A Neurosurgery Resident Solved a 22-Year Math Problem by Letting ChatGPT Run for 16 Hours

The problem sat at the intersection of matrix theory and functional analysis in numerical linear algebra — a niche but foundational question that had resisted two decades of effort by experts who had dedicated their careers to it.

From Brain Ultrasound to a Mathematical Breakthrough

Jin was not trying to make mathematical history. He was researching transcranial ultrasound — modeling how ultrasound waves penetrate the skull — which led him into matrix analysis and, eventually, into Crouzeix's conjecture . All of his higher-level mathematics was self-taught through his ultrasound research; his undergraduate degree was in geology from Peking University, and his medical degree came from Peking Union Medical College. He had no formal training in advanced mathematics .

The 16-Hour Autonomous Run

Jin adapted the prompt strategy OpenAI had used when solving the Cycle Double Cover conjecture. He designed a multi-agent, adversarial setup with four instructions :

1. Work offline: The model operated in a physically disconnected environment, preventing it from pulling existing proofs or solutions from the web.
2. Explore in parallel: The system spawned many subagents exploring genuinely different proof strategies, preventing premature convergence on a single approach.
3. Adversarial auditing: Candidate proofs were subjected to adversarial checks, forcing the model to find and fix its own errors.
4. No stopping early: The system was instructed not to stop until a complete proof survived all checks.

Jin started the system and walked away. After roughly 16 hours of continuous operation — during which GPT-5.6-Sol executed thousands of cycles of generation, hypothesis testing, rejection, and rebuilding — it produced a complete proof .

Rather than the heavy estimates experts had expected, the argument used a careful sampling strategy to reduce the problem to a simple positivity condition . It was, by the accounts of those who reviewed it, an elegant and unexpected approach.

Verification by the Experts

Numerical analyst Alex Townsend of Cornell University had spent the prior year routinely asking ChatGPT to attack Crouzeix's conjecture, watching it stall at the same missing lemma every single time . On July 30, 2026, he asked again — but this time, ChatGPT returned a link to a preprint posted July 27 titled "The Numerical Range Is a 2-Spectral Set" .

Townsend and Anne Greenbaum of the University of Washington reviewed the preprint with skepticism. After several hours, they concluded the proof was correct . Michel Crouzeix himself — the mathematician who had posed the conjecture 22 years earlier — reviewed the manuscript and confirmed its correctness .

Open Source and Independent Verification

Jin open-sourced his entire research pipeline, making available on GitHub the final paper, the exact prompt, all successive manuscript iterations, a Lean 4 formalization of the proof, and an axiom audit .

Remarkably, eight days after Jin's preprint appeared, a separate independent proof by Emiel Lorist and Felix Schwenninger — who had also used ChatGPT 5.6 for strategy exploration — was posted on arXiv, independently corroborating the result .

What This Means for AI and Science

The story of Jin's proof is notable for several reasons. It shows that a non-expert with no formal training in a field can, by designing the right autonomous workflow for a frontier AI model, make a genuine contribution to a problem that stumped career specialists. It also demonstrates the power of well-designed autonomous reasoning systems: Jin did not use AI as a search tool or chatbot — he designed a system that could explore, check, and refine its own work over many hours without human intervention .

The problem is not a toy or a puzzle; it is a 22-year-old open problem whose proof has been reviewed and accepted by the mathematician who posed it, alongside other leading experts in the field .