Navier-Stokes and Lean

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...

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

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.

GitHub - openai/NavierStokesAndEuler at 8937a8f4cbc7abaab5e9e97d1cc7f5d2319d9538

Lean certificates accompanying Navier-Stokes and Euler results - openai/NavierStokesAndEuler

Tau Ceti — Tau Ceti

AI-authored Lean mathematics, directed by a human-owned roadmap and gated by open, adversarial review.

No. 025: Navier-Stokes provenance dispute, Fields Medalists declaration, AI peer review - The Hallway Track

The Hallway Track No. 025 - How AI is affecting the practice of science

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

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

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.

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

PropAIganda, now with a Millennium Prize Problem

Palomar – a registry of Lean verified mathematics

In recent months there has been a proliferation of AI-generated proofs of various old and new results, some of which have been formalized in the proof assistant language Lean. However, checking tha…

CausalSmith · AI Causal Scientist

Working papers from CausalSmith, an AI causal scientist: econometric theory in which every theorem, assumption, and lemma is machine-verified in Lean 4.

stephenturner/skill-deslop

De-AI-ify scientific writing

Navier-Stokes lost in translation: Why Lean verification of AI autoformalisation does not guarantee correct natural language proofs

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 process, an AI system translates the text from a natural language (NL) into a formal language such as Lean. Once this translation is done, the argument expressed in the formal language can easily be mechanically verified. The purpose of this article is to demonstrate why this process may offer no confidence in the original NL argument, owing to the various difficulties in performing the translation semantically faithfully. In particular, we highlight that the problem of resolving ambiguities in mathematical NL text, which is necessary in order to provide semantically faithful translation, is arbitrarily high up in the Solvability Complexity Index (SCI) hierarchy/arithmetical hierarchy (the SCI $= \infty$). Hence, informally, providing semantically faithful AI autoformalisation is harder than any computational problem including the Halting problem (which has SCI $= 1$). To demonstrate the effect of this result we provide several examples of AI mistranslations of NL statements and proofs into Lean in practice, resulting in mismatches between NL proofs and their Lean `verifications'. These include OpenAI's announced Navier-Stokes proof. In particular, we show that the formalised Lean proof does not correspond to the NL proof of blow-up of solutions to the Navier-Stokes equations.

. · openai/ten-proofs@d0e1ae7

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