Tristan Buckmaster, a mathematician at New York University, had just spent months working on what he believed could be a life-defining mathematical proof. Then, during a Sept. 6 meeting with OpenAI employees, he claims that he was issued a stark warning: "Why would you ruin your career?"

That exchange, which Buckmaster described in a detailed public statement this week, sits at the heart of a ballooning controversy around a famous set of mathematical equations called Navier-Stokes.

OpenAI says its artificial intelligence (AI) model has solved an outstanding problem tied to this 200-year-old set of fluid dynamics equations. The set of equations appears on the list of one of the seven Millennium Prize Problems posed by the Clay Mathematics Institute, and its solution comes with a $1 million reward. Buckmaster had been working on the same problem. And he has questions about how the company got there so fast.

The OpenAI breakthrough

The Navier-Stokes equations were formulated in the early 1800s by Claude-Louis Navier and George Gabriel Stokes. They're basically Newton's second law of motion for fluids, describing how any fluid moves: Push air with a fan blade, and the equations predict how that air will flow; or heat the atmosphere with sunlight, and they help forecast tomorrow's weather. They describe how planes fly and how blood moves through your arteries.

The problem is that nobody has proved whether the equations always produce finite answers. Could there be rare conditions that make the equations go haywire? This is called a "blowup," or singularity, which happens when a quantity that should represent something physical (like fluid speed or pressure) spirals toward impossible infinities at certain points in space and time.

Imagine the swirl of water circling a bathtub drain. The Navier-Stokes equations describe how it spins faster as it tightens toward the center. A blowup would mean the math predicts that spin is accelerating forever ‪—‬ an infinite speed at a single point. The bathtub can't do that, but if the equations can, that means the model we trusted for years is wrong and we wouldn't even know where it breaks.

Whether the Navier-Stokes equations can blow up is one of the most famous unsolved questions.

To find the answer, OpenAI used a technique called "forcing" ‪—‬ giving the simulated fluid a smooth, controlled nudge from the outside. Imagine flicking a spinning top with your finger to see if it would fall over ‪—‬ or, in the case of the Navier-Stokes equations, blow up.

OpenAI set roughly 10,000 AI agents to simultaneously work on the problem. The agents worked for about 88 hours straight, with 100 of them starting with the related Euler equations. This is a simpler cousin of Navier-Stokes that discards viscosity to describe fluid motion without internal friction.

After roughly 50 hours, OpenAI's agents found blowups in the Euler equations, OpenAI representatives wrote in a statement. This discovery convinced them that Navier-Stokes was within reach, and they turned their attention to the bigger problem.

OpenAI described the singularity as a vortex spiraling tighter and tighter, stretching and thinning like pulled taffy. The result was a fluid that, mathematically, spins itself into infinity.

OpenAI ran the proof through a program called Lean, software that checks every step of a mathematical argument. The agents "sent 2.7 million messages and used approximately 130 billion output tokens," OpenAI representatives wrote in the statement. At a Sept. 8 news conference, company representatives said a similar effort performed by a customer would have cost $15 million.

The controversy

The result has created a blowup of its own within mathematics, centered around claims that OpenAI's models may have been given a head start by using the work of Tristan Buckmaster and Levent Alpöge, a mathematician at rival AI company Anthropic.

The two mathematicians had been working on the same strategy (smooth forcing) applied to those same Euler equations. The approach itself was developed by Diego Córdoba, a mathematician at the Institute of Mathematical Sciences in Madrid, and Luis Martínez-Zoroa, another mathematician at CUNEF University in Madrid. Almost no one else in the field was using it.

Buckmaster said that on Aug. 15 that he and Alpöge achieved blowup results for the Euler equations with smooth forcing. On Sept. 3, Buckmaster wrote that he received word through Alpöge that OpenAI had caught wind of their progress and adopted the same method.

"When I heard 'forced,'" Buckmaster wrote in his public statement, "it was a bright red flag."

His concern was that the smooth-forcing approach wasn't obvious or widespread. "The ideas making this line of attack possible are due to Córdoba and Martínez-Zoroa," Buckmaster wrote, crediting the two mathematicians who developed the technique. "Almost nobody else I know of was working on it."

And yet, according to OpenAI, the company launched its Navier-Stokes effort on Sept. 1 ‪—‬ the same day it heard "rumors of other Millennium Prize resolutions." Buckmaster claimed, in his public statement, that those rumors were about his and Alpöge's work specifically.

Sam Altman, OpenAI's CEO, acknowledged that the company had learned of Buckmaster and Alpöge's work before making its announcement. "We were curious if ours could do it too," he wrote on X. He added that "the approaches appear to be different" — a claim Buckmaster disputes.

OpenAI's statement names both Buckmaster and Alpöge, credits their independent resolution of the forced Euler problem, and says the company offered to co-publish.

"I did not get an answer"

Buckmaster also revealed that he had been using OpenAI's Codex coding assistant throughout his research, putting drafts of the work into the tool. OpenAI trains its models on user interactions, and Buckmaster said he asked the company directly whether his Codex sessions had been accessed.

"I asked whether the model had been trained on, or had access to, our sessions in Codex, into which we had been putting all our drafts," he wrote. "I asked again, about training, and I did not get an answer."

OpenAI's enterprise agreements govern how customer data can be used. But it's unclear whether Buckmaster was using Codex under an enterprise agreement or as an individual consumer — a distinction that may determine whether his drafts were available for training.

In a statement on X, OpenAI representatives denied that company employees or agents saw the pair's work "by any means" until it was publicly released. However, "we cannot rule out that de-identified data derived from their usage of our products helped improve our models," they added.

Then, there is the question of authorship. Buckmaster alleges that Sébastien Bubeck, who leads OpenAI's math research, twice pushed for Alpöge to be cut from the paper that the two mathematicians were preparing. Buckmaster emphasized that Alpöge works at Anthropic, a rival AI company.

It was at that same Sept. 6 meeting that Buckmaster claimed that OpenAI presented him with two options, both of which required cutting Alpöge from authorship. He declined and said he told Bubeck that if OpenAI released its result as proposed, he would go public. Bubeck allegedly replied, "If you don't want me to be nice, then I don't have to be nice."

Bubeck has flatly denied the allegations, calling them "false and inflammatory" in a statement on X. "We did not use their prompt or proofs to prompt our models," Bubeck told Wired. Mark Chen, OpenAI's chief research officer, said at last night's news conference that he was "a little disappointed with the allegations."

"I have not seen OpenAI's proof," Buckmaster wrote. "I do not know what their model did, or how. I do not know whether our data was used. I am not accusing anyone of anything." He has called for an independent investigation into how OpenAI's work began.

In a closing line, he wrote, "I would much rather be talking about mathematics, Luis and Diego's ideas, and what this all means for the rest of us."

The community response

Córdoba, one of the original developers of the forcing technique, said he was stunned by how quickly the result had materialized. "We're a little bit in shock," he told Scientific American.

Other mathematicians have also responded to the news, with some expressing open dismay at how the results were released.

"We have now seen that even the rumor of someone working on a massive problem can trigger a massive amount of AI-powered effort to flatten it before the original research project has time to reach its full potential," Terence Tao, a professor of mathematics at UCLA, wrote on Mastodon. "The incentives may now be pointing in the direction of no longer sharing any promising research directions with the broader community, which would reverse centuries of traditions of open science and do serious long-term damage to the future of the field."

OpenAI's proof still awaits formal peer review, and the Clay Mathematics Institute has not yet issued an official verdict confirming or denying OpenAI's proof. But the episode has already surfaced troubling questions for the mathematics community. For example, when a company trains its models on the interactions of millions of users, who really owns the answers that emerge? And what does AI's involvement mean for the future of the field?

Those questions, unlike the Navier-Stokes problem, may not have a clean proof.

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Over dinner at a gaming convention around 2010, board game designer James Ernest turned to his friend Eric Harshbarger and asked a question: Could you design a set of dice so that each player in a group — whether two people or any number — could grab one, roll and have a perfectly equal shot at going first? The rule is simple: Roll your dice, and whoever lands the highest number goes first. The hard part is making that outcome truly fair for every player, no matter how many are at the table.

There could be no ties and no rerolls.

Harshbarger, a mathematician at Auburn University in Alabama, didn't have an answer that night. But the question stuck. Over the next 15 years, he and a loose network of collaborators casually worked to crack what became known as the "go first dice" problem.

Now, they finally have the answer: a set of five 60-sided dice, collectively engraved with every number between 1 and 300, with no repeats. To mark the achievement, Harshbarger built five giant, wooden replicas of these hexecontahedrons (60-sided 3D shapes), each carved from a different type of wood. They are now on permanent display in Auburn's new mathematics building.

The problem with dice

To understand why it was so hard to crack the problem, it helps to understand the simple part.

"The easy thing is to avoid ties; you just put different numbers on all the dice," Harshbarger told Live Science. "The problem comes in how you distribute those different numbers across the dice so that the probability is equal not only for the whole set but for any subset."

That second condition is what makes the problem devilish. Ernest's request wasn't just that eight players could roll fairly; it was that any subset of players could grab any dice from the bag roll and still have equal odds. If five people are playing, each takes a die. Three people? Same thing. The numbers on each die had to be arranged so that the fairness held no matter how many people showed up to play.

Harshbarger looped in his childhood friend Robert Ford, a mathematician at Dalton State College. Within weeks, they had a three-player solution: numbers 1 through 18, spread across three standard six-sided dice in exactly the right arrangement. Four 12-sided dice, Ford later worked out entirely by hand, could accommodate four players fairly.

By 2012, Harshbarger was giving talks about the four-player set at math conferences, and The Guardian covered the story. His inbox erupted. He started selling handmade sets from home — buying blank, 12-sided dice; etching numbers onto each face with a laser cutter in his workshop; and then inking every number by hand. At his peak, he was dropping off 30 envelopes at the post office at a time, several times a week, shipping sets to customers around the world.

But as the team dug deeper into the math, they discovered that the dice were doing something even more remarkable than they'd realized: The configurations determined not only who went first but also the entire turn order, with every possible sequence of players equally likely to come up.

"If four people rolled, the chance that players finished in the order A, B, C, D was just as likely as C, B, D, A, or any of those combinations," Harshbarger said. "'Go first dice' is actually a bit of a misnomer, because they do much more than just determine who goes first; they actually determine the ordering of all the players."

This property, which the team calls permutation fairness, became the standard they chased for every set of dice from that point on.

More combinations than atoms in the universe

Four players was one thing. Five was another entirely.

Mathematically, the team knew a five-player same-shape set was feasible. But finding it meant searching an incomprehensibly large space of possible number arrangements.

"We're talking literally more than the number of atoms in the universe ‪—‬ like 10 to the 128th power combinations," Harshbarger said. "There's just no way, even if we had a billion billion years and all the computers, all the AI. We still couldn't do it today."

They couldn't brute-force their way to an answer. They needed mathematical shortcuts: symmetries and patterns to shrink the search space. But even then, years passed without a practical solution. Every path they found hit the same wall: dice too large to hold, too many sides to manufacture.

"My goal was a set for five players that can actually be manufactured, that you can hold in your hand, that board gamers could buy and roll," Harshbarger said. "Even for a problem that is silly and pointless as far as gameplay — just mathematically, can you buy these things and roll them, and they work? You can't do that with 180-sided dice."

Then, in mid-2023, Canadian software engineer Paul Meyer emailed Harshbarger out of the blue. He had been studying patterns in Harshbarger's four-player data and had written a program to exploit them. He hadn't expected it to work right away.

It did. Meyer found a configuration of five 60-sided dice that satisfied every condition.

"I double-checked his work and went, 'Oh my goodness; we've been searching for so long for this. This is amazing,'" Harshbarger said.

From workshop to gallery wall

The four-player set had long since been picked up by retailers — Maths Gear in the U.K. and Math Art Fun in the U.S. — so Harshbarger had stopped hand-making those years earlier. Now, the five-player version was real and small enough to manufacture.

But the story had one more turn. Auburn University was building a new home for its mathematics department and was looking for sculptural ideas to fill it. Harshbarger proposed building giant versions of the five new dice out of wood. The department agreed.

He spent months in his wood shop, constructing five oversize dice. Each is from a different type of wood: pine, poplar, oak, walnut or mahogany. The five sculptures now stand in Auburn's new math building, which opened this fall.

For Harshbarger, the whole point is making math impossible to walk past.

"When people see giant dice or little dice, they're fascinated just by the geometry," he said. "The hope is, they see these things and go, 'Oh, this is math too, and this is interesting.' Some of the most fun math problems are ones that are easily understood and not easily answered."