Moment-Constrained Vector Reconstruction of Random-Matrix Statistics in Finite Hilbert Spaces

Abstract

Random-matrix statistics are usually imposed at the level of matrix entries or spectral correlations. Here we formulate a complementary inverse problem: can a matrix with prescribed random-matrix moments be generated from a structured set of latent vectors? We introduce a pair-resolved vector ansatz consisting of two vector families, P and Q, construct a complex-symmetric non-Hermitian matrix M = a1P P T + a2QQT . The transpose is intentionally not a conjugate transpose; hence the reconstructed bilinear overlap matrices are not Hermitian Gram matrices once the algebraic parameters become complex. The free parameters of the vectors are fixed by complex algebraic constraints matching diagonal and off-diagonal random-matrix moments, together with a mixed-overlap condition suppressing systematic correlations between the two bilinear sectors. A fast machine-precision solve for N = 8 returns six complex branches. We therefore supplement moment matching with reproducible branch diagnostics: residual error, approximate vector orthogonality, non-Hermiticity, imaginary spectral weight, inverse participation ratio, maximum component weight, and eigenvector conditioning. Optional entanglement and low-weight Pauli-moment diagnostics can be added when N = 2n . This protocol constitutes a finite-dimensional inverse reconstruction of hidden vectorspace representations behind apparent random-matrix behavior. It is static and algebraic: it probes moment-induced delocalization, non-Hermitian branch structure, and complex spectral statistics, but it does not by itself establish dynamical chaos in the sense of sensitive dependence on nearby initial conditions.