OpenAI is basically using the Millennium Prize Problems as benchmarks to grind for higher performance scores, huh...

The Navier-Stokes equations saga of OpenAI is far from over...

The New York Times' latest follow-up has uncovered previously undisclosed details of this conflict.

The most critical one is that OpenAI updated its statement on user data:

Over the past two months, Buckmaster's Codex prompts could not have affected the internal model in any way, and any user input after July 3rd could not have entered this system either.

This is far clearer than OpenAI's previous statement that "it cannot be completely ruled out that anonymous user data once helped improve the model", and it also draws a timeline for whether the August research results were incorporated into the model.

However, this is still a unilateral statement from OpenAI, and there is no public technical material available for independent external verification at present.

Before the data dispute is settled, OpenAI dropped another bombshell:

After conquering the Navier-Stokes equations, the company has made "substantial progress" on another Millennium Prize Problem and is preparing to announce the results.

The latest rumor points precisely to the Hodge conjecture.

Wow, the paper for the first problem is still under review, and the second one is about to be submitted.

"Why are you ruining your own career?"

Let's briefly review the whole story of this controversy and the latest revelations.

Tristan Buckmaster, a mathematician at New York University, said that after he rejected the cooperation proposal put forward by OpenAI and stated that he might make their communication public, Sébastien Bubeck, head of research at OpenAI, said to him:

Why are you ruining your own career?

Buckmaster retorted how making the facts public could ruin a scholar's career. The other party then said:

If you don't want me to be nice, then I don't have to be nice.

Bubeck later admitted that the first sentence did come from him, but denied that it was a threat.

He then posted a screenshot of his chat with Levent Alpöge, Buckmaster's collaborator.

The screenshot shows that before the call, Bubeck proposed to coordinate the two parties' releases, recognize the research priority of the two, and was willing to share the proof and human prompts.

However, these messages can only prove that OpenAI expressed willingness to cooperate before the call, and cannot restore the call that really ignited the conflict.

What The New York Times supplemented this time is the more complete terms of that negotiation.

At that time, OpenAI had completed the Navier-Stokes proof, while Buckmaster and Alpöge had only advanced to the simpler Euler equations.

OpenAI offered Buckmaster several options:

• The two mathematicians will publish the Euler equation results first, and OpenAI will announce the Navier-Stokes proof the next day;
• Buckmaster will serve as the lead author to reorganize the paper generated by the OpenAI model;
• Provide Buckmaster with sufficient computing power to allow him to continue to complete his own research;
• OpenAI will also recommend that the $1 million Millennium Prize honor be attributed to Buckmaster and his collaborators.

It should be noted that OpenAI is not going to pay the $1 million directly, and there is no evidence that the two parties discussed a non-disclosure agreement.

This means that OpenAI is willing to publicly recommend the attribution of the prize.

But Buckmaster still felt that OpenAI was "buying him off".

The crux lies with his collaborator Alpöge.

Although Alpöge works at Anthropic, this research is his personal side project and was not initiated by Anthropic.

But Bubeck confirmed directly in an interview with The New York Times that OpenAI really did not want Alpöge to appear in the final paper:

In our view, how can an Anthropic employee participate in an internal OpenAI project?

Buckmaster believes that the research of the two used both Anthropic's Claude and OpenAI's Codex. The most reasonable approach should be for the two companies to set aside competition and jointly handle this achievement.

He described that OpenAI could not understand why he did not accept the deal, nor why he was unwilling to "throw Levent under the bus".

Bubeck was equally puzzled by this.

He told The New York Times that after OpenAI completed other mathematical proofs earlier, it had reached similar cooperation arrangements with external mathematicians, so he was surprised by Buckmaster's refusal to cooperate.

As for the "career-ruining" warning, Bubeck provided the following context:

Buckmaster stated that as soon as OpenAI released the proof, he would immediately raise a series of accusations to the media.

What he really wanted to ask was why risk damaging his own career with unsubstantiated accusations.

Bubeck admitted that it was "extremely poor wording" and said he took it back and apologized on the spot.

But Buckmaster obviously did not feel the "kindness" in it.

On Sunday, he turned his office at New York University into a temporary "war room":

The blackboard was filled with the timeline of events, and several colleagues sat beside him, discussing with him how to respond.

He also called Peter Sarnak, a mathematician at the Institute for Advanced Study in Princeton. After listening to the terms proposed by both parties, Sarnak said:

I have never heard of such bargaining.

He also praised Buckmaster for "acting in the way of a mathematician".

Near midnight on Monday, Buckmaster hastily published three papers that had not yet been fully polished, as well as a 1870-word statement of events.

He did not leave the office to go home until 2:30 in the morning.

A few hours later, OpenAI officially announced the completion of the Navier-Stokes proof.

What really stings mathematicians about this controversy may not just be the data issue.

Buckmaster believes that what is ignored in the quarrel is the path paved by human mathematicians for this breakthrough.

Several years ago, Spanish mathematicians Diego Córdoba and Luis Martínez-Zoroa proposed a new strategy to attack related fluid equations:

Through a series of precise external force "kicks", the fluid is continuously pushed to an extreme state, eventually forcing a singularity where the velocity approaches infinity.

Buckmaster was responsible for setting the problem, and Alpöge input the problem into Anthropic's model.

The two then continuously provided feedback and revised based on the model output, gradually solving more complex equations.

Buckmaster therefore emphasized:

The main intellectual credit should go to Luis and Diego.

According to the past research rhythm, this method could have been continued by different mathematicians, and it might take another ten years to reach Navier-Stokes.

But now, OpenAI has mobilized tens of thousands of Agents to compress this journey into just a few days.

On the one hand, predecessors spent several years proposing the core strategy, and on the other hand, AI used massive computing power to complete the final sprint.

So who exactly should this breakthrough be attributed to?

There is no answer to this question yet.

But what we can see is that OpenAI has already started working on the next one...

The next one may be the Hodge conjecture, and the model may be called Aeon

The latest revelation says that OpenAI has set its next target on the Hodge conjecture.

Long before the Navier-Stokes results were announced, Elliot Glazer, head of mathematics at Epoch AI, sorted out the chaotic sources of news at that time and mentioned that a mathematics professor heard from a so-called "credible channel" that:

It was not the Navier-Stokes equations that were conquered, but the Hodge conjecture.

At that time, this news was regarded as a rumor among rumors.

Now that OpenAI has confirmed that it has made "substantial progress" on another Millennium Problem, this old clue has been dug up again.

Although it remains to be confirmed, various rumors have made it the most highly anticipated answer so far.

So what exactly is the Hodge conjecture about?

Simply put, it tries to build a bridge between two worlds of mathematics.

One is topology, which focuses on how many "holes" there are in the whole high-dimensional space.

The other is algebraic geometry, which studies the geometric shapes characterized by polynomial equations.

The Hodge conjecture holds that in a specific class of complex geometric spaces, those special "holes" discovered by topology do not exist out of thin air, and there should be geometric structures composed of algebraic equations behind them. In other words:

Behind the invisible and intangible topological features in a space, there should be a "skeleton" that can be written out with equations.

This conjecture was proposed by British mathematician William Hodge in the 20th century, and was later included in the seven Millennium Prize Problems. It connects topology, complex geometry and algebraic geometry, and is also one of the most core and intractable problems in modern mathematics.

And it is still that unnamed internal OpenAI model that may challenge it.

OpenAI has only revealed that the model started training on August 28, and its mathematical ability has significantly exceeded GPT-6 Astra. Even during the Navier-Stokes project, it had not finished training.

Outside rumors say that its code name may be Aeon, but this name has not been confirmed by OpenAI.

It is this model that "keeps training while doing exercises" that is connected to a system composed of tens of thousands of Agents, trying multiple unsolved Millennium Prize Problems at the same time.

OpenAI revealed that the entire round of testing generated 4.9 million Agent messages and about 300 billion output Tokens. For the Navier-Stokes problem alone, about 10,000 parallel Agents were used, generating 2.7 million messages and consuming about 130 billion output Tokens.

It took only 88 hours from the start of the Agents to finding the solution, and then GPT-6 Astra spent another 17 hours completing Lean formalization and verification.

Navier-Stokes is only the first answer sheet publicly delivered by this brute-force problem-solving system.

Now, the second problem is already on the way.

OpenAI and Anthropic have been treating century-old conjectures as test sets

Putting the news of the past half month together, you will find that this is not an accidental mathematical breakthrough:

It is more like an AI problem-solving competition ignited by rumors and rapidly escalating.

The protagonists of the incident are of course OpenAI and Anthropic, the old rivals, and both are targeting the "ultimate boss" in the mathematical world such as the Millennium Prize Problems.

(Among the seven Millennium Prize Problems, only the Poincaré conjecture has been officially solved so far. If OpenAI's Navier-Stokes proof finally passes the review, there will be five left.)

How exaggerated this battle is, you can see clearly from a table:

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The most exciting part is of course that many problems that have plagued mankind for decades or even hundreds of years may be advanced rapidly.

But for mathematicians, the other side is also very real:

They raise questions, accumulate routes, and hand over unpublished drafts to AI tools, only to eventually find that the company behind the tool is racing against them.

Buckmaster said that he doesn't care about the $1 million prize, he just loves mathematics and wants to be part of this piece of history.

Now, he is indeed part of history, but in a way completely different from what he expected.

Facing The New York Times, the mathematician said:

It's very scary to fight against a trillion-dollar company. We are all afraid of OpenAI.

Reference links:

[1]https://www.nytimes.com/2026/09/10/science/tristan-buckmaster-openai-math-navier-stokes.html?unlocked_article_code=1.AFE.TK7P.Y7WvGPkfHaSk&smid=bs-share

[2]https://x.com/AndrewCurran_/status/2098083604853342688

[3]https://x.com/SebastienBubeck/status/2097379411691516310

This article is from the WeChat official account "QbitAI", author: Yishui, published with authorization from 36Kr.