OpenAI Flags Progress on Another $1 Million Math Problem Before the First Is Verified

Three days after publishing a Lean-verified proof claiming to resolve the Navier-Stokes Millennium Prize problem, OpenAI told the New York Times on Wednesday that it has already made "substantial progress" on a second problem from the same seven-problem set and is working through how to share the results publicly. The company has not named the problem, but circulating reports from sources familiar with the effort point to the Hodge conjecture, a decades-old open question in algebraic geometry. Separately, unconfirmed social-media accounts suggest Anthropic may be pursuing the Birch and Swinnerton-Dyer conjecture, though no official statement from that company has emerged.

None of this is certified mathematics yet. The Clay Mathematics Institute — which established the seven Millennium Prize Problems in 2000 and controls the $1 million award attached to each — still lists the Navier-Stokes problem as unsolved, and its president said evaluation would be "deliberately unhurried." Meanwhile, the first announcement has entangled OpenAI in a credit dispute involving an NYU mathematician and an Anthropic researcher who had been pursuing the same family of problems before OpenAI moved in.

What OpenAI's Navier-Stokes Proof Actually Claims — and What It Does Not

The Navier-Stokes equations, written down in the 19th century, describe how fluids such as air and water move. The central open question — whether the equations can produce a point of infinite velocity, or "blow up," in finite time — has resisted mathematicians for roughly 90 years. OpenAI's proof, published on September 8 alongside a 165-page analytical paper and a Lean 4 formalization, argues the answer is yes: given a smooth external force applied to an initially still fluid, a vortex can spiral inward, elongate, and accelerate until velocity diverges at its center.

The distinction between "forced" and "unforced" variants of the problem determines whether this result qualifies for the Clay prize. The official problem statement, written by mathematician Charles Fefferman, contains four options: A and B ask whether smooth, divergence-free initial conditions — with no external force applied — can evolve into a singularity. Options C and D allow an external force. OpenAI's proof addresses C and D. Most working mathematicians consider A and B the harder and more physically meaningful version of the question, because real fluids are not externally driven into singularity by a hand-chosen force. OpenAI acknowledged this and said it would not claim the $1 million prize.

The Lean formalization matters. Lean is a formal proof assistant that verifies whether each logical step in an argument follows from the previous one under the definitions encoded in the system. An independent researcher can download OpenAI's Lean code and confirm it compiles without errors. What Lean cannot verify is whether the Lean theorem statements faithfully capture the Clay problem — that mapping from mathematical English to formal code is done by humans and requires expert scrutiny. The Clay Institute has not completed that scrutiny, and mathematicians say the evaluation process will take at least two years.

How 10,000 Agents Searched for a Proof

OpenAI deployed a coordinating system of AI agents running on an unnamed internal model it describes as significantly more capable than GPT-6 Astra, the company's most advanced publicly available system. The training of this internal model began August 28, roughly a week before the proof run started.

The architecture operates in layers. Agents are organized into communicating subgroups of varying sizes; groups can share partial results with each other, and OpenAI used its Codex model to help consolidate useful ideas from across groups. The agents had two external capabilities: the ability to read from a cached snapshot of the internet and the ability to execute code in isolated environments. Neither capability involved access to any live user data, OpenAI said.

The run proceeded in two stages. A smaller group of approximately 100 agents spent around 50 hours producing a related result: a proof that the unforced Euler equations — the frictionless relative of Navier-Stokes with no external force applied, which removes the viscosity term — can blow up in finite time. Researchers fed that intermediate result into the Navier-Stokes effort. The larger group of approximately 10,000 concurrent agents then spent 88 hours and generated 2.7 million internal messages and roughly 130 billion output tokens before arriving at the singularity construction. GPT-6 Astra then spent an additional 17 hours formalizing and verifying the argument in Lean. The compute cost for the full project was estimated in the millions of dollars, with some secondary reporting placing the figure near $22 million.

The blowup mechanism the agents found is geometrically intuitive once stated: a vortex at rest begins spinning under an applied force, progressively elongating and tightening. As the spatial region containing the most extreme motion shrinks, its kinetic energy contribution stays bounded — but the velocity at the center grows without limit. OpenAI's researcher Ven Chandrasekaran explained the result by noting that such behavior is physically impossible in real fluids, which suggests the equations themselves may not be a perfect mirror of physical reality under extreme conditions.

A Credit Dispute That Lean Cannot Settle

Levent Alpöge, a mathematician employed by Anthropic, and Tristan Buckmaster, a professor at NYU's Courant Institute, had spent approximately a year using a combination of Anthropic and OpenAI's own tools — including OpenAI's Codex — to attack the same family of fluid equations. Their approach extended a "cascade" technique developed by Diego Córdoba and Luis Martínez-Zoroa, who had themselves spent years building a framework for constructing fluid singularities analytically. On September 7, the day before OpenAI's announcement, Alpöge and Buckmaster published proofs showing forced blowup for three related equations: the incompressible porous medium equation, the Boussinesq system, and the three-dimensional Euler equations.

In a statement released simultaneously, Buckmaster said OpenAI had become aware of their unpublished work and adopted the same approach. He alleged that Sébastien Bubeck, who leads OpenAI's mathematics team, pressured him during a phone call to exclude Alpöge's name from any joint announcement, specifically because of Alpöge's Anthropic employment. According to TechCrunch's reporting of Buckmaster's account, when Buckmaster refused and threatened to go public, Bubeck replied "Why would you ruin your career?" and later "If you don't want me to be nice, then I don't have to be nice."

Bubeck denied the account in a press briefing. OpenAI said its agents had not seen any of Buckmaster and Alpöge's work "through any means" before it was made public and that no specific user data was accessed. In the original September 8 publication, OpenAI acknowledged it "cannot rule out that de-identified data derived from their usage of our products helped improve our models." Then, in a September 10 update after conducting an investigation, the company stated it had "confirmed that Buckmaster's Codex prompts over the two months preceding this announcement and paper on September 8, 2026, could not have influenced the system in any way, including through training."

Buckmaster had used Codex extensively throughout the project, feeding entire drafts of mathematical arguments into the model. OpenAI's investigation concluded those sessions had no influence on training — but the company does not publish training data logs, and no external audit mechanism exists to independently verify that conclusion. The Lean verification of the proof can confirm its internal logical validity; it cannot confirm where the ideas originated.

Terence Tao, a Fields Medal laureate at UCLA who had written on September 7 that Alpöge and Buckmaster's results were "a remarkable achievement" and that singularity now looked "very feasible," later told CNN that the race dynamic was "turning the subject into a meaningless production quota game that ultimately is of very little benefit, either to mathematics or to the world." He compared an AI proof without human insight into the reasoning to watching only the first and last ten minutes of a film.

What the Hodge Conjecture and Birch-Swinnerton-Dyer Are Actually Asking

The Hodge conjecture, proposed by Scottish mathematician William Hodge at the 1950 International Congress of Mathematicians, sits in algebraic geometry. High-dimensional geometric objects called complex algebraic varieties — defined by solutions to systems of polynomial equations — can be analyzed using topological tools that decompose their structure into abstract components called Hodge classes. The conjecture asks whether every such class can be expressed as a combination of genuinely algebraic pieces: sub-objects defined by polynomial equations.

Informally, the Hodge conjecture asks whether every "topological shadow" cast by an algebraic variety is itself algebraic in origin. Experts are deeply divided on whether the conjecture is true. Some believe it holds for all dimensions; others have searched for counterexamples, arguing that high-dimensional counterexamples might exist precisely because human intuition breaks down in those settings. A counterexample would require constructing a specific variety with a Hodge class that cannot be written as any algebraic cycle — the kind of constructive task where large-scale computational search could in principle identify candidate structures. Whether an agent system could find a full proof or disproof of the general statement is unknown, and the math community received the Hodge rumors with considerably more skepticism than the Navier-Stokes claims.

The Birch and Swinnerton-Dyer conjecture, proposed by British mathematicians Bryan Birch and Peter Swinnerton-Dyer in the 1960s using Cambridge's early EDSAC-2 computer, connects two very different objects in number theory: elliptic curves and their associated L-functions. An elliptic curve is a geometric object defined by a cubic equation; its "rank" measures how many rational-number solutions it has, and computing that rank for arbitrary curves remains extremely difficult. The conjecture asserts that the rank of the curve is encoded in the behavior of the curve's L-function at a specific point — linking a hard arithmetic quantity to a computable analytic one. A proof would resolve the congruent number problem, a question about which integers can be expressed as areas of right triangles with rational sides, which has been open for over a thousand years.

Neither the Hodge conjecture nor BSD has any independently confirmed breakthrough as of this writing. The reported progress — Hodge from OpenAI, BSD potentially from Anthropic — remains unannounced and unverified.

A Structural Problem the Proofs Cannot Solve

If OpenAI's second announcement holds up mathematically, the pace of collapse of the Millennium Prize list will be extraordinary: from six unsolved problems as of August 2026 to potentially three within weeks. But the Navier-Stokes episode has surfaced a structural problem that no Lean verification can address.

When researchers use frontier AI labs' tools to develop unpublished work, the lab possesses everything needed to launch a competing effort the moment it hears a rumor. Buckmaster and Alpöge did not know their Codex sessions might become a road map. The question Simon Willison posed publicly captures the issue precisely: if a researcher uses an AI lab's coding assistant to partially solve a Millennium Prize problem, what is the probability that the work influences training in a way that allows the lab to complete the solution independently and publish first? OpenAI's own investigation concluded that influence could not have occurred in this case — but that conclusion rests on an internal inquiry with no external audit.

The Clay Institute has a two-year waiting period before it will certify any result, and its existing rules give it discretion to withhold a prize if credit or correctness cannot be established. For the Navier-Stokes problem, both questions remain live. The proof itself awaits expert review. The credit dispute awaits a resolution that the current attribution system — designed for disputes between individual human researchers, not between a researcher and a lab that trained on the researcher's keystrokes — may not be equipped to deliver. What comes next for mathematics may depend less on which problem falls next than on whether the field can develop new norms fast enough to protect the research process that makes those problems worth solving.