field $\Phi_0$ on Minkowski spacetime, and let $N \in \mathbb{N}$.

-   Let $D$ be a subset of Minkowski spacetime and let $\boldsymbol{\beta}: x \mapsto \boldsymbol{\beta}(x)$ be a (smooth, if $D$ is open) map assigning to each $x \in D$ a future-directed timelike vector $\boldsymbol{\beta}(x)$. Then we say that [*$\omega$ is a local thermal equilibrium state of order $N$ at sharp temperature for the temperature vector field $\boldsymbol{\beta}$*]{} (for short, $\omega$ is a $[D,\boldsymbol{\beta},N]$-LTE state) if $$\begin{aligned}
     \label{lte-s}
     \langle \eth_{\mu_1 \cdots \mu_n} :\Phi_0^2:(x) \rangle_\omega & =
      \langle \eth_{\mu_1 \cdots \mu_n} :\Phi_0^2:(0) \rangle_{\boldsymbol{\beta}(x)}\end{aligned}$$ holds for all $x \in D$ and $0 \le n \le N$. Here, we have written $\langle \,. \rangle_{\boldsymbol{\beta}(x)} =
    \omega_{\boldsymbol{\beta}(x)}(\,.\,)$ for the KMS-state of $\Phi_0$ defined with respect to the timelike, future-directed vector $\boldsymbol{\beta}(x)$. The balanced derivatives of the Wick square on the right hand side are evaluated at the spacetime point $0$; since the KMS state on the right hand side is homogeneous and isotropic, one has the freedom to make this choice.

-   Let $D$ be a subset of Minkowski spacetime and let $m: x \mapsto \varrho_x$ be a map which assigns to each $x \in D$ a probability measure compactly supported on $V_+$, the open 