A 22-Year-Old Mathematical Puzzle, Solved by a Union Hospital Intern?
In an astonishing development, a longstanding mathematical problem known as the Crouzeix conjecture, which had challenged experts in numerical linear algebra for 22 years, appears to have been solved by Shanmu Jin, a neurosurgery resident and postdoctoral researcher at Peking Union Medical College Hospital. With no formal advanced mathematical training—his background is in geology and medicine—Jin relied on self-study and, crucially, the AI model GPT-5.6.
The conjecture, proposed by mathematician Michel Crouzeix in 2004, concerns a fundamental constant (speculated to be 2) bounding the relationship between matrix norms and polynomial values over numerical ranges. It is critical for applications in matrix function analysis and numerical methods. Despite dedicated efforts by leading mathematicians, the best proven constant had only been reduced to 2.414.
Jin approached the problem using sophisticated prompt engineering with GPT-5.6. He adapted a known prompting strategy, isolating the AI from external resources to force original reasoning, employing multiple divergent "sub-agents" to explore different paths, and enforcing rigorous adversarial review of proposed proof steps. After about 16 hours of autonomous, unsupervised operation, the AI produced a novel and elegant proof. The key insight involved a clever "sampling strategy" that reduced the problem to a simple positivity condition—a solution described as elegant and unexpected by experts.
The proof was verified by the conjecture's originator, Michel Crouzeix, and other specialists like Alex Townsend and Anne Greenbaum, who expressed astonishment at its validity. Jin has made the entire process open-source, including the prompt, drafts, and formal verification code.
Remarkably, just eight days after Jin's preprint appeared, mathematicians Emiel Lorist and Felix Schwenninger published an independent, concise 5-page proof using a different approach, also developed with the aid of ChatGPT 5.6. Jin welcomed this complementary work.
This event marks a potential turning point, demonstrating how AI can enable experts from non-traditional backgrounds to solve deep theoretical problems and dramatically accelerate scientific discovery, heralding what some are calling a new "golden age" for interdisciplinary research.
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