As of: 2026-10-12 22:30 CEST.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code). Human editor: Adam Bicz.
Canonical page: https://postcutoff.com/v/caleb-writes-code-openai-navier-stokes-explained/

# OpenAI solving Navier-Stokes explained..

Caleb Writes Code, 12 September 2026, YouTube. 239,003 views as of 12 October 2026.
Kind: Review.
Watch: https://www.youtube.com/watch?v=7DncQnIjZmA

## Why it is here

Explainer of OpenAI's Navier–Stokes approach that sets the dispute aside (~240k views).

## Description (written by Gemini from the video)

**Summary**  
Caleb from the channel *Caleb Writes Code* breaks down the Navier–Stokes existence and smoothness Millennium Prize Problem, showing how the governing equations evolved from Newton's laws and Euler's equations. He then examines OpenAI's claimed counterexample (a finite-time singularity/blow-up), detailing how massive compute and multi-agent systems were deployed, while analyzing the controversy, priority disputes, and cheating allegations involving researchers Tristan Buckmaster and Levent Alpoge.

**What is shown**  
- **[00:00]** Whiteboard overview connecting Newton's second law ($F = m\vec{a}$) to fluid dynamics and OpenAI's Navier–Stokes claim.  
- **[00:24]** Illustrated elevator physics example calculating force and acceleration for solid bodies ($1000\text{ kg}$, $9.8\text{ m/s}^2$, $9800\text{ N}$).  
- **[01:08]** Breakdown of fluid mechanics vs. solid bodies, detailing the Navier–Stokes equation terms: density ($\rho$), acceleration ($\frac{\partial v}{\partial t} + (v \cdot \nabla)v$), pressure gradient ($-\nabla p$), external forces/gravity ($\rho \vec{g}$), and viscosity ($\mu \nabla^2 \vec{v}$).  
- **[01:53]** Diagrams illustrating local and advective fluid acceleration using garden hose and funnel analogies.  
- **[03:20]** Timeline of equation development (Euler, Navier, Stokes) and practical engineering applications (blood vessels, weather forecasting, aircraft aerodynamics, ocean currents).  
- **[04:00]** Formulation of the Millennium Prize problem: proving smooth solutions exist globally in 3D space versus finding a blow-up counterexample.  
- **[04:39]** Slide showing an evaluation chart: *"Performance of GPT-6 Astra and our Internal Model on a curated set of open math problems"* across test-time compute.  
- **[06:30]** 3D visual render of OpenAI's counterexample structure: a spiral vortex featuring inward spiraling and axial stretching leading to singularity.  
- **[07:12]** Breakdown of research history: Euler's unforced equation vs. Buckmaster and Alpoge's 1-year effort finding a forced blow-up solution.  
- **[08:13]** OpenAI's timeline: running 120 agents for 50 hours to achieve an unforced Euler blow-up, followed by an 88-hour run finding a Navier–Stokes counterexample.  
- **[09:04]** Screenshots of leaked RCS text messages with Levent Alpoge, OpenAI's *"Concurrent work"* acknowledgment statement, and public research drafts discussing priority and Codex telemetry concerns.  
- **[10:25]** Clay Mathematics Institute Millennium Prize rules documentation regarding prize attribution and co-authorship.

**Claims & numbers**  
- The presenter says the Navier–Stokes equations were progressively formulated over history starting roughly 181 years ago.  
- The presenter states the Clay Mathematics Institute designated a $7 million prize fund ($1 million per problem) for seven Millennium Prize problems in 2000.  
- The presenter claims OpenAI used 10,000 concurrent agents powered by internal models, consuming 130 billion tokens to search for the solution.  
- The presenter reports that mathematicians Tristan Buckmaster (NYU) and Levent Alpoge (Anthropic) spent approximately one year developing a forced blow-up solution for Euler's equations.  
- The presenter states OpenAI utilized 120 agents working concurrently over 50 hours to find an unforced Euler blow-up solution, and subsequently found a Navier–Stokes blow-up candidate in 88 hours.  
- The presenter notes allegations that OpenAI leveraged researchers' Codex scratchpad/message histories to replicate and accelerate work on the problem.

**Notable quotes**  
- *"Is there a condition where math starts to break down under fair situations, or does it always produce a smooth solution in 3D space?"* [04:02]  
- *"And they used 10,000 agents working concurrently, powered by OpenAI's internal models, spending 130 billion tokens in total."* [04:35]  
- *"OpenAI was able to find a way to blow up Euler's equation without being forced, and this only took them 50 hours of work with 120 agents working concurrently..."* [08:21]

**Assessment**  
This is an independent educational commentary and news review rather than a product demonstration. The presenter uses hand-drawn chalkboard animations, third-party paper diagrams, and leaked correspondence to explain the mathematical concepts and summarize the public controversy surrounding OpenAI's Navier–Stokes claim.

_Described by gemini-3.8-flash on 2026-10-12 from the video's audio and frames._

## Related

- 2026-09-08: [OpenAI claims a Millennium Prize problem](https://postcutoff.com/e/2026-09-08-openai-navier-stokes-blowup/)

## People in it

- [Levent Alpöge](https://postcutoff.com/person/levent-alpoge/), Mathematician, member of technical staff, Anthropic