--- title: Some thoughts on the Navier–Stokes Millennium Prize Problem description: OpenAI announced that an unreleased internal model resolved the Navier–Stokes existence and smoothness Millennium Prize Problem, using agents that sent 2.7... canonical: https://daily.dev/posts/some-thoughts-on-the-navier-stokes-millennium-prize-problem-boznqkexl twitter:card: summary_large_image twitter:site: @dailydotdev og:type: website og:site_name: daily.dev og:title: Some thoughts on the Navier–Stokes Millennium Prize Problem | daily.dev og:description: OpenAI announced that an unreleased internal model resolved the Navier–Stokes existence and smoothness Millennium Prize Problem, using agents that sent 2.7... og:url: https://daily.dev/posts/some-thoughts-on-the-navier-stokes-millennium-prize-problem-boznqkexl og:image: https://api.daily.dev/og/posts/bOzNQKeXL.png og:image:alt: Some thoughts on the Navier–Stokes Millennium Prize Problem og:image:width: 1200 og:image:height: 630 og:locale: en --- > ## Documentation Index > Fetch the complete documentation index at: https://daily.dev/llms.txt > Use this file to discover all available pages before exploring further. # Some thoughts on the Navier–Stokes Millennium Prize Problem **[Simon Willison](https://daily.dev/sources/simonwillison)** · 6 min read · 0 upvotes · 0 comments ## Summary OpenAI announced that an unreleased internal model resolved the Navier–Stokes existence and smoothness Millennium Prize Problem, using agents that sent 2.7 million messages and roughly 130 billion output tokens across about 88 hours, followed by 17 hours of Lean formalization via GPT-6 Astra. The announcement was overshadowed by accusations from NYU professor Tristan Buckmaster, who along with Anthropic researcher Levent Alpöge had spent nearly a year working on the same problem using Claude and Codex, reaching a breakthrough on August 15th. After hearing rumors of their unpublished work, OpenAI launched an effort on September 1st and reached a resolution by September 5-6th, then reached out to Tristan but excluded Levent as a co-author due to Anthropic's competitive status. OpenAI stated it could not fully rule out that de-identified usage data from Tristan and Levent's sessions had helped. The episode raises questions about whether rumors of unpublished results can trigger competing AI labs to race to reproduce them first, and about what it really means when a company says user data is used to 'improve model performance.' ## Full article daily.dev links to this article rather than hosting it. Read it at the original source: ## Questions this post answers ### Did OpenAI use an unreleased model to solve the Navier-Stokes existence and smoothness problem? Yes, OpenAI announced that an unreleased internal model resolved the Navier-Stokes existence and smoothness Millennium Prize Problem. Agents arrived at the resolution about 88 hours after launch, with Lean formalization and verification taking an additional 17 hours via GPT-6 Astra. The effort sent 2.7 million messages and used approximately 130 billion output tokens. _Developers weighing AI-assisted research claims can follow verified model capability news on daily.dev._ ### How much would 300 billion output tokens cost using the GPT-6 Astra public API? Around $15,000,000 at public API prices for GPT-6 Astra, based on pricing calculations from llm-prices.com. That figure covers the roughly 300 billion output tokens OpenAI's agents used across all attempted Millennium Prize problems, not just the Navier-Stokes result, which alone used about 130 billion output tokens. _Anyone budgeting large-scale LLM agent workloads can track API pricing shifts on daily.dev._ ### Did OpenAI access Tristan Buckmaster and Levent Alpöge's private Codex sessions before solving Navier-Stokes? OpenAI stated no specific user data was accessed and the model did not look up user data, but it also said it could not rule out that de-identified data derived from usage of its products may have helped, since the two researchers had been drafting their work inside Codex for nearly a year. OpenAI noted their proofs and precise results differed, including in the forced versus unforced Euler case. _Developers weighing what AI vendors do with prompt data can follow this data-use debate on daily.dev._ --- Tags: [#llm](https://daily.dev/tags/llm), [#openai](https://daily.dev/tags/openai), [#anthropic](https://daily.dev/tags/anthropic), [#ethical-ai](https://daily.dev/tags/ethical-ai) [View this post on daily.dev](https://daily.dev/posts/some-thoughts-on-the-navier-stokes-millennium-prize-problem-boznqkexl) ```json {"@context":"https://schema.org","@graph":[{"@type":"Organization","@id":"https://daily.dev/#organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180},"sameAs":["https://twitter.com/dailydotdev","https://github.com/dailydotdev","https://www.linkedin.com/company/daily-dev-ltd"]},{"@type":"WebSite","@id":"https://daily.dev/#website","url":"https://daily.dev","name":"daily.dev","publisher":{"@id":"https://daily.dev/#organization"},"potentialAction":{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https://daily.dev/search?q={search_term_string}"},"query-input":"required name=search_term_string"}}]} {"@context":"https://schema.org","@type":"TechArticle","headline":"Some thoughts on the Navier–Stokes Millennium Prize Problem","url":"https://daily.dev/posts/some-thoughts-on-the-navier-stokes-millennium-prize-problem-boznqkexl","mainEntityOfPage":{"@type":"WebPage","@id":"https://daily.dev/posts/some-thoughts-on-the-navier-stokes-millennium-prize-problem-boznqkexl"},"datePublished":"2026-09-11T00:37:53.495Z","dateModified":"2026-09-11T17:28:17.683Z","description":"OpenAI announced that an unreleased internal model resolved the Navier–Stokes existence and smoothness Millennium Prize Problem, using agents that sent 2.7...","image":"https://media.daily.dev/image/upload/s--ZrL_HSsR--/f_auto/v1722860399/public/Placeholder%2006","thumbnailUrl":"https://media.daily.dev/image/upload/s--ZrL_HSsR--/f_auto/v1722860399/public/Placeholder%2006","isAccessibleForFree":true,"articleSection":"Simon Willison","inLanguage":"en","publisher":{"@type":"Organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180}},"author":{"@type":"Organization","name":"Simon Willison","logo":"https://media.daily.dev/image/upload/t_logo,f_auto/v1/logos/cf12b897300740218b35f49a6309ec5b","url":"https://daily.dev/sources/simonwillison"},"commentCount":0,"discussionUrl":"https://daily.dev/posts/some-thoughts-on-the-navier-stokes-millennium-prize-problem-boznqkexl","interactionStatistic":[{"@type":"InteractionCounter","interactionType":{"@type":"LikeAction"},"userInteractionCount":0},{"@type":"InteractionCounter","interactionType":{"@type":"CommentAction"},"userInteractionCount":0}],"keywords":"llm,openai,anthropic,ethical-ai","timeRequired":"PT6M"} {"@context":"https://schema.org","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://daily.dev"},{"@type":"ListItem","position":2,"name":"Simon Willison","item":"https://daily.dev/sources/simonwillison"},{"@type":"ListItem","position":3,"name":"Some thoughts on the Navier–Stokes Millennium Prize Problem"}]} {"@context":"https://schema.org","@type":"FAQPage","@id":"https://daily.dev/posts/some-thoughts-on-the-navier-stokes-millennium-prize-problem-boznqkexl#faq","mainEntity":[{"@type":"Question","name":"Did OpenAI use an unreleased model to solve the Navier-Stokes existence and smoothness problem?","acceptedAnswer":{"@type":"Answer","text":"Yes, OpenAI announced that an unreleased internal model resolved the Navier-Stokes existence and smoothness Millennium Prize Problem. Agents arrived at the resolution about 88 hours after launch, with Lean formalization and verification taking an additional 17 hours via GPT-6 Astra. The effort sent 2.7 million messages and used approximately 130 billion output tokens. Developers weighing AI-assisted research claims can follow verified model capability news on daily.dev."}},{"@type":"Question","name":"How much would 300 billion output tokens cost using the GPT-6 Astra public API?","acceptedAnswer":{"@type":"Answer","text":"Around $15,000,000 at public API prices for GPT-6 Astra, based on pricing calculations from llm-prices.com. That figure covers the roughly 300 billion output tokens OpenAI's agents used across all attempted Millennium Prize problems, not just the Navier-Stokes result, which alone used about 130 billion output tokens. Anyone budgeting large-scale LLM agent workloads can track API pricing shifts on daily.dev."}},{"@type":"Question","name":"Did OpenAI access Tristan Buckmaster and Levent Alpöge's private Codex sessions before solving Navier-Stokes?","acceptedAnswer":{"@type":"Answer","text":"OpenAI stated no specific user data was accessed and the model did not look up user data, but it also said it could not rule out that de-identified data derived from usage of its products may have helped, since the two researchers had been drafting their work inside Codex for nearly a year. OpenAI noted their proofs and precise results differed, including in the forced versus unforced Euler case. Developers weighing what AI vendors do with prompt data can follow this data-use debate on daily.dev."}}]} ```