Beijing Neurosurgeon Proves Crouzeix’s Conjecture Using AI

When Beijing-based neurosurgeon Jin Shanmu pulled up a chair at his computer, he was not looking to make mathematical history. He was simply trying to crack a problem related to brain ultrasounds.

Jin, a postdoctoral researcher and resident at the Peking Union Medical College Hospital, managed to prove Crouzeix’s conjecture, a problem in numerical linear algebra.

His tool? A 16-hour autonomous run on GPT-5.6-Sol, operating on the ChatGPT Work platform.

Proposed by French mathematician Michel Crouzeix, the conjecture posits that the norm of applying any function to a matrix is no larger than twice the function’s maximum value on that matrix’s numerical range.

While abstract, it has long been an intriguing problem in matrix analysis – a field Jin stumbled into while undertaking research on transcranial ultrasounds.