GitHub - openai/NavierStokesAndEuler at 8937a8f4cbc7abaab5e9e97d1cc7f5d2319d9538

Valerio Capraro on Twitter / X

BREAKING: OpenAI’s solution to Navier–Stokes does not match its Lean verification.The most important article to read today is not one of OpenAI’s 700 AI-generated math papers.It is this other paper, making a deep and worrying point:A Lean-verified proof does not… pic.twitter.com/bU7TQdv594— Valerio Capraro (@ValerioCapraro) October 8, 2026

NIK on Twitter / X

>sam altman: "we tried this because there were rumor on the internet">rumor: math professor close to solving navier-stokes>the professor: using codex for a year>openai: has all his logs>checked every codex session tagged to N-S>found the most promising one>spun up 10,000… pic.twitter.com/RInlP88gmA— NIK (@ns123abc) September 8, 2026

Navier-Stokes lost in translation: Why Lean verification of AI...

Autoformalisation is increasingly used to verify mathematical texts, including those generated by AI, as in OpenAI's announced proof of blow-up of solutions to the Navier-Stokes equations. In this...

Navier-Stokes and Lean

I was working on this week's post on Lean after reading Kevin Hartnett's book  The Proof in the Code: How a Truth Machine Is Transforming Ma...

tristanbuckmaster (@tristanbuckmaster@mastodon.social)

Today, Levent Alpöge and I have made public three results: finite-time blowup with smooth forcing for incompressible porous media, for Boussinesq, and for 3d incompressible Euler. https://cims.nyu.edu/~tristanb/statement.pdf https://cims.nyu.edu/~tristanb/euler.pdf https://cims.nyu.edu/~tristanb/ipm.pdf https://cims.nyu.edu/~tristanb/boussinesq.pdf https://github.com/tristanbuckmaster/fluid_lean

No, AI hasn’t officially solved Navier–Stokes. Yet

PropAIganda, now with a Millennium Prize Problem

OpenAI solves Navier–Stokes, but research misconduct concerns remain | Olga Holtz posted on the topic | LinkedIn

OpenAI says its AI has solved a Millennium Prize Problem: Navier–Stokes. A human mathematician asks whether OpenAI committed research misconduct. As a mathematician working at the intersection of mathematics and AI, I find this truly messy while also potentially historic. Three things deserve attention. 1. The fine print IS the mathematics. The claimed construction starts with a three-dimensional incompressible fluid at rest. Under a smooth external force, its velocity becomes unbounded in finite time, even while its kinetic energy stays bounded. Crucially, the force itself remains smooth through the singularity. Forcing is explicitly permitted in Clay’s official breakdown alternatives C and D. Its presence does not disqualify the result. But this construction does not establish blow-up for unforced Navier–Stokes. That distinction matters. 2. This was a research system, not one clever prompt. OpenAI reports roughly 10,000 concurrent agents in the successful group and a result after about 88 hours. The Navier–Stokes effort used approximately 130 billion output tokens; Lean formalization and verification took another 17 hours. Human researchers redirected the effort and used tools to consolidate intermediate insights. The salient part here, as I already wrote earlier, is a new organization of mathematical discovery using an enormous amount of compute, not a faster chatbot. The published Lean formalization is important: there is something concrete for others to check. Let‘s see independent scrutiny of both the proof and whether the formal statement matches the intended theorem. 3. Proof verification is not provenance verification. The result follows the line of attack designed by Spanish mathematicians Diego Córdoba and Luis Martínez Zoroa, who appeared to be close to solving the problem themselves. Tristan Buckmaster has raised concerns about research priority and possible use of private Codex material (his statement is linked to the comment by Guillermo Núñez Ponasso below). Those concerns are not automatic proof of misconduct, yet: OpenAI recognizes Buckmaster and Levent Alpöge’s priority on forced Euler, a related but different problem. It denies accessing their unpublished work. But it also cannot rule out de-identified data from their OpenAI product usage helping improve its models, though it calls that unlikely. Is it though? That ‚leakage’ is quite possible, in my view. If correct, this is an extraordinary result. Let’s get independent verification. And let’s protect unpublished research — but exactly how to protect it from an AI lab if using its tools for this research? Recall that a verified theorem can settle a mathematical question. It cannot, by itself, settle scientific credit or establish that the discovery process was fair. Are you comfortable using an AI lab’s tools when that lab could also become your research competitor? https://lnkd.in/g4HzzUNp #AI #Mathematics #AIforScience #MillenniumProblems #NavierStokes | 54 comments on LinkedIn

On the Navier–Stokes Millennium Prize Problem

We’re sharing an AI-generated solution to the Navier–Stokes Millennium Prize Problem, including a writeup and a formal proof in Lean.

AI Has Solved One of Math’s $1 Million Millennium Prize Problems | Quanta Magazine

Mathematicians at OpenAI showed that the Navier-Stokes equations, which describe how fluids flow, can sometimes “blow up.” But the massive result is not without controversy.

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Lean certificates accompanying proofs in mathematics and theoretical computer science - . · openai/ten-proofs@94bc0fe

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Lean certificates accompanying proofs in mathematics and theoretical computer science - . · openai/ten-proofs@e9dce9a

[10 05 2025 The Curve] Open Models

Open Models in 2025 Nathan Lambert Interconnects // Ai2 The Curve // 5 October 2025

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Lean certificates accompanying proofs in mathematics and theoretical computer science - . · openai/ten-proofs@d0e1ae7

Navier-Stokes Equation - Clay Mathematics Institute

This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding.

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Lean certificates accompanying proofs in mathematics and theoretical computer science - . · openai/ten-proofs@5a102c1