Quantitative Chollet Inequalities for Matrices of Rank at Most Four: Spectral bounds and an order-nine tight-frame construction

Abstract

Chollet's permanent conjecture asks whether per(A o B) <= per(A) per(B), where o denotes the entrywise (Hadamard) product, for complex Hermitian positive semidefinite matrices. We present computer-assisted proofs of two restricted forms with explicit constants strictly smaller than one. For every integer n >= 10 and every such matrix A of rank at most four, the self-conjugate ratio per(A o conjugate(A)) / per(A)^2 is bounded by 999991742359 / 10^12 when the denominator is nonzero. For order nine, we obtain the bound 999815240367 / 10^12 for correlation matrices whose four nonzero eigenvalues all equal 9/4. The first argument combines complex-sphere integrals, spectral subspace tilts, projection bounds, exact finite covers, and an analytic infinite tail. The second uses a quadratic relation among nine Gram vectors, a positive operator on the ten-dimensional space of quadratic forms, and rigorously bounded entropy. All decisive finite calculations use rational arithmetic and full closed-domain certificates. The unrestricted conjecture, including general non-tight order-nine rank-four matrices, is outside these results.