Zero-inverse-temperature-parameter limit with forbidden pairs retained

Regularity bounds $\varphi$ above outside a bounded set, so $B$ is bounded. Define $\chi =+\mathrm{\infty }$ on $B$ and $0$ elsewhere. Then $\chi$ is even, stable with constant $0$, and has zero far tail. Its finite-volume beta-zero partition is exactly the specified hard-admissibility partition.

For each bounded region $A$, the event that $A$ contains a forbidden pair is a bounded measurable local event. Its probability is continuous under ${\tau }_L$ by [F6]. Exhausting space by countably many bounded cubes shows that ${\mathcal{K}}_B$ is ${\tau }_L$-closed. Apply [F4] to the bounded tests $f_M=a\min (N_C,M)$ and let $M\to \mathrm{\infty }$. Monotone convergence and [F7] give $I(P)\ge a\rho (P)-(e^a-1)$, hence $\rho (P)\le (I(P)+e^a-1)\mathrm{/}a$. Choosing $a>2\max (\log z,0)$ makes $I(P)-\rho (P)\log z$ coercive in $I(P)$. By [F2], every finite sublevel is compact and lower semicontinuous, so the infimum on ${\mathcal{K}}_B$ is attained whenever the admissible class is nonempty. The empty configuration law gives a finite candidate.

Fix $P\in {\mathcal{K}}_B$ with $I(P)<\mathrm{\infty }$. Condition its restriction to a cube of side $n$ on configurations whose distinct points have separation at least $1\mathrm{/}q$ and whose finite pair energies are at most $q$. As $q\to \mathrm{\infty }$, these events increase to probability one because configurations are locally finite, simple, and hard-admissible. Choose $q=q(n)$ so the conditioning probability tends to one. Tile space by these conditioned blocks separated by an empty corridor wider than the range of $B$ and the fixed tail cutoff, take independent copies, then average the lattice phase uniformly. No cross-block pair is forbidden. The conditional entropy inequality $D(P(\cdot \mid E)\mathrm{\parallel }Q)\le [D(P\mathrm{\parallel }Q)+1]\mathrm{/}P(E)-\log P(E)$ follows by expanding the conditional density $1_E(dP\mathrm{/}dQ)\mathrm{/}P(E)$ and using $-u\log u\le 1\mathrm{/}e$. Additivity on independent blocks and the vanishing corridor fraction give $\mathrm{lim\hspace{0.17em}sup}I(P_n)\le I(P)$; local count truncation and finite first moments give $P_n\to P$ in ${\tau }_L$. Each $P_n$ has finite original energy density.

For a finite-volume beta-zero Gibbs law $G_L^b$, direct integration of its density gives $-\log Z_{Q,L}=D(G_L^b\mathrm{\parallel }{Q}_L)-(\log z)E_{G_L^b}{N}_L\mathrm{.}$ Stationarize independent $L$-blocks and average their phase. The entropy chain rule bounds the resulting specific entropy by the displayed finite-volume entropy per volume, and the intensity is $L^{-d}{E}_{G_L^b}{N}_L$. The empty configuration gives $Z_{Q,L}\ge e^{-L^d}$; applying the Poisson entropy count inequality with a coefficient greater than $\max (\log z,0)$ in the displayed identity bounds both intensity and entropy density uniformly. A forbidden-pair event supported in a fixed bounded cube can occur in the block law only when that cube crosses a block face, whose random-phase probability is $O(1\mathrm{/}L)$. Hence hard admissibility passes to every local limit. Compactness [F2], lower semicontinuity, and the entropy identity therefore give the pressure upper bound by the negative minimum in the statement. For the reverse bound, use the approximating finite-energy laws from step 1.3. Their independent separated blocks are admissible; the spatial likelihood theorem [F3] supplies typical blocks with likelihood cost $I(P_n)$, while empirical counts converge to $\rho (P_n)$. Changing from Poisson measure to the beta-zero weight gives the lower rate $-I(P_n)+\rho (P_n)\log z$. Let $n\to \mathrm{\infty }$ and optimize over $P$. Corridors change the normalized log weight by their vanishing volume fraction, so free and periodic rates agree.

Put $\alpha =\log z$, $p_L=L^{-d}\log Z_{Q,L}$, and $d_L=L^{-d}D(G_L^{\mathrm{p}\mathrm{e}\mathrm{r}}\mathrm{\parallel }{Q}_L)$. The exact identity in step 2.1 reads $d_L=\alpha {\rho }_L-p_L$. For $a>\max (\alpha ,0)$, the entropy count inequality gives $d_L\ge a{\rho }_L-(e^a-1)$; the convergent pressure rates therefore bound both ${\rho }_L$ and $d_L$. The grand density and periodic hard-admissibility condition are invariant under translations modulo the box, so [F8] gives relatively compact continued laws. Every limit $P$ lies in ${\mathcal{K}}_B$ by the bounded local forbidden-pair tests from step 1.2. Along a subsequence realizing the entropy liminf, [F8] and count-tail uniformity give $I(P)-\alpha \rho (P)\le \mathrm{lim\hspace{0.17em}inf}(d_L-\alpha {\rho }_L)=-\lim p_L={\min }_{{\mathcal{K}}_B}(I-\alpha \rho )\mathrm{.}$ The opposite inequality is the definition of the minimum. Thus every accumulation point is a minimizer. If there is just one, relative compactness forces convergence of the entire sequence in ${\tau }_L$, including every stated tame test.

For the activity-$z$ Poisson law $Q_z$ and a bounded cube $A$, the finite-sector Poisson formula gives the expected number of unordered forbidden pairs as $\frac{z^2}{2}{\int }_{A^2}{\mathbf{1}}_B(x-y)dxdy$. This follows by cancelling $N(N-1)$ against $N!$ in the nonnegative sector sum; the remaining exponential sum cancels its Poisson normalization. If $\mathrm{\mid }B\mathrm{\mid }=0$, Tonelli makes this expectation zero. Exhausting space by integer cubes shows that $Q_z$ is hard-admissible almost surely. For all sufficiently large $L$, the same null-set calculation covers every pair of distinct representatives and their countably many periodic images; self-image offsets $Lk$, $k\mathrm{\ne }0$, lie outside the bounded set $B$. Thus both finite grand laws are unrestricted Poisson restrictions, and their periodically continued laws converge on every bounded region to $Q_z$ by [F7]. If $\mathrm{\mid }B\mathrm{\mid }>0$, choose $A$ large enough that $\mathrm{\mid }A\cap (A+h)\mathrm{\mid }>0$ for every $h\in B$. Changing variables gives ${\int }_{A^2}{\mathbf{1}}_B(x-y)dxdy={\int }_B\mathrm{\mid }A\cap (A+h)\mathrm{\mid }dh>0$. This finite positive expectation implies positive probability of a forbidden pair, so $Q_z\mathrm{\notin }{\mathcal{K}}_B$. Finally $B=\{0\}$ excludes no distinct pair at all.