Complete Positivity of Additive Matrices and Harmonic Decompositions of Quartics

We give a spectral criterion for complete positivity of an additive matrix family in every dimension when the diagonal parameter is at most four. The proof constructs a finite nonnegative factor at the spectral boundary from two laws with prescribed means and covariances. The same criterion yields a harmonic decomposition of every nonnegative quartic and an exact test for when a product of two positive definite quadratic forms is a sum of real fourth powers. For probability measures on the sphere with vanishing fourth harmonic moments, we classify the feasible second moments, identify the boundary support and obtain sharp bounds for isotropic regularization. The accompanying code reproduces the calculations.