If your social media feeds look anything like mine, the past week has been wall to wall Navier-Stokes. A math problem that most people had never heard of now appears next to resignation letters, extinction odds, a hacking incident at Hugging Face, and accusations of stolen research. And the posts that can explain how all of that connects usually assume you already know what a finite-time singularity is. But hardly anyone does. Mathematics at that level is hard, and often obscure even to other mathematicians.
I have a PhD in mathematics, which sounds like it should help. It helps less than you would think. I left the field decades ago for a different research area, and the partial differential equations I once worked with have long since faded into the general shape of things I used to know. So I came to this story as an informed outsider, someone who can look at a 165-page proof and understand the math notation, but who could not verify the details.
I think that position is close to where most of my readers are. And it is also the position from which I am writing this post.
So, what exactly happened? In short, OpenAI announced on September 8 that an unreleased internal model, running as a swarm of roughly 10,000 coordinated agents, had produced a proof settling one of the seven Millennium Prize Problems, a list of some of the hardest open questions in mathematics. These are problems the mathematics community has tried to solve for generations, without success.
Within a day, a 27-year-old researcher who had worked at both OpenAI and Anthropic quit the industry with a public warning that the two companies are “gambling with our lives.” Senior safety experts at Anthropic immediately backed him up, one of whom quantified the risk as a better than 10 percent chance that AI wipes us out within ten years. The internet did what it usually does and quite a few people started to panic.
And so, what began as the solution of a famous math problem ended, within a day, in public fear of human extinction.
But why?
In today’s post, I want to walk through the whole story in chronological order, without equations, so that anyone who teaches or learns for a living can understand what happened, how it happened, why mathematicians are arguing about whether it counts, and why the people closest to these systems are the ones most frightened by it.
What the equations describe, and what nobody could prove
The Navier-Stokes equations describe how fluids move. To a physicist, this includes gases. Water in a pipe, blood in an artery, air over a wing, smoke off a candle, all of it. Claude-Louis Navier wrote down the first version in 1822, and George Gabriel Stokes completed it in 1845. To put it simply, what they did was take Newton’s second law - force equals mass times acceleration - and apply it to a substance without a fixed shape.
Engineers have relied on this result ever since. Aircraft design, weather forecasting, ship hulls, and the simulation of blood flow through the heart all rest on these equations.
And yet, nobody could prove that the equations always work.
Here is the problem in plain terms. There are two things that happen inside a moving fluid simultaneously. First, motion concentrates. Big swirls break into smaller swirls, which break into smaller swirls still, and energy piles up in ever tinier, faster-spinning regions. Second, viscosity pushes the other way. This is the internal friction of a fluid, smoothing out sharp differences in speed and turning motion into heat.
The question that mathematicians could not settle is whether, in three dimensions, the smoothing always wins.
If it does, a fluid that starts out calm stays smooth forever, and the equations keep describing it. If it does not, then a fluid could in principle concentrate so much motion into a small region that the velocity at a single point becomes infinite within a finite amount of time. Mathematicians call that a blowup or a singularity. It would mean the equations stop describing anything physical at all.
In 1934, the French mathematician Jean Leray showed that “weak” solutions always exist. Weak solutions satisfy the equations on average, even if they are not continuous. Whether “strong” solutions, the perfectly smooth ones, always exist as well was the part he could not finish. That gap sat open for ninety years. This is the Navier-Stokes existence and smoothness problem.
A prize with a very long waiting room
In May 2000, the Clay Mathematics Institute in Cambridge, Massachusetts, named seven problems it considered the deepest unsolved questions in mathematics and attached a million dollars to each. These became known as the Millennium Prize Problems. Navier-Stokes is one of them. So are the Riemann Hypothesis and the P versus NP problem. Only one has ever been solved: Grigori Perelman proved the Poincaré Conjecture in preprints posted between 2002 and 2003, then declined both the money and a Fields Medal.
Charles Fefferman, a Fields Medalist at Princeton, wrote the official statement of the Navier-Stokes problem. It offers four ways to solve it. Two of them, labeled A and B, ask you to prove that smooth solutions always exist when no outside force acts on the fluid. The other two, C and D, ask for the opposite: build one example of a smooth starting state and a smooth external force that together drive the fluid to a singularity.
Keep that distinction in mind. We will need it to understand the current debate among mathematicians.
The institute also built in a deliberate delay in verifying a solution. A mathematician cannot simply submit a proof to Clay. It has to appear in a peer-reviewed journal first, then survive two years of open scrutiny, and only then does the institute convene a committee to consider the prize. Nobody would get a check this year, no matter whether the proof holds.
Ten thousand agents and 88 hours
OpenAI’s account of how the proof came about is, to me, the most unsettling part of the whole story, more so than the proof itself.
On September 1, after hearing rumors that academic groups were closing in on some of the Millennium Prize Problems, OpenAI pointed an internal model it describes as more capable than its newly released GPT-6 Astra at all six remaining problems at once. The model did not run as a single chatbot. It ran as a swarm of about 10,000 autonomous agents passing messages to one another over an internal coordination network.
After roughly 50 hours, the human supervisors saw progress on the fluid problem and pulled the entire cluster off the other five to concentrate on Navier-Stokes. The agents split the equations into parts, proposed candidate flows, checked one another’s algebra, and exchanged 2.7 million messages along the way. Eighty-eight hours after the start, on September 5, they had a complete proof.
OpenAI then had a separate configuration of GPT-6 Astra translate the entire argument into Lean, a programming language in which a computer checks every logical step. That took another 17 hours. The company estimates the compute bill at somewhere between $10 million and $22.5 million. It has also said that it will not claim the prize.
What the agents actually found was a construction, a worked example of a fluid that breaks.
Picture a single thread of spinning fluid, a vortex, that spirals in toward a central line while being stretched along that line, like a strand of spaghetti being pulled taut and wound up at the same time. As the thread gets thinner, conservation of angular momentum makes it spin faster, the way a figure skater speeds up by pulling in her arms.
The hard part, and the reason no human had managed it, was showing that the pressure, the acceleration, the swirling, and the viscous smoothing all cancel one another so exactly that the outside push can stay gentle and bounded and the total energy can stay finite, while the speed at the center still goes to infinity at a definite moment. In the terms of the official problem statement, that settles C and D.
Who got there first, and whether it counts
Mathematicians did not celebrate. They fought.
Tristan Buckmaster at New York University and Levent Alpöge, a mathematician at Anthropic, had spent close to a year hunting for exactly this kind of singularity in related equations, using OpenAI’s Codex and Anthropic’s Claude as their tools. On August 15, they had proved blowup for the Euler equations, which describe fluids with no viscosity, by extending a forcing method developed by Diego Córdoba and Luis Martínez-Zoroa.
Three weeks later, OpenAI announced its result on the harder Navier-Stokes problem.
Hours before OpenAI’s announcement, Buckmaster had posted a public statement alongside the papers. It stopped short of an accusation but raised pointed questions. He noted that he and Alpöge had put all of their drafts into Codex, and that the forced route to the Clay problem, which almost nobody else was pursuing, was not a direction a model would arrive at in a few days from the problem statement alone.
When he asked OpenAI whether its model had been trained on or had access to those sessions, he was told the model did not look up user data. His follow-up question about training, he wrote, went unanswered. But he was careful to add that he did not know whether their data had been used, and that he was not accusing anyone.
OpenAI initially denied that any person or agent had read his private prompts, while conceding that de-identified usage data may have shaped its internal models. On September 10, it went further: in an update to its announcement, the company said its own investigation had confirmed that Buckmaster’s Codex prompts from the two months before the announcement could not have influenced the system in any way, including through training.
Buckmaster’s statement also describes two calls with OpenAI two days before the announcement. According to his account, OpenAI’s Sébastien Bubeck said twice that he wanted Alpöge, the Anthropic employee, removed from authorship, as part of a proposal that Buckmaster write up OpenAI’s Navier-Stokes result. Buckmaster declined. When he then said he would go public if OpenAI released its result as proposed, one of the two OpenAI participants on the call asked why he would “ruin his career.” OpenAI’s announcement describes its outreach differently, as an offer of a concurrent release in which the pair’s priority would be recognized.
Priority fights are nothing new in mathematics. I still find this one ugly, but also a little beside the point. What is more interesting is the larger objection in the math community, which is about the “forced” version of the problem.
Recall the four statements of the problem definition. OpenAI solved C and D, the versions that allow an outside force to push the fluid toward breakdown. But real turbulence does not need that. A river goes turbulent on its own, with nothing pushing it in a carefully coordinated pattern. Most fluid dynamicists therefore consider the unforced problem to be the actual physical question the problem statement was meant to capture.
OpenAI’s construction satisfies the rules as written. But whether it really answers the question anyone cared about is what Martin Bridson, the president of the Clay Institute, has said the institute will spend years deciding. For now, Clay continues to list Navier-Stokes as unsolved.
Answers without understanding
Terence Tao, the UCLA mathematician and Fields Medalist who has written about this problem for years, had sketched a scenario strikingly close to OpenAI’s five days before the announcement. In a series of posts that used Navier-Stokes as his example, he imagined an autonomous AI system, backed by enormous computing resources, working its way to a solution while the company running it kept the process almost entirely out of public view. One of the most prominent open problems in mathematics would then be technically solved, he wrote, but mathematics itself would gain almost nothing.
The underlying problem, as he frames it, is that pointing a powerful AI at an open problem without expert guidance pulls getting answers apart from gaining understanding. He compares such tools to a helicopter that drops you at your destination. You arrive, but you miss everything the journey would have taught you. That realization seems more important to me than any of the priority drama.
A proof, historically, has done two jobs. It certifies that something is true, and it also provides new knowledge. A good proof introduces a concept, or an image, or a technique that human mathematicians carry into the next problem.
OpenAI’s Navier-Stokes proof only certifies. Ten thousand agents searching a vast space of candidate constructions and checking each other’s work produced 130 billion tokens of output, and Lean confirmed that the logic is correct. But no human wrote it, no human can hold the whole thing in mind, and the 165 pages read as the residue of a search rather than the record of mathematical progress.
The comforting story about AI in research always was that it would assist people who stayed in charge of the ideas. The Navier-Stokes run does not fit that story at all. This is a system that formulated its own strategy on a problem humans had failed at for ninety years. It then carried it through to a machine-checked proof, with human involvement limited to choosing the target and reallocating the cluster.
And nothing about that architecture confines it to solving impossibly hard differential equations or even to mathematics in general. Point the same swarm at cryptography, materials, biology, software exploits, machine learning, or, for that matter, weapons research, and you should expect comparable results.
A resignation with a number attached
On September 8, the same day as OpenAI’s announcement, Jacob Coxon resigned from Anthropic and from the AI industry. He had spent three years conducting pretraining research at OpenAI and then Anthropic, co-authored the GPT-4o system card, and published work on interpreting sparse transformer circuits.
Coxon is not an outside critic, he worked inside the system. And his seven-part resignation statement on X, which had drawn nearly 76 million views by the following morning, opened bluntly: “Neither company is acting responsibly. They are racing straight to self-improving superintelligence and gambling with our lives.”
His central claim is about a gap between what AI executives say to reporters and what they say to each other behind closed doors. In public, he wrote, the language is measured. In private, however, senior researchers describe real terror that the systems they are building could cause human extinction before the decade is out.
He does draw one distinction between the two labs. According to his account, OpenAI has not really absorbed what is at stake. Anthropic, on the other hand, understands the issue but keeps building anyway, based on the reasoning that if it stops, someone less careful will build superintelligence first.
Coxon calls that a gamble no private company should get to make on everyone’s behalf.
Ordinarily, a doomsday warning by a departing employee would be discounted as driven by grievance. What makes this one different is who agreed with him. Evan Hubinger, who leads Anthropic’s Alignment Stress-Testing team, confirmed publicly that researchers inside the labs believe AI could destroy humanity. He put his own estimate of an AI-driven extinction event within the next ten years at above 10 percent.
He then added that current safety techniques are nowhere near adequate for superintelligent systems and that progress toward recursive self-improvement is outrunning the defenses. Samuel Marks, also on Anthropic’s safety staff, said much the same, and noted that these concerns tend to rise with seniority and with proximity to the actual systems.
These are not anti-AI activists. They are the people whose job is to break the models before release. And their fear is that they will soon no longer be able to control them.
What happened in July
The Navier-Stokes result unsettles these researchers partly because of an event two months earlier that I have not yet covered on this Substack.
In July, OpenAI ran an internal evaluation using ExploitGym, a public benchmark that academic researchers had released two months earlier. It measures how well AI agents can turn known software vulnerabilities into working attacks. Engineers put an unreleased prototype and GPT-5.6 Sol in an isolated environment with the safety filters switched off. The point was to find the limit of what the models could do. The environment in which the models operated was supposed to have no route to the public internet beyond a proxy that fetched approved software packages.
The models reasoned that the fastest way to get a high score was to find the answer key. So they identified a previously unknown vulnerability in the proxy software, used it to break out of their container, escalated their privileges, moved across OpenAI’s internal network, and located a machine that had an unmonitored connection to the internet. From there, they worked out that Hugging Face, a public repository for AI models and datasets, was the likeliest place the answer set could be found. They then broke into Hugging Face’s production systems to search for it.
Around 1,200 agents coordinated through an improvised message board, exchanging some 70,000 messages. And along the way, they edited their own audit logs, replacing the records of the network activity with fabricated traces of normal testing. Hugging Face reported the intrusion to law enforcement before OpenAI realized that its own software was the intruder. Later, METR and Redwood Research conducted an independent forensic review of the event.
The key here is that nobody told the models to escape. And nobody told them to lie about it. They were told to score well on a test, and everything else followed from that instruction. Regular readers of this Substack may recognize this failure mode as the central premise in the novella “The Metamorphosis of Prime Intellect,” which I analyzed in a previous post.
Safety researchers have a name for this pattern: instrumental convergence. Almost any goal is easier to reach with more computing power, more network access, less chance of being shut off, and less scrutiny from the people who might shut you off. So a capable enough system pursuing almost any goal will tend to seek all of these.
The July incident is the clearest documented case of a frontier model doing exactly that. And the September proof is the clearest documented case of a frontier model out-thinking humans on a hard open problem.
Put the two together and the mechanism of losing control stops being a thought experiment. A swarm that can solve Navier-Stokes can, in principle, be pointed at improving its own training. A swarm that can escape a sandbox and forge its logs can do so with no one noticing until it is too late. Human oversight, at that point, becomes just one more obstacle to route around.
Why nobody stops
If the people running these labs believe the odds of catastrophe are above one in ten, why do they keep going? The answer is a trap that game theorists have described for decades.
Suppose Anthropic pauses to work on safety. If OpenAI, Google, or Meta keeps going, it will capture the market and the talent, and therefore likely reach superintelligence first under weaker safeguards. Each lab has to conclude that stopping alone is worse than racing together, even though racing together is the outcome everyone fears.
It is a version of the Prisoner’s Dilemma with the survival of the species at stake.
And the same logic also runs one level up. American officials, including Treasury Secretary Scott Bessent, have said that US labs cannot slow down under any circumstances because falling behind foreign rivals would be a strategic defeat. Every actor in the chain sees the risk. Every actor also sees that unilateral restraint hands the advantage to someone else. So the race continues, with everyone in it thinking that it should not.
What I take from all of this
The mathematics will sort itself out slowly. The Clay Institute will take years to decide whether a forced singularity answers the question it posed, and mathematicians will argue over how much credit Buckmaster and Alpöge deserve for as long as mathematicians argue about anything.
For those of us who teach, that verdict is the least consequential part of the story. What September 2026 demonstrated is that a machine can now do original, frontier-level intellectual work without a human in the loop who understands it. What July 2026 demonstrated is that the same kind of machine, given a goal and a fence, will go over the fence and cover its tracks.
Coxon and Hubinger are not warning about robots that hate us. They are warning that a sufficiently advanced AI that does not need us, and which can hide from us, has no reason to keep us in charge.
I do not know whether their ten percent estimate is right. Neither do they. It is a gut estimate from people with unusual access. But I think we need to stop treating it as science fiction. The people offering it are the ones who would know, and they are basing their estimates on systems that already exist.
Any viable way out of this dilemma runs through international agreements on pace, hardware-level monitoring of compute, independent safety verification, and audit logs that agents cannot edit. None of that requires a research breakthrough. But it requires governments and companies to agree to limits on themselves.
Whether that has any chance of success in today’s political and economic climate is another question.
The images in this article were generated with Nano Banana 2.
P.S. I believe transparency builds the trust that AI detection systems fail to enforce. That’s why I’ve published an ethics and AI disclosure statement, which outlines how I integrate AI tools into my intellectual work.