Trace-Gibbs bounded correlation measure and balance

Assume the Axiom of Choice. Let $\mathcal{H}\mathrm{\ne }\{0\}$ be separable, $K=K^{\ast }$ densely defined and bounded below with energy unit J, and let $e^{-sK}$ be trace class for every $s$ in an open interval $I\subset (0,\mathrm{\infty })$ measured in J${}^{-1}$. Fix $\beta \in I$, $Z_{\beta }=\mathrm{Tr}(e^{-\beta K})$, ${\sigma }_{\beta }=Z_{\beta }^{-1}{e}^{-\beta K}$, and $U(t)=e^{-itK\mathrm{/}\mathrm{\hslash }}$ for $\mathrm{\hslash }>0$ in J s and $t$ in s. For bounded $A,B$, let $C_{AB}$ and $S_{AB}$ be the correlation function and finite spectral measure defined in Absolute convergence of trace-Gibbs bounded correlation series. Write $S_{BA}(-\cdot )$ for the pushforward under $\omega \mapsto -\omega$. Then for every bounded Borel test function $f$ with compact support, $\int f(\omega )S_{BA}(-d\omega )=\int f(\omega )e^{-\beta \mathrm{\hslash }\omega }{S}_{AB}(d\omega )\mathrm{.}$ The weighted measure on the right extends to a finite complex measure: the reindexed coefficient family is absolutely summable and has the same total-variation bound as $S_{BA}$. For Hermitian $A=B$, $S_{AA}$ is a positive finite measure. This is a discrete-spectrum equilibrium correlation identity; it asserts no temporal decay, continuum limit, or ordinary zero-frequency transport coefficient.