---
id: "2026-11-03-alon-seymour-thomas-separator-conjecture-gpt-6-astra-two-proofs"
url: "https://postcutoff.com/e/2026-11-03-alon-seymour-thomas-separator-conjecture-gpt-6-astra-two-proofs/"
as_of: "2026-11-09T21:13:00+01:00"
date: "2026-11-03"
date_precision: day
category: science
importance: 4
confidence: medium
status: [Awaiting review]
verification: Lean-verified
sources: 2
editor: human editor
human_review: null
version: "2026-11-03"
---

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/2026-11-03-alon-seymour-thomas-separator-conjecture-gpt-6-astra-two-proofs/

# K_h-minor-free separators O(h√n): two AI-assisted proofs

Full title: Alon–Seymour–Thomas separator conjecture (1990): second AI-derived proof, by Conroy, Le and Spalding-Jamieson, whose ideas are 'due to GPT-6 Astra' and formalized in Lean

Two preprints claim the 1990 Alon–Seymour–Thomas conjecture that every n-vertex K_h-minor-free graph has a balanced separator of size O(h√n). Xinheng Lin (arXiv 2610.40270, 30 Oct) gives a 21h√n bound developed with ChatGPT 6 Pro; Jonathan Conroy, Hung Le and Jack Spalding-Jamieson (arXiv 2611.00663, 3 Nov) give an elementary proof whose "key technical ideas" are credited to GPT-6 Astra, with a Lean formalization made by GPT-6 Astra and Claude Opus 5.5. The human authors of the second paper "claim no role in developing the technical ideas".

## Key facts
- Conjecture: Alon, Seymour, Thomas, 'A separator theorem for nonplanar graphs' (JAMS 1990) proved O(h^{3/2}√n) and conjectured O(h√n)
- arXiv 2611.00663 (Conroy, Le, Spalding-Jamieson): proof builds on Le's 2026 framework; new idea is a short hitting walk on vertex expanders (length O(k/ψ(G)) hitting k large subsets)
- AI disclosure 2611.00663 (verbatim): 'The key technical ideas of this paper are due to GPT-6 Astra, with some simplification due to Claude Opus 5.5. A Lean formalization of our main result was produced using a combination of GPT-6 Astra and Claude Opus 5.5 … The human authors claim no role in developing the technical ideas.' The rewritten human proof has 'a slightly worse constant than the formalization'
- arXiv 2610.40270 (Xinheng Lin): separator of size ≤ min{n, 21h√n} for h ≥ 3; 'developed under the author's guidance with the assistance of ChatGPT 6pro … GPT 6.1 Sol (Max) was also used to check the proof repeatedly'
- Verification status: two independent unrefereed preprints; 2611.00663 has a Lean formalization of the main result (definitions checked by the authors); no expert commentary or HN discussion found by 3 Nov

## What happened

Within four days two independent preprints claimed the O(h√n) separator bound for K_h-minor-free graphs conjectured by Alon, Seymour and Thomas in 1990. Lin's paper (listed in the 30 Oct catalogue) was written with ChatGPT 6 Pro. The Conroy–Le–Spalding-Jamieson paper says the ideas came from GPT-6 Astra, with simplifications by Claude Opus 5.5, and that the main result was formalized in Lean.

## Why it matters

The separator theorem underlies many algorithms on minor-closed graph classes. Two separate AI-assisted proofs, one with a Lean check, make this one of the stronger AI-originated claims on a well-known conjecture so far, though neither is refereed.

## What is disputed or not yet verified

- Verification: Preprints; Lean formalization for 2611.00663

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 187 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 126 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 217 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 156 days after its cutoff

## Sources

1. [arXiv 2611.00663: K_h-minor-free graphs have balanced separators of size O(h√n)](https://arxiv.org/abs/2611.00663) (arxiv.org, paper)
2. [arXiv 2610.40270: proof of the Alon–Seymour–Thomas conjecture (Xinheng Lin)](https://arxiv.org/abs/2610.40270) (arxiv.org, paper)

## Changes

- 2026-11-03 (filed): Created. Promotes the Lin lead of 30 Oct

## Related

- 2026-11-03: [AI-assisted results, 3 Nov: arXiv of 3 Nov and Hexagon of 2–3 Nov](https://postcutoff.com/e/2026-11-03-ai-assisted-results-november-3-listing/index.md)
- 2026-10-30: [AI-assisted results, 30 Oct: arXiv announcement of 29–30 Oct](https://postcutoff.com/e/2026-10-30-ai-assisted-results-october-30-listing/index.md)