---
id: "2026-11-04-glauber-dynamics-colorings-1-plus-1-over-e-gpt-6-astra-pro"
url: "https://postcutoff.com/e/2026-11-04-glauber-dynamics-colorings-1-plus-1-over-e-gpt-6-astra-pro/"
as_of: "2026-11-09T21:13:00+01:00"
date: "2026-11-04"
date_precision: day
category: science
importance: 4
confidence: medium
status: [Awaiting review]
verification: Preprint only
sources: 1
editor: human editor
human_review: null
version: "2026-11-04"
---

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/2026-11-04-glauber-dynamics-colorings-1-plus-1-over-e-gpt-6-astra-pro/

# Colourings Glauber dynamics above (1+1/e)Δ (GPT-6 Astra Pro)

Full title: Glauber dynamics for colourings mixes rapidly at q ≥ (1+1/e+γ)Δ: preprint whose main ideas 'emerged from extensive discussions with ChatGPT-6 Astra Pro'

Xiongxin Yang (arXiv 2611.04699) claims that for large maximum degree Δ the Glauber dynamics for proper q-colourings of any graph has spectral gap Ω(1/n) whenever q ≥ (1+1/e+γ)Δ ≈ 1.368Δ, far below the long-standing general-graph thresholds near 11/6 ≈ 1.83 (Vigoda 1999 and later small improvements). The paper says the main ideas came from ChatGPT-6 Astra Pro. Unrefereed single-author preprint.

## Key facts
- Claim: for every γ > 0 and sufficiently large Δ, spectral gap Ω_γ(1/n) and mixing time O_γ(n² log q) when q ≥ (1+1/e+γ)Δ
- 1+1/e comes from the hard-core uniqueness threshold ≈ e/Δ on the infinite Δ-regular tree; proof via a Bochner-type inequality using 'hidden hard-core models' in the colour classes
- Context: the conjecture is rapid mixing for q ≥ Δ+2; for general graphs the known threshold had stayed close to 11/6 Δ since Vigoda (1999)
- Disclosure: 'The main ideas of this paper emerged from extensive discussions with ChatGPT-6 Astra Pro. ChatGPT-6 Astra Pro also suggested that a finer analysis … should improve the constant 1 + 1/e further … All mathematical claims have been independently verified by the author'
- Verification: preprint; no expert comment or HN thread found on 4 Nov

## What happened

A single-author preprint posted on 3 Nov 2026 claims a large improvement on one of the best-known problems in sampling: when Glauber dynamics for random proper colourings mixes fast on every graph of maximum degree Δ. The claimed threshold (1+1/e)Δ ≈ 1.368Δ (for large Δ) would replace general-graph bounds that had stayed near 1.8Δ for over 25 years. The paper credits ChatGPT-6 Astra Pro with the main ideas and says the author simplified the arguments and checked all claims.

## Why it matters

If it holds, this is a major step towards the Δ+2 conjecture, and another case of a frontier model supplying the central idea of a theoretical computer science result. Status is unverified: no specialist reading has appeared yet.

## What is disputed or not yet verified

- Verification: Unrefereed preprint

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 188 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 127 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 218 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 157 days after its cutoff

## Sources

1. [arXiv 2611.04699: Glauber Dynamics for Sampling Colorings: Rapid Mixing above (1+1/e)Δ](https://arxiv.org/abs/2611.04699) (arxiv.org, paper)

## Changes

- 2026-11-04 (filed): Created from the arXiv PDF

## Related

- 2026-11-04: [AI-assisted results, 4 Nov: arXiv of 4 Nov and Hexagon of 2–3 Nov](https://postcutoff.com/e/2026-11-04-ai-assisted-results-november-4-listing/index.md)
- 2026-10-26: [3-colorable graphs: O(n^0.1537) colors with GPT-6 Astra](https://postcutoff.com/e/2026-10-26-three-coloring-n-0-1537-gpt-6-astra/index.md)