Neurosurgery resident uses GPT-5.6 Sol to prove Crouzeix’s conjecture in a 16-hour autonomous run
Result confirmed
Importance: major (4 of 5)The takeaway
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|.
Status
- Claim
Result confirmed
- Our reporting
- Medium confidence
- Verification
- Expert-checked
- Importance
- Major (4 of 5)
- Last verified
- 30 September 2026
Your AI and this story
- GPT-6 Astra88 days after its cutoff
- Claude Opus 5.527 days after its cutoff
- Gemini 4 Hydrogen118 days after its cutoff
- Grok 4.757 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 27 days before 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.
Sources
3 sources from 3 sites. Numbers match the chips in the text.
3 sources: 1 primary, 1 press, 1 reaction
Primary
Press
Reactions
- Alex Townsend: SIAM News essay on the Crouzeix conjecture (PDF)alextownsend.net, discussion
Changes
- Added the independent Lorist–Schwenninger proof (arXiv 2608.03841)
- Filed