Mathematicians say a neurosurgery resident used ChatGPT, powered by GPT-5.6, to solve Crouzeix's conjecture, a major open problem in numerical linear algebra
On July 30, 2026, the first author asked ChatGPT 5.6 to solve Crouzeix's conjecture, as he had been routinely doing for the preceding year.
Earlier GPT-3.5 testing found stronger performance on problems predating its knowledge cutoff than on newer ones, making a reported result on an open conjecture a meaningful escalation in the type of mathematical work being attributed to ChatGPT.
First-order effects
The resident and the mathematicians reporting the result must turn the ChatGPT-produced solution into a proof that specialists can inspect, since acceptance of a conjecture’s resolution rests on verification rather than the model’s output alone.
ChatGPT and GPT-5.6 gain a prominent research-use case if the solution withstands review: assisting work on a major open problem rather than only routine mathematics tasks.
Second-order effects
Mathematicians evaluating Crouzeix’s conjecture face a new workflow in which checking an AI-generated argument becomes as important as generating candidate approaches.
Competing frontier-model teams will be judged more closely on whether their systems can produce verifiable advances on unsolved questions, not merely solve established benchmark problems.
Third-order effects
If the Erdős, Jacobian and Crouzeix reports yield validated results, mathematical research will increasingly separate idea generation by models from proof certification by human specialists and formal or conventional review.
A growing record of AI-assisted conjecture claims would make independent verification a central measure of research-model usefulness, limiting the value of headline claims that cannot be checked.
The trend: Frontier AI is moving from solving known technical tasks toward generating candidate solutions for open mathematical problems, with verification becoming the decisive bottleneck.
Latest crazy story from the frontiers of math and AI — a neurosurgery resident, with no training in advanced math, uses ChatGPT 5.6 to solve a major open problem in numerical linear algebra. https://alextownsend.net/... [image]
I think it is becoming clear that if you care enough and have the perseverance to keep trying (both very important qualities in good scientists), you now have a good chance at solving some minor or major long-standing pure math problem using AI even if you aren't a professional
@tonylfeng ... Why was a doctor interested in a question about matrices? Because of his research on transcranial ultrasound! (See screenshot below from the news article https://alextownsend.net/...) [image]
Another remarkable story in the age of AI solutions: “A neurosurgery resident with no training in advanced math used ChatGPT 5.6 to solve a major open problem in numerical linear algebra.” We will continue to see an exponential increase in such solutions and discoveries!
RSI watch: this neurosurgeon got interested in the problem via his research in brain imaging. Chinese math education is much better than in the US. GPT 5.6 is really amazingly strong at math and physics. RSI: numerical linear algebra is highly relevant to model optimization.
Written by associate professor at Cornell and emeritus professor at UW: “Both authors of this article, as well as Michel Crouzeix himself, have checked the proof thoroughly and believe that Dr. Jin's manuscript is correct.” Important enough that an entire workshop was held on it …
Neurosurgeons and joggers attacking long-standing math problems because of AI - we are in a surreal realm. Now let's make that possible for physical science.