Fixed-particle local ensemble equivalence at interior density

Assume AC, the ultrafilter extension principle, DC, and HB, and the superstable regular model of Superstable regular continuum grand pressure and variational formula, $\beta >0$, and an interior attainable density $\rho >0$. Let $N_L\mathrm{/}{L}^d\to \rho$, let $C_{L,N_L}$ be the free-boundary positional canonical law with exactly $N_L$ particles, and let $R_{L,\omega }$ be its periodized empirical field from Periodized empirical field of a finite configuration. Choose any finite supporting activity $z>0$ of $F_{\beta }$ at $\rho$ and set ${\mathcal{M}}_{z,\beta }:={\mathrm{argmin}}_{P\in {\mathcal{P}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}^1}\{I(P)+\beta \mathrm{\Phi }(P)-\rho (P)\log z\}\mathrm{.}$ For every open ${\tau }_L$ neighborhood $U$ of ${\mathcal{M}}_{z,\beta }$, $C_{L,N_L}^{\ast }\{\omega :R_{L,\omega }\mathrm{\notin }U\}⟶0.$ Here $C^{\ast }(E)$ is the outer probability defined in Conditioning grand empirical fields on an exact particle number; convergence in probability uses this convention, agreeing with ordinary probability for measurable events. If ${\mathcal{M}}_{z,\beta }=\{P_{\ast }\}$, then $R_{L,\omega }\to P_{\ast }$ in probability in ${\tau }_L$. This is local observable equivalence for a uniformly sampled bulk location; it makes no claim about the raw finite-box law at a fixed origin or about uniqueness for every interaction.