By now, most people have heard that OpenAI has likely solved a Millennium Prize Problem; specifically, a proof of finite-time blowup for the forced three-dimensional Navier-Stokes equations, announced yesterday afternoon at 1:20 p.m. ET.@OpenAI, September 8, 2026, 1:20 p.m. ET As someone who has followed the story somewhat closely, along with the broader debate around AI in mathematics, I wanted to lay out a timeline of events and (hopefully) put this result in the context of the past year.
Trust me bro
At the beginning of 2026, AI in mathematics was still treated as something of a novelty. There were some interesting results, mostly on Erdős problems, mostly with commercially available AI models, and mostly from a few peoplePrimarily @AcerFur and @Liam06972452. on Twitter. After one such problem was solved in January, Terence Taothe Fields Medal-winning, MacArthur-recognized, field-defining mathematician, who kept a page tracking AI-assisted mathematical contributions,AI contributions to Erdős problems remarked that he was less interested in the solution itself than in "the emerging AI-powered capability to rapidly write and rewrite expositions of the solution."@tao, January 7, 2026.
The results began to look like less of a novelty in May, when OpenAI announced that an internal model had disproved the Erdős unit distance conjecture,An OpenAI model has disproved a central conjecture in discrete geometry, May 20, 2026. an open problem in discrete geometry. The result came with a companion paper by nine mathematicians that digested and simplified the argument.Alon et al., Remarks on the disproof of the unit distance conjecture, May 20, 2026. This seemed like an encouraging model for AI-assisted mathematics: the model produced a result, while mathematicians worked together to make it legible and useful to the field. The proof was treated not as the end of the process, but as the beginning of one. This human-centered focus, however, did not last very long.
By the summer, the character of the results had changed quite a bit. On July 19, Levent Alpöge (a mathematician and member of technical staff at Anthropic) posted a counterexample to the Jacobian conjecture:@__alpoge__, July 19, 2026.
hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final
((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0)
Notably, this was a black-box proof: although the counterexample itself was easy to verify, no paper accompanied it, nor was there any explanation of how it had been found. A better understanding emerged only afterward, through the work of other mathematicians (with assistance from AI models)@davikrehalt, July 20, 2026.. Alpöge subsequently posted a resolution of the Hadamard matrix conjecture on August 12@__alpoge__, August 12, 2026. and, on August 23, a proof that $S^6$ admits a complex structure.@__alpoge__, August 23, 2026.
The labs followed a similar trend. On August 1, OpenAI published ten results attributed to an internal version of Astra,Ten advances in mathematics and theoretical computer science, August 1, 2026. among them the first explicit construction of a non-sofic group. Unlike the unit distance result, this construction (and the nine other results) came without a companion paper by mathematicians with domain expertise. Instead, the proofs were entirely AI-written. Again, understanding had to come from outside. Francesco Fournier-Facio, a mathematician at Cambridge, independently worked out the criterion underneath the OpenAI results and used it to produce a broader family of examples, including torsion-free ones.A torsion-free non-sofic group, August 3, 2026.
Similarly, on August 10, Anthropic announced that an unreleased model had raised the lower bound on the proportion of zeros of the zeta function lying on the critical line from 41.6% to 67.2%, after a member of technical staff asked it to "take a real stab" at the Riemann hypothesis and then mostly sent it words of encouragement.Learning more about Claude's mathematical capabilities, August 10, 2026. Like several other announcements, this one came without a companion paper to help mathematicians understand the argument beyond assessing its correctness.
Hype moments and aura
Around this time, some mathematicians started pushing back. Their issue wasn't that the proofs were wrong; in fact, most were accompanied by (compiling) Lean certificates. One complaint was the writing. In a talk at IPAM,Reflections on the Foundations of Interpretability workshop, IPAM, UCLA. Tao described how AI-generated proofs would spend pages on trivial details and then a single line on the key idea. The other complaint was that mathematics was being used as a benchmark for AI companies, strip-mined@tao, September 2, 2026. for hype moments and aura by people with no particular interest in the field itself. IA recipient of the 2022 Fields Medal.n a blog post sarcastically titled "Care for a little more AI?", Hugo Duminil-Copin lamented that:Care for a little more AI?, Proofs and Prompts, August 30, 2026.
...the current use of AI does not empower us, it petrifies us. These artificial discoveries risk decapitating entire fields before they have had time to develop to their full potential. Even worse, they nuke the mathematical landscape, making it increasingly difficult to inhabit after each blast.
Tao (previously a vocal proponent of AI in math)How Terry Tao Became an Evangelist for AI in Math, Quanta, June 8, 2026. soon began making a similar case. He developed his argument in a sequence of Mathstodon posts from September 2 to September 5.
In the first post, Tao described the idea of contamination,A term from AI evaluation, where a problem whose solution is already public is no longer useful as a test. cited Duminil-Copin's article, and suggested that:@tao, September 2, 2026, 11:15 p.m. ET.
It may become necessary to declare certain classes of mathematical problems off-limits to automated solvers, in order to preserve their broader value to the mathematical ecosystem (for instance, through the training of future mathematicians). Admittedly this can be hard to enforce when such tools are both powerful and widely available.
In a second post (whose importance will be obvious later), he used Navier-Stokes to explain what could be lost if an important problem were solved without insight into the process that produced the answer:@tao, September 3, 2026, 11:37 a.m. ET.
[T]here is now a scenario in which an autonomous AI harness, backed by an enormous amount of computational resources, performs this entire iteration internally, and ends up producing the final ansatz, and thence the solution to the Navier-Stokes regularity problem, while the AI company running the harness keeps the process to arrive at that ansatz almost completely out of public view. 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.
Later that day, he put the concern more generally:@tao, September 3, 2026, 2:53 p.m. ET.
But the currently fashionable practice of pointing a powerful AI tool at the task of answering a problem $X$, unguided by any human expert in the field $X$ resides in, has created an unprecedented divergence between the production of answers, and the production of insight, to the point where the two questions have become negatively correlated.
He followed this with a case study from a few days earlier. On August 31, Julia StadlmannJ.L. Doob Research Assistant Professor at the University of Illinois. released a preprint lowering the best known bound on gaps between primes.arXiv:2608.31126, August 31, 2026. Tao described what happened next:@tao, September 3, 2026, 2:53 p.m. ET.
...we were treated to the unedifying spectacle of no fewer than three separate AI companies racing to announce their own improvement on the bounded gaps between prime result that I mentioned yesterday. These results are numerically stronger than Stadlmann's improvement, but I am very glad that Stadlmann was able to complete her analysis just in time before the problem became contaminated [...]
It's fun to make predictions
Unbeknownst to Tao, rumors about an AI lab solving two Millennium Prize Problems had been circulating before his post on Navier-Stokes. Motivated by these rumors and Tao's post, Andrew Curran tweeted:@AndrewCurran_, September 4, 2026, 10:26 p.m. ET.
It's fun to make predictions. Here's a new one: Anthropic has solved a Millennium Prize Problem. And I'll be even more specific. Claude has solved Navier-Stokes. It is out for expert review. And to give myself a hard deadline, they will announce it before the IPO.
The next morning, Tao added a clarification to his own thread:@tao, September 5, 2026, 11:19 a.m. ET.
[I]n response to recent rumors about a possible solution to the Navier-Stokes problem: I am not aware of any significant developments in this regard; the above discussion is hypothetical, but not completely implausible at the current level of development of AI technology.
The rumors continued regardless.
Let ten thousand agents bloom!
At 11:58 p.m. ET on Monday, September 7, Tristan BuckmasterProfessor of mathematics at NYU's Courant Institute. posted three results from a collaboration with Levent Alpöge: finite-time blowup with smooth forcing for incompressible porous media, for Boussinesq, and for 3D incompressible Euler, with Lean formalizations for each.@tristanbuckmaster, September 7, 2026, 11:58 p.m. ET. Alongside them, he posted a four-page statementstatement.pdf. describing the events leading up to the announcement.
According to his account, on September 3, with the rumors circulating, Buckmaster emailed a mathematician at OpenAI to correct the record. Over the following days, he was asked with increasing urgency to meet. On a call on Sunday, September 6, he was told that an internal OpenAI model had proved finite-time blowup for forced Navier-Stokes:
I asked when the first prompt was sent by them. This question was not answered directly by OpenAI for some time. Eventually it was agreed that it had been sent in the past few days, after information about our work had reached OpenAI. I asked whether the model had been trained on, or had access to, our sessions in Codex, into which we had been putting all our drafts for the whole of this project. I was told the model did not look up user data. I asked again, about training, and I did not get an answer.OpenAI says its researchers "did not see any of their work through any means until they released it publicly," but adds that "while unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models." On the Navier-Stokes Millennium Prize Problem.
He described two proposals:
Two proposals were offered to me. The first was that we post our Euler result, and that OpenAI post its Navier-Stokes result the next day. The second was that, after posting Euler, I alone write a paper presenting the Navier-Stokes result, acknowledging that an internal OpenAI model had resolved it. Sebastien twice asserted that he wanted Levent removed from authorship, and said it would all be simple if only it were not the case that, and it was so annoying that, Levent works at Anthropic. [...] I declined both offers.
OpenAI's announcement came thirteen hours later. Running ten thousand agents on an internal model for 88 hours, they had produced a proof and a Lean formalization that a fluid can develop a singularity in finite time. In other words, a "solution" to Navier-Stokes.@OpenAI, September 8, 2026. Shortly after, Sébastien Bubeck described how the effort had started:@SebastienBubeck, September 8, 2026, 1:39 p.m. ET.
We began working on the Millennium problems due to viral twitter rumors that Anthropic had resolved 2 Millenium [sic] problems.
He said they saw none of Buckmaster and Alpöge's work until it was released, and that they had assumed Anthropic was behind the rumors because Alpöge's posts use Anthropic internal models. He made sure to emphasize that:
although a group of people was involved in our efforts, we collectively had no research-level expertise in fluid dynamics and the Navier-Stokes problem, and therefore were unable to meaningfully contribute to the mathematical content.
The difference from May is shocking. The unit distance result had been posted alongside a companion piece written by nine outside domain experts, invited to review and explain the proof. The Navier-Stokes result, on the other hand, emerged from direct competition with two people already working on the problem. OpenAI focused on producing and verifying the result as quickly as possible, in an "unedifying" rush similar to the one Tao observed with the bounded-gaps result. The broader work of making the result understandable was largely neglected, and the lack of domain expertise was, if anything, touted as evidence of their accomplishment.
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Tao's post, ironically enough, seems to have become something of a self-fulfilling prophecy. By choosing Navier-Stokes as his example of how an AI-generated solution could inhibit the development of a field, he appears to have fueled existing rumors. Those rumors, in almost comically short order, helped set off a race to solve the problem, more or less in the way he described.
After OpenAI's announcement, he reflected on the state of mathematics:@tao, September 8, 2026, 4:32 p.m. ET.
We have now seen 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. The incentives may now be pointing in the direction of no longer sharing any promising research directions with the broader community, which would reverse centuries of traditions of open science and do serious long-term damage to the future of the field.
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For more details about the allegations that OpenAI plagiarized in-progress work from Buckmaster and Alpöge, read Simon Willison's account. If you're interested in what other mathematicians think, I recommend reading Mathematicians Need to Act and What's Not New. For a wider range of perspectives, listen to or read Quanta's What Is Math For in the Age of AI?, a live recording from the recent ICM that brings together three mathematicians to discuss what they value in their work.
If this is all a bit too depressing, I recommend reading The 92-Year-Old Mathematician and the Teenage Apprentice, published a few days ago, about Joan Birman and the fifteen-year-old she took on as a mentee when she was 92.
The cover painting is Buffalo Trail: The Impending Storm by Albert Bierstadt, 1869.