Partial Proof of a Conjecture with Implications for Spectral Majorization

In this paper we report on new results relating to a conjecture regarding properties of n× n, n≤ 6, positive definite matrices. The conjecture has been proven for n≤ 4 using computer-assisted sum of squares (SoS) methods for proving polynomial nonnegativity. Based on these proven cases, we report on the recent identification of a new family of matrices with the property that their diagonals majorize their spectrum. We then present new results showing that this family can extended via Kronecker composition to n>6 while retaining the special majorization property. We conclude with general considerations on the future of computer-assisted and AI-based proofs.