---
id: "2026-11-06-navier-stokes-unforced-leray-hopf-nonuniqueness-buckmaster-coiculescu-li"
url: "https://postcutoff.com/e/2026-11-06-navier-stokes-unforced-leray-hopf-nonuniqueness-buckmaster-coiculescu-li/"
as_of: "2026-11-09T21:13:00+01:00"
date: "2026-11-06"
date_precision: day
category: science
importance: 4
confidence: medium
status: [Awaiting review]
verification: "Lean proof, not yet reviewed"
sources: 4
editor: human editor
human_review: null
version: "2026-11-08"
---

As of: 2026-11-09 21:13 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/2026-11-06-navier-stokes-unforced-leray-hopf-nonuniqueness-buckmaster-coiculescu-li/

# Navier–Stokes: unforced Leray–Hopf non-uniqueness, AI-written

Full title: Buckmaster, Coiculescu and Li post a 'primarily AI-generated' proof that Leray–Hopf solutions of the unforced 3D Navier–Stokes equations are not unique

Hexagon 2611.00143 (Tristan Buckmaster, Matei Coiculescu, Chen Li, NYU; tools Claude Opus, Codex, Lean) claims that the incompressible Navier–Stokes equations on R³ without external force admit a continuum of distinct suitable Leray–Hopf weak solutions with the same compactly supported, axisymmetric no-swirl datum in L^{3,∞}. The construction uses a discretely self-similar background built from a train of thin vortex rings. Provenance: "Primarily AI-generated"; some parts checked by a human. Hou, Wang and Yang had claimed the unforced case earlier by a computer-assisted proof (arXiv 2509.25116).

## Key facts
- Reactions by 8 Nov: no response from Thomas Hou (whose computer-assisted proof of the unforced case, arXiv 2509.25116, came first) or other experts found by web search
- Claim (abstract): a continuum of distinct suitable weak solutions in the Leray–Hopf class with the same datum; solutions smooth at positive times, satisfying local and global energy equalities, separating in L² at rate t^{1/4+σ}
- Method (abstract): 'The proof follows the program of Jia and Šverák, with a discretely self-similar background in place of a self-similar one', built 'from a train of thin vortex rings' obtained by lifting a periodic row of planar vortices; a growing Floquet mode is constructed from the angular-two instability of the planar vortex
- AI disclosure (Hexagon provenance, verbatim): 'Text declaration: Primarily AI-generated. … Some parts of this contribution, but not all, have been checked and understood by a human. … The broad steps of the construction have been understood by the authors.' Tools: Claude Opus, Codex, Lean
- Lean: 'Lean formalization forthcoming at the public repository at https://github.com/mcoiculescu/unforced-leray-hopf-public'; the authors 'used an adaptation of Scott Armstrong's skill files for the Lean formalization process'
- Prior work: Albritton–Brué–Colombo proved non-uniqueness for the forced equations (Annals 2022); Hou, Wang and Yang claimed the unforced case with a computer-assisted proof of an unstable self-similar profile (arXiv 2509.25116, v2 March 2026). The Hexagon paper's bibliography cites Albritton–Brué–Colombo and Guillod–Šverák but, in the parsed list, not Hou–Wang–Yang
- Status: unrefereed deposit, 80 pages, v1 6 Nov; no expert reaction found as of 7 Nov (HN, web search)

## What happened

Tristan Buckmaster, Matei Coiculescu and Chen Li deposited the paper on Hexagon on 6 Nov, labelled "no human author claimed" and "primarily AI-generated". It carries out the Jia–Šverák strategy: find a background solution whose linearization has a growing mode, then build a second solution that branches off it. Here the background is discretely self-similar and is constructed explicitly, so the argument does not depend on numerically certified profiles.

The question was already reported answered: Hou, Wang and Yang (arXiv 2509.25116) gave a computer-assisted proof in 2025–26. The new deposit claims a different and more explicit route, written mainly by AI. The authors say a Lean formalization will follow.

## Why it matters

Uniqueness of Leray–Hopf solutions is one of the central questions about the Navier–Stokes equations, alongside (and separate from) the Millennium regularity problem. Buckmaster is a leading figure in fluid non-uniqueness, so the deposit carries weight, but it is unrefereed and only partly checked by its authors.

## What is disputed or not yet verified

- Verification: Unrefereed Hexagon deposit; Lean formalization announced, not yet public

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 190 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 129 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 220 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 159 days after its cutoff

## Sources

1. [Hexagon 2611.00143: Non-Uniqueness of Leray-Hopf Solutions to the Unforced Three-Dimensional Navier-Stokes Equations](https://hexagonmath.org/abs/2611.00143) (hexagonmath.org, paper)
2. [Announced Lean repository (forthcoming)](https://github.com/mcoiculescu/unforced-leray-hopf-public) (github.com, code)
3. [Hou, Wang, Yang: computer-assisted non-uniqueness for the unforced equations (arXiv 2509.25116)](https://arxiv.org/abs/2509.25116) (arxiv.org, paper)
4. [Albritton, Brué, Colombo: forced case (arXiv 2112.03116)](https://arxiv.org/abs/2112.03116) (arxiv.org, paper)

## Changes

- 2026-11-08: Re-checked reactions incl. Hou's: none found
- 2026-11-07 (filed): Created (science agent; Hexagon record and provenance read; prior Hou–Wang–Yang claim found by web search)

## Related

- 2026-10-17: [2D Euler non-uniqueness without force claimed three times, all with AI](https://postcutoff.com/e/2026-10-17-2d-euler-unforced-nonuniqueness-lp-vorticity-ai/index.md)
- 2026-09-08: [OpenAI claims a Millennium Prize problem](https://postcutoff.com/e/2026-09-08-openai-navier-stokes-blowup/index.md)