In Beijing, a neurosurgeon named Jin Shanmu set out to understand brain ultrasounds and arrived, unexpectedly, at the edge of mathematical history. By running an AI system autonomously for sixteen hours, he produced a proof of Crouzeix's conjecture — a problem in numerical linear algebra that had resisted resolution for decades. The achievement is less a story about one man's triumph than a quiet signal that the boundary between human curiosity and machine reasoning is shifting in ways we are only beginning to measure.
Chinese neurosurgeon solves decades-old math conjecture using ChatGPT
A proof to a conjecture that had eluded mathematicians for decades
Mark
So he was not actually trying to solve this conjecture at all?
Mimi
No. He was working on something completely different—brain imaging. The conjecture just appeared in his research path, like finding a door you did not know was there.
Mark
And he just... handed it to ChatGPT?
Mimi
Not quite. He set up a specific version of the system and let it run for sixteen hours. The AI did the actual work—the reasoning, the logical steps, all of it.
Mark
Does that feel like cheating to mathematicians?
Mimi
That is the question, is it not? He did not cheat. He proved something true. But he did it in a way that would have been impossible ten years ago. The proof is real. The method is new.
Mark
What does this mean for other unsolved problems?
Mimi
It suggests that there are entire categories of problems—abstract, logical, mathematical—that we might be able to solve if we ask the right questions and give the system enough time to work. We are only beginning to understand what that looks like.
Mark
Do you think mathematicians will start using AI this way?
Mimi
Some already are. But there is something about the tradition of mathematics that values the human mind working through a problem. This changes that equation. Whether that is good or bad depends on what you think mathematics is for.
El Pulso
- A decades-old mathematical conjecture about matrix norms — elegant in its statement, stubborn in its resistance — has finally been broken, not by a specialist but by a doctor chasing a different question entirely.
- Jin Shanmu's sixteen-hour autonomous AI run compressed what might have taken human mathematicians months of exploration into a single overnight session, unsettling assumptions about how mathematical discovery happens.
- The mathematics community is absorbing a result that carries two shocks at once: the conjecture itself is solved, and the solver was neither a trained algebraist nor, strictly speaking, a human working alone.
- The proof now invites a harder question — if abstract mathematical reasoning can be delegated to a machine set loose on logical possibility, which other unsolved problems are quietly waiting for the same treatment?
In Beijing, a neurosurgeon named Jin Shanmu set out to understand brain ultrasounds and arrived, unexpectedly, at the edge of mathematical history. By running an AI system autonomously for sixteen hours, he produced a proof of Crouzeix's conjecture — a problem in numerical linear algebra that had resisted resolution for decades. The achievement is less a story about one man's triumph than a quiet signal that the boundary between human curiosity and machine reasoning is shifting in ways we are only beginning to measure.
Jin Shanmu was not looking for a place in mathematical history. A neurosurgeon and postdoctoral researcher at Peking Union Medical College Hospital in Beijing, he was investigating how brain ultrasounds behave under certain conditions when the mathematics of his medical research pulled him into unfamiliar territory — the domain of Crouzeix's conjecture.
First proposed by French mathematician Michel Crouzeix, the conjecture belongs to numerical linear algebra. It makes a precise claim about matrix norms: that applying any function to a matrix should never produce a norm exceeding twice the maximum value that function reaches within the matrix's numerical range. For decades, the problem had sat unresolved, the kind of quiet puzzle that specialists carry with them without quite solving.
Rather than retreat to safer ground, Jin followed the thread. His instrument was not a whiteboard but GPT-5.6-Sol, running autonomously on the ChatGPT Work platform for sixteen uninterrupted hours. The system moved through the logical landscape of the problem — testing approaches, building arguments, tracing chains of reasoning — and when it finished, a proof existed where none had before.
The ripple through the mathematics community was not simply about the conjecture being solved. It was about what the solving revealed: an AI system had worked through genuinely abstract mathematical reasoning, constructing a rigorous logical structure without human guidance at each step. Jin was a doctor, not a linear algebraist. Yet his curiosity, his willingness to follow an unexpected thread, and access to a powerful autonomous system produced what specialists had not.
The moment raises questions that will outlast the proof itself. If machines can now navigate the landscape of abstract mathematical possibility and return with results, what other long-standing problems might yield — and what becomes of the solitary human mathematician when discovery no longer requires solitude?
Jin Shanmu was not hunting for a place in mathematical history when he sat down at his computer one afternoon in Beijing. The neurosurgeon, working as a postdoctoral researcher and resident at Peking Union Medical College Hospital, had a narrower problem in mind: understanding how brain ultrasounds behaved in certain conditions. What he found instead was a proof to a conjecture that had eluded mathematicians for decades.
The problem he stumbled into is called Crouzeix's conjecture, named after French mathematician Michel Crouzeix who first proposed it. In the language of numerical linear algebra, it concerns matrix norms—a measure of how large a matrix is in a particular sense. The conjecture states something elegant and specific: when you apply any function to a matrix, the resulting norm should never exceed twice the maximum value that function reaches within the matrix's numerical range. It sounds abstract because it is. But for mathematicians working in this corner of algebra, it has been a persistent puzzle, the kind of problem that sits in the back of the mind, waiting.
Jin's path to solving it was unconventional. While researching transcranial ultrasounds—the kind used to image the brain without surgery—he found himself drawn into questions about matrix analysis. The mathematics underlying his medical research led him deeper into territory where Crouzeix's conjecture lived. Rather than abandon the thread, he decided to pursue it.
His tool was not a whiteboard or a stack of papers. Instead, Jin turned to GPT-5.6-Sol, a version of ChatGPT running on the ChatGPT Work platform. He set the system to work autonomously, letting it run for sixteen hours straight. During that time, the AI worked through the logical landscape of the problem, testing approaches, building arguments, following chains of reasoning that might have taken a human mathematician weeks or months to explore manually.
When the run completed, Jin had what he came for: a proof. The conjecture that had resisted solution for so long had yielded. The achievement rippled through the mathematics community—not because it solved a problem that would change the world, but because it demonstrated something new about what artificial intelligence could do. This was not a system trained to recognize patterns in images or generate text. This was a system that had autonomously worked through abstract mathematical reasoning, building a rigorous logical structure from the ground up.
The story sits at an intersection that is becoming increasingly common: a researcher in one field—medicine, in this case—finding that the tools of artificial intelligence could solve problems in another field entirely. Jin was not a specialist in numerical linear algebra. He was a doctor investigating ultrasounds. Yet the combination of his curiosity, his willingness to follow a mathematical thread, and access to a powerful AI system produced a result that specialists in the field had not achieved.
What remains to be seen is whether this moment marks the beginning of a broader shift. If AI systems can work autonomously on abstract mathematical proofs, what other problems might yield? And what does it mean for mathematics itself when the path to discovery no longer requires a human mathematician sitting alone with a problem, but instead involves setting a machine loose on the landscape of logical possibility and waiting to see what it finds?
Citas Notables
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— Description of Crouzeix's conjecture
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