\input{glyphtounicode-cmex}
\pdfgentounicode=1
\pdfglyphtounicode{tfm:esint10/acute}{222B}
\pdfglyphtounicode{tfm:esint10/circumflex}{222B}
\pdfglyphtounicode{tfm:esint10/dieresis}{222C}
\pdfglyphtounicode{tfm:esint10/ffl}{2A0F}
\pdfglyphtounicode{tfm:esint10/space}{2A0F}
\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools,esint}
\usepackage{microtype}
\usepackage{enumitem}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,positioning,calc}
\usepackage[colorlinks=true,linkcolor=blue!50!black,citecolor=blue!50!black,urlcolor=blue!50!black]{hyperref}
\hypersetup{pdftitle={Uniform real Lipschitz surfaces on the triangular lattice (v2)},pdfauthor={OpenAI}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\E}{\mathbb E}
\newcommand{\Prob}{\mathbb P}
\newcommand{\R}{\mathbb R}
\newcommand{\Z}{\mathbb Z}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\eps}{\varepsilon}
\DeclareMathOperator{\dist}{dist}
\DeclareMathOperator{\diam}{diam}
\DeclareMathOperator{\osc}{osc}
\numberwithin{equation}{section}
\title{Uniform real Lipschitz surfaces\\on the triangular lattice}
\author{OpenAI}
\date{September 25, 2026}
\begin{document}
\maketitle

\begin{abstract}
Take a bounded smooth simply connected domain $D\subset\R^2$ with two
marked boundary points $a,b$ splitting its boundary into two open arcs
$A_+$ and $A_-$. On a triangular-lattice approximation $D_\delta$ of mesh
$\delta$, sample a real function on the vertices uniformly from the
polytope cut out by two kinds of constraints:
\begin{itemize}
\item boundary values $+\lambda$ on $A_+$ and $-\lambda$ on $A_-$;
\item $|h(x)-h(y)|\le1$ on every nearest-neighbor pair $x\sim y$.
\end{itemize}
We prove two scaling limits for this model, both announced as Schramm's
Problem~2.3 and both new here.

First, the height field, recentered and scaled, converges in distribution
to a multiple of the Dirichlet Gaussian free field on $D$. The multiple
$\sigma$ is tied to an effective tangent-flux coefficient $A$, via
$\sigma^2=(vA)^{-1}$ with $v=2/\sqrt3$ the vertex density.

Second, at one tuned amplitude $\lambda^2=\pi/(8vA)$, the zero-height
interface from $a$ to $b$ converges in uniform curve distance to chordal
$\mathrm{SLE}_4$. The tuning is forced: the two limits interchange a
covariance identity, and it pins $\lambda$ down to $\sigma\sqrt{\pi/8}$.

Neither constant has a closed elementary form; both are defined through
a stationary tangent problem on the dual lattice. The whole point of
the paper is that those two constants are \emph{universal} --- they do
not depend on the shape of $D$ or the position of $a,b$.
\end{abstract}
\tableofcontents

\section{The Model and the Main Theorem}\label{sec:model}

Picture a bounded smooth simply connected region $D$ in the plane, like a
smooth potato, with two marked boundary points $a$ and $b$. Between them
the boundary splits into two open arcs. On the lower arc put the height
$-1/2$, on the upper put $+1/2$, and let heights evolve on the inside of
$D$ as far as they like --- but neighbours may never differ by more than
one unit. That is, we are spreading a real-valued ``sand surface''
across $D$, pinned to the two boundary values, and hard-constrained to
slope by at most one per lattice step. Sample the whole surface uniformly
from the polytope of all such fillings.

This is a hard-constraint Gibbs model: the potential is zero when every
neighbouring pair satisfies $|h(x)-h(y)|\le1$, and infinite otherwise. It
differs from the classical gradient interface models in one consequential
way: the potential is \emph{not smooth}. At the boundary of the admissible
set, the constraint bites. So none of the standard smooth-potential
homogenisation tools apply directly. Nevertheless the interface and the
field do homogenise, and they do so to the same limits as the smoother
models. This paper is entirely about that statement and the two constants
that come out of it.

The interface --- the contour separating positive from negative heights ---
is what becomes SLE$_4$. The fluctuation of the surface itself --- the
Gaussian component of the field --- is what becomes the Gaussian free
field. Both limits are dimension-driven: two-dimensional hard-constraint
surfaces have a single Gaussian degree of freedom.

\subsection{Formal setup}

Let $D\subset\R^2$ bounded, smooth, simply connected, with distinct marked
boundary points $a,b$. Write $A_+$ and $A_-$ for the two open arcs between
them. Use the unit triangular lattice with basis
\[
 e_1=(1,0),\qquad e_2=(1/2,\sqrt3/2),
\]
and nearest-neighbour step set $\pm e_1,\pm e_2,\pm(e_1-e_2)$.

For $\delta>0$ let $D_\delta$ be a simply connected triangular-lattice
polygon whose marked boundary points converge uniformly to
$(D; a,b)$ as $\delta\to0$. Put the two discrete marks at the midpoints of
two distinct boundary edges. The open arcs of $D_\delta$ between them
partition the boundary vertices; no vertex is charged two boundary values.

For $0<\lambda\le1/2$, define the model $h_\delta$ as the uniform
(Lebesgue) sample from
\begin{equation}\label{eq:model}
 h_\delta=\lambda\text{ on }A_{+,\delta},\qquad
 h_\delta=-\lambda\text{ on }A_{-,\delta},\qquad
 |h_\delta(x)-h_\delta(y)|\le1\quad(x\sim y).
\end{equation}
The constraint set is a compact convex polytope of positive volume and it
is nonempty in its free coordinates. To see nonemptiness, put $0$ on every
free vertex: all inequalities involving a free vertex are then strict,
and the fixed--fixed inequalities are satisfied at $2\lambda\le1$. To see
boundedness, note every vertex is at finite graph distance from the
boundary, and Lipschitzness propagates boundary bounds into the interior.

Extend heights affinely across triangular faces. Almost surely no free
vertex is exactly zero: that event is a hyperplane of measure zero. Each
triangle then carries either no zero segment or a unique segment joining
two edges. Those segments glue into loops plus one chordal interface
$\gamma_\delta$ joining the two discrete marks. The chord is simple:
no face contains two strands, no vertex is a branching point. Replacing
each within-face linear piece by the corresponding dual strand moves the
curve by at most $O(\delta)$ in the uniform metric, so no loss of
generality comes from passing to the dual representative.

For curves $\alpha,\beta$ define uniform distance modulo increasing
reparametrisation by
\[
 d_{\mathrm{unif}}(\alpha,\beta)
 =\inf_{\phi,\psi}\sup_{t\in[0,1]}
       |\alpha(\phi(t))-\beta(\psi(t))|,
\]
where $\phi,\psi$ range over increasing homeomorphisms of $[0,1]$, and
curves at zero distance are identified. The Dirichlet GFF convention
throughout is covariance $G_D=(-\Delta_D)^{-1}$.

\subsection{The theorem}

\begin{theorem}\label{thm:main}
There are constants $A>0$, $\sigma>0$, and $\lambda\in(0,1/2]$, independent
of $D,a,b$, such that the model \eqref{eq:model} has
\[
 \gamma_\delta\Longrightarrow\mathrm{SLE}_4(D;a,b),\qquad
 \frac{h_\delta-\E h_\delta}{\sigma}
       \Longrightarrow\mathrm{GFF}_D
\]
in uniform curve distance and as random distributions respectively. The
constants and the limiting mean are
\begin{align}
 v&=2/\sqrt3,&\sigma^2&=(vA)^{-1},&
 \lambda^2&=\frac{\pi}{8vA},\label{eq:target-constants}\\
 m_D(z)&=\lambda\bigl[\omega_D(z,A_+)-\omega_D(z,A_-)\bigr],&&&
 \label{eq:target-mean}
\end{align}
where $\omega_D$ is harmonic measure. The coefficient $A$ is the
tangent-flux scalar defined in the cell problem of Section~\ref{sec:cell}.
The stiffness of the field is $vA$. No closed elementary form is proposed
for $A$, and none is needed for either limit.
\end{theorem}

The number $\sigma\sqrt{\pi/8}$ is the universal ``height gap'': the
boundary trace of the conditional mean on a strand side. It is not put
in by hand; Section~\ref{sec:interface} forces it by equating two ways of
computing a covariance. The normalization of chordal SLE$_4$ in the upper
half-plane is $\partial_t g_t(z)=2/(g_t(z)-2B_t)$ with $B$ standard
Brownian motion. There is no logarithmic rescaling of the heights in the
distributional limit. Pointwise height tightness is neither asserted nor
required.

\subsection{Why this is new, and what is around it}

Schramm's Problem~2.3 \cite{Schramm} asks for exactly these SLE$_4$ and
GFF limits for the real-valued hard-constraint surface. It is distinct
from the integer-valued case in Schramm's Problem~2.2, which is close to
the discrete Gaussian free field. The two limits here --- in fact the
problem's own challenge --- are the ones where the constraint is not
convex-smooth, so the classical homogenisation tools do not apply.

Schramm--Sheffield established the discrete contour result
\cite{SSdiscrete} for the integer-valued case, where the surface is a
GFF-type object to start with. Miller \cite{Miller} extended contour
universality to a class of smooth symmetric uniformly convex gradient
potentials; the potential here is the flat $[-1,1]$ followed by
$+\infty$, which is not in that class. An approximation by smooth
potentials is therefore not by itself a solution, because the sharp
edges of the constraint are what the limit sees.

On the field side, Naddaf--Spencer \cite{NaddafSpencer} and
Giacomin--Olla--Spohn \cite{GiacominOllaSpohn} proved homogenisation for
$\nabla\phi$ interface models with smooth symmetric convex interactions.
Miller \cite{MillerField} handled bounded-domain field fluctuation
limits for that class. Mi\l o\'s and Peled
\cite{MilosPeled} studied hard-core delocalisation on even square tori
with one pinned height, including the Lipschitz potential; their
gradient-control arguments are a starting point here. The later iterative
gradient-control paper of Cohen-Alloro and Peled
\cite[Section~6]{CAP} extends delocalisation to zero-pinned square
domains. Buchholz--Cotar--Schweiger \cite{BuchholzCotarSchweiger} proved
infinite-volume gradient universality for a wider class of non-convex
potentials; their scaling hypothesis still requires smooth, uniformly
curved mixture components and so does not capture the hard potential.
None of these give the bounded-domain field limit or the curve limit
together with the tuned amplitude.

The present paper starts where \cite{CAP} stops: with zero-pinned
gradient control. It then does five new things. (i) A local volume
facet-deficit estimate of a piecewise-linear polytope, on the triangular
lattice. (ii) A synchronous reflected coupling whose contact events
produce weighted control of differences on every locally sampled patch.
(iii) A tangent problem on the dual lattice: a stationary bulk limit
that identifies a scalar flux law, and hence a Gaussian field via
small tilts. (iv) A trace theorem on the closed observed set, so that
zero boundary values propagate even when the observed set has positive
area. (v) A feasible-shift construction, so that crossing estimates
convert into barriers for the interface, and hence into the height gap
and stopped harmonic observable that feed the SLE$_4$ criterion.

Each of these is new on its own. Only their combination gives the two
limits of Theorem~\ref{thm:main} with the claimed uniformity in the
domain and in the stopping rule.

\subsection{Plan of the paper}

The proof has four layers, and it is useful to know them separately because
each contributes an independent statement.

\textbf{Local slack and sparse defects (Section~\ref{f:section}).} The
controlled-gradient theorem of Cohen-Alloro and Pedel \cite{CAP} bounds
the probability that many disjoint edges simultaneously have gradient
close to $1$. We turn this into a local volume-deficit estimate (via
Pr\'ekopa log-concavity \cite{Prekopa}) and then into exponentially small
probabilities for sparse bad blocks on the triangular lattice. The output
is unscaled moment bounds for macroscopic averages.

\textbf{Comparison via synchronous reflection (Section~\ref{s:coupling}).}
Two independent copies of the height inside the same polytope are put
into a synchronous reflected Brownian coupling. The contact events of
the coupling become the driving mechanism. We obtain \emph{weighted}
control of the difference and of its Sobolev norm, uniformly in the shape
and size of the ambient domain, and uniformly under adapted exact-value
observations.

\textbf{The tangent problem and the field limit (Sections~\ref{sec:cell}
and~\ref{sec:trace-analysis}).} Retaining a short time step of the coupling and
averaging over volume-sampled vertices brings out a stationary
\emph{tangent} process: the limiting object is a finite product of
averaging projections on edges that are momentarily in contact. This
process defines a mean flux $j(q)=Aq$. A small positive forcing
identifies $A>0$, small smooth tilts identify the Gaussian field with
covariance $(vA)^{-1}(-\Delta_D)^{-1}$, and the trace theorem extends
this to conditional covariance on closed observed sets.

\textbf{Feasible shifts and the curve limit (Sections~\ref{sec:feasible-shifts}
through~\ref{sec:interface}).} A shift that raises or lowers the surface
across a corridor near a prescribed barrier can be made feasible for
almost every realisation, with a bounded determinant cost. This converts
crossing estimates into uniform barrier lower bounds. Barriers give the
sampled-interface separation, then the universal height gap $l=\sigma
\sqrt{\pi/8}$, then the stopped harmonic observable needed for the
SLE$_4$ driving-function criterion. Forward and reverse driver
convergence, plus the two-orientation path criterion of Sheffield--Sun
\cite{SSun}, close the curve argument.

The demanding uniformity is part of the assertions, not an optional
strengthening. Random observations are used only with their exact
conditional Gibbs kernels. The curve argument needs estimates
\emph{uniform over stopping rules}, not merely pointwise in a fixed
stopping rule. The natural order of proof is therefore:
unconditional covariance and trace, then labelled harmonic accesses,
then the driving criterion. That is the order below.

\section{Conventions and Notation}\label{sec:conventions}

Two different but equivalent coordinate systems are used below. In
local arguments, lattice spacing is $1$ and $n$ denotes the macroscopic
size of the region; discrete gradients do not carry a factor of $n$.
In the trace and continuum arguments, the lattice is already embedded
at physical mesh $n^{-1}$, and $x$ denotes the physical coordinate.
Both are the same convention, viewed at different scales.

Continuum tests use the Riemann pairing
\[
 \langle h,f\rangle_n=\frac1{vn^2}\sum_x f(x/n)h_x.
\]
In the continuum sections, $x$ is the physical coordinate and the same
pairing is written $\frac1{vn^2}\sum_x f(x)h_x$; the two forms agree
because $x/n$ in one is $x$ in the other.

Constants may depend on a fixed compact test region and on explicitly
fixed scale ratios, but never on $n$ or on the microscopic shape of
admissible observed strings.

An \emph{oriented sign string} prescribes the signs adjacent to a
finite piece of interface. Each such vertex has an oppositely signed
neighbour, hence its height lies in $[-1,1]$. This is the key rigidity
of a strand conditioning: the surface cannot run away from the
interface, because it is anchored on both sides. By contrast, a
newly revealed \emph{exact value} is a pin, and arbitrary values in
$[-1,1]$ are not uniformly bounded-control sites. All annealed
estimates on pins go through the conditional-kernel argument of
Section~\ref{s:coupling}; no quenched estimate for an arbitrary pin
vector is claimed or needed.

The \emph{reflected bath} has Brownian variance parameter $2$ in every
free coordinate and normal reflection on the convex polytope of admissible
heights. A density tilt by $\exp(\sum_x b_x h_x)$ adds drift $b$.
Finite-dimensional existence, continuity, invariance of the stated
density, and the stationary forward/reverse martingale decomposition are
the standard stochastic tools used below. Exact-value conditional laws
are taken on their almost-sure positive-volume fibres after the recorded
coordinates have been removed; null-volume conditionings are discarded.
Reflection and its martingale decomposition are applied on the free
coordinates. The finite-body references and the uniform forcing
construction are in Section~\ref{s:coupling}.

\subsection*{Notation summary}

\begin{itemize}[leftmargin=2em,itemsep=1pt]
\item $v=2/\sqrt3$: vertex density of the triangular lattice.
\item $Q_r=x_0+[-rn,rn]^2$: local box of side $2rn$ in lattice units;
      $\widehat Q_r=n^{-1}Q_r$: the same in scaled units.
\item $\int_{n,Q}F=n^{-2}\sum_{x\in Q}F(x)$ for a vertex field;
      same notation for edge fields, summed over unoriented edges.
\item $\nabla_e f=f(y)-f(x)$ for $e=(x,y)$: unscaled lattice gradient.
\item $\Prob_{\mathrm{vol}}$: annealed law obtained by first sampling the
      experiment, then an independent uniform vertex (or edge) in the
      indicated box.
\item $H_x=|C_x|$: size of the transmission cluster of $x$; see
      Section~\ref{s:coupling}.
\item $D_x$: square dissipation of cluster $C_x$; see
      Section~\ref{s:coupling}.
\item $A$: tangent-flux scalar, defined in Section~\ref{sec:cell}.
\item $\sigma^2=(vA)^{-1}$: field variance, fixed by Theorem~\ref{b:gff}.
\item $l=\sigma\sqrt{\pi/8}$: universal height gap of
      Corollary~\ref{h:assembly}.
\end{itemize}

\section{Local Slack and Macroscopic Averages}\label{f:section}

The comparison argument in Section~\ref{s:coupling} needs two things from the
raw model: spatial tightness of macroscopic averages, and an exponential
moment bound strong enough to survive small exponential tilts. Both come
from a single local-volume estimate: for a well-shaped triangular block,
the polytope of admissible fillings has enough volume that perturbations of
the shell heights are controlled by a log-concavity deficit. We then
upgrade this deficit to exponentially sparse bad blocks, on which every
edge is close to the constraint. The zero-pinned input is
Cohen-Alloro--Peled \cite[Theorem~1.1]{CAP}; the current section is what
that becomes on the triangular lattice, uniformly in sign-string
conditioning.

\subsection{The finite-volume laws and the moment statement}

Identify the triangular lattice with $\Z^2$ under basis $e_1,e_2$
and step set $\pm e_1,\pm e_2,\pm(e_1-e_2)$. Distances are graph distances.
Let $V$ be a finite free set, and fix height zero on its exterior vertex
boundary and on any additional pins. Write $\mu_0$ for uniform Lebesgue
measure on the remaining heights satisfying the edge constraints. Extend
a zero-pinned sample by zero to the whole lattice: the defining polytope
is compact and has positive volume on its free coordinates. A
\emph{bounded-data} law $\mu_b$ has the same fixed set and prescribed values
$b$ with $|b|\le B$, considered only when its polytope has positive volume.
All constants below are independent of the number and shape of the components
of $V$.

\begin{proposition}[Macroscopic exponential moments]\label{f:averages}
Fix $c,C>0$ and $B<\infty$. Let $R\ge1$ and let deterministic real weights
$q_x$ obey
\begin{equation}\label{f:weights}
 \sum_x|q_x|\le C,\qquad \max_x|q_x|\le CR^{-2},\qquad
 \operatorname{supp}q\subset Q,\quad \operatorname{diam}Q\le CR,
\end{equation}
with $Q$ an ambient box. Assume either $\sum_xq_x=0$, or there is a
connected set of pins $K$ of diameter at least $cR$ with $K\cup Q$ in a
box of diameter $CR$. Then for a sample $h\sim\mu_b$ there are $a>0$,
$M<\infty$ depending only on $(c,C,B)$ such that
\begin{equation}\label{f:average-mgf}
 \E\exp\left(a\left|\sum_xq_xh(x)\right|\right)\le M.
\end{equation}
The conclusion is uniform over mixtures of bounded pin values whenever
conditioning on those values leaves exactly the stated uniform law.
\end{proposition}

The zero-mass alternative covers differences of comparable-scale averages
even when the pin set is far away. An ordinary ball average, or a smooth
average of comparable scale, satisfies \eqref{f:weights}. Restriction of
original positive weights to the domain also satisfies the bound; after
renormalisation one rechecks $\max|q_x|\le CR^{-2}$. We first prove the
proposition for zero pins; the proof also identifies exactly which
deterministic inputs later comparison fields will need.

\subsection{A local filling-volume deficit}

Fix an even integer $L$ and tile the lattice by $Q_z=Lz+\{0,\dots,L\}^2$.
Distinct block interiors share no edge. For shell data $s$ on $\partial Q$
let $K_Q(s)$ be the fitting polytope and put
\[
 F_Q(s)=\log\operatorname{vol}_{(L-1)^2}K_Q(s),
\]
with $F_Q=-\infty$ on empty fibres. The inclusion
$\tfrac12(K_Q(s_1)+K_Q(s_2))\subset K_Q(\tfrac12(s_1+s_2))$ together with
Brunn--Minkowski makes $F_Q$ concave.

Concavity gives a nonnegative midpoint loss. On convex-good blocks we show
that this loss controls the four side-midpoint perturbations. It cannot
control a common height shift, which leaves filling volume unchanged; the
later coarse-grid argument absorbs the constant via the pin anchor or the
zero-mass hypothesis.

For each face let $v$ be its midpoint and $r$ the inward coordinate step.
The inward path is $v+kr$, $0\le k\le L/2$. Of the two tangential unit
steps, take $t$ with $r\cdot t=1/2$. The \emph{compressed ray} is
$v+\ell t$, $0\le\ell\le L/2$; its graph distance to $v+kr$ is
$\max(k,\ell)$. Figure~\ref{fig:triangular} illustrates one instance.

For fixed $0<\delta\le\delta_0\le1/16$ and integer $k_0\ge4/\delta_0$
call block $Q$ \emph{convex-good} for $h$ if
\begin{align}
 |h(x)-h(y)|&\le1-\delta &&(x\sim y\text{ in }Q),\label{f:near}\\
 |h(x)-h(y)|&\le(1-\delta_0)\operatorname{dist}(x,y)
 &&(x,y\in Q,\ \operatorname{dist}(x,y)\ge k_0),\label{f:far}
\end{align}
and in addition the four non-tie tests hold: for each face center $v$,
with the radial coordinate path from $v$ to the block center and $b_1$ the
first vertex of the unique compressed ray on that face, require
$|h(b_1)-h(v)|\ge s_0$. Take $L\ge8(k_0+2)/\delta_0$.

\begin{lemma}[Triangular block deficit]\label{f:block}
For these fixed parameters there is $c_*>0$ such that for convex-good $Q$
with $s=h|_{\partial Q}$ and $\chi:\partial Q\to\R$, whenever both
$K_Q(s\pm\chi)$ have positive volume, every pair $v,w$ of face centers
satisfies
\begin{equation}\label{f:block-deficit}
 g_Q:=F_Q(s)-\tfrac12\bigl(F_Q(s+\chi)+F_Q(s-\chi)\bigr)
 \ge c_*|\chi(v)-\chi(w)|^2.
\end{equation}
\end{lemma}

\begin{proof}
Expose one sign of each radial edge with uniform slack. Write
$x=v+ke_1$, $y=v+(k+1)e_1$ after a lattice symmetry.

\emph{Case $k\ge k_0$.} Prescribe $a(x)=h(x)$ and $a(y)=h(x)+1$. Both
points are at distance $\ge k_0$ from the shell and the change at $y$ is
at most two. For small $\beta\le\delta/2$,
\[
 |a(z)-h(b)|\le(1-\delta_0)\operatorname{dist}(z,b)+2
 \le(1-\beta)\operatorname{dist}(z,b),\quad z=x,y,\ b\in\partial Q.
\]

\emph{Case $k<k_0$.} Put $b_l=v+le_2$ on the compressed ray,
$f_l=h(b_l)-h(v)$, $\tau=\operatorname{sign}f_1$, and prescribe
$a(x)=h(v)$, $a(y)=h(v)+\tau$. The distances are
$\operatorname{dist}(x,b_l)=\max(k,l)$,
$\operatorname{dist}(y,b_l)=\max(k+1,l)$. For $l\ge1$,
\[
 s_0-(1-\delta)(l-1)\le\tau f_l\le(1-\delta)l,
\]
hence
\begin{equation}\label{f:triangular-slack}
 |\tau-f_l|\le l-\min\{s_0+\delta(l-1),1+\delta l\}
 \le(1-\min\{s_0,\delta\})l.
\end{equation}
The difference at $x$ is at most $(1-\delta)l$. On the opposite ray the
distances are $k+l$ and $k+1+l$; the inequality
$1+(1-\delta)l\le(1-\delta/2)(k+1+l)$ holds for $k+l\ge1$ (the missing
case is the designated edge itself). For another face with
$l=\operatorname{dist}(v,b)\ge L/2$, both prescribed differences are at
most $1+(1-\delta_0)l$ and both distances are at least $l-k-1$; the
choice of $L$ provides the required relative slack there too.

Take $\beta<\min\{\delta/2,s_0/2,1/4\}$; give the designated edge length
one and every other edge length $1-\beta$. All prescribed pairs are
Lipschitz in this weighted metric; the distance across the designated
edge is one since any alternative path has length at least two. Extending
by the minimum of upper distance cones produces a filling with designated
gradient $\tau$ and remaining gradients of modulus at most $1-\beta$.

\emph{Slice log-concavity.} Let $K_\pm\subset\R^m$ be positive-volume
convex bodies with projection width at most $A$ in a unit direction $u$,
top facet area at least $c_0$ times volume, and
$K\supset(K_-+K_+)/2$. Then
\begin{equation}\label{f:facet-input}
 \log\frac{|K|}{\sqrt{|K_-||K_+|}}
 \ge\kappa\bigl[(b_+-u\cdot m_+)-(b_--u\cdot m_-)\bigr]^2,
\end{equation}
with centroid $m_\pm$ and top projection endpoint $b_\pm$, and
$\kappa>0$ only depending on $A,c_0$. Proof: let $f_i$ be the projection
density of uniform measure on $K_i$ and $q_i$ the quantile.
Parallel-slice Brunn--Minkowski makes $f_i$ log-concave with $f_i(b_i)\ge c_0$.
If $f_i(q_i(p))<c_0$, log-concavity forces every density value left of it
to be at most the same value, so $p\le A f_i(q_i(p))$; else $q_i'(p)\le
c_0^{-1}$. Either way, with $C=\max(A,c_0^{-1})$,
\[
 0<q_i'(p)\le C/p.
\]
The mid-slice of $K$ at $\tfrac12(q_-+q_+)$ contains the mid-sum of the
corresponding slices. Slice Brunn--Minkowski plus the change of variables
gives
\[
 \frac{|K|}{\sqrt{|K_-||K_+|}}\ge\int_0^1\frac{q_-'+q_+'}{2\sqrt{q_-'q_+'}}\dd p
 \ge1+\frac1{8C^2}\int_0^1p^2(q_-'-q_+')^2\dd p,
\]
the last by subtracting one from the integrand, using
$2\sqrt{q_-'q_+'}(\sqrt{q_-'}+\sqrt{q_+'})^2\le8C^2/p^2$. With
$b_i-u\cdot m_i=\int_0^1pq_i'(p)\dd p$, Cauchy--Schwarz and $|d|\le A$,
$\log(1+x)\ge x/(1+x)$ produce \eqref{f:facet-input} with
$\kappa=(8C^2+A^2)^{-1}$.

Apply \eqref{f:facet-input} to $K_\pm=K_Q(s\pm\alpha\chi)$ with
$\alpha\le\beta/(8L)$. Endpoint feasibility gives
$|\chi(b)-\chi(c)|\le2L$. Shift every interior coordinate of an exposed
filling by $\pm\alpha\chi(v)$ and use the new shell data. Every
undesignated edge retains slack $\beta/2$, each facet contains a ball
of radius $\beta/8$, body volumes are $\le(4L)^{(L-1)^2}$, projection
widths are at most two; hypotheses of \eqref{f:facet-input} hold uniformly.

Define $U=m_+-m_-$ on the interior and $U=2\alpha\chi$ on the shell. For an
interior radial edge use $\tau(e_y-e_x)/\sqrt2$: the top projection
endpoints are $1/\sqrt2$, so the centroid-gap difference is
$-\tau(U(y)-U(x))/\sqrt2$. For an edge from face center $v$ to interior
$y$ use $\tau e_y$: top endpoints are $1+\tau(s(v)\pm\alpha\chi(v))$, so
the centroid-gap difference is $-\tau(U(y)-U(v))$. Thus \eqref{f:facet-input}
bounds every $|U(y)-U(x)|^2$ by a multiple of the common midpoint gap
$g_\alpha$. The bodies and centroids are the same for all radial edges;
varying $\tau$ changes no squared increment. Telescoping $\le L$ radial
edges between $v,w$ and $g_\alpha\le g_Q$ proves the lemma.
\end{proof}

When $f_1=0$ and $k=0$, either choice $a(y)=h(v)\pm1$ saturates the edge
$yb_1$; the non-tie requirement repairs this actual triangular obstruction.
It is not supplied by the square-lattice facet argument.

\subsection{Sparse exceptions and short stationary steps}

\begin{lemma}[Zero-pinned sparse blocks]\label{f:sparse}
For every $\rho>0$ the fixed parameters above and a fixed enlargement
$Q_z^*$ can be chosen so the following holds uniformly over zero-pinned
finite volumes. Declare every block within a fixed block distance of a pin
to be a pin block. Among remaining blocks call a block bad when it fails
\eqref{f:near}, \eqref{f:far} on $Q_z^*$, or one of the four non-tie tests.
Then for every finite set $A$ of block indices,
\begin{equation}\label{f:animal}
 \Prob(A\text{ entirely bad})\le\rho^{|A|}.
\end{equation}
Fixed thickenings of the bad set satisfy such a bound with an arbitrarily
small parameter, after adjusting initial parameters.
\end{lemma}

\begin{proof}
The controlled-gradient theorem \cite[Theorem~1.1]{CAP}, on a graph of
maximum degree six with an arbitrary nonempty zero set, gives for distinct
edges $e_1,\dots,e_j$ and $0<u\le1/8$:
\[
 \Prob\bigl(|\nabla_{e_i}h|\ge1-u\text{ for all }i\bigr)
 \le(C_6u)^{j/C_6}.
\]
For each pair in an enlarged block fix one shortest path. A path of length
$m$ violating linear slack with $\delta_0=\eps/2$ has $\ge m/2$ gradients
of modulus $\ge1-\eps$. Choose $\eps$ to dominate $2^m$ subsets, $k_0$ to
dominate the polynomial number of pairs, $L$ a fixed large multiple of
$k_0/\delta_0$, then $\delta$ to dominate the edge count. Disjoint
enlargements give distinct witnesses; apply the controlled-gradient bound
separately to short-edge and long-path witnesses and use the geometric
mean for the intersection. Each test then fails with arbitrarily small
probability $\rho$.

For a non-tie test condition on its enlarged shell. If a long-slack
filling exists, prescribe $h(x), h(x)\pm1/2$ at a tested edge. Distance
to the shell dominates $1/\delta_0$, so upper-cone extension produces two
feasible fillings. Contracting the conditional body toward either with
ratio $1/4$ gives fraction $4^{-m}$ in the intervals
$[1/8,5/8]$ and $[-5/8,-1/8]$; the difference marginal is log-concave,
so its supremum is bounded by $C_L$ and
\[
 \Prob\bigl(|h(x)-h(y)|<s_0,\text{ long slack}\mid\text{shell}\bigr)\le C_Ls_0.
\]
If no long-slack filling exists the event is empty. Conditional laws in
disjoint enlarged interiors are independent given the complement. Choose
$s_0$ small, combine witnesses, colour the block grid by finitely many
residue classes: \eqref{f:animal}. For a thickening extract a separated
positive fraction of witnesses in the original set and reuse counting.
\end{proof}

\begin{lemma}[Stationary step displacement]\label{f:step}
Let $(H_t)_{0\le t\le T}$ be a stationary reversible reflected diffusion
in a finite convex polytope, Brownian covariance $2I$, invariant uniform
law. The same holds with constant drift $b$ and invariant density
proportional to $e^{b\cdot h}$. For distinct free coordinates
$x_1,\dots,x_j$,
\begin{equation}\label{f:step-tail}
 \Prob\left(\sup_{t\le T}|H_t(x_i)-H_0(x_i)|>r\text{ for all }i\right)
 \le\bigl(Ce^{-r^2/(CT)}\bigr)^{j/2}.
\end{equation}
\end{lemma}

\begin{proof}
Normal reflection has a forward Brownian martingale part plus a
finite-variation part (reflection plus drift). Reversibility provides the
same decomposition backwards from $T$. The forward increment minus the
negative reversed increment cancels the finite-variation terms; each
coordinate displacement is half the sum of the two martingale increments.
Each direction has independent variance-two Brownian coordinates. If all
$j$ displacements exceed $r$, at least $j/2$ are witnessed in one
direction; a union over subsets and the Brownian maximal inequality prove
\eqref{f:step-tail}. Failed block persistence is witnessed by one
coordinate exceeding a fixed constant. Separated blocks have distinct
witnesses; take $T$ small enough to absorb the finite count. All block
tests persist once endpoint displacements are below the appropriate
fraction of the original margins.
\end{proof}

\subsection{Filling holes and testing macroscopic averages}

\begin{lemma}[Sparse-hole extension]\label{f:hole}
Let a random bad set on the square grid satisfy \eqref{f:animal} with
sufficiently small parameter. Fill bounded complementary components of
each bad star-connected cluster and retain outermost filled hulls, with a
fixed enlargement layer. There is an extension procedure which, for a
field $u$ on good sites, fills each hole by its value at one surrounding
good site; write $v$ for the result. Here good sites lie outside filled
hulls and good edges join neighbouring good sites. For a site $z$
incident to a hole, let $d_z$ be its largest incident hole diameter, zero
if none; let $d_e=\max(d_x,d_y)$ for $e=\{x,y\}$. Uniformly in
deterministic $z$,
\begin{equation}\label{f:hole-tail}
 \Prob(d_z\ge r)\le Ce^{-cr}.
\end{equation}
For every $1<p<2$ on a square of side $N$ with a fixed buffer,
\begin{equation}\label{f:hole-weighted}
 \|\nabla v\|_{\ell^p}
 \le E^{1/2}\left(\sum_e W_e^{2/(2-p)}a_e^{-p/(2-p)}\right)^{(2-p)/(2p)},
 \quad W_e\le C_p(1+d_e)^{2p},
\end{equation}
where $E=\sum_{e\text{ good}}a_e|\nabla u(e)|^2$, $a_e>0$. If the
sufficiently high powers of $d_e$ and $a_e^{-1}$ have bounded spatial
moments, then
\begin{equation}\label{f:hole-scaled}
 \|\nabla v\|_{\ell^p}\le CN^{2/p-1}\sqrt E.
\end{equation}
For a globally $K$-Lipschitz $u$, $|u(z)-v(z)|\le CK(1+d_z)$.
For every fixed $q,D<\infty$ the spatial average of $(1+d_z)^q$ is bounded
deterministically outside an event of probability $O(N^{-D})$; in a box
of side $n$ such bounds hold on all subboxes of radius $\ge n^{1/2}$ after
altering the exponent. Filling can be chosen equivariantly under block-grid
translations. Reciprocal-weight moments are separate hypotheses in
\eqref{f:hole-weighted}; they do not follow from sparsity alone.
\end{lemma}

\begin{proof}
Follow the filled outer grid contour of a hole and its exterior good rim.
After a fixed thickening this is a nearest-neighbour rim path. A hole of
diameter $d$ has at most $C(d+1)^2$ interior sites and boundary edges, and
each required rim path has at most that length. Take the lexicographically
first rim site as fill value. For an edge entering a hole, telescope
along the rim; the inequality $|\sum_{i=1}^m t_i|^p\le m^{p-1}\sum_i|t_i|^p$
together with the boundary-edge count gives
\[
 \sum|\nabla v|^p\le\sum_{e\text{ good}}W_e|\nabla u(e)|^p,
 \quad W_e\le C_p(1+d_e)^{2p}.
\]
Outer rims have bounded overlap; H\"older with exponents $2/p$ and $2/(2-p)$
gives \eqref{f:hole-weighted}. Without weights the required hole exponent
is $4p/(2-p)$ plus one further H\"older split. A filled site is within
distance $C(d+1)$ of every other site of its hole, giving the pointwise
error.

If a deterministic site belongs to a hull of diameter $\ge r$, a bad
star-animal of size $m\ge cr$ has a site within $Cm$ of it. There are
$O(m^2)$ root choices and at most $C_0^m$ animals; applying \eqref{f:animal}
(or a fixed-fraction separated subset) proves \eqref{f:hole-tail}, hence
uniformly bounded fixed moments.

For the stronger spatial statement: clusters with size in $[k,2k)$
contribute at most $Ck^{q+2}$ each to the sum of hole moments. For $r$
disjoint cluster witnesses in an $O(N)$ box, root/animal counting and
\eqref{f:animal} bound the probability by
\[
 \frac{(CN^2)^r}{r!}C_0^{2kr}\rho^{ckr}\le\left(C\frac{N^2}{r}e^{-c'k}\right)^r.
\]
Take $r=\lceil N^2k^{-(q+4)}\rceil$. For $k\le A\log N$,
$\exp(-cN^2/k^{q+3})$; larger clusters occur with probability $\le CN^2
e^{-cA\log N}=O(N^{-D})$ after choosing $A$ large. Each dyadic size range
contributes summably to $N^{-2}\sum_z(1+d_z)^q$; fixed small clusters
give a deterministic bound. Holes crossing the macroscopic buffer are
excluded by the same argument; polynomially many subboxes yield the
simultaneous statement. Lexicographic selection commutes with translation.
\end{proof}

\begin{lemma}[Discrete trace against a spread measure]\label{f:trace-average}
Let $\nu$ be positive of bounded mass in a box of side $O(N)$ with
$\nu(B(x,r))\le C((r+1)/N)^\alpha$ for $0\le r\le CN$. If $1<p<2$ and
$\alpha>2-p$, then
\[
 |\nu(v)-\nu(1)\overline v|
 \le CN^{1-2/p}\|\nabla v\|_{\ell^p},
\]
where $\overline v$ is the ordinary average.
\end{lemma}

\begin{proof}
Discrete $L^1$ Poincar\'e on dyadic subcells clipped to the box:
\[
 |v(x)-\overline v|\le C\sum_e\frac{|\nabla v(e)|}{1+\operatorname{dist}(x,e)}.
\]
For $p'=p/(p-1)$, a dyadic contribution to the averaged kernel is
$F_r(e)=Cr^{-1}\nu(B(e,Cr))$, with $\sup_{|e|\approx r}|F_r(e)|
\le CN^{-\alpha}r^{\alpha-1}$ and $\|F_r\|_{\ell^1}\le Cr$, so
$\|F_r\|_{\ell^{p'}}\le CN^{-\alpha/p}r^{1+(\alpha-2)/p}$. Dyadic sum
bounded by $CN^{1-2/p}$ iff $\alpha>2-p$. H\"older closes.
\end{proof}

A connected set of coarse pins of diameter $\ge cN$ carries a probability
measure $\nu$ satisfying the ball bound for $\alpha=1$: project onto a
coordinate axis of length $\ge c'N$, lift normalized measure, equal
weights. Connectivity ensures no gaps. Ball masses $\le C(r+1)/N$.

\begin{lemma}[Uniform average coercivity]\label{f:average-coercivity}
Assume $X$ is globally $K$-Lipschitz on the block grid with fixed $K$.
Let $u=X$ or $u=T_M(X)$ with $T_M(t)=\operatorname{sign}(t)(|t|-M)_+$.
Put $E=\sum_{e\text{ good}}|\nabla u(e)|^2$. For deterministic $w$
supported in an $O(N)$ box with $\sum|w|\le C$, $\max|w|\le CN^{-2}$,
either $\sum w=0$, or $u=0$ on a connected coarse pin set of diameter
comparable to $N$ at distance $O(N)$. There is an event of probability
$\ge1/2$ determined only by the bad set, on which simultaneously for every
such field,
\begin{equation}\label{f:coercivity}
 \left|\sum_zw_zX_z\right|\le C_0+C_1\sqrt E.
\end{equation}
\end{lemma}

\begin{proof}
Use Lemma~\ref{f:hole} with $p=3/2$. A Markov bound on the twelfth
spatial hole moment plus exclusion of holes crossing a fixed buffer gives
$\|\nabla v\|_{3/2}\le CN^{1/3}\sqrt E$. The filling error tested against
$|w|$ is bounded by the same spatial moment. In the anchored case use the
pin measure $\nu$ just constructed. Filling may overwrite a pin, but
Lipschitzness gives $|\nu(v)|\le CK\sum_z\nu(z)(1+d_z)$, whose expectation
is bounded via \eqref{f:hole-tail}; one more Markov bound makes it fixed.
Then Lemma~\ref{f:trace-average} gives \eqref{f:coercivity}. In the
zero-mass case the ordinary mean cancels. Truncation error $\le M\sum|w|$.
Choose both thresholds so that failure probabilities sum to $\le1/4$; the
buffer event fails at most $1/4$ for large $N$. Small $N$ absorbed by
deterministic Lipschitzness and the anchor/zero-mass hypothesis. All error
bounds depend on hole sizes, not on $X$, producing the simultaneous
statement.
\end{proof}

\subsection{From deficits to an exponential moment}

\begin{proof}[Proof of Proposition~\ref{f:averages} for zero pins]
For $R/L$ large integrate interiors of pin-free blocks in a fixed box
around the statistic and anchor. Retain every other vertex and every pin
block vertex; write $s$ for their heights. The unnormalized marginal
density is
\begin{equation}\label{f:product}
 p(s)=\mathbf1_{\mathcal A}(s)\prod_Q e^{F_Q(s|_{\partial Q})},
\end{equation}
with $\mathcal A$ the residual edge constraints. This is even,
log-concave, compactly supported, upper semicontinuous; $Z=\int p>0$.

For feasible $s\pm\chi$ with positive density, set $X_z=\chi(v_z)$ at a
face-center representative. The difference of two endpoint fillings
divided by two extends $\chi$ as a 1-Lipschitz field vanishing at pins,
so neighbours differ by $\le CL$. Choose $M$ above every representative
and shared face center in or adjacent to pin blocks; use $u=T_M(X)$, which
vanishes on pin blocks. For neighbouring ordinary good blocks $z,w$
(sharing a face center),
\[
 |u_z-u_w|^2\le2c_*^{-1}(g_z+g_w).
\]
For a good block next to a pin block with common face center $f$,
$|\chi(f)|\le M$ so $|u_z|^2\le|\chi(v_z)-\chi(f)|^2\le g_z/c_*$.
Treating pin blocks as zero uses
\begin{equation}\label{f:energy-deficits}
 \sum_{e\text{ good}}|\nabla u(e)|^2\le C\sum_Q g_Q.
\end{equation}

Partition into half-open blocks and replace the statistic by the retained
linear functional $\ell(s)=\sum_zw_zs(v_z)$, $w_z=\sum_{x\text{ in
block }z}q_x$. On every original configuration,
\begin{equation}\label{f:representative-error}
 \left|\sum_xq_xh(x)-\ell(s)\right|\le CL\sum_x|q_x|.
\end{equation}
Coarse weights meet Lemma~\ref{f:average-coercivity}; exact zero mass is
preserved. That lemma together with \eqref{f:energy-deficits} gives, on a
full-configuration event of probability $\ge1/2$ and simultaneously for
every feasible $\chi$,
\[
 |\ell(\chi)|\le C_0+C_2\sqrt{\sum_Qg_Q}.
\]
Take $t>C_0$ fixed, $D=((t-C_0)/C_2)^2>0$. The logarithmic midpoint
deficit of \eqref{f:product} is $\sum_Qg_Q$, so
\begin{equation}\label{f:pointwise-overlap}
 \sqrt{p(s+\chi)p(s-\chi)}\le e^{-D}p(s)
 \quad\text{whenever }\ell(\chi)=t.
\end{equation}

Define $A_t(s)=\sup_{\ell(\chi)=t}\sqrt{p(s+\chi)p(s-\chi)}$. Support
compactness restricts the supremum to a compact set; upper semicontinuity
gives measurability. Log-concavity gives $0\le A_t\le p$. The good event
may depend on discarded midpoint coordinates but the simultaneous form
of \eqref{f:pointwise-overlap} holds for its retained vector. Thus
$Z^{-1}\int A_t\le1-\rho_0$ with $\rho_0=(1-e^{-D})/2>0$.

If $\ell=0$ on free coordinates there is nothing to prove. Otherwise pick
coordinates $s=(r,\sigma)$ with $r=\ell(s)$, absorbing the Jacobian into
the complementary measure. For the density $a$ of $\ell(S)$,
Pr\'ekopa--Leindler in $\sigma$ \cite[Theorems~3,~6]{Prekopa} gives
$\sqrt{a(r-t)a(r+t)}\le Z^{-1}\int A_t(r,\sigma)\dd\sigma$. Integrating
gives overlap $\le1-\rho_0$. Since $a$ is even and log-concave, overlap
$\ge\int\min\{a(r-t),a(r+t)\}\dd r=1-\int_{-t}^t a$, so
$\Prob(|\ell(S)|\le t)\ge\rho_0$. Its survival function is log-concave
(again Pr\'ekopa--Leindler), so $\Prob(|\ell(S)|>jt)\le(1-\rho_0)^j$ for
$j\ge1$, a positive uniform exponential moment. Equation
\eqref{f:representative-error} transfers this to the original statistic.
For bounded $R/L$ or anchor distance, or for $q$ cancelling, a
deterministic bound suffices.
\end{proof}

\subsection{Bounded values, sign strings, and small tilts}

\begin{lemma}[One-sample bounded-data comparison]\label{f:bounded}
There is a coupling of $h_b\sim\mu_b$ and $h_0\sim\mu_0$ with
$|h_b-h_0|\le B$ at every vertex, whenever $|b|\le B$. Consequently
Proposition~\ref{f:averages} holds for bounded data and their admissible
mixtures.
\end{lemma}

\begin{proof}
Run monotone single-site uniform heat baths with common update variables
for pin values $-B,b,+B$. Extreme chains are exact translates since their
intervals are translates. The middle chain stays between them because
both conditional-interval endpoints are increasing functions of the
neighbours. A stationary subsequential coupling of the compact polytopes
gives the asserted coupling with uniform marginals. Each tested statistic
changes by $\le B\sum|q_x|$. Condition on mixed pin values and integrate
the same bound.
\end{proof}

A \emph{string site} prescribes a sign adjacent to an interface piece,
each such site having an oppositely signed neighbour. Its height lies in
$[-1,1]$. Additional intervals $0\le h\le\eps$ on positive sites are
permitted. Freezing all string values leaves a uniform law on the
remaining Lipschitz polytope; Lemma~\ref{f:bounded} therefore proves the
moment assertion with $B=1$, provided the anchor or zero-mass hypothesis
holds for the statistic in question. A finite collection of components
can be handled together with the common pinned set. Cutting along pins
and zero-extending the comparison run on the ambient lattice; neither
requires a regular shape.

The same argument transfers long-distance slack from zero pins to bounded
data at additive cost $2B$. Beyond a sufficiently large fixed distance
this is absorbed into a relaxed linear slack. It does not transfer
microscopic strict gradients by itself.

\begin{lemma}[Small linear tilts]\label{f:tilts}
Under Proposition~\ref{f:averages}, let $Y=\sum_xb_xh(x)$ with support in
an $O(R)$ box and $|b_x|\le cR^{-2}$. Assume its box has the stated
connected pin anchor, or $\sum_xb_x=0$. For sufficiently small fixed
$c>0$ the tilted density $e^Y/\E e^Y$ has uniformly bounded $L^2$ norm.
Then for any event $A$, $\Prob_Y(A)\le C\Prob(A)^{1/2}$. All fixed
moments of admissible statistics remain bounded under this tilt; any
arbitrary-parameter animal estimate is inherited.
\end{lemma}

\begin{proof}
\eqref{f:average-mgf} applied to $Y/c$ bounds $\E e^{2Y}$; Jensen plus
$|\E Y|$ bounded away from zero bounds $\E e^Y$ below. Cauchy--Schwarz.
For moments apply to the square of the desired moment. Combine finite
tests by H\"older with a further reduction of $c$.
\end{proof}

\begin{lemma}[Strict filling with sign intervals]\label{f:strict-filling}
In a finite patch prescribe shell values, interior fixed pins in
$[-1/2,1/2]$, and intervals containing zero at any other prescribed
vertex (one-sided or of the form $[0,\eps]$). Suppose a feasible filling
$h$ obeys
\begin{equation}\label{f:shell-room}
 |h(x)-b(s)|\le\operatorname{dist}(x,s)-2
\end{equation}
for every movable endpoint $x$ of a tested central edge and shell vertex
$s$. If the edge is not fixed--fixed, there is a feasible filling whose
gradient on it is at most $1/2$ in absolute value.
\end{lemma}

\begin{proof}
Interval extension criterion: data $[\alpha_z,\beta_z]$ admit a 1-Lipschitz
selection iff $\alpha_z\le\beta_w+\operatorname{dist}(z,w)$ for every pair.
Lower-cone function $g(y)=\sup_z\{\alpha_z-\operatorname{dist}(z,y)\}$ is
1-Lipschitz, above every lower endpoint and below every upper endpoint.
Infinite endpoints omitted.

At a movable $x$ the intervals propagated from non-shell data contain
both $h(x)$ and $0$. Feasibility gives $h(x)$; nonfixed intervals contain
$0$; a fixed pin $p$ has $|h(p)|\le1/2$ and distance $\ge1$ from $x$, so
its propagated interval contains $0$. The datum at $x$ (if present) also
contains $0$.

Case both endpoints $u,v$ movable: put $c=\operatorname{median}(0,h(u),h(v))$
and prescribe $g(u)=g(v)=c$. Then $c$ lies between $0$ and each old value,
respects every propagated interval, and $|c-h|\le|h(u)-h(v)|\le1$.
\eqref{f:shell-room} preserves endpoint--shell inequalities. Old data was
consistent; new endpoint difference is zero; extend by the criterion.

Case one endpoint is a fixed pin: prescribe $0$ at the other. $|h|\le3/2$;
\eqref{f:shell-room} preserves shell inequalities; propagated intervals
contain $0$; result gradient $\le1/2$. Same extension criterion.
\end{proof}

\begin{corollary}[Sparse contacts with signs]\label{f:conditioned-sparse}
Fix a domain, fixed pin values in $[-1/2,1/2]$, and deterministic sign
strings as above, allowing intervals $[0,\eps]$. For every prescribed
animal parameter there are a fixed box size and $s>0$ such that, except on
a bad set with that animal bound, tested central edges involving nonfixed
vertices have $|\nabla h|\le1-s$. String sites are nonfixed. Long-distance
slack and this edge slack persist with relaxed constants during a
sufficiently short stationary reflected step. Constants are uniform over
string shape and interval lengths.
\end{corollary}

\begin{proof}
Freeze string values (now in $[-1,1]$). Lemma~\ref{f:bounded} couples the
conditional field to a zero-pinned field within distance one. The
controlled-gradient witness argument still gives arbitrary-parameter
long-distance slack, absorbing the additive loss by raising the fixed
threshold. Uniform over string values, so integrating preserves the joint
form.

Condition only on the exterior of each enlarged patch. On a shell
admitting the above long slack, central endpoints obey
\eqref{f:shell-room}; apply Lemma~\ref{f:strict-filling}. If the
conditional body has dimension $m$, contraction toward its strict filling
with ratio $1-2s$ sends the body into $\{|\nabla h|\le1-s\}$. Relative
volume of the near-contact region $\le1-(1-2s)^m\le2ms$, and this
survives arbitrarily thin sign intervals since their effect on the body's
total volume cancels in relative volume. Degenerate intervals treated as
fixed coordinates.

Disjoint patches conditionally independent. For a specified collection
either $\ge1/2$ have long-slack failures or $\ge1/2$ have near contacts
with good long slack; pick preliminary parameters and then $s$ small
enough to absorb the subset count. A finite colouring removes separation.
Lemma~\ref{f:step} gives the short-step version. Tilts go through
Lemma~\ref{f:tilts}, subject to the anchor/zero-mass hypothesis.
\end{proof}

\begin{remark}[Fixed values after further reveals]\label{f:reveal-obligation}
The proof uses the numerical bound $1/2$ on fixed interior pins and
intervals containing zero on movable sign sites. After further freezes
these must be re-checked. In particular, arbitrary revealed values in
$[-1,1]$ cannot replace $[-1/2,1/2]$ in Lemma~\ref{f:strict-filling};
uniform quenched contact sparsity under such reveals is not asserted. The
adapted comparisons of Section~\ref{s:coupling} instead use
Lemma~\ref{c:freezing} (annealed marginal) with good marks recomputed
after rejoining.
\end{remark}

\subsection{Distributional tightness}

\begin{corollary}[Negative Sobolev tightness]\label{f:tightness}
Suppose rescaled free sets lie in a fixed bounded box, exterior data are
bounded, and a connected exterior pin set of macroscopic diameter is
available in a fixed larger box. The centered fields, made constant on
fundamental cells of mesh $n^{-1}$ and zero outside, are tight in
$H^{-s}$ on a fixed surrounding torus for every $s>0$.
\end{corollary}

\begin{proof}
Embed the cells in a dyadic square with $2^J\asymp n$ cells per side;
physical size stays bounded above and below. Expand piecewise constant
in the orthonormal 2D Haar basis (finite at the cell scale). A coefficient
on a physical cell of side $r=R/n$ is $r$ times a zero-total statistic
satisfying \eqref{f:weights}. For zero pins the tests include all ambient
lattice vertices including the zero extension, so the cancellation holds
across the boundary. Bounded data differ from the same zero-pinned sample
by a uniformly bounded field. \eqref{f:average-mgf} gives a second moment
$\le Cr^2$ for each centered coefficient; the coarse scaling coefficient
is bounded by the anchored bound.

With $F_j$ the level-$j$ wavelet sum: $O(r_j^{-2})$ coefficients so
$\E\|F_j\|_2^2\le C$; each parent-cell integral of $F_j$ vanishes.
Fractional Poincar\'e and duality give $\|F_j\|_{H^{-t}}\le Cr_j^t\|F_j\|_2$
for $0<t\le1$. Minkowski in $L^2(\Prob;H^{-t})$ and the geometric sum of
$r_j^t$ give a uniform second moment in $H^{-t}$. Compactness of
$H^{-t}\to H^{-s}$ on the fixed torus closes.
\end{proof}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=1.05,>=Latex]
\foreach \i in {0,...,4} {
  \foreach \j in {0,...,3} {
    \coordinate (v\i\j) at ({\i+0.5*\j},{0.8660254*\j});
    \fill[gray] (v\i\j) circle (1.1pt);
  }
}
\foreach \i in {0,...,3} {
  \foreach \j in {0,...,3} {
    \draw[gray!35] ({\i+0.5*\j},{0.8660254*\j}) --
      ({\i+1+0.5*\j},{0.8660254*\j});
  }
}
\foreach \i in {0,...,4} {
  \foreach \j in {0,...,2} {
    \draw[gray!35] ({\i+0.5*\j},{0.8660254*\j}) --
      ({\i+0.5*(\j+1)},{0.8660254*(\j+1)});
  }
}
\foreach \i in {0,...,3} {
  \foreach \j in {0,...,2} {
    \draw[gray!35] ({\i+1+0.5*\j},{0.8660254*\j}) --
      ({\i+0.5*(\j+1)},{0.8660254*(\j+1)});
  }
}
\draw[blue!65!black,very thick] (v00)--(v03);
\draw[red!65!black,very thick] (v20)--(v30);
\draw[green!45!black,thick,dashed] (v20)--(v11)--(v02)--(v03);
\draw[orange!80!black,thick,dotted] (v30)--(v21)--(v12)--(v03);
\node[below left] at (v00) {$v$};
\node[left] at (v01) {$v+e_2$};
\node[above] at (v03) {$v+3e_2$};
\node[below] at (v20) {$x_2$};
\node[below] at (v30) {$x_3$};
\draw[->] (4.5,0.35)--(5.2,0.35) node[right] {$e_1$};
\draw[->] (4.5,0.35)--(4.85,0.956) node[above right] {$e_2$};
\end{tikzpicture}
\caption{The compressed ray in the triangular exposure argument. Here
$x_k=v+ke_1$, and the highlighted test edge is $x_2x_3$. Both drawn paths
from its endpoints to $v+3e_2$ have length three. For a ray vertex
$v+\ell e_2$, the two distances are $\max(k,\ell)$ and
$\max(k+1,\ell)$, rather than $k+\ell$ and $k+1+\ell$. The sign of
$h(v+e_2)-h(v)$ selects which facet can be exposed with uniform slack;
its near-zero case must be excluded in that argument.}
\label{fig:triangular}
\end{figure}


\section{Synchronous Comparison and the Difference Field}\label{s:coupling}

Heights drift inside the polytope as the surface evolves. To compare two
surfaces we must run them on parallel time with a common Brownian driver,
so that the relative motion is only what the constraints do. The output of
this section is Theorem~\ref{c:comparison}: a comparison kernel with
bounded local energy moments, a vanishing original--filled $L^1$ error,
and microscopic tightness, uniform in the conditioning class specified
there. The engineering runs in three passes. First, the reflected bath is
constructed and the two replicas are joined. Second, the annealed
marginal-preservation lemma lets us condition on a queried transcript
without leaving the class. Third, the projection recursion on a
transmission cluster is analysed and summed over the cluster decomposition.

\subsection{The reflected bath}\label{ss:reflected-bath}

Fix a finite free set $V$ in the triangular lattice, a compact convex
polytope $K\subset\R^V$ defined by edge inequalities
$|h(x)-h(y)|\le1$ for free--free edges and by fixed
individual-position constraints at pins and sign walls. On the free
coordinates, $K$ is full-dimensional. The \emph{reflected bath}
$h_t$ on $K$ has Brownian variance parameter $2$ in every free
coordinate with normal reflection at $\partial K$. A density tilt by
$\exp(\sum_xb_x h_x)$ adds constant drift $b$; the invariant density is
proportional to $e^{b\cdot h}$ restricted to $K$.

The forward and reverse martingale decomposition is Lyons--Zheng
\cite{LyonsZheng} on the Neumann Dirichlet form. The coordinate functions
belong to its domain, so the identity applies even though they are not
Neumann-adapted. Convex Skorokhod uniqueness identifies the process with
the reflected Brownian motion driven by variance-two noise and drift $b$,
the identification being that of \cite[Section~2, Corollaries~3--4]{Slominski}.
This is separate from the Bass--Hsu semimartingale identification
\cite{BassHsu}, which is used only in the drift-zero case.

A small smooth tilt is conditioned on later; its effect on every
probability statement goes through a fixed moment of the density, not
a uniform density bound. Null-volume fibres of exact-value conditionings
are discarded by convention; positive-volume fibres keep the same polytope
structure and lose the pinned coordinates.

\subsection{Adapted freezing and annealed marginals}\label{ss:freezing}

The interface exploration of Section~\ref{sec:interface} queries both
surfaces at a common sequence of vertices, with each query and its
stopping decision determined by the previous answers and independent
auxiliary randomness. The transcript leaves two conditional Gibbs laws,
one per replica, pinned at the queried heights. We can replace every such
pair by a stationary synchronous coupling without changing either
replica's annealed marginal. This is the only place where conditioning is
allowed, and it is only annealed.

\begin{lemma}[Adapted freezing]\label{c:freezing}
Conditional on an adapted exact-value transcript, the unqueried fields
have the product of their Gibbs laws with those values pinned. Replacing
this product by a stationary synchronous coupling preserves each annealed
marginal and makes the coupled process stationary. The same
marginal-preservation statement holds for jointly sampled boundary data
with the conditional Gibbs kernels specified above.
\end{lemma}

\begin{proof}
For a fixed finite query history, every decision indicator is a function of
recorded values and auxiliary randomness. Disintegrate the original product
density over these values. The decision indicators do not introduce a factor
depending on an unqueried coordinate, so the remaining density is the
product of the two conditional Gibbs densities. Induction on the query
count, followed by partitioning over the stopping count, proves the claim.
Any coupling of these two kernels has the same individual conditional
marginals. Integration over the transcript recovers the original
marginals. A conditional invariant joining, obtained by the time-reversal
identity, makes the coupled process stationary.
\end{proof}

The lemma supplies one structural fact and only one: the invariant
marginals survive. It supplies no bound for a fixed transcript. All
estimates below are averaged over the transcript. The phrase
``uniform in the observations" everywhere in this paper means this
annealed uniform.

\subsection{Conditional moment pass-through}\label{ss:conditional}

\begin{lemma}[Conditional cluster moments]\label{c:conditional}
There is a deterministic sequence of conditional cluster moments.
Explicitly, for $q\ge1$ Jensen gives
\begin{equation}\label{c:conditional-moments}
 n^{-2}\sum_x\bigl(\E[H_x^k\mid\mathcal I]\bigr)^q
 \le \E\!\left[n^{-2}\sum_xH_x^{kq}\mid\mathcal I\right].
\end{equation}
Lower positive moment orders follow from any larger order. The
deterministic spatial-moment bounds of \ref{f:hole} pass to the conditional
moments with an arbitrarily fast polynomial exceptional probability.
\end{lemma}

\begin{proof}
\eqref{c:conditional-moments} is Jensen. To pass the deterministic spatial
moments: first truncate the local cluster sizes at a power of $n$, discard
the tail with the arbitrarily high cluster moments of \ref{f:step}, and
apply the polynomial exceptional bound. No bound involving the total
ambient volume is used. Mere finiteness of random all-order moment
constants does not suffice for the expectation arguments below, so the
deterministic exceptional bounds are essential here.
\end{proof}

\subsection{Averaging structure and cluster displacement}\label{ss:cluster}

On each projection substep the two surfaces share a common driver. The
relative motion on a transmission cluster is therefore generated only by
the projections onto the polytope facets momentarily in contact. The
following lemma records the structure of these projections and the
displacement bound they produce.

\begin{lemma}[Averaging structure and cluster displacement]\label{c:cluster}
On a zero-drift step, normal reflection of a difference acts on each
transmission cluster by compositions of averaging projections and their
convex mixtures. Free-input matrices are symmetric positive contractions
at each projection, with nonnegative entries and row and column sums at
most one. Pins supply their fixed difference; common individual walls
supply zero. For a tail with zero pinned input, insert coordinate
contractions after the averaging updates. If a cluster has $H$ free
vertices and total square loss $D$, then
\begin{equation}\label{c:displacement}
 \sup_{t\le T}\max_{x\in C}|u_t(x)-u_0(x)|
 \le C(H+1)\sqrt D.
\end{equation}
The same estimate holds for an untruncated conservative difference, and
is invariant under addition of a constant.
\end{lemma}

\begin{proof}
Use the finite-dimensional construction by projecting after each common
Brownian increment. Interpolate the two preprojection inputs, and also
interpolate a low/upper truncation of the preprojection value (for a
one-sided tail). For $u=(z-b)_+$, convexity and the pin bound $z|_P\le b$
give $u'\le Au$ coordinatewise; write $u'=RAu$ with diagonal
$0\le R\le I$. The lower tail is identical after sign change. Each
elementary matrix $A$ or $R$ is positive self-adjoint contraction,
hence
\[
 \|v-Av\|_2^2\le\|v\|_2^2-\|Av\|_2^2.
\]
Add zero as a possible initial level. Every intermediate coordinate is a
substochastic mixture of initial levels, with lost mass put at zero.
Jensen's identity and the column deficits show that its variance about
this mixture is at most the total loss $D$. It is therefore within
$\sqrt D$ of some initial level. Each elementary jump is at most
$\sqrt D$. A gap greater than $4\sqrt D$ between successive initial
levels cannot be crossed. There are at most $H+1$ levels, which proves
\eqref{c:displacement} uniformly in the projection mesh. Passing to the
reflected limit proves the assertion under any of the standard
constructive Skorokhod recipes.

We use $D_C=\sum_{x\in C}u_0(x)^2-\sum_{x\in C}u_T(x)^2$ in the
zero-drift case. With two tails add their losses. With drift, $D_C$
denotes the sum of nonnegative losses at projection substeps; the
separate source terms are estimated at the end of the proof of
Lemma~\ref{c:force}. General doubly stochastic matrices cannot replace
the projections here: a permutation has no square loss.
\end{proof}

\subsection{Accessible contacts and the finite-volume force}\label{ss:force}

Fix a force step of length $T_F$. We call a lattice edge
\emph{initially strict} when its slack is at least $s_0$ throughout
$[0,T_F]$ in at least one replica. The \emph{accessible set} of the step
is the set of free vertices incident to at least one edge that fails this
condition. On a well-formed initial configuration, the accessible set
decomposes into finite transmission clusters whose contact topology is
governed by the projections just analysed.

\begin{lemma}[Finite-volume force]\label{c:force}
Fix the force step $T_F$ and a testing box $B$. Let $\mathcal I$ be
the sigma-field generated by the initial pair and the recent transcript.
There are constants $c>0$, $\omega>0$ depending only on $T_F$ and the
fixed scale thresholds, and a uniform event $\mathcal F_B\in\mathcal I$
of conditional probability at least $p_0>0$, on which the sum of
cluster dissipations and mean-drift contributions obeys
\begin{equation}\label{c:force-loss}
 \sum_{C:C\cap B^+\ne\varnothing}D_C
 +\omega\sum_{C:C\cap B^+\ne\varnothing}H_C(\bar b_C)^2
 \ge c\,p_0\,g^2,
\end{equation}
where $g>0$ depends on the drift and on the fixed test region.
\end{lemma}

\begin{proof}
Use the orthonormal decomposition $|C|^{-1/2}\mathbf 1_C$ plus its
complement. On the complement the edge polytope is compact, the Brownian
covariance is $2I$, and the Skorokhod map is continuous in the driver.
On the constant coordinate the arithmetic mean carries variance
$2/|C|$ Brownian motion with drift $\bar b_C$. Since the decomposition is
orthogonal, the two blocks decouple and the contribution of the mean
drift is exactly $H_C\bar b_C^2$ up to a fixed factor.

Now suppose $g$ is a fixed nonnegative test with positive integral on
$B^+$, and consider the class of force steps on which every accessible
cluster has $D_C=0$ and $\bar b_C=0$. On this class the difference field
is a common projection-constant on every cluster, hence a constant on
$B^+$ on any drift-compatible configuration. Iterating over a fixed
telescoping sequence of testing boxes with geometric ratio, we contradict
the drift having positive $g$-average on any individual box. Hence the
complement has a uniform event $\mathcal F_B$ of probability at least
$p_0$ on which the (dissipation) + (drift) sum is positive against $g$.
The $p_0$ is chosen before $n$; skorokhod continuity and the compact
polytope then let the sum be dominated by $c p_0 g^2$ with a fixed
constant. Summing over the finite number of clusters meeting $B^+$
finishes.
\end{proof}

Equation~\eqref{c:short-step} records the short-step sparsity of these
events in a usable form.
\begin{equation}\label{c:short-step}
 \Prob\{\text{a tested vertex $x$ and every adjacent vertex $y$
 has $|h(x)-h(y)|\ge1-s$, on some
 subinterval of $[0,T_F]$}\}
 \le e^{-c s\,d(x,P)},
\end{equation}
with $d(x,P)$ the graph distance to the pin set. On a fixed collection
of separated blocks the joint failure probability is bounded by
$\exp(-c s\sum_x d(x,P))$; the sum of graph distances is linear in the
number of blocks. This is used in the insulation lemma below.


\subsection{Insulation across slack shells}\label{ss:insulation}

\begin{lemma}[Insulation]\label{c:insulation}
Outside a sparse set of initially selected boxes, every interior forcing
event in Lemma~\ref{c:force} has conditional probability at least $p_0/2$
of occurring in the uncut process, given the initial pair and transcript.
\end{lemma}

\begin{proof}
Cut the edges across an initially strictly slack shell. For the cut
exterior process, let $A_B$ be the event that an exterior endpoint moves
by more than $s/4$. If every endpoint of a cut edge moves by less than
$s/4$ in the original stationary process, none is contacted.

Until a first cut contact the cut and uncut solutions agree; uniqueness
makes them agree throughout this quiet event. In particular $A_B$ implies
a shell-displacement event of the original process. Equation
\eqref{c:short-step} gives joint sparse bounds for these events on
separated boxes. Lemma~\ref{c:conditional} with $q=1/2$ shows that boxes
with $\Prob(A_B\mid\mathcal I)>1/2$ are still arbitrarily sparse. On the
remaining initial configurations, exterior quietness has probability
$\ge1/2$ and depends only on exterior noises; the controlled interior
noises are independent of them. Their intersection has probability
$\ge p_0/2$, and no cut edge is contacted there. Thus the cut controlled
path is a path of the original process.
\end{proof}

\subsection{Coercivity and the cutoff estimate}\label{ss:coercivity}

\emph{Coercivity inequality.} To sum the local force tests without
counting a cluster loss once per vertex, associate to each testing box
$B$ the quantity
\[
 d_B=\sum_{C:C\cap B^+\ne\varnothing}\frac{D_C}{H_C}.
\]
A cluster meets at most $C_LH_C$ such boxes, so
\begin{equation}\label{c:allocation}
 \sum_Bd_B\le C_L\sum_CD_C.
\end{equation}
Let $H_B$ be the maximum incident cluster size in $B^+$. Increasing $m$
in \eqref{c:force-loss} absorbs the division by $H_C$. Conditional
Cauchy--Schwarz on the forced event $F$ gives
\[
 \E[d_B\mid\mathcal I]\ge
 \frac{c p_0^2 g^2}{1+\E[H_B^m\mid\mathcal I]},\qquad
 \Prob(F\mid\mathcal I)^2\le\E[H_B^m\mid\mathcal I]
 \,\E[\mathbf1_F H_B^{-m}\mid\mathcal I].
\]
Choose the edge and wall weights to be a fixed small multiple of
$[1+\E(H_B^m\mid\mathcal I)]^{-1}$. Telescoping along the finite facet
paths controls each tested original edge or anchor. Equation
\eqref{c:conditional-moments} supplies the reciprocal moments and
\eqref{c:allocation} supplies the coercivity inequality.

\emph{Cutoff estimate.} To bound the dissipation without assuming pointwise
tightness of $z$, let $\eta$ be a deterministic cutoff, equal to $1$ on an
inner box, zero outside an outer box. Let $d_{\rm lat}$ be the gap in
lattice units, so the cutoff has lattice Lipschitz constant
$C/d_{\rm lat}$. For a cluster put $\eta_C=\max_C\eta$. Then
Lemma~\ref{c:cluster} and contraction give
\[
 \sum_{x\in C}|u_T(x)^2\!-\!u_0(x)^2|
 \le C H_C^m\sqrt{D_C}\,\|u_0\|_{\ell^2(C)}.
\]
Since $|\eta_x^2-\eta_C^2|\le C\eta_C\operatorname{diam}(C)/d_{\rm lat}$,
Young's inequality yields
\begin{equation}\label{c:cutoff-cluster}
 \sum_{x\in C}\eta_x^2(u_T(x)^2-u_0(x)^2)
 \le-\tfrac12\eta_C^2D_C
      +C d_{\rm lat}^{-2}H_C^m\sum_{x\in C}u_0(x)^2.
\end{equation}
Using the maximum cutoff avoids division by a possibly zero minimum on
clusters crossing the support boundary. Sum and take expectations;
stationarity cancels the left side. Combining with the force bound
and $d_{\rm lat}\asymp n(s-r)$ gives, for $r<s<R$,
\begin{equation}\label{c:caccioppoli}
 \E\mathcal E_n(Q_r)
 +\E\sum_{C:C\cap Q_r\ne\varnothing}D_C
 \le C(s-r)^{-2}\E N_n(Q_s)+C,
\end{equation}
where $N_n(Q)=\int_{n,Q}H_x^m(u_0(x)^2+1)$.

For a free comparison choose at each time the constant minimizing
$\sum_x\eta_x^2(z_t(x)-c)^2$; the minimum at the next time is no larger
than the value at the previous minimizer. Its expectation is stationary,
so \eqref{c:cutoff-cluster} gives the same estimate. This uses local
oscillations bounded by $Cn$, not absolute heights or ambient diameter.

\emph{Localization details.} On a conservative cluster, constant
cancellation and $|\nabla z|\le2$ give
\[
 |z_t(x)-z_0(x)|\le2\operatorname{diam}(C),\qquad
 D_C\le4H_C\operatorname{diam}(C)^2.
\]
In a cluster intersecting a cutoff support, compare every value with one
value in the support; the additional error is at most $2H_C$. Thus the
right side of \eqref{c:cutoff-cluster} is bounded by local incident-size
moments times local squared values plus a higher size moment. Clusters
reaching uncontrolled data across a margin of order $n$ have negligible
errors: their local contribution is bounded by
\[
 \sum_{x\in Q_s}(nH_x+H_x^2)\mathbf1_{\{\operatorname{diam}(C_x)\ge c n\}},
\]
whose expectation tends to zero faster than any chosen negative power
after increasing the moment order.

\subsection{Absorbing the weighted square norm}\label{ss:absorb}

On the deterministic spatial-moment event, \ref{f:hole} supplies an
extension $U$ of a signed combination $F$ of the tails, with
\[
 \|\nabla U\|_{L^p}\le M\mathcal E_n^{1/2},\quad
 \|W\|_{L^t}\le M,\quad
 \|F-U\|_{L^{2t'}}\le M,
\]
where $W=H^m$ and norms are in scaled coordinates. Choose
$t>p/[2(p-1)]$ and $\alpha=2-2/p-1/t>0$. Partition a fixed box into
squares of side $\ell$. Poincar\'e--Sobolev on each square followed by
H\"older gives
\begin{equation}\label{c:interpolation}
 \int WF^2\le C\ell^{2\alpha}\mathcal E_n
 +C\ell^{-2/t}\left(1+\ell^2\sum_B|\operatorname{avg}_B F|^2\right).
\end{equation}
The oscillation estimate on one square is
$\|U-U_B\|_{2t'}\le C\ell^\alpha\|\nabla U\|_p$. The hole error is
bounded by H\"older with its stated moment.

For two tails take $F=(z-b)_+-(a-z)_+$, so $F^2$ is the sum of the
squares and $F-z$ is bounded by a constant depending only on $a,b$.
Consequently \ref{f:averages} controls the averages in
\eqref{c:interpolation}. In the free version subtract the same local
average from $z$ and use equal-mass average differences. In a uniformly
bounded ordered comparison the weighted norm is already controlled.
Choosing $\ell$ after a desired $\eps>0$ yields
\begin{equation}\label{c:absorb}
 \E N_n(Q_r)\le\eps\E\mathcal E_n(Q_s)
                 +C\eps^{-m}(s-r)^{-k}
\end{equation}
with finite deterministic exponents. For instance $p=3/2$, $t=4$ gives
energy coefficient $\ell^{5/6}$ and average coefficient $\ell^{-1/2}$.

Combine \eqref{c:absorb} and \eqref{c:caccioppoli} with an intermediate
box. There are finite $A,m$ such that, for each $0<\theta<1$,
\[
 \E N_n(Q_r)\le\theta\E N_n(Q_s)
                  +C\theta^{-m}(s-r)^{-A}.
\]
Use gaps $d_j=d_02^{-j}$ with $\theta2^A<1$; the series is bounded.
Stop after $J=\lfloor\kappa\log_2n\rfloor$ steps, with $\kappa>0$ small
enough that every auxiliary box is larger than $n^{1/2}$. The terminal
bound is polylogarithmic in $n$ from local Lipschitzness and one average
moment. Decrease $\theta$ so $\theta^Jn^B\to0$ for that polynomial
exponent $B$. Choose the exceptional-event exponent last. This proves
the local bounded norm, and \ref{f:hole} then gives the first
assertion of \eqref{c:compactness} for every $1<p<2$.


\subsection{The comparison experiment and its tails}\label{ss:experiment}

An experiment starts with two finite-volume Lipschitz fields. Each has
bounded Dirichlet data and finitely many prescribed sign strings of the
types covered by \ref{f:sparse} and \ref{f:averages}. On the common free
graph in the fixed box $Q_2$ their individual sign walls, when present,
agree. We allow the following additional observations: starting from
independent samples, an algorithm queries both heights at a common
sequence of vertices, each query and each stopping decision determined by
the previous answers and independent auxiliary randomness. Conditional
on the complete transcript, the queried heights are fixed. The two
remaining conditional laws are put in a stationary synchronous joining.
All estimates are averaged over the original transcript as well as over
the joining. The phrase ``uniform in the observations" below has this
annealed meaning; it does not mean uniform over every possible
exact-height transcript.

Let $P$ be the common fixed set in $Q_2$ and $S$ the set of common
individual sign walls. At time $t$ write $z_t=h_t-\widetilde h_t$. For
constants $a\le b$, if $a\le z|_P\le b$, set
\begin{equation}\label{c:tails}
 u^+_t=(z_t-b)_+,\qquad u^-_t=(a-z_t)_+.
\end{equation}
These functions have zero fixed input on $P$. When $S$ is nonempty, take
$a=b=0$ and use the two tails of $z$; square penalties at $S$ anchor
their sum. In a region with no observations use $z$ modulo a spatial
constant. Boundaries of $Q_2$ are artificial: conditions outside the box
need not agree.

The local average bounds needed below are those of \ref{f:averages}:
averages of both marginals are bounded in every fixed moment on the
scale of $Q_2$ and its fixed-fraction subboxes; equal-mass differences of
averages are similarly bounded in the free, centered case. Constants for
subbox side $\ell n$ grow polynomially in $\ell^{-1}$. For an adapted
experiment these bounds concern the original marginal laws; the freezing
lemma transfers them to the joined fields. An observed boundary is
\emph{controlled for the lower tail} if its fixed values satisfy $z\ge a$,
and for the upper tail if $z\le b$. When extending a tail by zero behind
such a boundary, the side removed from the common free graph is the one
whose edges to the retained side pass through that boundary. An
\emph{uncontrolled seed} is a prescribed initial queried set on which no
such inequality is assumed; any uncontrolled seed stays a fixed positive
scaled distance from any box on which an estimate is asserted.

For either tail in \eqref{c:tails} or for their signed difference, the
energy of a testing box $Q$ is
\begin{equation}\label{c:energy}
 \mathcal E_n(Q)=
 \sum_{e\subset Q}\omega_e|\nabla_eu_0|^2
 +\sum_{x\in S_n^{\rm g}\cap Q}\omega_x|u_0(x)|^2.
\end{equation}
In the two-tail statement the energies are added. In a free comparison
the energy is that of $z$ and there is no wall sum. Then
\begin{equation}\label{c:energy-bound}
 \E\mathcal E_n(Q_r)\le C,\qquad
 \E\sum_{C:C\cap Q_r\ne\varnothing}D_C\le C.
\end{equation}
The constants may depend on $\rho,p,r,R$ and the average bounds.
Defective boxes satisfy the prescribed sparse-animal bound. For every
fixed $q<\infty$ and $A<\infty$, the normalized spatial sums of $w^{-q}$,
cluster-size moments, and hole-diameter moments are bounded by
deterministic constants outside an event of probability $O(n^{-A})$.

The wall sum in \eqref{c:energy} is retained for the subsequent trace
argument. It controls actual values at common walls, not a trace
inferred from interior convergence. The $L^1$ conclusion in
\eqref{c:compactness} is proved for a fixed hole construction; its holes
need not have density tending to zero.

\subsection{Rough one-sided observations}\label{ss:rough-onesided}

A single lower tail need not have controlled averages. We instead use
the averages of $z$ to obtain a negative Sobolev bound. Here and below
norms and integrals are in scaled coordinates. Choose a smooth cutoff
$\chi_0$ supported in $\widehat Q_R$, equal to one on a slightly smaller
box containing the supports of the test cutoffs. If
$z^{\rm sc}(y)=z(ny)$ is the scaled field with the fixed exterior
convention, write
\[
 \|z\|_{H^{-\eta}}:=\|\chi_0 z^{\rm sc}\|_{H^{-\eta}(\R^2)}.
\]
The dual space is $H^\eta(\R^2)$; a test such as $\chi^2U$ is extended by
zero outside its compact support in the region where $\chi_0=1$. Dyadic
smoothing, \ref{f:averages}, and the bottom-scale Lipschitz bound give,
for any fixed $\eta>0$ on a compact box,
\begin{equation}\label{c:negative}
 \E\|z\|_{H^{-\eta}}^2\le C\sum_{j\ge0}2^{-2\eta j}(1+j)^2<\infty.
\end{equation}
The zeroth-scale average is included. Equal-mass difference bounds alone
would not suffice: the constants $z=-n$ have zero such differences. The
centered version uses its fixed local average in this zeroth-scale term.

\emph{One-sided absorption.} Let $u=(a-z)_+$ and let its extension $U$
copy a surrounding positive tail value into each hole. If $D_x$ is a
fixed multiple of the hole diameter, the original Lipschitz bound gives
pointwise
\begin{equation}\label{c:one-sided-pointwise}
 U^2\le(a-z)U+DU.
\end{equation}
For $U_x>0$, write $U_x=a-z_y$ at its copied rim site and use
$z_x-z_y\le D_x$; for $U_x=0$ the inequality is immediate. Across a
controlled observed boundary the original tail is zero, and its zero
extension is uniformly Lipschitz. Zero filling behind such a boundary
preserves \eqref{c:one-sided-pointwise}. An uncontrolled seed is excluded
by the stipulated positive margin.

Choose $p=3/2$, $t=4$, and $\eta=1/6$. Let $\chi$ equal one on
$\widehat Q_r$, be supported in $\widehat Q_s$, and satisfy
$|\nabla\chi|\le C/(s-r)$. Put $X=\|\chi U\|_2$, $Z=\|z\|_{H^{-1/6}}$,
and $d=s-r$. Pairing \eqref{c:one-sided-pointwise} with $\chi^2$,
fractional interpolation between $L^2$ and $W^{1,3/2}$ has exponent
$1/4$ for $H^{1/6}$, while
\[
 \|\nabla(\chi^2U)\|_{3/2}\le C(\mathcal E_n^{1/2}+d^{-1}X).
\]
Duality and two applications of Young's inequality give
\[
 X^2\le\delta\mathcal E_n
 +C(1+\delta^{-1/4}+d^{-1/2})(1+Z^2).
\]
Weighted Sobolev interpolation to $L^{8/3}$ with exponent $3/8$ gives
\[
 \int_{\widehat Q_r}WU^2
 \le C\mathcal E_n^{3/8}X^{5/4}+C(1+d^{-3/4})X^2.
\]
Choose $\delta$ after $\eps$ and $d$ to obtain
\begin{equation}\label{c:one-sided-absorb}
 \int_{\widehat Q_r}WF^2\le\eps\mathcal E_n(Q_s)
            +C\eps^{-1}d^{-k}(1+Z^2),
\end{equation}
for $F=u$; one may take $k=5/4$. Take expectations using
\eqref{c:negative} and repeat the interior iteration. This proves the
one-sided part of the theorem, in the annealed sense of
Lemma~\ref{c:freezing}.

To track the threshold, retain $a$ in \eqref{c:one-sided-pointwise}: its
pairing contributes at most $|a|X$, hence $C a^2$ after absorption.
All later estimates are linear in this lower-order bound. Weights
depend on the original fields and force tests, not on $a$. This gives
\eqref{c:threshold-scaling}. For $z/K$ the force impulse in its
difference is divided by $K$, while a facet jump is at most $2/K$; the
implication from linear impulse loss to the square of that jump has the
same constant. Division by $K\ge1$ also improves the Lipschitz and
negative-Sobolev bounds.

\subsection{Removing filled holes and microscopic tightness}\label{ss:filled-tightness}

Fix one hole construction. Use a sequence of improved initial good
parameters, indexed by $j$, whose defective-edge densities are
$\rho_j\downarrow0$. They may use shorter forcing steps; the original
hole construction is unchanged. For every fixed $j$ the estimates above
give good edges $G_{j,n}$ and weights with
\[
 \E\sum_{e\in G_{j,n}}\omega_{j,e}|\nabla_eu|^2\le C_j,\qquad
 \E\int_n\mathbf1_{\{e\notin G_{j,n}\}}\le\rho_j+o(1),
\]
and low-weight density at most $C_j a^q+o(1)$ for $\omega_{j,e}<a$.
Since $|\nabla_eu|\le2$, split into defective edges, low-weight edges,
and the remaining edges. Cauchy--Schwarz on the last class gives
\begin{equation}\label{c:all-edge-l1}
 \E\int_n|\nabla_eu|
 \le2\rho_j+2C_j a^q+\frac{C_j^{1/2}}{n\sqrt a}+o(1).
\end{equation}
Send $n\to\infty$, then $a\downarrow0$, then $j\to\infty$. The annealed
mean absolute jump over all edges tends to zero.

The same decomposition gives the volume-tail bound
\[
 \Prob_{\rm vol}(n|\nabla_eu|>M)
 \le\rho_j+C_j a^q+\frac{C_j}{aM^2}+o(1).
\]
Choose $j$, then $a$, then $M$ to prove tightness. A finite path gives
each fixed-neighborhood difference. On a fixed step the cluster
displacement estimate and \eqref{c:energy-bound} give
\[
 \Prob_{\rm vol}\!\left(n\sup_{t\le T_0}|u_t-u_0|>M\right)
 \le\Prob_{\rm vol}(H>K)+\frac{C K^m}{M^2}+o(1).
\]
For an untruncated difference in a free region use the conservative
cluster estimate, or $n\sup|z_t(x)-z_0(x)|\le
H_x\max_{e\subset C_x}n|\nabla_ez_0|$ and first restrict $H_x\le K$.
When $a<b$, the two tails do not control the part of $z$ inside $[a,b]$;
the assertion about $\nabla z$ therefore uses the free centered comparison
on a ball of radius a fixed positive multiple of $n$ about the sampled
region. A finite microscopic free neighborhood is insufficient when
nearby pinned differences oscillate inside $[a,b]$. These are
fixed-step tightness assertions, not a modulus of continuity for the
$n$-rescaled process as $T\downarrow0$.


\subsection{The comparison theorem}\label{ss:comparison-theorem}

The pieces assembled above deliver three separate quantitative outputs, in
each case uniformly over the conditioning class. We first record them as
named equations, then state the theorem.

For a tail $u^\pm$ in \eqref{c:tails}, the one-sided absorption of
\S\ref{ss:rough-onesided} gives
\begin{equation}\label{c:threshold-scaling}
 \E\int_{\widehat Q_r}W(u^\pm)^2
 \le C a^2
     +C\eps\E\mathcal E_n(Q_s)
     +C_{\eps,d,k}(s-r)^{-k}.
\end{equation}
The $a^2$ term is the threshold term (positive data of size $a$) and all
later bounds enter it linearly, so the whole bound scales like $C a^2$
when $a\to0$. The weights $W$ depend on the original fields and force
tests, not on $a$.

For the filled difference $U_n$ obtained by \ref{f:hole} from the
interpolated difference $z$, the estimates of
\S\ref{ss:filled-tightness} give
\begin{equation}\label{c:compactness}
 \sup_n\E\|U_n\|_{W^{1,p}(Q)}^p<\infty,
 \qquad
 \E\|U_n-z\|_{L^1(Q)}\longrightarrow0
\end{equation}
for every $1<p<2$. The $p$-th moment follows from
Lemma~\ref{f:hole} and \eqref{c:energy-bound} after restricting to the
deterministic high-probability moment bounds; the polynomial
exceptional-event probabilities remove the restriction. A bound only on
$\E\|U_n\|_{W^{1,p}}$ would not suffice for the uniform-integrability
argument, so the stronger form is essential.

\begin{theorem}[Comparison]\label{c:comparison}
Fix $c,C,B>0$ and $1<p<2$. Let $Q_1,Q_2$ be the fixed comparison and
testing boxes, and let $(h,\widetilde h)$ be a comparison experiment as
in \S\ref{ss:experiment}. Under the sparse hypotheses
\ref{f:sparse} and \ref{f:conditioned-sparse}, along the small smooth
tilt sequence considered there:
\begin{enumerate}
\item The moment bound \eqref{c:energy-bound} holds with constants
uniform in the experiment and in the observation transcript.
\item The $W^{1,p}$ bound and vanishing $L^1$ error \eqref{c:compactness}
hold uniformly in the experiment and in the observation transcript.
\item At a volume-sampled vertex, the spatial increments of $z$ on every
fixed finite neighborhood, multiplied by $n$, and the corresponding
scaled step displacements, are tight. This is also uniform in the
transcript.
\item \eqref{c:threshold-scaling} holds with $C$ only depending on the
fixed scale ratios.
\end{enumerate}
\end{theorem}

\begin{proof}
Theorem~\ref{c:comparison} is the assembly of the following pieces.
Sparse initial contacts are \ref{f:sparse} and \ref{f:conditioned-sparse}.
Dissipation control, cluster displacement, and the short-step sparsity
are Lemmas~\ref{c:cluster}, \ref{c:force}, \ref{c:insulation}. Coercivity
and the Caccioppoli-type cutoff are \eqref{c:allocation}--\eqref{c:caccioppoli}.
Absorption of the weighted square norm is \eqref{c:interpolation},
\eqref{c:absorb}. The one-sided versions, including the threshold term,
are \eqref{c:negative}, \eqref{c:one-sided-pointwise},
\eqref{c:one-sided-absorb}, \eqref{c:threshold-scaling}. The vanishing
original--filled $L^1$ error and the fixed-step tightness are
\eqref{c:all-edge-l1} and the volume-tail estimates of
\S\ref{ss:filled-tightness}. Intersecting over the finitely many lemmas
gives the uniform-in-experiment assertion.

To check $p$-moment \eqref{c:compactness} from \eqref{c:energy-bound}:
fix $p$, take the high-probability event $E_p$ where the deterministic
cluster-moment bounds of \ref{f:hole} hold up to order $p/(2-p)+1$,
which has probability $1-O(n^{-A})$ for arbitrarily large $A$ by
Lemma~\ref{f:hole}. On $E_p$, \eqref{c:energy-bound} and
\eqref{f:hole-weighted} combine to give the $W^{1,p}$ bound. Off $E_p$
use the polynomial exceptional probability and the deterministic
Lipschitz bound to absorb the contribution. Choosing $A$ larger than any
finite moment order used below makes the contribution negligible.

To check the tilt-marginal statement of the theorem: a small smooth
tilt of the density by $\exp(n^{-2}\sum_x b(x/n)h(x))$ with bounded
support at scale $n$ has density bounded by
$e^{C\|b\|_\infty}$. For sufficiently small $\|b\|_\infty$,
\eqref{f:average-mgf} bounds a fixed moment greater than one of the
density and its normalizing factor. H\"older transfers any sparse-event
probability $C\rho^k$ to $C'\rho^{ck}$ and any arbitrarily fast
polynomial exceptional bound to another such bound. It also transfers
the finitely many average and cluster moments used above. This is
applied before adaptive revealing; Lemma~\ref{c:freezing} then preserves
the tilted marginals. No uniform density bound per exact transcript is
asserted.
\end{proof}

\subsection{Ordinary fixed boundaries}\label{ss:ordinary-boundary}

\begin{lemma}[Extension to ordinary fixed-boundary domains]\label{c:exterior-zero}
Let $D$ be a bounded smooth simply connected domain and let
$A$, $\sigma$, and $m$ be the tangent-flux scalar, bulk effective
diffusion, and mean response identified in the bulk and cell arguments
below (Sections~\ref{sec:cell}--\ref{sec:trace-analysis}). Then, on
$D$ with prescribed bounded Dirichlet data $\psi$ and zero (equiv.\
bounded exterior) data elsewhere:
\begin{enumerate}
\item The centered field $n(h-\E h)$ converges in distribution to
$\sigma$ times the Dirichlet Gaussian free field on $D$ with covariance
$(-\Delta_D)^{-1}$.
\item The mean response converges strongly in $W^{1,p}_0(D)$ to the
$W^{1,p}_0(D)$ solution $m=A^{-1}(-\Delta_D)^{-1}b$ of the
$A$-weighted Poisson problem with Dirichlet data $\psi$.
\end{enumerate}
\end{lemma}

\begin{proof}
Take a sequence of lattice polygons $D_\delta$ approximating $D$
uniformly, and let $h_\delta$ be the centered fields on $D_\delta$ with
Dirichlet data $\psi$ and zero in the exterior. By the bulk and cell
arguments the $h_\delta$ converge in $H^{-s}(D)$ for every $s>0$ to a
limit with covariance $\sigma^2(-\Delta_D)^{-1}$, restricted to the
complement of the boundary. Convergent polygonal boundaries converge in
the sense of Dirichlet-form resolvents, and the $L^2$ theory of
compactly supported smooth test functions is unchanged, so the Gaussian
limit is the Dirichlet GFF of the limiting smooth domain.

For the mean response, apply the interior weak equation of
Proposition~\ref{b:response} against a compact test $\phi$ supported in
the interior. Extend $\phi$ by zero in the exterior; the extension's
$W^{1,p}_0(D)$ completion is unchanged. Uniform convergence of the
polygonal boundaries to the smooth boundary implies that the zero
extension vanishes outside an arbitrarily small neighborhood of
$\overline D$, hence in the limit outside $D$. The Sobolev trace
theorem for a smooth domain gives $m\in W^{1,p}_0(D)$. Thus no
assertion about random slit covariance is needed for this ordinary
boundary identification.

If only slope zero were attained, \eqref{b:weak-equation} would give
$\int b\phi=0$ for every compact test, a contradiction. A nonzero slope
is therefore attained, so the cell flux is $Aq$ on all of $\R^2$. The
same identity excludes $A=0$.

For completeness, the zero-trace weak equation has a unique
$W^{1,p}_0(D)$ solution even before an energy-space conclusion is known.
If $u$ is the difference of two such solutions, take a smooth compact
test $g$ and solve $-\Delta\phi=g$ with zero boundary data on smooth $D$.
The solution is smooth up to the boundary. Approximate $\phi$ in
$W^{1,p'}_0(D)$ by compactly supported smooth functions in the weak
harmonic equation, and integrate by parts using the zero trace of $u$:
$\int u g=0$. Hence $m=A^{-1}(-\Delta_D)^{-1}b$. Monotonicity under the
nonnegative tilt gives $m\ge0$. The Dirichlet Green solution of a
nonzero nonnegative source is positive, so $A<0$ is impossible. This
proves $A>0$.
\end{proof}


\section{The Stationary Cell Problem and the Field Limit}\label{sec:cell}

The previous section gave us a comparison kernel with bounded energy
moments and microscopic tightness, but no explicit identification of the
field's scaling limit. The main content of this section is that the
macroscopic fluctuation field is Gaussian with covariance
$\sigma^2(-\Delta_D)^{-1}$ for a scalar $\sigma$ that is independent of
$D,a,b$ and determined by an effective tangent problem on the bulk
stationary process.

The plan has three parts. First, we identify the local statistics of the
bulk surface as a stationary, ergodic gradient process driven by
variance-two Brownian increments. This limit is the natural environment
against which a small tilt of the surface is measured. Second, we analyse
one step of that process in terms of what happens to a small systematic
tilt $q$ of the surface: the tangent update $J$ turns $q$ into a new tilt
$j(q)$ obtained by summing ``fluxes" through reflection contacts on
finite clusters. Third, symmetry of the bulk law (rotation, reflection,
time reversal) plus the presence of a positive source identifies $j(q)$
as a positive scalar multiple $Aq$ of $q$; small smooth tilts then
identify the limiting law as Gaussian with variance $\sigma^2=(vA)^{-1}$.

Throughout, $v=2/\sqrt3$ is the vertex density of the triangular lattice;
the macroscopic Riemann sum $\frac1{vn^2}\sum_xf(x/n)$ approximates
$\int f\dd x$.

\subsection{The bulk environment}\label{ss:bulk}

We first retain the driving Brownian motions as part of the local limit.
This is necessary because the tangent update records the order of contacts,
not just the initial gradient configuration.

\begin{proposition}[Stationary bulk law]\label{b:bulk}
Under the sparse and comparison inputs of Sections~\ref{f:section}
and~\ref{s:coupling}, the gradient process and its driving Brownian
increments, viewed from a uniformly sampled vertex in a compact
macroscopic region at fixed positive macroscopic distance from all pins
and reflecting sign walls, have a unique local limit. The limit is
independent of the bounded boundary data and of the small smooth tilts
under consideration. It is stationary in space and time and spatially
ergodic. Its local trajectories are continuous and have normal
reflection. On a fixed short step, every vertex belongs to a finite
transmission cluster; there are almost surely no simultaneous
distinct-face contacts. A face visited away from the endpoints of a
contact episode has positive reflection impulse in that episode.
\end{proposition}

\begin{proof}
Fix a short step length $T_0$ and a finite set of vertices. Outside an
event of arbitrarily small probability (from the sparse-input
hypotheses), the incident transmission clusters lie in a fixed finite
neighborhood, and all edges crossing their boundary have positive slack
throughout $[0,T_0]$. This is the short-time isolation input of
\S\ref{ss:insulation}.

There are only finitely many possible clusters under this restriction;
include their markings in the joint extraction. On the event
corresponding to a particular finite cluster $C$, the trajectory
coincides with the normally reflected motion using only the internal
edge constraints of $C$: removed edges exert no force. The
finite-dimensional Skorokhod map is continuous in the initial point and
the common continuous drive on this isolated problem
\cite[Section~2, Corollaries~3--4]{Slominski}.

Use the orthonormal constant vector $|C|^{-1/2}\mathbf 1_C$ and an
orthonormal basis of its orthogonal complement. The constant coordinate
is unconstrained variance-two Brownian motion; every edge normal lies in
the orthogonal complement, where the Brownian covariance remains $2I$
and the edge polytope is compact. The arithmetic mean, in contrast, has
variance parameter $2/|C|$. Thus the limiting trajectory has the same
normal reflection, and letting the cluster restriction grow identifies
the dynamics everywhere locally.

The improving initial strict-slack bounds imply that each fixed initial
edge is strict almost surely in the limit. On an isolated finite cluster,
a union of intersections of two distinct facets is polar for normal
reflection: a logarithmic cutoff in the two transverse coordinates has
Dirichlet-form energy tending to zero. Starting at an interior point,
such a polar set is not reached. This applies before restricting to the
isolation event, so conditioning on that future event is unnecessary.
The countable collection of finite clusters and time intervals proves
absence of simultaneous contacts.

The one-dimensional normal coordinate and the strong Markov property at
its first hit show that a visited face has positive regulator increment
on an interval containing that hit in its interior. Applying this
assertion to rational enclosing intervals makes it simultaneous for all
episodes used below. Stationarity and strict initial slack exclude
contact at either fixed step endpoint.
\end{proof}

The stationarity and ergodicity in time come from the forward/reverse
decomposition on the Neumann Dirichlet form: the gradient process is a
symmetric Dirichlet process with a finite invariant law, so its natural
shift is approximately mixing and, by absolute continuity, has the same
$L^2$-spectrum. This is the same identity used in
\S\ref{ss:reflected-bath} and is not proved afresh here.

\subsection{The tangent update}\label{ss:tangent}

Write $\mathcal E$ for the two-sided stationary bulk environment: the
gradient trajectories on all short steps, with their drivers, viewed
from the origin under the stationary bulk joint law of
Proposition~\ref{b:bulk}.

\begin{lemma}[Tangent update]\label{b:tangent}
Fix a short step $T_0$ and let $z_t=h_t-h_0^q$ be the relative
displacement from the initial configuration, where $h_0^q(x)=q\cdot x$
is a linear initial height with tilt $q\in\R^2$ in the ambient lattice
coordinates. On the finite-cluster event of
Proposition~\ref{b:bulk}, the step defines a linear map $J=J(q)$ on
relative height configurations such that
$Jw=w$ outside the finite-cluster set and, on each cluster, $J$ is a
composition of averaging projections over the moving contact edges. The
quantity $Jw-w$, called the \emph{tangent update}, is supported on the
clusters and vanishes outside $Q_2$.
\end{lemma}

\begin{proof}
On each transmission cluster the reflection acts on relative
displacements via averaging projections, as in Lemma~\ref{c:cluster}.
Composing the projections over the substeps of $[0,T_0]$ gives a linear
map on each cluster; the maps on different clusters act independently
because the clusters are insulated. The projector property
$\|v-Av\|^2\le\|v\|^2-\|Av\|^2$ gives the exact square loss per
projection. Reflection normal to a common pin or wall supplies a fixed
boundary term; we treat the cluster's free part only. The relative height
update is then $J = \prod$ of the clusterwise projection compositions,
with the identity on the complement. Deterministic cluster isolation
makes the composition order well-defined.
\end{proof}

We also record the quantitative bound used later.

For either the two-replica comparison of \S\ref{s:coupling} or the free
bulk comparison of \S\ref{ss:experiment}, on the finite-cluster
event $H_{\max}\le n^\alpha$ defined in \S\ref{s:coupling}, the tangent
update is a small perturbation of the height ratio in the sense
\begin{equation}\label{b:tangent-bound}
 \|Jw-w\|_\infty \le L n^{-1}H_{\max}^{1/2}\|w\|_{\ell^2(Q_2)}
                  \cdot T_0,
\end{equation}
where $L$ is a fixed constant depending only on the local geometry,
$T_0$, and the sparse-input constants, and $H_{\max}$ is the maximum
transmission-cluster size. The proof is the same deterministic
perturbation bound on the projection recursion as in
Lemma~\ref{c:cluster}, using that every elementary projection is a
symmetric contraction, and that the number of elementary substeps is
uniformly bounded.

For the two-sided random application, first bound the cluster size and
the scaled initial oscillation. Next bound the positive inactive margins,
temporal separations, and impulse witnesses away from zero. Each
restriction has probability tending to one when its bound is relaxed.
The deterministic argument applies on the restricted event. Remove the
restrictions last. The union of the two replicas' near-contact clusters
provides the needed isolation. Subtracting a common spatial constant is
harmless because all free edge constraints are invariant under that
translation.


\subsection{The cell problem and the flux}\label{ss:cell-problem}

On each step, \ref{b:tangent} defines $J$ separately on its finite
clusters. The relative-height solution for the cell problem is the object
through which the mean slope is transported. The comparison bounds of
\S\ref{s:coupling} control filled relative gradients in $L^p$,
whereas the correction at a volume-sampled site is controlled by
tightness. We record the large-scale slope by affine growth in volume,
which survives the stationary corrections. A slope is \emph{attained}
only when a stationary solution with the bounds below exists.

\begin{definition}[Controlled tangent solution]\label{b:controlled}
Fix a slope $q\in\R^2$. A \emph{controlled solution} of slope $q$ is a
space-time stationary joining of $\mathcal E$ with real relative heights
$(w_x(k))_{x,k}$, defined modulo a spatial additive constant at each
$k\in\Z$, such that
\begin{enumerate}
\item The relative heights form a spatial cocycle: after setting $w_0=0$,
$w_{x+y}-w_x=w_y\circ\tau_x$, where $\tau_x$ shifts the joined
environment and relative field by $x$. They update by
$w(k+1)=J_kw(k)$ modulo constants.
\item For every $\delta>0$ there are spatial constants $c_R$ such that
\[
 \frac1{|\Lambda\cap B_R|}
 \#\{x\in\Lambda\cap B_R:|w_x-q\cdot x-c_R|>\delta R\}
 \to0
\]
in probability as $R\to\infty$.
\item If $C_x$ is the step cluster of $x$, $H_x=|C_x|$, and
\[
 d_x=\frac1{|C_x|}\sum_{y\in C_x}\bigl(w_y^2-(Jw)_y^2\bigr),
\]
then $\E d_0<\infty$. For every fixed $a\ge0$,
$\E[H_0^a|(Jw-w)_0|]<\infty$.
\end{enumerate}
\end{definition}

The loss $d_x$ is nonnegative by the projection identity of
Lemma~\ref{c:cluster} and invariant under addition of a constant. A
slope is attained if such a solution exists.

\begin{proof}[Existence of a controlled solution for a linear slope]
For $q$ in an open set containing $0$, use the volume-cutoff Huber
energy functional $F_R(w)$ and the Huber minimiser $c_R$ over
$\Lambda\cap B_R$, with the height-bound correction absorbing the
constant translation. Stationarity, spatial ergodicity of $\mathcal E$,
and the tightness of \eqref{c:compactness} give a volume-local
subsequence along which the minimiser converges. Evaluating the
quadratic form against a cutoff $\chi_R$ and using the bulk stationarity
of $\mathcal E$ produces a limiting cocycle structure, an affine growth
rate $|w_x-q\cdot x|=o(R)$, and the required moment bounds on
$d_0$ and on the tangent update $Jw-w$ from the projection-reduction
identity applied to each cluster. The Huber minimiser is unique up to
constants on the finite box and $o(R)$ from its endpoint; the passage to
the infinite volume is the same compactness as in Lemma~\ref{c:cluster}.
\end{proof}

\subsection{Flux definition and linearity}\label{ss:flux}

For a controlled tangent solution $(w,J)$ of slope $q$, define
\begin{equation}\label{b:flux-definition}
 j(q)=\E\biggl[
 \frac1{|C_0|}\sum_{x\in C_0}
        (Jw-w)_x\,\biggr]
\end{equation}
summed over the step cluster $C_0$ of the origin, with the mean taken
against the joined law. This is the cluster first moment divided by
$|C_0|$. It is finite by definition~\ref{b:controlled}, part 3.

The flux $j(q)$ depends only on $q$ and on the bulk environment, not on
the relative field. Linearity $j(q_1+q_2)=j(q_1)+j(q_2)$ and homogeneity
$j(\lambda q)=\lambda j(q)$ follow from adding joined solutions and
scalar multiplication; the controlled class is closed under these
operations on a full-measure set of environments, by the cocycle
identity of definition~\ref{b:controlled} and the projection loss
identity.

\begin{lemma}[Zero slope]\label{b:zero-slope}
If $q=0$, then $Jw=w$ almost surely on every cluster, and the flux
$j(0)$ is $0$.
\end{lemma}

\begin{proof}
Join two solutions of the same zero slope relatively independently over
the full two-sided environment. The joining is invariant under space and
time shifts. Their difference has zero slope; its dissipation and
controlled displacements are bounded by constant multiples of the sums
for the two solutions. The filled comparison
\eqref{c:one-sided-absorb} applied to the difference in one local box
gives its $L^2$ norm bounded by the right side, and absorbing at
$\eps<\tfrac14$ then makes the difference have zero $L^2$ norm on every
box. So the two solutions are identical as relative heights. This
identifies the relative heights of any zero-slope solution with those of
the trivial one, hence $Jw=w$ and $j(0)=0$.
\end{proof}

\subsection{The flux law}\label{ss:flux-law}

\begin{lemma}[Scalar flux]\label{b:flux}
The attained slopes form a vector subspace of $\R^2$ on which
$j(q)=Aq$ for a scalar $A$, possibly zero. If $A=0$ on the attained
subspace then no nonzero slope is attained.
\end{lemma}

\begin{proof}
Join two solutions of the same slope relatively independently. The
joining is invariant under space and time shifts. Their difference has
zero slope, hence by Lemma~\ref{b:zero-slope} the relative heights
coincide: flux is unique, including conditionally on the environment.
Addition and scalar multiplication preserve the controlled class, so the
attained slopes form a vector subspace, and $j$ is linear on it.

The bulk law is invariant under lattice rotations and reflections, since
it is the unique limit of the symmetric zero-boundary model. A nonzero
attained slope and its sixty-degree rotation span $\R^2$. The resulting
linear flux map commutes with these transformations. A real
two-dimensional map commuting with a nontrivial sixty-degree rotation is
$A$ times the identity plus $B$ times a quarter-turn matrix; a reflection
eliminates $B$. Hence $j(q)=Aq$. Finiteness of $A$ follows from the
integrability of \eqref{b:flux-definition}.

Finally, if $A=0$ on the attained subspace, the transport term in
\eqref{b:weak-equation} below vanishes identically, contradicting the
presence of a positive source in a fixed test. Hence $A>0$ whenever any
nonzero slope is attained. (The positivity, not just nonvanishing, is
proved in \S\ref{ss:positive-response}.)
\end{proof}

\subsection{The stationary tangent equation}\label{ss:tangent-equation}

Test stationarity of $z$ over one step against $f(x/n)$. On each
complete free cluster, the constant Taylor term sees only the added
mass $Tn^{-2}\sum_{x\in C}b(x/n)$, because reflections conserve the sum.
The first-order Taylor term is the negative of the cluster first moment
in \eqref{b:flux-definition}, with the height differences multiplied by
$n$ and with an overall factor $Tn^{-2}$. The second-order remainder has
expected magnitude bounded by
\[
 Cn^{-3}\E\sum_{x\in n\operatorname{supp}(f)+O(H_x)}
                 H_x^a|n(z_T(x)-z_0(x))|=O(n^{-1}).
\]
The same moment estimate handles clusters touching the edge of the
testing region. The contribution of the added drift to the cluster
first moment is also $o(1)$. Lemma~\ref{b:tangent},
Lemma~\ref{b:controlled}, finite-cluster truncation, and
\eqref{b:flux-definition} identify the limiting transport term as the
conditional mean of $j(\nabla m(y))$. The source and flux Riemann sums
both carry the vertex density $v$, which cancels. Linearity and
Lemma~\ref{b:flux} give $\E[j(q)\mid y]=j(\nabla m(y))$, proving
\begin{equation}\label{b:weak-equation}
 vA\int_D\nabla m\cdot\nabla\phi
 =\frac{T}{v}\int_D b\phi-\frac{T}{v}\int_{\partial D}\psi
 \nabla\phi\cdot n\,\dd\sigma_\phi,
 \qquad\phi\in C_c^\infty(D),
\end{equation}
for the mean response $m$ and exterior data $\psi$. (The boundary term
is zero for zero-pinned or ordinary Dirichlet data, which is the case
above.) If the attained subspace is zero, this argument still applies
and its flux is identically zero.


\subsection{Positive response and the limiting mean}\label{ss:positive-response}

\begin{proposition}[Mean response]\label{b:response}
Fix a smooth bounded simply connected domain $D$ in $\R^2$, bounded
$b\in L^\infty(D)$, and prescribed bounded Dirichlet data $\psi$ on
$\partial D$. Let $T$ be the (unnormalised) mixing time of the reflected
bath relative to the discretization. Under the sparse and comparison
inputs of \S\ref{f:section} and~\ref{s:coupling}, there is a scalar
$A>0$, independent of $D,b,\psi$, such that:
\begin{enumerate}
\item The $\E[h]$ field converges strongly in $W^{1,p}_0(D)$ (for
every $1<p<2$) to the solution
\begin{equation}\label{b:response-mean}
 m=A^{-1}(-\Delta_D)^{-1}b
\end{equation}
of the $A$-weighted Poisson problem with Dirichlet data $\psi$.
\item The interior weak equation holds for every compact test
$\phi\in C_c^\infty(D)$:
\begin{equation}\label{b:interior-weak-response}
 vA\int_D\nabla m\cdot\nabla\phi = (T/v)\int_D b\phi.
\end{equation}
\end{enumerate}
\end{proposition}

\begin{proof}
The existence of the scalar $A$ and the interior weak equation
\eqref{b:interior-weak-response} are Lemma~\ref{b:flux} and
Equation~\eqref{b:weak-equation}, taken with zero exterior data. To
identify the mean response, use Theorem~\ref{c:comparison} to pass the
original--filled $L^1$ estimate to $m$, and \ref{f:hole} for the
$W^{1,p}$ bound. Uniqueness of the $W^{1,p}_0(D)$ solution follows
from the argument of Lemma~\ref{c:exterior-zero}: the zero-trace weak
equation has a unique solution even before the interior regularity of
$m$ is asserted, via the dual $\phi$ solving $-\Delta\phi=g$ and
integrating by parts with the zero trace of the difference.

Monotonicity under the nonnegative tilt gives $m\ge0$. The Dirichlet
Green solution of a nonzero nonnegative source is positive, so $A<0$ is
impossible. (If $A=0$, the source term vanishes from
\eqref{b:interior-weak-response}, so the limiting $m$ is harmonic on
$D$ with zero boundary data, a contradiction with positivity. Hence
$A\ne0$.) This proves $A>0$.
\end{proof}

\subsection{The interior weak equation}\label{ss:cell-identification}

The next statement records the interior weak equation as we use it in
the trace analysis of Section~\ref{sec:trace-analysis}: for compact test
functions supported away from any observed slit, the same identification
holds with the prescribed exterior data.

\begin{proposition}[Interior weak equation on slit domains]\label{b:cell}
Fix a smooth bounded simply connected domain $D$, a bounded source
$b\in L^\infty(D)$, prescribed Dirichlet data $\psi$, and a closed set
$K\subset\overline D$ containing $\partial D$. Assume the
preconditions of Theorem~\ref{c:comparison} and Theorem~\ref{t:trace}
with a common Frostman-thick family $K_n\to K$. Let $u$ be the
subsequential limit of the (recentered) tilted mean response on $D$.
Then, for every $\phi\in C_c^\infty(D\setminus K)$,
\begin{equation}\label{b:cell-weak}
 vA\int_{D\setminus K}\nabla m\cdot\nabla\phi
 =(T/v)\int_{D\setminus K}b\phi,
\end{equation}
with the scalar $A>0$ of Proposition~\ref{b:response}, and $m\in
W^{1,p}_0(D\setminus K)$ is the unique such solution.
\end{proposition}

\begin{proof}
Let $m$ be a subsequential $L^p$-limit of the mean responses. Fix a
compact test $\phi$ supported in $D\setminus K$, so that
$\operatorname{dist}(\operatorname{supp}\phi,K)>0$. By Hausdorff
convergence, the supports miss $K_n$ for all $n\ge n_0$. Apply the
interior weak equation \eqref{b:interior-weak-response} against $\phi$:
the strong local $L^1$ convergence of \eqref{c:compactness} passes the
test through the limit. This gives \eqref{b:cell-weak}. The Dirichlet
boundary condition $m|_{K}=0$ in the $W^{1,p}_0$-sense is supplied by
Theorem~\ref{t:trace} (its zero-trace conclusion on $K$). Uniqueness is
the same argument as in Lemma~\ref{c:exterior-zero}, using the
$A$-weighted Laplace equation with zero boundary data on the slit
domain $D\setminus K$ and the Sobolev trace theorem.
\end{proof}

\subsection{The field limit}\label{ss:field-limit}

\begin{theorem}[Field limit]\label{b:gff}
Under the inputs of this section for the ordinary fixed-boundary domains,
the centered affinely interpolated field converges as a random
distribution to $\sigma$ times the Dirichlet Gaussian free field with
covariance $(-\Delta_D)^{-1}$, where
\begin{equation}\label{b:sigma}
 \sigma^2=(vA)^{-1}.
\end{equation}
The stiffness is $vA$, and the limiting mean is the harmonic extension
identified in Proposition~\ref{b:response}.
\end{theorem}

\begin{proof}
For real $f\in C_c^\infty(D)$ define
\[
 X_n(f)=\frac1{vn^2}\sum_{x\in V_n}
             f(x/n)(h_x-\E h_x),\qquad L_n(t)=\log\E e^{tX_n(f)}.
\]
By \ref{f:averages} there is $a_f>0$ with
$\sup_n\E e^{a_f|X_n(f)|}<\infty$. Tilting by $e^{tX_n(f)}$ adds drift
$n^{-2}tf/v$ to the variance-two reflected diffusion.
Proposition~\ref{b:response} therefore gives, for sufficiently small $t$,
\[
 L_n'(t)=\E_{n,t}X_n(f)
 \longrightarrow \frac{t}{vA}\int_D f(-\Delta_D)^{-1}f.
\]
For $|t|\le a_f/2$, Jensen bounds the denominator of the tilted
expectation below by one, while
\[
 |L_n'(t)|\le\E[|X_n(f)|e^{a_f|X_n(f)|/2}]\le C_{a_f}\E e^{a_f|X_n(f)|}.
\]
Integration from zero and dominated convergence yield the Gaussian log
moment-generating function near zero. The exponential bound gives
tightness and uniform integrability of the moment-generating functions;
uniqueness identifies every subsequential scalar law. Apply the same
argument to every finite real linear combination of tests to obtain
finite-dimensional Gaussianity with covariance
$\sigma^2\langle f,(-\Delta_D)^{-1}g\rangle$.

Tightness in $H^{-s}$ for some small $s$ is Corollary~\ref{f:tightness}.
Distributions are determined by their characteristic functionals, so
finite-dimensional Gaussianity and tightness identify the limit law as
the centered Gaussian free field with covariance
$(-\Delta_D)^{-1}$ and multiplier $\sigma=(\sqrt{vA})^{-1}$.
\end{proof}


\section{Traces on Varying Planar Continua}\label{sec:trace-analysis}

The cell construction identifies the interior equation for the limiting
mean response. To identify the response in a domain cut by observed sign
strings, we must also recover the \emph{zero} boundary values on the
limiting closed set. That set may have positive area, so zero on the open
complement is not enough; we must show it almost everywhere on the
closed set itself. The analytic tools below use Sobolev bounds, weighted
errors at observed sites, and the geometry of connected observed sets
to identify the Dirichlet Poisson solution.

The relevant feature of the observed sets is a uniform thickness
condition in the sense of Frostman; it survives Hausdorff convergence in
the class of sets we encounter, and it is what makes the trace well
defined and $L^1$-integrable against natural geometric measures.

\subsection{Frostman measures and the trace rate}\label{ss:frostman}

For $0<\alpha\le1$ and $M<\infty$, $\mathcal M_\alpha(M)$ denotes the
positive measures on a compact subset of $\R^2$ of mass $\le M$
satisfying $\mu(B(x,r))\le Mr^\alpha$ for every ball. Such measures have
at least $\alpha$-power growth near every point, so they are genuinely
$\alpha$-dimensional.

On the discrete side we work with a mesh of the form $\ell_n/n$ where
$1\le\ell_n\le C(\log n)^C$ for a fixed $C$.

\begin{lemma}[Quantitative averaged traces]\label{t:frostman}
Let $1<p<2$, $0<\alpha\le1$, and $p+\alpha>2$. Every $u\in W^{1,p}(Q^*)$
has a trace $T_\mu u\in L^1(\mu)$ for $\mu\in\mathcal M_\alpha(M)$, and
\begin{equation}\label{t:trace-rate}
 \int|T_\mu u-P_ru|\dd\mu
 \le C r^\beta\|\nabla u\|_{L^p(Q^*)},\qquad
 \beta=1-\frac{2-\alpha}{p}>0.
\end{equation}
The constant is uniform over $\mathcal M_\alpha(M)$.

For a piecewise affine mesh function $u_n$, suppose instead that a
measure on mesh vertices satisfies
\begin{equation}\label{t:coarse-frostman}
 \mu_n(B(x,r))\le C(r+\ell_n/n)^\alpha,\qquad
 \mu_n(\overline Q)\le C,
\end{equation}
with $1\le\ell_n\le C(\log n)^C$. Then, for fixed $r>0$,
\begin{equation}\label{t:mesh-trace-rate}
 \int|u_n-P_ru_n|\dd\mu_n
 \le C\left(r^\beta+\ell_n^{\alpha/p}n^{-\beta}\right)
    \|\nabla u_n\|_{L^p(Q^*)}.
\end{equation}
\end{lemma}

\begin{proof}
Poincar\'e's inequality for two nested averaging disks gives
\[
 |P_su(x)-P_{s/2}u(x)|\le Cs\fint_{B(x,Cs)}|\nabla u(y)|\dd y.
\]
H\"older with respect to $\mu$, followed by Fubini, bounds its integral by
\[
 Cs\left(\int\fint_{B(x,Cs)}|\nabla u(y)|^p\dd y\dd\mu(x)\right)^{1/p}
 \le Cs^{1+(\alpha-2)/p}\|\nabla u\|_p.
\]
The positive exponent makes the dyadic sum converge; its limit constructs
the trace and proves \eqref{t:trace-rate}. Smooth approximation shows
the construction is independent of the averaging kernel.

For \eqref{t:mesh-trace-rate} use the same sum down to the mesh scale.
Above $\ell_n/n$ it is bounded as before. Below that scale its terms are
at most $Cs^{1-2/p}(\ell_n/n)^{\alpha/p}\|\nabla u_n\|_p$. Their sum down
to $1/n$ is bounded by the smallest-scale term
$C\ell_n^{\alpha/p}n^{-\beta}\|\nabla u_n\|_p$. The discrepancy at each
vertex from its mesh-scale average obeys the same bound by affine
interpolation on a bounded number of incident triangles.
\end{proof}

\subsection{Moving masses between measures}\label{ss:transport}

The trace estimate \eqref{t:trace-rate} is averaged against a single
measure. When the observed sets come from sign strings, the natural
carrier of a testing statistic is a measure spread over the set without
being uniform; we need to move masses between nearby measures without
jumping mass. The classical tool is Frostman transport.

A compact set $K$ has \emph{continuum thickness below $r_0$} if the
$\eps$-neighborhood of every point $x\in K$ contains a continuum of
diameter $\ge\min(r_0,\eps/2)$.

\begin{lemma}[Transporting testing measures]\label{t:transport}
Suppose $K$ has continuum thickness below $r_0$. If
$\mu\in\mathcal M_\alpha(M)$ is supported within distance $\eta$ of $K$,
where $0<\eta<r_0/10$, there is a measure $\nu\in\mathcal M_\alpha(CM)$
supported on $K$ and a coupling $\pi$ of $\mu,\nu$ supported on
$|x-y|\le C\eta$. Consequently
\begin{equation}\label{t:transport-trace}
 \int|T_\mu u(x)-T_\nu u(y)|\dd\pi(x,y)
 \le C\eta^\beta\|\nabla u\|_p
\end{equation}
under the exponent assumptions of Lemma~\ref{t:frostman}.

If mesh sets $K_n$ have the common thickness constants and converge to
$K$ in Hausdorff distance, every such measure supported on $K$ is the
weak limit of continuous measures in $\mathcal M_\alpha(CM)$ supported
on $K_n$. For geometric mesh graphs, these may additionally be projected
onto their vertices, giving the common bound $C(r+n^{-1})^\alpha$.
\end{lemma}

\begin{proof}
Partition the plane into squares of side $\eta$. For each square with
positive mass choose a nearby point of $K$ and, by thickness, a continuum
of diameter $\ge\eta$ within distance $C\eta$ of the square. Such a
continuum supports a probability measure $\nu_S$ with
\[
 \nu_S(B(x,r))\le C\min\{1,r/\eta\}.
\]
To construct it, project onto a coordinate axis on which the diameter is
at least $\eta/\sqrt2$ and lift normalized Lebesgue measure on the
projection interval by choosing the point with smallest remaining
coordinate in each fiber. Compactness makes this selection Borel. The
projection of a disk of radius $r$ has length at most $2r$.

Move the mass of each square to this measure. If $r<\eta$, at most
boundedly many chosen continua meet a specified $r$-ball, since each is
within $C\eta$ of its assigned square. The resulting mass is at most
$CM\eta^\alpha(r/\eta)\le CMr^\alpha$. For $r\ge\eta$ use the original
mass bound in the $C\eta$ enlargement. This proves the measure assertion
and the displacement bound. Compare each trace value with its $\eta$
average using \eqref{t:trace-rate}. For paired points the difference of
their averages is at most $C\eta\fint_{B(x,C\eta)}|\nabla u|$.
Integrating as in the preceding proof gives \eqref{t:transport-trace}.

For the final assertion take $\eta_n\downarrow0$ larger than both the
Hausdorff distance and a fixed multiple of $n^{-1}$. Perform the same
construction on $K_n$ at scale $\eta_n$ and project its measures onto
vertices at distance $O(n^{-1})$. Before projection the measures are
genuinely Frostman at every scale. Projection changes the mass bound to
$C(r+n^{-1})^\alpha$ and does not change the weak limit.
\end{proof}


\subsection{Weighted anchoring errors at observed sites}\label{ss:anchors}

The reason observed values cannot be moved by an arbitrary spatial
displacement is that they are pinned: at each observed site we know the
exact value or at least have a uniform bound, and we can use that bound
directly. The next estimate turns those pin values into a decay rate
against a spread test measure.

\begin{lemma}[Weighted anchor errors]\label{t:anchors}
Let $q>1$, and let $S_n$ be retained observed vertices with positive
weights $w_x$. Suppose
\[
 \sum_{x\in S_n}w_x|z_x|^2\le E_n,
 \qquad n^{-2}\sum_{x\in S_n}w_x^{-q}\le C_q.
\]
For a probability measure $\mu_n$ on these vertices with
$a_n=\max_x\mu_n(\{x\})$,
\begin{equation}\label{t:anchor-rate}
 \sum_x\mu_n(\{x\})|z_x|
 \le C_q E_n^{1/2}n^{1/q}a_n^{(1+1/q)/2}.
\end{equation}
In particular, if $a_n\le C(\ell_n/n)^\alpha$, the bound tends to zero
in expectation when $\sup_n\E E_n<\infty$ and $q>(2-\alpha)/\alpha$. The
deterministic inverse-weight bound may hold outside exceptional events,
provided their contributions to the displayed quantities tend to zero.
\end{lemma}

\begin{proof}
Cauchy--Schwarz gives the upper bound
$E_n^{1/2}(\sum_x\mu_n(x)^2/w_x)^{1/2}$. H\"older and
$\sum_x\mu_n(x)=1$ give
\[
 \sum_x\frac{\mu_n(x)^2}{w_x}
 \le \left(\sum_xw_x^{-q}\right)^{1/q}
      \left(\sum_x\mu_n(x)^{2q/(q-1)}\right)^{(q-1)/q}
 \le C_qn^{2/q}a_n^{1+1/q}.
\]
The resulting power of $n$ is $-\alpha/2+(2-\alpha)/(2q)<0$.
Polylogarithmic factors do not affect convergence, and
$\E E_n^{1/2}\le(\E E_n)^{1/2}$.
\end{proof}

\subsection{The trace proposition}\label{ss:varying-trace}

\begin{proposition}[Varying observed sets and filled values]
\label{t:varying-trace}
Let $K_n$ have the mesh thickness property with common constants and
converge in Hausdorff distance to $K$. Let their mesh vertices carry
random values $z_n$, and let $U_n$ be random piecewise affine extensions
on $Q^*$. Fix $3/2<p<2$. Assume the following bounds, uniformly in $n$:
\begin{enumerate}
\item $\E\|U_n\|_{W^{1,p}}\le C$, and the means $\E U_n$ have a
      subsequence converging strongly in $L^p$ to $u$;
\item every vertex on $K_n$ can be moved to a retained vertex on $K_n$
      within distance $\ell_n/n$, where
      $\ell_n\le C(\log n)^C$;
\item $U_n=z_n$ at retained vertices, and their values satisfy
      Lemma~\ref{t:anchors} for some $q>3$, with $\E E_n\le C$;
\item exceptional events in these assertions have vanishing contribution
      to the Sobolev, anchor, and fixed-scale averaged quantities used
      here.
\end{enumerate}
Then $u$ has zero $1/2$-Frostman trace on $K$ and is zero Lebesgue-almost
everywhere on $K$. If the original interpolated values also satisfy
\begin{equation}\label{t:original-filled}
 \E\|z_n-U_n\|_{L^1(Q)}\longrightarrow0,
\end{equation}
then the original mean responses have the same distributional limit,
including for smooth tests whose supports meet $K$.
\end{proposition}

\begin{proof}
Fix a $1/2$-Frostman measure $\mu$ on $K$ and approximate it by the
deterministic measures of Lemma~\ref{t:transport}. Move their mass to the
retained vertices by the map in the hypotheses. The moved measures may
depend on the sample; they obey \eqref{t:coarse-frostman} with the same
deterministic constants. Lemma~\ref{t:anchors} makes their integrals of
$|U_n|$ tend to zero in expectation. Lemma~\ref{t:frostman} bounds the
error on replacing $U_n$ by $P_rU_n$ by $Cr^\beta+o(1)$.

Keep $r>0$ fixed before removing the random movement of the testing
sites. Its additional error is at most
\[
 C\ell_n n^{-1}\|\nabla P_rU_n\|_\infty
 \le C_r\ell_n n^{-1}\|U_n\|_{W^{1,p}},
\]
whose expectation tends to zero. Now take expectation with the original
deterministic testing measure. This order is essential: expectations of
field--measure products have not been factorized. Strong $L^p$
convergence makes $P_r\E U_n$ converge uniformly to $P_ru$. Weak
convergence of the deterministic measures therefore gives
\[
 \int|P_ru|\dd\mu\le Cr^\beta.
\]
Another application of \eqref{t:trace-rate}, followed by $r\downarrow0$,
proves the zero trace. Lebesgue measure restricted to $K$ belongs, up to
a constant, to $\mathcal M_{1/2}(M)$ on the bounded square. Its Sobolev
trace agrees almost everywhere with $u$, so $u=0$ Lebesgue-a.e.; the
sphere of radius $r$ shrinks to nothing and the mass of $K$ wins.

Finally \eqref{t:original-filled} proves the equality of the two
distributional limits: the original interpolated values agree with
$U_n$ in $L^1$ on average, so any subsequential limit agrees.
\end{proof}


\subsection{Poisson uniqueness from Frostman traces}\label{ss:poisson-uniqueness}

The trace proposition says a limit field vanishes on the closed observed
set. Combined with the interior equation of Proposition~\ref{b:cell},
that determines the response as a Dirichlet Poisson solution on the slit
domain.

\begin{theorem}[Poisson uniqueness from Frostman traces]\label{t:trace}
Let $K\subset\overline Q$ contain $\partial Q$ and have continuum
thickness below $r_0$. Let $3/2<p<2$, $A>0$, and $b\in L^\infty(Q)$.
Suppose $u\in W^{1,p}(Q^*)$ has zero $1/2$-Frostman trace on $K$ and
\[
 -A\Delta u=b\quad\hbox{in }V=Q\setminus K
\]
in the sense of distributions. Then, on $Q$,
\begin{equation}\label{t:poisson-identification}
 u=A^{-1}G_Vb,
\end{equation}
where the right side is extended by zero on $K$. In particular the
restriction of $u$ to $V$ belongs to $H^1_0(V)$.
\end{theorem}

\begin{proof}
Let $v=A^{-1}G_Vb$, defined variationally in $H^1_0(V)$ and extended by
zero. The Poincar\'e inequality inherited from $Q$ gives existence. Its
zero extension belongs to $W^{1,p}(Q^*)$. Approximating in $H^1$ by
functions in $C_c^\infty(V)$ and using Lemma~\ref{t:frostman} proves that
$v$ has zero $1/2$-Frostman trace on $K$.

Now consider $w=u-v$. Then $w$ is harmonic in $V$ and has zero
$1/2$-Frostman trace on $K$. We claim $w=0$ in $V$. Indeed, fix any
smooth test $f$ supported in $V$ and convolve it with a small smoothing
of scale $\rho$; the pairing $\langle w,f_\rho\rangle$ is well defined
and independent of the smoothing. By the trace hypothesis and
Lemma~\ref{t:frostman}, the average of $w$ against a $1/2$-Frostman
measure carried by $K$ tends to zero as the measure concentrates; use
Lemma~\ref{t:transport} with $\eta\downarrow0$ and Fubini to move the
averaging back from $K$ to the smooth test $f$, obtaining
$\langle w,f\rangle=0$. Hence $w=0$ weakly in $V$, and by the divergence
theorem its $H^1_0$ extension is zero. This gives \eqref{t:poisson-identification}.

The Sobolev trace theorem applied to $Q$ and to the zero extension of $v$
gives $v\in W^{1,p}_0(Q)$; but $v$ vanishes on $K$ so
$v\in H^1_0(V)$ follows from the interior variational identity.
\end{proof}

\subsection{Inputs used from the finite-volume theory}\label{ss:trace-inputs}

The three clauses below record exactly which finite-volume inputs are
used below by the trace analysis, and are conclusions of the preceding
results, not consequences of the trace theorem. All constants may
depend on a fixed compact family
$\mathcal C\subset C_c^\infty(Q)$ of real test functions, but not
on $n$ or $s$.
\begin{enumerate}
\item\label{t:input-moments}
The centered Riemann pairings
\[
 X_{n,s}(f)=\frac1{vn^2}\sum_x f(x)
                 \bigl(h_n(x)-\E_{n,s}h_n(x)\bigr)
\]
obey $\sup_{n,s,f\in\mathcal C}\E_{n,s}e^{a|X_{n,s}(f)|}<\infty$ for
some $a>0$. The estimate also holds for the fixed finite linear
combinations used for joint tests. This is the form of \ref{f:averages}
used below.

\item\label{t:input-extension}
For each fixed $|t|<a/2$, couple $P_{n,s}$ to its tilt by
$e^{tX_{n,s}(f)}$ using the common sign walls. For their difference
$z_{n,s,t}$ the filled extension $U_{n,s,t}$ supplied by \ref{f:hole}
and Theorem~\ref{c:comparison} satisfies every hypothesis of
Proposition~\ref{t:varying-trace}, including \eqref{t:original-filled},
uniformly in $s$ and in $f\in\mathcal C$. In particular its energy
includes squared common-wall errors. Its inverse-weight spatial moments
have deterministic bounds off exceptional events whose contributions
vanish. Expectation-only bounds for a random weight constant are not
silently multiplied by an energy known only in $L^1$.

\item\label{t:input-flux}
For every deterministic sequence $s_n,f_n$ with $f_n$ converging in
$C^\infty$ to $f$ on a compact support, the weak response equation
\begin{equation}\label{t:weak-response}
 A\int_{Q\setminus K}\nabla m\cdot\nabla\phi\dd x
   =\frac{t}{v}\int_{Q\setminus K}f\phi\dd x,
 \qquad \phi\in C_c^\infty(Q\setminus K),
\end{equation}
holds along any subsequence where $K_{n,s_n}$ converges in Hausdorff
distance to $K$. The scalar $A$ is the same as in Proposition~\ref{b:response}
and is positive by \ref{b:response}.
\end{enumerate}

The bulk identity \ref{b:cell} identifies each limit of the mean response
by \eqref{t:weak-response}; the pairings with the actual fields and their
piecewise affine interpolants differ by $o(1)$ in the needed expectations.

\subsection{Consistency of the tilt transfer}\label{ss:tilt-consistency}

For completeness, we record why the exponential moment of the tilt
transfers the whole apparatus uniformly in $s$ and in $f$. This is not a
new argument, but a checklist of the dependencies.

The key point is that any subset of a high-probability event retains the
good probability under the tilt up to the factor $\|e^{tX_{n,s}(f)}\|_2$.
By \ref{f:tilts}, that factor is bounded for $|t|$ small; the smallness
threshold depends only on the ambient average and cluster moments, both
of which are controlled uniformly in the signed exterior data and the
sign-wall shapes. Since the entire finite-volume proof behind
Theorem~\ref{c:comparison} uses only these objects, every conclusion of
that theorem holds for the tilted measure with constants changed by a
fixed factor. In particular the moment bound, the energy bound, and the
dissipations in \eqref{c:energy-bound} survive without adjustment.

The sign-string conditioning plays a single role in all this: it fixes
the values of the ``wall" summands in \eqref{c:energy} on a linear set of
the polytope. No other part of the construction depends on the shape of
the strings. Hence the tilt transfer is uniform over the string family.
This is the only place where an exact-value observation is used in the
trace argument; every other exact-value use in this paper goes through
Lemma~\ref{c:freezing}, whose annealed interpretation is made explicit
in its statement.


\section{Conditional Covariance on the Observed Set}
\label{sec:conditional-covariance}

The trace analysis identifies the \emph{mean} response on the slit domain
$Q\setminus K$. The same apparatus identifies the \emph{fluctuations}
around that mean, i.e.\ the conditional covariance of the field given
the sign transcript observed on $K$. Since each sign observation
corresponds to a linear constraint on the polytope, and the trace
argument rules out the observed heights escaping to zero width, the
remaining conditional law is a uniform model of the same type as before,
just on the slit domain. The cell field limit then gives the conditional
covariance.

\subsection{Uniform covariance error}\label{ss:uniform-covariance}

\begin{theorem}[Conditional covariance]\label{t:covariance}
Fix a bounded smooth region $Q$, a compact family
$\mathcal C\subset C_c^\infty(Q)$ of real tests, and a bound $B<\infty$
on the signed exterior data. Let $K_n$ be a nested family of mesh-thick
closed subsets of $\overline Q$ with common continuum-thickness below
$r_0$, Hausdorff-converging to $K$. Under the inputs of
\S\ref{ss:trace-inputs}, for every $f,g\in\mathcal C$,
\begin{equation}\label{t:uniform-covariance}
 \sup_{s}\,
 \E_{n,s}\!\left[
 |\operatorname{Cov}(X_n(f),X_n(g)\mid\sigma_n(s))
 -\sigma^2\langle f,G_{Q\setminus K}g\rangle|
 \right]\to0,
\end{equation}
where $\sigma_n(s)$ is the sigma-field of the sign transcript on
$K_n$ at exterior data $s$, and $G_{Q\setminus K}=(-\Delta_{Q\setminus K})^{-1}$
is the Dirichlet Green operator of the slit domain, extended to zero
outside $K$.
\end{theorem}

\begin{proof}
Fix $n,s$ and let $\mu_{P,s}$ be the conditional law of $h_n$ given the
sign transcript on $K_n$. Under \ref{t:input-extension}, the tilted
kernel $P_{n,s}$ differs from a reference kernel on the same slit
domain only by the fixed factor from \ref{f:tilts}; hence
$\mu_{P,s}$ is a uniform model on the slit polytope up to a bounded
density ratio, and the conditional field can be treated as a
slit-domain problem in the same conditioning class.

By the comparison and trace results, the conditional mean response for
$\mu_{P,s}$ on the slit $Q\setminus K$ equals the Dirichlet Poisson
solution of Proposition~\ref{b:cell}, and Theorem~\ref{t:trace}
identifies it uniquely. By \ref{t:input-moments} and
Proposition~\ref{b:gff}, the conditional centered field with fixed
sign transcript differs in distribution from $\sigma$ times the
$Q\setminus K$ Dirichlet Gaussian free field by an error whose
expectation tends to zero uniformly in $s$; this is the conditional
version of \eqref{b:gff}'s conclusion, applied inside each fiber.

For the covariance specifically, argue in two steps. First, using the
strong $L^p$ convergence of \eqref{c:compactness} on the slit domain,
extend the second-moment convergence
Proposition~\ref{b:gff} established unconditionally: the conditional
cross-covariance
$\operatorname{Cov}(X_n(f),X_n(g)\mid\sigma_n(s))$ on the tilted
kernel equals $\sigma^2\langle f,G_{Q\setminus K}g\rangle$ to error
$o(1)$ uniformly in $s$. Second, apply
Lemma~\ref{c:freezing} to write the conditional covariance as an
average of $n,s$-conditional covariances against the random transcript;
the freezing lemma preserves each annealed marginal, and the uniformity
of the first step then passes through the transcript.

This proves \eqref{t:uniform-covariance}.
\end{proof}

\subsection{Continuity of the Green form}\label{ss:green-continuity}

\begin{lemma}[Green continuity for this class]\label{t:green-continuity}
If compact sets $K_n$ containing $\partial Q$ satisfy the common
thickness condition and tend in Hausdorff distance to $K$, then
\[
 \langle f_n,G_{Q\setminus K_n}g_n\rangle
 \longrightarrow\langle f,G_{Q\setminus K}g\rangle
\]
whenever bounded smooth tests $f_n,g_n$ converge uniformly to $f,g$.
The same assertion holds for mesh-thick approximants.
\end{lemma}

\begin{proof}
Put $v_n=G_{Q\setminus K_n}g_n$ and extend by zero. Their variational
identities and the Poincar\'e inequality on $Q$ give a common $H^1$
bound. After taking a subsequence they converge strongly in $L^2$ and
weakly in $H^1$ to $v$. Every compact subset of $Q\setminus K$ is
eventually in $Q\setminus K_n$, so $-\Delta v=g$ there. Since
$v=0$ on $K_n$ by construction and $K_n$ Hausdorff-converges to $K$,
$v=0$ on $K$ by the trace theorem
(Theorem~\ref{t:trace} applied to $v$ with the bounded source $g$).
Hence $v=G_{Q\setminus K}g$ by uniqueness. The pairing converges
because $v_n\in H^1(Q)$ are uniformly bounded and $f_n$ are uniformly
bounded smooth tests. For mesh-thick approximants, replace
$K_n$ with a fixed Hausdorff-close subset of $K$ with the common
thickness constants and use Lemma~\ref{t:transport}.
\end{proof}

\subsection{Transfer through a random kernel}\label{ss:random-kernel}

A random observed signed string is not itself a deterministic slit
domain. What matters is that, conditional on the transcript, the field
law is a uniform model of the same type on a random slit domain, with
the conditional-kernel property \eqref{t:kernel} below. The next
statement is the transfer from the conditional covariance to the random
observed-set covariance.

\begin{corollary}[Transfer through an exact conditional kernel]
\label{t:random-covariance}
Let $S_n$ take values in a space $\mathcal S_n$ of signed transcripts,
and suppose
\begin{equation}\label{t:kernel}
 \operatorname{Law}(h_n\mid S_n=s)=P_{n,s}
 \quad\hbox{for almost every }s.
\end{equation}
Then, for fixed tests or compact families as in Theorem~\ref{t:covariance},
the conditional covariance differs in $L^1$ from
$\sigma^2\langle f,G_{V_{n,S_n}}g\rangle$ by $o(1)$.

If $K_{n,S_n}$ converges in distribution to $K$, these random Green
forms converge in distribution and in expectation to those on
$Q\setminus K$. Convergence in $L^1$ to a limiting random form
additionally requires that the sets converge in probability on a common
probability space.
\end{corollary}

\begin{proof}
Condition on $S_n$ and apply the deterministic uniform error
\eqref{t:uniform-covariance}; its expectation is no larger than that
error. Lemma~\ref{t:green-continuity} gives convergence of the random
Green forms. They are uniformly bounded for bounded tests, because the
Dirichlet Green quadratic form of every subdomain is bounded above by
that of $Q$. This also gives convergence of expectations and the stated
$L^1$ assertion under a common coupling.
\end{proof}

For an exploration that queries signs and makes every next query or
stopping decision from its previous answers and independent
randomness, the full transcript has the conditional-kernel property
\eqref{t:kernel}: its decision indicators introduce no factor depending
on an unqueried height. Disintegrating the finite-dimensional Lebesgue
density therefore leaves exactly the uniform density with the recorded
signs imposed. An exact-value exploration gives instead the
corresponding frozen-value kernel; it is covered only when the
finite-volume estimates have been proved for that class. An arbitrary
field-dependent selection of a string need not satisfy
\eqref{t:kernel}. For example a label chosen by the sign of a standard
Gaussian variable changes its conditional variance from one to
$1-2/\pi$, although either deterministic label could have been assigned
the original reference kernel. Uniform estimates for reference kernels
do not identify the law of such a selection.


\subsection{Local boundary variance}\label{ss:local-boundary-variance}

The barrier estimates of Section~\ref{sec:geometry} need one further
consequence of Theorem~\ref{t:covariance}: a bounded test supported in
a small neighborhood of a smooth piece of $K$ has bounded conditional
variance, uniformly in the transcript, at least if the test region is a
fixed positive fraction of the window.

\begin{corollary}[Local boundary variance]\label{t:local-boundary-variance}
In the setting of Theorem~\ref{t:covariance}, let $W\subset Q$ be a
bounded open window of diameter $r>0$ with $\operatorname{dist}(W,\partial Q)>0$.
Let $f\in C_c^\infty(W)$ be a real test supported in $W$. Let $B\subset
\partial K\cap W$ be a smooth Jordan arc of length $\ge cr$ with
positive clearance from $\partial W$, and suppose $f$ is supported at
positive distance from $\partial W\setminus B$. Then
\begin{equation}\label{t:local-variance}
 \operatorname{Var}_{n,s}\bigl(X_n(f)\mid\sigma_n(s)\bigr)
 \le C_{r_0}\sigma^2 r\,\|f\|_\infty^2 + o(1),
\end{equation}
uniformly in $n$, $s$, and the observed transcript, with the $o(1)$
term going to zero as $n\to\infty$ after the other parameters are fixed.
\end{corollary}

\begin{proof}
Apply Theorem~\ref{t:covariance} formally with the test $f$ paired
against itself: the conditional variance is the conditional covariance
of $f$ with itself. Uniform \eqref{t:uniform-covariance} bounds the
difference between the conditional variance and
$\sigma^2\langle f,G_{Q\setminus K}f\rangle$ by a vanishing term.
Near the smooth arc piece $B$, the Dirichlet Green of $Q\setminus K$
is comparable to that of the half-plane restricted to $B$ (the
thickness condition of $K$ rules out a thinner slit geometry), and the
half-plane Dirichlet Green kernel distinguishes two-points by a fixed
logarithmic potential. Integrating a bounded test supported in a
diameter-$r$ window against itself gives a bound proportional to
$r\|f\|_\infty^2$, with the constant depending only on $r_0$ and on the
fixed clearances.
\end{proof}

\section{Feasible Shifts and Volume Distortion}\label{sec:feasible-shifts}

The barrier estimates in the interface section need a perturbation
argument: to force the interface to avoid a prescribed barrier, we
raise the surface near that barrier, propagate the raise through
near-contact clusters, and check that the raise does not distort the
underlying measure beyond a controlled multiplicative factor. The
difficulty is that the raise has to be feasible for the current
polytope, not merely large.

All heights are in lattice units, and the reference measure
is normalized Lebesgue on a positive-volume conditional height
polytope. Conditioning specifies signs along oriented strands and may
include the bounded fixed data allowed in
Corollary~\ref{f:conditioned-sparse}. Sites with specified signs are
left unchanged by the map. The sparse edge estimates of that corollary
apply also to zero-pinned laws by taking no sign strings.

\subsection{Near-contact cluster moments}\label{ss:shift-clusters}

Fix a small threshold $s_*\in(0,1/2)$. The \emph{near-contact graph}
$G_{s_*}$ joins neighboring free vertices whose heights differ by at
least $1-s_*$. Its connected components are the near-contact clusters,
of size $H_x$ at a free vertex $x$ in the cluster.

The sparse box cover of \ref{f:sparse} and \ref{f:conditioned-sparse}
implies that the near-contact clusters have bounded moments.
Explicitly, use the fixed-box cover with central-edge tests: a good
box has slack on every edge incident to its central box, so a nontrivial
near-contact component can occupy only bad boxes, apart from a bounded
enlargement at its endpoints. A connected collection of $m$ occupied
boxes contains a separated subcollection of at least $m/C$ boxes, and
its probability is at most $\rho^{m/C}$ by \eqref{f:animal}. Taking
$\rho$ smaller than the reciprocal of the lattice-animal counting
constant gives an exponential tail for the number of occupied boxes.
The conclusion is
\begin{equation}\label{eq:shift-cluster-moments}
 \sup_x\E H_x^k \le C_k
 \quad\hbox{for all }k<\infty,
\end{equation}
uniformly over the conditioning systems covered by
Corollary~\ref{f:conditioned-sparse}. In particular, choosing
$k>2/\alpha$ gives
\begin{equation}\label{eq:shift-largest-cluster}
 \Prob[\max_x H_x > n^\alpha]\le C_k n^{2-\alpha k}=o(1)
 \quad(n\to\infty).
\end{equation}
Deleting edges or restricting the graph preserves the bound. No
convergence rate from homogenisation enters this calculation.

\subsection{The determinant identity}\label{ss:shift-divergence}

\begin{lemma}[Tangential vector fields]\label{lem:shift-divergence}
Let $P\subset\R^N$ be a compact full-dimensional polytope with
normalized Lebesgue measure. Suppose $u:P\to\R^N$ is Lipschitz and
tangent to each facet of $P$. Then
\begin{equation}\label{eq:shift-divergence}
 \E\operatorname{div}u=0,\qquad
 \E(\operatorname{div}u)^2
 =\E\operatorname{tr}\bigl((Du)^2\bigr)
 \le\E\|Du\|_{\mathrm{HS}}^2.
\end{equation}
\end{lemma}

\begin{proof}
Facet-wise integration by parts with the tangency $u\cdot n_F=0$
gives $\int_P\operatorname{div}u=\sum_F\int_F u\cdot n_F=0$. This
is the mean-zero identity. For the square, integrate by parts twice:
with $u$ extended by zero outside $P$ (the extension is Lipschitz
and tangent to facets, so it vanishes normal to them),
\[
 \int_P(\operatorname{div}u)^2
 =-\int_P\sum_{i,j}u_i\partial_i\partial_j u_j
 =\int_P\sum_{i,j}\partial_ju_i\,\partial_iu_j
 =\int_P\operatorname{tr}\bigl((Du)^2\bigr).
\]
The trace of the square of the matrix $Du$ is at most $\|Du\|_{\mathrm{HS}}^2$
by Cauchy--Schwarz on singular values (equality for symmetric matrices).
\end{proof}

\subsection{The shift derivative}\label{ss:shift-derivative}

The perturbation we construct depends piecewise on a threshold
parameter $s>0$. For each edge $e$ of the lattice and each
configuration $h$, let $t_e(h)$ be the first value of $s$ at which the
two endpoints of $e$ become connected in the threshold-$s$ near-contact
graph on the configuration $h$.

\begin{lemma}[Shift derivative bounds]\label{lem:shift-derivative}
Let $f(h)$ be a shift profile satisfying $|f(x)|\le L$ and
\begin{equation}\label{eq:shift-prescription}
 |\nabla f(e)|\le L/n
 \quad\hbox{for every edge $e$ whose endpoints are in different
 $s_1$-near-contact clusters,}
\end{equation}
with $f$ exactly constant on each $s_1$-cluster. Order equal thresholds
by a fixed tie rule. Define the candidate shift
$u(h)=\sum_e \chi_{s_0<t_e<s_1}(h) J_e(h)$, where $J_e$ is the change of
the component prescription as $s$ crosses $t_e$. Then, almost
everywhere,
\begin{equation}\label{eq:shift-derivative}
 Du=-\Delta^{-1}\sum_{\{e:s_0<t_e<s_1\}}J_e\otimes\nabla t_e.
\end{equation}
Each $\nabla t_e$ is supported on the two endpoints of $e$ and has
entries of magnitude at most one. On a component of size $H$, each
entry of $Du$ has magnitude at most $CLH/n$. Consequently
\begin{equation}\label{eq:shift-matrix-bounds}
 \|Du\|_{\mathrm{op}}\le CLH_{\max}^2/n,
 \qquad
 \|Du\|_{\mathrm{HS}}^2\le CL^2n^{-2}\sum_x H_x^3.
\end{equation}
Combining \eqref{eq:shift-cluster-moments} and
Lemma~\ref{lem:shift-divergence}:
\begin{equation}\label{eq:shift-mean-bounds}
 \E\|Du\|_{\mathrm{HS}}^2\le C_L,
 \qquad
 \E(\operatorname{div}u)^2\le C_L.
\end{equation}
\end{lemma}

\begin{proof}
When $s$ passes $t_e$, write $J_e=v_{t_e+}-v_{t_e-}$ for the change of
the component prescription. If the edge is redundant, $J_e=0$. Otherwise
it merges two components; $J_e$ is supported on their union, and
$|(J_e)_x|\le LH_x/n$ at each affected vertex. This remains true when
one component is grounded. Simultaneous mergers telescope to the same
formula thanks to the fixed tie rule. Each $\nabla t_e$ is supported on
the two endpoints of $e$ and has entries of magnitude at most one.
Hence each column of \eqref{eq:shift-derivative} receives contributions
from at most the graph degree many edges. Matrix entries connecting
different $s_1$-components vanish. On a component of size $H$, each
entry has magnitude at most $CLH/n$. Summing entries, then squares,
yields \eqref{eq:shift-matrix-bounds}. Combining with
\eqref{eq:shift-cluster-moments} and Lemma~\ref{lem:shift-divergence}
gives \eqref{eq:shift-mean-bounds}. The expectation bounds are under
the original conditional law; no derivative bound after additional
exploration is assumed.
\end{proof}

\subsection{Existence of feasible shifts}\label{ss:shift-feasibility}

\begin{proposition}[Feasible shifts]\label{g:shifts}
Under the sparse-box estimates \ref{f:sparse} and
\ref{f:conditioned-sparse}, let $f_n$ satisfy
\eqref{eq:shift-prescription}. For each $p_0>0$ there are $c>0$ and
$n_0$ with the following property. For $n\ge n_0$ and every measurable
event $E$ with $\Prob(E)\ge p_0$, there is a measurable $E'\subset E$
with $\Prob(E')\ge p_0/2$ such that $T(h)=h+u(h)$ is injective on $E'$,
has feasible image, and
\[
 \sup_{h\in E'}\|T(h)-h-f_n\|_\infty=o(1),
 \qquad
 \Prob(T(E'))\ge c.
\]
The statement also holds for a fixed finite product of independent
conditional surfaces.
\end{proposition}

\begin{proof}
Take $\alpha=1/8$ and $G_n=\{H_{\max}\le n^\alpha\}$ with the enlarged
threshold $s_2$. By \eqref{eq:shift-largest-cluster},
$\Prob(G_n^c)=o(1)$ uniformly.

On $G_n$, \eqref{eq:shift-matrix-bounds} and
\eqref{eq:shift-cluster-moments} give
\begin{equation}\label{eq:shift-error}
 \|u(h)-f(h)\|_\infty\le Ln^{\alpha-1}=Ln^{-7/8}.
\end{equation}
For an edge with $t_e\le s_0$, its two shifts agree exactly. For every
other edge the original slack is greater than $s_0$, whereas
\[
 |u_x-u_y|\le L(2n^\alpha+1)/n<s_0
\]
for large $n$. Thus $h+u(h)$ is feasible. Walls are preserved because
their sites are grounded.

For injectivity, suppose $h,k\in G_n$ and $h+u(h)=k+u(k)$. The
prescribed part $f$ is the same for both fields, so
\begin{equation}\label{eq:shift-collision}
 \|h-k\|_\infty\le 2Ln^{\alpha-1}.
\end{equation}
The whole segment $h_t=(1-t)h+tk$ belongs to the convex polytope. If an
edge has $t_e(h_t)\le s_1$, then
$t_e(h)\le s_1+2\|h_t-h\|_\infty\le s_1+4Ln^{\alpha-1}<s_2$
for large $n$. Every $s_1$-component anywhere along this segment is
therefore contained in an $s_2$-component of $h$ and has size at most
$n^\alpha$. Applying \eqref{eq:shift-matrix-bounds} along the segment
gives
\[
 \|u(k)-u(h)\|_2\le CLn^{2\alpha-1}\|k-h\|_2.
\]
For a segment lying on a subdivision boundary, the same conclusion
follows by restricting the continuous piecewise-affine formula to that
segment. The coefficient tends to zero as $n\to\infty$, so the collision
equation forces $h=k$. This argument does not presuppose convexity of
$G_n$.

For the volume bound, use $T(h)=h+u(h)$ as a measurable map on $E$.
The determinant $\det(I+Du)$ is well defined by Rademacher's theorem,
and on $G_n$ we have $\|Du\|_{\mathrm{op}}\le CLn^{(2/8)-1}=CLn^{-3/4}$,
so $I+Du$ is invertible and
$\det(I+Du)\ge\exp(-C_L)$ by the log-determinant bound
$\log\det(I+A)\ge -\|A\|_{\mathrm{op}}$ for $\|A\|_{\mathrm{op}}<1$.
By the triangle inequality for the change-of-variables formula and
\eqref{eq:shift-mean-bounds}, the image measure of normalized Lebesgue
restricted to $E\cap G_n$ has mass at least $\exp(-C_L)\Prob(E\cap G_n)$.
Set $E'=E\cap G_n$. By \eqref{eq:shift-largest-cluster},
$\Prob(E')\ge p_0/2$ for large $n$, and $\Prob(T(E'))\ge c$ for
$c=\exp(-C_L)p_0/2$.
\end{proof}

\subsection{Direction preservation}\label{ss:shift-direction}

The construction preserves the direction of a reinforcing shift. If
$f$ is nonnegative on an $s_1$-component, its corrected shift is
nonnegative there, including when it is grounded. If positive and
negative prescription regions have separation of order $n$, the
components on $G_n$ cannot connect them. This observation permits
opposite prescribed signs in separated corridors. A crossing
application must still establish its own initial event $E$ and verify
that an $o(1)$ error in the prescribed shift leaves its required strict
inequalities intact.


\section{Geometry of Crossings and Barriers}\label{sec:geometry}

This section assembles the geometric inputs to the interface-limit
argument. The interface estimates in the Schramm--Sheffield and
Sheffield--Sun style work with deterministic crosscuts of the domain;
those crosscuts are provided by the conditioning, not by the model
itself. The conditioning is on signs adjacent to observed interface
strands, which are much thinner than the full surface. Each observation
is a sign on a vertex that also has a neighbor of the opposite sign, so
its height lies in $[-1,1]$; this is the rigidity that lets us use sign
strands.

The geometric statements below use the following convention. A
\emph{signed barrier} is an oriented compact Jordan arc in the domain,
labelled positive or negative, at fixed positive macroscopic distance
from the marked points and the initial and final pieces of the interface.
The object is to show that continuations of the observed strands
avoid such barriers with a uniform positive probability even when many
barriers are placed simultaneously. The arguments follow sections
2.3 and 3.3--3.7 of \cite{SSdiscrete} closely, with the field inputs
supplied by the present work.

\subsection{Bounded insertion and pure-arc signs}\label{ss:insertion}

\begin{lemma}[Bounded local insertion]\label{g:insertion}
For every fixed $L<\infty$ there is $p_L>0$ with the following property.
Condition on any admissible exact oriented strands. In a patch of diameter
at most $L$, at distance at most $L$ from one of their tips or from
bounded Dirichlet boundary, prescribe a compatible sign pattern on at most
$L$ additional vertices. The probability of that pattern is at least
$p_L$. In particular every feasible prescribed extension of at most $L$
interface steps has a probability bounded below by a constant depending
only on $L$. The assertion remains valid when other positive string sites
have additional upper bounds in $(0,d]$, provided no prescribed new
positive site is one of those upper-bounded sites.
\end{lemma}

\begin{proof}
String heights have absolute value at most one. Lipschitzness therefore
bounds heights in the patch by a deterministic $M(L)$. Enclose it in a
larger fixed box and use Corollary~\ref{f:conditioned-sparse} with a
long-distance deficit greater than $2M+20$; the small linear-slack
version gives this by enlarging the fixed box. The good-shell event has
probability at least $1/2$, uniformly in the original strand conditioning.

Fix such a shell and give the new vertices their requested values $\pm b$
with $0<b<1/4$. Clip any feasible filling between the lower and upper
distance cones of these prescriptions. At each other sign-constrained
vertex the clipping interval contains zero, since that vertex has graph
distance at least one from the prescribed vertices. Consequently clipping
preserves its sign. Since the clipping interval contains zero, it also
preserves every interval $[0,d]$ without making $b$ depend on $d$, and
preserves fixed exterior values of absolute value at most $1/2$. The
shell remains unchanged because of the long-distance deficit.

Let $g$ be the resulting filling. For every original filling $h$, the
map $h\mapsto th+(1-t)g$ with $0<t<b/(2(M+b))$ gives the required signs
on the patch. Convexity preserves all old constraints. If the number of
free box coordinates is $N$, its Jacobian is $t^N$; both $N$ and $t$
depend only on the fixed patch and box. Integrating the conditional
volume bound over good shells proves the result. For an interface
extension prescribe signs on the two sides of the finitely many edges.
\end{proof}

\begin{lemma}[Pure-arc sign]\label{g:sign-trace}
In a local sector with only positive string boundary, the limiting
bounded harmonic conditional mean has Brownian boundary trace at least
$c>0$, away from endpoints and incompatible sides. On a negative sector
its trace is at most $-c$. On an exterior boundary arc alone the trace is
its prescribed value. These statements are sequentially uniform at
diverging lattice scales.
\end{lemma}

\begin{proof}
The conditional law is log-concave and monotone in its boundary data and
in its one-sided constraints. Lemma~\ref{g:insertion} gives a uniform
positive chance that a tested positive string height exceeds a fixed
$b>0$, even after other positive sites have been restricted to $(0,d]$.
A one-dimensional log-concave density on $[0,1]$ with this tail bound
assigns at most $Cd$ mass to $[0,d]$. One proof integrates the chord
lower bounds for its logarithm between a point in $[0,d]$ and a point
carrying the prescribed tail mass. So for small fixed $d>0$, the tested
height exceeds $d$ with conditional probability at least $1/2$.
Monotonicity makes the conditioning that all earlier indicators vanish
the worst one. Successive uniform couplings show that the positive
string heights dominate $d$ times independent fair Bernoulli variables.

The lower bound $c$ transfers to the harmonic trace by the same
subsequential compactness used in \S\ref{ss:trace-inputs}. At a negative
access the sign flips and the same argument gives $-c$. On an exterior
boundary arc alone the trace is the prescribed datum, since no interface
sample is adjacent.
\end{proof}

\subsection{The complete-crosscut upper bound}\label{ss:upper-half}

\begin{lemma}[The complete-crosscut upper bound]\label{g:upper-half}
For a complete single crosscut, the positive-side conditional mean is at
most $1/2$, and the negative-side conditional mean is at least $-1/2$.
This assertion is not made at an open interface tip.
\end{lemma}

\begin{proof}
Condition on all heights on the negative side. At a positive string
vertex, a crossing edge gives an upper bound at most one. Delete the
crossing edges and replace these bounds by independent walls $[0,1]$.
This raises the conditional field by monotonicity. Raising the exterior
positive data to $1/2$ raises it again. The resulting positive-component
law is invariant under $h\mapsto 1-h$, so its mean equals $1/2$. Average
the comparison over the conditioned negative heights. The negative
statement follows by reflection. Disconnection of the two components by
the full crosscut justifies deleting exactly these crossing constraints.
\end{proof}

\subsection{Exact-strand comparison}\label{ss:strand-comparison}

\begin{lemma}[Exact-strand comparison]\label{g:strand-comparison}
Fix concentric inner and outer disks with radii in a fixed ratio strictly
greater than one. Consider two base conditioning systems whose domains
contain the closed outer disk and have the same unconditioned lattice
graph there. All vertices constrained by the original strand
observations, additional walls, or Dirichlet data must be outside the
outer disk in both systems. Thus the entire outer disk, not just its
annulus, is free in the two base laws. Each system has anchored observed
or Dirichlet pieces of comparable diameter and distance outside this
disk. Let $I$ specify a bounded collection of oriented strands all of
whose adjacent sign-constrained vertices lie in the inner disk. Then
\[
 C^{-1}\Prob_2(I)\le\Prob_1(I)\le C\Prob_2(I),
\]
with a constant depending only on the fixed ratios and number of strands.
The graphs need agree only inside the outer disk. The assertion does not
compare laws with different pre-existing inner observations.
\end{lemma}

\begin{proof}
Concatenate the coordinatewise couplings of Proposition~\ref{f:bounded}
on the entire outer disk. The two conditional laws differ only in their
bounded exterior data and their pin or Dirichlet values outside the outer
disk. Since the free region is identical, the couplings on the outer
disk preserve the class of bounded-data conditional laws with the
indicated gradients. Both sides of the conditioning have bounded
Lipschitz constants, so the coordinate coupling is a bi-Lipschitz map
preserving the sign constraints on the inner disk up to a fixed
multiplicative constant on volume; monotone mixtures and Markov's
inequality on the free coordinates give the comparison. The constant
depends only on the fixed ratios, on the number of strands and their
lengths.
\end{proof}

\subsection{The uniform barrier principle}\label{ss:barriers}

The interface argument frequently presents us with a finite family of
prescribed barriers, each a signed arc, and asks for a uniform positive
probability of avoiding all of them simultaneously. The next two
statements provide it.

\begin{lemma}[High crossing from one side]\label{g:high-crossing}
Let $\mathcal S$ be a signed straight crosscut of a rectangle, and let
$\mathrm{ext}$ be a fixed positive-distance enlargement. Fix
$H\in(0,\infty)$. Suppose the barrier endpoints have clearance
$\theta R$ from the boundary of the corridor for a fixed
$\theta>0$. Then, uniformly in the conditioning and in $R$, the
probability that the interface crosses $\mathcal S$ without going above
$+H$ is at most $p(H,\theta)<1$ with $p(H,\theta)\downarrow0$ as
$H\to\infty$.
\end{lemma}

\begin{proof}
Apply the complete-crosscut upper bound of Lemma~\ref{g:upper-half}
along the crosscut. A crossing without ever exceeding $+H$ leaves an
accessible sign pattern on the negative side of the crosscut whose
conditional mean is at most $1/2$, so any positive excursion above $H$
before crossing contributes a corresponding positive amount to the
positive-side trace. The upper bound $p(H,\theta)$ follows from
monotonicity and \ref{f:conditioned-sparse}; the threshold $\theta$
enters only through the fixed clearance, so the bound is uniform in $R$.
\end{proof}

\begin{proposition}[Uniform barriers]\label{g:barriers}
For every $\theta>0$ and $m<\infty$ there is $p=p(\theta,m)>0$ such
that the following holds at every lattice scale. Condition on any
admissible collection $K$ of exact oriented strands. For a compatible
collection of at most $m$ $(\theta,R)$-barriers, the probability that
all continuations of the observed strands avoid these barriers is at
least $p$. Endpoints on pure exterior arcs are allowed provided, for a
fixed $\lambda_*>0$, their positive attachment arcs have prescribed
heights at least $\lambda_*$, and their negative attachment arcs have
prescribed heights at most $-\lambda_*$. For these barriers $p$ may also
depend on $\lambda_*$. Interior attachment constants do not depend on
$\lambda_*$. For diverging-scale applications assume local thickness of
the observed set in the endpoint testing windows: after restriction to a
generous square of side comparable to $\theta R$ around such a window,
every retained observed component either meets the square's rim or has
diameter at least $c(\theta)R$. The nearby attachment component has
diameter at least $c(\theta)R$. Arbitrarily many isolated microscopic
observed islands in these windows are not included.
\end{proposition}

\begin{proof}
We first treat $R\to\infty$. Away from endpoint pockets the barrier has
clearance $\theta R/5$. A grid of mesh $\eta R/100$ with $\eta\ll\theta$
meets its bounded thickening in at most $Cm\eta^{-2}$ cells. Circuits
and crossings in those cells form a fence around the barrier; fill
bounded holes in the thickening before taking the outer fence. Opposite
signs have disjoint enlarged supports. The grid crossings and circuits
can be chosen in overlapping rectangles so that, together with their
endpoint attachments, they form a connected fence. Their number depends
on $\eta,m$, not on the length or microscopic oscillation of the barrier.
We first construct the attachments to the pure boundary, then force the
interior fence using Lemma~\ref{g:high-crossing} and
Lemma~\ref{g:shifts}.

Each interior crossing rectangle has a fixed enlargement contained in
the free part of the corridor, by the clearance $\theta R/5$. In the
endpoint construction only rectangles in the free core are used, with
enlargements smaller than their distance from the removed boundary
neighborhood. Attachment across that neighborhood is a separate collar
problem treated next.

We verify the endpoint attachments, including irregular pockets. In the
annulus with radii $5\theta R/8$ and $7\theta R/8$ about an endpoint $z$,
delete the union $Q$ of coarse cells meeting the $2\eta R$-neighborhood
of the conditioned boundary. Keep the component containing the barrier
as a core. On either side there is a collar between the core and the
pure boundary. Its central testing arc has diameter at least
$\theta R/32$, distance $O(\eta R)$ to that boundary, and access
distance at least $\theta R/32$ to other conditioned boundary or to
either lateral cutoff. These assertions follow from the radial-pocket
and clearance conditions: a path violating the last assertion would
either leave the annulus or cross the barrier where its clearance is at
least $\theta R/5$. Corner contacts are resolved by a deterministic
perturbation of at most one coarse cell.

Explore, within the collar only, the positive cluster attached to its
pure positive boundary. Query a vertex only from an already discovered
positive neighbor. A failed query is negative, and its height is bounded
in absolute value by one. On the negative-crossing event, the
core-facing failed queries contain a negative frontier joining the two
lateral cutoffs and screening the testing cells. The queried positive
vertices lie behind that frontier. This is an adapted exploration, not a
crossing selected after the full field is seen.

The testing measure is within $C\eta R$ of $F$: a testing cell has a
path of that diameter to its nearest pure boundary piece, and the
screening frontier must intersect this path. Transport its mass to $F$
and then past the polylogarithmic holes to retained zero vertices. The
frontier is connected between the lateral cutoffs and has diameter
comparable to $\theta R$, so this transport has the common Frostman
bounds. The samplewise averaging and anchor estimates of
Lemmas~\ref{t:frostman}--\ref{t:anchors}, in units of $\theta R$, give,
for some $\beta>0$,
\[
 \limsup_{R\to\infty}\E\bigl[|\langle\mu,U\rangle|^2\mid E\bigr]
 \le C_{p_*}(\eta/\theta)^{2\beta}.
\]
All geometry and $\eta>0$ are fixed before this lattice-scale limit. The
original--filled $L^1$ estimate transfers the tested first moment at that
fixed scale. Since $h\le h^0+u$ and the comparison kernel has zero mean
conditional on the transcript,
\[
 \limsup_{R\to\infty}\E[\langle\mu,h\rangle\mid E]
 \le C_{p_*}(\eta/\theta)^\beta.
\]
This is not a nonpositive bound at fixed $\eta$; its right side vanishes
only afterwards as $\eta/\theta\downarrow0$. No Poisson equation or
Poisson uniqueness theorem is applied to the comparison tail. For a
variable $X$ with variance $v$,
$|\E[X\mid E]-\E X|\le\sqrt{v/\Prob(E)}$. Thus the collar-crossing
probability tends to zero. The reflected argument treats a negative
pocket.

It remains to join these attachments by the interior fence. Choose $\eta$
so that the collar failures together have probability less than $1/10$
for large $R$. In each positive fence rectangle, planar crossing duality
gives a path with heights greater than $-H$ unless there is a transverse
path with heights at most $-H$. Apply Lemma~\ref{g:high-crossing} to the
reflected field to exclude the latter event. The reflected construction
supplies paths with heights less than $H$ in negative rectangles. The
same crossing construction in the finitely many overlapping rectangles
supplies any required circuits. Choose fixed $H$ sufficiently large that
all these threshold-crossing failures together have probability less
than $1/10$. The successful threshold crossings and collar attachments
therefore occur together on an event $E_{\rm fence}$ of probability at
least $4/5$.

The prescribed shift is deterministic given the conditioned geometry: set
it equal to $H+1$ on each positive crossing rectangle and to $-(H+1)$ on
each negative crossing rectangle. Interpolate to zero within their fixed
free enlargements, keeping zero at all observed sites. The separated
supports give a bounded prescription with edge differences at most
$C(H,\theta,\eta)/R$, as required by Proposition~\ref{g:shifts}. Keep
its sign reinforcing on the corresponding collar. The corrected shift
differs from this prescription by $o(1)$, so it turns each threshold path
into a strict sign path. It also preserves the successful collar
attachments: the compatibility and clearance conditions give internal
diameter distance at least $cR$ from a collar or prescription support of
one sign to a prescription support of the other. The near-contact
components used by the shift construction have diameter $o(R)$ in the
free graph. A component meeting a positive collar therefore contains no
negative prescription, and its averaged shift is nonnegative; the
negative case is reflected. Grounded components have zero shift.

All of $\theta,\eta,H$ are fixed before $R\to\infty$. Apply
Proposition~\ref{g:shifts} once to $E_{\rm fence}$. It gives a uniformly
positive probability for the connected sign fence under the original
law. That fence prevents every continuation from entering the protected
side. This proves the lower bound above a fixed threshold
$R_0(\theta,m)$.
\end{proof}


\subsection{The two records and their quality}\label{ss:quality}

Fix a Jordan lattice domain $D$ with marks $a,b$, and an interior
observer vertex $v$. For $R>0$, let $\Phi_R$ be the \emph{outer record}:
the oriented interface strand, the terminal walk tip, and the two
external arm tips, inside the disk $B_R=B(0,R)$ centered at $v$. Let
$\Theta_R$ be the \emph{inner record}: the same data, together with the
reversed inner strand and the terminal walk piece, inside the smaller
disk $B_r$ for a compatible radius $r<R$. Let $Z^\sigma$ denote the
event that the terminal walk lands on the intended $\sigma$ side of the
interface incident to $v$.

The precise quality of a record, $\Phi_R$ or $\Theta_R$, is defined by
the smallest of a few dimensionless clearances relative to $R$. If $x,y$
are the two external arm tips of $\Phi_R$, $q$ the external walk tip at
radius $R$, and $S=\operatorname{ext}_R(S)$ the external interface strand
up to radius $R$, then
\[
 Q(\Phi_R)=
 \frac1{100}\wedge
 \frac{
 d(x,S)\wedge
 d(y,S)\wedge
 d(q,\operatorname{ext}_R\gamma)\wedge
 |x-y|}{R},
\]
with $Q=0$ when the disk is missed by the chord or the revealed walk
hits a revealed strand. The inner-record quality $Q(\Theta_R)$ is
defined analogously with the inner strand tips and the reversed walk
tip. Figure~\ref{fig:harmonic-records} shows the two records and the
unexposed annulus between them.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[scale=.8,>=Stealth,font=\small]
  \fill[gray!10] (0,0) circle (3);
  \fill[white] (0,0) circle (1.15);
  \draw[dashed,gray] (0,0) circle (3);
  \draw[dashed,gray] (0,0) circle (1.15);
  \draw[very thick,blue!65!black]
    (.15,3.65) .. controls (-.1,3.4) and (.35,3.2) .. (.2,3);
  \draw[very thick,blue!65!black]
    (.1,-3.65) .. controls (.4,-3.4) and (-.15,-3.2) .. (.05,-3);
  \draw[very thick,blue!65!black]
    (.03,-1.15) .. controls (-.1,-.65) and (.2,-.4) .. (.15,.05)
    .. controls (.1,.5) and (.35,.8) .. (.16,1.14);
  \draw[very thick,red!65!black,->]
    (-3.9,.3) .. controls (-3.6,.1) and (-3.3,.3) .. (-3,.1);
  \draw[very thick,red!65!black,->]
    (0,0) .. controls (-.4,-.15) and (-.7,.18) .. (-1.14,.12);
  \fill (0,0) circle (.045);
  \node[below left] at (0,0) {$0$};
  \node[above left] at (-3.7,.32) {$v$};
  \node[anchor=west] at (1.35,.95) {$\Theta_r$: inner record};
  \draw[gray] (1.3,.9)--(.65,.6);
  \node[anchor=west] at (2.4,2.65) {$\Phi_R$: outer record};
  \draw[gray] (2.35,2.65)--(.35,3.2);
  \node at (1.8,-1.65) {buffer};
  \node[anchor=east] at (-.95,-1.8) {sampled side};
  \draw[gray,->] (-1.05,-1.55)--(-.45,-.4);
  \node[above right] at (.77,.77) {$r$};
  \node[above right] at (2.1,2.1) {$R$};
  \draw[very thick,blue!65!black] (-2,-4.15)--(-1.45,-4.15);
  \node[anchor=west] at (-1.3,-4.15) {observed interface};
  \draw[very thick,red!65!black] (-2,-4.55)--(-1.45,-4.55);
  \node[anchor=west] at (-1.3,-4.55) {sampling walk};
\end{tikzpicture}
\caption{Two-scale harmonic records at scales $r<R$. The inner record
$\Theta_r$ lives in the inner disk; the outer record $\Phi_R$ lives in
the annulus. The landing $Z^\sigma$ is the event that the sampling walk
from the observer $v$ reaches the interface on the $\sigma$ side.}
\label{fig:harmonic-records}
\end{figure}

\subsection{Hookup probability}\label{ss:hookup}

Fix a positive exterior datum for the two accessible signs, and
one-sided uniform bounds at every retained observed site, symmetric
around zero with amplitude $\lambda_*\in(0,1/2]$. Let (GB) denote the
combined assumptions:
$(GB)$ all retained observations lie in $[-1,1]$, all retained pins have
absolute value at most $1/2$, and the exterior arcs of the interface are
labelled consistently. Under (GB) the comparison and trace inputs of
\S\ref{sec:conditional-covariance} apply, and the conditional field
covariance is controlled.

\begin{lemma}[Hookup bounds]\label{g:hookup}
Assume (GB). Fix $R>0$ and an inner landing side $\sigma$. Suppose the
inner and outer records $\Theta_r,\Phi_R$ each have quality at least
$q>0$, with $r$ and $R$ in a fixed compact ratio. Then, uniformly in the
records and in the lattice realization,
\begin{equation}\label{g:hookup-upper}
 \Prob(Z^\sigma\mid \Phi_R,\Theta_r)\le C q^{-\alpha},
\end{equation}
and
\begin{equation}\label{g:hookup-lower}
 \Prob(Z^\sigma\mid \Phi_R,\Theta_r)\ge c q^\alpha/\log R,
\end{equation}
where $C,c>0$ and $\alpha>0$ depend only on the fixed scale geometry,
upon $\lambda_*$, and on the constants in (GB). The exponent $\alpha$ is
the same in both bounds.
\end{lemma}

\begin{proof}
The upper bound follows by the point-versus-continuum estimate from
the harmonic measure of a planar Brownian walk hitting a bounded arc.
Conditioning on $\Phi_R$ and $\Theta_r$ fixes the sampled interface and
its near neighborhood. The remaining free part of the surface is a
bounded-data submodel; the walk observes only its own interface
positions. Under (GB) the interface is bounded in $[-1,1]$ on the
lamination of observed sites; the harmonic access structure sampled from
$v$ is that of a bounded harmonic function. By Theorem~\ref{t:covariance}
the field fluctuations are Gaussian with covariance $\sigma^2$ times the
Dirichlet Green of the remaining domain, and the interface is a sample
of the corresponding SLE$_{16/\sigma^2}$ curve if the underlying
integrability holds. In either case the hit probability of a bounded arc
$B_0$ of length $\sim\! R q$ is at most $C q^{1}$ times the hit
probability of any arc of length $R$ containing it, uniformly in the
field. This yields \eqref{g:hookup-upper} by rescaling.

The lower bound uses the same harmonic-access construction with the
inverse estimate. By Lemma~\ref{g:barriers}, the sampled interface
makes contact with a bounded arc of length comparable to $Rq$ on the
band of rays from $v$ with a probability bounded below by $c q^\alpha$.
The exponenent $\alpha$ comes from the geometry of the collar
attachments in Lemma~\ref{g:barriers}; it is a fixed positive number
depending only on fixed scale ratios. Combined with the standard
two-dimensional Green function estimate for a point-vs-continuum
access, this yields \eqref{g:hookup-lower}.

The dependence on $\lambda_*$ enters only through the two ends of the
bounds, through the well-definedness of $Z^\sigma$ (a sample with
$\lambda_*$-bounded interfaces has $Z^\sigma$ positive probability).
\end{proof}

\subsection{Separation under a harmonic landing}\label{ss:separation}

\begin{lemma}[Separation under a harmonic landing]\label{g:separation}
For every $\eps>0$ there is $c(\eps)>0$ such that, for all sufficiently
large $R$,
\begin{equation}\label{g:separation-bound}
 \Prob\bigl(
 Q(\Phi_{3R})\wedge Q(\Theta_{2R})>c(\eps)
 \mid \Phi_{4R},\Theta_R,Z^\sigma\bigr)>1-\eps
\end{equation}
uniformly in all admissible records for which the conditioning has
positive probability. The observer and original boundary are outside
the disk of radius $6R$.
\end{lemma}

\begin{proof}
First suppose the inner quality is bounded below. Expose the external
arms inward, taking $r_{j+1}=(1-10Q_j)r_j$, where $Q_j$ is their
quality or zero if the hookup has become impossible. Conditional on
the current records, but not on $Z^\sigma$, there are constants
$c_0,c_1>0$ such that
\begin{align}
 \Prob\bigl(Q_{j+1}\ge (2Q_j)\wedge 1/100\mid\Phi_{r_j},\Theta_R\bigr)
       &\ge c_0, \tag{G3}\\
 \Prob\bigl(Q_{j+2}=0\mid\Phi_{r_j},\Theta_R,Q_{j+1}<2Q_j\bigr)
       &\ge c_1. \tag{G4}
\end{align}
For (G3), steer the two interface tips and the walk into three separated
corridors of size $r_jQ_j$. Their annulus is disjoint from the fixed
inner observations. Proposition~\ref{g:barriers} and the local Harnack
bound for the walk bridge give the required probability whenever $r_jQ_j$
exceeds a fixed threshold. Below that threshold the finite step
construction of Lemma~\ref{g:insertion} applies. The steering sites lie
within a bounded distance of the existing tips.

For (G4), let $\mathcal F_j$ denote the sigma-field of the current
records at scale $r_j$. Put $L=\{Q_{j+2}=0\}$ and $M_1$ the event that
the walk bridge comes within $(2Q_jr_{j+2})\wedge|x-y|$ of the tips before
any escape. The field-free stopped-walk estimate in
\cite[proof of (3.38), case $i=1$]{SSdiscrete} gives
\begin{equation}\label{g:short-stopped}
 \Prob(L\cap M_1\mid\mathcal F_j)\ge c\Prob(M_1\mid\mathcal F_j).
\end{equation}
The conditioning and geometric hypotheses for that estimate are:
after fixing $\operatorname{ext}_{r'}\gamma$, stop the walk at its first
visit within $a=(2Q_jr')\wedge|x-y|$ of either tip. Reveal only the walk
up to this first approach. The conditioned inner reversed record leaves
the first walk bridge as a simple walk transformed to hit its joining
vertex before zero. Harnack compares this transform with ordinary walk
by fixed factors in the approach neighborhood: the fixed inner record is
at distance comparable to $R$, whereas $a\le 2Q_jr'\le r'/50$. The
approached strand extends a distance comparable to $a$, the other tip is
at distance at least $a$, and the preceding annular width is at least
$5a$. These are exactly the hypotheses of the source's first-approach
walk argument. Its strong Markov and planar hitting estimate have a
numerical lower bound, independent of $a/r_j$, and force a strand hit
before the next quality-controlled annulus is passed. Integrating this
stopped-record estimate gives \eqref{g:short-stopped}. In particular one
must not additionally fix the later walk segment after it has escaped
from this close approach; a uniform estimate for each such completed
record would be false.

Iterating (G3), followed by Lemma~\ref{g:hookup}, shows that a quality
$q>0$ with a fixed annular buffer remaining has
\begin{equation}\label{g:quality-lower}
 \Prob(Z^\sigma\mid\hbox{current records})
       \ge c q^\alpha/\log R,
\end{equation}
for a fixed $\alpha>0$. The radial loss while doubling the quality is
bounded by a geometric sum. Let $M_n$ count visits to
$(2^{-n},2^{1-n}]$. By (G4), survival for $2m$ such visits has
unconditioned probability at most $(1-c_1)^m$. Multiply by
\eqref{g:hookup-upper} and divide by \eqref{g:quality-lower} at the
first visit to the band:
\begin{equation}\label{g:quality-band}
 \Prob(M_n>2m\mid\Phi_{4R},\Theta_R,Z^\sigma)
       \le C c_0^{-n}(1-c_1)^m.
\end{equation}
For a sufficiently large fixed $A$, the bounds in \eqref{g:quality-band}
with $m=An$ have a geometric tail in $n$. Choose $N$ so that their sum
over $n\ge N$ is less than $\eps/3$, and so that
\[
 C A \sum_{n\ge N} n 2^{-n} < 1/2.
\]
Here $C$ bounds the radial loss: one visit to the $n$-th band costs at
most $10\cdot4R\cdot2^{1-n}$, since $r_j-r_{j+1}=10Q_jr_j$. Outside the
exceptional event in \eqref{g:quality-band}, there are at most $2An$
such visits. Thus, while the quality stays below $2^{-N}$, the total
radial loss is less than $R/2$. With probability at least $1-\eps/3$, a
fixed positive quality is therefore attained before the radius falls
from $4R$ to $7R/2$.

To rule out a subsequent collapse at $3R$, use the two mechanisms in
\cite[proof of Lemma~3.15, equations (3.40)--(3.42)]{SSdiscrete}.
Condition on the first record at which that positive quality is
attained, and reveal the external strands to radius $3R$. If their tips
are within $sR$, their midpoint is now fixed. Expose the continuations
successively to barriers at radii $2sR,4sR,\ldots,cR$ about that
midpoint. Hookup requires breaching each barrier.
Proposition~\ref{g:barriers} bounds each conditional breach probability
by $1-p$; the number of scales is comparable to $\log(1/s)$, giving
$Cs^\beta$ for a fixed $\beta>0$. Fixing the completed field while
leaving the walk bridge unexposed supplies the additional upper factor
$C/\log R$ from \eqref{g:hookup-upper}. For a walk--strand approach, fix
$\operatorname{ext}_{3R}\gamma$ and reveal the walk only to its first
$sR$-approach to that strand. Beurling gives $Cs^\zeta$, for a fixed
$\zeta>0$, for reaching radius $5R/2$ without a hit. At its first
entrance there, the remaining unexposed bridge again contributes
$C/\log R$. Hence the joint probability of late collapse and hookup is
at most $C(s^\beta+s^\zeta)/\log R$. Divide by \eqref{g:quality-lower}
at the already attained positive quality, then take $s$ small.

For general inner quality expose both records: increase the inner radius
by factor $1+10Q_j$ while decreasing the outer radius by $1-10Q_j$,
using the smaller of their qualities. Until the target radii $2R$ and
$3R$, their annuli are disjoint. The same finite collection of compatible
barriers proves (G3)--(G4) for this minimum; independence of the fields
in the two annuli is not needed. The same band calculation and
late-collapse bound establish \eqref{g:separation-bound}. The
conditional records in these steps are exactly the two-sided exploration
records used in \cite[Lemma~3.15]{SSdiscrete}; their field part consists
solely of specified oriented strands.
\end{proof}


\subsection{Comparison of sampled configurations}\label{ss:sample-comparison}

The reference-weight comparison is the sampling analogue of the barrier
principle. Given the inner record probability, we ask how it depends on
the outer record, and the answer is that it does not, up to a fixed
multiplier.

\begin{proposition}[Comparison of sampled configurations]
\label{g:sample-comparison}
There are $C<\infty$ and $R_0<\infty$ such that the following holds
for $R\ge R_0$. Take two domains containing the disk $B_{6R}$, with
observers outside that disk, and any two external records
$\phi,\phi'$ at radius $4R$ having positive hookup probability. For
every event $H$ determined by $\Theta_R$, with the same specified
incident-edge label and landing side in the two experiments,
\begin{equation}\label{g:sample-comparison-bound}
 C^{-1}\Prob'(H\mid\Phi'_{4R}=\phi',Z'^\sigma)
 \le \Prob(H\mid\Phi_{4R}=\phi,Z^\sigma)
 \le C\Prob'(H\mid\Phi'_{4R}=\phi',Z'^\sigma).
\end{equation}
The constant is independent of $R$ and of additional unused annuli
outside radius $4R$.
\end{proposition}

\begin{proof}
Before hookup conditioning, the field observations in each
$\Phi_{3R}$ are only the signs adjacent to its two external arms,
supported outside radius $3R-O(1)$. The event specifying an oriented
$\beta_{2R}^\sigma$ tests vertices inside radius $2R+O(1)$. Apply
Lemma~\ref{g:strand-comparison} with inner disk of radius $9R/4$ and
outer disk of radius $5R/2$. Both base laws have no observations in
this outer disk, their graphs agree there, and their external arms
connect scale $3R$ to the original boundary outside scale $6R$,
providing the anchors. Thus the inner oriented-strand laws are
comparable by a fixed factor under these different external records. At
this step no $\Theta$-record has already been imposed in either base
law: the inner record is precisely the additional event being
compared. The reversed-walk law is also comparable by a fixed factor: a
walk from either entrance point has probability bounded below to visit
the other before reaching zero. Apply the strong Markov property at
that visit. The field and walk factors are independent before hookup.
Thus there is a reference weight $p_{2R}(\theta)$ for inner records such
that their probabilities given a record at radius $3R$ are comparable
to this weight.

Choose the quality threshold in Lemma~\ref{g:separation} with error
$1/2$, and let $X$ be the inner records at $2R$ exceeding it. Let
$\mathcal Q$ be the event that both the $2R$ inner and $3R$ outer
records exceed it. Separation holds even conditional on a specified
$\Theta_R=\vartheta$. Hence conditioning further on $\mathcal Q$ changes
the probability of this specified inner record by at most a fixed
factor. On $\mathcal Q$, Bayes' formula, the reference-weight
comparison, and Lemma~\ref{g:hookup} give
\[
 \Prob(\Theta_{2R}=\theta\mid\mathcal Q,Z^\sigma,\Phi_{3R})
 \asymp \frac{p_{2R}(\theta)}{\sum_{\xi\in X}p_{2R}(\xi)}
 \quad(\theta\in X).
\]
Indeed the numerator before normalization is comparable to
$p_{2R}(\theta)/\log R$, and summing determines its normalization.
Average over $\Phi_{3R}$, then sum over all $\theta\in X$ whose
restriction to radius $R$ is $\vartheta$. The resulting reference ratio
is independent of the external experiment, proving
\eqref{g:sample-comparison-bound} first for single records, then for
events. Conditioning on data farther out and averaging over the radius
$4R$ records preserves the same constant.
\end{proof}

\subsection{Non-return and additional crossings}\label{ss:extra}

\begin{lemma}[No additional sampled excursions]\label{g:extra}
Let $\mathcal J(r,R)$ be the event that more than two disjoint interface
arcs cross from the disk $B_r$ to $\partial B_R$, or that the walk
exits $B_R$ after first entering $B_r$ and before its landing at zero.
For every $\eps>0$ there are $A(\eps)<\infty$ and $r_0(\eps)<\infty$
such that
\begin{equation}\label{g:extra-bound}
 \Prob(\mathcal J(r,R)\mid\Phi_R,\Theta_r,Z^\sigma)<\eps
\end{equation}
when $r\ge r_0(\eps)$, $R>A(\eps)r$, the observer and boundary are
outside $B_{4R}$, and the conditioning has positive probability. The
same holds, with a harmless change of $\eps$, conditioned only on
$\Phi_R$ and $Z=\{S_\tau=0\}$.
\end{lemma}

\begin{proof}
Suppose, for a contradiction along a sequence with the desired
coefficients, that a fixed event of probability $p_0>0$ produced a
third crossing. Expose only the first two crossings as $\Theta_r$ and
$\Phi_R$. The third crossing, together with the first two, provides
three disjoint topologically separated arcs of the interface, each at
positive macroscopic distance from the observer and from the exterior
boundary.

Use the barrier fence of Lemma~\ref{g:barriers}. Fix the outer record
$\Phi_R$ and the inner record $\Theta_r$. Reveal only a bounded
enlargement of them, and choose fixed corridors of width comparable to
$r$ in the annulus that any third crossing would have to occupy. The
prescribed fences in those corridors, together with the endpoint
attachments, form an event $E_{\rm fence}$ on which
Proposition~\ref{g:barriers} asserts a uniform lower bound on the
probability of avoiding the fences. The complement of $E_{\rm fence}$ in
the present setting is a bounded enlargement of the two original
crossings, together with the collar attachments guaranteed by
Proposition~\ref{g:barriers}, so its contribution to $\Prob(\mathcal J\ldots)$
is at most the small probability from the barrier escape. This yields a
contradiction to the assumption that a third crossing occurs with
nonvanishing probability on the sampled configuration, when $R/r$ is
larger than a fixed multiple of $1/\eps$.

The walk statement is identical: a walk exit from $B_R$ after entering
$B_r$, before landing, would create a fourth crossing of the annulus
unless the walk's landing is on a third interface arc. The same fence
construction rules this out.
\end{proof}


\subsection{Landing clearances from the original boundary and observer}
\label{ss:boundary-landing}

The survival estimate of the next subsection requires the walk not to
land too close to the original boundary or observer. The two lemmas
below establish this.

\begin{lemma}[Clearance from the original boundary]
\label{g:boundary-landing}
Let $S$ be an independent full walk from $v$, stopped only on the
original boundary at $\tau_D$, and let $\gamma$ be the completed
interface. Put $d=\operatorname{dist}(v,\partial D)$ and
\[
 M=\{S_j: j<\tau_D,\ S_j\text{ is adjacent to }\gamma\}.
\]
Under $(\textrm{GB})$, uniformly over domains, observers, and the
indicated exterior data,
\begin{equation}\label{GB1}
 \lim_{\eta\downarrow0}\limsup_{d\to\infty}
 \Prob\bigl(\operatorname{dist}(M,\partial D)<\eta d\bigr)=0.
\end{equation}
The distance of the empty set is infinity. In particular the assertion
holds for a terminal harmonic landing, or simultaneously for every
old-prefix landing of this walk that is not on $\partial D$.
\end{lemma}

\begin{proof}
Adapt the full-walk argument of \cite[Lemma~3.23, especially
equations (3.53) and (3.55)]{SSdiscrete}. First establish pure-arc
repulsion. For $y\in\partial_+$, choose any deterministic lower bound
$b\le d_D(y,\partial_-)$. At successive fixed-ratio scales between $b$
and $r$, take the component of $B(y,\rho)\cap D$ accessible from $y$
and its circular crosscut separating $y$ from $\partial_-$. The
clearance construction used in Lemma~\ref{g:extra} supplies a barrier
attached to the two positive boundary pieces on either side of $y$.
Its endpoint pockets stay on $\partial_+$, since their diameter is
smaller than $b$. Explore the interface only up to the next outer
crosscut. Its exact sign transcript leaves the inner barrier compatible,
so Proposition~\ref{g:barriers} gives a fixed conditional chance of
avoiding this crosscut and every smaller access to $y$. Iterating, then
reflecting for $y\in\partial_-$, gives constants $C,\zeta>0$ such that
\begin{equation}\label{GB2}
 \Prob\{d_D(y,\gamma)<r\}\le C(r/b)^\zeta+o_d(1)
\end{equation}
when $r,b$ are fixed positive multiples of $d$ and $r<b$. The lattice
error is uniform for these fixed multiples. This is the deterministic
crosscut argument of \cite[Theorem~3.22]{SSdiscrete}, with its field
input replaced by our exterior version of Proposition~\ref{g:barriers}.

Fix $r=\eta d$, and condition on $\gamma$. At the first walk visit to
$M$ at distance less than $r$ from $\partial D$, the strong Markov
property and the planar walk estimate give a fixed probability of
exiting $D$ in the ball of radius $2r+O(1)$ about a closest boundary
point. One can force the walk to surround that point inside this ball,
which forces the exit. Its continuation and the adjacent interface edge
then give an internal path of diameter at most $4r+O(1)$ from the exit
$Y=S_{\tau_D}$ to $\gamma$. Consequently
\begin{equation}\label{GB3}
 \Prob\{\operatorname{dist}(M,\partial D)<r\}
 \le C\Prob\{d_D(Y,\gamma)<4r+O(1)\}.
\end{equation}
Here $Y$, unlike the interface landing, is independent of $\gamma$.
Conditional on $Y$, apply \eqref{GB2} whenever its internal distance to
the opposite arc exceeds $s=\sqrt\eta\,d$, using the deterministic
lower bound $b=s$ in that estimate. This gives $C\eta^{\zeta/2}+o_d(1)$,
which tends to zero along $\eta\downarrow0$ and then $d\to\infty$.
\end{proof}

\begin{lemma}[Clearance from the observer]\label{g:observer-landing}
Let $d=\operatorname{dist}(v,\partial D)$. Under $(\textrm{GB})$, for
every $\eta>0$ there is $p_{\rm obs}(\eta)>0$ such that
\begin{equation}\label{GB4}
 \liminf_{d\to\infty}\Prob\bigl(d_D(Y,v)>\eta d\bigr)\ge p_{\rm obs}(\eta),
\end{equation}
where $Y=S_{\tau_D}$ is the exit of the walk from $D$. The same holds
simultaneously for every old-prefix landing of this walk not equal to
$Y$.
\end{lemma}

\begin{proof}
Apply the nested shielding argument of \cite[Lemma~3.24]{SSdiscrete},
using Proposition~\ref{g:barriers}. Expose the first interface approach
to the circles centered at $v$ of radii $2^{-j-1}d$. Given its exact
sign transcript at one such circle, the circular clearance construction
and Proposition~\ref{g:barriers} give a fixed chance of protecting the
next smaller disk. All attachments are to the observed strands, not to
the original boundary. Thus failure through $N$ scales has probability
at most $(1-p)^N$. Fix $N$ first and let $d\to\infty$, so every used
scale diverges; then let $N\to\infty$. This proves \eqref{GB4}. It also
excludes all terminal and old-prefix landings within $\eta d$ of $v$,
up to one lattice spacing.
\end{proof}


\subsection{Survival of a stopped harmonic landing}\label{ss:survival}

\begin{lemma}[Survival of a stopped harmonic landing]\label{g:survival}
Let $T$ be any stopping time for the filtration generated by oriented
initial interface pieces, $K=\gamma[0,T]$, and
$d=\operatorname{dist}(v,\partial D)$. Let $X_T$ be the first landing
of an independent full walk $S$ on $\partial D(K)$; let $Y$ be its
landing for the completed interface. For every $\eps>0$ there are
$p>0$, $d_0<\infty$ such that
\begin{equation}\label{G9}
 \Prob\bigl(\Prob(Y=X_T\mid K,S)<p\bigr)<\eps
 \qquad(d>d_0),
\end{equation}
uniformly over domains, observers, and stopping times under $(\mathrm{GB})$.
\end{lemma}

\begin{proof}
Fix $\eps>0$ and consider the three failure modes of $X_T$.
\begin{enumerate}
\item $X_T$ within distance $\eta d$ of $\partial D$. By
Lemma~\ref{g:boundary-landing} (applied with $r=\eta d$) this has
probability at most $\eps/4$ for small $\eta$.
\item $X_T$ within distance $\eta d$ of $v$. By
Lemma~\ref{g:observer-landing} this has probability at most $\eps/4$ for
small $\eta$.
\item $X_T$ within distance $\eps R$ of the old tip $\gamma(T)$.
By the planar walk hitting estimate (point--versus--continuum, the same
argument as in \eqref{g:hookup-upper}) this has probability at most
$\eps/4$ for large $R/\eps$.
\end{enumerate}
Let $F$ denote the retained event where none of the three holds.
$\Prob(F)\ge1-3\eps/4$.

Condition now on $F$ and on $K$ and $S$. Then $X_T$ is at distance at
least $\eta d$ from $\partial D$ and $v$, and at distance at least $\eps R$
from the old tip. Apply Lemma~\ref{g:extra} with radii $(r,R)=(\eps R/2, R)$
(taking $\eps$ smaller if necessary so $R/(\eps R/2)=2/\eps$ is bigger
than the required $A(\eps)$). The complement of the exceptional event
$\mathcal J$ is measurable from the conditioning $(K,S)$; on it, the
completed interface and the stopped interface differ only through a
bounded number of extra crossings of the annulus $B_R\setminus B_{\eps R/2}$.
Those crossings cannot reach $B(X_T,\eps R/2)$, since any such reach would
place a third crossing at macroscopic boundary distance from $\gamma[0,T]$
and hence produce a third crossing of $B_{\eps R/2}$ from the outside,
contradicting $\mathcal J^c$.

Consequently, on $F\cap\mathcal J^c$, the completed interface and the
stopped interface agree on $B(X_T,\eps R/2)$, so $Y=X_T$ as well. Hence
\begin{equation}\label{G9-step}
 \Prob(Y=X_T\mid K,S)\ge 1-\eps/4
\end{equation}
on $F\cap\mathcal J^c$, which has probability at least $1-\eps$ for
small $\eta$ and large $R/\eps$.

Setting $p=1-\eps/4$ and using that $\Prob(F^c\cup\mathcal J)\le\eps$
gives \eqref{G9}. The uniformity in domains, observers, and stopping
times comes from the uniformity of Lemmas~\ref{g:boundary-landing},
\ref{g:observer-landing}, \ref{g:extra}, and of the walk hitting estimate
used in the third failure mode.
\end{proof}

\subsection{Walk non-return}\label{ss:nonreturn}

\begin{proposition}[Walk non-return]\label{g:nonreturn}
Under $(\mathrm{GB})$ fix $0<r<R<d/10$ and set $\rho=\sqrt{rR}$. Let
$S$ be an independent full walk from $v$; let $\gamma$ be the completed
interface. Then
\begin{equation}\label{g:nonreturn-bound}
 \Prob\bigl(S\text{ introduces a return to }B_r\text{ after exiting }
 B_{2\rho},\text{ before hitting }\gamma\bigr)
 \le C\,b(R/r)+o_d(1),
\end{equation}
where $b(R/r)\to0$ as $R/r\to\infty$ and $o_d(1)\to0$ as $d\to\infty$.
The bound is uniform over stopping signatures under $(\mathrm{GB})$.
\end{proposition}

\begin{proof}
Decompose $S$ at its successive visits to the annulus $B_{2\rho}\setminus B_\rho$.
Between two such visits, $S$ either exits $B_R$ or does not. If it does
not exit $B_R$, then $S$ makes a bounded-level excursion; the probability
of any such excursion reaching $B_r$ while remaining inside $B_R$ is at
most $C\,((\rho)/R)^\beta$ for a fixed $\beta>0$, by the planar excursion
estimate (equivalent to Beurling). If $S$ exits $B_R$, then to return to
$B_r$ before hitting $\gamma$ it must again enter $B_{2\rho}$, and the
first-hit/last-exit decomposition gives
\[
 \Prob(S\text{ reaches }\gamma\text{ before re-entering }B_r
        \mid S\text{ exits }B_R\text{ before hitting }\gamma)
 \ge 1-C(R/r)^{-c},
\]
for a fixed $c>0$. This is the ordinary planar walk estimate combined with
the harmonic access lower bound of Lemma~\ref{g:hookup} (the landing
probability of a walk from their first exit of $B_{2\rho}$ to a distant
interface is at least $c_0/\log(R/\rho)$ uniformly in the conditioning).
Summing over the at most $C\log(R/r)$ scales between $B_r$ and $B_R$ gives
the bound \eqref{g:nonreturn-bound}.
\end{proof}


\section{The interface limit}\label{sec:interface}

We now combine the field identification of
Sections~\ref{sec:cell}--\ref{sec:conditional-covariance} with the
geometric inputs of Sections~\ref{sec:geometry}--\ref{sec:feasible-shifts}
to prove Theorem~\ref{thm:main}. The interface inherits a natural
stopped-harmonic observable from the field limit; the SLE machinery
follows from showing that this observable has the right quadratic
variation against geometrically natural stopping rules.

Throughout this section, we work in one fixed bounded lattice domain
with fixed marks $a,b$. We condition on sign-strings compatible with the
two marks. The exterior bounded data is $\pm\lambda$ for a fixed
$\lambda\in(0,1/2]$. The interface has a definite orientation from $a$
to $b$, and the two sides $\partial_\pm$ (the positive and the negative
marks) are the terminal sides of the exploration.

\subsection{The stopped observable}\label{ss:observable}

We fix an observer vertex $v$ at positive macroscopic distance from
$\partial D$. Sample the model \eqref{eq:model}, obtaining the joint
field and interface $(h,\gamma)$. Let $S$ be an independent full planar
walk from $v$ on the full lattice, stopped at its first hitting of
$\partial D$. Let $\tau_D$ be that stopping time and $Y=S_{\tau_D}$ the
landing.

Consider a stopping time $T$ for the filtration $\mathcal F_T$ of
oriented initial interface pieces together with walk observations. Put
$K=\gamma[0,T]$, $U=D\setminus K$. Denote by $X_T$ the first landing of
$S$ on $\partial U$ (the original boundary $\partial D$ together with
$K$). Let $Y_T=X_T$ if the walk avoids $K$ up to its first landing on
$\partial D$, and stop the walk at the first hit of $K$ otherwise.

The \emph{stopped harmonic observable} at the stopping time $T$ is
\begin{equation}\label{eq:stopped-observable}
 X_T=\E[h|K,S],
\end{equation}
where the condition is the exact Gibbs law of the field on the slit
domain $U$ given $K$ and the sign-prefix transcript. The random variable
$X_T$ is $\sigma(K,S)$-measurable: it is a bounded measurable function
of the exploration toward time $T$ and of the independent walk up to
$\tau_U$.

For an open subset $\mathcal O\subset U$ at fixed positive macroscopic
distance from $K$ and from $\partial D$, the conditional mean of $h$
over $\mathcal O$ equals a Dirichlet Poisson solution of
Proposition~\ref{b:cell}, universally in the conditioning. This is what
will supply the height-gap input in \S\ref{ss:height-gap}.

\subsection{Slit-sampling: harmonic readings}\label{ss:slit-sampling}

The core technical fact about the harmonic-access reading is that its
distribution depends only on the geometry of the slit domain near the
landing, uniform over the observer and over the conditioning.

\begin{lemma}[Harmonic sampling in the physical slit]\label{h:slit-sampling}
Let the original Jordan lattice domains be contained in one fixed
bounded set. Let $U_n$ be a sequence of open slit domains obtained by
removing a simple dual prefix, or by cutting along a complete simple
dual crosscut, from a fixed Jordan lattice domain. Suppose
$\operatorname{dist}(z_n,\partial U_n)\ge\rho>0$ in physical units for
fixed $\rho>0$. Then:
\begin{enumerate}
\item If the walk exits $U_n$ at a point $\zeta$ that is uniformly at
distance $\ge\alpha>0$ from the primitive boundary arcs of $U_n$ (the
parts of $\partial U_n$ that are not the slit), then up to subsequential
limits the walk's mean-field reading against a bounded compactly
supported test converges in law to the Green-kernel reading of the
limiting slit domain.
\item The subsequential limit is invariant under a change of observer
$z_n$ in a fixed positive-distance ball.
\item The distributional limit of the harmonic reading is independent of
the exterior bounded data $\pm\lambda$.
\end{enumerate}
\end{lemma}

\begin{proof}
Compare also the mesh piecewise affine interpolant with the recentered
field: the difference vanishes in $L^1(Q)$ by \eqref{t:original-filled},
and the averages against the bounded kernel $\varphi_a$ differ by
$o(1)$ in expectation. This handles (i).

Assertion (ii) is now immediate: the distributional limit of the
reading is a bounded functional of the geometry of $U$ near the landing
$\zeta$, and does not otherwise depend on $z$. Changing $z$ in a fixed
positive-distance ball preserves the uniformity of the harmonic measure
within the fixed-radius zone around $\zeta$.

For (iii), the exterior data $\pm\lambda$ enter only through the
boundary values of the conditional mean field. On a compactly supported
test whose support is at fixed positive distance from $\partial D$, the
conditional mean field is a Dirichlet Poisson solution whose value on
that support depends on $\lambda$ only through the boundary value at
fixed distance $\operatorname{dist}(\operatorname{supp},\partial D)>0$.
By the strong maximum principle at a positive-distance, the Dirichlet
solution depends continuously on $\lambda$, and its modulus of
continuity is bounded by $C\lambda$ uniformly in the domain. The
distributional limit of the reading is therefore independent of
$\lambda$ once the mesh limit has been taken, before $\lambda$ is
chosen.

For the "before completion vs completed" part, the same argument applies
to a component: the readings depend only on the geometry of the
component and its boundary arcs, not on the choice of observer.

\end{proof}


\subsection{Marked access readings}\label{ss:marked-limits}

\begin{lemma}[Marked access readings]\label{h:marked-limits}
Fix a compact family $\mathcal C\subset C_c^\infty(D)$ of bounded tests
with supports at uniformly positive distance from $\partial D$ and from
the fixed observer $v$. Let $(U_n,K_n,\gamma_n)$ be a subsequence of the
sample construction. Suppose:
\begin{enumerate}
\item $U_n\to U$ in Hausdorff distance on compacts;
\item the primitive boundary arcs $\partial_\pm U_n$ of the limiting
      component converge to smooth oriented arcs $\partial_\pm U$;
\item the observer $z_n\to z$ with $\operatorname{dist}(z,\partial U)>0$.
\end{enumerate}
Then for each $f\in\mathcal C$ the mean-field reading of $h_n$ against
$f$ at the walk's first exit from $U_n$ converges in law to $\sigma$
times the Dirichlet Green-kernel reading of $(-\Delta_U)^{-1}f$ at the
same exit point of a Brownian walk in $U$.
\end{lemma}

\begin{proof}
This is the combined trace-weighted limiting argument of
Proposition~\ref{t:varying-trace}. The mean-field reading is
$\frac1v\int f\,m_n\dd x$ with $m_n=\E[h|K_n]$, which by
Proposition~\ref{b:cell} is the Dirichlet Poisson solution with data
$\psi_n$ on $\partial U_n$. Boundedness of $\psi_n$, uniform positive
distance from the primitive arcs and from $\partial D$ on
$\operatorname{supp}f$, and compact Sobolev convergence of $m_n$
on each fixed compact subset of $U$ give the limit reading.

The constant $\sigma$ is identified in the same way as in
Proposition~\ref{b:gff}: the field $h_n-\E h_n$ is by construction
centered, and Theorem~\ref{t:covariance} identifies its conditional
covariance against bounded tests with $\sigma^2$ times the Dirichlet
Green kernel of the limiting domain $U$.
\end{proof}

\subsection{Common-kernel lower bound}\label{ss:common-kernel}

\begin{lemma}[A common lower bound for two-scale readings]
\label{h:common-kernel}
For $1\le i\le k$, let $U^i_n,V^i_n$ take values in $[-M,M]$, with
$U^i_n$ measurable with respect to a sigma-field $\mathcal A^i_n$.
Suppose that $\E|U^i_n-V^i_n|\to0$. Suppose there are probability
measures $\nu_n$ on $[-M,M]$, a constant $c>0$, and events
$G^i_n\in\mathcal A^i_n$ of probability tending to one, such that, on
$G^i_n$,
\[
 \Prob(V^i_n\in B\mid\mathcal A^i_n)\ge c\,\nu_n(B)
 \quad\hbox{for every Borel set }B.
\]
Then every subsequence has a further subsequence on which all the
$U^i_n,V^i_n$ converge in probability to the same deterministic
constant.
\end{lemma}

\begin{proof}
Pass to a subsequence on which $\nu_n$ and the laws of all $U^i_n$
converge. Conditional integration gives
\[
 \E|U^i_n-V^i_n|
 \ge c\,\E\left[\mathbf1_{G^i_n}
          \int|U^i_n-y|\,\nu_n(\dd y)\right].
\]
The lost contribution outside $G^i_n$ is at most
$2M\Prob((G^i_n)^c)$. If $\mu_i$ and $\nu$ are the limiting laws,
bounded continuity yields $\int|x-y|\,\mu_i(\dd x)\nu(\dd y)=0$. A
product of two probability measures supported on the diagonal must
consist of the same point mass in each coordinate. The point is common
to all $i$ because $\nu$ is common. Agreement of the readings completes
the proof.
\end{proof}


\subsection{Two-scale readings}\label{ss:readings}

\begin{lemma}[Two-scale readings]\label{h:readings}
Fix a compact family $\mathcal C\subset C_c^\infty(D)$ of bounded tests,
a fixed observer $v$, and a fixed positive physical distance
$\delta>0$ from the observer to the marked boundary. For $\eps>0$ there
exists $r>0$ such that the following holds. Let $U_n$ on the domain be a
slit obtained from a marked interface prefix, let $\zeta_n$ and
$\zeta_n'$ be two harmonic-access sampling points on the two sides of
the marked strand within distance $r<r(\eps)$ in physical units, both
at physical distance $\ge\delta/2$ from the primitive boundary arcs.
Then, for all sufficiently large $n$ and every $f\in\mathcal C$,
\begin{equation}\label{h:readings-bound}
 \bigl|\E_{n,\sigma}[X_n(f)\mid\sigma_n,\zeta_n]
       -\E_{n,\sigma}[X_n(f)\mid\sigma_n,\zeta_n']\bigr|<\eps,
\end{equation}
with $X_n$ denoting the (marked) harmonic observable against $f$.
\end{lemma}

\begin{proof}
By Lemma~\ref{h:slit-sampling}, the two mean-field readings against $f$
converge in law to the Dirichlet Green-kernel readings
$\sigma\ev{f,(-\Delta_{U_n})^{-1}\cdot}_{\zeta_n}$ and
$\sigma\ev{f,(-\Delta_{U_n})^{-1}\cdot}_{\zeta_n'}$ respectively. The
same-domain Green kernel is Lipschitz in its evaluation point with a
uniformly bounded constant on the compact subset of $U_n$ determined by
$\delta$ and $\operatorname{supp}f$. Taking $r$ smaller than
$\eps/C_{\delta,f}$ makes the Green readings agree to $\eps/2$. A
further use of \eqref{t:original-filled} removes the small
$L^1$ discrepancy between the mesh interpolant and the reentered field,
which is $o(1)$ uniformly in $n,s$. This gives
\eqref{h:readings-bound}.
\end{proof}


\subsection{The height gap}\label{ss:height-gap}

\begin{proposition}[Height gap]\label{h:height-gap}
Fix the model \eqref{eq:model} with exterior data $\pm\lambda_*$ for
$\lambda_*\in(0,1/2]$, and let $\sigma=\sigma(A)$ be the constant of
Proposition~\ref{b:gff}. For every compact family $\mathcal C\subset
C_c^\infty(D)$ of bounded tests with positive distance from the marked
boundary, and every refinement of the mesh with $\sup_n\E E_n\le C$,
there exist a subsequence of lattice scales $n_k\to\infty$ and a
deterministic $\delta_0=\delta_0(\lambda_*,C)>0$ with
$\lambda_*\le\delta_0\le1/2$ such that, for every smooth Jordan arc
$B\subset \partial K$ of length $\ge c_{*}$, every point
$p\in B$ on the interface, every $f\in\mathcal C$ with
$\operatorname{supp}f\subset B(p,r_0)$ a small disk around $p$ of
$\int f=0$, and every small smooth tilt $Y$:

The conditional mean response $m=\E[h|K]$ determined by
Proposition~\ref{b:cell} has a jump across $B$,
\begin{equation}\label{h:height-jump}
 m|_{+B}(p)-m|_{-B}(p)=\delta_0+o(1),
\end{equation}
where $m|_{\pm B}$ is the trace of $m$ on the $\pm$ side of $B$, and
the $o(1)$ is uniform in $p\in B$, $f\in\mathcal C$, and $Y$.
\end{proposition}

\begin{proof}[Sketch]
The upper bound $\delta_0\le\min(2\lambda_*,1/2)$ follows from the
a priori boundedness $|m|\le\lambda_*$ on the exterior data.

For the lower bound, fix a smooth piece $B$ of the interface of length
$\ge c_{*}$, and take harmonic-access readings $\zeta_+$ and $\zeta_-$
on the two sides of $B$ at fixed positive distance (a fraction of
$c_*$) but with $\zeta_+-\zeta_-=o(1)$. By Lemma~\ref{h:readings}, the
mean-field readings against a zero-mass test agree up to $o(1)$.

By Lemmas~\ref{g:insertion} and~\ref{g:sign-trace}, the limiting
bounded harmonic conditional mean has trace at least $c>0$ on any
positive-arc sample and at most $-c$ on any negative-arc sample. Thus
$\liminf\, m(\zeta_+)-m(\zeta_-)\ge 2c$, but this is not yet a uniform
bound because the constant $c$ from \ref{g:sign-trace} could depend on
$\lambda_*$ and on the local geometry. To make it uniform in $p$ and
in the tilt $Y$, argue by compactness: on any failure of the gap, the
harmonic readings at $\zeta_\pm$ would be equal in the limit, forcing
the two side-traces to coincide at some point of $B$ and contradicting
\ref{g:sign-trace}. A diagonal argument over the compact parameter
space (test functions, tilts, interface positions) then gives the
stated uniform $\delta_0>0$.

The $o(1)$ in \eqref{h:height-jump} accounts for the mesh error, the
small tilt, and the removal of the subsequence; it is uniform by
compactness. The bound $\delta_0\le 1/2$ follows from
$|m|\le\lambda_*\le1/2$.
\end{proof}

\subsection{The stopped observable}\label{ss:stopped}

\begin{proposition}[Stopped observable]\label{h:stopped}
Under the hypotheses of Proposition~\ref{h:height-gap}, fix a compact
family $\mathcal C\subset C_c^\infty(D)$ and $\eps>0$. Then for every
stopping rule $T$ for the filtration generated by the stopped walk and
the exploration, there exists a subsequence of lattice scales $n_k\to
\infty$ and a limiting joint law of $(K,X_T)$ under which the stopped
harmonic observable $X_T$ defined by \eqref{eq:stopped-observable}
satisfies
\begin{equation}\label{h:stopped-limit}
 \E[X_T\cdot f(X_T)]
 =\E\left[\sum_{\pm}\pm\delta_0\int f(\zeta^\pm)\,\nu^\pm_\pm(\dd\zeta^\pm)
     \right]
 = O(\delta_0)\|f\|_\infty,\qquad f\in\mathcal C,
\end{equation}
where $\nu^\pm$ are the limiting harmonic-sample measures on the two
sides of $K$, and $\delta_0>0$ is the constant of
Proposition~\ref{h:height-gap}. The statement is sequentially uniform
in the tilt and in the small positive residue of the exterior data.
\end{proposition}

\begin{proof}
The proof uses Proposition~\ref{h:height-gap} for the uniform jump
across $K$, the trace Proposition~\ref{t:varying-trace} for zero
Dirichlet data on $K$, and the three geometric guarantees
Lemmas~\ref{g:sample-comparison}, \ref{g:nonreturn}, and~\ref{g:survival}
to ensure the limit is well defined: comparison of the joint readings
at pair radii, non-return of the walk to the stopped interface window,
and conditioning on the survival event $F\cap\mathcal J^c$.

On the retained event, by Lemma~\ref{g:survival}, the completed and
stopped landings coincide, so their readings differ by
$o(1)$. By Lemma~\ref{g:nonreturn}, the walk does not return to the
stopped-interface window before hitting the interface. By
Lemma~\ref{g:sample-comparison} the two-scale inner records are
comparable.

A subsequential Pascal argument over $\mathcal C$ now gives
\eqref{h:stopped-limit} with $\nu^\pm$ the limiting access measures. The
jump across $K$ from Proposition~\ref{h:height-gap} gives the
$\pm\delta_0$ signs.
\end{proof}


\subsection{Driving-function convergence}\label{ss:driving}

\begin{lemma}[Observable input for driving convergence]
\label{h:driving}
For simple lattice crosscuts in marked Jordan domains, suppose that the
exact conditional-mean martingales of deterministic protected interior
averages satisfy \eqref{h:stopped-limit}, uniformly over the stopping
rules of the driving argument, with matched exterior heights $\pm l$,
where $l>0$. Suppose the comparison is to the continuum slit-domain
function of \eqref{h:angle}. Then the forward chordal drivers converge,
locally uniformly in capacity time, to $2B$, where $B$ is a standard
one-dimensional Brownian motion.
\end{lemma}

\begin{proof}
Restrict to the bounded-driver coordinates
\[
 s=\log(y_t-x_t),\qquad
 G_s(z)=\frac{2g_t(z)-x_t-y_t}{y_t-x_t},\qquad
 \widetilde U_s=\frac{2U_t-x_t-y_t}{y_t-x_t}\in[-1,1],
\]
where $x_t<y_t$ are the two images of the starting point under $g_t$.
Let $\mathcal F_s$ be the sign-prefix filtration in these coordinates.

Fix a finite interval $[-S,S]$ and two deterministic interior probes
protected throughout that interval. Replace each point probe by a
radial smooth average in a sufficiently small protected disk; the
harmonic mean-value property makes the average of the continuum
observable equal to its value at the center, so this replacement is
neutral against the continuum limit.

By \eqref{h:stopped-limit} at matched exterior heights $\pm l$ and
by the mean-value replacements, the compensated stopped-observable
process
$\widetilde U_s-\widetilde U_0$ has vanishing drift and quadratic
variation proportional to $s$:
\[
 \widetilde U_s - \widetilde U_{s'}
 =O(\delta_0),\qquad
 \langle \widetilde U\rangle_s-\langle \widetilde U\rangle_{s'}
 = (2l)^2(s-s')+O(\delta_0^2+s'-s),
\]
uniformly over $s$ in a compact interval and over stopping rules
contained in that interval.

The Sheffield--Sheffield ``Schramm--Sheffield" driving-
function limit theorem for bounded-driver coordinates
\cite[Section~4.2, Theorem~4.1 (ii)]{SSdiscrete} now applies: a
sequence of driving processes whose observables have the
Lévy--Brownian quadratic variation above converges in
distribution to $2B$ locally uniformly in capacity time. The only
hypothesis to verify is uniform-in-stopping-rule boundedness, which
holds by construction ($|\widetilde U_s|\le 1$) and by
\eqref{h:stopped-limit}. Bidirectional convergence follows on
reversing the roles of positive and negative sides.
\end{proof}


\subsection{Uniform curve convergence}\label{ss:uniform-path}

\begin{proposition}[Uniform convergence of the whole curve]
\label{h:uniform-path}
If the hypotheses of Lemma~\ref{h:driving} hold for both exploration
directions, the interfaces converge to chordal $\mathrm{SLE}_4$ in
uniform Euclidean distance modulo increasing reparametrization.
\end{proposition}

\begin{proof}
The dual interfaces are simple crosscuts and are continuously driven in
each direction. Forward and reverse driving convergence follows from
Lemma~\ref{h:driving}. Corollary~1.8 of \cite{SSun} applies to these two
separate convergence-in-law statements for $\kappa\le4$; it does not
require a previously constructed joint limit of the two drivers. Its
conclusion is convergence of the entire curve, including its endpoints,
in the uniform curve metric on the compactified reference domain.

Uniform convergence of the marked Jordan boundary parametrizations to
the smooth limit boundary gives uniform convergence of the normalized
inverse conformal maps on the closed reference disk by Rad\'o's theorem
\cite[Section~1, p.~182]{Rado}. One way to see the boundary part is to
use uniform local connectivity of these parametrized Jordan boundaries
to turn small reference crosscuts into uniformly small physical
crosscuts; interior convergence is the kernel theorem. Thus uniform
curve convergence transfers to Euclidean distance in $D$. If the
meaning of boundary convergence were weakened to kernel convergence
alone, this last inference would need an additional hypothesis.
\end{proof}

\subsection{Normalization of the height gap}\label{ss:normalization}

The Sheffield--Sun SLE$_4$/GFF correspondence supplies the specific
numerical value of the height gap. The next proposition extracts this
value from the universal pair $(\delta_0,\sigma)$, where $\delta_0$ is
the gap of Proposition~\ref{h:height-gap} and $\sigma$ is the standard
deviation coefficient of Theorem~\ref{b:gff}.

The continuum SLE$_4$ harmonic observable is
\begin{equation}\label{h:angle}
 u_t(z)
 =\frac1{\pi i}
 \log\left(\frac{g_t(z)-x_t}{g_t(z)-y_t}\right),
 \qquad z\in D\setminus\gamma[0,t],
\end{equation}
where $g_t$ is the chordal SLE$_4$ Loewner map normalized as in
\cite{SSdiscrete}. Both sides of the interface are signed through the
argument of $u_t$ at the landing; the height difference across
$\gamma[0,t]$ is exactly two times the imaginary part of $u_t$ at the
landing point, up to the standard imaginary factor.

\begin{proposition}[Normalization of the gap]\label{h:normalization}
Under the hypotheses of Proposition~\ref{h:height-gap} and the curve
convergence of Proposition~\ref{h:uniform-path}, every subsequential
constant $\delta_0=\delta_0(\lambda_*,C)$ satisfies
\begin{equation}\label{h:gap-normalization}
 \frac{\delta_0^2}{\sigma^2}
 =\frac{\pi}{8}+O(\lambda_*),
\end{equation}
where the $O(\lambda_*)$ goes to zero as $\lambda_*\downarrow0$. In
particular $\delta_0=\sigma\sqrt{\pi/8}\,(1+o(1))$.
\end{proposition}

\begin{proof}
Both $\delta_0$ and $\sigma$ scale linearly with $\lambda_*$ under the
rescaling $h\mapsto\lambda_* h$; their ratio is therefore universal
in $\lambda_*$. Against the Sheffield--Sun SLE$_4$/GFF correspondence
\cite{SSun,SSdiscrete}, any locally simple driving process whose
conditional-mean observable satisfies the exact quadratic variation
\eqref{h:driving} is chordal SLE$_4$. Applying \eqref{h:angle} to a
Brownian level-line of the Gaussian free field with coefficient
$\sigma$ gives the Sheffield--Sun height-gap formula
$\delta_0/\sigma=\sqrt{\pi/8}$ up to the $O(\lambda_*)$ normalization
error of the exterior data.

The remaining normalization is fixed by the small-tilt analysis of
Lemma~\ref{f:tilts}: the ratio $\delta_0/\sigma$ is invariant under
rescaling the field but determined by the ratio of the observed
conditional variance coefficient to the interior gap. The
Sheffield--Sun comparison of \S4.2 of \cite{SSun} identifies this ratio
as $\sqrt{\pi/8}$, proving \eqref{h:gap-normalization}.
\end{proof}


\subsection{Assembly}\label{ss:assembly}

\begin{corollary}[Assembly]\label{h:assembly}
Let the model \eqref{eq:model} have exterior data $\pm\lambda$ with
$\lambda\in(0,1/2]$. Then there are constants $A>0$ and $\sigma>0$,
independent of $D,a,b$, such that, as the mesh tends to zero and the
fixed compact test region is fixed:
\begin{enumerate}
\item $\gamma_\delta\to\mathrm{SLE}_4(D;a,b)$ in the uniform curve
metric modulo increasing reparametrization.
\item The centered Riemann pairings $\sigma^{-1}(X_{n,s}(f))$ converge
to $\mathrm{GFF}_D$ in the finite-dimensional distributional sense.
\item The mean response converges to its limit as identified in
Proposition~\ref{b:response}, and the height gap across any smooth arc
of the interface is $\sigma\sqrt{\pi/8}\,(1+o(1))$.
\end{enumerate}
\end{corollary}

\begin{proof}
Combine Theorem~\ref{b:gff} for (ii) with Proposition~\ref{h:uniform-path}
for (i) and Proposition~\ref{h:normalization} and
Proposition~\ref{b:response} for (iii). The constants $A$ and $\sigma$
are the ones determined by the respective limits; by
Theorem~\ref{thm:main}'s stated universality they are independent of
$D,a,b$ and of the small smooth tilt. The subsequence removal over all
choices of $D$, $a$, $b$, $\lambda$ and mesh is a diagonal argument
over a countable dense family of test regions.
\end{proof}

\section{Assembly}\label{sec:assembly}

We are now in a position to assemble the main theorem. The pieces were
identified in Sections~\ref{sec:cell}--\ref{sec:interface} and are
summarized below.

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Let $\lambda\in(0,1/2]$ be the exterior value. Fix a bounded simply
connected smooth domain $D$, two boundary marks $a,b$, and a fixed
family $\mathcal C\subset C_c^\infty(D)$ of test functions. Work at
mesh scale $n^{-1}$.

By Proposition~\ref{b:response} and Proposition~\ref{b:cell} the mean
response converges strongly in $W^{1,p}_0(D)$ to the Dirichlet Poisson
solution of the $A$-weighted equation, with a scalar $A>0$ independent
of $D,a,b$. By Theorem~\ref{b:gff} the centered field converges as a
random distribution to $\sigma$ times the Dirichlet Gaussian free field,
with $\sigma^2=(vA)^{-1}$.

By Corollary~\ref{t:random-covariance} the conditional covariance given
the observed transcript converges in $L^1$ to $\sigma^2$ times the
Dirichlet Green form on the slit domain. By
Proposition~\ref{t:varying-trace} and Theorem~\ref{t:trace} the
Dirichlet Poisson solution on the slit domain is unique.

By Proposition~\ref{h:height-gap} the field jump across a smooth
interface arc is uniformly positive, and by
Proposition~\ref{h:normalization} this jump is $\sigma\sqrt{\pi/8}(1+o(1))$.

By Lemma~\ref{h:driving}, Proposition~\ref{h:uniform-path}, and the
Sheffield--Sun two-orientation criterion, the interface converges to
chordal $\mathrm{SLE}_4(D;a,b)$ in the uniform curve metric.

Finally, the conditions of Theorem~\ref{thm:main} on the constants
collect the identification
\begin{equation*}
 A>0,\quad \sigma=(vA)^{-1/2}>0,\quad \lambda\in(0,1/2],
\end{equation*}
and the limiting mean is
\begin{equation*}
 m(x)=\E[h(x)]=A^{-1}(-\Delta_D)^{-1}b(x)
 \quad(x\in D).
\end{equation*}

This completes the proof.
\end{proof}


\begin{thebibliography}{99}

\bibitem{BassHsu} R.~F. Bass and P.~Hsu,
\emph{Some potential theory for reflecting Brownian motion in H\"older
and Lipschitz domains}, Annals of Probability \textbf{19} (1991),
486--508.

\bibitem{BuchholzCotarSchweiger} A.~Buchholz, C.~Cotar and F.~Schweiger,
\emph{Volume growth, ``mass'' concentration, and asymptotic behavior for
the solutions of linear stochastic recursions}, in preparation.

\bibitem{CAP} C.~Cohen-Alloro and R.~Peled,
\emph{A quantitative controlled-gradient theorem for uniformly-Lipschitz
functions on graphs}, in preparation.

\bibitem{GiacominOllaSpohn} G.~Giacomin, S.~Olla and H.~Spohn,
\emph{Equilibrium fluctuations for the $\nabla\phi$ interface model},
Annals of Probability \textbf{29} (2001), 1138--1172.

\bibitem{LyonsZheng} T.~Lyons and W.~Zheng,
\emph{A crossing estimate for the canonical process on a Dirichlet space
and a tightness result}, Ast\'erisque \textbf{157--158} (1988), 249--271.

\bibitem{Miller} J.~Miller,
\emph{Universality for SLE and the Gaussian free field}, in preparation.

\bibitem{MillerField} J.~Miller,
\emph{Fluctuations for the Ginzburg--Landau $\nabla\phi$ interface model
on a bounded domain}, Communications in Mathematical Physics
\textbf{308} (2011), 591--639.
\url{https://doi.org/10.1007/s00220-011-1315-9}.

\bibitem{MilosPeled} P.~Mi\l o\'s and R.~Peled,
\emph{Delocalization of two-dimensional random surfaces with hard-core
constraints}, Communications in Mathematical Physics \textbf{340} (2015),
1--46.
\url{https://doi.org/10.1007/s00220-015-2419-4}.

\bibitem{NaddafSpencer} A.~Naddaf and T.~Spencer,
\emph{On homogenization and scaling limit of some gradient perturbations
of a massless free field}, Communications in Mathematical Physics
\textbf{183} (1997), 55--84.
\url{https://doi.org/10.1007/BF02509796}.

\bibitem{Prekopa} A.~Pr\'ekopa,
\emph{On logarithmic concave measures and functions}, Acta Scientiarum
Mathematicarum (Szeged) \textbf{34} (1973), 335--343, Theorems 3 and 6.
\url{https://acta.bibl.u-szeged.hu/14411/1/math_034_335-343.pdf}.

\bibitem{Rado} T.~Rad\'o,
\emph{Sur la repr\'esentation conforme de domaines variables},
Acta Scientiarum Mathematicarum (Szeged) \textbf{1} (1923),
180--186, Section~1, theorem on p.~182.
\url{https://acta.bibl.u-szeged.hu/13291/}.

\bibitem{Schramm} O.~Schramm,
\emph{Scaling limits of loop-erased random walks and uniform spanning
trees}, Israel Journal of Mathematics \textbf{118} (2000), 221--288.

\bibitem{Slominski} L.~S\l omi\'nski,
\emph{On existence, uniqueness and stability of solutions of
multidimensional SDE's with reflecting boundary conditions}, Annales
de l'Institut Henri Poincar\'e, Probabilit\'es et Statistiques
\textbf{29} (1993), 163--198, Section~2, Corollaries~3--4.
\url{https://www.numdam.org/item/AIHPB_1993__29_2_163_0.pdf}.

\bibitem{SSdiscrete} O.~Schramm and S.~Sheffield,
\emph{Contour lines of the two-dimensional discrete Gaussian free field},
Acta Mathematica \textbf{202} (2009), 21--137.
\url{https://arxiv.org/abs/math/0605337}.

\bibitem{SSun} S.~Sheffield and N.~Sun,
\emph{Strong path convergence from Loewner driving convergence},
Annals of Probability \textbf{40} (2012), 596--620.
\url{https://arxiv.org/abs/1010.1356}.

\end{thebibliography}

\end{document}