Canonical energy and fluctuation responses

For the canonical preparation Canonical thermal preparation, write $H:=H_{N,\lambda }:{\mathrm{\Gamma }}_N\to \mathbb{R}\cup \{+\mathrm{\infty }\}$ for its measurable Hamiltonian and $m:=m_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ for its reference. With these data fixed independently of $T$, let $Z(\beta )=\int e^{-\beta H}dm$ be finite and positive on an open interval of $\beta >0$. If $H$ may equal $+\mathrm{\infty }$, put $S_H:=\{x:H(x)<\mathrm{\infty }\}$, restrict $m$ to this measurable set, and evaluate energy moments on $S_H$; the canonical weight is zero outside it, and the restricted measure remains sigma-finite. In this statement $U$ and $\mathrm{Var}(H)$ are the expectation and variance of $H{\mathrm{\mid }}_{S_H}$. Assume the local envelope hypotheses of Dominated exponential normalization and covariance hold for the finite-valued statistic $X=H{\mathrm{\mid }}_{S_H}$ and $\theta =-\beta$ on the restricted measure space. Under the canonical preparation Canonical thermal preparation, the mean energy $U=⟨H⟩$ obeys ${\mathrm{\partial }}_{\beta }\log Z=-U,{\mathrm{\partial }}_{\beta }U=-\mathrm{Var}(H),C_V:={\mathrm{\partial }}_{T}U=\frac{\mathrm{Var}(H)}{k_B{T}^2}\ge 0,$ where $\beta =(k_{B}T{)}^{-1}$ and $C_V$ is the model heat capacity at the fixed controls. This is a finite-model response identity, not an empirical claim.