---
title: "On the Navier–Stokes Millennium Prize Problem"
slug: on-the-navier-stokes-millennium-prize-problem
url: https://listedarticles.com/articles/on-the-navier-stokes-millennium-prize-problem
canonical_url: https://simonwillison.net/2026/Sep/8/on-navier-stokes/
content_type: blog_post
language: en
published_at: 2026-09-08T00:00:00.000Z
updated_at: 2026-09-16T16:11:12.578Z
author: "Simon Willison"
author_url: https://simonwillison.net
authored_by: agent
publisher: "Simon Willison"
publisher_url: https://simonwillison.net
topics: ["Mathematics", "AI", "LLMs", "OpenAI", "Research Ethics", "AI Safety"]
license: all-rights-reserved
word_count: 299
reading_minutes: 1
citation: "Simon Willison, Simon Willison. \"On the Navier–Stokes Millennium Prize Problem.\" 8 Sept 2026. https://simonwillison.net/2026/Sep/8/on-navier-stokes/ (all-rights-reserved)"
# The full text follows. The web page shows an extract and sends readers
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---

# On the Navier–Stokes Millennium Prize Problem

> Simon Willison documents OpenAI's claim to have resolved the Navier-Stokes Millennium Prize Problem using an internal model in under four days, and the ethical controversy that followed. An NYU mathematician and an Anthropic researcher had been working on the problem for nearly a year using Claude and Codex, and allege their preliminary results reached OpenAI before its effort began.

> **Indexed summary.** This entry is an agent-written synopsis of an article first published at [simonwillison.net](https://simonwillison.net/2026/Sep/8/on-navier-stokes/). Read the original for the full text.

OpenAI's agents sent 2.7 million messages and used approximately 130 billion output tokens to resolve the Navier-Stokes existence and smoothness problem, with Lean formalization taking a further 17 hours via GPT-6 Astra. The announcement was immediately shadowed by mathematician Tristan Buckmaster's account that he and Anthropic researcher Levent Alpoge had been working on the same problem for almost a year.

## Key points

- OpenAI launched its effort on 1 September 2026 after hearing rumours that two Millennium Prize problems had been solved using LLMs, and completed it on 5 September, about 88 hours after the first agents started.
- Tristan Buckmaster (NYU) and Levent Alpoge (Anthropic) had a breakthrough on 15 August; the mathematical rumour mill connected their unpublished work to OpenAI's team.
- Buckmaster asked when OpenAI's first prompt was sent and received a delayed, indirect answer; he also asked whether the model was trained on their Codex sessions and received no clear reply.
- OpenAI offered to co-announce with Buckmaster and recognised their priority, but excluded Alpoge as a co-author due to OpenAI's competitive relationship with Anthropic.
- Willison draws a parallel to the security world: just as knowing a vulnerability exists can trigger LLM-powered exploit searches, knowing a mathematical proof exists may now trigger well-funded LLM efforts to get there first.
- The episode raises unresolved questions about whether de-identified usage data from Codex sessions influenced the model that solved the problem.

## Why it matters

The Navier-Stokes result signals that AI systems can now contribute to problems at the frontier of mathematics, but the accompanying controversy reveals that research norms, credit attribution, and data provenance have not kept pace with these capabilities.

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*Source: [On the Navier–Stokes Millennium Prize Problem](https://simonwillison.net/2026/Sep/8/on-navier-stokes/)*