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Matrix Proof Found The GPT-5.6 model helped find a path to proving the 22-year-old Kroizë hypothesis on matrices. On July 27, Shangmu Jin posted a manuscript with an exact boundary for matrix functions. On August 14, mathematicians Alex Townsend and Anne Greenbaum reported that, along with the hypothesis author Michel Kroizë, they had thoroughly checked the proof. The Kroizë hypothesis states that if a polynomial p, a formula of additions and multiplications, nowhere exceeds a magnitude M on the numerical range of a matrix A, then the matrix p(A) has a norm no higher than 2M. In 2004, Michel Kroizë formulated a general variant of the question, and his 2007 estimate gave a multiplier of 11.08; in 2017, Kroizë and Palenzia reduced it to 1 + √2, approximately 2.414. To get exactly two, a different approach was needed in the proof. In the previous approach, the target value was estimated along with an additional expression, which kept the universal bound above two. The current manuscript by Jin cites a specific contribution from ChatGPT: the model suggested where to take the values of the auxiliary function and add a point at zero. This choice removes the additional expression, and comparing two weighted sums of squares gives a multiplier of two, as reported in SIAM News. 🔗 Read original →