Full phase shell marginal satisfies the positional variational bound

Put D_n=D(M_n^ph || C_n^ph), and let B_n be the positional marginal. The conditional chain rule gives $D_n=D(B_n\mathrm{\mid }\mathrm{\mid }{C}_n^{p}os)+E_{B_n}D(b_n(\mathrm{.}\mathrm{\mid }q)\mathrm{\mid }\mathrm{\mid }{\nu }_b^{\otimes N_n}),$ so D(B_n||C_n^pos)≤D_n by the conditional chain rule [F8]. The canonical positional density relative to full Poisson probability Q_Λn is 1_{N=N_n} exp(-bU_n,per)/Z_Q,n^pos, where Z_Q,n^pos=exp(-V_n)Z_n,N_n^pos. Integrating the log likelihood against B_n gives $D(B_n\mathrm{\mid }\mathrm{\mid }{Q}_{{\mathrm{\Lambda }}_n})+b{E}_{B_n}{U}_{n,per}=D(B_n\mathrm{\mid }\mathrm{\mid }{C}_n^{pos})-\log Z_{Q,n}^{pos}\le D_n-\log Z_{Q,n}^{pos}\mathrm{.}$ By [F1] and [F4], V_n^-1 log Z_Q,n^pos→F_b(ρ)-1; hence the right side divided by V_n has limsup at most 1-F_b(ρ).

For every sufficiently large $n$, the periodized-image series is absolutely convergent by [F3]. Given a finite $q\subset {\mathrm{\Lambda }}_{L_n}$ in the shell, form $q_M={\bigcup }_{a\in \{0,\dots ,M-1{\}}^d}(q+L_{n}a)$. Since $L_n$ is an integer, its unit-cell occupation pattern repeats exactly $M^d$ times. In the pair sum for $H_{\varphi }(q_M)\mathrm{/}{M}^d$, each fixed image offset $k\in {\mathbb{Z}}^d$ has multiplicity factor ${\prod }_{j=1}^d(1-\mathrm{\mid }{k}_j\mathrm{\mid }\mathrm{/}M{)}_{+}$, which tends to $1$ and is bounded by $1$. For finite periodic energy, absolute convergence lets us first restrict to finitely many offsets with arbitrarily small omitted absolute tail; the factors converge on that finite set and keep the omitted tail bounded, so $H_{\varphi }(q_M)\mathrm{/}{M}^d\to U_{n,\mathrm{p}\mathrm{e}\mathrm{r}}(q)$. If any image term is $+\mathrm{\infty }$, the desired lower bound is automatic. Applying [F2] to $q_M$, dividing by $M^d$, and taking the limit proves $U_{n,\mathrm{p}\mathrm{e}\mathrm{r}}\ge A{\sum }_i{N}_{C_i}^2-B{N}_n$; the same replication applied to the stability bound proves $U_{n,\mathrm{p}\mathrm{e}\mathrm{r}}\ge -B_0{N}_n$. On the full shell from [F7], $U_{n,\mathrm{p}\mathrm{e}\mathrm{r}}={\mathcal{H}}_n-\sum K\le V_n(e_b+{\delta }_n)$. The periodized cell bound yields $E_{B_n}{\sum }_i{N}_{C_i}^2\mathrm{/}{V}_n\le A^{-1}(e_b+{\delta }_n+B{N}_n\mathrm{/}{V}_n)$, uniformly bounded. Because $L_n$ is an integer, its torus is tiled by $V_n$ unit cells; torus translation invariance makes all $E_{B_n}{N}_{C_i}^2$ equal, so each is bounded by the preceding sum divided by $V_n$. Cauchy–Schwarz then gives a uniform bound $E_{B_n}{N}_{C_i}{N}_{C_j}\le C$ for every pair of cells. For the negative pair terms, each bounded-distance cell has finitely many neighboring cells and each term is bounded below by $-2B_0$ by [F2]; for remaining offsets, [F3] supplies the summable envelope $\mathrm{\Psi }$. Hence the expected sum of negative pair magnitudes is $O(V_n)$, and so is the total negative part of $U_{n,\mathrm{p}\mathrm{e}\mathrm{r}}$. The shell upper bound on $E{U}_{n,\mathrm{p}\mathrm{e}\mathrm{r}}$ then gives $E_{B_n}{U}_{n,\mathrm{p}\mathrm{e}\mathrm{r}}^{+}=O(V_n)$. Since the sum of positive pair terms equals $U_{n,\mathrm{p}\mathrm{e}\mathrm{r}}$ plus the sum of negative pair magnitudes, its expectation is also $O(V_n)$. Therefore $E_{B_n}\mathrm{\mid }{U}_{n,\mathrm{p}\mathrm{e}\mathrm{r}}\mathrm{\mid }=O(V_n)$.

Let ${\tilde{g}}_n=\tilde{g}(L_n)$ be the real corridor width supplied by [F3] at $L_n$, and set $g_n=\lceil {\tilde{g}}_n\rceil$. Then $g_n\to \mathrm{\infty }$ and $g_n\mathrm{/}{L}_n\to 0$, and $a_n=L_n+g_n$ is an integer. By [F11], take independent tile configurations with law $B_n$ on ${\mathbb{Z}}^d$ and independently choose a uniform phase in $[0,a_n{)}^d$. The map from these data to the union configuration is measurable: for each bounded Borel observation set, only finitely many tile indices can meet it, and each resulting count is a finite sum of Borel indicators of the finite cell-coordinate lists. Thus the constructed law $P_n$ is well-defined and locally finite. Translating the phase preserves its uniform law and only relabels the independent, identically distributed tiles, so $P_n$ is translation-stationary. Put ${\rho }_n=N_n\mathrm{/}{V}_n$. Its intensity is $N_n\mathrm{/}{a}_n^d={\rho }_n{V}_n\mathrm{/}{a}_n^d\to \rho$. For a fixed local-tame test $f$ supported in a bounded set $A$, choose $r$ larger than its coordinate diameter. The random phase puts $A$ wholly inside an occupied tile with probability at least $(L_n-2r{)}^d\mathrm{/}{a}_n^d\to 1$. On this event torus translation invariance of $B_n$ gives exactly its local law. For any Borel set $E\subseteq {\mathrm{\Lambda }}_{L_n}$, torus invariance and the exact count $N_n$ give $E_{B_n}{N}_E=\frac{1}{V_n}{\int }_{{\mathbb{T}}_{L_n}^d}{E}_{B_n}{N}_{E+y}dy=\frac{N_n\mathrm{\mid }E\mathrm{\mid }}{V_n}\mathrm{.}$ The first equality uses invariance; for each fixed configuration, the integral in the numerator sums over its $N_n$ points, and each point lies in $E+y$ for a set of translations of measure $\mathrm{\mid }E\mathrm{\mid }$. Apply this identity to each occupied portion of $A$ on the bad phase set; gap portions contain no points. The phase-averaged expectation of $(1+N_A)$ on that set is at most its probability times $1+{\rho }_n\mathrm{\mid }A\mathrm{\mid }$. Since $\mathrm{\mid }f\mathrm{\mid }\le C(1+N_A)$ and ${\rho }_n$ is bounded, $P_n(f)-B_n(f)\to 0$.

Fix $n$ and a lattice phase, and restrict to a large observation cube. Its intersections with complete occupied tiles have independent laws $B_n$; its gap pieces are empty; and only $O_n(\text{surface area})$ tile intersections meet the observation boundary. The Poisson restrictions on these disjoint pieces are independent by [F10]. For each complete occupied tile, the relative entropy is $D(B_n\mathrm{\parallel }{Q}_{{\mathrm{\Lambda }}_{L_n}})$; restriction contraction [F8] bounds each boundary-tile contribution by this same finite value. An empty gap piece $G$ has entropy $D({\delta }_{\mathrm{\varnothing }}\mathrm{\parallel }{Q}_G)=-\log Q_G(\mathrm{\varnothing })=\mathrm{\mid }G\mathrm{\mid }$. Product additivity follows by iterating the chain rule [F8]. For the mixture over the uniform phase, integrate the variational objective from [F8] for any bounded test and bound it by the phase-average conditional entropy; taking the supremum shows the mixture entropy is at most that average. Divide by observation volume and let the cube grow: the complete-tile density tends to $a_n^{-d}$, the gap-volume density to $(a_n^d-L_n^d)\mathrm{/}{a}_n^d$, and the boundary contribution to zero. The specific-entropy limit exists by [F5], so $I(P_n)\le \frac{D(B_n\mathrm{\parallel }{Q}_{{\mathrm{\Lambda }}_{L_n}})+a_n^d-L_n^d}{a_n^d}\mathrm{.}$ For the energy comparison, assign half of each pair term to each endpoint cell. Step 2.1 gives $E_{B_n}{N}_{C_i}{N}_{C_j}\le C$ uniformly, a bound $O(V_n)$ on the total negative pair magnitude, and a bound $O(V_n)$ on the total positive pair sum: the latter is the total energy plus the negative pair magnitude, hence is bounded by the positive part of the total energy plus that magnitude. Fix $R>R_0$. Periodic-image pairs of torus separation at most $R$ that are absent from the finite-tile free energy $U_{L_n,\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}$ cross a tile face; their endpoints lie in a boundary strip of width $R+1$ containing $O(R{L}_n^{d-1})$ unit cells. The negative contribution per such cell is bounded by the finite number of near offsets, $\varphi \ge -2B_0$ from [F2], and the pair-count bound; the expected positive contribution per cell is bounded by the total positive pair-sum estimate and torus translation invariance. Thus the expected absolute fixed-range discrepancy divided by $V_n$ is $O(R\mathrm{/}{L}_n)$. For remaining pairs, [F3] bounds the expected absolute energy per volume by $C{\sum }_{\mathrm{\mid }k{\mathrm{\mid }}_{\mathrm{\infty }}>R-c_d}\mathrm{\Psi }(k)$, tending to zero as $R\to \mathrm{\infty }$. Cross-tile pairs have separation at least $g_n$; the same tail bound with cutoff $g_n$ tends to zero as $g_n\to \mathrm{\infty }$. Taking first $n\to \mathrm{\infty }$ at fixed $R$ and then $R\to \mathrm{\infty }$ therefore gives $\mathrm{\Phi }(P_n)=E_{B_n}{U}_{n,\mathrm{p}\mathrm{e}\mathrm{r}}\mathrm{/}{a}_n^d+o(1)$. Combining the entropy bound, step 1.1, and $a_n^d\mathrm{/}{V}_n\to 1$ gives ${\mathrm{lim\hspace{0.17em}sup}}_n[I(P_n)+b\mathrm{\Phi }(P_n)]\le 1-F_b(\rho )\mathrm{.}$ Here the gap fraction vanishes because $g_n\mathrm{/}{L}_n\to 0$.

Stability gives $\mathrm{\Phi }(P_n)\ge -B_0\rho (P_n)$, so step 4.1 bounds $I(P_n)$. Compactness [F5] supplies a convergent subsequence from every subsequence. If $P$ is a cluster point, intensity is continuous in ${\tau }_L$ because the unit-cell count is a local-tame test, hence $\rho (P)=\rho$. Lower semicontinuity and step 4.1 give $I(P)+b\mathrm{\Phi }(P)\le 1-F_b(\rho )$. Subtract $\alpha \rho$ and apply the variational lower bound and exact minimum [F6]; $P$ must attain the grand variational minimum. The singleton hypothesis forces $P=P_{\ast }$. Thus every cluster point is $P_{\ast }$, compactness gives $P_n\to P_{\ast }$, and step 3.1 gives $B_n[f]\to P_{\ast }(f)$ for every local-tame $f$. By [F9], ${\bar{B}}_n$ is stationary with intensity $N_n\mathrm{/}{V}_n$, and torus invariance gives ${\bar{B}}_n(f)=B_n[f]$; therefore ${\bar{B}}_n\to P_{\ast }$ in ${\tau }_L$. For the exact-$N_n$ canonical law $C_n^{\mathrm{p}\mathrm{o}\mathrm{s}}$, [F4] says its periodized empirical field $R_{L_n,\omega }$ converges in outer probability in ${\tau }_L$ to $P_{\ast }$. For a local-tame $f$ with $\mathrm{\mid }f\mathrm{\mid }\le c(1+N_A)$, [F9] makes $X_n:=R_{L_n,\omega }(f)$ measurable and gives $\mathrm{\mid }{X}_n\mathrm{\mid }\le c(1+N_n\mathrm{\mid }A\mathrm{\mid }\mathrm{/}{V}_n)$, uniformly in $n$. Thus $X_n\to P_{\ast }(f)$ in probability and the variables are uniformly bounded; for each $ϵ>0$, $\mathrm{\mid }E{X}_n-P_{\ast }(f)\mathrm{\mid }\le ϵ+(M+\mathrm{\mid }{P}_{\ast }(f)\mathrm{\mid })\mathrm{Pr}(\mathrm{\mid }{X}_n-P_{\ast }(f)\mathrm{\mid }>ϵ)$ for a common bound $M$, so $E{X}_n\to P_{\ast }(f)$. Torus translation invariance and [F9] identify $E{X}_n=C_n^{\mathrm{p}\mathrm{o}\mathrm{s}}[f]$. Hence the exact-$N_n$ canonical positional local expectations also converge to $P_{\ast }$.