---
id: "2026-07-27-crouzeix-conjecture-proved"
url: "https://postcutoff.com/e/2026-07-27-crouzeix-conjecture-proved/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-07-27"
date_precision: day
category: science
importance: 4
confidence: medium
status: [Result confirmed]
verification: Expert-checked
sources: 3
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/2026-07-27-crouzeix-conjecture-proved/

# Neurosurgery resident uses GPT-5.6 Sol to prove Crouzeix's conjecture in a 16-hour autonomous run

A preprint posted 27 Jul 2026 proves Crouzeix's conjecture (2004): for every square matrix A and polynomial f, ‖f(A)‖ ≤ 2·max over the numerical range W(A) of |f|. The proof came from one uninterrupted 16-hour autonomous GPT-5.6 Sol run prompted by Shanmu Jin, a self-taught neurosurgery resident. Michel Crouzeix, Anne Greenbaum and Alex Townsend checked it.

## Key facts
- Previously best known constant: 1+√2 (Crouzeix–Palencia 2017); conjectured optimal constant 2
- Single 16-hour autonomous run of GPT-5.6 Sol
- Checked by Crouzeix himself, Anne Greenbaum and Alex Townsend (SIAM News essay)
- Independent proof: Emiel Lorist and Felix Schwenninger, 'A solution to Crouzeix's conjecture' (arXiv 2608.03841, 4 Aug 2026); their AI disclosure says GPT-5.6 Sol Pro in ChatGPT 'was used to explore proof strategies for this note'

## What happened

A non-mathematician set GPT-5.6 Sol on the problem. The model produced a complete proof in one long run, which the conjecture's originator and other specialists confirmed.

## Why it matters

Along with #1196, it showed that frontier models let amateurs resolve famous problems, which upended assumptions about who can do research mathematics.

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 88 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 27 days after its cutoff
- Gemini 4 Hydrogen (training cutoff March 2026): 118 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 57 days after its cutoff

## Sources

1. [arXiv 2608.03841: A solution to Crouzeix's conjecture (Lorist, Schwenninger)](https://arxiv.org/abs/2608.03841) (arxiv.org, paper)
2. [SCMP: Chinese doctor stuns maths world cracking decades-old problem using ChatGPT](https://www.scmp.com/tech/tech-trends/article/3363966/chinese-doctor-stuns-maths-world-cracking-decades-old-problem-using-chatgpt) (scmp.com, press)
3. [Alex Townsend: SIAM News essay on the Crouzeix conjecture (PDF)](https://alextownsend.net/essays/SIAMNews_CrouzeixConjecture.pdf) (alextownsend.net, discussion)

## Changes

- 2026-09-30: Added the independent Lorist–Schwenninger proof (arXiv 2608.03841)
- 2026-09-29 (filed): Created

## Related

- 2026-09-30: [Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help](https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/index.md)
- 2026-07-09: [OpenAI broadly releases GPT-5.6 after government-gated preview](https://postcutoff.com/e/2026-07-09-gpt-5-6-sol-terra-luna/index.md)
- 2026-05-03: [Amateur with GPT-5.4 Pro 'vibe-maths' a 60-year-old Erdős conjecture on primitive sets](https://postcutoff.com/e/2026-05-03-erdos-1196-primitive-sets/index.md)