Infrared qualification of a scale-invariant covariance ($C(r)-C(r_0)=-\frac{A}{2{\pi }^2}\log (r/r_0)$)

An exact $P(k)=A/k^3$ has divergent unsubtracted point variance. Its finite isotropic covariance difference is the displayed logarithm, unchanged by simultaneous dilation of $r,r_0$. Thus the nonzero-mode scaling statement has a natural subtraction interpretation. Exact dilation invariance of a continuous finite-variance isotropic covariance itself would instead force a constant covariance, corresponding only to a zero-mode spectral measure. Cutoffs or a zero-mode subtraction must accompany the nontrivial spectrum.