The Surgeon Who Took On a 22‑Year Math Monster

I. That Morning on arXiv

On July 30, 2026, Alex Townsend, professor of applied mathematics at the University of Washington, opened the preprint server arXiv as he had done on countless ordinary mornings.

For more than a year, Townsend had developed a near‑compulsive routine: every few weeks he would ask GPT‑5.6 to prove the Crouzeix conjecture, a famous open problem in numerical linear algebra. Hundreds of attempts, hundreds of failures. The AI’s answers either collapsed under logical holes or stalled at a critical lemma it could not cross. Humanity had been stuck on this problem for 22 years, and the AI seemed no smarter than its human counterparts.

That day, the pattern broke.

On arXiv, Townsend found a paper titled “The Numerical Range Is a 2‑Spectral Set.” The Crouzeix conjecture, which had frustrated numerical linear algebra for two decades, had been proved.

The author field held a name that made him stop: not a mathematician, not even a degree in mathematics. It was Shanmu Jin, a postdoctoral researcher and neurosurgery resident at Peking Union Medical College Hospital in Beijing.

Townsend stared at the screen. His hand hovered over the keyboard, but he did not type.

II. The Constant That Tormented Mathematicians for 22 Years

To understand the shock, you need to grasp the conjecture itself.

In modern applied mathematics and engineering, matrices are the universal language of change: quantum states, Google’s PageRank, the parameter updates in large language models—all are projections of huge matrices on the complex plane.

But non‑normal matrices behave oddly. They are unpredictable. To tame them, French mathematician Michel Crouzeix proposed in 2004 that for any polynomial p and any matrix A, the spectral norm is always bounded by the maximum of the function on the matrix’s “numerical range,” and the optimal constant is exactly 2.

The statement is short and geometrically neat. Short, however, does not mean easy.

In 2007, Crouzeix himself, after immense effort, could only prove the inequality with a constant of 11.08. A decade later, in 2017, the American Institute of Mathematics hosted a week‑long workshop in San Jose, bringing together the world’s best. Crouzeix and his collaborator Palencia pushed the constant down to 1+√2, about 2.414.

Then progress stopped.

For 22 years, no one found the next step. It was not a shortage of ideas or computing power. It was the lack of one precise angle—the right leverage point. That angle eventually came from a hand that holds a scalpel.

III. A Person Who Should Not Have Been There

Shanmu Jin’s résumé has nothing to do with mathematics.

He majored in geology as an undergraduate, then shifted to medicine and earned a medical doctorate. His formal mathematics training amounted to the basic courses required for any science student.

“Everything beyond that, I learned by myself,” Dr. Jin wrote casually in an email to Townsend and Greenbaum.

Why would a neurosurgeon take on such a hard problem?

It began with a clinical challenge: transcranial ultrasound research. To send ultrasound through the human skull, he found himself deep in matrix analysis. While self‑studying, he came across the Crouzeix conjecture. Its clean geometric form drew him in.

He decided to try—not with pencil and paper, but with a large language model.

IV. 16 Hours, Tens of Thousands of Failures

The schedule of a neurosurgery resident at Peking Union is punishing. Dr. Jin’s time was sliced into surgery and call shifts. The only feasible approach was to set up the AI and let it run unattended.

The secret was prompt engineering.

He did not treat the AI as a chatty assistant. Instead, he turned it into a virtual mathematics institute. He borrowed and adapted the prompt OpenAI had used for the Cycle Double Cover conjecture, and built a careful strategy:

Physical disconnection – the prompt cut off internet access. He did not want the AI to recycle failed human attempts; he forced it to reason from first principles.

Multi‑branch exploration – he asked ChatGPT to spawn many “sub‑agents,” each pursuing genuinely different paths. The instruction was blunt: do not prematurely converge on the same tempting idea.

Adversarial auditing – every candidate proof had to survive repeated attacks by other agents. A route was discarded only when another agent found a concrete counterexample.

No surrender – the prompt ordered the system not to give up until it produced a proof that could withstand extreme logical scrutiny.

After setting this up, Dr. Jin hit Enter and went to his ward rounds.

GPT‑5.6 Sol ran on the ChatGPT Work platform for about 16 hours, iterating through tens of thousands of hypotheses, rejections, and rebuilds.

When mathematicians later read the final theorem, they were struck by its elegance: the AI had not used the brute‑force estimates that everyone had expected. Instead, it found a clever “sampling strategy” that reduced the problem to a surprisingly simple “positivity condition.”

It was like pushing a heavy door with your shoulder for years, then realising you only need to press a small latch.

Townsend’s fingers finally hit the keyboard. He typed two words.

V. “Stunned”

At first, the top mathematicians viewed the preprint with scepticism. Outsiders claiming to solve world‑class problems usually end in embarrassment.

Within hours, scepticism gave way to astonishment.

Townsend and Greenbaum, both authorities in the field, described their reaction in two words: “Stunned.”

Then the conjecture’s originator, Michel Crouzeix, went through the proof line by line. His verdict: Dr. Jin’s manuscript is correct. The 22‑year‑old case was closed.

Dr. Jin also made everything open source: the final paper, the prompt, the iteration drafts, the Lean 4 formalisation, and the audit reports. All in the open, for anyone to check.

But he did not stop.

VI. Another Door, Eight Days Later

Just eight days after the preprint appeared—on August 4, 2026—two other mathematicians, Emiel Lorist of Delft University of Technology and Felix Schwenninger of the University of Twente, posted an independent proof, only 5 pages long.

Its title: “A Solution to Crouzeix’s Conjecture.”

That 5‑page proof took a completely different route. It combined the classical double‑layer potential representation with a perturbation lemma about 2‑expansions. To traditional mathematicians, it felt more familiar and easier to digest.

Coincidentally: Lorist and Schwenninger acknowledged in their AI disclosure that they, too, had used ChatGPT‑5.6 while exploring their strategy.

A problem that resisted solution for 22 years had now, with AI assistance, generated two distinct and equally sound proofs within eight days.

When Dr. Jin heard the news, he felt no annoyance. He was genuinely pleased—he saw it as a welcome complement.

VII. The Telescope

In the summer of 2026, something shifted.

Mathematics has had physicians who produced important results before—but all of them had received serious mathematical training in their era. This may be the first time a long‑standing open conjecture was definitively solved by a doctor without specialised training, using an AI.

Steven Strogatz, professor at Cornell and a well‑known mathematics writer, posted on social media: “A neurosurgeon without advanced math training, using ChatGPT‑5.6, solved a major open problem in numerical linear algebra. This is remarkable on so many levels—I myself spent a week trying to solve this problem back in 2012.”

A few years ago, the idea that an AI could supply a decisive proof idea for a major conjecture would have seemed far‑fetched. Now it has happened.

It resembles the two leaps in astronomy: the first, when astronomers had to do all orbital calculations by hand, and only a few prodigies could see the stars clearly; the second, when Galileo built the telescope.

AI will accelerate discovery and collapse disciplinary boundaries. But it also forces a new question about identity: when a neurosurgeon can pluck a mathematical crown jewel in his spare time, what does it mean to be a mathematician or a scientist?

The telescope is here. What we do with it is now the open question.