==== TABLE OF CONTENTS ====
Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//paper.tex
Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//sections/cell.tex
Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//sections/coupling.tex
Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//sections/foundations.tex
Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//sections/geometry.tex
Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//sections/interface.tex
Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//sections/shift.tex
Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//sections/trace-analysis.tex
Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//sections/trace.tex
Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//triangular-figure.tex
===========================

==== BEGIN Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//paper.tex ====
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\hypersetup{pdftitle={Uniform real Lipschitz surfaces on the triangular lattice},pdfauthor={OpenAI}}
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\title{Uniform real Lipschitz surfaces\\on the triangular lattice}
\author{OpenAI}
\date{September 25, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove the Gaussian free field and SLE$_4$ scaling limits for uniformly
sampled real nearest-neighbor Lipschitz heights on the triangular lattice,
resolving Schramm's Problem~2.3. On approximations of smooth simply connected
domains, the centered height field converges as a random distribution to a
multiple of the Dirichlet Gaussian free field. At one tuned two-arc boundary
amplitude, the zero-height interface converges in uniform curve distance to
chordal SLE$_4$. We identify the relation between the field variance and the
boundary height in terms of an implicit stationary tangent-flux coefficient.
\end{abstract}
\tableofcontents

\section{The Model and the Theorem}\label{sec:model}
Let $D\subset\R^2$ be a bounded smooth simply connected domain, with
distinct marked boundary points $a,b$. Write $A_+$ and $A_-$ for its
intervening open boundary arcs. The unit triangular lattice has basis
\[
 e_1=(1,0),\qquad e_2=(1/2,\sqrt3/2),
\]
and nearest-neighbor steps $\pm e_1,\pm e_2,\pm(e_1-e_2)$. For mesh
$\delta>0$, take a simply connected triangular-lattice polygon $D_\delta$
whose marked boundary converges uniformly to $(D;a,b)$. Place the two
discrete marks at the midpoints of distinct boundary edges. The resulting
open arcs partition the boundary vertices, so no vertex receives two
boundary values.

For $0<\lambda\le1/2$, let $h_\delta$ have normalized Lebesgue measure on
the set of real vertex heights satisfying
\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 polytope is bounded because every vertex is at finite graph distance
from the prescribed boundary. In its free coordinates it has nonempty
interior: assigning zero to every free vertex leaves all inequalities
involving a free vertex strict. Fixed--fixed inequalities are already
satisfied by $2\lambda\le1$.

Extend heights affinely across faces. Almost surely no free vertex has
height zero, since that event is a hyperplane of Lebesgue measure zero.
Each triangle then has either no zero segment or a unique segment joining
two edges. These segments form loops and one chordal interface
$\gamma_\delta$ joining the discrete marks. The chord is simple: no face
contains two strands and no vertex is a branching point. Replacing its
within-face placement by the corresponding dual-lattice strand changes
uniform distance by $O(\delta)$.

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

\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 the stated curve metric and as random distributions, respectively.
The constants and 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}
Here $\omega_D$ is harmonic measure and $A$ is the tangent-flux coefficient
defined in the cell construction. The stiffness is $vA$; no elementary
closed form for it is proposed.
\end{theorem}

The normalization of chordal SLE$_4$ in the half-plane is
$\partial_tg_t(z)=2/(g_t(z)-2B_t)$ with standard real Brownian motion $B$.
There is no logarithmic rescaling of the heights in the distributional
limit. Pointwise height tightness is neither asserted nor required.

\subsection{History and scope}
Schramm's Problem~2.3 asks for SLE$_4$ and GFF limits for this real-valued
hard-constraint model \cite{Schramm}. It is distinct from the integer-valued
model in the preceding problem. Schramm--Sheffield establish the discrete
Gaussian contour result \cite{SSdiscrete}; Miller extends contour
universality to smooth symmetric uniformly convex gradient potentials
\cite{Miller}. The hard potential here is constant on $[-1,1]$ and infinite
outside. It does not satisfy those convexity hypotheses. An approximation
by smooth potentials is therefore not, by itself, a solution.

The identification of effective stiffness and Gaussian fluctuations
belongs to the gradient-field homogenization tradition of Naddaf and
Spencer \cite{NaddafSpencer} and Giacomin, Olla and Spohn
\cite{GiacominOllaSpohn}. For smooth symmetric uniformly convex interactions,
Miller proved a bounded-domain field limit by coupling boundary-conditioned
fields so that their difference is harmonic in the bulk with high probability
\cite{MillerField}. Mi\l o\'s and Peled \cite{MilosPeled}
studied hard-core fluctuations, including the uniform Lipschitz
potential, on even square tori with one pinned height. Their
delocalization and controlled-gradient arguments provide related
finite-volume methods. Cohen-Alloro and Peled's later gradient control
also extends the delocalization argument to square-lattice domains with
zero Dirichlet data \cite[Section~6]{CAP}. These hard-core delocalization
results do not give the
bounded-domain field or contour limits considered here. More recently, Buchholz, Cotar and
Schweiger \cite{BuchholzCotarSchweiger} proved infinite-volume
gradient-field universality for a class of nonconvex potentials.
Their scaling theorem retains smooth, uniformly curved mixture
components and does not cover the flat/infinite hard potential.

The convex-volume foundation uses Pr\'ekopa's integral log-concavity
theorem \cite{Prekopa}; the quantitative facet deficit, triangular
exposure and average anchoring are proved here. The controlled-gradient
estimate of Cohen-Alloro and Peled \cite{CAP} supplies the zero-pinned
starting point. Extending it to bounded data and sign observations
requires separate finite-volume arguments. Those extensions are needed
because an interface exploration continually changes the conditioned
height polytope.

For the finite reflected dynamics we use S\l omi\'nski's fixed-domain
Skorokhod approximation \cite{Slominski}, the Neumann semimartingale
identification of Bass and Hsu \cite{BassHsu}, and the stationary
forward--reverse decomposition of Lyons and Zheng \cite{LyonsZheng}.
The uniform changing-polytope force argument and weighted constant-drift
identification are proved locally. At the curve stage, we adapt the
harmonic-observable method of Schramm and Sheffield \cite{SSdiscrete}
and use the two-orientation path criterion of Sheffield and Sun
\cite{SSun}. The hard-constraint model must supply its own conditional
covariance, height gap and stopping-time-uniform observable before
those deterministic and diffusion arguments apply.

\subsection{Plan of the proof}
The proof has four parts. First, local volume curvature and sparse
defects give unscaled moments of macroscopic averages
(Section~\ref{f:section}). Second, synchronous
reflected couplings turn Brownian contact events into weighted control of
differences and their errors at observed sites
(Section~\ref{s:coupling}). Third, the stationary
tangent update identifies a scalar flux law and, through small tilts, a
Gaussian field (Section~\ref{sec:cell}). Analytic trace estimates on
varying observed sets then give conditional Dirichlet covariance
(Sections~\ref{sec:trace-analysis} and~\ref{sec:conditional-covariance}).
Finally, feasible shifts with controlled volume cost turn crossing
estimates into barriers (Section~\ref{sec:feasible-shifts}). These
barriers and conditional field estimates give sampled-interface separation,
then the height gap and the stopped harmonic observable
(Sections~\ref{sec:geometry} and~\ref{sec:interface}).

The demanding uniformity is part of the assertions, not an optional
strengthening: random observations are used only with their exact conditional
Gibbs kernels, and the curve argument needs estimates uniform over the
stopping rules used to test its driving function. In particular,
conditional covariance, survival reweighting, and convergence of labelled
harmonic accesses are established before the driving criterion is applied.

\section{Conventions for the Estimates}\label{sec:conventions}
For local arguments rescale the lattice to spacing one and write $n$ for
the macroscopic length. Discrete edge gradients do not include a factor
$n$. Continuum tests use the Riemann pairing
\[
\langle h,f\rangle_n=\frac1{vn^2}\sum_x f(x/n)h_x.
\]
In the continuum trace sections, the lattice is instead already embedded
at physical mesh $n^{-1}$; there $x$ denotes the physical location and the
same pairing is written with $f(x)$. These are the same normalization.
Constants may depend on a fixed compact test region and on explicitly fixed
scale ratios, but not on $n$ or the microscopic shape of admissible observed
strings. An oriented sign string prescribes the signs adjacent to an
interface piece. Each such vertex has an opposite-sign neighbor, hence its
height belongs to $[-1,1]$. Newly revealed exact values are instead pins;
their annealed estimates must be justified by conditioning, not by treating
arbitrary revealed values as uniformly bounded.

The reflected bath has Brownian variance parameter two in free coordinates
and normal reflection on the convex height polytope. A density tilted by
$\exp(\sum_xb_xh_x)$ adds drift $b$. Finite-dimensional normal-reflection
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 fibers after the recorded coordinates are removed;
null-volume conditionings are discarded. Reflection and its martingale
decomposition are applied on these free coordinates. The finite-body
references and the separate uniform forcing argument appear in
Section~\ref{s:coupling}.

\input{sections/foundations}
\input{sections/coupling}
\input{sections/cell}
\input{sections/trace-analysis}
\input{sections/trace}
\input{sections/shift}
\input{sections/geometry}
\input{sections/interface}

\section{Proof of the Main Theorem}\label{sec:assembly}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
The ordinary fixed-boundary domains satisfy the hypotheses of
Proposition~\ref{f:averages}, Theorem~\ref{c:comparison}, and
Lemma~\ref{c:exterior-zero}. Proposition~\ref{b:response} therefore
defines a positive, domain-independent scalar $A$, identifies the mean,
and supplies the small-tilt response. Theorem~\ref{b:gff} gives the
distributional field limit with $\sigma^2=(vA)^{-1}$.

All interface observations used here are dual sign prefixes, complete
crosscuts, or the explicitly separated inner and outer records.
Field estimates are applied to their sign-conditioned kernels before
the harmonic-landing reweighting. The prefix and crosscut kernels
used for conditional covariance satisfy the connected-set thickness
condition of Theorem~\ref{t:covariance}. Local barrier estimates use
Corollary~\ref{t:local-boundary-variance}, with the endpoint-window
condition checked in Proposition~\ref{g:barriers}, including for the
two-sided records. Extra exact values occur only in the annealed
comparison experiments of Lemma~\ref{c:freezing}. Thus no uniform
quenched assertion about arbitrary frozen values is needed.

Propositions~\ref{g:sample-comparison} and~\ref{g:nonreturn}, together
with Lemma~\ref{g:survival}, give the two-scale and stopped sampling
estimates. Lemmas~\ref{h:slit-sampling} and~\ref{h:marked-limits}
retain the oriented harmonic accesses. The resulting height gap and
stopped observable are Propositions~\ref{h:height-gap} and~\ref{h:stopped}.
In each finite collection of experiments the positive exterior heights
have a positive lower bound, as required by the exterior barriers.

The reference-first extraction in Section~\ref{sec:interface} then
permits matching the exterior height to the extracted gap. Forward
and reverse driving convergence and the strong-path criterion give
the stated uniform curve limit. Proposition~\ref{h:normalization}
identifies every extracted gap as $\sigma\sqrt{\pi/8}$;
Corollary~\ref{h:assembly} removes the subsequences and places this
constant in $(0,1/2]$. This proves all assertions, including
\eqref{eq:target-constants} and~\eqref{eq:target-mean}.
\end{proof}

\begin{thebibliography}{99}
\setlength{\itemsep}{2pt plus 1pt minus 0.5pt}
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\emph{A contour line of the continuum Gaussian free field},
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\end{thebibliography}
\end{document}
==== END Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//paper.tex ====

==== BEGIN Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//sections/cell.tex ====
\section{The Stationary Cell Problem and the Field Limit}
\label{sec:cell}

The comparison estimate controls how two constrained fields respond to a
small change of law. In a bulk synchronous coupling the common Brownian noises
cancel, leaving the imposed relative drift and the averaging action of
edge contacts. We retain the order of those contacts in a stationary bulk
limit, then use the average transport of relative heights to identify the
macroscopic equation. A positive source will show that its coefficient
is a positive scalar, and exponential tilting will identify the Gaussian
field law.

Throughout this section lattice vertices have unit spacing:
$V_n\subset\Lambda$, where $\Lambda$ is the triangular lattice, and
$x\in V_n$ represents the physical point $x/n$. The vertex density is
$v=2/\sqrt3$. All bulk sampling regions have a fixed positive
macroscopic buffer from fixed pins and reflecting sign walls. Trace
estimates up to walls are separate from these bulk estimates.

Here are the quantitative comparison bounds used in the construction.
For a stationary synchronous difference $z=h-\widetilde h$, a fixed
short time interval has finite transmission clusters $C$, whose sizes
have all fixed moments under volume sampling. Write $e_C^2$ for their
quadratic loss, denoted by $D_C$ in the conservative comparison of
Section~\ref{s:coupling}. On each fixed interior macroscopic set $Q$,
\begin{equation}\label{b:energy-input}
 \E\sum_{C:C\cap nQ\ne\varnothing}e_C^2\le C_Q.
\end{equation}
The version with a drift difference $n^{-2}b(x/n)$ has an additional
uniformly bounded error. The filled differences $U_n$ obey, for every
$10$ below the one supplied by short-step sparsity, and join an edge
when its minimum slack during the step is at most $s$. Choose $s$ off
the atoms of the distributions of these minima; only countably many
values are excluded. The resulting components are finite with the stated
moments. On every fixed finite neighborhood their membership is stable
under local path convergence, except on a null event. This convention
also applies to the union of the two approximating replicas' edges.
It avoids retaining an unspecified auxiliary partition in the cell law.
For an oriented edge facet write
its normal as $a_e$; the sign does not affect the matrix
\[
 P_e=I-\frac{a_ea_e^{\mathsf T}}{|a_e|^2}.
\]
This matrix averages the two incident coordinates and fixes all others.
It is an orthogonal projection, is doubly stochastic, and preserves constants.

\begin{lemma}[A stable finite product]\label{b:tangent}
On a finite isolated cluster, consider a normal-reflection trajectory with
interior endpoints, no simultaneous distinct-face contacts, and positive
impulse in each visited-face episode. For two compared trajectories whose
initial states are sufficiently close to the reference initial state and
which use the same driving path, sufficiently close to the reference drive,
their synchronous difference update is exactly a finite product
\[
 J=P_{e_k}\cdots P_{e_1}.
\]
Consequently, if the initial relative differences multiplied by $n$ are
tight, their limiting update is $w'=Jw$ modulo constants. An additional
relative drift bounded by $Cn^{-2}$ makes a vanishing error after this
multiplication by $n$.
\end{lemma}

\begin{proof}
For each facet $e$, its contact times form a compact subset of $[0,T]$.
Different nonempty contact sets are disjoint and thus have positive
distance. Since the number of facets is finite, there can be only finitely
many switches of contact label. Repetitions of the same label can be grouped.
Choose a finite partition with endpoints off all facets so that each
contact interval allows only its designated facet, and each remaining
interval has no contact. All undesignated inequalities on each such
interval have positive minimum slack.

Continuity of the finite Skorokhod map preserves these inactive margins
under small perturbations. Positive impulse supplies a strict running-minimum
witness in the normal coordinate. Therefore both sufficiently close
synchronous paths reach the designated face within that interval. The
one-dimensional reflection formula then removes their normal difference
exactly; their tangential difference stays unchanged. Their difference at
the end of the interval is $P_e$ times its input difference. Contact-free
intervals preserve the difference. Composition proves the exact identity.

The conclusion is stronger than an error $o(1)$, which would not survive
multiplication by $n$. On each fixed regular path neighborhood the identity
is exact. Normal monotonicity bounds a finite-dimensional perturbation
from a differential drift by its integrated Euclidean magnitude, at most
$C(C)Tn^{-2}$. Its product with $n$ tends to zero.

For the 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.
\end{proof}

Write $\mathcal E$ for the two-sided stationary bulk environment, including
the gradient trajectories and drivers. On each step, Lemma~\ref{b:tangent}
defines $J$ separately on its finite clusters. We now describe the class of
relative height solutions for which the mean flux will be unique.
The comparison bounds control filled relative gradients in $L^p$, whereas
the correction at a volume-sampled site is controlled by tightness.
We therefore record the large-scale slope by affine growth in volume,
which survives these stationary corrections. A slope is called attained
only when a stationary solution with the bounds below exists.

\begin{definition}[Controlled tangent solution]\label{b:controlled}
A controlled solution of slope $q\in\R^2$ is a space-time stationary
joining of $\mathcal E$ with real relative heights $(w_x(k))$, defined
modulo a spatial additive constant at each $k\in\mathbb Z$, such that:
\begin{enumerate}
\item The relative heights form a spatial cocycle: after choosing $w_0=0$,
they obey $w_{x+y}-w_x=w_y\circ\tau_x$, where $\tau_x$ shifts the joined
environment and relative field by $x$. They update by
$w(k+1)=J_k w(k)$ modulo constants.
\item For every $\delta>0$ there are spatial constants $c_R$ for which
\[
 \frac1{|\Lambda\cap B_R|}
 \#\{x\in\Lambda\cap B_R:
       |w_x-q\cdot x-c_R|>\delta R\}
 \longrightarrow0
\]
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}
The loss defining $d_x$ is nonnegative and invariant under addition of
a constant. A slope is called attained if such a solution exists.
\end{definition}

\subsection{Extraction and growth of the relative heights}

\begin{lemma}[Volume-sampled tangent solutions]\label{b:extraction}
Let $z$ be a stationary comparison on scale $n$ covered by
Theorem~\ref{c:comparison}, with relative drift $n^{-2}b(x/n)$.
Sample $x$ in a fixed compact macroscopic region separated by a positive
macroscopic distance from all fixed pins and reflecting sign walls.
Extract local limits of
$n(z_{x+i}-z_x)$, jointly with $x/n$ and the bulk environment. Conditional
on the macroscopic position, the limit is a stationary spatial cocycle,
has update $w'=Jw$ modulo constants, and has affine volume growth with a
possibly random slope $q$. Conditional on almost every value of $q$, it is
a controlled solution on the original environment law. If the expected
macroscopic difference converges to $m$, then
\[
 \E[q\mid x/n=y]=\nabla m(y)
\]
in the distributional sense in the interior.
\end{lemma}

\begin{proof}
The comparison estimates give tightness on every finite neighborhood and
every finite set of step times. A diagonal extraction preserves the exact
additivity of the differences. Shifting the root by a fixed lattice vector
changes its macroscopic position by $o(1)$ and changes its sampling law
only by a vanishing boundary error. The conditional local law is therefore
spatially stationary. Time stationarity passes directly. Its environmental
marginal is the same ergodic law at every macroscopic position by
Proposition~\ref{b:bulk}. Lemma~\ref{b:tangent} identifies the update.

For integrability, distribute $n^2 e_C^2$ equally among the vertices of
$C$. Its expectation at a volume-sampled site is bounded by
\eqref{b:energy-input}. Cluster displacement is bounded by a fixed
polynomial in cluster size times the square root of cluster loss.
For any $11$.
Let $\mathcal I$ be the invariant sigma-field of this joint spatial law.
The spatial ergodic theorem makes the rescaled gradients converge weakly
to their conditional mean $q$ given $\mathcal I$. To see that the potential
has affine growth, subtract its mean on each box and rescale by $R$.
Poincare's inequality and the spatial $L^p$ bound give weak
$W^{1,p}$ compactness and strong $L^p$ compactness. Every compact limit
has distributional gradient $q$, and is therefore affine. This proves
the asserted volume growth for $\widehat w$.

The correction $w_x-\widehat w_x$ is a finite stationary random variable:
the hole has finite diameter and every finite collection of original
scaled differences is tight. No integrability of this correction is
required to transfer volume growth. For fixed $\delta>0$, stationarity
and bounded convergence show that the expected fraction of sites where
$|w_x-\widehat w_x|>\delta R$ tends to zero. The original cocycle thus
has the same slope. The identical argument for its stationary step
displacement shows that $q$ is time invariant.

For any Borel slope event $B$, the function
$\Prob(q\in B\mid\mathcal E)$ is spatially invariant. Ergodicity of
$\mathcal E$ makes it constant. Conditioning on $q$ therefore leaves
the environmental marginal unchanged. Stationarity and the controlled
bounds disintegrate for almost every slope value, proving the claimed
deterministic-slope solutions.

Finally, the filled gradients are uniformly integrable in volume sampling.
Discrete summation by parts against a smooth macroscopic test identifies
their expected weak limit with $\nabla m$. The expected filled and original
potentials have the same limit by their vanishing $L^1$ difference.
The conditional spatial mean of the extracted filled gradients is $q$,
so its expectation is precisely the stated derivative.
\end{proof}

We next prove uniqueness at each attained slope. Two ingredients are
needed: dissipation removes a sublinear stationary difference, and the
actual contact graph connects the lattice. The latter is a statement about
the environment, not about any particular solution joined to it.

\begin{lemma}[Forward recurrence of contact tests]\label{b:recurrence}
Fix an edge. Suppose that there are forward-past measurable events $G_j$
with $\Prob(G_j)\to1$. For each $j$, suppose a finite measurable cover of
$G_j$ assigns to each class a fixed finite path of genuine free facets
connecting the edge endpoints. Assume each assigned facet can be forced
during a bounded time interval with conditional probability at least
$p_j>0$, given the entire forward history at the start of that interval.
Then the future contact graph connects the edge endpoints almost surely.
If these assumptions hold for every lattice edge, that graph is connected.
\end{lemma}

\begin{proof}
Use a time grid whose spacing accommodates a force attempt and is a
multiple of $T$. Choose measurably one class at an overlap. Maintain a
separate visit counter for each class and cycle through its assigned
facets. On infinitely many visits to that class, each facet is attempted
infinitely often. Conditional Borel--Cantelli, applied to the forward
filtration, makes each such facet occur infinitely often: its conditional
success probability at an attempted step is at least $p_j$.

Stationarity and Poincare recurrence imply
\[
 \Prob(\text{infinitely many visits to }G_j)\ge\Prob(G_j).
\]
Some class is visited infinitely often on this event, and its contact
path connects the prescribed endpoints. Let $j\to\infty$, and then
intersect the resulting probability-one events over the countably many
edges. Neither temporal ergodicity nor conditioning on a temporal
invariant event is used.
\end{proof}

We next verify the forward-past premise in the infinite-volume law.
This requires more than independence of future drives: finite-volume
insulation marks may depend on distant initial data.

\begin{lemma}[Passage of force opportunities to the bulk]
\label{b:force-transfer}
The finite-cover force and insulation estimates of
Lemmas~\ref{c:force} and \ref{c:insulation}, with the improving
good-configuration probabilities in \ref{f:conditioned-sparse} and
\eqref{c:short-step}, imply the hypotheses of
Lemma~\ref{b:recurrence} for the bulk environment.
\end{lemma}
\begin{proof}
Fix a lattice edge and a quality index $j$. Use the finite cover in
Lemma~\ref{c:force}, choosing the first neighborhood at an overlap.
It has finitely many labels $1,\ldots,M_j$, each carrying a fixed path
of genuine free facets joining the edge endpoints. Artificial grounding
is excluded by the collar slack, and there are no genuine pins or walls
in this bulk test box. Let $G_{j,n}$ mark the configurations satisfying
the initial slack and insulation conditions, and let $L_{j,n}$ be their
cover label. The estimates give
$\Prob(G_{j,n}=0)\le\delta_j+o(1)$ with $\delta_j\downarrow0$.
The force duration and its positive probability $p_j$ may depend on $j$.

At each time of a grid accommodating the force duration, these marks
are functions of the full finite-volume initial configuration, before
future drives are sampled. For an assigned facet $e$ write $C_e$ for
the event of visiting that facet during the next force interval.
The force and insulation proof, conditional on the full initial data,
and independence of subsequent Brownian increments from the past give
\begin{equation}\label{b:finite-force-inequality}
 \E\bigl[\Psi\mathbf1_{\{G_{j,n}=1,L_{j,n}=\ell\}}
                    \mathbf1_{C_e}\bigr]
 \ge p_j\E\bigl[\Psi\mathbf1_{\{G_{j,n}=1,L_{j,n}=\ell\}}\bigr]
\end{equation}
for every bounded nonnegative function $\Psi$ of the forward past.
Here the initial bad-insulation event was discarded before forcing;
no conditioning on a future cluster or future success event is used.

Retain the finite-valued marks in the local extraction constructing
the bulk law, for every fixed quality, spatial translate, and time $kT$.
For attempts of a particular quality use a coarser subgrid whose spacing
accommodates its force duration.
A diagonal extraction is possible because these index sets are
countable. The resulting marked extension is stationary. Its unmarked
marginal is still the unique bulk environment of
Proposition~\ref{b:bulk}. The limiting marks need not be functions of
the limiting initial state, and we do not assert that they are.

Take $\Psi$ to be a bounded nonnegative continuous cylinder function
of the unmarked past. Visiting a specified facet in a fixed compact
time interval is closed in the uniform local path topology: the
maximum of its oriented edge gradient equals the facet bound. The
discrete mark event is clopen. Thus the left integrand in
\eqref{b:finite-force-inequality} is upper semicontinuous, and
Portmanteau gives
\[
 \E\bigl[\Psi\mathbf1_{\{G_j=1,L_j=\ell\}}\mathbf1_{C_e}\bigr]
 \ge \limsup_n
 \E\bigl[\Psi\mathbf1_{\{G_{j,n}=1,L_{j,n}=\ell\}}
                    \mathbf1_{C_e}\bigr]
 \ge p_j\E\bigl[\Psi\mathbf1_{\{G_j=1,L_j=\ell\}}\bigr].
\]
The last expectation passes exactly by continuity. Approximation of
finite-dimensional past laws by continuous functions, followed by a
monotone-class argument over past cylinders, extends the inequality
to every bounded nonnegative function of the full unmarked forward
past $\mathcal F_0$. Contact at a deterministic interval endpoint is
a null event by Proposition~\ref{b:bulk}, so the closed-event passage
does not create an endpoint-only contact for the later tangent argument.

Define
\[
 a_\ell=\Prob(G_j=1,L_j=\ell\mid\mathcal F_0),
 \qquad a=\sum_{\ell=1}^{M_j}a_\ell.
\]
Dropping the mark from the left side of the preceding inequality gives
$\Prob(C_e\mid\mathcal F_0)\ge p_j a_\ell$ for every facet of
the path with label $\ell$. Since $\E(1-a)\le\delta_j$,
$\Prob(a<1/2)\le2\delta_j$. On $\{a\ge1/2\}$ choose a
maximizing label, with a fixed tie rule. Every facet of its path then
has conditional contact probability at least $p_j/(2M_j)$.
These are the required forward-past measurable opportunity classes.
Choose conditional-expectation versions equivariantly under the
countable spatial and time shifts; their joint law is stationary.
Their probabilities approach one, so Lemma~\ref{b:recurrence} applies.
The auxiliary marks served only in the transfer; the resulting
contact-connectivity assertion concerns the original environment alone.
\end{proof}

\begin{lemma}[Zero-slope uniqueness]\label{b:zero-slope}
A controlled tangent solution of slope zero is spatially constant.
\end{lemma}

\begin{proof}
Let $a$ be a nonnegative compactly supported Lipschitz function, equal to
one on the unit ball. For $M=\eps R$, let
\[
 \Phi_M(s)=
 \begin{cases}s^2,&|s|\le M,\\2M|s|-M^2,&|s|>M,\end{cases}
 \qquad
 F_R(w)=\min_c\sum_x a(x/R)\Phi_M(w_x-c).
\]
Its Lipschitz constant in each input coordinate is bounded by
$2M a(x/R)$. Hence $F_R(Jw)-F_R(w)$ is integrable, although the two
terms need not separately have finite expectations. They have the same
distribution by time stationarity. Truncate each scalar minimum at level
$K$ and use
$|(s\wedge K)-(t\wedge K)|\le|s-t|$; dominated convergence gives
\begin{equation}\label{b:stationary-loss}
 \E[F_R(Jw)-F_R(w)]=0.
\end{equation}

Choose a minimizing center $c_R$. On a cluster $C$ set
$a_C=\min_{x\in C}a(x/R)$ and
\[
 D_C^\Phi=\sum_{x\in C}
    \{\Phi_M(w_x-c_R)-\Phi_M((Jw)_x-c_R)\}\ge0.
\]
Clusterwise Jensen and evaluation of the new minimum at $c_R$ give
\begin{equation}\label{b:huber-cutoff}
 F_R(Jw)-F_R(w)
 \le-\sum_C a_CD_C^\Phi
       +C\eps\sum_{C:C\cap\operatorname{supp}(a(\cdot/R))\ne\varnothing}
             \sum_{x\in C}H_x^k|(Jw-w)_x|.
\end{equation}
Indeed the cutoff oscillation on $C$ is at most
$C\operatorname{diam}(C)/R$, while $\Phi_M$ is $2\eps R$-Lipschitz.
The controlled displacement moments bound the expectation of the last
sum by $C R^2$; clusters reaching the support from outside are covered
by a further fixed power of $H_x$. Equations
\eqref{b:stationary-loss}--\eqref{b:huber-cutoff} therefore bound the
normalized expected convex loss by $C\eps$.

For fixed $\eps>0$, volume sublinearity puts all but a vanishing fraction
of the weighted sites within $o(R)$ of a common center. The Huber
minimizer is itself $o(R)$ from that center. To verify this, the derivative
from an exceptional fraction $\delta$ of sites is at most
$2\eps R\delta$ times total weight; the derivative from the remaining
sites has a fixed sign and larger magnitude if the center is displaced
by more than their oscillation plus $C\eps R\delta$. Thus such a
displacement cannot minimize the loss.

First bound cluster size. The entire input of a typical interior cluster
then lies in the quadratic region, with probability tending to one.
Stochastic averaging keeps its output there. On that cluster
$D_C^\Phi=\sum_C(w_x^2-(Jw)_x^2)$, independently of the center.
The stationary integrable quadratic loss allows the exceptional fraction
to be discarded by truncating that loss before taking $R\to\infty$.
The cluster-size restriction is then removed using its moments. It follows
that $\E d_0\le C\eps$. Send $\eps\downarrow0$.

Each factor of $J$ is an orthogonal averaging projection. Its quadratic
loss is nonnegative, and the losses telescope. Zero total loss therefore
makes every constituent projection fix its input. The relative heights
are constant in time, and each actual contact equates its incident values.
Lemma~\ref{b:recurrence} connects every pair of neighboring sites and
proves spatial constancy. This argument never conditions on the tangent
solution, which may depend on future drives.
\end{proof}

\subsection{Mean flux and the macroscopic equation}

For a controlled solution define the cluster-averaged flux at $x$ by
\begin{equation}\label{b:flux-definition}
 \mathcal J_x(w)=
 -\frac1{T|C_x|}\sum_{y\in C_x}(y-\overline C_x)(Jw-w)_y,
 \qquad
 \overline C_x=\frac1{|C_x|}\sum_{y\in C_x}y.
\end{equation}
Conservation makes this independent of the chosen cluster origin, and the
controlled moments make it integrable. Write $j(q)=\E\mathcal J_0(w)$
when the slope is $q$. The minus sign is the sign of down-gradient transport.

\begin{proposition}[The cell flux and weak equation]\label{b:cell}
Under the preceding inputs, $j(q)$ is well defined on the vector space of
attained slopes and is linear there. If any nonzero slope is attained,
every slope in $\R^2$ is attained and $j(q)=Aq$ for a finite scalar $A$.
For a stationary comparison with drift difference $n^{-2}b(x/n)$,
every subsequential expected macroscopic difference $m$ satisfies
\begin{equation}\label{b:weak-equation}
 \int j(\nabla m)\cdot\nabla f=\int b f,
 \qquad f\in C_c^\infty(U),
\end{equation}
in each limiting open region $U$ free of observations.
\end{proposition}

\begin{proof}
Join two solutions of the same 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. Lemma~\ref{b:zero-slope} makes their relative heights identical.
This proves uniqueness of flux, including conditional uniqueness given
the environment. Addition of joined solutions and scalar multiplication
preserve the controlled class. Hence the attained slopes form a vector
subspace and the flux 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 rotations and reflections. A real
two-dimensional map commuting with a nontrivial sixty-degree rotation
is a scalar matrix plus a scalar multiple of the quarter-turn matrix;
a reflection eliminates the latter. Thus $j(q)=Aq$. Its coefficient is
finite by the integrability of \eqref{b:flux-definition}.

We justify passage of fluxes to expectations. At a volume-sampled site let
$D_n=n^2e_C^2/|C|$ and $H_n=|C|$. A scaled step displacement, or the
scaled cluster first moment per site, obeys
\[
 |F_n|\le C H_n^a\sqrt{D_n}
\]
for a fixed $a$. For $11$, by
Lemma~\ref{c:exterior-zero}. To recall its geometric content, put the
original difference equal to zero on the boundary and exterior, and
retain the coercive edges incident to the common pins. Whenever a hole
meets this connected zero set, a path to infinity exits through a zero
rim vertex; choose that vertex as the hole's anchor. The usual rim
energy estimate holds for this anchor, so the extension is still bounded
in $W^{1,p}$ and remains exactly zero outside the polygon. Its expected
value is therefore weakly compact in that
space and strongly compact locally in
$L^p$. The original and filled expectations have the same limit.

Uniform convergence of the polygonal boundaries to the smooth boundary
implies that this zero extension vanishes outside an arbitrarily small
neighborhood of $\overline D$. Its limiting extension consequently
vanishes 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
$\int b f=0$ 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 $u$.
This gives $\int u g=0$. 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 $A>0$.

For arbitrary identical bounded boundary data, the same zero-extension
argument and weak identity identify every mean-response subsequence with
$A^{-1}(-\Delta_D)^{-1}b$. Compactness therefore gives full convergence.

For the untilted mean, compare the prescribed-boundary bath with the
zero-boundary bath on the same domain. The latter has mean zero by height
reflection symmetry. Proposition~\ref{b:cell}, with source zero, makes
every limiting prescribed mean harmonic. Monotone comparison with the
zero-boundary law shifted by $\pm\lambda$ bounds this mean between
$-\lambda$ and $\lambda$. Near a compact subarc, compare instead with
the zero-boundary law shifted by that arc's prescribed value. The
coupled difference has exact zero boundary on that subarc. A cutoff
supported away from the marks permits the same local zero-extension
argument, giving the stated trace. A bounded harmonic function in smooth
$D$ is uniquely determined by these two open-arc values; the two marked
points have zero harmonic measure. This identifies the mean and every
subsequence.
\end{proof}

\subsection{Gaussian fluctuations}

\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
\[
 \sigma^2=(vA)^{-1}.
\]
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 $a_f>0$ such that
$\sup_n\E e^{a_f|X_n(f)|}<\infty$. 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's inequality 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 these moment-generating functions;
uniqueness identifies every subsequential scalar law. Apply the same
argument to every finite real linear combination of tests to obtain
Gaussian finite-dimensional laws with covariance
$(vA)^{-1}(-\Delta_D)^{-1}$.

Here is the required distributional tightness. The anchored and
equal-mass average bounds in \ref{f:averages} give bounded second moments
for the coarse coefficients of a compactly supported wavelet expansion.
A normalized two-dimensional wavelet at spatial scale $2^{-j}$ has
coefficient equal to $O(2^{-j})$ times a smooth average difference.
Thus its variance is $O(2^{-2j})$ while there are $O(2^{2j})$
coefficients at that scale. Summation with weights $2^{-2sj}$ gives
a uniform local $H^{-s}$ second moment for each $s>0$, down to the
lattice scale. At finer scales the deterministic physical Lipschitz
bound $O(n)$ for the interpolant gives total squared coefficients
$O(n^2 2^{-2j})$, whose weighted tail is bounded as well. Compact
Sobolev embedding into a slightly more negative order and a countable
interior exhaustion give tightness as distributions.

Finally the Riemann pairing and affine pairing have vanishing difference.
An interior nodal hat function has integral $(vn^2)^{-1}$ and, by the
symmetry of its triangular star, zero first moment about its vertex.
Taylor expansion of $f$ therefore gives
\[
 \int f h^{\mathrm{aff}}-\frac1{vn^2}\sum_xf(x/n)h_x
 =O\!\left(n^{-2}\frac1{n^2}
                 \sum_{x\text{ near }n\operatorname{supp}(f)}|h_x|\right).
\]
Deterministic Lipschitzness and bounded boundary values give
$|h_x|=O(n)$ in the bounded rescaled domain, so this error is $O(n^{-1})$.
The same is true after centering. The scalar limits and tightness thus
identify the asserted affinely interpolated distributional limit.
\end{proof}
==== END Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//sections/cell.tex ====

==== BEGIN Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//sections/coupling.tex ====
\section{Stationary Reflected Comparisons}\label{s:coupling}

We prove the local comparison estimate used to pass from the constrained
microscopic dynamics to a continuum equation. The input consists of the
sparse-box estimates \ref{f:sparse} and \ref{f:conditioned-sparse}, the
extension over their holes in \ref{f:hole}, and the average bounds in
\ref{f:averages}. The principal new
point is a lower bound on dissipation obtained by forcing a contact inside
a fixed box. Its constants remain uniform when a constraint becomes
redundant, and after adapted exact-value observations.

\subsection{Experiments and the comparison statement}

We use the unit triangular lattice, with macroscopic scale $n$. Write
$Q_r=x_0+[-rn,rn]^2$, using any fixed pair of lattice coordinates, and
$\widehat Q_r=n^{-1}Q_r$ for its scaled version. For a vertex function,
$\int_{n,Q}F=n^{-2}\sum_{x\in Q}F(x)$; for an edge function the same
notation sums over unoriented lattice edges with midpoint in $Q$.
Omitting $Q$ means the currently specified localization box.
Constants may depend on fixed box ratios but not on $n$ or on the
diameter of the ambient finite domain. After choosing an orientation
$e=(x,y)$, a lattice gradient is the unscaled difference
$\nabla_e f=f(y)-f(x)$.
The notation $\Prob_{\rm vol}$ denotes the annealed law obtained by
first sampling the experiment and then an independent uniform vertex
in the indicated box, or an independent uniform edge for an edge
quantity. In a vertex-sampled gradient one fixes a lattice step
$\mathbf e$ and uses the edge $(X_n,X_n+\mathbf e)$.

An experiment starts with two finite-volume Lipschitz fields. Each has
bounded Dirichlet data and finitely many prescribed sign strings of the
type covered by \ref{f:sparse} and \ref{f:averages}. On the common free
graph in $Q_2$ 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, with
each query and its 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 then put in a stationary synchronous joining. All estimates for
such an experiment are averaged over the original transcript as well as
the joining. The phrase ``uniform in the observations'' below has this
annealed meaning; it does not mean uniform over every possible exact-height
transcript.

We also allow jointly sampled boundary data when, conditional on those
data, the two interiors have their specified Gibbs kernels. Replacing
their conditional product by a stationary synchronous joining must
preserve each annealed marginal. The sparse-contact and average bounds
are required for those annealed marginals, with the same constants as
in the theorem below. This includes coupled random pins; it does not
assert a quenched estimate for an arbitrary realized pin vector.
The fixed set and the component on which a kernel is sampled may be
part of these recorded data. For example, a zero-pinned kernel on a
recorded component has the required bounds uniformly in that component,
provided its tested averages have the stipulated anchor. Its mixture
therefore has the same bounds. The other marginal must still satisfy
its own annealed estimates; selecting a component supplies no such
estimate by itself.

Let $P$ be the common fixed set in $Q_2$ and let $S$ be 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 we
take $a=b=0$ and use the two tails of $z$; square penalties at $S$ will
anchor their sum. In a region with no observations, we can instead use
$z$ modulo a spatial constant. Boundaries of $Q_2$ are artificial:
conditions outside that box need not agree.

The local average bounds required below are precisely those supplied by
\ref{f:averages}: averages of both marginals are bounded in every fixed
moment on the scale of $Q_2$ and its fixed-fraction subboxes, or differences
of equal-mass averages are so bounded in the free, centered case. The
constants for subbox side $\ell n$ may grow polynomially in $\ell^{-1}$.
For an adapted experiment these bounds concern the original marginal
laws; the conditioning argument below transfers them to the joined fields.
In a one-sided experiment we only require $z|_P\ge a$, and use $u^-$. An
observed boundary is \emph{controlled for the lower tail} if its fixed
values satisfy $z\ge a$; for the upper tail the condition is $z\le b$.
When extending a tail by zero behind such a boundary, we mean the side
removed from the common free graph, whose edges to the retained side
all pass through that observed boundary. An \emph{uncontrolled seed}
is a prescribed initial queried set on which no such inequality is
assumed. Any uncontrolled seed or other uncontrolled boundary must
remain a fixed positive scaled distance from the box on which an
estimate is asserted.

For a fixed step length $T$ and contact tolerance $\delta>0$, a
\emph{transmission edge} is a free--free lattice edge whose constraint
has slack at most $\delta$ at some time in $[0,T]$ in either replica.
The components of their union on the free vertices, including
singletons, are the transmission clusters. Write $C_x$ for the
cluster containing $x$ and $H_x=|C_x|$; also write $H_C=|C|$.
Contacts with fixed vertices or individual walls supply boundary
input to a cluster and do not connect clusters through a fixed
vertex. For a zero-drift cluster with zero tail input, its square
dissipation is
\[
 D_C=\sum_{x\in C}u_0(x)^2-\sum_{x\in C}u_T(x)^2.
\]
For two tails their dissipations are added. With drift, $D_C$ means
the sum of the nonnegative square losses at projection substeps,
with the source contribution kept separately. Lemma~\ref{c:cluster}
proves nonnegativity and the projection interpretation. Dissipation
sums concern clusters insulated from uncontrolled data; the
remaining clusters contribute the localization errors estimated
below. The theorem chooses $T$ and $\delta$ through its fixed
testing parameters.

\begin{theorem}[Local reflected comparison]\label{c:comparison}
Consider the experiments just specified, with $|a|+|b|$ bounded. Assume
the conclusions of \ref{f:conditioned-sparse}, \ref{f:hole}, and \ref{f:averages}
for their original marginal laws. Individual reflecting constraints are
ordinary one-sided sign walls, not pairs $0\le h_x\le\eps$ with
$\eps$ allowed to tend to zero. Fix $00$. There are fixed testing parameters, a step
length $T_\rho>0$, good edges $G_n$ and good wall sites $S_n^{\rm g}$,
and positive weights $w_e,w_x\le1$ with the following properties.

For either tail in \eqref{c:tails}, or for their signed difference, let
\begin{equation}\label{c:energy}
 \mathcal E_n(Q)=
 \sum_{e\in G_n,\ e\subset Q}w_e|\nabla_e u_0|^2
 +\sum_{x\in S_n^{\rm g}\cap Q}w_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}
Here $C$ and $D_C$ have the preceding step-cluster meaning. The constants in
\eqref{c:energy-bound} 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 relevant finite collection of moment orders can be chosen in advance.

Let $U_n$ be the extension over holes supplied by \ref{f:hole}, with a
fixed choice of the testing parameters. It can be set to zero on the
exceptional event. In scaled coordinates,
\begin{equation}\label{c:compactness}
 \E\|U_n\|_{W^{1,p}(n^{-1}Q_r)}^2\le C,\qquad
 \E\int_{n,Q_r}|U_n-u_0|\longrightarrow0.
\end{equation}
In a free comparison subtract a local cutoff-weighted average from both
$U_n$ and $z_0$. Its gradient and dissipation estimates are unchanged.
All these centered estimates are uniform in the size of the ambient
domain. The same bounds hold for a rough one-sided adapted experiment
with only $z|_P\ge a$, for $u^-=(a-z)_+$, and with zero extension behind
a controlled observed boundary. They hold uniformly after replacing $z$
by $z/K$, $K\ge1$, with bounded rescaled thresholds.

\emph{Threshold-uniform specialization.}
In the one-sided case, for each threshold $a$ satisfying the required
condition $z|_P\ge a$, the weights may be chosen independently of $a$.
With the original marginal moment bounds fixed,
\begin{equation}\label{c:threshold-scaling}
 \E\mathcal E_n\bigl((a-z)_+;Q_r\bigr)
 +\E\|U_n^{(a)}\|_{W^{1,p}(n^{-1}Q_r)}^2
 \le C_0(1+a^2).
\end{equation}
Thus, when $a=K\ge1$ and the observed values satisfy $z|_P\ge K$,
the energy and squared Sobolev norm of $(1-z/K)_+$ are bounded by
$2C_0$, uniformly in $K$.

For an ordinary simply connected polygon domain with identical fixed
boundary data, the exterior-preserving construction in
Lemma~\ref{c:exterior-zero} gives a bounded $W^{1,p}(\R^2)$ extension
which is exactly zero outside the lattice polygon. This construction
uses connectivity of its zero exterior; it is not asserted for an
arbitrary disconnected internal observed set.

If $Q_R$ has no observations or individual walls, and $X_n$ is sampled
from normalized ambient volume in $Q_r$, then
$n\nabla_{(X_n,X_n+\mathbf e)}z_0$ for each fixed lattice step $\mathbf e$,
$n(z_0(X_n+i)-z_0(X_n))$ for each fixed lattice offset $i$, and
$n\sup_{t\le T_0}|z_t(X_n)-z_0(X_n)|$ are tight. Thus this
untruncated conclusion has a fixed positive macroscopic free buffer.
It also holds up to controlled observations when $a=b$, because the
two tails then control $z-a$. For general $aq\}\right)
 \le \frac{2}{q}\bigl(2\rho^{q/2}\bigr)^{|I|}.
\]
Also, if $0\le X_n\le n^B$ and
$\Prob(X_n>M)\le n^{-A}$, then
\[
 \Prob\{\E[X_n\mid\mathcal I]>M+1\}\le n^{B-A}.
\]
\end{lemma}
\begin{proof}
On the first event let $N=\sum_{i\in I}\mathbf1_{A_i}$ and $k=|I|$.
Then $\E[N\mid\mathcal I]>qk$ and $N\le k$, so
$\Prob(N\ge qk/2\mid\mathcal I)\ge q/(2-q)\ge q/2$.
The unconditional probability is at most
$2^k\rho^{\lceil qk/2\rceil}$ by choosing a witnessing subset.
For the second assertion use conditional Markov on
$\E[X_n\mathbf1_{X_n>M}\mid\mathcal I]$.
\end{proof}

In particular, for $q\ge1$, Jensen's inequality 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 moment order.
The deterministic spatial moment bounds from \ref{f:hole}, with an
arbitrarily fast polynomial exceptional probability, pass to the
conditional moments in \eqref{c:conditional-moments}. First truncate
local clusters at a power of $n$ and discard the tail using the
arbitrarily high cluster moments. No bound involving the total ambient
volume is used. Mere finiteness of random all-order moment constants
would not suffice for the expectation arguments below.

For later use we recall why short stationary steps have the requisite
spatial sparsity. Conditional on a fixed transcript, each marginal is
reversible normally reflected Brownian motion, with variance parameter
two in every free coordinate. We use the symmetric Neumann
identification of the fixed-polytope process \cite{BassHsu} and the
form-domain forward--reverse identity \cite[Section 1, equation
(1.5)]{LyonsZheng}. The forward--reverse martingale
decomposition bounds each coordinate's displacement during $[0,T]$
by a fixed multiple of the suprema of its forward and reversed Brownian
martingales. Within either direction the coordinate martingales have
independent Brownian coordinates. If $k$ separated boxes have a
displacement exceeding $d>0$, select one of their finitely many shell
sites in each box; at least $k/2$ witnesses occur in one time direction.
The reflection principle and a union bound give
\begin{equation}\label{c:short-step}
 \Prob(\hbox{all these boxes move by more than }d)
 \le C\bigl(C_L e^{-c d^2/T}\bigr)^{k/2}.
\end{equation}
The constants are independent of the pinned values and of the number
of free coordinates. Fixed coordinates contribute no displacement.
Mix over transcripts. Combining \eqref{c:short-step} with initial edge
slack in \ref{f:conditioned-sparse}, and taking $T$ small after the slack parameter,
shows that transmission edges are confined to clusters of sparse boxes.
Connected-animal counting gives all cluster-size moments, their spatial
concentration, and arbitrarily small defect density by improving the
fixed parameters and shortening the testing step. Small constant drifts
are covered by the same reversible decomposition for their tilted
invariant measures, or by the bounded drift correction.

Here is the constant-drift extension of the finite-body identification.
After removing exact pins, let $K\subset\R^m$, $m\ge1$, be the
positive-volume polytope of free coordinates and put
\[
 \mu_b(\dd h)=Z_b^{-1}e^{b\cdot h}\,\dd h,
 \qquad \mathcal E_b(f,g)=\int_K\nabla f\cdot\nabla g\,\dd\mu_b.
\]
Its form domain is $H^1(K^\circ)$. An all-pinned fiber is deterministic
and requires no reflected process.
The density is smooth and bounded above and below on this fixed compact
body. Integration by parts for a smooth function $g$ with zero normal
derivative gives
\[
 \int_K f(\Delta g+b\cdot\nabla g)\,\dd\mu_b
       =-\mathcal E_b(f,g).
\]
Thus the weighted Neumann form identifies a stationary reversible
process with generator $\Delta+b\cdot\nabla$. For the coordinate $h_i$,
writing $\rho_b$ for the density and $n_{\rm out}$ for the outward unit
normal, the same calculation gives
\[
 \mathcal E_b(h_i,g)
 =-b_i\int_K g\,\dd\mu_b
  +\int_{\partial K}g\,n_{{\rm out},i}\rho_b\,\dd\mathcal H^{m-1}.
\]
The coordinate additive-functional argument in \cite{BassHsu} therefore
gives drift $b_i\dd t$ and an inward normal regulator. Its coordinate
martingale brackets are $2\delta_{ij}t$, since the energy density is
$2\nabla f\cdot\nabla g$. The coordinate functions belong to the form
domain, so the Lyons--Zheng identity applies even though they need not
satisfy the generator's Neumann boundary condition. Convex Skorokhod
uniqueness identifies this process with the reflected bath driven by
variance-two Brownian motion and drift $b$. This weighted-form argument
is separate from the zero-drift assertion of \cite{BassHsu}.

\subsection{Projection updates and square dissipation}

\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
their pinned values. Except at finitely many parameter breakpoints,
the derivative of projection is computed on a fixed active face.
An active edge makes its two free derivatives equal. A free component
therefore receives the average of its input derivatives, unless it
touches a pin or individual wall, when it receives the prescribed
derivative. Several pins active on a parameter interval have equal
derivatives. Integrating over the interpolation parameter gives the
asserted convex mixture of orthogonal averaging projections and zero
blocks. All intermediate contacts belong to the union of endpoint
near-contact clusters: an infinitesimal projection cannot contact an
edge whose slacks in both endpoint fields are bounded below.

For $u=(z-b)_+$, convexity and the pin bound $z|_P\le b$ give
$u'\le Au$ coordinatewise. Write this as $u'=RAu$ with a diagonal
matrix $0\le R\le I$. The lower tail is identical after changing sign.
Each elementary matrix $A$ or $R$ is a 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 the initial levels, with lost mass put
at zero. Jensen's identity and the column deficits show that its
variance about that 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.
\end{proof}

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. For two tails, add their losses. With drift, $D_C$ denotes
the sum of the nonnegative losses at projection substeps; the separate
source terms are estimated at the end of the proof. General doubly
stochastic matrices cannot replace the projections in this lemma: a
permutation has no square loss.

\subsection{Accessible contacts in a fixed box}

The next construction supplies coercivity even when a nominated edge
inequality is redundant. Choose a fixed enlarged testing box, containing
at most $N$ free coordinates, and a collar $B$ separated from its central
test patch by the long-slack range. In the initial field use coordinates
$y_x=h_x-h_x(0)$. Adjoin a ground coordinate $y_*=0$ for all individual
constraints, and add the auxiliary bounds $|y_x|\le r_0$ on the collar.
These bounds restrict the paths we construct; they are not extra reflecting
walls of the original process.

\begin{lemma}[Facet paths and uniform forcing]\label{c:force}
Suppose every original edge involving a free coordinate has initial
slack at least $s>0$. Assume that in both directed orientations every
original-edge path between a central test vertex and the collar, without
an individual-constraint edge to ground, has residual slack greater
than $\Lambda>2$. Every free component is required to reach the collar
or an observed fixed vertex through original edges. Ordinary sign walls
are allowed, and a tested wall has initial slack at most one.

For every central edge there are at most $N+1$ genuine facet constraints
joining its endpoints, or grounding both endpoints at genuine pins or
walls. A tested individual wall is grounded by such a path. For any fixed
$T>0$ there are $p_0,I_0>0$, depending only on these fixed parameters,
such that every facet selected in these paths can receive normal impulse
at least $I_0$ in time $T$, on an event of the interior Brownian noises
of probability at least $p_0$, while all collar coordinates move by less
than $r_0$. Take $r_01/2$ are still arbitrarily sparse.
On the remaining initial configurations exterior quietness has
probability at least $1/2$ and depends only on exterior noises. The
controlled interior noises are independent of them. Their intersection
has probability at least $p_0/2$, 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}

At an upper-edge contact of the first replica, $h_i-h_j=1$ and feasibility
of the second implies $z_i-z_j\ge0$. The contribution to the upper-tail
square loss is $2(u_i-u_j)\dd L\ge0$. The lower tail follows by reversing
the sign. Pin contacts and common walls have the same dissipative order.
Let $g$ be the initial jump at a facet selected in Lemma~\ref{c:force}.
It has a fixed bound in absolute value: use two for a lattice edge or
a zero-input pin edge, and the string height bound for an individual
wall. On the forced event, either a relevant coordinate moves by
at least $|g|/4$, or the jump retains magnitude at least $|g|/2$
during the impulse. Lemma~\ref{c:cluster} in the first case, and the
impulse in the second, give
\begin{equation}\label{c:force-loss}
 D_C\ge c g^2 H_C^{-m}
\end{equation}
for a fixed $m$. If the initial jump has the wrong sign for this facet,
the large-movement alternative must occur before contact. Forcing one
replica suffices; no force of the second replica is imposed.

To sum the local tests without counting a cluster loss once per vertex,
associate to a testing box $B$ the quantity
\[
 d_B=\sum_{C:\,C\cap B^+\ne\varnothing}\frac{D_C}{H_C},
\]
where $B^+$ is its fixed enlargement. 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 all required reciprocal moments,
and \eqref{c:allocation} supplies the expected coercivity inequality.

We now bound the dissipation without already assuming pointwise tightness
of $z$. Let $\eta$ be a deterministic cutoff, equal to one on an inner
box, zero outside an outer box. Let $d_{\rm lat}$ denote the gap
between those boxes in lattice units, so the cutoff has lattice
Lipschitz constant $C/d_{\rm lat}$.
For a cluster put $\eta_C=\max_C\eta$. 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. Cluster multiplicities and
diameters can be included in a larger fixed moment of $H_x$. Combining
with the force bound and using $d_{\rm lat}\asymp n(s-r)$ gives,
for $rp/[2(p-1)]$ and put
$\alpha=2-2/p-1/t>0$. Partition a fixed box into squares of side
$\ell$. Poincare--Sobolev on each square followed by Holder 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}
Indeed the oscillation estimate on one square is
$\|U-U_B\|_{2t'}\le C\ell^\alpha\|\nabla U\|_p$;
sum using $2t'>p$. For the piecewise constant averages use
$\sum_B\ell^2|U_B|^{2t'}\le
\ell^{-2(t'-1)}(\sum_B\ell^2|U_B|^2)^{t'}$.
The hole error is bounded by Holder with its stated moment.

For two tails take $F=(z-b)_+-(a-z)_+$. Their squares add to $F^2$,
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
$\ell^{-1/2}$. No higher energy moment is used. Exceptional events
are discarded using their arbitrarily fast polynomial probability,
local Lipschitz bounds, and the moments of one local average.

For completeness, combine \eqref{c:absorb} and
\eqref{c:caccioppoli} with an intermediate box. There exist 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 successive gaps $d_j=d_02^{-j}$, with fixed
$\theta2^A<1$. The resulting series is bounded. Stop after
$J=\lfloor\kappa\log_2n\rfloor$ iterations, with $\kappa>0$ small
enough that every auxiliary interpolation box is still larger than
$n^{1/2}$. The terminal bound is polynomial in $n$, from local
Lipschitzness and one average moment. After fixing $\kappa$, decrease
$\theta$ so that $\theta^Jn^B\to0$ for that polynomial exponent $B$.
Choose the exceptional-event exponent last. This proves the local
bounded norm and \eqref{c:energy-bound}, and \ref{f:hole} then gives
the first assertion of \eqref{c:compactness} for every $10$ 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.

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.

Here is a quantitative absorption using only the first energy moment.
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 $d=s-r$.
This $d$ is the dimensionless scaled gap; the corresponding lattice
gap is $nd$, as in the cutoff calculation above.
Pair \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
\[
 X^2\le\delta\mathcal E_n
 +C(1+\delta^{-1/4}+d^{-1/2})(1+Z^2).
\]
The derivative of $\chi^2U$ contains $\chi U$, which explains why
no uncontrolled larger-box $L^2$ term is introduced. 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}
where $F=u$, and the filling error has been included by Holder.
One may take $k=5/4$ if the extension constants have no additional
gap dependence; only a finite polynomial exponent is used.
Take expectations using \eqref{c:negative}, and repeat the interior
iteration. This proves the one-sided part of the theorem. The
argument is annealed over the actual transcript by Lemma~\ref{c:freezing}.
To track the threshold, retain $a$ in
\eqref{c:one-sided-pointwise}: its pairing contributes at most
$|a|X$, and hence $C a^2$ after absorption. All later estimates are
linear in this lower-order bound. The weights depend on the original
fields and force tests, not on $a$. This proves
\eqref{c:threshold-scaling}. For $z/K$ the force impulse in its
difference is divided by $K$, while a facet jump is bounded by
$2/K$; thus 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 obtaining microscopic tightness}

Fix first one hole construction. To prove its error tends to zero,
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 already proved give good edges
$G_{j,n}$ and weights with
\[
 \E\sum_{e\in G_{j,n}}w_{j,e}|\nabla_eu|^2\le C_j,\qquad
 \E\int_n\mathbf1_{\{e\notin G_{j,n}\}}\le\rho_j+o(1),
\]
and low-weight density at most $C_j a^q+o(1)$ for $w_{j,e}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 $a0$ 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}
where $Q$ is an ambient coordinate box. Assume either $\sum_xq_x=0$, or
there is a connected set of pins $K$, of diameter at least $cR$, such that
$K\cup Q$ lies in a coordinate box of diameter at most $CR$. For a sample
$h$ from $\mu_b$, there are $a>0$ and $M<\infty$, 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 specified uniform law.
\end{proposition}

The zero-mass alternative covers differences of comparable-scale averages
in a free ball even when the nearest pin is much farther away. An ordinary
ball average, or a smooth average of that scale, satisfies
\eqref{f:weights}. Restriction of its original positive weights to the
domain also satisfies that condition. Renormalization after restriction
requires a new verification of the bound $\max|q_x|\le CR^{-2}$.
We first prove the proposition for zero pins. The proof will also identify
the exact deterministic estimates required of later comparison fields.

\subsection{A local filling-volume deficit}
Fix a large even integer $L$ and tile the lattice by
$Q_z=Lz+\{0,\ldots,L\}^2$. The interiors of distinct blocks have no
connecting edge. For shell data $s$ on $\partial Q$, let $K_Q(s)$ be the
polytope of fillings and put
\[
 F_Q(s)=\log\operatorname{vol}_{(L-1)^2}K_Q(s),
\]
with value $-\infty$ when the volume is zero. The inclusion
$\frac12(K_Q(s_1)+K_Q(s_2))\subset K_Q((s_1+s_2)/2)$ and
Brunn--Minkowski imply concavity of $F_Q$.

Concavity gives a nonnegative midpoint loss. We next show that, on blocks
with slack, this loss controls differences between the perturbations of
the shell heights at the four side midpoints. It cannot control a common
height shift, which leaves filling volume unchanged. The later coarse-grid
argument combines these differences; the connected pins or the zero-mass
hypothesis in Proposition~\ref{f:averages} remove the additive constant.

For each of the four sides, let $v$ be its midpoint and let $r$ be
the coordinate unit step pointing toward the block center. The inward
path is $v+kr$, $0\le k\le L/2$. Of the two tangential unit steps
on that side, let $t$ be the one 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 $\max(k,\ell)$. These choices define
the four tests below. Figure~\ref{fig:triangular} illustrates one of them.

For fixed $0<\delta\le\delta_0\le1/16$, an integer $k_0\ge4/\delta_0$,
and $s_0>0$, call a block \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 the following four non-tie conditions hold. For each face center $v$,
take the radial coordinate path from $v$ to the block center. Exactly one
ray on its starting face has compressed triangular distances to that path;
if $b_1$ is its first vertex, require $|h(b_1)-h(v)|\ge s_0$.
Take $L\ge8(k_0+2)/\delta_0$.

\input{triangular-figure}

\begin{lemma}[Triangular block deficit]\label{f:block}
For these fixed parameters there is $c_*>0$ such that, if $Q$ is
convex-good for $h$, $s=h|_{\partial Q}$,
$\chi:\partial Q\to\R$, and both $K_Q(s+\chi)$ and
$K_Q(s-\chi)$ have positive volume, then every two face centers $v,w$
satisfy
\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}
We first expose one sign of each radial edge with uniform slack in all
other constraints. Write $x=v+ke_1$, $y=v+(k+1)e_1$ after applying a lattice
symmetry. If $k\ge k_0$, prescribe $a(x)=h(x)$ and $a(y)=h(x)+1$.
Both points are at distance at least $k_0$ from the shell, and the change
at $y$ is at most two. Hence, for sufficiently small $\beta\le\delta/2$,
\[
 |a(z)-h(b)|\le(1-\delta_0)\operatorname{dist}(z,b)+2
 \le(1-\beta)\operatorname{dist}(z,b)
 \quad(z=x,y,\ b\in\partial Q).
\]
If $k1$. Extending by the minimum of the upper
distance cones gives a filling with designated gradient $\tau$ and all
other gradients at most $1-\beta$ in absolute value.

We need the following quantitative finite-dimensional consequence of
slice log-concavity, and give its proof.
If positive-volume convex bodies $K_-,K_+\subset\R^m$
have projection widths at most $A$ in a unit direction $u$, their top
facet areas divided by volume are at least $c_0>0$, 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}
Here $m_\pm$ are the centroids, $b_\pm$ the top projection endpoints,
and $\kappa>0$ depends only on $A,c_0$. This is a finite-dimensional lemma;
it has no restriction on the spatial dimension of the lattice.

To prove \eqref{f:facet-input}, let $f_i$ be the normalized density
of the projection of uniform measure on $K_i$, and let $q_i(p)$
be its quantile, $00$ the fixed parameters above and a fixed enlargement
$Q_z^*$ can be chosen so that 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 the remaining blocks declare a block bad when it
fails \eqref{f:near}, the long-distance condition \eqref{f:far} on $Q_z^*$,
or one of the four non-tie tests. Then
\begin{equation}\label{f:animal}
 \Prob(A\text{ is entirely bad})\le\rho^{|A|}
\end{equation}
for every finite set $A$ of block indices. The collar defining pin blocks
contains all enlargements $Q_z^*$. Fixed finite thickenings of the bad set
also satisfy such a bound with an arbitrarily small parameter, after
adjusting the 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 fixed set, states that
for distinct edges $e_1,\ldots,e_j$ and $0r
                  \text{ for all }i\right)
 \le\bigl(Ce^{-r^2/(CT)}\bigr)^{j/2}.
\end{equation}
Thus the zero-pinned good-block conditions persist with relaxed fixed
constants throughout a sufficiently short step, apart from arbitrarily
sparse exceptional blocks. The same implication holds for any other
stationary constrained law for which the initial block tests have the
joint sparse bound.
\end{lemma}
\begin{proof}
Normal reflection has a forward Brownian martingale part and a
finite-variation part consisting of boundary reflection and any constant
drift. Reversibility gives the same decomposition backwards
from $T$. Adding the forward increment and the negative reversed increment
cancels the finite-variation terms, so each coordinate displacement is half the
sum of the corresponding forward and reversed martingale increments.
Each of the two martingale vectors has independent variance-two Brownian
coordinates, although the two vectors need not be independent. If all $j$
coordinate displacements exceed $r$, at least $j/2$ are witnessed in one
of the two directions. A union over subsets and the Brownian maximal
inequality prove \eqref{f:step-tail}. A failed block persistence test is
witnessed by a coordinate displacement exceeding a fixed constant.
Separated blocks have distinct witnesses; their finite coordinate count
is absorbed by taking $T$ smaller. Edge slack, long-distance slack, and
non-ties all persist once their two endpoint displacements are below
the appropriate fixed fraction of their original margins.
\end{proof}

\subsection{Filling holes and testing macroscopic averages}
We next isolate the deterministic geometry. It is valid for comparison
fields as well as the displacements of shell data used below.
\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 the bounded complementary components
of each bad star-connected cluster and retain its outermost filled hulls.
A fixed layer of enlargement is allowed. There is an extension procedure
which, for a field $u$ on good sites, fills each hole by its value at one
surrounding good site. Here good sites lie outside the filled holes and
good edges join neighboring good sites. Write $v$ for the resulting field.
A hole is incident to $z$ if it contains $z$ or a neighbor of $z$;
let $d_z$ be the largest diameter of an incident hole, zero if none,
and put $d_e=\max(d_x,d_y)$ 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 $10$.
In particular, bounded spatial moments of sufficiently high powers of
$d_e$ and $a_e^{-1}$ imply
\begin{equation}\label{f:hole-scaled}
 \|\nabla v\|_{\ell^p}\le C N^{2/p-1}\sqrt E.
\end{equation}
If $u$ is the restriction of a globally $K$-Lipschitz field on the grid,
then $|u(z)-v(z)|\le CK(1+d_z)$.

For every fixed $q,D<\infty$, the spatial average of $(1+d_z)^q$ in a
square of side $N$ is bounded by a deterministic constant outside an event
of probability $O(N^{-D})$. In a box of side $n$, these bounds hold
simultaneously on all subboxes of radius at least $n^{1/2}$, after altering
the failure exponent. Filling can be chosen equivariantly under translations
of the block grid. No reciprocal-weight assertion follows from sparsity
alone; those moments are separate hypotheses in \eqref{f:hole-weighted}.
\end{lemma}
\begin{proof}
Follow the filled outer grid contour of a hole and its exterior good rim.
After a fixed thickening this gives nearest-neighbor rim paths. A hole
of diameter $d$ has at most $C(d+1)^2$ vertices and boundary edges, and
any required rim path has at most that length. Choose the lexicographically
first rim site as the filling value. For an edge entering a hole, telescope
along its rim; the inequality $|\sum_{i=1}^m t_i|^p\le
m^{p-1}\sum_i|t_i|^p$, followed by the boundary-edge count, gives
\[
 \sum|\nabla v|^p\le\sum_{e\text{ good}}W_e|\nabla u(e)|^p,
 \qquad W_e\le C_p(1+d_e)^{2p}.
\]
Outer rims have bounded overlap. Holder with exponents $2/p$ and
$2/(2-p)$ gives \eqref{f:hole-weighted}. For unweighted energy the required
hole exponent is $4p/(2-p)$; in the weighted case another Holder inequality
separates the hole and reciprocal-weight moments. A hole filling value lies
within distance $C(d+1)$ of each of its sites, proving the pointwise error.

If a deterministic site belongs to a hull of diameter at least $r$, a bad
star-animal of size $m\ge cr$ has a vertex within $Cm$ of it. There are
$O(m^2)$ root choices and at most $C_0^m$ such animals. Applying
\eqref{f:animal}, or extracting a fixed-fraction separated subset before
doing so, proves \eqref{f:hole-tail}. In particular all fixed moments
have uniformly bounded expectations.

Here is 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 such cluster witnesses in an $O(N)$ box, root and animal counting
and \eqref{f:animal} bound the probability by
\[
 \frac{(CN^2)^r}{r!}C_0^{2kr}\rho^{ckr}
 \le\left(C\frac{N^2}{r}e^{-c'k}\right)^r.
\]
For all sufficiently large fixed $k$, take
$r=\lceil N^2k^{-(q+4)}\rceil$. For $k\le A\log N$ and large $N$, the
resulting bound is
\[
 \exp(-cN^2/k^{q+3}).
\]
Larger clusters occur with
probability at most $CN^2e^{-cA\log N}$, which is $O(N^{-D})$ by
choosing $A$ large. On all dyadic size ranges the resulting contribution
to $N^{-2}\sum_z(1+d_z)^q$ is summable; clusters below the fixed cutoff
give a fixed deterministic bound. This proves the assertion. The same
argument excludes holes crossing the macroscopic buffer, and a union over
the polynomially many subboxes proves the stated simultaneous version.
Lexicographic selection commutes with translation; a random grid phase
can be adjoined when invariance under single lattice steps is desired.
\end{proof}

\begin{lemma}[Discrete trace against a spread measure]\label{f:trace-average}
Let $\nu$ be a positive measure of bounded mass in a square of side $O(N)$,
and assume $\nu(B(x,r))\le C((r+1)/N)^\alpha$ for $0\le r\le CN$.
If $12-p$, then
\[
 |\nu(v)-\nu(1)\overline v|
 \le C N^{1-2/p}\|\nabla v\|_{\ell^p},
\]
where $\overline v$ is the ordinary average over that square.
\end{lemma}
\begin{proof}
Discrete $L^1$ Poincare on successive dyadic squares, clipped to the
ambient square at its boundary, gives
\[
 |v(x)-\overline v|\le C\sum_e
       \frac{|\nabla v(e)|}{1+\operatorname{dist}(x,e)}.
\]
For $p'=p/(p-1)$, a dyadic contribution to the averaged kernel is
$F_r(e)=Cr^{-1}\nu(B(e,Cr))$. Its supremum is at most
$CN^{-\alpha}r^{\alpha-1}$ and its $\ell^1$ norm at most $Cr$.
Thus
\[
 \|F_r\|_{\ell^{p'}}\le
 C N^{-\alpha/p}r^{1+(\alpha-2)/p}.
\]
The dyadic series is bounded by $CN^{1-2/p}$ precisely under the stated
strict inequality on $\alpha$. Holder finishes the proof.
\end{proof}

A connected set of coarse pins of diameter at least $cN$ carries a
deterministic probability measure $\nu$ with the preceding ball bound for
$\alpha=1$: choose a coordinate projection of length at least $c'N$ and
one pin site over each attained integer coordinate. Connectivity ensures
there are no missing intermediate coordinates. Equal weights on the chosen
sites have ball masses at most $C(r+1)/N$.

\begin{lemma}[Uniform average coercivity]\label{f:average-coercivity}
Suppose $X$ is globally $K$-Lipschitz on the block grid, with fixed $K$.
Suppose either $u=X$, or $u=T_M(X)$ for fixed
$T_M(t)=\operatorname{sign}(t)(|t|-M)_+$. Let
$E=\sum_{e\text{ good}}|\nabla u(e)|^2$. For deterministic weights $w$
supported in an $O(N)$ square with $\sum|w|\le C$ and
$\max|w|\le CN^{-2}$, assume either $\sum w=0$, or $u=0$ on a connected
deterministic coarse pin set of diameter comparable to $N$ at distance
$O(N)$. There is an event of probability at least $1/2$, determined only
by the bad set, on which simultaneously for every such field $X$,
\begin{equation}\label{f:coercivity}
 \left|\sum_z w_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, and exclusion of holes crossing a fixed buffer, give
$\|\nabla v\|_{3/2}\le CN^{1/3}\sqrt E$. The filling error tested against
$|w|$ is bounded by the same spatial moment. In the anchored case choose
the deterministic pin measure $\nu$ just constructed. Although filling
may overwrite a pin value, the Lipschitz bound gives
\[
 |\nu(v)|\le CK\sum_z\nu(z)(1+d_z).
\]
The right side has bounded expectation by \eqref{f:hole-tail}; another
Markov bound makes it a fixed constant. Applying
Lemma~\ref{f:trace-average} to $\nu$ and $|w|$ proves
\eqref{f:coercivity}. In the zero-mass case the ordinary mean cancels and
no pin measure is needed. The truncation error is at most $M\sum|w|$.
Choose the two moment thresholds so their failure probabilities sum to
at most $1/4$; the buffer event has failure at most $1/4$ for large $N$.
Bounded $N$ is absorbed by deterministic Lipschitzness and the anchor or
zero-mass condition. All error bounds depend on hole sizes rather than
on $X$, which proves the simultaneous assertion.
\end{proof}

This last lemma is an unscaled estimate. Its bounded additive error on
the pin set does not assert convergence to zero Sobolev trace in a later
rescaled limit.

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

For feasible $s\pm\chi$ with positive density, put $X_z=\chi(v_z)$ at
a fixed face-center representative of each block. The difference of any
two full endpoint fillings, divided by two, extends $\chi$ as a
1-Lipschitz field vanishing on pins. Consequently neighboring $X$ values
differ by at most $CL$. Choose $M$ above the distance of every representative
and shared face center in or next to a pin block to a pin, and use
$u=T_M(X)$. It vanishes on pin blocks. For neighboring ordinary good blocks
$z,w$, their common face center and Lemma~\ref{f:block} give
\[
 |u_z-u_w|^2\le2c_*^{-1}(g_z+g_w).
\]
For a good block next to a pin block, the common face center $f$ satisfies
$|\chi(f)|\le M$, so $|u_z|^2\le|\chi(v_z)-\chi(f)|^2\le g_z/c_*$.
Treating pin blocks as good zero blocks therefore gives
\begin{equation}\label{f:energy-deficits}
 \sum_{e\text{ good}}|\nabla u(e)|^2\le C\sum_Q g_Q.
\end{equation}

Partition vertices into half-open blocks and replace the statistic by
the retained linear functional $\ell(s)=\sum_z w_zs(v_z)$, where
$w_z=\sum_{x\text{ in block }z}q_x$. 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}
The coarse weights satisfy the hypotheses of
Lemma~\ref{f:average-coercivity}, and exact zero mass is preserved.
That lemma and \eqref{f:energy-deficits} show, on a full-configuration
event of probability at least $1/2$, simultaneously for every feasible
$\chi$,
\[
 |\ell(\chi)|\le C_0+C_2\sqrt{\sum_Qg_Q}.
\]
Take fixed $t>C_0$ and $D=((t-C_0)/C_2)^2>0$. Since the logarithmic
midpoint deficit of \eqref{f:product} is $\sum_Qg_Q$, it follows that
\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 $A_t(s)=\sup_{\ell(\chi)=t}\sqrt{p(s+\chi)p(s-\chi)}$.
Compactness of the support restricts the supremum to a compact set;
upper semicontinuity gives measurability. Log-concavity implies
$0\le A_t\le p$. The good event may depend on discarded midpoint
coordinates: the simultaneous assertion \eqref{f:pointwise-overlap}
still holds for its retained vector. Thus
\[
 Z^{-1}\int A_t\le1-\rho_0,\qquad
 \rho_0=(1-e^{-D})/2>0.
\]
If $\ell$ is zero on the free coordinates there is nothing to prove.
Otherwise choose linear coordinates $s=(r,\sigma)$ with $r=\ell(s)$,
absorbing their constant Jacobian into the complementary measure.
For the density $a(r)$ of $\ell(S)$, Pr\'ekopa--Leindler in $\sigma$
\cite[Theorems 3 and 6]{Prekopa} gives
\[
 \sqrt{a(r-t)a(r+t)}\le Z^{-1}\int A_t(r,\sigma)\dd\sigma.
\]
Integrating yields overlap at most $1-\rho_0$. Since $a$ is even and
log-concave, this overlap is at least
$\int\min\{a(r-t),a(r+t)\}\dd r=1-\int_{-t}^t a$.
Hence $\Prob(|\ell(S)|\le t)\ge\rho_0$. Its survival function is
log-concave, again by Prekopa--Leindler, so
\[
 \Prob(|\ell(S)|>jt)\le(1-\rho_0)^j\quad(j\ge1).
\]
This gives a positive uniform exponential moment. Equation
\eqref{f:representative-error} transfers it to the original statistic.
For bounded $R/L$, distance to the anchor, or cancellation of $q$, gives
the required deterministic bound.
\end{proof}

\subsection{Bounded values, sign constraints, and small tilts}
\begin{lemma}[One-sample bounded-data comparison]\label{f:bounded}
There is a coupling of $h_b\sim\mu_b$ and one sample $h_0\sim\mu_0$ 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 $-B,b,+B$. The two extreme chains can be maintained as
exact translates, since their conditional intervals are translates.
The middle chain remains between them: both endpoints of a conditional
interval are increasing functions of the neighboring heights. A stationary
subsequential coupling on the finite compact polytopes gives the asserted
coupling with the uniform marginals. Each tested statistic changes by at
most $B\sum|q_x|$. Condition first on mixed pin values and integrate the
same uniform bound.
\end{proof}

A string here is a prescribed set of positive or negative sites, each
having an oppositely prescribed neighbor. Thus its heights lie in $[-1,1]$.
Additional intervals $0\le h\le\eps$ on positive string sites are allowed.
After all string values are frozen, the remaining conditional law is
uniform on the corresponding Lipschitz polytope. Lemma~\ref{f:bounded}
therefore proves the moment assertion with $B=1$ on each component, provided
the anchor or zero-mass hypotheses are checked for the tested statistic.
A finite collection of components can instead be considered together
with the common pinned set. Cutting along pins and zero-extending the
zero-pinned comparison use the ambient lattice; neither operation requires
a regular shape for the boundary.

The same comparison transfers long-distance slack from zero pins to bounded
data at additive cost $2B$. At distances beyond a sufficiently large fixed
threshold 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 $Y=\sum_x b_xh(x)$, with support
in an $O(R)$ box and $|b_x|\le cR^{-2}$. Assume its box has the specified
connected pin anchor, or that $\sum_xb_x=0$. For sufficiently small fixed
$c>0$, the tilted density $e^Y/\E e^Y$ has uniformly bounded $L^2$ norm.
Hence for every event $A$,
\[
 \Prob_Y(A)\le C\Prob(A)^{1/2}.
\]
All fixed moments of admissible average statistics remain bounded under
this tilt, and any established arbitrary-parameter animal estimate is
inherited, with a harmless constant prefactor and changed parameter.
\end{lemma}
\begin{proof}
Apply \eqref{f:average-mgf} to $Y/c$ to bound $\E e^{2Y}$, and use
Jensen together with the uniform bound on $|\E Y|$ to bound $\E e^Y$
away from zero. Cauchy--Schwarz gives the assertions; for moments apply
it with the square of the desired moment. Multiple bounded tests can be
combined by Holder and a further reduction of $c$.
\end{proof}

\begin{lemma}[Strict filling with sign intervals]\label{f:strict-filling}
In a finite patch, prescribe arbitrary shell values, interior fixed pins
in $[-1/2,1/2]$, and intervals containing zero at any other prescribed
vertices. The intervals may be one-sided or of the form $[0,\eps]$.
Suppose a feasible filling $h$ satisfies
\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 every shell
vertex $s$. If the edge is not fixed--fixed, there is a feasible filling
whose absolute gradient on that edge is at most $1/2$.
\end{lemma}
\begin{proof}
Recall the interval extension criterion. Data $[\alpha_z,\beta_z]$ at
specified vertices admit a 1-Lipschitz selection exactly when
$\alpha_z\le\beta_w+\operatorname{dist}(z,w)$ for every pair. Sufficiency
follows from the lower-cone function
$g(y)=\sup_z\{\alpha_z-\operatorname{dist}(z,y)\}$, which is Lipschitz,
lies above every lower endpoint and below every upper endpoint. Infinite
interval endpoints are omitted from the corresponding bound.

At a movable vertex $x$, the intervals propagated from all non-shell data
contain both $h(x)$ and zero. Indeed feasibility gives the first assertion.
For a nonfixed interval datum the second follows because its interval
contains zero. For a fixed pin $p$, its distance to $x$ is at least one,
whereas $|h(p)|\le1/2$, so its propagated interval contains zero as well.
The datum at $x$ itself, if present, also contains zero.

If both endpoints $u,v$ are movable, put
$c=\operatorname{median}(0,h(u),h(v))$ and prescribe $g(u)=g(v)=c$.
The value $c$ lies between zero and each old endpoint value; thus it
respects every propagated non-shell interval at both endpoints. Moreover
$|c-h(u)|,|c-h(v)|\le|h(u)-h(v)|\le1$.
Equation~\eqref{f:shell-room} preserves all endpoint--shell inequalities.
The new endpoint pair has difference zero, and every pair among the old
data was already consistent. The interval criterion therefore extends
the new prescriptions to a feasible filling.

If one endpoint is a fixed pin, prescribe zero at the other endpoint.
Its original height has absolute value at most $3/2$, so
\eqref{f:shell-room} again preserves all shell inequalities. The propagated
non-shell intervals contain zero, and the resulting edge gradient has
absolute value at most $1/2$. Apply the 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 described above, allowing additional 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, the tested
central edges involving nonfixed vertices have $|\nabla h|\le1-s$.
String sites are included among the nonfixed vertices. Long-distance
slack and this edge slack persist with relaxed constants during a
sufficiently short stationary reflected step. Constants are uniform
over the strings' shape and over the positive interval lengths.
\end{corollary}
\begin{proof}
First freeze all string values, making them pins in $[-1,1]$.
Lemma~\ref{f:bounded} couples the conditional field within coordinatewise
distance one of the corresponding zero-pinned field. The controlled-gradient
witness argument gives arbitrary-parameter long-distance slack, with
the additive comparison loss absorbed by increasing its fixed threshold.
This bound is uniform over the conditioned string values, so integrating
preserves its joint form.

Now condition only on the exterior of each enlarged patch; its interior
strings remain variable. On a shell for which a filling has the preceding
long slack, central endpoints satisfy \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 entire
body into $\{|\nabla h|\le1-s\}$. Therefore the conditional relative volume
of the near-contact region is at most
$1-(1-2s)^m\le2ms$. This remains true for arbitrarily thin sign intervals;
their effect on the body's total volume cancels in the relative-volume
estimate. Degenerate prescribed intervals are treated as fixed coordinates.

For disjoint enlarged patches the conditional laws are independent.
For a specified collection of bad boxes, either at least half have
long-slack failures, or at least half have near contacts with good long
slack. The joint bounds just proved apply to these two alternatives;
the subset count is absorbed by choosing the preliminary parameters
and then $s$ small enough. A finite coloring removes the separation
restriction. Lemma~\ref{f:step} gives the short-step statement. The tilt
transfer, when its statistic meets the anchored or zero-mass hypotheses,
is Lemma~\ref{f:tilts}.
\end{proof}

\begin{remark}[Fixed values after further reveals]\label{f:reveal-obligation}
The strict-filling proof uses the numerical
bound $1/2$ on fixed interior pins and uses intervals containing zero on
movable sign sites. After additional values are frozen, these hypotheses
must be checked again. In particular, arbitrary revealed values in
$[-1,1]$ cannot be substituted for the fixed $[-1/2,1/2]$ pins in
Lemma~\ref{f:strict-filling}. Uniform quenched contact sparsity under
such reveals is not asserted. The adapted comparisons below instead
use the annealed marginal-preservation argument of
Lemma~\ref{c:freezing}, with good marks recomputed after rejoining.
\end{remark}

\subsection{Distributional tightness and conditioning scope}
\begin{corollary}[Negative Sobolev tightness]\label{f:tightness}
Suppose the rescaled free sets lie in a fixed bounded box, their 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 lattice cells of mesh $n^{-1}$ and extended by zero outside,
are tight in $H^{-s}$ on a fixed surrounding torus for every $s>0$.
\end{corollary}
\begin{proof}
Embed the lattice cells in a dyadic coordinate square with $2^J$ cells
on each side, where $2^J$ is comparable to $n$. Its physical size remains
bounded above and below. Expand the piecewise constant field in the
orthonormal two-dimensional Haar basis; the expansion terminates at the
cell scale. A Haar coefficient on a cell of physical side $r=R/n$ is
$r$ times a statistic satisfying \eqref{f:weights} and having exactly
zero total mass. For zero pins the tests include all ambient lattice
vertices, including the zero extension, so this cancellation also holds
across the boundary. Bounded data differ from the same zero-pinned sample
by a uniformly bounded field. Proposition~\ref{f:averages} consequently
gives a second moment at most $Cr^2$ for each centered coefficient.
The coarse scaling coefficient is bounded by the anchored assertion.

Write $F_j$ for the sum of wavelets at level $j$. There are $O(r_j^{-2})$
coefficients there, so $\E\|F_j\|_2^2\le C$. Each parent-cell integral
of $F_j$ vanishes. For $00$. When interpolation is restricted to a polygon,
the boundary-cell contribution must also have vanishing area; this holds
for the stated smooth-domain approximation and bounded heights adjacent
to its fixed boundary.

Finally, all conditioning statements above specify the resulting kernel.
If a random exploration followed by freezing preserves a marginal law,
the corresponding annealed estimates are inherited by that marginal.
This does not give the same estimates uniformly for each realized revealed
set and every realized revealed value. Any later use of a quenched bound
must separately verify the fixed-value, geometric, and conditional-law
hypotheses of the result being invoked.
==== END Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//sections/foundations.tex ====

==== BEGIN Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//sections/geometry.tex ====
\section{Harmonic sampling and stopped-interface geometry}
\label{sec:geometry}

We use lattice units in this section. A strand observation specifies the
orientation of finitely many pieces of zero interface, hence only the signs
of adjacent heights. Its conditional law is uniform on the corresponding
convex height polytope. All assertions below concern admissible observations,
that is, observations of positive probability. Constants are uniform in
the number of lattice edges in an observed strand. The number of distinct
macroscopic strands in a comparison is bounded. The exterior heights have
absolute value at most \(1/2\).

The geometric constructions and the random-walk estimates used below are
those of \cite[Sections 2.3 and 3.3--3.7]{SSdiscrete}. We specify their
hypotheses and prove the replacements that concern the law of the field.
In particular, a result for Gaussian interfaces is not used as a theorem
about the present field.

\subsection{Local insertion, signs, and crossings}

\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 \(M=M(L)\). 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
\(\pm b\), where \(00\),
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 some
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 \(C d\) 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. Thus, 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.

Write $A$ for the full original string-height vector. Couple its positive
coordinates with independent fair pins $B_x\in\{0,d\}$ so that
$B_x\le A_x$ on the positive portion, and extend this coupling to the
remaining coordinates of $A$ by disintegration. Conditional on $(A,B)$,
sample the original Gibbs interior given $A$ and a comparison Gibbs
interior pinned to $B$ on the positive portion and to $d/2$ on the
remaining comparison boundary. Then replace this conditional product
by its stationary synchronous joining. Lemma~\ref{c:freezing} preserves
the original field marginal and the comparison mixture. The latter has
the required estimates uniformly in its pins, which lie in
$[0,d]\subset[-1/2,1/2]$. Its law is invariant under
$h\mapsto d-h$, because this map sends the kernel with pins $B$ to
the kernel with pins $d-B$, and the fair product law is unchanged.
Its mean is therefore $d/2$. The filled lower tail of the comparison
has zero fixed input on the positive portion. Theorem~\ref{c:comparison}
and Proposition~\ref{t:varying-trace} give zero trace for every limit
of its mean there; no harmonic equation for the tail is used.
Jensen's inequality bounds the negative part of the limiting mean
difference by this nonnegative zero-trace limit. The two field means are
bounded, harmonic away from the observed sets, and their remote
contribution is bounded by a constant times the harmonic probability of
reaching an endpoint or an incompatible side. This probability tends to
zero along Brownian approaches to the pure portion, by Beurling.
The lower trace is therefore at least \(d/2\). All bounds used are
uniform for each deterministic admissible conditioning sequence, which
gives the claimed sequential uniformity. Reflect the field for the
negative statement.
\end{proof}

\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 \(h\mapsto1-h\), 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 deletion of precisely these
crossing constraints.
\end{proof}

\begin{lemma}[High crossings]\label{g:high-crossing}
In a compact corridor of fixed shape and size \(n\), with an observed
component of comparable size and distance, the probability of a path
of heights at least \(K\) crossing a fixed positive fraction of that
corridor tends to zero as \(K\to\infty\), after \(n\to\infty\).
The same is true for the difference of two independent samples under
possibly different bounded exterior or strand conditionings.
In both statements the corridor has an enlarged neighborhood of
width \(cn\), for a fixed \(c>0\), containing no original fixed pins
or sign walls. In the two-field statement this neighborhood lies
in both domains with the same unconditioned lattice graph, and
each original law has the stated macroscopic anchor. The two
conditionings may differ outside this neighborhood.
\end{lemma}

\begin{proof}
We give the two-field proof. Explore every cluster of
\(h_1-h_2\geq K\) connected to a deterministic transverse seed.
Every queried vertex away from the seed has difference at least \(K-2\):
positive queries satisfy the threshold, and failed queries are adjacent
to a positive query. Freeze both samples on the common queried set,
then use the stationary synchronous joining of their conditional laws.
The exploration decisions used only recorded values; hence this
replacement preserves each annealed marginal. All statements below are
annealed over this transcript, not asserted for every possible
conditional pin value.
Use a middle testing subcorridor a fixed positive distance from
the transverse seed. The free-buffer hypothesis removes all
original pins and sign walls from its larger comparison window;
the newly queried common pins there satisfy the displayed
threshold inequality. Thus the one-sided comparison has no
uncontrolled original boundary in that window.

Away from the seed, set
\[
 u_K=\left(1-\frac{h_1-h_2}{K-2}\right)_+ .
\]
It has zero data on the revealed set. The local truncated comparison
estimate in Theorem~\ref{c:comparison}, applied after division by
\(K-2\), gives compact filled extensions with uniformly bounded local
Sobolev energy. Here the cutoff energy grows at most quadratically with
the truncation level: after division it is bounded by the squared cutoff
norm and the macroscopic second moments of \(h_1/(K-2)\) and
\(h_2/(K-2)\). Proposition~\ref{f:averages} controls these moments uniformly.
The same bounds hold after conditioning on an event of probability
bounded below, with its reciprocal as a constant.

Suppose a crossing had probability bounded below along \(K\to\infty\)
and \(n\to\infty\). On that event the revealed set contains a connected
continuum of fixed positive diameter away from the seed. Pass to a
subsequence jointly with this continuum and the filled extensions.
Macroscopic averaged field tightness gives
\((h_1-h_2)/(K-2)\to0\) as a distribution in this iterated limit.
Since \(u_K\geq1-(h_1-h_2)/(K-2)\), its limit is at least one almost
everywhere. The varying-set anchor argument of
Proposition~\ref{t:varying-trace} gives zero trace for that same limit.
Here one uses its samplewise inequalities, without a Poisson equation
for the truncated tail. Choose a probability measure on the crossing
continuum by coordinate projection, with ball masses bounded by $Cr$
in physical units. Move it along the revealed set past each hole to a
retained zero vertex. The maximum hole diameter is polylogarithmic
outside the negligible exceptional event, and the crossing component
cannot be contained in such a hole. Lemma~\ref{t:frostman} bounds the
integral of a radius-$r$ average of the filled tail by
$C(r^\beta+o(1))\|U_n\|_{W^{1,p}}$, for some $\beta>0$.
These inequalities hold for the random measure before expectation.
Joint compactness and strong local $L^p$ convergence permit the mesh
limit at fixed $r$; the bounded conditional Sobolev norm permits
$r\downarrow0$. This proves the zero trace. A \(W^{1,p}\) function which is
at least one almost everywhere cannot have zero trace on a continuum
carrying a fixed positive-diameter Frostman test with exponent
greater than \(2-p\). This is a contradiction.

For one field retain the entire original field on the deterministic
testing window. Given its transcript, compare its unvisited Gibbs
kernel with a symmetric field zero-pinned on the revealed set and
original exterior, and extend this comparison field by zero on its
fixed coordinates. Keep the original field, not its restriction to
a random unvisited component, in the full-window difference.
For each recorded set, deterministic ambient tests of the comparison
field have the bounds of Proposition~\ref{f:averages}; the revealed
crossing component provides a macroscopic anchor on the crossing
event. The original full field keeps its annealed bounds by
disintegration. Queried nonseed differences are at least \(K-1\),
and the same lower tail with \(K-1\) has zero input there. The
free buffer excludes all original walls from this comparison window.
The preceding argument therefore applies without a random mask
on the original field.
\end{proof}

\subsection{Barriers and exact strand comparison}

For a union \(K\) of observed strands, let \(D(K)\) be the domain obtained
by deleting closed lattice triangles meeting \(K\). Write
\[
 d_U(A,B)=\inf\{\operatorname{diam}\alpha:
       \alpha\subset\overline U\text{ joins }A\text{ to }B\}.
\]
The distinction between this internal diameter distance and Euclidean
distance is essential below.

A \((\theta,R)\)-barrier is a path \(\Upsilon\) of diameter in
\((\theta R,R]\), with an observed or exterior boundary component of
diameter at least \(\theta R\) within distance \(\theta^{-1}R\).
Its interior is in \(D(K)\). At each endpoint \(z\) on the boundary,
the first encountered circular crosscut at radius \(\theta R\) bounds
a pocket \(A_z\) with one simple boundary arc; the path meets every
smaller centered circle once inside this pocket. If a point of the
barrier is within \(\theta R/5\) of the observed boundary, it belongs
to one of these pockets and lies within \(\theta R/2\) of its endpoint.
These are the four geometric conditions of
\cite[Section 3.4]{SSdiscrete}. A signed barrier is compatible if the
boundary arc of each endpoint pocket has the indicated sign.
Oppositely signed barriers must have internal distance at least
\(2\theta R\). At bounded lattice scales no hexagon may meet both
prescribed signs.

\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
\(C m\eta^{-2}\) cells. Circuits and crossings in those cells form a
fence around the barrier; fill bounded holes in the thickening before
taking its 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. We will then force the interior fence using
Lemma~\ref{g:high-crossing} and Proposition~\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 the 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.

Radially select points on the testing arc, move them to centers of
adjacent core cells, and smear them in balls of radius \(\eta R/400\)
inside those cells. For the resulting probability test \(\mu\),
\[
 \mu(B(x,tR))\leq C(t+\eta)/\theta.
\]
For fixed \(\eta,\theta\), all these are smooth tests with uniformly
bounded rescaled norms. The number of coarse cells is bounded, so
subsequential uniformity applies to arbitrary varying old strands.
Beurling and the access separation show that the original mean of
the test is at least \(c+o(1)\) by Lemma~\ref{g:sign-trace}; for
an exterior attachment replace \(c\) by \(c\wedge\lambda_*\).
Its centered variance tends to zero as \(\eta/\theta\to0\) by
Corollary~\ref{t:local-boundary-variance}. Indeed the smearing improves
the displayed ball-mass bound to \(\mu(B(x,tR))\leq Ct/\theta\)
at every scale: for \(t<\eta\), the density bound is
\(C/(\theta\eta R^2)\), giving \(Ct^2/(\theta\eta)\leq Ct/\theta\).
Thus the tests are uniformly Frostman and, for fixed \(\eta\), form
a compact smooth family. The stated local-cut condition supplies the
corollary's thickness hypothesis; no bound on the ambient diameter
divided by \(R\) is imposed.

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.

Here is the comparison on this event, denoted by $E$. Include all
original string heights in the exact-value transcript; they may be
recorded before the adapted queries. Given the complete transcript,
restrict to the core-facing component cut off by its negative frontier
$F$. The original interior has its exact Gibbs kernel with the
recorded boundary values. As comparison kernel take the uniform law
on this same component with all fixed boundary values replaced by
zero, and denote its field by $h^0$. It is symmetric under height
reflection, so its conditional mean is zero. The zero-pinned laws
have sparse contacts and average bounds uniformly over these component
shapes, using $F$ as a connected anchor. The original estimates remain
annealed estimates under its original strand-conditioned marginal.
Disintegrating and rejoining the two kernels as in
Lemma~\ref{c:freezing} preserves these marginals. It does not require
microscopic slack uniformly for every recorded nonzero pin vector.

For the analytic estimate, retain $h$ on the entire deterministic
testing window, including its coordinates outside the selected
component $C$. Extend only $h^0$ by zero at every vertex outside $C$,
so it is a full ambient zero-pinned field for each recorded shape.
Rejoining may be performed with the full original conditional Gibbs
kernel: its other components retain their conditional baths.
Consequently the full $h$ still has its original annealed marginal.
Put $z=h-h^0$ on this whole window; no component indicator multiplies
$h$. Deterministic ambient zero-mass tests of $h^0$ satisfy
Proposition~\ref{f:averages} uniformly for every recorded shape,
and the coarse coefficient is anchored by $F$. The full original
$h$ has its original anchored and zero-mass bounds. The localized
dyadic argument in \eqref{c:negative} therefore gives
\[
 \E[\|\chi_0 z\|_{H^{-\alpha}}^2\mid E]
 \le C_{\alpha,p_*},\qquad \alpha>0,
\]
when $\Prob(E)\ge p_*$, with the deterministic cutoff $\chi_0$
supported in the fixed larger testing window. Only the original
marginal costs the reciprocal event probability; the zero-pinned
bound is uniform conditional on the recorded shape.

On $F$ the original values are nonpositive. The upper comparison tail
$u=(h-h^0)_+$ consequently has zero fixed input there. Extend it by
zero behind $F$ and apply the one-sided part of
Theorem~\ref{c:comparison}. Use the core-facing components of windows
of a fixed positive fraction of $\theta R$ around the central testing
arc. The lateral cutoffs and any uncontrolled seed cannot be reached
within the corresponding larger window component, by the access
separation above. Thus the cutoff is localized in that component;
all edges to the removed positive cluster pass through $F$.
The nearby original boundary anchors the original marginal, and
$F$ anchors the comparison marginal. The zero-extended tail is
uniformly Lipschitz across the frontier. If its hole extension has
a positive copied value $U(x)=u(y)$, the retained rim site $y$ lies
in $C$ and $u(y)=z(y)>0$. Full-window Lipschitzness of $z$ gives
\[
 U(x)^2\le z(x)U(x)+C(1+d_x)U(x).
\]
At $U(x)=0$ this is automatic. The inequality remains valid even
if a hole copies a positive value into the removed side; it does
not require an exact-zero filled extension on every internal
component. Pairing it with a deterministic cutoff uses the full
$z$ in Sobolev duality, as in \eqref{c:one-sided-pointwise}--
\eqref{c:one-sided-absorb}, rather than a randomly masked original
field. Retained frontier vertices still have value zero.
Thus the expected energy and
squared $W^{1,p}$ norm of the filled tail $U$, for $3/20$ costs only a factor $p_*^{-1}$ in these annealed bounds.

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 \(v\),
\[
 |\E[X\mid E]-\E X|\leq\sqrt{v/\Prob(E)}.
\]
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
\(R_0(\theta,m)\).

For \(R\leq R_0\), the barrier thickening and the required finite sign
fences lie in bounded patches at bounded distance from the anchoring
components. Lemma~\ref{g:insertion} forces them with probability
bounded below. Taking the smaller of the two bounds proves the
all-scale statement.

Here is the local-thickness check for the applications below. External
arms and adapted continuations are attached to the original exterior.
An internal record is a connected incident strand through the sampling
center, of diameter comparable to its record radius when its quality
is positive. An endpoint testing square can be chosen small enough
not to contain that center. If an internal record of smaller diameter
meets such a square, its connection to the center crosses the square's
rim. Otherwise its diameter is already comparable to the square's
scale. In the extra-excursion construction the endpoint squares lie
near radius \(\rho\), have size at most \(c\rho\), and are disjoint
from the center even when the inner record radius is much smaller
than \(\rho\). Thus no global lower bound on all observed strand
diameters is being presumed there. In the strand-swap application
the comparison annulus is free of all the inner observations.
\end{proof}

\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
\[
 C^{-1}\Prob_2(I)\leq\Prob_1(I)\leq C\Prob_2(I),
\]
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}
Take two independent samples from these base laws, conditioning the
first additionally on \(I\). The only observations inside the outer
disk are now this additional event in the first replica, and their
support lies inside the inner disk. Hence the annulus remains free
in both replicas. Each corridor used there has an anchor at comparable
distance under each law; an inner strand from \(I\) is also an
allowed anchor for the first replica.
Choose the two circuit supports in a closed middle subannulus
and cover them by finitely many corridors whose fixed
enlargements remain in that annulus. They satisfy the
free-buffer hypothesis of Lemma~\ref{g:high-crossing} in
both replicas. Lemmas~\ref{g:high-crossing}
and \ref{g:shifts} therefore give a
uniformly positive probability of two nested circuits in the free
annulus on which \(h_1-h_2\) has opposite strict signs.
We use the cluster-swapping construction of Sheffield
\cite[Chapter~8]{SheffieldSurfaces}, in the hard-constraint form
described below; see also \cite[Section~2]{CAP}.
For each edge declare it a dependency edge when swapping the
replicas at just one endpoint violates an edge constraint in one
of the replicas. This graph depends only on the unordered height
pairs. If the order of the replicas reverses across an edge, either
partial swap is feasible: each mixed endpoint difference lies
between differences of the original feasible samples. Hence a
dependency path cannot change the sign of \(h_1-h_2\).
No dependency component meeting the inner disk reaches outside
the two opposite-order circuits.

Swap both replicas at every vertex in a dependency component
meeting the inner disk. Every swapped component stays inside the
outer disk. The base laws impose no vertex observations there;
all their original observations therefore remain at untouched
vertices. A graph edge across a swapped-component boundary is
not a dependency edge, so the partial swap on that edge is feasible.
This proves preservation of both base polytopes. All vertices
tested by \(I\) are swapped, which transfers \(I\) from the first
replica to the second; the first replica is not required to retain
this additional event. The operation is an involution because the dependency
graph and selected components are invariant under swaps.
On each component-selection region it is a coordinate permutation,
so it preserves Lebesgue measure. If \(V_i\) is the volume of the
polytope with exterior conditioning \(i\), the uniform positive
fraction of swappable pairs gives
\[
 c\,V_1(I)V_2\leq V_1V_2(I).
\]
Reverse the experiments for the other inequality.
\end{proof}

\subsection{The harmonic-landing records}

Use rounded disks \(\mathfrak B_R\) made of dual hexagons.
Fix the origin and an incident dual edge \(e_\sigma^*\). Let \(S\)
be independent simple random walk from \(v\), let \(\tau_0\) be
its first visit to zero, and reverse its portion up to \(\tau_0\)
to obtain \(\breve S\). Let \(\tau\) be the first visit to a
vertex adjacent to the completed chord or to exterior boundary.
Define
\[
 Z^\sigma=\{S_\tau=0,\ e_\sigma^*\subset\gamma\}.
\]
Let \(\operatorname{ext}_R\gamma\) consist of the two initial
pieces grown from the marked endpoints until first entering
\(\mathfrak B_R\); it is the whole chord if the disk is missed.
Let \(\beta_R^\sigma\) be the part in \(\mathfrak B_R\) of the
zero interface through \(e_\sigma^*\), connected to that edge,
and put it equal to the empty path if there is no such interface.
Let \(\operatorname{ext}_R S\) end at the first entrance to
\(\mathfrak B_R\), and let \(\operatorname{int}_R\breve S\)
end at its first exit. Set
\[
 \Phi_R=(D,\partial_+,h_\partial,v,
              \operatorname{ext}_R\gamma,\operatorname{ext}_R S),
 \qquad
 \Theta_R=(\beta_R^\sigma,\operatorname{int}_R\breve S).
\]
The geometric path in \(\Theta_R\) is initially unoriented.
Conditioning on its orientation means specifying the adjacent signs.
Every subsequent field-law application below conditions on these
exact signs, then averages over the orientations.

Let \(x,y\) be the two external strand tips and \(q\) the external
walk tip at radius \(R\). When there has been no forbidden contact,
define
\[
 Q(\Phi_R)=\frac1{100}\wedge
 \frac{
 d(x,\operatorname{ext}_R S)\wedge
 d(y,\operatorname{ext}_R S)\wedge
 d(q,\operatorname{ext}_R\gamma)\wedge |x-y|}{R}.
\]
Put \(Q=0\) if the disk is missed by the chord or the revealed walk
hits a revealed strand. Define \(Q(\Theta_R)\) by the same formula
using the two inner strand tips and reversed-walk tip; put it equal
to zero for a closed inner interface or a forbidden walk contact.

Figure~\ref{fig:harmonic-records} depicts 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) {observed walk pieces};
\end{tikzpicture}
\caption{The two records leave an annulus available for separation
and hookup. The inner walk arrow indicates reversal from the harmonic
landing. The shaded annulus is unexposed; no interface connection there
has yet been required.}
\label{fig:harmonic-records}
\end{figure}

\begin{lemma}[Hookup probability]\label{g:hookup}
For \(9R/81\) is fixed:
\[
 \Prob(Z^\sigma\mid\Phi_{R'},\Theta_R)\leq C/\log R.
 \tag{G1}
\]
For every \(a>0\) there is \(c(a)>0\) such that the same probability
is at least \(c(a)/\log R\) when
\(Q(\Phi_{R'})\wedge Q(\Theta_R)\geq a\) and \(R\geq R_0(a)\).
\end{lemma}

\begin{proof}
The field part of \(\Phi_{R'}\) constrains only sites adjacent to
the external arms, hence outside radius \(R'-O(1)\).
The sites specifying \(\beta_R^\sigma\) lie inside radius \(R+O(1)\).
Choose comparison disks between these two radii, with
outer radius \((R+R')/2\) and inner radius \(R+(R'-R)/4\).
For sufficiently large \(R\), they have a free intervening annulus
of fixed relative width. The original base observations are all
outside the outer disk, and the external arms give the required
anchors. Thus Lemma~\ref{g:strand-comparison} applies to the two
orientations of the inner strand, comparing the exterior data with
their global sign reversal. Together with sign symmetry it gives a uniformly
positive probability that the two observed unoriented configurations
have compatible signs on the side containing the walk. In that case
join corresponding interface tips and the two walk tips by three
disjoint corridors, of width \(c(a)R\). Four signed barriers alongside
the two interface corridors have the required pure endpoint pockets.
Proposition~\ref{g:barriers} forces the interface joins within these
corridors with uniformly positive probability. Planarity forces the
outer arms to merge with the corresponding inner arms; there is no
other exit through the fences.

The remaining walk bridge must reach the joining vertex of the reversed
inner piece. Its conditioning to reach this vertex before zero has
probability bounded below. Harnack gives a positive probability to
follow the free third corridor to a fixed relative neighborhood of
that vertex; the two-dimensional point-hitting estimate gives
probability comparable to \(1/\log R\) to hit it there. The loop portion
between its first and last visits stays in that corridor with probability
bounded below, by the last-exit decomposition. This proves the lower
bound. Conditional on any completed field configuration, hitting the
same vertex before a nearby interface continuum has probability at
most \(C/\log R\); this proves the upper bound. These are precisely
the walk hypotheses of \cite[Lemma 3.14]{SSdiscrete}: radii in the
specified compact ratio range, observer outside the larger disk,
and two prescribed first/last walk pieces. All field estimates used
in its corridor construction have been supplied above.

We record the two augmented upper bounds used below. First, the upper
bound still holds after the whole field, hence \(\gamma\), is fixed;
the middle walk bridge remains unexposed. Second, put
\(H=(\Phi_{3r},\Theta_{2r})\), and expose that middle walk only until
its first exit from a circle of radius \(\rho/3>10r\), if this occurs
before its prescribed terminal piece. Call this stopped prefix
\(S_{**}\). On this escape event,
\[
 \Prob(Z^\sigma\mid H,S_{**},\hbox{escape})
       \leq C/\log r.                              \tag{GK}
\]
For verification, let \(\hat q\) be the joining vertex of the fixed
reversed inner piece. The first-hit/last-exit decomposition leaves a
walk from the current exit vertex to \(\hat q\), conditioned to hit
\(\hat q\) before zero, followed by its last-exit loop. Harnack bounds
the conditioning probability below by a fixed constant, since the
exit vertex is outside \(B_{10r}\) while
\(|\hat q|=2r+O(1)\). To realize \(Z^\sigma\), this first part
must hit \(\hat q\) before the connected inner interface. The same
point-versus-continuum estimate as above is at most \(C/\log r\);
the remaining loop can only reduce the probability. This argument
holds for every completed field consistent with the records, and
averaging gives (GK). It does not expose the later successful bridge.
\end{proof}

\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 \(R\),
\[
 \Prob\bigl(
 Q(\Phi_{3R})\wedge Q(\Theta_{2R})>c(\eps)
 \mid \Phi_{4R},\Theta_R,Z^\sigma\bigr)>1-\eps
 \tag{G2}
\]
uniformly in all admissible records for which the conditioning has
positive probability. The observer and original boundary are outside
\(\mathfrak B_{6R}\).
\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
\[
 \Prob(Q_{j+1}\geq(2Q_j)\wedge 1/100\mid\Phi_{r_j},\Theta_R)
       \geq c_0,
 \tag{G3}
\]
\[
 \Prob(Q_{j+2}=0\mid\Phi_{r_j},\Theta_R,Q_{j+1}<2Q_j)
       \geq c_1. \tag{G4}
\]
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), put \(\mathcal F_j=\sigma(\Phi_{r_j},\Theta_R)\),
\(q=Q_j>0\), \(r'=r_{j+1}\), and
\(F=\{Q_{j+1}<2q\}\). Reveal the two external arms to radius \(r'\),
but do not yet reveal the whole intervening walk. Write \(x,y\) for
their tips, \(w\) for the eventual walk tip at radius \(r'\), and
\(S^\dagger\) for the walk between its first radius-\(r_j\) and
radius-\(r'\) entrances. The following five events cover \(F\):
\[
\begin{split}
 M_0&=\{Q_{j+1}=0\},\\
 M_1&=\{d(S^\dagger,\{x,y\})<(2q r')\wedge|x-y|\},\\
 M_2&=F\cap\{r'Q_{j+1}=d(w,\operatorname{ext}_{r'}\gamma)\},\\
 M_3&=F\cap\{02q r'\), with the rounded-disk lattice margin
absorbed in these constants. Thus \(M_i\subset F\) and their union
is \(F\), including ties in the quality minimum.

Let \(L=\{Q_{j+2}=0\}\). The field-free stopped-walk estimate in
\cite[proof of (3.38), case \(i=1\)]{SSdiscrete} gives
\[
 \Prob(L\cap M_1\mid\mathcal F_j)
       \geq c\Prob(M_1\mid\mathcal F_j).             \tag{G4b}
\]
Here are the conditioning and geometric hypotheses for that estimate.
After fixing \(\operatorname{ext}_{r'}\gamma\), stop \(S^\dagger\)
at its first visit within
\(a=(2q r')\wedge|x-y|\) 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
\(a\leq2q r'\leq r'/50\). 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 (G4b). 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.

For \(M_2\) and \(M_4\) the same hitting bound starts at the current
walk tip and does apply after that radius-\(r'\) record is fixed.
On \(M_3\), a compatible barrier around the midpoint of \([x,y]\)
at radius \(2|x-y|\) forces the two strands to join before the next
inner circle, since its radial gap is \(10|x-y|\). This kills hookup
without meeting the fixed inner strand. Proposition~\ref{g:barriers}
applies, and bounded scales use Lemma~\ref{g:insertion}. The case
\(M_0\) is immediate. Hence (G4b) holds for every \(M_i\), with a
common \(c>0\). The events need not be disjoint: summing yields
\[
 5\Prob(L\cap F\mid\mathcal F_j)
 \geq\sum_{i=0}^4\Prob(L\cap M_i\mid\mathcal F_j)
 \geq c\Prob(F\mid\mathcal F_j).
\]
Division by \(\Prob(F\mid\mathcal F_j)>0\) proves (G4).

Iterating (G3), followed by Lemma~\ref{g:hookup}, shows that a quality
\(q>0\) with a fixed annular buffer remaining has
\[
 \Prob(Z^\sigma\mid\hbox{current records})
       \geq c q^\alpha/\log R                         \tag{G5}
\]
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 the upper bound in (G1), and divide by (G5) at the first
visit to the band. This yields
\[
 \Prob(M_n>2m\mid\Phi_{4R},\Theta_R,Z^\sigma)
       \leq C c_0^{-n}(1-c_1)^m.                     \tag{G6}
\]
For a sufficiently large fixed \(A\), the bounds in (G6) 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
\[
 C A\sum_{n\ge N}n2^{-n}<1/2.
\]
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 (G6),
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 (G1).
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 (G5) 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 (G2).
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}

\begin{proposition}[Comparison of sampled configurations]
\label{g:sample-comparison}
There are \(C<\infty\) and \(R_0<\infty\) such that the following holds
for \(R\geq R_0\). Take two domains
containing \(\mathfrak B_{6R}\), 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,
\[
 C^{-1}\Prob'(H\mid\Phi'_{4R}=\phi',Z'^\sigma)
 \leq \Prob(H\mid\Phi_{4R}=\phi,Z^\sigma)
 \leq C\Prob'(H\mid\Phi'_{4R}=\phi',Z'^\sigma).
 \tag{G7}
\]
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 (G7) 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}

\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 \(\mathfrak B_r\) to
\(\partial\mathfrak B_R\), or that the walk exits
\(\mathfrak B_R\) after first entering \(\mathfrak B_r\) and before
its landing at zero. For every \(\eps>0\) there are
\(A(\eps)<\infty\) and \(r_0(\eps)<\infty\)
such that
\[
 \Prob(\mathcal J(r,R)\mid\Phi_R,\Theta_r,Z^\sigma)<\eps
 \tag{G8}
\]
when \(r\geq r_0(\eps)\), \(R>A(\eps)r\), the observer and boundary are outside
\(\mathfrak B_{4R}\), 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}
Separation in inner and outer buffers makes all four records at
scales \(2r,3r,R/2,3R/4\) have quality at least \(c(\eps)\), except
with probability \(O(\eps)\). Put \(\rho\asymp\sqrt{rR}\).
Let \(H=(\Phi_{3r},\Theta_{2r})\). The walk portion \(S_*\) from its
first radius-\(3r\) entrance to the last visit, before zero, to the
endpoint of its radius-\(r\) reversed piece is unlikely to leave
\(\mathfrak B_{\rho/3}\) without hitting the external arms. The
ordinary Beurling estimate and first-hit/last-exit decomposition
bound this probability, conditional on \(H\), by
\(b(R/r)\to0\). If it escapes, reveal only its prefix \(S_{**}\)
up to first exit. The augmented bound (GK), not a bound conditioned
on the completed walk, gives
\[
 \Prob(S_*\not\subset\mathfrak B_{\rho/3},Z^\sigma\mid H)
       \leq C b(R/r)/\log r.
\]
Divide by the lower bound \(c(\eps)/\log r\) on separated records
and average over them. This proves confinement under the landing law.

For the outer construction, use only \(\operatorname{ext}_{3r}S\).
Let \(S'\) be its terminal path from its last exit of
\(\mathfrak B_{R/3}\) to its endpoint at radius \(3r\).
Choose measurably from \(S'\) an arc on the circle of radius \(\rho\)
whose union with \(S'\) separates the inner and outer circles.
Intersect this open circular arc with \(D(\operatorname{ext}_{3r}\gamma)\).
Its two components next to \(S'\) have other endpoints on the boundary
of this deleted-triangle domain, not on the interface itself;
call them \(\alpha_1,\alpha_2\). Thus an endpoint attachment does
not already count as an interface intersection. They are
\(\Phi_{3r}\)-measurable. In the component on either side, place
nested barriers separating such a piece from the inner circle.
The deterministic construction is the clearance construction in
\cite[proof of (3.48)]{SSdiscrete}: take a separating circular arc;
if its middle has adequate boundary clearance, attach shortest
endpoint segments, and otherwise repeat at the smaller distance
between its two facing pure boundary pieces. The procedure
terminates because the two boundary pieces are disjoint.
Each output has a fixed relative barrier parameter, although its
lattice diameter may be small. Barriers selected in distinct
dyadic annuli remain mutually separated.

After exposing the two interface arms to one barrier, the next
barrier satisfies Proposition~\ref{g:barriers}, including its
bounded-scale case. Thus crossing \(N\) barriers has probability
at most \((1-c)^N\), conditional on \(H\) but not on the landing.
For \(F=\{\gamma\cap(\alpha_1\cup\alpha_2)\ne\varnothing\}\),
fix the completed field while retaining the unexposed walk bridge.
The first augmented upper bound following (G1) gives
\[
 \Prob(F,Z^\sigma\mid H)
 =\E[1_F\Prob(Z^\sigma\mid H,\gamma)\mid H]
 \leq \frac{C}{\log r}\Prob(F\mid H).
\]
The same lower normalization gives \(C(\eps)(1-c)^N\) under the
landing law, where \(N\to\infty\) with \(R/r\). Repeat the whole
construction from the inner reversed record \(\Theta_{R/3}\),
without also revealing the forward intermediate bridge.

Outside these exceptional events, confinement identifies the terminal
forward path \(S'\) with the corresponding portion of the reversed
inner record. The same deterministic circular-arc selection then
makes the two crosscuts from the separate descriptions coincide. Their
boundary endpoints therefore attach to lattice edges incident to both
the inner incident strand and the two external arms. These strand
attachments are on opposite sides of the incident
edge: the other possibility contradicts the winding number of
the circular separator completed along the common walk path.
This is the deterministic Jordan-curve conclusion in
\cite[final paragraphs of Lemma 3.17]{SSdiscrete}; its hypotheses
are the two coinciding crosscuts, a simple interface, and the
common walk avoiding it before zero, all obtained above.
Thus each external arm has merged with a distinct inner arm,
and there can be no additional annular interface crossing.

The total error is \(C\eps+o_{R/r}(1)+C(\eps)(1-c)^N\).
Choose the buffer error first and then \(R/r\) large.
Average over \(\Theta_r\) and the six possible incident edges
for the external-only statement.
\end{proof}

\subsection{Boundary clearance, survival, and nonreturn}

The survival and nonreturn arguments adapt
\cite[Lemma~3.26, especially equation (3.62)]{SSdiscrete}, using the
barrier and sampled-excursion estimates proved above.
For the boundary-attaching arguments in this subsection, fix
\(\lambda_*>0\) and assume the exterior Dobrushin data satisfy
\[
 h_\partial\in[\lambda_*,1/2]\text{ on }\partial_+,
 \qquad
 h_\partial\in[-1/2,-\lambda_*]\text{ on }\partial_-.
 \tag{GB}
\]
All constants may depend on \(\lambda_*\). This includes the reference
data \(\pm1/2\), and the candidate matched data \(\pm l\) with
\(l\geq c\) from Lemma~\ref{g:sign-trace}. No uniform assertion as
\(l\downarrow0\) is required. The source's boundary condition
\cite[condition \((\partial)\), Section 3.7]{SSdiscrete} is more
permissive, allowing small wrong-sign heights; we do not need or invoke
that additional Gaussian statement.

\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 \((\mathrm{GB})\), uniformly over domains, observers, and the
indicated exterior data,
\[
 \lim_{\eta\downarrow0}\limsup_{d\to\infty}
 \Prob\bigl(\operatorname{dist}(M,\partial D)<\eta d\bigr)=0.
 \tag{GB1}
\]
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}
We adapt the full-walk argument of \cite[Lemma~3.23, especially
equations (3.53) and (3.55)]{SSdiscrete}. We first establish pure-arc
repulsion. For \(y\in\partial_+\), choose
any deterministic lower bound \(b\leq d_D(y,\partial_-)\).
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
\[
 \Prob\{d_D(y,\gamma)0\) there are \(p>0\), \(d_0<\infty\) such that
\[
 \Prob\bigl(\Prob(Y=X_T\mid K,S)d_0),                                      \tag{G9}
\]
uniformly over the domains, observers, and stopping times under
\((\mathrm{GB})\).
\end{lemma}

\begin{proof}
Conditional on an exact stopped prefix \(K\), the field has the
ordinary strand-conditioned law: whether the rule stopped at this
prefix is already decided by that prefix. The independent walk
adds no field conditioning. Put \(U=D(K)\).
For small fixed \(\delta>0\), except on an event tending uniformly
to zero with \(\delta\), the following four conditions hold:
\[
\begin{split}
 &d_U(X_T,\hbox{opposite-sign side})\geq2\delta d,\\
 &\operatorname{diam} S[0,\tau_T]<\delta^{-1}d,\\
 &\operatorname{diam} S[t_*,\tau_T]<\delta d/4,
 \quad t_*=\inf\{t:d(S_t,\partial U)\leq\delta^2d\},\\
 &d(v,K)\geq\delta^{1/2}d.
\end{split}                                                   \tag{G10}
\]
The middle two statements are the planar walk hitting estimate
uniform in \(U\). The last statement follows from
Lemma~\ref{g:observer-landing} and \(K\subset\gamma\).
For the observer-separated stopping times needed in the driving
argument, the last condition can instead be imposed deterministically.

For the first condition delete all crosscuts of diameter at most
\(4\delta d\) joining the two sign sides. The new boundary of the
component containing \(v\) lies in two balls of radius \(8\delta d\).
Indeed each deleted crosscut separates \(v\) from one of the two
sign-change points; any two crosscuts for the same point which
touch the retained component must intersect. All such crosscuts
are consequently within twice their diameter of a fixed one.
Apply the walk hitting estimate to those two balls, using the last
condition. This also explains why arbitrary narrowing of \(U\)
does not invalidate (G10).

Fix records satisfying (G10), and suppose the landing is positive.
The deterministic clearance construction used in
Lemma~\ref{g:extra} gives a compatible crosscut \(\Upsilon_0\)
in \(B(X_T,\delta d)\setminus B(X_T,\delta d/2)\), joining the
positive boundary pieces on either side of \(X_T\) and separating
it from the negative side. If no hexagon centered at a vertex of the
stopped walk meets this crosscut,
avoidance of this crosscut protects the entire walk.
Otherwise add the boundaries of the walk hexagons on its
unprotected side and take the boundary facing the negative side.
Its diameter is at most \(2\delta^{-1}d\); its new part is at
distance at least \(\delta^2d/2\) from \(\partial U\), by the
terminal-walk condition. Indeed a contacting walk hexagon occurs
before \(t_*\), since the terminal walk piece lies in
\(B(X_T,\delta d/4)\), whereas the crosscut lies outside
\(B(X_T,\delta d/2)\). That contact is at distance at least \(\delta^2d-O(1)\)
from \(\partial U\); therefore the boundary-attached original
crosscut has diameter at least \(\delta^2d-O(1)\) as well. The old endpoints retain their pure
pockets. It is therefore a barrier with parameter \(c\delta^3\)
and scale \(2\delta^{-1}d\).

Proposition~\ref{g:barriers} gives probability at least \(p_\delta>0\)
that the future interface avoids the resulting barrier. On this
event it is adjacent to no vertex of the stopped walk, so \(Y=X_T\).
In the first branch \(\Upsilon_0\) can have bounded lattice diameter;
this is covered by the all-scale part of that proposition.
Reflect signs for a negative landing. Choose \(\delta\) to make
the failure probability of (G10) smaller than \(\eps\), and then
take \(d\) sufficiently large.
\end{proof}

\begin{proposition}[Nonreturn at stopped harmonic landings]
\label{g:nonreturn}
For every \(\eps>0\) there are \(s>0\) and \(d_0<\infty\) such that
\[
 \Prob\bigl(d(X_T,\gamma\setminus\gamma[0,T])d_0),                                      \tag{G11}
\]
uniformly over all sign-filtration stopping times \(T\), with
\(d=\operatorname{dist}(v,\partial D)\), under \((\mathrm{GB})\).
\end{proposition}

\begin{proof}
Let \(\mathcal G=\sigma(K,S)\), \(A=\{Y=X_T\}\), and
\(q=\Prob(A\mid\mathcal G)\). Lemma~\ref{g:survival} implies,
for every \(\mathcal G\)-measurable event \(E\),
\[
 \Prob(E)\leq\eps_1+p^{-1}\Prob(E\cap A).              \tag{G12}
\]
Take \(E\) to be the event that the old prefix has more than two
crossings of an annulus about its walk landing. On \(A\), this
implies the corresponding event for the completed interface at
its completed harmonic landing. Lemma~\ref{g:extra}, with error
smaller than \(p\eps_1\), therefore removes all additional old
excursions from a small comparison window.

Exclude landings close to the old tip using the walk hitting bound.
Lemmas~\ref{g:boundary-landing} and \ref{g:observer-landing} exclude
landings close to the original boundary or observer, uniformly over
all prefixes. Choose the largest comparison window smaller than
one tenth of this retained clearance. This verifies the original
boundary and observer exclusions when Lemma~\ref{g:extra} is applied
after recentering at the landing.

In the remaining window exactly two incident old arms reach the
outer circle, and the old tip lies outside it. There are two
relevant complementary components. In each, select nested
compatible barriers attached to its pure old side. After each
failed shield, reveal the future arms up to that barrier; the
conditional field law is again an exact-strand law, so the next
shield succeeds with probability at least a fixed \(c>0\).
Failure through \(N\) scales has probability at most \((1-c)^N\).
When neither complementary component is traversed, the future
cannot enter the innermost Euclidean ball. The removal of
additional old excursions is what converts this internal-access
conclusion to Euclidean nonreturn.

Choose \(\eps_1\), obtain \(p\) from survival, choose the completed
excursion error below \(p\eps_1\), choose \(N\), and then let all
the finitely many retained scales tend to infinity in lattice
units. Finally choose their innermost relative radius \(s\).
The sum of the discarded probabilities can be made less than
\(\eps\), proving (G11).
\end{proof}

\begin{remark}[What the geometric result supplies]
The random mass of old-boundary accesses revisited by the future is
the conditional harmonic probability of the event in (G11).
Its expectation tends to zero, uniformly over stopping rules.
Consequently bounded harmonic boundary tests may discard this mass.
This is the estimate needed before a tower-property argument for a
stopped observable. Absolute continuity of access in each fixed domain
alone would not give its uniform form.
\end{remark}
==== END Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//sections/geometry.tex ====

==== BEGIN Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//sections/interface.tex ====
\section{Height Gap and Curve Identification}\label{sec:interface}

The field results identify harmonic conditional means away from prescribed
strands. To identify the curve, we must determine their boundary traces on
the strand and retain those traces when the strand is stopped. This section
separates the corresponding implications. Every application below requires
the uniform versions of the preceding field and geometric estimates; a
statement for one fixed conditioning would not suffice.

\subsection{Conditional means viewed from harmonic landings}

Throughout this section $\gamma_n$ denotes the ordered \emph{dual sign
strand}, rather than the affine zero segments. Write $\mathcal F^n_t$
for the sigma-field generated by its oriented prefix up to time $t$.
It reveals adjacent signs, rather than the height ratios encoded by
affine crossing positions. Let $T$ be a stopping time for this filtration.
For a deterministic smooth test $f$ define
\[
 X_n(f)=\int_{D_n}f(z)h_n(z)\dd z,\qquad
 M^n_T(f)=\E[X_n(f)\mid\mathcal F^n_T].
\]
Thus $M^n_t(f)$ is an exact martingale. A test chosen afresh after $T$ does
not define this martingale and will not be used in the driving argument.
For a complete strand, write
$m_n(z)=\E[h_n(z)\mid\gamma_n]$.
The same notation with a superscript $T$ denotes the stopped conditional
mean. Comparisons with bounded pinned data bound these profiles uniformly.
Their subsequential limits are harmonic on the complementary open sets:
Proposition~\ref{b:cell} supplies the interior weak equation,
Proposition~\ref{b:response} gives $A>0$, and the common-wall comparison
supplies compactness.

Let $v_n$ be an interior observer vertex, with a fixed positive physical
distance from the original boundary. An independent walk from $v_n$
samples the harmonic measure
of the complete strand and the original boundary. A landing on the
positive side is understood as an oriented access; two prime ends with the
same Euclidean location need not have the same sign. We condition on that
side and, when necessary, on the finitely many possible orientations of
the incident lattice edge. All statements about boundary samples below are
made under this harmonic sampling law. Lemma~\ref{g:extra} is needed to
exclude additional excursions near the incident strand.

For the global boundary-sampling statements, an experiment means a
Dobrushin domain with sign-compatible exterior heights
$+\lambda$ and $-\lambda$, where $\lambda$ is fixed as $n\to\infty$.
For any finite collection of experiments fix
$0<\lambda_0\le\min\lambda\le1/2$. Constants in estimates whose
barriers attach to original boundary may depend on $\lambda_0$.
The artificial bounded-data conditionings used for local comparison
are not additional global experiments to which boundary avoidance
is asserted.

Here is the exclusion required before using the interior records of
(G7)--(G8). Let $\xi_n$ be the walk landing and let
$d_n=\operatorname{dist}(v_n,\partial D_n)$ in lattice units.
Lemmas~\ref{g:boundary-landing} and~\ref{g:observer-landing} give
\begin{equation}\label{h:landing-exclusions}
 \lim_{\eta\downarrow0}\limsup_{n\to\infty}
 \Prob\left(\xi_n\hbox{ lies on a strand side},\
   \operatorname{dist}(\xi_n,\partial D_n\cup\{v_n\})<\eta d_n
       \right)=0.
\end{equation}
The boundary estimate is the consequence of exterior-attaching barriers
and the independent full-walk exit comparison proved there; it does
not use SLE or a matched height gap. Its stronger full-walk form
also applies to landings on an arbitrary stopped prefix.

When conditioning on a side or on an incident-edge label, first take
a subsequence along which its probability converges. If the limit is
zero, its entire contribution to a bounded harmonic test vanishes.
Otherwise its probability is bounded below along that subsequence,
so dividing \eqref{h:landing-exclusions} by that lower bound gives
the same exclusion for the conditioned sampling law. There are only
finitely many such labels at each stage. This is the only
normalization needed to pass the exclusion to the boundary samples
used below.

\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 the component containing an observer $z_n$
after removing a simple dual prefix, or after cutting along a
complete simple dual crosscut. Suppose
$\operatorname{dist}(z_n,\partial U_n)\ge\rho>0$ in physical units.
Label its prime-end boundary by the original positive and negative
arcs and the two oriented strand sides; empty arcs are allowed.
There are at most four cyclic arcs before completion and at most
two in a completed component.

Kill triangular-lattice walk on its first visit to the original
boundary or to a vertex adjacent to the explored strand. Original
boundary values take priority at a vertex of both types.
Let $\xi_n$ denote this landing with its incident-side label.
There is a coupling with Brownian exit $\xi_n^{\rm B}$ from $U_n$
such that, for every $a>0$,
\[
 \sup_{U_n,z_n}
 \Prob\{|\xi_n-\xi_n^{\rm B}|>a
           \text{ or their labels differ}\}\longrightarrow0 .
 \tag{H2}
\]
The supremum includes all such prefixes, hence all stopping rules.
Repeated Euclidean boundary points are distinguished by their
incident accesses.
\end{lemma}
\begin{proof}
We first recall the required deterministic screening argument.
For a simply connected polygonal slit $U$, two complementary
prime-end arcs, and $R>0$, let $L_R$ be the union of internal
paths of diameter at most $R$ joining those arcs. The frontier
of the observer component of $U\setminus L_R$ is contained in
two balls of radius $2R$. Indeed, classify a crosscut by which
of the two arc endpoints it separates from the observer.
Two crosscuts in one class touching the observer component
must intersect, and their diameter bound then puts that entire
class in one ball. The argument uses prime ends on a slit,
not an identification of its two sides. This is the planar
argument in \cite[equation (3.55)]{SSdiscrete}.
Beurling's estimate therefore gives
\[
 \Prob_z\{\text{Brownian motion enters }L_R\}
       \le C(R/\rho)^\alpha .
\]
Apply this to each arc against its complementary arc. For a
partition into $m$ cyclic arcs, entrance into the union of
short paths joining different labels has probability at most
$Cm(R/\rho)^\alpha$, for a fixed $\alpha>0$.

The lattice walk has the required physical incidence. Every
primal edge crossed by an explored dual edge has both endpoints
observed and absorbing. Thus the interpolated walk cannot cross
the dual strand before it is killed. An interior killing vertex
has a connector of diameter $C/n$ through its own dual hexagon
to an incident strand edge, on the side bearing its sign.
If several strand edges are incident, every such access has
that same sign. At original boundary the connector is trivial
and the incoming edge specifies its access. Consecutive explored
dual edges cross primal edges belonging to a common triangle;
their absorbing endpoints form a connected set attached to the
original absorbing boundary. This set lies within $C/n$ of
the physical boundary and reaches across the annuli used below.
It supplies the connected obstacle for the discrete Beurling
estimate, including at the last explored dual vertex.

Fix $n^{-1}\ll r\ll R\ll\rho$. Couple the unrestricted
interpolated walk, with its diffusive clock, and Brownian motion
so that their distance up to a fixed time cutoff is less than
$r/4$, except with probability $o_n(1)$. The ordinary planar
invariance principle gives this coupling independently of the
slit; the triangular walk is isotropic. Both exit-time tails
at the cutoff are bounded by those for a fixed enclosing ball.
For each process use its \emph{own} first entrance into the
$2r$ boundary layer. Beurling's estimate at these two marginal
stopping times confines each remaining tail to diameter $R$,
except with total probability $C((r+n^{-1})/R)^\alpha$.
No Markov assertion about the joint coupling filtration is made.

Let $\theta$ be the first time either coupled path is within
$r$ of the physical boundary. On successful coupling both
paths are still alive, both of their $2r$ layer times have
occurred, and their positions at $\theta$ can be joined by a
segment inside $U_n$. If the exit labels differ, their confined
tails, this segment, and the lattice terminal connector give
an internal path of diameter $CR$ joining different labels.
The Brownian path has therefore entered the corresponding
mixed-access set. The preceding screening bound gives
\[
 \Prob\{\text{label mismatch or exit distance}>CR\}
 \le o_n(1)+C(r/R)^\alpha+Cm(R/\rho)^\alpha
       +\text{cutoff error}.                              \tag{H3}
\]
Take the time cutoff large, then $R$ small, then $r/R$ small,
and finally $n$ large. This proves (H2).
In particular the proof compares the walk directly with the
physical dual slit, not with an unmarked Hausdorff approximation
or a thickened domain with possibly many label changes.
\end{proof}

\begin{lemma}[Marked limits and local access]\label{h:marked-limits}
In the setting of Lemma~\ref{h:slit-sampling}, let the observers
converge, $z_n\to z$, and take a Hausdorff subsequence
of the closed removed sets consisting of the explored slit and the
original exterior. Retain the conformal maps from the disk to the
observer components, normalized by $f_n(0)=z_n$ and $f_n'(0)>0$,
and the inverse images of all boundary-label transition accesses.
After a further subsequence on which these marks converge,
bounded harmonic functions with the prescribed finite
label values converge on compact subsets to the Poisson
extension of the limiting marked arc data. Their harmonic
boundary samples converge with these marks. At a surviving
common local boundary piece of two such domains, the limit
preserves the incident-side identification.
\end{lemma}
\begin{proof}
The protected observer and the common enclosing ball make the
normalized inverse maps a nondegenerate normal family. Kernel
convergence identifies their limit with the observer component.
Extract also all transition angles on the unit circle, preserving
their cyclic order. The associated bounded step functions converge
at every angle except the finitely many limiting transitions.
Poisson integration proves compact convergence; normalized
arclength gives zero mass to those transitions. Colliding
transitions merely delete an arc of vanishing harmonic mass.
This retains the boundary marks rather than reconstructing
them from the limiting closed set.

For approach samples, use a free Brownian path independent of
all domain and field records. At fixed positive distance from
the boundary, kernel convergence and compact harmonic convergence
give convergence of its smoothed readings. The boundary-layer
tail estimate in the preceding proof lets that distance decrease
to zero after taking the mesh limit. Equivalently, the normalized
conformal maps give the limiting Brownian exit angle.

Finally suppose the two finite domains and their labels agree
on the component of a ball of radius $R$ next to a common
oriented strand. Their bounded label-Poisson extensions differ,
at a point in the concentric ball of radius $R/2$ and at distance
$r\ll R$ from that strand, by at most
$CM(r/R)^\alpha$: the difference is zero on the common local
boundary, and Beurling bounds escape from the ball.
The same bound holds if the common boundary is an original
arc. It is uniform in the domains, so it persists in marked
kernel limits. The mixed-access screening in the preceding
proof excludes conflating distinct sides in this procedure.
This proves the asserted local identification, without
claiming continuity of prime-end labels from closed sets alone.
\end{proof}

For the global experiments, the original domain approximations supply
the common bounded enclosure. Before applying these lemmas to
complete-strand samples, fix $\rho>0$ below the observer's original
boundary clearance and discard the bounded contribution from
configurations whose observer-to-strand distance is less than $\rho$
in physical units. Lemma~\ref{g:observer-landing} makes this contribution
vanish as $\rho\downarrow0$ after the mesh limit. On the retained
configurations both lemmas apply with this fixed clearance. For a
positive-probability side or orientation branch, divide the discarded
probability by its lower bound as above. This localization estimates
reading errors under the original sampling law; the lower comparison
in (G7) remains on all its admissible records.

For a bounded harmonic limiting profile $m$ and an independent Brownian
motion $W$ in its component, $m(W_t)$ has an almost sure limit as $W$
approaches its exit. Denote that limit by $Y$. It is a random boundary
value, not yet a deterministic height gap. The following elementary lemma
specifies the comparison needed to make it deterministic.

\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
\[
 \E|U^i_n-V^i_n|\longrightarrow0.
\]
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 $G^i_n$,
\[
 \Prob(V^i_n\in B\mid\mathcal A^i_n)\ge c\,\nu_n(B)
 \quad\text{for every Borel set }B.
\]
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}

The constant $c$ in this lemma need not approach one. It must, however,
stay positive as unused scale buffers grow. A different reference law for
each external configuration would not provide the stated conclusion.

\begin{lemma}[Locality of bounded conditional means]\label{h:locality}
Assume the sequentially uniform comparison and trace estimates of
Theorem~\ref{c:comparison} and Theorem~\ref{t:trace}, and the local
weak equation of Proposition~\ref{b:cell}. Consider two conditioning
systems with uniformly bounded mean profiles and the same lattice
domain and original fixed data inside a ball of radius $R$. Suppose
they share a connected oriented strand across that ball and all sign
observations inside it. For averaged observations at distance of order $r$
from that strand, supported in the concentric ball of radius $R/2$,
where $r\to\infty$ in lattice units and $r/R\to0$,
the two conditional means differ by a quantity tending to zero.

There is also an exterior version. If the two systems agree outside
a ball of radius $\rho$, their original exterior data agree, and their
unchanged strands connect that ball to radius $R$, then at distance
of order $r$ from its center, with $\rho\ll r\ll R$, the limiting
mean difference is at most $CM(\rho/r)^{1/2}$. Here $M$ bounds the
individual profiles. In both assertions use translates and dilates
of the fixed radial probability kernel from
Section~\ref{sec:trace-analysis}. A sampling ball centered at $y$
has radius $s$ less than one tenth of both $r$ and the distance from
$y$ to the boundary of the common free region. For each fixed $a>0$,
retain $s\ge ar$ while taking the mesh limit in coordinates rescaled
by $r$; the cutoff may then decrease to zero.
\end{lemma}
\begin{proof}
For the first assertion compare the two fields with common reflecting
sign walls. On shared strands their limiting mean difference has zero
Brownian-tested trace by the comparison and trace estimates. The
common original fixed data give zero differences at the corresponding
pins, so the same trace argument gives zero trace on the common
original boundary pieces. Away from these observed boundaries the
difference is harmonic by the weak equation and $A>0$. Its
absolute value on the other boundary pieces is at most $2M$.
The maximum principle therefore bounds it by $2M$ times the
probability of escaping the common ball before hitting the connected
common strand. Beurling bounds that probability by $C(r/R)^{1/2}$.
For arbitrary sequences with $r\to\infty$ and $r/R\to0$, rescale
by $r$ and first restrict to a common ball of radius $Lr$ about a
nearest strand point, with $L$ fixed. Centrality puts this ball inside
the shared region for all sufficiently large scales. Local $L^1$
compactness transfers its $CM L^{-1/2}$ bound to the fixed-cutoff
averages. First take the mesh limit, then $L\to\infty$, and finally
remove the averaging cutoff. Sequential uniformity gives the
assertion for every admissible conditioning sequence.

For the second assertion apply the identical argument in the common
exterior region, with the small changed window as the only uncontrolled
boundary. The probability of reaching that window from distance $r$
before hitting a connected common arm is at most $C(\rho/r)^{1/2}$.
The bounds persist under averaging in a sufficiently small ball about
the observation point. In both assertions the zero trace is a
conclusion of the common-wall comparison, not an equality of the
sampled wall heights.
\end{proof}

\begin{proposition}[The two readings required for the height gap]
\label{h:readings}
Assume the harmonic-mean compactness and common-sign trace comparison of
Proposition~\ref{b:cell}, Theorem~\ref{c:comparison}, and
Theorem~\ref{t:trace}. Assume also the landing exclusions
\eqref{h:landing-exclusions} and the sampled-configuration estimates
of Proposition~\ref{g:sample-comparison} and Lemma~\ref{g:extra}.
Then, along any joint subsequence of finitely many experiments, the
following construction gives the hypotheses of
Lemma~\ref{h:common-kernel}.
The limiting variables are the Brownian boundary samples of the
conditional mean profiles.
\end{proposition}
\begin{proof}
Fix the positive-probability side and orientation branches described
above. We construct an inner reading $V_n$ from the incident strand
and an outer reading $U_n$ from the external arms. Both approximate
the same boundary sample. The comparison of sampled configurations
will then bound the conditional law of $V_n$ below by one common
reference law, independently of the external record.

For a target error, first use \eqref{h:landing-exclusions} to discard
strand landings within distance $\eta d_n$ of the original boundary
or observer, choosing $\eta>0$ so that their probability is small.
Recenter a retained landing at zero. Choose the lattice radii
\[
 1\ll r_{\rm in}\ll R_{\rm core}\ll R_{\rm in}\ll R_{\rm mid}
       \ll R_{\rm conn}\ll r_{\rm out}\ll n.                 \tag{H1}
\]
Require also $r_{\rm out}<\eta d_n/20$. Thus the largest disks used by
(G7)--(G8), including their factors $6$ and $4$, lie inside the original
domain and exclude the observer. Clearance is measurable in the outer
record, which includes the domain and observer; retaining it imposes no
additional condition on the inner strand. Write
$\mathcal A_n=\sigma(\Phi_{R_{\rm mid}})$ under the specified landing
event $Z^\sigma$, and let $G_n\in\mathcal A_n$ be the retained
clearance event. The diagonal choices below make $\Prob(G_n)\to1$.

\emph{Agreement of complete-profile readings.}
First take mesh limits at fixed positive physical approach distances.
In each limiting component the bounded harmonic martingale has a
boundary limit. Its values at first entrances into balls of radii
tending to zero about the eventual landing have that same limit:
these entrance times approach the exit time pathwise. They need not
be stopping times after the future-dependent center is specified.

At an interior approach point use a radial smooth average in a ball
smaller than one tenth of its distance to the strand and of the
approach radius, with the fixed probability kernel defining $P_s$ in
Section~\ref{sec:trace-analysis}. The limiting harmonic average equals
the value at its center. Impose a positive lower cutoff on the averaging radius
before taking the mesh limit, then remove it. At each fixed positive
physical approach scale the sampling point is interior almost surely,
so this cutoff loses arbitrarily little probability. The comparison
compactness and the vanishing original--filled $L^1$ error give
convergence of the conditional means in $L^1$ on every compact subset
of the limiting free regions, on a joint subsequential realization.
At a fixed radius cutoff $a>0$, the sampling supports have distance
at least $9a$ from the strand; the retained landing clearance and (H1)
also keep them a fixed positive distance from the original exterior.
Hausdorff convergence therefore places all these supports in one
compact subset $C_a$ of the limiting free regions. Since the kernels
have supremum norm at most $\|\varphi\|_\infty a^{-2}$, the discrepancy
of their readings is bounded uniformly by
$\|\varphi\|_\infty a^{-2}\|m_n-m\|_{L^1(C_a)}$, which tends to zero.
This transfers both readings to the lattice, including for kernels
selected from the records.
A diagonal choice of the radii in (H1) therefore makes the readings of
the complete profile at $r_{\rm in}$ and $r_{\rm out}$ agree in $L^1$.
The scales are chosen slowly; no convergence rate is asserted.

\emph{The reading determined by the inner record.}
Retain the incident oriented strand and terminal walk piece in
$\Theta_{R_{\rm in}}$. Define its canonical mean in the disk of radius
$2R_{\rm in}$, with zero exterior heights and precisely those oriented
strand observations. This conditioning is feasible by assigning
sufficiently small signed heights. The construction is the same
measurable function of the inner record in every experiment.

Apply (G8) at $(R_{\rm core},R_{\rm in})$ and average its conditional
bound. Except on an event of vanishing probability under $Z^\sigma$,
the walk does not leave $R_{\rm in}$ after its first entrance at
$R_{\rm core}$. Its first entrance at $r_{\rm in}$ therefore belongs
to the recorded terminal piece, and reversing that piece determines
the sampling point. Any complete-chord piece in $B(0,R_{\rm core})$
other than the incident strand $\beta_{R_{\rm in}}^\sigma$ would
give two additional crossings of the same annulus. Outside the same
type of exceptional event, the canonical and complete conditionings
thus agree throughout $B(0,R_{\rm core})$.

Choose the inner radial kernel using the distance to the recorded
strand and the cutoff just described. Its canonical mean reading,
denoted $V_n$, is determined by $\Theta_{R_{\rm in}}$ alone.
If this record does not contain the required approach point, or its
kernel cutoff fails, set $V_n=0$. This is the same bounded function
of the inner record in every experiment; it does not use $G_n$.
Lemma~\ref{h:locality}, with
$r_{\rm in}/R_{\rm core}\to0$, makes its difference from the
complete-profile inner reading tend to zero. The extra-excursion
estimate supplies both the common strand neighborhood and the
presence of the chosen approach point in the recorded walk.

\emph{The reading determined by the outer record.}
Retain the external arms and initial walk in $\Phi_{R_{\rm mid}}$.
Delete every oriented dual-edge observation whose adjacent primal
sign pair meets $B(0,R_{\rm conn})$; retain the remaining external-arm
edge observations and both vertices of each pair. Every retained sign
vertex still has an oppositely constrained neighbor.

Apply (G8) at $(R_{\rm mid},R_{\rm conn})$ and average its bound.
Except on an event of vanishing probability under $Z^\sigma$, the
unrecorded connection of the two external arms stays inside
$B(0,R_{\rm conn})$: the arms already supply two disjoint annular
crossings, and an excursion of their connection would supply two more.
Hence the modified and complete conditionings agree outside
$B(0,R_{\rm conn}+C)$, with an absolute lattice constant $C$.
All retained observations are determined by $\Phi_{R_{\rm mid}}$.
The first walk entrance at $r_{\rm out}$ is also in that record,
since it precedes the first entrance at $R_{\rm mid}$.

Use a radial kernel smaller than one tenth of $r_{\rm out}$ and
of the distance to the recorded external strands, with a positive
lower cutoff as above. Its support stays outside the changed window.
The mean under the modified conditioning, tested with this kernel,
is $\mathcal A_n$-measurable; denote it by $U_n$. Set $U_n=0$
when its required outer clearance or kernel cutoff fails.
In the common exterior component, limiting mean differences are bounded
harmonic functions with zero trace on the unchanged strands and original
boundary. The only uncontrolled boundary is the deleted window, where
the difference has absolute value at most $2M$. A connected common arm
crosses the intervening annulus. The maximum principle and Beurling's
estimate therefore bound the influence of this window at the sampling
ball by
\[
 C M(R_{\rm conn}/r_{\rm out})^{1/2}.
\]
The bounded enlargement by $C$ is negligible at these scales.
Sequentially uniform compactness and trace comparison transfer the
bound to the lattice with an error tending to zero. Combining the two
localizations with complete-profile agreement gives
$\E|U_n-V_n|\to0$.

\emph{One common lower kernel and the order of limits.}
For each $n$, fix one admissible reference external record at radius
$4R_{\rm in}$. Equation (G7) bounds every inner-record law below by
$C^{-1}$ times this reference law, for every admissible external record
at that radius in every experiment. Since
$R_{\rm mid}>4R_{\rm in}$, the record at $R_{\rm mid}$ is a
restriction of the record at $4R_{\rm in}$. Averaging the bound over
the latter conditional on $\Phi_{R_{\rm mid}},Z^\sigma$ preserves
$C^{-1}$. Push this one reference law through the canonical inner-mean
function defining $V_n$ to obtain $\nu_n$. Lattice symmetries identify
the finite list of landing orientations. The exceptional events used
to make the readings agree do not restrict this lower comparison, and
no factor is paid for unused annuli.

Here is the scale order for a finite collection of limiting Brownian
boundary readings and a requested error less than $\eps>0$.
Choose the clearance exclusion before any physical buffers. Choose the
outer physical approach scale for both the boundary-reading error and
the retained clearance. Next choose $R_{\rm conn}/r_{\rm out}$
for exterior locality and $R_{\rm mid}/R_{\rm conn}$ for its
excursion error. Choose $R_{\rm in}\ll R_{\rm mid}$, then
$R_{\rm core}/R_{\rm in}$ for the inner excursion error, and finally
$r_{\rm in}/R_{\rm core}$ for inner locality and the inner boundary
reading. These choices are in physical units before the mesh limit, so
every retained lattice scale diverges. Hold the scales and kernel
cutoffs fixed while taking that limit, and then remove the cutoffs.
Repeat for $\eps\downarrow0$ and take a diagonal subsequence.
The comparison constant in (G7) is independent of all these unused buffers.
Lemma~\ref{h:common-kernel} now applies.
\end{proof}

\begin{proposition}[Subsequential universal height gap]\label{h:height-gap}
Assume the uniform inputs of Proposition~\ref{h:readings} and the
trace bounds of Lemmas~\ref{g:sign-trace} and~\ref{g:upper-half}.
Every mesh subsequence has a
further subsequence and a deterministic $l\in(0,1/2]$ such that all
terminal conditional mean profiles in any prescribed finite collection
of the sign-compatible Dobrushin experiments specified above have
Brownian boundary trace $+l$ on a
positive strand side and $-l$ on a negative strand side.
\end{proposition}
\begin{proof}
Include a fixed reference Dobrushin experiment with original boundary
heights $\pm1/2$. The barrier estimate gives nonzero harmonic mass to
its path sides. Compactness supplies joint mean profiles and harmonic
samples. Proposition~\ref{h:readings} and
Lemma~\ref{h:common-kernel} make the sampled positive boundary value a
single deterministic constant, shared by all included experiments.
The reference experiment's positive lower trace bound gives $l>0$,
and its complete-crosscut comparison gives $l\le1/2$.
Sign reflection identifies the negative value as $-l$.

Apply this argument to a countable collection of interior balls, with
a diagonal subsequence. If a terminal profile had a set of incorrect
boundary values of positive harmonic measure from any ball in that
collection, its sampled boundary value would have positive probability
of differing from $l$. Thus the statement holds almost surely in
every component seen from that exhaustion. Original boundary arcs retain
their prescribed values by the boundary comparison, rather than by the
height-gap argument.
\end{proof}

\subsection{Passing to a stopped strand}

For an independent walk from $z$, let $\mu^z_{n,T}$ be harmonic measure on
the boundary of the component of $D_n\setminus\gamma_n[0,T]$ containing
$z$. The nonreturn input needed here is the following consequence of
Proposition~\ref{g:nonreturn}: for each compact set of protected observers,
\begin{equation}\label{h:nr}
 \lim_{s\downarrow0}\limsup_{n\to\infty}\sup_T
 \E\int_{\gamma_n[0,T]}
 \mathbf1_{\{\operatorname{dist}(x,\gamma_n(T,\mathrm{end}])0$, uniformly in $T_n$. This is the required
change of sampling law; mere absolute continuity is not used.

At physical radius $r$ let $V^{\rm old}_{n,r}$ be a small radial
average of the stopped conditional mean at the first walk entrance
into $B(\xi,r)$. Its averaging radius is less than one tenth of
the distance to the observed strand and one tenth of $r$.
Define $V^{\rm end}_{n,r}$ in the same way with the complete strand
and $\upsilon$. Positive cutoffs on the averaging radius are removed
after the mesh limit, as in Proposition~\ref{h:readings}.
Let $b(\xi)$ be $+l$ or $-l$ according to the old oriented side,
and put
\[
 Z_n=|V^{\rm old}_{n,r}-b(\xi)|
                \mathbf1_{\{\xi\notin\partial D_n\}} .
\]
This is measurable with respect to $\mathcal G_n$.
By Proposition~\ref{h:height-gap} and
Lemmas~\ref{h:slit-sampling}--\ref{h:marked-limits},
\[
 \lim_{r\downarrow0}\limsup_n
 \E\!\left[|V^{\rm end}_{n,r}-b(\upsilon)|
                \mathbf1_{\{\upsilon\notin\partial D_n\}}\right]=0.
 \tag{H4}
\]
This is the bounded harmonic martingale boundary limit in the
completed component, sampled by the lattice walk through (H2).
First discard arbitrarily small completed-observer clearances
using Lemma~\ref{g:observer-landing}; (H2) then applies with a
fixed protected observer.
The first entrance around its eventual landing need not itself
be a stopping time: these approach points tend to the exit
pathwise, which is all that this boundary-limit assertion uses.

Discard old-side landings within distance $2R$ of the original
boundary and those within distance $2R$ of the future strand.
Their total probability, denoted $e_n(R)$, satisfies
$\lim_{R\downarrow0}\limsup_n e_n(R)=0$ by
\eqref{h:landing-exclusions} and \eqref{h:nr}.
On the remaining event, and on $A_n$, the two readings use the
same walk segment, the same label, and the same averaging kernel
when $r0$. Choose $R$ so that the discarded mass is small compared
with $p$ times the desired final error. Then choose $r\ll R$
to make both (H4) and the locality error equally small.
Keep these physical scales and the kernel cutoffs fixed while
taking $n\to\infty$. This order proves that the old-side boundary
reading equals its prescribed label in $L^1$, uniformly in
the chosen sequence of stopping rules.

Original boundary accesses retain their original values.
For clarity, the argument proving (GB2)--(GB3) also yields
\[
 \lim_{R\downarrow0}\limsup_n
 \Prob\{d_{D_n}(S_{\tau_{D_n}},\gamma_n)0$. Suppose the comparison is to the continuum
slit-domain function \eqref{h:angle}. Then the forward chordal drivers
converge, locally uniformly in capacity time in law, to $2B$.
\end{lemma}
\begin{proof}
We specify which part of the harmonic-observable criterion is used.
Let $x_t0$ introduce the bounded-driver coordinates
\[
 s=\log(y_t-x_t),\qquad
 G_s(z)=\frac{2g_t(z)-x_t-y_t}{y_t-x_t},\qquad
 \widetilde U_s=\frac{2U_t-x_t-y_t}{y_t-x_t}\in[-1,1].
\]
Let $\mathcal F_s$ be the sign-prefix filtration at the corresponding
time, equivalently the filtration of the explored dual curve after
this deterministic coordinate change. In these coordinates, as in
\cite[Sections 4.2--4.5]{SSdiscrete},
fix a finite interval $[-S,S]$ and the 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. Thus
\eqref{h:stopped-observable} supplies the same two approximate
martingales as the point probes.

For stopping times $s_0\le s_1$ with
$s_1-s_0\le\eps^2$ and driver oscillation at most $\eps$, the two-probe
Taylor calculation gives, outside an event of arbitrarily small
prescribed probability,
\begin{align}
 \left|\E[\Delta\widetilde U+
       2\widetilde U_{s_0}\Delta s\mid\mathcal F_{s_0}]\right|
       &\le C_S\eps^3, \label{h:driver-first}\\
 \left|\E[(\Delta\widetilde U)^2-
       2(1-\widetilde U_{s_0}^{\,2})\Delta s
           \mid\mathcal F_{s_0}]\right|
       &\le C_S\eps^3. \label{h:driver-second}
\end{align}
Here $\Delta s=s_1-s_0$ and
$\Delta\widetilde U=\widetilde U_{s_1}-\widetilde U_{s_0}$.
To obtain these estimates from qualitative convergence, fix
$S,\eps$ and the exceptional probability first, and only then take
the mesh sufficiently fine. Conditional Markov inequality converts
the $L^1$ observable errors into the needed conditional errors.
No predetermined rate is required.

To apply the approximate-diffusion criterion, start the comparison
diffusion at the same bounded-driver value as the discrete process.
For a fixed comparison horizon $S$, choose the exceptional tolerance
$\eta<\delta^5/S^2$, where $\delta$ is the criterion's approximation
parameter, before choosing the mesh; Proposition~4.5 of
\cite{SSdiscrete} then gives its comparison on $[0,S-1]$.
For a limiting window $[-S,S]$, start instead at $-S'$ and let
$S'\to\infty$ after the mesh limit. The coupling argument in the
proof of Theorem~4.1 of \cite{SSdiscrete}, pp.~115--116, removes the
initial value and identifies the stationary whole-line
bounded-coordinate diffusion. Thus an arbitrary initial Jacobi law
is not substituted for the chordal limit.

The deterministic Taylor argument and the approximate-diffusion
argument of \cite[Propositions 4.2 and 4.5]{SSdiscrete} use precisely
these martingale and harmonic-observable inputs; no Gaussian
conditional law enters these steps. They identify the bounded-coordinate
diffusion with squared diffusion coefficient $2(1-u^2)$ and drift
$-2u$. The inverse coordinate change \cite[Section~4.6]{SSdiscrete} gives
the chordal driver $2B$.
The discrete harmonic-measure comparison in that source is replaced
here by the explicit continuum comparison assumed in
\eqref{h:stopped-observable}.
\end{proof}

For completeness, the normalization of this identification can also be
seen directly in any continuous Loewner subsequential limit. If the
angle at one protected point is a continuous martingale, the identity
\[
 U_t=\operatorname{Re}g_t(z)-
       \operatorname{Im}g_t(z)\cot\arg(g_t(z)-U_t)
\]
makes the driver a semimartingale after localization. Writing
$U=V+N$, where $V$ has finite variation and $N$ is a continuous local
martingale, and $Z_t=g_t(z)-U_t$, It\^o's formula gives the finite-variation
part of its angle as
\[
 -\operatorname{Im}(Z_t^{-1})\dd V_t+
 \operatorname{Im}(Z_t^{-2})
       \left(2\dd t-\tfrac12\dd[N]_t\right).
\]
Two protected points with different
$\operatorname{Re}(Z_t^{-1})$ force $\dd V_t=0$ and
$\dd[N]_t=4\dd t$. A dense set of probes provides such a pair locally.
L\'evy's characterization then gives $U=2B$.
This calculation identifies a continuous limit; the
approximate-diffusion argument is still needed to obtain driving
tightness.

\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.

In every traversed triangular face, the affine zero segment and the
corresponding ordered dual strand stay within $C/n$ of each other.
Their order agrees, so an increasing reparametrization gives the same
$C/n$ bound for the complete curves. Hence the choice of polygonal
representative does not alter the conclusion.
\end{proof}

\subsection{The variance normalization}

The height-gap argument so far allows $l$ to depend on the extracted
subsequence. The conditional covariance identifies it.

\begin{proposition}[Identification of the gap]\label{h:normalization}
Assume the joint conclusions of Proposition~\ref{h:stopped},
Theorem~\ref{t:covariance}, and the field limit
Theorem~\ref{b:gff}, with the uniform moments needed to pass conditional
second moments. Then every subsequential height gap satisfies
\[
 l^2=\frac{\pi\sigma^2}{8}.
\]
\end{proposition}
\begin{proof}
Use the experiment with original heights $\pm l$, whose interface limit
is $\mathrm{SLE}_4$ by the preceding propositions. Stop before compact
smooth tests approach the curve. The conditional means are the
corresponding harmonic observables, while their conditional covariance
is $\sigma^2$ times the Dirichlet Green kernel of the remaining domain.
The sequential uniformity of Theorem~\ref{t:covariance} is used to
apply it to the actual random stopped strings. The moments from
Proposition~\ref{f:averages} justify convergence of the averaged
conditional variances.

In the half-plane set $Z_t(z)=g_t(z)-2B_t$. It\^o's formula yields
\[
 \dd\log Z_t(z)=-\frac{2}{Z_t(z)}\dd B_t.
\]
The cross variation of the harmonic observables is therefore
\begin{equation}\label{h:mean-bracket}
 \frac{16l^2}{\pi^2}
       \operatorname{Im}\frac1{Z_t(z)}
       \operatorname{Im}\frac1{Z_t(w)}\dd t.
\end{equation}
For covariance $(-\Delta)^{-1}$ the half-plane Green function is
\[
 G_{\mathbb H}(z,w)=\frac1{2\pi}
       \log\left|\frac{z-\overline w}{z-w}\right|.
\]
Differentiating its conformal pullback gives
\begin{equation}\label{h:green-loss}
 -\partial_tG_{D_t}(z,w)=\frac2\pi
       \operatorname{Im}\frac1{Z_t(z)}
       \operatorname{Im}\frac1{Z_t(w)}.
\end{equation}
These normalizations and the observable-bracket/Green-loss comparison
agree with \cite[preprint version, Lemmas 2.5--2.6,
equations (2.3)--(2.4)]{SScontinuum}.
Only this calculation is used; the stronger conditional coupling
in that work is not an input.

Integrate both identities against a nonzero nonnegative smooth test
$f$ twice. No pointwise value of the limiting field is used. The law
of total variance equates the gained variance of the conditional-mean
martingale with the expected loss of the residual Green variance.
For a nonzero localized time interval this gives
\[
 \frac{16l^2}{\pi^2}\,
 \E\int\left(\int f(z)\operatorname{Im}(1/Z_t(z))\dd z\right)^2\dd t
 =
 \frac{2\sigma^2}{\pi}\,
 \E\int\left(\int f(z)\operatorname{Im}(1/Z_t(z))\dd z\right)^2\dd t.
\]
The common factor is finite and strictly positive. Cancelling it
proves the formula.
\end{proof}

\begin{corollary}[Universality]\label{h:assembly}
For the model of Section~\ref{sec:model}, the constant
\[
 \lambda_*=\sigma\sqrt{\pi/8}
\]
belongs to $(0,1/2]$, is independent of the marked domain, and gives
the curve conclusion of Theorem~\ref{thm:main}. The field stiffness
is $vA=\sigma^{-2}$, and its limiting mean is the harmonic extension
of the prescribed exterior values.
\end{corollary}
\begin{proof}
Apply the reference height-gap extraction to every subsequence.
Proposition~\ref{h:normalization} gives the same value $\lambda_*$
on each extraction; its membership in $(0,1/2]$ comes from the
reference complete-crosscut bounds. Include, in each finite comparison,
the fixed experiment with exterior heights $\pm\lambda_*$.
Every subsequence then has a further subsequence to which the driving
and uniform-path conclusions apply with the same matched value.
This proves full convergence. The field assertions are the
identifications in Theorem~\ref{b:gff} and
Proposition~\ref{b:response}.
\end{proof}
==== END Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//sections/interface.tex ====

==== BEGIN Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//sections/shift.tex ====
\section{Feasible Shifts and Volume Distortion}
\label{sec:feasible-shifts}

The geometric arguments will need to raise or lower heights slowly across
a corridor while retaining an event of uniformly positive probability.
Simply adding the prescribed shift can violate a nearly saturated edge
constraint. We correct the shift by averaging it over clusters of such
edges, so that the two corrected endpoint shifts agree at every exact
edge contact. A
determinant identity for vector fields tangent to the height polytope
then controls the volume lost by this height-dependent correction.
The resulting proposition transfers any event of probability bounded
below to a feasible shifted event of probability bounded below; the
geometric application must supply that initial event.

Constraint-preserving perturbations and their volume cost also underlie
Richthammer's deformed translations for hard-core particles
\cite[Sections~5.4 and~6.5--6.7]{Richthammer} and the addition algorithm
of Mi\l o\'s and Peled for hard-constraint surfaces
\cite[Sections~1.1.2 and~2]{MilosPeled}. The threshold-averaged component
construction below differs from those algorithms and supplies its own
tangential-vector-field determinant identity.

All heights in this section are in lattice units, and the law is
normalized Lebesgue measure 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 will
be left unchanged by the map. The sparse edge estimates of that
corollary apply also to zero-pinned laws by taking no sign strings.

\subsection{The cluster consequence of sparseness}

Fix a bounded-degree lattice graph and a set of at most $C n^2$ vertices.
For a sufficiently small fixed $s_*>0$, join neighboring vertices when
\[
 |h_x-h_y|\geq 1-s_*.
\]
Ignore edges whose two endpoints are fixed. The sparse edge estimate of
Corollary~\ref{f:conditioned-sparse}, including its zero-pin specialization,
implies that the size $H_x$ of the resulting component containing $x$
satisfies
\begin{equation}\label{eq:shift-cluster-moments}
 \sup_x \mathbb E H_x^k\leq C_k,\qquad k<\infty.
\end{equation}
The constants are uniform over the conditioning systems covered by
that corollary. To see the implication, use its overlapping fixed-box
cover: the central-edge tests include every edge with a nonfixed endpoint,
also near fixed pins. A nontrivial component can occupy only bad boxes,
apart from a bounded enlargement at its endpoints, since a good box has
slack on every edge incident to its central box. A connected collection
of $m$ occupied boxes contains a separated subcollection of at least
$m/C$ boxes. Its probability is at most $\rho^{m/C}$. Taking $\rho$
smaller than the reciprocal of the lattice-animal counting constant
gives an exponential tail for the number of occupied boxes, hence
\eqref{eq:shift-cluster-moments}.
Deleting edges or restricting the graph preserves this conclusion.
In particular, for any fixed $\alpha>0$,
\begin{equation}\label{eq:shift-largest-cluster}
 \mathbb P\left[\max_x H_x>n^\alpha\right]
 \leq C_k n^{2-\alpha k}=o(1)
\end{equation}
after choosing $k>2/\alpha$. No convergence rate from homogenization enters
this assertion.

\subsection{A determinant identity on polytopes}

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

\begin{proof}
Write $\ell_i\geq0$ for the normalized affine slack of facet $F_i$.
There is a finite constant $C_i$, depending here on the polytope, such that
$\operatorname{dist}(x,F_i)\leq C_i\ell_i(x)$ for $x\in P$. Indeed, express
$x$ as a convex combination of vertices, fix $v_0\in F_i$, and replace
each vertex $v\notin F_i$ in that combination by $v_0$; use the maximum of
$|v-v_0|/\ell_i(v)$ over these finitely many vertices.
Tangency and Lipschitz continuity give
\[
 |\nabla\ell_i\cdot u(x)|
 \leq C_i\operatorname{Lip}(u)\ell_i(x).
\]
Consequently, for sufficiently small positive or negative $t$,
$T_t=I+tu$ maps $P$ into itself and maps $\partial P$ into $\partial P$.
It is injective when $|t|\operatorname{Lip}(u)<1$. To prove surjectivity,
fix $y\in P$. The map $x\mapsto\Pi_P(y-tu(x))$, where $\Pi_P$ is
Euclidean projection, is a contraction of $P$ and has a fixed point $x$.
Put $n=y-tu(x)-x$. The projection inequality says that $n$ is an
outward normal to $P$ at $x$, so $n\cdot(y-x)\le0$. Tangency to every
facet containing $x$ gives $n\cdot u(x)=0$, because the normal cone
is generated by those facet normals. Consequently
$0\ge n\cdot(y-x)=|n|^2$, so $n=0$ and $T_t(x)=y$.

The Lipschitz area formula now gives
\[
 \int_P \det(I+tDu(x))\,dx=|P|.
\]
For the piecewise affine field constructed below, this identity is
elementary: subdivide $P$ into its finitely many affine regions,
apply the linear determinant formula on each region, and sum.
Injectivity makes the image interiors disjoint, while their boundaries
have zero volume; surjectivity makes their union all of $P$.
The determinant is positive for these $t$. Comparing the linear and
quadratic coefficients of this polynomial identity proves
\eqref{eq:shift-divergence}, since the quadratic coefficient is
$((\operatorname{div}u)^2-\operatorname{tr}((Du)^2))/2$.
Finally $|\operatorname{tr}(A^2)|\leq\|A\|_{\mathrm{HS}}^2$.
Thus no integration by parts across corners or nonsmooth interfaces is
required. The polytope-dependent restriction on $t$ has disappeared from
the resulting identity.
\end{proof}

\subsection{Construction and quantitative bounds}

Let $V_n$ contain the sites on which the prescribed shift may be nonzero.
Assume $|V_n|\leq Cn^2$, and adjoin all their neighbors. Declare all adjoined
vertices, all fixed sites, and all sites carrying sign or additional wall
conditions to be grounded. Include graph edges with at least one
nongrounded vertex, and omit edges with two grounded endpoints.
Grounding means only that the shift is zero there; a grounded height need
not be fixed under the original law.

Let $f=f_n$ be a deterministic prescription, zero at every grounded site
and outside $V_n$, such that
\begin{equation}\label{eq:shift-prescription}
 |f_x|\leq M,\qquad |f_x-f_y|\leq L/n
 \quad\hbox{for every lattice edge }xy.
\end{equation}
The constants $L,M$ may be arbitrarily large fixed constants. They do not
depend on $n$. Choose fixed
$00$ there are $c>0$ and $n_0$,
uniform over the allowed conditioning systems, with the following property.
For $n\geq n_0$ and every measurable event $E$ with $\mathbb P(E)\geq p_0$,
there is a measurable $E'\subset E$ with $\mathbb P(E')\geq p_0/2$ such that
$T(h)=h+u(h)$ is injective on $E'$, has feasible image, and
\[
 \sup_{h\in E'}\|T(h)-h-f_n\|_\infty=o(1),\qquad
 \mathbb P(T(E'))\geq c.
\]
The statement also holds for a fixed finite product of independent
conditional surfaces.
\end{proposition}

\begin{proof}
Take $\alpha=1/8$ and
$G_n=\{H_{\max}\leq n^\alpha\}$, using the enlarged threshold $s_2$.
By \eqref{eq:shift-largest-cluster},
$\mathbb P(G_n^c)=o(1)$ uniformly.
If $h\in G_n$, then \eqref{eq:shift-error} gives
$\|u-f\|_\infty\leq Ln^{\alpha-1}$.
For an edge with $t_e\leq s_0$, its two shifts agree exactly.
For every other edge the original slack is greater than $s_0$, whereas
\[
 |u_x-u_y|\leq L(2n^\alpha+1)/nK
 \ \hbox{or}\ \|Du\|_{\mathrm{HS}}^2>K\bigr]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}
where $1\le\ell_n\le C(\log n)^C$.  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}
Poincare's inequality for two nested averaging disks gives
\[
 |P_su(x)-P_{s/2}u(x)|
 \le Cs\fint_{B(x,Cs)}|\nabla u(y)|\dd y.
\]
Holder's inequality 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 that
the construction is independent of the chosen 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, namely
$C\ell_n^{\alpha/p}n^{-\beta}\|\nabla u_n\|_p$.
The difference between a vertex value and its mesh-scale average has the
same bound, by affine interpolation on the bounded number of incident
triangles and the same Holder calculation.
\end{proof}

\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 $0<\eta1$, 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
$E_n^{1/2}(\sum_x\mu_n(x)^2/w_x)^{1/2}$.
Holder 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}

\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 $3/23$, 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.  We may 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 $u$, which proves the area assertion.
Finally \eqref{t:original-filled} proves the equality of the two
distributional limits.
\end{proof}

\begin{lemma}[Uniform Brownian approach measures]\label{t:exit}
Let $K$ have continuum thickness, put $V=Q\setminus K$, and fix $z\in V$.
For sufficiently small $\eps>0$, let $V_\eps$ be the component containing
$z$ of $\{x\in V:\operatorname{dist}(x,V^c)>\eps\}$.  The exit law
$\mu_{z,\eps}$ of Brownian motion from $V_\eps$ satisfies
\begin{equation}\label{t:exit-frostman}
 \mu_{z,\eps}(B(x,r))\le C_z r^{1/2},
\end{equation}
uniformly in $\eps$.  The constant is uniform for $z$ a positive distance
from $K$, and depends on the common thickness constants and $Q$.
\end{lemma}
\begin{proof}
We use the planar Beurling estimate in its equivalent inward form: if an
obstacle contains a continuum crossing the annulus between radii $r$ and
$R$, the probability, from outside the outer circle, of reaching the
inner circle before the obstacle is at most $C(r/R)^{1/2}$.
The inward form follows from the usual outward estimate by inversion;
this is the Brownian estimate used in the boundary sampling arguments of
\cite{SSdiscrete}.

Each component of the closed $\eps$ neighborhood of $K$ contains a
continuum of diameter bounded below by the thickness scale.  The same is
true of the exterior obstacle.  If an $r$-ball meets the support of the
exit law, its obstacle component therefore crosses an annulus from
radius $O(r)$ to a fixed radius
$R0$, 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 Poincare inequality inherited from $Q$ gives existence.
Its zero extension is in $W^{1,p}(Q^*)$.  Approximating it 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 $K$.

Thus $w=u-v$ is harmonic on $V$ and has the same zero traces.  Fix $z\in V$
and use Lemma~\ref{t:exit}.  Transport $\mu_{z,\eps}$ to a measure on
$K$ by Lemma~\ref{t:transport}; its support is within distance $\eps$ of
$K$.  Since $\beta=1-3/(2p)>0$, \eqref{t:transport-trace} yields
\[
 \E_z|w(B_{\tau_{V_\eps}})|
 \le C_z\eps^{1-3/(2p)}\|\nabla w\|_p\longrightarrow0.
\]
The Sobolev trace on this interior boundary equals the smooth harmonic
representative of $w$.  For each fixed $\eps$, that representative is
bounded on the compact closure of $V_\eps$.  Optional stopping, or the
mean-value characterization of harmonic measure, gives
$w(z)=\E_z w(B_{\tau_{V_\eps}})=0$ in the limit.  This works in each
component of $V$.  The zero trace tested against Lebesgue measure on $K$
also gives $u=0$ almost everywhere there, completing
\eqref{t:poisson-identification} on all of $Q$.
\end{proof}

\begin{remark}[The boundary notion and the order of limits]
The proof does not promote a $W^{1,p}$ trace directly to an $H^1$ capacity
trace.  It first proves equality with the variational Green solution; the
$H^1$ assertion follows afterwards.  In a lattice application, fix
$\eps$ and use radial ball averages of radius less than $\eps/2$ before
passing to a limit.  Their limits equal the harmonic function at their
centers by the mean-value property.  No convergence of unaveraged lattice
point values is required.

Thickness cannot simply be dropped.  On the punctured unit disk,
$\log(1/|x|)$ is harmonic, has zero outer trace, and belongs to
$W^{1,p}$ for every $p<2$.  The singleton has zero $p$-capacity, and its
neighborhood hitting probabilities decay only logarithmically, so
\eqref{t:exit-frostman} fails.  Conversely, the usual slit-tip singularity
$r^{-1/2}\sin(\theta/2)$ has gradient in $L^p$ only for $p<4/3$ and
does not satisfy the hypotheses above.
\end{remark}

==== END Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//sections/trace-analysis.tex ====

==== BEGIN Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice//sections/trace.tex ====
\section{Conditional Dirichlet covariance}\label{sec:conditional-covariance}

The response argument in Section~\ref{sec:cell} identified the Gaussian
covariance for ordinary fixed-boundary domains. We now identify the
conditional covariance when internal oriented strands have also been
observed. A common sign wall does not make the two coupled heights equal
there. Instead, the squared common-wall errors in the comparison estimate,
together with thickness of the observed set, give zero trace for the
limiting response. The preceding analytic section then identifies the
response as the Dirichlet Green solution, and exponential tilting gives
the covariance.
This argument uses no interface convergence or height-gap theorem.

\subsection{The uniform input and the conditioning class}

For each mesh $n^{-1}$ let $\mathcal S_n$ be a family of deterministic
oriented sign-string configurations with ordinary one-sided sign walls,
fixed Dirichlet values in $[-1/2,1/2]$, and positive-volume conditional
height polytopes.  The sign strings are of the type specified in
\ref{f:conditioned-sparse}; no further exact-height transcript is fixed.
Denote their conditional
laws by $P_{n,s}$, $s\in\mathcal S_n$.  Realize each observed set as a
closed geometric set $K_{n,s}\subset\overline Q$ containing $\partial Q$:
join the observed lattice vertices along their strings, include the
exterior fixed region, and use a fixed $O(n^{-1})$ convention at faces.
Such conventions have the same Hausdorff limits.  Require the common
mesh thickness constants of Definition~\ref{t:thickness}.  Write
$V_{n,s}=Q\setminus K_{n,s}$.

The following are the exact forms of the preceding estimates used here.
They make the scope of the uniform assertion explicit.  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|