Lemma: Positive mass-density Gram matrix

The integrals in (eq:mass-Gram) are finite, and $K$ is real symmetric positive semidefinite. There are real coefficients ${\ell }_{\alpha a}$ such that

Moreover,

Proof. The Coulomb singularity is locally integrable in three dimensions; compact smooth profiles make the integrals finite. For $q={\sum }_a{c}_a{\mu }_a$, the Coulomb quadratic form is nonnegative: its Fourier representation has nonnegative multiplier $4\pi /\text{abs}{\mathbf{k}}^2$. The integral is finite at zero and infinity for a smooth compact profile. Thus $c^{\mathsf{T}}Kc\ge 0$. A real spectral factorization gives the coefficients. Expanding the squared profile difference proves (eq:EG-rate).