{"id":3167,"date":"2026-09-15T13:05:35","date_gmt":"2026-09-15T10:05:35","guid":{"rendered":"https:\/\/www.lbscience.org\/en\/2026\/09\/15\/pocket-money-for-openai\/"},"modified":"2026-09-19T19:57:06","modified_gmt":"2026-09-19T16:57:06","slug":"pocket-money-for-openai","status":"publish","type":"post","link":"https:\/\/www.lbscience.org\/en\/2026\/09\/15\/pocket-money-for-openai\/","title":{"rendered":"Pocket Money for OpenAI?"},"content":{"rendered":"

The Navier\u2013Stokes equations describe the behavior of fluids, such as liquids, and are therefore highly important and can be used to describe a range of phenomena. You can read about them in our post, which presents a physics-centric perspective on the subject and reviews the state of research at the time it was written, in 2021 [1].<\/p>\n

The Navier\u2013Stokes equations are included in the list of seven \u201cMillennium Problems\u201d defined by the Clay Mathematics Institute in 2000. Solving any one of the seven problems carries a prize of one million dollars [2]. In the case of the Navier\u2013Stokes equations, the prize will be awarded for solving at least one of four questions concerning specific cases of the equations, and now two of them have been solved.<\/p>\n

The equations describe flow over time, i.e., they represent the fluid\u2019s behavior over time according to a particular initial state, which includes the velocity at every point. Until now, it was known that in three dimensions, for every initial state, a solution to the equations can be found that remains valid for a finite period of time. However, it was not known whether a solution always exists that remains valid \u201cindefinitely\u201d. The result now published shows that in three dimensions, there are initial conditions for which the equations have no solution for infinite time.<\/p>\n

According to an OpenAI report from September 8, 2026, the model that solved the problem is an internal company model that is not yet available to the general public [3]. The problem itself was solved using a combination of ten thousand AI agents, over approximately 88 hours. A further 17 hours were devoted to producing a formal proof and verifying its correctness with a computer. The company published the formal proof, but as of the time of writing, it has not released the model\u2019s full \u201creasoning\u201d process that led to the solution.<\/p>\n

And now for the not-so-scientific part: alongside this breakthrough, questions have arisen concerning credit for the mathematical progress. A rumor recently spread through the mathematics community that a solution had been found to one of the Navier\u2013Stokes problems proposed for the prize, but no official information had been published. There was even a rumor that such a solution had been found by Anthropic. In light of this, according to the announcement [3], OpenAI decided to test whether its internal model could tackle one of these problems, or similar but easier problems. After successfully addressing a similar but less general problem, it moved on to the Navier\u2013Stokes problems.<\/p>\n

At the same time, shortly before OpenAI\u2019s official announcement, two mathematicians, Tristan Buckmaster and Levent Alp\u00f6ge, posted their progress on these problems on Mastodon, and also credited additional mathematicians [4]. According to the post, they solved, among other things, the less general problem\u2014the one on which OpenAI\u2019s model had practiced. This is a tremendous achievement and a step toward solving the corresponding Navier\u2013Stokes problems, but it does not qualify for the monetary prize.<\/p>\n

Among other tools, the two mathematicians used OpenAI tools for their research, raising concerns that OpenAI may in fact have used information generated through their work [5]. The company maintains that no direct use was made of this information, but it cannot rule out the possibility that information from the mathematicians\u2019 work entered the training process of the model that solved the problem. Could such pieces of information, generated during the mathematicians\u2019 work and incorporated into the model\u2019s training, have helped it solve the mathematical problem? For now, the question remains open.<\/p>\n

The current breakthrough joins a recently emerging list of open mathematical problems solved by \u201cartificial intelligence\u201d [6]\u2014a development that raises philosophical, ethical, and practical questions about the practice of mathematics in the age of AI.<\/p>\n

Hebrew editing: Smadar Raban
\nEnglish editing: Elee Shimshoni<\/p>\n

\n

References<\/strong>:<\/p>\n

    \n
  1. Post about the Navier\u2013Stokes equations<\/a><\/li>\n
  2. Clay Mathematics Institute publication presenting the \u201cMillennium Problems\u201d concerning the Navier\u2013Stokes equations<\/a><\/li>\n
  3. OpenAI\u2019s announcement on Navier\u2013Stokes<\/a><\/li>\n
  4. Tristan Buckmaster\u2019s post regarding the progress of his and Alp\u00f6ge\u2019s research<\/a><\/li>\n
  5. Tristan Buckmaster\u2019s statement<\/a><\/li>\n
  6. Post about an AI disproof of an Erd\u0151s conjecture<\/a><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

    The Navier\u2013Stokes equations describe the behavior of fluids, such as liquids, and are therefore highly important and can be used to describe a range of phenomena. 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