Jin is a postdoctoral researcher and neurosurgery resident at Peking Union Medical College Hospital. He has no formal advanced mathematics training — his bachelor's degree is in geology, and he holds a medical doctorate . He originally encountered the problem while self-studying the mathematical theory needed for transcranial brain ultrasound research, where operator norms and numerical ranges arise naturally .

The problem he solved is Crouzeix's conjecture, proposed by French mathematician Michel Crouzeix in 2004. The conjecture states that for any square complex matrix A and polynomial p, the operator norm ||p(A)|| is at most twice the maximum value of |p| over the numerical range W(A) . The sharp constant of 2 had been open for 22 years; earlier bounds had only reached roughly 2.414 .

How Jin Shanmu Set Up the Autonomous Run

Jin's approach was not a simple ChatGPT query. He designed an autonomous reasoning system, borrowing from OpenAI's previous success on the Cycle Double Cover conjecture . His prompt strategy included four key instructions:

1. Air-gapped environment: The model was instructed to operate in a physically disconnected (air-gapped) environment to force purely original reasoning, preventing it from relying on existing solutions or known proofs .
2. Multi-agent parallel exploration: He launched multiple agent instances to explore different solution paths in parallel, preventing the system from prematurely converging on a single flawed approach .
3. Adversarial auditing: He applied adversarial auditing to candidate strategies — the AI was told to critique its own reasoning and test its proofs against counterexamples .
4. No early stopping: He commanded the model not to stop until a complete proof was obtained .

After starting the run, Jin left the system to work. "He locked the model off the web, started it, and left. Sixteen hours later it had the proof," according to reports . During those 16 hours, GPT-5.6-Sol executed thousands of cycles of hypothesis generation, testing, and refinement .

The Proof and Its Verification

The result was a proof that Crouzeix's conjecture is true: the constant 2 is correct. The proof was posted as a preprint on July 27, 2026, titled The Numerical Range Is a 2-Spectral Set .

The verification process was rigorous. Michel Crouzeix himself — the mathematician who originally posed the conjecture in 2004 — reviewed the argument and confirmed it is correct . Numerical analysts also examined the proof . Jin additionally published the full research materials on GitHub, including the final paper, prompts, iteration drafts, Lean 4 formal proof code, and audit reports .

What This Means for AI-Driven Research

Jin's achievement is notable not just because of the problem's difficulty, but because of his background and methodology. A neurosurgery resident with no formal advanced math degree, using an AI model autonomously for 16 hours, solved a problem that had frustrated mathematicians for two decades.

The story highlights a new mode of research: humans design the experimental conditions and reasoning framework, while AI models execute large-scale, autonomous exploration of mathematical space. It is a concrete demonstration that AI can serve as a genuine research partner in mathematics — not just a tool for computation or search, but a system capable of generating novel proofs.

Whether this approach becomes a standard method in mathematical research remains to be seen. But for now, Jin Shanmu's result stands as one of the most striking examples of AI-augmented discovery in 2026.

Jin is a postdoctoral researcher and neurosurgery resident at Peking Union Medical College Hospital. He has no formal advanced mathematics training — his bachelor's degree is in geology, and he holds a medical doctorate . He originally encountered the problem while self-studying the mathematical theory needed for transcranial brain ultrasound research, where operator norms and numerical ranges arise naturally .

The problem he solved is Crouzeix's conjecture, proposed by French mathematician Michel Crouzeix in 2004. The conjecture states that for any square complex matrix A and polynomial p, the operator norm ||p(A)|| is at most twice the maximum value of |p| over the numerical range W(A) . The sharp constant of 2 had been open for 22 years; earlier bounds had only reached roughly 2.414 .

How Jin Shanmu Set Up the Autonomous Run

Jin's approach was not a simple ChatGPT query. He designed an autonomous reasoning system, borrowing from OpenAI's previous success on the Cycle Double Cover conjecture . His prompt strategy included four key instructions:

1. Air-gapped environment: The model was instructed to operate in a physically disconnected (air-gapped) environment to force purely original reasoning, preventing it from relying on existing solutions or known proofs .
2. Multi-agent parallel exploration: He launched multiple agent instances to explore different solution paths in parallel, preventing the system from prematurely converging on a single flawed approach .
3. Adversarial auditing: He applied adversarial auditing to candidate strategies — the AI was told to critique its own reasoning and test its proofs against counterexamples .
4. No early stopping: He commanded the model not to stop until a complete proof was obtained .

After starting the run, Jin left the system to work. "He locked the model off the web, started it, and left. Sixteen hours later it had the proof," according to reports . During those 16 hours, GPT-5.6-Sol executed thousands of cycles of hypothesis generation, testing, and refinement .

The Proof and Its Verification

The result was a proof that Crouzeix's conjecture is true: the constant 2 is correct. The proof was posted as a preprint on July 27, 2026, titled The Numerical Range Is a 2-Spectral Set .

The verification process was rigorous. Michel Crouzeix himself — the mathematician who originally posed the conjecture in 2004 — reviewed the argument and confirmed it is correct . Numerical analysts also examined the proof . Jin additionally published the full research materials on GitHub, including the final paper, prompts, iteration drafts, Lean 4 formal proof code, and audit reports .

What This Means for AI-Driven Research

Jin's achievement is notable not just because of the problem's difficulty, but because of his background and methodology. A neurosurgery resident with no formal advanced math degree, using an AI model autonomously for 16 hours, solved a problem that had frustrated mathematicians for two decades.

The story highlights a new mode of research: humans design the experimental conditions and reasoning framework, while AI models execute large-scale, autonomous exploration of mathematical space. It is a concrete demonstration that AI can serve as a genuine research partner in mathematics — not just a tool for computation or search, but a system capable of generating novel proofs.

Whether this approach becomes a standard method in mathematical research remains to be seen. But for now, Jin Shanmu's result stands as one of the most striking examples of AI-augmented discovery in 2026.