The Navier-Stokes Theorem: Big Tech Between Understanding, Verifying, and Pacing
I waited a few weeks for the dust of controversy and hype to settle before trying to tell what really happened with Navier-Stokes. In the first few days, headlines split into two opposing camps: "Artificial intelligence has solved a Millennium Prize Problem" on one side, "No, they just stole the work of two mathematicians" on the other. Both reflect the tone of the event, not the facts. And the facts, lined up, tell a story far more interesting than any headline. There is also a thread linking this story to another that arrived just days later—the essay in which Dario Amodei called for pacing AI development: the same question about what in these corporate narratives is verifiable and what remains mere promise. We will get there after the math.
A Machine, a File, and an Linger Doubt
On September 8, OpenAI published what it described as a solution generated by an internal model to one of mathematics' most famous open questions: can the Navier-Stokes equations, under certain conditions, "blow up"? The answer came in two forms: a paper of over 160 pages and a file verified by the Lean software, which checks every logical step like a relentless proofreader—never tiring and never getting distracted.
At the very same time, however, some of the world's most authoritative mathematicians argued that the most important piece—truly understanding what is happening inside those equations—has not yet arrived. Terence Tao, Fields Medalist and likely the most followed living mathematician online, wrote so five days before the solution was made public, and before even knowing it existed. It was not a reaction; it was a prediction that came true. Tao feared a scenario where a solution "largely generated by AI" would end up contaminating the problem as a source of further progress, rather than advancing it.
This article attempts to hold both aspects together without choosing sides: what has actually been verified, what remains storytelling, and which questions remain open.
What We Are Talking About
The Navier-Stokes equations are, essentially, Newton's second law applied to fluids. They describe how water, air, blood in arteries, and the atmosphere move. They are used daily to design airplanes, forecast the weather, and study blood circulation. The mathematical problem concerning them, however, has nothing to do with tomorrow's weather: the question is whether a fluid starting calm and smooth, with finite energy, can remain smooth forever, or whether at some point a spot might appear where velocity grows without bound in finite time. Imagine a droplet inside a tightening vortex, shrinking down to a point of zero dimension spinning at infinite speed: that is what mathematicians call a "blowup," and it is precisely what, according to real physics, should never happen because the fluid's internal friction always tends to smooth out motion.
In two dimensions, it has long been known that this never happens. In three dimensions, the question has remained open for nearly ninety years, since Jean Leray's pioneering work in 1934, and in the year 2000 was listed among the seven Millennium Prize Problems of the Clay Mathematics Institute, each carrying a $1 million prize. The only time the prize was awarded was in 2010, for the Poincaré conjecture, and the winner, Grigori Perelman, declined it.
OpenAI claims the "blowup" side: a setup where, with a smooth external force applied to the fluid and finite energy maintained throughout the dynamics, a finite-time singularity still develops. The technical difficulty is not decorative: the collapse must emerge from the fluid's own motion, not from something unnatural forced into it.
Why does anyone care if it doesn't change how we predict tomorrow's weather? Because, as Tao writes, the problem is important not for its direct physical application, but for what it generates along the way: decades of attempts to answer it have produced now-fundamental mathematical tools, from Leray-Hopf weak solutions to the Beale-Kato-Majda blowup criteria. In other words, the problem has always been a theorem generator, not just a prize to collect.
Lean Verifies the Logic, Not the Question
The most misunderstood part of this whole saga concerns that Lean-verified file. Lean is a proof assistant—a program born at Microsoft in 2013 that checks step-by-step whether every deduction follows correctly from previous ones. It is why the announcement sounds more solid than a mere press release: the proof is not the opinion of a language model, but a file mechanically verified by a machine.
The problem is that Lean checks whether the proof derives from the written statement, not whether the written statement is the right one. If the translation of the problem into formal language—what exactly constitutes a "solution," "finite energy," or "smoothness"—is imprecise, the proof can be flawless while proving something completely different from what is intended. Lance Fortnow, a complexity theorist, notes a side effect: Lean is becoming a sort of postmark, a way to claim priority over a theorem before even writing it in a human-understandable way. He notes that just days earlier Anthropic had formalized Fermat's Last Theorem in Lean, and the news had "barely made a ripple"—a sign that the genre is getting saturated.
On this point comes the most useful contribution following the first wave of headlines: an essay by Silvia De Toffoli and Eamon Duede published as a guest post on Tao's blog. Their thesis is that there exist two distinct notions of proof: logical proof—mechanically verifiable deductive validity, which a Lean formalization fully satisfies and which guarantees certainty—and intelligible proof, which concerns understanding, or grasping why something is true. Historically, the two always walked together because no human could build a vast logical proof without first grasping the ideas making it true. With AI, they write, these two notions can now diverge dramatically. The sentence summarizing the core issue best is theirs: OpenAI has given an answer, but it is far from clear whether it has offered a fruitful solution. And this is not an isolated opinion: the Clay Institute itself writes on the official problem page that a proof matters because it yields not only certainty, but understanding. It is the written rule of the contest, thrown into crisis by the contest itself.
Ten Thousand Attempts and a Pre-Mapped Path
The most interesting part of the story, however, is how such a result was achieved, because it reveals more about what the machine actually did than any corporate statement. According to OpenAI's reconstruction, the company deployed roughly ten thousand concurrent agents, divided into teams exploring different approaches: some searched for a proof that the fluid blows up, others for the opposite disproof, without asking the system which of the two was true. The total compute cost, estimated by Simon Willison across the three hundred billion tokens used on all open problems attempted during those days (not just the 130 billion for Navier-Stokes alone), would sit around $15 million at list prices.
Yet the path those ten thousand agents traveled in eighty-eight hours was not opened by a machine; it was forged by two human mathematicians years prior. Diego Córdoba and Luis Martínez-Zoroa had been exploring forced collapse constructions for related equations since 2023. Their work enabled Tristan Buckmaster, a professor at New York University, and Levent Alpöge, a researcher also working for Anthropic, to achieve similar results for other equations over a year of self-funded personal collaboration. Córdoba captures this with a phrase that says more than any technical breakdown: if their work hadn't existed, AI would not have solved the problem. And Tao himself, on September 3—three days before the announcement—had publicly outlined the four-step strategy leading to the solution, describing it as a "reasonably well-defined strategy." The machine didn't find the path: it ran it in eighty-eight hours instead of months.
This does not mean it is mere brute force. Bryna Kra, a mathematician specializing in another long-standing problem (Nivat's conjecture), relates that she and her colleagues had intuitively felt for years where the connection between two different approaches lay, without being able to pin it down. A machine, she writes, doesn't need years to absorb distinct areas of mathematics before discovering connections, and it can try thousands of combinations without ever getting discouraged. This is praise, not condemnation: the machine excels at the mechanizable part—searching, combining, and verifying along a map already drawn by humans. It has not yet shown an ability to generate the next question—the one that decides where to dig next.
There is, however, a more troubling side to all this, representing perhaps Tao's strongest argument. If such an iterative process—which normally generates invaluable insights into real fluid behavior—is conducted entirely by a corporation that keeps the journey almost completely hidden from public view, then even if the problem is technically solved, almost nothing is added to collective knowledge. The real distinction, under this reading, is not brute force versus human ingenuity, but published process versus hidden process. OpenAI made the destination public, not the journey.
Twenty-Five Fields Medalists, a Shared Concern
On September 11, twenty-five Fields Medalists, including Tao himself, signed a hastily drafted statement without the customary consultation process, "because the urgency of the situation" required it. The text does not condemn the use of AI in itself, but denunciates a misalignment between the goals of corporations developing these models and those of the mathematical community: solving problems as a promotional showcase risks damaging the discipline itself. The increasingly rapid production of true-or-false claims, they write, could destroy fertile ground instead of birthing new ideas, and the rush with which these results are announced also raises serious credit attribution concerns.
The European Mathematical Society, in a September 10 statement signed by President Jan Philip Solovej, delivers what is perhaps the most incisive critique of all: it is not enough to say AI cannot do math; the real problem is that the model used for this result remains internal to OpenAI and accessible to no one else. In a spirit of open science, EMS writes, this is a problem the mathematical community must address. It is not an objection to the merit of the result, but an objection to access.
Contested Credit
The most delicate aspect concerns what exactly transpired between September 3 and September 8, and here the rule applies that contested claims must be attributed to those making them without taking sides. Buckmaster relates, in a statement published hours before the official announcement, that he was contacted by OpenAI after reaching out to an OpenAI researcher, concerned that rumors of his work with Alpöge had triggered a competitive race. During a September 6 call, he was described a proof for "forced" Navier-Stokes—the exact direction he and Alpöge had quietly chosen: he called it an alarm bell, because it is not a direction one arrives at in a few days starting from the problem statement alone. Buckmaster also recounts being told, regarding the possibility of staying off the publication, "why ruin your career?"—a phrase he experienced as a veiled threat.
Sébastien Bubeck, an OpenAI researcher present on that call, published a different version of the same episode: he said he made that remark in a completely different context, discussing who should be primary author on an OpenAI rewriting of the proof, and that he apologized on the spot for the poor choice of words. Sam Altman wrote that his team acted with integrity, that they initially believed they were working toward a joint announcement, and that upon discovering Buckmaster and Alpöge had solved a different problem (Euler, not Navier-Stokes), they offered to let them publish first. Alpöge, on his own profile, confirms the main facts but attaches opposite meaning to the same events, explicitly asserting that he has always used and wished to publicly thank AI tools, including OpenAI's, in his work.
The two versions of that phone call remain incompatible, and there is no way to establish which is correct. What is verifiable, however, according to an audit by Emily Riehl on X reported on September 9, 2026, is the bibliography: Wayback Machine checks confirm that OpenAI's public paper was updated on September 8 at 19:09 GMT, adding the missing citations to Córdoba and Martínez-Zoroa that were absent from the first version. That is not an opinion; it is a timestamp.
The Paradox of "Pacing"
A detail links this affair to a nearly parallel event just days later. On September 12, Dario Amodei, CEO of Anthropic, published an essay titled We Must Pace the Frontier, arguing that the industry must deliberately slow down the pace at which it enhances model capabilities. Within hours, Sam Altman wrote that he agreed, and Elon Musk replied with two words: "Dario is right." It is the exact same word—pacing—that appeared four days earlier on OpenAI's Navier-Stokes result page, where the company wrote that it aimed to guide and pace its progress.
This is no decorative detail. Amodei himself explains that how much American companies can afford to slow down depends on how far ahead they are relative to China, devoting a substantial portion of the text to urging the government to block advanced chip sales to Beijing and crack down on model distillation—the practice of training smaller models to mimic more powerful ones. If executed well, he writes, these measures would widen the American advantage over the next three to five years.
Critical readings were quick to follow from very different quarters. The Register described it as an attempt to secure favorable regulatory conditions disguised as a safety appeal. An independent analysis notes that the essay asks the government for two very concrete things—stopping chip sales to China and cracking down on distillation—while the part about international cooperation serves more to project an image of responsibility than to produce a realistic agreement with Beijing, which Amodei himself, under this reading, knows to be unattainable. Even cautious industry observers note that a genuine commitment to slow down would mean halting training runs, not merely adding external observers with publication rights: observers can be ignored; training runs cannot. Even Chinese state media commentary, which obviously has its own perspective, reached the same conclusion, viewing the essay as a tool to preserve American technological dominance rather than a proposal for shared safety.
Not everyone views the essay as mere PR. Demis Hassabis, CEO of Google DeepMind, publicly called it a step in the right direction, adding that details remain to be ironed out but the fundamental framing is correct. Zvi Mowshowitz, one of the most followed analysts on AI safety and generally strict with industry firms, called it "an excellent proposal," arguing that the first concrete commitment—external evaluators with permanent access—is not symbolic at all: in Amodei's text, these observers get desks, badges, company laptops, and crucially, the right to publish conclusions without company editorial oversight, with redactions permitted only for safety or legal secrecy, never to hide unfavorable results. It is a detail that critics dismissing the essay as mere rhetoric tend to omit: if genuinely respected, it places verification power outside the company, something no lab had done in this form before.
It is the same pattern threading through this article, only on a different scale. OpenAI used the Navier-Stokes result to inform the world about the pace of its progress—a phrase that simultaneously projects an image of a dangerously powerful company and sets the stage for requesting to be regulated on its own terms. The question raised by critics of Amodei's essay is the same one the math community asks about OpenAI's announcement: is the entity setting the pace actually slowing down, or merely choosing better words to describe its race?
What Becomes Scarce
The thread holding this entire story together, including the paradox of pacing, is not "who is right," but a simpler question: who verifies what? Lean verifies deductions, not statements. Corporations verify models, not public interest. The Clay Institute—which to date has validated nothing and keeps the problem listed as "active" rather than "solved"—verifies with intentionally slow deliberation. Nobody, for now, verifies the choice of which problem is truly worth tackling, and nobody, judging by criticisms of Amodei's essay, verifies whether those asking to slow down are actually slowing down or merely selecting their words more carefully.
Bryna Kra closes her reflection with the question that perhaps matters most: what do we value in mathematics now that a certain type of discovery is no longer a scarce resource? It is a question that can be asked outside mathematics to anyone promising to slow down while continuing, in the same breath, to explain how unstoppable their technology already is. Daniel Litt, a mathematician at the University of Toronto, offers perhaps the most constructive answer in the dossier, proposing that academic value be measured through seminars and public discussions rather than paper output alone—because someone who merely pushed a button without understanding will struggle to pass a rigorous discussion before peers. It is a principle valid outside academia as well: transparency is measured by what is shown step-by-step, not what is promised in words. Litt closes with a phrase worth taking away: there is still infinitely much to learn; we have always been at the beginning, and we always will be. That applies to mathematics, and perhaps to those using mathematics as a pawn in a larger game.
Note: some cited sources (Mastodon posts, PDF documents, X posts) were cross-verified via independent outlets; where original content is no longer publicly accessible, references remain traceable through reporting by outlets including Nature, Quanta Magazine, and Axios.