteen hundreds. They describe how fluids like water and air move, and engineers use them to design aircraft wings and to forecast the weather. The Millennium Prize formulation presents a question that seems straightforward on the surface. Take a smooth, well-behaved fluid, and the puzzle is whether the equations stay orderly forever or allow a breakdown, with velocity and energy racing to infinity in finite time. Mathematicians call that a blow up.

Consider what takes place with the equations of physics at the core of a black hole. The whole story runs through the route that proves the equations can blow up. The route came from two Spanish mathematicians, Diego Córdoba and Luis Martínez-Zorita. A few years ago, they developed a specific strategy, a sequence of carefully timed kicks applied to a fluid, intended to push it toward a blow up. Almost no one else in the field was working in that direction.

It was, in the words of one commentator, a rare secret in the mathematics world. The protagonist of our story is a New York University mathematician named Tristan Buckmaster. He and a collaborator, Levent Alpoge, who works at Anthropic, picked up the Spanish approach and extended it. Buckmaster described the collaboration as purely personal, with no institutional agreements. The pair would set up a problem, Alpoge would feed it into a model, and refine the output, again and again, for most of a year.

They used several large language models, including Anthropic's Claude and OpenAI's Codex. And crucially, the full draft sessions of their OpenAI Codex chats were stored on OpenAI's servers. For most of that year, the progress was slow. Then on August fifteenth, something changed. The pair noticed a step change in the output.

By August twenty second, their work had been formally verified. To be clear, this was not a complete solution to the Millennium problem. It was a verified step along a road that might lead to one. Then, on Thursday, September third, a rumor circulated. People were saying that Anthropic had resolved a major open problem.

Buckmaster wrote directly to a prominent OpenAI mathematician, Sebastian Bubeck, to clarify that the work was his and Alpoge's, not Anthropic's. What Buckmaster did not know, and what OpenAI would later confirm, was that the company had started its own sprint on the same problem on September first. The effort used a multi-agent system of ten thousand agents working in parallel, powered by an internal model described as significantly more capable than OpenAI's public flagship. The system ran for roughly eighty eight hours, exchanged two point seven million messages, and consumed about one hundred and thirty billion output tokens. A seventeen hour pass was then used to formalize the result.

The output, a roughly one hundred page proof, was ready by Tuesday, September eighth, when OpenAI publicly announced it. The proof, if correct, shows that a smooth fluid at rest under a smooth force of finite energy can develop a singularity. The fluid spirals into a vortex, stretches, and the equations break. The math itself, if it holds, answers the Millennium question. Yes, a smooth Navier-Stokes fluid can blow up.

But the question that has consumed the field since is who deserves the credit for finding the route. Buckmaster has laid out his case in a lengthy public statement. He rests it on two pillars.

The Millennium Prize and Seven Problems (5:30)

The first pillar is timing. OpenAI only began work on the problem in early September, after the rumor of an Anthropic breakthrough reached the company. The same rare route that Buckmaster and Alpoge had been quietly developing for a year was the route OpenAI's agents followed, and they followed it in under a week. The second pillar is data. Buckmaster asked whether the internal model had been trained on, or had access to, the Codex sessions in which the pair had been putting all of their drafts.

He was told the model does not look up user data. He asked again, specifically about training. He did not get an answer. Then, Buckmaster alleges, the conversation took a darker turn. He says OpenAI offered to let the pair publish first and claim the one million dollar prize, on one condition.

Alpoge had to be removed from the credits, because he works at Anthropic. Buckmaster refused. According to his account, he was then asked why he would want to ruin his career. He replied that he was an academic and could not see how going public would ruin his career. The next reply, he says, was a veiled threat.

If you do not want me to be nice, then I do not have to be nice. Near the end of his statement, he wrote, quote, I am not accusing anyone of anything. End quote. OpenAI's public position has shifted. The initial statement said neither researchers nor agents had seen Buckmaster's and Alpoge's work until it went public, but hedged that, while unlikely, the company could not rule out deidentified data derived from product usage improving their models.

By Wednesday evening, after press questions, the position had tightened. It was now, in the company's words, categorically impossible for the Codex prompts to have influenced the system. OpenAI cited a training data cutoff date of July third. Sam Altman posted that the team had acted with integrity and generosity throughout. Sebastian Bubeck has published a long text he sent to Alpoge, stating the pair would receive full credit and characterizing the company's posture as offers of help, not pressure.

Bubeck told The New York Times that having an Anthropic employee on an internal OpenAI project would look strange. The story reached Hacker News, Reddit, and X within hours. The dominant reaction was suspicion. A commenter quipped that anything not kept at gunpoint is destined to end up in training. The mathematician Terence Tao praised the Alpoge-Buckmaster work as remarkable, while warning about artificial intelligence companies treating problems as marketing events, likening it to using excavators to loot an archaeological site.

A few days later, Tao co-authored an open letter stating that artificial intelligence may solve problems, but risks losing the understanding and insight that come from the human process. The Clay Mathematics Institute called the events an exciting day and promised a deliberately unhurried and absolutely rigorous review. As of this recording, Navier-Stokes remains officially unsolved. Step back, and the question gets bigger than one proof. The question Buckmaster asked on the call, and did not get answered, was whether his work was used to train the model that beat him.

OpenAI's final answer is no, citing a cutoff date. But the original statement hung its hat on deidentified data. The coincidence of OpenAI choosing the exact same rare route, and the lead researcher's inability to answer in real time, point at something uncomfortable. If an artificial intelligence system can be shaped, even indirectly, by work its own users put in, then any researcher who uses a commercial system is competing against a possible future version of themselves. Researchers, lawyers, doctors, and engineers pour unpublished and sometimes sensitive work into artificial intelligence tools every day.

In most current commercial systems, that data can be used for training unless the user affirmatively opts out. As one commentator, Melissa Lee, has put it, researchers need confidence that unpublished work entered into a system will not feed that company's competing efforts. Once again, the legal framework is having to catch up with the technology. Buckmaster has said that mathematicians are now afraid to discuss open problems in public, for fear of being scooped by a company that treats them as benchmark tasks. So where does this leave the math.

If OpenAI's proof holds, the answer to the Navier-Stokes Millennium question is yes, the equations can blow up. That is a real mathematical result, and it extends work that began with the two Spanish mathematicians. The Clay Mathematics Institute's review is the next gate. We will now know exactly where the equations break down. But the wider question is not about a fluid.

It is about auditability, about whether users of commercial artificial intelligence systems can ever know, with confidence, whether their own inputs are part of what they are competing against.

The Blow-Up Route Explained (11:15)

Until that question has a clear answer, every new result produced by a frontier lab will arrive with the same shadow over it. The math, in other words, may be settled before the trust is. Buckmaster's full statement is in the description if you want to read it for yourself. We will be watching to see what the Clay Mathematics Institute says next.