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\title{First-order phase transition in Onsager's model of hard rods in $\mathbb R^3$}
\author{Generated by GPT-6 Astra \\ Prompted by Qidong He \\ Results not human-verified}
\date{\today}
\begin{document}
\maketitle
\begin{abstract}
We prove a first-order transition for hard rods in $\mathbb R^3$ whose
orientations vary continuously over the sphere. For every sufficiently
small positive ratio of diameter to length, the grand-canonical pressure
has a nondifferentiable point, with a jump in number density at the
intermediate Onsager scale. Following the phase-pressure strategy of
Lebowitz, Mazel, and Presutti, we connect coexistence in the homogeneous Onsager functional to the microscopic pressure through an exact auxiliary-field representation, exponential decay of correlations in the rod gas at fixed auxiliary field, and estimates of the free-energy cost of interfaces between connected families of minimizing angular profiles.
\end{abstract}


\section{The pressure transition and the proof strategy}\label{sec:main}

Let $d\omega$ be probability area measure on $\mathbb S^2$. A rod of
axial length one and diameter $\varepsilon>0$, with centre $x$ and
axis $u$, occupies
\[
 \mathcal R_\varepsilon(x,u)
 =x+[-1/2,1/2]u+B(0,\varepsilon/2).
\]
The orientation varies continuously over $\mathbb S^2$, with $u$ and
$-u$ describing the same rod. For a bounded region
$\Lambda$, activity $z>0$, and admissible exterior configuration
$\xi$, define
\begin{equation}\label{mod:partition}
 Z_{\Lambda,\varepsilon}(z\mid\xi)
 =\sum_{N\ge0}\frac{z^N}{N!}
 \int_{(\Lambda\times\mathbb S^2)^N}
 \1_{\mathrm{hc}}(X_1,\ldots,X_N;\xi)
 \prod_{j=1}^N dx_j\,d\omega(u_j).
\end{equation}
The indicator requires disjoint rod interiors. Centres in $\Lambda$
are integrated; the exterior rods are fixed. For
$\Lambda_s=[-s/2,s/2]^3$, write
\begin{equation}\label{mod:physical-pressure}
 P_\varepsilon(a)=\lim_{s\to\infty}\frac{1}{|\Lambda_s|}
 \log Z_{\Lambda_s,\varepsilon}(\varepsilon^{-1}e^a\mid\varnothing).
\end{equation}

\begin{theorem}[Nondifferentiability of the pressure]\label{trn:corner}
The pressure $P_\varepsilon$ exists, is finite, and is convex in $a$.
There are $a_t\in\mathbb R$, $\varepsilon_0>0$, and constants
$\Delta,c,C>0$ such that for every $0<\varepsilon<\varepsilon_0$
there is $a_\varepsilon$, with $a_\varepsilon\to a_t$ as
$\varepsilon\downarrow0$, satisfying
\begin{equation}\label{mod:goal}
 P'_{\varepsilon,+}(a_\varepsilon)
 -P'_{\varepsilon,-}(a_\varepsilon)
 \ge\frac{\Delta}{2\varepsilon},\qquad
 \frac{c}{\varepsilon}\le
 P'_{\varepsilon,\pm}(a_\varepsilon)\le\frac{C}{\varepsilon}.
\end{equation}
The one-sided derivatives are the limiting number densities from
the corresponding sides of the transition.
\end{theorem}

Thus the transition occurs at number density of order
$\varepsilon^{-1}$ and occupied volume fraction of order $\varepsilon$.
For axial length $\ell$ and diameter $d$, these scales become
$(\ell^2d)^{-1}$ and $d/\ell$, respectively.

\subsection*{The strategy and its relation to LMP}

Lebowitz, Mazel, and Presutti (LMP)~\cite{LMP98,LMP99} transferred
mean-field coexistence to a continuum system with a large but finite
interaction range. Their method combines a variational description
of the reference phases, estimates within each phase, control of
interfaces, and an adjustment of the chemical potential that makes
the corrected phase pressures equal. We follow this sequence. The
model-specific work replaces their Kac interactions by hard exclusion
and their scalar density minima by families of angular profiles.
The contour step uses Pirogov--Sinai theory~\cite{PS75,PS76},
particularly Zahradn\'ik's truncation and stability argument
\cite{Zahradnik84}.

The conclusion here concerns the pressure. Once the physical pressure
is identified with the maximum of phase pressures whose density
responses are separated, their crossing proves the theorem. This
avoids the further construction and analysis of distinct physical
Gibbs measures carried out by LMP. The argument still needs locality
of the reference gas and boundary estimates for restricted free
energies; auxiliary Gibbs states are used to establish those estimates.

The following eleven steps are the subjects of the next eleven
sections, in this order.
\begin{enumerate}[label=\textbf{\arabic*.},leftmargin=*]
\item \textbf{Identify the homogeneous phases.}
The Onsager functional has an isolated isotropic minimizer and
nonisotropic minimizers at one chemical potential. Their masses are
separated. Analytic coordinates divide the complete minimizing set
into finitely many connected components.

\item \textbf{Introduce an exact auxiliary field.}
Attach a normalized Gaussian variable to each spatial and angular
cell. Integrating these variables restores the original partition
function exactly. Conditional on the field, a quadratic count penalty
provides the reference system used in the subsequent estimates.

\item \textbf{Derive the variational limit.}
The thin-rod overlap operator and a random graph representation give
the finite-volume Onsager functional for the reference gas.
This supplies the microscopic approximation that plays the role of
coarse graining in LMP.

\item \textbf{Prove locality of the reference gas.}
The quadratic penalty makes the insertion generator coercive.
Its inverse yields covariance decay uniformly in the field, the
volume, and fixed exterior rods. This replaces LMP's restricted
reference-system uniqueness estimates.

\item \textbf{Construct a positive field action.}
Directional averaging splits the leading free energy into nonnegative
terms. A transport rigidity theorem identifies their common zeros.
Minimization over density profiles then gives a local field action
and quantitative separation of its minimizing components.

\item \textbf{Restore all particle counts.}
A projection empties cells exceeding the reference count bound.
A convergent expansion reconstructs their contribution and propagates
the reference estimates to the exact rod integral.

\item \textbf{Recover the exact field interaction.}
Integrate the field response to express the exact interaction as the
positive leading action plus a small summable error. A common additive
normalization allows pressures with different reference profiles to
be compared.

\item \textbf{Control boundary costs.}
Interpolate density profiles within a connected minimizing component
and compare restricted field integrals. This treats the continuous
families of angular minima. It gives surface bounds and a comparison
of partition functions across a separating layer.

\item \textbf{Assign contours and their costs.}
Blocks where a phase cannot be continued form contours. Rigidity and
isoperimetry supply their positive cost. The exact regional expansion
retains the boundary terms needed to combine neighboring regions.

\item \textbf{Identify the physical pressure.}
Following LMP's use of auxiliary phase pressures, truncate large
contour weights and prove uniform surface estimates. An induction
then shows that the physical pressure is the maximum of these phase
pressures.

\item \textbf{Locate the crossing.}
Differentiate the original Gaussian representation. The homogeneous
mass gap gives separated increments of the complete phase pressures,
while their leading values agree. Adjusting the chemical potential
forces a crossing and hence the asserted density jump.
\end{enumerate}


\section{Homogeneous reference phases}
\label{fnd:homogeneous}
Let $d\omega$ be probability area measure on $S^2$ and define
\[
 (Kf)(u)=\int_{S^2}|u\times v|f(v)\,d\omega(v),\qquad
 \Omega_a(f)=\int_{S^2}f(\log f-1-a)\,d\omega+\langle f,Kf\rangle.
\]
This is the grand-canonical form of Onsager's second-virial
functional~\cite{Onsager}. Profiles are nonnegative integrable
functions; the entropy may be $+\infty$. Write
$m_a=\inf\Omega_a$ and $\mathcal W_a=\{f:\Omega_a(f)=m_a\}$.

For bounds on fixed-mass Onsager minimizers, see
Vollmer~\cite[Section~7.1]{Vollmer}. The following
lemma gives the bounds needed with variable mass.

\begin{lemma}[Bounds and compactness of the minimizers]\label{fnd:wellbounds}
The set $\mathcal W_a$ is nonempty. Every minimizer is even and satisfies
\begin{equation}\label{fnd:euler}
 \log f+2Kf=a,\qquad e^{a-2e^a}\le f\le e^a.
\end{equation}
The minimizers are uniformly Lipschitz on the sphere. Their union over
a compact parameter interval is relatively compact in the uniform
norm, and limits at a convergent sequence of parameters are minimizers
at the limiting parameter.
\end{lemma}
\begin{proof}
If $\rho=\int f\,d\omega$, Jensen's inequality and positivity of the
kernel give
\[
 \Omega_a(f)\ge\rho\log\rho-(1+a)\rho.
\]
A minimizing sequence consequently has bounded mass and bounded
positive entropy. It is uniformly integrable and has a weakly
convergent subsequence in $L^1$. The entropy is lower semicontinuous.
The interaction is continuous under this convergence: approximate its
continuous kernel uniformly on $S^2\times S^2$ by finite sums of
products of continuous functions, and use the uniform mass bound.
This proves existence. Replacing $f(u)$ by
$[f(u)+f(-u)]/2$ preserves its interaction and mass and strictly lowers
its entropy unless $f$ is even.

For positivity and the Euler equation, fix a minimizer $f$, set
$V=2Kf$, and consider
$L(g)=\int g(\log g-1-a+V)\,d\omega$.
Its unique minimizer is $g=e^{a-V}$. If $L(g)<L(f)$, convexity and the
quadratic expansion give, for $f_t=(1-t)f+tg$,
\[
 \Omega_a(f_t)-\Omega_a(f)
 \le t[L(g)-L(f)]+t^2\langle g-f,K(g-f)\rangle<0
\]
for small $t>0$, a contradiction. Thus $f=e^{a-2Kf}$. Since
$0\le Kf\le\int f\le e^a$, the bounds follow. The kernel is
Lipschitz in its first argument, uniformly in the second. The Euler
equation gives a uniform Lipschitz bound for $f$. Arzel\`a--Ascoli and
continuity of the functional on the resulting bounded positive
profiles prove the compactness assertions.
\end{proof}

The spectrum below is the Onsager-kernel spectrum computed in
\cite[Corollary~10]{Vollmer}, divided by $4\pi$ because $d\omega$
is probability measure.

\begin{lemma}[Angular spectrum]\label{fnd:spectrum}
On spherical harmonics of degree $l$, the eigenvalue of $K$ is
\begin{equation}\label{fnd:eigenvalues}
 \kappa_0=\frac\pi4,\qquad
 \kappa_{2n}=-\frac{\pi}{2(2n-1)(2n+2)}
 \left(\frac{\binom{2n}{n}}{4^n}\right)^2\quad(n\ge1),
 \qquad \kappa_{2n+1}=0.
\end{equation}
In particular $\inf_l\kappa_l=\kappa_2=-\pi/32$, consecutive even
eigenvalues are distinct, and $|\kappa_{2n}|=O(n^{-3})$.
\end{lemma}
\begin{proof}
The Funk--Hecke formula gives
$\kappa_l=\frac12\int_{-1}^1\sqrt{1-t^2}P_l(t)\,dt$, where $P_l$
is the Legendre polynomial. Put $w(t)=\sqrt{1-t^2}$ and denote the
integrals of $wP_l$ and $P_l/w$ by $I_l$ and $J_l$. Two integrations
by parts in Legendre's equation give
$l(l+1)I_l=2I_l-J_l$; the boundary terms vanish and the remaining
endpoint singularities are integrable. Odd integrals vanish. The
generating function gives
\[
 \sum_{l\ge0}J_lz^l
 =\int_0^\pi(1-ze^{i\theta})^{-1/2}
                   (1-ze^{-i\theta})^{-1/2}\,d\theta
 =\pi\sum_{n\ge0}\left(\frac{\binom{2n}{n}}{4^n}\right)^2z^{2n}.
\]
Here conjugate off-diagonal terms combine into cosines with zero
integral. This proves \eqref{fnd:eigenvalues}. Finally,
\[
 \frac{|\kappa_{2n+2}|}{|\kappa_{2n}|}
 =\frac{4n^2-1}{4(n+1)(n+2)}<1,
\]
and the central binomial coefficient bound gives the stated decay.
\end{proof}

The isotropic critical point is the constant $r(a)>0$ determined by
$\log r+2\kappa_0r=a$; it increases continuously with $a$.

Onsager's degree-two trial profiles show how global minimality can
be lost before local stability~\cite[pp.~645--646]{Onsager}; see
also Kayser and Ravech\'e~\cite{KR}. The following proof uses an
explicit finite-amplitude comparison in the grand-canonical functional.

\begin{theorem}[Homogeneous coexistence]
\label{fnd:coexistence}
There is $a_t$ for which $\mathcal W:=\mathcal W_{a_t}$ contains
an isolated isotropic profile $r_I=r(a_t)<5$ and at least one
nonisotropic profile.
\end{theorem}
\begin{proof}
On the convex order interval $0<f\le M$, the Hessian obeys
\[
 D^2\Omega_a(f)[g,g]\ge
       \left(M^{-1}-\frac\pi{16}\right)\|g\|_2^2.
\]
If $e^a<16/\pi$, all minimizers lie in a strictly convex interval,
so the minimizer is unique and rotation invariant. More generally,
$r(a)$ is isolated among nearby critical points whenever
$r(a)<16/\pi$: choose $r(a)<M<16/\pi$ and apply the same inequality
in a small uniform neighborhood, uniformly for nearby $a$.

To find loss of global minimality before this threshold, take
$g(u)=1+P_2(u\cdot n)$. It has mass one, lies between $1/2$ and $2$,
and $\int P_2^2\,d\omega=1/5$. At density five,
\[
 \Omega_a(5g)-\Omega_a(5)=5\left(H-\frac\pi{32}\right),\qquad
 H=\int_0^1\frac{1+3t^2}{2}\log\frac{1+3t^2}{2}\,dt
   =\log2-1+\frac{2\pi}{9\sqrt3}.
\]
This is strictly negative. Indeed, the series $\log2=2\sum_{n\ge0}3^{-2n-1}/(2n+1)$
gives $\log2<1733/2500$. Together with $\sqrt3>19/11$ and
$\pi<22/7$, this yields
\[
 H-\frac\pi{32}
 <\frac{1733}{2500}-1+
   \frac{22}{7}\left(\frac{22}{171}-\frac1{32}\right)
 =-\frac{8021}{11970000}<0.
\]
Let $a_5=\log5+10\kappa_0$. The same strict trial comparison holds
near $a_5$, while $r(a_5)=5<16/\pi$.

Let $S=\{a:r(a)\in\mathcal W_a\}$ and set
$a_t=\sup\{a:(-\infty,a]\subset S\}$. Compactness in
Lemma~\ref{fnd:wellbounds} implies continuity of $m_a$ and closedness
of $S$. Thus $a_t\in S$ and $a_t<a_5$. There are $a_n\downarrow a_t$
with $a_n\notin S$. Choose minimizers $f_n$ and take a uniformly
convergent subsequence. Its limit $f_*$ lies in $\mathcal W$ and
cannot equal $r_I$: otherwise local strict convexity would force
$f_n=r(a_n)$ eventually. Hence a nonisotropic well coexists with the
isolated isotropic one.

\end{proof}

\begin{lemma}[Density separation]\label{fnd:densitygap}
There is $\Delta>0$ such that
\begin{equation}\label{fnd:gap}
 \int_{S^2}f\,d\omega\ge r_I+\Delta
 \qquad(f\in\mathcal W\setminus\{r_I\}).
\end{equation}
\end{lemma}
\begin{proof}
We eliminate the interaction from Onsager's equal-pressure and
equal-chemical-potential equations~\cite[Eq.~(95)]{Onsager}.
For $f\in\mathcal W$, write $f=\rho g$, $\int g=1$, and $t=\rho/r_I$.
The integrated Euler equation and equality of the minimum values give
\[
 \log\rho+\int g\log g+2\rho\langle g,Kg\rangle=a_t,
 \qquad \rho+\rho^2\langle g,Kg\rangle=r_I+\kappa_0r_I^2.
\]
Eliminating the interaction yields
\[
 \int g\log g=-\log t+2(1+\kappa_0r_I)(1-t^{-1}).
\]
The right side is strictly increasing on $(0,1]$ and vanishes at one,
whereas the left side is nonnegative. Thus $t\ge1$, and equality
forces $g=1$. Finally $\mathcal W\setminus\{r_I\}$ is compact because
$r_I$ is isolated. Minimizing its positive mass difference proves
\eqref{fnd:gap}.
\end{proof}

\subsection{The geometry of the well set}

Let $\mathcal P_N$ project onto the even harmonic degrees at most
$2N$, and put $\mathcal Q_N=I-\mathcal P_N$. The uniformly convergent
zonal expansion and Lemma~\ref{fnd:spectrum} show that
\begin{equation}\label{fnd:tailK}
 \|\mathcal Q_NK\|_{C\to C}
 \le\sum_{n>N}(4n+1)|\kappa_{2n}|\longrightarrow0.
\end{equation}

The reduction below is a global finite-mode version of the
Lyapunov--Schmidt construction used for Onsager equilibria in
\cite[Section~3]{Vollmer}: a uniform contraction on the harmonic
tail covers the entire minimizing set.

\begin{proposition}[Global analytic well coordinates]\label{fnd:analytic}
For fixed $a$, there are a finite-dimensional space $E$, an open set
$U\subset E$, a compact real analytic zero set $Z\Subset U$, and an
analytic injective map $\Psi:U\to C_{\rm even}(S^2)$ such that
$\Psi(Z)=\mathcal W_a$. 
\end{proposition}
\begin{proof}
Set $j_-=a-2e^a-1$, $j_+=a+1$, and let
$T_J(t)=\exp(\max(j_-,\min(t,j_+)))$. Choose $N$ so large that
$q_*=2e^{j_+}\|\mathcal Q_NK\|_{C\to C}<1$.
For every $v\in E=\mathcal P_NC_{\rm even}(S^2)$ the contraction
\[
 w\longmapsto-2\mathcal Q_NK T_J(v+w)
\]
on $\mathcal Q_NC_{\rm even}(S^2)$ has a unique fixed point $w(v)$.
The fixed point depends globally Lipschitz continuously on $v$.
Let $U$ consist of the $v$ for which $j_-<v+w(v)<j_+$ everywhere.
On $U$ the clipping is inactive. The derivative in $w$ of
$w+2\mathcal Q_NKe^{v+w}$ is within $q_*$ of the identity, so the
analytic implicit function theorem makes $w$ analytic on $U$.

For every well, \eqref{fnd:euler} and the one-unit margin in $J$ give
$w(\mathcal P_N\log f)=\mathcal Q_N\log f$. Define
\[
 F(v)=\mathcal P_N[v+w(v)+2Ke^{v+w(v)}-a],\qquad
 G(v)=\Omega_a(e^{v+w(v)})-m_a.
\]
Both are real analytic. Their complete common zero set in $U$ is
exactly $Z=\mathcal P_N\log\mathcal W_a$: a common zero produces a
positive profile of energy $m_a$, and every well satisfies both
equations. Compactness and the positive lower bound in
Lemma~\ref{fnd:wellbounds} show $Z\Subset U$.
The map $\Psi(v)=e^{v+w(v)}$ is injective because
$\mathcal P_N\log\Psi(v)=v$.

\end{proof}

\begin{corollary}[Finitely many well components]\label{fnd:components}
The compact set $\mathcal W_a$ has finitely many connected components.
\end{corollary}
\begin{proof}
The analytic set $Z$ has finitely many connected components
\cite[Corollary 2.7]{BM}. Its continuous injection $\Psi$ is a
homeomorphism onto $\mathcal W_a$.
\end{proof}

\begin{lemma}[Paths within a well component]\label{fnd:paths}
Any two wells in one component can be joined by a Lipschitz path
with values in that component, in the uniform norm.
\end{lemma}
\begin{proof}
A connected compact semianalytic set is rectifiably connected
\cite[Theorem 6.10]{BM}. The derivative of $\Psi$ is bounded near
$Z$, so its images of these curves are rectifiable in the uniform
norm. Parametrize each image by arc length.
\end{proof}

\section{An exact auxiliary field representation}\label{sec:model}
Set $\beta=\varepsilon^{-1}$ and $\zeta=e^{a_t}$, where $a_t$ is
provided by Theorem~\ref{fnd:coexistence}. The parameter $\beta$ measures
rod thinness. For the source $h=a-a_t$, write
\begin{equation}\label{mod:pressure}
 p_\varepsilon(h)=P_\varepsilon(a_t+h),\qquad z=\beta\zeta e^h.
\end{equation}

\subsection{Cells, volumes, and reference measure}

Partition space into congruent fine cubes \(B_i\) of volume \(v_s\)
and partition the sphere into finitely many shape-regular cells
\(U_a\), of probability areas \(\omega_a>0\). A marked cell
\(C=(i,a)\) has measure \(v_C=v_s\omega_a\).
Let \(P\) denote orthogonal averaging on these marked cells.
A coarse cube contains \(L^3\) fine spatial cubes and has physical
volume
\[
 V_L=v_sL^3.
\]
Volumes in pressure exponents are Euclidean volumes. Surface
estimates may absorb the fixed mesh factors in their constants.

Choose a preliminary source interval \(|h|\leq H\), a pin strength
\(\alpha\), and numbers
\[
 \rho>e\zeta e^H,\qquad
 R>H+2\alpha\rho+2.
\]
The choices of \(\alpha\) and the mesh are specified by the operator
estimates below. The count threshold in \(C\) is
\(K_C=\lceil\rho\beta v_C\rceil\). It is imposed only on a reference
system. The original rod model in \eqref{mod:partition} remains
unrestricted.

\subsection{Normalized Gaussian augmentation}

The completion of the square is the Gaussian auxiliary-field method
of Stratonovich and Hubbard~\cite{Stratonovich,Hubbard}. We use
normalized conditional densities on a bounded field interval, with
the particle count reset above the threshold.

For a physical configuration \(\eta\), let \(n_C=N_C(\eta)\) and
\(\widehat n_C=n_C{\bf1}_{\{n_C\leq K_C\}}\). Attach to each marked
cell an independent field \(\phi_C\in[-R,R]\), conditionally on
\(\eta\), with density
\begin{equation}\label{mod:gaussian}
 \frac{1}{c_{C,h}(\widehat n_C)}
 \sqrt{\frac{\beta v_C}{4\pi\alpha}}\,
 \exp\left\{
  -\frac{\beta v_C}{4\alpha}
  \left(\phi_C-h-\frac{2\alpha\widehat n_C}{\beta v_C}\right)^2
 \right\}.
\end{equation}
The normalizer \(c_{C,h}(n)\) is the Gaussian probability of
\([-R,R]\). For sufficiently large \(\beta\), all the means in
\eqref{mod:gaussian} stay a fixed positive distance from the
endpoints. Thus, uniformly in \(0\leq n\leq K_C\),
\[
 c_{C,h}(n)=1+O(e^{-c\beta}),
\]
with the same conclusion for any fixed number of \(h\)-derivatives,
after reducing \(c>0\).

Put
\[
 g_{C,h}(\phi)
 =\sqrt{\frac{\beta v_C}{4\pi\alpha}}\,
   e^{-\beta v_C(\phi-h)^2/(4\alpha)}.
\]
Expansion of the square shows that, after extracting \(g_{C,h}\),
the exact cell weight is
\begin{equation}\label{mod:cellweight}
 w_{C,h,\phi}(n)=
 \begin{cases}
  (\beta\zeta)^n
   \exp\{\phi n-\alpha n^2/(\beta v_C)\}/c_{C,h}(n),
       &0\leq n\leq K_C,\\[2mm]
  (\beta\zeta e^h)^n/c_{C,h}(0),&n>K_C.
 \end{cases}
\end{equation}
In particular, the second branch is independent of \(\phi\).
All geometric integrals and factorials remain those of
\eqref{mod:partition}.

Use the probability reference
\(d\lambda_C=d\phi_C/(2R)\) for the fields. If
\(R_{\mathrm{rod},\Lambda}(\phi\mid\xi)\) denotes the rod integral
with weights \eqref{mod:cellweight}, its full field density is
\begin{equation}\label{mod:fullfield}
 F_\Lambda(\phi\mid\xi)
 =(2R)^{|I(\Lambda)|}
  \prod_{C\in I(\Lambda)}g_{C,h}(\phi_C)\,
  R_{\mathrm{rod},\Lambda}(\phi\mid\xi).
\end{equation}
Here \(I(\Lambda)\) is the set of marked cells over \(\Lambda\).
Throughout the field construction, a region is a union of complete fine
spatial cubes and contains every angular cell above each of them.
Since every conditional law \eqref{mod:gaussian} integrates to one,
Tonelli's theorem gives the exact identity
\begin{equation}\label{mod:exactreset}
 Z_{\Lambda,\varepsilon}(\beta\zeta e^h\mid\xi)
       =\int F_\Lambda(\phi\mid\xi)\,d\lambda_\Lambda(\phi).
\end{equation}
This equality holds for every finite volume and every admissible
exterior configuration.

The explicit source contribution is
\begin{equation}\label{mod:source}
 \log\frac{g_{C,h}(\phi_C)}{g_{C,0}(\phi_C)}
 =\frac{\beta v_C h\phi_C}{2\alpha}
       -\frac{\beta v_C h^2}{4\alpha}.
\end{equation}
Its quadratic term sums to
\(-\beta h^2|\Lambda|/(4\alpha)\), because
\(\sum_a\omega_a=1\).


\section{The finite-volume variational limit}
\label{mic:section}

We identify the free energy of the quadratically pinned gas. The
spatial region and marked-cell partition remain fixed throughout
the limit of vanishing diameter.

Let $D\subset\mathbb R^3$ be a bounded box and set
$\Omega=D\times S^2$, with measure $d\nu=dx\,d\omega(u)$, where
$\omega(S^2)=1$. Functions on $\Omega$ are extended by zero in the
spatial variable. Write $b_\varepsilon(X,Y)$ for the overlap indicator
of the rods with marks $X=(x,u)$ and $Y=(y,v)$, and define
\[
 (Af)(x,u)=\int_{-1/2}^{1/2}f(x+tu,u)\,dt,\qquad
 (Kf)(x,u)=\int_{S^2}|u\times v|f(x,v)\,d\omega(v),
 \qquad T=2A^*KA.
\]
The operator $T$ preserves positivity, is self-adjoint, has spatial
range one, and has row integral $\pi/2$ on the full space. Fix a finite
partition $\{C_i\}_{i=1}^m$ of $\Omega$, put $v_i=\nu(C_i)>0$, and
let $P$ be orthogonal averaging on these cells. For a measurable
activity profile $0<a\le a_{\max}<\infty$, define
\begin{equation}
 \mathcal F_a(q)=\int_\Omega q\left(\log\frac qa-1\right)d\nu
       +\frac12\langle q,Tq\rangle+\alpha\|Pq\|_2^2.
 \label{mic:free-energy}
\end{equation}
At diameter $\varepsilon$, the reference partition function has activity
$a/\varepsilon$, hard exclusion, the factor
$\exp[-\alpha\varepsilon\sum_iN_i^2/v_i]$, and caps
$N_i\le\lceil\rho v_i/\varepsilon\rceil$.

\begin{theorem}[Pinned finite-volume limit]
\label{mic:finite-volume}
Suppose $T+2\alpha P\ge-\kappa I$ and
$\kappa a_{\max}<1$. Then
\begin{equation}
 \lim_{\varepsilon\downarrow0}\varepsilon\log Z_{\varepsilon,\mathrm{cap}}
  =-\min_{\substack{q\ge0\\\int_{C_i}q\,d\nu\le\rho v_i}}
       \mathcal F_a(q).
 \label{mic:cap-limit}
\end{equation}
\end{theorem}

\subsection{The overlap operator and common neighbors}

The leading excluded volume is Onsager's angular
interaction~\cite{Onsager}. The following estimate strengthens this
geometric limit to convergence in operator norm.

\begin{lemma}
\label{mic:overlap}
If $B_\varepsilon$ is the integral operator with kernel
$b_\varepsilon$, then
\[
 \|B_\varepsilon/\varepsilon-T\|_{2\to2}\le C\varepsilon,
 \qquad \sup_X\int b_\varepsilon(X,Y)\,d\nu(Y)\le C\varepsilon.
\]
\end{lemma}

\begin{proof}
For fixed orientations $u,v$, the excluded displacement set is
\[
 [-1/2,1/2]u+[-1/2,1/2]v+\overline B(0,\varepsilon).
\]
It contains the normal prism over the parallelogram, of thickness
$2\varepsilon$. Its volume is
\[
 2\varepsilon|u\times v|+2\pi\varepsilon^2
                       +\frac{4\pi}{3}\varepsilon^3.
\]
This follows by integrating the planar parallel-body formula over
normal heights; the parallelogram has perimeter four. Parallel
orientations follow by continuity. The remainder outside the prism has volume
$O(\varepsilon^2)$ uniformly in $u,v$, so Schur's test bounds its
operator norm by $C\varepsilon^2$.

For $k\in\mathbb R^3$ write $s_k(u)=\operatorname{sinc}(k\cdot u/2)$,
where $\operatorname{sinc}(t)=\sin(t)/t$. The Fourier symbol of the
prism divided by $\varepsilon$ is the angular kernel
\[
 2|u\times v|s_k(u)s_k(v)
       \operatorname{sinc}(\varepsilon k\cdot n(u,v)),
 \qquad n(u,v)=\frac{u\times v}{|u\times v|}.
\]
The symbol of $T$ omits the last factor. Set
\[
 J(r)=\int_{S^2}s_k(u)^2\,d\omega(u)\le\frac C{1+r},
 \qquad r=|k|.
\]
The angular Hilbert--Schmidt norm of the difference is at most
$C\min(\varepsilon^2r^2,1)J(r)\le C\varepsilon$.
Plancherel and the remainder estimate prove the norm convergence.

\end{proof}

\begin{lemma}[Compactness of the limiting interaction]\label{mic:compactness}
The compression of $T$ to a bounded spatial region is compact.
\end{lemma}
\begin{proof}
Cut the symbol of $T$ off to $|k|\le R$. Its
localization on the left to $D$ has Hilbert--Schmidt norm squared
\[
 (2\pi)^{-3}|D|\int_{|k|\le R}
        \|\widehat T(k)\|_{\mathrm{HS}(L^2(S^2))}^2\,dk<\infty.
\]
Right localization preserves this property, and the Fourier tail has
norm at most $C/(1+R)$, proving compactness.
\end{proof}

For $d\mu=w\,d\nu$ with bounded $w\ge0$, the limiting operator
$T_\mu$ on $L^2(\mu)$ is defined by conjugation: the isometry
$\mathcal U_\mu f=\sqrt w f$ maps $L^2(\mu)$ onto the subspace of
$L^2(\nu)$ supported on $\{w>0\}$, and
\[
 \mathcal U_\mu T_\mu\mathcal U_\mu^{-1}
       =M_{\sqrt w}T M_{\sqrt w}.
\]
The same conjugation applies to $B_\varepsilon/\varepsilon$, proving
norm convergence to the compact operator $T_\mu$.

If the partitions refine with $P\to I$ strongly, compactness gives
$\|T-PTP\|\to0$. Since
$PTP+2\alpha P\ge(2\alpha-\|T\|)P$, the coercivity hypothesis
of Theorem~\ref{mic:finite-volume} holds on a sufficiently fine
partition whenever $2\alpha>\|T\|$.

\begin{lemma}[Common-neighbor bound]
\label{mic:neighbors}
Let $\mu$ have bounded density on $\Omega$. For $X=(x,u)$ and
$Y=(y,v)$ put $s=|u\times v|$, $d=|(y-x)\times u|$, and
\[
 c_\varepsilon(X,Y)=\int b_\varepsilon(X,Z)b_\varepsilon(Y,Z)\,d\mu(Z).
\]
Then
\begin{align}
 c_\varepsilon(X,Y)&\le
 \begin{cases}
 C\varepsilon,&s\le\varepsilon,\\
 C\varepsilon^2s^{-1}(1+\log(s/\varepsilon)),&s>\varepsilon,
 \end{cases}\label{mic:neighbor-angle}\\
 c_\varepsilon(X,Y)&\le
     C\varepsilon\min\{1,(s+\varepsilon)/d\}.
 \label{mic:neighbor-distance}
\end{align}
The last quotient is interpreted as infinity when $d=0$.
\end{lemma}

\begin{proof}
Fix the direction $w$ of the common neighbor. Its possible centers
lie in two slabs of thickness $2\varepsilon$ inside a fixed bounded
ball. The sine of the angle between their normals is
\[
 \frac{|\det(u,v,w)|}{|u\times w|\,|v\times w|}
 \ge |\det(u,v,w)|.
\]
Their intersection has volume at most
$C\min\{\varepsilon,\varepsilon^2/|\det(u,v,w)|\}$.
Integration over $w$, using $\det(u,v,w)=s n(u,v)\cdot w$, gives
\eqref{mic:neighbor-angle}.

If both intersections occur, subtraction of the two intersection
relations gives
$y-x=t u-t'v+\tau w+e$, with $|t|,|t'|\le1/2$ and
$|e|\le2\varepsilon$. Dotting with $u\times w$ yields
$|((y-x)\times u)\cdot w|\le Cs+C\varepsilon$.
The allowed directions form a spherical belt of measure at most
$C\min\{1,(s+\varepsilon)/d\}$. For each of them the center volume
is at most $C\varepsilon$, proving \eqref{mic:neighbor-distance}.
\end{proof}

For $r>\varepsilon$, splitting at $s=\sqrt{\varepsilon r}$ gives
\begin{equation}
 \sup_{\max(s,d)\ge r}\frac{c_\varepsilon(X,Y)}\varepsilon
  \le C\sqrt{\frac\varepsilon r}\bigl(1+\log(1/\varepsilon)\bigr).
 \label{mic:neighbor-separated}
\end{equation}
For independent marks of bounded density, the probability that both
$s<r$ and $d<r$ is at most $Cr^4$: the angular cap and the transverse
spatial cylinder each contribute two powers of $r$.

\subsection{Sampling the interaction uniformly over trial profiles}

Operator norm control makes the interaction approximation uniform
over weights chosen after sampling the point cloud.

\begin{lemma}[Covariance sampling]
\label{mic:covariance-sampling}
Let $v_1,\ldots,v_m$ be independent copies of a random vector $v$ in a
separable Hilbert space with $\|v\|^2\le L$. Set
$Q=m^{-1}\sum_jv_j\otimes v_j$, $A=\mathbb E(v\otimes v)$, and
$k_m=(1+2\lceil\log m\rceil)^{1/2}$. Then, with
$b_m=2k_m\sqrt{L/m}$,
\[
 \mathbb E\|Q-A\|\le b_m\|A\|^{1/2}+b_m^2,
 \qquad \mathbb E\|Q\|\le(\|A\|^{1/2}+b_m)^2.
\]
\end{lemma}

\begin{proof}
Let $\sigma_j$ be independent signs taking values $\pm1$ with equal
probability, independently of the vectors. Symmetrization and
Tropp's matrix Rademacher inequality~\cite[Theorem 4.1]{TroppExpected} give
\begin{align*}
 d:=\mathbb E\|Q-A\|
 &\le\frac2m\mathbb E\mathbb E_\sigma
             \left\|\sum_j\sigma_j(v_j\otimes v_j)\right\|\\
 &\le 2k_m\sqrt{\frac Lm}\,\mathbb E\|Q\|^{1/2}
 \le b_m\sqrt{\|A\|+d}.
\end{align*}
The matrices act on the sample span, of dimension at most $m$, and
$(v\otimes v)^2\le L(v\otimes v)$. Solving the quadratic gives both
bounds. Symmetrization follows by adjoining an independent sample
and applying Jensen's inequality to these Bochner integrable
trace-class operators.
\end{proof}

\begin{lemma}[Empirical operator approximation]
\label{mic:empirical}
Suppose a real symmetric kernel $R_\varepsilon$ on a probability space
$(\Omega,\mu)$ satisfies
\[
 \|R_\varepsilon\|_\infty\le C_0/\varepsilon,
 \qquad \sup_y\int|R_\varepsilon(x,y)|^2\,d\mu(x)\le C_1/\varepsilon.
\]
For independent $Z_1,\ldots,Z_n$ of law $\mu$, let $B_n$ have entries
$R_\varepsilon(Z_j,Z_k)/n$ off the diagonal and zero on the diagonal.
Then, for even $n$,
\begin{equation}
 \mathbb E\|B_n\|
 \le\|R_\varepsilon\|_{2\to2}
       +C\sqrt{\frac{\log(2n)}{n\varepsilon}}
       +\frac C\varepsilon e^{-cn\varepsilon}.
 \label{mic:sample-bound}
\end{equation}
\end{lemma}

\begin{proof}
Write $n=2m$ and first take independent samples $X_1,\ldots,X_m$ and
$Y_1,\ldots,Y_m$. For $v_j=R_\varepsilon(\cdot,Y_j)$, the covariance
operator $\Sigma_Y=m^{-1}\sum_jv_j\otimes v_j$ has expectation
$R_\varepsilon R_\varepsilon^*$ and $\|v_j\|^2\le L=C_1/\varepsilon$.
Lemma~\ref{mic:covariance-sampling} gives
\[
 \mathbb E\|\Sigma_Y\|\le(r_\varepsilon+b_1)^2,
 \quad r_\varepsilon=\|R_\varepsilon\|,
 \quad b_1=2k_m\sqrt{L/m}.
\]
Define
$r_Y(x)=m^{-1/2}(R_\varepsilon(x,Y_1),\ldots,R_\varepsilon(x,Y_m))$.
Its conditional covariance $G_Y$ has the same nonzero eigenvalues as
$\Sigma_Y$, by the identities $G_Y=J_Y^*J_Y$ and
$\Sigma_Y=J_YJ_Y^*$ for $J_Yc=m^{-1/2}\sum_jc_jv_j$.

Put $S_Y(x)=\|r_Y(x)\|^2$. For each fixed $x$, its summands have
mean at most $L$, bound $M=C_0^2/\varepsilon^2$, and variance at most
$ML$. Scalar Bernstein therefore gives
\[
 \mathbb P_Y\{S_Y(x)>2L\}\le e^{-cm\varepsilon},
 \qquad
 \mathbb E_Y\mu\{x:S_Y(x)>2L\}\le e^{-cm\varepsilon}.
\]
Truncate $r_Y$ on this exceptional set. The truncated vectors have
squared norm at most $2L$, and their covariance is bounded above by
$G_Y$. Applying Lemma~\ref{mic:covariance-sampling} conditionally on
$Y$ to the vectors at $X_1,\ldots,X_m$ shows that the matrix
$M_{ij}=R_\varepsilon(X_i,Y_j)/m$, with exceptional rows set to zero,
has expected norm at most $r_\varepsilon+b_1+b_2$, where
$b_2=2k_m\sqrt{2L/m}$. The deleted rows have expected squared
Frobenius norm at most $M e^{-cm\varepsilon}$. Thus
\[
 \mathbb E\|M\|\le r_\varepsilon
       +C\sqrt{\frac{\log(2m)}{m\varepsilon}}
       +\frac C\varepsilon e^{-cm\varepsilon}.
\]

For a fixed full cloud, choose uniformly a set $S$ of $m$ indices,
and retain in $B_n$ only entries crossing from $S$ to $S^c$. Call
the resulting symmetric matrix $C_S$. Each off-diagonal entry is
retained with probability $n/[2(n-1)]$, so
\[
 B_n=\frac{2(n-1)}n\mathbb E_SC_S.
\]
The norm of $C_S$ is the norm of its off-diagonal block, whose entries
are half those of $M$. Jensen's inequality gives
$\mathbb E\|B_n\|\le(n-1)\mathbb E\|M\|/n$, proving
\eqref{mic:sample-bound}.
\end{proof}

Apply this lemma to $R_\varepsilon=b_\varepsilon/\varepsilon-V$,
where $V$ is a fixed bounded symmetric finite rank kernel approximating
$T_\mu$ on $L^2(\mu)$. Such kernels are dense among compact self-adjoint
operators, because bounded functions are dense in $L^2(\mu)$.
Lemma~\ref{mic:overlap} supplies the required row-square bound.
If $n\varepsilon/\log n\to\infty$, then
\[
 \limsup_{\varepsilon\downarrow0}\mathbb E\|U_n-V_n\|
       \le\|T_\mu-V\|,
 \qquad
 (U_n)_{jk}=\frac{b_\varepsilon(Z_j,Z_k)}{n\varepsilon}\ (j\ne k),
 \quad (V_n)_{jk}=\frac{V(Z_j,Z_k)}n.
\]
Here the diagonal correction is at most $\|V\|_\infty/n$. Consequently
the quadratic form error is uniform over all $f\in[0,1]^n$, since
$n^{-1}\|f\|_2^2\le1$.

\subsection{The graph representation}

For a finite graph $G$, let
$Z_G(\lambda)=\sum_{I\subset V(G):\,I\text{ independent}}\lambda^{|I|}$.
We use the graph case of
Jenssen--Perkins--Potukuchi--Simkin~\cite[Theorem 1.4]{JPPS}.
If the maximum degree $\Delta$ tends to infinity, the maximum common
neighbor count is $o(\Delta)$, and $p\Delta$ stays in a compact
subinterval of $(0,\infty)$, then, for $\lambda=p/(1-p)$,
\begin{equation}
 \log Z_G(\lambda)
 =\max_{0\le x_j\le p}
 \left\{\sum_j\left(x_j-x_j\log\frac{x_j}p\right)
                -\sum_{\{j,k\}\in E(G)}x_jx_k\right\}
       +o(np)+O(np^2).
 \label{mic:graph-formula}
\end{equation}
This follows by multiplying the independent-set probability by
$(1+\lambda)^n$. Decreasing a coordinate above $p$ improves both
terms of the objective, justifying the displayed restriction.

Fix a vector of cell tilts $t=(t_i)$, set $a^{(t)}=ae^{-t_i}$ on $C_i$,
and include the self-pin in
\[
 a_{\varepsilon,t}=a^{(t)} e^{-\alpha\varepsilon/v_i}\quad\hbox{on }C_i,
 \qquad A_{\varepsilon,t}=\int a_{\varepsilon,t}\,d\nu,
 \qquad d\mu_{\varepsilon,t}=a_{\varepsilon,t}\,d\nu/A_{\varepsilon,t}.
\]
Take an even sample size $n\asymp\varepsilon^{-1-b}$, where
$1/2<b<1$, and sample the marks with this probability law. In addition
to every physical overlap edge, insert independently a pin edge for
each same-cell pair with probability $p_i=1-e^{-2\alpha\varepsilon/v_i}$.
Set $\lambda=A_{\varepsilon,t}/(n\varepsilon)$. Averaging the
pin-edge avoidance weight gives
$\exp[-\alpha\varepsilon\sum_iN_i(N_i-1)/v_i]$; the self-pin
supplies the missing diagonal part of $N_i^2$.

Choose $\gamma\in((1+b)/2,1)$ and put $r=\varepsilon^\gamma$.
By Lemma~\ref{mic:neighbors}, the probability of any pair with both
$s<r$ and $d<r$ is at most $Cn^2r^4=o(1)$. On the complement,
all common-neighbor probabilities are $o(\varepsilon)$ uniformly.
Conditional Bernstein bounds and a union bound give physical
maximum common-neighbor count $o(n\varepsilon)$ and maximum degree $O(n\varepsilon)$. Conditioning on the physical graph,
mixed physical--pin and pin--pin common-neighbor counts are
$O(n\varepsilon^2+\log n)=o(n\varepsilon)$. Pin degrees are
$O(n\varepsilon)$, and a cell of positive sampling mass supplies
vertices of degree at least $cn\varepsilon$ with probability tending
to one. Hence $cn\varepsilon\le\Delta\le Cn\varepsilon$, and
$p\Delta$ is bounded above and below. These are precisely the
hypotheses of \eqref{mic:graph-formula} for graphs.

Matrix Bernstein~\cite[Theorem 1.4]{Tropp}, conditional on the marks, gives
\[
 \left\|\frac{\text{pin adjacency}-\text{conditional mean}}
                      {n\varepsilon}\right\|
 =O_{\mathbb P}\left(\sqrt{\frac{\log n}{n\varepsilon}}
                       +\frac{\log n}{n\varepsilon}\right)=o_{\mathbb P}(1).
\]
Indeed each centered edge matrix has norm at most one and the variance
parameter is $O(n\varepsilon)$. The conditional mean, before physical
overlaps are removed, converges to the finite rank kernel
$2\alpha\sum_i\mathbf1_{C_i}\otimes\mathbf1_{C_i}/v_i$.
Removing physical-overlap entries changes its normalized row norm by
$O(\varepsilon)$, because the pin probabilities are $O(\varepsilon)$
and physical degrees are $O(n\varepsilon)$.

Set $x_j=pf_j$ in \eqref{mic:graph-formula}. The preceding operator
estimates and finite rank approximation give
\begin{align}
 \varepsilon\log Z_G(\lambda)\longrightarrow
 \sup_{0\le f\le1}\bigg\{
 A_t\int(f-f\log f)\,d\mu_t
 -\frac12\langle a^{(t)}f,T(a^{(t)}f)\rangle_{L^2(\nu)}
 -\alpha\|P(a^{(t)}f)\|_{L^2(\nu)}^2
 \bigg\},\label{mic:graph-limit}
\end{align}
where $A_t=\int a^{(t)}\,d\nu$ and $d\mu_t=a^{(t)}\,d\nu/A_t$.
Here $np\varepsilon\to A_t$.

To justify the variational maximum, express
a bounded finite rank kernel in finitely many bounded features.
The empirical joint laws of these features and weights in $[0,1]$
have subsequential weak limits with the prescribed feature marginal.
Conditional averaging of the weight given the features preserves the
interaction moments and increases the concave entropy, giving the
upper bound. Evaluating
any fixed measurable continuum weight on the independent cloud gives
the lower bound by the law of large numbers. The residual interaction
is uniformly small by Lemma~\ref{mic:empirical}. Finally the sampling
density $\mu_{\varepsilon,t}$ differs relatively from $\mu_t$ by
$O(\varepsilon)$ at the fixed partition, which does not alter these
limits.

Putting $q=a^{(t)}f$ in \eqref{mic:graph-limit} gives
$-\inf_{q\ge0}\{\mathcal F_a(q)+\sum_it_i\int_{C_i}q\}$.
Clipping to $a^{(t)}$ decreases the entropy on $\{q>a^{(t)}\}$ and
both nonnegative interactions, so the constraint $q\le a^{(t)}$
does not change the minimum.

\subsection{Annealing and the cap constraint}

Let $Z_{\varepsilon,t}$ and $\mathbb P_{\varepsilon,t}$ be the
partition function and Gibbs law of the uncapped pinned gas with
activity $a^{(t)}/\varepsilon$. Write
$(n)_N=n(n-1)\cdots(n-N+1)$. Averaging $Z_G$ over positions and pin
edges multiplies its $N$-particle integral by $(n)_N/n^N$ and discards
terms above $n$.
Thus $\mathbb E Z_G\le Z_{\varepsilon,t}$. Repulsiveness bounds the
factorial moments of the particle count by those of a Poisson variable
of mean $A_t/\varepsilon$: insert prescribed particles and omit their
repulsive factors. For a sufficiently large fixed $K$,
\[
 \mathbb P_{\varepsilon,t}\{N>K/\varepsilon\}\le e^{-c/\varepsilon}.
\]
On $N\le K/\varepsilon$, the binomial correction is bounded below by
$\exp[-C/(n\varepsilon^2)]$. Consequently
\[
 0\le\log Z_{\varepsilon,t}-\log\mathbb E Z_G
       \le C/(n\varepsilon^2)+o(1)=o(\varepsilon^{-1}).
\]
Changing one sampled point changes only edges incident to its vertex.
Deleting that vertex bounds the oscillation of $\log Z_G$ by
$\log(1+\lambda)$. Group the independent pin uniforms by their first
endpoint; each group has the same oscillation bound. The exponential
bounded-differences inequality~\cite{McDiarmid} for these $2n$
independent blocks gives
\[
 0\le\log\mathbb E Z_G-\mathbb E\log Z_G
           \le Cn\lambda^2=O((n\varepsilon^2)^{-1})=o(\varepsilon^{-1}).
\]
The variables $\varepsilon\log Z_G$ are uniformly bounded, so their
convergence in probability also implies convergence of expectations.
This proves the following limit, without the coercivity hypothesis.

\begin{proposition}[Uncapped variational limit]\label{mic:uncapped}
For every fixed bounded tilt vector,
\begin{equation}
 \phi(t):=\lim_{\varepsilon\downarrow0}\varepsilon\log Z_{\varepsilon,t}
  =-\inf_{q\ge0}\left\{\mathcal F_a(q)+\sum_it_i\int_{C_i}q\,d\nu\right\}.
 \label{mic:tilted-limit}
\end{equation}
\end{proposition}

\begin{lemma}[Uniqueness of the constrained minimizer]
\label{mic:unique-minimizer}
Under the coercivity hypothesis of Theorem~\ref{mic:finite-volume},
$\mathcal F_a(q)+\sum_i t_i\int_{C_i}q$ has a unique minimizer
for each $t_i\ge0$, with or without the cell-mass constraints.
\end{lemma}
\begin{proof}
Clipping restricts the minimization to $0\le q\le a$.
On this convex set the entropy curvature is at least $1/a_{\max}$,
so \eqref{mic:free-energy} is strongly convex under the stated
coercivity margin. The order interval is weakly compact in $L^2$;
relative entropy is weakly lower semicontinuous, and compactness of
$T$ makes its quadratic form weakly continuous on bounded sets.
This proves existence and uniqueness.
\end{proof}

\begin{lemma}[Multipliers for finitely many caps]\label{util:cap-multipliers}
Let $F$ be convex on a convex class $\mathcal C$, and let
$m:\mathcal C\to[0,\infty)^k$ be affine. Suppose
\[
 V(b)=\inf\{F(q):q\in\mathcal C,\ m(q)\le b\}
\]
is finite for every $b\in(0,\infty)^k$. For such a $b$, if $q_b$ attains $V(b)$,
there is $t\in[0,\infty)^k$ such that $q_b$ minimizes
$F(q)+t\cdot m(q)$ on $\mathcal C$ and
$t_i(m_i(q_b)-b_i)=0$ for every $i$.
\end{lemma}
\begin{proof}
The finite convex nonincreasing function $V$ has a supporting
subgradient $p\le0$ at $b$~\cite{Rockafellar}. For $q\in\mathcal C$ and $\eta>0$,
use $V(m(q)+\eta\mathbf1)\le F(q)$ in its supporting inequality
and let $\eta\downarrow0$. With $t=-p$ this gives
\[
 F(q)+t\cdot(m(q)-b)\ge V(b).
\]
At $q_b$, every term in $t\cdot(m(q_b)-b)$ is nonpositive,
so the inequality forces each to vanish. Equality there proves
the tilted minimality.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{mic:finite-volume}]
Apply Lemma~\ref{util:cap-multipliers} on $0\le q\le a$, with
$F=\mathcal F_a$, $m_i(q)=\int_{C_i}q$, and $b_i=\rho v_i$.
The zero competitor and entropy lower bound make its value function
finite. It gives nonnegative multipliers $t_i$ and complementary
slackness. Since $t_i\ge0$, clipping extends tilted minimality to all
$q\ge0$.
Its uniqueness remains true in a small open neighborhood of the tilt,
because clipping gives $q\le ae^s$ there and the coercivity margin
is strict. Thus $\phi$ is differentiable there, with
$\partial_{t_i}\phi=-\int_{C_i}q$. Applying exponential Markov bounds
using \eqref{mic:tilted-limit} at $t\pm s e_i$ proves exponential
concentration of $\varepsilon N_i$ about this mass.

The cap upper bound follows by multiplying by $e^{-\sum_it_iN_i}$
and using $N_i\le\lceil\rho v_i/\varepsilon\rceil$; the ceiling
contributes only $O(1)$ to the logarithm. For the lower bound solve
the constrained problem at $\rho'<\rho$. Under its multiplier tilt,
the concentration event lies strictly below all target caps. Undo
the tilt on that event and use complementary slackness to obtain the
lower bound $-\mathcal F_a(q_{\rho'})$. As $\rho'\uparrow\rho$,
the comparison function $(\rho'/\rho)q_\rho$ and bounded convergence
give $\mathcal F_a(q_{\rho'})\to\mathcal F_a(q_\rho)$.
\end{proof}


\section{Locality of the reference gas}
\label{mic:locality-section}

We now control covariances uniformly in the volume and exterior rods.
This provides the locality supplied by the restricted-ensemble
Dobrushin analysis in LMP~\cite[Sections~5--6]{LMP99}. For the gas
conditioned on the auxiliary field, we use a coercive insertion
operator. The method is the continuum Glauber coercivity argument,
in particular the Mayer-kernel criterion of Kondratiev, Kuna, and
Ohlerich~\cite[Theorem~4.2 and Corollary~4.3]{KKO}. Here the cell pin
makes the Mayer operator almost positive, and we estimate separately
the terms produced by active count caps.

Use shape-regular marked cells with spatial side $\ell$, angular
diameter comparable with $\ell$, and measures $v_c\asymp\ell^5$.
The cells and $\alpha$ are fixed before $\varepsilon\downarrow0$.
Let $\beta=\varepsilon^{-1}$ and allow any measurable one-particle
activity $0\le z(X)\le M\beta$, including arbitrary fixed
accessibility multipliers. Pair interactions are
\[
 U_\varepsilon(X,Y)=
 \begin{cases}
 +\infty,&b_\varepsilon(X,Y)=1,\\
 2\alpha\varepsilon/v_c,&X,Y\in C_c\text{ and }b_\varepsilon(X,Y)=0,\\
 0,&\text{otherwise}.
 \end{cases}
\]
The self-pin $e^{-\alpha\varepsilon/v_c}$ is included in $z$.
Impose caps $N_c\le K_c$. A bounded measurable perturbation
$J=\sum_AJ_A$ is permitted, with supports consisting of complete
cells, and norm
\begin{equation}
 \kappa_{m,1}=\sup_i\sum_{A\ni i}|A|e^{m\operatorname{diam}(A)}
                                      \|J_A\|_\infty.
 \label{mic:weak-norm}
\end{equation}
Distances are measured on the spatial cell lattice; angular cells
over the same spatial cube have distance zero. Constants may depend
on the fixed mesh, $\alpha$, and $M$. Write $\mu$ for the finite-volume
Gibbs law, $D_XF(\eta)=F(\eta+X)-F(\eta)$ for insertion difference,
and
\[
 r_X(\eta)=z(X)\mathbf1_{\{N_{c(X)}(\eta)<K_{c(X)}\}}
    \exp\left[-\sum_{Y\in\eta}U_\varepsilon(X,Y)-D_XJ(\eta)\right]
\]
for the insertion rate. Here $c(X)$ is the marked cell containing $X$.
Forbidden insertions have rate zero.

\begin{theorem}[Uniform covariance]
\label{mic:uniform-covariance}
Choose $\alpha>\|T\|+1$. For a sufficiently fine fixed mesh, all
sufficiently small $\varepsilon$, and $\beta\kappa_{m,1}$ sufficiently
small, there are $C,m>0$, independent of the domain and accessibility
mask, such that
\begin{equation}
 |\operatorname{Cov}_\mu(F,G)|
 \le C\sum_{i,j}e^{-m d(i,j)}\mathcal D_i(F)\mathcal D_j(G),
 \qquad
 \mathcal D_i(F)^2=
 \mathbb E_\mu\int_{C_i}r_X(\eta)|D_XF(\eta)|^2\,d\nu(X).
 \label{mic:covariance}
\end{equation}
Here $C_i$ comprises all angular cells over spatial cell $i$.
The observables are square integrable and have finite displayed norms.
\end{theorem}

\subsection{A lower bound for the Mayer operator}

The limiting operator maps $L^2$ boundedly into spatial and angular
$H^1$. For the spatial derivative this follows from
$\|\widehat T(k)\|\le C/(1+|k|)$. For angular derivatives,
\[
 \int_{S^2}|\nabla_us_k(u)|^2\,d\omega(u)
 \le\begin{cases}Cr,&r\ge1,\\Cr^4,&r\le1,\end{cases}
\]
by the substitution $t=ru\cdot\widehat k/2$. Combining this estimate
with $J(r)\le C/(1+r)$ for the other sinc factor bounds the angular
Hilbert--Schmidt norm of the differentiated symbol uniformly.
The weak derivative of $|u\times v|$ is bounded. Cell Poincare
inequalities therefore imply
\begin{equation}
 \|(I-P_\ell)T\|\le C\ell,\qquad
 \|T-P_\ell TP_\ell\|\le2C\ell.
 \label{mic:mesh-error}
\end{equation}
For $X\in C_c$ and $\varepsilon\le\ell$,
\begin{equation}
 \int_{C_c}b_\varepsilon(X,Y)\,d\nu(Y)\le C\varepsilon\ell^4.
 \label{mic:within-cell}
\end{equation}
Indeed, the center displacement has the form $tu-sv+e$. In a single
spatial cell $t-s=O(\ell)$, while a single angular cell gives
$|u-v|=O(\ell)$. The relevant planar area is $O(\ell^2)$ and its
thickening has volume $O(\varepsilon\ell^2+\varepsilon^2\ell+
\varepsilon^3)$. Integrating over the angular cell gives
\eqref{mic:within-cell}.

The exact pair Mayer kernel is
\[
 B_\varepsilon^{\rm pin}
 =b_\varepsilon+(1-b_\varepsilon)
     \sum_c\vartheta_c\mathbf1_{C_c}\otimes\mathbf1_{C_c},
 \qquad \vartheta_c=1-e^{-2\alpha\varepsilon/v_c}.
\]
For $\varepsilon\le c v_{\min}/\alpha$,
\[
 \sum_cv_c\vartheta_cP_c\ge\alpha\varepsilon P_\ell.
\]
By \eqref{mic:mesh-error}, $T+\alpha P_\ell\ge-C\ell I$.
The pin-overlap subtraction has row norm at most
$C\alpha\varepsilon^2/\ell$, by \eqref{mic:within-cell}.
Lemma~\ref{mic:overlap} now gives
\begin{equation}
 B_\varepsilon^{\rm pin}\ge-\varepsilon\delta_\varepsilon I,
 \qquad
 \delta_\varepsilon\le C(\ell+\varepsilon+\alpha\varepsilon/\ell).
 \label{mic:mayer-lower}
\end{equation}
Its row integral is at most $C(1+\alpha)\varepsilon$, and its spatial
range is fixed. Compression to a finite union of complete cells
preserves these estimates.

\subsection{The insertion commutator and active caps}

Since $r_X\le M\beta e^{2\kappa_{0,1}}$, the reversible
birth--death generator

\[
 (LF)(\eta)=\int r_Y(\eta)D_YF(\eta)\,d\nu(Y)
                    +\sum_{Y\in\eta}[F(\eta-Y)-F(\eta)]
\]
has bounded birth rates. Canceling factorials in the grand-canonical density gives the
Georgii--Nguyen--Zessin identity~\cite[Theorem~2]{NZ} and hence
\[
 \mathcal E(F,G)=\mathbb E_\mu\int r_XD_XF D_XG\,d\nu(X)
                         =\langle F,-LG\rangle_\mu.
\]
One-forms are measured in $\mu(d\eta)r_X(\eta)d\nu(X)$; their norm
is denoted by $\|\cdot\|_1$.

Hold an inserted particle $X$ fixed and let $L^X$ evolve the background,
with birth rates $r_Y(\eta+X)$ and deaths of particles already in
$\eta$. Direct subtraction yields
\begin{align}
 D_X(-L)F&=(-L^X+I)D_XF
        +\int[r_Y(\eta)-r_Y(\eta+X)]D_YF(\eta)\,d\nu(Y),\nonumber\\
 D(-L)&=\mathcal H D,\qquad
 \mathcal H=-L^X+I+\mathcal K.
 \label{mic:commutator}
\end{align}
The identity
\[
 r_X(\eta)r_Y(\eta+X)=r_Y(\eta)r_X(\eta+Y)
\]
makes $L^X$ reversible for the one-form measure and $\mathcal K$
self-adjoint. In particular, $-L^X\ge0$.

First take $J=0$. Away from an active cap, the curvature kernel in
$\mathcal K$ is $B_\varepsilon^{\rm pin}$. Applying
\eqref{mic:mayer-lower} to the function $r_Xg_X$, and using
$r_X^2\le M\beta r_X$, bounds its quadratic form below by
$-M\delta_\varepsilon\|g\|_1^2$. If
$N_c(\eta)=K_c-1$, the additional same-cell block is
\[
 e^{-2\alpha\varepsilon/v_c}
 \mathbf1_{C_c}(X)\mathbf1_{C_c}(Y)
               (1-b_\varepsilon(X,Y))r_Y(\eta).
\]
In its quadratic form the term $1$ is a positive rank-one operator.
The subtracted overlap term has norm at most $CM\ell^4$, by
\eqref{mic:within-cell}. Retaining the positive block gives
\begin{equation}
 \langle g,\mathcal H g\rangle_1
 \ge(1-M\delta_\varepsilon-CM\ell^4)\|g\|_1^2.
 \label{mic:coercivity}
\end{equation}
Choose the mesh first so that the $\ell$ terms are small, and then
choose $\varepsilon$ small at that mesh. The lower bound is a positive
constant independent of the volume and mask.

For a bounded perturbation, an allowed double birth introduces the
factor $e^{-D_XD_YJ}$. Its difference from one is bounded by
\[
 C\sum_{A\ni c(X),c(Y)}\|J_A\|_\infty,
 \qquad |D_XD_YJ|\le4\sum_{A\ni c(X),c(Y)}\|J_A\|_\infty.
\]
The other insertion-ratio factors lie in $[0,1]$. Weighted Schur
summation bounds the resulting one-form operator by
$C\beta\kappa_{m,1}$. The previous pair and cap bounds apply with
the factor $e^{2\kappa_{0,1}}$ in the rate bound. Requiring
$C\beta\kappa_{m,1}$ smaller than a fixed reserve in
\eqref{mic:coercivity} therefore gives $\mathcal H\ge cI$.

\subsection{Weighted inverse and covariance}

We obtain spatial decay by the exponential conjugation method of
Combes and Thomas~\cite{CT}.
For a spatial-cell set $A$, multiply one-forms by
$w_X=e^{m d(c_{\rm spatial}(X),A)}$. This weight commutes with $L^X$
and with each positive active-cap block, because it is constant on
the whole spatial cell. The pair part has row mass $O(\varepsilon)$,
rate $O(\beta)$ and finite spatial range. Its weighted commutator is
therefore $O(m)$ at the fixed mesh. The weak part is controlled by
\eqref{mic:weak-norm}. For a sufficiently small $m>0$, both
conjugations by $w$ and $w^{-1}$ have real part bounded below by
$c'>0$. In finite volume the weights are bounded and invertible,
so the conjugated operators are invertible. Their accretivity gives
the uniform inverse bound $1/c'$. If $P_i$ selects insertions in
spatial cell $i$, this proves
\begin{equation}
 \|P_i\mathcal H^{-1}P_j\|_{1\to1}\le Ce^{-m d(i,j)}.
 \label{mic:inverse-blocks}
\end{equation}

The caps bound the number of rods in every finite volume. Together
with bounded total birth rates and reversibility, this makes the
generators bounded on their respective $L^2$ spaces; the identities
therefore extend from bounded functions by density. Integration by parts and
\eqref{mic:commutator} give
\[
 \|LF\|_2^2=\langle DF,\mathcal H DF\rangle_1
                              \ge c\mathcal E(F,F).
\]
Successive deletions reach the empty configuration. The
Georgii--Nguyen--Zessin identity therefore propagates a zero insertion
gradient through every particle-number sector, so the Dirichlet
kernel consists of constants. The spectral theorem gives a gap at
least $c$.

For centered $G$, solve $(-L)u=G$. The commutator gives
$Du=\mathcal H^{-1}DG$, and hence
\[
 \operatorname{Cov}_\mu(F,G)=\langle DF,\mathcal H^{-1}DG\rangle_1.
\]
This is the birth--death counterpart of the Helffer--Sj\"ostrand
covariance representation~\cite{HS}.
Inserting \eqref{mic:inverse-blocks} proves
Theorem~\ref{mic:uniform-covariance}.

In particular, marked-cell counts satisfy
\begin{equation}
 |\operatorname{Cov}_\mu(N_i,N_j)|\le C\beta e^{-m d(i,j)}.
 \label{mic:count-covariance}
\end{equation}
If the activity on any measurable set of marks is multiplied by
$e^{-t}$, its counting observable has insertion norm in a cell at
most $C\sqrt\beta e^{-t/2}$ times the square root of the mark measure
in that cell. This bound remains valid with a fixed bounded
perturbation, because the insertion rate retains the factor $e^{-t}$.
It is integrable over $t\ge0$ and will be used for boundary deletion.
For a bounded local observable $V$ supported on $A$,
\begin{equation}
 |\operatorname{Cov}_\mu(N_i,V)|
       \le C\beta\|V\|_\infty\sum_{j\in A}e^{-m d(i,j)},
 \label{mic:count-observable}
\end{equation}
since $|D_XV|\le2\|V\|_\infty$ on its support.



\section{Spatial rigidity and a positive field action}\label{fnd:rigiditysection}
Write $D=u\cdot\nabla_x$ for the transport derivative.

The next lemma is the polynomiality property of Killing tensors
in Euclidean space~\cite[Section~IV]{Thompson}, stated here for
distributional coefficients.

\begin{lemma}[Polynomial transport coefficients]\label{fnd:killing}
Let $B\subset\mathbb R^3$ be connected and open. Suppose $H(x,z)$ is
a homogeneous polynomial of degree $l$ in $z\in\mathbb R^3$, with
distributional coefficients on $B$, and
$(z\cdot\nabla_x)H=0$. Every coefficient is a polynomial in $x$ of
degree at most $l$.
\end{lemma}
\begin{proof}
For $D_z=z\cdot\nabla_x$ one has
$[D_z,\partial_{z_j}]=-\partial_{x_j}$. Induction gives
\[
 D_z^j\partial_z^\gamma H
 =(-1)^j j!\sum_{\substack{\xi\le\gamma\\|\xi|=j}}
 \binom\gamma\xi\partial_x^\xi\partial_z^{\gamma-\xi}H
 \qquad(0\le j\le|\gamma|).
\]
Taking $|\gamma|=j=l+1$ gives $\partial_x^\gamma H=0$. Mollifying
any coefficient on a relatively compact ball therefore produces
polynomials of degree at most $l$ on smaller balls. This
finite-dimensional polynomial space is closed in distributions,
so the coefficient itself is locally such a polynomial. Polynomials
agreeing on an open overlap agree identically. Connectedness of $B$
then gives one polynomial on the whole domain.
\end{proof}

\begin{theorem}[Local transport rigidity]\label{fnd:transport}
If $r$ is measurable on $B\times S^2$, $r(x,\cdot)\in\mathcal W_a$
for almost every $x$ in a connected open set $B$, and $Dr=0$ in
distributions, then $r(x,u)=f(u)$ almost everywhere for a single
$f\in\mathcal W_a$.
\end{theorem}
\begin{proof}
The well bounds give $0<m\le r\le M$. Spatial mollification
preserves the transport equation and these bounds. The ordinary
chain rule for the mollified functions, followed by local $L^2$
convergence and Lipschitz continuity of the logarithm on $[m,M]$,
proves $D\log r=0$. The Euler equation then gives $DKr=0$.

Let $r_l$ be the angular harmonic component of degree $l$; odd
components vanish. Multiplication by $u_j$ maps degree $l$ into
degrees $l-1,l+1$. At any odd degree $j$, the equations $Dr=DKr=0$
therefore read
\[
 A_j+B_j=0,\qquad
 \kappa_{j-1}A_j+\kappa_{j+1}B_j=0,
\]
where $A_j$ and $B_j$ are the degree-$j$ projections of $Dr_{j-1}$
and $Dr_{j+1}$. Distinct consecutive even eigenvalues imply
$A_j=B_j=0$. Thus $Dr_l=0$ for every even $l$. 

Represent $r_l$ by its homogeneous harmonic polynomial $H_l(x,z)$.
The transport identity on the sphere extends by homogeneity to all
$z$. Lemma~\ref{fnd:killing} makes its spatial coefficients
polynomials of degree at most $l$. In the coordinates of
Proposition~\ref{fnd:analytic},
\[
 v(x)=\mathcal P_N\log r(x)=a-2\mathcal P_NKr(x)
\]
is consequently the restriction of a polynomial map
$\mathbb R^3\to E$. It belongs to $Z$ almost everywhere on $B$,
and hence everywhere there by continuity.

Fix $x_0\in B$ and a direction $e$. The polynomial curve
$v_e(t)=v(x_0+te)$ lies in $Z$ near zero. On the component $I$
containing zero of $\{t:v_e(t)\in U\}$, the analytic functions
$F(v_e(t))$ and $G(v_e(t))$ vanish identically. Thus $v_e(I)\subset Z$.
A finite endpoint of $I$ would have its image in compact $Z\Subset U$,
contradicting maximality of $I$. Hence $I=\mathbb R$. The polynomial
curve is bounded and therefore constant. Since $e$ was arbitrary,
$v$ is constant, and the identity $r=\Psi(v)$ proves the claim.
\end{proof}

Following LMP's averaged-density representation~\cite[Section~3]{LMP99},
we separate the action into an entropy loss under averaging and a
homogeneous excess. Rods require averaging along their axes, for
which Theorem~\ref{fnd:transport} gives the equality case.

Put $h(t)=t\log t-t$. With the longitudinal averaging operator $A$, define
\begin{align*}
 D_q(x)&=\int_{S^2}\left[\int_{-1/2}^{1/2}h(q(x+tu,u))\,dt
                                      -h(Aq(x,u))\right]d\omega(u),\\
 V_a(f)&=\Omega_a(f)-m_a.
\end{align*}
Jensen's inequality gives $D_q\ge0$; $V_a\ge0$ for all nonnegative
profiles, without an evenness restriction.

\begin{corollary}[Zero local action]\label{fnd:zeroaction}
If a bounded nonnegative profile $q$ satisfies
$\int_B[D_q+V_a(Aq)]\,dx=0$ on a connected open set $B$, then
$q(x,u)=f(u)$ almost everywhere on $B\times S^2$ for one
$f\in\mathcal W_a$.
\end{corollary}
\begin{proof}
Set $r=Aq$. The zero potential gives $r(x,\cdot)\in\mathcal W_a$
almost everywhere. Strict convexity in the equality case of Jensen's
inequality gives $q(x+tu,u)=r(x,u)$ for almost every $(x,u,t)$ with
$x\in B$. Choose $\chi\in C_c^\infty((-1/2,1/2))$ of integral one.
In distributions,
\[
 r=\int\chi(t)q(x+tu,u)\,dt,\qquad
 Dr=-\int\chi'(t)q(x+tu,u)\,dt=0.
\]
Theorem~\ref{fnd:transport} gives $r=f$. For
$B'\Subset B$ and $0<b\le1/2$ with $B'+B(0,b)\subset B$, a change
of variables gives
\[
 2b\int_{B'\times S^2}|q-f|^2
 \le\int_{B\times S^2}\int_{-b}^b|q(x+tu,u)-f(u)|^2\,dt\,dx\,d\omega=0.
\]
An exhaustion proves the assertion on $B$.
\end{proof}


\subsection{Minimizing the density profile at a fixed field}
\label{fnd:latentsection}
Fix $a=a_t$ and $\zeta=e^{a_t}$. Retain the marked cells
$C_{i,b}=B_i\times U_b$, their measures $v_{i,b}=v_s\omega_b$,
and the cell projection $P$. Fields take values in $[-R,R]$.

Set $S=T+2\alpha P$.
Both summands have nonnegative kernels and bounded row sums; $T$ has
fixed finite spatial range. The operator estimate required below is
\begin{equation}\label{fnd:operatorhyp}
 S\ge-\delta I,\qquad M=\zeta e^{R+1},\qquad
 c=M^{-1}-\delta>0.
\end{equation}
Choose $\alpha\ge\|T\|$. The mesh estimate
\eqref{mic:mesh-error} gives $\|T-PTP\|\le C\ell$, where $\ell$ is
the fine marked-cell mesh. Since $PTP\ge-\|T\|P$,
\[
 S\ge(2\alpha-\|T\|)P-C\ell I\ge-C\ell I.
\]
After fixing $\alpha$ and $R$, choose the mesh so that
$\delta=C\ell<M^{-1}$. Choose
\begin{equation}\label{fnd:caphyp}
 \rho>\sup_{f\in\mathcal W}\|f\|_\infty,
 \qquad R>2\alpha\sup_{f\in\mathcal W}\|f\|_\infty.
\end{equation}
Admissible density profiles satisfy $q\ge0$ and
$\int_Cq\le\rho v_C$ on every marked cell.

\begin{proposition}[The capped minimizing profile]
\label{fnd:response}
Let $D$ be a finite union of complete spatial cells. For bounded
measurable sources $-R_-\le\phi\le R+1$, where $R_-<\infty$ is
fixed, the functional on profiles extended by zero outside $D$,
\[
 G_{D,\phi}(q)=\int_{D\times S^2}q(\log q-1-a-\phi)
                         +\frac12\langle q,S_Dq\rangle
\]
has a unique capped minimizer $q_{D,\phi}$. It satisfies uniform
bounds $0<m_*\le q_{D,\phi}<M$ on $D$, and cell multipliers
$\lambda_C\ge0$ satisfy
\begin{equation}\label{fnd:kkt}
 \log(q/\zeta)-\phi+Sq+\lambda_C=0\quad\hbox{on }C,
 \qquad \lambda_C\left(\int_Cq-\rho v_C\right)=0.
\end{equation}
\end{proposition}
\begin{proof}
Replacing $q$ by $q\wedge M$ decreases both terms: the derivative
$\log(q/\zeta)-\phi$ is positive above $M$, and $S$ has a
nonnegative kernel. On $[0,M]$,
write the entropy as a convex function plus $q^2/(2M)$ and combine
the latter with $S\ge-\delta I$. This proves strong convexity with
constant $c$, weak lower semicontinuity, existence by weak compactness
in $L^2$, and uniqueness.

Apply Lemma~\ref{util:cap-multipliers} on $[0,M]$ to $G$ and
the cell masses. The zero competitor and the entropy lower bound
make the value function finite. This gives nonnegative multipliers
$\lambda_C$ and complementary slackness.
For that minimizer $q_*$, set $V=-a-\phi+Sq_*+\lambda$. The
pointwise minimizer of $\int[h(q)+Vq]$ on $[0,M]$ is
$g=\min(M,e^{-V})>0$. If $g\ne q_*$, its negative first-order
improvement dominates the quadratic remainder along the segment
joining the two profiles, contradicting minimality. Thus $q_*=g$.
Positivity of $q_*=g$ gives $Pq_*>0$ on every marked cell. Hence
\[
 \log M+V=R+1-\phi+Sq_*+\lambda
          \ge2\alpha Pq_*>0.
\]
Thus $q_*<M$ and \eqref{fnd:kkt} holds. If
$\lambda_C>0$, saturation and $q\le\zeta e^{R+1-\lambda_C}$ give
\[
 0\le\lambda_C\le(\log\zeta+R+1-\log\rho)_+.
\]
The bounded row sum of $S$ now gives the uniform positive lower bound.

\end{proof}

We apply the exponential conjugation method of Combes and
Thomas~\cite{CT} to the constrained Euler equation.

\begin{lemma}[Weighted response]\label{fnd:weighted-response}
If $W$ is a positive spatial weight constant on complete spatial cells
and $|\log W_i-\log W_j|\le m\,d(i,j)$ for sufficiently small $m>0$,
then minimizers at two sources obey
\begin{equation}\label{fnd:weighted}
 \|W(q_{D,\phi_1}-q_{D,\phi_2})\|_2
 \le\frac2c\|W(\phi_1-\phi_2)\|_2.
\end{equation}
\end{lemma}
\begin{proof}
Subtract \eqref{fnd:kkt} for two sources and test with
$W^2(q_1-q_2)$. The cap term is nonnegative because a positive multiplier occurs
only at a saturated cap:
\[
 \sum_CW_C^2(\lambda_C^1-\lambda_C^2)
                     \left(\int_Cq_1-\int_Cq_2\right)\ge0.
\]
The entropy difference quotient is at least $M^{-1}$. Because $W$
commutes with $P$, only $T$ changes under conjugation. Finite range
and bounded row sums imply
$\|WTW^{-1}-T\|\le C(e^{Cm}-1)\le c/2$ for small enough rate $m$.
Coercivity and Cauchy--Schwarz give \eqref{fnd:weighted}.
\end{proof}

\begin{corollary}[The profile in the whole space]\label{fnd:allspace}
For every field in the compact product box there is a unique bounded
positive solution in the whole space $q_\phi$ of the capped Euler equations.
It commutes with lattice translations and is locally Lipschitz under
changes in finitely many field coordinates. Agreement of
fields in a radius-$d$ neighborhood of a target cell implies
\[
 \|q_\phi-q_\psi\|_{L^2(C)}\le C e^{-md}.
\]
Coordinatewise convergence of fields implies local $L^2$ convergence
of the profiles.
\end{corollary}
\begin{proof}
For $D\subset D'$, the restriction of $q_{D',\phi}$ to $D$ solves
the capped problem at the effective source
$\phi-T_{D,D'\setminus D}q_{D',\phi}$. This perturbation is
nonpositive, bounded, and supported in a fixed boundary strip.
Apply \eqref{fnd:weighted} with a weight decreasing away from a
target cell. The squared weighted source norm is an exponentially
decaying sum over the strip, bounded by $Ce^{-2md}$ after decreasing
$m$ to absorb lattice shell counts. Thus nested free-box minimizers
are Cauchy locally. The local Euler equations and multipliers pass
to the limit.

For two bounded solutions in the whole space use the same decaying weight
on their difference. All terms are integrable, the cap contribution
is nonnegative, and coercivity forces equality. This also proves
lattice translation covariance. Weighted response proves locality and
finite-coordinate continuity. To obtain product continuity, first
make the exterior contribution small using locality, then use
coordinate convergence in the resulting fixed finite neighborhood.
\end{proof}

\subsection{Local charges and an exact positive interaction}
Define
\begin{equation}\label{fnd:charges}
 e_i(\phi)=\int_{B_i}[D_{q_\phi}+V_a(Aq_\phi)]\,dx
 +\frac1{4\alpha}\int_{B_i\times S^2}|\phi-2\alpha Pq_\phi|^2.
\end{equation}
The charges are nonnegative, bounded and continuous in the product
topology. Their response to a field change decays exponentially
with distance, by the profile locality and the Lipschitz continuity
of entropy on the uniform positive profile interval.
For a homogeneous well $f$ and $r=2\alpha Pf$, the Euler equation
shows $q_r=f$ by Corollary~\ref{fnd:allspace}; the caps are inactive
by \eqref{fnd:caphyp}. Consequently $e_i(r)=0$.

\begin{proposition}[Positive interactions on lattice balls]\label{fnd:atoms}
There are nonnegative continuous interactions $\Phi^0_{i,n}$, with
declared support the entire fine lattice ball $B_n(i)$, such that
\[
 e_i=\sum_{n\ge0}\Phi^0_{i,n},\qquad
 \|\Phi^0_{i,n}\|_\infty\le Ce^{-m(n-1)}\quad(n\ge1).
\]
Every atom vanishes at a homogeneous well field. For every fixed $p$
and $0<a'<m/2$,
\[
 \sum_{i,n:\,0\in B_n(i)}|B_n(i)|^p
 e^{a'\operatorname{diam} B_n(i)}\|\Phi^0_{i,n}\|_\infty<\infty.
\]
\end{proposition}
\begin{proof}
Set $e_i^{[n]}(\phi)=\inf\{e_i(\psi):\psi=\phi\text{ on }B_n(i)\}$.
Compactness gives attainment, and exponential locality gives
$0\le e_i-e_i^{[n]}\le Ce^{-mn}$.
Continuity follows by uniform finite-range approximation of $e_i$.
The localizations increase with $n$. Take
$\Phi^0_{i,0}=e_i^{[0]}$ and
$\Phi^0_{i,n}=e_i^{[n]}-e_i^{[n-1]}$ for $n\ge1$.
Their nonnegativity, bounds, and root values follow immediately.
There are $O((n+1)^3)$ possible centers with $0\in B_n(i)$;
$|B_n(i)|=O((n+1)^3)$ and its diameter is $2n$.
The asserted norm is therefore finite.
\end{proof}

\subsection{Alignment with the well set}
Write $\|\cdot\|_i$ for the norm of $L^2(B_i\times S^2)$.
If $f\in\mathcal W$,
the Euler equation cancels the linear terms and yields
\begin{align*}
 D_Q(x)+V_a(AQ(x))
 &=\int_{S^2}\int_{-1/2}^{1/2}
 \bigl[h(Q(x+tu,u))-h(f(u))\\[-2mm]
 &\hspace{36mm}-\log f(u)(Q(x+tu,u)-f(u))\bigr]dt\,d\omega(u)\\
 &\quad+\langle AQ(x)-f,K(AQ(x)-f)\rangle.
\end{align*}
For $z\ge0$, $f>0$, the entropy bracket is at most $(z-f)^2/f$,
by $\log t\le t-1$. With $m_{\mathcal W}=\inf_{f\in\mathcal W,u}f(u)>0$,
\begin{equation}\label{fnd:localquadratic}
 0\le\int_{B_i}[D_Q+V_a(AQ)]
 \le C\sum_{|j-i|\le r_*}\|Q_j-f\|_{L^2(B_j\times S^2)}^2,
\end{equation}
where $r_*$ is a fixed fine-lattice range.

In the field coordinates
\[
 s_{i,b}=\sqrt{\omega_b}\phi_{i,b},\qquad
 (\Pi Q_i)_b=\frac{2\alpha\sqrt{\omega_b}}{v_s\omega_b}
                           \int_{B_i\times U_b}Q_i,
\]
one has $\|\Pi\|\le2\alpha/\sqrt{v_s}$ and the exact identity
\begin{equation}\label{fnd:pinningsquare}
 \frac1{4\alpha}\int_{B_i\times S^2}|\phi-2\alpha PQ|^2
 =\frac{v_s}{4\alpha}|s_i-\Pi Q_i|^2.
\end{equation}
Copying whole spatial cells preserves every marked-cell cap.
The normalized Lebesgue field measure is pushed forward under this
fixed diagonal coordinate change. Projected wells lie in its
interior by \eqref{fnd:caphyp}, giving uniform polynomial lower
bounds for small-ball probabilities.

\begin{lemma}[Alignment on a fixed patch]\label{fnd:alignment}
\label{fnd:nearestalignment}
For a fixed connected inner patch $I$, choose a connected outer patch
$O$ containing a neighborhood of $I$. There is a bounded,
nondecreasing concave modulus $\omega$, with $\omega(t)\to0$ as $t\downarrow0$,
such that, writing $E_O=\sum_{i\in O}e_i(\phi)$,
\[
 \min_{f\in\mathcal W}\sum_{i\in I}\|q_{\phi,i}-f\|_i^2
 +\sum_{\substack{i,j\in I\\i\sim j}}\|f_i-f_j\|_{L^2(S^2)}^2
 +\sum_{i\in I}|s_i-\Pi f_i|^2
 \le\omega(E_O).
\]
Here $i\sim j$ means neighboring spatial cells and $f_i$ is any
nearest well to $q_{\phi,i}$ in $\|\cdot\|_i$.
\end{lemma}
\begin{proof}
If the outer charge sum is zero, Corollary~\ref{fnd:zeroaction}
makes $q_\phi$ one common well on the inner patch; the pinning square
then gives $\phi=2\alpha Pf$ there. If the asserted vanishing modulus
failed, choose fields whose outer energy tends to zero and whose
inner error stays bounded below. Compactness of the countable
product field box gives a coordinatewise convergent subsequence.
Corollary~\ref{fnd:allspace} and continuity of the charges pass zero
energy and local profile convergence to the limit, a contradiction.
For any nearest well $f_i$,
$\|f_i-f\|_i\le2\|q_{\phi,i}-f\|_i$, so the same modulus controls
all such choices. A measurable nearest-well selection can be used:
the minimum is over a compact metric set, the distance is continuous,
and the Kuratowski--Ryll-Nardzewski selection theorem applies
\cite{KRN}.

Let $\psi(t)$ be the nondecreasing supremum of the bounded total
inner error at energy at most $t$, and choose $C\ge\sup\psi$.
The function
$\omega(t)=\inf_{s>0}[\psi(s)+Ct/s]$ is a bounded, nondecreasing
concave majorant of $\psi$ and tends to zero at zero. This proves
the quantitative form used under averaging.

\end{proof}

\begin{lemma}[Separation after cell projection]\label{fnd:projected-components}
The restriction of $P$ to $\mathcal W$ is injective. The images of
distinct well components are compact sets at positive mutual distance.
\end{lemma}
\begin{proof}
If $Pf=Pg$ for two wells, both solve the same
homogeneous pinned Euler equation. The angular operator
$2K+2\alpha P_{\rm ang}$ obeys the same lower bound as $S$;
this follows by testing the spatial inequality on profiles constant
in larger and larger boxes and dividing by volume. Subtracting the
two Euler equations and testing with $f-g$ gives at least
$c\|f-g\|_2^2$, but the expression is zero. Thus $f=g$.
The cell projection is therefore a continuous injection on compact
$\mathcal W$, and is a homeomorphism onto its image. Its finitely
many compact components have positive mutual distance.
\end{proof}

\subsection{Changes of field and comparison with trial profiles}
For an arbitrary trial $Q$, let $\ell_i(Q,\phi)$ be the expression
in \eqref{fnd:charges} with $Q$ replacing $q_\phi$.

\begin{proposition}[Exact change of action]\label{fnd:cocycle}
If $\phi-\psi$ is supported on finitely many marked cells, then
\begin{align}\label{fnd:cocycleformula}
 H^0(\phi;\psi)&:=\sum_i[e_i(\phi)-e_i(\psi)]\nonumber\\
 &=\int_0^1\sum_C(\phi_C-\psi_C)
 \left[\frac{v_C\{\psi_C+t(\phi_C-\psi_C)\}}{2\alpha}
              -\int_Cq_{\psi+t(\phi-\psi)}\right]dt.
\end{align}
The sum is absolutely convergent. It equals the sum of changes of the interactions
$\Phi^0_{i,n}$ from Proposition~\ref{fnd:atoms}.
\end{proposition}
\begin{proof}
Use the same finite free box $D$ for both profiles and extend both
profiles and fields by zero outside it. The cancellation
\[
 D_q+V_a(Aq)
 =\int_{S^2}\int_{-1/2}^{1/2}h(q(x+tu,u))\,dt\,d\omega
  -a\int_{S^2}Aq\,d\omega+\langle Aq,KAq\rangle-m_a
\]
removes the $h(Aq)$ terms. Integration over all spatial centers,
translation invariance of the longitudinal average, and expansion
of the pinning square give the exact identity
\begin{align*}
 &\sum_i\{\ell_i(q_{D,\phi},\phi_D)
                    -\ell_i(q_{D,\psi},\psi_D)\}\\
 &\qquad=G_{D,\phi}(q_{D,\phi})+\frac1{4\alpha}\int_D\phi^2
       -G_{D,\psi}(q_{D,\psi})-\frac1{4\alpha}\int_D\psi^2.
\end{align*}
The common constant baseline outside $D$ cancels. Weighted response
bounds each centerwise difference by an exponentially summable
function of its distance from the changed field set, uniformly in
$D$. To estimate entropy use the common positive interval inside
$D$; outside it both profiles vanish. Dominated convergence gives
the all-space charge sum.

The minimizing inequalities at two sources and the continuity
of the minimizer give the envelope formula of Danskin~\cite{Danskin}:
the minimized full action
$\min_qG_{D,\phi}(q)+(4\alpha)^{-1}\int_D\phi^2$ is differentiable,
with coordinate derivative
$v_C\phi_C/(2\alpha)-\int_Cq_{D,\phi}$. Integrate along the
straight segment of fields and pass to the limit to obtain
\eqref{fnd:cocycleformula}. For the ball interaction, atoms on balls
missing the changed set have zero difference. At distant centers,
only radii reaching the set contribute; their exponential norms
are summable. This justifies the double sum and proves equality
of the two action differences.
\end{proof}

\begin{proposition}[Comparison with a trial profile]\label{fnd:globalcomparison}
Suppose $\phi'-\phi$ is finitely supported, and $Q'$ is capped,
$0\le Q'\le M$, and agrees with $q_\phi$ outside finitely many
complete cells. Then
\begin{equation}\label{fnd:comparison}
 \sum_i[e_i(\phi')-e_i(\phi)]
 \le\sum_i[\ell_i(Q',\phi')-\ell_i(q_\phi,\phi)].
\end{equation}
\end{proposition}
\begin{proof}
Put $q'=q_{\phi'}$ and $\Delta=Q'-q'$. Locality gives
$\Delta\in L^1\cap L^2$. Subtract the Euler equation for $q'$ from
the action difference (all integrals below converge) to obtain
\begin{align*}
 \sum_i[\ell_i(Q',\phi')-\ell_i(q',\phi')]
 &=\int[h(Q')-h(q')-\log q'(Q'-q')]
      +\frac12\langle\Delta,S\Delta\rangle
      -\sum_C\lambda_C'\int_C\Delta\\
 &\ge\frac c2\|\Delta\|_2^2\ge0.
\end{align*}
The cap term is nonnegative, because a positive multiplier means
$q'$ saturates its cap, while $Q'$ is admissible. Bounded
multipliers and $L^1$ decay justify the sum. Combining this with
Proposition~\ref{fnd:cocycle} gives \eqref{fnd:comparison}.
\end{proof}

\begin{corollary}[A homogeneous trial profile]\label{fnd:roottrial-bound}
For a homogeneous root $r=2\alpha Pf$ and a finite region $D$,
\begin{equation}\label{fnd:roottrial}
 0\le H_D^0(\phi_D\mid r)
 \le\frac1{4\alpha}\sum_{C\subset D}v_C|\phi_C-r_C|^2,
 \qquad H_D^0(\phi_D\mid r)=\sum_i e_i(\phi_Dr_{D^c}).
\end{equation}
\end{corollary}
\begin{proof}
Use $Q'=f$ in Proposition~\ref{fnd:globalcomparison}. Its local Onsager action is zero, and only the displayed pinning
square remains. Nonnegativity gives the lower bound.
\end{proof}


\section{Restoring all particle counts}
\label{res:section}

We use the fixed spatial and angular mesh chosen above.  Its marked cells are denoted by $C_i$, their one-particle
measures by $v_i$, and their particle counts by $N_i$.  Constants may
depend on this mesh.  In particular $0<v_-\leq v_i\leq v_+<\infty$.
The graph $G$ joins any two cells in which rods can overlap.  It has
bounded degree, finite spatial range, and polynomial volume growth of
degree three.  All neighborhoods and distances below refer to this
full graph, including when the integration domain has holes.

Write $z_h=\beta\zeta e^h$, $K_i=\lceil\rho\beta v_i\rceil$,
and use the parameters and Gaussian normalizers of
\eqref{mod:gaussian}--\eqref{mod:cellweight}. In particular,
\begin{equation}\label{res:normalizertail}
 0\leq-\log c_{i,h}(n)\leq Ce^{-c\beta},\qquad 0\leq n\leq K_i.
\end{equation}
With $\nu_i$ denoting one-particle measure on $C_i$, the two cell
weights relative to $d\mathfrak L_i=\sum_{n\geq0}d\nu_i^{\otimes n}/n!$
are
\begin{equation}\label{res:localweights}
 a_i^\phi=\frac{(\beta\zeta)^{N_i}}{c_{i,h}(N_i)}
 e^{\phi_iN_i-\alpha N_i^2/(\beta v_i)}\quad(N_i\leq K_i),
 \qquad
 b_i=\frac{z_h^{N_i}}{c_{i,h}(0)}\quad(N_i>K_i).
\end{equation}
The empty-cell weight is $E_i=c_{i,h}(0)^{-1}$. All estimates below
allow additional one-particle accessibility multipliers in $[0,1]$.

\subsection{The empty-cell projection}

For a finite set $D$ of marked cells and an admissible
exterior rod configuration $\xi$, let $\mathcal R_D^{\rm full}$ be
the rod partition function with the weights
\eqref{res:localweights}, and let $\mathcal R_D^{\rm cap}$ be its
restriction to $N_i\leq K_i$ in every cell.  The exterior $\xi$ is
not restricted by a cap.  Define $\pi\omega$ by emptying every cell
whose count exceeds its cap.  Thus
\begin{equation}\label{res:resetcount}
 N_i(\pi\omega)=\widehat N_i(\omega),
 \qquad \widehat N_i=N_i\mathbf1_{\{N_i\leq K_i\}}.
\end{equation}

A nonempty $G^2$-connected set $S\subset D$ has dependency support
$U_S=N_G[S]$.  Two such sets are compatible when their dependency
supports are disjoint, equivalently when their $G$-distance is at
least three.  For a capped configuration $\sigma$ set
\begin{align}
 B_S(\sigma;\xi)
 &=\int_{N_i>K_i,\ i\in S}
       \prod_{i\in S}b_i(\omega_i)
       \chi_S(\omega_S;\sigma_{D\setminus S},\xi)
       \,d\mathfrak L_S(\omega_S),\notag\\
 w_S(\sigma;\xi)
 &=\frac{\mathbf1_{\{\sigma_S=\varnothing\}}}
          {\prod_{i\in S}E_i}\,B_S(\sigma;\xi),\label{res:polymers}\\
 \Xi_D^{\rm e}(\sigma;\xi)
 &=\sum_{\mathcal F\ {\rm compatible}}\prod_{S\in\mathcal F}w_S,
 \qquad K_D^{\rm e}=-\log\Xi_D^{\rm e}.\notag
\end{align}
Here $\chi_S$ is the product of hard-core factors with at least one
particle in $S$; the empty polymer family contributes one.

\begin{proposition}[Exact projection]\label{res:projection}
For every $D,\xi,\phi$,
\begin{equation}\label{res:partitionidentity}
 \mathcal R_D^{\rm full}(\phi\mid\xi)
 =\mathcal R_D^{\rm cap}(\phi\mid\xi)
       \mathbb E_{D,\phi,\xi}^{\rm cap}e^{-K_D^{\rm e}}.
\end{equation}
The pushforward of the full law by $\pi$ is the capped law tilted
by $e^{-K_D^{\rm e}}$.  The correction $K_D^{\rm e}$ is independent
of $\phi$.
\end{proposition}

\begin{proof}
Decompose the full integral according to its actual bad set
$B=\{i:N_i>K_i\}$.  The $G^2$-components of $B$ form a unique
compatible family.  Their closed $G$-neighborhoods are disjoint,
so the bad integrals factor conditional on the retained rods.
In the corresponding term on the right of
\eqref{res:partitionidentity}, the indicators force the cells in
$B$ to be empty.  Integrating them contributes exactly
$\prod_{i\in B}E_i$, canceling the denominator in
\eqref{res:polymers}.  All remaining factors agree with the original
sector.  This proves the identity sector by sector.  Inserting any bounded
function of $\pi\omega$ proves the pushforward assertion.  Finally
$b_i/E_i=z_h^{N_i}$ and the hard factors have no field dependence.
\end{proof}

\begin{lemma}[Exceptional-count tails]\label{res:tails}
There is $q_\beta=e^{-c\beta}$, after reducing $c>0$ if necessary,
such that
\begin{equation}\label{res:tailmoments}
 0\leq w_S\leq q_\beta^{|S|},\qquad
 \sum_{n>K_i}n^m\frac{(\beta\zeta e^{H}v_i)^n}{n!}
       \leq C_m\beta^m q_\beta
\end{equation}
for every fixed integer $m\geq0$.  These estimates are uniform in
the exterior rods and all accessibility multipliers.
\end{lemma}

\begin{proof}
Drop the hard constraints and accessibility multipliers.  The empty
weights cancel the factors $c_{i,h}(0)^{-1}$.  Writing
$r=e\zeta e^{H}/\rho<1$, the inequality $n!\geq(n/e)^n$ gives
\[
 \sum_{n>K_i}\frac{(\beta\zeta e^{H}v_i)^n}{n!}
 \leq\frac{r^{K_i+1}}{1-r}.
\]
Since $v_i\geq v_-$, this is exponentially small in $\beta$.
The ratio of successive summands is at most
$\zeta e^{H}/\rho<1$ beyond the cap.  Summing the geometric
tail with the extra polynomial $n^m$ gives the second estimate.
\end{proof}

\subsection{The expansion with a weak interaction}

We shall allow a bounded measurable interaction
$J=\sum_AJ_A$ on the actual rod configuration and the field.
Every declared support contains all rod and field arguments.
Supports are connected in a fixed finite-range enlargement of $G$.  For $a>0$ write
\begin{equation}\label{res:strongnorm}
 \|J\|_a=\sup_x\sum_{A\ni x}e^{a|A|}\|J_A\|_\infty=\kappa.
\end{equation}
The exponent $a$ is chosen with a fixed reserve for polynomial
factors and spatial decay.

Expand $e^{-J_A}=1+f_A$, with
$\eta_A=e^{\|J_A\|_\infty}-1$.  A bad-site atom at $i$ has support
$i^+=N_G[i]$ and majorant $q_\beta$; a weak atom $A$ has support
$A$ and majorant $\eta_A$.  An object $\gamma$ is a nonempty set of distinct atoms connected
by overlap of their supports.  Write
\begin{equation}\label{res:occurrencesize}
 U_\gamma=\bigcup_{a\in\gamma}B_a,
 \qquad \ell_\gamma=\sum_{a\in\gamma}|B_a|,
 \qquad z_\gamma^\#=\prod_{a\in\gamma}w_a.
\end{equation}
For $\gamma=(S,\mathcal F)$ its exact weight is
\begin{align}
 z_\gamma(\sigma;\xi)
 &=\frac{\mathbf1_{\{\sigma_S=\varnothing\}}}
          {\prod_{i\in S}E_i}
   \int_{N_i>K_i,\ i\in S}\prod_{i\in S}b_i(\omega_i)
       \chi_S(\omega_S;\sigma_{D\setminus S},\xi)\notag\\[-2mm]
 &\hspace{25mm}\times
       \prod_{A\in\mathcal F}
       f_A(\omega_S\sigma_{D\setminus S}\xi,\phi)
       \,d\mathfrak L_S(\omega_S).\label{res:mixedweight}
\end{align}
When $S$ is empty this is simply the product of the selected weak
factors on the retained configuration.  The factors in
\eqref{res:mixedweight} are evaluated on the reconstructed actual
configuration.  The bound
\begin{equation}\label{res:mixedmajorant}
 |z_\gamma|\leq z_\gamma^\#
       =q_\beta^{|S|}\prod_{A\in\mathcal F}\eta_A
\end{equation}
follows from Lemma~\ref{res:tails}.

\begin{lemma}[Projection with a weak interaction]\label{res:mixedprojection}
Let $K_D(J)$ be minus the logarithm of the gas of objects
\eqref{res:mixedweight}, with disjoint dependency supports. Then
\begin{equation}\label{res:mixedidentity}
 \mathcal R_D^{{\rm full},J}
  =\mathcal R_D^{\rm cap}\,\mathbb E_D^{\rm cap}e^{-K_D(J)}.
\end{equation}
The full law projects to the capped law tilted by $e^{-K_D(J)}$.
\end{lemma}
\begin{proof}
Expand the weak factors and partition each selected atom set into
its overlap components. Distinct components have disjoint dependence
and no hard factor connecting their bad rods, so their integrals
factor. The empty weights cancel as in Proposition~\ref{res:projection}.
Inserting a bounded function of the projected configuration proves
the assertion about its law.
\end{proof}

The logarithmic expansion uses Penrose's partition identity~\cite{Penrose67}
and the Koteck\'y--Preiss convergence criterion~\cite{KP86}, in the
formulation by iteration over trees in Fern\'andez and Procacci~\cite[Section~4]{FP}.

\begin{lemma}[Mixed cluster expansion]\label{res:mixedcluster}
Suppose that, for the required fixed $s\geq0$,
\begin{equation}\label{res:atomnorm}
 \rho_s:=\sup_x\sum_{a:x\in B_a}w_ae^{(s+2)|B_a|}\leq\tfrac14.
\end{equation}
The gas has an absolutely convergent logarithm.
Grouping the logarithm by
connected union supports gives
\begin{equation}\label{res:Knorm}
 K_D(J)=\sum_U K_{D,U}(J),\qquad
 \sup_x\sum_{U\ni x}e^{b|U|}(1+|U|)^p
       \|K_{D,U}(J)\|_\infty\leq C_{b,p}(q_\beta+\kappa)
\end{equation}
whenever \eqref{res:atomnorm} holds with sufficient reserve beyond
$b$.  A cluster marked by one prescribed bad site retains a factor
$q_\beta$; a cluster marked by a weak atom retains a factor
$\kappa$; the two marks together retain $q_\beta\kappa$.
\end{lemma}

\begin{proof}
For an ordered tuple $C=(\gamma_1,\ldots,\gamma_n)$, let
$\varphi^T(C)$ be the alternating sum over connected spanning
subgraphs of its incompatibility graph. Set
$\ell(C)=\sum_j\ell_{\gamma_j}$ and $U(C)=\bigcup_jU_{\gamma_j}$.
The tree bound is
\begin{equation}\label{res:treegraph}
 |\varphi^T(C)|\leq\sum_{T\ {\rm tree}}
 \prod_{\{i,j\}\in E(T)}\mathbf1_{\{U_{\gamma_i}\cap U_{\gamma_j}
 \ne\varnothing\}}.
\end{equation}
To prove it, order edges and group connected graphs by their minimum
spanning tree. Each group is a Boolean interval, whose alternating
sum is zero or has absolute value one.

For atom trees, iteration from zero in
\[
 T_a=w_ae^{u|B_a|}\exp\Bigl\{\sum_{b:B_b\cap B_a\ne\varnothing}T_b\Bigr\}
\]
is bounded by $w_ae^{(u+1)|B_a|}$ whenever
$\sup_x\sum_{a:x\in B_a}w_ae^{(u+1)|B_a|}\leq1$.
Every connected atom set has a spanning tree. Allowing repeated
species in the tree sum therefore gives
\begin{equation}\label{res:atomtree}
 \sum_{\gamma:a\in\gamma}z_\gamma^\#e^{u\ell_\gamma}
 \leq w_ae^{(u+1)|B_a|},\qquad
 \sum_{\gamma:x\in U_\gamma}z_\gamma^\#e^{u\ell_\gamma}
 \leq\sum_{a:x\in B_a}w_ae^{(u+1)|B_a|}.
\end{equation}
Apply the same recursion to object trees. Its supersolution is
$z_\gamma^\#e^{s\ell_\gamma+|U_\gamma|}$, since
\[
 \sum_{\gamma':U_{\gamma'}\cap U_\gamma\ne\varnothing}
 z_{\gamma'}^\#e^{s\ell_{\gamma'}+|U_{\gamma'}|}
 \leq |U_\gamma|\rho_s.
\]
Rooting at an object containing $x$ and using \eqref{res:treegraph}
yields
\begin{equation}\label{res:anchoredcluster}
 \sum_{n\geq1}\frac1{n!}\sum_C|\varphi^T(C)|
 \prod_jz_{\gamma_j}^\#e^{s\ell(C)}
 \mathbf1_{\{x\in U(C)\}}\leq\rho_s.
\end{equation}
If $n_i(C)$ counts bad-atom occurrences at $i$, rooting at such an
occurrence instead gives
\begin{equation}\label{res:badmarked}
 \sum_{n\geq1}\frac1{n!}\sum_C|\varphi^T(C)|n_i(C)
 \prod_jz_{\gamma_j}^\#e^{s\ell(C)}
 \leq q_\beta e^{(s+2)|B_i|}.
\end{equation}
Extra occurrence marks cost powers of $\ell(C)$, absorbed by the
reserved exponent. For a weak mark, let $\kappa_s$ be the weak-atom
part of $\rho_s$ and multiply weak majorants by
$\lambda_0=1+(4\kappa_s)^{-1}$. Their sum and the bad-atom sum remain
below $3/4$. The inequality
$k\leq(\lambda_0^k-1)/(\lambda_0-1)$ applied to the nonnegative
majorant series gives $C\kappa_s$ for one weak mark, and, using
\eqref{res:badmarked}, $Cq_\beta\kappa_s$ for both marks.
The case $\kappa_s=0$ is immediate.

Because $|U(C)|\leq\ell(C)$, these bounds prove
\eqref{res:Knorm}. A cluster joining two sites has occurrence size
at least a fixed multiple of their distance, so the reserved
exponent also gives spatial decay. For finitely many atoms the
series is the formal logarithm of the gas polynomial. The same
bounds hold in a complex activity disk larger than the unit disk;
analytic continuation identifies its exponential with that
polynomial. Positivity of the original fiber integral selects the
real logarithm. Absolute convergence then permits countable atom
families and marked derivatives.
\end{proof}

For fixed norm exponents, \eqref{res:atomnorm} follows from
$q_\beta\to0$ and sufficiently small $\kappa$ in the stronger norm
\eqref{res:strongnorm}.  For $J=0$, $K_D(0)=K_D^{\rm e}$ is field independent. The same
bad-atom formula defines its coefficients on the full lattice:
selected cells are empty in the background, and inserted rods
interact with all retained rods. This definition also allows
uncapped background counts and has the same uniform majorants.

Evaluate both expansions at $\eta=\sigma_D\xi$, retaining the original
exterior rods. The finite expansion consists of exactly those clusters
whose bad-atom sites lie in $D$. Its difference from the ordinary
full-lattice conditional sum consists of clusters with an exterior
bad-atom site and dependency union meeting $D$. A term in this
difference with nonzero oscillation at $i$ connects $i$ to $D^c$.
The marked tree bound with an exponential distance reserve therefore
gives, for the differences $\Delta_U$ of the conditional coefficients,
\begin{equation}\label{res:boundarycluster}
 \sum_{U\ni i}\operatorname{osc}_i\Delta_U
       \leq Cq_\beta e^{-m d(i,D^c)}.
\end{equation}

\subsection{Uniform response and reconstruction}

We use Theorem~\ref{mic:uniform-covariance} in the following form.  If a capped reference is perturbed by $Q$ with
$\beta\|Q\|_a$ sufficiently small, then
\begin{equation}\label{res:cappedcovariance}
 |\operatorname{Cov}_{\rm cap,Q}(F,G)|
 \leq C\sum_{i,j}e^{-m d(i,j)}
     \mathcal D_i(F)\mathcal D_j(G),\qquad
 \mathcal D_i(F)^2=
 \mathbb E\int_{C_i}r_X|D_XF|^2\,dX.
\end{equation}
Insertion rates satisfy $r_X\leq C\beta$.  If particle activity in
a measurable set is multiplied by $e^{-t}$, its restricted insertion
quadratic form retains that factor.  All these assertions are uniform
under hard accessibility masks and admissible exterior rods.
By \eqref{res:normalizertail} and \eqref{res:Knorm}, they apply to
the exact projected law whenever
\begin{equation}\label{res:smallness}
 \beta(q_\beta+\kappa)\leq c_0
\end{equation}
with the fixed constant $c_0$ chosen small enough.

\begin{lemma}[Reset-count covariance]\label{res:reset-covariance}
In the bare full reference,
\begin{equation}\label{res:resetcovariance}
 |\operatorname{Cov}(\widehat N_i,\widehat N_j)|
       \leq C\beta e^{-m d(i,j)}.
\end{equation}
\end{lemma}
\begin{proof}
The projected reset counts are the capped counts. Their insertion
gradients equal one in their own cells and zero elsewhere, so
\eqref{res:cappedcovariance} applies.
\end{proof}

\begin{lemma}[Response to deleting particle states]\label{res:deletion}
Let $A$ be any measurable set of one-particle states.  Multiply both
branches of \eqref{res:localweights} by $e^{-tN_A}$, keeping $h$,
the Gaussian attachment and its normalizers fixed.  If
$I_A=\{j:\nu_j(A\cap C_j)>0\}$, then
\begin{equation}\label{res:deletionbound}
 \left|\frac{d}{dt}\,\beta^{-1}\mathbb E_t\widehat N_i\right|
 \leq C(1+\beta q_\beta)e^{-t/2}
       \sum_{j\in I_A}e^{-m d(i,j)}.
\end{equation}
\end{lemma}

\begin{proof}
Under deletion, the same projection formula holds with correction
$K_t$.  Differentiation of a bad integral inserts $-N_A$.  Whenever
$N_A\geq1$,
$N_Ae^{-tN_A}\leq e^{-t/2}N_A$.
Lemma~\ref{res:tails} and the marked cluster estimate therefore give
\begin{equation}\label{res:deletionmark}
 \partial_tK_t=\sum_{j\in I_A}\sum_{U\ni j}D_{t,U}^{(j)},
 \qquad
 \sum_{U\ni j}e^{b|U|}(1+|U|)^p
     \|D_{t,U}^{(j)}\|_\infty
       \leq C_{b,p}\beta q_\beta e^{-t/2}.
\end{equation}
If $T_A$ counts retained particles
in $A$, differentiation of the normalized capped law yields
\[
 \frac{d}{dt}\mathbb E_t\widehat N_i
       =-\operatorname{Cov}_{\rm cap,K_t}
          (N_i,T_A+\partial_tK_t).
\]
The insertion norm of a deleted count in cell $j$ is at most
$C\sqrt\beta e^{-t/2}$.  For a bounded local $V_U$, its insertion
difference is at most $2\|V_U\|_\infty$, giving
\[
 |\operatorname{Cov}(N_i,V_U)|
 \leq C\beta\|V_U\|_\infty
             \sum_{x\in U}e^{-m d(i,x)}.
\]
Apply this to \eqref{res:deletionmark}; connected support and its
exponential reserve move $x$ to the anchor $j$.  The two terms cost
$C\beta e^{-t/2}$ and $C\beta^2q_\beta e^{-t/2}$, respectively,
which proves \eqref{res:deletionbound}.
\end{proof}

For an exterior configuration $\xi$, the interior one-particle
states forbidden by $\xi$ lie in a fixed boundary strip.  Deleting
those states and integrating \eqref{res:deletionbound} from zero to
infinity shows that
\begin{equation}\label{res:boundarymean}
 \left|\beta^{-1}\mathbb E_D^\xi\widehat N_i
       -\beta^{-1}\mathbb E_D^{\rm free}\widehat N_i\right|
       \leq Ce^{-m d(i,D^c)}.
\end{equation}
The same argument works for every bounded local observable of the
projected configuration: its insertion differences are bounded by
twice its supremum norm.  Its unscaled boundary error has a constant
finite for each fixed $\beta$.  Conditioning larger boxes on the
original exterior rods then gives convergence of all projected local
marginals in total variation.

To recover the actual configuration, condition on the projected
configuration $\sigma$.  A nonempty retained cell is fixed.  An
empty retained cell contains either vacuum or an entire bad
configuration, whose configuration space is
$\Omega_i^{\rm bad}=\bigsqcup_{n>K_i}C_i^n/\mathfrak S_n$ and
whose mark measure relative to the vacuum is
\begin{equation}\label{res:badmarkmeasure}
 d\nu_i^{\rm bad}(\omega_i)
   =\sum_{n>K_i}\frac{z_h^n}{n!}
       d\nu_i^{\otimes n}(\omega_i),\qquad
 \nu_i^{\rm bad}(\Omega_i^{\rm bad})\leq q_\beta.
\end{equation}
The hard constraints are imposed separately and only reduce this measure.  Under a weak interaction
$B$ of norm at most $\kappa$, the energy change relative to the vacuum in
one cell is at most $2\kappa$, so its nonvacuum conditional mass is
at most $q_\beta e^{2\kappa}$.

The following is Dobrushin's comparison argument~\cite[Theorem~3]{Dobrushin1970}
applied to the dilute conditional system.

\begin{lemma}[Fiber comparison]\label{res:fiber}
The conditional bad-cell system has a unique infinite-volume law.
If $c_{ij}$ is its total-variation influence matrix, then
\begin{equation}\label{res:influence}
 c_{ij}\leq Cq_\beta\left[
 \mathbf1_{\{i\sim_Gj\}}+
          \sum_{A\ni i,j}\|B_A\|_\infty\right],
 \qquad \sup_i\sum_je^{m d(i,j)}c_{ij}<\tfrac12.
\end{equation}
For bounded $F$ supported on a finite set $U$, changing the retained
configuration only at distance at least $r$ from $U$ changes its
conditional expectation by at most
$C|U|\|F\|_\infty e^{-mr}$, uniformly in exterior rods.
\end{lemma}

\begin{proof}
Changing a hard neighbor changes only the nonvacuum part of the
one-cell conditional law, of mass $O(q_\beta)$.  For a weak term
meeting both $i$ and $j$, the change in its energy relative to the vacuum
is at most $4\|B_A\|_\infty$.  The total-variation change is
therefore at most that energy oscillation times $Cq_\beta$.
This proves the first estimate.  Connected supports and the
exponential reserve in \eqref{res:strongnorm} give the second for
large $\beta$.

For the comparison statement, couple single-cell heat-bath updates
by maximal coupling.  If two specifications differ by conditional
discrepancies $b_i$, their time-averaged disagreement probabilities
obey $d_i\leq b_i+\sum_jc_{ij}d_j$ in the limit along a subsequence.
Such a subsequence exists since these are finitely many numbers in
$[0,1]$; the marginal laws remain their prescribed finite-volume
Gibbs laws throughout the coupling.  Iteration gives
$d\leq(I-C)^{-1}b$.  The weighted row bound in
\eqref{res:influence} makes the resolvent converge and gives
exponential decay of its entries.  A boundary change, or a change
of retained rods outside a large ball, supplies discrepancies only
near that change, together with the summable weak tails.  Thus local
marginals are uniformly Cauchy in total variation as boundaries
recede.  Their consistent limits give the asserted law and
quasilocality.
\end{proof}

\begin{lemma}[The bare reference law]\label{res:bare-reference}
For every field with $|\phi_i|\leq R$, the full bare reference has a unique
infinite-volume Gibbs law $\nu_\phi$. Finite-volume expectations of
bounded local observables converge uniformly over admissible exterior
rods, with an error decaying exponentially in the distance to the
boundary; the prefactor may depend on $\beta$ and the observable.
\end{lemma}
\begin{proof}
The deletion estimate gives convergence in total variation of the
projected local marginals. Approximate a conditional fiber expectation
by retaining its arguments in a finite neighborhood of the observable.
Lemma~\ref{res:fiber} bounds the approximation error exponentially;
projected convergence then constructs the actual local limits.
Consistency gives a law on configurations. Local finite-range
conditional distributions pass to the limit in total variation, so
this law satisfies the DLR equations. Conditioning any Gibbs law on
larger boxes and using the uniform boundary estimate proves uniqueness.
\end{proof}

\subsection{Covariance for actual rods}

\begin{proposition}\label{res:actualcovariance}
Let $B$ be a weak interaction satisfying \eqref{res:smallness},
including any coefficientwise partial sum or interpolation of such
an interaction.  For bounded actual-rod observables $F_U,G_V$
supported on finite sets $U,V$,
\begin{align}
 |\operatorname{Cov}_{{\rm full},B}(F_U,G_V)|
 &\leq C\beta\|F_U\|_\infty\|G_V\|_\infty
       \sum_{u\in U,v\in V}e^{-m d(u,v)},\label{res:actualcov}\\
 |\operatorname{Cov}_{{\rm full},B}(F_U,\widehat N_j)|
 &\leq C\beta\|F_U\|_\infty
       \sum_{u\in U}e^{-m d(u,j)}.\label{res:mixedcountcov}
\end{align}
The estimates hold in finite volumes uniformly in frozen admissible
exteriors and accessibility masks.
\end{proposition}

\begin{proof}
Let $K(s,t)$ be minus the logarithm of the positive fiber integral
with the added energy $sF_U+tG_V$. Differentiating the projected
partition function twice gives
\begin{equation}\label{res:twosource}
 \operatorname{Cov}_{\rm full,B}(F_U,G_V)
   =\operatorname{Cov}_{\rm cap,K}(K_s,K_t)
       -\mathbb E_{\rm cap,K}K_{st}.
\end{equation}
Conditional differentiation of the positive fiber integral yields
\[
 K_s=\mathbb E_{{\rm fiber},B}F_U,
 \qquad K_t=\mathbb E_{{\rm fiber},B}G_V,
 \qquad K_{st}=-\operatorname{Cov}_{{\rm fiber},B}(F_U,G_V).
\]
Perturbing the fiber by $tG_V$ changes a one-cell conditional law
only in $V$, by at most $Cq_\beta|t|\|G_V\|_\infty$.  Divide the
comparison estimate of Lemma~\ref{res:fiber} by $|t|$ and let
$t\to0$ to get
\begin{equation}\label{res:fibercov}
 |K_{st}|\leq Cq_\beta\|F_U\|_\infty\|G_V\|_\infty
        \sum_{u\in U,v\in V}e^{-m d(u,v)}.
\end{equation}
An allowed retained insertion in cell $x$ changes $F_U$ directly
only if $x\in U$.  It also changes bad-cell eligibility at $x$,
hard accessibility nearby, and weak terms meeting $x$.  All changes
of fiber conditional laws have size $O(q_\beta)$ with the same
exponential profile.  Hence
\begin{equation}\label{res:fibergradient}
 |D_XK_s|\leq C\|F_U\|_\infty
 \left[\mathbf1_{\{x\in U\}}+
        q_\beta\sum_{u\in U}e^{-m d(x,u)}\right].
\end{equation}
Applying \eqref{res:cappedcovariance} to the two gradients
in \eqref{res:fibergradient}, and convolving exponential kernels,
bounds the first term of \eqref{res:twosource} by the right side of
\eqref{res:actualcov}, with a smaller exponent.  Equation
\eqref{res:fibercov} bounds its second term.  If $G=\widehat N_j$,
that source is constant on every fiber: $K_t=N_j(\sigma)$ and
$K_{st}=0$.  Its capped insertion gradient is one in cell $j$,
which proves \eqref{res:mixedcountcov}.
\end{proof}

For an infinite weak background $B$, define
$B^{(R)}=\sum_{A\subset B_R(0)}B_A$ and tilt the bare reference
$\nu_\phi$ by $e^{-B^{(R)}}$.  These are bounded scalar tilts.
For $S>R$, interpolation of the additional terms, followed by
\eqref{res:actualcov}, gives for local $F_U$
\begin{equation}\label{res:partialbackground}
 \left|\mathbb E_{\nu_\phi^{B^{(S)}}}F_U
       -\mathbb E_{\nu_\phi^{B^{(R)}}}F_U\right|
 \leq C_{\beta,U}\kappa\|F_U\|_\infty
       e^{-c\operatorname{dist}(U,B_R(0)^c)}.
\end{equation}
Taking the supremum over $\|F_U\|_\infty\leq1$ constructs the
local limits in total variation.  Their projective consistency
defines $\nu_\phi^B$, independently of the exhaustion.  Terms whose
support meets a fixed region have uniformly summable tails, and
the bare specification has finite range.  Approximation by functions
of finitely many cells therefore passes the DLR identity to this
limit.  This construction also covers lexicographic partial backgrounds and
preserves \eqref{res:actualcov}--\eqref{res:mixedcountcov}.

\subsection{Weak free energies at a frozen boundary}

\begin{lemma}[Deletion response for actual observables]\label{res:actual-deletion}
Let $F_U$ be bounded and supported on a finite set $U$, and let
$N_B$ count particles in a finite set $B$ of marked cells. Multiply
their activities by $e^{-t}$. For $t\geq0$ and $0\leq s\leq1$,
under the weak background $sJ_D$,
\begin{equation}\label{res:actualdeletioncov}
 |\operatorname{Cov}_{{\rm full},t,sJ_D}(F_U,N_B)|
 \leq C\beta(1+\beta q_\beta)e^{-t/2}\|F_U\|_\infty
       \sum_{x\in U,y\in B}e^{-m d(x,y)}.
\end{equation}
\end{lemma}
\begin{proof}
Use a scalar source $uF_U$ and write
$S=\partial_uK_t(0)=\mathbb E_{\rm fiber}F_U$.  Let $T_B$ count
retained particles in $B$ and $N_B^{\rm bad}$ the reconstructed
particles there. Differentiating the fiber and capped
expectations gives the exact identity
\begin{equation}\label{res:actualdeletionidentity}
 \operatorname{Cov}_{\rm full}(F_U,N_B)
  =\operatorname{Cov}_{\rm cap}
       (S,T_B+\partial_tK_t)
       +\mathbb E_{\rm cap}
          \operatorname{Cov}_{\rm fiber}(F_U,N_B^{\rm bad}).
\end{equation}
The insertion norm of $T_B$
retains $e^{-t/2}$.  For $\partial_tK_t$, differentiate a bad
atom in the mixed expansion.  Weak-only objects have zero derivative,
and \eqref{res:tailmoments} supplies one count factor $C\beta$.
The cluster estimate with a bad-site mark gives \eqref{res:deletionmark} with the same
anchors in $B$, uniformly under the weak background.  Thus
\eqref{res:cappedcovariance} and \eqref{res:fibergradient} bound
the first term of \eqref{res:actualdeletionidentity} by the right
side of \eqref{res:actualdeletioncov}.  For the fiber term, the
derivative of a deleted cell's conditional law has total mass at
most $C\beta q_\beta e^{-t/2}$ by the marked tail.  The influence
resolvent then gives the same bound with coefficient
$C\beta q_\beta$.
\end{proof}

\begin{lemma}[Response of a logarithmic tilt]\label{res:tilt-response}
Let $\mu$ be a probability measure and $A,B$ real measurable functions,
with $A$ bounded.
Write $\mu_{t,s}$ for the normalized tilt of $\mu$ by
$e^{-tB-sA}$. If $\mathbb E_\mu e^{-(t\pm\eta)B}<\infty$
for some $\eta>0$, then
\begin{equation}\label{res:incrementresponse}
 \frac{d}{dt}\left[-\log\mathbb E_{\mu_{t,0}}e^{-A}\right]
 =-\int_0^1\Cov_{\mu_{t,s}}(A,B)\,ds.
\end{equation}
\end{lemma}
\begin{proof}
Differentiating the numerator and denominator gives
$\mathbb E_{\mu_{t,1}}B-\mathbb E_{\mu_{t,0}}B$.
Since $\partial_s\mathbb E_{\mu_{t,s}}B=-\Cov_{\mu_{t,s}}(A,B)$,
integration in $s$ proves the identity. The exponential moments
justify differentiation under the integrals.
\end{proof}

\begin{proposition}\label{res:frozenfreeenergy}
Let $J_D=\sum_{A\subset D}J_A$, and let $\mu_D^\xi$ be the bare
full reference at fixed fields and exterior rods $\xi$.  Extend the
fields outside $D$ in any bounded way and let $\nu_\phi$ be the
infinite bare reference.  Then
\begin{equation}\label{res:frozencomparison}
 \left|-\log\mathbb E_{\mu_D^\xi}e^{-J_D}
       +\log\mathbb E_{\nu_\phi}e^{-J_D}\right|
       \leq C\beta(1+\beta q_\beta)\kappa A(D),
\end{equation}
where $A(D)$ is its lattice boundary area.  The estimate is uniform
for finite cubical unions, including regions with holes.
\end{proposition}

\begin{proof}
Choose a strip $B\subset D$ along every boundary face, of fixed
physical width greater than the hard interaction range.  Then
$|B|\leq CA(D)$.  Delete all particle activities in $B$ by $e^{-t}$,
holding $h$, the normalizers and $J_D$ fixed, and put
$V_\xi(t)=-\log\mathbb E_{\mu_{D,t}^\xi}e^{-J_D}$.
Apply Lemma~\ref{res:tilt-response} with $A=J_D$ and $B=N_B$;
finite-volume count tails supply its exponential moments.
Summing \eqref{res:actualdeletioncov} over the terms of $J_D$ gives
$|V_\xi'(t)|\leq
C\beta(1+\beta q_\beta)\kappa e^{-t/2}|B|$.
At $t=\infty$ the strip is empty.  The remaining rods are beyond
hard interaction range of $\xi$, and $J_D$ has no arguments outside
$D$; consequently $V_\xi(\infty)$ is independent of $\xi$.
Integrating proves that all $V_\xi(0)$ lie in an interval of width
at most the right side of \eqref{res:frozencomparison}.
Finally the bare DLR identity gives
\[
 \mathbb E_{\nu_\phi}e^{-J_D}
 =\mathbb E_{\nu_\phi\vert D^c}
       \mathbb E_{\mu_D^\xi}e^{-J_D}.
\]
The negative logarithm of this average lies between the extreme
values of $V_\xi(0)$, completing the proof.
\end{proof}

\subsection{A common field interaction for all regional weak gases}

Let $J$ be a weak interaction covariant under a fixed sublattice of
translations, with regional restrictions
\begin{equation}\label{res:regionalJ}
 J_D=\sum_{U\subset D}J_U.
\end{equation}
If $J$ is obtained by expanding a gas and grouping each logarithmic
cluster by the union of its supports, restricting the original gas
to supports contained in $D$ gives exactly \eqref{res:regionalJ}.

Choose a translation-covariant lexicographic order and anchor each
support at its least site.  Define
\begin{equation}\label{res:increments}
 J_i=\sum_{\operatorname{anchor}(U)=i}J_U,
 \qquad J_{<i}=\sum_{\operatorname{anchor}(U)<i}J_U,
 \qquad
 v_i(\phi)=-\log\mathbb E_{\nu_\phi^{J_{<i}}}e^{-J_i}.
\end{equation}
The state for the interaction $J_{<i}$ is defined by the preceding
construction by finite tilts.  Since
$\|J_i\|_\infty\leq\kappa$, we have $|v_i|\leq\kappa$.
For a finite $D$, use $J_{i,D}$ and $J_{<i,D}$ with the additional
support restriction $U\subset D$, and define $v_{i,D}$ in the same
way.  Finite telescoping of normalized tilts gives
\begin{equation}\label{res:telescoping}
 V_D(\phi):=-\log\mathbb E_{\nu_\phi}e^{-J_D}
       =\sum_{i\in D}v_{i,D}(\phi).
\end{equation}

Put $t_\beta=C\beta\kappa$, with $C$ large enough for the estimates below.

\begin{lemma}[Comparison of regional increments]\label{res:regional-increments}
For $d_i=\dist(i,D^c)$,
\begin{equation}\label{res:incrementboundary}
 |v_{i,D}-v_i|\leq C(\kappa+\beta\kappa^2)e^{-cd_i}
 \leq t_\beta e^{-cd_i}.
\end{equation}
\end{lemma}
\begin{proof}

For background changes, use Lemma~\ref{res:tilt-response} with
$\mu=\nu_\phi^{B_0}$ and tilt observable $B=\Delta B$.
Apply it to finite sums first, then pass to summable interactions
by \eqref{res:actualcov}. If $d_i=\operatorname{dist}(i,D^c)$, the
strong norm and connected support imply
\[
 \|J_i-J_{i,D}\|_\infty\leq C\kappa e^{-cd_i},
 \qquad
 \sum_{U\ni x,\,U\not\subset D}e^{b|U|}\|J_U\|_\infty
       \leq C\kappa e^{-c\operatorname{dist}(x,D^c)}
 \quad(x\in D),
\]
with bound $C\kappa$ outside $D$.  The profile of the anchor source
$J_i$ is at most $C\kappa e^{-c d(x,i)}$.  Applying
\eqref{res:incrementresponse} and convolving the exponential
profiles yields
\eqref{res:incrementboundary}.
For every finite cubical union,
$\sum_{i\in D}e^{-cd_i}\leq CA(D)$: assign each site to a nearest
boundary face and sum the polynomial number of sites at each
distance from that face.  Equations \eqref{res:telescoping} and
\eqref{res:incrementboundary} therefore show
\begin{equation}\label{res:anchoredregional}
 |V_D-\sum_{i\in D}v_i|\leq Ct_\beta A(D).
\end{equation}

\end{proof}

\begin{lemma}[Local dependence of the increments]\label{res:local-increments}
If $\phi$ and $\phi'$ agree on $B_r(i)$, then
\begin{equation}\label{res:incrementlocality}
 |v_i(\phi)-v_i(\phi')|\leq Ct_\beta e^{-cr}.
\end{equation}
\end{lemma}
\begin{proof}
First hold the weak functions fixed. Changing $\phi_j$ by $t$
tilts the bare law by $e^{t\widehat N_j}$: this finite tilt satisfies
the new conditional specification and Lemma~\ref{res:bare-reference}
identifies its law. The identity passes through finite weak tilts
and their local limits. Lemma~\ref{res:tilt-response}, with
$A=J_i$ and $B=-\widehat N_j$, therefore gives
\[
 \partial_{\phi_j}v_i
 =\int_0^1\operatorname{Cov}_{\nu_\phi^{J_{<i}+tJ_i}}
                 (J_i,\widehat N_j)\,dt.
\]
The mixed covariance estimate bounds the sum over changed fields
by $Ct_\beta e^{-cr}$.  Then change the weak source functions to
their $\phi'$ values.  Terms that change have support meeting the
complement of $B_r(i)$.  Their contribution to $J_i$ is bounded
directly by $C\kappa e^{-cr}$, using
$|\log\mathbb E e^{-A}-\log\mathbb E e^{-A'}|\leq\|A-A'\|_\infty$.
Changes in the past background are bounded by
\eqref{res:incrementresponse} with the same profiles as above.
This proves \eqref{res:incrementlocality}.

\end{proof}

We now convert the local increments into an interaction by shell
telescoping, following Kozlov's reconstruction method~\cite[Sections~2 and~4]{Kozlov74}.

\begin{proposition}[Regional interaction]\label{res:regionalinteraction}
There are bounded measurable field potentials $\Psi^J_{i,r}$,
covariant under the same translations and supported on full
radius-$r$ balls $B_r(i)$, such that
\begin{equation}\label{res:radiusnorm}
 \sup_x\sum_{i,r:x\in B_r(i)}
       e^{c'r}(1+r)^p\|\Psi^J_{i,r}\|_\infty
       \leq C_{p,c'}t_\beta,
 \qquad t_\beta=C\beta\kappa,
\end{equation}
for fixed $p$ and sufficiently small $c'>0$.  Their ordinary
conditional Hamiltonian, the sum of all terms whose full support
meets $D$, satisfies
\begin{equation}\label{res:regionalfield}
 \left|V_D(\phi_Dr_{D^c})
       -H_D^J(\phi_D\mid r)\right|
       \leq Ct_\beta A(D).
\end{equation}
\end{proposition}

\begin{proof}
Fix a reference field $r_*$ for telescoping and put
\[
 v_i^{[r]}(\phi)=v_i(\phi_{B_r(i)}r_{*,B_r(i)^c}),\qquad
 \Psi^J_{i,0}=v_i^{[0]},\qquad
 \Psi^J_{i,r}=v_i^{[r]}-v_i^{[r-1]}\quad(r\geq1).
\]
Their exact sum is $v_i$, and \eqref{res:incrementlocality} bounds
the $r$th shell by $Ct_\beta e^{-cr}$.  Polynomial volume growth
proves \eqref{res:radiusnorm}.  In an ordinary conditional sum,
every term anchored in $D$ is included; terms anchored outside $D$
that meet $D$ have radius at least their distance from $D$.
Summing their exponential tails costs at most $Ct_\beta A(D)$.
Combine this with \eqref{res:anchoredregional} to obtain
\eqref{res:regionalfield}.
\end{proof}

The weak potentials above are contributions to the exponent.  If
the field action is written $\beta H$, their contribution to its
energy interaction is $\Psi^J/\beta$.


\section{The effective field interaction}
\label{sur:section}

The auxiliary field lives on the fine spatial lattice. A site carries
the angular-cell vector $\phi_i\in K=[-R,R]^{n_\omega}$, with normalized Lebesgue
reference measure $\lambda$; $n_\omega$ is the number of angular cells. Write $v_s$ for the physical volume of a
spatial cell and $v_a$ for its marked-cell volumes, so that
$\sum_a v_a=v_s$. All meshes, caps, and field bounds in this section
are fixed before $\beta=\varepsilon^{-1}$ tends to infinity. The positive
interaction $\Phi^0$ is expressed in these original coordinates by
$s_{i,a}=\sqrt{\omega_a}\phi_{i,a}$.

\subsection{From finite boxes to an exact field interaction}

We follow LMP in separating a leading variational functional from
a local remainder~\cite[Sections~3--5]{LMP99}.

Let $\mathcal R_{\beta,D}^{h}(\phi\mid\xi)$ be the rod-only fixed-field
partition function of the reset model. The exact reset gives
\begin{equation}\label{sur:convex-partition}
 \mathcal R_{\beta,D}^{h}(\phi\mid\xi)
 =\int w_{\beta,h,D}(\eta\mid\xi)
       \exp\!\left\{\sum_{i\in D,a}\phi_{i,a}\widehat N_{i,a}(\eta)\right\}
       d\eta,
 \qquad
 \widehat N_{i,a}=N_{i,a}{\bf1}_{\{N_{i,a}\le K_a\}},
\end{equation}
where $w$ is nonnegative and independent of $\phi$. Thus
$\log\mathcal R$ is convex. This formula also defines it on the enlarged
box $K^+=[-R-1,R+1]^{n_\omega}$. The enlargement is used for difference quotients;
the physical field law is still supported on $K$.

The finite-box variational limit, the microscopic locality estimate,
and the response estimate for the minimizing profile give the following facts.
First, in each fixed free box,
\begin{align}
 P_{\beta,D}^{h}(\phi)&:=\beta^{-1}\log\mathcal R_{\beta,D}^{h}(\phi)
       \longrightarrow P_{0,D}(\phi),\label{sur:fixed-box-limit}\\
 P_{0,D}(\phi)&=-\min_q\left\{
 \int_Dq\bigl(\log(q/\zeta)-1\bigr)
 +\tfrac12\langle q,T_Dq\rangle+\alpha\|P_Dq\|_2^2-\int_D\phi q
 \right\},\notag
\end{align}
with the marked-cell mass caps imposed on $q$. The limiting
minimizer is unique and continuous in $\phi$. The normalizer and
empty-cell estimates make the convergence uniform in the fixed source
interval. Second, for the full reset reference,
\begin{align}
 \left|\beta^{-1}{\mathbb E}_{D,\phi,\xi}\widehat N_{i,a}
      -\beta^{-1}{\mathbb E}_{\phi}\widehat N_{i,a}\right|
   &\le Ce^{-m_0d(i,D^c)},\label{sur:micro-locality}\\
 \beta^{-1}\left|\operatorname{Cov}_{\phi}
                  (\widehat N_{i,a},\widehat N_{j,b})\right|
   &\le Ce^{-m_0d(i,j)}.\notag
\end{align}
These bounds are uniform in the original admissible exterior rods.
Third, the density minimizers in free boxes converge exponentially locally
to the unique bounded all-space minimizer $q_\phi$.

The expectations in \eqref{sur:micro-locality} use the unique
whole-space bare reference of Lemma~\ref{res:bare-reference}.

\begin{lemma}[Uniform force convergence]\label{sur:force}
Put
\begin{align*}
 g_{\beta,h;i,a}(\phi)
 &=\frac{v_a}{2\alpha}(\phi_{i,a}-h)
           -\beta^{-1}{\mathbb E}_{\phi}\widehat N_{i,a},\\
 g_{0;i,a}(\phi)&=\frac{v_a}{2\alpha}\phi_{i,a}
                         -\int_{C_{i,a}}q_\phi.
\end{align*}
Uniformly over all fields in $K^{\mathbb Z^3}$ and the fixed source
interval,
\begin{equation}\label{sur:force-error}
 d_\beta:=\sup_{\phi,h,i,a}
 \left|g_{\beta,h;i,a}(\phi)-g_{0;i,a}(\phi)
                         +\frac{hv_a}{2\alpha}\right|\longrightarrow0.
\end{equation}
\end{lemma}
\begin{proof}
In a fixed box, we use the convergence theorem for gradients of convex
functions~\cite[Theorem~25.7]{Rockafellar}, with the following uniform estimate.
The capped counts make the functions in \eqref{sur:fixed-box-limit}
equi-Lipschitz on $K^+$. A finite net upgrades their pointwise
convergence to uniform convergence. If its error in a fixed box is
$\eta_{\beta,D}$ and the limiting gradient has modulus $\omega_D$,
convexity gives, for $0<t<1$ and $\phi\in K$,
\[
 |\partial_{i,a}P_{\beta,D}^{h}(\phi)
       -\partial_{i,a}P_{0,D}(\phi)|
 \le 2\eta_{\beta,D}/t+\omega_D(t).
\]
The two derivative bounds are the forward and backward difference
quotients. Let $\beta\to\infty$ first and then $t\downarrow0$.
Use a free box whose center is distance $\ell$ from its boundary.
The locality estimates for the microscopic gas and minimizing density bound the difference
between the global forces by the fixed-box derivative error plus
$2Ce^{-m\ell}$. Choose $\ell$ first and $\beta$ second. This proves \eqref{sur:force-error}.
\end{proof}

\begin{lemma}[Locality of the force error]\label{sur:force-locality}
The error force $E_{\beta,h}$ in this formula is quasilocal:
if $\phi=\psi$ on $B_r(i)$, then
\begin{equation}\label{sur:error-local}
 |E_{\beta,h;i}(\phi)-E_{\beta,h;i}(\psi)|\le Ce^{-mr}.
\end{equation}
\end{lemma}
\begin{proof}
Integrating the covariance bounds and variational response bounds along successive
field coordinates proves \eqref{sur:error-local} for finitely supported
changes. Summability and exhaustion extend the bound to arbitrary exterior changes.
\end{proof}

For finite-volume fields the force is the gradient of minus
$\beta^{-1}$ times the logarithm of the full fixed-field density.
Boundary convergence passes the vanishing of every finite-coordinate
loop integral to the global force. The variational force has the same
property. Consequently $E_{\beta,h}$ is closed and defines the
finite-change cocycle
\begin{equation}\label{sur:error-cocycle}
 \mathcal C_E(\phi;\psi)=\int_0^1\sum_{i\in A}
 E_i\bigl(\psi+t(\phi-\psi)\bigr)\cdot(\phi_i-\psi_i)\,dt,
 \qquad A=\{i:\phi_i\ne\psi_i\},
\end{equation}
whenever $A$ is finite.

Reconstructing potentials from quasilocal conditional energies goes
back to Sullivan~\cite{Sullivan73} and Kozlov~\cite{Kozlov74}.
The construction below adapts Kozlov's lexicographic regrouping
\cite[Section~4]{Kozlov74}; exponential locality gives the weighted norm.

\begin{proposition}[Small interaction representing the error]
\label{sur:error-interaction}
Fix a homogeneous well field $r_*$. There is a translation-covariant
interaction $\Delta_{\beta,h}$ on complete lattice balls whose cocycle
is \eqref{sur:error-cocycle} and for which, whenever $a<m$ and $p$ is fixed,
\begin{equation}\label{sur:radius-norm}
 \|\Delta_{\beta,h}\|_{a,p}:=
 \sup_x\sum_{i,r:\,x\in B_r(i)}
 e^{ar}(1+r)^p\|\Delta_{i,r}\|_\infty\longrightarrow0.
\end{equation}
Every term vanishes on $r_*$.
\end{proposition}
\begin{proof}
Order the lattice lexicographically and set
\[
 U_i(\phi)=\int_0^1 E_i\bigl(
   \phi_{j\prec i},r_*+t(\phi_i-r_*),r_{*,j\succ i}\bigr)
                                      \cdot(\phi_i-r_*)\,dt.
\]
This is well defined on the infinite past by quasilocality. Let
$U_i^{(r)}$ replace every field outside $B_r(i)$ by $r_*$ and put
$\Delta_{i,0}=U_i^{(0)}$ and
$\Delta_{i,r}=U_i^{(r)}-U_i^{(r-1)}$ for $r\ge1$. Declare the support
to be the entire ball. The force bound and locality separately give
\[
 \|\Delta_{i,r}\|_\infty\le C\min\{d_\beta,e^{-mr}\}.
\]
Finite lexicographic telescoping proves the cocycle identity for
fields equal to $r_*$ outside a finite box. Exhaustion then proves
\[
 \mathcal C_E(\phi;\psi)=
 \sum_{i,r}\bigl[\Delta_{i,r}(\phi)-\Delta_{i,r}(\psi)\bigr]
\]
for arbitrary finite changes, since centers far from the changed set
have an exponentially summable contribution. Finally,
\[
 \|\Delta_{\beta,h}\|_{a,p}
 \le C\sum_{r\ge0}(1+r)^{p+3}e^{ar}
                      \min\{d_\beta,e^{-mr}\}\longrightarrow0.
\]
Dominated convergence applies because $a<m$. Splitting the sum at
$m^{-1}\log(1/d_\beta)$ also gives the bound
$Cd_\beta^{1-a/m}(1+|\log d_\beta|)^{p+4}$.
\end{proof}

Retain the positive leading interaction $\Phi^0$ and define
\begin{equation}\label{sur:canonical-interaction}
 \Psi_{\beta,h}=\Phi^0+\Delta_{\beta,h}
 -\sum_i\frac{h}{2\alpha}\sum_av_a(\phi_{i,a}-r_{*,a}).
\end{equation}
Its force is exactly $g_{\beta,h}$. The complete signed perturbation
$\mathcal E_{\beta,h}=\Psi_{\beta,h}-\Phi^0$ satisfies
\begin{equation}\label{sur:source-norm}
 \|\mathcal E_{\beta,h}\|_{a,p}\le C|h|+\delta_{{\rm micro},\beta},
 \qquad \delta_{{\rm micro},\beta}\longrightarrow0.
\end{equation}

Let $F_D$ denote the full fixed-field density, including its Gaussian
factor relative to $\lambda$. For a coarse union $D$, let $A(D)$ be
its number of boundary faces multiplied by $L^2$.

\begin{lemma}[Pressure at a fixed homogeneous field]\label{sur:fixed-pressure}
For every homogeneous field $r\in K$, the limit
\[
 b_r=\lim_D(v_s\#D)^{-1}\log F_D(r\mid\mathrm{free})
\]
exists along aligned boxes. Changing the admissible exterior rods
changes the logarithm by at most $C_\beta A(D)$, uniformly in $r$.
\end{lemma}
\begin{proof}
We use the argument with separated boxes for the thermodynamic limit
\cite[Section~2]{Ruelle63}.
Divide each cell weight by its empty weight and remove the Gaussian
prefactor. The normalized empty weight is one. Dropping exclusions,
\eqref{mod:cellweight} bounds the one-cell sum by
$2\exp[C\beta v_a]$, uniformly in $r$ and $|h|\le H$.
Emptying a complete-cell strip therefore costs at most a constant
per cell at fixed $\beta$. A strip wider than the rod range removes
all exterior exclusions. Dropping exclusions between boxes gives the
product upper bound; separating boxes by empty strips gives the
product lower bound. Tiling a large aligned box and letting the tile
side increase proves existence. The removed cell prefactors have an
exact volume density and can then be restored.
\end{proof}

For a finite region $D$, let $H_D^\Psi$ be the sum of terms whose
declared supports meet $D$, evaluated with the specified exterior.
Define its homogeneous energy density by equal allocation to support
sites: $e(r)=v_s^{-1}\sum_{A\ni0}\Psi_A(r)/|A|$.
\begin{lemma}[Common pressure normalization]\label{sur:normalization}
With $b_*=b_{r_*}$,
\begin{equation}\label{sur:common-scalar}
 b_r+\beta e(r)=b_*=:b_0.
\end{equation}
\end{lemma}
\begin{proof}
Integrate the difference between the finite-box and global forces
along $r_*+t(r-r_*)$ on $D$. The boundary decay in
\eqref{sur:micro-locality} sums to $CA(D)$, so
\[
 \log\frac{F_D(r\mid\mathrm{free})}{F_D(r_*\mid\mathrm{free})}
 =-\beta\bigl[H_D^\Psi(r_D\mid r_*)-H_D^\Psi(r_*\mid r_*)\bigr]
    +O_\beta(A(D)).
\]
The first spatial moment of the interaction is summable, so the
energy difference equals $(e(r)-e(r_*))v_s\#D+O(A(D))$.
Every interaction term vanishes at $r_*$, hence $e(r_*)=0$.
Divide by volume and apply Lemma~\ref{sur:fixed-pressure}.
\end{proof}

Define
\begin{equation}\label{sur:root-boundary}
 \sigma_D(r)=H_D^\Psi(r\mid r)-e(r)v_s\#D.
\end{equation}
We retain $\sigma_D(r)$ when changing the root.

For frozen admissible rods $\xi$, subtract the canonical action from
$-\beta^{-1}\log F_D(\phi\mid\xi)$ and subtract its value at the root.
The resulting boundary action satisfies
\[
 |\partial_{i,a}B_D^\xi(\phi)|\le Ce^{-md(i,D^c)}.
\]
If $O$ contains a boundary band of width $cL$, summing this tail on a
coarse cubical union gives
\begin{equation}\label{sur:outer-boundary}
 \sup_\phi\left|B_D^\xi(\phi)
      -B_D^\xi(\phi_O r_{D\setminus O})\right|
 \le Ce^{-m'L}A(D).
\end{equation}
The retained function of the outer fields is part of the boundary weight.

\section{Surface estimates for restricted fields}\label{sur:surface-section}

\subsection{Local interpolation}

Write $Q(\phi)=q_\phi$ for the minimizing profile and use the trial
action $\ell_i$ of \eqref{fnd:charges}. In the original angular field
coordinates, set $|z|_\omega^2=\sum_a\omega_a z_a^2$ and
$(\Pi_0 Q_i)_a=2\alpha(v_s\omega_a)^{-1}\int_{B_i\times U_a}Q_i$.
Its pinning term is $(v_s/4\alpha)|\phi_i-\Pi_0 Q_i|_\omega^2$.
We use the quadratic estimate \eqref{fnd:localquadratic} and the global
comparison \eqref{fnd:comparison}; write $r_0$ for their fixed local
profile range. Copying complete cells and inserting wells preserve
all marked-cell caps.

Fix a component $\mathcal W_q$, put $\mathcal M_q=\Pi_0\mathcal W_q$,
and choose a root $f_q$ with field $r_q=\Pi_0 f_q$. All field
neighborhoods use the norm $|\cdot|_\omega$.
Lemma~\ref{fnd:nearestalignment} gives measurable nearest wells $f_i$
and a bounded concave modulus $\omega_0$ controlling squared local
profile errors and neighboring-well differences. Set
\[
 \omega(t)=C\{\omega_0(t)+\sqrt{\omega_0(t)}\}.
\]
This bounded concave modulus controls both the squared errors and the
linear neighboring-well differences used in randomized rounding below.
The results \ref{fnd:components}, \ref{fnd:paths}, and
\ref{fnd:projected-components} give finitely many separated projected
components and rectifiable paths within each component.

Let $E$ be a signed perturbation with exponentially summable norm at
most $\delta$, in energy units before multiplication by $\beta$.
Coarse blocks have side $L$. Assume that the declared supports of $E$
are connected and that $E$, its supports,
and the reference measure are covariant under translations by
$L\mathbb Z^3$. This holds for the canonical perturbation and for the regional
interactions of Proposition~\ref{res:regionalinteraction}.
The phase restriction allows at most
$\theta L^3$ fine spins outside a fixed neighborhood of
$\mathcal M_q$ in each block. A radius-$s$ ball about any
projected well has $\lambda$-mass at least $cs^{n_\omega}$ and lies inside the
field box for sufficiently small $s$.

For a finite vector $X$ with probability product reference $\lambda_X$, write
$\mathcal H(X\mid W)=-\mathbb E D(\mathcal L(X\mid W)\Vert\lambda_X)$;
omit $W$ for unconditional entropy.

\begin{lemma}[Entropy after replacing coordinates]\label{sur:replacement-entropy}
Let $X$ be a finite vector determined by source randomness $S$, and let
$U$ retain a deterministic subset of its coordinates, possibly permuted
in a way that preserves their reference measures. If fresh coordinates
$V$ satisfy $\mathcal H(V\mid S,W)\ge-a$, then
\[
 \mathcal H(U,V\mid W)\ge\mathcal H(X\mid W)-a.
\]
\end{lemma}
\begin{proof}
The chain rule and $\mathcal H(\cdot\mid\cdot)\le0$ give
$\mathcal H(U\mid W)\ge\mathcal H(X\mid W)$. Since $U$ is determined
by $S$, conditioning gives
$\mathcal H(V\mid U,W)\ge\mathcal H(V\mid S,W)$. Adding these two
bounds proves the claim; these are the entropy inequalities of
\cite[Proposition~1]{RR67}.
\end{proof}

For $m$ fresh coordinates sampled independently, conditional on $S,W$,
from $\lambda$ restricted and normalized to radius-$\beta^{-2}$ balls
about projected wells, the ball-mass bound
above gives $a=Cm\log\beta$, for $\beta\ge2$.

The elementary cubical bound used repeatedly is
\begin{equation}\label{sur:band-volume}
 \#\{i:d_\infty(i,\partial D)\le t\}
 \le C(t+1)A(D),\qquad 1\le t\le cL.
\end{equation}
For one coarse face the relevant prism, edge, and corner volumes
are bounded by $C(tL^2+t^2L+t^3)\le CtL^2$; summing over faces proves
the assertion for arbitrary coarse unions.

\begin{lemma}[Buffer energy]\label{sur:buffer}
Let $A>0$ be an area bound in fine-lattice units. Let $B$ be a buffer
of width comparable to $L$ in a restricted Gibbs field with arbitrary
fixed boundary fields. Assume that $B$ and its radius-$r_0$
neighborhood lie in a union $J$ of freely integrated
complete coarse blocks, with $\#J\le CLA$ and at most $CA$ fine sites
in its fixed-width boundary strip. Then
\begin{equation}\label{sur:buffer-bound}
 {\mathbb E}\sum_{i\in B}e_i
 \le CA+CLA\left(\delta+\frac{\log\beta}{\beta}+\beta^{-4}\right).
\end{equation}
Consequently a fine moving cut through a distance interval of length
$cL$ samples mean patch energy at most
\begin{equation}\label{sur:eta}
 \eta_L=C\left(L^{-1}+\delta+\frac{\log\beta}{\beta}+\beta^{-4}\right).
\end{equation}
\end{lemma}
\begin{proof}
Replace the fields in $J$ by independent uniform fields in balls of
radius $\beta^{-2}$ about $r_q$. Use a profile trial equal to
$f_q$ inside $J$ and to the old minimizing profile outside. Only a
fixed-width interface has a bounded new density-action cost, of order $A$.
Keep the negative old terms centered in $B$ in
\eqref{fnd:comparison} and drop the other negative terms.
The weak change costs at most $C\delta LA$. Lemma~\ref{sur:replacement-entropy}
bounds the conditional entropy loss by $CLA\log\beta$. Conditional Gibbs
optimality yields \eqref{sur:buffer-bound}. Whole-block replacement
preserves the phase restriction. A fixed-width patch belongs to
only a bounded number of the $cL$ cut positions. Averaging and
\eqref{sur:band-volume} give \eqref{sur:eta}.
\end{proof}

\begin{lemma}[Interpolation through a well component]
\label{sur:interpolation}
For source laws satisfying Lemma~\ref{sur:buffer}, choose a
$\nu$-net in $\mathcal W_q$ of cardinality $N_\nu$ and paths
to $f_q$ of length at most $D_\nu$. A moving interface can be joined
to a root plateau through $R_1$ fine layers with expected energy
cost at most $A\kappa_E$, where $\kappa_E\le\kappa$ and
\begin{equation}\label{sur:kappa}
 \begin{split}
 \kappa\le C\bigg[&\theta+\nu^2+\frac{D_\nu^2}{R_1}
 +\left(1+\frac{R_1N_\nu}{\nu}\right)\omega(\eta_L)\\
 &+L\delta+\frac{L\log\beta}{\beta}+L\beta^{-4}\bigg]
 \end{split}
\end{equation}
also includes its entropy cost divided by $\beta A$.
The coefficient of $\theta$ is independent of the subsequent net,
paths, and $R_1$.
\end{lemma}
\begin{proof}
Fix a separator made of coarse-block faces. Assign the blocks on each
side to the source that will be retained there. On either side choose
a cut at fine distance between $L/4$ and $L/3$ from the separator,
and put its interpolation strip on that same side, between the cut
and the separator. Require $R_1+r_0<L/6$. Copy that source beyond
the strip and put the root profile between the strips. The root
plateau therefore crosses the fixed separator, and every coarse
block retains coordinates from only its assigned source. The
hypothesis of Lemma~\ref{sur:buffer} is imposed on the available
bands on both sides. For a single source the other side is already
at the root.

First map a nearest well $f_i$ outside $\mathcal W_q$ to $f_q$, leaving
wells in $\mathcal W_q$ fixed. This map is Lipschitz on $\mathcal W$:
different components have a positive separation and the set has finite
diameter. Thus the transformed wells $h_i$ retain the neighboring
rigidity bound.

Let the net be $v_1,\ldots,v_{N_\nu}$. For the assignment step we use
a fixed-order variant of the random-radius partitions of
C\u{a}linescu, Karloff, and Rabani~\cite{CKR}; see also the construction
in \cite[Lemma~3.1]{MN07}. Choose $u$ uniformly in
$[\nu,2\nu]$ and assign $h$ to the first $v_j$ with $\|h-v_j\|\le u$.
For two wells,
\[
 {\mathbb P}_u(n(h)\ne n(h'))
       \le\frac{CN_\nu}{\nu}\|h-h'\|.
\]
Indeed, a different first successful index requires $u$ to lie between
two distances to at least one net point; the total interval length is
at most $N_\nu\|h-h'\|$. Parametrize each chosen path on $[0,1]$ with
Lipschitz constant at most $D_\nu$. Across the strip use
$Q_i'=\gamma_{n(h_i)}(s_i)$, where $s_i$ increases from zero to one
in $R_1$ layers. Equal assignments have neighboring discrepancy at
most $CD_\nu/R_1$. Unequal assignments cost a bounded amount and
have the preceding probability. The quadratic bound
\eqref{fnd:localquadratic}, patch rigidity, and concavity of $\omega$
give
\[
 CA\left[\frac{D_\nu^2}{R_1}
       +\frac{R_1N_\nu}{\nu}\omega(\eta_L)\right].
\]
The initial interface costs $CA[\nu^2+\omega(\eta_L)+\theta]$.
Write $E_i^{\rm patch}$ for the sum of charges on the fixed enlarged
patch around $i$. Projected component separation gives
\[
 {\bf1}_{\{f_i\notin\mathcal W_q\}}
 \le {\bf1}_{\{\phi_i\notin\mathcal N(\mathcal M_q)\}}
                  +C\omega(E_i^{\rm patch}).
\]
Only the fixed-width initial interface pays this cost. Each minority
site is charged $O(L^{-1})$ after averaging over cuts, and majority
in a buffer of volume $O(LA)$ gives $C\theta A$. Inside the strip all
$h_i$ already belong to $\mathcal W_q$, so no factor $R_1\theta$ arises.

Conditional on the source fields, sample the fresh spins independently
from $\lambda$ restricted and normalized to radius-$\beta^{-2}$ balls
about their trial projected wells. There are $O(LA)$ such spins. Their pinning
cost is $O(LA\beta^{-4})$, their conditional entropy is at least
$-CLA\log\beta$, and the weak cost is $O(\delta LA)$.
The fixed source assignment makes each output block retain a subset
of one old block. All fresh spins lie in the phase neighborhood,
so the majority constraint is preserved. Extra erased cells lie on
the root plateau and have only the pinning cost just estimated.
Average over cuts and $u$ at the level of the expected energy, then
choose deterministic parameters attaining the bound. The retained
coordinate sets are consequently deterministic for the entropy
arguments below.
\end{proof}

One first fixes $\theta$, then $\nu$, the finite paths and $R_1$, then
$L\gg R_1+r_0$, and finally small $\delta$ and large $\beta$.
The right side of \eqref{sur:kappa} can be made arbitrarily small.

\subsection{Rooted restricted partition functions}

Let $Z_D^{q,r_q}$ be the restricted field partition on a coarse union
$D$ with root field $r_q$ outside. Its leading rooted energy is
\[
 H_D^0(\phi_D\mid r_q)=\sum_i e_i(\phi_Dr_{q,D^c})\ge0.
\]
The weak part is the ordinary conditional Hamiltonian
$H_D^E=\sum_{A:A\cap D\ne\varnothing}E_A$, including its root value.

\begin{lemma}[Partition bound near a homogeneous well]\label{sur:well-partition}
For sufficiently large $\beta$, uniformly over finite coarse unions $D$,
\[
 -C(\log\beta+\beta\delta)\#D
 \le\log Z_D^{q,r_q}\le C\beta\delta\#D.
\]
The same bounds hold after omitting nonnegative leading terms or
weak interaction terms.
\end{lemma}
\begin{proof}
The leading energy is nonnegative, the weak energy has absolute value
at most $C\delta\#D$, and the reference is a probability measure;
these give the upper bound. For the lower bound, integrate each field
in a radius-$\beta^{-2}$ ball about $r_q$. The resulting product set
satisfies the phase restriction and has reference mass at least
$\exp[-C\#D\log\beta]$. Its leading energy is at most
$C\beta^{-4}\#D$ by \eqref{fnd:roottrial}. Omitting nonnegative
leading terms preserves both energy bounds; omitting weak terms cannot
increase their absolute norm.
\end{proof}

For a signed interaction, allocate every term $E_A$ equally among its
$|A|$ declared sites. If $B_L$ is one coarse block and $\mu$ is
coarse-translation invariant, define
\[
 e_0=\frac1{L^3}\sum_{i\in B_L}\mathbb E_\mu e_i,
 \qquad
 e_E=\frac1{L^3}\sum_{i\in B_L}\sum_{A\ni i}
                     \frac{\mathbb E_\mu E_A}{|A|}.
\]
Absolute summability gives $|e_E|\le C\delta$.

\begin{lemma}[Restricted pressure]\label{sur:restricted-pressure}
Along complete-block cubes $D$, the limit
\[
 \pi_q=\lim_D\frac{\log Z_D^{q,r_q}}{\#D}
\]
exists and is independent of the bounded exterior field.
\end{lemma}
\begin{proof}
We again use the proof of the thermodynamic limit by division into
boxes~\cite[Section~2]{Ruelle63}.
Normalize the reference on the allowed configurations of each block.
For any coarse union $D$, the norm of interactions meeting both $D$
and $D^c$ is
\[
 b(D)=\sum_{A:A\cap D\ne\varnothing,\,A\cap D^c\ne\varnothing}
                 \|\Phi_A^0+E_A\|_\infty=O(A(D)).
\]
Deleting those terms changes $\log Z_D$ by at most $\beta b(D)$.
Tiling a large cube by fixed smaller cubes gives matching upper and
lower limits as the smaller side tends to infinity. The same bound
removes the exterior field. Restore the block-reference normalization.
\end{proof}

\begin{lemma}[Restricted Gibbs state]\label{sur:restricted-state}
There is a restricted Gibbs state $\mu$ invariant under translations
by coarse blocks.
\end{lemma}
\begin{proof}
This is the finite-volume compactness construction of Gibbs states,
adapted to the allowed-block reference; compare
\cite[Appendix~A]{LR69}.
Relative to the normalized allowed-block reference, summability
bounds each fixed marginal density in $L^\infty$, uniformly in the
containing volume. Diagonal weak-* compactness passes bounded
measurable cylinder tests. Uniform interaction tails pass the DLR
kernels. Averaging translates and repeating the limit gives an
invariant Gibbs state.
\end{proof}

\begin{lemma}[Entropy bounds]\label{sur:entropy-bounds}
The state $\mu$ has an entropy density $s$. For every union $C$ of
complete blocks,
\begin{equation}\label{sur:entropy-chain}
 \mathcal H_\mu(X_C)\ge s\#C,
 \qquad
 \mathcal H_\mu(X_C\mid X_{C^c})\le s\#C.
\end{equation}
\end{lemma}
\begin{proof}
We use the entropy inequalities and lexicographic construction of
Robinson and Ruelle~\cite[Propositions~1--2]{RR67}.
In a lexicographic block order, conditioning on larger finite pasts
converges to conditioning on the infinite past. The chain rule on
large cubes gives its entropy per site as $s$. The conditioning sets
for a marginal of $C$ are contained in that past; those for the
conditional law given $C^c$ contain it.
\end{proof}

The next identity is the restricted-block form of the Gibbs
variational principle~\cite[Theorem~3]{Ruelle67}; the passage from
Gibbs conditional laws to this identity follows the boundary comparison
in \cite[Appendix~A]{LR69}.

\begin{lemma}[Pressure identity]\label{sur:variational}
The entropy and mean energies of $\mu$ satisfy
\begin{equation}\label{sur:variational-pressure}
 \pi_q=s-\beta(e_0+e_E).
\end{equation}
\end{lemma}
\begin{proof}
For a complete-block box $D$, conditional Gibbs densities with two
exterior fields have ratios between $e^{-4\beta b(D)}$ and
$e^{4\beta b(D)}$. Comparing with their mixture bounds the mutual
information between $X_D$ and $X_{D^c}$ by $4\beta b(D)$, so their
conditional and marginal entropy densities agree. The identity
\[
 \mathcal H_\mu(X_D\mid X_{D^c})
 =\beta\,\mathbb E_\mu H_D(X_D\mid X_{D^c})
        +\mathbb E_\mu\log Z_D(X_{D^c})
\]
then gives \eqref{sur:variational-pressure} after division by $\#D$.
The equal allocation of interaction terms changes the mean energy
only by the boundary contribution $O(b(D))$.
\end{proof}

Lemma~\ref{sur:well-partition} gives the pressure bounds below;
\eqref{sur:variational-pressure}, $s\le0$, and $|e_E|\le C\delta$
then bound the mean leading energy:
\begin{equation}\label{sur:small-pressure}
 -C(\log\beta+\beta\delta)\le\pi_q\le C\beta\delta,
 \qquad
 0\le e_0\le C\left(\delta+\frac{\log\beta}{\beta}\right).
\end{equation}
For a finite fine-site set $J$ containing a complete-block union $C$,
nonpositivity and the chain rule also give
\begin{equation}\label{sur:irregular-entropy}
 \mathcal H_\mu(X_J\mid X_{J^c})
 =\mathcal H_\mu(X_{J\setminus C}\mid X_{J^c})
    +\mathcal H_\mu(X_C\mid X_{C^c})\le s\#C.
\end{equation}
Gibbs optimality on $J$ follows by disintegrating the kernel on an
enclosing complete-block region, keeping each partial-block constraint.

The next bound has the role of LMP's surface error in comparing
restricted partition functions~\cite[Statement~3.5 and Section~6]{LMP99}.
Here the boundary modification interpolates within an entire connected
component of minimizing profiles.

\begin{proposition}[Rooted surface bound]\label{sur:rooted}
Under the preceding hypotheses and scale choices,
\begin{equation}\label{sur:rooted-estimate}
 \left|\log Z_D^{q,r_q}-\pi_q\#D\right|
       \le\beta\kappa A(D)
\end{equation}
uniformly over finite unions of complete coarse blocks, with $\kappa$
of the form \eqref{sur:kappa}.
\end{proposition}
\begin{proof}
For the lower bound take $X\sim\mu$. Copy its deep part into $D$,
interpolate to a root plateau in the boundary band, and use root
fields outside $D$. Apply \eqref{fnd:comparison} relative to
the global root, using a trial density profile equal to $f_q$ outside
$D$. The plateau makes all outside-centered trial costs vanish.
Copied local terms are a subset of nonnegative old terms centered
in $D$, so
\[
 {\mathbb E}H_D^0(Z_D\mid r_q)\le e_0\#D+\kappa_EA(D).
\]
The equal site allocation of signed weak terms gives
$e_E\#D+C\delta LA(D)$: terms wholly in the copied core agree,
and terms meeting its boundary band have total norm $O(\delta LA(D))$.
Apply Lemma~\ref{sur:replacement-entropy} to $X_D$, using the complete
field $X$ as source randomness, and then \eqref{sur:entropy-chain}:
\[
 \mathcal H(Z_D)\ge s\#D-CLA(D)\log\beta.
\]
The finite-volume variational principle proves the lower estimate.

For the upper bound let $Y$ have the rooted restricted Gibbs law on
$D$, independently of $X\sim\mu$. Insert its deep core into $X$,
using the interpolation above. Let $J$ be the actual finite set of
changed fine sites and choose a complete-block core $C\subset J$
such that every range-$r_0$ stencil centered in $C$ lies among the
changed cells and
\[
 |\#D-\#C|\le CLA(D).
\]
Every cell of $J$ is copied from $Y$ or is fresh. A partially cut
block retains $X$ only outside $J$; the whole-block source assignment
prevents it from also retaining $Y$. Apply
\eqref{fnd:comparison} relative to $X$, keep the negative old
terms centered in $C$, and drop the other negative terms. The copied
$Y$ costs are bounded by its entire nonnegative rooted energy. Thus
\begin{align*}
 {\mathbb E}H^0(Z;X)&\le {\mathbb E}H_D^0(Y_D\mid r_q)
                                  -e_0\#C+\kappa_EA(D),\\
 {\mathbb E}H^E(Z;X)&\le {\mathbb E}H_D^E(Y_D\mid r_q)
                                  -e_E\#C+C\delta LA(D).
\end{align*}
Apply Lemma~\ref{sur:replacement-entropy} to $Y_D$, with source
randomness $(X,Y)$ and conditioning $X_{J^c}$. Independence gives
\[
 \mathcal H(Z_J\mid X_{J^c})
 \ge\mathcal H(Y_D)-CLA(D)\log\beta.
\]
Conditional Gibbs
optimality of $\mu$ and \eqref{sur:irregular-entropy} now imply
\[
 \log Z_D^{q,r_q}
 =\mathcal H(Y_D)-\beta{\mathbb E}H_D(Y_D\mid r_q)
 \le\pi_q\#C+\beta\kappa A(D).
\]
Replacing $\#C$ by $\#D$ costs at most
$C\beta LA(D)(\delta+\log\beta/\beta)$ by
\eqref{sur:small-pressure}, already included in \eqref{sur:kappa}.
If an eroded core is empty, the affected region has at most
$CLA(D)$ sites; Lemma~\ref{sur:well-partition} gives both bounds directly.
\end{proof}

For the chosen perturbation $E$, the pressure in physical units is
\begin{equation}\label{sur:physical-branch}
 p_q(E)=b_0+\pi_q/v_s.
\end{equation}
The root boundary normalization is \eqref{sur:root-boundary}.

For a regional weak gas, add the potentials of
Proposition~\ref{res:regionalinteraction} divided by $\beta$ to the
weak energy interaction. Their additional norm is at most
$Ct_\beta/\beta$. Equation~\eqref{res:regionalfield} compares the
actual regional correction with its ordinary conditional Hamiltonian
at a logarithmic cost $Ct_\beta A(D)$. Applying
Proposition~\ref{sur:rooted} with this enlarged weak interaction
therefore gives the same rooted surface estimate for the actual
auxiliary field integral, on every coarse union, including punctured
regions. The homogeneous value of the correction is included in
its branch pressure.

\subsection{Collar kernels and arbitrary caps}

Let $S$ be a finite union of coarse blocks. Partition its complement
into a bounded inner region and an outer region, called the two caps,
so that every connected set joining them meets $S$. Prescribe fields
$a,b$ on these caps and impose only the phase-$q$ majority constraint
in $S$. Let $T$ be a union of complete coarse blocks containing the
inner cap and disjoint from the outer cap.

If $P$ is the set of coarse cubes incident to faces of $\partial T$,
assume its three-neighborhood lies in $S$. Its two-neighborhood is
then a complete-block reset region for Lemma~\ref{sur:buffer}.
Writing $A=L^2$ times the number of separator faces, the intervening
bands have volume $O(LA)$. Cuts at distance between $L/4$ and $L/3$
have area $O(A)$, and a site belongs to an $R_1$-thick moving strip
for at most $CR_1$ cut positions. These bounds follow by counting
prisms over separator faces. Assume that all declared interaction
supports are connected.

\begin{proposition}[Collar cross ratio]\label{sur:collar}
Let $K(a,b)>0$ be the restricted collar kernel with arbitrary bounded
inner and outer cap fields. Under this geometry,
\begin{equation}\label{sur:cross-ratio}
 \left|\log K(a,b)+\log K(a',b')
       -\log K(a,b')-\log K(a',b)\right|
       \le\beta A\kappa,
\end{equation}
with $\kappa$ as in \eqref{sur:kappa}, uniformly in the caps and in
the collar's size and topology.
\end{proposition}
\begin{proof}
Take independent source fields $X,Y$ from the kernels with caps
$(a,b)$ and $(a',b')$. Swap the complete bounded side $T$. On each
side of its boundary place a moving interpolation strip, with a
root plateau between them wider than $r_0$. Outside $T$ an output
retains its original source; inside it retains the opposite source.
Fresh fields occupy the intervening erased band. The source
assignment is made by whole coarse blocks, so each output block
retains at most one source and satisfies the majority condition.

For the first output use a global trial density equal to $Q(X)$
outside the finite modified set, to $Q(Y)$ in its copied interior,
and to the interpolation profiles in between. Reverse the roles
for the second output. The trials preserve all cell caps.
Add the two inequalities \eqref{fnd:comparison}. Each trial
agrees with its own source profile outside a finite set, and every
stencil copied entirely from a source cancels across the pair.
The plateau has only the small field-pinning cost. Dropping
the remaining negative old terms and applying
Lemmas~\ref{sur:buffer}--\ref{sur:interpolation} bounds the leading
change by $A\kappa_E$.

The weak changes are also added before they are bounded. Terms
wholly in a copied side cancel; every remaining connected support
meets the erased band. Their total norm is at most $C\delta LA$,
independent of the volume of the swapped inner cap. Choose one
deterministic pair of cuts and one net radius after averaging this
expected energy estimate.

The retained collar coordinates are a deterministic permutation of
a marginal of $(X_S,Y_S)$. Lemma~\ref{sur:replacement-entropy}, with
the complete source pair as randomness, bounds the joint output
entropy below by $\mathcal H(X_S)+\mathcal H(Y_S)-CLA\log\beta$.
Their joint law is a trial for the product target law.

An interaction missing $S$ lies on one cap side, by connectedness
of its declared support. Its contribution cancels because the
complete bounded side was swapped. The sum of the two full
cocycles therefore equals the difference of the target and source
collar Hamiltonians. The Gibbs variational principle gives
\[
 \log K(a',b)+\log K(a,b')
 \ge\log K(a,b)+\log K(a',b')-\beta A\kappa.
\]
Interchanging the source and target cap pairings gives the reverse
inequality. Positivity follows from bounded action and the
positive-mass small-ball trial. 
\end{proof}



\section{Contours and interface costs}
\label{con:geometry}

We use fine spatial lattice units in this section. A fine cell has physical
volume $v_s$, and a coarse cube contains $L^3$ fine cells. Thus
\begin{equation}\label{con:volume-unit}
 V_L=v_sL^3,\qquad \operatorname{Vol}(D)=V_L n(D),
\end{equation}
where $n(D)$ counts coarse cubes. Surface areas $\mathcal A(D)$ are measured
in fine-grid units. Constants may absorb the fixed mesh conversion in
surface estimates; the factor $V_L$ is retained in every bulk identity.

Let $Q_v=L(v+[0,1]^3)$, $v\in\mathbb Z^3$, be closed coarse cubes. Faces
are assigned measurably to one adjacent cube when integrating variables.
Topology always refers to closed cubes. Two indices are star adjacent if
their sup-norm distance is one. Thus a union of cubes is connected exactly
when its indices are star connected. Define
\[
 N_k(B)=\{v:\operatorname{dist}_\infty(v,B)\le k\}.
\]
Every spatial variable includes the entire angular fiber and all rod and
field variables assigned to that spatial cell. We choose the physical
coarse side longer than the microscopic interaction range.

Use the compact projected sets $\mathcal M_1,\ldots,\mathcal M_m$
and the norm $|\cdot|_\omega$ defined above. Choose $\rho_*>0$ smaller than one quarter of their minimum
separation. A fine site has label $q$ if its field lies within $\rho_*$
of $\mathcal M_q$, and label zero otherwise. A coarse cube has
$\tau_v=q$ when at least $(1-\theta)L^3$ fine sites have label $q$,
where $0<\theta<1/2$.

Set $\tau_v=0$ when no phase has this majority, and define
\[
 \Theta_v=q\quad\Longleftrightarrow\quad
 \tau_w=q\ \hbox{for every }\|w-v\|_\infty\le1;
 \qquad \Theta_v=0\ \hbox{otherwise}.
\]
Two star-adjacent nonzero $\Theta$ labels coincide. Write
$B=\{v:\Theta_v=0\}$. A contour support is a star component of the full,
unclipped set $N_k(B)$. Its bad set $B_\Gamma$ consists of the bad cubes
whose $k$-neighborhood lies in that component, and its support is
$A_\Gamma=N_k(B_\Gamma)$.

\subsection{Complementary components and regional domains}

\begin{lemma}[Connected complementary boundaries]\label{con:connected-boundaries}
For a finite connected union $A$ of closed coarse cubes, the open
complementary components correspond to face-connected components of
unused indices. Each complementary boundary, and its set of touching
cubes on either side, is connected in the respective Euclidean or star
sense.
\end{lemma}
\begin{proof}
A path in the open complement can be perturbed away from grid edges and
vertices, so it crosses only open faces; the converse follows by joining
face neighbors. Compactify space to $S^3$. For a complementary component
$D$, the sets $F=S^3\setminus D$ and $G=\overline D$ are connected finite
subcomplexes of a common triangulation: $F$ consists of $A$ and the other
components, whose closures meet $A$.
If vertices $x,y$ belonged to different components of $F\cap G$, choose
chains $a\subset F$, $b\subset G$ with boundary $y-x$.
The following is the reduced Mayer--Vietoris argument
\cite[Section~2.2, pp.~149--150]{Hatcher}.
Since $H_1(S^3)=0$, write $a-b=\partial(z_F+z_G)$ by assigning each
triangle of a filling chain to $F$ or $G$. Then
$a-\partial z_F=b+\partial z_G$ is a chain in $F\cap G$ joining $x$ to
$y$, a contradiction. Thus $\partial D=F\cap G$ is connected.
Its closed boundary faces have a connected intersection graph, which
connects the touching cubes by star steps. An edge or vertex contact
also belongs to this graph: among incident cubes, a face path from a
$D$ cube to an $A$ cube first leaves $D$ across a boundary face.
\end{proof}

\begin{lemma}[Containment of inherited regions]\label{con:inherited-containment}
If $D$ is a bounded complementary component of a connected closed
support and $A'\Subset D$ is another such support, every bounded
complementary component of $A'$ lies in $D$.
\end{lemma}
\begin{proof}
The connected set $S^3\setminus D$ avoids $A'$ and contains infinity,
so it lies in the unbounded component of $S^3\setminus A'$.
\end{proof}

A \emph{physical recursion domain} is a box or a bounded complementary
component of a connected closed support.

\begin{lemma}\label{con:labels-gap}
Every complementary boundary of a contour support has a single nonzero
facing label. Components with different facing labels have index distance
at least $2k+2$.
\end{lemma}
\begin{proof}
A cube outside the support and touching it has distance at least $k+1$
from every bad cube. Lemma~\ref{con:connected-boundaries} connects these
facing cubes by star steps, so their nonzero labels coincide. A star
path between differently labeled facing cubes contains a bad cube,
and hence has length at least $2k+2$. For arbitrary cubes in differently
labeled components, take a coordinate path in their bounding box. Its
first exit and last entry give facing cubes no farther apart.
\end{proof}

\begin{lemma}[Regional gap]\label{con:regional-distance}
If $D$ is a complementary face component of $A$, then for $v\in D$,
\[
 \operatorname{dist}_\infty(v,D^c)
 =\operatorname{dist}_\infty(v,A).
\]
For a child support $A'=N_k(B')\subset D$, the regional gap is exactly
\begin{equation}\label{con:regional-gap}
 \operatorname{dist}_\infty(A',D^c)\ge2
 \quad\Longleftrightarrow\quad
 \operatorname{dist}_\infty(B',D^c)\ge k+2.
\end{equation}
\end{lemma}

\begin{proof}
For the distance identity, choose a nearest cube of $D^c$. If it is
unused, a coordinate path inside the corresponding sup-norm box must
first leave $D$ through $A$: face-adjacent unused cubes remain in one
component. The reverse inequality follows from $A\subset D^c$. Finally,
for any set $C$,
\[
 \operatorname{dist}_\infty(N_k(B'),C)
 =\max\{\operatorname{dist}_\infty(B',C)-k,0\},
\]
by the triangle inequality and a shortest star path. This proves
\eqref{con:regional-gap}.
\end{proof}

Write $U_q(D)$ for the restriction $\tau=q$ throughout $D$. The physical
regional restriction $R_q(D)$ is $\Theta=q$ in the first $k+1$ inside
layers. For contour indexing, extend the outside labels by $q$; the actual
adjacent outside stencil agrees with this extension, while exterior rods
remain arbitrary. On a physical recursion domain,
\eqref{con:regional-gap} is equivalent to regional admissibility. Indeed,
the connected complement of $D$ lies in the child's unbounded component.
The gap excludes bad cubes from the first $k+1$ layers. A coordinate path
from a cube in this band to a nearest outside cube stays in the band until
its first exit. Its nonzero labels propagate the outside value $q$ back
to the starting cube. The converse follows immediately from the absence
of bad cubes in the band.

On arbitrary finite unions, including punctured sets, we will also use
an auxiliary gas indexed by support inclusion.

\subsection{Removed regions, shielding, and the contour identity}

A contour datum records $B_\Gamma$, $A_\Gamma$, the $\tau$ pattern on the
support, and the facing label of each complementary component. Filling every complementary component with its facing label must
produce exactly the recorded bad set $B_\Gamma$. Fix an
exterior phase $q$. Let $D_j$ denote its bounded components with facing
labels $p_j\ne q$. Define
\[
 V_\Gamma=A_\Gamma\cup\bigcup_jD_j,
 \qquad K_\Gamma=\{\hbox{support cubes touching a $q$ component}\},
 \qquad U_\Gamma=V_\Gamma\setminus K_\Gamma.
\]
The set $K_\Gamma$ is the full star-facing layer, including boundaries of
$q$ holes. Two contours are strongly compatible when their closed
supports do not touch.

\begin{lemma}[Nesting of removed regions]\label{con:laminar}
The removed regions of strongly compatible $q$ contours are nested or
disjoint. Distinct maximal regions have star distance at least two.
\end{lemma}
\begin{proof}
A connected support avoiding another lies in one complementary component.
If this component is bounded, Lemma~\ref{con:inherited-containment}
keeps all its interiors there: the removed regions are nested in a
non-$q$ component and disjoint in a $q$ hole. If bounded interiors
$H_1,H_2$ overlap while the second support avoids $H_1$, connectedness
gives $H_1\subset H_2$; its boundary and then the first support lie in
$H_2$, again giving nesting. Finally the boundary of each removed region
lies in its support, so disjoint removed regions cannot touch unless
their supports touch.
\end{proof}

\begin{lemma}[Microscopic separation]\label{con:shielding}
Every microscopic factor meeting $U_\Gamma$ depends on exterior
variables only through $K_\Gamma$.
\end{lemma}
\begin{proof}
Lemma~\ref{con:labels-gap} separates the inherited non-$q$ components
from all $q$ components. The layer $K_\Gamma$ covers the remaining
contacts, and a coarse side exceeds the microscopic interaction range.
\end{proof}

Let $x_v$ contain the rods and fields of a coarse cube. Write $\rho_U$
for the nonnegative conditional microscopic density, including every hard
exclusion. On compatible $q$ data $x_K$ define the joint integrals
\begin{align}
 N_\Gamma(x_K)&=\int\rho_U(x_U\mid x_K)
 \mathbf1_{\{\text{recorded pattern on }A\}}
 \prod_j\mathbf1_{R_{p_j}(D_j)}\,dx_U,\\
 D_\Gamma(x_K)&=\int\rho_U(x_U\mid x_K)
 \mathbf1_{U_q(A\setminus K)}
 \prod_j\mathbf1_{R_q(D_j)}\,dx_U,
 \qquad W_\Gamma=N_\Gamma/D_\Gamma.\label{con:raw-weight}
\end{align}
The empty rod configuration and fields near the phase root give $D_\Gamma>0$.
Shielding makes $W_\Gamma$ a function of $x_K$ alone and makes it
independent of the ambient regional domain. Extend it by zero on
incompatible collar fields. The extension set is source independent.

\begin{lemma}[Interior contour indices]\label{con:interior-indices}
After a maximal $q$ contour has been selected, its residual virtual
contours in each $D_j$ are precisely the regional $q$ contours there.
The indices for distinct $D_j$ form a Cartesian product.
\end{lemma}
\begin{proof}
Compatibility with the parent is equivalent to
\eqref{con:regional-gap}. This gap also ensures compatibility with
siblings and exterior contours and admissibility in the parent domain.
The denominator in \eqref{con:raw-weight} prescribes phase $q$ in every $D_j$.
\end{proof}

The cancellation of interior partition functions is the contour
reweighting used by Disertori--Giuliani~\cite[Section~4.3, Lemma~1]{DG}
and Disertori--Giuliani--Jauslin~\cite[Section~4.2, Lemma~2]{DGJ}.
Here the quotients remain conditional on the collar variables, and the
identity is established for each exterior configuration.

\begin{proposition}[Exact regional identity]\label{con:exact-identity}
Let $\mathcal C_q(D)$ be the admissible $q$ contour data in a physical
regional domain. Set
\[
 G_q(D;x)=\sum_{\mathcal F\subset\mathcal C_q(D)\ \mathrm{compatible}}
                \prod_{\Gamma\in\mathcal F}W_\Gamma(x_{K_\Gamma}),
\]
allowing nested removed regions. For every compatible exterior
configuration $\xi$,
\begin{equation}\label{con:forest}
 \int_{U_q(D)}\rho_D(x_D\mid\xi)G_q(D;x_D)\,dx_D
 =\int_{R_q(D)}\rho_D(x_D\mid\xi)\,dx_D.
\end{equation}
\end{proposition}

\begin{proof}
Physical configurations have a unique outside decomposition. Extract
the components of the full thickened bad set, start in the exterior $q$
sea, and retain all $q$ contours reached through that sea and its $q$
holes, stopping at each non-$q$ inherited component. To justify enclosure,
a contour with exterior-facing phase $p\ne q$ faces a finite $p$ sea
component. Its connected outer boundary lies in the disjoint union of
supports and hence in one support. This enclosing support is different
from the original one, since the chosen facing cube was in its unbounded
component. Repeating outward terminates by strict containment and
finiteness. The retained removed regions are disjoint; everything outside
them has $\tau=q$. Conversely, the recorded support patterns, regional
interior restrictions, and $q$ exterior reconstruct exactly this
decomposition. The full regional bands prevent new bad stencils or merged
$k$-neighborhoods at the interfaces.

Induct on the number of cubes of $D$. Group a virtual family by its
maximal removed regions. By Lemma~\ref{con:interior-indices}, the residual family
indices are a product of regional $q$-family indices in the inherited
components. Freeze every variable outside the first such component,
including rods in touching siblings, and apply the conditional induction
identity there. Repeat for all components. Each transformed component
retains its $q$ boundary band, so the next conditional identity has the
required exterior stencil. All selected weights are measurable outside
the changed component. The resulting integral over a selected $U$ is
exactly $D_\Gamma$; multiplication by $W_\Gamma$ replaces it by
$N_\Gamma$. Shielding permits successive cancellation for the disjoint
maximal regions. The resulting sum is the physical outside decomposition.
Every inherited domain is strictly smaller.
\end{proof}

Following Zahradn\'ik's truncation construction
\cite{Zahradnik84}, used in the continuum setting
by LMP~\cite[Section~3]{LMP99}, we clip contour weights to define
controlled phase pressures before proving stability.

For fixed thresholds $t_\Gamma$, classify a contour as active at a
reference configuration when $W_\Gamma>t_\Gamma$. In each virtual family
select active contours maximal under inclusion of removed regions. Their
removed regions are disjoint. All descendants inside selected inherited
components are unrestricted; every remaining contour is stable, including
ancestors surrounding selected regions. Put
$E=D\setminus\bigcup_{\Gamma\in\mathcal F}V_\Gamma$.
The selected contribution is exactly
\begin{equation}\label{con:active-decomposition}
 \int_{U_q(D)}\rho_D
 \prod_{\Gamma\in\mathcal F}
       \bigl(W_\Gamma\mathbf1_{\{W_\Gamma>t_\Gamma\}}\bigr)
 G_{q,E}^{\mathrm{stable},\mathcal F}
 \prod_{\Gamma\in\mathcal F,j}G_q(D_j)\,dx_D.
\end{equation}
The stable gas retains the original parent admissibility and
compatibility conditions. Its actual supports lie in $E$, even when a
stable contour's filled hull surrounds a hole of $E$. The active indicators depend on $K_\Gamma$, and the stable gas depends on $E$.
The same conditional cancellations therefore apply to
\eqref{con:active-decomposition}. Dropping the active indicators and
enlarging the stable gas gives the positive bound
\begin{equation}\label{con:positive-domination}
 \text{selected contribution}\ \le
 \int\rho_D\,\mathbf1_{R_{\mathcal F}(\phi)}
       G_{q,E}^{\mathrm{aux}}(x_E)\,dx_D,
\end{equation}
where the auxiliary gas contains all compatible $q$ contours with
$A_\Gamma\subset E$ and activities
$w_\Gamma=\min\{W_\Gamma,t_\Gamma\}$. 


\subsection{A positive charge on each contour}\label{con:peierls-geometry}

The contour cost supplies the energy bound in Peierls' energy--entropy
argument~\cite{Peierls36}; here it follows from local rigidity and
lattice isoperimetry.

Call a fine site defective when its label is zero or it has a nearest
neighbor with a different nonzero label. The fixed-patch rigidity
estimate supplies a finite patch $P_i$ and $e_*>0$ such that
\begin{equation}\label{con:defect-charge}
 i\text{ defective}\quad\Longrightarrow\quad
 \sum_{x\in P_i}e_x(\phi)\ge e_*.
\end{equation}
Here $e_x$ are the nonnegative leading charges. To obtain a uniform
$e_*$, take the finite union of closed defect events with distances
$\ge\rho_*$ for zero labels and $\le\rho_*$ for prescribed nonzero
labels. If the infimum of the patch charge vanished, product compactness
and exponential locality would give a limiting zero-charge patch. Local
rigidity makes all marked cells one common well, contradicting the
separation of those closed events.

Use the positive ball atoms of Proposition~\ref{fnd:atoms}, writing
$\Phi_{x,r}=\Phi^0_{x,r}$. Their localization error is $Ce^{-cr}$. Fix $R_0$ so that
$|P_i|Ce^{-cR_0}\le e_*/2$. Then
\[
 F_i=\sum_{x\in P_i}\sum_{r=0}^{R_0}\Phi_{x,r}\ge e_*/2
\]
at each defect, and its support lies in a fixed ball $B_S(i)$.

\begin{lemma}[Relative lattice isoperimetry]\label{con:relative-iso}
For $H\subset\{1,\ldots,M\}^3$ and its internal nearest-neighbor edge
boundary $\partial_QH$,
\[
 |\partial_QH|\ge(1-2^{-1/3})
                  \min\{|H|,|Q\setminus H|\}^{2/3}.
\]
\end{lemma}

\begin{proof}
Replace $H$ by its complement so $a=|H|\le M^3/2$. If $P_j$ counts its
nonempty coordinate-$j$ fibers, the Loomis--Whitney projection inequality
\cite[Theorem~2]{LW49}
$a^2\le P_1P_2P_3$ implies some $P_j\ge a^{2/3}$. The projection
inequality follows by applying Cauchy--Schwarz twice to
$\sum_{x,y,z}a_{xy}b_{yz}c_{zx}$, with the three projection indicators;
its square is at most
$(\sum a_{xy})(\sum b_{yz}^2)(\sum c_{zx}^2)$.
At most $a/M$ nonempty fibers are full. Each remaining fiber has an
internal crossing edge, whence
\[
 |\partial_QH|\ge a^{2/3}-a/M
 \ge(1-2^{-1/3})a^{2/3}.
\]
\end{proof}

\begin{proposition}[Protected Peierls charge]\label{con:protected}
There is $c_0>0$, independent of $L,\theta,\beta$ and the contour shape,
such that for each contour of size $n=|A_\Gamma|$ there is a fixed set
of leading atom labels, depending only on $B_\Gamma$, whose sum
$P_{B_\Gamma}$ satisfies
\begin{equation}\label{con:peierls}
 \operatorname{supp}P_{B_\Gamma}\subset N_2(B_\Gamma),\qquad
 P_{B_\Gamma}\ge\tau L^2n,\qquad
 \tau=\frac{c_0\theta^{2/3}}{(2k+1)^3}.
\end{equation}
For strongly compatible contours these protected atom sets are disjoint.
\end{proposition}

\begin{proof}
First suppose $\tau_v=0$ and write $M=L^3$. If the number $a_0$ of
zero-labeled sites is at least $\theta M/2$, it is a positive integer and
\[
 a_0\ge\max\{1,\theta M/2\}
       \ge2^{-2/3}(\theta M)^{2/3}.
\]
Otherwise at least $3M/4$ sites have nonzero labels. A most frequent
phase has population $a_q\ge3M/(4m)$ and complement larger than
$\theta M$. Hence
$\min(a_q,M-a_q)\ge c_m\theta M$. Relative isoperimetry applied to that
phase gives at least $c\theta^{2/3}L^2$ crossing edges. Each crossing has
a defective endpoint, and a site is incident to at most six edges.

If $\Theta_b=0$, either one of its 27 neighboring coarse cubes has
$\tau=0$, or two have different nonzero labels. In the latter case the
fine cube $N_1(b)$ contains at least $L^3/2$ sites of one phase and at
least that many of its complement. Relative isoperimetry again gives
$c\theta^{2/3}L^2$ defects. Summing over $B_\Gamma$ and using the
multiplicity 27 gives
\[
 |\{\text{defects in }N_1(B_\Gamma)\}|
       \ge c\theta^{2/3}L^2|B_\Gamma|/27.
\]
Color fine sites by coordinate residues modulo $2S+1$. One color has
at least $(2S+1)^{-3}$ of these defects, and their $B_S(i)$ are disjoint.
Summing the corresponding $F_i$ uses no atom twice. Since
$n\le(2k+1)^3|B_\Gamma|$, this proves the lower bound.

To make the protected Hamiltonian independent of the field, include,
once per label, every $\Phi_{x,r}$ with $r\le R_0$ and center within the
patch radius of $N_1(B_\Gamma)$. This fixed set contains every possible
color selection and preserves the lower bound by positivity. For
$L\ge S+2$ its full supports lie in $N_2(B_\Gamma)$. Distinct full
$k$-neighborhood supports have bad-set distance at least $2k+2$; their
$N_2$ neighborhoods are disjoint for $k\ge24$.
\end{proof}

\subsection{Seams, restoration, and separated collars}

Let $D_j$ be inherited components of a support or of a family with total
support size $N$. Every boundary face has a support cube on the other
side. Therefore
\begin{equation}\label{con:area-bound}
 \mathcal A_{\rm in}:=\sum_j\mathcal A(D_j)\le6L^2N.
\end{equation}
Suppose $\|\Phi_{i,r}\|_\infty\le C_0e^{-c_0r}$ and let $m(i,r)$ be
the number of inherited components met by its declared ball.

\begin{lemma}[Seam estimate]\label{con:seams}
For every fixed $p\ge0$,
\begin{equation}\label{con:positive-seam}
 \sum_{m(i,r)\ge2}(1+m(i,r))^p\|\Phi_{i,r}\|_\infty
 \le C_p\mathcal A_{\rm in}/L.
\end{equation}
If a signed interaction $\psi$ has connected declared supports and local
norm at most $\delta$, then
\[
 \sum_{X:\,m(X)\ge2}m(X)\|\psi_X\|_\infty
 \le C\delta\mathcal A_{\rm in}.
\]
\end{lemma}

\begin{proof}
For $r<L/4$, a ball meets at most eight coarse cubes. Two of these in
different complementary components cannot share a face; if both meet
the ball, they meet each other only at an edge or vertex. In the sup norm,
the intersection of their $r$-neighborhoods lies in the $r$-neighborhood
of that edge or vertex. It belongs to the skeleton of an inherited
boundary face. The number of possible fine centers is bounded by
\[
 C\sum_j|\partial D_j|
       \{L(r+1)^2+(r+1)^3\}
 \le C\frac{\mathcal A_{\rm in}}L(r+1)^2.
\]
Here $|\partial D_j|$ counts coarse boundary faces. Summation against
$e^{-c_0r}$ proves the short-radius estimate, with bounded $m\le8$.
For $r\ge L/4$, a ball meeting two components crosses a boundary face.
One face has at most
$CL^2(r+1)(1+(r+1)/L)^2$ possible centers, and
$m\le C(1+r/L)^3$. The remaining sum is at most
\[
 C_p\mathcal A_{\rm in}\sum_{r\ge L/4}
 (r+1)(1+(r+1)/L)^{2+3p}e^{-c_0r}
 \le C_pe^{-cL}\mathcal A_{\rm in}.
\]

A connected weak support crossing $\partial D_j$ contains the endpoints
of a fine boundary edge. Charge it to one such edge and sum the local
norm. Summation over $j$ counts a term meeting $m$ interiors at most
$m$ times. A fixed finite-range connected-support convention can be
enlarged to a nearest-neighbor connected hull at a fixed norm cost.

\end{proof}

Deleting or restoring terms of total norm $M$ changes a log integral by
at most $\beta M$. A term meeting $m$ interiors is restored in at most
$m$ interior factors, $m$ root divisors, and a bounded number of core or
exterior factors. Its total multiplicity is $C(1+m)$, covered by the
first two estimates. Thus the total logarithmic cost is at most
$C\beta(L^{-1}+\delta)\mathcal A_{\rm in}$.

Take $k\ge24$. Inside $D_j$ put
\[
 \ell_j(v)=\operatorname{dist}_\infty(v,D_j^c),\quad
 S_j=\{1\le\ell_j\le8\},\quad I_j=\{\ell_j\ge9\}.
\]
Use the level-four/five faces as separator $\Sigma_j$. The regional
restriction makes $S_j$ $p_j$-correct. The one-Lipschitz property of
$\ell_j$ separates $I_j$ from $D_j^c$. Each separator face lies a fixed
number of coarse steps from an original boundary face, so
\begin{equation}\label{con:collar-geometry}
 |\Sigma_j|\le C\mathcal A(D_j),\qquad
 |S_j|+|N_t(\Sigma_j)|\le CL\mathcal A(D_j),
\end{equation}
for $L/4\le t\le L/3$. Moreover
\[
 |N_{t+R}(\Sigma_j)\setminus N_t(\Sigma_j)|
 \le C(R+1)\mathcal A(D_j).
\]
Indeed a single coarse face contributes $O(L^3)$ to the tube volume and
$O((R+1)L^2)$ to the enlargement; a union's increment lies in the union
of the individual increments. Any fixed alignment margin fits inside
the collar for large $L$, with distance $cL$ from both caps.

For a selected family let $V$ be the union of removed regions and let
$E$ include its complement and all $q$ holes. Define
\[
 S_0=\{v\in A:1\le\operatorname{dist}_\infty(v,E)\le8\}.
\]
The same level-four/five construction gives its outer separator. Every
inherited non-$q$ component lies beyond it. Propagation from the facing
$q$ layer makes $S_0$ $q$-correct, and
\[
 \operatorname{dist}_\infty(S_0,D_j)\ge2k+2-8.
\]
Leading balls meeting both outer and inner collars therefore have radius
at least $cL$ and total norm $Ce^{-cL}\mathcal A_{\rm in}$; weak terms
cost $C\delta\mathcal A_{\rm in}$. Delete these along with the
multi-interior terms. Every remaining declared support meets at most one
collar. Connectedness of supports assigns every other term to exactly
one cap. Conditional collar integrations now factor.

The collar switch exchanges the complete bounded sides
$\{\ell_j\ge5\}$ and $\{v\in V:\operatorname{dist}_\infty(v,E)\ge5\}$
of the inner and outer separators. These contain the inherited interiors
and exclude $q$ holes. Omitted cap coordinates may be extended arbitrarily
when comparing cut and full kernels, since all terms reaching them were
deleted. If $I_j=\varnothing$, the cross ratio is one.

Collar kernels impose $\tau=q$. Once these labels are fixed, the
remaining $\Theta$ restrictions concern individual caps: a radius-one
stencil cannot cross a width-eight collar, and touching sibling collars
have equal labels by Lemma~\ref{con:labels-gap}.

For a conditional single-contour quotient, freeze $K=K_\Gamma$ and work
in $U=V_\Gamma\setminus K$. Set $\ell(v)=\operatorname{dist}_\infty(v,K)$
and use
\[
 O=\{\ell\le4\},\qquad S=\{5\le\ell\le12\},\qquad
 V'=U\setminus(O\cup S).
\]
The level-eight/nine separator has the same tube bounds. The collar lies
in the $q$ support buffer, every $D_j$ remains in $V'$, and every protected
atom lies outside all these collars. For each separator let $P$ be its incident coarse cubes. Their distance
levels are $4,5$ for inner and outer collars, and $8,9$ for the shifted
collar. Since the distance function is one-Lipschitz,
$N_3(P)\subset S$: the possible levels are respectively $[1,8]$ and
$[5,12]$. In the outer case these cubes remain in $A$, because positive
distance excludes $E$ and the component gap excludes every $D_j$.
The region $N_2(P)$ is a union of complete coarse blocks inside the
collar. The two interpolation strips lie on their respective sides of
the separator, at fine distances between $L/4$ and $L/3$. Their alignment
patches remain in the collar once $R_1+r_0<L/6$. Thus all three collars
satisfy the buffer and block conditions used in the surface estimates.

The boundary-cocycle estimate
$|\partial_iB^\xi|\le Ce^{-c\operatorname{dist}(i,U^c)}$ gives
\begin{equation}\label{con:boundary-truncation}
 \|B^\xi(\phi)-B^\xi(\phi_O r_{O^c})\|_\infty
 \le Ce^{-c'L}\mathcal A(U).
\end{equation}
To sum the gradient bound, assign each fine site to a nearest boundary
face; at distance $d$ there are at most $C\mathcal A(U)(d+1)^2$ such
sites. The sum starts at $d\ge cL$.
Before detaching inner collars in $V_{\rm aug}=S\cup V'$, also delete
terms meeting both $S_j$ and $V_{\rm aug}^c$. Their leading norm is
$Ce^{-cL}\sum_j\mathcal A(D_j)$ and their weak norm is
$C\delta\sum_j\mathcal A(D_j)$. This ensures that an inner root divisor
has its own phase on its whole exterior. Every deleted term meets a
collar or an inherited component, whereas the protected atoms lie in
$N_2(B_\Gamma)$. The Peierls charge survives all cuts.

\subsection{Removed volume and contour counting}

\begin{lemma}\label{con:counting}
For a contour of support size $n$,
\[
 n(V_\Gamma)\le Cn^{3/2}.
\]
The number of size-$n$ contour data whose removed region contains a
specified coarse site is at most $C_1^n$.
\end{lemma}

\begin{proof}
By the same Loomis--Whitney inequality, a finite lattice set $d$ has
projection sizes satisfying
$|d|^2\le P_1P_2P_3$. Each occupied coordinate line contributes two
boundary faces, giving
$|\partial d|\ge2(P_1+P_2+P_3)\ge6|d|^{2/3}$.
Every inherited boundary face meets a support cube, so
\[
 \sum_j|D_j|\le C\sum_j|\partial D_j|^{3/2}
 \le C\Bigl(\sum_j|\partial D_j|\Bigr)^{3/2}\le Cn^{3/2}.
\]
Add the support. Every bounded complementary component lies in its
support's coordinate bounding box, since a ray beyond an extreme
coordinate reaches infinity without meeting the support. A star-connected
support of size $n$ has coordinate span at most $n-1$. A removed region
containing the specified site therefore has only $O(n^3)$ possible
support anchors. A deterministic spanning-tree search encodes each
anchored support by a depth-first walk of length $2(n-1)$ with 26 choices
per step. Finite support labels and at most $6n$ complementary-component
labels add only exponential factors. Absorb the anchor polynomial.
\end{proof}


For a contour put $n_\Gamma=|A_\Gamma|$ and let its declared support
$S_\Gamma$ be all marked cells above $A_\Gamma$. Thus
$|S_\Gamma|\le d_Ln_\Gamma$. For a tuple
$C=(\Gamma_1,\ldots,\Gamma_m)$ set
$s(C)=\sum_jn_{\Gamma_j}$ and $S(C)=\bigcup_jS_{\Gamma_j}$.
Let $\varphi^T(C)$ be its connected incompatibility coefficient,
as in \eqref{res:treegraph}.

\begin{lemma}[Anchored contour clusters]\label{con:cutoff-locality}
For every fixed $b\ge0$, there is $\mathcal T_0(L,b)$ such that
\begin{equation}\label{con:cluster-reserve}
 \sup_x\sum_{m\ge1}\frac1{m!}\sum_C
 |\varphi^T(C)|\,e^{b s(C)}
 \prod_{j=1}^m e^{-\mathcal T n_{\Gamma_j}}
 \mathbf1_{\{x\in S(C)\}}
 \le C_L e^{-\mathcal T/2}
\end{equation}
for $\mathcal T\ge\mathcal T_0(L,b)$, uniformly over regional
contour index sets.
\end{lemma}
\begin{proof}
The counting bound gives at most $C_1^n$ contours of size $n$ containing
a prescribed coarse site. A contour has at most $27n_\Gamma$
incompatibility anchors. For majorants
$a_\Gamma=e^{-(\mathcal T-b)n_\Gamma}$,
\[
 \sum_{\Gamma'\not\sim\Gamma}a_{\Gamma'}e^{n_{\Gamma'}}
 \le27n_\Gamma\sum_{n\ge1}C_1^n e^{-(\mathcal T-b-1)n}
 \le n_\Gamma
\]
when $\mathcal T$ is large. This is the Koteck\'y--Preiss
criterion~\cite[Section~2, Theorem]{KP86}. The rooted tree bound
from Lemma~\ref{res:mixedcluster} bounds the left side of
\eqref{con:cluster-reserve} by
$C_L\sum_{n\ge1}C_1^n e^{-(\mathcal T-b-1)n}$.
Increasing $\mathcal T_0(L,b)$ bounds this sum by
$C_Le^{-\mathcal T/2}$.
\end{proof}

For $|w_\Gamma|\le e^{-\mathcal T n_\Gamma}$, group the convergent
logarithm by connected union supports. Since
$|S(C)|\le d_Ls(C)$ and $\diam S(C)\le C_Ls(C)$, increasing $b$
controls the required exponential support weights and polynomial
factors. The absolute majorants, uniformly in the rods and fields,
give interaction norm at most $\kappa=C_Le^{-\mathcal T/2}$.


\begin{lemma}[Localization of additional contour species]
\label{con:species-localization}
Let $D_j$ be disjoint physical recursion domains in a coarse union
$E$, and put $C=E\setminus\bigcup_jD_j$. An auxiliary $q$ contour on
$E$ absent from the union of the regional $q$ contour sets on $D_j$
has support meeting
\[
 S=C\cup\bigcup_j\{v\in D_j:\dist_\infty(v,D_j^c)<2\}.
\]
\end{lemma}
\begin{proof}
A connected support avoiding $S$ lies in one $D_j$ and has distance
at least two from $D_j^c$. The equivalence
\eqref{con:regional-gap} makes it regionally admissible there;
Lemma~\ref{con:inherited-containment} keeps all its inherited regions
inside $D_j$. Conversely every regional index is an auxiliary index.
\end{proof}

\begin{lemma}[Cost of adding species]\label{con:species-cost}
Suppose two clipped contour gases have nested index sets, identical
activities on common indices, and every additional species has support meeting a coarse set $S$. Then
\[
 \|\log G_2-\log G_1\|_\infty\le C_L\kappa\,\#S.
\]
\end{lemma}
\begin{proof}
This is the local cluster comparison underlying
\cite[Section~2, estimate~(4)]{KP86}. Their logarithmic cluster series
differ only in clusters containing
an additional species. Mark one such species and a site of its support
in $S$. The anchored estimate in Lemma~\ref{con:cutoff-locality},
summed over these sites, gives the bound. It is pointwise in the rods
and fields and hence also bounds the ratio of the corresponding joint
partition functions.
\end{proof}

\section{The physical pressure as a maximum of phase pressures}
\label{con:canonical}

Following LMP~\cite[Sections~3 and~6]{LMP99}, we first define auxiliary
pressures using clipped contours and then bound the physical partition
functions in terms of those pressures. Roots, phase neighborhoods,
regional restrictions, and the threshold $\mathcal T$ are independent
of the source.

\subsection{Normalization and quantitative inputs}

Fix the source $h$. Let $\nu_r$ be the unique whole-space bare rod
state at constant field $r$ (Lemma~\ref{res:bare-reference}), and
$\nu_{r,\mathrm{out}}$ its marginal outside $D$. Write
$\nu_{r,\phi;D}$ for the bare state at the complete field
$\phi_Dr_{D^c}$. The fixed-field density $F_D(r\mid\xi)$ includes the
Gaussian prefactor; denote its pressure per physical volume by $b_r$.
For a field restriction $R$ and a bounded joint perturbation $J$, define
\[
 C_D^{R,J}(r)=e^{b_r\operatorname{Vol}(D)}
 \mathbb E_{\nu_{r,\mathrm{out}}}
       \frac{Z_D^{R,J}(\xi)}{F_D(r\mid\xi)},\qquad
 V_D^J(\phi)=-\log\mathbb E_{\nu_{r,\phi;D}}e^{-J}.
\]
The conditional Hamiltonian $H_D^\Psi$ sums every interaction term
whose declared support meets $D$, with exterior field $r$. Recall
\[
 b_r+\beta e(r)=b_0,\qquad
 \sigma_D(r)=H_D^\Psi(r\mid r)-e(r)\operatorname{Vol}(D).
\]

\begin{lemma}[Exact normalized field integral]\label{con:field-change}
With the probability reference measure $\lambda_D$,
\begin{equation}\label{con:canonical-representation}
 \mathfrak Z_D^{R,J}(r):=e^{-\beta\sigma_D(r)}C_D^{R,J}(r)
 =e^{b_0\operatorname{Vol}(D)}
  \int_R e^{-\beta H_D^\Psi(\phi\mid r)-V_D^J(\phi)}\,d\lambda_D.
\end{equation}
\end{lemma}
\begin{proof}
Let $\mathcal G_D$ be the Gaussian prefactor in
\eqref{mod:fullfield} and set
$\mathcal L_D=\sum_{i\in I(D)}(\phi_i-r_i)\widehat N_i$.
At fixed $h$, the normalizers in \eqref{mod:cellweight} cancel, giving
\[
 \frac{F_D(\phi\mid\xi)}{F_D(r\mid\xi)}
 =\frac{\mathcal G_D(\phi)}{\mathcal G_D(r)}
   \mathbb E_{\nu_r(\cdot\mid\xi)}e^{\mathcal L_D}.
\]
Average over $\nu_{r,\mathrm{out}}$. The normalized finite tilt
$e^{\mathcal L_D}\nu_r/\mathbb E_{\nu_r}e^{\mathcal L_D}$ satisfies
the bare specification at $\phi_Dr_{D^c}$; uniqueness identifies it
with $\nu_{r,\phi;D}$. Thus the perturbation contributes the factor
$e^{-V_D^J(\phi)}$. The logarithmic derivative of
$\mathcal G_D(\phi)\mathbb E_{\nu_r}e^{\mathcal L_D}/\mathcal G_D(r)$
is $-\beta g_{\beta,h}$, which is also the derivative of
$-\beta H_D^\Psi$. Integrating from $r$ to $\phi$ gives
\[
 \frac{\mathcal G_D(\phi)}{\mathcal G_D(r)}
 \mathbb E_{\nu_r}e^{\mathcal L_D}
 =e^{-\beta[H_D^\Psi(\phi\mid r)-H_D^\Psi(r\mid r)]}.
\]
Insert this identity into $C_D^{R,J}$ and multiply by
$e^{-\beta\sigma_D(r)}$. The remaining constant is
$e^{(b_r+\beta e(r))\operatorname{Vol}(D)}$.
\end{proof}

Set $\mathfrak Z_q^{\rm phys}(D)=\mathfrak Z_D^{R_q(D),0}(r_q)$.
The auxiliary partition uses $R=U_q(D)$ and
$J=-\log G_{q,D}^{\rm aux}$. Write $I$ for the corresponding field
integrals, so $\mathfrak Z=e^{b_0\operatorname{Vol}}I$.
For a selected-family term $Z_{D,\mathcal F}(\xi)$, define
\[
 \mathfrak I_{\mathcal F}
 =e^{b_{r_q}\operatorname{Vol}(D)-\beta\sigma_D(r_q)}
   \mathbb E_{\nu_{r_q,\mathrm{out}}}
       \frac{Z_{D,\mathcal F}(\xi)}{F_D(r_q\mid\xi)}.
\]

The interaction $\Psi=\Phi^0+\psi$ has nonnegative leading atoms
vanishing at constant wells and $\|\psi\|\le\delta_\beta$. Its boundary
cocycle satisfies \eqref{con:boundary-truncation}; its homogeneous
trial bound is \eqref{fnd:roottrial}.

Apply the restricted-pressure construction to the field interaction
obtained from the clipped contour gas, and denote its physical pressure
by $p_q$. Lemma~\ref{con:cutoff-locality} and the rooted estimate
\ref{sur:rooted} give, on every finite coarse union $G$,
\begin{equation}\label{con:aux-surface}
 \left|\log I_q^{\rm aux}(G)-(p_q-b_0)\operatorname{Vol}(G)\right|
 \le\sigma_0\mathcal A(G),\qquad |p_q-b_0|\le r_\beta.
\end{equation}
Here $r_\beta=O(\log\beta+\beta\delta_\beta+t_\beta)$.
If the field pressure is measured per fine site as $\pi_q$, then
$p_q=b_0+\pi_q/v_s$.

The positive bare collar kernels satisfy, by \ref{sur:collar},
for arbitrary bounded caps,
\begin{equation}\label{con:crossratio}
 K_S(v,w)=\frac{K_S(v,r_q)K_S(r_q,w)}{k_S}e^{U_S(v,w)},
 \quad k_S=K_S(r_q,r_q),\quad
 |U_S|\le\beta\epsilon_K\mathcal A(S).
\end{equation}

The marked estimates \ref{res:frozenfreeenergy} and
\ref{res:regionalinteraction} bound changes outside the source region
$E$ in $V_D^{J_E}$ by $Ct_\beta\mathcal A(E)$. Adding species meeting
$H$ costs $C\kappa$ times its fixed-neighborhood volume. Frozen-rod
comparisons add this weak boundary cost to the bare cocycle. Moreover,
\begin{equation}\label{con:weak-consistency}
 V_D^{J_E}(\phi_Er_{D\setminus E})=V_E^{J_E}(\phi_E).
\end{equation}
At fixed $L$, $t_\beta$ is a fixed $L$-dependent constant times a
polynomial in $\beta$ times $\kappa$.

Put $P=\max_qp_q$ and fix
$\sigma=C_\sigma(\sigma_0+t_\beta+\beta\delta_\beta)$, where $C_\sigma$
covers the fixed constants for area conversion, boundary terms on
parent domains, and addition of contour species. Define
\begin{align}
 \eta_{\rm cut}&=C/L+C\delta_\beta+Ce^{-cL},\\
 \mathcal E_\beta&=
 C[\sigma+\beta\epsilon_K+\beta\eta_{\rm cut}
                 +\beta e^{-cL}+t_\beta]L^2
       +C(r_\beta+t_\beta)V_L,\\
 B_*&=\beta\tau L^2-\mathcal E_\beta.\label{con:error-budget}
\end{align}
The induction requires $B_*\ge3\mathcal T$ and $B_*$ sufficiently large
compared with $\log C_1$. The sufficient conditions
$\mathcal T=\beta\tau L^2/16$ and
$\mathcal E_\beta\le\beta\tau L^2/2$ give $B_*\ge8\mathcal T$.

The root divisors satisfy
\begin{equation}\label{con:root-divisor}
 \log k_S\ge-C(\log\beta+\beta\delta_\beta)\operatorname{Vol}(S).
\end{equation}
Apply Lemma~\ref{sur:well-partition} on $S$ with both caps at $r_q$;
the bound also allows the omitted nonnegative leading terms.

\subsection{Single-contour and simultaneous-family estimates}

\begin{lemma}[Allocation of interactions to collars]
\label{con:collar-factorization}
Partition the integrated coordinates into an exterior cap $E$, a core
cap $C$, inner caps $I_j$, and collars $S_0,S_j$.  The collar $S_0$
separates $E$ from the other caps, and $S_j$ separates $I_j$ from the
other caps.  All components of each cap are treated together.  Delete
the interaction terms specified in Section~\ref{con:peierls-geometry},
so that a remaining declared support meets at most one collar.  A
support missing all collars is then assigned to its unique cap; a
support meeting a collar is assigned to that collar.  Assign by full
declared supports, even when a term is constant at the fields in
question.  With the fixed exterior coordinates included in the
appropriate cap, this gives the exact decomposition
\[
 H^{\rm cut}=H_E+H_C+\sum_jH_{I_j}
       +h_0(S_0;E,C)+\sum_jh_j(S_j;C,I_j).
\]
Suppose the restrictions, after fixing the prescribed majority label
on every collar, are products of collar-majority indicators and
cap-only indicators.  Allow any additional nonnegative cap weights,
and absorb these and $e^{-\beta H_\alpha}$ into $a_\alpha$.
Write $K_0^{\rm cut}(c,e)$ and $K_j^{\rm cut}(i,c)$ for the collar
integrals of $e^{-\beta h_0}$ and $e^{-\beta h_j}$.

\end{lemma}

\begin{proof}
A connected support meeting distinct caps crosses a collar. If it
meets only $S_j$, its caps are $C,I_j$; if it meets only $S_0$, they
are $C,E$. This proves the energy decomposition. A radius-one
correctness stencil cannot cross a width-eight collar, so after fixing
the collar labels its indicator depends on one cap. Touching inner
collars have equal labels by Lemma~\ref{con:labels-gap}.

\end{proof}

\begin{lemma}[Factorization through collar kernels]\label{con:kernel-factorization}
With the preceding allocation, let $I^{\rm cut}$ be its joint field
integral, including any nonnegative weights on the separate caps.
For each collar let $r_e$ be its phase root, and let $M_e$ bound the
total energy norm of terms restored to compare its cut kernel with
the full kernel, uniformly in every cap extension.  If the full
kernel has cross-ratio bound $\epsilon_e$, then
\[
 K_e^{\rm cut}(v,w)\le
 e^{\epsilon_e+4\beta M_e}
 \frac{K_e^{\rm cut}(v,r_e)K_e^{\rm cut}(r_e,w)}
      {k_e^{\rm cut}},\qquad
 k_e^{\rm cut}=K_e^{\rm cut}(r_e,r_e).
\]
Consequently the cut integral obeys
\begin{equation}\label{con:allocated-collar-product}
 I^{\rm cut}\le
 e^{\sum_e(\epsilon_e+4\beta M_e)}
 \frac{F_E^{\rm cut}F_C^{\rm cut}\prod_jF_j^{\rm cut}}
      {k_0^{\rm cut}\prod_jk_j^{\rm cut}},
\end{equation}
where
\begin{align*}
 F_E^{\rm cut}&=\int a_E(e)K_0^{\rm cut}(r_q,e)\,d\lambda_E(e),\\
 F_C^{\rm cut}&=\int a_C(c)K_0^{\rm cut}(c,r_q)
                 \prod_jK_j^{\rm cut}(r_{p_j},c)\,d\lambda_C(c),\\
 F_j^{\rm cut}&=\int a_j(i)K_j^{\rm cut}(i,r_{p_j})
                                                  \,d\lambda_{I_j}(i).
\end{align*}
\end{lemma}

\begin{proof}
The energy comparison gives
$|\log K_e^{\rm cut}-\log K_e|\le\beta M_e$ at each cap pairing.
The four pairings in a cross ratio thus contribute at most
$4\beta M_e$.  Tonelli's theorem first integrates the collars and
writes the cut integral as
\[
 \int a_E(e)a_C(c)\prod_ja_j(i_j)
       K_0^{\rm cut}(c,e)\prod_jK_j^{\rm cut}(i_j,c)
             \,d\lambda_E\,d\lambda_C\prod_jd\lambda_{I_j}.
\]
Apply the cross-ratio inequality pointwise.  The remaining factors
depend on separate caps, and Tonelli gives
\eqref{con:allocated-collar-product}.  The cap weights may include boundary cocycles.
For an empty inner cap the root factor cancels its divisor.
\end{proof}

In each factor, terms assigned to an incident collar are integrated
with that collar's copied variables. Its root divisor retains the
same terms at the root. A term meeting $m$ inherited domains is restored
in at most $C(1+m)$ factors; Lemma~\ref{con:seams} bounds the total cost.

\begin{lemma}\label{con:single-contour}
Suppose, in every strictly smaller physical inherited domain $H$ and for
every phase $p$,
\[
 \mathfrak Z_p^{\rm phys}(H)
 \le e^{P\operatorname{Vol}(H)+\sigma\mathcal A(H)}.
\]
Then every $q$ contour admissible in $D$ satisfies uniformly in its
compatible frozen collar variables
\begin{equation}\label{con:W-bound}
 \log W_\Gamma\le-B_*n_\Gamma
                    +(P-p_q)\operatorname{Vol}(V_\Gamma).
\end{equation}
\end{lemma}

\begin{proof}
Keep numerator and denominator of \eqref{con:raw-weight} as joint
microscopic integrals on the same $U=(A\setminus K)\cup\bigcup_jD_j$.
In the denominator apply the exact regional identity successively and
conditionally inside the $D_j$, then clip the resulting activities.
Positivity gives a lower bound with the original regional cutoff
species. Their full regional gaps make cross-sibling compatibility
automatic. Their union is one compatible subgas. By
Lemma~\ref{con:species-localization}, every added auxiliary species
meets the core or the first inherited boundary layer. The core and these
layers contain at most $C_Ln_\Gamma$ sites, by the boundary bounds above.
Lemma~\ref{con:species-cost} therefore bounds the comparison cost by
$Ct_\beta V_Ln_\Gamma$.

Use the shifted collar $O,S,V'$ constructed above and retain only
auxiliary species with actual supports in $V'$, again a positive
denominator restriction. Integrating all rods on $U$ gives, for every
restriction $R$ and source $J$, the common normalization
\[
 Z_U^{R,J}(\xi)=
 \underbrace{F_U(r_q\mid\xi)
             e^{\beta H_U^\Psi(r_q\mid r_q)}}_{\mathcal A_U(\xi,r_q)}
 \int_R e^{-\beta H_U^\Psi(\phi\mid r_q)
            -\beta B_U^{\xi,r_q}(\phi)-V_U^{J,\xi}(\phi)}d\lambda_U.
\]
The prefactor is the same for numerator and denominator and cancels.
The frozen-boundary weak comparison replaces $V_U^{J,\xi}$ by the
canonical correction with a small surface error and leaves the bare
$B_U^{\xi,r_q}$ unchanged. Truncate that cocycle to $O$, with error
$C\beta e^{-cL}L^2n_\Gamma$. Set the $O,S$ arguments in the denominator
weak correction to the root, paying $Ct_\beta\mathcal A(V')$.
Equation~\eqref{con:weak-consistency} leaves exactly
$V_{V'}^{J_{V'}}$, including its scalar. Both sides now have the same
bare collar kernel.

Let $a_\xi(w)$ denote the entire positive outer-cap amplitude, including
the possibly large retained cocycle. Equation~\eqref{con:crossratio}
gives
\[
 \int a_\xi(w)K_S(v,w)d\lambda_O(w)
 =A_\xi K_S(v,r_q)e^{\varepsilon(v)},\quad
 A_\xi=\frac{\int a_\xi(w)K_S(r_q,w)d\lambda_O(w)}{k_S},
 \quad |\varepsilon|\le\beta\epsilon_K\mathcal A(S).
\]
The same $A_\xi$ cancels. On $V_{\rm aug}=S\cup V'$, restore the $S$
reference fields and add auxiliary species meeting $S$, using the marked
comparison. The denominator is bounded below by the auxiliary field
integral on $V_{\rm aug}$ times a small exponential loss. Equation~\eqref{con:aux-surface} applies also when this domain is punctured.

Choose the inner collars $S_j\subset D_j$. Delete the wrong-root
long tails and all multi-collar interactions. Lemma~\ref{con:seams}
counts every restoration; the protected charge is untouched. Allocate
terms by full declared supports and apply
Lemma~\ref{con:kernel-factorization}. Keep both cap factors and each root divisor,
then restore the deleted terms. If
$A_0=V_{\rm aug}\setminus\bigcup_jD_j$, this yields
\[
 I_N(V_{\rm aug})\le e^{E_{\rm in}}
       \frac{I_{\rm core^+}\prod_j I_{p_j}^{\rm phys}(D_j)}
            {\prod_jk_j},\qquad
 \operatorname{Vol}(\mathrm{core^+})
 =\operatorname{Vol}(A_0)+\sum_j\operatorname{Vol}(S_j).
\]
The core integral is at most
$\exp[-\beta\tau L^2n_\Gamma+Cr_\beta
\operatorname{Vol}(\mathrm{core^+})]$ by positivity and
Proposition~\ref{con:protected}. Apply the induction hypothesis to the
unchanged $D_j$, \eqref{con:root-divisor} to the divisors, and
\eqref{con:aux-surface} to the denominator. The only nonlocal bulk
difference is exactly
\begin{align*}
 &(P-b_0)\sum_j\operatorname{Vol}(D_j)
       -(p_q-b_0)\operatorname{Vol}(V_{\rm aug})\\
 &\qquad=(P-p_q)\operatorname{Vol}(V_{\rm aug})
                   -(P-b_0)\operatorname{Vol}(A_0).
\end{align*}
The last term is a core error, and the first is at most
$(P-p_q)\operatorname{Vol}(V_\Gamma)$. All surfaces are at most
$CL^2n_\Gamma$ and all core and copied volumes at most $CV_Ln_\Gamma$.
The budget \eqref{con:error-budget} absorbs these errors and proves
\eqref{con:W-bound}.
\end{proof}

\begin{lemma}\label{con:family-bound}
Under the same smaller-domain hypothesis, a selected maximal active
$q$-family $\mathcal F$, with $N=\sum_\Gamma n_\Gamma$ and exterior
$E=D\setminus\bigcup_\Gamma V_\Gamma$, has canonical contribution
\begin{equation}\label{con:F-bound}
 \mathfrak I_{\mathcal F}
 \le\exp\{P\operatorname{Vol}(D)+\sigma\mathcal A(D)
                  -B_*N-(P-p_q)\operatorname{Vol}(E)\}.
\end{equation}
\end{lemma}

\begin{proof}
Apply \eqref{con:positive-domination}, integrate all rods together, and
use the common canonical normalization once. The result is bounded by
\[
 e^{b_0\operatorname{Vol}(D)}
 \int_{R_{\mathcal F}}
 e^{-\beta H_D^\Psi(\phi\mid r_q)-V_D^{J_E}(\phi)}d\lambda_D.
\]
Set fields outside $E$ to the root in the weak correction, paying
$Ct_\beta\mathcal A(E)$, and use \eqref{con:weak-consistency}.
Let $S_0\subset A=\bigcup_\Gamma A_\Gamma$ be the joint outer collar,
and $S_j\subset D_j$ the inherited inner collars. Put
$C=A\setminus S_0$ and $I_j=D_j\setminus S_j$. Allocate the restrictions
to product-majority collars and cap-only indicators. Delete every term
meeting multiple collars and the required long tails. Conditional on
the caps, Lemma~\ref{con:kernel-factorization} factors the collar
integrations and applies the cross ratios with complete root divisors.
Restore all terms with their multiplicities.
The exterior acquires a copy of $S_0$, the core copies of all $S_j$, and
each inherited domain remains unchanged.

Write $E^+=E\cup S_0$. In the exterior factor first restore the $S_0$
reference fields while keeping $J_E$, then add species meeting $S_0$.
The errors are $Ct_\beta\mathcal A(E)+C\kappa\operatorname{Vol}(S_0)$;
the full weak scalar is preserved. We obtain
\[
 e^{-b_0\operatorname{Vol}(D)}\mathfrak I_{\mathcal F}
 \le e^{E_{\mathcal F}}
 \frac{ I_q^{\rm aux}(E^+)
       (\prod_j I_{p_j}^{\rm phys}(D_j))I_{\rm core^+}}
      {k_0\prod_jk_j},
\]
where
\[
 \operatorname{Vol}(E^+)=\operatorname{Vol}(E)+\operatorname{Vol}(S_0),
 \quad
 \operatorname{Vol}(\mathrm{core^+})
       =\operatorname{Vol}(A)+\sum_j\operatorname{Vol}(S_j).
\]
The copied field volumes obey
\[
 \operatorname{Vol}(E^+)+\sum_j\operatorname{Vol}(D_j)
 +\operatorname{Vol}(\mathrm{core^+})
 -\operatorname{Vol}(S_0)-\sum_j\operatorname{Vol}(S_j)
 =\operatorname{Vol}(D).
\]
Apply the auxiliary surface estimate, the physical induction, the
root-divisor trial, and the protected charge. The bulk expression is
\begin{align*}
 &b_0\operatorname{Vol}(D)
 +(p_q-b_0)[\operatorname{Vol}(E)+\operatorname{Vol}(S_0)]
 +(P-b_0)\sum_j\operatorname{Vol}(D_j)\\
 &=P\operatorname{Vol}(D)-(P-p_q)\operatorname{Vol}(E)
   +(p_q-b_0)\operatorname{Vol}(S_0)
   -(P-b_0)\operatorname{Vol}(A).
\end{align*}
The last two terms have magnitude at most $Cr_\beta V_LN$.
Furthermore $\mathcal A(E^+)\le\mathcal A(D)+CL^2N$ and
$\sum_j\mathcal A(D_j)\le CL^2N$. The constant $\sigma$ covers the parent boundary terms; the other
errors enter
\eqref{con:error-budget}, leaving $-B_*N$.
\end{proof}

\subsection{The numerical tree and the physical pressure}

The simultaneous induction below adapts Zahradn\'ik's stability
argument~\cite[Theorem~1.7 and Section~1.8]{Zahradnik84}:
an active contour must be large relative to the pressure deficit,
and summing such contours yields the physical pressure bound.

\begin{proposition}[Finite-volume pressure bounds]\label{con:physical-bounds}
For the canonical model constructed above, let the error budget
\eqref{con:error-budget} satisfy $B_*\ge3\mathcal T$, with $B_*$
sufficiently large. Then for every physical regional domain $D$ and
phase $q$,
\begin{equation}\label{con:physical-sandwich}
 e^{p_q\operatorname{Vol}(D)-\sigma\mathcal A(D)}
 \le\mathfrak Z_q^{\rm phys}(D)
 \le e^{P\operatorname{Vol}(D)+\sigma\mathcal A(D)}.
\end{equation}
\end{proposition}

\begin{proof}
Induct simultaneously over all phases and physical domains, ordered by
their number of coarse cubes. The cutoff pressures and $P$ are already
defined. The smaller-domain hypothesis gives both preceding lemmas.
Put $a=(P-p_q)V_L\ge0$. If a contour is active, then
\[
 -\mathcal T n<\log W_\Gamma\le-B_*n+Ca n^{3/2},
\]
so for $a>0$ its size is at least
\[
 N(a)=\max\left\{1,
       \left\lceil c\left(\frac{B_*-\mathcal T}{a}\right)^2\right\rceil\right\}.
\]
At $a=0$ there are no active contours. Extracting the common factor
$e^{P\operatorname{Vol}(D)+\sigma\mathcal A(D)}$ from
\eqref{con:F-bound} leaves a sum of weights
$e^{-B_*\sum n_\Gamma}e^{-a n(E)}$ over disjoint removed regions.

Expose the first unexamined coarse site. Either it belongs to $E$, with
weight $e^{-a}$, or to one removed region of size $n\ge N(a)$, with at
most $C_1^n$ choices by Lemma~\ref{con:counting}. Erase that entire
removed region and continue; its $q$ holes remain available. Induction
on the number of remaining sites bounds this sum by one provided
\[
 e^{-a}+\sum_{n\ge N(a)}C_1^ne^{-B_*n}\le1.
\]
For $a\ge1$ the geometric tail is smaller than $1-e^{-1}$ for large
$B_*$. For $0<a\le1$, the hypotheses give a tail at most
$C\exp[-cB_*^3/a^2]\le a/2\le1-e^{-a}$, uniformly in that interval.
The case $a=0$ has only the empty active family. This proves the upper
bound. 

For the lower bound, clip the exact regional gas before enlarging it.
Positivity makes the physical partition at least this original regional
cutoff partition. Apply Lemma~\ref{con:species-localization} with one domain and empty
core. Added species meet its first boundary layer, whose size is bounded
by $C\mathcal A(D)/L^2$. Lemma~\ref{con:species-cost} gives a surface
cost, proving the lower bound by the choice of $\sigma$.

\end{proof}

\begin{theorem}[Pressure maximum]\label{con:pressure-maximum}
For the parameter range in Proposition~\ref{con:physical-bounds},
\begin{equation}\label{con:maximum}
 p_\varepsilon(h)=\max_qp_q.
\end{equation}
\end{theorem}
\begin{proof}
 At fixed $\beta$, emptying a fixed-width boundary strip costs at most
activity times strip volume and removes the physical exterior. The
fixed-field reference admits the same comparison by its ideal-gas cell
bound and positive empty-cell weight. Each conditional Gaussian law
assigns a root neighborhood probability at least $e^{-C\beta}$,
uniformly in the reset count, so imposing the regional $q$ band also
costs a surface term. Thus $F_D(r\mid\xi)$ has bulk
$e^{b_r\operatorname{Vol}(D)}$, uniformly in $\xi$, which cancels in
the canonical ratio; $\sigma_D(r)$ is a surface correction.  Divide \eqref{con:physical-sandwich} by volume along boxes.
The upper bound gives a pressure limsup at most $P$, and the lower
bound for a maximizing phase gives a liminf at least $P$. This proves
\eqref{con:maximum}.
\end{proof}



\section{The crossing and the density jump}\label{trn:section}

We first record the elementary pressure facts, then compare the source
responses of the phase pressures. All pressures are per Euclidean
volume. The mesh, count caps, field interval, phase neighborhoods,
and contour index sets remain fixed as $h$ varies.

\begin{lemma}[Thermodynamic pressure]\label{mod:thermo}
For each $\varepsilon>0$, the limit defining $p_\varepsilon(h)$ exists,
is finite, and is convex in $h$.
\end{lemma}
\begin{proof}
We use the separated-box construction for the thermodynamic limit
\cite[Section~2]{Ruelle63}.
Dropping exclusions between boxes gives a product upper bound.
Boxes separated by an empty strip wider than $1+\varepsilon$ give a
product lower bound. Deleting a set $S$ of allowed centres changes
$\log Z_\Lambda$ by at most $z|S|$: integrate its particles with all
exclusions dropped for the upper bound, and leave it empty for the
lower bound. Tile large cubes with boxes of fixed side, then let that
side increase. The upper and lower bounds converge to the same limit;
the strip estimate also removes exterior boundary effects. Finally,
$0\le\log Z_\Lambda\le z|\Lambda|$, and finite-volume log partition
functions are convex in $h$.
\end{proof}

\begin{lemma}[Density response]\label{mod:density-response}
At every differentiability point of $p_\varepsilon$, the finite-volume
number densities converge to $p_\varepsilon'$. The one-sided derivatives
are the corresponding limits as $h$ approaches through differentiability
points from the left or right.
\end{lemma}
\begin{proof}
The first two derivatives of $\log Z_\Lambda$ are $\mathbb E N_\Lambda$
and $\Var N_\Lambda$. Convex difference-quotient bounds give convergence
of the first derivatives wherever the limiting pressure is
differentiable. Monotonicity of the derivative gives its one-sided limits;
these are the convex convergence principles of
\cite[Theorems~24.1 and~24.5]{Rockafellar}.
\end{proof}

\subsection{Source response of the joint model}

Let $D$ be a union of complete fine spatial cells. For a field event $E$
and a joint perturbation $J_h=\sum_AJ_{A,h}$, let
$Z_D^{E,J}(h\mid\xi)$ be the integral of the joint rod--field density
\eqref{mod:fullfield} over $E$, multiplied by $e^{-J_h}$.
The exterior rods $\xi$ and the event $E$ are independent of $h$.
We assume
\begin{equation}\label{trn:joint}
 \sup_i\sum_{A\ni i}\|J_{A,h}\|_\infty\le\kappa,
 \qquad
 \sup_i\sum_{A\ni i}\operatorname{Lip}_h(J_{A,h})<\infty.
\end{equation}
Fix $0<h_*\le H$. The constants below are uniform for $|h|\le h_*$.

\begin{lemma}[Source score]\label{trn:score}
For almost every $h$, the logarithmic derivative of the positive
joint density is
\begin{equation}\label{trn:score-id}
 \mathcal S_h=
 \sum_{i\in I(D)}\left[
 \frac{\beta v_i}{2\alpha}(\phi_i-h)
 +N_i-\widehat N_i-\partial_h\log c_{i,h}(\widehat N_i)\right]
 -\partial_hJ_h.
\end{equation}
\end{lemma}
\begin{proof}
Differentiate the physical activity factor and the attached Gaussian
in \eqref{mod:gaussian}. Their count terms combine to
$N_i-\widehat N_i$. The reset count is independent of $h$.
The Lipschitz assumption gives the derivative of $J_h$ almost everywhere.
\end{proof}

\begin{lemma}[Exceptional counts]\label{trn:rare-counts}
If $\rho>e\zeta e^{h_*}$, then, uniformly over $E$ and compatible
exterior rods with $Z_D^{E,J}>0$,
\begin{equation}\label{trn:bad}
 \mathbb E_h^{E,J}\sum_{i\in I(D)}(N_i-\widehat N_i)
 \le C\beta|D|e^{-c\beta}.
\end{equation}
\end{lemma}
\begin{proof}
Condition on the fields and the rods outside cell $i$. Emptying that
cell changes $J_h$ by at most $2\kappa$. In \eqref{mod:cellweight},
the empty term and every term with $n>K_i$ have the same Gaussian
normalizer. Divide by the empty term of the conditional partition
function and drop the hard exclusions. For $F\ge0$,
\begin{equation}\label{trn:tail}
 \mathbb E\bigl[F(N_i)\1_{\{N_i>K_i\}}\mid\phi,\eta_{i^c}\bigr]
 \le e^{2\kappa}\sum_{n>K_i}F(n)
             \frac{(\beta\zeta e^{h_*}v_i)^n}{n!}.
\end{equation}
Apply the scalar tail bound in Lemma~\ref{res:tails} with $m=1$
and $H$ replaced by $h_*$. It gives $C\beta e^{-c\beta}$ per
marked cell. There are at most $|D|/v_-$ such cells, proving the claim.
\end{proof}

The Gaussian means stay at distance at least $d_0>0$ from the endpoints.
The endpoint densities therefore give
\[
 1-c_{i,h}(m)\le2e^{-\beta v_i d_0^2/(4\alpha)},\qquad
 |\partial_hc_{i,h}(m)|
 \le C\sqrt{\beta v_i}\,e^{-\beta v_i d_0^2/(4\alpha)}.
\]
In particular, $c_{i,h}(m)\ge1/2$ for large $\beta$, and
\begin{equation}\label{trn:normalizer}
 \sum_{i\in I(D)}|\partial_h\log c_{i,h}(\widehat N_i)|
 \le C\beta|D|e^{-c\beta}.
\end{equation}
These bounds justify differentiation under the finite-volume integral,
so $\partial_h\log Z_D^{E,J}=\mathbb E_h^{E,J}\mathcal S_h$ almost everywhere.

\subsection{Differentiating the cutoff gas}

The numerator and denominator of $W_\Gamma$ are joint integrals on
the same region with the same frozen collar. Their restrictions are
independent of $h$. If $n_\Gamma$ is the number of support cubes and
$V_\Gamma$ its removed region, the geometric bound gives
$|V_\Gamma|\le CV_Ln_\Gamma^{3/2}$. Applying the preceding estimates
with $J=0$ to both integrals yields
\begin{equation}\label{trn:raw-deriv}
 |\partial_h\log W_\Gamma|\le C\beta V_Ln_\Gamma^{3/2}.
\end{equation}
A zero numerator has an allowed set of zero reference measure and
vanishes for every $h$. Extend the cutoff by zero off its fixed
compatible collar set. For an $h$-independent threshold $\mathcal T$,
\begin{equation}\label{trn:clip-deriv}
 w_\Gamma=\min\{W_\Gamma,e^{-\mathcal T n_\Gamma}\},\qquad
 |\partial_hw_\Gamma|
 \le C\beta V_Ln_\Gamma^{3/2}e^{-\mathcal T n_\Gamma}
\end{equation}
almost everywhere. Its declared marked-cell support is unchanged, with
$|S_\Gamma|\le d_Ln_\Gamma$.

\begin{lemma}[Differentiated cutoff expansion]\label{trn:cutoff}
For every fixed $a>0$, the interaction $J_h=-\log G_h$ of the cutoff
gas, grouped by declared union supports, satisfies
\begin{align}
 \sup_i\sum_{A\ni i}e^{a|A|}\|J_{A,h}\|_\infty
 &\le C_Le^{-\mathcal T/2},                                 \label{trn:J}\\
 \sup_i\sum_{A\ni i}e^{a|A|}\operatorname{Lip}_h(J_{A,h})
 &\le C_L\beta e^{-\mathcal T/4},                            \label{trn:dJ}
\end{align}
when $\mathcal T$ exceeds a constant depending on $a$, the mesh, and $L$.
\end{lemma}
\begin{proof}
Since $|S(C)|\le d_Ls(C)$, \eqref{trn:J} follows from
\eqref{con:cluster-reserve} with $b=ad_L$.
Differentiating a cluster term and using \eqref{trn:clip-deriv}
bounds its absolute majorant by
$C\beta V_L\sum_jn_{\Gamma_j}^{3/2}\le C\beta V_Ls(C)^{3/2}$
times the original majorant. Absorb this factor with
$s^{3/2}\le Ce^{\mathcal T s/4}$, and apply the same lemma with
threshold $3\mathcal T/4$. Its bound
$C_L\beta e^{-3\mathcal T/8}\le C_L\beta e^{-\mathcal T/4}$
proves \eqref{trn:dJ}. This is the rooted derivative expansion of
\cite[Section~2, equations~(2.6)--(2.7)]{FP}.
Uniform absolute convergence permits integration between source
values and gives the stated Lipschitz norm.
\end{proof}

\subsection{Increment bounds for the phase pressures}

Let $[m_q^-,m_q^+]$ contain the masses of the wells in component $q$.
On the $q$-majority event, the good-cell tolerance and minority fraction
imply
\begin{equation}\label{trn:majority}
 m_q^- -\omega_0\le
 M_D(\phi):=\frac1{2\alpha|D|}\sum_{i\in I(D)}v_i\phi_i
 \le m_q^+ +\omega_0.
\end{equation}
Indeed, if a good spatial cell has mass error at most $\eta/(2\alpha)$
and at most a fraction $\theta$ of cells are exceptions, one may take
$\omega_0=\eta/(2\alpha)+R\theta/\alpha$. Apply this estimate block
by block; boundary blocks have vanishing volume fraction.

\begin{lemma}[Microscopic realization of a branch]\label{trn:branch-realization}
Let $Z_{D,q}^{\rm cut}(h)$ be the complete joint partition function
restricted to $q$-majority fields and perturbed by the regional cutoff
interaction $J_{q,D}$. With a fixed compatible exterior configuration,
\[
 \lim_{D\uparrow\mathbb R^3}|D|^{-1}\log Z_{D,q}^{\rm cut}(h)=p_q(h).
\]
\end{lemma}
\begin{proof}
The bare boundary cocycle, the frozen weak comparison
\eqref{res:regionalfield}, and the rooted surface estimate give
\[
 \log Z_{D,q}^{\rm cut}(h)
 =\bigl(b_0(h)+\pi_q(h)/v_s\bigr)|D|+O_\beta(A(D)).
\]
Divide by the volume of aligned boxes and use
\eqref{sur:physical-branch}.
\end{proof}

\begin{lemma}[Branch increments]\label{trn:branch-increments}
Put $Q_q(h)=\beta^{-1}p_q(h)+h^2/(4\alpha)$. There is
$\sigma_\beta=Ce^{-c\beta}+C_Le^{-\mathcal T/4}\to0$ such that, for $h_1<h_2$,
\begin{equation}\label{trn:increments}
 (m_q^- -\omega_0-\sigma_\beta)(h_2-h_1)
 \le Q_q(h_2)-Q_q(h_1)
 \le(m_q^+ +\omega_0+\sigma_\beta)(h_2-h_1).
\end{equation}
\end{lemma}
\begin{proof}
The source score, exceptional-count bound, and cutoff estimate imply
\begin{equation}\label{trn:branch-score}
 \frac{\partial_h\log Z_{D,q}^{\rm cut}}{\beta|D|}
 =\mathbb E_{D,q,h}M_D-\frac h{2\alpha}+\mathcal E_{D,q}(h),
 \qquad |\mathcal E_{D,q}(h)|\le\sigma_\beta.
\end{equation}
Conversion from the marked-cell norm uses the positive minimum of $v_i$.
Integrate in finite volume, use \eqref{trn:majority}, and pass to the
pressure limit supplied by the preceding lemma.
\end{proof}

\subsection{Choosing the scales}

Reduce the homogeneous mass gap so that $0<\Delta\le r_I$.
Choose the parameters in the following order.
\begin{enumerate}[label=\arabic*.]
\item Fix $H>0$, then choose $\alpha>\|T\|+1$,
$\rho>e\zeta e^H$, and $R>H+2\alpha\rho+2$.
With $M=\zeta e^{R+1}$, choose the spatial and angular mesh to satisfy
the coercivity and covariance estimates. Its marked-cell volumes are
now fixed; the margin absorbs the ceiling in $K_i$ for large $\beta$.
\item Choose disjoint phase neighborhoods preserving the mass gap,
with good-cell mass error at most $\Delta/16$. Fix a rigidity patch,
a radius retaining half its positive defect charge, and $k\ge24$.
The full leading interaction is retained.
\item The protected charge has coefficient $\tau=c_P\theta^{2/3}$.
Choose $\theta$ so that both the minority mass error and $C_m\theta/\tau$
are sufficiently small. The constant $C_m$ is independent of the
subsequent net and interpolation choices.
\item Choose a well net of radius $\nu\le1$, its finite path family,
and the interpolation width $R_{\rm int}$, so that $C\nu$ and
$CD_\nu^2/R_{\rm int}$ are small fractions of $\tau$. Choose $\eta_0>0$
such that
\[
 C\left(1+\frac{R_{\rm int}N_\nu}{\nu}\right)\omega(\eta_0)
 \le c_0\tau,
\]
where $c_0$ is a sufficiently small fixed constant. Then choose $L$
large enough for the geometric margins, $C/L\le\eta_0/3$, and
$C/L+Ce^{-cL}\ll\tau$.
\item The weak interaction obeys
$\delta_\beta(h)\le C|h|+\delta_{{\rm micro},\beta}$, with
$\delta_{{\rm micro},\beta}\to0$.
Choose $0<h_*\le H$ so that $CLh_*$ and $Ch_*$ fit the preceding
error bounds and $h_*/(2\alpha)\le r_I/4$.
Finally increase $\beta$ to control the microscopic remainder,
$L\log\beta/\beta$, and the Gaussian and exceptional-count errors.
\end{enumerate}
All choices except $\beta$ are now fixed. Set
\begin{equation}\label{trn:T}
 \mathcal T=\frac{\beta\tau L^2}{16}.
\end{equation}
This threshold satisfies Lemma~\ref{trn:cutoff} for large $\beta$.
Equations~\eqref{res:tailmoments} and~\eqref{trn:J} give
$\beta(q_\beta+\kappa)\to0$, verifying~\eqref{res:smallness}.
For the error $\mathcal E_\beta$ in \eqref{con:error-budget},
$r_\beta/\beta\le Ch_*+o(1)$ and $t_\beta/\beta\to0$.
After division by $\beta L^2$, the volume term is $O(Lh_*)+o(1)$;
the other terms are controlled by the successive choices above.
Taking their finitely many allocated fractions sufficiently small gives
\begin{equation}\label{trn:B}
 B_*=\beta\tau L^2-\mathcal E_\beta
 \ge\tfrac12\beta\tau L^2=8\mathcal T.
\end{equation}

\begin{lemma}[Equal limiting intercepts]\label{trn:intercept-comparison}
At zero source,
\begin{equation}\label{trn:intercepts}
 \max_q\left|\frac{p_q(0)-b_0(0)}\beta\right|
 \le C\left(\frac{\log\beta}{\beta}+\delta_{{\rm micro},\beta}
           +\frac{t_\beta}\beta+\beta^{-4}\right)\longrightarrow0.
\end{equation}
\end{lemma}
\begin{proof}
At $h=0$, the canonical perturbation and the regional cutoff
correction have combined energy norm at most
$C(\delta_{{\rm micro},\beta}+t_\beta/\beta)$, by
\eqref{sur:source-norm} and Proposition~\ref{res:regionalinteraction}.
Apply \eqref{sur:small-pressure} with this norm and use
$p_q=b_0+\pi_q/v_s$. Division by $\beta$ gives the bound.
\end{proof}

\subsection{Proof of the main theorem}

The source adjustment is the LMP coexistence mechanism
\cite[Section~6, proof of Statement~3.3]{LMP99}: separated density responses force a crossing
of the corrected phase pressures.

\begin{proof}[Proof of Theorem~\ref{trn:corner}]
The reserve \eqref{trn:B} satisfies the quantitative conditions of
Proposition~\ref{con:physical-bounds}. Theorem~\ref{con:pressure-maximum}
therefore gives
\begin{equation}\label{trn:maximum}
 p_\varepsilon(h)=\max_qp_q(h),\qquad |h|\le h_*.
\end{equation}
The isotropic component has mass $r_I$ and the others have mass at
least $r_I+\Delta$. Choose the tolerances above so that
$\omega_0\le\Delta/8$, and take $\beta$ large enough that
$\sigma_\beta\le\Delta/16$. Equation~\eqref{trn:increments} gives
\begin{align}
 (r_I-3\Delta/16)(h_2-h_1)
 &\le Q_I(h_2)-Q_I(h_1)
 \le(r_I+3\Delta/16)(h_2-h_1),                       \label{trn:iso-slope}\\
 Q_q(h_2)-Q_q(h_1)
 &\ge(r_I+13\Delta/16)(h_2-h_1),\quad q\ne I.       \label{trn:nem-slope}
\end{align}
There are finitely many components, and their increment bounds imply
continuity. Thus $D_\beta(h)=\max_{q\ne I}Q_q(h)-Q_I(h)$ is continuous.
Choosing a maximizing component at $h_1$ shows that
\begin{equation}\label{trn:strict-cross}
 D_\beta(h_2)-D_\beta(h_1)\ge\frac{5\Delta}{8}(h_2-h_1).
\end{equation}
By \eqref{trn:intercepts}, $D_\beta(0)\to0$. Consequently $D_\beta$
has a unique zero $h_\varepsilon\in(-h_*,h_*)$ for large $\beta$, and
$|h_\varepsilon|\le8|D_\beta(0)|/(5\Delta)\to0$.

To the left of the zero the physical pressure equals the isotropic
branch; to the right it equals the maximum of the other branches.
Convexity gives one-sided derivatives. Their secant bounds are
\begin{align*}
 p'_{\varepsilon,-}(h_\varepsilon)
 &\le\beta\left(r_I+\frac{3\Delta}{16}
                         -\frac{h_\varepsilon}{2\alpha}\right),\\
 p'_{\varepsilon,+}(h_\varepsilon)
 &\ge\beta\left(r_I+\frac{13\Delta}{16}
                         -\frac{h_\varepsilon}{2\alpha}\right).
\end{align*}
Therefore
\begin{equation}\label{trn:corner-jump}
 p'_{\varepsilon,+}(h_\varepsilon)-p'_{\varepsilon,-}(h_\varepsilon)
 \ge\frac{5\Delta}{8\varepsilon}\ge\frac{\Delta}{2\varepsilon}.
\end{equation}
The isotropic lower increment bound, $\Delta\le r_I$, and
$h_*/(2\alpha)\le r_I/4$ give a positive lower bound proportional to
$\beta$ for both derivatives. The upper increment bounds give the
matching upper bound. Setting $a_\varepsilon=a_t+h_\varepsilon$
proves the theorem for $P_\varepsilon(a)=p_\varepsilon(a-a_t)$.
\end{proof}

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\end{thebibliography}
\end{document}