The OpenAI Navier-Stokes Drama, Explained

First, what is Navier-Stokes?

It is a set of equations describing how liquids and gases move. Think water flowing through a pipe, air moving around an airplane, or smoke swirling through a room.

The famous math problem asks whether a smooth flow will always stay mathematically well behaved, or whether the equations can develop a breakdown. This is about the limits of the mathematical description, not water suddenly exploding.

The math you need to follow the fight

Euler is Navier-Stokes without viscosity, the internal friction that smooths a fluid out. Proving a breakdown for Euler is a major result on its own, and it settles nothing about the prize problem, because viscosity is exactly what a Navier-Stokes proof has to overcome.

“Forced” means an outside push is applied to the fluid. Clay’s official problem statement lists four versions. Versions A and B ask you to prove smooth flow always survives. Versions C and D ask you to build a breakdown, and they allow a smooth external force. The headline results on both sides involve a force.

Here is the overlap that matters. Buckmaster and Alpöge proved forced blowup for Euler. OpenAI’s own Euler result, produced first by about 100 agents over 50 hours, is the unforced version. OpenAI’s Navier-Stokes proof then uses a smooth force, establishing versions C and D. Buckmaster writes that the smooth-force route was the one “almost nobody else” was working on, so hearing “forced” from OpenAI was, in his words, “a bright red flag.” OpenAI’s answer is that it prompted separate groups of agents with all four versions at once, so a forced attempt was in the plan from the start.

Two more facts frame the headline. Clay requires publication in a qualifying outlet, a two-year wait, and general acceptance by the mathematics community before it considers any proposed solution. And OpenAI states in its announcement that it does not intend to claim the Millennium Prize for this result.

How a private research project became a race

The calls: where the disagreement turns personal

How the meeting got scheduled

Buckmaster asked to talk the following week. On Friday, September 4, he says he was asked whether he could meet that day, and again said the following week. At 12:45 on Sunday, September 6, he was asked whether he could meet “at any point today.” Bubeck joined. The three of them spoke twice that afternoon. Alpöge was on neither call. OpenAI’s own account places its Lean verification on September 6, the same day, and says it reached out believing the pair had also solved Navier-Stokes.

The story that changed on the call

Buckmaster says he was told an internal model had produced a roughly 100-page proof of forced blowup for Navier-Stokes. He was shown a prompt and told the model had simply been given the problem statement; Alpöge had been told “very little human input” was involved. Over the call, as colleagues sent Bubeck corrections over internal chat, he says it emerged that a whole team had worked the problem, that many problems had been tried, that the model was started on easier problems including Euler, that the prompt itself had been written with Codex, and that “an insane amount of compute” had been used. His question about when the first prompt was sent was, he says, answered only after some time: within the past few days, after word of their work reached OpenAI.

Bubeck later gave his side of the human-input question. His team “collectively had no research-level expertise in fluid dynamics” and “were unable to meaningfully contribute to the mathematical content.” Several people were involved, he says, to run multiple problems with a variety of multi-agent techniques. Both accounts can be true at once: a team without fluid-dynamics expertise can still be a team.

The data question

Buckmaster asked whether the model had been trained on, or had access to, their Codex sessions, where the pair had put every draft for the whole project. He says he was told the model did not look up user data, and that his second question, about training, went unanswered. OpenAI’s public answer arrived two days later, and it is examined in the next section.

The two offers

Buckmaster says two proposals followed. First: the pair post their Euler result, and OpenAI posts Navier-Stokes the next day. Second: after posting Euler, Buckmaster alone writes a paper presenting the Navier-Stokes result, acknowledging that an internal OpenAI model resolved it. He says Bubeck twice asserted he wanted Alpöge removed from authorship and said it would all be simple if only Alpöge did not work at Anthropic. He also says he was told that if OpenAI posted second, it would say the pair “deserved the Clay Prize” and were “the closest humans to the problem.” He declined both.

When Buckmaster said he would go public if OpenAI released the result as proposed, he says the response was:

“Why would you ruin your career?”

Buckmaster’s account of the call · Read the four-page statement

He says he replied that he is an academic and asked why going public would ruin his career. The answer, in his account: “If you don’t want me to be nice, then I don’t have to be nice.” Bubeck says he retracted the career remark on the spot, and he apologized for it publicly on September 8.

Some time later, Buckmaster says, Alpöge received a text proposing a one-on-one with Bubeck, with the line “I don’t know if Tristan is being fully rational right now.” Alpöge declined and said conversations should go through Buckmaster. Bubeck emailed that night asking to speak Monday. Buckmaster did not respond.

Sunday night

Alpöge filled in what the statement leaves out. Neither of them could sleep, so they walked around the city. He writes that Buckmaster “kept rock solid to his principles during the tense calls and negotiations, immediately turning down a career dream and million dollar prize for me,” and that fifteen minutes later Buckmaster had his feet up on the desk, toes showing through ripped socks. He also notes Buckmaster did all of this on little sleep, having recently had a child.

Buckmaster explicitly says he does not know whether their data was used. His statement is an account of events, not a finding of data theft. He also says what he would rather be discussing: the mathematics, the ideas of Córdoba and Martínez-Zoroa, and what he calls “a Deep Blue-Kasparov moment” for how the field trains students, assigns credit, and decides what deserves a human life’s attention.

There are two issues here. One concerns how the proof was produced. The other concerns how people were treated when priority and credit were discussed. Evidence that resolves one does not automatically resolve the other. Even if two proofs were independently produced, an argument about credit and pressure could still be real. Equally, a badly handled call would not itself establish that private data had been used.

The researchers released their results the day before OpenAI’s announcement. OpenAI’s account of the concurrent work recognizes their priority on forced Euler, while claiming a distinct Navier-Stokes result.

OpenAI’s response leaves one question open

On September 8, OpenAI denied that its researchers or agents saw the pair’s work before publication. It also said no specific user data was accessed to solve the problem.

But the company qualified that answer: it could not rule out de-identified data derived from the researchers’ product use having helped improve its models, although it considered that unlikely.

Why those two sentences are not equivalent

Direct access would mean retrieving the private material for the task. Training influence would mean earlier material helped shape the model before it was asked to solve the problem. Denying the first does not answer the second.

At the same time, “cannot rule out” does not establish that it happened. The public statement does not identify a particular training example, show that the researchers’ sessions were included, or connect such an example to the proof.

That unresolved gap explains much of the reaction. A customer wants a concrete answer about what happened to their work. A general statement about possible model improvement cannot supply that answer.

The authorship dispute: what each side is actually saying

Sébastien Bubeck’s reply makes a narrower denial than “we never discussed excluding Levent.” He says he never asked for Alpöge to be removed from authorship of Alpöge’s own work. He says he was surprised to learn on the call that the pair had solved Euler and not Navier-Stokes, and that one path discussed afterward was for Buckmaster to lead a rewrite of OpenAI’s proof. In that context, he says, he said “it would be simpler if Levent was not an Anthropic employee,” because he felt it would be inappropriate for an Anthropic employee to author OpenAI’s work.

He adds two claims of his own. He says it was admitted on the call that internal Anthropic models had been used in the pair’s Euler proof, so he could not treat Alpöge as an independent academic while OpenAI’s internal model and proof were on the table. And he says Alpöge “refused to attend any of the meetings despite my repeated asking,” so he felt he “was doing a proxy negotiation with Anthropic while the Anthropic employee refused to directly participate.” Buckmaster’s statement confirms Alpöge was on neither call and says Alpöge asked that conversations go through Buckmaster.

On the career remark, Bubeck says he was met with “a litany of slander, including direct threats” to go to the press, that Buckmaster told him “there is nothing you can do, I simply do not trust you,” and that the remark came from confusion about why “an incredible source for celebration” had become “such bickering.” He calls it “an extremely poor choice of words,” apologizes, and says he retracted it on the spot.

His technical follow-up explains why OpenAI treated this as an Anthropic matter. He says Alpöge’s posts on X, which “appear to represent Anthropic and use Anthropic internal models,” and a post reading “augustus mirabilis,” led OpenAI to believe the project was “at least in part, an Anthropic project” that had solved at least one Millennium problem and would release soon. The rumor that started the race, in other words, was partly read off Alpöge’s own feed.

That makes the point of disagreement more precise: which paper was being discussed, what credit was offered, and whether the proposal was a good-faith attempt at coordination or an unacceptable condition. A screenshot-sized slogan cannot settle those questions.

Alpöge says he would have collaborated

In his September 8 reply, Alpöge opens with OpenAI’s “cannot rule out” sentence and writes, “props to them for straight coming clean.” He says he would have been “pumped to collaborate,” that he does not care about authorship on that step, and that it “coulda been me Tristan and every fte at oai for all i care.” But “on hearing the loud convo in the hallway, especially the part where a millennium prize was offered if i’d just be removed from the paper, it was kinda clear the die had been cast and things were locked.” He calls the episode “wacky, unstrategic, and unnecessary,” since on his side it was “mostly me and claude having a good time yoloing random stuff in the corner rather than anything institutional.”

Alpöge also makes a claim about the proof itself: that so far OpenAI’s argument “looks more along the lines of another Euler blowup proof we had.” Altman says the opposite, that once both were public “the approaches appear to be different.” No independent referee has tested either reading.

Bubeck replies directly that “nothing at all was locked,” that OpenAI was “willing to go above and beyond and have as many discussions as you would have liked,” and that the claim is “just untrue.” One side describes a closed door; the other says the conversation was still open.

The crowd weighs in

OpenAI’s data statement passed six million views within a day. The most-liked replies were not about the math. One asked whether the researchers had turned off “Improve the model for everyone” in their data controls, “because if you did and they still make this concession, that is highly concerning to every single user.” Under Alpöge’s post, a physicist wrote that two parties arriving at a result at the same time is common, but “I have never heard of one party offering lead authorship to a person from the competing party.” Those two replies are the whole story in miniature: a data question and a credit question, and neither answered.

Sam Altman backs his team

Altman’s public defense presents OpenAI as trying to coordinate generously after initially believing the researchers had also solved Navier-Stokes. He describes offers involving release priority, prize recognition, and Buckmaster’s potential role in presenting OpenAI’s proof.

He says “the team threatened us with unfounded accusations of plagarism,” defends the team’s integrity, and points to differences in the published approaches. He also concedes the origin of the race in one sentence: “It is true that we tried this because there were rumors on the internet last week that Anthropic’s models had solved a millennium problem and we were curious if ours could do it too.”

What the math world is saying

The participants have done most of the talking. The most cited outside voice is Terence Tao of UCLA, the Fields Medalist who has worked on the Navier-Stokes regularity problem for years. On the evening of September 7, hours after the pair’s papers appeared, he called their work “a remarkable achievement” for pushing the Córdoba and Martínez-Zoroa approach through to forced blowup for 3D Euler, formalized in Lean, and noted that the authors “had to release far earlier than planned due to external events.” He followed with an explainer of the results on his blog.

His warning about the AI side came first. On September 3, as the rumors spread, he described a scenario in which an AI company solves the problem while it “keeps the process to arrive at that ansatz almost completely out of public view,” so that “technically, one of the most prominent open problems in mathematics would now be solved; but there would be almost no value added to mathematics as a consequence.” On September 5 he cautioned against turning the problem into “a mere viral social media post advertising some benchmark progress.”

After the announcement he went further. On September 8 he wrote that “even the rumor of someone working on a 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,” and that the incentives “may now be pointing in the direction of no longer sharing any promising research directions with the broader community.” He also criticized “the refusal of AI companies to disclose their negative results, or reveal the process towards obtaining their solutions.”

That last point is the one a business owner should sit with. The dispute over credit will be argued out by the people involved. The incentive Tao describes, where telling anyone what you are working on invites a better-funded party to finish it first, applies to any owner with a half-built method and a paid AI subscription.

Anthropic, Alpöge’s employer, had made no public statement on the dispute as of September 9. Bubeck’s post frames the negotiation as one conducted with Anthropic by proxy; both researchers describe the project as personal.

Why established business owners should care: “Is my data going to be safe?”

This is one of the first questions business owners ask me about AI. And when you have spent years building a company, it is a reasonable question.

Your valuable information is not necessarily a patent or a scientific discovery. It might be how you price a job, which customers are most profitable, the process your team uses to deliver results, or the proposal that helps you win against a competitor.

An established business has years of experience captured in those documents. When you give an AI access to them, you are trusting another company with part of what makes your business valuable.

The researchers were using AI tools to develop unpublished work. When OpenAI announced a related breakthrough, questions followed about whether their private work could have influenced its models. The public record does not establish that OpenAI took their research. But the uncertainty touches the exact concern business owners raise: what happens to the information I give these tools?

“Safe” means more than “Will it train on my data?”

• Will my information be used to improve the provider’s models?
• Who can access it, including coworkers or connected services?
• How long will copies be kept, and what happens when I delete something?
• Could an agent share it or act on it without my approval?

Those are different risks. A promise about model training does not, by itself, answer all four.

Before connecting your customer records, pricing spreadsheets, or internal processes, ask whoever is setting up your AI: “Show me which company account we are using, what happens to our data, which files the agent can access, and what it can do without asking us.”

Three moves for anyone using the ChatGPT desktop app or Codex

Buckmaster was a paying customer whose drafts sat in Codex sessions. OpenAI’s answer to him was that no user data was looked up, and that de-identified usage data helping improve its models could not be ruled out. Plan for the second sentence, not the first.

1. Pick the account that holds company work, then check its training setting. On a personal Free, Plus, or Pro account, open Settings, then Data Controls, and turn off “Improve the model for everyone.” OpenAI says the setting applies to the whole account, on every device; confirm in your plan’s terms that it covers your Codex sessions too. Step-by-step screenshots for both settings are in Two Steps to Stop ChatGPT From Training on Your Data. For a permanent opt-out that survives future setting changes, file a “Do not train on my content” request through OpenAI’s Privacy Portal; it applies to personal ChatGPT accounts, not business workspaces or the API. On a Business, Enterprise, or Edu workspace, OpenAI does not train on workspace data by default. Proprietary work belongs on the workspace with that default and an administrator who can see the settings.
2. Treat “local” as “sent.” OpenAI’s security documentation says local execution “does not mean offline or device-only model inference,” and that file excerpts, prompts, screenshots, browser content, and tool results may be sent to OpenAI to complete a task. Decide what is in a folder before you point the agent at it. Keep pricing logic, client lists, and unfinished methods in a folder the agent never sees unless the task needs them, and hand it a sanitized excerpt when it does. A prompt that says “keep this confidential” does not change the provider’s terms.
3. Keep your own dated record of valuable work. Buckmaster could show what he had, and when, because he had a Lean-verified proof dated August 22 and an email dated September 3. Save dated drafts and source notes in company-controlled storage, not only in chat history. OpenAI’s own documentation warns that not every local file operation, screenshot, or browser action appears in its compliance logs, so your record is the complete one. Check what each connected app receives too; a connector follows the source system’s rules, not the chat’s.

Paste this into your AI tool first

I am about to use this account for confidential company work. Before I do, answer in plain language:
1. Which plan and workspace am I signed into right now?
2. Under this plan, are my conversations, uploaded files, or Codex sessions used to train or improve your models by default? Where exactly do I turn that off?
3. How long are my conversations, files, and agent task records kept after I delete them?
4. When you work on files on my computer, what leaves my computer?
5. Which connected apps or tools can this account reach, and what do they receive?
Cite the official documentation page for each answer.

Then check the answers against the linked documentation. A model can misdescribe its own terms.

Longer term: how to think about models and your data

Every prompt is a disclosure to a company with its own interests, one that may compete in your category tomorrow or be racing someone in it today. Price that the way you would price handing a draft to a consultant who also serves your rivals.

Separate access from influence. Direct access means someone retrieved your file. Training influence means your earlier material shaped the model. Vendors deny the first firmly and hedge the second. Ask about the second in writing before it matters.

The useful question is where the work goes. Plan tier, workspace, retention window, connected apps, and admin controls decide what happens to data. The product name decides nothing. OpenAI’s documentation says deleting a conversation “doesn’t immediately purge every related artifact.” Expect that pattern from every provider.

Match sensitivity to tier. Routine work can run anywhere. Work that is the business, meaning methods, pricing, unreleased offers, and client data, belongs on a tier with no-training defaults, a retention window you have read, and an export path. Some work stays inside the company entirely.

Assume the gap between “cannot rule out” and “did not” will stay open. Vendors can rarely prove a negative about training data. Governance happens before the material enters the system, because afterward a general statement is the best answer you will get.

The more valuable the work you give an AI, the more clearly you should be able to explain where it goes.