= Infrared qualification of a scale-invariant covariance
{title2=$C(r)-C(r_0)=-\frac{A}{2\pi^2}\log(r/r_0)$}
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.
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