On July 27, 2026, a preprint appeared with the proof of a problem that had been open for 22 years. The author is not a professional mathematician: Shanmu Jin is a neurosurgery resident at Peking Union Medical College Hospital, in Beijing. He graduated in geology. Mathematics beyond the basics, by his own account, was self-taught — he started studying it because of a transcranial ultrasound research project.
The decisive argument didn't come from a conversation with ChatGPT. It came from an autonomous run of about 16 hours of GPT-5.6 Sol, which Jin started and never interrupted once.
Michel Crouzeix, the French mathematician who formulated the conjecture in 2004, thoroughly verified the manuscript together with Alex Townsend and Anne Greenbaum. All three consider the proof correct.
The story made headlines for the obvious reason — "AI solves problem humans couldn't." But what matters to those who use AI for work lies elsewhere: in the design of the process. And that part can be copied.
Why 22 years
The Crouzeix conjecture is a matrix analysis problem. In 2007 the author himself proved that a certain constant — 11.08 — always works. In 2017, with Palencia, the bound dropped to 1 + √2. Proving that the exact value is 2, as the conjecture states, resisted two decades of attempts by full-time specialists.
Keep this detail in mind: it wasn't an obscure problem waiting for someone to look at it. It was a known problem, tackled by competent people, without a solution.
What he did differently: the prompt
Jin started from a prompt that OpenAI had used on another conjecture, the Cycle Double Cover conjecture, and adapted it. The design involved four decisions that explain the result:
- He cut off web access. The system was forbidden from consulting the internet and external contexts. If the answer couldn't be looked up, it had to be built.
- He had it create several competing subagents, each independently developing a different proof strategy.
- He banned early convergence. The instruction was explicit: the subagents could not all rush toward the same attractive idea. On a hard problem, the obvious path has already been tried by everyone — and failed.
- He required adversarial auditing. Every candidate proof was attacked until one survived verification.
Notice what's not on the list: none of this is about having access to a better model. It's about not letting the AI do what it does by default — which is to quickly produce the most plausible answer and stop there.
The public repository includes the prompt used, the successive manuscripts, a formalization in Lean, and an axiom audit. The method was published together with the result.
What the AI didn't do
This part tends to disappear from the summary, and it's the most important one for anyone who tries to repeat the recipe.
The autonomous run produced a result. It was the human who had to recognize that the result was relevant, convert it into a coherent proof, and sign off on it, taking responsibility for the manuscript. Someone without the mathematical foundation Jin built on his own would have looked at the same 16-hour output without knowing there was something there.
Due skepticism is also warranted: the work is still a preprint that has not undergone peer review. It has been examined by heavyweight specialists, including the conjecture's original author, but it has not yet gone through the mathematical community's formal process.
And there's a detail that puts the feat in perspective without diminishing it: eight days after Jin's preprint, Emiel Lorist and Felix Schwenninger published an independent proof, just five pages long and with a different argument — also backed by ChatGPT 5.6. In other words, the problem was ripe, and the tool helped more than one group at the same time.
How to bring this to your company
You probably don't have a matrix analysis conjecture on your desk. But you do have problems no one has solved: a process that always stalls at the same point, a proposal that never closes, a system nobody really understands.
The way almost everyone uses AI for this is to ask for the answer and accept the first one that comes back. The method that worked here is the opposite:
- Ask for different paths, not the best answer. "Give me four approaches that don't resemble each other" yields more than "what's the best way to do this."
- Block the shortcut. If the problem has a standard answer that has already failed at your company, say so and forbid the AI from proposing it.
- Have it attack its own answer. "Now find the three reasons this will fail" separates a good idea from a pretty one.
- Give the process time. Sixteen hours of exploration is not the same as sixteen questions. A long, uninterrupted run is a different mode of use.
- Keep someone who understands the subject at the end of the line. Without that person, you can't tell a good result apart from convincing text — and that's where most AI implementation projects for companies get lost.
The honest summary
What happened wasn't "the AI got smarter." The model was available to any subscriber. What changed the outcome was a self-taught doctor designing a process that forced the machine to disagree with itself for 16 hours, and having enough knowledge to recognize what came back.
The replicable part of this story isn't the solved problem. It's the method — and it's published.
Facts gathered from Alex Townsend's essay for SIAM News, coverage by the South China Morning Post, Yicai, and tech news from August 2026. Jin's manuscript remains an unpublished preprint that has not undergone peer review.
Perguntas frequentes
What was the Crouzeix conjecture and why was it hard?
It is a problem in matrix analysis formulated by Michel Crouzeix in 2004. He proved in 2007 that a constant of 11.08 always works and, in 2017, with Palencia, brought the universal bound down to 1 + √2. Proving that the exact value is 2 resisted 22 years of attempts by full-time specialists — it was not an obscure problem, but a known one that people had attacked.
Did artificial intelligence solve the problem on its own?
No. The roughly 16-hour autonomous run of GPT-5.6 Sol produced the key result, but a human still had to recognise that the result mattered, turn it into a coherent proof and take responsibility for the manuscript. Without the mathematical grounding Shanmu Jin built on his own, that same output would have gone unnoticed.
Who is Shanmu Jin?
He is a postdoctoral researcher and neurosurgery resident at Peking Union Medical College Hospital in Beijing. He graduated in geology and says his mathematical training beyond the basics was self-taught, prompted by research on transcranial ultrasound.
What did the prompt that produced the proof look like?
It was adapted from a prompt OpenAI had used on another conjecture and rested on four decisions: deny access to the public web and outside context, spawn several competing subagents each developing a different proof strategy, forbid early convergence on the same attractive idea, and require adversarial audits until one proof survived verification. The prompt is published in the repository along with the successive manuscripts, a Lean formalisation and an axiom audit.
Has the proof been officially confirmed?
Michel Crouzeix, who posed the conjecture, checked the manuscript thoroughly together with Alex Townsend and Anne Greenbaum, and all three consider it correct. Even so, the work remains a non-peer-reviewed preprint — it has not gone through the mathematical community's formal process. Eight days later, Emiel Lorist and Felix Schwenninger published an independent five-page proof with a different argument, also supported by AI.
How do I apply this method in my company?
Ask for different routes rather than the best answer; explicitly block the standard solution that already failed; tell the AI to attack its own proposal and list why it would fail; use long exploration runs instead of many short questions; and keep someone who knows the subject to judge the output. Without that last item you cannot tell a good result from convincing text.