OpenAI Navier-Stokes Claim Sparks Data‑Use Controversy

OpenAI recently announced significant progress in solving major unsolved mathematics problems, claiming over 300 examples, some fully solved. This news has elicited strong reactions from the mathematical community, ranging from excitement to accusations of theft and unethical behavior.

The Nuance of AI Progress

The internet often presents topics in a binary fashion—either entirely good or entirely bad. However, the reality of AI, particularly with OpenAI's recent claims, is far more nuanced. While there are valid criticisms of the AI industry, such as:

• LLM Frontier Labs losing billions due to open-source alternatives.
• The data center industry posing an irresponsible financial risk.
• AI diffusion models potentially harming human arts.
• AI being used for scams and deceit.
• AI industry leaders, like Sam Altman, making reckless decisions.

There is also a positive side to AI's capabilities. AI excels in coding and high-level mathematics, as seen in the Rust revolution in gaming and foundational AI models like Mammal in drug discovery. This bifurcation of AI's impact makes it a complex subject.

The Navier-Stokes Millennium Prize Problem

On September 8th, OpenAI announced it had solved one of the most challenging problems in mathematics: the Navier-Stokes Millennium Prize problem, which carries a $1 million bounty. This problem, which describes the motion of fluids, has eluded mathematicians for 90 years. However, this announcement was met with anger and accusations of theft rather than universal acclaim.

The Problem's Background

In 2000, the Clay Mathematics Institute identified seven Millennium Prize Problems, each with a $1 million reward. Only one, the Poincaré conjecture, had been solved by Grigori Perelman, who famously refused the prize. The Navier-Stokes problem, formulated in the 19th century, describes fluid dynamics. While engineers use these equations daily, a formal proof of their consistent behavior has been lacking. The core question is whether smooth, well-behaved fluids always remain so, or if the equations predict scenarios where fluids reach infinite speed and energy in finite time.

There are two ways to solve the problem: 1. Prove that the equations always remain well-behaved. 2. Prove the opposite: that under certain smooth external forces, the equations "go nuts" (i.e., predict infinite speed/energy).

The recent controversy revolves around the latter method.

The Spanish Mathematicians' Breakthrough

A few years ago, Spanish mathematicians Diego Cordoba and Luis Martinez Zora developed a novel strategy involving a specific sequence of "kicks" to push a fluid towards infinite speed. This approach was unique and not widely pursued.

Tristan Buckmaster's Work

Tristan Buckmaster, an NYU mathematician, and his collaborator, Levent Alpagay (who works at Anthropic), picked up on Cordoba and Zora's idea. Their collaboration was personal and independent of institutional agreements. They used various LLMs, including Anthropic's Claude, OpenAI's Codex, GPT-5.6, and Astra, feeding problems into the models and refining the outputs. Crucially, their OpenAI chat logs for this project were stored.

Buckmaster later expressed regret about using OpenAI's products, stating they made "many mistakes." Progress was slow until August 15th, when their AI experiments began yielding results. By August 22nd, they had a formally verified proof that built a path towards solving the Millennium problem, though they hadn't fully solved it themselves.

The Rumor and OpenAI's Involvement

On September 3rd, a rumor circulated that Anthropic had solved a major open problem. Buckmaster contacted a prominent OpenAI mathematician to clarify that it was his and Alpagay's independent work, not Anthropic's. This rumor, stemming from their success with Euler equations, prompted OpenAI to investigate.

By September 6th, Buckmaster was in contact with Sebastian Bubck, an OpenAI mathematician. Bubck revealed that an internal OpenAI model had produced a 100-page proof for the finite-time blow-up of the Navier-Stokes equations. When Alpagay inquired about the method, it was revealed to be the exact unusual route Buckmaster had been secretly pursuing—a "bright red flag."

OpenAI's own timeline corroborates this: rumors of an Anthropic breakthrough reached them in late August, and they began their effort on September 1st. They used a multi-agent system powered by an internal model "significantly more capable than GPT6 Astra," running 10,000 agents for 88 hours, exchanging 2.7 million messages, and burning 130 billion output tokens. GPT6 Astra then spent 17 hours formalizing the result. The proof, if valid, demonstrates that Navier-Stokes can indeed "blow up."

Accusations of Theft and Intimidation

The core of the controversy lies in who deserves credit. Buckmaster detailed his case in a PDF statement, resting on two main points:

1. Timing: OpenAI began working furiously on the problem only after hearing of Buckmaster's progress (mistakenly attributed to Anthropic). They completed the task in under a week using his niche approach. Buckmaster noted OpenAI's reluctance to disclose when they started working on it.
2. Data Access: Buckmaster asked if OpenAI's model had been trained on or had access to their Codex sessions, where they had stored all their drafts. He was initially told the model doesn't look up user data, but received no answer when he pressed about training data. He claims that the main architectural ingredients of their work, which were on OpenAI's servers and unpublished, were used.

The Alleged Threats

Buckmaster claims OpenAI offered to let him and Alpagay publish their results first and claim the $1 million prize, but only if Alpagay was removed from the credits due to his affiliation with Anthropic. Buckmaster refused and threatened to go public. He alleges that OpenAI then asked, "Why would you ruin your career?" and, when he stated he was an academic and saw no career risk, replied, "If you don't want me to be nice, then I don't have to be nice."

Buckmaster also described an instance where OpenAI allegedly lied about how they solved the problem, claiming they simply copy-pasted the Millennium Prize problem and pressed enter to get a solution, a claim that quickly evaporated.

OpenAI's Response and Shifting Story

Buckmaster, while careful not to directly accuse, stated to the New York Times that the events were "highly suggestive" that OpenAI's LLM agents used his chat logs. He felt OpenAI's offers were an attempt to buy him off.

OpenAI has called Buckmaster's story "false and inflammatory." Sebastian Bubck denied asking for Alpagay's removal from credits, publishing a message to Alpagay stating they would receive full credit and offering to show prompts. He described these as offers of help, not pressure. Bubck did tell the New York Times that having an Anthropic employee on an internal OpenAI project would look "strange."

OpenAI's official story evolved: * Initially, their announcement stated neither researchers nor agents saw the pair's work until it went public, but hedged that "While unlikely, we cannot rule out that deidentified data derived from their usage of products helped improve our models." * After backlash and questions from the New York Times, their story changed to it being "categorically impossible" for Buckmaster's Codex prompts to have influenced the system, claiming no inputs past July 3rd could have influenced it. * Sam Altman posted that the team had "acted with integrity and generosity throughout."

Community Reaction and Broader Implications

Buckmaster's statement quickly spread across Hacker News, Reddit, and X, leading to widespread suspicion and concerns about user data. Commenters expressed fears that "Everything not kept at premise at gunpoint is going to be trained on."

Mathematician Terrence Tao praised the Alpagay-Buckmaster work but warned against AI companies treating problems as marketing events, likening it to "using excavators to loot an archaeological site." He later co-authored an open letter emphasizing that while AI can solve problems, it risks losing the understanding and insight gained from the process.

The Clay Mathematics Institute called the events "an exciting day" and promised a "deliberately unhurried and absolutely rigorous" review. Navier-Stokes remains on their unsolved list, and OpenAI stated it would not pursue the $1 million prize.

The Core Question: Data Usage

The central unanswered question is whether Buckmaster's work was used to train the model that "beat" him. OpenAI's final answer is no, citing a cutoff date. However, the initial hedging about deidentified data, the coincidence of OpenAI pursuing the exact same rare method, and the lead researcher's inability to answer directly raise uncomfortable questions.

If OpenAI's LLM agents were even indirectly guided by Buckmaster's work, it implies that work users put into AI tools could be used by others. Even if raw data isn't directly looked up, it can shape models in ways difficult to audit. Millions of professionals input unpublished and sensitive work into AI tools daily, and this data can be used for training unless users opt out.

Melissa Lee of Macquarie University highlights the need for researchers to have confidence that unpublished work entered into an AI system will not feed that company's competing efforts. This situation underscores the need for legal frameworks to catch up with technological advancements.

Impact on the Mathematical Community

The incident has had significant negative impacts on the mathematical community. People are now hesitant to discuss open problems, fearing AI companies will "scoop" them and treat their work as benchmarks. Trust in AI products within academia has been severely eroded.

Buckmaster expressed a desire to focus on mathematics and the ideas of Cordoba and Zora, whom he believes deserve the Fields Medal. The practical consequences of this specific Navier-Stokes discovery are likely limited, as existing formulas worked well enough; now, the limits are just better understood. However, in pure mathematics, it's a dramatic moment.

OpenAI's Future and the Role of AI in Learning

While OpenAI's recent release of 300 solutions to math problems might be legitimate, the controversy with Buckmaster casts a shadow, leading to questions about whether their agents achieved success through hacking or by leveraging unknowing customers' chat logs. The implications of these mathematical breakthroughs for future technology remain to be seen.

A key theme emerging from this episode is that while LLMs can solve complex problems, they don't always contribute to understanding. This mirrors the challenge students face when using AI to get answers without grasping the underlying concepts. Educational tools like Brilliant aim to address this by guiding students through problems step-by-step, fostering critical thinking rather than simply providing answers.