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)
Science result
- Field
- mathematics / matrix analysis / operator theory
- Problem
- Crouzeix's conjecture (open since 2004)
- Result
- Proof that the numerical range W(A) is a 2-spectral set for every matrix A.
- AI system
- GPT-5.6 Sol
- Human role
- Autonomous: non-specialist prompted; experts verified
- Verification
- Expert-checked (Crouzeix, Greenbaum, Townsend); preprint
- Status
- confirmed
- Why surprising
- A doctor with no formal maths training settled a well-known conjecture in numerical analysis by letting a model run overnight.
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.
Changelog
- 2026-09-29: created
Related events
- OpenAI broadly releases GPT-5.6 (Sol, Terra, Luna) after government-gated preview ★★★★
- Amateur with GPT-5.4 Pro 'vibe-maths' a 60-year-old Erdős conjecture on primitive sets; Tao co-authors the paper ★★★★
- Sendov's 1958 conjecture on polynomial roots proved with GPT-5.6 Pro; Tao simplifies and formalises it ★★★★
Sources (2)
- discussionAlex Townsend: SIAM News essay on the Crouzeix conjecture (PDF)
- pressSCMP: Chinese doctor stuns maths world cracking decades-old problem using ChatGPT
id: 2026-07-27-crouzeix-conjecture-proved · updated 2026-09-29 · open in the interactive timeline