# training-data 68

Published articles for training-data 68.

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## Some thoughts on the Navier-Stokes Millennium Prize Problem

DevFeed: [Some thoughts on the Navier-Stokes Millennium Prize Problem](<https://devfeed.tech/articles/some-thoughts-on-the-navier-stokes-millennium-prize-problem-30512.md>)

Original publisher: [Read original article](<https://simonwillison.net/2026/Sep/8/on-navier-stokes/>)

Author: Simon Willison

Published: 2026-09-08T23:55:12Z

Content type: opinion

Language: en

Sources: [Simon Willison](<https://devfeed.tech/sources/simon-willison.md>), [Simon Willison's Weblog](<https://devfeed.tech/sources/simon-willison-s-weblog.md>)

Topics: [OpenAI](<https://devfeed.tech/topics/openai.md>), [Mathematics](<https://devfeed.tech/topics/mathematics.md>), [Claude](<https://devfeed.tech/topics/claude.md>), [codex](<https://devfeed.tech/topics/codex.md>), [Math and Logic](<https://devfeed.tech/topics/math-and-logic.md>), [Lean](<https://devfeed.tech/topics/lean.md>)

Tags: [agents](<https://devfeed.tech/tags/agents.md>), [ai](<https://devfeed.tech/tags/ai.md>), [ai-2-235](<https://devfeed.tech/tags/ai-2-235.md>), [ai-ethics](<https://devfeed.tech/tags/ai-ethics.md>), [ai-ethics-342](<https://devfeed.tech/tags/ai-ethics-342.md>), [claude](<https://devfeed.tech/tags/claude.md>), [codex](<https://devfeed.tech/tags/codex.md>), [generative-ai](<https://devfeed.tech/tags/generative-ai.md>), [generative-ai-1-981](<https://devfeed.tech/tags/generative-ai-1-981.md>), [llms](<https://devfeed.tech/tags/llms.md>), [llms-1-947](<https://devfeed.tech/tags/llms-1-947.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [mathematics-22](<https://devfeed.tech/tags/mathematics-22.md>), [openai](<https://devfeed.tech/tags/openai.md>), [openai-463](<https://devfeed.tech/tags/openai-463.md>), [paper](<https://devfeed.tech/tags/paper.md>), [training-data](<https://devfeed.tech/tags/training-data.md>), [training-data-68](<https://devfeed.tech/tags/training-data-68.md>)

### AI overview

This commentary examines OpenAI's reported resolution of the Navier-Stokes existence and smoothness problem with an unreleased model, alongside accusations that the effort may have drawn on information from related work by mathematicians using Claude and Codex. It also describes questions about timing, data access, authorship, and OpenAI's subsequent use of agents and Lean formalization.

### Source excerpt

On the Navier-Stokes Millennium Prize Problem introduces an impressive result from OpenAI, who used an unreleased model to produce a resolution to the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize Problems that have been subject to a $1,000,000 prize since May 24th, 2000. The discovery is somewhat overshadowed by accusations of skulduggery from Tristan Buckmaster, an NYU mathematics professor who was collaborating on related problems with Levent Alpöge, an accomplished mathematician who currently works for Anthropic. Tristan's complaint accompanied a hastily published version of their own results. Here's the PDF describing what happened. The very short version is that Tristan and Levent worked on the problem for almost a year, making extensive use of Claude and Codex (mainly GPT-5.6 Sol), then had a breakthrough on August 15th. The mathematical rumour mill kicked into gear and Tristan and Levent heard that OpenAI had heard that Anthropic had resolved "a major open problem", so they reached out and learned that OpenAI had a team working on a related problem, with a similar approach. Quoting Tristan: I asked when the first prompt had been 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. It gets more complicated from there. The OpenAI team offered to wait for Tristan to publish, or to have him author a paper about their result, but were clear that Levent would not be invited as a co-author due to OpenAI's competitive relationship with his employer. Here's how OpenAI described their work: On Tuesday, September 1, we h