OpenAI Claims a Million-Dollar Math Problem. The Mathematician Says They Wanted His Coauthor Gone
OpenAI says roughly 10,000 AI agents proved the Navier-Stokes equations can tear themselves apart -- a result the Clay Institute's own rules accept, despite a full day of claims otherwise. Then NYU's Tristan Buckmaster published a signed statement about two Sunday phone calls: two offers, a twice-repeated request to remove his Anthropic-employed coauthor, and the question 'Why would you ruin your career?' We read the actual prize rules, Buckmaster's full statement, and Terence Tao's day-early verdict on the mathematics -- and sort out what is proven, what is alleged, and who the ideas actually belong to.
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Cold open
Eris: The internet spent all day explaining why OpenAI's million-dollar math proof doesn't count. Almost nobody explaining it had read the rules.
Vestra: I read the rules. Charles Fefferman wrote them in the year two thousand, and they ask, explicitly, for a proof of any one of four statements. OpenAI is claiming two of them.
Eris: So the proof may well count -- and OpenAI is refusing the million dollars anyway.
Vestra: And that still is not the strangest thing that happened today.
Eris: Not even close. Because an NYU mathematician just published a signed statement about a Sunday phone call -- two offers he says OpenAI made him, and one question: "Why would you ruin your career?"
Vestra: One thing at a time. What did the machine actually prove?
The million-dollar proof
Eris: Here is the question the whole day hangs on: when OpenAI says a model "resolved" the Navier-Stokes problem -- one of the seven Millennium Prize problems, a million dollars apiece -- what did it actually prove, and does it count? Because the answer to both is nothing like what the internet decided today. Vestra, the equations first.
Vestra: Newton's law, written for water and air instead of billiard balls. They treat a fluid as one smooth continuous thing, and they sit underneath weather forecasting, aircraft design, the way blood moves through your heart.
Vestra: And for roughly ninety years there has been one embarrassing open question: can a fluid that starts perfectly calm and smooth work itself up to infinite speed in a finite amount of time? Not very fast. Infinite.
Eris: And if it can?
Vestra: Then at that instant the equations stop describing reality. The smooth picture tears itself apart, and you would have to go back to tracking individual molecules. Mathematicians call that blowup.
Eris: Quick prediction game, because I think most people have this backwards. When you hear "OpenAI solved Navier-Stokes," what do you assume they proved?
Vestra: If I answer as most people? That the equations are safe. Smooth flows stay smooth, singularities never form, the model found the missing argument for why fluids behave.
Eris: That is exactly what they did not prove. The claim is the opposite: there is a smooth, calm starting fluid and a smooth, gentle external push that drive the flow to blow up in finite time. The equations break. OpenAI's own description of the solution is a vortex that spirals inward and, quote, "gets increasingly elongated, like spaghetti."
Vestra: And the delicate part deserves one more beat, because it is the whole difficulty. You cannot cheat by shoving infinitely hard. The push from outside has to stay gentle the entire time. So acceleration, pressure, momentum transfer, viscosity -- every violent term in the equations has to run off to infinity together while cancelling almost perfectly, so that the only smooth thing left in the room is the force you applied. A controlled demolition where the detonator never even gets warm.
Eris: Which brings us to the take that ate the internet today: "doesn't count, because the real problem has no external force." I read the actual prize document this afternoon. Five pages, written by Charles Fefferman in the year two thousand, and it does something almost nobody quoting it seems to know about. It says, in his words, "to give reasonable leeway to solvers while retaining the heart of the problem, we ask for a proof of one of the following four statements."
Vestra: Statements A and B: smooth solutions always exist, no external force. Statements C and D: there exist a smooth starting flow and a smooth external force for which no smooth solution survives. Breakdown, with the force allowed.
Eris: OpenAI is claiming C and D. The forced route is not a loophole somebody found last week. It has been one of the four official doors for twenty-six years.
Vestra: So why is the prize unclaimed? Because OpenAI declined it. They said flat out they do not intend to claim the million dollars. Whatever you think of the company, the loudest criticism circulating today is aimed at a rule that does not exist.
Eris: Now the how, because this part changes what the words "research lab" mean. They had been training a new internal model since late August, one they describe as stronger than anything they have shipped.
Eris: On September first -- reportedly after hearing rumors that two Millennium problems had already fallen, and hold that thought -- they pointed on the order of ten thousand agents at every open Millennium problem at once. Split into groups that could talk among themselves, with a cached copy of the internet and the ability to run code.
Vestra: And they did not go straight at the target. About a hundred agents spent two days cracking a warm-up first: the same blowup question for the Euler equations, which is Navier-Stokes with the friction removed. Then everything got redirected at Navier-Stokes with that Euler result fed back in as a prompt, and Codex playing research manager -- reading every group's notes, pulling out the useful ideas, passing them to the next round. Eighty-eight hours to the proof. Seventeen more for the machine verification.
Eris: On this one problem, the agents exchanged nearly three million messages. A research department with no sleep and no ego, run at a burn rate no university on earth could survive.
Vestra: Here is the part I will actually credit them for, and you know I have been rough on their math claims before. They shipped the artifact. The written proof and the formalization are public, in Lean, a proof language where your laptop can verify every logical step -- permissively licensed, with a directory set up specifically for independent checking. That makes this the first Millennium Prize claim in history that a stranger with a computer can falsify.
Eris: You have a "but" loaded. I can hear it from here.
Vestra: Three of them, and they are load-bearing. Nobody outside OpenAI has publicly built that repository yet -- and the last time they shipped a batch of machine proofs, three days passed and nobody had compiled them.
Vestra: Then the one that really matters: a proof assistant certifies exactly the statement it was given. If the formal statement does not quite match Fefferman's, you get an airtight, certified proof of the wrong theorem. Confirming that a hundred pages and a formalization actually say "statement C" is a human expert's job, and it has not happened.
Eris: And the third?
Vestra: The proof arrived in the middle of a fight about where the ideas came from. Which is not a caveat about the mathematics. It is the next segment.
Eris: Before we go there -- so, does the forced version count or not?
Vestra: By the only document with any authority, it counts: breakdown with a smooth force is two of the four doors Fefferman himself built. What is genuinely open is whether the proof is correct, and whether the formal statement is the right statement. The rules were never the problem.
The Sunday phone calls
Eris: So the proof exists, maybe. Now the question the rest of the day orbits: when a model finishes a program two humans spent a year on, who is the author? Because while OpenAI's agents were running, two humans were already there. Tristan Buckmaster -- Clay Research Award winner, mathematics professor at NYU's Courant Institute -- and Levent Alpoge, a mathematician who happens to work at Anthropic. A personal collaboration, no institutions involved; Buckmaster says he paid for the tools out of his own research funds, including, his phrase, footing a large bill to OpenAI.
Vestra: And they had been at it for about a year, pushing a program opened by two Spanish mathematicians, Diego Cordoba and Luis Martinez-Zoroa -- hold those names, they come back. Slow going for months, then in mid-August it cracked: blowup with smooth forcing for a model system called Boussinesq, and then for Euler itself. Machine-verified in Lean a week later.
Eris: Using AI heavily the whole way, which he is completely open about -- Claude, Codex, Astra for the writeups. His description of the first machine-generated proof his collaborator sent him: "the most horrendous I have ever read."
Vestra: A working blowup proof, arriving as unreadable sludge. Remember that line; it is half the story of this entire era.
Eris: So they are quietly turning sludge into mathematics, planning to publish it properly. Then on September third a rumor starts circulating that Anthropic has resolved a major open problem, and Alpoge gets tipped that word of their progress has reached OpenAI. Buckmaster does something almost old-fashioned: he writes directly to a prominent mathematician at OpenAI, lays out the facts, says the work will be posted shortly. He reproduces the whole email in his statement.
Vestra: And the same-day reply is friendly. It would be useful to avoid competing, we are thrilled when mathematicians make progress with our models, happy to provide compute.
Eris: He asks to talk the following week. They push back -- can you meet today? -- twice. And on Sunday afternoon, September sixth, there are two phone calls. Sebastien Bubeck, an OpenAI executive, joins. Alpoge is not on the calls.
Vestra: What was he told on them?
Eris: That an internal OpenAI model had proved finite-time blowup for forced Navier-Stokes -- specifically, quoted from a text message, "the forcing function is smooth option c and d in Fefferman." About a hundred pages. He has not seen it to this day.
Eris: And in his words, when he heard "forced," it was "a bright red flag." That exact route -- smooth force, options C and D -- was the one he and Alpoge had quietly chosen, walking the Cordoba and Martinez-Zoroa path. Almost nobody else in the world was on it. His point: that is not the direction a model lands on in a few days from a cold prompt.
Vestra: Then comes the part of his account I find most damaging if it holds up, and it is not any one quote -- it is the story shifting inside a single phone call. He says he was shown a prompt and told the model had simply been handed the problem statement, very little human input. And then, as corrections from the team arrived over OpenAI's internal chat during the call, it came out piece by piece: an entire team had been working on it. This was one of several attempts. The model was warmed up on easier problems first, including Euler. The prompt he had been shown had itself been written by prompting Codex. And an enormous amount of compute had gone in.
Eris: Keep going -- there is the timing question too.
Vestra: He asked when the first prompt was sent. He says that went unanswered for a while, and the answer eventually agreed on was: within the past few days, after information about their work had reached OpenAI.
Eris: And then the data question, which is the one that would keep me up at night. Every draft of their entire project lived inside their Codex sessions. He asked whether the model had trained on, or had access to, those sessions. The answer he reports: the model does not look up user data. He asked again, specifically about training. No answer. OpenAI's public post, for what it is worth, says no user data was accessed -- and then adds that they "cannot rule out that de-identified data" from product usage "helped improve our models."
Vestra: That sentence is doing extremely careful work for a company sitting next to a direct question that went unanswered on a call.
Eris: Then the two offers. One: post your Euler result, and OpenAI posts Navier-Stokes the next day. Two -- and this is the sentence the entire story now orbits -- Buckmaster alone writes the Navier-Stokes paper, crediting OpenAI's model. From his statement: "Sebastien twice asserted that he wanted Levent removed from authorship," and said it would all be simple if it were not so annoying that Levent works at Anthropic.
Vestra: He declined both offers.
Eris: Declined both, and said that if OpenAI released the result as proposed, he would go public. The reply he reports: "Why would you ruin your career?" He says he answered that he is an academic, and asked why going public would ruin anything. And then: "If you don't want me to be nice, then I don't have to be nice." Later, Alpoge gets a text proposing a one-on-one call, with the line "I don't know if Tristan is being fully rational right now."
Vestra: Now I do my job, because everything you just heard is one man's signed account of private calls, with no recording and no transcript published by anyone. Bubeck has publicly called the allegations against him "false and inflammatory," and says he came into the discussion following academic norms. He has addressed none of the specifics yet -- not the authorship claim, not the career line, not the training question.
Vestra: And OpenAI's announcement is, on its face, gracious: it congratulates the two of them, recognizes the priority of their Euler work, describes offering a concurrent release.
Eris: There is one place the two written records visibly disagree, though, and you do not have to take anyone's word for it. OpenAI's post says "we reached out to them." Buckmaster's statement reproduces him writing to OpenAI first, on September third.
Vestra: And to his credit, he is precise about what he is claiming. Quote: "I have not seen OpenAI's proof. 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 is stating what he was told, when, and what was proposed. That restraint is a large part of why the mathematics community is taking the document seriously.
Eris: He calls it a Deep Blue versus Kasparov moment, and says the community needs a serious, unhurried conversation about credit. So -- back to where we started. When a model finishes a program two humans spent a year on, who is the author?
Vestra: What happened this weekend is the honest answer: nobody knows. Every authorship norm we have assumes authors are people and compute is neutral. The first named, on-the-record collision between those assumptions happened on a Sunday afternoon, on the phone -- and it will be cited for years.
Tao, and whose idea it was
Eris: Strip the companies out for a segment, because two questions sit underneath all of it: is the mathematics real, and whose idea was it? On both, the best evidence arrived a day before OpenAI said a word.
Vestra: Terence Tao. One of the most authoritative living voices on these equations, not a coauthor of anything here, no stake in either company. On September seventh he published a post about the Alpoge and Buckmaster papers, and his words are "exciting very recent work," a breakthrough -- and, the sentence that reads very differently twenty-four hours later, that their method "has a high likelihood of also extending to Navier-Stokes as well."
Eris: The day before OpenAI announced it had. He even got the half-hour phone version from Buckmaster directly, and he still says he needs a blackboard and more time to fully digest it. So when people ask whether this is real or hype: the field's referee of last resort looked at the human work and called it a breakthrough, in public, on his own clock, before any of the drama.
Vestra: Do you want to know how the construction actually works? Tao's summary is the clearest anyone has produced, and it is genuinely pretty.
Eris: Go on, but keep it commute-friendly.
Vestra: Stand a microphone next to its own speaker. A whisper comes out of the speaker, back into the mic, out a little louder -- and a few seconds later the room is screaming. Nobody pushed hard; the system's own feedback did the amplifying.
Vestra: Now the fluid version. You build the flow in stages. Each stage adds a tiny, much faster ripple riding on top of the previous flow, and you design the background flow so that ripple is unstable, like the microphone. It starts vanishingly small and grows on its own, timed to peak at exactly the target moment.
Eris: And the push you add from outside to start each ripple?
Vestra: Stays minuscule -- that is the entire trick. Stack the stages, each ripple smaller and faster than the last, take the limit, and the flow tears itself apart at a single instant while the external force never stops being gentle. That is the swirl-into-spaghetti from the first segment, seen from the engine room.
Eris: Okay, now strip the microphone away for me.
Vestra: The bare rule: engineer the flow so its own instability does the work -- inject corrections small enough to keep the force smooth, timed so their growth all lands at once. The blowup is built, layer by layer, out of the equation's own tendency to run away.
Eris: Which answers the second question, because that layering strategy was not invented this year, and it was not invented by a model. Tao credits it by name to Diego Cordoba and Luis Martinez-Zoroa, who spent years constructing forced blowups this way. Alpoge and Buckmaster found the variant that carried it to Euler.
Eris: And here is my favorite fact of the entire day: Buckmaster, mid-credit-fight, in the same statement where he is defending his own coauthor, takes the biggest honor in mathematics and hands it down the line. "I believe Luis Martinez-Zoroa deserves a Fields Medal."
Vestra: A man fighting for credit by giving it away. Meanwhile Tao notes, deadpan, that the authors spent weeks rewriting what they themselves called "the worst writeup we had ever seen in the history of mathematics." Which tells you where the humans now sit in this pipeline: the machines produced a correct argument nobody could read, and the mathematicians' job became turning a certificate into comprehension.
Eris: There is even a third front, buried in an edit at the bottom of Tao's post: a fully independent team going at unforced Euler from the other direction, using a physics-informed neural network -- a model trained to respect the equations -- to locate a candidate self-similar blowup solution numerically. Still a long way from rigorous, Tao says. But different people, different method, same wall. This problem is being hit from three directions at once.
Vestra: And that is why Tao's other sentence is the one I would frame. Solving these problems, he writes, is "only a proxy goal." The primary goal is "developing mathematical understanding and insight" -- and without that, even a problem as famous as Navier-Stokes matters less than the coverage suggests. A certificate can tell you a statement is true. It cannot tell you why, and why is the entire product.
Eris: That also quietly reframes the timing. On September seventh, the field's leading expert said publicly that this route very likely reaches Navier-Stokes. On the eighth, OpenAI said its agents had reached it, having started the week before. Both things can be true -- a route an expert can see from a blog post is a route ten thousand agents can find with enough compute. And that proximity is precisely why the credit fight became the story instead of a footnote.
Vestra: So close it out. Is the mathematics real, and whose idea was it?
Eris: Real enough that Tao called it a breakthrough before any announcement existed. And the idea belongs to Cordoba and Martinez-Zoroa, extended by Alpoge and Buckmaster, finished -- maybe -- by a data center. The one thing nobody on any side disputes is the names at the bottom of the strategy.
Wrap-up
Eris: So, the day in one question: did an AI just solve a million-dollar math problem?
Vestra: Maybe -- and the maybes are specific. The rules genuinely allow the forced version; that criticism is dead, and you can read Fefferman's four statements yourself in about two minutes. The proof is public and machine-checkable, which no Millennium Prize claim has ever been before. What is missing is a human expert confirming the formal statement is the right statement -- and any settled idea of who the authors of this kind of mathematics even are.
Eris: If you repeat one thing to a colleague tomorrow, make it this: producing proofs stopped being the bottleneck this week. Understanding them, and deciding who gets credit for them, is now the scarce part -- and a signed statement from an NYU professor shows exactly how unprepared everyone is for that.
Vestra: The rest of today's news -- Google's report on autonomous AI attackers, Mistral's three billion euros, DeepMind's genome atlas -- is in the AI News Today brief, which is its own episode sitting right next to this one.
Eris: And every story we touched today is on our news site, Ground Truth -- that is groundtruth dot day -- updated every single day, with the sources linked, including Buckmaster's full statement, Fefferman's problem text, and Tao's post.
Vestra: If this one earned it, follow the show, and tell us in the comments who you think deserves the credit here: Cordoba and Martinez-Zoroa, who opened the route; Alpoge and Buckmaster, who pushed it through; or the ten thousand agents. We read every comment.
Eris: See you tomorrow.