Proposition: Explicit compact-test covariance bound

Every compact smooth real reduced test satisfies

In particular, its thermal infrared integral is finite.

Proof. At $0<\omega \le {\omega }_c$, use (eq:testcoefficient) and Lemma lem:modebound:

At $\omega \ge {\omega }_c$, integrate once by parts in $t$:

Use $1+2n_{\mathrm{H}}(\omega )\le 1+2/(\beta \omega )$ for both channel bounds. Summing the two channel estimates and performing the elementary integrals gives (eq:compactcovariancebound).