Finite-range inhomogeneous Ising kernels form a specification

Since $\mathrm{\Delta }$ is finite and every site couples only to sites within the fixed finite range $R$, the boundary $C_{\mathrm{\Delta }}$ and the set of pairs meeting $\mathrm{\Delta }$ are finite. Every $E_{\mathrm{\Delta }}({\sigma }_{\mathrm{\Delta }}\mid \eta )$ is therefore real and finite. The configuration set $\{-1,+1{\}}^{\mathrm{\Delta }}$ is nonempty and finite, so $Z_{\mathrm{\Delta }}^{\eta }$ is a finite sum of strictly positive real numbers. When $\mathrm{\Delta }=\mathrm{\varnothing }$, that set has its single empty configuration, $E_{\mathrm{\Delta }}=0$, $Z_{\mathrm{\Delta }}^{\eta }=1$, and ${\gamma }_{\mathrm{\Delta }}$ is the identity kernel. In every case, ${\gamma }_{\mathrm{\Delta }}(\cdot \mid \eta )$ is a normalized probability distribution.

Write $X_{{\mathrm{\Delta }}^c}=\{-1,+1{\}}^{{\mathrm{\Delta }}^c}$ with its cylinder-generated Borel sigma-algebra, and let $r_{\mathrm{\Delta }}:X\to X_{{\mathrm{\Delta }}^c}$ be coordinate restriction. For a global input $x\in X$, the full-configuration kernel notation means ${\gamma }_{\mathrm{\Delta }}(f\mid x):={\gamma }_{\mathrm{\Delta }}(f\mid r_{\mathrm{\Delta }}x)$. For a bounded measurable $f$ on $X$, each replacement map $\eta \in X_{{\mathrm{\Delta }}^c}\mapsto {\sigma }_{\mathrm{\Delta }}\eta \in X$ is measurable and each weight in [F1] depends on only the finite boundary $C_{\mathrm{\Delta }}$. Thus the finite weighted sum defining ${\gamma }_{\mathrm{\Delta }}(f\mid \eta )$ is measurable. If $B\subseteq X$ is exterior-measurable, write $B=r_{\mathrm{\Delta }}^{-1}(D)$ for a measurable $D\subseteq X_{{\mathrm{\Delta }}^c}$. Since $r_{\mathrm{\Delta }}({\sigma }_{\mathrm{\Delta }}\eta )=\eta$, one has ${\mathbf{1}}_B({\sigma }_{\mathrm{\Delta }}\eta )={\mathbf{1}}_D(\eta )$ and therefore ${\gamma }_{\mathrm{\Delta }}({\mathbf{1}}_B\mid \eta )={\mathbf{1}}_D(\eta )$ by normalization. Equivalently, for the full-configuration input $x$, ${\gamma }_{\mathrm{\Delta }}({\mathbf{1}}_B\mid x)={\mathbf{1}}_D(r_{\mathrm{\Delta }}x)={\mathbf{1}}_B(x)$, proving properness. Its cylinder probabilities depend only on $C_{\mathrm{\Delta }}$ and the finitely many coordinates named by the cylinder, so it has finite range.

Let $\mathrm{\Delta }\subseteq \mathrm{\Lambda }$ be finite. Split all interaction terms meeting $\mathrm{\Lambda }$ into terms meeting $\mathrm{\Delta }$ and terms disjoint from $\mathrm{\Delta }$. For fixed spins $\xi =s_{\mathrm{\Lambda }\setminus \mathrm{\Delta }}$ and exterior $\eta$, the latter terms contribute a finite $C(\xi ,\eta )$ independent of $s_{\mathrm{\Delta }}$, so $E_{\mathrm{\Lambda }}(\zeta \xi \mid \eta )=E_{\mathrm{\Delta }}(\zeta \mid \xi \eta )+C(\xi ,\eta )$ and $Z_{\mathrm{\Lambda }}^{\eta }={\sum }_{\xi \in \{-1,+1{\}}^{\mathrm{\Lambda }\setminus \mathrm{\Delta }}}{e}^{-\beta C(\xi ,\eta )}{Z}_{\mathrm{\Delta }}^{\xi \eta }$. Thus the ${\gamma }_{\mathrm{\Lambda }}$-marginal weight of each $\xi$ is $e^{-\beta C(\xi ,\eta )}{Z}_{\mathrm{\Delta }}^{\xi \eta }\mathrm{/}{Z}_{\mathrm{\Lambda }}^{\eta }$, and applying ${\gamma }_{\mathrm{\Delta }}$ gives $({\gamma }_{\mathrm{\Lambda }}{\gamma }_{\mathrm{\Delta }}f)(\eta )=\frac{1}{Z_{\mathrm{\Lambda }}^{\eta }}{\sum }_{\xi }{e}^{-\beta C(\xi ,\eta )}{\sum }_{\zeta \in \{-1,+1{\}}^{\mathrm{\Delta }}}f(\zeta \xi \eta )e^{-\beta E_{\mathrm{\Delta }}(\zeta \mid \xi \eta )}=\frac{1}{Z_{\mathrm{\Lambda }}^{\eta }}{\sum }_{\xi ,\zeta }f(\zeta \xi \eta )e^{-\beta E_{\mathrm{\Lambda }}(\zeta \xi \mid \eta )}=({\gamma }_{\mathrm{\Lambda }}f)(\eta )\mathrm{.}$ The factor depending on $\xi$ remains in the outer sum, giving exactly the ${\gamma }_{\mathrm{\Lambda }}$ weights. Hence ${\gamma }_{\mathrm{\Lambda }}{\gamma }_{\mathrm{\Delta }}={\gamma }_{\mathrm{\Lambda }}$.

Steps 1.1 and 2.2 prove that the general position-dependent kernels satisfy the specification axioms of [F2]. Now assume the translation-invariant subcase in the Statement. Choose a finite set $D$ containing exactly one of $v,-v$ for each nonzero displacement pair with $J(v)\mathrm{\ne }0$. Use the shape $\{0\}$ with potential $-h{s}_0$ and, for each $v\in D$, shape $\{0,v\}$ with potential $-J(v)s_0{s}_v$. Every field site is counted once, and every unordered interacting pair is counted once because $D$ contains one orientation per pair. The CSM finite-volume and conditional energies in [F3] are therefore exactly $H_{\mathrm{\Lambda }}$ and $E_{\mathrm{\Lambda }}$ in [F1], so the kernels coincide. Applying [F4] confirms that this same kernel is a valid CSM finite-range specification under the matching hypotheses.