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Neurosurgery resident solves Crouzeix's conjecture with GPT-5.6

A neurosurgery resident with no formal advanced mathematics training has solved Crouzeix's conjecture, a 22-year-old open problem in numerical linear algebra, using a 16-hour autonomous run of GPT-5.6 Sol in ChatGPT Work mode. Dr. Shanmu Jin, a postdoctoral researcher and neurosurgery resident at Peking Union Medical College Hospital, prompted ChatGPT 5.6 on July 30, 2026 to work on the conjecture. The model ran autonomously for approximately 16 hours, exploring proof strategies through a branc

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Neurosurgery resident solves Crouzeix's conjecture with GPT-5.6

A neurosurgery resident with no formal advanced mathematics training has solved Crouzeix's conjecture, a 22-year-old open problem in numerical linear algebra, using a 16-hour autonomous run of GPT-5.6 Sol in ChatGPT Work mode.

Dr. Shanmu Jin, a postdoctoral researcher and neurosurgery resident at Peking Union Medical College Hospital, prompted ChatGPT 5.6 on July 30, 2026 to work on the conjecture. The model ran autonomously for approximately 16 hours, exploring proof strategies through a branching process where multiple agent instances proposed, challenged, and reconstructed arguments. The key theorem emerged from this process, which Jin then developed into a complete proof.

Neurosurgery and mathematics merged through AI

Mathematicians Alex Townsend (Cornell) and Anne Greenbaum (University of Washington), along with Michel Crouzeix himself, have reviewed the proof and believe it is correct. Townsend had spent the prior year periodically asking GPT-5.6 to attempt the conjecture, with the model either returning bogus proofs or stalling at missing lemmas.

Crouzeix's conjecture, posed in 2004, states that for any square matrix A and any polynomial p, the norm of p(A) is bounded by twice the maximum of p on the numerical range of A. The conjecture sits at the intersection of matrix analysis, operator theory, and polynomial approximation.

Jin's background is unusual for a conjecture solver. He holds an undergraduate degree in geology and an M.D., with all mathematics beyond standard science courses self-taught. He encountered the problem through research on transcranial ultrasound, which led him to matrix analysis and the conjecture's deceptively simple statement.

A second independent proof emerged almost simultaneously from a different research group, which Crouzeix and Townsend describe as a genuinely different approach. Jin welcomed the parallel result.

The work demonstrates that frontier models can now contribute original research-level mathematics, not just assist with known techniques. The 16-hour autonomous run represents a shift from AI as a coding assistant to AI as a research collaborator capable of sustained, multi-step reasoning on open problems.

Sources

Alex Townsend, SIAM News essay "The Neurosurgery Resident Who Proved Crouzeix's Conjecture" (2026): https://alextownsend.net/essays/SIAMNews_CrouzeixConjecture.pdf

Techmeme aggregation: https://www.techmeme.com/260813/p9#a260813p9

AI Midday: https://aimidday.com/neurosurgery-resident-solves-crouzeixs-conjecture-with-chatgpt-5-6

36kr coverage: https://eu.36kr.com/en/p/3937408047053960

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