OpenAI announces AI proof for Navier-Stokes Millennium Prize, faces academic allegations

OpenAI's Claims Of Solving Million-Dollar Math Problem Marred By Allegations From Academic Authored by Jacob Burg via The Epoch Times, OpenAI announced on Sept. 8 that one of its internal artificial intelligence (AI) models had found the solution to a generations-long mathematical problem that deals with the natural mechanics of fluids like water and air. The Navier-Stokes equations govern how humans understand weather, ocean currents, blood flow, and aircraft and vehicle aerodynamics, and have been around for nearly 200 years. Mathematicians have spent decades trying to understand how smooth three-dimensional fluids can break down, and whether all the equations converge into sensible solutions despite their widespread success in various applications. This question, known as the Navier-Stokes existence and smoothness problem, or Navier-Stokes for short, was selected in 2000 as one of the Clay Mathematics Institute's Millennium Prize Problems - considered the seven most important mathematical problems. Anyone who solves one of the problems is offered $1 million in prize money. OpenAI said an internal model "significantly more capable than GPT-6 Astra" solved the Navier-Stokes existence and smoothness problem. If so, it would mark one of the most significant advances in AI technology to date, even after previous AI models had solved other critical math problems. "This is a Deep Blue-Kasparov moment," New York University (NYU) mathematician Tristan Buckmaster wrote in a statement released on Monday, referring to the moment a supercomputer beat world chess champion Garry Kasparov nearly 30 years ago. "The community needs to have serious and unhurried discussion about where to go from here," he added. However, Buckmaster said in the same statement that he had been working on the problem with mathematician and Anthropic employee Levent Alpöge, and had made significant progress last month after using Anthropic's Claude and OpenAI's Codex. Before they could publish their work, Buckmaster said he contacted a prominent mathematician at OpenAI on Sept. 3 after rumors spread that Anthropic had solved a major open problem, and after Alpöge said he had received tips that information regarding the pair's progress on Navier-Stokes had been given to OpenAI. OpenAI told Buckmaster that an internal model had produced a proof after the firm gave it a prompt to solve the problem. Buckmaster then asked when the prompt was sent, but said he did not receive an immediate response. "Eventually it was agreed that it had been sent in the past few days, after information about our work had reached OpenAI," he said. The NYU mathematician asked OpenAI whether the model had been trained on or had access to his Codex sessions with Alpöge, since the pair had inputted their project drafts into the AI model. "I was told the model did not look up user data. I asked again, about training, and I did not get an answer," Buckmaster said. Buckmaster alleges that OpenAI offered him two paths forward: He and Alpöge could post their results about part of the problem, known as Euler, a day before OpenAI published results on Navier-Stokes; or Buckmaster alone could publish a paper on Navier-Stokes, acknowledge that an internal OpenAI model had solved it, and remove Alpöge as an author since he works for the company's competitor, Anthropic. In the first instance, OpenAI would cede the prize to Buckmaster, saying that they would have been the "closest humans to the problem." OpenAI, Buckmaster, and Alpöge did not respond to requests for comment by publication time. In its announcement on Tuesday, OpenAI said, "While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models." OpenAI mathematician and AI researcher Sébastien Bubeck wrote on X on Tuesday that he did not ask to remove Alpöge from authorship of his own work, but had said "it would be simpler" if he was not an Anthropic employee "because I felt it would be inappropriate for an Anthropic employee to author OpenAI's work." Sam Altman, CEO of OpenAI, also jumped into the fray on Tuesday, claiming that Alpöge refused to meet with his team, and that the Anthropic mathematician and Buckmaster had taken a different approach in working to solve Navier-Stokes. "It is true that we tried this because there were rumors on the internet last week that Anthropic's models had solved a millennium problem and we were curious if ours could do it too," Altman said. The comments did not address whether OpenAI had accessed Buckmaster's Codex logs, but Wired reported that the firm gave a closed press briefing on Tuesday where Bubeck and other OpenAI executives denied inspecting the logs or using them to inform their work. "We, whether it's the researchers or the agents, did not see any of their work until it was released publicly last night," Bubeck told reporters. Tyler Durden Wed, 09/09/2026 - 09:00

OpenAI said that an internal model 'significantly more capable than GPT-6 Astra' had produced a proof showing that the three-dimensional Navier-Stokes equations can develop a singularity in finite time, tackling one of mathematics' seven unsolved Millennium Prize Problems. This problem has allegedly remained unresolved for 90 years. How the breakthrough came togetherThe company said its effort began on September 1, after researchers heard rumours that two of the Millennium Prize problems had already been solved and turned its newest internal model toward the remaining ones. OpenAI executives told reporters the project consumed enormous computing resources, costing in the millions of dollars and involving roughly 10,000 agents working in parallel. The company later clarified that the effort began after hearing a rumour that it eventually traced back to Levent Alpoge, an Anthropic employee, and Tristan Buckmaster, a mathematics professor at NYU. OpenAI says the proof has been verified using the programming language Lean, which effectively guarantees its correctness.We’re sharing a solution to the Navier-Stokes Millennium Prize Problem, one of the deepest problems at the frontier of mathematics.The proof was produced by a group of agents, using an OpenAI next-generation model significantly more capable than GPT-6 Astra.The problem… pic.twitter.com/8zol3BPTL4— OpenAI (@OpenAI) September 8, 2026 A rival claim surfaces firstThe night before OpenAI's announcement, Buckmaster had posted that he and Alpoge had proved a related result, that the frictionless Euler equations can 'blow up', building on an approach pioneered by mathematicians Diego Cordoba and Luis Martinez-Zoroa. Buckmaster later said he and Alpoge had spent close to a year making slow progress on the problem before accelerating rapidly from mid-August using several AI models from both companies.drama summary for those confused:- Aug 15th: Tristan Buckmaster & Levent Alpöge make progress on a few important math problems- they do NOT have a proof for the $1,000,000 Millenium Prize problem. BUT, they do claim to have a proof for a similar (non-Millenium) Navier Stokes… https://t.co/b2CyqYPsf9 pic.twitter.com/PZIOfEyf4E— Konsti Wohlwend (@konstiwohlwend) September 8, 2026 Allegations of data access and pressure tacticsBuckmaster publicly questioned whether OpenAI had pursued a research direction it learned about from his and Alpoge's work, and raised concerns about whether private material from OpenAI's Codex tool could have played a role. According to his account, he asked OpenAI directly whether its model had been trained on or had accessed the Codex sessions where he and Alpoge had stored their drafts, and says he did not receive a clear answer on the training question.He further alleged that OpenAI researcher Sebastien Bubeck pushed to have Alpoge removed from authorship of any joint paper because of his employment at Anthropic, OpenAI's biggest competitor. Buckmaster said that when he threatened to make the exchange public, Bubeck responded by asking why he would want to 'ruin your career'.OpenAI's responseOpenAI said it did not access Buckmaster or Alpoge's specific user data while pursuing the proof. Bubeck denied the allegations in a post on social media, calling them false and inflammatory, and said in a press briefing that he had approached the discussion following standard academic norms while emphasising that OpenAI recognised the priority of Buckmaster and Alpoge's earlier work. OpenAI has said it reached out to the pair after completing its own proof and Lean verification, offering a concurrent release and recognition of their priority in a joint announcement.We congratulate Levent Alpöge and Tristan Buckmaster on their remarkable mathematical work.We (the researchers and the agents) did not see any of their work through any means until they released it publicly — in particular, no specific user data was accessed in order to solve… https://t.co/otJKRHnQgb— OpenAI (@OpenAI) September 8, 2026 Tao warns of a chilling effect on mathematicsFields Medalist Terence Tao, widely regarded as one of the world's leading mathematicians, raised broader concerns about AI labs using long-standing open problems as demonstrations of their models' capabilities. He said he had hoped AI would serve mathematicians the way a microscope serves biologists, but worried it was increasingly being used to produce answers without the accompanying insight into why a particular approach worked - insight he says often matters more to the field's progress than the answer itself. He also argued that AI companies rarely disclose the failed approaches their models tried, even though such dead ends often seed future breakthroughs, and warned that treating open problems purely as targets to be 'solved' could erode incentives for mathematicians to share unfinished work.Still more absolute from Terence Tao: “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… pic.twitter.com/vUrpxRgQA3— Gary Marcus (@GaryMarcus) September 8, 2026