Linear stability and divisibility gaps for Gaussian maxima

We study Gaussian maxima arising from isotropic families of unit vectors and show that their excess expected maximum is controlled by the distance from an orthonormal basis. We apply this stability result to correlation matrices with equal nonzero eigenvalues, obtaining an exact minimum when the matrix size is divisible by the rank and a quantitative gap otherwise. In dimension two, we determine the leading asymptotic term and describe the structure of near-optimal configurations. We also obtain covariance-block recovery results and a finite-dimensional lower bound for the higher-rank limiting problem.