Persistence diagrams of random matrices via Morse theory: universality and a new spectral diagnostic

ArXi:2603.27903v1 Announce Type: cross We prove that the persistence diagram of the sublevel set filtration of the quadratic form f(x) = x^T M x restricted to the unit sphere S^{n-1} is analytically determined by the eigenvalues of the symmetric matrix M. By Morse theory, the diagram has exactly n-1 finite bars, with the k-th bar living in homological dimension k-1 and having length equal to the k-th eigenvalue spacing s_k = \lambda_{k+1} - \lambda_k.