---
id: "2026-09-07-anandkumar-euler-singularity-r3"
url: "https://postcutoff.com/e/2026-09-07-anandkumar-euler-singularity-r3/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-09-07"
date_precision: day
category: science
importance: 3
confidence: medium
status: [Awaiting review]
verification: Unverified claim
sources: 4
editor: human editor
human_review: null
version: "2026-10-04"
---

As of: 2026-11-13 13:49 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-09-07-anandkumar-euler-singularity-r3/

# Caltech team (Anandkumar) reports a stable self-similar singularity candidate for the unforced 3D Euler equations on R³, found with PINNs and LLM help

On 7 Sep 2026, the evening before OpenAI's Navier–Stokes announcement, Anima Anandkumar's Caltech group posted a self-similar singular profile for the unforced incompressible 3D Euler equations on all of R³. Physics-informed neural networks found it, and LLMs helped simplify the bounds and formalise derivations in Lean. The arXiv papers (2609.10867, 2609.10860) describe "evidence" and a stability framework that is conditional on certifying explicit constants, so this is not yet a complete proof.

## Key facts
- Authors: Adarsh Ganeshram, Valentin Duruisseaux, Anima Anandkumar (+ Robert J. George on the stability paper)
- Setting: incompressible Euler on unbounded R³, no forcing; axisymmetric self-similar ansatz at blow-up rate 0.5 (matching a prediction by Constantin et al., arXiv 2602.17570)
- Method: PINN finds approximate profile; second-order optimisers (SS-eSOAP, SS-Broyden); certified via spline representation with interval arithmetic
- AI use (guest post): 'we used the OpenAI and other models extensively to simplify our bounds as well as formalize the derivations in Lean'
- arXiv 2609.10867 (111 pp.) abstract: 'We provide evidence of a finite-time singularity'; 2609.10860 (113 pp.): stability proof closes 'conditional on rigorous certification of the estimates and constants'
- The authors complain that mainstream media followed OpenAI's press release and did not acknowledge their work

## What happened

In the same week as the Buckmaster–Alpöge forced blow-up results (7 Sep) and OpenAI's forced Navier–Stokes claim (8 Sep), a third group posted a singularity for the *unforced* Euler equations on the whole space. They used AI-driven numerical discovery followed by computer-assisted proof techniques. Their guest post on Tao's blog stresses AI as a "complementary" tool, "built to propose solutions that did not compete with humans".

## Why it matters

Unforced Euler blow-up on R³ is a famous open problem in its own right, and it is a stepping stone toward the unforced Navier–Stokes question. The claim is still partly conditional, so its status should be tracked.

## What is disputed or not yet verified

- Verification: Interval-arithmetic certification of the profile; partial Lean formalisation; stability conditional

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 130 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 69 days after its cutoff
- Gemini 4 Hydrogen (training cutoff March 2026): 160 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 99 days after its cutoff

## Sources

1. [arXiv 2609.10867: Self-Similar Singularity of the Euler Equations on R³](https://arxiv.org/abs/2609.10867) (arxiv.org, paper)
2. [arXiv 2609.10860: Stability Framework for the Singularity of the Euler Equations on R³](https://arxiv.org/abs/2609.10860) (arxiv.org, paper)
3. [Anandkumar group page on the Euler result](https://tensorlab.cms.caltech.edu/users/anima/euler.html) (tensorlab.cms.caltech.edu, official)
4. [Anima Anandkumar (guest post on Tao's blog): Stable singularity of the Euler equations on R³](https://terrytao.wordpress.com/2026/09/10/stable-singularity-of-the-euler-equations-on-r3/) (terrytao.wordpress.com, discussion)

## Changes

- 2026-09-29 (filed): Created. Marked pending because the arXiv abstracts describe the stability proof as conditional

## Related

- 2026-09-08: [OpenAI claims a Millennium Prize problem](https://postcutoff.com/e/2026-09-08-openai-navier-stokes-blowup/index.md)
- 2025-09-17: [DeepMind and mathematicians use neural networks to find new unstable singularities in fluid equations](https://postcutoff.com/e/2025-09-17-deepmind-unstable-singularities-fluids/index.md)
- People: [Terence Tao](https://postcutoff.com/person/terence-tao/), [Levent Alpöge](https://postcutoff.com/person/levent-alpoge/)