FoxTPNL / Wikimedia Commons (CC BY 4.0) The company says it didn't access unpublished research stored in its own platform, but mathematicians involved aren't fully satisfied with that answer.
OpenAI says it cracked one of the hardest unsolved problems in mathematics. Two researchers think the company’s AI might have peeked at their homework along the way.
On September 8, OpenAI announced that its internal AI model had produced a solution to the Navier-Stokes existence and smoothness problem, one of seven Millennium Prize Problems established by the Clay Mathematics Institute. Only one of those problems, the Poincaré conjecture, had ever been solved before. If verified, this would represent the second solution in the program’s 26-year history, and the first generated primarily by artificial intelligence.
What OpenAI claims to have done #
The solution was reportedly generated by approximately 10,000 autonomous AI agents working in coordination over 88 hours, beginning around September 1. The computational effort was followed by a 17-hour period of formal verification using Lean, a proof assistant language, conducted with the recently released GPT-6 Astra model.
The result specifically demonstrates the possibility of a finite-time singularity, or “blow-up,” in three-dimensional solutions to the Navier-Stokes equations. In plainer terms: the AI found a scenario where the equations describing fluid flow break down, producing infinite values in finite time. That matters because the Millennium Prize challenge essentially asks whether such breakdowns can occur.
The computational bill reportedly ran as high as $15 million. OpenAI has published the proof and supporting documentation but declared it will not claim the $1 million prize associated with the problem.
The result addresses specific permitted statements, labeled “C” and “D,” within the Millennium Prize formulation rather than the entirety of the problem. The proof has not yet been published in a peer-reviewed journal, and the broader mathematical community is still working through the verification process.
The data access controversy #
The mathematical achievement, impressive as it may be, isn’t what’s generating the most heated debate. That distinction belongs to a dispute involving NYU mathematician Tristan Buckmaster and Anthropic researcher Levent Alpöge.
Buckmaster and Alpöge had been working on related fluid dynamics problems for nearly a year. They released their own findings on or around September 7, just one day before OpenAI’s announcement. The deeper concern involves Codex, OpenAI’s coding and research platform. The researchers have raised questions about whether OpenAI’s AI systems may have inadvertently accessed their unpublished work stored within Codex during the course of their own discussions on the platform. Buckmaster has been careful with his language, clarifying that he is not directly accusing OpenAI of misconduct, but has publicly criticized the company’s assurances about data security and raised pointed concerns about authorship.
OpenAI chief research officer Mark Chen responded by emphasizing that no user data was searched to produce the solution, stating that the company maintained full compliance with its data policies. OpenAI has firmly disputed any allegations of improper data use.
While OpenAI states that no specific user data was accessed in the process of solving the problem, it has not ruled out the possibility of broader training exposure. The distinction between “we didn’t look at your files” and “your files were never part of any training pipeline” is the kind of gap that keeps privacy lawyers employed.
Why the Lean proof matters, and why it’s not enough #
OpenAI’s case for the validity of its result leans heavily on the formal Lean verification. Lean proofs are machine-checkable, meaning each logical step can be independently validated by software. This provides a layer of confidence that traditional mathematical proofs, written in natural language, don’t automatically offer.
But formal verification confirms internal logical consistency. It doesn’t settle whether the right problem was actually solved, or whether the specific statements addressed fully satisfy the Clay Mathematics Institute’s criteria for awarding the prize. The Institute itself has historically taken years to evaluate claimed solutions. Grigori Perelman’s proof of the Poincaré conjecture, for instance, was published in 2002-2003 but wasn’t officially recognized until 2010.
OpenAI spent roughly $15 million to potentially solve a problem with a $1 million prize it won’t even collect. The real cost may end up being measured in researcher trust.
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