Shanmu Jin Reports AI-Assisted Proof of Crouzeix's Conjecture

On August 12, 2026, Steven Strogatz highlighted a July 27 preprint by Shanmu Jin, a postdoctoral researcher and neurosurgery resident at Peking Union Medical College Hospital, that claims a proof of Crouzeix's conjecture. In an August 11 account, Alex Townsend and Anne Greenbaum report that they and Michel Crouzeix checked the argument thoroughly and believe it is correct.

Crouzeix's conjecture states that, for every complex matrix $A$ and polynomial $p$, $\lVert p(A)\rVert \leq 2\max_{z\in W(A)}|p(z)|$, where $W(A)$ is the numerical range of $A$. Townsend and Greenbaum report that the decisive theorem emerged during an approximately 16-hour autonomous run of GPT-5.6 Sol in ChatGPT Work mode after Jin supplied a branching, adversarially checked research prompt without intervening during the run. Jin's public repository includes the prompt, successive manuscripts, a Lean 4 formalization, and an axiom audit.

The repository continues to describe the manuscript as a candidate proof and says formal peer review is pending. It also reports that computational and adversarial audits have found no specific mathematical error. A separate five-page proof by Emiel Lorist and Felix Schwenninger was posted on August 4; Townsend and Greenbaum say it uses a different argument and that ChatGPT 5.6 assisted its proof-strategy exploration.