On September 8, a Tuesday, OpenAI put out a paper claiming it had solved Navier-Stokes. If that name means nothing to you, don’t worry, it didn’t mean much to me either until last week. It’s one of the seven Millennium Prize Problems, the kind of math questions so hard that the Clay Mathematics Institute put a million dollar bounty on each one back in 2000. Only one has ever been solved before this, by a Russian mathematician named Grigori Perelman, and he turned down both the money and the medal that came with it.
What makes this one weird isn’t really the math. It’s the timeline. OpenAI says it only started working on this specific problem on September 1st, after its researchers heard a rumor. Seven days later they had a proof, ran it through 10,000 AI agents, and announced it to the world. Meanwhile a mathematician at NYU named Tristan Buckmaster came out and said, more or less, hang on, I’ve been working on this exact problem with a researcher at Anthropic, and what OpenAI just published looks a lot like our approach.
I’ve been reading about this on and off for three days now, mostly on my phone during lunch, and I still don’t have a fully settled opinion. Part of me wants to just be impressed. Part of me keeps appearing back to Buckmaster’s account and getting stuck there.
I’ll walk through what the problem actually is, how OpenAI says it did this, and then the messier part: the credit fight that’s currently splitting the math and AI world in half. Stick with me through the equations bit, I promise it’s not as bad as it sounds on paper.
What Navier-Stokes Actually Asks
Here’s the problem statement, stripped of the jargon. Navier-Stokes are a set of equations from the 1800s that describe how fluids move. Water in a pipe, air over a wing, blood in your arteries, all of it. Engineers use these equations every day and they work fine in practice. But nobody could prove, mathematically, whether the equations always behave. Meaning: could you start with a perfectly smooth, calm fluid and have the equations predict that at some point in the future, the fluid reaches infinite speed at a single point? That’s called a blow up or a singularity, and whether it can happen was an open question for about 90 years.
It sounds almost silly when you put it that way. Obviously water doesn’t actually move at infinite speed anywhere. But the question isn’t about real water, it’s about whether the math describing the water is well behaved. If the equations can blow up, that tells you something important about the limits of the model itself, and it matters for physics, weather prediction, basically anything downstream that leans on these equations being trustworthy.
OpenAI’s claim is that their system proved a singularity does form in the three dimensional version of the equations, in finite time, under specific starting conditions. Not that water actually explodes somewhere in your kitchen sink. That the math, taken on its own terms, permits it.
Think of it like this. Say you had a rule for predicting traffic flow on a highway, and the rule worked great for every real situation you’d ever tested it on. Cars slow down, speed up, merge, all normal. But nobody had checked whether, mathematically, that same rule could predict a car reaching the speed of light if you fed it some very specific, very unusual starting arrangement of vehicles. That’s roughly the shape of the Navier-Stokes question. The rule works beautifully in every practical case engineers throw at it. Whether it could theoretically break, under conditions nobody has ever actually seen, is a separate and much harder question.
For 90 years, nobody could prove it either way. Not “we tried and it didn’t blow up.” Actually prove, with full mathematical rigor, that it either always stays smooth no matter what you start with, or that there’s some starting condition somewhere that breaks it. Nine decades of some of the smartest people in the field poking at this and not landing a conclusive answer either way. That’s the size of the thing OpenAI is now claiming its agents cracked in a week.
The Specs: 10,000 Agents, 88 Hours, One Unreleased Model
This is the part that actually surprised me the most, more than the math itself if I’m honest.
You can read the research paper here: Navier Stokes Paper
OpenAI didn’t use a single model sitting and thinking for months. Their setup was a swarm. First they tested the approach on a related, slightly easier problem, the Euler equations, using close to 100 agents over about 50 hours, which produced a disproof of something called Euler regularity. That worked well enough that they scaled up hard. For Navier-Stokes itself, they set loose roughly 10,000 agents, all coordinating on pieces of the same proof, powered by an internal model OpenAI describes as more capable than GPT-6 Astra, their public flagship at the time. That model isn’t available to anyone outside the company.
The agents ran for 88 hours straight, exchanging close to 5 million messages between each other while checking and rebuilding pieces of the proof as a group. After that, it took another 17 hours to formalize the whole thing in Lean, a formal proof language that lets a computer independently check every logical step, no hand waving allowed. A computer scientist at OpenAI named Ven Chandrasekaran described the result at their briefing as fluids that start out completely normal reaching infinite speed in a finite window.
Five million messages is a wild number to sit with. I tried picturing what that actually looks like and honestly couldn’t, itself a strange feeling given I use these tools daily. One detail worth flagging: OpenAI’s release says the agents had access to tools including the ability to read from a cached version of the internet and run code. Fairly ordinary for an agentic system in 2026. But that one line, cached from when, cached from where, is exactly the detail Buckmaster’s side of the story keeps circling back to.
The Part Everyone’s Actually Talking About
Here’s where it gets a mess, and where I think most of the real story is.
Buckmaster wasn’t some random guy weighing in from the sidelines. He’d been working, in his words, in a “personal collaboration” with Levent Alpöge, an Anthropic mathematician, for close to a year, building on ideas from two other mathematicians, Diego Cordoba and Luis Martinez-Zoroa. On September 7th, a day before OpenAI’s announcement, the two of them published their own four-page statement along with three results of their own, on related but narrower fluid-blowup problems. Buckmaster explicitly credits Martinez-Zoroa in that statement, at one point saying he thinks the man deserves a Fields Medal. He does not extend the same generosity to OpenAI.
The statement lays out a fairly damning sequence. On September 3rd, worried that a rumor about “someone” solving a major open problem was actually about his and Alpöge’s private work, Buckmaster reached out to a mathematician at OpenAI to clarify the project was personal, not affiliated with either company. A call finally happened on September 6th, with OpenAI researcher Sebastien Bubeck on the line. Buckmaster says he was shown a prompt and told the internal model had simply been given the problem statement, with almost no human input. Over the course of that same call, according to his account, it came out piece by piece that an entire team had actually been working the problem for a while, that Euler was tried first, and that even the prompt he’d been shown had itself been written by prompting Codex.
Then comes the part I think matters most here. Buckmaster says he asked directly whether OpenAI’s models had been trained on, or had seen, his and Alpöge’s progress, since he’d been using OpenAI’s own Codex to proofread and validate parts of their work for months. He says OpenAI did not answer that question. Not denied it, not confirmed it, just didn’t answer. If your own drafts were sitting inside a tool built by the same company that then raced you to the finish line on the exact same problem, in a week, I don’t think it’s unreasonable to want a straight yes or no on that.
Buckmaster also says OpenAI offered him two options once Euler was published: either OpenAI would publish Navier-Stokes the next day, or Buckmaster alone could write up the Navier-Stokes result crediting an internal OpenAI model, with Alpöge left off entirely because he works for a rival lab. He declined both. OpenAI’s Bubeck has since disputed the framing of that second point on social media, though not the underlying facts of the call itself.
I don’t think there’s a clean verdict here yet, and honestly I’m suspicious of anyone claiming there is one already. To be fair to OpenAI, its own release doesn’t hide that a rumor is what triggered the whole push, it says so plainly. What’s contested is narrower than “who did the math first.” It’s whether a company whose tools sat inside a rival’s private workflow for months gets to call its own result independent.
Terence Tao, a mathematician at UCLA who wasn’t involved in any of this, called Buckmaster and Alpöge’s work remarkable when it came out. He hasn’t, as far as I’ve seen, weighed in on the credit dispute itself. That silence from the wider math community is its own kind of answer for now, everyone seems to be waiting to actually see OpenAI’s full proof before saying anything more.
Why This Actually Matters (And What’s Still Unsolved)
Setting the credit dispute aside for a second, because it deserves that, what does this mean.
Before September, exactly one Millennium Prize Problem had ever been solved: the Poincaré Conjecture, cracked by Perelman back in 2003, and he famously walked away from both the prize money and the Fields Medal. Six remained open. Navier-Stokes was one of them. The others still standing are the Riemann Hypothesis, the Hodge Conjecture, the Birch and Swinnerton-Dyer Conjecture, P vs NP, and the Yang-Mills existence and mass gap problem.
Riemann has been sitting there since 1859, so over 160 years now, and it’s about the pattern behind prime numbers, which sounds abstract until you remember basically all internet encryption leans on primes being hard to predict. P vs NP is more familiar to anyone who’s touched computer science, it’s the question of whether every problem whose solution is easy to check is also easy to solve, and a lot of cryptography quietly assumes the answer is no. Yang-Mills comes out of physics, tied to how particles get mass, and it’s considered by several people I’ve read on this to be harder to even state cleanly than Navier-Stokes was, let alone solve. If OpenAI or anyone else’s agents start seriously pointing compute at Riemann next, that’s the one I’d genuinely want to see formalized in Lean before believing a word of the press release.
If an AI system really did produce a valid proof here, checked and formalized in Lean where every step is machine verifiable, that’s not nothing. Lean verification is specifically the part that should make skeptics pause, because it’s not OpenAI just saying trust us, it’s a system where an independent formal checker confirms the logic holds line by line. That part I take seriously regardless of how the credit question shakes out. It’s worth remembering too that OpenAI has said it doesn’t plan to actually claim the million dollar prize itself, which for what it’s worth suggests they know full well how much scrutiny a Clay Institute submission would invite.
But I want to be straight about the other side of this too. The math community hasn’t finished picking through it. Formal verification confirms the logical steps are consistent, it doesn’t by itself confirm the framing or the significance of what was proven matches what’s being claimed in a press release. That distinction gets lost fast when a company has an incentive to make the announcement sound as big as possible.
My Honest Take
I lean toward this being a real result, mathematically, with a credit story attached to it that’s genuinely uglier than the math. Formalized proofs in Lean don’t fake themselves, and 10,000 agents burning through 5 million messages isn’t something you fake your way into a press release. Buckmaster and Alpöge’s own results, on the narrower problems, are already Lean-formalized too, which is exactly why I trust their side of the story as much as I do. They didn’t just make claims, they showed their work in a form anyone can check.
But I also can’t shake the detail about the unanswered question. A company saying nothing when asked directly whether its models had access to your private research is not the same as saying no. Maybe there’s an innocent explanation. Maybe there isn’t. Either way, that silence is doing a lot of the talking right now, more than any part of the actual math.
Where this leaves things for now: watch what Buckmaster and Alpöge publish next, and watch whether OpenAI says anything more specific about what its agents actually had access to during that week. Until one of those things happens, I’d hold off on calling this settled, in either direction.
Random thing that’s stuck with me through all of this, not really related to any of the math: I keep thinking about how none of the people actually arguing in public here, Buckmaster, Tao, the OpenAI team, seem to be enjoying this the way I expected. You’d think solving a 90 year old problem would just feel like winning. Instead everyone involved sounds tired, careful with their words, and a little bit burned. Maybe that’s just what it looks like now when a genuine scientific result and a corporate press cycle happen at the same time. I don’t have a neat way to end this one. Ask me again in a month once Buckmaster and Alpöge have published something more complete, and I might have a different answer than I do right now.