GPT-5.6 helped find a path toward proving Crouzeix's 22-year-old matrix conjecture

GPT-5.6 helped find a path toward proving Crouzeix's 22-year-old matrix conjecture

On 27 July, Shanmu Jin posted a manuscript establishing the exact bound for functions of matrices. On 14 August, mathematicians Alex Townsend and Anne Greenbaum reported that they had carefully checked the proof with Michel Crouzeix, who formulated the conjecture. On 4 August, another group published an independent route to the same theorem.

Matrices are arrays of numbers that define transformations of vectors. For a matrix A, its numerical range is a region in the complex plane formed by the inner products of Ax with x over all unit vectors x. Crouzeix's conjecture states that if a polynomial p, meaning a formula built from addition and multiplication, has magnitude at most M everywhere in this region, then the matrix p(A), obtained by substituting A into the formula, has norm at most 2M. In other words, it can increase the length of any vector by a factor of no more than 2M.

Michel Crouzeix formulated the general version of the problem in 2004. His 2007 estimate gave a factor of 11.08. In 2017, Crouzeix and Palencia reduced it to 1 + √2, approximately 2.414. Reaching exactly two required a different step in the proof.

The earlier approach estimated the target quantity together with an additional expression, and this combination kept the universal bound above two. In the current manuscript, Jin identifies a specific contribution from ChatGPT: the model suggested which points to use when evaluating an auxiliary function and proposed adding the point at zero. This choice removes the additional expression. A comparison of two weighted sums of squares then gives the factor of two. Jin states that he remains responsible for the mathematical claims and references in the manuscript.

The published prompt given to the model instructed it to develop several distinct lines of proof in parallel and search for counterexamples to each candidate. The repository also contains the manuscript source files, a Lean formalization (a representation of the proof in a language that allows a computer to verify each formal step), and a report listing the axioms used.

In their SIAM News article, Townsend and Greenbaum write that they and Crouzeix examined the manuscript carefully and consider it correct. On 4 August, mathematicians Emiel Lorist and Felix Schwenninger posted a five-page independent proof. They also used ChatGPT while searching for a strategy.