==== TABLE OF CONTENTS ====
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//01-introduction.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//02-finite-rows.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//02-inputs.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//03-radial.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//03-thermodynamic-kernels.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//04-angular-plane.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//04-spectral.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//05-electric.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//05-normalization.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//06-chiral-gaussianity.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//06-coupling.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//07-calibration.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//07-probabilistic-endgame.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//08-main-proof.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//08-occupation.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//angular.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//barrier.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//chordal.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//endpoint-field.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//endpoint-loops.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//introduction.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//main.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//observable.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//peeling.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//preliminaries.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//rotation-repair.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//sources/companions/results/gaussian-free-field-balanced-six-vertex-model/build/sections/01-introduction.tex
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//references.bib
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//sources/companions/results/critical-square-lattice-random-cluster-interfaces/build/references.bib
Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//sources/companions/results/gaussian-free-field-balanced-six-vertex-model/build/references.bib
===========================

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//01-introduction.tex ====
\begin{abstract}
We prove that the rescaled counting measure of a critical square-lattice
Fortuin--Kasteleyn Dobrushin interface in the unit square converges to the
Minkowski-content measure of its Schramm--Loewner limit, for every fixed
cluster weight $1\le q<4$. A single deterministic constant times the
predicted power of the mesh gives the normalization. Convergence is joint
with the ordered curve and includes the total mass, giving the scaling
limit of the interface's total number of steps.
\end{abstract}

\section{Introduction}\label{sec:introduction}

A scaling limit of an interface describes its shape, but does not by
itself describe how many lattice steps it takes to traverse that shape.
The latter question requires a measure on the limiting curve. For
Schramm--Loewner evolution (SLE), the appropriate measure is its
dimensional Minkowski content. The problem here is to identify that
measure as the limit of the unweighted number of steps of a critical
Fortuin--Kasteleyn (FK) interface on the square lattice.

The distinction between shape and occupation is quantitative. A
visitation exponent predicts the appropriate power of the mesh, but
does not rule out a slowly varying or nonconvergent factor. We prove
convergence with a single constant multiplying the power law. We also
control the mass near the boundary, so the result concerns the whole
interface, including its total number of steps.

\subsection{The lattice model and the main result}

Fix $q\in[1,4)$ and put
\begin{equation}\label{eq:parameters}
 p_q=\frac{\sqrt q}{1+\sqrt q},\qquad
 \kappa=\frac{4\pi}{\arccos(-\sqrt q/2)},\qquad
 d=1+\frac{\kappa}{8},\qquad x=2-d.
\end{equation}
Thus $4<\kappa\le6$ and $1/4\le x<1/2$.
Let $D=(0,1)^2$, with marked corners $a=(0,0)$ and $b=(1,1)$.
The counterclockwise boundary arc from $a$ to $b$, consisting of the
bottom and right sides, is denoted by $F$. The complementary arc is $W$.

For $n\ge1$, the graph has vertex set
$n^{-1}\Z^2\cap\overline D$ and all nearest-neighbor edges in
$\overline D$. Identify all vertices on $W$, including $a,b$, in a
single block; all other vertices are distinct. Denote this partition by
$\xi$. Remove the boundary edges on $W$ from the sampled set of edges,
and call the remaining set $E_n'$. The probability of a set of open
edges $A\subset E_n'$ is
\[
 \Pp_n(A)=\frac{1}{Z_n}
 p_q^{|A|}(1-p_q)^{|E_n'|-|A|}q^{k_\xi(A)},
\]
where $k_\xi(A)$ counts connected components after the prescribed
identification, including isolated vertices. The removed edges on $W$
are declared open for drawing the interface. Edges on $F$ remain random.

We use the Dobrushin medial strand from $a$ to $b$. Medial vertices are
midpoints of primal edges; medial edges join midpoints of consecutive
sides of a lattice square. At an open primal edge, pair the four medial
half-edges in the two turns around the adjacent dual faces. At a closed
primal edge, use the turns around the primal endpoints. The boundary
turns separate the primal wired arc from the complementary dual wired
arc. Two distinguished terminal half-edges occur at the boundary changes.
The pairing determines the strand and its order, including the order at
a medial vertex visited twice.

Let $e_{n,1},\ldots,e_{n,N_n}$ be the full medial-edge traversals of
this strand, including boundary traversals but excluding the two
terminal half-edges. Let $z_{n,j}$ be the midpoint of $e_{n,j}$.
Define
\[
 \Pi(u,v)=\bigl(\max(0,\min(1,u)),\max(0,\min(1,v))\bigr).
\]
Project the drawn strand by $\Pi$ and join its terminal points to $a,b$
by straight segments. The resulting oriented curve is $\eta_n$.
The terminal connectors lie within $O(n^{-1})$ of the marked corners
and carry no counting mass. For $c>0$, set
\begin{equation}\label{eq:counting-measure}
 \nu_{n,c}=c n^{-d}\sum_{j=1}^{N_n}\delta_{\Pi(z_{n,j})}.
\end{equation}
This convention counts traversals, not Euclidean length. In fact, a full
medial edge cannot recur on this strand: after resolving the pairings,
the strand is a component of a graph of degree two, with its two terminal
ends. A medial vertex can nevertheless be visited twice. This distinction
will matter when we replace counting by local hit indicators.

For continuous oriented curves $\gamma_1,\gamma_2:[0,1]\to\overline D$,
use the distance
\begin{equation}\label{eq:curve-metric}
 d_{\mathrm{curv}}(\gamma_1,\gamma_2)
 =\inf_{\phi_1,\phi_2}\sup_{0\le t\le1}
 |\gamma_1(\phi_1(t))-\gamma_2(\phi_2(t))|,
\end{equation}
where the infimum is over increasing homeomorphisms of $[0,1]$.
We identify curves at distance zero. This topology retains traversal
order, unlike a topology on trace sets alone.

Let $\eta$ be chordal $\mathrm{SLE}_\kappa$ in $(D;a,b)$, oriented
from $a$ to $b$. Its $d$-dimensional Minkowski-content measure is
normalized by
\begin{equation}\label{eq:content-measure}
 \mu_{\eta,r}
 =r^{-x}\Pi_*\bigl[
 \1\{\dist(z,\tr\eta)<r\}\,\mathrm dA(z)\bigr],
 \qquad \mu_{\eta,r}\longrightarrow\mu_\eta
 \quad\text{in probability as }r\downarrow0.
\end{equation}
Here the area measure is on the whole plane, not only on $D$.
The convergence is weak convergence of finite measures on
$\overline D$. The existence of this whole-trace measure, including
the stated projection convention, is justified in
Lemma~\ref{lem:whole-content} from the SLE content and Green-function
theorems.

\begin{theorem}\label{thm:main}
For each fixed $q\in[1,4)$ there is a deterministic constant
$c(q)\in(0,\infty)$ such that, along all positive integers $n$,
\[
 \bigl([\eta_n],\nu_{n,c(q)}\bigr)
 \ \Longrightarrow\
 \bigl([\eta],\mu_\eta\bigr).
\]
The first coordinate has the ordered curve topology
\eqref{eq:curve-metric}, and the second has the weak topology of finite
measures on $\overline D$. The two limiting coordinates belong to the
same SLE sample. In particular,
\[
 c(q)n^{-d}N_n\ \Longrightarrow\ \mu_\eta(\overline D)
\]
jointly with the curve and all continuous spatial tests of its measure.
\end{theorem}

The value of $c(q)$ depends on the stated discrete and continuum units;
its numerical evaluation is not needed. Every estimate below is for a
fixed $q$. We make no claim of uniformity as $q\uparrow4$.

\subsection{Context and inputs}

Schramm introduced SLE to describe conformally invariant limits of
planar lattice interfaces \cite{Schramm}. Beffara proved that the SLE
trace has dimension $1+\kappa/8$ for $\kappa\le8$ \cite{Beffara}.
Dimension predicts the exponent in \eqref{eq:counting-measure}; it
does not identify a counting-measure limit. Lawler and Sheffield
introduced a Doob--Meyer construction of natural parametrization
\cite{LawlerSheffield}; Lawler and Zhou established its existence
throughout $0<\kappa<8$ \cite{LawlerZhou}. Lawler and Rezaei proved
that the resulting natural measure agrees, up to normalization, with
$d$-dimensional Minkowski content \cite{LawlerRezaeiContent}. We use
their interior content theorem and Euclidean Green-function estimates
\cite{LawlerRezaeiBasic}, and supply the whole-square boundary argument
in Section~\ref{sec:occupation}.

There are two particularly relevant precedents for passing from a
lattice curve to its natural measure. For critical triangular-lattice
site percolation, Garban, Pete, and Schramm constructed an interface
measure measurable from the limiting SLE$_6$ curve, with conformal
covariance exponent $7/4$ \cite{GPS}. Their normalization at mesh
$\eta$ is $\eta^2/\alpha_2^\eta(\eta,1)$, where
$\alpha_2^\eta(\eta,1)$ is the alternating two-arm probability.
Holden, Li, and Sun identified the resulting measure with Minkowski
content and proved convergence in natural parametrization \cite{HLS}.
The later sharp alternating-arm asymptotics of Du, Gao, Li, and Zhuang
\cite{DGLZ} permit that normalization to be replaced by a constant
times $\eta^{7/4}$. These are triangular-site percolation results,
not occupation results for square-lattice FK interfaces.

For square-lattice loop-erased random walk, Lawler and Viklund proved
convergence to SLE$_2$ when each step receives a deterministic constant
times the $5/4$ power of the mesh \cite{LawlerViklund}. Their result
uses model-specific Green-function estimates and is stated for bounded
simply connected domains with analytic boundary. In the FK setting,
the Ising case already has ordered-trace convergence to SLE$_{16/3}$
\cite{IsingInterfaces}. These precedents distinguish the geometric
trace limit from the microscopic estimates required for natural
measure convergence.

For the present FK model we take chordal ordered-trace convergence
from the companion manuscript \cite[Theorem \texttt{chordal:trace}]{FKCompanion}.
This is the trace input specified in Input~\ref{inp:trace}. We separately
identify the additional companion results used in the argument:
buffered FK comparison and passage estimates, the whole-plane loop
limit, and the square-row representation. They are cited at their
stated scopes in Section~\ref{sec:inputs}; none supplies the occupation
asymptotic proved here. The public crossing and arm-separation inputs
are due to Duminil-Copin, Manolescu, and Tassion \cite{DMT}, building
on the FK-Ising separation arguments of Chelkak, Duminil-Copin, and
Hongler \cite{CDCH}. The electric normalization is fixed by the
six-vertex height covariance \cite{DKLM,GFFCompanion}.

One parameter issue requires care. The angular-current proofs in the
GFF companion are presented in a parameter interval different from
the FK interval. In Section~\ref{sec:spectral} we apply their algebraic
arguments to the FK companion's row representation and verify the
required hypotheses there. We do not import an angular-spectrum
theorem outside its stated range.

\subsection{How the normalization is determined}

There are two distinct obstacles. First, one must obtain information
more precise than an arm exponent. Second, one must compare that
information with visits to a single medial edge. The proof separates
these tasks.

We orient loops and insert phases depending on their winding around
marked lattice points. Summing the orientations gives scalar loop
weights; for small charges these weights are positive. Regeneration
across surrounding open circuits gives quantitative loss of dependence
on the exterior and zero-free complex neighborhoods for the associated
partition functions. These estimates are developed in
Section~\ref{sec:radial}.

The height potential associated with small electric charges is nearly
harmonic, with a summable power error. The proof uses two directions
of row slicing: the untwisted current controls the potential away from
the line of insertions, and twisted rows control its differences near
that line. Step-two lattice potential theory then gives a Coulomb
correlation formula with a power remainder. Complex continuation
extends it to charges that impose arm events, and a reflection symmetry
removes the remaining linear ambiguity in the local normalization.
Sections~\ref{sec:spectral} and~\ref{sec:electric} contain this part of
the argument.

To reach a single lattice edge, we condition two opposite-color arms
to reach a large radius. Section~\ref{sec:coupling} constructs a
coupling that makes the exterior configurations identical with power
accuracy. The successful separating band has exactly two medial
strands crossing it, so its exterior continuation is independent of
the microscopic starting configuration. This last property is stronger
than separation of two chosen arms; it is what makes the coupling
useful for counting a distinguished interface.

Section~\ref{sec:calibration} compares two electric charges at a
mesoscopic separation with charges at neighboring lattice points.
The coupling applies even though the remaining electric factor is
signed. A nonzero mesoscopic amplitude controls the relative error,
and a summable comparison between scales $n$ and $N\in[n,2n]$ gives a
genuine microscopic amplitude. A second comparison, with a disk probe,
produces deterministic constants for replacing lattice visits by
neighborhood area.

Finally, Section~\ref{sec:occupation} performs this replacement in
$L^2$. Ordered-trace convergence identifies the disk probes, while
SLE Minkowski content identifies their small-radius limit. Integrable
boundary-strip estimates complete the passage to measures on the
closed square.

The potentially reusable ingredients are the complex stability of
radially tilted loop partition functions, the two-direction estimate
for the electric height potential, and the separating-band coupling
used to convert mesoscopic correlation amplitudes into microscopic
normalization. Their roles are quantitative: they remove errors that
an exponent-only argument cannot detect.
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//01-introduction.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//02-finite-rows.tex ====
\section{Finite rows and their scalar products}\label{sec:finite}

The first step is finite-dimensional.  We express every fixed word of
marked rows as a finite sum of determinants on the Bethe roots, with all
normalizing factors specified.  The construction of those roots and the
estimates needed to pass to infinite volume are deferred to
Section~\ref{sec:thermodynamic}.  In particular, no completeness assertion
for Bethe eigenvectors is used in this section.

\subsection{The physical row and the standard monodromy}

The algebra below holds for $0<\lambda<\pi$.  Its application to the part
of Theorem~\ref{thm:main} not covered by the known interval uses
$\pi/3<\lambda<\pi$.  Set
\begin{equation}\label{eq:spectral-coordinates}
 \lambda=\arccos(-\Delta),\qquad \eta=\lambda/\pi,\qquad
 v=\pi/2,\qquad x=i\eta z,\qquad w=e^{iz}.
\end{equation}
The physical time angle is $z=0$, increasing time points east, and the
space angle is $v$, increasing space points north.  Thus a common
imaginary translation of the $z$ arguments is a common real translation
of their $x$ arguments.  At a path step, the spin is $+1$ when the arrow
points along its left normal; the height increment is minus that spin.
These conventions fix the sign of every marked row.

Let $L$ be even and let $b_0,\ldots,b_{L-1}$ be real numbers.  On
$\mathcal H_L=(\C^2)^{\otimes L}$ use the orthonormal spin basis, and put
\[
 Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\qquad
 X=\begin{pmatrix}0&1\\1&0\end{pmatrix}.
\]
An auxiliary copy of $\C^2$ is denoted by the subscript $a$.  Products
over the sites below have increasing site index from left to right.  Define
\begin{align}
 R(u)&=
 \begin{pmatrix}
 \sin(\lambda-u)/\sin\lambda&0&0&0\\
 0&\sin u/\sin\lambda&1&0\\
 0&1&\sin u/\sin\lambda&0\\
 0&0&0&\sin(\lambda-u)/\sin\lambda
 \end{pmatrix},\label{eq:physical-R}\\
 M_{\rm ph}(x)&=\prod_{j=0}^{L-1}
 R_{aj}\bigl(\lambda/2+i(x-b_j)\bigr),\qquad
 T^D(x)=\tr_a\bigl(D_aM_{\rm ph}(x)\bigr),\label{eq:physical-row}
\end{align}
where $D$ is an arbitrary complex $2\times2$ matrix.  Its row index is
on the right tile of the horizontal seam and its column index on the
left.  The four indices in \eqref{eq:physical-R} send the top and left
step spins to the bottom and right step spins.  When $b_j=0$ and $x=0$,
dividing all vertex weights by $\sin(\lambda/2)/\sin\lambda$ gives
$a=b=1$ and $c=2\cos(\lambda/2)$, hence $\Delta=-\cos\lambda$.

Write $[u]=\sinh u$ and $s_\lambda=[i\lambda]=i\sin\lambda$.
The standard monodromy used in the algebraic Bethe ansatz is
\begin{align}
 L_{aj}(x-b_j)&=
 \begin{pmatrix}
 [x-b_j+i\lambda/2]/s_\lambda&0&0&0\\
 0&[x-b_j-i\lambda/2]/s_\lambda&1&0\\
 0&1&[x-b_j-i\lambda/2]/s_\lambda&0\\
 0&0&0&[x-b_j+i\lambda/2]/s_\lambda
 \end{pmatrix},\notag\\
 M(x)&=\prod_{j=0}^{L-1}L_{aj}(x-b_j)
      =\begin{pmatrix}\mathsf A(x)&\mathsf B(x)\\
                         \mathsf C(x)&\mathsf L(x)\end{pmatrix}_a.
 \label{eq:standard-monodromy}
\end{align}
In particular, the physical mixed diagonal entry has the opposite sign.
Let
\begin{equation}\label{eq:parity-gauges}
 \mathcal P_L=\prod_{j\ {\mathrm{even}}}Z_j,
 \qquad \mathcal Q_L=\prod_{j=0}^{L-1}Z_j.
\end{equation}

\begin{lemma}[Finite row algebra]\label{lem:row-algebra}
For even $L$, real $b_j$, and all complex spectral arguments,
\begin{equation}\label{eq:monodromy-gauge}
 M_{\rm ph}(x)=\mathcal P_L M(x)\mathcal P_L\mathcal Q_L,
 \qquad
 T^D(x)^*=T^{X\overline D X}(\bar x).
\end{equation}
The rows with a common diagonal twist commute.  More generally, put
\[
 \mathcal R(u)=
 \begin{pmatrix}
 [u+i\lambda]/s_\lambda&0&0&0\\
 0&[u]/s_\lambda&1&0\\
 0&1&[u]/s_\lambda&0\\
 0&0&0&[u+i\lambda]/s_\lambda
 \end{pmatrix},\qquad
 \mathcal R_{\rm ph}(u)=Z_a\mathcal R(u)Z_b.
\]
Then
\begin{equation}\label{eq:physical-RTT}
 \mathcal R_{{\rm ph},ab}(p-q)M_{{\rm ph},a}(p)M_{{\rm ph},b}(q)
 =M_{{\rm ph},b}(q)M_{{\rm ph},a}(p)
   \mathcal R_{{\rm ph},ab}(p-q).
\end{equation}
Where the intertwiner is invertible, exchanging two rows conjugates
their joint auxiliary mark by $\mathcal R_{\rm ph}(p-q)$.
For the homogeneous chain, $T^I(i\lambda/2)$ is the cyclic spin
translation and $T^I(-i\lambda/2)$ is its inverse.
\end{lemma}

\begin{proof}
The local identity is
$R_{aj}(\lambda/2+i(x-b_j))=Z_aL_{aj}(x-b_j)Z_j$.
Both local matrices commute with $Z_aZ_j$.  Moving the auxiliary $Z$'s
through the product, and using that $L$ is even, gives
\eqref{eq:monodromy-gauge}.  This also proves that the gauge is
independent of the roots, spectral arguments, and inhomogeneities.

If $N_{\alpha\beta}(x)$ is one auxiliary block of a physical local
matrix, direct conjugation of its entries gives
$N_{\alpha\beta}(x)^*=N_{\bar\alpha\bar\beta}(\bar x)$, where a bar on
a spin index exchanges its two values.  In an entry of a monodromy,
different local factors act on different quantum sites.  They therefore
commute when the adjoint reverses their order.  Consequently
\[
 (M_{{\rm ph},\alpha\beta}(x))^*
 =M_{{\rm ph},\bar\alpha\bar\beta}(\bar x).
\]
Taking the marked trace proves the second identity in
\eqref{eq:monodromy-gauge}; no reversal of the spatial inhomogeneities
is involved.

The standard six-vertex Yang--Baxter identity gives
$\mathcal R_{ab}(p-q)M_a(p)M_b(q)
=M_b(q)M_a(p)\mathcal R_{ab}(p-q)$.
It is a local identity of the three displayed six-vertex matrices:
spin conservation reduces its nonconstant entries to the addition
formula for $\sinh$.  Moreover,
\[
 M_{{\rm ph},a}(p)M_{{\rm ph},b}(q)
 =Z_b\mathcal P_L M_a(p)M_b(q)\mathcal P_L Z_b,
\]
and the analogous reversed product has $Z_a$ in place of $Z_b$.
Substitution yields \eqref{eq:physical-RTT}.  Taking auxiliary traces
proves the assertion about marks.  For a common diagonal matrix $D$,
$D\otimes D$ commutes with the intertwiner, so the two twisted rows
commute.  Finally $R(0)$ is the permutation matrix, and tracing a chain
of such permutations gives the cyclic translation.  The other endpoint
follows by the adjoint identity.
\end{proof}

\subsection{Bethe vectors, counting functions, and finite regularity}

Let $|0\rangle=|+\rangle^{\otimes L}$.  It is annihilated by
$\mathsf C(x)$, and its two diagonal eigenvalues are
\begin{equation}\label{eq:vacuum-functions}
 a(x)=\prod_{j=0}^{L-1}\frac{[x-b_j+i\lambda/2]}{s_\lambda},
 \qquad
 l(x)=\prod_{j=0}^{L-1}\frac{[x-b_j-i\lambda/2]}{s_\lambda}.
\end{equation}
The scalar functions in the creation-entry commutation rules are
\begin{equation}\label{eq:fgbeta}
 f(u)=\frac{[u+i\lambda]}{[u]},\qquad
 g(u)=\frac{s_\lambda}{[u]},\qquad
 \beta(u)=\frac{g(u)}{f(u)}=\frac{s_\lambda}{[u+i\lambda]}.
\end{equation}
In particular $\beta(0)=1$ and $1/f(0)=0$ by continuation.
For a finite list $U$ set
$|U\rangle=\prod_{u\in U}\mathsf B(u)|0\rangle$.
The creation entries commute, so this vector is symmetric in $U$.
Fix an integer $0\le n\le L$.  For a list
$Y=(y_1,\ldots,y_n)$ of distinct roots define
\begin{align}
 F_Y(x)&=\prod_{y\in Y}\frac{[y-x+i\lambda]}{[y-x]},\notag\\
 q_Y(x)&=\frac{l(x)}{a(x)}
          \prod_{y\in Y}\frac{[x-y+i\lambda]}{[x-y-i\lambda]},
 \label{eq:bethe-functions}\\
 \tau_t(x;Y)&=e^{it}a(x)F_Y(x)
                   \bigl(1+e^{-2it}q_Y(x)\bigr).
 \label{eq:bethe-eigenvalue}
\end{align}
Here $t$ is real near zero.  We say that $Y$ is on shell for $t$ when
\begin{equation}\label{eq:bethe-equations}
 q_Y(y_j)=-e^{2it}\qquad(1\le j\le n).
\end{equation}
The apparently singular eigenvalue in \eqref{eq:bethe-eigenvalue} then
has removable poles at the roots.

For real $u$ put
\begin{equation}\label{eq:bare-kernels}
 s(u)=\frac{\sin\lambda}{\pi(\cosh(2u)-\cos\lambda)},\qquad
 K(u)=\frac{\sin(2\lambda)}
               {\pi(\cosh(2u)-\cos(2\lambda))}.
\end{equation}
At $\lambda=\pi/2$, the second function is identically zero.  Let $S$
and $\Theta$ be their odd primitives, vanishing at zero.  Define
\begin{equation}\label{eq:counting-function}
 \xi_Y(u)=\sum_{j=0}^{L-1}S(u-b_j)-\sum_{k=1}^n\Theta(u-y_k).
\end{equation}
Consecutive real roots are solutions of
\begin{equation}\label{eq:consecutive-counting}
 \xi_Y(y_j)=j-\frac{n+1}{2}+\frac t\pi,
 \qquad y_1<\cdots<y_n.
\end{equation}

\begin{lemma}[Finite Bethe data]\label{lem:bethe-data}
Let $L$ be even, $b_j$ real, and $0<\lambda<\pi$.
A real solution of \eqref{eq:consecutive-counting} satisfies
\eqref{eq:bethe-equations}.  If its Bethe vector is nonzero, then
$|Y\rangle$ is an eigenvector of
$e^{it}\mathsf A(x)+e^{-it}\mathsf L(x)$ with eigenvalue
$\tau_t(x;Y)$, and $\mathcal P_L|Y\rangle$ is an eigenvector of the
physical twisted row with eigenvalue $(-1)^n\tau_t(x;Y)$.
The Jacobian of \eqref{eq:consecutive-counting} is
\begin{equation}\label{eq:Gaudin-matrix}
 G_{jk}=\xi_Y'(y_j)\delta_{jk}+K(y_j-y_k),\qquad
 \xi_Y'(u)=\sum_{a=0}^{L-1}s(u-b_a)-\sum_{k=1}^nK(u-y_k).
\end{equation}
If $\det G\ne0$, the roots have a real analytic continuation for real
$t$ near the given value, and on that branch
\begin{equation}\label{eq:twist-root-derivative}
 G\frac{\dd Y}{\dd t}=\frac1\pi\bbone.
\end{equation}
For $\lambda\ge\pi/2$, $G$ is positive definite at every real root
list.  Whenever $\xi_Y'(y_j)>0$, write
\begin{equation}\label{eq:Gaudin-weights}
 \ell_j=\frac1{\xi_Y'(y_j)},\qquad
 (K\ell)_{jk}=K(y_j-y_k)\ell_k;
 \qquad G\diag(\ell)=I+K\ell.
\end{equation}
These assertions concern finite matrices; they give no uniform bound on
$(I+K\ell)^{-1}$ as $L$ increases.
\end{lemma}

\begin{proof}
Logarithmic differentiation gives $q_Y'/q_Y=2\pi i\xi_Y'$.
Comparing the limits as $u\to+\infty$ fixes the integration constant:
\begin{equation}\label{eq:q-counting-phase}
 q_Y(u)=(-1)^n e^{2\pi i\xi_Y(u)}\qquad(u\in\R).
\end{equation}
Indeed $S(+\infty)=(1-\eta)/2$ and
$\Theta(+\infty)=(1-2\eta)/2$, while
$q_Y(+\infty)=e^{-i\lambda L+2i\lambda n}$ and $L$ is even.
Equation~\eqref{eq:consecutive-counting} now gives the Bethe equations.
The two diagonal action formulas proved in
Lemma~\ref{lem:finite-actions} below give the coefficient replacing a
root $p$ by $x$ as
\[
 g(x-p)\left(
 e^{it}a(p)\prod_{y\ne p}f(y-p)
 -e^{-it}l(p)\prod_{y\ne p}f(p-y)\right).
\]
The Bethe equation, with the self factor $-1$ removed, makes the
parenthesis zero.  The unchanged coefficient is
\eqref{eq:bethe-eigenvalue}.  The physical eigenvalue follows from
\eqref{eq:monodromy-gauge}, since $\mathcal Q_L=(-1)^n$ on the sector
with $n$ minus spins.

Differentiating the left side of
\eqref{eq:consecutive-counting} with respect to every root gives
\eqref{eq:Gaudin-matrix}; the $K(0)$ term cancels the derivative of the
self interaction.  The analytic implicit-function theorem gives the
local branch, and differentiation with respect to $t$ gives
\eqref{eq:twist-root-derivative}.  When $K\le0$,
\[
 G_{jj}=\sum_a s(y_j-b_a)+\sum_{k\ne j}|K(y_j-y_k)|
       >\sum_{k\ne j}|G_{jk}|.
\]
The matrix is real symmetric, so strict diagonal dominance proves
positive definiteness.  The formula for $G\diag(\ell)$ follows
entry by entry.
\end{proof}

In the remainder of the paper, a \emph{regular real on-shell list}
means a list of distinct real roots satisfying the Bethe equations,
with $\xi_Y'(y_j)>0$ for every $j$ and $\det G_Y\ne0$.
Section~\ref{sec:thermodynamic} constructs lists with these properties.

We also record the full-line kernels that will be used to compare the
finite matrices with convolution.  With
$\widehat f(p)=\int_{\R}e^{-ipu}f(u)\dd u$,
\begin{align}
 \widehat s(p)&=\frac{\sinh((\pi-\lambda)p/2)}{\sinh(\pi p/2)},&
 \widehat K(p)&=\frac{\sinh((\pi-2\lambda)p/2)}{\sinh(\pi p/2)},
 \label{eq:kernel-transforms}\\
 h:=\|K\|_{L^1(\R)}&=|1-2\eta|<1,&
 r(u):=(I+K*)^{-1}s(u)&=\frac1{2\lambda\cosh(u/\eta)}.
 \label{eq:dressed-density}
\end{align}
Here $K*$ denotes convolution with $K$.  To verify the transforms,
integrate $e^{-ipu}\sin\theta/[\pi(\cosh(2u)-\cos\theta)]$ around a
rectangle of height $i\pi$, for $0<\theta<2\pi$.  The two residues at
$i\theta/2$ and $i(\pi-\theta/2)$ give
$\sinh((\pi-\theta)p/2)/\sinh(\pi p/2)$; coincident or zero values
follow by continuity.  Take $\theta=\lambda,2\lambda$.
The kernel $K$ has constant sign, proving the $L^1$ formula, and
$\widehat s/(1+\widehat K)=1/(2\cosh(\lambda p/2))$ proves the
formula for $r$.  In particular $\int r=1/2$.
We write
\begin{equation}\label{eq:full-line-resolvent}
 R_0*=K*(I+K*)^{-1};
 \qquad R_0+K*R_0=K.
\end{equation}
The inverse exists, for example on bounded functions, by the Neumann
series since $h<1$.

\subsection{Exact actions on products of creation entries}

For the next formulas all arguments are initially distinct and avoid
the zeros of their displayed denominators.  They are identities of
meromorphic functions; coincident cases are obtained by taking the
limit of the complete expression.  We use the scalar functions
$f,g,\beta$ defined in \eqref{eq:fgbeta}.

\begin{lemma}[Single-entry actions]\label{lem:finite-actions}
For a finite spectral set $U$, put $E=U\cup\{v'\}$.  Then
\begin{align}
 \mathsf B(v')|U\rangle&=|E\rangle,\notag\\
 \mathsf A(v')|U\rangle
 &=\sum_{p\in E}a(p)\beta(v'-p)
          \prod_{u\in E\setminus\{p\}}f(u-p)
             |E\setminus\{p\}\rangle,\label{eq:A-action}\\
 \mathsf L(v')|U\rangle
 &=\sum_{p\in E}l(p)\beta(p-v')
          \prod_{u\in E\setminus\{p\}}f(p-u)
             |E\setminus\{p\}\rangle,\label{eq:L-action}\\
 \mathsf C(v')|U\rangle
 &=\sum_{\substack{p,q\in E\\p\ne q}}
       a(p)l(q)\beta(v'-p)\beta(q-v')f(q-p)
       \prod_{u\in E\setminus\{p,q\}}f(u-p)f(q-u)
             |E\setminus\{p,q\}\rangle.\label{eq:C-action}
\end{align}
The last sum is over ordered pairs: $p$ belongs to the upper diagonal
factor and $q$ to the lower diagonal factor.  Thus a word of $m$ entries
is an explicit sum over paths that add the new argument at each step
and remove zero, one, or two elements.  If the final set has the
original cardinality, its sets of removed original roots and surviving
added arguments have equal size, at most $m$.
\end{lemma}

\begin{proof}
The standard RTT relation gives commuting creation entries and
\begin{align*}
 \mathsf A(p)\mathsf B(u)
  &=f(u-p)\mathsf B(u)\mathsf A(p)
        +g(p-u)\mathsf B(p)\mathsf A(u),\\
 \mathsf L(p)\mathsf B(u)
  &=f(p-u)\mathsf B(u)\mathsf L(p)
        +g(u-p)\mathsf B(p)\mathsf L(u),\\
 [\mathsf C(p),\mathsf B(u)]
  &=g(p-u)\bigl(\mathsf A(u)\mathsf L(p)
                    -\mathsf A(p)\mathsf L(u)\bigr).
\end{align*}
Move the diagonal entry successively through the creation entries and
apply its vacuum value.  Combining terms that replace the same root
uses
\[
 f(u-h)-f(u-p)=g(p-u)\frac{[p-h]}{[u-h]}.
\]
This gives \eqref{eq:A-action}; reversing differences gives
\eqref{eq:L-action}.  These formulas include the unchanged term $p=v'$
because $\beta(0)=1$.

For completeness, the two-removal induction can be checked before any
on-shell specialization.  Strip one creation at $u$ from the product,
use the commutator, and apply the two diagonal formulas.  For fixed
upper and lower removals $h,k$, divide the coefficients by
\[
 a(h)l(k)f(k-h)\prod_{s\ {\mathrm{remaining}}}f(s-h)f(k-s).
\]
The required addition of the three resulting contributions is
\begin{align*}
 \beta(p-h)\beta(k-p)
 &=\frac{\beta(p-h)\beta(k-p)}{f(u-h)f(k-u)}\\
 &\quad+g(p-u)\left(
       \frac{\beta(u-h)\beta(k-p)}{f(k-u)}
       -\frac{\beta(p-h)\beta(k-u)}{f(k-p)}\right).
\end{align*}
Substituting \eqref{eq:fgbeta} and multiplying by the displayed
denominators reduces this identity to the addition formula for
$\sinh$.  The initial case is $\mathsf C(p)|0\rangle=0$; for one
creation the two ordered removals give the displayed commutator on the
vacuum.  The induction proves \eqref{eq:C-action}, including cases
where $p$ or $q$ is the newly added argument.  Finally, there are only
$m$ newly added arguments.  If the final cardinality equals the initial
one, the number of absent original roots equals the number of
surviving added arguments and is therefore at most $m$.
\end{proof}

We now have an exact expansion of a finite row word into off-shell
Bethe vectors.  The remaining finite-dimensional step is to take their
scalar products with the initial vector and cancel the vacuum factors.

\subsection{Slavnov's identity with its normalization factors}

Let $Y$ be on shell for $t$ and have $n$ distinct real roots.  For a
list $U$ of length $n$, define the algebraic scalar product
\begin{equation}\label{eq:ABA-scalar}
 S(Y,U)=\langle0|\prod_{y\in Y}\mathsf C(y)
                         \prod_{u\in U}\mathsf B(u)|0\rangle.
\end{equation}
It is not defined using a Hermitian adjoint; that identification will
be proved separately.  Write $C(Y,U)_{jk}=1/[U_k-y_j]$, and put
\begin{align}
 m_u^t(y)&=-\frac1{2\pi i}\bigl[(1+e^{-2it}q_Y(u))\coth(u-y)
                 \notag\\
 &\hspace{24mm}-\coth(u-y-i\lambda)
                 -e^{-2it}q_Y(u)\coth(u-y+i\lambda)\bigr],
 \label{eq:finite-source-column}\\
 M^t(Y,U)_{jk}&=m_{U_k}^t(y_j).
 \label{eq:reduced-scalar-matrix}
\end{align}
All root coincidences in these matrices mean analytic limits along
the full formula.  In particular
\begin{equation}\label{eq:source-Gaudin-limit}
 M^t(Y,Y)=G_Y.
\end{equation}

\begin{lemma}[Exact scalar products and deleted columns]
\label{lem:scalar-product}
In the normalization \eqref{eq:standard-monodromy},
\begin{equation}\label{eq:Slavnov-exact}
 S(Y,U)=e^{-int}\left(\prod_{y\in Y}l(y)\right)
 \frac{\det_{j,k}\partial_{y_j}\tau_t(U_k;Y)}
      {\det C(Y,U)}.
\end{equation}
Consequently, if $\det G_Y\ne0$, then $S(Y,Y)\ne0$ and
\begin{equation}\label{eq:Gaudin-norm-exact}
 S(Y,Y)=(2\pi i s_\lambda)^n
       \left(\prod_{y\in Y}a(y)l(y)\right)
       \left(\prod_{j\ne k}f(y_j-y_k)\right)\det G_Y.
\end{equation}
Suppose $U$ is obtained by replacing the roots indexed by
$H=(p_1,\ldots,p_k)$ by $V=(v'_1,\ldots,v'_k)$, in those same slots.
Set
\[
 A_Y(p)=a(p)\prod_{y\in Y\setminus\{p\}}f(y-p)\quad(p\in Y).
\]
Then the normalized scalar product is exactly
\begin{equation}\label{eq:deleted-scalar-product}
 \frac{S(Y,U)}{S(Y,Y)}
 =(-s_\lambda)^{-k}
   \frac{\prod_{v'\in V}a(v')F_Y(v')}{\prod_{p\in H}A_Y(p)}
   \frac{\det\bigl((G_Y^{-1}m_{v'}^t)_p\bigr)_{H,V}}
        {\det\bigl(1/[v'-p]\bigr)_{H,V}}.
\end{equation}
If the positive weights \eqref{eq:Gaudin-weights} are defined, the
last numerator equals
\begin{equation}\label{eq:W-determinant}
 \det\bigl((G_Y^{-1}m_{v'}^t)_p\bigr)_{H,V}
 =\left(\prod_{p\in H}\ell_p\right)
       \det\bigl(W_{v'}(p)\bigr)_{H,V},
 \qquad W_{v'}=(I+K\ell)^{-1}m_{v'}^t.
\end{equation}
Empty products and determinants have value one.
\end{lemma}

\begin{proof}
We apply the trigonometric scalar-product theorem to the fundamental
six-vertex monodromy \eqref{eq:standard-monodromy}, with highest vector
$|0\rangle$, vacuum functions $a,l$, and diagonal twist
$\kappa=e^{-2it}$.  In this unnormalized monodromy convention the
prefactor is $\prod_{y\in Y}l(y)$ for the eigenvalue
$\mathsf A+\kappa\mathsf L$; this is
\cite[Proposition~3.1, equations~(3.8)--(3.9)]{KMST2005}.
Replacing that eigenvalue by $\tau_t=e^{it}\tau_\kappa$ multiplies its
$n$ derivative columns by $e^{it}$ and therefore gives the compensating
factor $e^{-int}$ in \eqref{eq:Slavnov-exact}.  The untwisted
derivative-over-Cauchy version, with its equal-spin-normalized
monodromy, is \cite[Theorem~3.1, equations~(3.4)--(3.5)]{KMT1999}.
These are finite scalar-product identities; they make no assertion
about completeness or about a thermodynamic limit.

Differentiation of \eqref{eq:bethe-eigenvalue} gives
\[
 \partial_{y_j}\tau_t(u;Y)
     =e^{it}a(u)F_Y(u)(-2\pi i)m_u^t(y_j).
\]
The factor $e^{int}$ cancels that in
\eqref{eq:Slavnov-exact}.  When $u\to y_k$, the product
$(1+e^{-2it}q_Y(u))\coth(u-y_k)$ tends to
$-2\pi i\xi_Y'(y_k)$, because
$q_Y'(y_k)=-2\pi i e^{2it}\xi_Y'(y_k)$.
The two remaining coth terms give $K(y_k-y_j)$ after division by
$-2\pi i$.  This proves \eqref{eq:source-Gaudin-limit}, including its
diagonal.

At a root $p$ the residue of the extracted vacuum factor is
\begin{equation}\label{eq:vacuum-residue}
 \lim_{u\to p}[u-p]a(u)F_Y(u)=-s_\lambda A_Y(p).
\end{equation}
Cancel these poles with the corresponding Cauchy poles.  If every
column tends to its root, the remaining determinant is $\det G_Y$,
giving \eqref{eq:Gaudin-norm-exact}.  For partial coincidence first
cancel the unchanged columns.  The determinant of the matrix obtained
from $G_Y$ by replacing columns $H$ by $m_{v'}^t$, divided by
$\det G_Y$, is
$\det((G_Y^{-1}m_{v'}^t)_p)_{H,V}$: multiply the matrix on the left by
$G_Y^{-1}$ and expand along its unchanged unit columns.  Expanding the
Cauchy determinant along its unchanged poles leaves exactly
$\det(1/[v'-p])_{H,V}$ with the same slot order.  Each deleted root
loses the factor in \eqref{eq:vacuum-residue}.  This proves
\eqref{eq:deleted-scalar-product}, including its sign and constant.
Finally $G_Y^{-1}=\diag(\ell)(I+K\ell)^{-1}$ proves
\eqref{eq:W-determinant}.

All factors in the norm outside $\det G_Y$ are nonzero: real $y,b$
give $[y-b\pm i\lambda/2]\ne0$, and distinct real roots give
$[y-y']\ne0$ and $[y-y'\pm i\lambda]\ne0$.
The scalar-product theorem is first used at generic variables.
At any regular on-shell specialization, the nonsingular counting
Jacobian permits a local analytic perturbation of the roots along
with the parameters.  Meromorphic continuation and the pole
cancellations above then give the asserted formulas there as well.
In particular, no exclusion at a root of unity is necessary.
\end{proof}

\subsection{Hilbert duality and exact cancellation along a row word}

\begin{lemma}[The Hilbert dual]\label{lem:hilbert-dual}
For even $L$ and real inhomogeneities,
\begin{equation}\label{eq:B-Hilbert-adjoint}
 \mathsf B(x)^*=-\mathsf C(\bar x).
\end{equation}
Thus for a distinct real on-shell list with $\det G_Y\ne0$, the
physical unit vector
\begin{equation}\label{eq:physical-Bethe-vector}
 \Omega_Y=\frac{\mathcal P_L|Y\rangle}{\|\mathcal P_L|Y\rangle\|}
\end{equation}
is well defined, and
\[
 \|\mathcal P_L|Y\rangle\|^2=(-1)^nS(Y,Y)>0.
\]
If $Y,Z$ are two such real on-shell lists of the same size, possibly
for different twists, then
\begin{equation}\label{eq:two-orientation-overlap}
 |\langle\Omega_Y,\Omega_Z\rangle|^2
       =\frac{S(Y,Z)S(Z,Y)}{S(Y,Y)S(Z,Z)}.
\end{equation}
These identities also hold for real staggered $b_j$, without a Perron
interpretation of the vectors.
\end{lemma}

\begin{proof}
Polarize the adjoint identity in Lemma~\ref{lem:row-algebra} over the
four auxiliary marks.  It gives
$\mathsf B_{\rm ph}(x)^*=\mathsf C_{\rm ph}(\bar x)$.
Using the gauge identity and commuting $\mathcal P_L$ with
$\mathcal Q_L$ gives
$\mathsf B(x)^*=\mathcal Q_L\mathsf C(\bar x)\mathcal Q_L$.
Since $\mathsf C$ changes the number of minus spins by one, it
anticommutes with $\mathcal Q_L$, proving
\eqref{eq:B-Hilbert-adjoint}.  Creation entries commute, so for real
roots
\[
 (\mathcal P_L|Y\rangle)^*
       =(-1)^n\langle0|\prod_{y\in Y}\mathsf C(y)\mathcal P_L.
\]
Lemma~\ref{lem:scalar-product} makes this norm nonzero, and positivity
is then Hilbert-space positivity.  Applying the same identity to both
cross scalar products proves \eqref{eq:two-orientation-overlap}.
Absorbing the twist into the monodromy merely replaces
$\mathsf B$ by $e^{it}\mathsf B$ and $\mathsf C$ by
$e^{-it}\mathsf C$, preserving the adjoint relation and the normalized
ratio.
\end{proof}

There is a useful explicit form of the cancellation in
\eqref{eq:two-orientation-overlap}.  Denote the twists of $Y,Z$ by
$t_Y,t_Z$, respectively.  Put
\[
 \Delta(Y)=\prod_{j<k}[y_k-y_j],\qquad
 \mathcal H(Y,U)=\prod_{j,k}[y_j-U_k+i\lambda],\qquad
 c_n=(2\pi i)^n(-1)^{n(n-1)/2}.
\]
The trigonometric Cauchy identity rewrites
\eqref{eq:Slavnov-exact} as
\begin{equation}\label{eq:Slavnov-separated}
 S(Y,U)=c_n\left(\prod_{y\in Y}l(y)\right)
             \left(\prod_{u\in U}a(u)\right)
       \frac{\mathcal H(Y,U)}{\Delta(Y)\Delta(U)}
             \det M^{t_Y}(Y,U).
\end{equation}
Consequently all one-set factors cancel separately, leaving
\begin{equation}\label{eq:overlap-determinant-finite}
 |\langle\Omega_Y,\Omega_Z\rangle|^2
 =\frac{\mathcal H(Y,Z)\mathcal H(Z,Y)}
        {\mathcal H(Y,Y)\mathcal H(Z,Z)}
   \frac{\det M^{t_Y}(Y,Z)\det M^{t_Z}(Z,Y)}
        {\det G_Y\det G_Z}.
\end{equation}
This is a finite identity, including the full shifted pair product.
Along a finite regular twist branch the overlap is nonzero near zero,
since its value at zero is one.  Its local logarithm and derivatives
there are therefore well defined.  Neither changing the phases of the
unit vectors nor changing the logarithm branch changes the quadratic
coefficient.  No neighborhood uniform in $L$ is asserted here.

We finally make the cancellation for an arbitrary fixed row word just
as explicit.  In the remainder of this subsection $t=0$, $Y$ is a
regular real on-shell list, and the spectral arguments $v'_1,\ldots,v'_m$
are distinct and generic.  Define normalized physical rows by
\begin{equation}\label{eq:normalized-finite-row}
 A_L^D(x)=\frac{T^D(x)}{(-1)^n\tau_0(x;Y)},\qquad
 P_L=A_L^I,\qquad J_L=A_L^Z,
\end{equation}
on the open set where the denominator is nonzero.  Then
$P_L(x)\Omega_Y=\Omega_Y$.  Holomorphic continuation and norm bounds
for these rows will be proved using the estimates of
Section~\ref{sec:thermodynamic}.

For each path in Lemma~\ref{lem:finite-actions}, let $T$ be the set
remaining just after its current removal and define
\begin{align}
 U_Y(T,p)&=
  \frac{\prod_{u\in T\setminus Y}f(u-p)}
       {\prod_{y\in Y\setminus(T\cup\{p\})}f(y-p)},&
 L_Y(T,p)&=
  \frac{\prod_{u\in T\setminus Y}f(p-u)}
       {\prod_{y\in Y\setminus(T\cup\{p\})}f(p-y)},
 \label{eq:path-quotients}\\
 \chi_Y(p)&=
  \begin{cases}1,&p\in Y,\\q_Y(p),&p\notin Y.\end{cases}
 \label{eq:path-lower-factor}
\end{align}
Associate with an action at $v'$ the correction factor
\begin{equation}\label{eq:path-correction}
 \gamma=
 \begin{cases}
  1,&\mathsf B\text{ (no removal)},\\
  \beta(v'-p)U_Y(T,p),&\mathsf A\text{ (remove }p),\\
  \chi_Y(p)\beta(p-v')L_Y(T,p),&\mathsf L\text{ (remove }p),\\
  \chi_Y(q)\beta(v'-p)\beta(q-v')f(q-p)
        U_Y(T,p)L_Y(T,q),&\mathsf C\text{ (remove }p,q).
 \end{cases}
\end{equation}
These are explicit rational functions of hyperbolic sines.  The only
dependence on the original root set outside the factors displayed in
\eqref{eq:path-quotients} is through $q_Y$ at added arguments.

\begin{corollary}[Exact finite-word expansion]\label{cor:finite-word-expansion}
Expand each auxiliary mark as
\[
 T^{D_j}=\sum_{\alpha,\beta}(D_j)_{\beta\alpha}
 M_{{\rm ph},\alpha\beta}.
\]
For each resulting entry word, let $n_j$
be the number of minus spins after its $j$th action, with $n_0=n$.
Entry words with $n_m\ne n$ have zero expectation in $\Omega_Y$.
For a path of a word with $n_m=n$, let $H\subset Y$ be the roots
absent at the end and let $V$ be its surviving added arguments, ordered
in the same replacement slots.  Then $|H|=|V|=:k\le m$, and its
contribution to the normalized physical expectation is
\begin{equation}\label{eq:exact-path-determinant}
 \left(\prod_{j=1}^m(D_j)_{\beta_j\alpha_j}\right)
 (-1)^{\sum_{j=1}^m(n_{j-1}-n)}
 (-s_\lambda)^{-k}
 \frac{\prod_{j=1}^m\gamma_j}
      {\prod_{j=1}^m(1+q_Y(v'_j))}
 \frac{\det(W_{v'}(p))_{H,V}}
      {\det(1/[v'-p])_{H,V}}
 \prod_{p\in H}\ell_p.
\end{equation}
The expectation is the sum of these contributions.  In particular,
the determinant size is bounded by the word length.  For diagonal
entry words the parity sign in \eqref{eq:exact-path-determinant} is
one.  Among distinct removed original roots the correction product
contains only reciprocal $f$ factors, with one for each pair.  Each
removed original root also contributes one factor $\beta$ joining it
to the row argument at the step that removes it.
\end{corollary}

\begin{proof}
For this proof extend the vacuum base to added arguments by
\[
 \mathcal A_Y(p)=
 \begin{cases}
  A_Y(p),&p\in Y,\\
  a(p)F_Y(p),&p\notin Y.
 \end{cases}
\]
The coefficient of an upper removal is
$\mathcal A_Y(p)\beta(v'-p)U_Y(T,p)$.
The corresponding lower coefficient is
$\mathcal A_Y(p)\chi_Y(p)\beta(p-v')L_Y(T,p)$:
on a root the equality of the two vacuum bases is precisely
\eqref{eq:bethe-equations} at $t=0$, including its self factor $-1$;
off a root their ratio is $q_Y(p)$.
For two removals the extra factor is $f(q-p)$, exactly as in
\eqref{eq:C-action}.  Thus the coefficient of a path before taking its
scalar product is the product of its corrections $\gamma_j$ times one
$\mathcal A_Y(p)$ for every removed element.

Every original root is either retained or removed once.  Similarly,
each new spectral argument is either retained in $V$ or removed once,
possibly in the very action that introduced it.  Consequently the
product of vacuum bases along the path is
\[
 \left(\prod_{p\in H}A_Y(p)\right)
 \left(\prod_{v'\in\{v'_1,\ldots,v'_m\}\setminus V}
                       a(v')F_Y(v')\right).
\]
Multiply by \eqref{eq:deleted-scalar-product}.  The deleted-root bases
cancel, and the surviving arguments supply the missing vacuum bases.
Dividing by $\prod_j\tau_0(v'_j;Y)$ leaves exactly
$\prod_j(1+q_Y(v'_j))^{-1}$ and $(-s_\lambda)^{-k}$.
Equation~\eqref{eq:W-determinant} supplies the determinant and weights.

The Hilbert dual contributes the same $(-1)^n$ to the scalar product
and norm, so it cancels by Lemma~\ref{lem:hilbert-dual}.  Each physical
entry, after division by $(-1)^n$, contributes
$(-1)^{n_{j-1}-n}$ from $\mathcal Q_L$ on its input sector; adjacent
$\mathcal P_L$ factors cancel.  This proves the explicit parity sign.
Different final spin sectors are orthogonal, proving the zero
assertion.

Finally consider a pair of deleted original roots.  If removed at
different steps, the earlier root occurs in the denominator of
\eqref{eq:path-quotients} at the later removal.  If removed together,
the two denominators contribute $f(q-p)^{-2}$ and the extra factor
$f(q-p)$ leaves one reciprocal.  There are no other factors involving
two original roots.  The factor $\beta$ at each original removal is
explicit in \eqref{eq:path-correction}.  This proves the last claims.
\end{proof}

For real $u$, $|f(u)|^2=1+\sin^2\lambda/\sinh^2u\ge1$.
Thus the reciprocal factors between deleted roots cause no collision
growth, and for a bounded generic added argument the corresponding
$\beta$ factor decays like $e^{-|p|}$ as its real root $p$ escapes.
These observations alone do not control simultaneous escape in the
determinant of \eqref{eq:exact-path-determinant}.  The next section
establishes the necessary density, inverse, and determinant estimates;
it will apply the exact finite formula just proved.
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//02-finite-rows.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//02-inputs.tex ====
\section{Model conventions and inputs}\label{sec:inputs}

Throughout the proof $q\in[1,4)$ is fixed. Constants may depend on $q$,
on a compact set of charge parameters, and on the stated geometric
separations. No estimate is asserted uniformly as $q\uparrow4$.
Write
\[
 m=\sqrt q=2\cos\lambda,\qquad
 \rho=\frac{\lambda}{\pi}\in(0,1/3],\qquad
 \kappa=\frac{4\pi}{\pi-\lambda}.
\]
We use unit square tiles in the intermediate arguments. Their corners
alternate between primal and dual sites, and the primal edges are tile
diagonals. Thus the primal mesh is $\sqrt2$ times the tile-edge length.
Each tile has one of the two noncrossing pairings of its four side
midpoints. An actual primal or dual path uses open edges of that color;
boundary identifications are not path edges. This distinction is essential
when a crossing must reach a wall or separate two arm clusters.

The two arcs in each tile can be drawn disjointly, with their order at
successive ports retained. This rounded drawing and the usual medial
drawing have uniform distance $O(1)$ in tile units under corresponding
parametrizations. In physical coordinates their distance is $O(n^{-1})$.
When a strand visits a medial vertex twice, the two occurrences retain
their traversal order. For a simply connected domain, the usual
Dobrushin completion pairs the boundary ports along the wired and free
arcs. More generally, a positive cap completion wires each designated
primal boundary interval separately and adds no primal identification
on a dual interval. The switch law with weight $m$ per completed loop
is then exactly the critical FK law, up to a deterministic factor.
Indeed, Euler's formula gives
\[
 \#\{\text{completed loops}\}=2k_\xi(\omega)+|\omega|+C,
 \qquad m^2=q,\qquad \frac{p_q}{1-p_q}=m.
\]
Consequently all conditional laws obtained by exposing switches have
the FK domain Markov property with their induced boundary partitions.
These conventions are those of \cite[Section ``Tiles, positive laws,
and localization,'' Lemma ``The positive loop law'']{FKCompanion}.

\subsection{Trace and local FK estimates}

\begin{standinginput}[Chordal trace convergence]\label{inp:trace}
For the marked uniform polygon approximations of a bounded marked
Jordan domain used in Theorem~\ref{thm:main}, the critical Dobrushin
interface converges in law to chordal $\mathrm{SLE}_\kappa$ in the
uniform Euclidean oriented-curve topology modulo increasing
reparametrization.
\end{standinginput}

This is precisely \cite[Theorem ``Dobrushin convergence for
$1\le q<4$,'' label \texttt{chordal:trace}]{FKCompanion}. We use it
as the trace input, without deriving any assertion about occupation
measures from trace convergence alone.

Here and below, a buffer has fixed positive relative width and at least
a sufficiently large fixed number of mesh layers. A collar is a full
annulus, or one of the fixed rectangular or straight-wall sector
geometries explicitly described below. The graph and deterministic
wire portions are fixed inside the collar; only the partition transmitted
from the unobserved side varies. Only the number of prescribed
deterministic through-wire blocks is bounded, and no additional
identification jumps the collar. The exterior-induced partition on
the unobserved side is arbitrary: its number of blocks is not bounded.
Nor is any bound imposed on the number of random open components
crossing the collar. The full annuli used for inward comparison have
no prescribed through-wires.

\begin{standinginput}[FK localization]\label{inp:fk}
The following estimates hold for critical square-lattice FK at the
fixed parameter $q$.
\begin{enumerate}[label=(\roman*),leftmargin=*]
\item An actual crossing of a discrete simple-polygon quadrilateral
with bounded extremal distance has probability bounded below uniformly
in the boundary partition. The dual statement holds as well. Arm
separation, quasi-multiplicativity, and localization to fixed separated
landing intervals hold for each fixed color word, in particular for
one arm and for a primal--dual pair.
\item If two positive laws agree on a collar between radii $r$ and
$16r$, their marginals beyond its fixed side have mutually bounded
densities. Equivalently, expectations of every nonnegative function
on the remote edge set are comparable. This holds for a full annulus,
for a bounded number of sectors separated by straight coordinate walls
with matching deterministic wire portions, and for a fixed-shape
rectangular bulk collar. Coincident banks of a slit are treated as
distinct boundary incidences.
\item Through $k$ sufficiently separated full annuli, the ratio of
remote expectations improves to $1+O((1-c)^k)$, or both expectations
are zero. The assertion holds in either radial direction and after
positive modifications supported on the unobserved side.
\item If $X$ counts disjoint raw strand passages through a fixed
buffered annulus or one of the preceding sectors, then, for every
fixed $B<\infty$,
\[
 \E\bigl[e^{BX}\mid\text{exterior}\bigr]\le C_B.
\]
The bound is uniform in scale and in the exterior-induced boundary
partition. It also holds for a fixed finite family of such counts.
\item Let $r_j$ be dyadic radii and $N_j$ the number of raw nests wholly
inside $r_j$, in a geometry extending beyond the largest cutoff by a
buffer. A bulk nest is a closed loop surrounding the marked point.
The increment satisfies $0\le N_j-N_{j-1}\le X_j$, where the buffer
of $X_j$ is disjoint from $B_{r_{j-k_0}}$ for a fixed $k_0$.
For $b>0$, set $F_j=\E b^{N_j}$ and
$\dd\nu_j=F_j^{-1}b^{N_j}\dd\Pp$. Then
\[
 C_b^{-1}\le\frac{F_j}{F_{j-1}}\le C_b,
 \qquad \E_{\nu_j}e^{BX_i}\le C_{b,B}\quad(i\ge j).
\]
These statements remain valid conditionally on exterior data of the
specified geometry, and with a fixed family of buffered passage counts
adjoined at or beyond the outer tilt level. The constants can be chosen
uniformly for $b$ in a compact subset of $(0,\infty)$.
\end{enumerate}
\end{standinginput}

Part~(i) is \cite[Theorem~1.2 and Propositions~6.2, 6.3, and~6.5]{DMT}.
The last three propositions apply to fixed color words, including the
two words needed here. Parts~(ii)--(v) are respectively the lemmas
``Bounded densities across a collar,'' ``Relative comparison,''
``Exponential passage moments,'' and ``Local tilt bounds'' in
\cite[Section ``Tiles, positive laws, and localization'']{FKCompanion};
their source labels are \texttt{prelim:collar},
\texttt{prelim:relative-collar}, \texttt{prelim:passages}, and
\texttt{prelim:tilts}. The compact-uniform assertion in Part~(v)
follows from those proofs by bounding $|\log b|$ on the compact set.
We require Part~(iii) only in full annuli. In particular, these inputs
do not assert comparison across arbitrary irregular walls or under
complex-valued tilts.

The separation argument with high probability used in
Section~\ref{sec:coupling} is the actual-cluster construction of
\cite[Lemma~5.9]{CDCH}, whose original statement concerns FK-Ising.
For our range of $q$, its conditional crossing steps are supplied by
\cite[Theorem~1.2]{DMT}, as in the proof of that paper's
Proposition~6.2. Section~\ref{sec:coupling} specifies the extra buffers
and the arm continuations needed for the coupling; we do not identify
ordinary separation up to constants with that stronger conclusion.

The free and wired plane laws coincide, are invariant under lattice
symmetries, and are exchanged by primal--dual symmetry at these
self-dual weights; see \cite[Theorem~1.1]{DCLM} and its square-lattice
specialization in \cite[Section ``Crossing estimates and finite plane
loops'']{FKCompanion}. Buffered circuits of both colors show that
every plane medial strand is almost surely a finite loop. A translation
by one tile may interchange the two colors, but preserves this loop law.

\subsection{Plane loops and angular rows}

\begin{standinginput}[The nested plane loop limit]\label{inp:loops}
The entire loop collection of the free infinite-volume critical
square-lattice FK law converges, through the full mesh sequence, to
nested whole-plane $\mathrm{CLE}_\kappa$. Convergence is in spherical
loop-matching topology, with curves compared modulo their traversal
parametrizations, and includes all nested generations. The limit is
invariant under similarities and, more generally, M\"obius maps.
\end{standinginput}

This is the separately cited Theorem ``Nested full-plane convergence
for $q<4$,'' label \texttt{loop:plane}, in
\cite[Section ``From squares to the free plane law and the
sphere'']{FKCompanion}. Its statement concerns exactly the plane law
used here. It is stronger than Input~\ref{inp:trace}, and is used for
the finite-resolution nesting limits in Sections~\ref{sec:radial}
and~\ref{sec:calibration}. Passage moments from Input~\ref{inp:fk}
provide the required uniform integrability when nesting functionals
are unbounded.

Orienting a completed contractible loop and assigning total turning
weights $e^{i\lambda}$ and $e^{-i\lambda}$ recovers its unoriented
weight $m$. Summing the local orientations gives the homogeneous
six-vertex weights
\[
 a=b=1,\qquad c=2\cos(\lambda/2),\qquad
 \Delta=\frac{2-c^2}{2}=-\cos\lambda.
\]
The induced height changes by one across an arrow. The plane arrow
law is the zero-slope law obtained by first taking the longitudinal
limit of balanced even-width tori and then letting the width tend to
infinity. Here balance means zero total arrow flux across a transverse
section. The identification of finite local arrow and phase words
with the FK loop expansion is part of the plane-kernel construction
in \cite[Section ``Row algebra and the infinite-plane state'']{FKCompanion}.
It agrees with the balanced-plane law in \cite[Theorem~2.2 and
Equation~(18)]{DKLM}.

\begin{standinginput}[Physical angular representation]\label{inp:rows}
At $0<\lambda\le\pi/3$, the normalized homogeneous rows have a plane
Hilbert-space representation with vacuum $\Omega$. Finite ordinary
spin and diagonal phase words have the expectations of the preceding
physical plane law. With the angular coordinate normalized so that
the strip is $|\Rea z|<\pi/2$, marked rows $A^D(z)$ are bounded and
holomorphic, satisfy $\|A^D(z)\|\le2\|D\|$, and obey
\[
 A^D(z+ib)=U_bA^D(z)U_b^{-1},\qquad U_b=e^{ibB},
\]
where $B$ is self-adjoint. For a diagonal unitary twist, the normalized
commuting row family has scalar spectral functions which, under
$w=e^{iz}$, are inner in the right half-plane. They continue without
zeros through a common-width band at each strip edge. In particular,
their zeros lie in a fixed cone $\Rea w_j\ge c_*|w_j|$, and their
general factorization has the form
\[
 f(w)=\eta\exp(-a_*w-b_*/w)\prod_j\beta_j(w),\qquad
 a_*,b_*\ge0,\quad |\eta|=1,
\]
where $\beta_j$ are normalized half-plane Blaschke factors and
$\sum_j\Rea w_j/(1+|w_j|^2)<\infty$. The sum may initially be infinite.
\end{standinginput}

These are the plane-kernel, angle-generator, and fixed-band
continuation lemmas, labels \texttt{bulk:plane-kernels},
\texttt{bulk:boost}, and \texttt{bulk:edge-band}, together with
Equation \texttt{bulk:inner-factorization}, in
\cite[Section ``Angular transfer and its two leading modes'']{FKCompanion}.
The argument in \cite[Section~4]{GFFCompanion} for the current
cocycle, finite-inner spectrum, spectral-width tails, and variogram
is stated there for $\pi/3<\lambda<\pi$. Those stronger conclusions
are not included in Input~\ref{inp:rows}. In
Section~\ref{sec:spectral} we transfer the needed arguments to the
representation above and prove the conclusions at our parameter.

\subsection{The covariance normalization}

Only a separated-increment covariance, rather than a pointwise
Gaussian limit for the height, is needed to determine the electric
constant. Let $h_\delta$ be the height of the preceding balanced plane
law on mesh $\delta$, with an arbitrary additive constant, and put
$G(u,v)=-(2\pi)^{-1}\log|u-v|$.

\begin{standinginput}[Separated height covariance]\label{inp:covariance}
For four distinct points $u,u',v,v'$ and lattice approximants tending
to them,
\begin{align*}
 &\lim_{\delta\downarrow0}
 \E\bigl[(h_\delta(u')-h_\delta(u))
          (h_\delta(v')-h_\delta(v))\bigr]\\
 &\qquad=\sigma^2\bigl(G(u',v')-G(u',v)-G(u,v')+G(u,v)\bigr),
 \qquad \sigma^2=\frac{2}{\pi-\lambda}.
\end{align*}
The convergence is locally uniform away from endpoint collisions.
\end{standinginput}

Indeed, $c=2\cos(\lambda/2)\in[\sqrt3,2)$, so the covariance
part of \cite[Definition~2.7 and Theorem~2.8]{DKLM} applies directly
to this physical law and unit-height convention. Its multiplier is
$1/\arcsin(c/2)=2/(\pi-\lambda)$. The same normalization is recorded
in the main theorem of \cite{GFFCompanion}. In
Section~\ref{sec:electric} we use Input~\ref{inp:covariance} only
for four distinct collinear points. Thus this use needs neither the
smaller-$c$ part of that companion's GFF theorem nor an extension
of the DKLM theorem beyond its stated range.
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//02-inputs.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//03-radial.tex ====
\section{Radial regeneration and normalized correlations}\label{sec:radial}

The first normalization problem is local. A loop surrounding a marked
point can be arbitrarily small, so convergence of macroscopic loops does
not directly give convergence of an insertion which weights every such
loop. We prove that the dependence on these small loops cancels in ratios
of partition functions. The cancellation must remain bounded under a
complex perturbation of the weights, because Section~\ref{sec:electric}
will use analytic continuation.

We work in unit tile coordinates. For a rounded loop $\ell$ and a tile
vertex $u$, put
\[
 R_u(\ell)=\max_{z\in\ell}|z-u|.
\]
Enclosure refers to the disjoint switch drawing from
Section~\ref{sec:inputs}. A circuit state $\gamma$ consists of an actual
primal-open circuit surrounding $u$, with all its vertices wired, and
the ordinary FK law in its interior. Its size is $R$ if its inscribed
and outer radii about $u$ lie between fixed positive multiples of $R$.
No further regularity of the circuit is required. At a circuit edge,
the two switch arcs belong to its respective sides. Consequently the
interior loop configuration separates exactly from the exterior, also
when a loop uses an inward arc beside the circuit.

Let $N_\gamma$ count its interior loops enclosing $u$, and define
\begin{equation}\label{eq:radial-partition}
 F(\gamma;v)=\E_\gamma v^{N_\gamma},
 \qquad
 \frac{\dd\Pp_{\gamma,v}}{\dd\Pp_\gamma}
       =\frac{v^{N_\gamma}}{F(\gamma;v)},\qquad v>0.
\end{equation}
A plane state of size $R$ uses the plane FK law and counts precisely
the loops enclosing $u$ with $R_u(\ell)\le R$. We use the notation
$F(\gamma;v)$ for this state as well. An exchange of the two colors
gives the corresponding statements for dual circuits. In comparisons
the color of $u$ is fixed, or transported by a lattice symmetry together
with the two color names.

\begin{proposition}[Radial estimates]\label{prop:radial}
Fix $1\le q<4$, fixed radius-comparison constants, and a compact interval
$V\subset(0,\infty)$. The following bounds are uniform over the above
circuit and plane states and over $v\in V$.

For states $\gamma,\gamma'$ of sizes $R,R'\ge r_0$,
\begin{equation}\label{eq:radial-comparison}
 \left|\log\frac{F(\gamma;v)}{F(\gamma';v)}\right|
       \le C\bigl(1+|\log(R/R')|\bigr).
\end{equation}
There is $\delta>0$ such that every $F(\gamma;ve^w)$ is nonzero for
$|w|<\delta$. Its logarithm, continued from its real value at $w=0$,
has absolute value at most $C\log(2+R)$. Moreover
\begin{equation}\label{eq:radial-complex-ratio}
 \left|\log\frac{F(\gamma;ve^w)}{F(\gamma;v)}
       -\log\frac{F(\gamma';ve^w)}{F(\gamma';v)}\right|
 \le C|w|\bigl(1+|\log(R/R')|\bigr).
\end{equation}

For two states of comparable size $R$, their positively tilted laws
can be coupled so that their configurations agree inside a common
surrounding primal-open circuit containing $B(u,s)$, except with
probability $C(s/R)^c$, for $r_0\le s\le R/C$. Conditional on the common
circuit, the common continuation has its circuit law tilted by $v$.

Finally, let $F_{\ge h}(\gamma;v)$ count only the eligible loops with
$R_u(\ell)\ge h$. For comparable starts of size $R$,
\begin{equation}\label{eq:radial-truncation}
 \frac{F_{\ge\epsilon R}(\gamma;v)/
             F_{\ge\epsilon R}(\gamma';v)}
      {F(\gamma;v)/F(\gamma';v)}=1+o(1)
\end{equation}
uniformly as $\epsilon\downarrow0$ and $\epsilon R\to\infty$.
The same real comparison and regeneration bounds hold with any such
lower cutoff on the counted radii.
\end{proposition}

\subsection{Passage bounds at a circuit and under a tilt}

The FK estimates used here are the bounded-density collar lemma,
exponential passage moments, and local tilt bounds in
\cite[Section~2, ``Comparison through a collar'' and ``Passages and
positive local tilts'']{FKCompanion}; their precise scope is recorded
in Input~\ref{inp:fk}. We first explain the extension to an irregular
wired circuit, where a bulk-buffer assertion alone would not suffice.

A newly counted enclosing loop whose maximum radius crosses a specified
cutoff has a passage in one of a fixed number of buffers at that scale:
it either goes around $u$ in the band or leaves the band. The same
observation charges every enclosing loop whose status changes when a
subannulus is resampled. Loops which do not enclose $u$ need not be
charged. Thus, for passage counts $X_j$ on $l$ geometrically spaced
bands, finite coloring of their buffers and conditional iteration give
\begin{equation}\label{eq:radial-band-moments}
 \E\exp\left(t\sum_{j=1}^l X_j\right)
       \le \exp(C_t(l+1)),\qquad t<\infty.
\end{equation}
The bound is conditional on data beyond the buffered cuts.

At the outer wall of a primal-wired circuit, count large enclosing
loops using their dual sides. Those sides have free boundary at the
wall. In a cropped annular or rectangular passage region, the ordered
medial separators force distinct actual dual crossing components:
two neighboring sides may use one component, so a factor two and
bounded end effects are allowed. The crop ends lie strictly inside
the region traversed by the separators. An inward switch arc beside
an open circuit edge still has an actual interior dual gap on its
other side; it cannot use a fictitious path through the exterior.

Explore complete actual dual components from one end of the crop,
including unsuccessful components. After a component is removed, its
incident unexplored boundary edges are closed in the explored color.
The remaining law is dominated by filling the cropped box and wiring
its cut boundary. The original wall is free in this color, and
conditioning farther away can identify cut-boundary vertices only.
Filling the missing vertices and, if necessary, wiring the entire
boundary can only increase actual crossing probabilities. A blocker
in the filled box has uniformly positive probability by the crossing
bounds of Input~\ref{inp:fk}, including the wall version of
\cite[Theorem~1.2]{DMT}. Successive component discoveries therefore
give a geometric tail. To obtain an arbitrary fixed exponential rate,
subdivide the passage distance into $J$ separated slices, cover each
by buffered small boxes, and apply the same exploration to the boxes
intersected with the circuit interior. Every original passage crosses
a small buffer in each slice. If $X$ is the original count and $X_i$
the $O(J^2)$ small-buffer counts, then $JX\le\sum_iX_i$.
Their buffers have a coloring with $k$ colors independent of $J$.
Conditional iteration in each color class, followed by H\"older
across classes, therefore bounds
$\E e^{cJX/k}$ using the common first exponential rate $c$ of
the $X_i$. The prefactor may depend on $J$, which is harmless:
choose $J$ large and fixed to obtain any prescribed rate.
The finitely many scales where these boxes do not fit are covered by
a deterministic finite-size bound. This proves
\eqref{eq:radial-band-moments} also in the outer circuit collar.
The argument uses the specified wired wall, not an arbitrary partition
on a jagged boundary.

Now first count nests only up to a radius a fixed factor inside the
inscribed radius of the circuit. The local tilt bounds compare
successive cutoff masses and give every fixed passage moment at the
last weighted level. They are uniform for $v\in V$: in their proof,
the H\"older exponents can be chosen using
$\sup_{v\in V}|\log v|$, on either side of $1$. Moving the cutoff
back a bounded number of levels leaves the passage buffers outside
the tilt. H\"older's inequality then restores those levels. The
preceding wall estimate restores the remaining boundedly many outer
levels at bounded cost. Negative powers have the same comparison,
by Jensen's inequality followed, when needed, by this cutoff argument.
Bounded densities across a full bulk collar now compare the interior
cutoff mass to that for any other comparable state. This proves
\eqref{eq:radial-comparison}, first for comparable sizes and then
by multiplying comparisons across scales.

These statements remain uniform if counting begins at a lower
maximum-radius threshold $h$. Start the cutoff induction with no
weight below $h$. In particular, under a tilt supported below the
inner end of $l$ bands, their joint passage moments still obey
\eqref{eq:radial-band-moments}; the innermost boundedly many buffers
use the local tilt bounds, and all others are separated from its
support. Adding the weights in those $l$ bands changes the density,
including its normalization, by a factor whose fixed moments cost
$\exp(C(l+1))$. Hence \eqref{eq:radial-band-moments} holds under the
full positive tilt up to the outer end of those bands as well.
These are the real estimates needed for
regeneration.

\subsection{Stopping circuits and exponential waiting times}

Fix $K$ sufficiently large and use levels $r_i=r_0K^i$, $i\ge0$,
with $r_0$ large enough for all crossing constructions. Each trial
band has radii comparable to $r_i$ and $3r_i$ and has full collars
on both sides. The value of $K$ leaves these enlarged trial regions
disjoint and leaves an unused inward collar after a starting circuit.
Assign a state the level just below its inscribed radius; states at
one level have sizes within a fixed factor. A plane cutoff is assigned
in the same way, and its first trial lies well inside that cutoff.

Inspect trials inward until finding a surrounding primal-open
circuit, choosing the outermost circuit in that band. The usual
outer exploration is a stopping exploration: no edge strictly inside
the selected circuit need be queried. Intersecting candidate circuits
can be merged by taking their outer boundary. Failed tests concern
previous, disjoint bands. At level zero use a terminal symbol with
continuation mass one. Conditional on a stopped circuit, the ordinary
continuation is its wired FK law. If $D$ counts the weighted enclosing
loops outside that circuit, then
\begin{equation}\label{eq:radial-transition}
 F(\gamma;v)=\E_\gamma\!\left[v^D F(\gamma_1;v)\right],
\end{equation}
where $\gamma_1$ denotes the stopped state, including the terminal
case. For a plane start, $D$ also respects its original maximum-radius
cutoff. The assignment of circuit-side arcs makes this factorization
exact. It follows that the stopped process remains a regeneration
process under the tilt.

Let $J$ be the number of levels crossed in one transition. Assigning
the loops counted by $D$ to their maximum-radius bands gives, under
any of the positive tilts under discussion,
\begin{equation}\label{eq:radial-jump-cost}
 \E_{\gamma,v}[e^{tD};J=j]\le e^{C_t(j+1)},\qquad t<\infty.
\end{equation}
We next prove
\begin{equation}\label{eq:radial-jump-tail}
 \Pp_{\gamma,v}(J>l)\le Ce^{-cl}.
\end{equation}
This needs more than a crossing bound after arbitrary neighboring
edges have been exposed, because such an exposure need not leave a
favorable conditional law under the tilt.

For each of the first $l$ trials, designate its central band as a
region to be resampled by the ordinary FK conditional law. All counts
used below have buffers in that trial's enlarged collar. A trial is
called typical if the old passage counts which can charge changed
enclosing loops are at most $H$, and, for every boundary partition
which changes outside its collar, the ordinary conditional redraw
has probability at least $b$ of producing a surrounding circuit and
at most $H$ new charging passages. Here $b>0$ and $H<\infty$ will be
fixed independently of $l$ and of the lattice scale. The second
condition depends only on collar edges outside the resampled band,
so remains true after other trial bands are changed.

We justify that typicality fails with arbitrarily small ordinary
conditional probability. Expose the collar up to its far cuts, and
let $Y$ count its actual components joining the band-facing cuts to
the far cuts, on both radial sides, plus one. Only these components
can acquire new identifications when the far partition is changed.
For fixed collar edges, any two redraw kernels thus have density
ratio at most $q^{C Y}$. Complete-component exploration in cropped
subcollars, as above, gives a uniform exponential tail for $Y$.
First fix a large bound $Y_0$. Given $Y\le Y_0$, the worst conditional
probability, over the induced far partitions, of more than $H$ new
passages is at most $q^{CY_0}$ times that probability under the
actual partition, or its conditional mixture. The latter probability
has arbitrarily small expectation as $H\to\infty$, by the ordinary
conditional passage estimate. Markov's inequality then makes the
collar-edge event on which this worst probability exceeds a fixed
tolerance arbitrarily rare. Success without the passage restriction
has conditional probability at least $2b>0$ by RSW. Choose the
tolerance below $b$, and then $H$ large. The old passage test is
handled by the same moment bound.

All these estimates hold conditional on data outside the enlarged
trial collar, uniformly in its induced partition. For any prescribed
set of trials their failures therefore have probability at most a
chosen small constant to the size of that set, by conditional
iteration. A binomial sum proves, at any desired exponential rate,
that fewer than $l/3$ typical trials is unlikely under ordinary FK.
To transfer this statement to the fully tilted law, first retain
only the tilt below the inner end of these $l$ trials, omitting
boundedly many end trials for the buffers. The same ordinary
conditional argument still applies: the retained weight counts
loops wholly inside that lower cutoff, so is measurable outside
each enlarged trial collar. Conditional on the exterior of a collar,
it is constant, and the conditional law in the collar is ordinary
FK. The remaining change of density
has second moment at most $e^{C(l+1)}$, by
\eqref{eq:radial-band-moments} and the real mass comparisons.
Cauchy--Schwarz, choosing the preceding ordinary rate larger than
$C$, gives
\begin{equation}\label{eq:radial-typical}
 \Pp_{\gamma,v}(|T|<l/3)\le Ce^{-cl},
\end{equation}
where $T$ is the set of typical trials. This proof applies equally
when a lower radius threshold is present.

For completeness, the insertion argument converting
\eqref{eq:radial-typical} into \eqref{eq:radial-jump-tail} is as
follows. Write $A_l$ for failure at all $l$ trials, and for a fixed
subset $S$ of the trials put $A_S=A_l\cap\{S\subset T\}$. Redraw
the bands in $S$ successively with their ordinary FK heat baths,
retaining only outcomes with success and at most $H$ new passages.
Each retention has probability at least $b$, even after the preceding
redraws. At one accepted redraw, at most $CH$ old or new enclosing
loops change their weight. Consequently the ratio of input to output
tilt densities is at most
$M=\exp(CH\sup_{v\in V}|\log v|)$ per redraw.
Ordinary heat baths preserve the ordinary law, including their
successive composition. If $E_S$ is the event with exact success
set $S$ among these $l$ trials, integration of the accepted transitions
therefore gives
\[
 \Pp_{\gamma,v}(E_S)
       \ge (b/M)^{|S|}\Pp_{\gamma,v}(A_S).
\]
The events $E_S$ are disjoint. With $\delta=b/M$, summing over $S$
yields the useful integrated inequality
\begin{equation}\label{eq:radial-insertion}
 1\ge
 \E_{\gamma,v}\left[\1_{A_l}(1+\delta)^{|T|}\right].
\end{equation}
The set $T$ stays inside the expectation; no deterministic choice of
typical trials is required. Equations~\eqref{eq:radial-typical} and
\eqref{eq:radial-insertion} prove \eqref{eq:radial-jump-tail}.

Combining the tail with \eqref{eq:radial-jump-cost} gives a joint
exponential moment: there exist $a,t>0$ such that
\begin{equation}\label{eq:radial-joint-moment}
 \sup_{\gamma,v}\E_{\gamma,v}e^{aJ+tD}<\infty.
\end{equation}
Indeed fix an exponent $T_0>0$ in
\eqref{eq:radial-jump-cost} and apply H\"older on $\{J=j\}$:
\[
 \E[e^{tD};J=j]
 \le \E[e^{T_0D};J=j]^{t/T_0}
             \Pp(J=j)^{1-t/T_0}.
\]
For $t/T_0$ small, the resulting exponent in $j$ is negative; then
choose $a>0$ smaller than its absolute value.

\subsection{Coupling states at different levels}

At a common level, the next-band transitions of any two states have
a common part of uniformly positive mass. To see this, use bounded
densities through the unused collar to minorize both ordinary
band marginals by a fixed fraction of one reference marginal, and
restrict to its successful circuits. Given the next screen, its
continuation mass compares to any reference mass at that level by
\eqref{eq:radial-comparison}. The number of newly charged loops can
be restricted to $D\le H_1$ at arbitrarily small discarded ordinary
probability, by the fixed-band passage estimate. On this restriction
$v^D$ is bounded above and below uniformly. Subtract the two discarded
parts from the reference marginal and then take its common minimum.
The normalizations also compare to the same continuation mass, so
the common part retains a fixed mass $p_0>0$ under the tilted laws.
In particular a jump of one level has probability bounded below.

For starts at the same or neighboring levels, advance whichever
state has the higher remaining level; when the levels agree, attempt
coupling on this common part. Use a fixed mass strictly smaller than
the available overlap. After a failed attempt the residual transition
moments are bounded by the original moments divided by $1-p_0$.
This avoids conditioning estimates depending on an overlap which
might itself be close to one.

Here is the quantitative reason this procedure also works for
asynchronous levels. If their current difference is $d\ge1$ and
the higher state jumps by $J$, the new difference is $|d-J|$.
For sufficiently small $a>0$, \eqref{eq:radial-joint-moment} gives
an exponential drift outside a fixed finite set:
\[
 \E e^{a|d-J|}
 \le e^{ad}\E[e^{-aJ};J\le d]
          +\E[e^{a(J-d)};J>d]
 \le \theta e^{ad},\qquad d\ge d_0,
\]
with $\theta<1$. The last tail term is exponentially small relative
to $e^{ad}$, while $J\ge1$ bounds the first coefficient below one.
From $\{0,\ldots,d_0\}$, finitely many one-level jumps bring the
levels together with uniformly positive probability, followed by
the common-part coupling. The exponential drift bounds return
times, and the failed-attempt residual moments remain bounded.
Geometrically summing these returns proves an exponential moment
for the number of transitions until meeting. More explicitly, the
kernel of one unsuccessful excursion from this finite set has mass
strictly below one at discount one and a bounded exponential moment;
the same is true at a sufficiently small positive time exponent.
Summing its powers proves the assertion.

Iterating \eqref{eq:radial-joint-moment} and reducing its exponents
then bounds total distance and charged-loop cost up to meeting. If
$L$ denotes the largest number of levels descended by either chain,
and $D_1^*,D_2^*$ their respective costs before the common state,
\begin{equation}\label{eq:radial-meeting-moment}
 \E\exp\{c(L+D_1^*+D_2^*)\}\le C.
\end{equation}
One may require each chain to take one transition before permitting
meeting; the bound is unchanged. The terminal symbol provides the
same construction at the finitely many bottom levels.

If the initial size is $R$, reaching a radius comparable to $s$
without meeting requires order $\log(R/s)$ descended levels.
Equation~\eqref{eq:radial-meeting-moment}, with a smaller exponent
to cover a final overshoot, bounds this probability by $C(s/R)^c$.
On meeting, both chains have exactly the same surrounding circuit
and the same conditional tilted continuation. Their configurations
can therefore be made identical inside it. This proves the inward
coupling assertion in Proposition~\ref{prop:radial}.
Terminal exhaustion, though a valid meeting for partition-function
identities, is counted as failure in this spatial coupling; the
coupling attempts here are stopped before the radii drop below $s$.

\subsection{Complex stability and removal of the inner cutoff}

Fix a real $v\in V$. For a state $\gamma$ at level $i$ define
\[
 G_\gamma(w)
     =\frac{F(\gamma;ve^w)}{F(\gamma;v)}
     =\E_{\gamma,v}e^{wN_\gamma}.
\]
At finite scale these are holomorphic functions. We prove by induction
on the level that they do not vanish and that, for states at the same
or neighboring levels,
\begin{equation}\label{eq:radial-induction}
 |\log G_\gamma(w)-\log G_{\gamma'}(w)|\le B|w|.
\end{equation}
All logarithms start at zero at $w=0$. The terminal levels have
uniformly bounded size and give the initial assertion directly.

To compare two new starts, use the preceding coupling, requiring one
transition in each chain so that their meeting state lies below the
highest level being proved. Regeneration gives
\[
 G_{\gamma_j}(w)=
        \E[e^{wD_j^*}G_{\gamma_*}(w)],\qquad j=1,2,
\]
with the same random meeting state $\gamma_*$. Divide both sides by
$G_\sigma(w)$ for any reference state $\sigma$ at the preceding
level. Chaining the inductive bounds through the lower levels gives
\[
 Q(w)=\frac{G_{\gamma_*}(w)}{G_\sigma(w)},\qquad
 |\log Q(w)|\le B|w|(L+2).
\]
By \eqref{eq:radial-meeting-moment}, for each fixed $B$ and all
sufficiently small $|w|$, both divided expectations lie in
$\{z:|z-1|<1/4\}$. Their difference has the sharper bound
\begin{align*}
 \left|\E[(e^{wD_1^*}-e^{wD_2^*})Q(w)]\right|
 &\le |w|\E\left[(D_1^*+D_2^*)
   e^{|w|(D_1^*+D_2^*)+B|w|(L+2)}\right]\\
 &\le C_0|w|.
\end{align*}
Here $C_0$ is independent of $B$, provided $B|w|$ is smaller than
a fixed fraction of the exponent in
\eqref{eq:radial-meeting-moment}. The logarithm is uniformly
Lipschitz on the displayed disk. Thus choose $B$ larger than the
resulting fixed multiple of $C_0$, and only then choose $\delta>0$
so that the disk and exponent conditions hold for $|w|<\delta$.
This closes both the nonvanishing induction and
\eqref{eq:radial-induction}. Telescoping levels proves
\eqref{eq:radial-complex-ratio}. The real comparison to a terminal
state bounds $|\log F(\gamma;v)|$ by $C\log(2+R)$, and another
telescoping sum proves the asserted logarithmic bound in the complex
neighborhood.

It remains to compare full and thresholded masses. Put $h=\epsilon R$
and let $l$ be the number of levels between $R$ and $h$. In each
regeneration path stop at its first screen whose level is at most
$J$ levels above $h$, where $1\ll J\le l/2$. Call this stop typical
if its level is between one and $J+O(1)$ levels above $h$.
The probability of an atypical stop is $Ce^{-cJ}$ under both the
full and thresholded tilts: sum the jump-tail bound over its possible
origins, each level being visited at most once. On a typical path
every loop already charged encloses the stopping screen or lies
outside it, and hence has maximum radius above $h$. The two
weighting rules have therefore agreed up to this stop.

Write
\[
 Q_h(\sigma)=F_{\ge h}(\sigma;v)/F(\sigma;v)
\]
for the continuation quotient. Normalize by its value $q_h$ at any
reference screen at the threshold level. Real comparisons, including
their thresholded version, give on typical stops
\begin{equation}\label{eq:radial-cutoff-quotient}
 e^{-C(J+1)}\le Q_h(\sigma)/q_h\le e^{C(J+1)}.
\end{equation}
Couple the two full-tilt paths from the original comparable states.
Their stopped screens agree except with probability $Ce^{-cl}$,
after changing $c$ to account for $J\le l/2$. On typical stopped
paths regeneration gives the exact identity
\[
 I_\gamma:=\E_{\gamma,v}
       [\1_{\mathrm{typ}}Q_h(\sigma)]
   =\frac{F_{\ge h}(\gamma;v)}{F(\gamma;v)}
            \Pp_{\gamma,v,\ge h}(\mathrm{typ}),
\]
where the probability on the right uses the thresholded tilt.
The difference of the two $I_\gamma$, divided by $q_h$, is
at most $Ce^{C(J+1)-cl}$. The lower bound in
\eqref{eq:radial-cutoff-quotient} converts this into a relative
error $Ce^{C'(J+1)-cl}$.

The omitted atypical paths also have small \emph{relative} weight.
Indeed their fraction of the full mass is their probability under
the full tilt, and their fraction of the thresholded mass is their
probability under the thresholded tilt. Both are $Ce^{-cJ}$.
The factorization is used only on typical paths, where the already
charged counts agree; an overshoot below $h$ does not require such
a factorization. Taking $J$ a sufficiently small positive fraction
of $l$ proves \eqref{eq:radial-truncation} and completes the proof
of Proposition~\ref{prop:radial}.

\subsection{Several marked points}

Fix $p\ge2$ labeled tile vertices $\mathbf u=(u_1,\ldots,u_p)$.
For $\varnothing\ne I\subsetneq\{1,\ldots,p\}$ let $N_I$ be the number of plane loops
enclosing exactly the points with labels in $I$. Assign relative
weights $\mathbf v=(v_I)$, with $v_\varnothing=v_{\{1,\ldots,p\}}=1$,
and define
\[
 C(\mathbf v;\mathbf u)
       =\E\prod_{\varnothing\ne I\subsetneq\{1,\ldots,p\}}
                       v_I^{N_I}.
\]
This is well defined even for complex weights. Every separating loop
intersects a fixed finite collection of tile paths joining the points;
only finitely many switch strands use those paths. Let $b_0\in(1,2)$
be a fixed continuity cutoff for the limiting loop law, and write
\[
 F_n(v)=\E v^{\#\{\ell:u\text{ inside }\ell, R_u(\ell)\le b_0n\}}.
\]
Translation and duality make this function independent of the choice
of tile vertex $u$. Rounding and color conventions are transported
by these symmetries.

\begin{proposition}[Normalized multipoint correlations]
\label{prop:normalization}
Suppose $p$ is fixed, $\max_j|u_j|\le Cn$, and
$a=\min_{i\ne j}|u_i-u_j|\ge r_0$. Around every parameter whose
singleton weights $v_{\{j\}}$ are strictly positive real numbers,
the normalized correlation
\begin{equation}\label{eq:normalized-correlations}
 \mathcal C_n(\mathbf v;\mathbf u)
   =\frac{C(\mathbf v;\mathbf u)}
          {\prod_{j=1}^p F_n(v_{\{j\}})}
\end{equation}
is holomorphic and bounded in a fixed complex neighborhood of the
singleton weights, uniformly when the other weights range over a
bounded set, with bound $C(n/a)^C$. For $a\ge cn$, if $u_j/n$ converge
to distinct points and the point parities are fixed,
\eqref{eq:normalized-correlations} has a limit locally uniformly in
these complex parameter neighborhoods.

When all nontrivial weights lie in a fixed compact subset of
$(0,\infty)$ and $a\ge cn$, the normalized correlations are bounded
above and below by positive constants. Their tilted laws are local
at each point in the following quantitative sense. Consider two
possibly different collections with bounded point counts, all weights
in the stated positive compact set, and sizes comparable to the
same $n$, satisfying the same fixed geometric bounds. Align one
chosen point in each by a lattice symmetry, transporting its color,
and suppose its singleton weight is the same in both. The two laws admit
a coupling with identical configuration inside a common surrounding
circuit containing $B(u,s)$, with error $C(s/n)^c$. For minimum
separation $a$ the analogous error is at most
$C(n/a)^C(s/a)^c$, whenever $r_0\le s\le a/C$.
\end{proposition}

\begin{proof}
Take disjoint patches $P_j$ centered at $u_j$, of radii a small fixed
multiple of $a$, and with disjoint larger collars. At a positive
singleton base point $v_j=v_{\{j\}}$, use the positive auxiliary
weight
\[
 U=\prod_j v_j^{M_j},
 \qquad M_j=\#\{\ell:u_j\text{ inside }\ell,\ \ell\subset P_j\},
 \qquad Z_U=\E U.
\]
Write $\Pp_U=U\Pp/Z_U$. The collar comparison and
Proposition~\ref{prop:radial} give
\begin{equation}\label{eq:radial-proxy-mass}
 Z_U\asymp\prod_jF_a(v_j).
\end{equation}
Patches can be round, with a bounded whole-tile rounding; the mass
comparison is unaffected by a fixed change in their radii.

Every weighted loop not assigned to these local nests can be charged
to a passage at patch scale or larger. To make the count explicit,
join the points by a fixed spanning tree of paths of total length
$O(n)$. A separating loop hits a tree path. If this contact occurs
inside a patch but the loop is not contained there, follow the loop
to its first departure through a patch-scale buffer. Otherwise choose
its nearest approach to a marked point and use a buffer at the
corresponding geometric scale. Its diameter is comparable to or
larger than that scale. A loop enclosing several points has such
a witness as well. A bounded number of buffers at each scale
suffices because $p$ is fixed; choose them where the distance to the
nearest point is comparable to the scale. Their overlap has a
bounded coloring. Thus the total witness count $X$ satisfies
\begin{equation}\label{eq:radial-macro-moments}
 \E_Ue^{tX}\le C_t(n/a)^{C_t},\qquad t<\infty.
\end{equation}
For the lowest buffers, move the local cutoff back a bounded number
of levels and use the local tilt bounds before restoring it by
H\"older. Higher buffers are separated from the local tilts.
Conditioning through the disjoint patch collars treats all $p$
patches at once. This proves \eqref{eq:radial-macro-moments} directly
from the band estimates; it applies to arbitrarily large excursions,
since the finite tree already provides their witnesses.

At the positive singleton base, the absolute value of the remaining
weight is bounded by $K^X$, for a fixed $K$, even if some nonsingleton
weights vanish. Equations~\eqref{eq:radial-proxy-mass} and
\eqref{eq:radial-macro-moments}, followed by scale comparison from
$a$ to $n$, prove the asserted real-base upper bound. If all weights
are bounded above and below positively, the remaining weight is
also bounded below by $K^{-X}$. Jensen's inequality and
\eqref{eq:radial-macro-moments} then give the lower comparison.

For the complex bound, in every patch search inward for the first
stopping circuit with the same clear collars as before. Under
$\Pp_U$, its waiting time $L_j$ and the number $D_j$ of local charged
loops completed before it have a joint exponential moment, uniformly
in scale. The proof of \eqref{eq:radial-jump-tail} applies with a
patch cutoff in place of the initial circuit: its ordinary crossing
and collar estimates are conditional through the same free space.
The argument also applies to lower-threshold local tilts. Conditional
on these stopping circuits $\gamma_j$, both the true correlation
and the auxiliary weight factor into their exterior factors and
the continuation masses $F(\gamma_j;v_j)$.

Write the perturbed singleton weight as $v_je^{w_j}$. Before the
stops it changes the local weight by at most
$\exp(\sum_j|w_j|D_j)$ in absolute value. After dividing by the
reference factors in \eqref{eq:normalized-correlations},
\eqref{eq:radial-complex-ratio} bounds the continuation corrections
by
\[
 \exp\left(C\sum_j|w_j|
       [1+L_j+\log(2+n/a)]\right).
\]
The remaining nonsingleton factors are bounded by $K^X$. Apply
H\"older, use every fixed moment in
\eqref{eq:radial-macro-moments}, and then choose $\max_j|w_j|$ small
enough for the joint exponential moments of $L_j,D_j$.
This proves the uniform complex bound, including exhaustion at the
terminal scale. Nonvanishing of the denominator is supplied by
Proposition~\ref{prop:radial}.

We now prove convergence, retaining $a\ge cn$. First let the
singleton weights be real and positive; other weights may be
arbitrary fixed complex numbers in their bounded set. Remove
singleton weighting from all loops of maximum radius below
$\epsilon n$ about their point, and truncate each reference $F_n$
by exactly this prescription. Denote the resulting normalized
quantity by $\mathcal C_{n,\epsilon}$. The preceding bounds imply
\begin{equation}\label{eq:radial-normalized-truncation}
 \lim_{\epsilon\downarrow0}\limsup_{n\to\infty}
 |\mathcal C_n-\mathcal C_{n,\epsilon}|=0.
\end{equation}
Here are the details ensuring cancellation of the small-loop masses.
The first-screen waiting times in all patches can be restricted to
any fixed large number $L_0$ of bands with arbitrarily small additive
error, before or after thresholding. Use their exponential tails
and H\"older with the bounded moments of the absolute exterior
factor; the uniform threshold version gives the same estimate for
the truncated expression. On the retained stops the screens have
size at least a fixed multiple of $nK^{-L_0}$. For sufficiently
small $\epsilon$, every completed exterior loop with nontrivial
weight is unchanged by thresholding: a loop enclosing the point
outside its screen encloses that screen, and every other exterior
factor is already macroscopic. The only changes are therefore in
the continuation masses. Their ratios to the reference $F_n$ agree
to $1+o(1)$ by \eqref{eq:radial-truncation}, uniformly at these
boundedly many levels. The integral of the absolute exterior factor
is bounded. First let $n\to\infty$, then $\epsilon\downarrow0$,
and finally $L_0\to\infty$. This proves
\eqref{eq:radial-normalized-truncation}.

For fixed $\epsilon>0$, the truncated expression has a limit by
the plane nested-loop convergence in Input~\ref{inp:loops}. We
specify the continuity and integrability requirements. Loops deciding
the truncated test have spherical diameter bounded below after
rescaling, and the cited topology retains their individual curves.
A fixed continuum point lies on no macroscopic loop almost surely:
RSW screening bounds the probability that a loop arriving from a
fixed distance reaches a shrinking disk about it. Enclosure is
therefore a continuous winding test at each prescribed point.
Choose maximum-radius thresholds outside the at most countable
set of atoms of the relevant radius variables. Loops separating
the marked points but escaping an arbitrarily large fixed ball
can first be discarded. They must meet the finite joining tree,
so successive annular screens make the discarded probability tend
to zero uniformly in the mesh. Passage moments give uniform
integrability, including for complex exterior factors. Thus
convergence of the truncated expectations follows from the stated
plane theorem, without a bounded-domain loop limit. The real
truncated reference masses stay bounded away from zero: their
counts have uniformly bounded moments at fixed $\epsilon$, and
Jensen's inequality applies when the weight is below one.

The cutoff $b_0$ can be chosen once for the finitely many limiting
reference tests. Plane homogeneity gives the same choice at
translated points. Equation~\eqref{eq:radial-normalized-truncation}
then gives convergence of the untruncated normalized expression
for real positive singletons. The complex bounds give a normal
family. Two subsequential holomorphic limits agree on the real
singleton box, with arbitrary fixed remaining weights; repeated
one-variable uniqueness makes them identical throughout each
complex neighborhood. Hence the convergence is locally uniform.

Finally assume all weights are positive and bounded above and
below. The true tilted mass compares both ways to $Z_U$ by the
$K^{\pm X}$ bounds. Under that true law, H\"older and
\eqref{eq:radial-macro-moments} preserve an exponential tail for
the first patch screen, with the additional factor $(n/a)^C$
when the separation varies. Conditional on that screen, the inward
law is exactly the singleton tilt at its point. Apply the circuit
coupling from Proposition~\ref{prop:radial}, allowing for its random
initial level by the first-screen tail. This yields the final
coupling assertion. It also couples local observables obtained by
averaging orientation marks on loops contained in the common screen:
a mark strictly inside the screen cannot lie on an exterior loop.
\end{proof}

\begin{corollary}[Macroscopic tests with local insertions]
\label{cor:macroscopic-test}
Let $u_j/n$ converge to distinct points, with their point parities
fixed, and let $P_j$ be disjoint round patches centered at $u_j$,
of radii fixed positive multiples of $n$, with disjoint larger
collars. Fix $v_j>0$ and define
\[
 U_n=\prod_jv_j^{\#\{\ell:u_j\text{ inside }\ell,\ \ell\subset P_j\}}.
\]
Suppose $T_n$ is a uniformly bounded discretization,
$\sup_n\|T_n\|_\infty<\infty$, of a fixed bounded functional $T$ of the
rescaled plane loop collection, determined in a fixed bounded region
by loops of diameter at least a fixed $\delta>0$. Suppose that this
functional is almost surely continuous for the plane loop limit,
with its marked-point, patch-boundary and contact tests taken at
continuity radii. Here discretization means that
$T_n(\omega_n)\to T(\omega)$ whenever the relevant rescaled
loop collections $\omega_n$ converge to a continuity configuration
$\omega$. Then the following sequence converges as $n\to\infty$:
\begin{equation}\label{eq:radial-macroscopic-test}
 \frac{\E[U_nT_n]}{\prod_jF_n(v_j)}.
\end{equation}

The conclusion also applies to tests obtained by first excluding
arbitrarily large excursions, provided the discarded part tends
to zero in probability under the normalized $U_n$ law uniformly
in $n$. Boundedness of $T_n$ may be replaced by
$|T_n|\le K^{X_n}$, where $X_n$ is the sum of a fixed finite family of
macroscopic passage counts satisfying
\begin{equation}\label{eq:radial-test-integrability}
 \sup_n\frac{\E[U_ne^{tX_n}]}{\E U_n}<\infty
 \qquad\text{for every fixed }t<\infty,
\end{equation}
also after the inner-radius truncations of the local weights.
The bounded truncations of the test must satisfy the preceding
continuity requirement.
\end{corollary}

\begin{proof}
Choose $\eta>0$ smaller than the patch radii and, taking account of
the fixed band-radius factors, small enough that every screen sought
below $\eta n$ has diameter less than $\delta n$.
Under the normalized $U_n$ law, regeneration between radii
$\eta n$ and $\epsilon n$ supplies a surrounding circuit at each
marked point, except with probability $C(\epsilon/\eta)^c$.
The same statement holds for thresholded local weights. A loop
used by the test has diameter at least $\delta n$, so cannot be
contained inside these circuits; it cannot cross them either.
Thus the events where a tested macroscopic loop enters the
$\epsilon n$ neighborhoods of the marked points can be discarded
with an error tending to zero. Under
\eqref{eq:radial-test-integrability}, H\"older's inequality makes
the same discarded contribution small for the unbounded test.

Search for the first screens below $\eta n$. Conditional on these
screens the test, after this exclusion, is determined by their
exterior. Restrict their waiting times to a fixed number of bands,
use their exponential tails for the remainder, and truncate the
local weights and all reference factors at the same radius
$\epsilon n$. Exactly the continuation-quotient argument proving
\eqref{eq:radial-normalized-truncation} shows that this changes
\eqref{eq:radial-macroscopic-test} by $o(1)$, uniformly for large
$n$ as $\epsilon\downarrow0$. For fixed $\epsilon$, all remaining
weights and tests concern macroscopic loops. Input~\ref{inp:loops}
and the stated continuity assumptions give convergence. Large
excursions are removed before this step and restored afterwards
using their assumed small probability and uniform integrability.

For an unbounded test, truncate at $|T_n|\le M$. The mass comparison
\eqref{eq:radial-proxy-mass} bounds
$\E U_n/\prod_jF_n(v_j)$ above and below. Condition
\eqref{eq:radial-test-integrability} therefore makes the discarded
normalized expectations tend to zero as $M\to\infty$, uniformly
in $n$ and in the inner cutoff. This proves the extension.
For the passage counts used below, condition
\eqref{eq:radial-test-integrability} follows from
\eqref{eq:radial-macro-moments}: their buffers are outside the local
patch cores, with a bounded cutoff rollback at the innermost buffers.
\end{proof}
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//03-radial.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//03-thermodynamic-kernels.tex ====
\section{Thermodynamic row kernels}\label{sec:thermodynamic}

The finite formulas of Section~\ref{sec:finite} reduce a row word to
sums over finitely many selected Bethe roots.  To pass to a plane
kernel we must control those sums even when several selected roots
approach the sparse ends of the root configuration.  We first
construct the real roots, then prove the inverse and determinant
estimates that justify this passage.  All constants in this Section
may depend on the fixed parameter $\pi/3<\lambda<\pi$.  They need not
remain bounded as $\lambda\uparrow\pi$.

\subsection{The three limiting procedures}

We use the notation $s,K,r,G,\xi$ of Section~\ref{sec:finite}, and put
\[
 h=\int_{\R}|K(u)|\dd u=|1-2\eta|<1,\qquad
 R_0=K*(I+K*)^{-1}.
\]
Here $K*$ denotes convolution on the line; the last expression means
the convolution kernel of the indicated operator.  Since $h<1$, its
Neumann series converges on bounded functions.  There is also a
number $a>0$ such that the same series converges with the weight
$e^{a|u|}$: choose $a$ so that
$\int e^{a|u|}|K(u)|\dd u<1$.  In particular $R_0$ and each of its
fixed derivatives are exponentially integrable.

The following limits have different purposes.
\begin{enumerate}
\item In the \emph{homogeneous plane limit}, all $b_j=0$ and $L$
 tends to infinity through even integers.  If $\lambda>\pi/2$ we
 use $n=L/2$.  If $\pi/3<\lambda\le\pi/2$, we use
 $n=L/2-\lfloor L^\epsilon\rfloor$, where $\epsilon>0$ is
 sufficiently small and fixed.  Its permitted size is specified in
 the proof of Lemma~\ref{lem:row-normalization}.
\item In the \emph{array limit}, which we use only for
 $\lambda>2\pi/3$, put
 \[
 b_j=\left(j-\frac{L-1}{2}\right)\frac d2,\qquad
 B=\frac{Ld}{2},\qquad n=L/2.
 \]
 First $B\to\infty$ through the allowed even widths, and then
 $d\downarrow0$.  A common real center $a_B$ of the row arguments
 may vary with $B$, provided
 $|a_B|\le(1-\gamma)B/2$ for a fixed $\gamma>0$.
 Differences of arguments range over a fixed compact set.
\item In the \emph{regular-filling limit}, also used subsequently
 only for $\lambda>2\pi/3$, the chain is homogeneous.  First
 $L\to\infty$ with $n/L\to\rho<1/2$, and then
 $\rho\uparrow1/2$.  We assert comparison with the plane state in
 this procedure for equal-time local matrices.
\end{enumerate}
For example, uniform array convergence to a function $F$ means
\begin{equation}\label{eq:array-limit-convention}
 \lim_{d\downarrow0}\limsup_{B\to\infty}
 \sup_{\substack{|a_B|\le(1-\gamma)B/2\\
                  \boldsymbol{x}\in\mathcal K}}
 |F_{d,B}(a_B+\boldsymbol{x})-F(\boldsymbol{x})|=0
\end{equation}
for every fixed compact parameter set $\mathcal K$ under discussion.
It is not necessary to assert a limit for every fixed $d$.
A uniform bound in this procedure means that there are $C,d_0>0$
such that for every $0<d<d_0$ there is $B_0(d)$ for which the bound
holds whenever $B\ge B_0(d)$.  The corresponding convention for
regular filling is $\lim_{\rho\uparrow1/2}\limsup_{L\to\infty}$.
Word lengths, matrices, derivative orders, and compact parameter
sets are fixed before these limits.  No estimate below is asserted
uniformly in an unbounded word length.

\subsection{Real roots, counting cells, and the sparse ends}

\begin{lemma}[Construction and density of the roots]\label{lem:root-limits}
The zero-twist consecutive-root equations of
Lemma~\ref{lem:bethe-data} have distinct real solutions with the
following properties.
\begin{enumerate}
\item If $\lambda>\pi/2$, the solutions at half filling exist for
 every finite homogeneous or real-array chain.  Their counting
 derivative $\rho_L=\xi'$ is positive.  If
 \[
 p_L(u)=\sum_{j=0}^{L-1}r(u-b_j),
 \]
 then $\rho_L-p_L$ and their integrated discrepancy are uniformly
 bounded on the entire line.
\item In the homogeneous chain, let
 \[
 r_Q=(I+K_Q)^{-1}s,\qquad H_Q=(I+K_Q)^{-1}1,
 \]
 where $K_Q$ integrates over $[-Q,Q]$.  For all $Q\ge0$,
 \begin{equation}\label{eq:regular-density-comparison}
 c r(u)\le r_Q(u)\le C r(u),\qquad
 c\le H_Q(u)\le C\quad (|u|\le Q).
 \end{equation}
 The mass $\int_{-Q}^Q r_Q$ increases to $1/2$, and
 \[
 \frac12-\int_{-Q}^Q r_Q\asymp e^{-Q/\eta}.
 \]
 Choose $Q=Q_L$ by $\int_{-Q}^Q r_Q=n/L$.
 If $Q$ stays bounded with a positive limiting filling, or if
 $Lr(Q)$ tends to infinity at least as a fixed positive power of
 $L$, the roots exist for all sufficiently large $L$.  They differ
 from the mid-quantiles of $Lr_Q$ by a bounded distance in counting
 coordinates, and
 \begin{equation}\label{eq:regular-counting-density}
 \xi'(u)=Lr_Q(u)+O(1)
 \end{equation}
 on the interval, including a bounded counting extension at its
 endpoints.
\item In all three limiting procedures the measures
 \[
 \nu_L=\sum_j\ell_j\delta_{y_j},\qquad
 \ell_j=\xi'(y_j)^{-1},
 \]
 have uniformly bounded mass on intervals of length one.  In a
 region whose slightly enlarged neighborhood has density at least
 $M$, their error against bounded Lipschitz tests on a fixed
 bounded interval is at most
 \begin{equation}\label{eq:local-quadrature}
 \frac{C}{M}\bigl(\|\phi\|_\infty+\Lip(\phi)\bigr).
 \end{equation}
 Thus $\nu_L$ converges locally to Lebesgue measure in the dense
 portions of the limits.
\end{enumerate}
At zero twist the homogeneous vectors constructed from these roots
are the sector Perron vectors.  The Gaudin matrices are nonsingular,
so each constructed finite root list has a real-analytic twist branch
near zero.
\end{lemma}

\begin{proof}
Suppose first that $K<0$ and write $k=-K>0$.  The Jacobian of the
counting equations is
\[
 G_{ii}=\sum_j s(y_i-b_j)+\sum_{j\ne i}k(y_i-y_j),
 \qquad G_{ij}=-k(y_i-y_j)\quad(i\ne j).
\]
Its quadratic form is the sum of a positive diagonal form and
$\sum_{i<j}k(y_i-y_j)|u_i-u_j|^2$.  It is strictly positive,
so the equations are strictly convex gradient equations.
Start at $\lambda=\pi/2$, where $K=0$ and each counting level has
one real preimage, and continue in a compact parameter interval.
There can be no coincident roots because their prescribed counting
levels are distinct.

There can be no escaping roots either.  If $k_0$ roots tend to
$+\infty$, sum their equations.  The interaction primitives within
that set cancel by oddness.  Dividing by $k_0$, the limiting excess
of the left side over the prescribed levels is
\[
 (1-\eta)(L/2-n+k_0)>0.
\]
The same argument applies at $-\infty$.  All site parameters here
are finite in each continuation problem.  Nonsingularity and this
compactness continue the solution to the required parameter.

At half filling the total mass of $\rho_L=\xi'$ is $n$.
The two outer counting half-levels occur at infinity.  Consequently
the signed measure and its primitive
\[
 e_L=\sum_j\delta_{y_j}-\rho_L(u)\dd u,\qquad
 F_L(u)=e_L((-\infty,u])
\]
satisfy $|F_L|\le1/2$ and $F_L(\pm\infty)=0$.
The density equation and convolution inversion give
\begin{equation}\label{eq:counting-sawtooth}
 \rho_L-p_L=-R_0*e_L=-R_0'*F_L.
\end{equation}
Integrating once also bounds the difference of the tail integrals
by $\|R_0\|_1/2$.  This proves the first assertion.  Moreover
$|\rho_L'|\le C\rho_L$: both $s$ and $k$ are positive kernels
with bounded logarithmic derivatives.

We give the details at a sparse end, since they will be needed for
uniform inverses.  Number the roots from the right as
$x_1>x_2>\cdots$ and write
$T(u)=\int_u^\infty\rho_L(v)\dd v$.  Then
$T(x_j)=j-1/2$.  For either $q=s$ or $q=k$, put
$\bar q(u)=\int_u^\infty q(v)\dd v$.
The displayed hyperbolic formulas for the kernels imply
\begin{align}
 q(u)&\le C\bar q(u) &&(u\in\R),\notag\\
 c\bar q(u)&\le q(u),\qquad
 \bar q(u+t)\le C e^{-2t}\bar q(u)
 &&(u,t\ge0).
 \label{eq:kernel-tail-comparison}
\end{align}
At a root $x_j$ lying beyond every bare center, its own kernel and
the $j-1$ farther-out kernels contribute at most
$h/2+h(j-1)=hT(x_j)$ to this tail.  The remaining centers are
inward.  Their tail is at least $(1-h)T(x_j)$, so
\begin{equation}\label{eq:edge-index-density}
 c(j-\tfrac12)\le\rho_L(x_j)\le C(j-\tfrac12).
\end{equation}
This bounds each fixed-index reciprocal density away from both
zero and infinity.

Terminal gaps cannot become long.  If $x_{j+1}$ also lies beyond
the bare centers and $\Delta=x_j-x_{j+1}$, compare the inward
tail at $x_j$ with its tail at $x_{j+1}$ using
\eqref{eq:kernel-tail-comparison}.  It gives
\[
 (1-h)(j-\tfrac12)
 \le C e^{-2\Delta}(j+\tfrac12),
 \qquad
 \Delta\le\frac12\log\frac{3C}{1-h}.
\]
Conversely $1=\int_{x_{j+1}}^{x_j}\rho_L$ and the upper tail
comparison give $\Delta\ge c/(j+1)$.
Thus each fixed-index gap has positive finite subsequential limits.

The explicit tail of $p_L$, together with its bounded integrated
error, locates any sufficiently large fixed edge index at
$e_L^{\rm edge}+O(1)$, where
\begin{equation}\label{eq:nominal-edges}
 e_L^{\rm edge}=\eta\log L
 \quad\hbox{or}\quad
 e_{d,B}^{\rm edge}=B/2+\eta\log(1/d)
\end{equation}
in the homogeneous and array cases, respectively.  For example,
choose an index larger than the integrated error bound; the
$p_L$ tail is comparable to
$\exp(-(u-e_L^{\rm edge})/\eta)$ near the displayed edge.
That root lies beyond the bare centers in the limiting procedures.
The gap estimate then places the last root no more than a bounded
distance beyond that edge.  The left end is identical.
For each fixed density threshold $M$, the portions with density
below $M$ start within a distance $C_M$ of the nominal edges and
contain only a bounded number of root indices.  Both the number and
the distance may depend on $M$.

These statements have their asserted array uniformity.  In a
purported failure sequence we may take $d\downarrow0$ and $B$
arbitrarily large depending on $d$.  Every fixed edge index is then
eventually beyond all bare centers, and
\eqref{eq:edge-index-density} applies to every such index.
One should not instead take a fixed-$d$ edge limit and assert that
its reciprocal weights tend to zero inward: the subsequent
$d\downarrow0$ limit is essential for that assertion.

For the regular-filling construction, comparison with the full-line
equation gives
\begin{equation}\label{eq:finite-Q-correction}
 r_Q-r=(I+K_Q)^{-1}K_{\rm out}r,
\end{equation}
where $K_{\rm out}$ integrates over $\R\setminus[-Q,Q]$.
If $K\ge0$, then $h<1/3$ in the parameter range at issue and the
right side has sup norm at most $h r(Q)/(1-h)$.
If $K<0$, write it as the negative of the exit kernel
$(I-k_Q)^{-1}k_{\rm out}$ applied to $r$.
That positive kernel has mass at most $h$: the constant $h$ is a
supersolution of its mass equation, since
$h-k_Qh-k_{\rm out}1=(1-h)k_Q1\ge0$.
The error is therefore at most $h r(Q)$.
These estimates prove the comparison for $r_Q$, since
$r(u)\ge r(Q)$ on the interval.  The Neumann series gives the
comparison for $H_Q$; for $K\ge0$ use
$\|H_Q-1\|_\infty\le h/(1-h)<1/2$.
Differentiating \eqref{eq:finite-Q-correction} in $u$ shows that
each fixed derivative of its right side is $O(r(Q))$.
Consequently all fixed derivatives of $r_Q$ are bounded by
$C_jr_Q$.

Differentiation with respect to the interval endpoint, followed by
symmetry of the kernel, gives
\begin{equation}\label{eq:filling-derivative}
 \frac{\dd}{\dd Q}\int_{-Q}^Q r_Q(u)\dd u
       =2r_Q(Q)H_Q(Q)>0.
\end{equation}
Indeed $(I+K_Q)\partial_Qr_Q
=-r_Q(Q)(K(\,\cdot-Q)+K(\,\cdot+Q))$; pair this with $H_Q$.
The mass tends to $\int_\R r=1/2$ by
\eqref{eq:finite-Q-correction}.  Integrating
\eqref{eq:filling-derivative} from $Q$ to infinity proves the
asserted mass-deficit estimate.

Let $y_j^0$ be the mid-quantiles of $Lr_Q$.
The continuum counting function has derivative $Lr_Q$ on the
interval and, by oddness, gives the required levels at these
quantiles.  Midpoint quadrature against a function of bounded
variation bounds the discrete residual of each root equation by
a constant.  The same holds for every fixed derivative of the
counting function: apply that quadrature to translates of the
interaction kernel or its derivatives.

Put $m_L=Lr(Q)$, and use displacement coordinates
$\zeta_j=Lr_Q(y_j^0)(y_j-y_j^0)$.  At the quantiles the Jacobian in
these coordinates is
\[
 I+\left(\frac{K(y_i^0-y_j^0)}{Lr_Q(y_j^0)}\right)_{ij}
       +O(m_L^{-1})
\]
in the sup row norm.  The middle matrix has absolute row norm
at most $h+o(1)$ by quadrature and the relative derivative bounds
just proved.  Its inverse is uniformly bounded.
On any fixed-radius ball in the $\zeta$ coordinates, the physical
displacements are $O(m_L^{-1})$.  Neighbor sums of kernel
derivatives, weighted by $1/(Lr_Q)$, are bounded.  Comparing the
bare-minus-root density to $Lr_Q$ before displacing the roots
also bounds the change of the divided diagonal entries by
$O(m_L^{-1})$.  Thus the Jacobian varies by $o(1)$ on that ball.

For clarity, Newton's map here may be taken with the inverse
Jacobian frozen at the quantiles.  Choose the ball radius larger
than twice that inverse bound times the residual bound.
For large $L$ its derivative has norm less than $1/2$, so it maps
the ball into itself and is a contraction.  The fixed point gives
the exact roots and \eqref{eq:regular-counting-density}.
Positivity of the resulting counting derivative forces their
ordering.  The same estimates hold on compact continuation
intervals joining $\lambda$ to $\pi/2$.  The Jacobian is
nonsingular; equivalently its symmetrization has positive
eigenvalues, since the absolute row bound for the perturbation of
the identity is below one.

It remains to justify quadrature and the state identification.
In a counting cell of density at least $M$, its length $\Delta_j$
is $O(M^{-1})$.  The relative derivative bound gives
$|\ell_j-\Delta_j|\le C\Delta_j^2$.  Integrate a Lipschitz test
cell by cell and include the two partial boundary cells to obtain
\eqref{eq:local-quadrature}.  Uniform local mass follows in the
dense region from the same argument and at the ends from
\eqref{eq:edge-index-density} and the gap bounds.  In regular
filling every cell is dense, including the bounded counting
extension of the interval.

The Gaudin determinant and Lemma~\ref{lem:hilbert-dual} make
each Bethe vector nonzero.  In the homogeneous chain the logarithmic
row derivative at the swap has, up to an additive scalar, local
link matrix equal to the swap times $-\eta R'(0)$.  Its hopping
coefficient on an opposite-spin pair is $-\eta/\sin\lambda<0$.
Within each nontrivial particle sector the hopping graph is
connected, so the sector minimum is simple and has a positive
vector.  At $\lambda=\pi/2$ this is the free nearest-neighbor
exclusion Hamiltonian.  In ordered particle coordinates its
eigenvectors are antisymmetrized plane waves with circle sign
$(-1)^{n-1}$, and the minimum fills the consecutive momenta about
zero.  These are precisely the consecutive Bethe roots: their
physical shift phase is
$e^{ip}=-F_{\{y\}}(i\lambda/2)$, and differentiation of the
eigenvalue gives $-2\pi\eta s(y)=-\cos p$ at the free point.
The sign gauge thus centers the occupied momenta at zero.

The real-root branch remains this simple minimum by continuation.
The physical transfer matrix is primitive in the sector: its
diagonal is positive, and successive allowed auxiliary swaps move
any particle to a neighboring hole, connecting the sector.
It commutes with the link Hamiltonian.  The positive simple
minimum is consequently its Perron vector.  Taking the long
cylinder limit gives its normalized vacuum expectations.
Finally nonsingularity of $G$ gives the finite analytic twist
branch by the implicit function theorem.  This last statement
does not require a twist neighborhood uniform in $L$.
\end{proof}

\subsection{Uniform inverses on pointwise function spaces}

Lebesgue $L^\infty$ equivalence classes cannot be used when an
operator evaluates a function at the roots.  We use $C_b(\R)$
with its pointwise sup norm, or the space of bounded continuously
differentiable functions with norm
$\|u\|_\infty+\|u'\|_\infty$.  The root-node spaces carry their
ordinary finite-dimensional sup norms.

\begin{lemma}[Gaudin and mixed-measure inverse bounds]
\label{lem:gaudin-inverse}
In the limits specified above,
\begin{equation}\label{eq:Gaudin-sup-inverse}
 \|(I+K\ell)^{-1}\|_{\infty\to\infty}\le C,\qquad
 (K\ell)_{ij}=K(y_i-y_j)\ell_j.
\end{equation}
There is also a small $a>0$ for which the inverse has uniformly
bounded row norm after conjugation by either $e^{ay_i}$ or
$e^{-ay_i}$.

Let $\chi_-,\chi_0,\chi_+$ be a smooth nonnegative partition of
unity.  The middle function has compact support, the transitions
lie in regions of a fixed sufficiently high density, and the two
transition regions recede from one another.  Put
\[
 D_L=\nu_L-\dd u,\qquad D^\bullet=\chi_\bullet D_L.
\]
The mixed operator
\begin{equation}\label{eq:mixed-operator}
 I+R_0*(D^0+D^+)
\end{equation}
has a uniform lower bound in the pointwise sup norm.
The analogous assertion holds with the two ends interchanged.
\end{lemma}

\begin{proof}
For regular filling, local quadrature up to the endpoints bounds
the absolute row norm of $K\ell$ by $h+o(1)<1$.
This proves \eqref{eq:Gaudin-sup-inverse} there.
For $K<0$ at half filling, first truncate the exponentially small
kernel tail at a fixed distance and then choose a high density
threshold.  Local mass and \eqref{eq:local-quadrature} show that
every row sufficiently far inside the dense portion has absolute
row sum at most a fixed $q<1$.  The edge-index bounds confine all
other rows to a fixed number of indices at either end.

If the inverse bound failed, there would be vectors $W_L$ with
$\|W_L\|_\infty=1$ and
$(I-k\ell)W_L\to0$ in sup norm.  A near-unit coordinate must
lie among those fixed edge indices.  Retain one end and extract
all fixed-index gaps and weights.  The bounds of
Lemma~\ref{lem:root-limits} make the limiting weights positive
and finite; at every fixed index they satisfy
$c/j\le\ell_j\le C/j$.  The limiting points have no finite
accumulation, since otherwise this lower bound would contradict
the uniform local mass bound.
In the array case this extraction is along $d\downarrow0$ with
$B$ sufficiently large, exactly as in
\eqref{eq:array-limit-convention}.

Put $v_j=\ell_jW_j$ in the resulting one-sided limit.
It is nonzero, and $v_j\to0$ as $j\to\infty$.
Uniform local mass and exponential tails justify passing the
equation to this limit.  It reads
\[
 (\rho_j-k(0))v_j=\sum_{i\ne j}k(x_j-x_i)v_i.
\]
After division by its positive left coefficient, this is an
averaging equation with strictly positive off-diagonal weights
whose sum is at most one.  The latter follows from finite Gaudin
row dominance and Fatou's lemma; a strictly positive limiting
deficit is not required.  The absolute maximum of $v$ is attained,
and some other index has smaller modulus.  Its positive weight
strictly decreases this maximum, a contradiction.

We spell out the mixed case.  The measure
\[
 \widetilde\nu=\dd u+D^0+D^+
       =\chi_-\,\dd u+(1-\chi_-)\nu_L
\]
is nonnegative.  Exponential decay and bounded local variation
justify Fubini, and $R_0+K*R_0=K$ gives the exact identity
\begin{equation}\label{eq:undo-dressing}
 (I+K*)(I+R_0*(D^0+D^+))u
       =u+K*(\widetilde\nu u).
\end{equation}
The line convolution $I+K*$ is boundedly invertible because
$h<1$.

For $K<0$, suppose the undressed operator had normalized
approximate null functions $u_L$.  They satisfy
$u_L=k*(\widetilde\nu u_L)+o(1)$ uniformly.
The quantity
$\sup_x\int k(x-y)\widetilde\nu(\dd y)$ is uniformly bounded.
It follows that the supremum of $|u_L|$ on
$\supp\widetilde\nu$ is bounded below.  Divide by that supremum.
The support supremum is now one and the full supremum remains
bounded.  On the support, every row except a fixed number of
right-edge atoms has sum at most $q<1$: the replaced side is
Lebesgue, the blend is dense, and the remaining noncontracting
part is the original discrete right end.

Thus a near-unit support value survives at one of those atoms.
The modified left side is arbitrarily far away from each retained
right-edge index, so its contribution disappears in the extracted
equation.  The preceding harmonic-maximum argument applies.
There is no additional approximate mode between the atoms:
$k*(\widetilde\nu u_L)$ has a uniformly bounded derivative,
so the approximate equation makes $u_L$ equicontinuous up to a
vanishing sup error.  In the regular-filling case the entire
mixed measure instead has absolute kernel row norm $h+o(1)$,
by quadrature, and contracts directly.
Equation~\eqref{eq:undo-dressing} proves the claimed lower bound.

Finally exponential conjugation changes the discrete kernel row
norm by at most
\[
 \sup_i\sum_j |K(y_i-y_j)|\ell_j
             |e^{a(y_i-y_j)}-1|\le C|a|.
 \]
This follows by summing over unit intervals and using local mass
and exponential kernel decay.  A sufficiently small $a$ therefore
preserves the inverse bound by the inverse identity.
The same perturbation estimate preserves the mixed lower bounds.
The weighted middle contraction is proved explicitly in
Lemma~\ref{lem:weighted-flags}.
\end{proof}

\subsection{An analytic normalization for all marked rows}

The inverse estimates control the scalar-product columns.
To continue the resulting kernels from one convenient spectral
region to the whole strip, we also need a common operator bound.
Write
\[
 \mathcal S_\lambda=\{x\in\C:|\Im x|<\lambda/2\}.
\]
This is the angle strip $|\Re z|<v$ under $x=i\eta z$.

\begin{lemma}[Harmonic row normalization]\label{lem:row-normalization}
There is a real harmonic function $U$ on $\mathcal S_\lambda$
such that, for every finite real-inhomogeneity chain,
\begin{equation}\label{eq:harmonic-row-bound}
 \|T^D(x)\|\le2\|D\|
                 \exp\left(\sum_jU(x-b_j)\right).
\end{equation}
Choose a nonvanishing analytic function $\mathcal N_L$ whose
modulus is the exponential on the right.  In the homogeneous plane
and array limits, the untwisted vacuum eigenvalue $\tau_L$ satisfies
\begin{equation}\label{eq:vacuum-modulus-normalization}
 |\tau_L(x)/\mathcal N_L(x)|\longrightarrow1
\end{equation}
locally uniformly in the strip, with the moving-bulk convention
\eqref{eq:array-limit-convention} in the array case.
Consequently, for each compact $\mathcal K\subset\mathcal S_\lambda$,
\begin{equation}\label{eq:normalized-row-bound}
 \limsup\sup_{x\in\mathcal K}\|A_L^D(x)\|\le2\|D\|.
\end{equation}
In particular the vacuum denominator has no zeros on such a compact
for all sufficiently large systems in the indicated sense.
\end{lemma}

\begin{proof}
On $\Im x=\lambda/2$, local inversion says that each site matrix
divided by its equal-spin entry is unitary.  Taking the auxiliary
trace costs at most $2\|D\|$, and gives the boundary estimate with
$|a(x)|$.  Crossing gives the reflected lower boundary estimate.
For $0<\Im x<\lambda/2$ define
\begin{equation}\label{eq:harmonic-normalizer}
 U(x)=\log\left|\frac{[x+i\lambda/2]}{[i\lambda]}\right|
       +\int_\R r(u)
            \log\left|\frac{[u-x+i\lambda]}{[u-x]}\right|\dd u.
\end{equation}
The integrand has no singularities in that open half-strip and
decays exponentially at both real ends.  Its boundary value on
$\Im x=\lambda/2$ is zero, because the two sinh moduli coincide.
Reflect \eqref{eq:harmonic-normalizer} conjugately to the lower
half-strip.  The two values agree on the real line.  Differentiating
the logarithmic sinh kernels at that line, with the denominator
term interpreted by its Poisson-kernel limit, gives the upper
normal derivative
\[
 \pi(s-K*r-r)=0.
\]
Thus the two harmonic functions glue harmonically.  Their growth
at the real ends is at most linear.

For unit vectors $u,v$, apply the strip maximum principle to
$\log|\langle u,T^D(x)v\rangle|-\sum_jU(x-b_j)$.
The boundary bound is $\log(2\|D\|)$ and the growth is at most
linear.  One may justify the unbounded-strip principle by subtracting
$\varepsilon\cosh(a\Re x)\cos(a\Im x)$, with
$0<a<\pi/\lambda$, applying the principle in finite rectangles,
and then sending $\varepsilon$ to zero.
Taking the supremum over $u,v$ proves
\eqref{eq:harmonic-row-bound}.  The strip is simply connected, so
a harmonic conjugate gives the nonvanishing $\mathcal N_L$.

It remains to compare it with the actual eigenvalue.
It suffices first to work in a fixed small open neighborhood inside
the upper swap point $i\lambda/2$.
There $q_Y(x)\to0$.  Homogeneously, the site ratio $l/a$ has an
arbitrarily small modulus per site if the neighborhood is chosen
close enough to the swap.  The root ratios are uniformly bounded
per root, since their denominator stays away from zero.
As $n\le L/2$, the small site ratio gives exponential decay in $L$.
For the array, $\lambda>2\pi/3$ and the neighborhood can be chosen
so that each site and root ratio has modulus at most one.
Indeed the difference of the squared sinh moduli is governed,
respectively, by $\sin(2\Im x)\sin\lambda>0$ and
$\sin(2\Im x)\sin(2\lambda)<0$.
There are order $1/d$ sites at bounded distance from the moving
bulk center, with site ratios uniformly below one.  Hence
$q_Y(x)=O(e^{-c/d})$ after $B\to\infty$, uniformly in that bulk.

Put
\[
 H_x(u)=\log\left|\frac{[u-x+i\lambda]}{[u-x]}\right|,
 \qquad
 \widetilde H_x=(I+K*)^{-1}H_x.
\]
These are smooth tests in $u$ on the chosen neighborhood.
The exponentially weighted Neumann series gives uniform exponential
decay for $\widetilde H_x$ and all its fixed derivatives, also
after real recentering.  The density equation gives the exact
identity
\begin{equation}\label{eq:normalizer-discrepancy}
 \sum_{y\in Y}H_x(y)-\int_\R p_L(u)H_x(u)\dd u
       =\int_\R\widetilde H_x(u)e_L(\dd u).
\end{equation}
At half filling, integrate by parts against the bounded counting
sawtooth $F_L$.  In counting coordinates its mean over every
complete cell is zero.  On a dense fixed window, the relative
derivative bound for $\rho_L$ makes the physical cell averages
converge to that mean.  Thus $F_L$ tends weakly to zero there.
The exponential tail of the test and the uniform bound $|F_L|\le1/2$
discard the complement of an increasing window.  This proves
that \eqref{eq:normalizer-discrepancy} tends to zero, including
the array limit after bulk recentering.

For the homogeneous near-half-filled sequence, use the true
outer counting half-levels as cell endpoints.  They are a bounded
counting distance from $\pm Q$ by
\eqref{eq:regular-counting-density}, and the positive comparison
extends to them.  The cellwise primitive is zero at the endpoints
and is bounded by $1/2$ inside.  The same midpoint argument
therefore works within these cells.
Outside them, the root measure vanishes and
\[
 |\xi'(u)|\le C Lr(Q)=O(L^\epsilon).
\]
To see this uniform exterior bound, replace the root sum of any
translate of $K$ by its $Lr_Q$ integral.  Midpoint quadrature and
bounded counting displacement cost $O(1)$ by variation and
reciprocal-density summation.  The remaining exterior expression
$L(s-K_Qr_Q)$ is $O(Lr(Q))$, by the full-line equation and
\eqref{eq:finite-Q-correction}.  Its pairing with the dressed
test outside the cells is consequently
$O(L^\epsilon e^{-aQ})$ for a fixed $a>0$.
Since $Q=\eta(1-\epsilon)\log L+O(1)$, it tends to zero when
\begin{equation}\label{eq:epsilon-choice}
 0<\epsilon<\frac{a\eta}{1+a\eta}.
\end{equation}
Decreasing $a$ if necessary gives one such choice on the compact
continuation interval used to construct the roots.

The first-term eigenvalue formula, $q_Y\to0$, and
\eqref{eq:normalizer-discrepancy} now prove
\eqref{eq:vacuum-modulus-normalization} on this open neighborhood.
On the whole strip, $\tau_L/\mathcal N_L$ is bounded by two.
Every subsequential holomorphic limit therefore has modulus one
on the neighborhood and is a unimodular constant by the maximum
modulus principle.  Normal-family compactness gives local uniform
modulus convergence throughout the strip.  The same argument
works for any purported failing moving-bulk array sequence.
Dividing \eqref{eq:harmonic-row-bound} by $|\tau_L|$ proves the
last assertions.
\end{proof}

\subsection{Weighted flags for finite dressed columns}

Fix a word length $m$ and generic arguments $v'_1,\ldots,v'_m$
in a small complex neighborhood of $i\lambda/2$.  Generic here
means that the finite difference factors in
Corollary~\ref{cor:finite-word-expansion} have no poles or forbidden
coincidences.  This is a nonempty open condition.
We may recenter all real coordinates in the array case.
The columns in \eqref{eq:W-determinant}, at zero twist, are denoted
by $W_{v'}$.  Their source columns are
\begin{equation}\label{eq:thermodynamic-source-column}
 m_{v'}(u)=
 \frac{(1+q_Y(v'))\coth(v'-u)-\coth(v'-u-i\lambda)
                       -q_Y(v')\coth(v'-u+i\lambda)}
      {-2\pi i}.
\end{equation}
Extend each finite node solution to the line by
$W=m-K*(\nu_LW)$.  Lemma~\ref{lem:gaudin-inverse}, local mass,
and the kernel derivative bounds give a uniform bound for $W$
and each fixed derivative on the line.  Undoing this convolution
equation gives
\begin{equation}\label{eq:finite-dressing-equation}
 W=W^0-R_0*(D_LW),\qquad W^0=(I+K*)^{-1}m.
\end{equation}
All these statements are uniform over unit linear combinations
of the fixed finite column list.

The pole spaces needed for this equation are particularly small.
Let $(d_j)$ be the increasing list, with multiplicities, formed
from
\begin{equation}\label{eq:decay-pole-list}
 \left\{\frac{2k-1}{\eta}:k\ge1\right\}
 \quad\cup\quad
 \left\{\frac{2k}{1-\eta}:k\ge1\right\}.
\end{equation}
At coincident entries, retain both copies; the list bounds the
maximum possible pole orders.  For every $j$,
\begin{equation}\label{eq:pole-count}
 d_j\ge2j-1.
\end{equation}
Indeed, if $n_1,n_2$ count the entries of the two progressions
strictly below $T=2j-1$, then
$2n_1-1<\eta T$ and $2n_2<(1-\eta)T$.
Adding gives $n_1+n_2<j$.

For completeness, the Fourier calculation behind the list uses
\[
 1+\widehat K(p)
 =\frac{2\sinh((\pi-\lambda)p/2)\cosh(\lambda p/2)}
        {\sinh(\pi p/2)}.
\]
In particular
\[
 \widehat R_0(p)=
 \frac{\sinh((\pi-2\lambda)p/2)}
      {2\sinh((\pi-\lambda)p/2)\cosh(\lambda p/2)}.
\]
The Fourier transform of each coth difference in
\eqref{eq:thermodynamic-source-column} has denominator
$\sinh(\pi p/2)$, as follows by shifting the integration contour
by $i\pi$ and accounting for the coth residues.
This denominator cancels upon dressing.  The remaining possible
poles for both $W^0$ and $R_0$ are exactly those bounded by
\eqref{eq:decay-pole-list}; the apparent singularity at zero is
removable.  On any horizontal Fourier contour avoiding these
poles, the transforms decay exponentially in the real direction.
The generic arguments range in a compact analytic neighborhood,
so this decay is uniform.  Multiplication by any fixed power of
the Fourier variable preserves integrability.

Thus, for $\beta>0$ not in the list, shifting the contour and
subtracting the crossed residues gives
\[
 R_{0,\beta}=R_0-P_\beta,\qquad
 W^0_\beta(c)=W^0(c)-p_\beta(c),
\]
where the global exponential polynomials $P_\beta,p_\beta(c)$
belong to
\begin{equation}\label{eq:residue-space}
 \mathcal E_\beta
 =\Span\{u^q e^{-du}:d<\beta,\ 0\le q<m_d\}.
\end{equation}
Here $m_d$ is the multiplicity in the list, and
$\dim\mathcal E_\beta\le r_\beta:=\#\{j:d_j<\beta\}$.
Using two nearby pole-free contours gives, for some $\delta>0$,
\begin{equation}\label{eq:shifted-kernel-envelope}
 e^{\beta u}|R_{0,\beta}^{(q)}(u)|
       \le C_{\beta,q}e^{-\delta|u|}
 \quad(q=0,1,2),
\end{equation}
and the corresponding bounds for the free columns.
Any other fixed derivative order is available in the same way.
The residue polynomials are subtracted on the whole line, without
cutoffs; this preserves their finite-dimensional translation span.

\begin{lemma}[Finite-chain weighted flags]\label{lem:weighted-flags}
Let $E=\C^m$ be the coefficient space of the fixed column list.
For each admissible $\beta>0$ there is a subspace
$F_{\beta,L}\subset E$ of codimension at most $r_\beta$
such that
\begin{equation}\label{eq:weighted-node-decay}
 |W(c)(u)|\le C_\beta e^{-\beta u}\|c\|_2
 \quad\hbox{at every positive root }u,\quad c\in F_{\beta,L}.
\end{equation}
For a fixed finite list of increasing shifts, these subspaces may
be chosen nested.  There are analogous flags at the negative end.
The constants are uniform in the three limiting procedures,
with regular filling understood as $Q\to\infty$ after $L\to\infty$.
Restriction to any column coordinate subspace preserves the
codimension bound and all unit-vector estimates.
\end{lemma}

\begin{proof}
Choose the partition in Lemma~\ref{lem:gaudin-inverse} so that
\[
 \supp D^-\subset(-\infty,a_L],\qquad
 \supp D^+\subset[b_L,\infty),\qquad a_L<0<b_L.
 \]
The word center is deep between these boundaries.  The middle
support is compact and dense.  At a fixed density threshold,
Lemma~\ref{lem:root-limits} gives
\begin{equation}\label{eq:last-node-split}
 \max Y\le b_L+C_0,\qquad \min Y\ge a_L-C_0,
\end{equation}
with a threshold-dependent constant uniform in the limits.
For regular filling take the transitions just inside $[-Q,Q]$.
For the array both the last root and the transition contain the
same $\eta\log(1/d)$ shift in \eqref{eq:nominal-edges}.
All three signed pieces have uniformly bounded variation mass
on intervals of length one.

Use the pointwise Banach space
\[
 X_\beta=\left\{u\in C^1(\R):
       \|u\|_\beta:=\sup_x e^{\beta x}
                  (|u(x)|+|u'(x)|)<\infty\right\}.
\]
The quadrature error on each unit-scale test is as small as desired
on the middle support and its taper.  To bound
$R_{0,\beta}*(D^0u)$, insert a smooth unit-scale partition in the
integration variable and apply \eqref{eq:local-quadrature}.
For the output and its first derivative, the test derivatives
involve $R_{0,\beta},R_{0,\beta}',R_{0,\beta}''$ and $u,u'$.
The envelopes \eqref{eq:shifted-kernel-envelope} sum after
weighting to give
\[
 \|R_{0,\beta}*(D^0u)\|_\beta
       \le C_\beta\varepsilon_{\rm quad}\|u\|_\beta.
\]
Choose the density threshold so this norm is below $1/2$ for the
fixed finite list of shifts, including zero.

For $c\in E$ solve in $X_\beta$
\begin{equation}\label{eq:shifted-middle-equation}
 (I+R_{0,\beta}*D^0)U_\beta(c)
     =W^0_\beta(c)-R_{0,\beta}*(D^-W(c)).
\end{equation}
The true-column bound and local variation imply
\[
 \|R_{0,\beta}*(D^-W(c))\|_\beta
       \le C_\beta e^{\beta a_L}\|c\|_2.
\]
For example multiply the convolution integrand by
$e^{\beta(x-y)}e^{\beta y}$, use $y\le a_L$, and sum the
exponentially decaying weighted kernel envelopes over unit
intervals.  The same argument applies to the first derivative.
Consequently
\begin{equation}\label{eq:shifted-solution-bound}
 \|U_\beta(c)\|_\beta\le C_\beta\|c\|_2.
\end{equation}
This estimate holds for every coefficient vector before imposing
any residue condition.

Define the linear map
\begin{equation}\label{eq:residue-condition-map}
 \Lambda_\beta(c)
  =P_\beta*(D^0U_\beta(c)+D^-W(c))-p_\beta(c).
\end{equation}
Its values lie in $\mathcal E_\beta$.  Indeed convolution of an
exponential polynomial with a source preserves that span:
expand each $(x-y)^q$.  The middle source is compactly supported,
and the left moments converge because $d>0$ and the left
variation mass is locally bounded.  Rank-nullity therefore gives
$\operatorname{codim}\ker\Lambda_\beta\le r_\beta$.
No inverse or condition-number bound for $\Lambda_\beta$ is used.

Let
\[
 W^p(c)=(I+R_0*D^0)^{-1}
             (W^0(c)-R_0*(D^-W(c))),
\]
using the shift-zero contraction on $X_0$.
Restoring the residues in \eqref{eq:shifted-middle-equation} gives
\[
 (I+R_0*D^0)U_\beta(c)
  =W^0(c)-R_0*(D^-W(c))+\Lambda_\beta(c).
\]
For $c\in\ker\Lambda_\beta$, this identity first proves
$U_\beta(c)\in X_0$: in the rearranged equation the middle
source $D^0U_\beta$ is a finite compactly supported measure,
so its convolution and first derivative are bounded at both
ends.  The other two terms are already bounded.
This membership assertion for a fixed chain does not yet require
a uniform estimate for that compact source.  Only now apply the
shift-zero inverse to identify $U_\beta(c)=W^p(c)$ and obtain
its uniform unweighted bound.

Conversely, if $W^p(c)\in X_\beta$, rewrite its equation with
the shifted kernels.  The defect is an element of
$\mathcal E_\beta$ which also belongs to $X_\beta$.
It is zero: every nonzero exponential polynomial in
\eqref{eq:residue-space} violates the $\beta$-weighted bound at
$+\infty$.  Weighted uniqueness then gives
$W^p(c)=U_\beta(c)$ and $\Lambda_\beta(c)=0$.
Hence
\begin{equation}\label{eq:flag-membership}
 F_{\beta,L}:=\ker\Lambda_\beta
       =\{c:W^p(c)\in X_\beta\}.
\end{equation}
The spaces are nested, because $W^p(c)$ is already unweighted
bounded at $-\infty$ and a larger positive weight at $+\infty$
implies every smaller one.  Estimates
\eqref{eq:shifted-solution-bound} hold uniformly for every unit
vector in these subspaces.

For such a vector, subtract the true and middle equations:
\[
 (I+R_0*(D^0+D^+))(W(c)-W^p(c))
       =-R_0*(D^+W^p(c)).
\]
The right side has sup norm at most $C_\beta e^{-\beta b_L}$.
The infinite Lebesgue portion of $D^+$ causes no extra factor:
sum its uniformly bounded local variation against
$e^{-\beta(b_L+j)}$ over $j\ge0$.
The mixed lower bound therefore gives the same estimate for
$W-W^p$.  At every actual positive root,
\eqref{eq:last-node-split} converts it to
$C_\beta e^{-\beta u}$, which proves
\eqref{eq:weighted-node-decay}.
Reflecting proves the negative-end assertion.
Finally restriction of \eqref{eq:residue-condition-map} to a
coordinate subspace still has rank at most $r_\beta$, and a unit
vector there is a unit vector in $E$.  All the asserted
restriction properties follow.
\end{proof}

\subsection{Joint determinant tails and the limiting kernels}

The flags control determinants, rather than merely their individual
columns.  This distinction is needed when several roots escape
together.

\begin{lemma}[Joint tail bound]\label{lem:joint-determinant-tails}
For a fixed generic row word and any one of its nonzero finite
action paths, let $H=(p_1,\ldots,p_k)$ be the deleted original
roots and $V$ its surviving row arguments.
After extracting the quadrature weights $\prod_{i=1}^k\ell_{p_i}$
from \eqref{eq:exact-path-determinant}, the remaining summand is
bounded in modulus by
\begin{equation}\label{eq:joint-tail-majorant}
 C\exp\left(-\varepsilon\sum_{i=1}^k|p_i|\right)
\end{equation}
for some $\varepsilon>0$.  The constant and exponent may depend
on the fixed word and generic compact parameter set, but are
uniform in the limiting procedures.  In particular, the sum over
paths with $\max_i|p_i|>R$ is $O(e^{-\varepsilon'R})$ uniformly,
for some $\varepsilon'>0$.
\end{lemma}

\begin{proof}
Choose pole-free numbers
$\beta_j\in(2j-2,2j-1)$ for $1\le j\le k$.
By \eqref{eq:pole-count}, the corresponding flag has
codimension at most $j-1$.
For a positive-row minor with $r$ columns, order its rows as
$p_1\ge\cdots\ge p_r\ge0$.
Choose an orthonormal basis of its coefficient space backwards:
the $j$th basis vector belongs to $F_{\beta_j,L}$.
This is possible because these spaces are nested and have
dimension at least $r-j+1$.  A unitary basis change does not alter
the determinant modulus.  Applying
\eqref{eq:weighted-node-decay} to its columns and using the
rearrangement inequality in the determinant expansion gives
\begin{equation}\label{eq:positive-minor-bound}
 |\det W|\le C\exp\left(-\sum_{j=1}^r\beta_jp_j\right)
   =C\exp\left(-\int_0^\infty
                \sum_{j=1}^{k_+(t)}\beta_j\dd t\right),
\end{equation}
where $k_+(t)=\#\{i:p_i>t\}$.
For a determinant with both signs, first expand by its positive
and negative rows.  Every Laplace minor has a column coordinate
subspace, on which the same codimension and unit-vector estimates
hold.  Choose its flag basis separately.  Thus no simultaneous
adaptation of the two tail flags is required.  The finite Laplace
sum gives the product of the positive bound and its negative
counterpart.

The other factors have a precise growth budget.
The hyperbolic Cauchy determinant in
\eqref{eq:exact-path-determinant} contributes at most
\[
 C\exp\left(k\sum_i|p_i|-\sum_{i<j}|p_i-p_j|\right)
\]
after the root-pair cancellations.  To check those cancellations
also near collisions, for each pair of deleted original roots
Corollary~\ref{cor:finite-word-expansion} supplies a reciprocal
$f$ factor.  Combined with its Cauchy denominator, its modulus
is bounded by
\[
 |[p_i-p_j\pm i\lambda]|^{-1}
       \le C_\lambda e^{-|p_i-p_j|}.
\]
All other fixed-argument factors are bounded on the generic
parameter set.  There is at least one removal factor
$\beta(v'-p_i)$ or $\beta(p_i-v')$ for each original root
deleted, and it is at most $Ce^{-|p_i|}$.
The factors $q_Y(v')$ are bounded and the row denominators
$1+q_Y(v')$ are bounded away from zero in the spectral region
used here.  The total remaining cost is therefore
\[
 C\exp\left((k-1)\sum_i|p_i|
                         -\sum_{i<j}|p_i-p_j|\right).
\]
With $k_-(t)=\#\{i:-p_i>t\}$, its exponent is exactly
\begin{equation}\label{eq:Cauchy-tail-cost}
 \int_0^\infty
 \bigl(k_+(t)(k_+(t)-1)+k_-(t)(k_-(t)-1)\bigr)\dd t.
\end{equation}
One verifies this identity first on a region with the signs and
ordering fixed, or by writing every distance as an integral of
indicator functions.

Set $\varepsilon=\min_{j\le k}(\beta_j-(2j-2))>0$.
For every $r\le k$,
$\sum_{j=1}^r\beta_j\ge r(r-1)+\varepsilon r$.
Combining \eqref{eq:positive-minor-bound} at both ends with
\eqref{eq:Cauchy-tail-cost} proves
\eqref{eq:joint-tail-majorant}.  The case $k=0$ is the empty
determinant and needs no tail estimate.
Uniform local quadrature mass now bounds the sums of the
majorant by convergent geometric series on unit boxes.
Discarding boxes with some coordinate beyond $R$ proves the
last assertion, with a harmless smaller exponent.
\end{proof}

\begin{proposition}[Limits of fixed row words]\label{prop:word-kernel}
For each fixed integer $m\ge0$ and fixed $2\times2$ matrices
$D_1,\ldots,D_m$, the homogeneous plane coefficients
\begin{equation}\label{eq:word-kernel-definition}
 \left\langle\Omega_L,
     A_L^{D_m}(x_m)\cdots A_L^{D_1}(x_1)\Omega_L\right\rangle
\end{equation}
converge locally uniformly on $\mathcal S_\lambda^m$ to a
holomorphic function
$\mathcal K_{D_m,\ldots,D_1}(x_m,\ldots,x_1)$.
It is invariant under every common complex translation for which
the translated arguments remain in the strip.
The same functions are the array limits in
\eqref{eq:array-limit-convention}.

The limiting Gram row actions have norm at most $2\|D\|$.
In the regular-filling procedure the expectations of every fixed
equal-time local matrix product converge to those of the
homogeneous plane construction.
\end{proposition}

\begin{proof}
First keep all arguments in the generic neighborhood used above.
The finite row expansion has finitely many action patterns, each
with at most $m$ deleted original roots.  Words with a net change
of particle number have coefficient zero.
On each bounded rapidity window, the line extensions of the
dressed columns and their derivatives are uniformly bounded.
Local quadrature and their convolution equation therefore give,
along any subsequence, a smooth local limit satisfying
\[
 W+K*W=m^{(0)}_{v'},
 \qquad
 m^{(0)}_{v'}(u)=
       \frac{\coth(v'-u)-\coth(v'-u-i\lambda)}{-2\pi i}.
\]
The tails in this equation are negligible by exponential kernel
decay and bounded local mass.  Since $h<1$, the bounded solution
is unique.  The elementary identity
$m^{(0)}_{v'}(u)=s(u-v'+i\lambda/2)$ shows that dressing
replaces $s$ by $r$.  Equivalently its Fourier transform is
$e^{-ip(v'-i\lambda/2)}/(2\cosh(\lambda p/2))$.
This gives
\begin{equation}\label{eq:continuum-dressed-column}
 W_{v'}(u)\longrightarrow
 \frac{1}{2\lambda
          \cosh((u-v'+i\lambda/2)/\eta)}.
\end{equation}
For the array, the same conclusion holds along every bulk-recentered
sequence in its prescribed double-limit sense.

Apply \eqref{eq:exact-path-determinant} on a bounded rapidity
window.  Its weights are exactly the product quadrature
measure.  Local quadrature, \eqref{eq:continuum-dressed-column},
and $q_Y(v')\to0$ turn its sum into the corresponding line
integral.  The apparent near-collision singularities have already
been canceled by the root-pair factors; deleting diagonal tuples
has no effect in the limit, since the maximal atom on such a
window tends to zero.  Lemma~\ref{lem:joint-determinant-tails}
then permits the window to grow to the whole line.
This proves convergence of every path and hence of the word.
It also proves independence of the choice between homogeneous
and array limits.  All surviving ingredients depend only on
rapidity differences.  Changing all variables of integration
by a common real amount therefore proves real-translation
invariance in the $x$ coordinate.

Lemma~\ref{lem:row-normalization} bounds all the finite words
locally uniformly on the full strip.  Normal-family compactness
and the identity theorem extend the convergence from the generic
open set to $\mathcal S_\lambda^m$, with uniqueness of every
subsequential limit.  The derivative in the common translation
direction vanishes on that open set and hence everywhere.
Integrating it along translations staying in the strip gives the
stated complex-translation invariance.
For any finite linear combination of word vectors, apply the
finite operator inequality \eqref{eq:normalized-row-bound} to
its Gram norm.  Passing to the limit gives the bound
$2\|D\|$ for the induced row action and also shows that it
respects zero Gram vectors.

For the remaining equal-time assertion, first fix $Q<\infty$
and send $L\to\infty$ with filling
$\int_{-Q}^Qr_Q$.  The same finite path calculation is valid,
now with dense nodes on $[-Q,Q]$ and the interval dressing
operator.  Then send $Q\to\infty$.
The inverse bounds, local convergence, and joint tail bound
are uniform in this second limit, so the same full-line integrals
result.

This calculation can be made on small contours surrounding
$i\lambda/2$, using nested distinct radii to keep all arguments
generic.  For the regular-filling assertion we only use
$\lambda>2\pi/3$; choose the contour radii so small that all the
fixed shifted poles stay outside them.
On these contours the homogeneous $q_Y$ still tends to zero
exponentially, and the eigenvalue first term is analytic and
nonzero.  Thus the normalized words are holomorphic on the
enclosed neighborhood for sufficiently large $L$.
The same determinant bounds apply on the contours, including
their portions just outside the strict strip.
Cauchy's formula now gives the coincident endpoint values.
At the swap, marked rows implement an arbitrary matrix on one
site together with a shift.  Products implement arbitrary
matrices on adjacent finite sets, and translation supplies any
finite set of sites.  The half-filled homogeneous calculation
has the same endpoint contours and integrals.  This proves
equality of the equal-time local states in the stated order of
limits, without asserting an unrestricted row-strip limit at a
fixed nonzero magnetization.
\end{proof}

\subsection{Clustering of diagonal row words}

\begin{proposition}[Separated diagonal words]\label{prop:word-clustering}
Partition a fixed finite list of diagonal marked rows into groups.
Within each group fix its internal arguments in the strict angle
strip.  Translate the arguments of group $g$ by a common imaginary
amount $it_g$, and let every $|t_g-t_{g'}|$ tend to infinity.
Then the plane word kernel factors into the product of the
individual group kernels.  Convergence is locally uniform in
the internal strict arguments.  The order within each group is
retained.
\end{proposition}

\begin{proof}
We first use generic arguments near the upper boundary in the
$x$ coordinate.  Their real parts may be separated by arbitrary
translations.  The integral formula of
Proposition~\ref{prop:word-kernel} applies there by analytic
continuation of its difference integrals; keep the imaginary
parts in a fixed small generic box and avoid the finitely many
coincidence poles during continuation.

For a path with deleted roots $H$ and surviving arguments $V$,
write $N_H(t)$ and $N_V(t)$ for the numbers of their real parts
to the left of $t$.  These two lists have equal length.
The quotient of the continuum determinant
\eqref{eq:continuum-dressed-column} and the original hyperbolic
Cauchy determinant, including the root-pair cancellations, is
bounded by
\begin{equation}\label{eq:matching-cluster-bound}
 C\exp\left(-(\eta^{-1}-1)
                   \int_\R(N_H(t)-N_V(t))^2\dd t\right).
\end{equation}
Here the constant is uniform as the fixed internal lists separate.
To verify the exponent, use
\[
 \cosh((p-v'+i\lambda/2)/\eta)
       =i\sinh((p-v')/\eta)
\]
and apply the sinh Cauchy identity at scales $1/\eta$ and $1$.
The exponent in their ratio is $-(\eta^{-1}-1)$ times
\[
 \sum_{p\in H,v'\in V}|p-\Re v'|
       -\sum_{p<p'}|p-p'|
       -\sum_{v'<v''}|\Re v'-\Re v''|.
\]
This expression is exactly the integral in
\eqref{eq:matching-cluster-bound}.  The bounded imaginary offsets
and fixed within-group generic factors change only the constant.
The reciprocal $f$ factors remove the near-root-collision
denominators as in Lemma~\ref{lem:joint-determinant-tails}.

The integral in \eqref{eq:matching-cluster-bound} controls ordered
matching distance.  Indeed $N_H-N_V$ is integer-valued, so its
square is at least its absolute value, and
\[
 \int_\R|N_H-N_V|\dd t
     =\sum_j|p_j-\Re v'_j|
\]
for the two ordered lists.  Since $\eta<1$, the estimate
localizes all integration variables integrably around the
surviving arguments, uniformly in their separations.

For diagonal words every action has one removal.  Each removal
also has its factor $\beta$ decaying exponentially in the
distance between the adding argument and the removed argument
or root.  If a path couples two different groups, either this
factor or \eqref{eq:matching-cluster-bound} makes its integral
vanish as the gap grows.  More explicitly, first restrict every
matched root to a fixed radius of its matching argument using
the uniform exponential bound.  On this restricted region any
removal between distinct groups has distance tending to infinity.
Discard these paths and then increase the matching radius.
The surviving paths have their additions and removals inside
individual groups; in each group the numbers of deleted original
roots and surviving added arguments are equal.

On a bounded matching region the two determinants are block
diagonal in the limit, so their quotient tends to the product of
the group quotients.  Cross-group correction products tend to
one.  In fact $f(u)\to e^{\pm i\lambda}$ as
$u\to\pm\infty$, and at every removal the numbers of extra
surviving arguments and missing original roots in another group
agree.  Their limiting phases cancel.  The constants
$(-[i\lambda])^{-1}$ occur once per deleted column and multiply
in the same way.  Termwise limits, the integrable matching bound,
and the finite sum over paths therefore give exactly the product
of the separate group answers.

If different groups were interlaced in the original word, they
may first be gathered while retaining each internal order.
The normalized intertwiner in RTT tends exponentially to an
invertible diagonal matrix at large real separation.
Its conjugation of a pair of diagonal marks therefore differs
from the unchanged pair by an exponentially small matrix.
The bounded limiting rows make the corresponding exchange error
tend to zero.

Finally the row norm bound makes these translated kernels a
normal family as functions of the internal strict arguments.
Every subsequential holomorphic limit equals the product on the
generic open box just treated.  The identity theorem determines
it throughout the product strip, proving the proposition and
its local uniformity.
\end{proof}

The construction now supplies all row-kernel inputs for the
angular Hilbert space: holomorphy, the common-translation
symmetry, bounded marked actions, and diagonal clustering.
The physical identification and endpoint continuation are
treated next.
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//03-thermodynamic-kernels.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//04-angular-plane.tex ====
\section{The plane representation and its angular spectrum}\label{sec:angular}

The word kernels constructed in Section~\ref{sec:thermodynamic} now become
operators on a Hilbert space.  We identify their boundary values with local
spins and spatial translations, and their physical-angle coefficients with
the prescribed plane law.  The resulting spectral description will provide
both the normalization calculation and the separation into two chiral
limits.  Throughout this section
\[
 \pi/3<\lambda<\pi,\qquad \eta=\lambda/\pi,\qquad v=\pi/2,
 \qquad w=e^{iz}.
\]
All constants may depend on the fixed parameter.  The arguments below use
the ordinary homogeneous-row Hilbert norm.  No metric associated with a
diagonal cut, and no positive-association statement for $c<1$, is used.

\subsection{From word kernels to operators}

Let $\mathcal W$ be the complex vector space spanned by formal strict-row
words
\[
 A^{D_m}(z_m)\cdots A^{D_1}(z_1)\Omega,
 \qquad |\Re z_j|<v,
\]
including the empty word $\Omega$.  Give two words the limiting scalar
product obtained by reflecting the first word with the adjoint rule of
Lemma~\ref{lem:row-algebra}.  Proposition~\ref{prop:word-kernel} proves
existence of all these simultaneous limits.  The form is positive, since
every finite Gram matrix is a limit of positive finite-chain Gram matrices.
We quotient by its null space and complete; the resulting Hilbert space is
denoted by $\mathcal H$.

\begin{proposition}[The angular representation]\label{prop:angular-space}
On $\mathcal H$ the rows act as bounded holomorphic operator families, with
\[
 \|A^D(z)\|\le 2\|D\|,\qquad
 A^D(z)^*=A^{X\overline D X}(-\overline z).
\]
They satisfy the finite RTT exchange identities wherever their auxiliary
intertwiner is invertible.  There is a strongly continuous unitary group
$U_b$, $b\in\R$, fixing $\Omega$, such that
\begin{equation}\label{eq:angular-covariance}
 U_bA^D(z)U_b^*=A^D(z+ib).
\end{equation}
Words with real arguments in any nonempty open subinterval of $(-v,v)$
span a dense subspace.  The family $P=A^I$ consists of commuting normal
operators and has a simultaneous representation by almost surely
holomorphic scalar functions $f$ on the strip, with
\begin{equation}\label{eq:angular-scalar-star}
 f(-\overline z)=\overline{f(z)},\qquad |f(z)|\le2.
\end{equation}
In the half-plane coordinate this reality condition reads
$f(\overline w)=\overline{f(w)}$.
\end{proposition}

\begin{proof}
The uniform normalized row bounds of Lemma~\ref{lem:row-normalization},
applied before taking each Gram limit, show that left multiplication by a
row preserves null vectors and has the asserted bound.  Thus it extends
to the completion.  Adjoint and RTT identities pass to all word pairings
and then to bounded operators.  Matrix coefficients on words are
holomorphic by Proposition~\ref{prop:word-kernel}; local boundedness
extends this to all matrix coefficients and hence to operator
holomorphy.

Translate all arguments of a word by $ib$.  In its Gram product the
adjoint arguments are translated by the same amount, because
$-\overline{z+ib}=-\overline z+ib$.  Common-translation invariance of the
word kernels therefore makes this map an isometry.  Translation by $-ib$
is its inverse.  Continuity of the kernels gives strong continuity on
words, and the isometry bound extends it to $\mathcal H$.  This proves
\eqref{eq:angular-covariance}.  Notice that positive dilation of $w$
corresponds to one of the two signs of this unitary parameter.

If a vector is orthogonal to every word whose real arguments lie in a
fixed open interval, successive applications of the identity theorem to
the word arguments show that it is orthogonal to all strict words.  This
proves density.  Commutation of the untwisted rows and their adjoint rule
put them in one abelian von Neumann algebra.  Generate that algebra using
a countable dense set of arguments.  A multiplication representation,
together with the uniformly bounded Taylor series on a countable cover
of the strip, gives scalar analytic representatives on one common set
of full measure.  This also proves
\eqref{eq:angular-scalar-star}.  The spectral measure class is invariant
under imaginary translation by \eqref{eq:angular-covariance}.
\end{proof}

\subsection{Finite division and the two boundary lines}

Boundary rows have a direct spin interpretation.  Establishing that
interpretation in the limiting space requires more than convergence of
scalar eigenvalues.  We give the finite estimate and the common-radius
continuation argument explicitly.  This is the finite local mechanism
used in the angular-transfer construction of
\cite[Section ``Angular transfer and its two leading modes'']{AngularRepository2026};
the proof here checks it for the present parameter range.

\begin{lemma}[Boundary division and scalar continuation]
\label{lem:scalar-continuation}
There is $r_\lambda>0$, independent of the even chain length, for which
the two homogeneous quotients
\[
 T^D(z)T^I(z)^{-1},\qquad T^I(z)^{-1}T^D(z)
\]
are holomorphic and bounded by $C_\lambda\|D\|$ on discs of radius
$r_\lambda$ about $z=v$ and $z=-v$.  The limiting rows have strong
boundary values
\begin{equation}\label{eq:angular-endpoints}
 A^D(v)=SD_0,\qquad
 A^D(-v)=(XD^tX)_0S^{-1},
\end{equation}
where $S$ is the unitary site shift and $S\Omega=\Omega$.
Almost every spectral function extends analytically and without zeros
to bands of a common positive width about $\Re z=\pm v$, has modulus
one on these lines, and satisfies $|f|\le1$ between them.  Consequently
it is inner in the right half-plane, and every zero $\zeta$ satisfies
\begin{equation}\label{eq:angular-cone}
 \Re\zeta\ge c_0|\zeta|
\end{equation}
for a fixed $c_0>0$.  At each specified strict argument the spectral set
on which $f$ vanishes has measure zero.
\end{lemma}

\begin{proof}
Write $u=\eta(v-z)$, so $u=0$ at the upper endpoint, and put
$b(u)=\sin u/\sin\lambda$ and $r(u)=1-b(u)^2$.  Multiply the row
$T^D(u)$ by the unmarked row $\widetilde T^I(-u)$ with reversed
auxiliary order.  Folding the second auxiliary index by transposition
expresses the product as the two-auxiliary partial trace of
\[
 (D\otimes I)G_0(u)\cdots G_{L-1}(u),\qquad
 G_j(u)=R_{aj}(u)R_{a'j}(-u)^{t_{a'}}.
\]
The auxiliary line spanned by
$\Phi=2^{-1/2}(|++\rangle+|--\rangle)$ is invariant.  Inversion of
$R$ makes the action on this line equal to $r(u)I$.  At $u=0$ the
swap identity makes the range of $G_j(0)$ lie in
$\C\Phi\otimes\C^2$.  Thus, in the same auxiliary decomposition for
every site,
\begin{equation}\label{eq:folded-block}
 r(u)^{-1}G_j(u)=
 \begin{pmatrix}I&B_j(u)\\0&C_j(u)\end{pmatrix},\qquad
 \|B_j(u)\|\le C_\lambda,\quad
 \|C_j(u)\|\le C_\lambda|u|.
\end{equation}
These bounds follow directly from the entries of $R$, since
$\sin\lambda\ne0$.  For example, on a sufficiently small disc of
radius proportional to $\sin\lambda$ one has $|r|\ge3/4$ and
$\|C_j\|\le1/4$.  The upper-right block of the product in
\eqref{eq:folded-block} is bounded by a geometric series, while its
lower-right block has norm at most $4^{-L}$.  Taking the unmarked
auxiliary trace therefore gives
\[
 r(u)^{-L}T^I(u)\widetilde T^I(-u)=I+F_L(u),
 \qquad \|F_L(u)\|\le3\cdot4^{-L}.
\]
The off-diagonal triangular block has zero auxiliary trace.  The marked
partial trace is uniformly bounded by $C_\lambda\|D\|$.  The displayed
identity is invertible, so both finite square factors are invertible and
\[
 T^D(u)T^I(u)^{-1}
 =\bigl(r(u)^{-L}T^D(u)\widetilde T^I(-u)\bigr)(I+F_L(u))^{-1}
\]
has the required uniform bound.  Reversing the local product gives the
same argument with an invariant covector and proves the other quotient.
Crossing transfers it to the lower endpoint.  None of these steps uses
the sign of $\cos\lambda$.

The common finite vacuum is an eigenvector of $T^I$ also in these discs
by analytic continuation.  Its eigenvalue is nonzero there.  Applying
the quotient to that vacuum gives uniform analytic bounds on
$A_L^D(z)\Omega_L$, including all fixed Taylor orders.  Pairing with
strict words and taking normal-family limits yields a unique bounded
holomorphic vacuum vector on the disc, since its values on the interior
are already fixed by Proposition~\ref{prop:word-kernel}.

Here is a careful way to include endpoint vectors without assuming their
membership in $\mathcal H$.  First take a subsequential Gram enlargement
containing finite words of bounded endpoint rows and shifts.  Finite
swap and crossing identities give \eqref{eq:angular-endpoints} in this
enlargement.  Uniform Taylor estimates compare endpoint vacuum vectors
with nearby strict vectors in norm, before the chain limit as well as
after it.  Their limits are therefore the continued vectors just
constructed.

Choose a closed interval $I\subset(-v,v)$ with nonempty interior such
that every RTT intertwiner between either endpoint and an angle in
$I$ is invertible.  Such an interval exists: the excluded angles form a
finite set.  More explicitly, the possible zeros of the intertwiner
determinant occur at $s\in\pi\Z\pm\lambda$, where
$s=\eta(z-\alpha)$.  When $\lambda>\pi/2$ the possible interior
exceptions at the endpoints are
$\alpha=\pm(v-(\pi-\lambda)/\eta)$; avoiding these points suffices.
There is then a common endpoint disc on which all these inverses are
bounded.  Commute an endpoint-near row through a fixed word with angles
in $I$.  RTT expresses the result as a finite sum of bounded strict
rows acting on continued marked vacuum vectors.  The number and bounds
of the terms may depend on the word, but the radius of analyticity
does not.  The same finite-chain estimates give norm convergence to the
endpoint action.  Density of the $I$-words and the uniform interior row
bound extend this to strong convergence on all of $\mathcal H$.
In particular both $S$ and $S^{-1}$ preserve $\mathcal H$; their
restrictions are unitary inverses.  Successive strong approximations
give the same Gram limits for products containing several endpoint rows.

To pass from this vector continuation to scalar continuation with one
radius, choose countably many dense $I$-words $h_j$ and a finite measure
equivalent to the scalar spectral measure.  On a fixed endpoint disc
write the continued vector $P(z)h_j$ as
$\sum_{k\ge0}a_{jk}(z-v)^k$.  For $r'<R$ smaller than its common
radius, Cauchy's estimate gives
\[
 \|a_{jk}\|_{L^2}\le M_{j,R}R^{-k},\qquad
 \sum_{k\ge0}(r')^k\|a_{jk}\|_{L^2}<\infty.
\]
The series of fiber norms is therefore summable almost everywhere.
Intersecting over the countable words and radii produces pointwise
holomorphic vector series on the same disc.  On the interior part each
series equals $f(z)h_j$.  Its continuation is consequently proportional
to $h_j$.  At almost every spectral point at least one $h_j$ is nonzero,
and the resulting scalar ratios agree by the identity theorem.  This
constructs the scalar continuation on a common disc.

Equation~\eqref{eq:angular-endpoints} gives unitary values at both
endpoints.  Covariance and strong convergence give unitary boundary
values at $\pm v+ib$.  Applying this first on a countable dense set of
$b$'s and then using continuity shows that $|f|=1$ on each continued
boundary diameter.  Schwarz reflection gives
$f(z)\overline{f(2v-\overline z)}=1$ there, and the analogous formula
at $-v$.  Thus the continued functions are nonzero.  Rational imaginary
translations cover the boundary lines by discs of one radius, with a
common null set.  The bounded strip maximum principle now yields
$|f|\le1$.  Under $w=e^{iz}$ the excluded boundary bands give
\eqref{eq:angular-cone}.  Finally each scalar function has isolated
zeros.  Fubini on an imaginary-translation orbit gives a null zero set
at almost every point of that orbit, and invariance of the spectral
measure class transfers the assertion to every specified point.
\end{proof}

\subsection{Identification with the physical plane}

We next fix the probabilistic meaning of this representation.  The
physical row angle is $z=0$, or $w=1$, and the physical weights are
$cR(\lambda/2)$, with $a=b=1$.  Multiplying every vertex weight by $c$
does not affect any normalized finite-volume law.

\begin{proposition}[Physical law, both axes, and reflection]
\label{prop:physical-plane}
The row representation gives the local probabilities of the plane law
$\P_c$ in Theorem~\ref{thm:main}.  In particular its prescribed iterated
balanced-torus limit exists when $c<1$.  Quantization with time pointing
east and quantization with time pointing north describe this same law.
For an ordinary observable supported on one side of either coordinate
cut, the squared norm of its prepared state is its expectation paired
with its complex-conjugate reflected observable, with the corresponding
spin reflection.  Products of neutral height observables may be prepared
using paths confined to that side.
\end{proposition}

\begin{proof}
At fixed even transverse length, primitivity of the physical transfer
in the chosen sector identifies the long-torus limit with its normalized
Perron vector.  A finite collection of arrows is read by physical row
steps on a seam and by spins on finitely many equal-time sections.
Repeated seam marks are combined, and the spins are expressed by
\eqref{eq:angular-endpoints}.  Its joint probabilities are therefore
finite linear combinations of strict and endpoint word coefficients.
Their limits exist by Proposition~\ref{prop:word-kernel} and
Lemma~\ref{lem:scalar-continuation}.  When $c<1$ the root construction
used the exactly balanced sector, so these are precisely the two limits,
in their prescribed order, defining $\P_c$.

For completeness we identify the state also when the root construction
uses sectors tending to half filling.  Let $W_c$ be the physical vertex
weight and, for a stationary probability measure $\rho$ on allowed
arrow configurations, put
\begin{equation}\label{eq:angular-free-energy}
 \mathcal F_c(\rho)=-h(\rho)-\E_\rho\log W_c,
\end{equation}
where $h$ is specific entropy per vertex.  The equivalent gradient
formulation fixes the height parity and forgets an additive constant;
these changes have zero entropy cost per unit area.  Spatial translation
acts on gradients: after translating an anchored height representative,
one subtracts its new reference value.  Thus fixing the reference parity
does not remove any physical translation from the gradient action.
The domain and reference-height convention are those of
\cite[Section 2.3 and Appendix B]{Sheffield2005}; the entropy--energy
decomposition is also explicit in
\cite[Sections 2.3.2--2.3.4]{LammersTassy2024}.
Since the alphabet
is finite and $0<c<\infty$, the entropy and local energy in
\eqref{eq:angular-free-energy} are finite.  Hard constraints are included
by restricting to allowed configurations.

Write $U(0)$ for the homogeneous harmonic normalization in
Lemma~\ref{lem:row-normalization}, and set
$p_*=U(0)+\log c$.  The same boundary-unitarity and strip argument used
there applies to a row with specified auxiliary input and output spins:
it is a matrix-unit auxiliary mark.  A rectangle of width $a$ and
height $b$, with any boundary data, therefore has partition sum at most
\[
 \exp\{abp_*+C(a+b)\}.
\]
Indeed each open row has operator norm at most $2e^{ap_*}$, and
summing the boundary spins costs only an exponential in its perimeter.
The finite entropy variational inequality on the rectangle then gives
$h(\rho)+\E_\rho\log W_c\le p_*$ for every stationary allowed state.

For the near-balanced tori, choose their longitudinal lengths sufficiently
large to realize the Perron asymptotic.  Their constrained log partition
functions per vertex tend to $p_*$ by
Lemma~\ref{lem:row-normalization}.  Tile each torus by fixed-size squares.
Entropy subadditivity, with a negligible uncovered fraction, bounds its
entropy per vertex above by the entropy of a fixed-square marginal
divided by the square area, plus a vanishing boundary error.  Pass to the
local limit at this fixed square size, then let the square size grow.
The exact finite identity ``entropy plus mean log weight equals log
partition function'' proves the reverse inequality
$h(\rho)+\E_\rho\log W_c\ge p_*$.  Hence the limiting state attains
the global minimum of \eqref{eq:angular-free-energy}.  Its expected
spatial tilt is zero by the sector sizes.  The other expected tilt is
zero as well: at the physical angle the current mark is skew-adjoint,
whereas its finite physical expectation is real.  This observation
specifies the tilt; it is not being used as a uniqueness argument.

The equilibrium characterization following equation (6) in
\cite{DKLM2026}, for $1\le c\le2$, identifies the
finite-specific-free-energy minimizer with the balanced plane law.
We spell out the variational-domain reduction needed to apply this
characterization, since a zero expected tilt alone would not suffice.
For a fixed spatial translation lattice $\mathcal L$, let
$\mathcal P_{\mathcal L}$ denote invariant laws of allowed gradients and
write
\[
 \sigma_{\mathcal L}(a)
  =\inf\{\mathcal F_c(\rho):\rho\in\mathcal P_{\mathcal L},\ S(\rho)=a\}.
\]
Here $S(\rho)$ is the expected physical slope.  Both the preceding
perimeter bound and the tiling proof apply on this domain, using the
average energy over a fundamental cell if necessary.  We keep the same
$\mathcal L$ when forming the infimum and the ergodic decomposition.
The surface tension is unchanged on passing from full translations to
a finite-index sublattice: average any sublattice-invariant law over
the finitely many translation cosets.  This preserves its slope and
its entropy--energy density and makes it fully invariant; the reverse
inequality between the infima follows from inclusion of the domains.
Thus one may use the full physical gradient translation action.  If a
periodic height-potential representation uses a parity-preserving
sublattice, the argument is performed throughout on that sublattice
and is then projected back to arrows.

Specific free energy is affine under this ergodic decomposition
\cite[Chapter 3]{Sheffield2005}.  Thus every component of a globally
minimizing law is itself a global minimizer: each component has free
energy at least the global infimum, and their average equals it.
For $c\ge1$, strict convexity of the six-vertex surface tension on the
interior of its slope domain is
\cite[Theorem 13.16 of the 2020 preprint]{LammersTassy2024}.
Its stated condition is $c^2\ge\max(1,1)$, and the accompanying
simply-attractive-potential description supplies the height-model
setting.  Arrow reversal makes $\sigma_{\mathcal L}$ even.  The origin
is an interior slope: height functions formed as sums of two
one-dimensional unit-increment paths give allowed stationary gradient
laws at every slope in $[-1,1]^2$, all of finite specific free energy.
Hence strict convexity implies
$\sigma_{\mathcal L}(a)>\sigma_{\mathcal L}(0)$ for every nonzero
interior $a$.  A boundary slope cannot be another minimizer: otherwise
convexity along the segment from $0$ to that slope would contradict
this strict inequality at an interior point.  Every minimizing ergodic
component therefore has slope zero and is a fixed-slope minimizer.
The variational principle makes it a minimal gradient phase.

At this last step we use the zero-slope uniqueness supplied by the
combined equilibrium characterization in \cite{DKLM2026}, with its
Sheffield and delocalization inputs and its range $1\le c\le2$.
This is not an assertion of uniqueness for every Gibbs state, nor an
extension to $c>2$.  The domain reduction above explains why the
possibly nonergodic auxiliary state is covered, even when a formulation
of the underlying height result is stated for minimal phases.
The gradient-to-arrow correspondence is one-to-one after forgetting
the additive height constant, so their common law is $\P_c$.

It remains to compare the two axes in the range $c<1$, where we have not
used an equilibrium uniqueness theorem.  Fix a rectangle.  Orient its
input path upward along the left side and then eastward along the top
side, and express all spins relative to those directed steps.  Let $K$
be the tensor of its unmarked physical tiles, viewed as a matrix from
the input sides to the other two sides.  It is a product of physical
$R(\lambda/2)$ matrices and permutations of tensor factors.  The local
eigenvalues before the harmless factor $c$ are
$1/c,1/c,1/c+1,1/c-1$.  Thus $K$ is invertible for $c<1$.

For a finite surrounding cylinder, cut on all four sides and contract
the exterior with its normalized vacuum data.  This gives a linear
functional $L$ of the inside matrix, with $L(K)=1$.  The normalization
includes the exterior transfer eigenvalues appropriate to that cylinder.
Putting elementary matrices $D_1,\ldots,D_q$ on the input path tests
the functional on $K(D_1\otimes\cdots\otimes D_q)$: the row index of
each mark is inside and its column index outside.  Since tensor products
of elementary matrices span the input matrix algebra and $K$ is fixed
and invertible, the limiting marked path values determine every entry
of the limiting exterior functional.  They also prove that those
entries have limits.

In the eastward representation the successive path angles are $v,0$;
in the northward representation, with space pointing west, they are
$0,-v$.  To compare them, first replace the first pair by
$(v-\varepsilon,\varepsilon)$ and translate both angles by $-v$.
The resulting pair is $(-\varepsilon,-v+\varepsilon)$, still strict.
Common-translation invariance identifies the marked word values.
Strong boundary convergence then permits $\varepsilon\downarrow0$.
The spin and matrix-index conventions agree: at a reversed collapsed
step the mark is $XD^tX$, exactly as in
\eqref{eq:angular-endpoints}.  Crossing only relabels those spins; the
interior physical tensor remains $K$.  Thus the two normalized limiting
functionals agree on $KD$ for every elementary tensor $D$, and hence on
every inside matrix.  All local arrow probabilities agree.  This
argument compares normalized exterior functionals; it requires no
equality between eigenvalues of cylinders of different circumferences.

Finally, in either homogeneous axial quantization, transposing the
finite row product at a coordinate cut reflects its tiles and changes
the spin mark by the stated adjoint rule.  The finite Gram square is
therefore the expectation of the observable and its conjugate reflected
copy.  Pass to the local and Gram limits to obtain the assertion.
Neutral height differences can be read along paths on the prescribed
side of the cut, so the same identity applies to their finite products.
\end{proof}

\subsection{Finite inner functions and a measure on dilation orbits}

We now restrict when indicated to the closed diagonal-word subspace
$\mathcal D\subset\mathcal H$: it is generated by words all of whose
marks are diagonal.  It is invariant under $P,J$ and their adjoints,
where $J=A^Z$ and $Z=\diag(1,-1)$.  Endpoint diagonal rows show that it
contains the states needed for ordinary height insertions.  Write
$P(w)$ and $J(w)$ in half-plane coordinates from now on.

\begin{lemma}[Inner spectra and the current measure]\label{lem:inner-spectra}
On $\mathcal D$, almost every spectral function is either $1$ or a
finite Blaschke product in the right half-plane, with
\begin{equation}\label{eq:finite-inner-symmetry}
 f(0)=f(\infty)=1,\qquad
 f(\overline w)=\overline{f(w)}.
\end{equation}
Its zeros obey \eqref{eq:angular-cone}.  In particular its degree is
even; write it as $2n_f$ when it is nonconstant.  Every zero produces
a fixed-width, fixed-depth dip of $|f(e^s)|$ near its logarithmic radius.

There is a possibly infinite positive measure $\mu$ on the nonconstant
functions and a measurable spectral vector $\mathsf h$ such that
\begin{equation}\label{eq:current-cocycle}
 J(w)\Omega=(1-f(w))\mathsf h,
 \qquad \mu(\dd f)=\|\mathsf h(f)\|^2\,\sigma(\dd f).
\end{equation}
Here $\sigma$ is any scalar measure used for the direct-integral
representation; the resulting measure $\mu$ does not depend on that
choice.  It is invariant under dilation of the argument.  On the
cross-section where the smallest zero radius is one, it has the form
\begin{equation}\label{eq:orbit-disintegration}
 \mu=\dd s\,\nu(\dd F),\qquad f(w)=F(e^s w),\qquad
 \kappa:=\nu(1)<\infty.
\end{equation}
If $\mathcal L(F)$ is the difference between its largest and smallest
logarithmic zero radii, then
\begin{equation}\label{eq:orbit-width-tail}
 \nu\{\mathcal L>t\}\le Ce^{-2\eta t}\quad(t\ge1).
\end{equation}
For a fixed diagonal word $h$,
\begin{equation}\label{eq:diagonal-boost-tail}
 \|(1-P(e^s))h\|\le C_h e^{-\eta|s|}.
\end{equation}
For products of fixed words placed at scales $r_i$, the corresponding
bound outside their fixed internal argument ranges is
$C\sum_i e^{-\eta|s-\log r_i|}$, uniformly when these ranges separate.
\end{lemma}

\begin{proof}
As an auxiliary argument is dilated far from all arguments in a fixed
word, the scalar-normalized RTT intertwiner tends to an invertible
diagonal matrix, with error $O(e^{-\eta|s|})$.  Its limiting diagonal
matrix commutes with the diagonal joint auxiliary marks.  Move $P(e^s)$
through the word to the vacuum, where it equals the identity.  The
exchange errors are finite sums of bounded marked-row products, proving
\eqref{eq:diagonal-boost-tail}.  Moving through several separated groups
gives the stated sum, since the number of factors is fixed.  These are
operator-norm bounds on each finite exchange error, not just bounds on
its vacuum expectation.

On a countable total set in $\mathcal D$, the spectral integral of
$|1-f(e^s)|^2$ over $s$ is consequently finite.  The general inner
factorization allowed by Lemma~\ref{lem:scalar-continuation} is
\begin{equation}\label{eq:general-inner-factorization}
 f(w)=\gamma e^{-aw-b/w}\prod_j\beta_j(w),\qquad
 a,b\ge0,\quad |\gamma|=1,
\end{equation}
where $\beta_j$ is a constant-phase multiple of
$(w-\zeta_j)/(w+\overline\zeta_j)$ and
$\sum_j\Re\zeta_j/(1+|\zeta_j|^2)<\infty$.
Indeed continuation through every nonzero finite point of the imaginary
axis excludes singular measure there; the only remaining singular
atoms are at $0$ and $\infty$, giving $b/w$ and $aw$.
If $a>0$ or $b>0$, then $|f(e^s)|$ tends to zero at one end, contrary
to the integrability just proved.  For a zero $\zeta$ with
$r=|\zeta|$, the identity
\[
 \left|\frac{t-\zeta}{t+\overline\zeta}\right|^2
 =1-\frac{4t\Re\zeta}{|t+\overline\zeta|^2},\qquad t>0,
\]
and \eqref{eq:angular-cone} imply
$1-|f(e^s)|\ge\varepsilon_0$ whenever
$|s-\log r|\le\delta_0$, with fixed positive
$\varepsilon_0,\delta_0$.  Infinitely many zeros would escape to one
end and supply infinitely many disjoint such intervals, again
contradicting integrability.  The resulting finite product has limits
at both ends, and integrability forces both limits to be one.
Reality was already established.  Nonreal zeros occur in conjugate
pairs, and the endpoint quotient is $(-1)^{\deg f}$, so the degree is
even.  This proves \eqref{eq:finite-inner-symmetry} and the dip assertion.

Differentiate commutation of equal diagonal-twist transfers at twist
zero and apply it to $\Omega$.  Since $P(w)\Omega=\Omega$, it gives
\begin{equation}\label{eq:twist-cocycle}
 (1-P(w'))J(w)\Omega=(1-P(w))J(w')\Omega.
\end{equation}
There is no component of $J(w)\Omega$ on $f=1$.  Indeed, let $Q$ be
the projection to that spectral set, which contains $\Omega$.
The endpoint identities and their dilates glue $QJ(w)Q$ to the left
half-plane by $QJ(-w)Q=-QJ(w)Q$, because $P(w)=I$ on $Q$.
It is bounded on the punctured plane, so the singularities at zero
and infinity are removable.  Liouville's theorem and oddness make it
zero.  On every
nonconstant fiber $|f(1)|<1$, so one may set
$\mathsf h=J(1)\Omega/(1-f(1))$.  Equation~\eqref{eq:twist-cocycle}
gives \eqref{eq:current-cocycle}, first at a countable set of arguments
and then everywhere, including endpoints by strong convergence.
Covariance, with $U_b\Omega=\Omega$, makes this quotient transform
without a scalar multiplier.  Its squared fiber norm therefore gives
a dilation-invariant measure.

Every nonconstant finite product has a unique smallest zero radius.
Rescale it to one to obtain a measurable cross-section.  Translation
invariance in its logarithmic dilation coordinate gives
\eqref{eq:orbit-disintegration}, initially as a sigma-finite
disintegration.  Each orbit has a dip of fixed width at its first zero,
where $|1-f(1)|\ge\varepsilon_0$.  Since
\[
 \int |1-f(1)|^2\dd\mu=\|J(1)\Omega\|^2<\infty,
\]
this disintegration is locally finite and $\kappa<\infty$.

Applying \eqref{eq:diagonal-boost-tail} to $J(1)\Omega$ gives
\begin{equation}\label{eq:two-dip-bound}
 I(l):=\int |1-f(1)|^2|1-f(e^l)|^2\dd\mu
 \le Ce^{-2\eta l}\qquad(l\ge1).
\end{equation}
On an orbit of width $\mathcal L$, integrate first over the dip window
of its smallest zero and then over $l$ placing the second argument in
the dip window of its largest zero.  Both intervals have a fixed width
and both factors a fixed lower bound.  Thus
\[
 c\,\nu\{\mathcal L>t\}
 \le \int_{t-2\delta_0}^{\infty} I(l)\dd l,
\]
after changing the constant for bounded $t$.  This proves
\eqref{eq:orbit-width-tail}.
\end{proof}

\subsection{Low spectral bands and the two leading current modes}

The next estimate is an operator statement on the full space
$\mathcal H$.  It therefore applies also when the states on either side
of a current are different.  Define
\[
 H_0=-\log|P(1)|,\qquad Q_u=\bbone_{[0,u]}(H_0),\qquad
 Q_u^\sigma=Q_u\bbone_{\{\sgn P(1)=\sigma\}},\quad \sigma=\pm1.
\]
The logarithm is defined by spectral calculus; the fixed-argument
no-zero assertion makes its value finite almost everywhere.  Put
\begin{equation}\label{eq:angular-N-definition}
 N(w)=\tfrac12\{J(w)P(w)+P(w)J(w)\}.
\end{equation}
This current averages the two positions of a mark among two consecutive
steps.  The factor $1/2$ will be retained throughout.

\begin{lemma}[Low-band current estimates]\label{lem:low-band}
There are constants $A,a,C,u_0>0$ such that, for $0<u<u_0$,
$Q_uJ(w)Q_u$ extends to a bounded holomorphic function on the full
annulus $Au<|w|<a/u$.  If
\begin{equation}\label{eq:angular-Laurent}
 Q_uJ(w)Q_u=\sum_{j\in\Z}w^j J_{u,j},
\end{equation}
then $\|J_{u,j}\|\le C(Cu)^{|j|}$, with the exponent zero for $j=0$.
Set $\theta=\eta/2>0$.  Uniformly on the closed right unit semicircle,
\begin{align}
 \|Q_u^\sigma J(w)Q_u^\sigma\|&\le Cu,
 &\|Q_u^\sigma J(w)Q_u^{-\sigma}\|&\le Cu^\theta,
 \label{eq:angular-J-bands}\\
 Q_u^\sigma N(w)Q_u^\sigma
 &=\sigma Q_u^\sigma(wJ_{u,1}+w^{-1}J_{u,-1})Q_u^\sigma+O(u^2),
 \label{eq:angular-N-leading}\\
 \|Q_u^\sigma N(w)Q_u^{-\sigma}\|&\le Cu^{1+\theta},
 &\|Q_uN(w)Q_u\|&\le Cu.
 \label{eq:angular-N-bands}
\end{align}
The estimates remain valid for unequal input and output cutoffs, using
their maximum for $u$.  With $m$ ordinary time rows on each side,
$N$ costs $O(m^{-1})$, its remainder after the leading-mode expression
costs $O(m^{-1-\theta})$, and a sign-opposite undoubled $J$ costs
$O(m^{-\theta})$.

Finally, if $S=P(i)$ and $K$ is the principal argument of $S^2$, then
\begin{equation}\label{eq:angular-phase-bound}
 S^2=e^{iK},\qquad |K|\le C H_0.
\end{equation}
\end{lemma}

\begin{proof}
Evaluate \eqref{eq:general-inner-factorization} at $1$.  On the band
$H_0\le u$ it gives
\[
 a+b+\frac12\sum_j
 \log\left(1+\frac{4\Re\zeta_j}{|1-\zeta_j|^2}\right)\le u.
\]
The cone condition implies the zero budget
\begin{equation}\label{eq:angular-zero-budget}
 a+b+\sum_j\min(|\zeta_j|,|\zeta_j|^{-1})\le Cu.
\end{equation}
Indeed $\Re\zeta_j\ge c_0|\zeta_j|$ and
$|1-\zeta_j|\le1+|\zeta_j|$.  Each zero is therefore either of
radius at most $Cu$ or of reciprocal radius at most $Cu$.
Comparing each factor at $w$ and at $1$ shows, on
$Au<|w|<a/u$, $\Re w\ge0$, that
\[
 \left|\log\frac{f(w)}{f(1)}\right|
 \le Cu(|w|+|w|^{-1}).
\]
Choose $A$ large and $a$ small once and for all.  The function and its
inverse are then bounded on this half-annulus.  More precisely,
\begin{equation}\label{eq:angular-band-sign}
 P(w)Q_u^\sigma=\sigma Q_u^\sigma
       +O\bigl(u(|w|+|w|^{-1})\bigr),
\end{equation}
and the same estimate holds for the inverse restricted to the band.

The endpoint formulas give
\[
 J(i)=SZ_0,\qquad J(-i)=-Z_0S^{-1},\qquad
 J(-i)=-P(i)^{-1}J(i)P(i)^{-1}.
\]
The other endpoint gives the analogous identity on the other ray.
Dilate these uncompressed identities using
\eqref{eq:angular-covariance}, and only then compress.  This is
important because $Q_u$ need not commute with the dilation group.
The formula
\begin{equation}\label{eq:angular-current-gluing}
 J_u(-w)=-P(w)^{-1}J_u(w)P(w)^{-1},\qquad
 J_u(w)=Q_uJ(w)Q_u,
\end{equation}
defines the left half of the annulus; both inverses here and below are
only on $Q_u\mathcal H$.  Boundedness and strong boundary convergence
make the two definitions agree on both imaginary rays.  Scalar Morera
applied to every matrix coefficient gives holomorphic gluing.  Thus
$\|J_u(w)\|\le C$ on the full annulus.  Cauchy's formula on circles
near its inner and outer boundaries proves
\eqref{eq:angular-Laurent} and its coefficient bounds.  In particular,
uniformly on $|w|=1$,
\begin{equation}\label{eq:angular-odd-expansion}
 \tfrac12(J_u(w)-J_u(-w))
 =wJ_{u,1}+w^{-1}J_{u,-1}+O(u^3)=O(u).
\end{equation}
The nonconstant even modes are $O(u^2)$, but this observation alone
does not control the constant coefficient.

For $X_{\sigma\tau}(w)=Q_u^\sigma J(w)Q_u^\tau$,
\eqref{eq:angular-current-gluing} and
\eqref{eq:angular-band-sign} give
\[
 X_{\sigma\tau}(-w)=-\sigma\tau X_{\sigma\tau}(w)
                 +O(u)\|X_{\sigma\tau}(w)\|.
\]
On an equal-sign block its even part is therefore bounded by
$Cu\|X_{\sigma\sigma}(w)\|$.  Its odd part is $O(u)$ by
\eqref{eq:angular-odd-expansion}.  Absorbing gives
$\|X_{\sigma\sigma}\|\le Cu$ and then an $O(u^2)$ bound for its
whole even part, including the constant coefficient.  This proves the
first estimate in \eqref{eq:angular-J-bands}.

For opposite signs use an additional RTT comparison at $r=\sqrt u$.
The normalized intertwiner for arguments $r,w$ is
$R_\infty+O(r^\eta)$ with bounded inverse, uniformly for $w$ on the
closed right unit semicircle; $R_\infty$ is diagonal and invertible.
It commutes with the joint mark $I\otimes Z$.  Conjugating this mark
and expanding the $O(r^\eta)$ error in tensor matrix units, each
producing a bounded marked-row product, proves
\[
 \|[P(r),J(w)]\|\le Cr^\eta.
\]
Strong approximation includes the endpoints.  On $Q_u^\sigma$,
$P(r)=\sigma I+O(\sqrt u)$ by
\eqref{eq:angular-band-sign}.  Compression of the commutator yields
\[
 (2-C\sqrt u)\|X_{\sigma,-\sigma}(w)\|
 \le Cu^{\eta/2},
\]
which proves the second estimate in
\eqref{eq:angular-J-bands}.  Inserting
$P(w)=\sigma I+O(u)$ in \eqref{eq:angular-N-definition} now proves
\eqref{eq:angular-N-leading}--\eqref{eq:angular-N-bands}: equal signs
add, whereas opposite signs cancel.  In the latter case the odd part
of $X$ is itself $O(u^{1+\theta})$ by the gluing identity.  Thus the
leading expression can equivalently be written
\[
 \tfrac12\{\sgn(P(1))Q_u, wJ_{u,1}+w^{-1}J_{u,-1}\}
       +O(u^{1+\theta})
\]
on the whole band.  Higher modes need not vanish at a fixed cutoff;
the displayed error is what removes them after rescaling.

For unequal bands embed both in their larger band.  To obtain the
damped estimates, decompose each side into $H_0\le m^{-1}$, dyadic
intervals above $m^{-1}$ up to $u_0$, and the remaining spectrum.
On a band with lower endpoint $t$, $m$ physical rows contribute
$e^{-mt}$.  For any $\beta>0$ the double sum
\[
 \sum_{t,t'} C\max(t,t',m^{-1})^\beta e^{-cm(t+t')}
 \le C_\beta m^{-\beta}
\]
is finite with the asserted bound; unrestricted bounded rows handle
the exponentially damped remainder bands.  The same proof works when
the damping rows range in a fixed compact strict-angle interval, since
Schwarz--Pick gives a uniform comparison of their modulus loss with
$H_0$.

On a sufficiently small band, \eqref{eq:angular-band-sign} at $w=i$
gives $|\arg f(i)^2|\le C H_0(f)$, by applying the estimate with
cutoffs decreasing to the fiber value.  On its complement the principal
argument is at most $\pi$, which is also bounded by a fixed multiple
of $H_0$.  Spectral calculus proves
\eqref{eq:angular-phase-bound}.
\end{proof}

\subsection{The endpoint Hamiltonian and the sum rule}

The contraction $P(1)$ measures time damping.  We also need the generator
obtained by differentiating the unitary endpoint, because it is local
on spin states.  These are different operators: we keep the notation
$H_0$ for the former and $E$ for the latter.

\begin{lemma}[The local endpoint generator]\label{lem:endpoint-Hamiltonian}
The nonnegative self-adjoint spectral operator
\begin{equation}\label{eq:angular-endpoint-energy}
 E=S^{-1}\partial_zP(z)\big|_{z=v},\qquad
 E(f)=-\left.\partial_s\arg f(ie^s)\right|_{s=0},
\end{equation}
agrees on local-spin vectors with the vacuum-subtracted nearest-neighbor
Hamiltonian whose link is
\begin{equation}\label{eq:angular-local-link}
 h_{j,j+1}=-\eta\,\operatorname{swap}_{j,j+1}R'_{j,j+1}(0)
 =\begin{pmatrix}
 \eta\cot\lambda&0&0&0\\
 0&0&-\eta/\sin\lambda&0\\
 0&-\eta/\sin\lambda&0&0\\
 0&0&0&\eta\cot\lambda
 \end{pmatrix}_{j,j+1}.
\end{equation}
In particular $[h_{j,j+1},Z_j+Z_{j+1}]=0$ and $E\Omega=0$.
The closure of the local-spin subspace reduces $E$, and its real-time
evolution is the local Hamiltonian evolution.  For all sufficiently
small $u$,
\begin{equation}\label{eq:angular-E-H-comparison}
 \bbone_{\{E\le u\}}\le\bbone_{\{H_0\le Cu\}}.
\end{equation}
With $Z_1=S^{-1}Z_0S$, the exact adjacent-pair identity is
\begin{equation}\label{eq:angular-adjacent-current}
 N(i)=\tfrac12 S^2(Z_0+Z_1).
\end{equation}
\end{lemma}

\begin{proof}
Differentiate the inner factors along the imaginary boundary.  The
singular factors in \eqref{eq:general-inner-factorization} contribute
$a+b$, and a zero $\zeta$ contributes
\begin{equation}\label{eq:angular-zero-energy}
 \frac{2\Re\zeta}{|i-\zeta|^2}.
\end{equation}
All contributions are nonnegative.  Scalar continuation makes the
derivative finite almost everywhere and justifies the sum, or one may
first differentiate finite partial products and use positivity.
Since $\partial_z=iw\partial_w$, this is exactly the spectral
derivative in \eqref{eq:angular-endpoint-energy}.  The cone condition
bounds \eqref{eq:angular-zero-energy} below by a constant times
$\min(|\zeta|,|\zeta|^{-1})$.  Thus small $E$ implies the small/large
zero budget in \eqref{eq:angular-zero-budget}.  On that budget the
negative logarithm of each factor at $1$ is bounded above by the same
quantity.  This proves \eqref{eq:angular-E-H-comparison}.

At finite length, differentiate one factor in the swap row and divide
by its shift.  The inserted link is
$-\eta\operatorname{swap}R'(0)$, proving
\eqref{eq:angular-local-link}.  Dividing by the vacuum eigenvalue
subtracts its vacuum energy.  For a fixed local matrix $O$, the finite
action on $O\Omega_L$ is therefore
\[
 \left[\sum_jh_{j,j+1},O\right]\Omega_L,
\]
where only links meeting the support of $O$ occur.  Express
$O\Omega_L$ by endpoint rows, translating the support by a fixed shift
if necessary.  The finite division estimate and the RTT continuation
used in Lemma~\ref{lem:scalar-continuation} give uniform vector Taylor
bounds on a common endpoint disc for this fixed word.  Cauchy's formula
therefore passes its derivative, in norm and in all Gram pairings, to
the spectral derivative.  Consequently
\begin{equation}\label{eq:angular-local-commutator}
 E(O\Omega)=\left[\sum_jh_{j,j+1},O\right]\Omega.
\end{equation}
Iterating is legitimate because the commutator remains local.
If $O$ is supported on an interval of length $\ell$, each new
commutator enlarges the support by at most two.  Its norm is bounded
by $2\|h\|$ times the number of interacting links.  Hence
\[
 \|E^kO\Omega\|\le C_O C_O^k k!.
\]
Local vectors are analytic for the symmetric operator specified by
\eqref{eq:angular-local-commutator}.  Analytic-vector uniqueness makes
its closure self-adjoint on their closed span.  Since $E$ extends this
operator, the span reduces $E$ and the two self-adjoint operators agree
there.  This also identifies real-time evolution.  Finally
$J(i)=SZ_0$ and $P(i)=S$ in
\eqref{eq:angular-N-definition} give
\eqref{eq:angular-adjacent-current}.
\end{proof}

\begin{lemma}[Energy and winding sum rule]\label{lem:sum-rule}
Let $p$ denote the phase of the unit shift $S=P(i)$, modulo $2\pi$, and
let $\operatorname{sym}$ average a measure at $p$ and $-p$.  Set
\[
 D_0=-\langle\Omega,[Z_0,[Z_0,h_{0,1}]]\Omega\rangle.
\]
Away from $p=0$, the projection of the current measure obeys
\begin{equation}\label{eq:angular-sum-rule}
 2\operatorname{sym}\bigl((p)_*(E\mu)\bigr)
       =D_0\frac{\dd p}{2\pi}.
\end{equation}
On each orbit of degree $2n_F$, the phase winds monotonically through
$n_F$ complete turns.  In particular
\begin{equation}\label{eq:angular-degree-integral}
 \int n_F\dd\nu(F)<\infty.
\end{equation}
\end{lemma}

\begin{proof}
Since $Z_0\Omega=S^{-1}J(i)\Omega$, the spectral measure of the local
density vector is
\begin{equation}\label{eq:angular-density-measure}
 \|Z_0\Omega\|^2(\dd f)=|1-f(i)|^2\dd\mu(f).
\end{equation}
Fourier transform the vacuum double commutators of the translates
$Z_j$.  By $E\Omega=0$, their transform is twice the symmetrized
energy-weighted spectral measure of $Z_0\Omega$.  On the other hand
\eqref{eq:angular-local-commutator} and conservation of
$Z_0+Z_1$ leave only the three coefficients at separations $0,1,-1$,
namely $2D_0,-D_0,-D_0$.  Their transform is
$|1-e^{ip}|^2D_0$.  Thus
\[
 2\operatorname{sym}\bigl((p)_*(E|1-f(i)|^2\mu)\bigr)
 =|1-e^{ip}|^2D_0\frac{\dd p}{2\pi}.
\]
Testing on compact subsets of the circle without zero permits division
by the squared factor and proves \eqref{eq:angular-sum-rule}.

On a nonconstant finite-product orbit, direct differentiation gives
$-\dd p/\dd s=E(F(e^s\cdot))>0$.  The boundary phase change on the
positive imaginary ray is one half of the full boundary change: this
follows either by pairing conjugate zeros in
\eqref{eq:finite-inner-symmetry} or by integrating
\eqref{eq:angular-zero-energy} with its dilated argument.  It is
$2\pi n_F$.  At each nonzero phase there are consequently $n_F$
crossings, counted with multiplicity, and none has zero derivative.
The change of variables from $s$ to phase gives
\[
 (p)_*(E\,\dd s\,\nu)=\left(\int n_F\dd\nu(F)\right)\dd p
\]
away from zero, first with truncated orbit measures and then by monotone
convergence.  Equation~\eqref{eq:angular-sum-rule} implies finiteness;
in these conventions it also gives
$4\pi\int n_F\dd\nu=D_0$.  The endpoints of an orbit and its finitely
many zero-phase crossings have zero $\dd s$ measure, so no missing
phase atom is involved in this computation.
\end{proof}

\subsection{The height variogram}

The measure constructed above already determines height covariances.
The following bound will also supply the logarithmic majorants needed
for nonsmooth tests and regularity.

\begin{proposition}[Exact variogram and logarithmic growth]
\label{prop:variogram}
After an axial symmetry, any nonzero lattice displacement may be
written $(k,j)$ with $k\ge1$ and $|j|\le k$.  Its height variance is
\begin{equation}\label{eq:angular-variogram}
 \begin{aligned}
 \E_c\bigl(h(k,j)-h(0,0)\bigr)^2
 &=2\Re\int\bigl(1-f(1)^k f(\pm i)^{|j|}\bigr)\dd\mu(f)\\
 &=4\kappa\log k+O(1),
 \end{aligned}
\end{equation}
where the sign of $i$ is chosen according to the vertical direction.
The error is uniform over these displacements.  In particular the
variance is bounded by $C(1+\log(1+|k|+|j|))$.
\end{proposition}

\begin{proof}
First consider a directed path of physical or endpoint steps, and write
their spectral multipliers as $t_1,\ldots,t_m$.  Reflection changes
the sign of a current.  Equations~\eqref{eq:current-cocycle} and
\eqref{eq:angular-endpoints} therefore express the covariance of two
distinct increments, in their path order, as the negative real part of
the integral of
\[
 (1-t_a)t_{a+1}\cdots t_{b-1}(1-t_b),\qquad a<b,
\]
with the conjugate term supplying the real covariance.  A single
increment has variance one.  At an endpoint this equals
$2\Re\int(1-f(i))\dd\mu$, because $|f(i)|=1$ and
\eqref{eq:angular-density-measure} integrates to
$\|Z_0\Omega\|^2=1$.  The same identity holds at the physical step
$w=1$: on each orbit move the ray from $i\R_+$ to $\R_+$ by Cauchy's
theorem for $(1-F(w))\dd w/w$.  Its small and large circular arcs
vanish, since $F(0)=F(\infty)=1$.  The integrated identity is
justified by the bounds below.

The elementary telescoping identity
\[
 1-\prod_{a=1}^m t_a
 =\sum_{a=1}^m(1-t_a)
  -\sum_{a<b}(1-t_a)t_{a+1}\cdots t_{b-1}(1-t_b)
\]
therefore gives the first equality of
\eqref{eq:angular-variogram}, taking $k$ horizontal and $|j|$
vertical steps.  It is equally valid by truncating the orbit measure
first and then passing to the absolutely convergent integrals below.

Here are bounds that both justify this passage and give the asymptotic.
On an orbit choose one zero radius as origin in logarithmic coordinates
and let $\mathcal L$ be the range of the log radii.  Beyond either
extreme by distance $d$, the factor formulas and the endpoint
normalizations give
\begin{equation}\label{eq:angular-orbit-tail}
 |1-F(e^s)|+|1-F(\pm i e^s)|\le Cn_F e^{-d},
\end{equation}
or the trivial constant bound when it is smaller.  The factors on the
imaginary ray have modulus one; those on the positive ray have modulus
at most one.  Hence the integrand in
\eqref{eq:angular-variogram} is absolutely integrable on each orbit.
Outside the zero range enlarged by $\log k+\log(1+n_F)$, the inequality
$|1-a^kb^{|j|}|\le k|1-a|+|j||1-b|$, with $|j|\le k$, and
\eqref{eq:angular-orbit-tail} give a bounded integral.  The remaining
extra interval lengths cost at most
$C(\mathcal L+\log(1+n_F)+1)$.

For the main interval, use any single zero.  Its cone bound gives
\[
 |F(e^s)|^k\le
 \exp\{-c k e^{-|s-s_0|}\}
 \quad\text{for } |s-s_0|\le\log k,
\]
where $s_0$ is its logarithmic radius (with the sign dictated by the
chosen orbit coordinate).  The integral of the right side over this
interval is bounded uniformly in $k$.  Multiplication by
$F(\pm i e^s)^{|j|}$ does not change that bound.  Consequently the
integral of the real part of $1-F(e^s)^kF(\pm i e^s)^{|j|}$ is
\[
 2\log k+O\bigl(1+\mathcal L+\log(1+n_F)\bigr).
\]
The remainder is integrable against $\nu$ by
\eqref{eq:orbit-width-tail}, \eqref{eq:angular-degree-integral}, and
$\kappa<\infty$.  Multiplication by the factor $2$ in the variogram
proves \eqref{eq:angular-variogram}.  The same majorants justify the
orbitwise contour move and the telescoping passage above.  Axial
symmetry, already proved for the physical law, supplies the final
uniform displacement bound.
\end{proof}

We have thus obtained a physical representation with a finite measure
$\nu$ of dilation orbits, an as yet unevaluated constant $\kappa$, and
uniform low-energy current estimates.  Section~\ref{sec:normalization}
determines $\kappa$.  Section~\ref{sec:chiral} uses the same estimates,
including their higher-mode errors, to identify all limiting current
correlations.
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//04-angular-plane.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//04-spectral.tex ====
\section{Electric potentials and angular spectra}\label{sec:spectral}

The radial estimates control changes of boundary data.  To obtain a
convergent normalization, we also need an equation for the spatial
dependence of an electric insertion.  The equation proved below is
approximate harmonicity with a summable power error.  Its proof uses
two slicings of the square array.  Slicing parallel to the insertion
line gives an improved estimate from the vacuum spectrum; slicing
perpendicularly gives derivative bounds which carry that estimate up
to the line.

Throughout this section the lattice spacing is one in tile
coordinates, and $0<\lambda\le\pi/3$ is fixed.  We use the plane row
representation of Input~\ref{inp:rows}.  Some spectral arguments below
adapt those of \cite[Section~4, ``Finite inner functions and a measure
on dilation orbits'' and ``The height variogram'']{GFFCompanion}.
That section is stated for $\lambda>\pi/3$.  We therefore give the
arguments in the present representation, identifying the row
identities they require.  No assertion about an angular representation
outside its stated parameter range is used as an input.

\subsection{Rows, currents, and their spectral measure}

We first specify the row coordinates and the exchange identity.  Put
$v_0=\pi/2$.  A strip angle $z$, with $|\Re z|<v_0$, corresponds to
the half-plane argument $w=e^{iz}$; conversely
$z=-i\operatorname{Log}w$, using the logarithm on $\Re w>0$.
The argument of the local matrix in a homogeneous row is
\[
 u(z)=\rho(v_0-z),\qquad
 R(u)=\begin{pmatrix}
 \sin(\lambda-u)/\sin\lambda&0&0&0\\
 0&\sin u/\sin\lambda&1&0\\
 0&1&\sin u/\sin\lambda&0\\
 0&0&0&\sin(\lambda-u)/\sin\lambda
 \end{pmatrix}.
\]
On an even spatial circle of size $N$, with quantum spin space
$(\C^2)^{\otimes N}$, define the monodromy and marked row by
\[
 \mathcal M_{N,c}(z)=R_{c0}(u(z))\cdots R_{c,N-1}(u(z)),\qquad
 T_N^D(z)=\operatorname{tr}_c\bigl(D_c\mathcal M_{N,c}(z)\bigr).
\]
The index $c$ denotes a two-dimensional auxiliary space.  Let
$\Omega_N$ be the positive unit Perron vector of the untwisted
rows on the zero-total-spin sector, and let $\Lambda_N(z)$ be
its eigenvalue for real strip angles, analytically continued
using that common eigenvector.  The marked rows themselves act
on the full quantum spin space; in particular a local matrix
$O$ need not preserve the zero-total-spin sector.
For real strip angles, divide by $\Lambda_N(z)$ and take plane
word limits.  Denote the resulting row and its bounded holomorphic
continuation by $\widehat A^D(z)$.  We use
$A^D(w)=\widehat A^D(-i\operatorname{Log}w)$ below.  A finite row
word means a vector
\[
 \widehat A^{D_r}(z_r)\cdots\widehat A^{D_1}(z_1)\Omega,
 \qquad r<\infty,
\]
initially with real $z_j\in(-v_0,v_0)$; the empty word is the
vacuum $\Omega$.

The local Yang--Baxter identity, and its row consequence, are
\begin{align}
 R_{12}(u-u')R_{13}(u)R_{23}(u')
 &=R_{23}(u')R_{13}(u)R_{12}(u-u'),\notag\\
 \mathcal R_{12}(z,z')\mathcal M_{N,1}(z)\mathcal M_{N,2}(z')
 &=\mathcal M_{N,2}(z')\mathcal M_{N,1}(z)
       \mathcal R_{12}(z,z'),\label{eq:row-exchange}\\
 \mathcal R_{12}(z,z')&=R_{12}\bigl(\rho(z'-z)\bigr).\notag
\end{align}
These are the finite identities in the Row Algebra and the
Infinite-Plane State subsection of \cite{FKCompanion}.
In particular, if $\mathcal R$ is invertible and
\[
 \mathcal R(D\otimes E)\mathcal R^{-1}
 =\sum_\ell D_\ell\otimes E_\ell,
\]
then auxiliary tracing, followed by the plane limit, gives the
marked exchange rule
\begin{equation}\label{eq:marked-row-exchange}
 \widehat A^D(z)\widehat A^E(z')
 =\sum_\ell\widehat A^{E_\ell}(z')\widehat A^{D_\ell}(z).
\end{equation}
There are at most sixteen terms after expansion in tensor matrix
units.  The normalization cancels on the two sides.  Analyticity
extends this identity wherever its intertwiner is invertible.

Set
\[
 Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\qquad
 X=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
 K(\zeta)=\operatorname{diag}(\zeta,\zeta^{-1}),
 \qquad |\zeta|=1,
\]
and
\[
 P_\zeta(w)=A^{K(\zeta)}(w),\qquad
 P(w)=A^I(w),\qquad J(w)=A^Z(w).
\]
The Hilbert space completion of the row words is denoted by
$\mathcal H$, and $\Omega$ has norm one.  The rows are bounded and
holomorphic for $\Re w>0$, with $\|A^D(w)\|\le2\|D\|$.
Their reflection rule is
\begin{equation}\label{eq:row-adjoint}
 \widehat A^D(z)^*
 =\widehat A^{X\overline D X}(-\overline z),\qquad
 A^D(w)^*=A^{X\overline D X}(\overline w).
\end{equation}
This is Equation \texttt{bulk:adjoint} of \cite{FKCompanion},
continued holomorphically from real strip angles.  In particular,
$J(w)^*=-J(\overline w)$.
The $P_\zeta$ rows with the same twist commute and are normal;
$P_\zeta(1)$ is self-adjoint.  Every $P_\zeta(w)$ is a
contraction in the closed right half-plane away from zero.  At its two boundary
arguments the untwisted rows are the spatial shift and its inverse:
\begin{equation}\label{eq:spectral-endpoints}
 P(i)=S,\qquad P(-i)=S^{-1},\qquad
 A^D(i)=SD_0,\qquad A^D(-i)=(XD^tX)_0S^{-1}.
\end{equation}
Here $D_0$ acts at one site and $S\Omega=\Omega$.  Dilation of the
argument is implemented by a strongly continuous unitary group fixing
$\Omega$.  Also $P(w)\Omega=\Omega$.

These facts follow from the Plane Kernels, Angle Generator, and
Scalar Continuation in a Fixed Band lemmas in
\cite[Section ``Angular transfer and its two leading modes'']{FKCompanion}.
We record the analytic estimates in the latter proof which will
also justify differentiation at an endpoint.  For one radius
$r_\lambda>0$, independent of all sufficiently large even $N$,
the two quotients
\begin{equation}\label{eq:finite-endpoint-division}
 T_N^D(z)T_N^I(z)^{-1},\qquad T_N^I(z)^{-1}T_N^D(z)
\end{equation}
are holomorphic and bounded by $C_\lambda\|D\|$ on the discs
$|z-v_0|<r_\lambda$ and $|z+v_0|<r_\lambda$.
This is Equation \texttt{bulk:division} and the crossing argument
in that proof.  On these endpoint discs $T_N^I(z)$ is invertible,
so $\Lambda_N(z)$ is nonzero there.  We use nonvanishing of its
complex continuation only on these discs.  In particular, the marked vacuum vectors have
uniform holomorphic bounds on these discs.

For clarity, the dense set used to extend these vectors consists
of finite words whose real angles lie in $(-v_0/3,v_0/3)$.
It is dense because orthogonality to such words extends, one
argument at a time, by holomorphy.  Exchanging an endpoint row
through any one such word uses
Equation~\eqref{eq:marked-row-exchange}.  Its finitely many
intertwiners stay invertible on a common disc about that endpoint:
the word angles are separated from the endpoint by the middle
third of the strip.  The continuation radius is independent of
the word, although its vector bound may depend on the word.
This is the dense-set continuation step in the proof of the same
Scalar Continuation lemma.  The denominators of those intertwiners
do not depend on $D$.  Thus the scalar continuation discs, and
the constants in the low-band arguments below, are uniform for
the unit diagonal marks $K(\zeta)$.

We will need the following concrete consequence for a fixed local
spin matrix $O$.  For the finite vacuum $\Omega_N$ just specified,
there are
some $r_O>0$ and $C_O<\infty$,
\begin{equation}\label{eq:local-endpoint-bound}
 \sup_N\sup_{|z-v_0|<r_O}
 \left\|\frac{T_N^I(z)}{\Lambda_N(z)}O\Omega_N\right\|
 \le C_O.
\end{equation}
Only sufficiently large circles containing the support of $O$
are included.  To obtain this bound, translate the fixed support
into an interval, expand $O$ in tensor matrix units, and express
each resulting vector by a finite product of positive endpoint
rows, using Equation~\eqref{eq:spectral-endpoints}.  Translation
commutes with the unmarked row.  Move the row at $z$ to the
vacuum through those endpoint rows.  The intertwiners are now
near $R(0)$, the invertible swap, so their coefficients are
bounded on a sufficiently small disc.  The remaining endpoint
rows are bounded shifts with local marks, and the marked vacuum
vector is bounded by Equation~\eqref{eq:finite-endpoint-division}.
This proves Equation~\eqref{eq:local-endpoint-bound} without a
bound depending on $N$.  Cauchy's formula consequently bounds
the $k$th vector Taylor coefficient by $C_O r_O^{-k}$ after
decreasing $r_O$.  Plane word convergence and these bounds pass
endpoint derivatives and their Gram pairings to the plane.

The untwisted commuting rows have spectral functions $f$ satisfying
\begin{equation}\label{eq:spectral-inner-input}
 |f(w)|\le1\quad(\Re w>0),\qquad
 f(\overline w)=\overline{f(w)},\qquad |f(it)|=1\quad(t\ne0).
\end{equation}
They continue through the imaginary axis away from zero.  Their zeros
$a_j$ lie in a fixed cone $\Re a_j\ge c_0|a_j|$; the same cone and
continuation statements hold for a fixed twist.  The resulting inner
factorization has the form
\begin{equation}\label{eq:general-spectral-factorization}
 f(w)=\gamma e^{-a w-b/w}
       \prod_j\frac{w-a_j}{w+\overline{a_j}}\,\gamma_j,
 \qquad a,b\ge0,\quad |\gamma|=|\gamma_j|=1,
\end{equation}
with the usual summability condition on the zeros.  This is precisely
the factorization established in the Inner Factorization and Low
Spectral Bands subsection of \cite{FKCompanion}.

Let $\mathcal D$ be the closed subspace generated by words whose marks
are diagonal.  It is invariant under $P,J$ and their adjoints, and
contains finite height-phase insertions, by
Equation~\eqref{eq:spectral-endpoints}.  All untwisted spectral
statements in the next proposition are on this subspace.

The proposition supplies two different estimates for the electric
potential.  Dilation decay first forces the spectral functions to
be finite products, and the current cocycle then supplies their
measure.  Its width tail makes modes containing both very small
and very large zeros rare; this will give the extra power in the
off-axis Laplacian estimate.  The degree moment has a separate
role: it makes the logarithmic height variogram integrable, giving
the growth bound needed for potential inversion in
Section~\ref{sec:electric}.

\begin{proposition}[The current spectrum]\label{prop:spectral}
Almost every spectral function on $\mathcal D$ is either $1$ or a
finite Blaschke product with
\begin{equation}\label{eq:finite-spectral-inner}
 f(0)=f(\infty)=1,
 \qquad f(\overline w)=\overline{f(w)}.
\end{equation}
There is a spectral vector $\mathsf h$, not necessarily of finite
norm, such that
\begin{equation}\label{eq:current-cocycle}
 J(w)\Omega=(1-f(w))\mathsf h.
\end{equation}
The measure $\mu$ given by the squared fiber norm of $\mathsf h$
is invariant under dilation.  On the cross-section of products whose
smallest zero has modulus one, it decomposes as
\begin{equation}\label{eq:current-disintegration}
 \mu=\dd s\,\nu(\dd F),\qquad f(w)=F(e^s w),
 \qquad \nu(1)<\infty.
\end{equation}
If $\mathcal W(F)$ is the difference between the largest and smallest
logarithmic zero moduli, and $2n_F$ is the degree, then, for some
$\eta_0>0$,
\begin{equation}\label{eq:current-width-tail}
 \nu\{\mathcal W>t\}\le C e^{-2\eta_0t},\qquad
 \int n_F\,\nu(\dd F)<\infty.
\end{equation}
The physical plane height has logarithmic variance.  More precisely,
after an axial symmetry, write a nonzero displacement as $(k,j)$ with
$k\ge1$ and $|j|\le k$.  Then
\begin{equation}\label{eq:spectral-variogram}
 \begin{split}
 \E\bigl(h(k,j)-h(0,0)\bigr)^2
 &=2\Re\int\bigl(1-f(1)^k f(\pm i)^{|j|}\bigr)\,\mu(\dd f)\\
 &=4\nu(1)\log k+O(1),
 \end{split}
\end{equation}
where the sign agrees with the direction of the second coordinate.
The error is uniform over these displacements.
\end{proposition}

\begin{proof}
We begin with the decay furnished by
Equation~\eqref{eq:marked-row-exchange}.  If $w=e^s$ and $w'$ is
fixed, then $z=-is$ and the exchange parameter in
Equation~\eqref{eq:row-exchange} is
$\rho(z'-z)=\rho z'+i\rho s$.  Put
$b(u)=\sin u/\sin\lambda$.  For $u=u_0+i\rho s$, the displayed
matrix gives
\begin{equation}\label{eq:normalized-intertwiner-limit}
 \frac{R(u)}{b(u)}
 =\operatorname{diag}(-e^{\pm i\lambda},1,1,-e^{\pm i\lambda})
       +O(e^{-\rho|s|})
 \qquad(s\longrightarrow\pm\infty).
\end{equation}
Indeed $\sin(\lambda-u)/\sin u=-e^{\pm i\lambda}
+O(e^{-2\rho|s|})$, while the off-diagonal entries become
$1/b(u)=O(e^{-\rho|s|})$.  The limit is diagonal and invertible,
and the normalized inverse is bounded for large $|s|$.  Scalar
normalization does not change auxiliary conjugation in
Equation~\eqref{eq:marked-row-exchange}.  This calculation holds
for every fixed $0<\lambda<\pi$.

Move a row $P(e^s)$ through a fixed diagonal word $v$ to the vacuum.
The limiting diagonal intertwiner commutes with the joint diagonal
mark.  Each exchange therefore differs from a commuting exchange by
$O(e^{-\rho|s|})$ in operator norm.  The number of exchanges is fixed,
all marked rows are bounded, and $P(e^s)\Omega=\Omega$.  Thus, with
some $\eta_0>0$ which we may decrease below $\rho$,
\begin{equation}\label{eq:diagonal-dilation-decay}
 \|(1-P(e^s))v\|\le C_v e^{-\eta_0|s|}
 \quad\text{outside a fixed bounded interval of }s.
\end{equation}
The endpoints are included by their strong limits.  Thus the decay
uses the explicit exchange rule and the target-range row bounds.

For a countable total set of diagonal words, the spectral integral of
$|1-f(e^s)|^2$ over $s$ is finite by
Equation~\eqref{eq:diagonal-dilation-decay}.  Therefore it is finite
for almost every spectral function on $\mathcal D$.  In
Equation~\eqref{eq:general-spectral-factorization}, a positive $a$
or $b$ would make $f(e^s)$ tend to zero at one end; hence $a=b=0$.
For a zero $a_j$ of modulus $r_j$, the identity
\[
 \left|\frac{t-a_j}{t+\overline{a_j}}\right|^2
 =1-\frac{4t\Re a_j}{|t+\overline{a_j}|^2}
\]
and the cone condition give fixed constants $c_1,c_2>0$ such that
$1-|f(e^s)|\ge c_1$ on $|s-\log r_j|\le c_2$.
Infinitely many zeros would escape towards zero or infinity and
yield infinitely many disjoint such intervals.  Thus there are only
finitely many zeros.  Integrability forces both endpoint limits to
be one.  Reality pairs nonreal zeros with their conjugates, and the
quotient of the endpoint limits is $(-1)^{\deg f}$, so the degree
is even.  A constant spectral function must be one.  This proves
Equation~\eqref{eq:finite-spectral-inner}.

Differentiate commutation of two equally twisted rows with respect
to their common twist at $\zeta=1$ and apply it to $\Omega$.
Linearity in the auxiliary mark gives
\begin{equation}\label{eq:twist-derivative-cocycle}
 (1-P(w'))J(w)\Omega=(1-P(w))J(w')\Omega.
\end{equation}
There is no current on the spectral set $f=1$.  To see this, let $Q$
be its spectral projection.  This set is invariant under dilation,
so $Q$ commutes with the dilation group; also $QS=SQ=Q$.
The endpoint formulas give
$J(-i)=-S^{-1}J(i)S^{-1}$.  Compressing and then dilating yields
\[
 QJ(-ir)Q=-QJ(ir)Q\qquad(r>0).
\]
These are the matching boundary values for the continuation of
$QJ(w)Q$ to the left half-plane by odd reflection.
The continued function is bounded on the punctured plane, and thus
extends across zero and infinity.  Liouville's theorem and oddness
make it zero.  Since $\Omega$ belongs to this spectral set, this
also shows $QJ(w)\Omega=0$.  On every other fiber $|f(1)|<1$, so
define $\mathsf h=J(1)\Omega/(1-f(1))$ there.
Equation~\eqref{eq:twist-derivative-cocycle} proves
Equation~\eqref{eq:current-cocycle}, first on a countable dense set
of arguments and then by analyticity and endpoint continuity.

Dilation covariance fixes the vacuum and transforms this quotient
without a scalar factor.  Its squared fiber norm therefore defines
a dilation-invariant measure $\mu$.  Normalize a nonconstant product
by making its smallest zero modulus one.  Translation invariance in
the remaining logarithmic coordinate gives
Equation~\eqref{eq:current-disintegration}, initially with a
sigma-finite transversal measure.  Every normalized product has a
fixed-depth, fixed-width dip at that first zero, whereas
\[
 \int |1-f(1)|^2\,\mu(\dd f)=\|J(1)\Omega\|^2<\infty.
\]
Integration over the dip proves $\nu(1)<\infty$.

Apply Equation~\eqref{eq:diagonal-dilation-decay} to
$v=J(1)\Omega$.  In the current measure this says
\[
 I(l):=\int |1-f(1)|^2|1-f(e^l)|^2\,\mu(\dd f)
 \le C e^{-2\eta_0l}\qquad(l\ge1).
\]
For an orbit of width $\mathcal W$, first put the argument $1$ in
the dip at its smallest zero, and then put $e^l$ in the dip at its
largest zero.  Both choices occupy intervals of fixed positive
length and both factors are bounded below.  Consequently
\[
 c\,\nu\{\mathcal W>t\}
 \le\int_{t-C}^{\infty}I(l)\,\dd l,
\]
with an adjustment for bounded $t$.  This proves the width tail.

We next justify the degree integrability needed for the height
variance.  This step records the local energy argument rather than
presuming an integrability property of the transversal measure.
For this local computation use the full untwisted representation
on $\mathcal H$, and let $E$ be its nonnegative spectral endpoint
generator
\[
 E(f)=-\left.\frac{\dd}{\dd s}\arg f(ie^s)\right|_{s=0}.
\]
In the general factorization its two exponential factors contribute
$a+b$, and each zero contributes $2\Re a_j/|i-a_j|^2\ge0$.
On $\mathcal D$ the exponential factors have already been excluded.
At finite
circumference, differentiating the endpoint swap row gives the local
link
\[
 h_{j,j+1}=-\rho\,\operatorname{swap}_{j,j+1}R'_{j,j+1}(0)
 =\begin{pmatrix}
 \rho\cot\lambda&0&0&0\\
 0&0&-\rho/\sin\lambda&0\\
 0&-\rho/\sin\lambda&0&0\\
 0&0&0&\rho\cot\lambda
 \end{pmatrix}_{j,j+1}.
\]
For a local spin matrix $O$, the vacuum normalization subtracts the
vacuum energy, and therefore
\begin{equation}\label{eq:local-energy-action}
 E(O\Omega)=\left[\sum_j h_{j,j+1},O\right]\Omega.
\end{equation}
Only commutator terms whose links meet the support of $O$ survive.
Equation~\eqref{eq:local-endpoint-bound} gives a disc and uniform
vector Taylor bounds for this fixed $O$.  Cauchy's formula
therefore passes the first derivative, and its Gram pairings with
fixed local vectors, to the plane, proving
Equation~\eqref{eq:local-energy-action}.  Iterated commutators stay local
and obey $\|E^kO\Omega\|\le C_O^{k+1}k!$, since each commutator
enlarges the support by at most two sites.  These analytic vectors
identify the closure of the local commutator operator with the
spectral generator on the local-spin subspace.  Thus
Equation~\eqref{eq:local-energy-action} also justifies its quadratic
forms and the following Fourier computation.

Write $p=\arg f(i)$ modulo $2\pi$, and let $\operatorname{sym}$
average a measure at $p$ and $-p$.  The spectral measure of
$Z_0\Omega=S^{-1}J(i)\Omega$ is $|1-f(i)|^2\mu$.
Set
\[
 D_0=-\langle\Omega,[Z_0,[Z_0,h_{0,1}]]\Omega\rangle.
\]
Since $[h_{0,1},Z_0+Z_1]=0$, the Fourier transform of the vacuum
double commutators has only three coefficients, at $0,1,-1$,
equal to $2D_0,-D_0,-D_0$.  By
Equation~\eqref{eq:local-energy-action}, the same transform is
twice the symmetrized energy-weighted spectral measure of
$Z_0\Omega$.  Dividing by $|1-e^{ip}|^2$ on compact subsets of
$p\ne0$ gives
\begin{equation}\label{eq:current-energy-sum-rule}
 2\operatorname{sym}\bigl(p_*(E\mu)\bigr)
 =D_0\frac{\dd p}{2\pi}\qquad(p\ne0).
\end{equation}
On a degree-$2n_F$ orbit the boundary phase is strictly decreasing
and makes $n_F$ full turns along the positive imaginary ray.  The
change of variables $-\dd p=E\,\dd s$ therefore gives
\[
 p_*(E\,\dd s\,\nu)=\left(\int n_F\,\nu(\dd F)\right)\dd p
 \qquad(p\ne0).
\]
One may first restrict to measurable sets of orbits with bounded
degree and width and to a bounded range of $s$, and then exhaust
these sets by monotone convergence.  Comparing with
Equation~\eqref{eq:current-energy-sum-rule} proves
$4\pi\int n_F\,\nu(\dd F)=D_0<\infty$.

It remains to recover the height variance.  A directed path of
physical or endpoint steps has arguments $w_j\in\{1,i,-i\}$ and
multipliers $t_j=f(w_j)$.  By
Equations~\eqref{eq:row-adjoint} and \eqref{eq:current-cocycle},
the left vacuum current is $-(1-f(w))\mathsf h^*$ and the right
vacuum current is $(1-f(w))\mathsf h$.  Thus, for $a<b$,
\begin{equation}\label{eq:two-current-coefficient}
 \left\langle\Omega,
 J(w_a)\left(\prod_{a<j<b}P(w_j)\right)J(w_b)\Omega\right\rangle
 =-\int(1-t_a)t_{a+1}\cdots t_{b-1}(1-t_b)\,\dd\mu.
\end{equation}
For any directed tile step, its transverse spin is $+1$ when the
arrow points along the left normal to that step.  The height on
the left of an arrow exceeds the height on its right by one, so
the increment along the directed step is minus its transverse
spin.  This convention applies to the physical step and to both
endpoint directions; the two minus signs cancel in the product.
Taking the real part
of Equation~\eqref{eq:two-current-coefficient} gives their physical
covariance in path order.  At an endpoint the
single-increment variance one equals
$2\Re\int(1-f(i))\,\dd\mu$, by $|f(i)|=1$ and
$\|Z_0\Omega\|=1$.  The corresponding physical-step identity
follows by moving the integration ray for
$(1-F(w))\dd w/w$ from $i\R_+$ to $\R_+$ on each orbit.
The small and large circular arcs vanish because
$F(0)=F(\infty)=1$; the integrable bounds below justify the
integration over orbits.  Now use the telescoping identity
\[
 1-\prod_{a=1}^m t_a
 =\sum_a(1-t_a)
  -\sum_{a<b}(1-t_a)t_{a+1}\cdots t_{b-1}(1-t_b).
\]
This proves the first equality in
Equation~\eqref{eq:spectral-variogram}, initially with truncated
orbit integrals.

Outside the zero range by logarithmic distance $d$, the factor
formula gives
\[
 |1-F(e^s)|+|1-F(\pm i e^s)|\le C n_F e^{-d},
\]
or the trivial uniform bound if it is smaller.  Thus the integral
outside the zero range enlarged at each end by
$\log k+\log(1+n_F)$ is bounded uniformly.  The additional
interval lengths cost at most
$C(1+\mathcal W+\log(1+n_F))$.  On an interval of radius
$\log k$ about any one logarithmic zero modulus $s_0$, the cone
condition gives
\[
 |F(e^s)|^k\le\exp\{-c k e^{-|s-s_0|}\}.
\]
Its integral on that interval is bounded uniformly in $k$.
Multiplication by $F(\pm i e^s)^{|j|}$ does not change the bound.
Consequently the real part of the orbit integral in
Equation~\eqref{eq:spectral-variogram} is
\[
 2\log k+O(1+\mathcal W+\log(1+n_F)).
\]
The error is integrable by Equation~\eqref{eq:current-width-tail}
and degree integrability.  These same majorants justify the ray
move and telescoping passage.  Axial symmetry of the plane law,
which follows from the physical plane identification in
Input~\ref{inp:rows}, supplies the asserted bound for every
displacement.
\end{proof}

\subsection{Electric insertions and the estimate away from their line}

Fix $p$ distinct tile vertices $u_1,\ldots,u_p$ on one tile axis,
in a disk of radius $Mn$, where $M$ and $p$ are fixed and $n\ge2$.
For real charges $t=(t_1,\ldots,t_p)$ with $\sum_jt_j=0$, define
\begin{equation}\label{eq:electric-characteristic}
 U_t=\exp\left(i\sum_jt_jh(u_j)\right),\qquad
 C(t;u)=\E U_t.
\end{equation}
Neutrality makes the expression independent of the additive height
constant.  Its loop expansion gives relative weight
\begin{equation}\label{eq:electric-loop-weight}
 v_I(t)=\frac{\cos(\lambda+\sum_{j\in I}t_j)}{\cos\lambda}
\end{equation}
to a loop enclosing exactly the point set $I$.  Indeed a diagonal
height string contributes $e^{it}$ when a counterclockwise loop
encloses charge $t$, while its two winding weights before the
insertion are $e^{i\lambda}$ and $e^{-i\lambda}$.  Summing their
orientations gives Equation~\eqref{eq:electric-loop-weight}.
The same identity follows for endpoint strings from
Equation~\eqref{eq:spectral-endpoints}; ice conservation permits
rerouting the strings.  These finite-word identities use the
plane-kernel identification in Input~\ref{inp:rows}.  Spin reversal
also gives $C(t;u)=C(-t;u)$.

Choose $b>0$ sufficiently small in terms of $p$ and $\lambda$, and
assume $\max_j|t_j|\le b$.  All factors in either winding
expansion are then positive.  The number of charged loops has mean
$O(1+\log n)$ by the buffered passage estimates: assign loops to
geometric scales about the marked points until scale $Mn$, and use
a bounded number of buffers at each scale.  Since
$\log v_I(t)\ge-Cb$, Jensen's inequality gives
\begin{equation}\label{eq:small-charge-lower-bound}
 C(t;u)\ge c n^{-Cb}.
\end{equation}
The constants do not require a lower bound on the separation of
distinct marked vertices.

Fix a tile vertex $z_0$ and define the complex-valued potential
\begin{equation}\label{eq:electric-potential}
 H(z)=\frac{\E[(h(z)-h(z_0))U_t]}{C(t;u)}.
\end{equation}
Changing $z_0$ adds a constant.  Its unit increments are bounded
uniformly in $n$: condition on the unoriented loops in the positive
tilt of Equation~\eqref{eq:electric-loop-weight}.  A marked spin
lies on one loop, and its orientation average is a ratio of two
orientation sums with denominator bounded away from zero for these
charges.  The numerator is uniformly bounded.  Proposition~\ref{prop:spectral}
also gives, for each fixed insertion experiment,
\begin{equation}\label{eq:potential-growth}
 |H(z)-H(z')|\le C_{t,u}\sqrt{1+\log(2+|z-z'|)}.
\end{equation}
This is Cauchy--Schwarz in the positive physical six-vertex law,
using $|U_t|=1$ and the logarithmic variance.

For a function on tile vertices, write
\[
 \Delta_2H(z)=H(z+2e_1)+H(z-2e_1)
             +H(z+2e_2)+H(z-2e_2)-4H(z).
\]
The factor two removes the signs of the slowly damped row modes.

\begin{lemma}[Off-axis Laplacian estimate]\label{lem:off-axis}
There are constants $c_*>0$ and $C<\infty$, depending on the fixed
parameter and on $p,M$, such that, if $z$ has distance $r\ge10$
from the axis containing the insertions, then
\begin{equation}\label{eq:off-axis-laplacian}
 |\Delta_2H(z)|\le C n^{Cb}r^{-2-c_*}.
\end{equation}
For each fixed experiment, unit increments in either coordinate
direction tend to zero when their base points tend to infinity
along either normal to the insertion axis.
\end{lemma}

\begin{proof}
Take rows parallel to the insertion axis, with time increasing
towards $z$.  On the insertion slice, $U_t$ is a finite product of
on-site diagonal unitaries, since all height differences can be
written along that slice.  The exterior vector therefore has norm
at most one.  The other side is the vacuum.  The $r+O(1)$ ordinary
rows between the two carry the damping $P(1)^k$, while translation
along the slice carries powers of the unitary $P(i)$.

For completeness, current placement in the stencil can be handled
before any inverse is introduced.  Start all height differences
at one location a bounded number of rows before the stencil.  A
step issued from the vacuum has spectral vector
$(1-f(w))\mathsf h$.  Telescoping a path of such steps gives the
difference of its endpoint translation multipliers.  Thus the
Laplacian stencil is, up to a common sign, the multiplier
\begin{equation}\label{eq:laplacian-fiber}
 f(1)^k f(i)^j B(f)\mathsf h,\qquad
 B(f)=f(1)^2+f(1)^{-2}+f(i)^2+f(i)^{-2}-4,
\end{equation}
where $k=r+O(1)$ and $j\in\Z$.  With the vacuum on the bra side,
the same calculation uses the conjugate multipliers.  The inverse
time powers in this notation cancel against already present
ordinary rows.  Thus it represents a finite sum of legitimate
nonnegative row products even when $f(1)$ is small.  Cauchy--Schwarz
and Equation~\eqref{eq:small-charge-lower-bound} reduce the lemma
to a bound on the norm of Equation~\eqref{eq:laplacian-fiber}.

Put $u_f=-\log|f(1)|$ and consider a small dyadic band
$u/2<u_f\le u$.  Evaluation of the factorization at $1$ and the
zero cone give the budget
\begin{equation}\label{eq:spectral-zero-budget}
 \sum_j\min(|a_j|,|a_j|^{-1})\le Cu.
\end{equation}
Each zero is either of modulus at most $Cu$ or of modulus at least
$(Cu)^{-1}$.  On the unit semicircle the logarithm of $f$ relative
to its approximate sign is $O(u)$.  Hence $B(f)=O(u^2)$ on the
whole band.  If all zeros are small, expansion of the factors gives
\[
 \log f(w)=a_f/w+O(u^2),\qquad |a_f|\le Cu,
 \qquad |w|=1,\ \Re w\ge0,
\]
with a real $a_f$; if all zeros are large, it instead gives
$\log f(w)=b_fw+O(u^2)$ with real $b_f$ and $|b_f|\le Cu$.
The logarithms are taken relative to the same sign; the squares
of $f$ in $B$ remove that sign.  The quadratic terms in $B$ cancel, because
$1^2+i^2=0$ and $1^{-2}+i^{-2}=0$.  Therefore
\begin{equation}\label{eq:pure-band-cancellation}
 |B(f)|\le Cu^3
 \quad\text{if the band has only small zeros or only large zeros}.
\end{equation}
The bounds on the remainders follow by summing the factor expansions;
Equation~\eqref{eq:spectral-zero-budget} bounds both their first
moments and the sums of their quadratic remainders.

We now quantify the cost of the remaining, mixed-zero band.  Along
one dilation orbit, a dyadic band occupies a bounded logarithmic
length on each exterior tail: there $u_f$ is comparable to the sum
of the small zero moduli, or to the sum of the reciprocal large
zero moduli, and these sums vary geometrically with dilation.
The intervening part has length at most $\mathcal W+C$.
In particular
\[
 \mu\{u/2<u_f\le u\}\le C\int(1+\mathcal W)\,\dd\nu\le C.
\]
A mixed-zero band requires
$\mathcal W\ge2\log(1/u)-C$.  By
Equation~\eqref{eq:current-width-tail}, its measure is at most
\[
 C\int_{\{\mathcal W\ge2\log(1/u)-C\}}
       (1+\mathcal W)\,\dd\nu
 \le C u^{2\eta_0},
\]
after decreasing $\eta_0$ if needed.  Combining this with the
$O(u^2)$ bound on that set and
Equation~\eqref{eq:pure-band-cancellation}, and then taking square
roots, proves
\begin{equation}\label{eq:laplacian-band-norm}
 \left(\int_{\{u/2<u_f\le u\}}|B(f)|^2\,\dd\mu\right)^{1/2}
 \le C u^{2+c_*}
\end{equation}
for some $c_*>0$.

On this band the time rows contribute $e^{-c r u}$.  Summing the
dyadic estimates in Equation~\eqref{eq:laplacian-band-norm} gives
$Cr^{-2-c_*}$.  On the complement of the small bands, expand the
stencil into its finitely many current increments before estimating.
The current measure of $\{u_f\ge u_0\}$ is finite for each $u_0>0$,
since $|1-f(1)|\ge1-e^{-u_0}$ there and
$\int|1-f(1)|^2\,\dd\mu<\infty$.  Cancel the at most two inverse
time powers in Equation~\eqref{eq:laplacian-fiber} first.  At least
$r-O(1)$ time rows remain, giving exponential damping on this
finite-measure set.  The constant spectral
function contributes no current.  This proves
Equation~\eqref{eq:off-axis-laplacian}.

The last assertion needs no power rate.  Each unit increment is
paired with a bounded exterior vector and contains arbitrarily many
ordinary rows acting on $J(1)\Omega$ or $J(\pm i)\Omega$.
On the nonconstant current spectrum $|f(1)|<1$, and the constant
spectrum has zero current.  Dominated convergence in its finite
current-vector norm therefore makes the increment tend to zero.
\end{proof}

\subsection{Twisted rows and finite differences near the axis}

The preceding proof uses a vacuum half-plane and thus loses its
damping at the insertion line.  A different slicing supplies bounds
there: an interval between insertions contains a constant diagonal
twist, and a step-two height increment is small on its low bands.

For a fixed unit twist define
\[
 J_\zeta(w)=A^{ZK(\zeta)}(w),\qquad
 H_\zeta=-\log|P_\zeta(1)|,\qquad
 Q_\zeta(u)=\1_{[0,u]}(H_\zeta).
\]
Split $Q_\zeta(u)$ further into the two signs of $P_\zeta(1)$.
For each fixed argument $w$, the joint spectral projection of
$\{f:f(w)=0\}$ is zero, by the Scalar Continuation in a Fixed
Band lemma, label \texttt{bulk:edge-band}, in \cite{FKCompanion}.
In particular this holds at $w=1$, so $H_\zeta$ is a densely
defined nonnegative self-adjoint operator.  The same zero budget as in
Equation~\eqref{eq:spectral-zero-budget}, now including the possible
exponential factors in Equation~\eqref{eq:general-spectral-factorization},
shows that on $Q_\zeta(u)$ both $P_\zeta(w)$ and its inverse are
uniformly bounded for
\begin{equation}\label{eq:twisted-annulus}
 A u<|w|<a/u,\qquad \Re w\ge0,
\end{equation}
and on the unit semicircle
\begin{equation}\label{eq:twisted-sign}
 P_\zeta(w)Q_\zeta^\sigma(u)
 =\sigma Q_\zeta^\sigma(u)+O(u),\qquad \sigma\in\{+1,-1\}.
\end{equation}
The constants $A,a$ and the errors are uniform over unit twists.

\begin{lemma}[Twisted finite differences]\label{lem:twisted-differences}
Suppose a row interval contains no insertion, has one fixed unit
twist, and leaves at least $l$ physical rows on each side of an
observation stencil.  For every fixed $k\ge1$, every $k$-fold
coordinate difference with steps of length two satisfies
\begin{equation}\label{eq:twisted-derivative-bound}
 |D_2^{(k)}H(z)|\le C_k n^{Cb}l^{-k},\qquad l\ge10k.
\end{equation}
The estimate is uniform in the twist and in the spatial position of
the stencil.  The directions of the differences may be mixed.
\end{lemma}

\begin{proof}
Write $U=P_\zeta(i)$.  The twisted endpoint formulas give
\[
 J_\zeta(i)=UZ_0,\qquad J_\zeta(-i)=-Z_0U^{-1},\qquad
 J_\zeta(-i)=-U^{-1}J_\zeta(i)U^{-1},
\]
and the analogous identity on the other ray.  This follows by
substituting $ZK(\zeta)$ in
Equation~\eqref{eq:spectral-endpoints}; $Z$ commutes with the
diagonal twist and changes sign under $X$.  Dilate the endpoint
identity before compressing.  The compressed current
$J_{\zeta,u}(w)=Q_\zeta(u)J_\zeta(w)Q_\zeta(u)$ consequently
extends to the other half of Equation~\eqref{eq:twisted-annulus} by
\begin{equation}\label{eq:twisted-current-gluing}
 J_{\zeta,u}(-w)
 =-P_\zeta(w)^{-1}J_{\zeta,u}(w)P_\zeta(w)^{-1},
\end{equation}
where the inverses are restricted to the low band.  The formulas
agree on both imaginary rays by strong endpoint continuity.
Morera's theorem applied to matrix coefficients gives a bounded
holomorphic function on the full annulus.  Its Laurent expansion
$\sum_j w^jJ_{\zeta,u,j}$ satisfies
\[
 \|J_{\zeta,u,j}\|\le C(Cu)^{|j|}.
\]
In particular its odd part on $|w|=1$ is $O(u)$.

On an equal-sign block,
Equations~\eqref{eq:twisted-sign} and
\eqref{eq:twisted-current-gluing} show that the even part is bounded
by $Cu$ times the whole block.  Absorbing this term proves that the
whole equal-sign block is $O(u)$.  A two-step height difference is,
up to its sign convention and a fixed factor, the doubled current
\[
 J_\zeta(w)P_\zeta(w)+P_\zeta(w)J_\zeta(w).
\]
For a spatial difference the precise endpoint compensation is
\[
 U^{-2}\bigl(J_\zeta(i)U+UJ_\zeta(i)\bigr)
 =U^{-1}Z_0U+Z_0.
\]
The compensating inverse is unitary and preserves the low bands.
On opposite-sign blocks its leading signs cancel, leaving $O(u)$
by boundedness of the current.  Thus the doubled current is $O(u)$
on the full low band.  This argument does not require a separate
decay bound for an undoubled opposite-sign current.

We describe why these row estimates apply to observations at any
spatial position.  In a finite row the auxiliary mark measures the
spin at the insertion seam.  Differentiate
$[P_\zeta(w),U]=0$ in $\theta$, where $\zeta=e^{i\theta}$.
Since $\partial_\theta P_\zeta(w)=iJ_\zeta(w)$ and
$\partial_\theta U=iUZ_0$, the result is the exact identity
\begin{equation}\label{eq:translated-current}
 U^{-1}J_\zeta(w)U
 =J_\zeta(w)+Z_0P_\zeta(w)-P_\zeta(w)Z_0.
\end{equation}
The local conservation law
$[Z_c+Z_0,R_{c0}]=0$ identifies the right-hand side as the row
with the measured auxiliary spin moved across tile $0$, while the
diagonal twist remains at the original seam.  Iteration gives the
probe at site $j$ by conjugation with $U^j$.  Equivalently, the
diagonal factors acquired in a translated seam move it back by ice
conservation.  A spatial height step is the on-site operator $Z$,
which commutes with these factors.  These are identities of a
finite tensor array with no wraparound; only finitely many
translations occur in every plane correlation.  They therefore
pass through the plane-kernel limit.  All compensating shifts are
unitary, so their number introduces no factor into the estimate.

On a low band, Equation~\eqref{eq:twisted-sign} gives
$P_\zeta(i)^2=I+O(u)$ and $P_\zeta(1)^2=I+O(u)$.
Accordingly each additional spatial two-step difference, acting
by translation on both sides of the probe, contributes another
factor $u$: for its translation $T$ and compressed probe $O$, use
$T^{-1}OT-O=T^{-1}[O,T-I]$.  A temporal two-step difference shifts two rows from
one side of the probe to the other and has the same property.
The inverses used in this low-band calculation are bounded there;
before compression, all expressions retain their common
nonnegative damping rows.  Induction on the number of differences
therefore gives $C_k u^k$ for a $k$-fold difference of $H$, before
division by $C(t;u)$, when both adjacent states have cutoff $u$.
For unequal cutoffs the estimate holds with their maximum, since
both projections lie in that larger low band.

The remaining rows turn these band estimates into spatial estimates.
Split each side into $[0,l^{-1}]$, dyadic bands above $l^{-1}$ up
to a small fixed threshold, and the remaining spectral projection.
A band of lower endpoint $u$ is damped by $e^{-lu}$ by its $l$
physical rows.  The low-band terms are bounded by the convergent
double dyadic sum
\[
 C_k\sum_{u,u'}\max(u,u',l^{-1})^k e^{-c l(u+u')}
 \le C_k l^{-k}.
\]
For the remaining bands, expand the finitely many differences and
keep $l-O(k)$ rows; their contribution decays exponentially.
All exterior strings are products of unit-twist rows and diagonal
unitaries.  Their norms are at most one.  Finally divide by
$C(t;u)\ge c n^{-Cb}$ to obtain
Equation~\eqref{eq:twisted-derivative-bound}.
\end{proof}

\begin{proposition}[A summable Laplacian error]\label{prop:laplacian}
For the neutral small-charge experiments of
Equation~\eqref{eq:electric-characteristic}, there are $c_1>0$ and
$C<\infty$ such that
\begin{equation}\label{eq:laplacian}
 |\Delta_2H(z)|\le C n^{Cb}
       \left(1+\min_j|z-u_j|\right)^{-2-c_1}
 \qquad(z\in\Z^2).
\end{equation}
The constants are uniform over the stated geometry and charges.
\end{proposition}

\begin{proof}
Let $r=\min_j|z-u_j|$, and first suppose $r$ is sufficiently large.
Choose $\delta>0$ so small that
$(2+c_*)(1-\delta)>2$, where $c_*$ is supplied by
Lemma~\ref{lem:off-axis}.  If the transverse distance of $z$ from
the insertion axis is at least $r^{1-\delta}$, that lemma gives
the result immediately with exponent
$(2+c_*)(1-\delta)-2>0$.

Otherwise move $z$ in the normal direction to a vertex $z'$ in
the same coordinate-parity coset whose transverse distance lies
between $r^{1-\delta}$ and $r^{1-\delta}+2$.  Its longitudinal
distance from every insertion is at least a fixed multiple of $r$.
The entire intervening stencil therefore has, in the slicing with
time along the insertion axis, an insertion-free interval with
$l\ge c r$ rows on both sides.  Each normal step of length two
changes $\Delta_2H$ by a sum of third differences, since each
central second difference is a translate of two successive
step-two differences.  By
Lemma~\ref{lem:twisted-differences}, summing at most
$C r^{1-\delta}$ steps gives
\[
 |\Delta_2H(z)-\Delta_2H(z')|
 \le C n^{Cb}r^{1-\delta}r^{-3}
 =C n^{Cb}r^{-2-\delta}.
\]
At $z'$ use Lemma~\ref{lem:off-axis}.  Taking
$c_1=\min\{\delta,(2+c_*)(1-\delta)-2\}>0$ proves the claim for
large $r$.  For bounded $r$, the uniform bound on unit increments
of $H$ proves the same estimate after enlarging $C$.
\end{proof}

The power in Equation~\eqref{eq:laplacian} makes its right-hand side
summable on each of the four coordinate-parity cosets.  Together
with the sublogarithmic growth in
Equation~\eqref{eq:potential-growth}, this is the input needed to
recover the potential from its discrete Laplacian.
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//04-spectral.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//05-electric.tex ====
\section{Electric correlations and their amplitudes}\label{sec:electric}

The Laplacian estimate of Proposition~\ref{prop:laplacian} determines
the interaction between separated electric charges.  Its integrable
power remainder is essential: it gives a convergent amplitude, rather
than only a logarithmic exponent.  We first obtain that interaction for
small charges, then continue it to the charges that suppress microscopic
loops around the marked points.

We retain the neutral axial correlation
\[
 C(\mathbf t;\mathbf u)
 =\E\exp\!\left(i\sum_{j=1}^p t_jh(u_j)\right),
 \qquad \sum_{j=1}^p t_j=0,
\]
and its potential $H_{\mathbf t}$ from Section~\ref{sec:spectral}.
In the positive winding convention a loop enclosing the set $I$ of
marked points has relative weight
\[
 v_I^+(\mathbf t)=
 \frac{\cos(\lambda+\sum_{j\in I}t_j)}{\cos\lambda}.
\]
The opposite convention has $v_I^-(\mathbf t)=v_I^+(-\mathbf t)$
and gives the same correlation.  In particular,
$C(\mathbf t;\mathbf u)=C(-\mathbf t;\mathbf u)$.
Write
\[
 v(t)=\frac{\cos(\lambda+t)}{\cos\lambda},\qquad
 I_\lambda=(-\pi/2-\lambda,\pi/2-\lambda).
\]
Thus $v(t)>0$ on $I_\lambda$.  As in Section~\ref{sec:radial},
$F_n(v)$ uses the fixed plane cutoff $b_0n$ and fixed lattice rounding.

\subsection{The potential kernel and local charge profiles}

Set $A=1/[2\pi(\pi-\lambda)]$.  The covariance calculation below
identifies this constant in the potential-kernel argument.

\begin{proposition}[Coulomb formula]\label{prop:coulomb}
Fix a number $p$ of marked points.  There are $b,c,\xi>0$ and a
real-analytic function $f$ on a neighborhood of $[-b,b]$, with
$f(0)=0$, such that,
for neutral real charges $\max_j|t_j|\le b$ and distinct points on a
tile axis satisfying
\[
 \max_j|u_j|\le Mn,\qquad
 \min_{i\ne j}|u_i-u_j|\ge n^{1-\xi},
\]
one has
\begin{equation}\label{eq:coulomb}
 \log C(\mathbf t;\mathbf u)
 =\sum_{j=1}^p f(t_j)
   +2A\sum_{i<j}t_it_j\log|u_i-u_j|+O(n^{-c}),
 \qquad f(0)=0.
\end{equation}
Here the same local function $f$ is used for every number of points,
on their common small-charge neighborhood; it is independent of the
geometry and of the point parities.
The constants can depend on $p,M$ and $q$.

For either choice of sign, the corresponding multiplicative statement
has analytic continuation along compact paths from zero in
\[
 \mathcal D_p^\pm
 =\{\mathbf t\in\R^p:\ \sum_jt_j=0,
                         \ \pm t_j\in I_\lambda\text{ for every }j\}.
\]
More precisely, on a sufficiently thin complex neighborhood of each
such compact path within the neutral hyperplane $\sum_jt_j=0$,
there is a holomorphic function $M_p$ such that
\begin{equation}\label{eq:coulomb-continued}
 \frac{C(\mathbf t;\mathbf u)}
 {\displaystyle\prod_{i<j}|u_i-u_j|^{2At_it_j}}
 =M_p(\mathbf t)+O(n^{-c}).
\end{equation}
The positive constants $c$ and $\xi$ may decrease with the path.
Near zero, $M_p(\mathbf t)=\exp(\sum_jf(t_j))$.
The error in \eqref{eq:coulomb-continued} is additive; a relative
error is asserted only where $M_p$ is nonzero.
\end{proposition}

We first derive \eqref{eq:coulomb} for real charges with an unspecified
coefficient in place of $A$; the continuation and the identification of
that coefficient will follow below.  Choose $b$ small enough that
all subset weights in both winding conventions are strictly positive.
The inward coupling of Proposition~\ref{prop:radial}, in its multipoint
form in Proposition~\ref{prop:normalization}, shows that the
unit increments of $H_{\mathbf t}$ near $u_j$ approach a profile
depending only on $t_j$.  Indeed, inside a common surrounding stopping
circuit a marked spin belongs to a loop inside that circuit.  Its
orientation average depends only on whether that loop encloses $u_j$,
and hence only on the single charge $t_j$.  No remote charge enters
this average.  Comparing configurations at separation scales $a$ and
$2a$ gives a power error; summing these errors over successive doublings
constructs a unique limiting increment profile.  Neutrality can always
be maintained by splitting the remote compensating charge into finitely
many small charges.  The same coupling compares two different remote
configurations, including different numbers of compensators: pad them
with zero charges to obtain a common finite number of marked points.
Thus the limit is independent of that choice.

Let $\phi_t(y)$ be the step-two Laplacian of this limiting potential,
with its charge at the origin.  Translation invariance, including an
odd tile translation with the corresponding color exchange, identifies
these profiles for every possible location of the charge.  They also
agree when the remote points are placed on the other tile axis: the
conditional loop law and the orientation average inside the matched
stopping circuit are identical.  This observation will allow us to
compare all four parity classes of the step-two Laplacian.

The profiles have a summable power tail.  To see this at a point $y$,
compare the limiting profile to an experiment whose separation is
$(1+|y|)^B$.  First choose $B$ so large that the inward coupling error
is smaller than $(1+|y|)^{-2-c_2}$ for some $c_2>0$.
Then decrease $b$ so that the factor $n^{Cb}$ in
\eqref{eq:laplacian}, at this value of $n$, does not exhaust the
power gain of that estimate.  It follows that
\begin{equation}\label{eq:profile-tail}
 |\phi_t(y)|\le C(1+|y|)^{-2-c_2}.
\end{equation}
All constants here can be chosen uniformly for the small charges under
consideration, since their positive weights lie in a fixed neighborhood
of one.

Put $g_{\mathbf t}(z)=\Delta_2H_{\mathbf t}(z)$.
The local convergence just established gives the following useful
quantitative form.  After decreasing $b$ and $\xi$, there is $c_3>0$
such that
\begin{equation}\label{eq:profile-approximation}
 \sum_{z\in\Z^2}
 \left|g_{\mathbf t}(z)-
              \sum_j\phi_{t_j}(z-u_j)\right|
 \log(n+|z|+2)=O(n^{-c_3}).
\end{equation}
Here and below the tile coordinate lattice is identified with $\Z^2$.
For completeness, take disjoint balls of radius $n^\varepsilon$ around
the $u_j$.  Within these balls the coupling has a fixed power gain,
with a possible loss $n^{C\xi}$ from the ratio between the configuration
scale and its least separation.  Choose $\xi$ and then $\varepsilon$
small enough that summing over $O(n^{2\varepsilon})$ sites, and
multiplying by a logarithm, still leaves a power gain.  Outside the
balls, \eqref{eq:laplacian} and \eqref{eq:profile-tail} give a bound
of order $n^{Cb-\varepsilon c_1}$, with a possibly smaller exponent
than $c_1$ and an extra logarithm.  Decrease $b$ once more to make this
a negative power.  The same estimates on the unbounded part of the
lattice include the displayed logarithmic weight.  This proves
\eqref{eq:profile-approximation}.

We now apply ordinary lattice potential theory, keeping its parity
issue explicit.  For each $a\in\{0,1\}^2$ let
$\Lambda_a=a+2\Z^2$.  Let $K$ be the potential kernel on $2\Z^2$,
normalized by
\[
 \Delta_2K=\delta_0,\qquad
 K(y)=k\log|y|+k'+O(|y|^{-1})\quad (|y|\longrightarrow\infty),
 \qquad k\ne0.
\]
The kernel normalization, asymptotic and increment estimates follow from
\cite[Proposition~4.4.2, Theorem~4.4.4 and Corollary~4.4.5]{LawlerLimic2010},
after rescaling each parity coset to the integer lattice and adjusting
the generator normalization.  The same kernel is used on each translated
lattice.  By
\eqref{eq:laplacian}, the convolution
\[
 U_a(z)=\sum_{w\in\Lambda_a}K(z-w)g_{\mathbf t}(w),
 \qquad z\in\Lambda_a,
\]
is absolutely convergent, has growth $O_n(1+\log(2+|z|))$, and has
Laplacian $g_{\mathbf t}$.  On the other hand, the exact variogram
bound in Proposition~\ref{prop:spectral} and Cauchy--Schwarz give
\[
 |H_{\mathbf t}(z)-H_{\mathbf t}(z_0)|
 \le |C(\mathbf t;\mathbf u)|^{-1}
       \bigl(\E|h(z)-h(z_0)|^2\bigr)^{1/2}
 =O_n\!\left(1+\sqrt{\log(2+|z-z_0|)}\right).
\]
Thus $H_{\mathbf t}-U_a$ is a sublinearly growing harmonic function
on $\Lambda_a$, and the lattice Liouville theorem makes it constant;
this is the sublinear-growth consequence of the gradient estimate
\cite[Theorem~6.3.8(a)]{LawlerLimic2010}, with the same rescaling.
The tail bound also gives
\[
 U_a(z)=
 \left(\sum_{w\in\Lambda_a}g_{\mathbf t}(w)\right)
                  (k\log|z|+k')+o(1).
\]
The sublogarithmic growth of $H_{\mathbf t}$ therefore forces
\begin{equation}\label{eq:zero-coset-mass}
 \sum_{w\in\Lambda_a}g_{\mathbf t}(w)=0
 \quad\text{for every }a.
\end{equation}
In particular $U_a(z)\to0$ at infinity.  The unit increments of
$H_{\mathbf t}$ tend to zero along the off-axis normals, as proved
in Section~\ref{sec:spectral}.  Evaluating neighboring parity classes
along those normals shows that their four additive constants agree.
Consequently the convolution formula determines all point differences
of $H_{\mathbf t}$, not only differences within one parity class.

Let
\[
 b_a(t)=\sum_{y\in\Lambda_a}\phi_t(y).
\]
Equation~\eqref{eq:profile-approximation}, followed by
\eqref{eq:zero-coset-mass} and $n\to\infty$, implies
\[
 \sum_j b_{a-u_j}(t_j)=0
\]
in every separated neutral experiment; the subscripts are taken modulo
two.  In a two-charge experiment, changing one point by one tile along
the axis shows that $b_a(t)$ is independent of the corresponding
coordinate parity.  Repeating the experiment on the other axis proves
independence of both parities.  Denote the common sum by $b_*(t)$.
Two charges give $b_*(-t)=-b_*(t)$, and three give
$b_*(s+t)=b_*(s)+b_*(t)$ whenever these charges are small.
The uniform bound from \eqref{eq:profile-tail} excludes discontinuous
additive functions.  Hence $b_*(t)=b_1t$ for a constant $b_1$.

Evaluate the convolution formula at $u_i$.  The contribution of its
own profile is
\[
 g_0(t_i)=\sum_{y\in2\Z^2}K(-y)\phi_{t_i}(y),
\]
which depends only on $t_i$.  Every other profile contributes
\[
 b_*(t_j)\bigl(k\log|u_i-u_j|+k'\bigr)
       +O(n^{-c}),
\]
after possibly decreasing $c$.  To justify the power error, split the
profile sum at a small power of $|u_i-u_j|$; use the kernel asymptotic
and its increment bound on the inner part, and
\eqref{eq:profile-tail} on the remaining part.  The error from replacing
$g_{\mathbf t}$ by its profiles is already controlled by
\eqref{eq:profile-approximation}.  Neither this argument nor the
self contribution depends on the point parity.

For $0\le l\le1$ differentiation of the characteristic function gives
the exact identity
\[
 \frac{\dd}{\dd l}\log C(l\mathbf t;\mathbf u)
       =i\sum_i t_iH_{l\mathbf t}(u_i).
\]
The common additive constant disappears by neutrality.  The cross
terms integrate to
$2A\sum_{i<j}t_it_j\log|u_i-u_j|$ with $2A=ikb_1$.
The self terms integrate separately at each point.  The terms involving
$k'$ are also single-point terms, since
$\sum_{i\ne j}t_it_j=-\sum_i t_i^2$.
Absorb them into $f(t_i)$, taking $f(0)=0$.
Uniformity of the preceding estimates for $l\in[0,1]$ proves
\eqref{eq:coulomb} for small real charges.

\subsection{Analytic continuation and the electric constant}

The complex estimates of Section~\ref{sec:radial} turn this local
calculation into a formula at finite charges.  First they determine
the scaling exponent of each positive nesting weight, without yet
determining its amplitude.  On thin complex neighborhoods of compact
subintervals of $I_\lambda$, the functions
\[
 \frac{\log F_n(v(t))}{\log n}
\]
form a locally bounded holomorphic family; logarithms are continued
from real positive weights.  Extract any convergent subsequence.
For small neutral systems, the all-positive comparison in
Proposition~\ref{prop:normalization} bounds the normalized correlation
above and below.  Equation~\eqref{eq:coulomb}, with separations of
order $n$, therefore shows that its limiting exponent $g$ satisfies
\[
 \sum_j g(t_j)=-A\sum_jt_j^2.
\]
The two- and three-charge identities imply
$g(t)=-At^2+Bt$ near zero.  Holomorphic uniqueness extends this identity
along $I_\lambda$.  Since
\[
 v(t)=v(-2\lambda-t),
\]
the same symmetry holds for $g$, and it forces $B=-2\lambda A$.
Every subsequential exponent is thus the same.  We have proved,
locally uniformly also in those complex neighborhoods,
\begin{equation}\label{eq:nest-log-exponent}
 F_n(v(t))=\exp\{\beta(t)\log n+o(\log n)\},
 \qquad \beta(t)=-A(t^2+2\lambda t).
\end{equation}

Divide $C(\mathbf t;\mathbf u)$ by the cross-distance product in
\eqref{eq:coulomb-continued}.  Neutrality gives
$\sum_j\beta(t_j)=-A\sum_jt_j^2$, so
Proposition~\ref{prop:normalization} and
\eqref{eq:nest-log-exponent} bound this quotient by $n^{o(1)}$
on compact complex neighborhoods with macroscopic separations.
If the least separation is only $n^{1-\xi}$, the bound is
$n^{o(1)+C\xi}$.  These are bounds for the quotient itself; no
nonvanishing claim about the correlation is needed here.

Here is the quantitative continuation argument.  Fix a reference
configuration with macroscopic separations and fixed parities.  Compare
its quotients at sizes $n$ and $N$, uniformly for $n\le N\le2n$.
Equation~\eqref{eq:coulomb} makes their difference $O(n^{-c})$ on
a small real neighborhood of zero.  Apply the two-constants theorem
in disks slit along a compact real segment, successively in the
$p-1$ independent charge variables.  A negative power on the segment
and an $n^{o(1)}$ bound on the other boundary give a negative power
on a smaller complex neighborhood.  A finite chain of overlapping
disks carries this estimate along any prescribed compact path in
$\mathcal D_p^+$.  The resulting exponent may be smaller, but remains
positive.  Summing the comparison over dyadic scales, and using its
uniformity for every $N\in[n,2n]$, gives a holomorphic limit $M_p$
through all integers with a power error.

Next subtract the reference quotient from the quotient of any allowed
configuration.  Their difference has the same small-charge power bound
by \eqref{eq:coulomb}, while its bound along the continuation path is
$n^{o(1)+C\xi}$.  Choose $\xi$ sufficiently small after fixing that
path.  The same finite sequence of two-constants estimates still leaves
a negative power.  This proves the geometry-uniform statement in
\eqref{eq:coulomb-continued}.  The opposite winding convention proves
the assertion on $\mathcal D_p^-$.  The initial amplitude is therefore
real analytic.  To obtain the same conclusion for $f$, compare the
logarithmic amplitudes of the neutral triples $(s,t,-s-t)$ and pairs
$(s+t,-s-t)$.  Their difference is the analytic function
$f(s)+f(t)-f(s+t)$.  Averaging this identity in $s$ over a fixed small
interval first shows that the continuous function $f$ is continuously
differentiable; differentiation at $s=0$ then expresses $f'(t)$ as a
constant minus an analytic function.  Hence $f$ is real analytic.

It remains to identify $A$.  Take four distinct collinear continuum
points $a,b,c,d$ and the neutral charges $(-s,s,-t,t)$ at their lattice
approximations.  The mixed derivative at zero of $\log C$ is minus
the covariance of the two height increments.  In
\eqref{eq:coulomb} it kills all single-point terms and yields
\[
 -2A\log\frac{|a-c|\,|b-d|}{|a-d|\,|b-c|}
\]
for that covariance in the limit.  Differentiation of the limit is
legitimate by the locally uniform analytic convergence just proved.
Input~\ref{inp:covariance}, with
$G(z,w)=-(2\pi)^{-1}\log|z-w|$ and
$c=2\cos(\lambda/2)$ in the six-vertex parametrization, gives
\begin{equation}\label{eq:stiffness}
 2A=\frac{\sigma(2\cos(\lambda/2))^2}{2\pi}
    =\frac{1}{\pi(\pi-\lambda)},
 \qquad A=\frac{1}{2\pi(\pi-\lambda)}.
\end{equation}
This is the separated-endpoint covariance assertion of
\cite[Theorem~1.1, Part~(iii)]{GFFCompanion}; at the present parameters
it lies in the established interval $\sqrt3\le c<2$ of
\cite[Theorem~2.8]{DKLM}.  Only the covariance statement at that scope
is used here.  This completes the proof of
Proposition~\ref{prop:coulomb}.

\subsection{Pure nesting amplitudes and nonzero special charges}

\begin{proposition}[Individual nesting amplitudes]\label{prop:pure-nest}
There is $\varepsilon_0>0$ and a nonvanishing holomorphic function
$a(t)$ near $[-\varepsilon_0,\varepsilon_0]$, positive for real $t$,
such that
\[
 F_n(v(t))=a(t)n^{\beta(t)}(1+o(1))
\]
locally uniformly on a complex neighborhood of that interval.
The limit uses the fixed cutoff and rounding in the definition of
$F_n$.  No rate is asserted for this individual amplitude.
\end{proposition}

\begin{proof}
Set
\[
 L_n(t)=\log F_n(v(t))-\beta(t)\log n,\qquad L_n(0)=0.
\]
For fixed macroscopic two- and three-point configurations,
Proposition~\ref{prop:normalization} gives a holomorphic limit of
$C/\prod_jF_n(v(t_j))$, nonzero near zero because its value at zero
is one.  Proposition~\ref{prop:coulomb} gives the corresponding
nonzero limit after division by $n^{\sum_j\beta(t_j)}$.
Taking the logarithms normalized at zero shows that
$\sum_jL_n(t_j)$ converges holomorphically for each such neutral
system.  Subtract the identities for $(s,t,-s-t)$ and $(s+t,-s-t)$.
It follows that
\[
 L_n(s)+L_n(t)-L_n(s+t)
\]
converges near $(0,0)$.  Differentiation at $s=0$ shows that
$L_n'(t)-L_n'(0)$ converges.  Hence, with the real number
$b_n=L_n'(0)$, the functions $L_n(t)-b_nt$ converge holomorphically
near zero.  Neutral correlations alone have left precisely this
linear ambiguity.

To determine $b_n$, choose one charge $t$ in a small interval about
$-2\lambda$ and choose sufficiently many other charges
$s_1,\ldots,s_k$ in the neighborhood just considered, with
$\sum_js_j=-t$.  They can be chosen strictly inside that neighborhood.
All these individual charges belong to $I_\lambda$, and the entire
neutral parameter set connects to zero inside $\mathcal D_{k+1}^+$.
For this fixed macroscopic configuration both the continued amplitude
in Proposition~\ref{prop:coulomb} and the limit in
Proposition~\ref{prop:normalization} are analytic and are not
identically zero.  Their restrictions cannot vanish on an open real
parameter set.  Choose real parameters in the indicated open set where
both are nonzero.  Taking absolute values and ordinary real logarithms
then shows that
\[
 L_n(t)+\sum_jL_n(s_j)
\]
converges.  Put $t'=-2\lambda-t$, which is near zero.
The identities $v(t')=v(t)$ and $\beta(t')=\beta(t)$ give
$L_n(t)=L_n(t')$ exactly.  Substituting the already convergent
remainders $L_n(z)-b_nz$ leaves
\[
 b_n\left(t'+\sum_js_j\right)=b_n(t'-t).
\]
The fixed coefficient $t'-t$ is nonzero, so $b_n$ converges.
Exponentiating the resulting locally uniform limit of $L_n$ proves
the proposition.
\end{proof}

We next specify the charges used to detect microscopic arms.  A
single group consists of one charge $\alpha=\pi/2-\lambda$ and
at least two small negative charges whose sum is $-\alpha$.
Choose sufficiently many compensators that each lies, with its
negative, in the interval of Proposition~\ref{prop:pure-nest}.
The number of charges is now fixed.  They need not all lie in the
small-system neighborhood used in the proof of
\eqref{eq:laplacian} for that number of points.

\begin{proposition}[Two special core charges]\label{prop:special-charges}
Two separately neutral groups of the preceding kind can be chosen
so that their combined amplitude in
\eqref{eq:coulomb-continued} is nonzero.  The formula then has a
relative power error, including when the two $\alpha$-points have
opposite colors and least separation $r\ge n^{1-\xi}$, for a
sufficiently small fixed $\xi>0$.

Place the first core at the origin, the second at odd axial separation
$r\asymp n^{1-\xi}$, and all compensators at distinct positions
$n z_j+O(1)$, with fixed nonzero axial $z_j$.  Write
$B_* =\sum_{j\,\mathrm{remote}}\beta(t_j)$.
There is a constant $M_*\ne0$, depending only on those charges,
their macroscopic positions, and the fixed microscopic conventions,
such that
\begin{equation}\label{eq:special-close-pair}
 C_r(n)=M_*n^{B_*-x}r^{2A\alpha^2}(1+O(n^{-c}))
       =M_*n^{B_*}r^{-2x_1}(r/n)^x(1+O(n^{-c})),
 \qquad x_1=-\beta(\alpha).
\end{equation}
The same conclusion is uniform for
$n^{1-\xi}\le r\le n^{1-\xi'}$, with fixed $0<\xi'\le\xi$,
after decreasing $c$.  For $r=an$ with fixed small $a>0$, the full
Coulomb product applies; its remaining geometric factor tends to
$M_*$ as $a\downarrow0$ and stays bounded above and away from zero.
\end{proposition}

\begin{proof}
For one group use the opposite winding convention.  Every subset
charge lies between $-\alpha$ and $\alpha$, so all scalar weights
are nonnegative.  The only possible zero is a loop enclosing every
compensator and excluding the $\alpha$-point.  Every singleton weight
is strictly positive: at the core this uses $\lambda>0$, and at
the compensators it follows from their smallness.  The group thus
lies strictly inside $\mathcal D_p^-$, where the continued formula
has a power error.

Its real correlation compares above and below to the product of its
opposite-convention singleton factors $F_n(v(-t_j))$.
For the lower comparison, impose surrounding primal circuits in the
separated point patches and connect them by primal paths outside the
smaller cores.  At least two compensators are used.  A loop enclosing
all compensators but not the core could not lie within one patch and
could not cross this connected collection of circuits and paths;
therefore the zero-weight loops are excluded.  The circuit and
connection event costs a fixed positive probability by RSW and
collar comparison.  More explicitly, first integrate the positive
singleton tilts in the smaller cores, leave full collars to the
connecting event, and then use the passage estimates and the local
tilt bounds of Section~\ref{sec:radial}.  All remaining charged
macroscopic loop counts have fixed exponential moments under this
proxy law.  Truncate them at a sufficiently large fixed number, losing
less than half the lower probability of the connecting event.
On the retained event every nonzero scalar factor is bounded below,
which proves the lower comparison.  The upper comparison follows
from the same passage moments.

By \eqref{eq:nest-log-exponent} this comparison fixes the logarithmic
exponent of the group correlation.  After division by its Coulomb
product the result is bounded below by $n^{-o(1)}$.
If its amplitude vanished, \eqref{eq:coulomb-continued} would instead
bound that quotient by $O(n^{-c})$, a contradiction.  Thus the
single-group amplitude is nonzero.  For two independently neutral
groups, the identity
\[
 M_{p+p'}(\mathbf t,\mathbf s)=M_p(\mathbf t)M_{p'}(\mathbf s)
\]
holds near zero by the single-point form of
\eqref{eq:coulomb}, and hence throughout the continuation with
each group separately neutral.  The combined amplitude is nonzero
as well.  The combined positive-convention scalar weights may have
either sign; nonvanishing of the combined amplitude has been proved
using the opposite convention and analytic continuation.

Finally separate the core-core factor $r^{2A\alpha^2}$ in the
Coulomb product.  Every other distance divided by $n$ is a fixed
nonzero number plus $O(r/n+n^{-1})$.  Neutrality and
\eqref{eq:stiffness} give
\[
 2x_1+2A\alpha^2
 =4A\alpha(\alpha+\lambda)
 =\frac{\pi-2\lambda}{2(\pi-\lambda)}=x.
\]
The exponent of $n$ in the remaining product is consequently
$B_*-x$.  When $r/n$ tends to zero at a power rate its geometric
factor has a power-rate limit $M_*$, proving
\eqref{eq:special-close-pair}.  Keeping $r/n=a$ instead gives the
last assertion directly from the uncontracted Coulomb product.
\end{proof}

\subsection{Arm estimates needed for microscopic comparison}

Let $\pi_1(R)$ be the plane probability that a point is connected in
its own color to distance $R$, with a fixed bounded microscopic inner
scale.  Let $\pi_2(r,R)$ be the plane probability of a primal arm and
a dual arm across the concentric square annulus with radii $r,R$.
All connections here are actual lattice paths.  Bounded changes of
these radii are harmless by the arm estimates in Input~\ref{inp:fk}.

\begin{lemma}[Sharp one-arm comparison]\label{lem:one-arm}
There are constants $0<c<C<\infty$ such that
\begin{equation}\label{eq:one-arm}
 cR^{-x_1}\le\pi_1(R)\le CR^{-x_1},
 \qquad x_1=-\beta(\alpha),
\end{equation}
for all radii above the fixed microscopic scale.
\end{lemma}

\begin{proof}
Use one of the neutral single-core groups above at mutual separations
of order $R$, now in the positive winding convention.  Its singleton
core weight is $v(\alpha)=0$.  Thus no loop wholly inside its core
patch may enclose the point.  In the switched-region description this
is exactly connection of the point's color to the patch rim, up to a
bounded resizing of a whole-tile box.  All weights of this single group
are nonnegative.  Localize the small positive compensator weights in
their disjoint patches.  Passage bounds through collars give the upper
comparison
\[
 C(\mathbf t;\mathbf u)
 \le C\pi_1(R)\prod_{j\,\mathrm{remote}}F_R(v(t_j)).
\]

For the reverse comparison, give the core arm a separated landing and
continue it through a fixed corridor to a circuit of the same color
around one compensator patch.  Any loop enclosing the core point must
then enclose that circuit and its compensator as well: otherwise it
would cross the connected color path.  Hence no remaining loop has
the zero core-only weight.  The landing extension, corridor, and
circuit cost a bounded factor by Input~\ref{inp:fk}, uniformly after
the positive tilts inside the compensator cores, since full collars
separate those tilts from the construction.  The macroscopic loop
counts still have every fixed exponential moment with a bounded cost
relative to the arm probability.  To apply that bound, first cut the
arm requirement back through a fixed-factor buffer, use arm
quasimultiplicativity and localization, and leave space between its
support and the passage buffers.  The local tilt bounds deal with the
compensator cutoffs.  A fixed truncation of these counts then retains
a positive fraction of the constructed event and bounds all nonzero
weights below.  Therefore
\[
 C(\mathbf t;\mathbf u)
 \asymp\pi_1(R)\prod_{j\,\mathrm{remote}}F_R(v(t_j)).
\]
The nonzero Coulomb amplitude makes the left side comparable to
$R^{\sum_j\beta(t_j)}$, and
Proposition~\ref{prop:pure-nest} supplies the compensator powers.
Canceling them proves \eqref{eq:one-arm}.
\end{proof}

\begin{lemma}[Logarithmic two-arm exponent]\label{lem:two-arm-log}
For every $\delta>0$ there is $C_\delta<\infty$ such that
\begin{equation}\label{eq:two-arm-log}
 C_\delta^{-1}(r/R)^{x+\delta}
 \le\pi_2(r,R)\le C_\delta(r/R)^{x-\delta},
 \qquad R\ge r\ge r_0,
\end{equation}
where $r_0$ is a fixed microscopic scale.
\end{lemma}

\begin{proof}
In a Dobrushin square of size $R$, place the test disk at its center.
For $r\le R/C$, the probability that the distinguished interface enters
this disk compares to $\pi_2(r,R)$, allowing fixed factors in the
radii.  A disk visit requires both color paths to distance comparable
to $R$, giving the upper comparison through full collars.  For the
lower comparison, separate both ends of the annular arms by the
landing-extension result in Input~\ref{inp:fk}.  Continue the outer
primal arm to a flat portion of the wired wall and the outer dual arm
to a flat portion of the complementary dual wall, away from the marked
corners, through disjoint corridors.  Up-to-wall crossings cost a
bounded factor.  Continue both inner ends into a fixed enlargement of
the disk.  The distinguished strand separating these two boundary
regions must then enter that enlargement.  These are the usual
landing and localization arguments of \cite[Section~6]{DMT}; the
paths reach the actual wall neighborhoods and do not use boundary
identification as an annular crossing.  Removing the wired perimeter
edges from the sampled set does not alter the argument, since their
endpoints are already identified and those edges are declared open
when drawing the strand.

Fix a large annulus ratio $K$ and let the inner radius tend to infinity.
Input~\ref{inp:trace} applies to these square approximations.
Open and closed disk-hitting events, with a fixed enlargement or
contraction of the radius, give upper and lower limiting bounds even
without selecting a radius of continuity.  The Euclidean SLE Green
function estimate \cite[Theorem~2.3, Equation~(12)]{LawlerRezaeiBasic}, applied at
the fixed interior center of the square by local conformal distortion,
gives a positive asymptotic proportional to $K^{-x}$ as
$K\to\infty$.  Its hypothesis $\kappa<8$ holds here.  The constants
in the preceding arm-to-disk comparisons do not depend on $K$.
Consequently there are constants $c,C>0$, independent of large $K$,
such that, for each fixed $K$ and all sufficiently large $r$,
\[
 cK^{-x}\le\pi_2(r,Kr)\le CK^{-x}.
\]

Let $Q$ be a quasimultiplicativity constant.  Choose $K$ so large
that the fixed factors $Q$, $c^{-1}$, and $C$ in a product over one
annulus are absorbed by $K^{\delta/2}$.  Iterate the last display
on annuli of ratio $K$, once their inner radii exceed the corresponding
threshold.  Quasimultiplicativity gives the powers $x\pm\delta/2$.
A final annulus of bounded ratio is handled by RSW, and the finitely
many scales below the threshold are absorbed in $C_\delta$ using
finite energy and fixed-factor arm extensions.  Enlarging the constant
and relaxing to $x\pm\delta$ proves \eqref{eq:two-arm-log} for all
the stated radii.  This argument supplies the logarithmic exponent;
the sharp two-arm comparison will be obtained from the microscopic
calibration in Section~\ref{sec:calibration}.
\end{proof}
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//05-electric.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//05-normalization.tex ====
\section{Determining the covariance below \texorpdfstring{\(c=1\)}{c=1}}\label{sec:normalization}

Throughout this section \(0<c<1\), so \(2\pi/3<\lambda<\pi\).
Our objective is to determine the orbit mass \(\kappa\) defined in
Section~\ref{sec:angular}. Two different finite systems give opposite
bounds. A homogeneous chain at fixed particle density gives a lower
bound through its response to a seam twist. A dense array of real
inhomogeneities gives an upper bound through the change of its vacuum
vector under the same twist. Neither comparison uses positive association.

\begin{proposition}[The covariance constant below one]\label{prop:normalization}
With the row and height conventions of Sections~\ref{sec:finite}
and~\ref{sec:angular},
\begin{equation}\label{eq:kappa-value}
 \kappa=\frac1{2\pi(\pi-\lambda)}.
\end{equation}
Consequently a limiting covariance \(-2\kappa\log|x-y|\) has squared
multiplier \(4\pi\kappa=2/(\pi-\lambda)=\sigma(c)^2\) relative to
\eqref{eq:green}.
\end{proposition}

\subsection{Homogeneous stiffness at fixed density}

For \(Q<\infty\), let \(K_Q\) denote convolution with \(K\) restricted to
\([-Q,Q]\), and write
\[
 r_Q=(I+K_Q)^{-1}s,\qquad H_Q=(I+K_Q)^{-1}\bbone.
\]
These are the density and dressed constant of Lemma~\ref{lem:root-limits}.
Choose the homogeneous finite chain with particle number
\(n/L\to\int_{-Q}^{Q}r_Q(u)\,\dd u\). The first limit in this subsection
is \(L\to\infty\) at this fixed \(Q\); only afterwards does \(Q\to\infty\).
The resulting local state is the same balanced plane state, by the
root-limit and physical-identification results already proved.

Insert the auxiliary seam \(e^{itZ}\), and let \(\mathcal H_L(t)\) be the
endpoint Hamiltonian before subtracting its ground energy. Denote its
continued sector ground energy by \(e_L(t)\), its unit ground vector at
zero twist by \(\Omega_L\), and
\[
 E_{\mathrm{exc},L}
   =\mathcal H_L(0)-e_L(0)
\]
on that magnetization sector. The untwisted sector minimum is simple.
All inverse-energy expressions below are restricted to
\(\Omega_L^\perp\) in that sector.

At the swap endpoint, the auxiliary spin is carried from site to site.
The seam therefore twists the closing link. Conjugating by site rotations
whose angles increase in increments \(t/L\) distributes this twist over
all links, since each link conserves the sum of its two spins. Let
\(b_j\) and \(d_j\) be respectively the first and second derivatives of
the local link under the single-link twist, with a common choice of
orientation. Put \(\mathcal J_L=\sum_j b_j\) and
\(D_{0,L}=L^{-1}\sum_j d_j\).
Ordinary second-order perturbation theory then gives
\begin{equation}\label{eq:kohn}
 L e_L''(0)=\langle D_{0,L}\rangle
 -\frac2L
 \left\|E_{\mathrm{exc},L}^{-1/2}
       (I-|\Omega_L\rangle\langle\Omega_L|)\mathcal J_L\Omega_L\right\|^2 .
\end{equation}
Indeed, after distributing the seam,
\(\mathcal H_L'(0)=L^{-1}\mathcal J_L\) and
\(\mathcal H_L''(0)=L^{-2}\sum_jd_j\).
The sign chosen for \(t\) has no effect on \eqref{eq:kohn}.

\begin{lemma}[Homogeneous stiffness]\label{lem:stiffness}
The iterated homogeneous stiffness satisfies
\begin{equation}\label{eq:stiffness-value}
 \lim_{Q\to\infty}\lim_{L\to\infty}L e_L''(0)
      =\frac2{\pi-\lambda}.
\end{equation}
\end{lemma}
\begin{proof}
The root-dependent part of the endpoint energy is
\(\sum_j e(y_j)\), where
\[
 e(u)=-2\pi\eta s(u).
\]
Differentiating the counting equations at zero twist gives
\[
 G y'=\frac{\bbone}{\pi},\qquad
 (G y'')_j=-L s'(y_j)(y'_j)^2
      +\sum_kK'(y_j-y_k)(y'_j-y'_k)^2.
\]
All derivatives here are taken on the regular real-root branch at fixed
finite \(L\). Lemma~\ref{lem:gaudin-inverse} and the fixed-\(Q\) quadratures
imply, at the nodes,
\[
 L y_j'\longrightarrow W(y_j),\qquad
 W=\frac{H_Q}{\pi r_Q}.
\]
Set \(\varepsilon_Q=(I+K_Q)^{-1}e'\).
The symmetry of \(G\) gives
\(L G^{-1}e'\to\varepsilon_Q/r_Q\).
Substituting the equation for \(y''\) into the second energy derivative
and expanding its square therefore gives
\begin{align}
 \lim_{L\to\infty}L e_L''(0)
 &=\int_{-Q}^{Q}
   \left\{(r_Q\varepsilon_Q'-\varepsilon_Qr_Q')W^2
       +2\varepsilon_Q W(r_QW)'\right\}\,\dd u \notag\\
 &=2\bigl(\varepsilon_Qr_QW^2\bigr)(Q).
 \label{eq:stiffness-boundary}
\end{align}
Here is the cancellation in the first equality. The direct squared
displacement term is \(\int r_Q e''W^2\).
The term \(-s'+K_Q'r_Q\) in the implicit derivative is
\(-r_Q'\), so it contributes \(-\int\varepsilon_Qr_Q'W^2\).
Transposing the other \(K_Q'\) term replaces \(e''\) by
\(\varepsilon_Q'\). The remaining cross term uses
\((I+K_Q)(r_QW)=1/\pi\).
The integrand is now the derivative of
\(\varepsilon_Qr_QW^2\). The density and \(W\) are even and
\(\varepsilon_Q\) is odd, proving the boundary expression.

We next evaluate its boundary factors, rather than differentiating a
bulk asymptotic. Write
\[
 r(u)=\frac1{2\lambda\cosh(u/\eta)}.
\]
If \(g\) solves the full-line equation with the relevant source, its
restriction is corrected to the interval solution by
\[
 g\ \longmapsto\
 g+(I+K_Q)^{-1}K_{\mathrm{out}}g,
\]
where \(K_{\mathrm{out}}g\) integrates over \(\R\setminus[-Q,Q]\).
Apply this map to \(r\) and to \(-2\pi\eta r'\).
On the outer tail near \(Q\) their ratio tends uniformly to \(2\pi\).
The contribution of the opposite tail is negligible relative to \(r(Q)\):
insert a sufficiently small exponential weight in the convergent Neumann
series for \(K_Q\), and use the separation of the two ends.
The same comparison gives \(r_Q(Q)\asymp r(Q)\), hence
\[
 \frac{\varepsilon_Q(Q)}{r_Q(Q)}\longrightarrow2\pi.
\]

After translating the endpoint \(Q\) to zero, the Neumann series for
\(H_Q\) converges term by term to its half-line series and is dominated
by the geometric series with ratio \(\|K\|_1<1\). Thus \(H_Q(Q)\) has
a limit. Symmetry of the resolvent, or differentiation of its series,
gives the exact identity
\begin{equation}\label{eq:dressed-constant-boundary}
 \frac{\dd}{\dd Q}\int_{-Q}^{Q}H_Q(u)\,\dd u=2H_Q(Q)^2.
\end{equation}
The bulk series also gives
\[
 \frac1{2Q}\int_{-Q}^{Q}H_Q(u)\,\dd u
       \longrightarrow\frac1{1+\widehat K(0)}.
\]
Since the right side of \eqref{eq:dressed-constant-boundary} has a limit,
averaging that identity identifies
\[
 H_Q(Q)^2\longrightarrow\frac1{1+\widehat K(0)}
       =\frac{\pi}{2(\pi-\lambda)}.
\]
Finally \eqref{eq:stiffness-boundary} equals
\(2\varepsilon_Q(Q)H_Q(Q)^2/[\pi^2r_Q(Q)]\).
The two boundary limits yield \eqref{eq:stiffness-value}.
\end{proof}

\subsection{An infrared comparison with the plane}

The inverse in \eqref{eq:kohn} is not a local observable.
We therefore regularize it before passing to the plane.
For \(u>0\) put
\[
 g_u(E)=\frac{1-e^{-uE^2}}{E},\qquad g_u(0)=0.
\]
On nonnegative energies \(0\le g_u(E)\le E^{-1}\).
Extend \(g_u\) as an odd function to the real line when using its
Fourier representation.

\begin{lemma}[Infrared comparison]\label{lem:infrared}
With \(\kappa\) as in Section~\ref{sec:angular},
\begin{equation}\label{eq:kappa-lower}
 \kappa\ge\frac1{2\pi(\pi-\lambda)}.
\end{equation}
\end{lemma}
\begin{proof}
Replace the inverse-energy quadratic form in \eqref{eq:kohn} by
the smaller form with \(g_u(E_{\mathrm{exc},L})\), keeping \(u\) fixed.
Translation invariance rewrites its factor \(2/L\) as the sum over
link separations of twice a symmetrized single-source pairing.

An explicit real-time representation is
\[
 g_u(E)=\int_0^\infty
       \operatorname{erfc}\!\left(\frac{t}{2\sqrt u}\right)
       \sin(tE)\,\dd t,\qquad
 \operatorname{erfc}(x)=\frac2{\sqrt\pi}\int_x^\infty e^{-s^2}\,\dd s.
\]
Integration by parts gives this identity. Its sine kernel is bounded at
zero, integrable, and has a Gaussian tail at large real times.
Oddness expresses the symmetrized pairing through expectations of
real-time commutators of the local currents.
The endpoint-Hamiltonian identification in
Lemma~\ref{lem:endpoint-Hamiltonian} makes this the same real-time evolution
as that generated by the finite-range link Hamiltonian.
For link separation \(r\), split the time integral at a fixed small
multiple of \(r\). In the shorter-time part the finite-range
Lieb--Robinson estimate gives exponential decay in \(r\); in the
longer-time part the filter tail is summable without using locality
\cite{LiebRobinson1972}. The estimates are uniform in the circle distance.
For bounded times the local evolutions are norm approximated on a bounded
neighborhood, so local convergence of the ground states gives convergence
of each such commutator.

It follows that the regularized term converges, first as \(L\to\infty\)
and then as \(Q\to\infty\), to a continuous plane momentum density at
zero. Denote this density by \(\mathcal S_u(0)\), using densities
relative to \(\dd p/(2\pi)\). This step uses only fixed \(u\);
no limit of an inverse-energy observable has been taken.
Consequently \eqref{eq:kohn} and Lemma~\ref{lem:stiffness} imply
\begin{equation}\label{eq:regularized-stiffness}
 \frac2{\pi-\lambda}\le D_0-\mathcal S_u(0).
\end{equation}
The current and local spin preserve the magnetization sector, so the
nonnegativity used in this comparison is sector nonnegativity. The
real-time commutator estimate itself needs no ground-state hypothesis.

At nonzero momentum, the continuity equation
\[
 i[E,Z_j]=b_j-b_{j-1}
\]
identifies the current spectral density. On the vacuum,
\(\|Z_0\Omega\|^2(\dd f)=|1-f(i)|^2\mu(\dd f)\).
Canceling the squared shift difference away from phase zero and applying
the sum rule in Lemma~\ref{lem:sum-rule} gives
\begin{equation}\label{eq:filtered-density}
 D_0-\mathcal S_u(p)
 =\text{density of }\ 
       2\,\mathrm{sym}\!\int E(f)e^{-uE(f)^2}\,\mu(\dd f).
\end{equation}
The equality extends to the limit \(p\to0\) because the left side
is continuous. Indeed, the summable commutator bounds show that the
fixed-\(u\) susceptibility measure has an absolutely convergent spatial
Fourier series, hence no atom at phase zero. The regulator annihilates
zero energy, and the \(f=1\) current component was removed in the
construction of \(\mu\). Thus the off-zero division has omitted no
additional contribution to this continuous density.

On a dilation orbit write \(p(s)=\arg f(ie^s)\).
Its absolute derivative is \(E\), and its winding number is \(n_f\)
when the Blaschke degree is \(2n_f\). For small nonzero \(p\) of either
sign, changing variables from \(s\) to \(p\) in
\eqref{eq:filtered-density} leaves \(n_f\) crossings, weighted by
\(e^{-uE^2}\), and the density convention contributes \(4\pi\).
As \(p\to0\), precisely one crossing escapes to the corresponding
orbit end, where \(E\to0\); the other crossings have positive limiting
energy. The sum is bounded by \(n_f\), and
\(\int n_f\,\dd\nu<\infty\) by the sum rule. Dominated convergence,
first at \(p\to0\) and then at \(u\to\infty\), therefore gives
\[
 \lim_{u\to\infty}\bigl(D_0-\mathcal S_u(0)\bigr)=4\pi\kappa.
\]
Combining this with \eqref{eq:regularized-stiffness} proves
\eqref{eq:kappa-lower}.
\end{proof}

\subsection{The exact overlap to be estimated}

We now use the other finite system: real site parameters on an array
of spacing \(d/2\), of length \(B=Ld/2\), at half filling.
Let \(\Omega_{L,t}\) be the unit real-root vector with seam twist
\(e^{itZ}\), continued near zero, and put
\begin{equation}\label{eq:fidelity-cost}
 I_L=\left\|(I-|\Omega_L\rangle\langle\Omega_L|)
           \partial_t\Omega_{L,t}\big|_{t=0}\right\|^2.
\end{equation}
These staggered vectors need no Perron interpretation.

Let \(A\) and \(C\) be the root lists at zero and at \(t\).
Use \([x]=\sinh x\), and let \(M_{AC}\) be the reduced scalar-product
matrix of Section~\ref{sec:finite}, with on-shell roots \(A\) and column
arguments \(C\). In \(M_{CA}\), use the twisted on-shell roots \(C\)
and their twisted Bethe ratio \(e^{-2it}q_C\). At \(t=0\) both coincident
matrices are the appropriate Gaudin matrices. Lemmas~\ref{lem:scalar-product}
and~\ref{lem:hilbert-dual}, followed by the hyperbolic Cauchy identity, give
\begin{equation}\label{eq:exact-overlap}
 |\langle\Omega_{L,0},\Omega_{L,t}\rangle|^2
 =
 \prod_{j,k}
 \frac{[A_j-C_k+i\lambda][C_j-A_k+i\lambda]}
      {[A_j-A_k+i\lambda][C_j-C_k+i\lambda]}
 \frac{\det M_{AC}\det M_{CA}}{\det G_A\det G_C}.
\end{equation}
Every factor depending on only one root set occurs once upstairs and
once downstairs. This includes the finite scalar-product prefactor and
the factor converting an algebraic dual into a Hilbert dual. Thus
\eqref{eq:exact-overlap} is a normalized Hilbert-space identity, not just
an identity of unnormalized Bethe vectors.

For each finite chain, the analytic implicit-function theorem applies
to the counting equations because \(G\) is nonsingular.
The roots remain distinct, the vector remains nonzero and its squared
normalized overlap is nonzero on a sufficiently small neighborhood of
zero. The local logarithm with value zero at \(t=0\) is therefore defined.
Differentiating a unit vector shows
\begin{equation}\label{eq:overlap-quadratic}
 \log|\langle\Omega_{L,0},\Omega_{L,t}\rangle|^2
       =-I_Lt^2+O_L(t^3).
\end{equation}
Only its quadratic coefficient is used. There is no need for a
complex neighborhood whose radius is uniform in \(L\).

\begin{lemma}[Staggered overlap asymptotic]\label{lem:overlap-asymptotic}
In the order \(B\to\infty\), then \(d\downarrow0\),
\begin{equation}\label{eq:overlap-density}
 \frac{I_L}{B}\longrightarrow\frac1{2\lambda(\pi-\lambda)}.
\end{equation}
The estimates below are uniform in any fixed-fraction interior of the array.
\end{lemma}

\subsection{Counting-coordinate accuracy and the finite trace limit}
\label{sec:finite-trace}

This subsection and Section~\ref{sec:overlap-symbol} together prove
Lemma~\ref{lem:overlap-asymptotic}. We first compare the finite quadratic
logarithmic traces, divided by the physical length \(B\), with their bulk
convolution traces; the next subsection evaluates the resulting symbols.
All Taylor coefficients are taken at finite \(L\) before \(B\to\infty\)
at fixed \(d\), and then \(d\downarrow0\), with \(B=Ld/2\).

We first justify the precision needed to differentiate
\eqref{eq:exact-overlap}. The leading root density alone would not suffice.
In a dense interior interval, Lemmas~\ref{lem:root-limits}
and~\ref{lem:gaudin-inverse} give
\[
 \xi'(u)=d^{-1}+O(1),\qquad
 |\xi^{(m)}(u)|\le C_m d^{-1}\quad(m\ge1),
\]
with fixed derivative order. In the coordinate \(z=\xi(u)\), roots have
unit midpoint spacing and derivatives of the inverse coordinate satisfy
\[
 \left|\frac{\dd^m u}{\dd z^m}\right|\le C_m d^m.
\]
For a smooth exponentially localized test \(F\), localized away from the
ends, this gives
\begin{equation}\label{eq:counting-derivative}
 \int\left|
       \frac{\dd^m}{\dd z^m}F(u(z))\right|\,\dd z
       \le C_m d^{m-1}.
\end{equation}
Cutoffs can be put arbitrarily many decay lengths from the test center;
their errors tend to zero as \(B\to\infty\) at fixed \(d\).

Midpoint summation minus integration can be integrated by parts against
bounded periodic primitives of the unit-spaced counting discrepancy.
Using \eqref{eq:counting-derivative} to arbitrarily high fixed order gives
an error \(O_N(d^N)\) for every fixed \(N\), besides the vanishing
end errors. Apply this to the dressed density equation and all its
fixed derivatives. The uniform site array has the same quadrature
accuracy, so convolution inversion improves the density estimate to
\begin{equation}\label{eq:all-order-density}
 \xi'(u)=d^{-1}+O_N(d^N)
\end{equation}
with the same accuracy for its fixed derivatives in the interior.

Differentiating the Gaudin equation for the twist gives the constant
interior displacement
\[
 y_j(t)-y_j(0)
   = d z_*+O_N(d^N),\qquad
 z_*=\frac{t}{2(\pi-\lambda)},
\]
to each fixed Taylor order at zero. For the first derivative this is
\[
 \frac{d}{\pi(1+\widehat K(0))}
       =\frac{d}{2(\pi-\lambda)}.
\]
For higher derivatives, the differentiated site-array source is
arbitrarily small by midpoint quadrature, and difference terms vanish
on a constant displacement. Induction with the Gaudin inverse proves the
assertion. The statement concerns spacings and root differences, not an
arbitrarily chosen absolute mesh origin. Errors over several spacings
can grow polynomially with index separation; the kernels against which
they are summed decrease exponentially.

We next establish the finite matrix estimates needed for the trace
comparison, including the sparse ends. Write
\[
 x_j=y_j(0),\quad \rho(u)=\xi_0'(u),\quad
 \rho_j=\rho(x_j),\quad \ell_j=\rho_j^{-1},\quad
 \mathsf D_0=\diag(\ell_j),\quad \mathcal A=G_0\mathsf D_0=I+K\ell.
\]
For each matrix \(M(t)\) among \(M_{AC}(t),M_{CA}(t),G_C(t),G_A\),
use \(N(t)=M(t)\mathsf D_0\), keeping \(\mathsf D_0\) fixed at zero twist.
Then \(N(0)=\mathcal A\), and \(\log\det N-\log\det M\) is constant.
Equivalently, \(N(0)^{-1}N^{(r)}(0)
=\mathsf D_0^{-1}G_0^{-1}M^{(r)}(0)\mathsf D_0\), so the finite trace identity is
\begin{equation}\label{eq:second-log-trace}
 \left.\frac{\dd^2}{\dd t^2}\log\det M(t)\right|_{t=0}
 =\Tr\bigl(\mathcal A^{-1}N''(0)
       -\mathcal A^{-1}N'(0)\mathcal A^{-1}N'(0)\bigr).
\end{equation}
In particular, no derivative of \(\ell_j\) occurs, and an edge weight
has not been replaced by \(d\).

Here and below constants are uniform for sufficiently small \(d\) and
\(B\ge B_0(d)\), as in Section~\ref{sec:thermodynamic}.
Let \(p(u)=\sum_\alpha r(u-b_\alpha)\).
The sawtooth estimate \eqref{eq:counting-sawtooth} gives
\(\rho=p+O(1)\), with bounded integrated error.
The \(d/2\)-spaced array and exponential decay of \(r\) give
\(p(u)\le C/d\) on the line. Between the outer bare sites,
at least \(c/d\) sites lie within distance one of \(u\), once \(B\ge2\);
hence \(\rho(u)\ge c/d\) there for small \(d\).
At an exterior root, \eqref{eq:edge-index-density} instead gives
\(\rho_j\ge c\). Consecutive counting levels differ by one, so
\begin{equation}\label{eq:overlap-finite-array-bounds}
 \begin{gathered}
 0<\rho(u)\le C/d,\qquad cd\le\ell_j\le C,
 \qquad x_{j+1}-x_j\ge cd,\\
 \sup_u\sum_j\ell_j e^{-\gamma|u-x_j|}\le C_\gamma
 \quad(\gamma>0).
 \end{gathered}
\end{equation}
The last estimate is the local quadrature-mass bound of
Lemma~\ref{lem:root-limits}, summed over unit intervals.
Since \(K<0\), \(\rho\) is a sum of positive kernels \(s\) and
\(k=-K\); their fixed derivatives are bounded by constant multiples
of themselves. Thus \(|\rho^{(m)}(u)|\le C_m\rho(u)\).

Put \(v_j=y'_j(0)\), \(w_j=y''_j(0)\) on the twisted branch, and
\(\sigma(u)=\sum_\alpha s(u-b_\alpha)\).
Twice differentiating the finite counting equations gives
\[
 G_0v=\pi^{-1}\bbone,\qquad G_0w=-H,\qquad
 H_i=\sigma'(x_i)v_i^2
       -\sum_jK'(x_i-x_j)(v_i-v_j)^2.
\]
Since \(G_0^{-1}=\mathsf D_0\mathcal A^{-1}\),
Lemma~\ref{lem:gaudin-inverse} first gives \(|v_j|\le C\ell_j\).
Using \(\sigma(x_i)+\sum_j k(x_i-x_j)=\rho_i\), we then obtain
\[
 |H_i|\le C\left(\ell_i^2\rho_i+
                         \sum_j k(x_i-x_j)\ell_j^2\right)\le C.
\]
Here \(\rho_i\ell_i=1\), \(\ell_i\le C\), and the last bound in
\eqref{eq:overlap-finite-array-bounds} were used. Consequently
\begin{equation}\label{eq:overlap-root-jets}
 |v_j|+|w_j|\le C\ell_j.
\end{equation}
For the moving counting density \(\rho_t=\xi_{C(t)}'\), derivatives
at fixed \(u\) are
\[
 \dot\rho(u)=\sum_jK'(u-x_j)v_j,\qquad
 \ddot\rho(u)=\sum_j\{K'(u-x_j)w_j-K''(u-x_j)v_j^2\}.
\]
These functions and their required fixed spatial derivatives are
\(O(1)\), by \eqref{eq:overlap-finite-array-bounds} and
\eqref{eq:overlap-root-jets}. This includes all terminal indices.

We remove the coincident-column singularity before differentiation.
For either overlap orientation, let \(Y(t)\) be its on-shell list,
\(U(t)\) its column list, and
\(Q(t,u)=e^{-2it_Y}q_{Y(t)}(u)\), where \(t_Y=0\) for \(M_{AC}\)
and \(t_Y=t\) for \(M_{CA}\). Thus \(Q(t,Y_j(t))=-1\).
Set
\[
 h_j=U_j-Y_j,\qquad
 L_j(t)=\int_0^1\rho_{Y(t)}(Y_j+\theta h_j)\dd\theta,
 \qquad \mathcal E(z)=\frac{e^z-1}{z},\quad \mathcal E(0)=1.
\]
The identity \(q'/q=2\pi i\rho_Y\) gives
\(Q(t,U_j)=-e^{2\pi i h_jL_j}\). With
\[
 H_\lambda(u)=\frac{\coth u-\coth(u+i\lambda)}{-2\pi i},
 \qquad \beta_j(t)=1-e^{2\pi i h_jL_j},
\]
the finite scalar-product matrix is exactly
\(M_{ij}=K(u_{ij})+\beta_jH_\lambda(u_{ij})\),
where \(u_{ij}=U_j-Y_i\). Its constant coth tails have canceled.
On the diagonal this becomes the analytic identity
\begin{equation}\label{eq:overlap-coalescent-column}
 M_{jj}=K(h_j)+L_j\mathcal E(2\pi i h_jL_j)
       [h_j\coth h_j-h_j\coth(h_j+i\lambda)].
\end{equation}
At zero twist, \(h_j=0\), \(L_j=\rho_j\), and
\(|h'_j|+|h''_j|\le C\ell_j\).
Differentiating the integral defining \(L_j\), using the preceding
density bounds, gives \(|L'_j|+|L''_j|\le C\): every spatial
density derivative is multiplied by a velocity or acceleration
\(O(\ell_j)\), and the fixed-position time derivatives are \(O(1)\).
For \(z_j=2\pi i h_jL_j\), this implies
\[
 z'_j(0)=2\pi i\rho_jh'_j(0)=O(1),\qquad
 z''_j(0)=2\pi i(\rho_jh''_j+2h'_jL'_j)\big|_{t=0}=O(1).
\]
The bracket in \eqref{eq:overlap-coalescent-column} is analytic
with value one at zero; it and \(\mathcal E(z_j)\) have bounded first two
Taylor derivatives. The product rule and \(\ell_j\rho_j=1\) therefore
give \(\ell_j(|M'_{jj}(0)|+|M''_{jj}(0)|)\le C\).
No reciprocal displacement or reciprocal root velocity is introduced.

For \(i\ne j\), ordinary differentiation at \(u=x_j-x_i\) gives
\[
 \begin{split}
 M'_{ij}&=K'(u)u'_{ij}+\beta'_jH_\lambda(u),\\
 M''_{ij}&=K''(u)(u'_{ij})^2+K'(u)u''_{ij}
       +\beta''_jH_\lambda(u)+2\beta'_jH'_\lambda(u)u'_{ij}.
 \end{split}
\]
Here \(\beta_j(0)=0\), \(|\beta'_j(0)|+|\beta''_j(0)|\le C\),
and \(u'_{ij},u''_{ij}\) are bounded by \eqref{eq:overlap-root-jets}.
For some fixed \(\gamma>0\),
\[
 |K^{(r)}(u)|\le C_re^{-\gamma|u|},\quad
 |H_\lambda(u)|\le C(1+|u|^{-1})e^{-\gamma|u|},\quad
 |H'_\lambda(u)|\le C(1+|u|^{-2})e^{-\gamma|u|}.
\]
Choose small \(\alpha>0\) and write
\(\|T\|_{\alpha,x}=\sup_i\sum_j
 e^{\alpha|x_i-x_j|}|T_{ij}|\).
The minimum gap bounds the two pole factors by \(C/d\) and \(C/d^2\).
Multiplication by the column weight \(\ell_j\), local quadrature mass,
and the diagonal estimate prove
\begin{equation}\label{eq:overlap-taylor-row-bounds}
 \|M'(0)\mathsf D_0\|_{\alpha,x}\le C/d,\qquad
 \|M''(0)\mathsf D_0\|_{\alpha,x}\le C/d^2
 \quad(M=M_{AC},M_{CA}).
\end{equation}
For the Gaudin denominator, put \(r_i(t)=\rho_t(y_i(t))\).
The formulas
\[
 r'_i=\dot\rho_i+\rho'_iv_i,\qquad
 r''_i=\ddot\rho_i+2\dot\rho'_iv_i+\rho''_iv_i^2+\rho'_iw_i
\]
give bounded first two derivatives. Differentiating
\(G_{ij}(t)=r_i(t)\delta_{ij}+K(y_i(t)-y_j(t))\) and summing with
the same column weights thus gives
\(\|G'_C(0)\mathsf D_0\|_{\alpha,x}+\|G''_C(0)\mathsf D_0\|_{\alpha,x}\le C\).
Lemma~\ref{lem:gaudin-inverse}, with both signs of its exponential
weight, gives \(\|\mathcal A^{-1}\|_{\alpha,x}\le C\).
Weighted absolute row norms are submultiplicative by the triangle
inequality. Hence the matrix inside the trace in
\eqref{eq:second-log-trace}, and each of its diagonal entries,
are bounded by \(C/d^2\), uniformly in \(B\).

The discarded count includes all exterior roots. Indeed, with
\(T(u)=\int_u^\infty\rho(v)\dd v\), the half-integer tail levels
and integrated sawtooth estimate give
\[
 \#\{j:x_j\ge x\}\le T(x)+1
 \le\sum_\alpha\bar r(x-b_\alpha)+C,
 \qquad \bar r(v)=\int_v^\infty r(u)\dd u.
\]
For \(x=B/2-R\), at most \(C(R/d+1)\) bare sites lie to its right,
each contributing at most \(1/2\). The other sites contribute at most
\(C\sum_{m\ge0}e^{-md/(2\eta)}\le C/d\).
Reflection therefore proves
\begin{equation}\label{eq:overlap-end-count}
 \#\{j:|x_j|\ge B/2-R\}\le C(R+1)/d.
\end{equation}
Deleting these diagonal entries costs at most
\(C(R+1)/(d^3B)\) after division by \(B\).
This tends to zero at fixed \(d\) if \(R=R(B)\to\infty\) and
\(R(B)/B\to0\); a fixed number of nested strips has the same property.

For the inverse comparison we use one common weighted row algebra,
including at the sparse ends:
\begin{equation}\label{eq:overlap-common-row-norm}
 \|T\|_{a,d}=\sup_i\sum_j e^{ad|i-j|}|T_{ij}|,
 \qquad T_n=I+(dK(d(i-j)))_{1\le i,j\le n}.
\end{equation}
Since \(|x_i-x_j|\ge cd|i-j|\), the actual-matrix bounds above hold
in this norm for sufficiently small fixed \(a>0\).
Choose \(a\) also so that \(\int e^{a|u|}|K(u)|\dd u<1\).
For small \(d\), \(\|T_n-I\|_{a,d}\le q<1\); the finite Neumann
series gives \(\|T_n^{-1}\|_{a,d}\le(1-q)^{-1}\), uniformly in \(n\).
Thus the exact identity
\[
 \mathcal A^{-1}-T_n^{-1}
       =\mathcal A^{-1}(T_n-\mathcal A)T_n^{-1}
\]
holds in the same weighted algebra for both inverses.
Two nested physical cutoffs separated by \(R\) inside the bare array
have at least \(cR/d-O(1)\) intervening root indices, since
\(\rho\ge c/d\) there. Any index path from a retained inner diagonal
to the outer strip therefore gains a factor \(Ce^{-c'R}\).

In the interior, \eqref{eq:all-order-density} and the twist-jet
estimates give \(\ell_j=d+O_N(d^N)\) and arbitrarily accurate
comparisons of root differences with \(d(j-i)\).
Apply these to the normalized Taylor matrices, using
\eqref{eq:overlap-coalescent-column} on the diagonal.
Fixed derivatives at distinct-root poles cost only powers of \(1/d\);
polynomial growth with index separation is absorbed by exponential
decay. Choosing a higher initial accuracy order therefore makes the
interior row errors of the Taylor matrices and of
\(\mathcal A-T_n\) \(O_N(d^N)\) for every fixed \(N\), besides errors
vanishing at fixed \(d\) as
the strips recede. Their global row bounds have only polynomial
costs in \(1/d\). The inverse identity and telescoping the fixed
products in \eqref{eq:second-log-trace}, with nested strips, give a
retained diagonal error at most
\[
 P(d^{-1})\bigl[O_N(d^N)+o_{B\to\infty,d}(1)+C_de^{-c'R(B)}\bigr]
\]
for a fixed polynomial \(P\). No intermediate row sum costs a factor
growing with \(B\). Since exactly \(n=B/d\) indices occur, the trace
per physical length costs only one additional factor \(1/d\).
Send \(B\to\infty\) first, and choose \(N>\deg P+1\) before
letting \(d\downarrow0\).

Finally, the weighted Neumann series compares \(T_n^{-1}\) with the
full-lattice convolution inverse: every omitted path from an interior
row crosses the matrix boundary, and so has the same exponentially
small bound. The same argument applies to the finite products.
The corresponding convolution trace integrand has
diagonal coefficient \(d(2\pi)^{-1}\int_{-\pi/d}^{\pi/d}
\widehat F(p)\dd p\). Multiplication by \(n/B=1/d\) gives precisely
the Fourier measure \(\dd p/(2\pi)\) per unit physical length.
No infinite-volume trace-class assertion is involved.

\subsection{Evaluation of the symbol}
\label{sec:overlap-symbol}
\begin{proof}[Proof of Lemma~\ref{lem:overlap-asymptotic}]
In the product part of \eqref{eq:exact-overlap}, the quadratic coefficient
is a sum of
\((\log[x+i\lambda])''\) times the two first displacements. Its integral
on the real line is \(2\), from the two limits of \(\coth(x+i\lambda)\).
The constant accelerations cancel between cross and within-set terms.
With root density \(1/d\) and displacement \(dz_*\), the coefficient
per unit length is \(2z_*^2\), with a vanishing two-limit error.

For the determinants, the bulk sequence of \(dM_{AC}\), at row-column
difference \(y=d(k+z_*)\), is \(d\) times
\begin{equation}\label{eq:overlap-toeplitz}
 \frac{(1-e^{2\pi i z_*})\coth y-\coth(y-i\lambda)
                         +e^{2\pi i z_*}\coth(y+i\lambda)}
      {-2\pi i}.
\end{equation}
For \(M_{CA}\) replace \(z_*\) by \(-z_*\); for the two Gaudin
denominators use \(z_*=0\). The coincident pole is continued by its
removable limit. The preceding trace comparison reduces the computation
to the quadratic coefficient of these bulk symbols.

Use Fourier exponent \(-ipd(k+z_*)\).
The simple pole contributes the shifted-sampling identity
\[
 \sum_{k\in\Z}\frac{e^{-ipd(k+z_*)}}{k+z_*}
    =\pi\bigl(\cot(\pi z_*)-i\sgn p\bigr),
 \qquad 0<|p|<\pi/d,
\]
with symmetric summation. Subtract \(1/y\) from the singular cotangent
before treating the remaining part. Its derivative is integrable and
analytic in every strip
\(|\Im y|<a<\min(\lambda,\pi-\lambda)\).
Contour shift therefore makes every nonzero alias exponentially small
in \(1/d\), including its fixed Taylor coefficients. This remains
uniform at the band ends since a nonzero alias has absolute frequency
at least \(\pi/d\); the value at \(p=0\) is irrelevant to the integral.

Write \(a_0=e^{-\lambda p}\) and \(b_0=e^{-(\pi-\lambda)p}\).
The continuous transform together with the sampled pole gives
\[
 \frac{(1-e^{2\pi i z_*})(1+a_0b_0)/2-b_0
                              +e^{2\pi i z_*}a_0}
      {1-a_0b_0}
       +\frac{1+e^{2\pi i z_*}}2
 =\frac{(1-b_0)(1+e^{2\pi i z_*}a_0)}
        {1-a_0b_0}.
 \]
Dividing by the value at zero twist, the relative symbol is
\[
 \frac{1+e^{2\pi i z_*}a_0}{1+a_0}.
\]
The offset Fourier phases cancel between the two orientations.
Their combined quadratic logarithmic coefficient is
\[
 -4\pi^2 z_*^2\frac{a_0}{(1+a_0)^2}.
\]
Since
\[
 \int_{\R}\frac{e^{-\lambda p}}{(1+e^{-\lambda p})^2}\,
       \frac{\dd p}{2\pi}=\frac1{2\pi\lambda},
\]
the determinant contribution per length is \(-2\pi z_*^2/\lambda\).
For each finite \(L\), let \([t^2]\) denote the Taylor coefficient at
\(t=0\) of the local logarithm fixed above. Adding the product contribution
proves the explicit coefficient limit
\[
 \lim_{d\downarrow0}\limsup_{\substack{B\to\infty\\ B=Ld/2}}
 \left|\frac1B[t^2]\log|\langle\Omega_{L,0},\Omega_{L,t}\rangle|^2
       +\frac1{2\lambda(\pi-\lambda)}\right|=0.
\]
The Taylor coefficient is taken before either array limit; no common
analytic neighborhood or uniform finite-\(t\) expansion is asserted.
By \eqref{eq:overlap-quadratic} that coefficient is \(-I_L\), which
proves the lemma.
\end{proof}

\subsection{Packing angular windows}

We finally connect the auxiliary staggered susceptibility to the plane.
Let \(\zeta_L\) be the spectral measure of the orthogonal derivative vector
in \eqref{eq:fidelity-cost}, for the simultaneous self-adjoint rows
\(P_L(e^s)\), divided by their designated untwisted vacuum eigenvalues.
Its total mass is \(I_L\). Denote its spectral coordinates by \(f_s\).
The differentiated eigen-equation expresses multiplication of that vector
by \(1-f_s\) as the marked row vector, with its vacuum component removed.
Consequently convergence of all finite Gram words gives
\begin{equation}\label{eq:weighted-measure-convergence}
 (1-f_s)^2\,\zeta_L
       \longrightarrow (1-f(e^s))^2\,\mu
\end{equation}
against bounded continuous functions of finitely many coordinates in a
fixed window. The convergence is uniform when the window center stays
in a fixed-fraction array interior.

We need a finite-array bound for two dips before summing over windows.
Write \(\psi_L\) for the orthogonal derivative vector and
\(P_s=P_L(e^s)\), \(J_s=J_L(e^s)\). The differentiated eigen-equation is
\[
 (1-P_{s'})\psi_L=iJ_{s'}\Omega_L-a_{s'}\Omega_L
\]
for a scalar \(a_{s'}\). Since \(P_s\Omega_L=\Omega_L\),
\[
 (1-P_s)(1-P_{s'})\psi_L=-i[P_s,J_{s'}]\Omega_L.
\]
The spectral theorem therefore gives the exact identity
\[
 \int(1-f_s)^2(1-f_{s'})^2\,\dd\zeta_L
       =\|[P_s,J_{s'}]\Omega_L\|^2.
\]
Lemma~\ref{lem:row-normalization} bounds every singly marked row,
uniformly at all centers in a fixed-fraction array interior, once the
array is sufficiently long and its spacing sufficiently small.
In finite RTT the normalized intertwiner at separation \(s-s'\)
is an invertible diagonal matrix plus
\(O(e^{-\eta|s-s'|})\); the diagonal matrix commutes with the mark
\(I\otimes Z\). Expanding the conjugation error in auxiliary matrix
units expresses the commutator as a fixed finite sum of two-row products.
The single-row bounds consequently imply
\begin{equation}\label{eq:finite-two-dip}
 \int(1-f_s)^2(1-f_{s'})^2\,\dd\zeta_L
       \le C e^{-2\eta|s-s'|}.
\end{equation}
The constant is independent of the array length, small spacing and both
bulk centers. In particular there is no additive convergence error whose
sum over a growing number of pairs could accumulate.

\begin{lemma}[Packing angular windows]\label{lem:detector-packing}
The plane orbit mass satisfies
\begin{equation}\label{eq:kappa-upper}
 \kappa\le\frac{\eta}{2\lambda(\pi-\lambda)}.
\end{equation}
\end{lemma}
\begin{proof}
Choose finitely many real samples in a window, and let
\(Q(f)=\sum_j(1-f_{s_j})^2\).
Any bounded continuous detector \(F\) supported away from the all-ones
point can be written
\[
 F=\sum_j \frac{F}{Q}(1-f_{s_j})^2,
\]
with each quotient extended continuously by zero near that point.
Thus \eqref{eq:weighted-measure-convergence} gives convergence for such
detectors without assuming convergence of the full unweighted mass.
Similarly, products of two detectors are bounded by finite sums of
two-dip factors. The constants depend on the fixed window and mesh of
samples, not on the array center in the permitted bulk.

The cone condition and the dip estimate for inner functions provide
a fixed dip width and depth around each zero scale, independent of the
degree. Choose a fixed fine sampling mesh and a continuous detector
\(0\le F_T\le1\) in a window of length \(T\), equal to one when a
prescribed interior dip is detected and zero near all ones.
Each dilation orbit contributes an interval of centers of length at
least \(T-C\). Hence
\[
 \int F_T\,\dd\mu\ge(T-C)\kappa.
\]
Take the detector to be a continuous cutoff of
\(\max_j|1-f_{s_j}|\) at a fixed positive dip threshold \(a\).
Then \(F_T\le C_a\sum_j(1-f_{s_j})^2\), with \(C_a\) independent of
\(T\). Equation~\eqref{eq:finite-two-dip} bounds the product for two
windows separated by a gap \(g\) by \(CT^2e^{-2\eta g}\).
For \(N\) regularly spaced windows, their pair integrals sum to at most
\[
 \frac{C N T^2e^{-2\eta g}}{1-e^{-2\eta(T+g)}}.
\]

Pack windows of length \(T\) and gaps \(g\) into a bulk interval of
logarithmic length \((1-\gamma)B/\eta\).
Their number \(N_B\) satisfies
\[
 \frac{N_B}{B}\longrightarrow
          \frac{1-\gamma}{\eta(T+g)}.
\]
For numbers \(a_j\in[0,1]\),
\(\sum_j a_j\le1+\sum_{j<k}a_ja_k\).
Integrating this inequality against \(\zeta_L\), using bulk-uniform
detector convergence and the finite two-dip estimate, gives
\[
 \frac{1-\gamma}{\eta(T+g)}(T-C)\kappa
 \le \lim_{d\downarrow0}\lim_{B\to\infty}\frac{I_L}{B}
       +\frac{CT^2 e^{-2\eta g}}{T+g}.
\]
For fixed \(T,g,\gamma\), all detectors use only a fixed finite number
of samples, so this application does not ask for convergence of a
growing word. The uniform estimates allow the inequalities to be summed
over their translated windows. The single-window convergence error,
after division by \(B\), is \((N_B/B)\varepsilon_{d,B,T}\), where
\(\lim_{d\downarrow0}\limsup_{B\to\infty}\varepsilon_{d,B,T}=0\).
Shortening the bulk interval at its ends by \(O(T+g)\) keeps every
sample in the permitted interior and does not change the limiting count.

Use Lemma~\ref{lem:overlap-asymptotic}, choose
\(g=A\log T\) with \(A\) sufficiently large, let \(T\to\infty\), and
then let \(\gamma\downarrow0\). This proves
\eqref{eq:kappa-upper}.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:normalization}]
Since \(\eta=\lambda/\pi\), the upper bound
\eqref{eq:kappa-upper} equals the lower bound
\eqref{eq:kappa-lower}. Thus \eqref{eq:kappa-value} holds.
The covariance conversion follows directly from \eqref{eq:green}.
\end{proof}
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//05-normalization.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//06-chiral-gaussianity.tex ====
\section{Chiral currents and Gaussian correlations}\label{sec:chiral}

We now turn the angular spectrum into limits of height derivatives. The
argument has two steps. Separating two angular scales gives two independent
chiral current forms. A second separation, with one scale tending to zero,
determines the current collision coefficient. Holomorphy in both axial
quantizations then determines every separated current correlation from this
coefficient.

We work at a fixed parameter in the new range and use the diagonal-word
Hilbert space $\mathcal D$ of Section~\ref{sec:angular}. Its vacuum is
$\Omega$. The commuting rows $P(w)$ have spectral functions $f$, the marked
row is $J(w)$, and
\[
 H_0=-\log|P(1)|,\qquad S=P(i),\qquad
 N(w)=\frac{J(w)P(w)+P(w)J(w)}2.
\]
On this space, Lemma~\ref{lem:inner-spectra} gives finite Blaschke products
with $f(0)=f(\infty)=1$ and real symmetry, including the constant product
$f=1$. Their zeros $\zeta$ lie in the fixed cone
$\Re\zeta\ge c_0|\zeta|$. The current vacuum identity and its measure are
\begin{equation}\label{eq:chiral-vacuum-data}
 J(w)\Omega=(1-f(w))\mathsf h,\qquad
 \mu(df)=\|\mathsf h(f)\|^2\sigma(df),\qquad
 \mu=ds\,\nu,\qquad \nu(1)=\kappa<\infty.
\end{equation}
Here $\sigma$ is a scalar spectral measure and $s$ is logarithmic
dilation along a nonconstant spectral orbit.
The symbol $\mathsf h$ is a spectral field; it need not be a vector of finite
norm. Every use of it below includes an integrable multiplier.

\subsection{The two chiral forms}

Define nonnegative spectral multiplication operators on $\mathcal D$ by
\begin{equation}\label{eq:chiral-energies}
 a_+(f)=-f'(0)=\sum_{\zeta}2\Re(\zeta^{-1}),\qquad
 a_-(f)=-\left.\frac{d}{dv}f(v^{-1})\right|_{v=0}
                         =\sum_{\zeta}2\Re\zeta.
\end{equation}
Both values are zero on $f=1$, and both are strictly positive on every
nonconstant fiber. Write $E_\pm(R)=\bbone_{a_\pm\le R}$ and
$\mathcal D_\pm^{\mathrm{fin}}=\bigcup_{R>0}E_\pm(R)\mathcal D$.
These are dense linear subspaces. A current form on this domain means a
sesquilinear form whose restriction to each bounded spectral band is a
bounded operator; no unbounded-operator closure is implicit in this usage.

\begin{lemma}\label{lem:chiral-forms}
There are consistent skew-adjoint forms $j_\pm$ on
$\mathcal D_\pm^{\mathrm{fin}}$ such that, for every $R>0$,
\begin{equation}\label{eq:chiral-form-bound}
 \|E_\pm(R)j_\pm E_\pm(R)\|\le CR,
 \qquad E_\pm(R)j_\pm\Omega=E_\pm(R)a_\pm\mathsf h.
\end{equation}
Their vacuum means vanish. For $s>0$, the damped forms extend to bounded
operators and satisfy
\begin{equation}\label{eq:chiral-damped-form}
                     \|e^{-sa_\pm}j_\pm e^{-sa_\pm}\|\le C/s.
\end{equation}
For every nonnegative measurable function $g$ on $(0,\infty)$,
\begin{equation}\label{eq:chiral-Haar-integral}
             \int g(a_\pm(f))\dd\mu(f)
                     =\kappa\int_0^\infty g(t)\frac{dt}{t}.
\end{equation}
\end{lemma}

\begin{proof}
We give the construction at zero. The construction at infinity uses the
coordinate $v=1/w$ and the same estimates. The reciprocal-radius budget
in \eqref{eq:chiral-energies}, together with the cone condition, gives
uniformly on the band $E_+(R)$
\begin{equation}\label{eq:chiral-disc-inverse}
 P(w)=I+O(R|w|),\qquad \|P(w)^{-1}\|\le C,
                  \quad \Re w\ge0,\quad |w|<c/R.
\end{equation}
For example, normalize each Blaschke factor to one at zero. Its difference
from one is at most $C|w|\Re(\zeta^{-1})$ in this disc, and the sum of
these bounds is at most $CR|w|$. Taking $c$ small bounds the product and
its inverse. All inverses in this proof are restricted to the stated band.

On $E_+(R)\mathcal D$ write $P_R(w)$ for the restriction of $P(w)$.
Compress $J$ by $E_+(R)$ and continue it to the other half-disc by the
reversal identity
\begin{equation}\label{eq:chiral-gluing}
 J_R(-w)=-P_R(w)^{-1}J_R(w)P_R(w)^{-1},
       \qquad J_R=E_+(R)JE_+(R).
\end{equation}
For this boundary gluing, first dilate the uncompressed endpoint reversal
identity and then compress by $E_+(R)$, which commutes with every $P(w)$.
Thus the argument of Lemma~\ref{lem:low-band} applies without moving a
cutoff through a boost. The resulting
function is bounded and holomorphic on the punctured disc. Its center is
removable: the negative Laurent coefficients vanish by the Cauchy bounds.
At the center \eqref{eq:chiral-gluing} says $J_R(0)=-J_R(0)$, so that
constant coefficient is zero. Define $E_+(R)j_+E_+(R)$ to be the linear
Taylor coefficient. The Cauchy estimate on a circle of radius proportional
to $R^{-1}$ proves the bound $CR$.

Compression from a larger band to a smaller band commutes with taking
this coefficient. This proves consistency and defines the form on the
union of bands. For positive real $w$, $J(w)^*=-J(w)$, so the coefficient
is skew-adjoint as a form. The same argument gives $j_-$.

Dilation multiplies $a_+$ by the dilation factor and $a_-$ by its
reciprocal. On each nonconstant orbit either substitution changes Haar
measure $ds$ to $dt/t$. Equation~\eqref{eq:chiral-vacuum-data} therefore
gives \eqref{eq:chiral-Haar-integral}. In particular,
$\int_{a_\pm\le R}a_\pm^2\dd\mu=\kappa R^2/2$.
Dividing the vacuum identity by $w$, or by $v$ at infinity, and taking
the band Taylor coefficient now gives the vacuum assertion in
\eqref{eq:chiral-form-bound}. Its inner product with $\Omega$ is zero:
$\Omega$ lies in the zero-energy band, and its compressed current mean
is bounded by $CR$ for every $R>0$.

For the damping assertion, divide the spectrum of $a_\pm$ into
$[0,s^{-1}]$ and $(2^{q-1}s^{-1},2^qs^{-1}]$, $q\ge1$.
The form bound controls a block between levels $q,r$ by
$Cs^{-1}\max(2^q,2^r)$. Damping on both sides supplies
$e^{-2^{q-1}}e^{-2^{r-1}}$ when the respective index is positive.
The sum of these block norms is at most $C/s$, which proves both
existence of the bounded extension and \eqref{eq:chiral-damped-form}.
The same argument shows that tails of these damped block sums tend to
zero in operator norm.
\end{proof}

\subsection{Separated scales and representative norms}

Let $\mathcal C$ be the span of strict diagonal words applied to $\Omega$.
For two positive scales $r_1,r_2$, represent an elementary tensor
$u_1\otimes u_2$ of such words in $\mathcal D$ by replacing every argument
in word $i$ by $r_iw$, multiplying the two resulting words in that order,
and applying the product to $\Omega$. Extend this prescription to finite
linear combinations and denote a chosen representative of $u$ by
$u_{r_1,r_2}$. Different word expressions for the same tensor have
difference tending to zero in norm as $|\log(r_1/r_2)|\to\infty$.
Indeed, Proposition~\ref{prop:word-clustering}, applied after adjunction,
gives
\begin{equation}\label{eq:chiral-Gram-limit}
       \langle u_{r_1,r_2},v_{r_1,r_2}\rangle
                          \longrightarrow\langle u,v\rangle
                  \quad(u,v\in\mathcal C\otimes\mathcal C).
\end{equation}
The asymptotic diagonal RTT exchange permits the two separated word
groups to be interchanged with an error tending to zero in norm.
Consequently a fixed bounded diagonal marked row at $r_iw$ acts in the
limit on factor $i$ alone, for every strict $w$.

We use convergence of representatives, not a bounded map from the entire
tensor completion. Precisely, a family $v_\delta\in\mathcal D$ converges
strongly to $v\in\mathcal D\otimes\mathcal D$ if, for every $\epsilon>0$,
there is a finite word tensor $u$ with
\[
 \|u-v\|<\epsilon,\qquad
 \limsup_{\delta\downarrow0}\|v_\delta-u_\delta\|<\epsilon.
\]
The scales defining $u_\delta$ will always be specified. Equation
\eqref{eq:chiral-Gram-limit} makes this definition independent of the
chosen word expressions.

\begin{lemma}\label{lem:chiral-representatives}
Suppose $T_\delta$ are uniformly bounded operators and, on every finite
word tensor, $T_\delta u_\delta$ converges strongly to $Tu$. Then $T$
extends to a bounded operator on the tensor completion, and
$T_\delta v_\delta\to Tv$ for every strongly converging family
$v_\delta\to v$. Such operator limits may be composed, without a
commutation assumption.

A family $v_\delta$ converges strongly to $v$ if its pairings with all
core representatives converge to the pairings with $v$ and its norms
converge to $\|v\|$.
\end{lemma}

\begin{proof}
If $\|T_\delta\|\le M$, then \eqref{eq:chiral-Gram-limit} implies
$\|Tu\|\le M\|u\|$ on the core. Approximate $v_\delta$ by $u_\delta$
as in the definition. The norm of
$T_\delta(v_\delta-u_\delta)$ is at most $M$ times that approximation
error, while $T_\delta u_\delta$ has the asserted limit. Taking the
approximation error to zero proves the first assertion and then composition.
For the last assertion, use
\[
 \|v_\delta-u_\delta\|^2
 =\|v_\delta\|^2+\|u_\delta\|^2
       -2\Re\langle v_\delta,u_\delta\rangle
                         \longrightarrow\|v-u\|^2
\]
and approximate $v$ by a core vector $u$.
\end{proof}

\begin{proposition}[Two-scale spectral limit]\label{prop:two-scale}
Take $r_1=\delta^{-1}$ and $r_2=\delta$, with $\delta\downarrow0$ through
arbitrary positive real values. On core representatives, the joint spectral
functions $(f(\delta^{-1}\,\cdot),f(\delta\,\cdot))$ converge in law in
the compact-open topology of the right half-plane to the two tensor-factor
functions $(F_1,F_2)$. Jointly with these functions,
\begin{equation}\label{eq:chiral-scaled-spectrum}
\begin{split}
 \delta^{-1}H_0&\longrightarrow A:=a_+\otimes I+I\otimes a_-,\\
 \delta^{-1}\arg S&\longrightarrow -a_+\otimes I+I\otimes a_-,
       \qquad \sgn P(1)\longrightarrow I
\end{split}
\end{equation}
in spectral law. The argument is the principal argument near zero.
There is an unbounded set of $R>0$ for which
\begin{equation}\label{eq:chiral-band-limit}
 Q_\delta(R):=\bbone_{H_0/\delta\le R}
                       \longrightarrow Q(R):=\bbone_{A\le R}
\end{equation}
strongly on representatives. Bounded spectral multipliers obtained jointly
from these limits have the same strong interpretation whenever their
discontinuity sets have zero limiting state mass.

If $m_\delta\ge0$ and $l_\delta$ are integers with
$\delta m_\delta\to x\ge0$ and $\delta l_\delta\to y$, then
\begin{equation}\label{eq:chiral-semigroup-limit}
 P(1)^{m_\delta}S^{l_\delta}\longrightarrow
       e^{-(x+iy)a_+}\otimes e^{-(x-iy)a_-}
\end{equation}
strongly on representatives.
\end{proposition}

\begin{proof}
The strong row actions preceding Lemma~\ref{lem:chiral-representatives}
give convergence of every polynomial in finitely many samples of the two
spectral functions and their complex conjugates. The functions are bounded
by one on the right half-plane; normal-family compactness therefore gives
the joint compact-open convergence. The limiting functions are the original
finite Blaschke spectral functions on the two factors, and are not
identically zero. For a vector $v$, its state spectral measure is
$\sigma_v(B)=\|\bbone_Bv\|^2$ on measurable sets of spectral functions.
These measures have total masses
$\|u_\delta\|^2\to\|u\|^2$; normalization to probability is harmless
when $u\ne0$.

The rescaled quantities in \eqref{eq:chiral-scaled-spectrum} require
control of \emph{every} zero, including zeros between the two scales.
For a fixed core representative, the exterior RTT estimate from
Lemma~\ref{lem:inner-spectra} gives, outside fixed word offsets,
\begin{equation}\label{eq:chiral-exterior-dip}
 \|(I-P(e^s))u_{r_1,r_2}\|^2
      \le C_u\sum_{i=1}^2e^{-2\eta|s-\log r_i|}.
\end{equation}
Every zero $\zeta$ forces a fixed-width interval about $\log|\zeta|$
on which $|1-f(e^s)|$ is bounded below by a positive constant. This
follows from the cone condition and the modulus loss of its single
Blaschke factor; all other factors have modulus at most one.
Integrating \eqref{eq:chiral-exterior-dip} outside slightly smaller
windows thus yields
\begin{equation}\label{eq:chiral-all-zero-event}
\begin{split}
 &\sigma_{u_{r_1,r_2}}
 \left\{\text{some zero lies outside }
     \bigcup_{i=1}^2\{\zeta:|\log(|\zeta|/r_i)|\le D\}\right\}\\
 &\hspace{40mm}\le C_u e^{-cD}.
\end{split}
\end{equation}
An additive fixed change in $D$ absorbs the dip width and the word offsets.
To justify the event bound explicitly, if any zero lies outside the enlarged
windows, one entire dip interval lies in the integration region. The
integral of $|1-f(e^s)|^2$ there is then at least a fixed positive number.
Markov's inequality after integration over the state spectral measure gives
\eqref{eq:chiral-all-zero-event}.

Here is the order of truncations. Given a desired exceptional mass, first
choose $D$ large enough in \eqref{eq:chiral-all-zero-event} and so that
the finite zero sets of $F_1,F_2$ lie in their annuli with correspondingly
high probability. Choose their boundary circles to carry no limiting zero
almost surely. Only countably many radii are excluded: first restrict to
a bounded zero count, then apply this assertion to its finite intensity
measure, and finally take the union over count bounds. On these fixed
annuli, the cone keeps the zeros in a compact subset of the open right
half-plane. The argument principle and compact-open convergence give joint
convergence of the complete zero multisets, including multiplicities.
Next choose a finite degree bound $M$ so that larger limiting counts have
arbitrarily small mass. Take the scale limit only after $D$ and $M$ have
been fixed. Zero counts and locations now lie in a compact, bounded-count
set, so every product expansion below is uniform. Finally let the allowed
exceptional mass tend to zero, increasing $D$ and $M$ if necessary.

Each limiting product has even degree: conjugate zeros pair, real zeros
contribute a factor $-1$ between the two ends, and both end values are one.
On the preceding high-mass event the complete limiting zero counts occur
in the chosen annuli. Multiplicity convergence therefore makes both
finite-scale clusters even with probability tending to one. This proves
the needed parity; it is not an assumption on a truncated cluster.

For clarity, write the large zeros as $\delta^{-1}\alpha_j$ and the
small zeros as $\delta\beta_j$. On the retained event each normalized
zero has modulus between $e^{-D}$ and $e^D$, and there are at most $M$
zeros. Normalize each factor to one at zero. The two even clusters then
give, for $w=1,i$,
\begin{equation}\label{eq:chiral-product-expansion}
 f(w)=1-\delta\left(
       w\sum_j2\Re\alpha_j^{-1}
          +w^{-1}\sum_j2\Re\beta_j\right)+O_{D,M}(\delta^2).
\end{equation}
The two sums converge jointly to $a_+(F_1)$ and $a_-(F_2)$.
Taking logarithmic modulus at $1$ and phase at $i$ gives
\eqref{eq:chiral-scaled-spectrum}. The middle sign is positive on this
event. Values of $H_0/\delta$ can be arbitrarily large on the discarded
event; convergence in law and bounded multiplier convergence require no
first-moment estimate for that quantity.

We next strengthen bounded spectral convergence to convergence of vectors.
Choose a countable total word core, including its rational linear
combinations, and exclude the union of the atoms of $A$ in all their state
measures. That union is countable; any other $R$ is admissible in
\eqref{eq:chiral-band-limit}. At such $R$, continuous cutoffs approximate
the indicator in each state $L^2$ norm. More generally, let $b_\delta$ be
a uniformly bounded spectral multiplier with the stated joint limit $b$.
The algebra of polynomials in a countable dense set of interior samples
and their conjugates generates the spectral functions and is dense in
the state $L^2$ space. Given a state $u$, choose such a polynomial $p$ with
$\|(b-p)u\|$ small. Joint convergence gives
\begin{equation}\label{eq:chiral-multiplier-norm}
 \|(b_\delta-p_\delta)u_\delta\|^2
          \longrightarrow\|(b-p)u\|^2.
\end{equation}
For indicators, take the continuous approximations first. Polynomial
actions already converge strongly. Equation
\eqref{eq:chiral-multiplier-norm} proves strong convergence of the
multiplier actions; polarization gives the corresponding mixed pairings.
Lemma~\ref{lem:chiral-representatives} extends these actions to all
strongly converging families and permits their compositions. No assertion
that a current preserves the exceptional-zero event is needed.

Finally, $|f(1)|\le1$ and $|f(i)|=1$. The bounded multipliers
$f(1)^{m_\delta}f(i)^{l_\delta}$ converge jointly by
\eqref{eq:chiral-scaled-spectrum} to
$e^{-xA+iy(-a_+\otimes I+I\otimes a_-)}$, which is
\eqref{eq:chiral-semigroup-limit}. The same vector argument applies.
\end{proof}

\begin{figure}[htbp]
\centering
\input{figures/chiral-windows}
\caption{The two windows in logarithmic radius. Their width and a bound
on the number of zeros are fixed before $\delta$ tends to zero. Outside
an event of arbitrarily small state mass, no zero lies between or beyond
the windows.}
\label{fig:chiral-windows}
\end{figure}

The exceptional-event estimate is the reason that local spectral convergence
can be used at the intervening physical scale $w=1$. It also permits the
energy cutoffs needed for current insertions. We next prove their convergence
in norm, before removing those cutoffs by time damping.

\subsection{Current insertions and damping}

\begin{lemma}\label{lem:current-limit}
For every admissible $R>0$ in Proposition~\ref{prop:two-scale} and every
$|w|=1$ with $\Re w\ge0$,
\begin{equation}\label{eq:chiral-current-limit}
 \delta^{-1}Q_\delta(R)N(w)Q_\delta(R)
 \longrightarrow
 Q(R)\bigl(w(j_+\otimes I)+w^{-1}(I\otimes j_-)\bigr)Q(R)
\end{equation}
strongly on representatives. All displayed operators are bounded at fixed
$R$. Finite products with propagators from
\eqref{eq:chiral-semigroup-limit} converge when successive positive time
gaps stay bounded below. In these vacuum products, the cutoffs on both
sides of each current may be removed after the scale limit, with error
tending to zero uniformly on compact sets of separated time coordinates.
\end{lemma}

\begin{proof}
Write $Q_\delta=Q_\delta(R)$ and $Q=Q(R)$. Glue
$J_\delta(w)=Q_\delta J(w)Q_\delta$ to the full annulus
\[
                  CR\delta<|w|<c/(R\delta)
\]
using Lemma~\ref{lem:low-band}. Let $C_{k,\delta}$ be its Laurent
coefficients. Fix $\rho>0$ sufficiently small compared with $R^{-1}$.
At the outer circle, the substitution $v=\delta w$ gives
\begin{equation}\label{eq:chiral-outer-coefficient}
 \delta^{-1}C_{1,\delta}
    =\frac1{2\pi i}\int_{|v|=\rho}
                  J_\delta(v/\delta)\frac{dv}{v^2}.
\end{equation}
At the inner circle, put $w=\delta v$ to obtain
\begin{equation}\label{eq:chiral-inner-coefficient}
 \delta^{-1}C_{-1,\delta}
    =\frac1{2\pi i}\int_{|v|=\rho^{-1}}
                              J_\delta(\delta v)\,dv.
\end{equation}
Both contours are counterclockwise. The rescaled contours are fixed, and
both integrands are uniformly bounded in operator norm at fixed $R,\rho$.

On the strict right semicircle the marked rows converge on their respective
tensor factors. The cutoffs converge strongly by
\eqref{eq:chiral-band-limit}. On the reflected semicircle the integrand is
the product specified by reversal, for example
\[
 -\bigl(Q_\delta P(v/\delta)^{-1}Q_\delta\bigr)
     J_\delta(v/\delta)
     \bigl(Q_\delta P(v/\delta)^{-1}Q_\delta\bigr).
\]
The inverse factors are uniformly bounded spectral multipliers on
$Q_\delta\mathcal D$ only. Their joint limits follow from
Proposition~\ref{prop:two-scale}, since on $Q\mathcal D^{\otimes2}$
the first energy is at most $R$ and \eqref{eq:chiral-disc-inverse}
applies. The second contour uses the analogous estimate at infinity.
Lemma~\ref{lem:chiral-representatives} therefore proves convergence of
these noncommuting products at every strict contour point.

Here is the norm justification for passing through the contour integrals.
For a representative $u_\delta$, let $v_\delta(t)$ be the integrand
applied to it, including the scalar contour weight, and let $v(t)$ be
its limit. The vectors are bounded uniformly in $t,\delta$. Core test
pairings of $\int v_\delta(t)dt$ converge by a single dominated integral.
Their squared norms converge because
\begin{equation}\label{eq:chiral-double-contour}
 \left\|\int v_\delta(t)dt\right\|^2
 =\iint\langle v_\delta(t),v_\delta(s)\rangle\,dt\,ds
 \longrightarrow
 \iint\langle v(t),v(s)\rangle\,dt\,ds.
\end{equation}
Pairings of strongly convergent representatives converge, so dominated
convergence applies to this double integral. The two imaginary-axis points
have contour measure zero. The norm criterion in
Lemma~\ref{lem:chiral-representatives} now gives strong convergence of
each coefficient action. By the definitions of the chiral forms, the
limits of \eqref{eq:chiral-outer-coefficient} and
\eqref{eq:chiral-inner-coefficient} are respectively
\[
                 Q(j_+\otimes I)Q,
                   \qquad Q(I\otimes j_-)Q.
\]

It remains to pass from these coefficients to $N$. Put
$\Sigma=\sgn P(1)$ and
\[
 T_\delta(w)=w\delta^{-1}C_{1,\delta}
                         +w^{-1}\delta^{-1}C_{-1,\delta}.
\]
The same-sign and opposite-sign estimates of Lemma~\ref{lem:low-band}
give, for some $\theta>0$,
\begin{equation}\label{eq:chiral-sign-removal}
 \delta^{-1}Q_\delta N(w)Q_\delta
       =\tfrac12\bigl(\Sigma T_\delta(w)+T_\delta(w)\Sigma\bigr)
                       +O_R(\delta^\theta)
\end{equation}
in operator norm, uniformly on the unit semicircle. Indeed the
anticommutator is $+T_\delta$ on the positive-sign block,
$-T_\delta$ on the negative-sign block, and zero between opposite signs.
On same-sign blocks the even Laurent part and all odd modes of order
larger than one vanish after rescaling, with the stated remainder.
The opposite-sign block of the rescaled $N$ has the same vanishing bound.
Proposition~\ref{prop:two-scale} gives $\Sigma\to I$ strongly on
representatives. The coefficient operators are uniformly bounded and have
the strong limits just proved. Composition in
\eqref{eq:chiral-sign-removal} proves \eqref{eq:chiral-current-limit}.
The low-band bounds include the endpoints $w=\pm i$.

We finish with a summable bound for removing cutoffs. For all $U>0$ and
all sufficiently small $\delta$,
\begin{equation}\label{eq:chiral-all-band-bound}
 \|\bbone_{H_0/\delta\le U}\,\delta^{-1}N(w)
                      \bbone_{H_0/\delta\le U}\|\le CU.
\end{equation}
If $U\delta$ is small this is Lemma~\ref{lem:low-band}; otherwise the
global bound on $N$ gives it by $\delta^{-1}\le C U$.
Decompose $H_0/\delta$ into bands $[0,1]$ and
$(2^{q-1},2^q]$, $q\ge1$. A current block between levels $q,r$ costs
at most $C\max(2^q,2^r)$. Each internal propagator with
$\delta m\ge b>0$ supplies at level $q\ge1$ the factor
$e^{-b2^{q-1}}$, since
$\|P(1)^m\bbone_{H_0/\delta>2^{q-1}}\|\le e^{-b2^{q-1}}$;
unit shifts do not affect this norm.
The outer states are the vacuum, of energy zero. For a product of $k$
currents, a bound for the sum over its $k-1$ internal indices is a constant
times
\begin{equation}\label{eq:chiral-dyadic-majorant}
 \sum_{q_1,\ldots,q_{k-1}\ge0}
       \prod_{j=1}^{k-1}(1+2^{q_j})^2e^{-b'2^{q_j}},
                         \qquad b'>0.
\end{equation}
This converges, and its tails with any $2^{q_j}>R$ tend to zero uniformly
in the mesh. The limiting forms have the identical bound using $A$ and
\eqref{eq:chiral-form-bound}. Fixed-cutoff convergence, followed by this
tail estimate, proves the product assertion. The constants are uniform
when the finitely many limiting time gaps have a fixed positive lower
bound.
\end{proof}

\subsection{Continuum derivative correlations}

In the remainder of this section $z=x+iy$ denotes a physical point, not
a row angle. At a mesh face define the centered height differences
\begin{align*}
 D_{x,\delta}h_\delta(z)
   &=\tfrac12\bigl(h_\delta(z+\delta)-h_\delta(z-\delta)\bigr),\\
 D_{y,\delta}h_\delta(z)
   &=\tfrac12\bigl(h_\delta(z+i\delta)-h_\delta(z-i\delta)\bigr),\\
 D_\delta^+&=\tfrac12(D_{x,\delta}-iD_{y,\delta}),\qquad
 D_\delta^-=\tfrac12(D_{x,\delta}+iD_{y,\delta}).
\end{align*}
The mesh inverse $\delta^{-1}$ is not included in these definitions.
One may evaluate at any point using the stipulated face convention.
All arguments below allow the chosen lattice faces to move with the mesh.

Take points whose real coordinates are pairwise distinct. In eastward
quantization, a centered horizontal or vertical difference is the mean of
two consecutive height steps. The path convention and $N=(JP+PJ)/2$
identify its insertion as $-N(1)$ or $-N(i)$, respectively. To read a
product, join these two-step segments in increasing time order, using
positive time steps and signed transverse steps. Between their endpoints
the propagators are $P(1)^{m_\delta}S^{l_\delta}$. The endpoint offsets
change $m_\delta,l_\delta$ by at most a fixed number of sites and do not
change their rescaled limits.

Lemma~\ref{lem:current-limit} and
\eqref{eq:chiral-semigroup-limit} show that the insertions of
$\delta^{-1}D_\delta^+$ and $\delta^{-1}D_\delta^-$ converge to
$-j_+\otimes I$ and $-I\otimes j_-$, respectively. For example,
the two combinations follow from
\[
 N(1)\rightsquigarrow j_+\otimes I+I\otimes j_-,
 \qquad
 N(i)\rightsquigarrow i j_+\otimes I-i I\otimes j_-.
\]
Currents in different factors commute. Their propagators and the tensor
vacuum factor as well. The limiting mixed correlations therefore factor
into a pure $+$ correlation and a pure $-$ correlation.

Write $a=a_+$ and $j=j_+$ for the moment. For increasing real parts,
$\Re z_{i_1}<\cdots<\Re z_{i_k}$, define
\begin{equation}\label{eq:chiral-ordered-function}
 F_k(z_1,\ldots,z_k)
 =\left\langle\Omega,\,
 (-j)e^{-(z_{i_k}-z_{i_{k-1}})a}(-j)\cdots
 e^{-(z_{i_2}-z_{i_1})a}(-j)\Omega\right\rangle.
\end{equation}
Set $F_0=1$ and $F_1=0$. The products in
\eqref{eq:chiral-ordered-function} are defined by inserting positive
real damping on both sides of each form; arbitrary extra end damping
may be inserted because $a\Omega=0$. Lemma~\ref{lem:chiral-forms}
makes the resulting factors bounded, and the dyadic tail bound shows
that the value is independent of this allocation. By reality of the
height field, the pure $-$ answer is $\overline{F_k(z_1,\ldots,z_k)}$.
All the preceding convergence is uniform on compact subsets of the
region of distinct real coordinates. Indeed, it holds along every
sequence of moving mesh faces converging to such a configuration,
and the cutoff and damping bounds are uniform on its compact subsets.

\begin{lemma}\label{lem:chiral-holomorphy}
Let
\[
 U_x=\{(z_1,\ldots,z_k):\Re z_i\ne\Re z_j\text{ for }i\ne j\},
 \quad
 U_y=\{(z_1,\ldots,z_k):\Im z_i\ne\Im z_j\text{ for }i\ne j\}.
\]
The function $F_k$ has a symmetric, jointly holomorphic extension to
$U_x\cup U_y$, and is the limit of the centered $+$ derivative
correlations on both sets.

Fix $z_2,\ldots,z_k$ with distinct real coordinates and distinct imaginary
coordinates. As a function of $z_1$, the extension is holomorphic away
from the grid points
\begin{equation}\label{eq:chiral-grid-points}
                    p_{jh}=\Re z_j+i\Im z_h,\qquad 2\le j,h\le k.
\end{equation}
Near any such point it is $O(|z_1-p_{jh}|^{-2})$, locally uniformly
as the fixed points vary while retaining these coordinate separations.
For $k\ge2$ it tends to zero as $|z_1|\to\infty$ with the other points fixed.
\end{lemma}

\begin{proof}
On a compact subset of an eastward ordering chamber, give each current
a fixed positive amount of damping on both sides. The remaining
semigroups have parameters with real parts bounded below. These
semigroups are holomorphic in operator norm there: for instance
$\|a^m e^{-sa}\|\le m!s^{-m}$ after reducing $s$ by a fixed factor.
The resulting bounded product is jointly holomorphic in all its
parameters. This proves joint holomorphy on $U_x$, rather than merely
separate holomorphy.

By Proposition~\ref{prop:physical-plane}, northward quantization describes
the same physical law. In its coordinates $z'=-iz$, the centered
differences transform as
$D_{x',\delta}=D_{y,\delta}$ and
$D_{y',\delta}=-D_{x,\delta}$, hence
$D_\delta^+=-iD_\delta^{\prime+}$. The possible face-origin shift is
one lattice spacing and is covered by moving-face convergence. The
northward formula is thus holomorphic on $U_y$. On $U_x\cap U_y$
both formulas are limits of the same physical moments, so they agree
and glue. Permuting the insertions does not change a physical product,
which proves symmetry.

With the other points as in the statement, failure of both axial
orderings occurs only at the grid points \eqref{eq:chiral-grid-points}.
Choose a small neighborhood of the fixed configuration in which all
its real gaps and imaginary gaps remain at least $b>0$. Near $p_{jh}$,
the eastward representation has only one possibly small gap,
$g_x=|\Re(z_1-p_{jh})|$, between $z_1$ and $z_j$.
The northward representation has only one possibly small gap,
$g_y=|\Im(z_1-p_{jh})|$, between $z_1$ and $z_h$.
Use the ordering with $g=\max(g_x,g_y)\ge|z_1-p_{jh}|/\sqrt2$.
Only the two currents bordering that gap need shrinking damping.
Give each damping $g/4$ on both sides: the inner dampings consume
$g/2$, and the outer dampings come from neighboring fixed gaps or
from a vacuum endpoint. All other currents receive fixed damping.
Equation~\eqref{eq:chiral-damped-form} gives a cost $C/g$ for each
of the two currents and a bounded cost for the rest. This proves
the uniform quadratic bound. A point sharing the other coordinate
is separated in the chosen ordering.

Finally, when $|z_1|$ is large, one coordinate makes $z_1$ extreme in
the corresponding ordering, with a gap $g$ of order $|z_1|$ to all
other points. Give the isolated current damping of order $g$ on both
sides, using free damping on its vacuum side. Keep fixed damping
around all remaining currents. The isolated factor costs $C/g$;
imaginary semigroup factors have norm one. Thus $F_k\to0$ at infinity.
\end{proof}

\subsection{The collision operator}

The remaining input for Gaussianity is the coefficient at a genuine
collision. A second separation of scales shows that a short pair of
chiral currents becomes a scalar, even between nonvacuum exterior states.

\begin{lemma}\label{lem:collision}
For $a=a_+$ and $j=j_+$, the bounded operators
\begin{equation}\label{eq:chiral-collision-operator}
 K_\epsilon=e^{-\epsilon a}(\epsilon j)e^{-\epsilon a}
                      (\epsilon j)e^{-\epsilon a}
\end{equation}
are uniformly bounded for $\epsilon>0$ and satisfy
\begin{equation}\label{eq:chiral-collision-limit}
                    K_\epsilon\rightharpoonup-\kappa I
                       \qquad(\epsilon\downarrow0).
\end{equation}
Consequently, for generic distinct exterior points,
\begin{equation}\label{eq:chiral-double-pole}
 \lim_{\epsilon\downarrow0}\epsilon^2
 F_k(z_j+\epsilon,z_2,\ldots,z_k)
   =-\kappa F_{k-2}(z_2,\ldots,\widehat z_j,\ldots,z_k).
\end{equation}
The coefficient on the right is independent of both colliding positions.
\end{lemma}

\begin{proof}
Split the middle damping of \eqref{eq:chiral-collision-operator} into
two halves. Each of the resulting damped current factors has norm bounded
by a constant, by \eqref{eq:chiral-damped-form} at scale $\epsilon/2$.
This gives uniform boundedness.

Apply the representative construction with scales $r_1=\epsilon$ and
$r_2=1$. Its Gram and row limits use only separation of the two scales,
so they are unchanged. The all-zero event estimate
\eqref{eq:chiral-all-zero-event}, with the same order of fixed windows,
bounded degree, and then scale limit, gives
\begin{equation}\label{eq:chiral-collision-energy}
                     \epsilon a(f)\longrightarrow a(F_1)
\end{equation}
jointly with the two spectral functions. To see the scaling directly,
a zero $\epsilon\alpha$ contributes $2\Re\alpha^{-1}$ to $\epsilon a$,
whereas a zero $\beta$ at the unshifted scale contributes
$2\epsilon\Re\beta^{-1}$. On the bounded-count annuli the latter
sum tends uniformly to zero. The exceptional-zero event has arbitrarily
small mass and is discarded before making this expansion.

Choose a non-atomic band endpoint and put
$B_\epsilon=\bbone_{\epsilon a\le R}$ and
$B=\bbone_{a\otimes I\le R}$. The spectral multiplier argument proves
$B_\epsilon\to B$ strongly. The definition of $j$ by its Taylor
coefficient gives, for a sufficiently small fixed $\rho<c/R$,
\begin{equation}\label{eq:chiral-collision-contour}
 \epsilon B_\epsilon jB_\epsilon
   =\frac1{2\pi i}\int_{|v|=\rho}
             B_\epsilon J(\epsilon v)B_\epsilon\,\frac{dv}{v^2}
   \longrightarrow B(j\otimes I)B.
\end{equation}
Indeed, the unscaled band is $a\le R/\epsilon$, whose Taylor disc
has radius proportional to $\epsilon/R$. On the right semicircle
the row acts on the first factor; on the left semicircle the bounded
compressed inverse and reversal apply. The single- and double-contour
argument \eqref{eq:chiral-double-contour} proves convergence in
representative norm. Thus \eqref{eq:chiral-collision-energy} and
\eqref{eq:chiral-collision-contour} give the cutoff limit of
$K_\epsilon$.

These cutoffs can be removed in operator norm uniformly in $\epsilon$.
Decompose $\epsilon a$ into $[0,1]$ and the dyadic bands above one.
A block of $\epsilon j$ between levels $q,r$ costs at most
$C\max(2^q,2^r)$. The three damping factors of $K_\epsilon$ give an
absolutely summable majorant
\[
 C\sum_{q_0,q_1,q_2\ge0}
   \max(2^{q_0},2^{q_1})\max(2^{q_1},2^{q_2})
                       e^{-b(2^{q_0}+2^{q_1}+2^{q_2})}.
\]
Absorbing the lowest-band constants into $C$ gives some fixed $b>0$.
The part with any $2^{q_i}>R$ tends to zero as $R\to\infty$.
The same estimate holds for the limiting tensor product. Therefore,
on separated-scale representatives,
\[
 K_\epsilon\longrightarrow
           \bigl(e^{-a}je^{-a}je^{-a}\bigr)\otimes I.
\]

A fixed unboosted word vector $u$ is exactly the representative of
$\Omega\otimes u$: its small-scale word is the identity. Hence for
fixed word vectors $u,v$ the last limit contracts the first factor to
\[
 \langle\Omega,je^{-a}j\Omega\rangle
   =-\int a(f)^2e^{-a(f)}\dd\mu(f)
   =-\kappa\int_0^\infty t e^{-t}\,dt=-\kappa.
\]
The sign uses the skew-adjointness of $j$; the integrals are justified
by \eqref{eq:chiral-Haar-integral}. This proves
\eqref{eq:chiral-collision-limit} on a dense set of test vectors.
Uniform boundedness extends it to every pair of vectors in $\mathcal D$.

For \eqref{eq:chiral-double-pole}, take $z_1=z_j+\epsilon$ in the
eastward ordering. The two colliding currents are consecutive. After
multiplication by $\epsilon^2$, their part of the product becomes
$K_\epsilon$, provided its two outer dampings are subtracted from the
adjacent exterior gaps. Those gaps remain positive for small $\epsilon$;
at an endpoint, insert damping freely on the vacuum. The two exterior
vectors converge strongly, because all their damping gaps remain positive.
They may therefore be replaced by their limits in the weak convergence
of the uniformly bounded $K_\epsilon$. The two insertion minus signs
cancel. Omitting the colliding currents recombines the adjoining
semigroups, giving exactly the right side of
\eqref{eq:chiral-double-pole}. No dependence on the collision position
remains.
\end{proof}

\subsection{Removal of artificial poles and Wick recursion}

\begin{proposition}\label{prop:gaussian-currents}
Let $k\ge1$, and let $z_1,\ldots,z_k$ have pairwise distinct real
coordinates or pairwise distinct imaginary coordinates. For
$\varepsilon_i\in\{+,-\}$ and moving mesh faces $z_{i,\delta}\to z_i$,
\begin{equation}\label{eq:chiral-Gaussian-current-limit}
 \lim_{\delta\downarrow0}
 \E\prod_{i=1}^k
       \delta^{-1}D_\delta^{\varepsilon_i}h_\delta(z_{i,\delta})
   =\sum_{\pi}\ \prod_{\{i,j\}\in\pi}
                  C_{\varepsilon_i,\varepsilon_j}(z_i,z_j),
\end{equation}
where the sum is over pairings, is zero for odd $k$, and
\begin{equation}\label{eq:chiral-contractions}
 C_{++}(z,w)=-\frac{\kappa}{(z-w)^2},\qquad
 C_{--}(z,w)=-\frac{\kappa}{(\bar z-\bar w)^2},\qquad
 C_{+-}(z,w)=C_{-+}(z,w)=0.
\end{equation}
Convergence is uniform on compact subsets of either separated-time
region. By linearity the corresponding assertion holds for all products
of the real centered derivatives $\delta^{-1}D_{x,\delta}$ and
$\delta^{-1}D_{y,\delta}$. These are the off-diagonal derivatives of the
Gaussian covariance
\begin{equation}\label{eq:chiral-log-covariance}
                              K_\kappa(z,w)=-2\kappa\log|z-w|.
\end{equation}
\end{proposition}

\begin{proof}
The preceding lemmas already give the limits as products of pure chiral
functions and their conjugates. It remains to determine $F_k$.
Fix generic spectators $z_2,\ldots,z_k$ as in
Lemma~\ref{lem:chiral-holomorphy}. Its quadratic bound makes every
grid puncture of $F_k$ a pole of order at most two: the Cauchy bound
for a Laurent coefficient of order $-m$, $m>2$, is $Cr^{m-2}$ on a
circle of radius $r$, and tends to zero.

First consider an artificial puncture
$p=p_{jh}=\Re z_j+i\Im z_h$ with $j\ne h$.
Take a fixed small circle $\Gamma$ about its position for the chosen
spectators. In a sufficiently small complex neighborhood of the
spectators, the circle encloses just this moving grid point and remains
in $U_x\cup U_y$. The contour moments
\[
 m_\ell=\frac1{2\pi i}\int_\Gamma
                  \zeta^\ell F_k(\zeta,z_2,\ldots,z_k)\,d\zeta
\]
are jointly holomorphic in the spectator variables, by the uniform
bounds and joint holomorphy on the circle. If the principal part is
$A/(\zeta-p)^2+B/(\zeta-p)$, residue calculation gives
\begin{equation}\label{eq:chiral-pole-moments}
 m_\ell=Bp^\ell+A\ell p^{\ell-1},\qquad
 m_0m_2-m_1^2=-A^2,\qquad
 m_0m_3-m_1m_2=-2pA^2.
\end{equation}
If $A$ is nonzero at any parameter value, the nonzero denominator
persists locally and
\[
             p=\frac{m_0m_3-m_1m_2}
                       {2(m_0m_2-m_1^2)}
\]
is holomorphic there. This is impossible, since
\[
 p=\frac{z_j+\bar z_j}{2}+\frac{z_h-\bar z_h}{2},
                         \qquad \partial_{\bar z_j}p=\tfrac12.
\]
Thus $A$ vanishes. If $B$ is nonzero, $p=m_1/m_0$ gives the same
contradiction. Every artificial puncture is removable.
This removal is joint in the parameters: inside $\Gamma$, the Cauchy
integral of $F_k(\eta,\mathbf z)/(\eta-\zeta)$ supplies a jointly
holomorphic extension and agrees with each one-variable removable
extension.

At a true collision, put
$c=(z_1+z_j)/2$ and $d=z_1-z_j$, retaining generic exterior points.
Locally the union $U_x\cup U_y$ contains every $d\ne0$.
The function
$H(c,d,\mathbf z)=d^2F_k$ is uniformly bounded there and therefore
extends jointly holomorphically to $d=0$. One can see this directly
by its Cauchy coefficients in $d$: boundedness removes the negative
coefficients, uniformly on smaller spectator neighborhoods.
Symmetry under exchanging the two colliding insertions makes $H$ even
in $d$. Lemma~\ref{lem:collision} identifies its constant term as
\[
 H(c,0,\mathbf z)
             =-\kappa F_{k-2}(\mathbf z),
\]
independent of $c$. Subtracting this term leaves an even holomorphic
function with zero constant term, hence a function divisible by $d^2$.
It follows that
\[
 F_k+\frac{\kappa F_{k-2}(\mathbf z)}{(z_1-z_j)^2}
\]
is holomorphic at this diagonal. In particular there is no simple
residue. Independence of the collision center is needed here: a varying
coefficient of $d^{-2}$ could otherwise create a simple term when
$z_j$ is held fixed.

Subtract all these true double-pole terms as a function of $z_1$.
The artificial points are removable, the true points have no remaining
principal parts, and the resulting function is entire.
Lemma~\ref{lem:chiral-holomorphy} gives decay at infinity, and the
subtracted rational terms also decay there. Liouville's theorem yields
\begin{equation}\label{eq:chiral-Wick-recursion}
 F_k(z_1,\ldots,z_k)
   =-\kappa\sum_{j=2}^k
       \frac{F_{k-2}(z_2,\ldots,\widehat z_j,\ldots,z_k)}
                           {(z_1-z_j)^2}.
\end{equation}
Together with $F_0=1$ and $F_1=0$, this is Wick's recursion with
contraction $-\kappa/(z_i-z_j)^2$. Continuity extends it from the
generic spectators to every strict axial chamber. Factorization of
the two chiral sectors and reality then give
\eqref{eq:chiral-Gaussian-current-limit}.

Finally,
$\partial_z\partial_w K_\kappa(z,w)=-\kappa/(z-w)^2$ and
$\partial_z\partial_{\bar w}K_\kappa(z,w)=0$ away from collisions.
The conjugate identity gives the other pure contraction. This proves
the covariance assertion and, by linearity, all the real derivative
formulas. The uniform moving-face convergence established above gives
the asserted local uniformity. Estimates across a time-ordering wall
and for undifferentiated fields will be supplied in
Section~\ref{sec:probability}; none is needed for this proposition.
\end{proof}
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//06-chiral-gaussianity.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//06-coupling.tex ====
\section{An outward coupling conditioned on two arms}
\label{sec:coupling}

The electric correlations will compare a pair of points at a growing
separation with the two sites adjacent to a single interface segment.
The comparison requires more than an estimate for an arm probability:
after conditioning the two colors to reach a distant rim, the exterior
configurations must become identical with high probability.  We prove
this statement here.  The same coupling will also compare a segment
with a small disk.

All distances in this section are in tile units.  A cut is the boundary
of a finite union of whole tiles.  A strand is cut at every port on this
boundary.  We use square cuts for the successive coupling trials and
allow either a square or a round cut for the final target.  Round cuts
are simple whole-tile approximations to circles, at bounded Hausdorff
distance from those circles.  In particular, reaching a cut means
reaching an exposed gap of the appropriate color.  Boundary
identifications do not supply edges of a color path.

\subsection{Three starting configurations}

Fix a center and a target cut of radius $R$.  We consider the following
starts, contained well inside that cut.
\begin{enumerate}[label=\textup{(\roman*)}]
\item A tile side $e$.  Its two end sites have opposite colors, and the
associated full medial segment crosses that side.  The event $S_R(e)$
requires an actual path of each color from its end site to the target
cut.
\item Two sites $u,v$ of opposite colors, with $|u-v|\asymp r$ and both
within distance $O(r)$ of the center.  The event $S_R(u,v)$ requires
their respective color paths to reach the target cut.
\item A round whole-tile disk of radius $r$.  The event $S_R(r)$ requires
both colors to cross the intervening annulus, starting at gaps of the
disk boundary.
\end{enumerate}
We sometimes write $S_R$ when the start is specified separately.  Every
path in this definition is restricted to the region between the start
and its first arrival at the target cut.

These definitions have useful descriptions in the strand picture.
For a point start, its color region reaches the target cut if and only
if no completed loop inside that cut surrounds the point.  Indeed, a
region that does not reach the cut has a boundary loop separating it
from the cut; conversely, an actual color path cannot cross a loop.
Thus in cases \textup{(i)} and \textup{(ii)}, $S_R$ says that no
completed loop surrounds either starting site.  In case \textup{(iii)},
$S_R(r)$ is equivalent to the existence of a strand subarc joining the
two rims.  To see the latter assertion, cut all strands at both rims.
An arc with both ends on the same rim cuts off a region incident only
to that rim.  If there are no through-arcs, at most one color can have
a region incident to both rims.  If there are through-arcs, the regions
between consecutive through-arcs give paths of both colors between
the rims.  The rounded drawing makes this a statement about disjoint
arcs, even when the straight medial drawing has a double visit.

Write $\pi_1(r)$ for the plane one-arm probability to radius $r$, and
$\pi_2(r,R)$ for the plane alternating two-arm probability across a
square annulus.  Their estimates from Lemmas~\ref{lem:one-arm}
and~\ref{lem:two-arm-log} are available below.  A fixed change of a
radius, or of the shape of a rim between the round and square choices,
changes the following comparisons by a constant only.

\begin{lemma}[Probabilities of the three starts]
\label{lem:start-probabilities}
There is a fixed minimum radius $r_0$ such that, for $R$ sufficiently
larger than the start,
\begin{equation}\label{eq:start-probabilities}
 \Pp(S_R)\asymp
 \begin{cases}
   \pi_2(r_0,R),&\text{for a tile-side start},\\
   \pi_1(r)^2\pi_2(r,R),&\text{for two sites at separation }\asymp r,\\
   \pi_2(r,R),&\text{for a disk of radius }r.
 \end{cases}
\end{equation}
The same constants apply in an ambient FK law with a full bulk collar
beyond the target cut, including after a nonnegative integrable
reweighting supported beyond that collar.  In the second line, the
constants can depend on the fixed comparability bounds for the two
sites.
\end{lemma}

\begin{proof}
For the upper bound in the second line, choose disjoint boxes of radius
$c r$ about the two sites, with $c>0$ small enough.  Each site must have
a one-arm connection to its own box boundary.  Outside a square of
radius $C r$ about the center there must also be an alternating pair
of arms extending to a fixed fraction of $R$.  Leave fixed-ratio
collars between these three observations.  Bounded-density comparison
through these collars, as recorded in Input~\ref{inp:fk}, bounds their
joint probability by
$C\pi_1(c r)^2\pi_2(C r,R/C)$.  One-arm extension and
quasimultiplicativity give the asserted upper bound.

For the lower bound, require a one-arm event in each small box with a
landing extension pointing into a designated corridor.  Require
alternating arms in the outer annulus with separated inner landings.
The two corridors can be chosen disjoint, one for each color.  Uniform
crossing estimates and same-color FKG connect the corresponding
landings at a positive fixed cost.  The collars give a lower comparison
for the events before this gluing, and quasimultiplicativity removes
the fixed changes of radii.  This proves the second line.  It also
explains why no favorable boundary partition is required: each
crossing estimate is uniform in the partition induced by the already
observed edges.

For a tile-side start, the initial part of the two paths can be imposed
inside a fixed box by finite energy, after which the same two-arm
extension argument applies.  Cropping gives the reverse inequality.
For a disk start, crop or extend at its two rims and use the annular
two-arm event directly.  These arguments take place before the
remaining bulk collar.  Conditioning on all data beyond that collar
proves the bounds uniformly in the exterior state.  Integration proves
the assertion for a nonnegative exterior reweighting.
\end{proof}

\begin{proposition}[Outward conditioned coupling]
\label{prop:outward}
Fix $q\in[1,4)$.  Consider one ambient FK law whose graph is the full
tile lattice throughout the region bounded by the target cut and in
a fixed-ratio bulk collar beyond it.  All imposed wires and
nontrivial identifications are supported outside this region and
collar.  The law may be multiplied by a common
nonnegative integrable function supported beyond the collar, with
positive finite normalizing expectation.  Let two starts of the types
above be contained in a disk of radius $s$, where $1\le s\le R/C$.
Their laws, conditioned on their respective events $S_R$, have a
coupling such that, except with probability
\begin{equation}\label{eq:outward-error}
 C(s/R)^c,
\end{equation}
the configurations agree on and outside an annular band lying between
the starts and the target cut.  There are exactly two strand subarcs
joining the rims of this band.  Each start reaches the same
transmitted primal region and the same transmitted dual region of
the band.  Consequently the exterior sees the same pairing of its
exposed strand ends in the two configurations.

Here $c>0$ and $C<\infty$ depend only on $q$, the fixed collar
geometry, and the fixed comparability constants of the starts.  The
assertion is uniform in the exterior reweighting.
\end{proposition}

The proof constructs an annular band which has positive probability
and transmits exactly one region of each color.  We first explain its
geometry and the gluing that preserves an arbitrary event on its
central edges.  We then show that a conditioned exploration has enough
opportunities to use this band.

\subsection{A band with exactly two through-arcs}

\begin{lemma}[The reference band]
\label{lem:reference-band}
At every sufficiently large scale $l$, there is a fixed-shape square
annular band $\mathcal B_l$, of inner and outer radii comparable to
$l$, and an event of probability bounded below in the plane FK law
with the following properties.  Both colors cross $\mathcal B_l$;
there are exactly two strand through-arcs; and each of the two
crossing color regions has separated landing extensions beyond both
rims.  The extension boxes can be chosen a positive relative distance
from the central band and from each other.  The constants and the
relative sizes of all these regions are independent of $l$.
\end{lemma}

\begin{proof}
Choose one radial lane on the top flat sides of two concentric squares
and another on their bottom flat sides.  All lane widths and the gaps
between lanes are fixed positive multiples of $l$.  The top lane will
carry primal crossings and the bottom lane dual crossings.  In each
lane choose three parallel rectangular strips: a main strip and a
flanking strip on each side.  Require longitudinal crossings of all
three strips.  The main crossing is extended beyond both square cuts
into prescribed, separated landing boxes.  Inside the central annulus,
require transverse same-color crossings joining each flanking
crossing to the main crossing.  A transverse crossing runs across the
full width of a flank strip, so it meets every longitudinal crossing
of that strip.  Crossing estimates and FKG give a positive fixed
probability for these requirements.  The regions for the two colors
are disjoint, so the colors can be treated successively using the
conditional crossing bounds.

In each flanking strip choose the extremal longitudinal crossing
closest to the main strip and expose it from the main-strip side.
The transverse joining paths can be stopped at their first contact
with this crossing.  Thus the resulting event still contains the
crossing event just imposed, but its verification does not inspect
the two corridors between opposing flanks of different colors.
This is the usual extremal-crossing exposure: the path and the
configuration on its searched side determine the extremal path,
whereas edges strictly on its other side retain an FK conditional
law.  The main roads and their tethers are connected within the
central annulus.

We next join the two color lanes around both sides of the annulus.
In either remaining corridor, the dual flank is one target side.
The other target side consists of dual cells adjacent to the
corridor side of the primal flank.  We give the lattice detail needed
for this second side, since replacing it by a wired identification
would not produce the required barrier.

Take the central square cuts along dual-lattice rim lines, adjusting
by boundedly many rows.  The primal flank starts half a mesh before
the first rim line and ends half a mesh after the second.  Retain its
portion ending at its first visit to the ending line and beginning
at its last visit to the starting line before that ending-line visit.
Its interior then avoids both lines.  Follow
the centers of the dual cells incident to the path on its corridor
side.  At a turn, continue through the centers in that side's sector
around the primal vertex.  Consecutive centers are nearest neighbors;
none of these dual steps crosses an edge of the primal path.  The
resulting walk reaches both dual rim lines.  Retain a crossing
subwalk with its interior off those lines and erase loops.  This
gives a simple dual side arc, still in the flank strip and incident
to the primal path.  All its edges are on the unsearched side of
the extremal exposure.  Together with the dual flank and the two
rim intervals, it bounds a dual-lattice quadrilateral in the
unrevealed corridor.

The extremal distance between the two target sides of this
quadrilateral is uniformly bounded above.  Indeed, a family of
square contours at intermediate radii crosses the fixed-width gap
between the lane strips.  Continue each contour to its first hits
of the two target sides.  There are order $l$ such disjoint
contours, each of length $O(l)$.  The corresponding parallel paths,
or the extremal-length variational formula, give a constant upper
bound independent of the jagged flank geometry.  The quadrilateral
crossing theorem in Input~\ref{inp:fk} therefore gives an actual dual
crossing from the dual flank to a dual cell incident to the primal
flank, with conditional probability bounded below.  Already open
edges on the dual flank only help.  The two corridors are treated
in succession, each with its induced boundary partition.

The two corridor crossings, the tethers, and the two main roads now
form a closed route around the inner square.  One portion is an
actual primal path and the other an actual dual path.  They are
joined by exactly two steps between adjacent sites of opposite
colors.  Its winding number about the inner square is one: its
portions lie successively in the top lane, the right corridor, the
bottom lane, and the left corridor.  Erasing detours within any of
these four regions does not change that winding number.  Such a
route is a barrier even if it was not initially chosen simple.

Figure~\ref{fig:barrier} shows this closed route and the two
adjacency steps at which it can meet medial strands.
An actual color path does not cross a medial strand.  Each of the
two steps between adjacent opposite-color sites crosses exactly
one medial segment.  Every strand through-arc of a slightly wider
central band must intersect the closed route, because the route
separates the two rims.  Distinct through-arcs are disjoint in the
rounded drawing, and therefore use different crossing segments of
the route.  There are at most two through-arcs.  The longitudinal
roads give crossings of both colors, so the elementary annular
description above gives at least two.  There are exactly two.
Enlarge the central band by fixed margins while keeping both ends
of the main roads and their landing boxes outside it.  This gives
the stated extensions without changing the barrier argument.
Every construction has used finitely many crossing requirements
of fixed geometry, so their total probability has a positive
scale-independent lower bound.
\end{proof}

\begin{figure}[tbp]
\centering
\input{figures/barrier}
\caption{The topological consequence of the reference-band
construction.  A primal path and a dual path form a route of winding
number one around the inner rim, with exactly two adjacency steps
$a_1,a_2$ between colors.  Only those steps can cross medial strands.
Hence at most two strand arcs can cross the band; the two radial
color roads force both to be present.  The picture is schematic:
the adjacency steps are enlarged, the flanking crossings used to
construct the route are omitted, and neither the color paths nor
the medial arcs are asserted to be straight or smooth.}
\label{fig:barrier}
\end{figure}

\subsection{Landing extensions and gluing}

We use a strengthened form of arm separation which retains the
crossing clusters.  For a fixed-ratio bulk annulus and every
$\varepsilon>0$, the probability in the plane FK law that some crossing arms admit no
separated landing extensions in their respective original color
clusters is at most $\varepsilon$, when the separation parameter is
chosen small enough.  This is precisely the event $B_n$ preceding
Lemma~5.9 of \cite{CDCH}.  The replacement arms there are required to
belong to the same respective primal or dual clusters.  The proof
uses the quadrilateral crossing bounds, including at the exposed
jagged sides, the small- and large-extremal-length bounds, and an
upper bound on the number of crossing components.  These are all
available for $1\le q<4$ by \cite[Theorem~1.2 and Section~6]{DMT};
indeed its proof of Proposition~6.2 invokes this argument.  Thus we
use the proof at its stated probabilistic inputs, rather than an
Ising-specific identity.

We need two geometric features of these extensions.  Their endpoints
stay away from the corners of the square cut, and the extension
boxes can be much smaller than the separation between different
landings.  In the proof of \cite[Lemma~5.9]{CDCH}, separation occurs
at the scale denoted there by $\delta^3 n$, whereas extensions are
constructed at smaller scales $\delta^k n$, $k\ge4$.  Only finitely
many such scales are needed for a specified error tolerance.  The
arms consequently protrude a positive relative distance beyond
their landing cut, inside disjoint boxes having a fixed-factor
margin between them.  All constants may depend on the error
tolerance.  Bounded lattice rounding is harmless once the trial
scale exceeds a corresponding fixed minimum.

Suppose a past has been exposed up to a square cut of radius $l$,
and the conditional, unconditioned-FK probability that the specified
start reaches a larger cut of radius $8l$ is at least $h>0$.
Here and below the factors $8$ and $16$ may be increased by fixed
amounts to accommodate collars.  Apply the preceding landing
statement on an annulus strictly between those two cuts.  Its
failure event can be assigned ordinary plane probability less than
$h/(2C)$, where $C$ is the collar-comparison constant.  Its
conditional probability given the past is then less than $h/2$.
Thus, with conditional probability at least $h/2$, the two
start-connected clusters have separated outer extensions before
radius $16l$.  The crossing portions are taken after their last
visits to the annulus's inner rim.  Replacing those portions within
their same clusters preserves connection to the start.  A finite
cover of the landing rim by small boxes allows the landing
locations to be restricted to one pair of boxes, at a cost depending
on $h$ only.

There is a corresponding construction facing inward from the
remaining annulus toward $R$.  Arms beginning at a scale comparable
to $K l$, where $K$ is a sufficiently large fixed number, can have
separated inward protrusions and can be required to reach a little
beyond the target cut.  Their probability is bounded below by
$c\pi_2(l,R)$.  To check the relative cost of the landing failure,
crop the remaining arms through a fixed-ratio buffer before the
annulus where the landing test is performed.  Collar comparison
and quasimultiplicativity bound the joint probability of that
failure and the remaining arms by
$C\varepsilon\pi_2(l,R)$.  Choose $\varepsilon$ small, or use the
equivalent well-separated-arm comparison.  This also shows that
the construction is uniform under conditioning on data separated
from it by the specified collars.

The two sets of protrusions can be joined through the free space
between them.  Group their locations on a sufficiently fine but
fixed grid along each cut.  For every resulting pair of location
choices, there are disjoint corridors joining the primal tabs and
the dual tabs, with widths bounded below by a positive multiple
of $l$.  The constants can be small functions of $h$; they do not
change the total number of geometric trial scales.  In a
protrusion box, require a transverse crossing at a level between
the landing cut and the protrusion's outermost required level.
It must meet the protruding arm.  Overlapping longitudinal and
transverse crossings then carry that contact through its corridor.
Only one color is involved in each corridor and its overlap boxes.
The conditional crossing estimates and FKG in that color give a
positive fixed gluing probability.  This is the tube construction
of \cite[Proof of Proposition~5.4]{CDCH}.

The distinction between the central band and its extensions is
essential here.  An arbitrary event $E$ may fix \emph{every edge}
of the central band, so no gluing edge is allowed to lie there.
The protrusions extend outside that band; their overlap boxes and
the connecting corridors lie strictly outside it.  Within an
overlap box the only requirement inherited from the extension is
a connection in that box's color, which is increasing in its
color edges.  Thus FKG applies after $E$ is fixed.  The auxiliary
extensions are integrated out when we take a measure on the
central-band configurations.  They are not additional band data
that must agree in the eventual coupling.

\subsection{Frequent opportunities under the conditioned law}

We now prove Proposition~\ref{prop:outward}.  Denote the common
ambient law, including its permitted exterior reweighting, by
$\Pp_0$, and write
\[
 Q_a=\Pp_0(\,\cdot\mid S_R(a))
\]
for a particular start $a$.  Until the target scale, all conditional
edge laws before the $S_R(a)$ conditioning are ordinary FK laws
with induced partitions, or mixtures of such laws.  Consequently
the collar and crossing comparisons just established apply to
$\Pp_0$ as well.

Choose a fixed large ratio $K$ leaving room for the past-extension
annulus, the reference band with its two collars, and the
outer-extension annulus.  Trial radii are
$l_i=K^i l_0$, where $l_0$ is a fixed large multiple of $s$,
increased to a fixed minimum mesh scale when necessary.  Stop the
trials before a fixed fraction of $R$.  The number $k$ of available
trials satisfies
\[
 k\ge \frac{\log(R/s)}{\log K}-C.
\]
When the right side is bounded, enlarging the constant in
\eqref{eq:outward-error} suffices.  We may therefore assume there
are many trials.

Let $\mathcal F_i$ contain all edges inside the radius-$l_i$ cut,
together with the identities of the start-connected color
regions there.  Call trial $i$ good for start $a$ if
\begin{equation}\label{eq:good-trial}
 \Pp_0(S_{8l_i}(a)\mid\mathcal F_i)\ge h.
\end{equation}
The threshold $h>0$ is chosen after $K$.  The preceding separation
and gluing constructions then use narrower corridors if necessary;
they require no increase of $K$ depending on $h$.

Under $Q_a$, with probability at least $1-Ce^{-c k}$, at least
$3k/4$ of the trials are good.  Here is the conditioning calculation.
The event $S_R(a)$ implies $S_{8l_i}(a)$ at every trial.  For a
specified collection $J$ of bad trials, expose success to
$8l_i$ before passing to the next selected trial.  Since badness is
$\mathcal F_i$-measurable, the tower property and the negation of
\eqref{eq:good-trial} give a factor at most $h$ for each $i\in J$.
Moreover, by \eqref{eq:start-probabilities} and
quasimultiplicativity,
\[
 \frac{\Pp_0(S_R(a))}{\Pp_0(S_{l_0}(a))}
       \ge C^{-1}e^{-C_0 k}.
\]
The microscopic or one-arm factors in
\eqref{eq:start-probabilities} cancel in this ratio.  Summing over
collections of at least $k/4$ bad trials gives
\begin{equation}\label{eq:good-trial-tail}
 Q_a\{\text{at least }k/4\text{ bad trials}\}
 \le C e^{C_0 k}2^k h^{k/4}.
\end{equation}
Choose $h$ so small that the right side decreases exponentially
in $k$.  This choice is uniform over all the permitted starts and
exterior laws.

\subsection{A common band measure and loss of the past}

At a good trial, place a scaled copy of the reference band between
the past-extension and outer-extension annuli.  Its central
configuration has a subprobability measure $\nu_l$ defined as
follows: take the plane FK law, restrict to the successful
reference-band construction of Lemma~\ref{lem:reference-band},
including the auxiliary protrusions, and then retain only the
central-band edges.  There is a fixed $v_0>0$ such that
$\nu_l(\text{all band data})\ge v_0$.

For every event $E$ of central-band data and every good past,
\begin{equation}\label{eq:band-minorization-raw}
 \Pp_0\{\text{band data}\in E,\ S_R(a)
                     \mid\mathcal F_i\}
       \ge c_h\,\nu_l(E)\,\pi_2(l,R).
\end{equation}
Indeed, the good past supplies the separated start-connected
extensions with probability bounded below in terms of $h$.
Require the reference-band event with its protrusions, and
separated arms continuing toward $R$.  Before gluing, these
observations are separated by full collars.  Bounded-density
comparison can therefore be applied successively from the two
sides of the central band.  It preserves the factor $\nu_l(E)$
for an arbitrary, possibly nonmonotone, set $E$, and gives the
factor $\pi_2(l,R)$ for the remaining arms.  There is still
unobserved space for the two-color tube construction.  Its
same-color overlap requirements leave $E$ fixed, as explained
above, and complete both paths from the start to the target.
This proves \eqref{eq:band-minorization-raw}.

To make the averaging explicit, condition beyond a collar
containing all these constructions, and let $\xi$ denote the
resulting exterior state.  Given the already fixed past, write
$p_\xi$ for the conditional probability of reaching $8l$ and
$\mu$ for the conditional distribution of $\xi$.  Goodness
means $\int p_\xi\,\mu(\dd\xi)\ge h$; it does not assert
$p_\xi\ge h$ for each $\xi$.  Uniform collar comparison makes
the landing-error probability at most $\varepsilon$ for every
$\xi$, with $\varepsilon<h/2$.  In the preceding construction,
keep the probability of the start-connected extensions as a
factor instead of replacing it by a constant.  The finitely
many landing-location choices all have the same lower gluing
bound.  The conditional lower bound is therefore
\[
 c\nu_l(E)\pi_2(l,R)(p_\xi-\varepsilon)_+.
\]
Its integral is at least
$c\nu_l(E)\pi_2(l,R)(h-\varepsilon)$, as required in
\eqref{eq:band-minorization-raw}.  This proves the assertion for
the exterior mixture without imposing pointwise goodness.
No identification jumps an annulus used in these comparisons.

In the reverse direction, any successful continuation from the
past must have two arms in the remaining annulus.  Cropping
through a collar and using \eqref{eq:start-probabilities} gives
\begin{equation}\label{eq:band-denominator}
 \Pp_0(S_R(a)\mid\mathcal F_i)\le C\pi_2(l,R).
\end{equation}
Dividing \eqref{eq:band-minorization-raw} by
\eqref{eq:band-denominator} shows that, under $Q_a$, the band
marginal after every good past dominates $c\nu_l$.  Hence two
good pasts, even for different starts, have a uniformly positive
common part on identical central-band data.

It remains to check that coupling this band is enough.  Cut its
strands at both rims.  There are two through-arcs.  All remaining
arcs have both ends on the same rim.  Whatever the configuration
inside the inner rim, completing its pairings joins the two
through-arcs to one another: they are the only two strand ends
left after the same-rim arcs and the interior pairings are
followed.  Interior loops can change in number, but their loop
weight is then a factor depending solely on that interior
completion.  The effective pairing seen on the outer rim is
fixed by the band.

The same fact can be expressed in FK terms.  Exactly one primal
region and one dual region transmit through the band.  An
exterior connection that enters the past must do so through the
transmitted region of its color, whose trace on the outer rim is
already determined by the band.  Thus the induced exterior
primal partition is independent of the earlier past.  On $S_R(a)$,
the starting regions have made contact with these two transmitted
regions.  Conditional on that contact, the remaining constraint
is only that the transmitted regions continue to the target
cut.  Paths that cross the band repeatedly use the same
transmitted region and impose no additional constraint.  The
domain Markov property now shows that the conditional law on
and outside the outer rim, given the common band and successful
continuation, is identical for the two pasts.  A common exterior
weight or an ambient wall farther out changes this common law
in the same way in both cases.

We can therefore complete the coupling as follows.  At each trial,
if both pasts are good, use the common submeasure $c\nu_l$ to
couple the central-band configurations with probability at least
$p>0$.  On success, fill the interior gaps according to their
respective conditional laws and sample the exterior from the
common conditional law just described.  On failure, use the
residual band marginals and fill the data up to the next trial
cut with their correct conditional laws.  No edges beyond that
next cut are inspected merely to decide whether a coupling
attempt failed.  This is a coupling of the successive conditional
kernels, so each complete marginal remains $Q_a$ throughout.

For completeness, this iteration does not require the trials to
be independent.  By \eqref{eq:good-trial-tail} and a union bound
over the two marginals, except with probability $Ce^{-c k}$
there are at least $k/2$ jointly good trials.  Their goodness is
known from the two pasts before the corresponding attempt.
At each such trial, conditional on the entire coupled past and
on all earlier failures, the overlap probability is at least
$p$.  Iterating at these predictable trial times bounds the
probability of $k/2$ failed good attempts by $(1-p)^{k/2}$.
The total failure probability is consequently at most
$Ce^{-c'k}\le C(s/R)^c$.  On success the band and its exterior
agree, and the two transmitted color regions have been reached
by both starts.  This proves Proposition~\ref{prop:outward}.

For a tile-side start, the common through-strand passes through
the associated medial segment: otherwise that segment would lie
on a completed inner loop surrounding one of its two end sites,
contrary to $S_R(e)$.  For a disk start, the common through-strand
meets the disk, since every disk-to-target strand must pass
through the band.  Thus an exterior rule selecting the
distinguished Dobrushin strand gives the same verdict for a
microscopic segment and for a disk after successful coupling.
This is the form used in the occupation comparison.
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//06-coupling.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//07-calibration.tex ====
\section{From electric correlations to microscopic normalization}
\label{sec:calibration}

The electric formula is accurate when marked points remain separated
by a mesoscopic distance. Counting an edge requires two opposite-color
marked points at distance one. The outward coupling bridges these
scales. We first obtain a pure-power amplitude for a neighboring pair,
then compare that pair with a disk. The latter comparison is the
normalization needed for occupation measures.

\subsection{A signed factor with positive local conditioning}

Fix the two neutral charge groups of
Proposition~\ref{prop:special-charges}. Each group has a core charge
$\alpha=\pi/2-\lambda$ and sufficiently many small negative
compensating charges. Place the compensators at distinct points
$u_j(n)$ on the axis by fixed, color-preserving lattice roundings
$u_j(n)=n\zeta_j+O(1)$, where the $\zeta_j$ are distinct fixed
locations away from the origin. The two cores lie within distance
$r=o(n)$ of the origin, in opposite colors. The first core is held
fixed. Write $e$ for the neighboring pair corresponding to a medial
segment, and write $r$ for the separated pair, with the appropriate
odd lattice separation. All rounding conventions preserve these
colors.

Let
\[
 B_* = \sum_{j\text{ remote}}\beta(t_j),\qquad
 \beta(t)=-A(t^2+2\lambda t),\qquad
 A=\frac{1}{2\pi(\pi-\lambda)}.
\]
Choose disjoint small patches of radius a fixed multiple of $n$ around
the remote points. Reweight only the loops wholly contained in these
patches, using their single-point weights
$v(t_j)=\cos(\lambda+t_j)/\cos\lambda$. These weights are positive.
Let $Z_*(n)$ be the mass of this local reweighting and let
$\Pp_n^*$ be its normalized probability law. The cutoff comparison
and the pure-nest asymptotic give
\begin{equation}\label{eq:proxy-mass}
 Z_*(n)\asymp\prod_{j\text{ remote}}F_n(v(t_j))
 \asymp n^{B_*}.
\end{equation}
Constants are uniform over the finitely many rounding classes used
below. We use the positive-winding expansion of the electric
correlation. Its remaining loop factor, denoted by $Y_r$, need not
be nonnegative.

Let $R$ be larger than the core separation by a fixed factor and
smaller than a sufficiently small multiple of $n$. The event $S_R(r)$
means that both core colors reach the tile rim of radius $R$.
The scalar weight of a loop enclosing just one core and no remote point
is zero, because $\cos(\lambda+\alpha)=0$. A completed loop inside
the $R$-disk enclosing both opposite-color cores contains another
loop separating them. Thus any nonzero configuration has $S_R(r)$.
Conversely, factoring out its indicator merely restricts the original
loop product. Consequently the full electric correlation satisfies
\begin{equation}\label{eq:core-factorization}
 C_r(n)=Z_*(n)\Pp_n^*(S_R(r))
             \E_n^*[Y_r\mid S_R(r)].
\end{equation}
The same identity holds for the neighboring pair $e$.

We record two uniform bounds used throughout this section. For every
fixed $j\ge1$,
\begin{align}
 \E_n^*[|Y_r|^j\mid S_R(r)]&\le C_j,
 \label{eq:macro-moments}\\
 \frac{\Pp_n^*(S_R(r))}{\Pp(S_R(r))}
 &=1+O((R/n)^{c_0}).
 \label{eq:proxy-local-probability}
\end{align}
Here $\Pp$ denotes the ordinary plane FK law; the same statements hold
with $r$ replaced by $e$. To justify the first bound, discard
configurations on which $Y_r=0$. Every non-unit factor remaining
on the other configurations belongs to a macroscopic loop.
A separating loop must enclose a remote point, and an ancestor
enclosing both cores contains such a separating loop. A loop charged
only at remote points but not included in the local reweighting has
left its remote patch. Each such loop therefore has a passage through
one of finitely many buffered regions of size comparable to $n$.
These buffers may be chosen away from the support of $S_R(r)$.
Even a loop with an arbitrarily large excursion has such a passage.
Conditional passage moments and the local-tilt estimate of
Input~\ref{inp:fk} control every fixed moment of the product.
Cutting the arm event back by a fixed factor leaves collars before
the passage buffers, so the estimates cost its ordinary probability,
not its reciprocal. This gives \eqref{eq:macro-moments} after
normalization. The relative collar comparison gives
\eqref{eq:proxy-local-probability}, because the nonnegative local
reweighting is supported at distance comparable to $n$.

\begin{lemma}\label{lem:signed-comparison}
Consider two opposite-color core pairs contained in a disk of radius
$s$, with the same remote charges and local reweighting. If
$Cs<R<cn$, then
\[
 \left|\E_n^*[Y_1\mid S_R(1)]-
             \E_n^*[Y_2\mid S_R(2)]\right|
 \le C(s/R)^{c_1}.
\]
The constants are independent of the microscopic separations of the
pairs.
\end{lemma}

\begin{proof}
Apply Proposition~\ref{prop:outward} to the two starts, with the common
nonnegative remote reweighting. On successful coupling the band and
the entire exterior agree. Exactly one global loop has the two
through-arcs in that band. It separates the opposite-color cores:
an inner closure separating them would contradict $S_R$. Its
enclosures of remote points are the same in both configurations.
Every other loop outside the band either encloses both cores or
neither, and the ancestors of the through-loop enclose both.
Inside the band no completed loop can enclose a core under $S_R$.
The factors $Y_1,Y_2$ therefore agree on successful coupling.
H\"older's inequality and \eqref{eq:macro-moments} bound their
contribution on coupling failure by a positive power of its probability.
Reducing the exponent in Proposition~\ref{prop:outward} proves the
claim. No positivity of $Y_i$ is used.
\end{proof}

\subsection{The neighboring-pair amplitude}

Set $x_1=-\beta(\alpha)$. The parameter identities are
\begin{equation}\label{eq:exponent-identity}
 2x_1+2A\alpha^2=x,
 \qquad
 x=\frac{\pi-2\lambda}{2(\pi-\lambda)}.
\end{equation}
For a sufficiently small fixed $\xi>0$, choose an odd separation
$r\asymp n^{1-\xi}$. Proposition~\ref{prop:special-charges} gives
\begin{equation}\label{eq:mesoscopic-core}
 \begin{split}
 C_r(n)
 &=M_* n^{B_*-x}r^{2A\alpha^2}(1+O(n^{-c_2}))\\
 &=M_* n^{B_*}r^{-2x_1}(r/n)^x(1+O(n^{-c_2})),
 \qquad M_*\ne0.
 \end{split}
\end{equation}
The amplitude depends on the fixed remote geometry, charges, and
rounding convention, but not on $n$ or $r$ in this regime. Indeed the
full Coulomb formula uses the actual pairwise distances; replacing
the remote distances from either core by their limiting values changes
its product by $O(r/n)+O(n^{-1})$. These errors can be included in
the power remainder. We will not use this equivalence with a fixed
$M_*$ when $r/n$ is held positive.

\begin{proposition}\label{prop:microscopic-amplitude}
For the fixed remote experiment there exists $c_*\ne0$ such that
\begin{equation}\label{eq:microscopic-amplitude}
 C_e(n)=c_*n^{B_*-x}(1+O(n^{-c_3})).
\end{equation}
In particular this limit holds through all positive integers $n$.
\end{proposition}

\begin{proof}
Take
\[
 r\asymp n^{1-\xi},\qquad R\asymp n^{1-b\xi},
 \qquad 0<b<1.
\]
All radii are chosen once for the comparison interval $[n,2n]$.
From \eqref{eq:core-factorization}, \eqref{eq:proxy-mass},
\eqref{eq:mesoscopic-core}, the one-arm bound, and
\eqref{eq:start-probabilities}, we obtain, for every small
$\delta>0$,
\begin{equation}\label{eq:expectation-lower}
 \left|\E_n^*[Y_r\mid S_R(r)]\right|
 \ge c_\delta(R/n)^x(r/R)^\delta.
\end{equation}
For example the upper estimate
$\Pp(S_R(r))\le C_\delta r^{-2x_1}(r/R)^{x-\delta}$
in the denominator gives precisely this inequality.
Its right side has order
$n^{-[bx+\delta(1-b)]\xi}$, whereas the signed comparison error
is $O(n^{-c_1(1-b)\xi})$. Choose $b>0$ sufficiently small and then
$\delta>0$ sufficiently small that
\[
 c_1(1-b)>bx+\delta(1-b).
\]
Lemma~\ref{lem:signed-comparison} is then a relative power-error
comparison of the two conditional expectations.

Apply the same argument with the remote geometry at any integer scale
$N\in[n,2n]$, using the \emph{same} core positions, $r$, and $R$.
Equation~\eqref{eq:proxy-local-probability} removes the remote tilt
from the arm probabilities with a power error. Uniformly in this
interval,
\begin{equation}\label{eq:fixed-radii-ratio}
 \frac{C_e(N)}{C_r(N)}
 =\frac{\Pp(S_R(e))}{\Pp(S_R(r))}(1+O(n^{-c_4})).
\end{equation}
The ordinary probability ratio on the right does not depend on $N$.
The mesoscopic formula therefore yields
\begin{equation}\label{eq:dyadic-amplitude-ratio}
 \frac{N^{x-B_*}C_e(N)}{n^{x-B_*}C_e(n)}
 =1+O(n^{-c_5}),\qquad n\le N\le2n.
\end{equation}
The denominator is nonzero for all sufficiently large $n$, since
\eqref{eq:expectation-lower} dominates the coupling error.

Write $a_n=n^{x-B_*}C_e(n)$. On a dyadic sequence, the consecutive
ratios of $a_n$ differ from one by a summable sequence. Their product
converges to a nonzero value, with tail error $O(n^{-c_5})$.
Equation~\eqref{eq:dyadic-amplitude-ratio} compares every intervening
integer to the preceding dyadic scale. Hence the entire sequence has
the same nonzero limit and the stated power error.
\end{proof}

\begin{proposition}\label{prop:sharp-arms}
For all radii above a fixed lattice scale,
\begin{equation}\label{eq:sharp-arms}
 \Pp(S_R(e))\asymp R^{-x},\qquad
 \pi_2(r,R)\asymp(r/R)^x.
\end{equation}
The estimates hold with fixed-factor changes of the inner and outer
rims and with a further bulk collar.
\end{proposition}

\begin{proof}
First choose $R=\theta n$, with $\theta>0$ a sufficiently small fixed
constant. Equations~\eqref{eq:core-factorization},
\eqref{eq:macro-moments}, and~\eqref{eq:microscopic-amplitude}
give the lower bound $\Pp(S_R(e))\ge c n^{-x}$.

For the reverse bound, compare the neighboring pair with an
opposite-color pair at separation $r\asymp aR$, where $a>0$ is a
small fixed constant. Use the full Coulomb formula, not the
$r=o(n)$ simplification with its limiting amplitude. Its magnitude
is uniformly comparable to
$n^{B_*}r^{-2x_1}(r/n)^x$ as first $n\to\infty$ and then
$a\downarrow0$. The nonzero amplitude and the remote-distance
factors vary continuously in this limit. It follows that
\[
 \left|\E_n^*[Y_r\mid S_R(r)]\right|
 \ge c_\delta\theta^x a^\delta.
\]
Choose $\delta<c_1$ and then $a$ small enough that
$Ca^{c_1}<\tfrac12 c_\delta\theta^x a^\delta$.
Lemma~\ref{lem:signed-comparison} gives a strictly positive lower
bound for the magnitude of the conditional expectation at $e$.
Equation~\eqref{eq:core-factorization} now gives
$\Pp(S_R(e))\le C n^{-x}$. Fixed-factor extensions cover all large
$R$, and finite energy covers the remaining bounded scales.

The first line of \eqref{eq:start-probabilities} identifies
$\Pp(S_R(e))$ with $\pi_2(C,R)$. Quasimultiplicativity therefore gives
\[
 \pi_2(r,R)\asymp
 \frac{\pi_2(C,R)}{\pi_2(C,r)}\asymp(r/R)^x.
\]
All comparisons used buffered events, so their collar and rounding
versions follow from the same fixed-factor estimates.
\end{proof}

\subsection{Calibration by disk probes}

A disk does not carry a prescribed electric charge. We therefore
introduce a diagnostic loop factor which agrees with the neighboring
pair's factor after successful outward coupling. Its purpose is only
to determine deterministic normalization constants.

Let $s=\epsilon n<R$. Under the same remote local reweighting, let
$S_R(s)$ be the disk two-arm event. If exactly one global loop touches
the disk and leaves the $R$-cut, assign that loop the factor it would
receive as a separator of the two cores, with its actual remote
enclosures. Give its ancestors the factors corresponding to enclosing
both cores, and give other loops no core charge. Divide out the local
remote weights already included in $\Pp_n^*$. Call the resulting
factor $Y_s$. In every other case set $Y_s=0$. In particular the
distinguished loop receives value zero when it encloses no remote
point. Define the deterministic quantity
\begin{equation}\label{eq:disk-diagnostic}
 W_s(n)=Z_*(n)\Pp_n^*(S_R(s))
                       \E_n^*[Y_s\mid S_R(s)].
\end{equation}
The moment bound \eqref{eq:macro-moments} also holds for $Y_s$:
every charged loop in a nonzero configuration spans between the disk
and a remote patch, or leaves a remote patch.

In the coupling of the disk start and the edge start, the factors
$Y_s$ and $Y_e$ coincide on success. The unique band through-loop
touches the disk, and no second loop can both touch the disk and leave
$R$. Replacing its inside portion preserves its remote enclosures;
the same is true for its ancestors. Thus
\begin{equation}\label{eq:disk-signed-comparison}
 \left|\E_n^*[Y_s\mid S_R(s)]-
             \E_n^*[Y_e\mid S_R(e)]\right|
 \le C(s/R)^{c_6}.
\end{equation}

\begin{lemma}\label{lem:disk-diagnostic}
For almost every fixed pair of scaled radii $0<\epsilon<\theta$
in the permissible range, with $s=\epsilon n$ and $R=\theta n$,
the limit
\begin{equation}\label{eq:diagnostic-limit}
 \lim_{n\to\infty}n^{-B_*}W_s(n)
\end{equation}
exists. Tile-rim roundings do not change the limit.
\end{lemma}

\begin{proof}
We use Corollary~\ref{cor:macroscopic-test}. First remove the nest weights
below radius $\delta n$ at each remote point, dividing by the
corresponding single-point nest partition functions. Their inward
coupling and cutoff insensitivity make the normalized error tend to
zero as $\delta\downarrow0$, uniformly in $n$.

There is a minor extra condition here: a loop deciding the disk
diagnostic might enter one of those small neighborhoods. It can be
discarded at a vanishing cost. Indeed the positive remote tilt produces
a surrounding screen before reaching the neighborhood with probability
tending to one. A loop arriving from the distant disk cannot cross
that screen. More annular trials reduce the exceptional probability,
and the moment bound for the remaining loop factors makes its weighted
contribution vanish. Thus, outside a vanishing exceptional event, the
diagnostic is determined by the truncated macroscopic configuration.

For fixed $\delta$, truncate excursions beyond a large fixed multiple
of $n$ and the number of charged macroscopic passages. The remaining
test uses finitely many loops of diameter bounded below after scaling.
It reads their ordered trajectories, their windings about the remote
points, and their intersections with the two disk rims. In a nonzero
diagnostic, ancestors of the distinguished loop can be recognized
among the remote-enclosing loops by whether they enclose the disk
center: any other such loop is disjoint from the disk, since otherwise
it too would touch the disk and leave $R$.

Input~\ref{inp:loops} gives convergence of these loop tests. Fixed
marked points are almost surely off all macroscopic loop traces.
For almost every choice of radii, no relevant loop is at a tangency
threshold, so intersection tests are continuity events. This follows
also by Fubini from the countable loop collection and its matching
topology. Bounded tile-rounding errors do not affect such tests.
The passage moment bounds remove the count and excursion truncations.
Finally let $\delta\downarrow0$. The pure-power asymptotics of the
remote nest factors identify their normalization with $n^{B_*}$,
proving \eqref{eq:diagnostic-limit}.
\end{proof}

\begin{proposition}\label{prop:disk-calibration}
There is a sequence $\epsilon\downarrow0$, a choice of radii
$R/n\asymp\epsilon^b$, and positive deterministic numbers
$L_\epsilon\asymp\epsilon^{-x}$ such that
\begin{equation}\label{eq:disk-calibration}
 \limsup_{n\to\infty}
 \left|
 \frac{n^x\Pp(S_R(e))}{L_\epsilon\Pp(S_R(\epsilon n))}-1
 \right|\le C\epsilon^{c_7}.
\end{equation}
The exponent $b>0$ may be chosen sufficiently small in advance to
satisfy any additional finitely many inequalities of the form
$(1-b)c>bx$, with $c>0$ fixed. The same constants apply to every
translate and rotation of the edge start, including color exchange.
\end{proposition}

\begin{proof}
Write $t=R/n$. Equations~\eqref{eq:core-factorization},
\eqref{eq:proxy-mass}, \eqref{eq:microscopic-amplitude}, and
\eqref{eq:sharp-arms} imply
\begin{equation}\label{eq:edge-expectation-size}
 \left|\E_n^*[Y_e\mid S_R(e)]\right|\asymp t^x.
\end{equation}
For $t\asymp\epsilon^b$, the relative error in
\eqref{eq:disk-signed-comparison} is bounded by
$C\epsilon^{(1-b)c_6-bx}$. Choose $b$ so that this exponent is
positive; it can simultaneously satisfy the stated finite collection
of further inequalities. The relative collar error in
\eqref{eq:proxy-local-probability} is $O(\epsilon^{bc_0})$.
Consequently
\begin{equation}\label{eq:disk-ratio-identity}
 \frac{n^x C_e(n)}{W_s(n)}
 =\frac{n^x\Pp(S_R(e))}{\Pp(S_R(s))}
                    (1+O(\epsilon^{c_7}))
\end{equation}
for a positive exponent $c_7$ after it is decreased if necessary.
Although the two correlations can be signed, their ratio in this
display is positive for small $\epsilon$, by the relative comparison.

Choose a sequence of continuity pairs for
Lemma~\ref{lem:disk-diagnostic}, with
$R/n\asymp\epsilon^b$. The sharp arm estimates show that the right
side of \eqref{eq:disk-ratio-identity} is comparable to
$\epsilon^{-x}$. Thus the limit of $n^{-B_*}W_s(n)$ is nonzero for
each sufficiently small chosen $\epsilon$. Define
\[
 L_\epsilon=
 \frac{c_*}{\displaystyle\lim_{n\to\infty}n^{-B_*}W_s(n)}.
\]
The numerator is the amplitude in
\eqref{eq:microscopic-amplitude}. This definition and
\eqref{eq:disk-ratio-identity} give positivity,
$L_\epsilon\asymp\epsilon^{-x}$, and
\eqref{eq:disk-calibration}.

All probabilities in \eqref{eq:disk-calibration} are ordinary plane
probabilities. Translation, square symmetries, and planar self-duality
identify them for the different edge locations and orientations, using
covariant tile-disk roundings. Therefore the calibration does not
depend on the checkerboard class or orientation of the probe.
\end{proof}
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//07-calibration.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//07-probabilistic-endgame.tex ====
\section{Moment bounds and convergence of the height field}\label{sec:probability}

The current limits of Section~\ref{sec:chiral} concern insertions at
separated times. We now pass to the height field itself. Three additional
estimates are needed: ordinary moments for spatial averages, compactness
in negative regularity, and a uniform estimate for removing an endpoint
smoothing. The last estimate also permits integration against the finite
signed measures in Theorem~\ref{thm:main}.

Throughout the new argument, fix \(0<c<\sqrt3\), and retain the plane state,
\(\kappa\), the vacuum \(\Omega\), the shift \(S=P(i)\), and the
nonnegative endpoint Hamiltonian \(E\) from Section~\ref{sec:angular}.
The Gaussian field appearing before its normalization is identified has
covariance
\begin{equation}\label{eq:prob-log-kernel}
 K_\kappa(x,y)=-2\kappa\log|x-y|.
\end{equation}
We denote this field modulo constants by \(\mathcal H_\kappa\).
All constants below may depend on the fixed parameter and the stated
moment order. None depends on an integer approximation to \(\delta^{-1}\):
\(\delta>0\) is always an arbitrary real mesh size.

\subsection{Ordinary moments from the endpoint Hamiltonian}

On an equal-time column, \(Z_l\) is the diagonal spin at site \(l\), with
values \(\pm1\). These are self-adjoint site operators; they should be
distinguished from marked time rows. A path height increment is the
negative of its step spin. Vacuum moments of the commuting site spins
are their ordinary moments under \(\P_c\).

\begin{lemma}[Ordinary spatial moments]\label{lem:ordinary-moments}
Fix \(0<A<\infty\). Let \(R\ge1\), and let \(g=(g_l)_{l\in\Z}\) be a
finitely supported real sequence such that
\begin{equation}\label{eq:prob-g-conditions}
 \sum_l|g_l|\le AR,\qquad
 \sum_l|g_{l+1}-g_l|\le A,\qquad
 \sum_l|g_{l+1}-g_l|^2\le A/R.
\end{equation}
Then, for every \(1\le b<\infty\),
\begin{equation}\label{eq:prob-ordinary-moments}
 \left\|\sum_lg_lZ_l\right\|_{L^b(\P_c)}\le C_{A,b}.
\end{equation}
In particular the conclusion holds if \(g\) has support of length at
most \(AR\), \(|g_l|\le A\), and
\(|g_{l+1}-g_l|\le A/R\). The same assertions hold with the two axes
interchanged.
\end{lemma}

\begin{proof}
Work throughout this proof in \(\mathcal H_{\mathrm{spin}}\), the closed
span of \(O\Omega\) with \(O\) a finite local spin matrix.
By Lemma~\ref{lem:endpoint-Hamiltonian}, this subspace contains
\(\Omega\) and reduces \(E\). Every finite local spin matrix and its
adjoint preserve the local generating vectors and hence their closed
span. In particular \(D=D_g=\sum_lg_lZ_l\) is bounded and self-adjoint
on \(\mathcal H_{\mathrm{spin}}\). The shift preserves this subspace
in both directions, since it translates local matrices and fixes
\(\Omega\). In this proof \(E\) denotes its self-adjoint restriction
to \(\mathcal H_{\mathrm{spin}}\),
\(B_M=\bbone_{\{E\le M/R\}}\) is its spectral projection there, and
all Hilbert-space norms are taken on this subspace.

First we prove a two-sided compression bound. For fixed \(M\),
Lemma~\ref{lem:endpoint-Hamiltonian} places the range of \(B_M\)
inside the ambient spectral band \(H_0\le C M/R\), when \(R\) is
sufficiently large. The full-space estimates of Lemma~\ref{lem:low-band}
therefore apply to these compressions without increasing their norms.
The identity
\[
 N(i)=\tfrac12S^2(Z_0+Z_1)
\]
and translation by the unitary shift therefore give
\(\|B_M(Z_l+Z_{l+1})B_M\|\le C_M/R\).
Pair the consecutive sites in the sum for \(D\). The coefficients of the
paired sums have total absolute value at most \(AR\); the remaining
single-site terms have total absolute coefficient at most
\(\sum_l|g_{l+1}-g_l|\). Consequently
\begin{equation}\label{eq:prob-two-sided}
 \|B_MDB_M\|\le C_{A,M}.
\end{equation}
For bounded \(R\), this follows instead from
\(\|D\|\le\sum_l|g_l|\).

The local-link description of \(E\) in
Lemma~\ref{lem:endpoint-Hamiltonian} gives
\begin{equation}\label{eq:prob-double-commutator}
 \|[D,[E,D]]\|\le C\sum_l|g_{l+1}-g_l|^2\le C_A/R.
\end{equation}
Indeed the diagonal part of a link commutes with \(D\), and its spin
exchange preserves \(Z_l+Z_{l+1}\); only the coefficient difference
\(g_l-g_{l+1}\) enters each commutator. These identities initially hold
on the local analytic core in \(\mathcal H_{\mathrm{spin}}\), which is
a core for the restricted \(E\). On that core, \([E,D]\) is a finite
sum of bounded local operators. To extend it, approximate a vector in
\(\Dom(E)\subset\mathcal H_{\mathrm{spin}}\) in the graph norm by local
core vectors and use \(EDu=DEu+[E,D]u\) there. Boundedness of \(D\)
and of this commutator, followed by closedness of \(E\), shows that
\(D\) preserves the restricted domain and that the identity extends to
it. The same holds for every fixed local power of \(D\), justifying
the following energy-projection and vacuum calculations on this subspace.

Set \(m_* =\|DB_1\|\), which is finite since \(D\) is bounded.
For a unit vector \(u\in B_1\mathcal H_{\mathrm{spin}}\), expanding the double
commutator yields
\[
 \langle Du,EDu\rangle
 =\tfrac12\langle u,[D,[E,D]]u\rangle
      +\Re\langle Du,DEu\rangle
 \le \frac{C_A+m_*^2}{R}.
\]
Here \(Eu\in B_1\mathcal H_{\mathrm{spin}}\) and \(\|Eu\|\le R^{-1}\).
The component of \(Du\) above energy \(2/R\) thus has squared norm at
most \((C_A+m_*^2)/2\). Its component below \(2/R\) is bounded by
\eqref{eq:prob-two-sided}, with \(M=2\). Taking a supremum gives
\[
 m_*^2\le C_A+\tfrac12m_*^2,
 \qquad\text{and therefore}\qquad
 \|DB_1\|=\|B_1D\|\le C_A.
\]

We next bound powers of \(D\) on the vacuum. Since \(E\Omega=0\),
\[
 2\langle D^k\Omega,ED^k\Omega\rangle
 =\langle\Omega,[D^k,[E,D^k]]\Omega\rangle.
\]
In the site-diagonal representation, let \(D^{\mathrm{sw},l}\) be the
value of \(D\) after exchanging the spins at \(l,l+1\). An exchange
term in this double commutator has, up to a bounded link coefficient
and sign, the multiplier
\(\bigl(D^k-(D^{\mathrm{sw},l})^k\bigr)^2\).
This nonnegative multiplier commutes with the exchange. Furthermore,
\[
 |D-D^{\mathrm{sw},l}|\le2|g_l-g_{l+1}|,
 \qquad
 \bigl|D^k-(D^{\mathrm{sw},l})^k\bigr|^2
 \le C_{A,k}|g_l-g_{l+1}|^2(1+|D|^{2k-2}).
\]
For the second inequality use the factorization of \(a^k-b^k\) and
\(|a-b|\le2A\). Taking the absolute value of an exchange expectation
is bounded by the expectation of its nonnegative multiplier, since the
exchange is a norm-one involution commuting with that multiplier.
Summing the links proves
\begin{equation}\label{eq:prob-power-energy}
 \langle D^k\Omega,ED^k\Omega\rangle
 \le \frac{C_{A,k}}R
       \bigl(1+\|D^{k-1}\Omega\|^2\bigr).
\end{equation}
The low-energy component satisfies
\(\|B_1D^k\Omega\|\le C_A\|D^{k-1}\Omega\|\); on its complement,
positivity of \(E-R^{-1}\) bounds the squared norm by
\(R\langle D^k\Omega,ED^k\Omega\rangle\).
Induction in \(k\) now bounds every \(\|D^k\Omega\|\).
These are the even ordinary moments of \(D\); interpolation gives all
finite orders. Axial symmetry gives the other orientation.
\end{proof}

\begin{corollary}[Neutral blocks]\label{cor:neutral-blocks}
Let \(b\in\R^2\), \(r>0\), and let \(f\) be a measurable real density
supported in a square of side \(Ar\) centered within distance \(Ar\)
of \(b\), with
\[
 \int f=0,\qquad |f|\le A r^{-2}.
\]
For every finite \(q\),
\begin{equation}\label{eq:prob-neutral-block}
 \|\langle h_\delta,f\rangle\|_{L^q}\le C_{A,q}
\end{equation}
uniformly in \(b,r,\delta\). Products of finitely many such blocks have
uniform absolute moment bounds, even when their supports overlap.
\end{corollary}

\begin{proof}
Suppose first \(r/\delta\ge1\), and put \(R=r/\delta\).
Integrate \(f\) on mesh cells, giving masses \(a_{jl}\) of size
\(O(R^{-2})\), with \(O(R)\) possible indices in each direction.
Let \(m_j=\sum_l a_{jl}\), and choose a mass-one discrete profile
\((p_l)\), of width \(O(R)\), bounded by \(C/R\).
The masses \(a_{jl}-m_jp_l\) are neutral in each column. Their discrete
primitives are bounded by \(C/R\), with consecutive differences
\(O(R^{-2})\). Summation by parts and
Lemma~\ref{lem:ordinary-moments}, applied after multiplication by \(R\),
bound the \(L^q\) norm of each column contribution by \(C_q/R\).
Their sum is bounded by Minkowski's inequality.
For the remaining masses \(m_jp_l\), note that
\(\sum_jm_j=0\), \(|m_j|\le C/R\); sum by parts in the other axis,
then average against \(p_l\). This gives the same bound.
If \(r<\delta\), the support meets only a bounded number of neighboring
faces. Neutrality, bounded total variation, and unit height jumps give
a deterministic bound. Finally apply Hölder's inequality to products,
using the required higher moments. No disjointness is used.
\end{proof}

\subsection{Smooth test functions}

\begin{proposition}\label{prop:smooth-tests}
For every finite family of smooth compactly supported densities
\(\phi_1,\ldots,\phi_m\) of integral zero, the vector
\(\bigl(\langle h_\delta,\phi_i\rangle\bigr)_i\) converges jointly in law
and in every joint moment to the corresponding pairings of
\(\mathcal H_\kappa\). The moment convergence is uniform when the tests
vary in a compact smooth family with a common compact support.
\end{proposition}

\begin{proof}
First assume that \(\int\phi(x,y)\,dy=0\) for every \(x\).
For fixed continuum time \(x\), put
\(F(x,y)=\int_{-\infty}^y\phi(x,t)\,dt\).
Discrete summation by parts writes the line pairing as
\[
 \int h_\delta(x,y)\phi(x,y)\,dy
 =-\sum_{l\in\Z}F(x,(l+1)\delta)
       \bigl(h(\lfloor x/\delta\rfloor,l+1)
             -h(\lfloor x/\delta\rfloor,l)\bigr).
\]
The primitive is compactly supported, bounded, and changes by \(O(\delta)\)
at consecutive sites. Lemma~\ref{lem:ordinary-moments} therefore bounds
this pairing in every \(L^q\), uniformly in \(x\) and \(\delta\).
The same remains true for any finite product of line pairings.

Replace the forward height differences in this expression by the
centered differences of Proposition~\ref{prop:gaussian-currents}, shifting
the coefficients accordingly. The coefficient error has size
\(O(\delta)\) and consecutive differences \(O(\delta^2)\), on
\(O(\delta^{-1})\) sites. Dividing this error by \(\delta\) puts it
under Lemma~\ref{lem:ordinary-moments}; the resulting \(L^q\) error is
\(O(\delta)\), uniformly in the line coordinate.

Expand a joint moment as an integral over its time coordinates.
For distinct fixed times, Proposition~\ref{prop:gaussian-currents}
applies to the transverse Riemann sums, locally uniformly on their
compact supports and including the rounding of faces. Their limit is
Wick's formula for transverse derivatives of \(K_\kappa\).
Integration by parts against the compact primitives gives the
undifferentiated test densities and the kernel
\eqref{eq:prob-log-kernel}. The line moment bounds dominate the integrand
uniformly. Equal-time walls have Lebesgue measure zero, so dominated
convergence completes the time integrations. The resulting products of
logarithmic kernels are integrable: in each Wick pairing the two-point
variables are disjoint from those of every other pair, and the logarithm
is locally integrable in two dimensions. Gaussian moments determine the
law. Axial symmetry gives the same result for horizontally line-neutral
tests.

For an arbitrary smooth neutral \(\phi\), let
\[
 q(x)=\int\phi(x,y)\,dy,\qquad
 p_R(y)=R^{-1}p(y/R),\qquad
 \phi_R(x,y)=\phi(x,y)-q(x)p_R(y),
\]
where \(p\) is a fixed smooth compact probability density.
Then \(\phi_R\) is vertically line-neutral, while \(q p_R\) is
horizontally line-neutral. For fixed \(R\), the already proved result
gives
\[
 \lim_{\delta\downarrow0}\E|\langle h_\delta,q p_R\rangle|^2
 =\iint K_\kappa(z,z')q(x)p_R(y)q(x')p_R(y')\,dz\,dz'.
\]
This tends to zero as \(R\to\infty\). To see this, set
\(y=Rt\), \(y'=Rt'\). The additive \(-2\kappa\log R\) contributes
zero because \(\int q=0\). The remaining logarithm is
\(\log\sqrt{((x-x')/R)^2+(t-t')^2}\), which converges almost everywhere
to \(\log|t-t'|\), independently of \(x,x'\).
On the fixed supports it is bounded in absolute value by
\(C+\log_+(1/|t-t'|)\). Dominated convergence and neutrality of \(q\)
prove the assertion. The identical approximation holds for
\(\mathcal H_\kappa\). Thus first taking the mesh limit at fixed \(R\),
and only then taking \(R\to\infty\), proves joint convergence in law.

For any fixed compact support, Corollary~\ref{cor:neutral-blocks} bounds
all higher moments of each fixed smooth pairing. Uniform integrability
therefore upgrades convergence in law to all joint moments. Finally,
small changes in a smooth test on a fixed support give small \(L^q\)
changes by the same corollary, after dividing by the sup norm of the
change. A finite covering of a compact family proves the asserted
uniformity.
\end{proof}

We have now identified the limit on smooth neutral tests. The next
argument gives tightness in the requested distribution spaces; it does
not attempt to deduce point-increment convergence from distributional
convergence.

\subsection{Negative regularity and the outward test convention}

Fix a smooth compactly supported probability density \(\rho\), and write
\(X_\delta=h_\delta-\langle h_\delta,\rho\rangle\).
For a compactly supported smooth cutoff \(\chi\), put
\(X_\delta^\chi=\chi X_\delta\).

\begin{lemma}[Pinned averages]\label{lem:pinned-averages}
Let \(0<\ell\le1\), and let \(\psi_{b,\ell}\) be supported in a square
of side \(A\ell\) about \(b\), with
\(|\psi_{b,\ell}|\le A\ell^{-2}\). For centers \(b\) in a fixed compact
set and every finite \(q\),
\begin{equation}\label{eq:prob-pinned-averages}
 \|\langle X_\delta^\chi,\psi_{b,\ell}\rangle\|_{L^q}
 \le C_{A,q,\chi,\rho}(1+|\log\ell|).
\end{equation}
The estimate is uniform in \(\delta>0\).
\end{lemma}

\begin{proof}
Let \(m=\int\chi\psi_{b,\ell}\), so \(|m|\le C\).
Subtract \(m\) times a smooth mass-one bump of scale \(\ell\) at \(b\).
The difference obeys Corollary~\ref{cor:neutral-blocks}.
Next telescope that bump through the scales
\(\ell,2\ell,\ldots,1\), and finally compare the scale-one bump with
\(\rho\). Each difference is neutral, has bounded total variation,
and is supported and bounded at its stated scale. There are
\(O(1+|\log\ell|)\) differences. The same argument works below the mesh
scale because the final clause of Corollary~\ref{cor:neutral-blocks}
has no lower bound on \(\ell/\delta\).
\end{proof}

\begin{proposition}\label{prop:negative-regularity}
For every bounded open \(U\subset\R^2\) and every \(-1<\alpha<0\),
the pinned fields converge in law, with tightness, in the ordinary
restriction spaces
\[
 \mathcal C^\alpha_{\mathrm{res}}(U):=B^\alpha_{\infty,\infty}(U),\qquad
 B^\alpha_{p,q}(U),\qquad W^{\alpha,p}(U),
 \quad 1\le p<\infty,\quad1\le q\le\infty,
\]
to the corresponding pinning of \(\mathcal H_\kappa\).
Here the Hölder laws are supported on the separable closure of smooth
functions in the displayed norm. Convergence also holds in the literal
outward-scaling completion defined by (16) of
\cite[Definition~2.6(iii)]{DKLM2026}. No boundary regularity of \(U\) is
assumed. An intrinsic dual interpretation of the negative Sobolev norms
is treated at the end of this section.
\end{proposition}

\begin{proof}
Choose a compactly supported \(\chi\) equal to one on a box containing
\(\overline U\) and \(\supp\rho\). Use compactly supported wavelets of
sufficient finite regularity, including their coarse-scale scaling
functions. Normalize the scale-\(2^{-j}\) test functions as densities
of size \(O(2^{2j})\), and denote their averages of \(X_\delta^\chi\)
by \(a_{j,k}\). There are \(O(2^{2j})\) relevant indices \(k\).
For \(\alpha<\beta<0\), choose a moment order \(b\) with
\(2+b\beta<0\). Lemma~\ref{lem:pinned-averages} gives
\begin{equation}\label{eq:prob-wavelet-sum}
 \E\sum_{j\ge0}\sum_k|2^{j\beta}a_{j,k}|^b
 \le C_b\sum_{j\ge0}2^{(2+b\beta)j}(1+j)^b<\infty.
\end{equation}
The coarse block has the same bound. This bounds the stronger
\(B^\beta_{\infty,\infty}\) norm in probability. On a fixed compact
support its bounded sets have uniformly vanishing tails in
\(B^\alpha_{\infty,\infty}\), since
\[
 \sup_{j>J,k}2^{j\alpha}|a_{j,k}|
 \le 2^{(\alpha-\beta)J}
          \sup_{j,k}2^{j\beta}|a_{j,k}|.
\]
For finitely many levels only finitely many coefficients remain.
This proves compactness by finite-dimensional truncation, and also puts
the fields and limits in the separable smooth closure.

The same gain proves the other asserted norm compactness. Indeed a
compactly supported distribution bounded in \(B^\beta_{\infty,\infty}\)
has scale blocks satisfying
\(\|\Delta_j X\|_{L^p}\le C2^{-j\beta}\) for every finite \(p\).
Thus
\[
 \sum_{j>J}2^{j\alpha}\|\Delta_jX\|_{L^p}
 \le C\sum_{j>J}2^{j(\alpha-\beta)}\longrightarrow0.
\]
This controls every \(\ell^q\) Besov block norm, including \(q=\infty\).
It also controls the negative Bessel-potential norm: on a scale block,
\((1-\Delta)^{\alpha/2}\) has an \(L^1\)-bounded convolution kernel
after extracting \(2^{j\alpha}\), so summing these bounds works for
all \(p\ge1\). The same summable block bound controls the usual
full-space negative Slobodeckij realization. Passing to its restriction
uses the defining quotient norm. Restricting a global extension to
\(U\) is continuous by definition; no extension operator from \(U\) is
being invoked. Proposition~\ref{prop:smooth-tests} identifies every
subsequential limit, and hence proves the full mesh convergence.

For clarity, retain also the source's printed outward convention.
Let \(Q_0=(-1,1)^2\), and let \(\mathcal T_1(Q_0)\) consist of
compactly supported neutral densities whose zero extension is
one-Lipschitz. For smooth functions its seminorm is
\begin{equation}\label{eq:prob-outward-norm}
 \|f\|_{\mathrm{out},\alpha,U}
 =\sup_{\substack{0<\varepsilon\le1,\ \varphi\in\mathcal T_1(Q_0)\\
                    \supp\varphi\subset U/\varepsilon}}
  \varepsilon^{-\alpha}
     \left|\int f(x/\varepsilon)\varphi(x)\,dx\right|.
\end{equation}
The absolute value is immaterial for the symmetric test family.
Constants vanish in this seminorm. Its completion is understood modulo
its kernel, as in the source.

Write \(\varepsilon=2^{-j}a\), \(1/2\le a\le1\).
The exact mesh identity, including the floor convention, is
\[
 h_\delta(x/\varepsilon)=h_{2^{-j}\delta}(x/a).
\]
After the change of variables \(x=ay\), the test becomes
\(a^2\varphi(ay)\), supported in \(2Q_0\), neutral, with a uniformly
bounded Lipschitz norm. The condition involving \(U\) only reduces this
family. For any fixed \(-1<\gamma<0\), local
\(\mathcal C^\gamma\) control bounds all these pairings. One can see
this directly from a wavelet expansion: cancellation of fine wavelets
against a Lipschitz test gains \(2^{-j}\), and the resulting series
converges because \(1+\gamma>0\).
The preceding moment argument, with a still stronger negative exponent
and arbitrarily high moment order, therefore gives random variables
\(Z_{j,\delta}\) such that
\[
 \sup_\delta\|Z_{j,\delta}\|_{L^b}\le C_b,
 \qquad
 \sup_{2^{-j-1}\le\varepsilon\le2^{-j}}(\text{quantity in }
             \eqref{eq:prob-outward-norm})
       \le C2^{j\alpha}Z_{j,\delta}.
\]
A local pin may be inserted in the rescaled field since every test is
neutral. In particular the supremum over \(j\ge J\) tends to zero in
\(L^b\), uniformly in \(\delta\), by the summability of
\(\sum_{j\ge J}2^{j\alpha}\).
On finitely many bands, the tested original coordinates lie in one
bounded box. Their test norms are bounded by constants depending on the
number of bands, so the stronger local compactness already proved applies.

These estimates also give the required smooth completion, rather than
only a bound on a larger space of functionals. Take a smooth compact
cutoff \(\chi_0\) equal to one on a sufficiently large fixed box and a
compact smooth mollifier \(\theta_\eta\), and use
\[
 f_{\delta,J,\eta}(x)
       =\chi_0(2^{-J}x)(X_\delta*\theta_\eta)(x).
\]
Convolution preserves the local negative Hölder bound, uniformly for
\(0<\eta\le1\), since its rescaled radius on band \(j\) is at most
\(2^{-j}\). For the outward bands \(j\ge J\), the cutoff becomes a bump
of scale at most \(2^{J-j}\) in the rescaled coordinates. The local
\(\mathcal C^\gamma\) test bound gives its contribution the additional
factor \(2^{(J-j)(2+\gamma)}\le1\); comparison of the fixed pin with the
rescaled pin costs at most \(C(1+j)\) by the same dyadic telescoping as
in Lemma~\ref{lem:pinned-averages}. Thus the cutoff approximations have
tails bounded by a constant times
\(\sum_{j\ge J}(1+j)2^{j\alpha}\), uniformly in probability and in the
mollification parameter. On finitely many bands send \(\eta\downarrow0\);
then send \(J\to\infty\). These smooth approximations and the vanishing tail prove
membership and tightness in the completion of
\eqref{eq:prob-outward-norm}. Its limits agree with the smooth-test limit
on each finite band. This proves the literal outward assertion too.
\end{proof}

\subsection{Removing smoothings at separated endpoints}

Let \(\varrho\) be a smooth probability density supported in a small
unit square, and define
\[
 H_{\delta,r}(b)=\int h_\delta(x)r^{-2}
                   \varrho((x-b)/r)\,dx.
\]
Differences \(H_{\delta,r}(b)-H_{\delta,2r}(b)\) obey
Corollary~\ref{cor:neutral-blocks}. At \(r\asymp\delta\), the residual
\(h_\delta(b)-H_{\delta,r}(b)\) is deterministically bounded by unit
height jumps. We will use both kinds of differences as individual
endpoint factors.

\begin{lemma}[A separated endpoint factor]\label{lem:separated-block}
Fix a moment order \(k\ge2\), a separation \(d_0>0\), and a bounded
region containing all factor centers. Let \(r_0=a_kd_0\), with \(a_k>0\)
sufficiently small. Consider at most \(k\) neutral height factors with
uniformly bounded total variation and the ordinary moment bounds of
Corollary~\ref{cor:neutral-blocks}, each supported within distance
\(Cr_0\) of its center. A factor may instead be a terminal residual of
scale comparable to \(\delta\). Suppose one selected factor has scale
\(r\le r_0\), \(r\ge c_0\delta\), and every other center is at distance
at least \(c_1d_0\) from its center. Then, for some \(\tau>0\),
\begin{equation}\label{eq:prob-separated-block}
 \left|\E\prod_i A_i\right|
 \le C(r/d_0)^\tau.
\end{equation}
The constants are uniform in the centers, the mesh, and all other scales
between \(r\) and \(r_0\). Other centers need not be mutually separated.
\end{lemma}

\begin{proof}
We first prove the estimate when the selected support has an empty axial
slab on both sides, each of width at least \(c_2d_0\), separating it from
all other factors. Insert time cuts in these two slabs.
The vectors defined by the exterior factors have bounded Hilbert norms.
Here is the connection with ordinary moments. Read all heights on one
side of a cut relative to a common face, along a common seam and spatial
paths contained on that side. Spatial steps use equal-time diagonal
spins; each time-seam step uses a time spin, with repeated occurrences
allowed when several paths share the step. In a finite homogeneous
system, reflection across the cut transposes the entire product and
complex-conjugates its scalar coefficients. The crossing identity
reverses the time-step spins and preserves the spatial-step spins,
exactly as for the reflected height differences. The squared norm is
therefore the ordinary expectation of the exterior product times its
conjugate reflected product. This finite identity passes to the plane
by Proposition~\ref{prop:physical-plane}. Hölder's inequality and the
ordinary moments of the individual factors bound that expectation.
The factors on a side may overlap; Hölder's inequality requires no
separation among them.

Inside the slab, use neutrality to express the selected factor as an
average of differences from a fixed reference face in its support.
Each difference has a path of \(O(r/\delta)\) steps, with a bounded
number of straight segments. Choose the direction of the difference
so that time runs forward; spatial steps may have either sign.
There remain \(m\ge c_3d_0/\delta\) unmarked time rows on both sides.
Pair consecutive steps along each straight segment. The paired current
identity and Lemma~\ref{lem:low-band} give the sandwiched norm bounds
\begin{equation}\label{eq:prob-damped-costs}
 \begin{aligned}
 \|P(1)^m(\text{two consecutive step sum})P(1)^m\|
       &\le C m^{-1},\\
 \|P(1)^m(\text{one step})P(1)^m\|
       &\le C m^{-\theta_0}
 \end{aligned}
\end{equation}
for some \(0<\theta_0\le1\). To verify the passage from band bounds to
these inequalities, split both sides into the bands
\(H_0\le m^{-1}\) and
\(2^{j-1}/m<H_0\le2^j/m\).
A block between two such bands has current bound a power of the larger
cutoff, while the time rows contribute
\(\exp(-c2^j-c2^{j'})\). The resulting double geometric-exponential
sum is \(O(m^{-1})\) for a paired current. For a single current use the
equal-sign \(O(u)\) and opposite-sign \(O(u^\theta)\) estimates,
with \(\theta_0=\min(1,\theta)\). Bands above a fixed cutoff are
exponentially damped and the uncompressed row norms are bounded.
Unit shifts commute with the damping and do not enlarge these norms.

There are \(O(r/\delta)\) pairs and only a bounded number of unpaired
steps. The bounded total variation of the selected factor and the
exterior norm bounds give
\[
 \left|\E\prod_iA_i\right|
 \le C\left(\frac r{d_0}
               +\left(\frac\delta{d_0}\right)^{\theta_0}\right)
 \le C(r/d_0)^{\tau_0},\qquad \tau_0=\min(1,\theta_0).
\]
This includes a terminal residual: its comparison paths have bounded
length and the same damping margins. Rounding the cuts changes their
widths by only a bounded number of sites.

It remains to create a free slab without losing the estimate when the
factors are returned to their original positions. Cluster the centers
by their first-coordinate projections: consecutive projections belong
to the same cluster if their gap is at most \(a d_0/k\), for a sufficiently
small fixed \(a\). Every cluster has width at most \(a d_0\).
Choose \(a_k\) much smaller than \(a/k\), so that the supports preserve
a gap of size \(c_kd_0\) at each cluster boundary.
In the cluster containing the selected center, every other center has
second-coordinate distance at least \(c_kd_0\) from it. Indeed its
first-coordinate distance is at most \(a d_0\), whereas its Euclidean
distance has the lower bound in the statement.

Translate the clusters strictly to the left as one group vertically,
and do the same independently for the clusters strictly to the right.
For each group there is an interval of translation amounts, of length
at least \(c_kd_0\) and at distance at most \(C_kd_0\) from zero, such
that every translation in the interval leaves a vertical gap of size
\(c_kd_0\) around the selected support. To construct one, exclude the
intervals of translations putting any of its at most \(k\) centers
within a fixed small multiple of \(d_0\) of the selected height.
Their total length is bounded by \(C_kd_0\); choose a larger bounded
translation range and take a complementary interval of the asserted
size. The choices for the two groups are independent. Restrict to
translations by even numbers of lattice sites. Throughout the resulting
rectangle of choices the selected factor has a free slab in the
northward time ordering, so the preceding estimate applies.

We transfer this estimate back through the first-coordinate gaps.
Write
\[
 K=\operatorname{Arg}(S^2),\qquad -\pi\le K\le\pi.
\]
This is bounded spectral functional calculus. The low-band estimate
\(S=\pm1+O(H_0)\) gives \(|K|\le CH_0\) for small \(H_0\); on the
complement the same inequality follows by increasing \(C\), since
\(|K|\le\pi\). Thus
\begin{equation}\label{eq:prob-shift-strip}
 |K|\le CH_0,
 \qquad
 \|P(1)^N e^{izK}\|\le1
       \quad\text{if}\quad |\Im z|\le N/C.
\end{equation}
For integer \(m\), \(e^{imK}=S^{2m}\).
Across a first-coordinate gap containing \(N\ge c_kd_0/\delta\)
unmarked rows, translating an entire exterior group by \(2m\) sites
therefore inserts \(P(1)^NS^{\pm2m}\) between fixed exterior vectors.
This identity follows by conjugating every site and seam insertion
on that side by the site shift, which fixes the vacuum. Reflection and
the ordinary moment estimates bound both exterior vectors uniformly.
Equation~\eqref{eq:prob-shift-strip} extends the pairing to a bounded
holomorphic function of \(z/(d_0/\delta)\) on a strip of fixed positive
width. If one of the exterior groups is absent there is no restoration
step for that group.

For completeness, the elementary analytic estimate used here is the
following. A holomorphic function bounded by one on a fixed-width strip,
whose absolute value is at most \(a\le1\) on a fixed-length real interval
within a fixed bounded distance of zero, has absolute value at zero at
most \(C a^\vartheta\), where \(\vartheta>0\) depends only on those
geometric constants. Take a half-disc based on a shorter part of the
interval. The two-constants theorem bounds the function by
\(a^{\omega}\) on an interior disc, since the harmonic measure of the
base there is bounded below by \(\omega>0\). Cover the remaining path
to zero by finitely many overlapping discs inside the strip.
The three-circles inequality transfers a positive power bound at each
step. The number and radii of these discs depend only on the fixed
strip, interval, and distance bounds.

Our free-slab estimate initially holds only at the allowed integer
values of \(m\). Cauchy's estimate on a slightly smaller strip bounds
the derivative in the normalized real coordinate. The gap between
successive allowed values is \(O(\delta/d_0)\), so throughout a shorter
real interval the bound becomes
\[
 C(r/d_0)^{\tau_0}+C\delta/d_0
 \le C'(r/d_0)^{\tau_0},
\]
using \(r\ge c_0\delta\) and \(\tau_0\le1\).
First restore the left group while keeping the right translation at
any one of its allowed values. The analytic estimate gives the same
bound with a possibly smaller positive exponent, uniformly over those
right translations. Then restore the right group. The two reductions
of the exponent still leave a fixed \(\tau>0\). This proves
\eqref{eq:prob-separated-block} in the original geometry.
\end{proof}

\begin{proposition}\label{prop:offdiagonal}
For every \(k\ge1\), all \(k\)-point height-increment moments converge
to the Wick increment moments of \(\mathcal H_\kappa\), uniformly on
compact sets of tuples whose endpoint sets belonging to different
increments are disjoint.
\end{proposition}

\begin{proof}
The first moment is zero by arrow-reversal symmetry. For \(k\ge2\),
fix such a compact set of tuples and let \(d_0>0\) be a lower bound on
all cross-pair endpoint distances. Endpoints within a single pair may
coincide. Choose \(r_0=a_kd_0\) as in
Lemma~\ref{lem:separated-block}, and put \(r_j=2^{-j}r_0\).
For each endpoint, telescope its bump averages down to the last scale
\(r_J\asymp\delta\), and include the residual from that average to the
specified face value. This is a finite exact identity, including for
points on grid lines under the fixed floor convention.

Expand the product of increments. A factor is either a coarse
increment at scale \(r_0\), or one of the two endpoint blocks at a
dyadic scale, including the terminal residual. All individual blocks
have uniformly bounded moments. The coarse increments also do: join
their endpoints by a bounded-length chain of overlapping scale-\(r_0\)
bumps and telescope, applying Corollary~\ref{cor:neutral-blocks} to
each difference. The number of links is bounded by the fixed compact
size and separation data.

In a summand containing an endpoint block, select one of minimum scale
\(r\). Decompose every other coarse increment by a bump chain avoiding
a disc of radius \(d_0/3\) about the selected endpoint. Such a chain
exists with bounded length because both endpoints of that other pair
are at least \(d_0\) away; a polygonal detour around the disc has length
bounded in terms of the same fixed data. After subdividing into steps
of size \(O(r_0)\), every factor other than the selected one has center
at distance at least \(c d_0\) from it. There are still at most \(k\)
factors in each resulting product; the number of resulting products is
bounded independently of the mesh and dyadic levels. Apply
Lemma~\ref{lem:separated-block} to each of them.

At fixed deepest dyadic index \(j\), there are at most
\(C_k(1+j)^{k-1}\) choices of the other levels and endpoint labels.
Consequently the error from keeping only scales coarser than \(r_{J_0}\)
is bounded by
\begin{equation}\label{eq:prob-desmoothing-tail}
 C\sum_{j\ge J_0}(1+j)^{k-1}2^{-\tau j},
\end{equation}
uniformly once the mesh is fine enough to reach that level. This tends
to zero as \(J_0\to\infty\). Notice that the terminal residual obeys
the same bound; dropping it before applying the estimate would not
justify point evaluation.

For fixed \(J_0\), Proposition~\ref{prop:smooth-tests} gives the joint
moments of the smoothed increments, uniformly in their centers.
The corresponding Gaussian Wick expressions converge uniformly as the
smoothing radius tends to zero: each paired kernel involves endpoints
from two different increments, so its arguments stay a positive
distance apart. Combine this with \eqref{eq:prob-desmoothing-tail}.
If the two endpoints within a pair coincide, that increment and its
smoothed versions are identically zero; no additional exclusion is
needed.
\end{proof}

\subsection{Ordinary finite-energy signed measures}

\begin{lemma}\label{lem:log-integrability}
Let \(\mu\) be a compactly supported finite signed measure for which
\(\iint G(x,y)\,\mu(dx)\mu(dy)\) is an ordinary finite signed-measure
integral. Then \(\mu\) has no atoms, and
\begin{equation}\label{eq:prob-absolute-energy}
 \iint\log_+\frac1{|x-y|}\,|\mu|(dx)|\mu|(dy)<\infty.
\end{equation}
If \(\mu,\nu\) both satisfy this assumption, the corresponding mixed
absolute logarithmic integral is finite as well.
\end{lemma}

\begin{proof}
The total variation of \(\mu\otimes\mu\) is
\(|\mu|\otimes|\mu|\). Ordinary integrability is integrability with
respect to this variation. On a fixed compact support, the negative
part of \(-\log|x-y|\) is bounded, which proves
\eqref{eq:prob-absolute-energy}. An atom would give positive product
mass to the logarithmic diagonal singularity, so there are no atoms.

For the mixed assertion take dyadic squares of side \(2^{-j}\) in a
box containing both supports. Up to an additive bounded kernel,
\(\log_+(1/|x-y|)\) is bounded above by a constant times
\(\sum_{j\ge0}\bbone_{\{|x-y|\le C2^{-j}\}}\).
Write \(a=|\mu|\), \(b=|\nu|\).
At a fixed level the mixed mass of these nearby pairs is at most
\(\sum_Q\sum_{Q'\sim Q}a(Q)b(Q')\), where each square has a bounded
number of neighbors. Cauchy--Schwarz bounds this by
\[
 C\left(\sum_Qa(Q)^2\right)^{1/2}
  \left(\sum_Qb(Q)^2\right)^{1/2}.
\]
The sums of \(\sum_Qa(Q)^2\) and \(\sum_Qb(Q)^2\) over all levels
are finite by their self-integrability: pairs in one square are within
\(\sqrt2\,2^{-j}\), and summing their indicators is bounded by
\(C(1+\log_+(1/|x-y|))\).
A second Cauchy--Schwarz inequality over \(j\) proves the claim.
\end{proof}

\begin{proposition}\label{prop:finite-energy}
For every finite family of compactly supported, neutral, finite signed
measures of the ordinary finite-energy class in
Theorem~\ref{thm:main}, the associated height pairings converge jointly
in law to the centered Gaussian vector with covariance obtained by
integrating \(K_\kappa\) against the pairs of measures.
\end{proposition}

\begin{proof}
Let \(V(a)=\E(h(a)-h(0))^2\) for lattice-face displacement
\(a\in\Z^2\), and put \(V(0)=0\).
Proposition~\ref{prop:variogram} and axial symmetry imply
\[
 V(a)=4\kappa\log(1\vee|a|)+O(1),
\]
with a uniform bounded error over all lattice displacements.
For distinct continuum points \(x,y\) in a fixed compact set, write
\[
 V_\delta(x,y)=
 V\bigl(\lfloor x/\delta\rfloor-\lfloor y/\delta\rfloor\bigr).
\]
For \(0<\delta\le1\) this gives
\begin{equation}\label{eq:prob-rounded-log-bound}
 \left|V_\delta(x,y)-4\kappa\log\delta^{-1}\right|
       \le C\bigl(1+|\log|x-y||\bigr).
\end{equation}
Indeed when \(|x-y|\) exceeds a fixed multiple of \(\delta\), the
lattice separation is comparable to \(|x-y|/\delta\).
In the other case its size is bounded, and
\(\log\delta^{-1}\le C+|\log|x-y||\).
This latter case includes distinct points rounded to the same face.

For any four points, stationarity of increments and zero mean give
\begin{align}\label{eq:prob-four-variograms}
 &\E[(h_\delta(x')-h_\delta(x))
            (h_\delta(y')-h_\delta(y))]\\
 &\qquad=\tfrac12\bigl(
 V_\delta(x',y)+V_\delta(x,y')
       -V_\delta(x',y')-V_\delta(x,y)\bigr).\nonumber
\end{align}
The four copies of \(4\kappa\log\delta^{-1}\) cancel exactly.
Equation~\eqref{eq:prob-rounded-log-bound} therefore supplies a majorant
by a constant and four continuum logarithmic singularities.

For a neutral \(\mu\ne0\), its Jordan parts have the same mass
\(M>0\). The measure
\(M^{-1}\mu^+(dx')\mu^-(dx)\) represents its pairing as an integral
of \(h_\delta(x')-h_\delta(x)\).
Use this representation for two pairings. By
Lemma~\ref{lem:log-integrability}, the four-logarithm majorant is
integrable against the resulting product measure. Its endpoint
coincidences have measure zero because the Jordan parts are nonatomic.
Proposition~\ref{prop:offdiagonal} for two increments and dominated
convergence thus give, for every such \(\mu,\nu\),
\begin{equation}\label{eq:prob-energy-covariance}
 \lim_{\delta\downarrow0}
       \E[\langle h_\delta,\mu\rangle
                  \langle h_\delta,\nu\rangle]
     =\iint K_\kappa(x,y)\,\mu(dx)\nu(dy).
\end{equation}
The zero measure causes no exception.

It remains to identify laws beyond second moments. Let
\(\mu_\epsilon=\mu*\varrho_\epsilon\), where the mollifier is smooth,
nonnegative and compactly supported. These measures are smooth and
neutral. Mollification converges in logarithmic energy:
\begin{equation}\label{eq:prob-energy-mollification}
 \iint K_\kappa(x,y)
       (\mu-\mu_\epsilon)(dx)(\mu-\mu_\epsilon)(dy)
       \longrightarrow0.
\end{equation}
To check this for the present ordinary integral, the logarithm convolved
in one or both variables converges pointwise off the diagonal. On the
fixed enlarged compact support its absolute value is bounded by
\(C(1+|\log|x-y||)\), uniformly in small \(\epsilon\).
For \(|x-y|\ge C\epsilon\) this follows by comparison of distances;
for \(|x-y|<C\epsilon\), scaling the bounded compact mollifier gives
\(C+|\log\epsilon|\le C'(1+|\log|x-y||)\).
Lemma~\ref{lem:log-integrability} permits dominated convergence in each
of the four energy terms, proving
\eqref{eq:prob-energy-mollification}.

Apply \eqref{eq:prob-energy-covariance} to
\(\mu-\mu_\epsilon\). It gives
\[
 \lim_{\epsilon\downarrow0}\limsup_{\delta\downarrow0}
 \|\langle h_\delta,\mu-\mu_\epsilon\rangle\|_{L^2}=0.
\]
At fixed \(\epsilon\), Proposition~\ref{prop:smooth-tests} identifies
the Gaussian limit. The same energy estimate approximates the Gaussian
pairing. Applying this argument to a finite family, or to each of its
linear combinations, proves the asserted joint law and cross covariance.
\end{proof}

\subsection{Intrinsic negative Sobolev norms on an arbitrary open set}

The restriction-space proof above already requires no regularity of
\(\partial U\). We give a further argument to cover the convention in
which a negative Sobolev norm is the norm dual to an intrinsic positive
Slobodeckij norm on \(U\). The distinction matters for irregular open
sets, where an extension theorem is unavailable.

For \(0<s<1\), \(1<q<\infty\), write
\[
 \|\phi\|_{W^{s,q}(U)}^q
 =\|\phi\|_{L^q(U)}^q+
   \iint_{U\times U}
       \frac{|\phi(x)-\phi(y)|^q}{|x-y|^{2+sq}}\,dx\,dy.
\]
For \(q=\infty\), use the sup norm and the essential supremum of
\(|\phi(x)-\phi(y)|/|x-y|^s\).
The cells in the following argument are intersections of ordinary
dyadic squares with \(U\); they may have arbitrarily small relative
area and arbitrarily irregular shapes.

\begin{lemma}[Averages on small-area cells]\label{lem:cell-averages}
Let \(Q\) be a square of side \(0<\ell\le1\) in a fixed bounded box,
and let \(A\subset Q\) be measurable with \(|A|>0\). Put
\(t=|A|/\ell^2\). Then for every finite \(b\),
\begin{equation}\label{eq:prob-cell-average}
 \left\|\frac1{|A|}\int_A X_\delta(x)\,dx\right\|_{L^b}
 \le C_b(1+|\log\ell|+|\log t|)
\end{equation}
uniformly in \(\delta\) and in the shape of \(A\).
\end{lemma}

\begin{proof}
We first refine the comparison of line weights. Suppose nonnegative
weights \(a_l\), supported on \(O(R)\) consecutive sites, satisfy
\[
 \sum_l a_l=m>0,\qquad a_l\le M/R,\qquad R\ge1.
\]
Let \(b_l\) be a mass-one profile bounded by \(C/R\), and assume that
the union of the supports of \(a\) and \(b\) lies in one interval of
length \(CR\). We claim
\begin{equation}\label{eq:prob-line-entropy}
 \left\|\sum_l(a_l-mb_l)h_l\right\|_{L^b}
 \le C_bm\bigl(1+\log_+(M/m)\bigr).
\end{equation}
Increasing a fixed constant in \(M\) if needed, assume \(M\ge m\).
For dyadic integer \(v\ge1\), let \(A_va\) replace the weights in
each consecutive block of \(v\) sites by their block average.
The difference \(d=A_va-A_{2v}a\) is neutral in every \(2v\)-block.
For its normalized primitive \(g_l=m^{-1}\sum_{i\le l}d_i\), this gives
\[
 \sum_l|g_l|\le Cv,\qquad
 \sum_l|g_{l+1}-g_l|\le C,\qquad
 \sum_l|g_{l+1}-g_l|^2\le C/v.
\]
The first bound sums a block length times its variation mass.
For the last, use \(\|d\|_\infty\le Cm/v\) and
\(\|d\|_1\le2m\). Lemma~\ref{lem:ordinary-moments} and summation by
parts give cost at most \(C_bm\) for each flattening step. This use of
the lemma explains why its statement retained the three summability
conditions, without a support-diameter hypothesis.

If \(\min(R,Rm/M)>1\), choose a dyadic \(v_0\) comparable from below
to that number; otherwise put \(v_0=1\).
When \(v_0>1\), one can start directly with \(a-A_{v_0}a\).
Its primitive has the same first two bounds, and
\[
 m^{-2}\|a-A_{v_0}a\|_2^2
 \le C m^{-2}\|a\|_2^2
 \le CM/(Rm)\le C/v_0.
\]
When \(v_0=1\), this initial difference vanishes. Continue flattening
until \(v\asymp R\), where comparison with \(mb\) has bounded cost
by the original width-\(R\) case of the lemma. The number of steps is
\(O(1+\log_+(M/m))\), proving \eqref{eq:prob-line-entropy}.

Now rescale the square \(Q\) to a unit square. For any probability
density \(f\le M\) on it, set \(m(x)=\int f(x,y)\,dy\).
Apply \eqref{eq:prob-line-entropy} to its mesh masses on each vertical
line, comparing them with \(m(x)\) times one bounded profile.
Minkowski's inequality bounds the total cost by
\[
 C_b\int m(x)\bigl(1+\log(M/m(x))\bigr)\,dx
 \le C_b(1+\log M).
\]
The last inequality is Jensen's inequality for the concave function
\(u\mapsto u\log(M/u)\), with \(\int m=1\) on an interval of length
one. Compare the remaining horizontal marginal with a bounded profile
by the same line estimate. We have therefore compared the original
density with a fixed scale-\(\ell\) product smoothing, at cost
\(C_b(1+\log M)\). Telescoping that smoothing to scale one and then
to the pin costs \(C_b(1+|\log\ell|)\), as in
Lemma~\ref{lem:pinned-averages}.
If \(\ell<\delta\), all comparisons inside a comparable square use
only a bounded number of faces and have a deterministic bound; the
remaining pin comparison is unchanged. Taking
\(f=\ell^2|A|^{-1}\bbone_A\) in rescaled coordinates gives
\(M=t^{-1}\) and proves \eqref{eq:prob-cell-average}.
\end{proof}

\begin{proposition}[Intrinsic dual convergence]\label{prop:intrinsic-dual}
Let \(U\subset\R^2\) be bounded and open, let \(0<s<1\), and let
\(1\le p<\infty\), with conjugate exponent \(q=p'\).
The pinned fields converge in law, with norm tightness, in
\((W^{s,q}(U))^*\) to the canonically pinned Gaussian field with kernel
\(K_\kappa\). The laws are supported on a separable closed subspace of
this dual. No extension property or regularity of \(\partial U\) is
required.
\end{proposition}

\begin{proof}
Take a fixed square containing \(U\), subdivided dyadically at side
lengths \(\ell_j\asymp2^{-j}\). For a square \(Q\) at level \(j\), put
\[
 U_Q=Q\cap U,\qquad t_Q=|U_Q|/|Q|,\qquad
 X_{\delta,Q}=|U_Q|^{-1}\int_{U_Q}X_\delta.
\]
Ignore cells of zero area. Let
\[
 Z_{\delta,j}=\sum_{Q\text{ at level }j}
                   X_{\delta,Q}\bbone_{U_Q},\qquad
 F_{\delta,j}(\phi)=\int_U Z_{\delta,j}\phi,
\]
and let \(P_j\phi\) denote cell averaging of \(\phi\) on the same
partition. Consistency of averaging implies
\begin{equation}\label{eq:prob-cell-difference}
 (F_{\delta,j+1}-F_{\delta,j})(\phi)
     =\int_U Z_{\delta,j+1}(\phi-P_j\phi).
\end{equation}

Suppose first \(p>1\), so \(q<\infty\), and choose
\(0<\zeta<sq\). Call a parent \(P\) of side \(\ell=\ell_j\) thick
when \(t_P\ge\ell^\zeta\). Jensen's inequality gives
\begin{align*}
 \int_{U_P}|\phi-\phi_P|^q
 &\le |U_P|^{-1}
            \iint_{U_P\times U_P}|\phi(x)-\phi(y)|^q\,dx\,dy\\
 &\le C\ell^{sq}t_P^{-1}
       \iint_{U_P\times U_P}
          \frac{|\phi(x)-\phi(y)|^q}{|x-y|^{2+sq}}\,dx\,dy.
\end{align*}
Summing over thick parents yields
\begin{equation}\label{eq:prob-thick-cell}
 \|\phi-P_j\phi\|_{L^q(\mathrm{thick})}
        \le C\ell^{s-\zeta/q}[\phi]_{W^{s,q}(U)}.
\end{equation}
On all cells, averaging is an \(L^q\) contraction, so the same difference
has \(L^q\) norm at most \(2\|\phi\|_{L^q}\).

Lemma~\ref{lem:cell-averages}, with \(b=p\), implies
\[
 \E\|Z_{\delta,j+1}\|_{L^p(U)}^p\le C_p(1+j)^p.
\]
For this inequality sum
\(|Q|t_Q(1+j+\log(1/t_Q))^p\) over children, and use
\(t(1+\log(1/t))^p\le C_p\) for \(0<t\le1\).
A child of a thin parent has \(t_Q\le4\ell^\zeta\).
The sharper elementary inequality
\[
 t(A+\log(1/t))^p\le C_p A^p t^{1/2},\qquad A\ge1,
\]
therefore gives
\begin{equation}\label{eq:prob-thin-cell}
 \E\|Z_{\delta,j+1}\bbone_{\mathrm{thin}}\|_{L^p(U)}^p
       \le C_p(1+j)^p\ell^{\zeta/2}.
\end{equation}
The sum of the areas \(|Q|\) is bounded by the area of the surrounding
square, independently of the geometry of \(U\).
Equations~\eqref{eq:prob-cell-difference}--\eqref{eq:prob-thin-cell}
and Hölder's inequality now give
\begin{equation}\label{eq:prob-dual-tail-step}
 \E\|F_{\delta,j+1}-F_{\delta,j}\|_{(W^{s,q}(U))^*}
 \le C(1+j)
       \left(\ell_j^{s-\zeta/q}+\ell_j^{\zeta/(2p)}\right).
\end{equation}
Both exponents are positive.

At \(p=1\), \(q=\infty\), the intrinsic Hölder seminorm gives directly
\[
 \|\phi-P_j\phi\|_\infty
       \le C\ell_j^s[\phi]_{C^s(U)}.
\]
There is no need to distinguish thin parents in this case. The first
moment version of the cell estimate gives
\begin{equation}\label{eq:prob-dual-endpoint}
 \E\|F_{\delta,j+1}-F_{\delta,j}\|_{(W^{s,\infty}(U))^*}
       \le C(1+j)\ell_j^s.
\end{equation}
Thus at every finite \(p\) the right-hand side, denoted by \(a_j\),
is summable over \(j\), uniformly in \(\delta\).

Let \(\mathcal E_0\) be the closed linear span in
\(\mathcal E=(W^{s,q}(U))^*\) of the countable collection of cell
indicator pairing functionals. It is a separable Banach space.
For each fixed mesh, \(X_\delta\) is a bounded function on the bounded
set \(U\), and its conditional cell averages converge to it in
\(L^p(U)\). Since \(L^p(U)\) maps continuously into \(\mathcal E\),
they also converge in \(\mathcal E_0\). Summing the preceding
estimates proves
\begin{equation}\label{eq:prob-dual-uniform-tail}
 \sup_\delta\E\|X_\delta-F_{\delta,J}\|_{\mathcal E}
       \le\sum_{j\ge J}a_j\longrightarrow0.
\end{equation}

Every coefficient \(X_{\delta,Q}\) is the pairing with the neutral
measure
\[
 \mu_Q=|U_Q|^{-1}\bbone_{U_Q}\,dx-\rho\,dx.
\]
This bounded compact density has ordinary finite logarithmic energy,
regardless of the shape or area of \(U_Q\). Proposition~\ref{prop:finite-energy}
therefore gives joint convergence of the coefficients at each finite
level. Couple their Gaussian limits using the countable family of
Gaussian pairings with \(\mu_Q\), and denote the corresponding
finite-level elements of \(\mathcal E_0\) by \(F_J\).
Finite-dimensional convergence and the lower semicontinuity of the norm
transfer the bound
\(\E\|F_{j+1}-F_j\|_{\mathcal E}\le a_j\).
The sum of these norms is almost surely finite and integrable, so
\(F_J\) converges almost surely and in \(L^1(\mathcal E_0)\) to
an \(\mathcal E_0\)-valued random variable \(F\).
Finite-level convergence and \eqref{eq:prob-dual-uniform-tail}
then prove \(X_\delta\Rightarrow F\) in \(\mathcal E_0\), including
norm tightness.

We finally identify the limit on the whole intrinsic test space, not
only on interior smooth tests. For \(\phi\in W^{s,q}(U)\), the measure
\[
 \mu_\phi=\phi\bbone_U\,dx-\left(\int_U\phi\right)\rho\,dx
\]
has ordinary finite logarithmic energy. Indeed \(q>1\) or \(q=\infty\),
and on a fixed compact set Hölder's inequality for the logarithmic
kernel gives
\[
 \iint |\log|x-y||\,|f(x)f(y)|\,dx\,dy
       \le C_q\|f\|_{L^1}\|f\|_{L^q}.
\]
For finite \(q\), \(P_J\phi\to\phi\) in \(L^q(U)\); for \(q=\infty\)
the intrinsic Hölder bound gives uniform essential convergence.
The associated neutral measures consequently converge in logarithmic
energy. Hence \(F_J(\phi)\) converges to the canonical pinned Gaussian
pairing with \(\mu_\phi\). This identifies \(F\) on every intrinsic
test and rules out an unidentified boundary functional in the larger
dual. The proof is complete.
\end{proof}

\begin{remark}[Other difference-norm conventions and the known interval]
If a negative Besov convention is dual to a positive intrinsic Besov
difference norm, choose \(0<t<s<1\). The positive test norm of order
\(s\) controls the intrinsic \(W^{t,q}\) norm: in a dyadic difference
expansion, the gain \(2^{-j(s-t)}\) is summable, for every sequence
exponent including its endpoints. The convergence just proved for
\((W^{t,q}(U))^*\) therefore gives the corresponding negative Besov
dual convergence of order \(-s\). This is in addition to the ordinary
restriction Besov conclusion of Proposition~\ref{prop:negative-regularity}.

For \(\sqrt3\le c\le2\), the smooth, finite-energy and off-diagonal
limits are already supplied by \cite[Theorem~2.8]{DKLM2026}.
The bounds required for the topology arguments above are available on
this interval from \cite[Theorem~4.5]{DKLM2026}, which is stated for
\(1\le c\le2\). We spell out the relevant implication to retain the
arbitrary-domain quantifier. Its even increment-moment bound is a sum
over pairings of products of logarithmic factors involving the minimum
cross-endpoint distance for that pair. The discrete expression contains
\(1\vee d_{ij}\) in that distance, so it remains finite when rounded
endpoints coincide. If the continuum cross-distance is \(D\), rounding
gives \(D\le\delta(d_{ij}+C)\), while increment lengths on our fixed
compact support are at most \(C/\delta\). Thus its logarithmic factor
is at most \(C(1+\log_+(1/D))\), also at rounded coincidences.
The logarithm of a minimum of four cross-distances is bounded by the
sum of the four corresponding positive logarithms.

For a probability density bounded by \(1/(\ell^2t)\), logarithmic
integration gives
\(C(1+|\log\ell|+|\log t|)\); for example integrate the bound
\(\min(1,C r^2/(\ell^2t))\) for the mass of a disc of radius \(r\)
over logarithmic scales. A fixed smooth pin has a uniform such bound.
In each moment pairing the endpoint variables are disjoint from those
of the other pairs, so these integral bounds multiply. Even moments
therefore give \eqref{eq:prob-cell-average} and
\eqref{eq:prob-pinned-averages} at every finite moment order.
Zero rounded increments contribute zero and can be removed before
forming a scale ratio. This verifies the inputs to both topology proofs
on the established interval, including \(p=1\), \(q=\infty\), and
cells \(Q\cap U\) of arbitrarily small positive area.
\end{remark}

The section has established all three convergence mechanisms with
covariance \(K_\kappa\). Since
\(K_\kappa=4\pi\kappa\,G\), the squared multiplier is
\(4\pi\kappa\). Proposition~\ref{prop:normalization} computes it for
\(c<1\); the conditional normalization theorem quoted in the
introduction applies after these convergence statements have been
proved for the remaining new interval. The final section assembles
these identifications with the established parameter range.
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//07-probabilistic-endgame.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//08-main-proof.tex ====
\section{Completion of the proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
For \(\sqrt3\le c\le2\), the theorem is
\cite[Theorem~2.8, with Definitions~2.6--2.7]{DKLM2026}, in precisely the
unit-height and Green-kernel conventions used here. The source's
regularity estimates also retain the lattice cutoff when distinct
continuum points round to the same face. Thus this interval includes
the endpoint \(c=2\).

Fix \(0<c<\sqrt3\). Sections~\ref{sec:finite}
and~\ref{sec:thermodynamic} construct the limits of every finite word
of rows and marks. Proposition~\ref{prop:physical-plane} identifies the
physical words with local arrow probabilities of the prescribed plane
law. For \(c<1\) the construction uses the exactly balanced sector
throughout and proves existence in the required iterated order. Where
an auxiliary near-balanced limit is used for \(c\ge1\), the equilibrium
identification gives the already existing balanced law. The same
proposition identifies the two axial descriptions of its local
observables.

The angular representation of Section~\ref{sec:angular} has orbit mass
\(\kappa<\infty\) and height variogram
\(4\kappa\log|x|+O(1)\). Proposition~\ref{prop:gaussian-currents}
establishes the holomorphic and antiholomorphic Wick rules, with
logarithmic covariance \(-2\kappa\log|x-y|\).
Propositions~\ref{prop:smooth-tests}, \ref{prop:negative-regularity},
\ref{prop:offdiagonal}, and~\ref{prop:finite-energy} then give all the
claimed convergences with squared GFF multiplier \(4\pi\kappa\).
Their estimates apply to arbitrary real mesh sizes and identify every
subsequential limit; hence the limits hold along the full
\(\delta\downarrow0\).

If \(0<c<1\), Proposition~\ref{prop:normalization} gives
\[
 4\pi\kappa=\frac2{\pi-\lambda}=\frac2{\arccos\Delta}.
\]
For \(1\le c<\sqrt3\), we have now proved a subsequential GFF limit in
the required senses for the balanced law, so the hypothesis of
\cite[Theorem~4.3]{DKLM2026} is satisfied. That theorem identifies its
squared multiplier as \(-1/f''(0)\), where \(f\) is the source's
free-energy function. By \cite[Theorem~4.4]{DKLM2026},
\(f''(0)=-\tfrac12\arccos\Delta\), giving the same multiplier.
This use of the conditional theorem comes after, and does not supply,
the convergence argument.

Finally, ordinary finite signed-energy tests have the joint covariance
in the statement by Proposition~\ref{prop:finite-energy}. All pinning
choices differ by an additive constant, the negative-regularity assertions
use the common chosen pin, and increments cancel it. The uniformity in
\eqref{eq:increment-main} and the arbitrary-open-set conclusions have
already been proved in their respective propositions. This completes
the theorem.
\end{proof}
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//08-main-proof.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//08-occupation.tex ====
\section{Occupation measures and the whole trace}
\label{sec:occupation}

Proposition~\ref{prop:disk-calibration} compares a microscopic segment
with a disk whose radius is a fixed fraction of the domain size.  We
first make this comparison inside second moments.  The resulting
approximation permits us to use ordered-curve convergence at a fixed
disk radius, and only afterwards let that radius tend to zero.  We
also control the boundary strip, since weak convergence of finite
measures on the closed square includes convergence of total mass.

\subsection{Segment probes and their replacement by disks}

Let $\mathcal E_n$ be the deterministic set of full medial segments
available in the discrete square, including the boundary collar of
Section~\ref{sec:introduction}.  A segment here is an edge of the drawn
medial graph, before applying $\Pi$.  Write $z_e$ for its projected
midpoint and $I_e$ for the indicator that it belongs to the distinguished
strand.  The pairing at every medial vertex splits that vertex into
two vertices of degree two.  In the resulting finite graph, the
component containing a distinguished terminal half-edge is a path,
with the other distinguished half-edge as its other end.  It cannot
repeat an edge: a repeated edge would put that component on a cycle,
which cannot contain its degree-one terminal vertices.  Consequently
the counting multiplicity in~\eqref{eq:counting-measure} is exactly
$I_e$, even when the unsplit medial vertex is visited twice.  Distinct
collar edges are still counted separately if their projections coincide.

There is a constant $A_0>0$, determined by the density of medial
midpoints in the stated lattice units, such that
\begin{equation}
 A_0n^{-2}\sum_{e\in\mathcal E_n}h(z_e)
 \longrightarrow \int_D h(z)\dd A(z)
 \qquad\bigl(h\in C(\overline D)\bigr).
 \label{eq:probe-riemann}
\end{equation}
Indeed the bulk midpoints form a periodic set of fixed density, and
there are only $O(n)$ additional boundary probes.  Define
\begin{equation}
 X_n(f)=A_0n^{-2}\sum_{e\in\mathcal E_n}f(z_e)n^x I_e
       =A_0n^{-d}\sum_{e\in\mathcal E_n}f(z_e)I_e.
 \label{eq:microscopic-test}
\end{equation}

Use the sequence of radii $\epsilon\downarrow0$ supplied by
Proposition~\ref{prop:disk-calibration}, together with its constants
$L_\epsilon\asymp\epsilon^{-x}$.  The physical scale of a unit tile is
$a_0/n$.  For a probe $e$ in the bulk, let $B_e^\epsilon$ be the
whole-tile disk of radius $s=\epsilon n$ used in that proposition,
translated to the probe.  Its physical radius is $a_0\epsilon$, up to
$O(n^{-1})$.  Let $J_e^\epsilon$ indicate that the distinguished strand
enters this disk; the occurrence of an unrelated loop does not count.
For $f\in C_c(D)$, put
\begin{equation}
 X_{n,\epsilon}(f)
   =A_0n^{-2}\sum_{e\in\mathcal E_n}f(z_e)L_\epsilon J_e^\epsilon.
 \label{eq:disk-test}
\end{equation}
For fixed $\epsilon$ sufficiently small, every disk occurring in this
sum is inside $D$ once $n$ is sufficiently large.

\begin{proposition}[Replacement in second moment]
\label{prop:l2-replacement}
For every real $f\in C_c(D)$,
\begin{equation}
 \lim_{\epsilon\downarrow0}\limsup_{n\to\infty}
 \E\bigl[|X_n(f)-X_{n,\epsilon}(f)|^2\bigr]=0,
 \label{eq:l2-replacement}
\end{equation}
where $\epsilon$ tends to zero through the sequence in
Proposition~\ref{prop:disk-calibration}.  The same approximation holds
for every finite vector of such tests.
\end{proposition}

\begin{proof}
Fix a compact set $K\Subset D$ containing $\supp f$, and abbreviate
\[
 H_e^0=I_e,\qquad H_e^1=J_e^\epsilon,
 \qquad w_0=n^x,\qquad w_1=L_\epsilon,
 \qquad V_e^a=w_aH_e^a\quad(a\in\{0,1\}).
\]
We first compare products at two probes $e,e'$ satisfying
$z_e,z_{e'}\in K$ and $|z_e-z_{e'}|\ge g$, where $g>0$ is fixed.
Constants in this comparison may depend on $K$ and $g$.  Choose
$\ell=c_{K,g}n$ in tile units, with $c_{K,g}>0$ small enough that
balls of radius several times $\ell$ about the two probes are disjoint
and contained in the square.  All auxiliary cuts below leave fixed
annular collars.  Let $\mathcal F$ be the edges outside the ball of
radius $\ell$ about $e$.

Take $R\asymp\epsilon^b n$ as in
Proposition~\ref{prop:disk-calibration}, where $b>0$ will be sufficiently
small, and write $h_\epsilon=R/n\asymp\epsilon^b$.
Let $S_R^a$ denote the local two-color reaching event from the segment
start when $a=0$ and from $B_e^\epsilon$ when $a=1$.  For sufficiently
small $\epsilon$, these events and their collars are inside the
$\ell$-ball.  On every exterior configuration of positive probability,
set
\[
 p_a^{\mathcal F}=\Pp_n(S_R^a\mid\mathcal F),
 \qquad
 Q_a^{\mathcal F}=\Pp_n(\,\cdot\mid\mathcal F,S_R^a),
 \qquad \Lambda_a=w_ap_a^{\mathcal F}.
\]
The conditional FK law has the partition induced by the exposed
exterior; the uniformity in that partition is essential here.

Write $p_a$ for the corresponding ordinary plane probability.  The
relative comparison through the full collar in Input~\ref{inp:fk}
gives, for some $c_{\rm mix}>0$,
\begin{equation}
 p_a^{\mathcal F}=p_a
       \bigl(1+O_{K,g}(h_\epsilon^{c_{\rm mix}})\bigr).
 \label{eq:occupation-relative-comparison}
\end{equation}
Proposition~\ref{prop:sharp-arms}, Equation~\eqref{eq:start-probabilities},
and Proposition~\ref{prop:disk-calibration} consequently imply
\begin{align}
 \Lambda_a&\le C_{K,g}h_\epsilon^{-x},
 \label{eq:occupation-lambda-bound}\\
 |\Lambda_0-\Lambda_1|
 &\le C_{K,g}h_\epsilon^{-x}
 \bigl(\epsilon^{c_{\rm cal}}
          +h_\epsilon^{c_{\rm mix}}+o_{n,\epsilon}(1)\bigr)
 \label{eq:occupation-lambda-difference}
\end{align}
for some $c_{\rm cal}>0$.  Here and below $o_{n,\epsilon}(1)$ tends to
zero as $n\to\infty$ for each fixed admissible $\epsilon$; it can be
chosen uniformly over the probes and exposed exteriors under
consideration.  Translation, rotation, and the color exchange in
Proposition~\ref{prop:disk-calibration} identify their plane ratios.

Fix a second probe type $v\in\{0,1\}$.  Let $T_{e'}^v$ be its
two-color reaching event to a cut of radius $\ell/C$ about $e'$, with
$C$ a sufficiently large fixed constant, defined using actual paths
inside that cut rather than connections supplied by exterior wiring.
This event is
$\mathcal F$-measurable and is necessary for $H_{e'}^v=1$.
Define
\[
 q_a^{\mathcal F}
   =\E_{Q_a^{\mathcal F}}[H_e^aH_{e'}^v].
\]
Proposition~\ref{prop:outward} couples $Q_0^{\mathcal F}$ and
$Q_1^{\mathcal F}$ with failure probability at most
$C(s/R)^{c_{\rm out}}$, for some $c_{\rm out}>0$.  On success there is
an identical intermediate band with exactly two through strands, and
the completions outside that band agree.  The two through strands have
one possible connection through the interior.  In the segment
experiment that connection contains $e$, while in the disk experiment
it enters $B_e^\epsilon$.  Whether the connection belongs to the
distinguished strand is therefore the same in both completions.
The identical outer connection also preserves its hit verdict at $e'$.
This last observation is needed because $H_{e'}^v$ itself need not be
$\mathcal F$-measurable.  It gives
\begin{equation}
 |q_0^{\mathcal F}-q_1^{\mathcal F}|
 \le C_{K,g}(s/R)^{c_{\rm out}}\1_{T_{e'}^v}.
 \label{eq:occupation-q-difference}
\end{equation}

A hit by the distinguished strand also requires continuation from
$R$ to a cut of radius comparable to $\ell$ about $e$.  By the start
comparisons, sharp arms, and their conditional collar versions,
\begin{equation}
 q_a^{\mathcal F}
    \le C_{K,g}h_\epsilon^x\1_{T_{e'}^v}.
 \label{eq:occupation-q-bound}
\end{equation}
One can obtain this bound by taking the ratio of the start probability
to radius $\ell/C$ and that to radius $R$ in the conditional FK law.
Both cuts have collars before the exposed exterior; the start factors
in Equation~\eqref{eq:start-probabilities} cancel, leaving
$\pi_2(R,\ell/C)\le C_{K,g}h_\epsilon^x$.

Since $H_e^a=1$ implies $S_R^a$, the conditional expectation of
$V_e^aH_{e'}^v$ is $\Lambda_aq_a^{\mathcal F}$.  Subtracting these
expressions and using
Equations~\eqref{eq:occupation-lambda-bound}--\eqref{eq:occupation-q-bound}
yields
\begin{align}
 &\left|\E\bigl[(V_e^0-V_e^1)H_{e'}^v\mid\mathcal F\bigr]\right|
 \nonumber\\
 &\quad\le C_{K,g}\1_{T_{e'}^v}
 \left[h_\epsilon^{-x}(s/R)^{c_{\rm out}}
       +\epsilon^{c_{\rm cal}}+h_\epsilon^{c_{\rm mix}}
       +o_{n,\epsilon}(1)\right].
 \label{eq:conditional-product-replacement}
\end{align}
Thus the continuation bound cancels the factor $h_\epsilon^{-x}$ in
the normalization error.  The coupling error retains this factor.
Finally, the necessary arms at the second probe satisfy
\[
 w_v\Pp_n(T_{e'}^v)\le C_{K,g}.
\]
Multiplication of Equation~\eqref{eq:conditional-product-replacement}
by $w_v$ and integration therefore show that
\begin{equation}
 \limsup_{n\to\infty}
 \sup_{\substack{z_e,z_{e'}\in K\\|z_e-z_{e'}|\ge g}}
 \left|\E[(V_e^0-V_e^1)V_{e'}^v]\right|
 \le C_{K,g}\eta(\epsilon),
 \label{eq:separated-product-replacement}
\end{equation}
where
\[
 \eta(\epsilon)=\epsilon^{(1-b)c_{\rm out}-bx}
                +\epsilon^{c_{\rm cal}}
                +\epsilon^{bc_{\rm mix}}.
\]
Choose $b$ sufficiently small that
$(1-b)c_{\rm out}-bx>0$, in addition to the requirements in
Proposition~\ref{prop:disk-calibration}.  Then
$\eta(\epsilon)\to0$.  Interchanging the probes gives the analogous
replacement in the second factor.

It remains to control nearby probes.  The first-moment arm bound on
$K$ is
\begin{equation}
 \E V_e^a\le C_K\qquad(a=0,1).
 \label{eq:interior-probe-first-moment}
\end{equation}
For two probes at distance $y\ge C\epsilon$, impose their necessary
arms separately to radii comparable to $y$, using disjoint collars.
Conditioning on the exterior of either collar leaves the other event
measurable, so the uniform conditional bounds multiply.  They give
\begin{equation}
 \E[V_e^aV_{e'}^v]\le C_K y^{-2x}.
 \label{eq:probe-pair-bound}
\end{equation}
When $a=v=0$, this holds down to a fixed multiple of the mesh; at
smaller distances use the bound $n^{2x}$.  Thus its right side can
there be replaced by $C_K(y\vee n^{-1})^{-2x}$.
If $y\le C\epsilon$ and at least one probe is a disk probe,
Equation~\eqref{eq:interior-probe-first-moment} instead gives
\begin{equation}
 \E[V_e^aV_{e'}^v]\le C_K\epsilon^{-x}.
 \label{eq:overlapping-disk-bound}
\end{equation}
For a mixed pair bound the disk factor by $L_\epsilon$ and use the
microscopic first moment; for two disk probes use the first moment of
either one.

The periodic probe set has at most $C(1+n^2u^2)$ points in a ball of
radius $u$.  Summing Equation~\eqref{eq:probe-pair-bound} in dyadic
distance annuli with $y<g$ shows that its contribution to the double
Riemann sum is at most
$C_K(g^{2-2x}+n^{2x-2})$.  The contribution of
Equation~\eqref{eq:overlapping-disk-bound}, after taking the upper
limit in $n$, is at most $C_K\epsilon^{2-x}$.  Both exponents are
positive.  Expanding the square in
Equation~\eqref{eq:l2-replacement}, use these bounds for pairs at
distance less than $g$ and
Equation~\eqref{eq:separated-product-replacement} for all other pairs.
For fixed $g$ and sufficiently small $\epsilon$ this proves
\[
 \limsup_n\E|X_n(f)-X_{n,\epsilon}(f)|^2
 \le C_K\|f\|_\infty^2
       \bigl(g^{2-2x}+\epsilon^{2-x}\bigr)
       +C_{K,g}\|f\|_\infty^2\eta(\epsilon).
\]
Let $\epsilon\downarrow0$ first and then $g\downarrow0$.
Applying the scalar result to each component proves the assertion for
finite vectors.
\end{proof}

\subsection{Minkowski content in the closed square}

We record the whole-domain consequence of the SLE content theorems,
including the boundary estimate needed to match the convention of
Equation~\eqref{eq:content-measure}.

\begin{lemma}[Whole-square content]
\label{lem:whole-content}
Let $\eta$ be chordal $\mathrm{SLE}_\kappa$ in $(D;a,b)$, with the
parameters of Equation~\eqref{eq:parameters}.  Its interior
$d$-dimensional Euclidean Minkowski-content measure extends by zero
on $\partial D$ to a finite random measure $\mu_\eta$ on
$\overline D$.  For every real $f\in C(\overline D)$,
\begin{equation}
 r^{-x}\int_\C f(\Pi z)
       \1_{\{\dist(z,\tr\eta)<r\}}\dd A(z)
 \longrightarrow \mu_\eta(f)
 \quad\hbox{in probability as }r\downarrow0.
 \label{eq:whole-content-convergence}
\end{equation}
Its expected total mass is finite, and some nonnegative
$f\in C_c(D)$ has $\E\mu_\eta(f)>0$.  If
$U_\delta=\{z\in\overline D:\dist(z,\partial D)\le\delta\}$,
then
\begin{equation}
 \E\mu_\eta(U_\delta)\le C\delta^{1-x}.
 \label{eq:continuum-boundary-strip}
\end{equation}
\end{lemma}

\begin{proof}
Here $4<\kappa\le6<8$ and $0<x<1$.  Theorem~3.1 of
\cite{LawlerRezaeiContent} establishes almost sure and $L^2$
convergence of the full-trace neighborhood-area approximations on
interior dyadic squares in the half-plane.  Together with its
measure construction and Theorem~1.1, this gives the interior content
measure and convergence against continuous tests of compact support.
The normalization there is precisely radius to the power $d-2$
times Euclidean neighborhood area.

For completeness, this interior conclusion is preserved by the
conformal map $F:\mathbb H\to D$ taking $0,\infty$ to $a,b$.
The inverse image of a compact subset of $D$ is a compact subset of
$\mathbb H$.  Given $\vartheta>0$, subdivide a neighborhood of that
compact set into patches with centers $w_j$ on which
$|F'|/|F'(w_j)|$ lies between $1-\vartheta$ and $1+\vartheta$.
For sufficiently small $r$, conformal distortion bounds the preimage
of a physical $r$-neighborhood on the $j$th patch between neighborhoods
with radii $(1\pm C\vartheta+o_r(1))r/|F'(w_j)|$.
Neighborhoods of the whole
trace may be used: a trace portion outside a larger patch cannot
affect a sufficiently small neighborhood of its inner patch.
Changing area by $|F'|^2$ and applying the half-plane convergence
squeezes the limit between the corresponding patchwise integrals.
Taking $r\downarrow0$ first and then $\vartheta\downarrow0$ identifies
the measure as
\[
 F_*\bigl(|F'|^d\mu_{\mathbb H}\bigr)
 \quad\hbox{on }D,
\]
and gives convergence in probability for compactly supported tests.
Continuous nonnegative tests can first be approximated on these
patches; the signed case follows by subtraction.

Theorem~2.3 of \cite{LawlerRezaeiBasic} applies to chordal SLE in any
simply connected domain.  It gives a Euclidean one-point asymptotic,
with an error bounded by a power of
$r/\dist(z,\partial D)$ when this ratio is below $1/10$.
Its Green function is a constant times the conformal radius to the
power $-x$, multiplied by the angular factor
$\sin^{8/\kappa-1}\theta$.  This factor is at most one, and Koebe's
theorem compares conformal radius to boundary distance.  Hence, with
$y=\dist(z,\partial D)$,
\begin{equation}
 \Pp\{\dist(z,\tr\eta)<r\}
       \le C(r/y)^x\qquad(0<r<y/10).
 \label{eq:sle-distance-upper-bound}
\end{equation}
The constants are uniform in $z\in D$, including near the corners.
The interior Green moment identity in
Theorem~3.1 of \cite{LawlerRezaeiContent}, transported by $F$, implies
positive expected content against every nonzero nonnegative
continuous interior test.  This uses the Green function with the
Euclidean content constant absorbed, as in that paper.

Let $M_r$ denote the random approximation measure on the left side
of Equation~\eqref{eq:whole-content-convergence}.  For $r<\delta/10$,
integrating Equation~\eqref{eq:sle-distance-upper-bound} over
$10r<y\le\delta$ gives
\[
 \E M_r\bigl(\{z\in D:10r<\dist(z,\partial D)\le\delta\}\bigr)
 \le C\int_0^\delta y^{-x}\dd y
 \le C\delta^{1-x}.
\]
The interior strip $y\le10r$ has area $O(r)$, so its contribution is
at most $Cr^{1-x}$ even without a probability bound.  Outside $D$,
only the $r$-neighborhood of $\overline D$ contributes.  Its area is
$O(r)$, and $\Pi$ maps it onto the boundary.  We have therefore shown
\begin{equation}
 \E M_r(U_\delta)
       \le C\delta^{1-x}+Cr^{1-x}.
 \label{eq:content-approximation-boundary}
\end{equation}

The interior Green bound is integrable over the square.  Local
convergence, Fatou's lemma, and an exhaustion of $D$ consequently
give a finite interior measure with finite expected total mass.
The same argument in boundary strips gives
Equation~\eqref{eq:continuum-boundary-strip}.  Extend this measure
by assigning zero mass to $\partial D$.  For a continuous test on
$\overline D$, multiply it by a continuous cutoff vanishing on
$U_\delta$ and equal to one outside $U_{2\delta}$.  Local convergence
applies to the cutoff test.  Equations~\eqref{eq:continuum-boundary-strip}
and~\eqref{eq:content-approximation-boundary}, followed by Markov's
inequality, make the discarded parts arbitrarily small, first as
$r\downarrow0$ and then as $\delta\downarrow0$.
This proves Equation~\eqref{eq:whole-content-convergence} with exactly
the prescribed projection and normalization.  The conformal map
extends to the marked corners by Jordan-domain boundary continuity,
so this argument includes both chordal endpoints.
\end{proof}

\subsection{Identification of the joint limit}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Fix an admissible $\epsilon>0$ and $f\in C_c(D)$.  Input~\ref{inp:trace}
implies, jointly with the ordered curve, that
\begin{equation}
 X_{n,\epsilon}(f)\ \Longrightarrow\
 L_\epsilon\int_D f(z)
       \1_{\{\dist(z,\tr\eta)<a_0\epsilon\}}\dd A(z).
 \label{eq:fixed-radius-joint-limit}
\end{equation}
To verify the approximation of the disk tests, consider any sequence
of curves converging in the ordered-curve metric.  Their trace sets
then converge in Hausdorff distance.  The tile
rounding, disk center choice, and rounded switch drawing change
distances by $O(n^{-1})$.  Sandwich the disk indicators between
Euclidean distance indicators with radii
$a_0\epsilon\pm o(1)$ and apply
Equation~\eqref{eq:probe-riemann}.
For every closed set $T\subset\C$ and every $r>0$, the level set
$\{z:\dist(z,T)=r\}$ has area zero: the distance function has gradient
of magnitude one almost everywhere off $T$, whereas its gradient
vanishes almost everywhere on any positive-area level set.
Thus the sandwich has the asserted limit, and the continuous mapping
theorem gives Equation~\eqref{eq:fixed-radius-joint-limit}.
This reasoning uses no simplicity of $\tr\eta$ and applies to any
finite collection of tests.

The deterministic quantities $L_\epsilon(a_0\epsilon)^x$ are bounded
above and below by positive constants.  Choose a subsequence of the
admissible radii, still denoted by $\epsilon$, on which
\begin{equation}
 L_\epsilon(a_0\epsilon)^x\longrightarrow L\in(0,\infty).
 \label{eq:occupation-limit-constant}
\end{equation}
By Lemma~\ref{lem:whole-content}, the right side of
Equation~\eqref{eq:fixed-radius-joint-limit} tends in probability to
$L\mu_\eta(f)$ on the same SLE probability space.  The finite-vector
version of Proposition~\ref{prop:l2-replacement} and the
converging-together argument now give
\begin{equation}
 \bigl([\eta_n],X_n(f_1),\ldots,X_n(f_k)\bigr)
 \ \Longrightarrow\
 \bigl([\eta],L\mu_\eta(f_1),\ldots,L\mu_\eta(f_k)\bigr)
 \label{eq:interior-joint-limit}
\end{equation}
for all real $f_1,\ldots,f_k\in C_c(D)$, along all integers $n$.
Indeed, at each fixed radius the approximation converges along the
entire mesh sequence; its replacement error tends to zero in the
successive order $n\to\infty$, $\epsilon\downarrow0$.
Extracting radii in Equation~\eqref{eq:occupation-limit-constant}
does not extract meshes.  This also shows directly why the limiting
measure and curve in Equation~\eqref{eq:interior-joint-limit} belong
to the same SLE sample.

To pass to the closed square, regard $X_n$ as the measure defined by
Equation~\eqref{eq:microscopic-test}.  If a probe is at physical
boundary distance $y\ge C/n$, a hit requires its two color arms to a
cut of radius comparable to $ny$ in tile units.  That cut and a full
collar can be placed inside the square.  The conditional arm bound
from Proposition~\ref{prop:sharp-arms} gives
\begin{equation}
 \E[n^x I_e]\le Cy^{-x}.
 \label{eq:lattice-near-boundary-first-moment}
\end{equation}
There are $O(n)$ probes in every strip of width $1/n$ adjacent to the
square boundary.  Consequently, for a sufficiently large fixed
integer $j_0$ and $\delta\ge j_0/n$,
\begin{align}
 \E X_n(U_\delta)
 &\le Cn^{x-1}
       +Cn^{-1}\sum_{j=j_0}^{\lceil\delta n\rceil}(j/n)^{-x}
 \nonumber\\
 &\le Cn^{x-1}+C\delta^{1-x}.
 \label{eq:lattice-boundary-strip}
\end{align}
The first term includes all probes in the remaining bounded-width
mesh strip and in the collar, using $I_e\le1$.  Projecting them does
not change their number.  The terminal connectors have no mass.
Thus all boundary contributions vanish in the successive limits
$n\to\infty$, $\delta\downarrow0$.

Use the same continuous interior cutoffs as in the proof of
Lemma~\ref{lem:whole-content}.  For every $f\in C(\overline D)$ the
discarded discrete part has first moment at most
$C\|f\|_\infty(\delta^{1-x}+n^{x-1})$, and the discarded limit has
first moment at most $CL\|f\|_\infty\delta^{1-x}$.
Equation~\eqref{eq:interior-joint-limit} therefore extends to every
finite set of continuous tests on $\overline D$, including $f=1$.
The same first-moment estimates give
$\sup_n\E X_n(\overline D)<\infty$, after absorbing finitely many
small meshes.  For each finite $M$, the set of nonnegative measures of
total mass at most $M$ is compact in the weak topology on the compact square,
so the measures are tight.  Together with curve tightness from
Input~\ref{inp:trace}, a countable determining family of continuous
tests identifies the joint limit as
\[
 ([\eta_n],X_n)\ \Longrightarrow\ ([\eta],L\mu_\eta).
\]
Finally, by the edge-counting identity preceding
Equation~\eqref{eq:microscopic-test},
$X_n=A_0n^{-d}\sum_{j=1}^{N_n}\delta_{\Pi(z_{n,j})}$.
Taking the single deterministic constant
\[
 c(q)=\frac{A_0}{L}\in(0,\infty)
\]
proves the asserted convergence in the product of the ordered-curve
and weak finite-measure topologies, including convergence of total
mass.
\end{proof}
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//08-occupation.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//angular.tex ====
\section{Angular transfer and its two leading modes}\label{sec:angular}

Throughout this section, $1\le q<4$, $d=\sqrt q=2\cos\lambda$,
$\rho=\lambda/\pi\in(0,1/3]$, $v=\pi/2$, and $Q=-e^{i\lambda}$.
The argument has two parts. We first construct a Hilbert space from
infinite-plane row correlations and prove an angular estimate there. We
then identify its scalar products with reflection across diagonal cuts
of the physical square array. Boundary conditions enter the resulting
estimate through the norms of the exterior states.

\subsection{Rows, twists, and the estimate}

In the spin basis $+,-$, set
\[
 J=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
 L_-=|-\rangle\langle+|,\quad L_+=|+\rangle\langle-|,
 \qquad K(\zeta)=\operatorname{diag}(\zeta,\zeta^{-1}).
\]
Here and below twists satisfy $|\zeta|=1$. In the ordered pair basis
$++,+-,-+,--$, let
\begin{equation}\label{bulk:R}
 R(u)=\begin{pmatrix}
 a(u)&0&0&0\\0&b(u)&1&0\\0&1&b(u)&0\\0&0&0&a(u)
 \end{pmatrix},\qquad
 a(u)=\frac{\sin(\lambda-u)}{\sin\lambda},\quad
 b(u)=\frac{\sin u}{\sin\lambda}.
\end{equation}
Time and space steps have unit length and angles $\alpha_t$ and
$\omega_j$. Their successive sums give a rhombic tile array; initially
$0<\omega_j-\alpha_t<\pi$. A spin is positive when its arrow points
along the left normal of the corresponding step. If $h_0,h_1$ are the
two time sides before and after advancing in space, and $s_0,s_1$ the
two space sides before and after advancing in time, the tile tensor
has output indices $(h_0,s_1)$, input indices $(h_1,s_0)$, and value
$R(\rho(\omega_j-\alpha_t))$.

On a spatial circle of even size $N$, write
\begin{equation}\label{bulk:finite-row}
 T_N^D(\alpha)=\operatorname{Tr}_c
 D_cR_{c0}(\rho(\omega_0-\alpha))\cdots
 R_{c,N-1}(\rho(\omega_{N-1}-\alpha)).
\end{equation}
The auxiliary index is $c$. At the seam, $D_{x,y}$ uses the spin $x$
from tile $-1$ and $y$ from tile $0$. Let $\Omega_N$ be the positive
unit Perron vector of the untwisted rows on the zero-total-spin
sector, and $\Lambda_N(\alpha)$ their eigenvalue. Normalized rows are
$A_N^D(\alpha)=T_N^D(\alpha)/\Lambda_N(\alpha)$.
Their common Perron data, and the limiting operators $A^D(\alpha)$,
are constructed below. In the homogeneous transverse array
$\omega_j=v$, put
\begin{equation}\label{bulk:MN}
 \begin{aligned}
 P_\zeta(z)=A^{K(\zeta)}(z),\qquad
 M(z)&=e^{i\tau\rho z}A^D(z),\\
 N(z)&=M(z)P_\zeta(z)+P_\xi(z)M(z),
 \end{aligned}
\end{equation}
where
\begin{equation}\label{bulk:charges}
 \zeta/\xi=Q^\epsilon,\quad \epsilon\in\{-1,1\},\qquad
 (D,\tau)=(L_-,\epsilon)\ \text{or}\ (L_+,-\epsilon).
\end{equation}

\begin{theorem}[Angular transfer estimate]\label{bulk:angular-estimate}
Fix a compact interval $I\subset(-v,v)$. Let $B_{\rm in}$ be a
product of $P_\zeta$ rows and $B_{\rm out}$ a product of $P_\xi$ rows,
at real angles in $[-v,v]$. Suppose each product contains at least
$n\ge1$ angles in $I$. Uniformly for $x,y\in[-v,v]$,
\begin{align}
 \|B_{\rm out}N(x)B_{\rm in}\|&\le Cn^{-1+\rho},
 \label{bulk:low-estimate}\\
 \|B_{\rm out}\{N(x)-e^{i\tau(x-y)}N(y)\}B_{\rm in}\|
 &\le Cn^{-1-\rho}.\label{bulk:high-estimate}
\end{align}
The constant depends on $q$ and $I$. Boundedly many additional pairs
of before-twist rows may be inserted immediately to the right of
$N$, and additional pairs of after-twist rows immediately to its
left. The two terms in \eqref{bulk:high-estimate} may use different
such insertions; on each side their counts must be even. The
constant may depend on a bound for these counts. Multiplication of
$D$ by a common scalar multiplies the bounds by its absolute value.

The same estimates hold in the physical square array sliced in a
diagonal time direction: its time angles are $\pm\gamma_0$, where
$\gamma_0=\pi/4$, and its space angles alternate between
$\pi-\gamma_0$ and $\gamma_0$. The exterior states may contain any
finite number of physical row and spin insertions outside the
sandwich. A scalar contraction is bounded by the displayed rate
times the two reflection norms, with the squares centered on a cut
included in its reflection kernel.
\end{theorem}

\subsection{Row algebra and the infinite-plane state}

The finite-dimensional identities used throughout are
\begin{align}
 R_{12}(x-y)R_{13}(x)R_{23}(y)
 &=R_{23}(y)R_{13}(x)R_{12}(x-y),\label{bulk:YB}\\
 R(u)R(-u)&=(1-b(u)^2)I,\label{bulk:inversion}\\
 J_cR(u)^{t_c}J_c&=R(\lambda-u).\label{bulk:crossing}
\end{align}
For completeness, the nontrivial one-minus block of
\eqref{bulk:YB} is symmetric precisely by the three sine identities
\[
 b(x-y)+a(x)b(y)=b(x)a(y),\qquad
 b(y)+a(x)b(x-y)=b(x)a(x-y),
\]
\[
 a(x)+b(x-y)b(y)=a(x-y)a(y).
\]
Spin reversal gives the one-plus block; the remaining blocks are
scalar. The other identities follow directly from \eqref{bulk:R}.
Sliding the intertwiner in \eqref{bulk:YB} through two complete rows
shows that transfers with a common diagonal twist commute: the
intertwiner preserves total spin and hence commutes with the tensor
square of that twist. For arbitrary insertions $D,E$, the same
operation conjugates $D\otimes E$ by a two-spin intertwiner.
Expanding the conjugated matrix in matrix units expresses the
exchanged rows as a finite sum with new insertions. Its coefficients
are analytic wherever that intertwiner is invertible. Division by
the scalar Perron eigenvalues preserves this exchange identity.

For strict angles, the untwisted transfer is primitive on each
total-spin sector. It has positive diagonal entries; moreover an
auxiliary spin can swap with any chosen opposite pair of site
spins, without swapping at the other sites. Such exchanges connect
the sector. Commutation therefore gives common positive left and
right Perron vectors. In the homogeneous array, auxiliary
transposition reverses the product in \eqref{bulk:finite-row}, since
the distinct site operators commute. Equation \eqref{bulk:crossing}
then gives
\begin{equation}\label{bulk:adjoint}
 A^D(\alpha)^*=A^{J\overline D J}(-\alpha)
 \qquad(\alpha\in\mathbb R).
\end{equation}
The corresponding finite-volume identity makes the left and right
vacua equal.

\begin{lemma}[Plane kernels]\label{bulk:plane-kernels}
Finite words of homogeneous normalized strict-angle rows have
limiting Gram matrices as $N\to\infty$. Their quotient and completion
give a separable Hilbert space $\mathcal H$ with unit vector $\Omega$
and bounded left row actions satisfying
\begin{equation}\label{bulk:row-bound}
 \|A^D(\alpha)\|\le2\|D\|.
\end{equation}
Bare vacuum coefficients are continuous in strict angles and are
unchanged by a common translation of all time angles, as long as
the translated angles remain in $(-v,v)$.
\end{lemma}

\begin{proof}
We first identify the plane state, including the effect of the
zero-spin projection. Take periodic lists of time and space angles
in a strict range, with the two direction ranges ordered and
transversely separated in a half-circle. Use tori whose period
sizes are even multiples of the list periods. The resulting
rhombic square array is embedded and has bounded angles. Indeed
the cyclic order at each vertex gives local injectivity, while
averages of runs of either family have uniformly positive length
and uniformly transverse directions. Thus index distances control
Euclidean distances in both directions, giving properness and
global embedding.

Expand each $R$ into its two adjacent-side pairings with oriented
turn weight $e^{i\rho W/2}$. The pairing at the corner between the
two positive steps, and at its opposite corner, has coefficient
$a(u)$; the other pairing has coefficient $b(u)$. For example, the
mixed entry follows from
$a(u)e^{iu}+b(u)e^{-i(\lambda-u)}=1$ and its conjugate. In a
single-pairing entry the two turns cancel. A contractible loop
therefore has weight $d$. An essential simple oriented torus loop
has total turn zero. All essential loops are parallel up to
orientation. Their number $k$ is even, by the even intersection
parities with both period cycles and primitivity of their common
homology class. The allowed orientation count after projection
onto zero spin on either chosen quantum section is
\begin{equation}\label{bulk:essential-count}
 \chi=2^k\quad\text{if the intersection is zero},\qquad
 \chi=\binom{k}{k/2}\quad\text{otherwise}.
\end{equation}
Since $d<2$, there is $c(d)>0$ such that
$c(d)d^k\le\chi\le2^k$ for every even $k$.

Replacing $\chi$ by $d^k$ gives the canonical isoradial FK law,
up to the bounded torus genus factor of Lemma
\ref{prelim:loop-law}. To check the parameter, the relative
open weight at a primal corner between the positive steps is
$\sqrt q\,b/a$, and at the other primal corner it is
$\sqrt q\,a/b$. If $\gamma$ is the angle at the primal corner, the
odds divided by $\sqrt q$ are
\[
 \frac{\sin(\rho\gamma)}{\sin(\rho(\pi-\gamma))}.
\]
This is the canonical weight, with the complementary-angle
convention of \cite[Theorem 1.1]{DCLM2019Isoradial}. That theorem applies
to these fixed doubly periodic bounded-angle isoradial grids. It
gives uniqueness and buffered crossing estimates, uniformly over
the boundary partition on the outer buffer. The bulk passage-count
argument of Lemma \ref{prelim:passages} consequently applies
on each such grid. No estimate as an angle degenerates is used.

Here is why the essential-orientation tilt does not change the
local limit, even for unbounded aspect ratios. Let $r\le R$ denote
the shorter and longer period sizes. Cover the torus by $O(R/r)$
buffered boxes of size a small fraction of $r$, with a bounded
number of disjoint-buffer colors, so that every essential loop
crosses one of them. Include $O(R/r)$ translates, by medium periods,
of a target annulus about the query. Let $Y$ be the sum of the
passage counts in this cover. For each fixed $c>0$, conditional
passage moments, iteration within each color, and H\"older's
inequality give
\begin{equation}\label{bulk:torus-moment}
 \mathbb E e^{cY}\le e^{C_cR/r},\qquad k\le Y.
\end{equation}
The density of the tilted law is bounded above by $C e^{c_0Y}$
before normalization and its normalizing denominator is bounded
below. Consequently
$\mathbb E_{\rm tilt}e^Y\le C e^{CR/r}$, and Jensen's inequality
gives $\mathbb E_{\rm tilt}Y\le CR/r$. Translation invariance then
bounds the expected count at the target annulus uniformly.

Condition on the exterior of its inner box. The number of exterior
excursions from that boundary to distances comparable to $r$ before
returning is tight. If it is at most $K$, changing the filling
can alter only $O(K)$ essential loops: every affected loop must
leave the injective chart along such an excursion. The common
homology class can change only when no essential loop remains
unaffected. Thus the ratios of the genus and orientation tilt over
all fillings are bounded in terms of $K,q$. For the binomial case
this also follows directly by comparing central binomial
coefficients with indices differing by $O(K)$; a switch between
the two cases in \eqref{bulk:essential-count} can occur only when
there are $O(K)$ essential loops. The conditional inner law is
therefore comparable with FK under the induced partition.
Successive buffered primal and dual circuits show that an arm or
medial strand from a fixed window reaches toric distances with
probability tending to zero under the tilted law as well.

Now condition outside a large fixed test box containing the query.
With probability tending to one, no exterior actual cluster or
medial path incident to this box reaches toric distances. On that
event changing its filling leaves both the essential factor and
the genus factor exactly unchanged. Its conditional law is
ordinary FK. Letting the test box grow, comparison of increasing
cylinder events with free and wired laws and uniqueness identifies
the local plane law.

The same conclusion holds with finitely many marked matrices
$D_i$. Given the unoriented pairing, the absolute marked
orientation ratio is at most $\prod_i2\|D_i\|$: only marked loops
are broken, each newly cut arc has at most two orientations,
unaffected contractible loops retain weight $d$, and any residual
flux constraint has no more orientations than the central count
in \eqref{bulk:essential-count}. When all marked strands stay in
a fixed window, they do not encounter the distant flux projection.
The absence of escaping strands therefore proves marked local
convergence too.

Sending the time length to infinity first gives the
Perron-normalized coefficient at fixed spatial circumference.
Sending the circumference to infinity, and diagonalizing the two
limits, gives the stated plane coefficient. Any finite word can
be included in a periodic list; unmarked filler rows disappear
from its normalized longitudinal vacuum coefficient.

There is also a transverse quantization. Take a circle in the time
index, transfer in decreasing space index, and let the auxiliary
space spin flow in increasing time. The same difference relation
makes the untwisted transverse transfers commute and gives common
Perron data. The seam marks of the original word become one
tensor-product quantum insertion. Taking cylinder limits in this
order shows that its plane coefficient is unchanged when the
strict transverse angles are changed. Rotating both families and
then returning the transverse angles to $v$ proves common
time-angle translation invariance. Continuity follows from the
finite-box FKG sandwich and uniqueness as a common finite periodic
medium varies; sufficiently large fixed circuits at the limiting
medium control long marked strands, and all remaining local
orientation factors are continuous.

Finite-volume positivity now gives the limiting Gram form. To
prove boundedness before completing, let $h$ be a formal word
vector and set $T=(A^D)^*A^D$, with the adjoint understood first
as the reflected formal row in \eqref{bulk:adjoint}. The
marked expansion bounds
$\|T^jh\|^2\le C_h(2\|D\|)^{4j}$. Iterated Cauchy--Schwarz applied
to the positive form $\langle h,Th\rangle$ gives
\[
 \|A^Dh\|^2
 \le \|h\|^{2-2^{1-m}}
       \|T^{2^{m-1}}h\|^{2^{1-m}}.
\]
Letting $m\to\infty$ gives \eqref{bulk:row-bound}; the same inequality
shows that null vectors remain null. Continuity of kernels, and
words with rational angles and matrix entries, give separability.
\end{proof}

We may initially enlarge this construction by finite-position
quantum spin insertions and shifts. Moving a shift through a word
only translates finitely many markings, so the same proof gives
the enlarged Gram space and bounds. In that space the endpoint
rows are already defined by
\begin{equation}\label{bulk:endpoint-rows}
 A^D(v)=SD_0,\qquad
 A^D(-v)=(JD^tJ)_0S^{-1},
\end{equation}
where $(S\sigma)_j=\sigma_{j+1}$. These formulas follow from
$R(0)$ being the swap and \eqref{bulk:crossing}; their vacuum
eigenvalue is one. We next show that these endpoints preserve the
original cyclic space $\mathcal H$.

\subsection{A self-adjoint angle generator}

For a formal word $h$, translate all its angles by $t$ and call the
result $V_th$, for $t$ in a neighborhood of zero where the word is
strict. Reflection in its Gram formula negates the angles in the
reflected word. Common translation invariance therefore gives
\begin{equation}\label{bulk:hankel}
 \langle V_th,V_sg\rangle=\langle h,V_{s+t}g\rangle
\end{equation}
whenever both sides are defined. These curves are continuous. If
$h$ is null, then
$\|V_th\|^2=\langle h,V_{2t}h\rangle=0$ for small $t$, so the
translation is locally well defined on actual vectors.

\begin{lemma}[Angle generator]\label{bulk:boost}
There is a self-adjoint operator $B$ on $\mathcal H$ such that
$V_th=e^{tB}h$ locally for each word $h$. Ordinary strict-angle words
form a graph core for $B$. Each family $A^D$ extends to a bounded
holomorphic operator function on $|\operatorname{Re}z|<v$, with
\begin{equation}\label{bulk:imaginary-covariance}
 A^D(x+ib)=U_bA^D(x)U_{-b},\qquad U_b=e^{ibB}.
\end{equation}
\end{lemma}

\begin{proof}
For each word $h$, the function
$f_h(t)=\langle h,V_th\rangle$ is continuous and exponentially
convex on a sufficiently small interval: every permitted Hankel
matrix is a Gram matrix by \eqref{bulk:hankel}. The interval
Bernstein--Widder theorem, in the form of
\cite[Theorem 3.5 and Corollary 3.6]{BuescuPaixao2018}, represents
$f_h$ as a Laplace transform of a positive measure with finite
integrals throughout that interval. In particular it is analytic
near zero. Inner products of finite differences of $V_th$ are
the corresponding differences of $f_h(s+t)$. Their limits show
that all Hilbert-space derivatives $h^{(k)}$ exist and satisfy
\begin{equation}\label{bulk:analytic-vector-bound}
 \|h^{(k)}\|^2=f_h^{(2k)}(0)
 \le C_h(2k)!r_h^{-2k}.
\end{equation}
The Taylor series recovers the vector curve: this follows by
comparing its inner products and its squared norm using
\eqref{bulk:hankel}.

On the span of all these derivative vectors, define
$B_0h^{(k)}=h^{(k+1)}$. The cross versions of
\eqref{bulk:hankel} make this well defined and symmetric. More
explicitly, if a linear combination of derivatives vanishes, its
proposed derivative has zero inner product with every word by
differentiating the cross identity; density makes that derivative
zero. The same identity gives symmetry between derivative
vectors. The domain is dense and invariant. Bound
\eqref{bulk:analytic-vector-bound}, also with $k$ replaced by $k+j$,
implies that every vector in this domain is analytic for this same
operator: for some $t>0$ its series
$\sum_{j\ge0}t^j\|B_0^jh^{(k)}\|/j!$ converges. Nelson's theorem
\cite[Definition 4.7 and Theorem 4.8]{Neeb2011} therefore makes $B_0$
essentially self-adjoint. Denote its closure by $B$.

Finite differences of ordinary words approximate $h^{(k)}$ both
in norm and after applying $B$, by the Taylor series just obtained.
Ordinary words are thus a graph core. The vector series gives
$V_th=e^{tB}h$ for small real $t$ on each word.

For fixed $D$, the row family is strongly continuous, by the Gram
continuity and uniform bound. The word identity
$A^D(x+t)V_th=V_t(A^D(x)h)$ yields, weakly on the graph core,
\[
 \frac{d}{dx}\langle g,A^D(x)h\rangle
 =\langle Bg,A^D(x)h\rangle-\langle g,A^D(x)Bh\rangle.
\]
Integrate this equality and approximate $g,h$ in graph norm to
extend it to $\operatorname{Dom}B$. The extension in
Equation~\eqref{bulk:imaginary-covariance} then satisfies the Cauchy--Riemann
equations between domain vectors, with continuous partial
derivatives. Its uniform bound $2\|D\|$ extends weak holomorphicity
from that dense set to all vectors, hence gives bounded operator
holomorphicity throughout the strip.
\end{proof}

Words whose angles all lie in the middle third $(-v/3,v/3)$ already
span densely: test orthogonality to them and apply analyticity
successively in every argument of a word. For a fixed twist,
analytic continuation of the commuting-row identity and
\eqref{bulk:adjoint} gives
\begin{equation}\label{bulk:P-star}
 [P_\zeta(z),P_\zeta(w)]=0,\qquad
 P_\zeta(z)^*=P_\zeta(-\overline z).
\end{equation}
Thus this family lies in an abelian von Neumann algebra. Its
simultaneous spectral variables can be chosen as scalar functions
$f(z)$, analytic almost everywhere on the strip, bounded by $2$,
and satisfying $f(-\overline z)=\overline{f(z)}$. To obtain one
common exceptional set, start with a multiplication representation
of the algebra generated by countably many arguments in a dense
set, and use the norm-bounded Taylor series on countably many
discs. Covariance \eqref{bulk:imaginary-covariance} says that
$f(z)\mapsto f(z+ib)$ preserves the spectral measure class.

\subsection{Continuation through the two edges}

\begin{lemma}[Scalar continuation in a fixed band]\label{bulk:edge-band}
Almost every spectral function $f$ extends analytically and
without zeros to bands of one fixed positive width about the
lines $\operatorname{Re}z=\pm v$. It has modulus one on those lines
and satisfies $|f(z)|\le1$ in the intervening strip. At any specified
point of the strip, the set of spectral functions vanishing there
has measure zero.
\end{lemma}

\begin{proof}
We first obtain a uniform finite-volume vacuum estimate. Temporarily
write $T_D(u)=T_N^D(v-u/\rho)$. For $u$ in a fixed sufficiently
small complex disc about zero, both
\begin{equation}\label{bulk:division}
 T_D(u)T_I(u)^{-1},\qquad T_I(u)^{-1}T_D(u)
\end{equation}
are holomorphic with norms at most $C\|D\|$, uniformly for all
sufficiently large $N$.

To prove the first assertion, multiply by the transfer
$\widetilde T_I(-u)$ having reversed auxiliary order. Fold its
auxiliary index $c'$ by transposition. The product is the
two-auxiliary partial trace of
$(D\otimes I)\prod_jG_j$, in increasing site order, where
\[
 G_j=R_{cj}(u)R_{c'j}(-u)^{t_{c'}}.
\]
The line spanned by $|++\rangle+|--\rangle$ is invariant in the
auxiliary space, and on this line $G_j$ is
$r(u)I$, where $r(u)=1-b(u)^2$, by inversion. At $u=0$ the range
of $G_j$ is entirely in that line tensored with the site, by the
swap identity. In the common triangular decomposition, $G_j/r(u)$
has identity on this line, a complementary diagonal block of
norm at most $C|u|$, and a bounded off-diagonal block. A product of
these matrices has uniformly bounded norm, since the
off-diagonal block is bounded by a geometric series. Its partial
trace without $D$ differs from the identity by at most
$C(C|u|)^N$: the off-diagonal block has zero partial trace, the
line block gives the identity, and the remaining diagonal product
has the stated norm. This partial trace is invertible on one
fixed disc, proving the first bound in \eqref{bulk:division}.
Reversing the product in $G_j$ gives the same argument with the
invariant covector, proving the second. Crossing gives the
corresponding bounds near $u=\lambda$.

The common vacuum is an eigenvector of $T_I(u)$ by analyticity,
and its eigenvalue is nonzero in these discs. Hence
$A_N^D(z)\Omega_N$ has a uniform vector bound in discs of a fixed
radius about $z=\pm v$. Pairing with each fixed limiting word,
normal-family compactness gives analytic subsequential limits;
their values on the interior fix them uniquely. The vector norm
bound identifies these limits with a bounded holomorphic
$\mathcal H$-valued function. Uniform derivative bounds also give
strong convergence at the real endpoint to the swap expression
\eqref{bulk:endpoint-rows}, initially in the enlarged space.

Now commute this endpoint row through a word with all angles in
$(-v/3,v/3)$. Each exchange uses an intertwiner whose inverse is
analytic on one fixed disc about the endpoint, independent of the
word: the middle-third angle restriction keeps its parameter
uniformly away from inversion zeros. The exchange expresses the
result as a finite sum of bounded interior rows applied to the
continued vacuum vectors. It follows that the endpoint preserves
$\mathcal H$ and that $A^D(z)h$ continues holomorphically on this
same disc for every middle-third word $h$. Its bound may depend
on the word, because the number of exchanges does. This does not
alter the radius. Strong convergence of interior rows to the
endpoint holds first on this dense set and then on all of
$\mathcal H$, by the uniform interior operator bound.

Choose a countable dense set of middle-third words $h_j$, and a
finite equivalent measure in the spectral representation of
$P_\zeta$. If $r$ is the common continuation radius, write the
continued vector as $F_j(z)=\sum_{k\ge0}a_{jk}(z-v)^k$ at the
right endpoint. For $r'<R<r$, Cauchy's estimate gives
\[
 \|a_{jk}\|_{L^2}\le M_{j,R}R^{-k},\qquad
 \sum_{k\ge0}(r')^k\|a_{jk}\|_{L^2}<\infty.
\]
Since the measure is finite, the corresponding pointwise series
of fiber norms is summable almost everywhere. Intersect these
full-measure sets over the countable core and a countable family
of smaller radii. We obtain pointwise holomorphic vector series
on the same disc for every $j$. The left endpoint is identical.
On its interior portion the vector equals $f(z)h_j$.
Analyticity consequently makes the continued fiber proportional
to $h_j$ everywhere. At almost every spectral point at least one
vector in the countable dense set is nonzero; ratios using any
such vector agree on the interior and hence throughout the disc.
This constructs the scalar continuation with one common radius.

The swap formulas show that $P_\zeta(v)$ and $P_\zeta(-v)$ are
unitary inverses. Strong boundary convergence and
\eqref{bulk:imaginary-covariance} give unitary horizontal limits
at $\pm v+ib$. For countably many $b$ within the continuation
discs, the scalar value there has modulus one almost everywhere.
Continuity gives modulus one on each full vertical diameter.
Schwarz reflection across that diameter now implies
$f(z)\overline{f(2v-\overline z)}=1$, and the analogous identity
at $-v$, throughout the symmetric disc. Thus there are no zeros
there. Rational imaginary translations preserve null sets and
cover both boundary lines by discs of the same radius. This gives
the asserted fixed bands.

The bounded-function maximum principle on the strip gives
$|f|\le1$. Finally each nonzero analytic $f$ has isolated zeros,
so Fubini's theorem shows that on a fixed vertical orbit almost
every argument has no zero for almost every spectral function.
The covariance of the spectral measure class transfers this
null-set assertion to any prescribed point of that orbit.
\end{proof}

\subsection{Inner factorization and low spectral bands}

Use $w=e^{iz}$ to map $|\operatorname{Re}z|<v$ to the right
half-plane. We keep the notation $f$ for the resulting function.
Lemma \ref{bulk:edge-band} says that $f$ is inner, continues through
every finite nonzero point of the imaginary axis, and that all its
zeros satisfy
\begin{equation}\label{bulk:zero-cone}
 \operatorname{Re}w_j\ge c|w_j|
\end{equation}
for one $c>0$ independent of the spectral point. The disk
factorization and its boundary-spectrum description
\cite[pp.~1367--1368, Equations (1)--(2)]{AiDongJiang2019}, transported
by $(w-1)/(w+1)$, therefore give
\begin{equation}\label{bulk:inner-factorization}
 f(w)=\eta e^{-aw-b/w}\prod_j\beta_j(w),\qquad
 a,b\ge0,\quad |\eta|=1.
\end{equation}
Here $\beta_j$ is $(w-w_j)/(w+\overline{w_j})$ with the canonical
constant phase, chosen positive at $w=1$ when $w_j\ne1$.
The zeros satisfy the half-plane Blaschke condition
$\sum_j\operatorname{Re}w_j/(1+|w_j|^2)<\infty$.
There is no other singular factor: analytic continuation across
the rest of the boundary excludes singular measure there, leaving
only atoms at $0$ and $\infty$. Under the Cayley map these two
atoms give exactly $b/w$ and $aw$.

Suppose $-\log|f(1)|\le s$, where $s>0$ is sufficiently small.
There is then no zero at $1$, and evaluation of
\eqref{bulk:inner-factorization} gives
\begin{equation}\label{bulk:zero-budget}
 a+b+\frac12\sum_j
 \log\left(1+\frac{4\operatorname{Re}w_j}{|1-w_j|^2}\right)\le s.
\end{equation}
The cone condition implies, with a fixed constant,
\begin{equation}\label{bulk:small-large}
 a+b+\sum_j\min\{|w_j|,|w_j|^{-1}\}\le Cs.
\end{equation}
Indeed, for $r=|w_j|$, the fraction inside the logarithm is at least
$4cr/(1+r)^2$, and the resulting logarithm is bounded below by a
positive constant times $\min(r,r^{-1})$.

Choose $c_1>0$ sufficiently small. On the half-annulus
\begin{equation}\label{bulk:annulus}
 c_1^{-1}s<|w|<c_1/s,\qquad |\arg w|\le\pi/2,
\end{equation}
$f$ and $f^{-1}$ are bounded by fixed constants. On the unit
semicircle,
\begin{equation}\label{bulk:near-sign}
 f(w)=f(1)+O(s).
\end{equation}
To see both assertions directly, \eqref{bulk:small-large} separates
every zero into $|w_j|\le Cs$ or $|w_j|^{-1}\le Cs$. For a small
zero the ratio $\beta_j(w)/\beta_j(1)$ differs from one by at most
$C(1+|w|^{-1})|w_j|$; for a large zero its error is at most
$C(1+|w|)/|w_j|$. Their logarithms are summable and uniformly
bounded on \eqref{bulk:annulus}, after decreasing $c_1$.
The exponential factor is controlled by
$a|w-1|+b|w^{-1}-1|$. On $|w|=1$ the same estimates are $O(s)$.
This proves the claims, including bounded invertibility.

For a twist $\zeta$, define the nonnegative spectral operator
\[
 H_\zeta=-\log|P_\zeta(0)|,\qquad
 E_\zeta(s)=\mathbf1_{[0,s]}(H_\zeta).
\]
The no-zero assertion of Lemma \ref{bulk:edge-band} makes this
definition unambiguous almost everywhere. Since $P_\zeta(0)$ is
self-adjoint, split $E_\zeta(s)$ further according to its sign,
denoting these projections by $E_\zeta^\sigma(s)$,
$\sigma=\pm1$. Equations \eqref{bulk:annulus}--\eqref{bulk:near-sign}
say that the compressed $P_\zeta$ is boundedly invertible on the
half-annulus and
\begin{equation}\label{bulk:sign-approx}
 P_\zeta(w)E_\zeta^\sigma(s)
 =\sigma E_\zeta^\sigma(s)+O(s)
 \quad (|w|=1,\ \operatorname{Re}w\ge0).
\end{equation}

\subsection{Reversal, Laurent modes, and the sandwich}

The endpoint formulas give the exact reversal identity
\begin{equation}\label{bulk:reversal}
 M(v)+P_\xi(v)M(-v)P_\zeta(v)=0.
\end{equation}
For example, if $D=L_-$, then
$K(\xi)DK(\zeta)=(\zeta/\xi)D$ and
\[
 e^{-i\tau\rho v}(\zeta/\xi)=-e^{i\tau\rho v}.
\]
The raising case is the analogous computation, with
$\tau=-\epsilon$. The identity at $-v$ follows using the inverse
endpoint twists. Imaginary covariance extends these relations
after every common imaginary shift: both $M$ terms acquire the
same scalar multiplier. Boundary convergence is strong.

Compress $M$ between the low bands of the two twists:
\[
 M_c(w)=E_\xi(s)M(w)E_\zeta(s).
\]
For $\operatorname{Re}w<0$ define it from the right half-plane by
\begin{equation}\label{bulk:gluing}
 M_c(-w)=-P_\xi(w)^{-1}M_c(w)P_\zeta(w)^{-1},
 \qquad \operatorname{Re}w>0,
\end{equation}
where the inverses are only on the stated low bands. The two
versions of \eqref{bulk:reversal} give equality on both imaginary
rays. Uniform bounds and strong boundary continuity allow weak
holomorphic gluing, for instance by Morera's theorem on small
rectangles crossing the rays. Thus $M_c$ is holomorphic on the
full annulus \eqref{bulk:annulus}. Throughout it,
\begin{equation}\label{bulk:Mc-bound}
 \|M_c(w)\|\le C|w|^{\tau\rho}.
\end{equation}

Write its Laurent expansion as $\sum_{j\in\mathbb Z}w^jM_{c,j}$.
Cauchy estimates on circles near the inner or outer edge give
\begin{equation}\label{bulk:Laurent}
 \|M_{c,j}\|\le C(C's)^{|j-\tau\rho|}.
\end{equation}
On $|w|=1$, summing these estimates yields
\begin{align}
 \|M_c(w)\|&=O(s^\rho),\label{bulk:Mc-small}\\
 \frac{M_c(w)-M_c(-w)}2
 &=w^\tau M_{c,\tau}+O(s^{1+\rho}),\qquad
 \|M_{c,\tau}\|=O(s^{1-\rho}).\label{bulk:odd-mode}
\end{align}
Here the nearest odd integer to $\tau\rho$ is $\tau$; the next
possible distance is $1+\rho$. This is where $0<\rho\le1/3$ is
used in the ordering of the relevant exponents.

Refine the left and right low bands by signs $\sigma_\xi,\sigma_\zeta$.
If the signs agree, \eqref{bulk:gluing} and
\eqref{bulk:sign-approx} imply
$M_c(-w)=-M_c(w)+O(s)\|M_c(w)\|$ on the unit semicircle.
By \eqref{bulk:Mc-small}, its even part is $O(s^{1+\rho})$.
On this block,
\[
 E_\xi^{\sigma_\xi}(s)N(w)E_\zeta^{\sigma_\zeta}(s)
 =2\sigma_\xi w^\tau M_{c,\tau}+O(s^{1+\rho}).
\]
If the signs disagree, the two leading signs in
$M_cP_\zeta+P_\xi M_c$ cancel, giving $O(s^{1+\rho})$ directly.
Consequently, on the full low bands,
\begin{align}
 \|E_\xi(s)N(x)E_\zeta(s)\|&\le Cs^{1-\rho},
 \label{bulk:low-band}\\
 \|E_\xi(s)\{N(x)-e^{i\tau(x-y)}N(y)\}E_\zeta(s)\|
 &\le Cs^{1+\rho}.\label{bulk:high-band}
\end{align}
All real $x,y\in[-v,v]$ correspond to the closed unit semicircle.
A product of an even bounded number of extra rows is $I+O(s)$ on
each sign block, by \eqref{bulk:sign-approx}. Its error multiplied
by \eqref{bulk:Mc-small} is $O(s^{1+\rho})$. This proves exactly
the same conclusions with the extra pairs stated in Theorem
\ref{bulk:angular-estimate}, even if the two terms use different
pairs.

To finish the sandwich estimate, decompose each $H_\zeta,H_\xi$
into $[0,n^{-1}]$, dyadic intervals above $n^{-1}$ up to a small
fixed threshold, and the remaining interval. Schwarz--Pick gives,
uniformly for $x$ in the fixed compact interval $I$,
\[
 1-|f(x)|\ge c_I(1-|f(0)|).
\]
One can obtain this directly from the pseudohyperbolic contraction
between $0$ and $x$; their hyperbolic distance stays bounded on
$I$. A band with lower endpoint $u$ below the fixed threshold is
therefore damped by the $n$ strict steps by $e^{-cnu}$; the
remainder is damped exponentially in $n$. All other rows are
contractions, including the endpoint rows. Apply
\eqref{bulk:low-band} or \eqref{bulk:high-band} with $s$ the larger
of the two band upper endpoints. The double dyadic sum of
\[
 C\max(u,u',n^{-1})^\beta e^{-cn(u+u')},
 \qquad \beta=1-\rho\ \text{or}\ 1+\rho,
\]
is at most $C_\beta n^{-\beta}$. Terms involving a remainder band
use the uniform unrestricted row bound and are exponentially
small. This proves \eqref{bulk:low-estimate} and
\eqref{bulk:high-estimate} in the homogeneous representation.

\subsection{Diagonal cuts and the physical reflection norm}

It remains to prove the final assertion of Theorem
\ref{bulk:angular-estimate}. Work in the diagonal time frame and
write
\[
 \gamma_0=\pi/4,\quad X=e^{-i\gamma_0},\quad
 Y=e^{i\gamma_0},\quad H=-X,\quad L=Y.
\]
Space steps alternate $H,L$, starting with $H$; time steps are
$X$ or $Y$. Put $R_*=R(\lambda/2)$ and let
\begin{equation}\label{bulk:G}
 G=\prod_j\bigl(\operatorname{swap}R_*\bigr)_{2j,2j+1}.
\end{equation}
The factors act on disjoint pairs. Each is positive definite:
its middle block has eigenvalues $1\pm b(\lambda/2)>0$, and the
two outer entries are $a(\lambda/2)>0$. Use $G$ as the
finite-circle metric. Directly from the collapsed tiles,
\begin{equation}\label{bulk:diagonal-rows}
 T_N^D(\gamma_0)=D_{N-1}SG,\qquad
 T_N^D(-\gamma_0)=(JD^tJ)_0S^{-1}G.
\end{equation}

Here is the geometry behind both the metric and these formulas.
The lower slice has vertices $p,p-X,p+Y-X,\ldots$.
Reflection in the cut through $p$, perpendicular to time, gives
upper order $L,H$. For lower spins $(c_0,c_1)$ and upper spins
$(d_0,d_1)$, the intervening square kernel is
\[
 (R_*)_{-c_0,d_0;-d_1,c_1}
 =(\operatorname{swap}R_*)_{d_0,d_1;c_0,c_1}.
\]
Shifting the slice origin by $Y$ or $X$ puts the seam at site $-1$
or $0$ respectively, with reversed conventions in the latter
case. Equivalently, every second indexed tile has collapsed.
A coincident-step tile gives $R(0)$, the swap; an opposite-step
tile gives $R(\lambda)$, which imposes
$h_0=-s_1,\ h_1=-s_0$, with weight one. Its duplicate sides are
therefore spin identifications. The adjoint for the $G$ metric
exchanges the two time steps and sends $D$ to $J\overline D J$.

Both untwisted operators $S^{\pm1}G$ are primitive on the
zero-spin sector. Over a complete shift cycle, paired bonds of
both parities are available. One can hold every pair or swap a
chosen adjacent opposite pair, and such swaps connect the sector.
The transfers commute, have a common positive right Perron
vector, and have the same eigenvalue: their relation by $S^2$
and uniqueness make the vector $S^2$-invariant. Their left
Perron data are its $G$-dual. Normalize the vacuum to unit
$G$ norm.

These degenerate indexed arrays are exactly physical square
tori; they need no limiting estimates for degenerate rhombi.
To check this in both quantizations, let $p_t$ follow a periodic
sequence of $X,Y$ steps with even period size, and set
\[
 q_{2j}=j(Y-X),\qquad q_{2j+1}=q_{2j}-X.
\]
The uncollapsed tiles of the array $p_t+q_j$ biject with the
physical squares centered at each time level. The collapsed
identifications join duplicate physical edges as just described.
There are no free cycles consisting solely of duplicates:
each collapsed pair uses a time edge, which by the alternating
space pattern is also incident to an uncollapsed tile. Hence
zero-spin projection on either an unmarked zigzag slice or a
periodic time path is precisely the corresponding physical flux
projection. Marks at the time seam mark that physical path.
The periods preserve parity and are uniformly transverse, so
the torus-to-plane argument of Lemma \ref{bulk:plane-kernels}
applies directly to these square kernels.

For clarity, the transverse period product is primitive too.
Prescribe arbitrary zero-sum spin sequences on the time paths
$p_t$ and $p_t+m(Y-X)$. Interpret them as increments of integer
heights with consistent sublattice parity, periodically bounded
on the two lifted paths. Each path is geodesic in the square
graph with steps $X,Y$, so its own heights are $1$-Lipschitz.
Between the two paths the graph distance is at least $|m|$:
if the time-level difference is $\ell$, the two lower bounds
are $|\ell|$ and $2|m|-|\ell|$. Thus for all sufficiently large
$|m|$ the prescribed heights together are $1$-Lipschitz.
Extend them by the minimum of the integer distance cones
$h(a)+\operatorname{dist}(a,\cdot)$ from prescribed vertices $a$.
This extension is time-periodic; its parity is fixed, so every
edge increment is exactly $\pm1$. It gives an allowed spin
configuration of positive weight across the transfer strip.
Therefore a sufficiently large power of either oriented
two-space-step transfer has all entries positive. The
difference relation makes this period product commute with
the untwisted transfer at strict transverse angle $v$ on the
same time chain; the intertwiners involved are invertible.
They consequently share Perron data. Seam-marked transverse
vacuum coefficients agree with those at that strict angle.

Construct the limiting Gram space with the $G$ metric, allowing
the two endpoint rows, local site matrices, and $S^2$.
The adjoint of a site matrix, conjugated by $G$, is still local
and bounded. The preceding physical cylinder limits and the
marked moment argument give the Gram limits and bounded row
actions. Let $\mathcal H_*$ be the closed subspace generated
by seam words in $A^D(\pm\gamma_0)$ alone. Their Gram kernels
are the homogeneous transverse kernels, by the adjoint formula,
the transverse Perron comparison, and then the plane limit.
Thus $\mathcal H_*$ identifies isometrically with the subspace
of $\mathcal H$ generated by those words.

This subspace also contains the physical states required in
the theorem. It is reducing for $P_1(\pm\gamma_0)$. Those
operators have dense range there, because their adjoints have
no kernel by the no-zero assertion at the specified interior
angles. The finite-row identities give
\[
 S^2P_1(-\gamma_0)=P_1(\gamma_0).
\]
Both shifts by two therefore preserve $\mathcal H_*$, first
on a dense range and then by continuity. Equation
\eqref{bulk:diagonal-rows} with arbitrary $D$ similarly shows
that site matrices at $-1$ and $0$ preserve it; shifting by two
gives every finite site. Every physical edge between tiles
lies on a time-level zigzag slice, so arbitrary finite physical
modifications are such site insertions interleaved with layer
transfers. They belong to $\mathcal H_*$. For ordinary
homogeneous slicing the same assertion follows from the
endpoint shifts; several time-edge insertions on a strict
row can also be moved to one seam by commuting through finite
strings of invertible $R$ matrices and expanding their matrix
elements.

Finally, reflection in a time-level cut, followed by complex
conjugation and arrow reversal, matches lower spins to upper
spins through the intervening kernel $G$. Normalize by the
unmarked vacuum, or equivalently by the unit $G$ vacuum and
one common Perron factor per layer. The squared norm of a
lower state is then exactly the vacuum expectation ratio
of its reflected configuration. Between two cuts, transfer
from the lower slice of the first to the lower slice of
the second, and glue to the reflected future there using
$G$. Strings immediately below and above the last cut act
on the respective states and leave this kernel unchanged.
The homogeneous estimate restricted to $\mathcal H_*$ is
therefore the asserted diagonal estimate. This completes the
proof of Theorem \ref{bulk:angular-estimate}.

\section{Local path calculus for the current}\label{sec:current}

The transfer estimate applies to a lattice current because its
path dependence can be removed locally. This section proves the
needed tensor identities and records the phase bookkeeping when
a current is placed between two diagonal cuts.

\subsection{A path primitive and its deformation rule}

Direct a simple path $\mathcal C$ of tile edges from its initial
point. Measure each crossing spin using the left normal of the
path step. At a crossing, a matrix has row index on the right tile
and column index on the left tile. Lift the tangent angle
continuously along a rounded version of the path, starting with a
specified lift. Define its primitive $\mathcal J(\mathcal C)$ by
summing over the choice of one marked step:
\begin{itemize}
\item at that step insert $e^{i\rho\alpha}L_-$, where $\alpha$
      is the lifted tangent angle;
\item on all preceding steps insert $K(Q)$;
\item on all later steps insert the identity.
\end{itemize}
Thus both strand arrows at the marked crossing point away from it.
All insertions are additional to the tile tensors. Other primitives
and other marked insertions are kept outside the deformation
corridor, unless their order is explicitly included.

\begin{lemma}[Local deformation]\label{cur:deformation}
In the positive winding expansion, the primitive is invariant,
pairing by pairing, under simple-path isotopies that continue its
initial lift, keep its endpoints in their regions, and avoid the
other insertions. The statement is a tensor identity for arbitrary
outside spins. A pure diagonal string obeys the corresponding
deformation rule without a source.
\end{lemma}

\begin{proof}
Draw the strands smoothly and disjointly, with transverse path
crossings. At a marked crossing use the tangent-angle lift rotated
by less than $\pi/2$ to the orthogonal crossing direction; this
agrees with the lattice convention at a tile edge. Sliding the
source on its strand while its tangent changes by $b$ changes the
sum of the turns of the two departing pieces by $-2b$.
The winding factor is consequently multiplied by $e^{-i\rho b}$,
which cancels the change of the marker gauge. All other spin
constraints and prefix factors correspond directly.

A generic isotopy also creates or removes a pair of crossings at
a tangency. Terms marked elsewhere are unchanged because the
two new diagonal factors cancel. The two terms marked at the
new crossings cancel one another. To check the sign, orient a
test tangent $\beta$ of the strand approximately in the path
direction near tangency, so crossings $1,2$ occur in that order
both along the path and along this test direction. With the
source at crossing $2$, let $c_1$ be the signed spin at crossing
$1$; the strand there travels opposite to $\beta$, away from the
source. The orthogonalized lifts satisfy
\begin{equation}\label{cur:tangency-lift}
 (\alpha_2-\beta_2)-(\alpha_1-\beta_1)=-c_1\pi.
\end{equation}
For example, when the first path tangent is clockwise from
$\beta$, the offsets are $-\pi/2,+\pi/2$ and $c_1=-1$; the other
case reverses the signs. The ratio of the marker gauge times
the source-piece winding factors is therefore $e^{-ic_1\lambda}$.
The extra prefix factor is $Q^{c_1}$, and
\[
 e^{-ic_1\lambda}Q^{c_1}=-1.
\]
The outside constraints and factors of the two terms are
identical. This proves the cancellation.

A generic isotopy has only isolated tangencies of this kind,
besides transverse slides, and can be chosen to keep all fixed
insertions disjoint. This proves the primitive identity. For a
pure diagonal string, only the first cancellation is needed;
equivalently it is the tile ice-conservation identity.
\end{proof}

\subsection{Boundary-tree coefficients}

We use the boundary factors from Section \ref{sec:exponents}.
For reference, if $\theta_j$ is the lifted inward normal at port
$j$, $h_j$ its real height parameter, and $c=\pm1$ is its
inward/outward arrow sign, they are
\[
 b_j(c)=(2\sin\lambda)^{-1/2}
       \exp\{ic(\rho\theta_j/2+\lambda h_j)\}.
\]
Consider a disk, order its boundary from a flat gap $p$, and
assume there are no obstructing extra strings. Start a path from
$p$ toward an interior vertex $z$ with initial tangent lift
strictly between the positive boundary tangent and that tangent
plus $\pi$. Every boundary arc bounds a child interval not
containing $p$; these arcs form the separator tree of the
complementary regions.

\begin{lemma}[Tree evaluation]\label{cur:tree-weights}
For an arc $e:i\to j$ bounding its child interval, put
$m=h_i-h_j$ and
\begin{equation}\label{cur:weights}
 \begin{aligned}
 w_e&=\frac{\cos((m-\tfrac12)\lambda)}{\sin\lambda},&
 k_e&=-\frac{\cos((m+\tfrac12)\lambda)}{\sin\lambda},\\
 a_e&=e^{-i\lambda(h_i+h_j)},&
 c_*&=\frac{e^{-i\lambda}}{2\sin\lambda}.
 \end{aligned}
\end{equation}
For each fixed pairing, the primitive to $z$ is the sum over a
marked edge of the separator-tree path from $p$ to the region of
$z$: use $k_e$ on earlier edges, $c_*a_e$ on the marked edge, and
$w_e$ on every other boundary arc. Unmarked internal loops have
weight $d$, and a marked closed loop contributes zero.

For a pure $K(Q)$ string to $z$, use $k_e$ on the full
separator-tree path and $w_e$ off it. An internal loop enclosing
$z$ then has weight $-2$ instead of $d$. The parity of the number
of such loops records the color change from the boundary-tree
region to the lattice color of $z$.
\end{lemma}

\begin{proof}
By Lemma \ref{cur:deformation}, for each pairing we may choose a
simple path crossing minimally the boundary arcs separating $p$
from $z$, followed, if necessary, by the nested closed loops.
Such paths exist in the disk regions cut out by the disjoint
arcs and loops. A sole outward source on a closed loop is
inconsistent with the remaining arrow constraints, so it gives
zero.

For a parent-to-child crossing, the boundary-ordered arc travels
along the positive normal of the path. Let $\beta$ be its
tangent lift, continued from $\theta_i$. The correct lift relation
is the exact identity
\begin{equation}\label{cur:exact-angle}
 \beta-\alpha=\pi/2,
\end{equation}
without a multiple of $2\pi$. To verify this, close a simple curve
by taking the boundary from $p$ to $i$, the arc to the crossing,
and the reversed path back to $p$. Its positive full turn, with
the left right-angle turns at the port and crossing and the
short turn at $p$, is $\beta-\alpha+3\pi/2$. It equals $2\pi$,
proving \eqref{cur:exact-angle}.

For an outward source the sum of strand turns is
$\theta_i+\theta_j-\pi-2\beta$. Multiplying its winding weight by
the two outward boundary factors and by $e^{i\rho\alpha}$ gives
\[
 \frac1{2\sin\lambda}
 e^{-i\lambda(h_i+h_j)}
 e^{i\rho(\alpha-\beta-\pi/2)}
 =c_*a_e.
\]
Summing the two orientations of an unmarked arc gives $w_e$.
If a prefix string crosses it, multiplication by $K(Q)$ changes
that sum to $k_e$. These are precisely the tree weights in
\eqref{cur:weights}.

For a pure string, the same calculation holds on every separating
boundary arc. A counterclockwise enclosing loop has intersection
factor $Q^{-1}$, so the sum of its two orientations is
$e^{i\lambda}Q^{-1}+e^{-i\lambda}Q=-2$.
Crossing each such medial loop switches the adjacent color,
which proves the final parity assertion.
\end{proof}

\subsection{Reflection and two-step increments}

Reflect in a line of angle $\phi$, and combine this operation
with complex conjugation and arrow reversal. Relative to the
mirrored step direction, spins are unchanged, while the left and
right tiles are exchanged. Consequently the reflected primitive
has the form
\[
 L_+,\ K(Q^{-1}),\qquad e^{i\rho\alpha'},
 \qquad \alpha'=2\phi-\alpha,
\]
up to a common scalar. It obeys the reflected local deformation
identities, using the opposite winding expansion. In the notation
of \eqref{bulk:charges}, it has $\epsilon=-1$ and still $\tau=1$.
Further reflections alternate between these two types. Moreover,
a high difference with coefficient
$e^{i(\arg d_1-\arg d_2)}$ becomes the same expression with the
reflected directions $d'_1,d'_2$.

A \emph{low stencil} is a primitive increment on a straight
two-step extension from a vertex $z$. A \emph{high stencil} is
the difference of two such increments in distinct tile-axis
directions $d_1,d_2$, with coefficient
$e^{i(\arg d_1-\arg d_2)}$ on the subtracted term.
Both extensions have one common arriving prefix with fixed lift.
The local path choices are simple and extensible.

\begin{proposition}[Extraction in a clear slab]\label{cur:slab-estimate}
Suppose a diagonal-time slab of macroscopic thickness about a
low or high stencil contains no other non-diagonal tensors or
endpoints. Diagonal chains may cross the slab. The stencil has an
intact neighborhood disjoint from all other chains except its
own prefix, and the slab has spare margins for routing and cuts.
Bounded lattice errors are permitted inside these margins.
Then the stencil contraction can be written as the corresponding
low or high expression in Theorem \ref{bulk:angular-estimate},
between exterior states independent of the choice of stencil
term. Only boundedly many compensating pairs of rows and a
common bounded scalar are introduced.

In particular, if the two clear parts of the slab contain at
least $n$ physical steps, its scalar contraction is bounded by
$Cn^{-1+\rho}$ for a low stencil, and by $Cn^{-1-\rho}$ for a
high stencil, times the two exterior reflection norms. Reflected
stencils satisfy the same statement.
\end{proposition}

\begin{proof}
There are two matters to check: the compared paths must have
common endpoints, and rerouting the other charges must contribute
the same phase to every possible source position.

Let $a,b$ be the two positive-time unit steps, namely $X,Y$ in
the diagonal slicing. An increment in a negative direction is
exactly the negative of the corresponding forward increment
ending at $z$. This rewriting is local, even if previous cuts
have been made. Choose a point on the common prefix in the
intact neighborhood, express each increment as a difference of
two complete primitives from that point, cancel their common
initial marked terms, and apply Lemma \ref{cur:deformation}
inside the neighborhood. Thus no earlier path or cut is altered.

Use the following common endpoints for the high stencil:
\begin{itemize}
\item For directions $a,b$, start at $z$ and append two
      after-twist steps in the other direction to end at
      $z+2a+2b$.
\item For directions $-a,-b$, start at $z-2a-2b$, prepend
      two before-twist steps in the other direction, and end
      at $z$.
\item For directions $a,-b$, use the trajectory starting at
      $z-2b$ with steps $b,b,a,a$.
\end{itemize}
The case $-a,b$ is the same with $a,b$ interchanged.
Backward differencing changes the relative sign exactly as
required to obtain the coefficient for the two positive
directions.

Approach the common start by the same prefix, with a final
positive-direction segment. A simple replacement can first bend
around the stencil and approach from the required side, even
when the original prefix arrived from later times. In a box
with spare room, choose lattice corridors preserving the
entrance and reserve a clear rectangle for the last positive
steps. All marker lifts now differ by the short angle between
$a$ and $b$ in their common time frame; no extra $2\pi$ turn
is introduced. The low case is the same construction with one
increment. The explicit prefix charge is $Q^\epsilon$, with
$\epsilon=1$ or $-1$ according to the reflected type.

We next consolidate diagonal chains in the clear band. Keep as
common background every chain except the explicit prefix
steps between the common start and the marker. In particular,
the charge arriving at the common start is included in this
background. Using real charges modulo $2\pi$, route it to carry
a total phase $\zeta$ on a common positive-time seam before
the start, and a phase $\xi$ from the start through the
comparison block and to the future seam. These are the net
oriented charges through ordinary sections, and
\[
 \zeta/\xi=Q^\epsilon.
\]
All endpoints are preserved. Joining and funneling the chains
can be done outside the sandwich cuts but inside the larger
vacant margins, including transverse lattice displacements.
Choose the cuts through level vertices of the positive-time
seam with many positive steps on each side. This construction
does not require a monotone path between the original remote
cut positions.

Inside the comparison rectangle, first route the $\xi$
background along one common skirt on the higher-space-coordinate
side of both positive-step paths, avoiding every possible
marker. In $X,Y$ coordinates it runs around the northwest side
of the small positive rectangle between the common endpoints.
Tile ice conservation then moves this skirt onto either
compared path, always on the same, left spin side of a broken
source edge. At a marker it therefore multiplies $L_-$ or
$L_+$ on a fixed side by the diagonal twist. This multiplication
is one common scalar.

It remains to justify that the change from the original
background routing to this common skirt has a common phase.
Their difference is a closed tile-edge chain in the larger
slab, hence a weighted sum of face boundaries. Tile ice
conservation cancels every face contribution except at a
source edge, whose signed spin jump is fixed and equal to
two relative to its positive step. All prospective source
edges lie in the same winding region of this common
closed-chain difference. Indeed other original chains avoid
the intact neighborhood; the chosen prefix approaches the
common start from the past without entering the positive
rectangle; the replacement before the start follows that
same final incoming line with net charge $\zeta$; and the
common skirt and outgoing seam stay outside the rectangle.
No edge of the routing difference separates the interiors
of two possible marker edges. Extend the coincident clear
incoming and outgoing parts by many lattice edges if needed.
The face coefficient seen by the source is consequently
identical in every stencil term. Summing conservation gives
one common phase. There are no contributions from other
defects, since all changes remain inside the clear band and
its margins. The same reasoning can be phrased as a sequence
of path slides consistently kept on one side of the entire
marker rectangle. Every phase remains unitary.

We have now obtained exactly the before- and after-twist
operators in \eqref{bulk:MN}, with their required relative
phase, the compensating even pairs described above, and
one common bounded scalar. The two exterior states are
independent of which term is used. Theorem
\ref{bulk:angular-estimate} gives the claimed bound.

For later repeated cuts, note also how these states glue.
A consolidated chain at a diagonal cut ends at a shared
level vertex. Reflection conjugates its charge to the
inverse in the mirrored direction, preserving the spin
convention. Joining after reversing path direction therefore
continues with the same charge and creates no endpoint at
the cut. At preliminary tile-axis cuts, route the chains
transversely over the line through its level vertices, with
no segment lying on the line inside the spare margin; the
same argument applies. Reflections preserve the high or
low type at every other intact stencil point. This proves
the reflected assertion and the compatibility needed for
successive extractions.
\end{proof}

The proposition is a local norm estimate. In later applications
the number of available clear slabs and the sizes of their
exterior norms are estimated separately by the positive
partition comparisons; the local extraction itself makes no
uniform bound on those norms.
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//angular.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//barrier.tex ====
\begin{tikzpicture}[x=0.86cm,y=0.86cm,>=Latex,
  every node/.style={font=\small}]
  \fill[black!5] (-3,-2) rectangle (3,2);
  \fill[white] (-1,-.65) rectangle (1,.65);
  \draw[black!55] (-3,-2) rectangle (3,2);
  \draw[black!55] (-1,-.65) rectangle (1,.65);
  \node at (0,0) {inner rim};
  \node[anchor=west] at (3.12,-1.8) {outer rim};
  \draw[black!55] (3.03,-1.8)--(3.1,-1.8);

  \draw[blue!70!black,line width=1.35pt]
    (0,.65)--(0,2.35);
  \draw[red!75!black,line width=1.35pt,densely dashed]
    (0,-.65)--(0,-2.35);
  \node[anchor=west,text=blue!70!black] at (.12,2.27)
    {primal road};
  \node[anchor=west,text=red!75!black] at (.12,-2.27)
    {dual road};

  \draw[blue!70!black,line width=1.6pt]
    (-1.62,1.00)--(-.8,1.22)--(0,1.25)--(.8,1.22)--(1.62,1.00);
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    (1.66,.95)--(2,.15)--(2,-1.2)--(-2,-1.2)--(-2,.15)--(-1.66,.95);
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  \draw[black,line width=1.1pt] (-1.62,1.00)--(-1.66,.95);

  \draw[black,line width=.8pt]
    (1,.45).. controls (1.26,.67) and (1.40,.79) .. (1.64,.975)
      .. controls (1.98,1.24) and (2.52,1.66) .. (2.9,2);
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    (-1,.45).. controls (-1.26,.67) and (-1.40,.79) .. (-1.64,.975)
      .. controls (-1.98,1.24) and (-2.52,1.66) .. (-2.9,2);
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  \node[anchor=south east] at (1.59,1.10) {$a_2$};
  \node[anchor=south west] at (-1.59,1.10) {$a_1$};
  \node[anchor=west] at (3.12,.55) {through-arcs};
  \draw[black!65,->] (3.05,.64)--(2.27,1.46);
  \node at (1.5,-1.65) {barrier};
\end{tikzpicture}
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//barrier.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//chordal.tex ====
\section{Transfer to marked Jordan quadrilaterals}\label{sec:jordan}

Throughout Sections~\ref{sec:jordan}--\ref{sec:chordal}, fix
$1\le q<4$ and write
\[
 d=\sqrt q=2\cos\lambda,\qquad \rho=\lambda/\pi,
 \qquad h=1-2\rho\in[1/3,1).
\]
The four-change calculation of Section~\ref{sec:fourchange}, whose
holomorphicity hypothesis is supplied in Section~\ref{sec:holomorphic},
gives the functions
\begin{equation}\label{jordan:functions}
 f(\chi)=\frac{I_C(\chi)}{I_B(\chi)+I_C(\chi)},\qquad
 g(\chi)=\frac{f(\chi)}{f(\chi)+d(1-f(\chi))},
 \quad 0<\chi<1,
\end{equation}
where
\begin{align*}
 I_B(\chi)&=\int_\chi^1
 u^{\rho-1}(u-\chi)^{\rho-1}(1-u)^{1-3\rho}\,du,
 \\
 I_C(\chi)&=\int_1^\infty
 u^{\rho-1}(u-\chi)^{\rho-1}(u-1)^{1-3\rho}\,du.
\end{align*}
The first function is the probability in the open-cap normalization;
the second is the probability of an \emph{actual open path} between the
two separately wired intervals.  Wiring changes component weights; it
does not supply an open path between different intervals.

\begin{proposition}[Jordan-quad transfer]\label{jordan:transfer}
Consider the marked, uniformly convergent square lattice polygon
approximations $(D_n;a_n,b_n,c_n,d_n)$ to a bounded Jordan quadrilateral
$(D;a,b,c,d)$ in the four-mark sense of Subsection~\ref{intro:crossings}.
Keep every edge of the polygon graph random, and separately wire the two closed arcs $a_nb_n$ and
$c_nd_n$, with no other primal identifications.  If
$\psi:D\to\mathbb H$ sends $(a,c,d)$ to $(0,1,\infty)$ and
$\chi=\psi(b)$, the probability of an actual open path between these
arcs tends to $g(\chi)$.
\end{proposition}

\begin{proof}
First consider a fixed rational orthogonal polygon in tile coordinates.
The deterministic caps of Section~\ref{sec:preliminaries} contribute
one or two additional loops according to the pairing.  Their weight
ratio is exactly the factor $d$ in \eqref{jordan:functions}, so the
four-change formula gives $g$.  If its designated color lies on the
dual grid, translate the complete tile domain, switches, and caps by
one tile step.  This preserves its loop law and makes that color the
prescribed primal grid, at a geometric cost $O(\delta_n)$.  Consequently
there are suitable approximations of every fixed test polygon on the
\emph{same} lattice as the target graph.

Choose a plane Schoenflies homeomorphism taking $\overline D$ to the
unit square, with the two electrodes on the left and right sides.
Uniform marked boundary convergence and winding number imply the
following strict inclusions for all sufficiently large $n$: $D_n$
contains every closed inset square and is contained in every prescribed
neighborhood of the square in these coordinates; its four marked arcs
are close to their corresponding sides.  Fix $\varepsilon>0$.
In physical coordinates polygonize the pullbacks of
\[
 [\varepsilon,1-\varepsilon]\times[-\varepsilon,1+\varepsilon]
 \quad\hbox{and}\quad
 [-\varepsilon,1+\varepsilon]\times[\varepsilon,1-\varepsilon].
\]
Call these the easier and harder test quads, respectively, and put
their electrodes on the nearly vertical sides.  Take the polygonization
error in Schoenflies coordinates smaller than $\varepsilon/10$, with
marks in flat side interiors, and retain this margin after all tile
translations.  Such simple rational orthogonal polygonizations exist:
first use radial shrinking in a Carath\'eodory disk parametrization to
obtain an analytic Jordan approximation, then a fine simple polygon;
replace each segment by a rectangular staircase inside its own thin
corridor and join adjacent staircases in disjoint small vertex
neighborhoods.  The marks can be placed in order in side interiors.
These constructions give marked uniform convergence as
$\varepsilon\downarrow0$.

\begin{figure}[tb]
\centering
\begin{tikzpicture}[scale=.82]
 \begin{scope}
  \draw[densely dashed,gray] (0,0) rectangle (3,3);
  \draw (0.4,-.3) -- (2.6,-.3) (0.4,3.3) -- (2.6,3.3);
  \draw[very thick] (0.4,-.3) -- (0.4,3.3)
                   (2.6,-.3) -- (2.6,3.3);
  \node at (1.5,1.5) {easier};
  \node[below] at (1.5,-.55) {upper comparison};
 \end{scope}
 \begin{scope}[xshift=5.2cm]
  \draw[densely dashed,gray] (0,0) rectangle (3,3);
  \draw (-.3,.4) -- (3.3,.4) (-.3,2.6) -- (3.3,2.6);
  \draw[very thick] (-.3,.4) -- (-.3,2.6)
                   (3.3,.4) -- (3.3,2.6);
  \node at (1.5,1.5) {harder};
  \node[below] at (1.5,-.55) {lower comparison};
 \end{scope}
\end{tikzpicture}
\caption{The Jordan-quad comparison in Schoenflies coordinates.
The dashed square is the limiting target; thick sides are separately
wired electrodes.  The easier quad extends beyond the free sides and
ends inside the target's electrode sides.  The harder quad has the
opposite inclusions.  Physical test polygons retain strict margins
after pullback and lattice approximation.}
\label{jordan:sandwich-figure}
\end{figure}

For the upper bound, restrict both laws to their common primal edges
in the easier quad.  Any target edge outside this common graph can
attach to it only on its two electrode sides: the easier horizontal
sides lie strictly outside the target.  Outside pieces cannot join
the two electrode sides, even through a target wire block, because
the substantial intervening strip lies wholly inside the easier quad
and the two target wire blocks lie in separate exterior zones.
Thus the target's induced boundary partition on the common graph is
weaker than separate wiring of the easier electrodes.  Domain Markov
and FK monotonicity give a coupling in which the easier common-edge
configuration dominates the target's.  Every actual target crossing
contains a common-edge crossing of the easier electrodes.  Indeed,
follow it from the initial exterior zone to the opposite zone, taking
its last exit from the first and first entry into the second; its
intermediate edges must belong to the common graph.  There is no
escape through the distant horizontal sides.

For the lower bound interchange the roles using the harder quad.
Its exterior pieces can attach to the target only on the designated
target arcs, and cannot link opposite sides outside the common graph.
Here it matters that the target contains \emph{all} lattice edges in
its closure, and that both graphs use exactly the same primal grid.
The target wires therefore dominate the induced harder-quad
partition.  Every harder crossing contains an actual shared-edge
piece joining the two target arcs.  This proves the lower comparison.
All transitions occur at ordinary graph vertices; shrinking lattice
edges have uniformly shrinking diameter in Schoenflies coordinates,
so the strict margins justify the discrete versions of the statements.

The conformal cross-ratios of the two test quads tend to $\chi$.
Indeed uniformly convergent parametrizations of Jordan curves imply
uniform local connectedness of their boundaries: points close in the
image have parameters close in the circular order, except for the
identified endpoints, because the limiting parametrization is an
embedding.  Kernel convergence and boundary extension of normalized
Riemann maps then give marked conformal convergence
\cite{Pommerenke1992}.  Since $g$ is continuous on $(0,1)$, let first
$n\to\infty$ and then $\varepsilon\downarrow0$ in the two comparisons.
\end{proof}

\begin{corollary}[The prescribed bond-percolation crossing]\label{jordan:cardy}
At $q=1$, with every edge, including every boundary edge, sampled
independently with probability $1/2$ and with no wiring, the probability
of the event $C_n$ defined in Subsection~\ref{intro:crossings} tends to
\[
 \frac{\displaystyle\int_0^\chi [t(1-t)]^{-2/3}\,dt}
      {\displaystyle\int_0^1 [t(1-t)]^{-2/3}\,dt}.
\]
\end{corollary}
\begin{proof}
For $q=1$ the boundary partition does not change the edge law, and
$d=1$ gives $g=f$.  The $\rho=1/3$ specialization of the four-change
integral calculation is precisely the displayed expression.  The
comparison in Proposition~\ref{jordan:transfer} concerned actual edge
paths throughout, so no external wire is counted as a crossing and
no interface-event continuity assertion is needed.
\end{proof}

\section{Alternating tests after exploration}\label{sec:movingtests}

The Jordan result alone does not apply to a domain cut out by a random
exploration.  We prove the needed extension directly under the
original lattice law.  Conditional errors at exceptional histories
will be controlled by an \emph{averaged} six-arm estimate.

\subsection{Geometry and conditional laws}

\begin{lemma}[A compatible simple drawing]\label{moving:drawing}
The prescribed polygon graph and its Dobrushin strand admit a tile
and collar representation in a Jordan polygon $V_n$ such that:
\begin{enumerate}[label=(\roman*)]
\item $V_n$ has the same marked uniform Jordan limit as $D_n$;
\item the rounded strand is simple, visits the same ordered local
pieces as the prescribed medial strand, and is at curve distance
$O(\delta_n)$ from it;
\item at every switch-revealing stop, the remaining domain is a simple
slit disk.  Under deterministic cap closure its unrevealed edges have
the original conditional FK law, with known open edges contracted
and known closed edges deleted;
\item primal and dual banks of the prefix have physical access to
their corresponding known edges and wire vertices within
$O(\delta_n)$.  Every completion of the unknown edges embeds its
actual paths in the slit disk without crossing the past.
\end{enumerate}
The construction can use identical domains and caps away from an
interval where two additional boundary changes are inserted.
\end{lemma}

\begin{proof}
Make one tile for each edge of the polygon graph, including boundary
edges.  Glue the four triangles directed toward each interior square
center.  The exterior triangle of a boundary edge is a separate tab,
with its own outer dual vertex; do not glue different exterior tabs
along their radial sides.  Each primal boundary vertex retains all
its interior incidences.  Flatten the exterior tabs toward their
bases, to less than half their original height, and add a still
thinner Jordan collar.  The tabs are then disjoint except at their
prescribed vertices.  All changes lie within $O(\delta_n)$ of the
original boundary, and tracing the tabs and collar along the original
parametrization proves (i).

Draw the switches and boundary caps as disjoint arcs.  A local wire
can be drawn behind its cap in the collar.  The unmatched ports
continue to the marked outer-boundary points.  The finite choices
inside each tile and the collar can be made deterministically.
At a wired boundary edge, capping its outer dual vertex joins the
inside ports in the same way for either switch state.  Declaring that
edge open for the drawing merely substitutes a cap-sized detour and
possibly deletes a microscopic loop.  On the free side, the cap
around a primal vertex is exactly the turn beside the dual wired
arc.  Inside the polygon, glued half-segments give the specified
medial edges and their quarter-turn pairings.  Match corresponding
ordered pieces in the two drawings.  Each substituted piece lies in
boundedly many neighboring tiles, so simultaneous parametrizations
have distance $O(\delta_n)$.  Zero-length pieces are handled by limits
of increasing reparametrizations.  This also covers a midpoint visited
with both prescribed pairings and proves (ii).

The tile/cap Euler identity gives exactly the conditional FK law
asserted in (iii).  On a queried primal-open tile the adjacent primal
corners are joined by its actual edge; on a queried closed turn the
bank stays at one primal corner.  Along a cap it stays at one vertex
or uses neighbors already belonging to that wire interval.  Thus the
bank is a known chain attached to the appropriate original interval,
and every queried primal-open edge accesses this bank within one tile.
The same statement holds for dual edges.  Designated outer vertices
access their arcs through the collar.  These are physical accesses,
not additional identifications.  They prove (iv), also when the last
begun tile is revealed in full.  Projecting back to the usual primal
or dual lattice moves points by $O(\delta_n)$ and duplicates no edge;
separate virtual copies of a vertex may become one physical vertex.
\end{proof}

In a four-open-change experiment, deterministic closure with separate
wires of one color differs by a density depending only on the two
pairing outcomes.  This density and its reciprocal are bounded by
constants depending on $q$.  At a switch-revealing stopping time the
odds change is still exactly
\begin{equation}\label{moving:odds}
 p\longmapsto \frac{p}{p+d(1-p)}.
\end{equation}
One obtains this by conditioning the \emph{full} finite-sample density,
not by postulating a Markov property for a limiting curve.  Consequently
conditional rare-event probabilities in a deterministic-cap law can
be averaged under open-cap histories at a bounded multiplicative cost.

\begin{lemma}[Small-interval behavior]\label{moving:asymptotic}
For $f$ in \eqref{jordan:functions},
\[
 f(1-\chi)=1-f(\chi),\qquad
 f(\chi)=c_q\chi^h(1+o(1))\quad(\chi\downarrow0),
 \qquad c_q>0.
\]
\end{lemma}
\begin{proof}
The substitutions $u^{-1}$ in $I_C$ and
$u=1-(1-\chi)t$ in $I_B$, followed by the Euler transformation of
the beta integral, give
\[
 I_B(\chi)=\left(\frac{1-\chi}{\chi}\right)^h I_C(1-\chi).
\]
Both $I_C(0)$ and $I_C(1-)$ are finite and strictly positive: at the
possible singularity $u=1$, the latter integrand has exponent
$-2\rho>-1$, while the integrand at infinity has exponent
$-1-\rho<-1$.  Substitution into \eqref{jordan:functions} proves both
claims, with $c_q=I_C(0)/I_C(1-)$.
\end{proof}

\begin{theorem}[Moving four-change test]\label{moving:slit}
Let a four-change strand in $V_n$ be stopped before completing its
strand, revealing only the switches it has encountered.  Let $U_n$
be its remaining simple slit disk, with the tip as the new starting
mark.  Consider histories for which a fixed basepoint has a fixed
positive disk in $U_n$, and the four prime-end marks in a disk chart
normalized at that basepoint remain mutually separated by a fixed
positive amount.  Then, on these histories,
\begin{equation}\label{moving:limit}
 \mathbb P_{\mathrm{open}}(E_{\mathrm P}\mid\mathcal F_n)
       -f(\chi_{U_n})\ \longrightarrow\ 0
 \quad\hbox{in probability}.
\end{equation}
Here $E_{\mathrm P}$ is the primal pairing event, equivalently the
actual bridge event after separate primal closure, and $\chi_{U_n}$
is its current ordered cross-ratio.  The assertion holds for arbitrary
mesh-dependent stopping rules satisfying this localization.
\end{theorem}

We supply the proof in several steps.  Every annulus used below lies
a fixed positive distance inside the original domain before its mesh
limit is taken.  The inputs are the bulk passage estimate of
Section~\ref{sec:preliminaries}, the vanishing full-lattice one-arm
bound with wired rims, and the six-arm estimate
\begin{equation}\label{moving:sixarm}
 \mathbb P(A_6(r,R))\le C(r/R)^{2+c_0},\qquad c_0>0,
\end{equation}
for the critical square-lattice FK law with this fixed $q<4$
\cite[Corollary~6.7]{DMT2021Fractal}.  An outer buffer and the bulk
boundary-condition comparison transfer this estimate from the
infinite-plane law to the original finite law.  Bounded open-cap
reweightings are harmless.  No six-arm estimate conditioned on an
arbitrary explored maze is asserted.

\begin{lemma}[Physical paths and chart oscillation]\label{moving:chart-modulus}
Let $u_n$ be the horizontal coordinate of the conformal rectangle
map of $U_n$, with the designated primal arcs at $0$ and $1$.
Under the localization in Theorem~\ref{moving:slit}, there is a common
modulus tending to zero for the oscillation of $u_n$ on a connected
path of vanishing physical diameter, including paths with accesses
to the prime boundary.  Consequently, for $\beta>0$, a possible
primal path starting in $\beta\le u_n\le1-\beta$ cannot reach a known
primal-open edge or a primal electrode inside a sufficiently small
physical ball.  ``Possible'' means that all still unknown primal
edges are filled open, before any external wire jump is permitted.
\end{lemma}
\begin{proof}
Away from the normalizing point, a connected set in a ball of radius
$r$ is hit before domain exit by Brownian motion from that point
with probability tending uniformly to zero as $r\downarrow0$.
Comparison with a fixed enclosing Euclidean annulus gives this
bound.  In the normalized disk, a connected set of a fixed positive
diameter has hitting probability bounded below; this follows, for
example, by a crosscut between separated angular sectors and the
Beurling estimate.  The rectangle maps of the disk form a compact
family because the four marks remain separated.  Thus a small
physical continuum cannot have a macroscopic rectangle-coordinate
oscillation.  Near the basepoint use interior distortion instead.
Endpoint accesses follow by interior approximation.

By Lemma~\ref{moving:drawing}, every known primal-open edge and fixed
primal wire attachment accesses $u_n=0$ or $1$ within
$O(\delta_n)$.  A short possible connection from the central band to
one of them would contradict the just-proved modulus.  Small buffers
in the band absorb individual tile errors.  In particular the two
opposite designated intervals have not already met through the
revealed bank under the stated nondegeneracy.
\end{proof}

\begin{lemma}[Deletion and protected-ball costs]\label{moving:deletion}
Fix a history and use separate deterministic primal closure.
Deleting a set of still random edges changes the actual crossing
probability by at most the conditional probability that an endpoint
of a deleted edge reaches a wire interval by an actual open path,
allowing one initial test step over the deleted edge.
For a central-band set contained in a ball of radius $r$, this cost
is bounded by a quantity tending to zero as $r\downarrow0$,
uniformly in the history and center, provided its required outer
radius is fixed and the mesh is sufficiently small.  The same holds
for any fixed finite number of such balls.
\end{lemma}
\begin{proof}
Couple the deleted and original laws monotonically.  Explore the
actual open clusters from all deletion endpoints in the stronger
configuration, revealing the corresponding edges in both laws.
FK partition monotonicity persists at every step.  After the clusters
are enclosed by their closed cuts, the conditional laws outside
agree and can be coupled identically.  If no explored cluster reaches
a wire interval, the crossing decision is unchanged.  This proves
the first assertion, including contact with a wire along a known bank.

Choose the fixed outer radius small enough for
Lemma~\ref{moving:chart-modulus}.  Every component of possible edges
inside this ball that meets the central-band set has no accessible
wire or known primal-open attachment.  Its uncertain boundary
connections can occur only at the ball rim.  Project to the physical
lattice, fill missing edges, and wire that rim.  Splitting physical
vertex copies only weakens the original induced partition.  The
full-lattice wired-rim law therefore dominates every relevant actual
arm simultaneously.  Cropping the ball slightly absorbs tile errors.
Its one-arm probability from radius $r$ to the fixed outer radius
tends to zero: buffered dual circuits from RSW block such an arm.
A finite union bound gives the last assertion.
\end{proof}

\subsection{A directed obstacle away from possible connections}

Fix a small trimming parameter $a_*>0$ and then
$0<\beta<a_*/20$.  A primal vertex is a \emph{candidate} if its chart
coordinate lies in $[\beta,1-\beta]$ and it has a possible connection
to an electrode.  Each candidate supplies a possible arm to a fixed
positive physical distance, by Lemma~\ref{moving:chart-modulus}.

\begin{lemma}[Flagged cells and their cost]\label{moving:flags}
In a fixed bulk block let $\pi$ be a simple known primal bank path.
Choose fixed scales $\ell\ll R_1$, away from its endpoints and the
original boundary.  Flag a deterministic grid cell of side $\ell$
if, in its annulus from $100\ell$ to $R_1$, the prefix already supplies
four cyclically alternating actual arms, its two primal arms being
known wired bank paths.  If a candidate within $O(\ell)$ of this cell
actually connects to an electrode, then six actual alternating arms
occur in a cropped annulus from $200\ell$ to $R_1/2$.
For a bounded region covered by the deterministic cells, the averaged
conditional attachment cost of all flagged cells is at most
\begin{equation}\label{moving:flag-sum}
 C\ell^{-2}(\ell/R_1)^{2+c_0}=O_{R_1}(\ell^{c_0}).
\end{equation}
The statement remains valid with one initial test step and with the
bounded cap changes in \eqref{moving:odds}.
\end{lemma}
\begin{proof}
The extra actual primal arm cannot make a lateral connection to
either known primal leg inside the wider annulus, even in the
all-unknown-primal-open completion, by
Lemma~\ref{moving:chart-modulus}.  On the side containing the extra
arm it is bracketed by the two known primal legs.  A planar crossing
alternative in each intervening strip supplies a dual radial
separator.  The original dual separator on the other side remains.
After cropping at both rims, these three primal and three dual arms
are disjoint and alternate.

For precision, thicken actual paths to their tile color territories.
An open transversal between two primal territories would give an
actual lateral bridge, with an error of at most one tile.  Its absence
forces a dual territorial radial crossing in the intervening
topological rectangle, which shadows a dual edge path in the cropped
annulus.  If the no-bridge test is made after every unknown primal
edge is opened, every dual separator so obtained consists of known
closed tiles.  Thus the four old legs and the separators used in
this argument are actual original edges; no auxiliary boundary
condition manufactures them.

Apply \eqref{moving:sixarm} in the original full-sample law to each
cell of the deterministic grid, then sum over $O(\ell^{-2})$ cells.
The center may have been selected by the past because the sum was
taken over the entire deterministic grid.  Bounded cap densities
control the averaged conditional probabilities under either history
law.  Markov's inequality now makes the conditional flagged-cell
cost small outside a set of histories of small probability.  This
is an annealed-to-conditional argument, with $R_1$ and all comparison
constants fixed before $\ell\downarrow0$.
\end{proof}

\begin{lemma}[Directed obstacle]\label{moving:obstacle}
After removing histories of arbitrarily small probability and allowing
an arbitrarily small conditional deletion cost, there is a Jordan
polygon $H$ containing the basepoint, selected from a finite list at
each fixed accuracy, with the following properties.
Its boundary follows the original boundary and explored prefix to
that accuracy; away from finitely many protected balls and flagged
cells it has fixed positive clearance from all central candidates.
As the accuracies decrease, $H$ and $U_n$ have the same pointed kernel
limits.  Artificial marks in $H$ may be chosen with disk angles
arbitrarily close to the corresponding angles in $U_n$.
\end{lemma}
\begin{proof}
We give the construction and its scale order.  Retain only prefix
excursions reaching depth $r$ from the limiting Jordan boundary,
including each complete segment between successive visits to depth
$h_b\ll r$ and the last incomplete segment if present.  Their number
is tight, uniformly in small $h_b$, because each visits one of a fixed
finite family of bulk annuli at depth $r/2$.  First bound this number,
then choose $h_b$ small.  Join their boundary-near endpoints to the
exterior of a slightly larger polygon through protected balls of
radius $O(h_b)$.  Lemma~\ref{moving:deletion} controls central
attachments in these finitely many balls.  No boundary arm exponent
for the rough Jordan curve is needed.

Break the retained excursions into blocks of diameter at most
$\epsilon\ll h_b$, each contained with margin in a bulk square $S$
of side $10\epsilon$.  Successive displacement thresholds and the
bulk passage estimate make the number of blocks tight.  After fixing
a block-count cutoff, snap its endpoints to a much finer deterministic
grid and protect them by balls of radius $e\ll\epsilon$.  The actual
known primal bank gives a path between the true endpoints in $S$;
loop erasure gives a simple path $\pi$, possibly trivial.  Take
\[
 \ell\ll R_1\ll e,
\]
so all flag annuli are bulk annuli.  Declare forbidden every closed
$\ell$-cell within $3\ell$ of a candidate, except flagged cells and
cells in enlarged endpoint protections.

There is a connector between the protected endpoints avoiding all
forbidden cells.  Otherwise planar cell separation gives a polygonal
circuit through a chain of touching forbidden cells and possibly the
exterior of $S$, separating the two endpoints.  It can be chosen
away from their protection balls and within a cell-sized error of
the cell chain.  Choose a candidate witness in each cell.  Two
successive witnesses can be joined avoiding $\pi$ inside a disk of
radius $O(R_1)$.  If they could not, a proper crosscut of the simple
path $\pi$ would separate them in a circle of radius between
$4R_1$ and $6R_1$.  Its full-path endpoints lie outside because the
cell is away from endpoint protections.  The crosscut passes within
$O(\ell)$ of the witness segment.  The two witnesses have possible
arms to that circle, on opposite sides of $\pi$, and cannot connect
locally to $\pi$ by Lemma~\ref{moving:chart-modulus}.  The two known
radial legs of $\pi$ and the two possible arms consequently interlace.
In each intervening strip the optimistic no-bridge test supplies a
\emph{known} dual radial separator, as in the proof of
Lemma~\ref{moving:flags}.  The cell is therefore flagged, a
contradiction.

Push the separating circuit through the witness points and these
avoiding joins.  Its displacement is $O(R_1)$, smaller than its
distance from the endpoints, so its winding distinction is preserved.
Along the exterior of $S$ avoidance is automatic because $\pi$ has
an interior margin.  We have obtained a separating circuit disjoint
from $\pi$, which is impossible.  Overlapping endpoint protections
or a trivial $\pi$ are handled by a connector within their union.

Select one polygonal connector for each allowed finite grid pattern.
Their union and the protected exterior attachments form an obstacle
$K$ attached to the complement of the larger polygon.  The connectors
need not form one simple curve.  They stay close to the retained
prefix, and every omitted prefix piece is boundary-near; conversely
the entire prefix is approached by this obstacle or the outer wall.
Within distance $\ell/2$ of $K$, candidates occur only in flagged or
protected error regions.  The basepoint component of the complement
is simply connected, because each obstacle component is attached to
the exterior.  It is a fixed polygonal domain, possibly with slits
and pinches.  An interior radial approximation in its Riemann map,
followed by a polygonal perturbation, gives a Jordan domain $H$ whose
boundary is within $\ell/10$ of the obstacle or outer wall.
The outer wall was chosen much farther than $\ell$ from the original
graph.  Hence the asserted clearance holds.  Every choice is from a
finite list at fixed scales and count cutoffs.

Normalize maps of $H$ and $U_n$ at the same basepoint, and transfer
marks by finite-grid approximations to their disk angles.  On a
Hausdorff subsequence of the original boundary and prefix, every
compact path in the limiting basepoint component eventually belongs
to both constructions, while every limiting boundary point is
approached by complements of both.  Thus their pointed kernels agree,
and their normalized interior charts have the same limit.  This is
an interior statement; no locally connected boundary for the
limiting slit domain is asserted.  The fixed basepoint margin and
separation of the marks persist throughout.
\end{proof}

\subsection{Alignment of the comparison charts}

\begin{proof}[Proof of Theorem~\ref{moving:slit}]
It suffices to prove an upper bound for the primal connection under
separate deterministic primal closure.  Interchanging the colors
and using the other deterministic closure gives the complementary
upper bound; \eqref{moving:odds} then yields the open-cap limit.

Use Lemma~\ref{moving:obstacle}, and write $u$ and $v$ for the rectangle
abscissas in $U_n$ and $H$.  Test on the Jordan subquad of $H$ between
$v=a_*$ and $v=1-a_*$, with these sides separately wired.  At fixed
accuracy there are finitely many such quads; approximate each on the
original color grid by the polygon construction of
Proposition~\ref{jordan:transfer}.  Their limiting crossing
probabilities differ from $g(\chi_{U_n})$ by arbitrarily small
trimming, chart-angle, and kernel errors, uniformly under the
nondegeneracy assumptions.

We justify comparison on shared edges.  Take a fine finite grid of
abscissa bins in the central part of $(0,1)$, with width much smaller
than both $\beta$ and $a_*$.  In each bin choose a nearby vertical
fiber for each chart.  Its central part is a compact internal curve
and is matched in the other chart by kernel convergence.  Its two
remaining tails can be confined to balls of vanishing radius,
with matched centers.  Here is the required length--area argument.
Remove small fractions of rectangle height at top and bottom.
For each fixed fraction, inverse maps and derivatives converge on
the compact retained rectangles.  Let that fraction decrease slowly.
For a bin $I$ and a deleted height interval $J$, Cauchy--Schwarz gives
\[
 \int_I\!\operatorname{length}\bigl(\Phi(\{x\}\times J)\bigr)\,dx
 \le (|I||J|)^{1/2}
       \left(\int_{I\times J}|\Phi'(z)|^2\,dA(z)\right)^{1/2},
\]
where $\Phi$ is the inverse rectangle map.  The last integral is
bounded by the common physical area bound.  Choose levels in each
bin controlling the tails of both charts.  Their initial endpoints
match on the retained compact pieces, so matched small endpoint
balls contain all four tails.  This argument does not require
uniform local connectedness of the explored boundaries.  Tile errors
are absorbed by decreasing the fractions slowly enough.
Lemma~\ref{moving:deletion} controls the conditional attachment cost
of these finitely many endpoint balls.

Delete still unknown edges touching a slightly narrower band,
initially $3\beta\le u\le1-3\beta$, if their endpoints fail containment
in $H$ with a sufficiently small fixed fraction of the clearance
$\ell$, or if their two abscissas differ by more than $a_*/10$.
These deletions have negligible conditional crossing cost.  To see
this, suppose an endpoint of such an edge reaches an electrode by
an actual path, allowing the initial test step.  Paths meeting a
flagged cell, protection ball, or fiber-endpoint ball while centrally
located already have negligible total cost.  For every other path,
trace it inward from the electrode after its last entry into the
broader band $2\beta\le u\le1-2\beta$.  It crosses a comparison
$u$-fiber on its main internal piece, giving an anchor in $H$ with
nearly equal abscissas.  It cannot subsequently leave $H$, since its
central vertices are candidates and the wall has clearance from
them.  Each later crossing of either chart's comparison fiber is
again on its main piece.  If $v$ drifted by more than a few bin
widths from $u$, it would cross a $v$-fiber in a nonmatching $u$-bin,
a contradiction.  The endpoint therefore has neither proposed
defect.  Vertices without a possible electrode connection can be
discarded at no crossing cost.  Smaller nested bands and clearance
constants accommodate complete edge segments and physical-grid
projection.

Now keep the surviving uncertain edges in
$4\beta\le u\le1-4\beta$ that also belong to the trimmed test graph.
A nonshared original piece can attach to these only on a designated
side of the test quad: clearance excludes its free horizontal sides,
and the abscissa buffers exclude an attachment from a noncentral
piece.  No nonshared piece, even with history wires, connects the
two designated sides.  Such a traversal must pass through the middle
shared strip by chart matching, while all identifications lie in the
two separate groups near $u=0,1$.  After matching physical vertices,
the induced original partition is thus dominated by the test law's
separate wiring.  Every modified original actual crossing contains
a shared crossing between the test sides, by taking its traversal
from the first exterior group to the second.  Domain Markov and
partition monotonicity give the desired test upper bound.

For completeness, the construction has the following quantifiers.
If convergence in probability failed with fixed error and probability,
choose first $a_*$ small, then $\beta$ and fiber precision smaller.
Choose obstacle accuracies tending to zero, with, at each stage,
a deep-excursion cutoff, boundary protection radius $h_b$, block
sizes and count cutoffs, endpoint protections $e$, the intermediate
radius $R_1$, and finally the flag scale $\ell$.  The conditional
protection costs and the averaged flag cost
\eqref{moving:flag-sum} are made small in that order.  For each fixed
stage choose the lattice mesh small enough for every geometric margin
and each of the finitely many Jordan test limits.  Select failing
histories outside all exceptional sets, and pass to a common kernel
subsequence.  The fiber argument and additional deletions along this
subsequence have uniform conditional error by
Lemma~\ref{moving:deletion}, and contradict the failure.  All six-arm
summations took place under the original law before this deterministic
selection of histories.  This proves \eqref{moving:limit}.
\end{proof}

\section{Identification of the Dobrushin strand}\label{sec:chordal}

\subsection{Two small tests determine the driver}

Map $V_n$ to $\mathbb H$, sending its two Dobrushin ends to $0$ and
$\infty$, with a convergent interior normalization.  The inverse maps
converge uniformly in disk coordinates, including the boundary, by
marked Jordan convergence.  Choose the orientation so that the
positive real side is primal-free.  The rounded simple strand has a
continuous Loewner driver $W_t^{(n)}$, with
\begin{equation}\label{chordal:loewner}
 \partial_t g_t^{(n)}(z)=\frac{2}{g_t^{(n)}(z)-W_t^{(n)}},
 \qquad g_0^{(n)}(z)=z,\qquad W_0^{(n)}=0.
\end{equation}
We suppress the superscript when discussing a single mesh.
Fix $T,R<\infty$ and stop at $T$ or the first exit of $|W_t|<R$.
All constants in this subsection may depend on these fixed bounds.

\begin{lemma}[Bounded martingale tests]\label{chordal:tests}
For every $x>2R+1$, put
\begin{equation}\label{chordal:martingale}
 M_t(x)=\left(\frac{xg_t'(x)}{g_t(x)-W_t}\right)^h.
\end{equation}
These processes and their reciprocals are bounded on the stopped
range, uniformly in the mesh.  For arbitrary stopping rules
$0\le\sigma_n\le\tau_n$ in that range and uniformly bounded
$\mathcal F_{\sigma_n}$-measurable tests $F_n$,
\begin{equation}\label{chordal:approx-martingale}
 \mathbb E\!\left[F_n\bigl(M_{\tau_n}(x)-M_{\sigma_n}(x)\bigr)\right]
 \longrightarrow0
\end{equation}
under the original single-strand Dobrushin law.
\end{lemma}
\begin{proof}
Insert on the original free side a small separate primal interval
whose endpoints in the initial coordinate approximate $x,x+s$.
Use the four-open-change experiment.  Until the specified stop its
cross-ratio is
\[
 \chi_t=\frac{g_t(x+s)-g_t(x)}{g_t(x+s)-W_t}.
\]
The real Loewner solutions remain separated from the driver and
from one another.  At each fixed $s>0$ the four marks, including
$\infty$, therefore stay nondegenerate.  A physical point corresponding
to $iM$ for sufficiently large fixed $M$ has a fixed unaffected disk;
it supplies the basepoint in Theorem~\ref{moving:slit}.  The trajectory
cannot end at either inserted mark within this localization.

Let $Z_t$ be the conditional pairing probability.  It is an exact
bounded martingale.  At any of the allowed stopping rules,
Theorem~\ref{moving:slit} gives $Z_t-f(\chi_t)\to0$ in probability,
and hence in mean, for fixed $s$.  Divide the martingale identity by
$f(s/(x+s))$ only at this fixed $s$.  Lemma~\ref{moving:asymptotic}
and analytic ODE estimates near $x$ give, uniformly in the stopped
range,
\begin{equation}\label{chordal:small-test}
 \frac{f(\chi_t)}{f(s/(x+s))}
     =\left(\frac{xg_t'(x)}{g_t(x)-W_t}\right)^h+o_s(1),
\end{equation}
after the mesh limit.  The expression on the right is bounded above
and away from zero independently of small $s$:
$g_t(x)-W_t$ is bounded below, $g_t(x)$ is bounded above, and
\[
 g_t'(x)=\exp\left(-\int_0^t\frac{2\,du}{(g_u(x)-W_u)^2}\right)
\]
has the same bounds.

We next compare the laws, after replacing the normalized conditional
probabilities by the bounded expressions in
\eqref{chordal:small-test}.  Separate deterministic four-change
closure adds just the small wire interval to the Dobrushin law.
Couple monotonically and explore clusters from that interval in the
stronger law.  The probability that its actual cluster reaches a
fixed small outer neighborhood tends to zero as the interval shrinks.
Inside the intervening annulus there are no wire attachments away
from the rims, since the old primal wire is a positive distance away.
Filling to the full-lattice wired-rim law dominates the arm, which
is blocked by the buffered RSW circuits.  Outside this neighborhood
the configurations can consequently be coupled identically.
The open-four and deterministic-four laws also have vanishing total
variation difference: their density depends only on the pairing,
and the exceptional pairing has probability
$g(s/(x+s))\to0$.  The stopped strands stay outside a fixed
neighborhood of the insertion point, by the separated real and
nearby complex solutions of \eqref{chordal:loewner}.  The drawing and
caps were chosen identical off that neighborhood.

Thus bounded stopped-history functions of $M$ transfer with vanishing
error, first as the mesh tends to zero and then as $s\downarrow0$.
We never divide a coupling error by $f(s/(x+s))$.  The same stopping
rules are applied in the coupled experiments, so the conclusion
includes mesh-dependent rules and yields
\eqref{chordal:approx-martingale}.
\end{proof}

\begin{proposition}[Driving convergence]\label{chordal:driver}
On every compact capacity interval, the Dobrushin driving functions
converge in law, uniformly, to $\sqrt\kappa B_t$, where $B$ is standard
Brownian motion and
\begin{equation}\label{chordal:kappa}
 \kappa=\frac{8}{1+h}
        =\frac{4\pi}{\arccos(-\sqrt q/2)}.
\end{equation}
\end{proposition}

\begin{proof}
We first work inside the preceding localization.  Set
\[
 X_t(x)=g_t(x),\qquad A_t(x)=x^h g_t'(x)^h,
 \qquad M_t(x)=A_t(x)(X_t(x)-W_t)^{-h}.
\]
The processes $X,A$ are uniformly bounded and uniformly Lipschitz
in time there, with $A$ bounded away from zero.  Take
$2R+1<x_1<x_2$.  Their real solutions stay separated: their gap
solves
\[
 \partial_t(X_t(x_2)-X_t(x_1))
 =-\frac{2(X_t(x_2)-X_t(x_1))}
 {(X_t(x_1)-W_t)(X_t(x_2)-W_t)}.
\]
It consequently has a positive lower bound on the stopped range.

Here is the two-test convexity argument in detail.
At a stopping time $\sigma$, freeze $X,A$ and write
$F_j(w)=A_\sigma(x_j)(X_\sigma(x_j)-w)^{-h}$.
Both derivatives are positive, and
\[
 \frac{F_1'(w)}{F_2'(w)}
 =\frac{A_\sigma(x_1)}{A_\sigma(x_2)}
   \left(\frac{X_\sigma(x_2)-w}{X_\sigma(x_1)-w}\right)^{h+1}
\]
has derivative uniformly bounded below by a positive constant on
$[-R,R]$.  Thus for
$a_\sigma=F_1'(W_\sigma)/F_2'(W_\sigma)$, a bounded predictable
quantity, integration of the derivative difference on either side
of $W_\sigma$ gives
\begin{equation}\label{chordal:two-test}
 F_1(w)-F_1(W_\sigma)
 -a_\sigma\bigl(F_2(w)-F_2(W_\sigma)\bigr)
 \ge c(w-W_\sigma)^2.
\end{equation}
The change of the frozen $X,A$ between $\sigma$ and $\tau$ costs at
most $C(\tau-\sigma)$.  Applying
\eqref{chordal:approx-martingale} to these two increments, with the
predictable coefficient incorporated into the test, yields
\begin{equation}\label{chordal:increment}
 \mathbb E[(W_\tau-W_\sigma)^2]
 \le C\mathbb E[\tau-\sigma]+o(1)
\end{equation}
for every sequence of allowed stopping rules.  Boundedness and
\eqref{chordal:increment} give the stopping-time tightness criterion
in the Skorokhod $J_1$ topology.  Since every path is continuous,
all subsequential limits are continuous, and the convergence may
be realized uniformly on compact stopped time intervals.

At inner localization levels chosen to be continuity levels of the
hitting operation, ODE continuity and
\eqref{chordal:approx-martingale} make each $M(x)$ a martingale in
the limit.  Such levels can increase to any outer localization.
The identity
\[
 W_t=X_t(x)-\left(A_t(x)/M_t(x)\right)^{1/h}
\]
and the positive bounds on $A,M$ show that $W$ is a continuous
semimartingale.  Write its decomposition as a local martingale plus
a continuous finite-variation process $B^{\mathrm{fv}}$, and let
$[W]$ be its quadratic variation.  With $Z=X-W$,
\[
 dX=2Z^{-1}\,dt,\qquad d\log A=-2hZ^{-2}\,dt.
\]
It\^o's formula gives the finite-variation part of $dM/M$ as
\[
 hZ^{-1}\,dB^{\mathrm{fv}}-4hZ^{-2}\,dt
       +\frac{h(h+1)}2Z^{-2}\,d[W].
\]
Since $M$ is a martingale, multiplication by $Z^2/h$ yields the
identity of signed measures
\begin{equation}\label{chordal:characteristics}
 (X-W)\,dB^{\mathrm{fv}}-4\,dt
       +\frac{1+h}{2}\,d[W]=0.
\end{equation}
Subtract the identities for $x_1,x_2$.  Their positive separation
forces $dB^{\mathrm{fv}}=0$, and then
$d[W]=8(1+h)^{-1}dt$.  L\'evy's characterization identifies the
localized law as Brownian with the rate in \eqref{chordal:kappa}.
The trigonometric identity follows from
$\arccos(-d/2)=\pi-\lambda$.

Finally, the probability of exiting a large outer driver bound
before $T$ is bounded in the limit by the Brownian probability of
hitting inner bounds approaching it.  These probabilities tend to
zero as the bounds increase.  Remove localization to obtain full
compact-time driving convergence.  Every subsequential law has been
identified, so the convergence holds along the full mesh sequence.
\end{proof}

\subsection{Physical compactness and avoidable crossings}

\begin{lemma}[Territorial arm counting]\label{chordal:territories}
In a buffered bulk annulus, $m$ disjoint radial medial sublegs force
at least $m$ alternating actual primal/dual arms in a further cropped
annulus.  If only one color is retained, one obtains at least one
crossing in a distinct intervening strip per two assignments of that
color to successive medial separators.
\end{lemma}
\begin{proof}
The rounded medial arcs divide tiles into primal and dual territories.
A path within a territory shadows a path of its color on the physical
lattice: within a tile it passes to the corresponding corner, and
passage between opposite corners uses precisely the edge supplied
by that switch.  The shadowing has an error bounded by one tile.
Actual paths stay inside their territories.

Take disjoint simple radial sublegs, perturbing both annular radii
within buffers to regular values.  Thin parallel crosscuts on their
two sides give ordered territorial crossings of opposite colors at
each separator.  Compress successive runs of equal color in the
cyclic list.  Each medial separator contributes a color transition,
so at least $m$ alternating crossings remain.  Shadow them by lattice
walks in the central crop.  Near the wider rims keep territorial tails
until they can be spliced to the walks; thus no positive mesh-width
clearance of the thin territorial paths at the rims is needed.
Opposite-color paths are disjoint.  Same-color members of the
alternating list lie in different strips separated by an
opposite-color territorial path and cannot meet.  Disjoint tails
from the inner wider rim preserve order at first hits of the outer
cropped rim; taking last passages to those hits gives the claimed
actual arms.  The same strip ordering, after ignoring the other
color's assignments, proves the last assertion.
\end{proof}

\begin{lemma}[Curve compactness and no triple visits]\label{chordal:compactness}
The physical Dobrushin strands are tight in the uniform Euclidean
curve topology modulo increasing reparametrization.  Every
subsequential limit has no nonconstant interval on the original
boundary, and no interior point is visited three times with
nonconstant excursions between consecutive visits.
\end{lemma}
\begin{proof}
Bulk passage counts in a fixed finite family of buffered annuli are
tight by Section~\ref{sec:preliminaries}.  We first show that a
macroscopic displacement confined to a shrinking neighborhood of the
original boundary has vanishing probability.  In fixed plane
Schoenflies coordinates for $D$, such a displacement crosses
longitudinally one of finitely many boundary subarcs of positive
length, after shortening the subpassage to avoid the two change
points.  On each subarc the boundary type is homogeneous.  Its
nonwired-color bank is an actual path using that color's edges.

Surround the subarc by a thin Jordan ribbon and place the permitted
path in a still thinner band with strict margins.  Fill to the full
lattice of the bank color and wire the ribbon's outer boundary.
This dominates the actual crossing: there is no nearby designated
wire of this color, and all exterior attachments to the shared graph
occur on its ordinary sides.  Longitudinal ribbon crossings become
unlikely as the ribbon narrows.  One concrete proof chooses any
number $N$ of ordered intermediate positions on the subarc, each
with its own disjoint fixed neighborhood.  At each position two
across-ribbon sections can be confined to arbitrarily small physical
balls.  A longitudinal crossing must then traverse a Euclidean
annulus of arbitrarily large logarithmic aspect ratio inside that
subribbon; its extremal length tends to infinity.  Choose transverse
blocker quads with bounded transverse extremal distance and with outer
margins beyond the permitted band.  The strong quad RSW theorem
\cite[Theorem~1.2]{DMT2021Fractal} gives an actual opposite-color
transverse blocker with conditional probability at least $c>0$,
under arbitrary exterior partitions.  Approximate the fixed Jordan
corridors by polygons with strict margins before taking the mesh
limit.  Successive conditioning over the disjoint quads bounds the
crossing probability by $(1-c)^N$.  Let $N$ grow and then narrow the
ribbon.  Every geometric scale here was fixed before the mesh limit.

At a fixed displacement accuracy, outside these unlikely boundary
travels, every excess collection of successive displacements forces
many passages through a finite annulus cover of the compact part
away from the ribbon.  The passage estimates therefore bound the
number of successive displacements at each accuracy.  Subdivide
curves at these displacement times and apply the usual
Arzel\`a--Ascoli argument modulo reparametrization.  This proves
tightness and also rules out a nonconstant interval of a limit
confined to the original boundary.

Three genuine visits to an interior point, with positive intervening
excursions, produce six disjoint medial legs to some fixed positive
outer radius.  Lemma~\ref{chordal:territories} turns them into six
alternating actual arms across a cropped annulus.  Cover the possible
contact points by a deterministic grid of mesh $\ell$.  The bulk
six-arm estimate and the $O(\ell^{-2})$ grid count give a bound
$O(\ell^{c_0})$.  First take the lattice mesh limit, then
$\ell\downarrow0$, and finally a countable union over interior
compact sets and rational excursion radii.  This excludes precisely
three visits separated by nonconstant excursions; constant waiting
intervals are irrelevant to the curve topology.
\end{proof}

\begin{lemma}[Macroscopic avoidability]\label{chordal:avoidability}
At a switch-revealing stop of the Dobrushin exploration, call a
component of $U_n\cap A(z;r,R)$ \emph{avoidable} if it does not
separate the tip access from the target access inside $U_n$.
The conditional probability of a future radial traversal in any
avoidable component is bounded by a quantity tending to zero as
$r/R\downarrow0$, uniformly over histories and centers, with $r,R$
fixed before the mesh limit and $\delta_n\ll r$.
\end{lemma}
\begin{proof}
An avoidable component cannot access both designated primal and
dual prime boundary arcs strictly between the rims.  A path joining
such accesses contains a crosscut separating the tip from the target.
The future simple strand traverses one component between rims;
choose a color whose designated boundary is absent there.  Its bank
supplies an actual arm of unqueried edges in a cropped annulus.
A known-open edge or wire jump of that color would give a short
physical access to its designated boundary there, contrary to the
choice of component.

To bound all such components at once, keep their random edges of
this color within a cropped grid annulus.  Unqueried tiles are uncut;
an adjacency of incident edges, or attachment to a known edge or
wire vertex, has a corresponding $O(\delta_n)$ path in $U_n$ until
it reaches a designated wire.  Thus away from the cropped rims the
induced partition contains only ordinary physical-vertex attachments.
Delete known edges of the opposite state, fill the rest to the full
lattice, and wire both rims.  FK monotonicity dominates all the
relevant actual arms simultaneously by one full-lattice arm event.
Buffered dual circuits give its vanishing bound.  Sum over the two
possible missing colors.  The argument uses the conditional FK law
at actual revealing stops; an optional last piece of length
$O(\delta_n)$ is absorbed by the buffers.
\end{proof}

\begin{corollary}[No return from a small target cap]\label{chordal:target}
After the strand first enters a shrinking initial conformal cap
around its target, the probability of a subsequent displacement of
fixed positive physical size tends to zero.
\end{corollary}
\begin{proof}
At the first entrance the previously untouched cap gives a route
from the tip to the target inside a small physical ball.  Stop just
before entrance, or at the entering piece and use its bank with a
one-step buffer.  Outside this inner ball every relevant annular
component is avoidable.  Apply Lemma~\ref{chordal:avoidability}.
The diameters of the initial conformal caps shrink uniformly because
the initial inverse disk maps converge uniformly up to the boundary.
\end{proof}

\subsection{The ordered trace, including zero-capacity intervals}

\begin{lemma}[Interior double visits in every time interval]
\label{chordal:dense-doubles}
For ordinary chordal $\mathrm{SLE}_\kappa$, $4<\kappa<8$, almost surely
every nonempty open capacity interval contains both visits to some
interior self-double point.
\end{lemma}
\begin{proof}
The double-point and boundary-hit dimension theorems of Miller--Wu
\cite[Theorem~1.1 and Corollary~1.7]{MillerWu2017} give, respectively,
\[
 1+\frac\kappa8-\frac6\kappa
 \quad\hbox{and}\quad 2-\frac8\kappa.
\]
Their difference is $(\kappa-4)^2/(8\kappa)>0$, so the trace has
an interior double point almost surely.  Let $E_T$ mean that both
visits occur strictly before $T$.  Scaling makes $\mathbb P(E_T)$
independent of $T>0$, while $\bigcup_{m\ge1}E_m$ has probability one.
Thus $\mathbb P(E_T)=1$ for every $T>0$.  At a deterministic time $a$,
the domain Markov property restarts ordinary SLE with additive
capacity clock.  An interior double point of the restarted trace
maps under $g_a^{-1}$ to an interior point of the original remaining
domain.  Apply this observation to all rational open intervals.
\end{proof}

\begin{lemma}[Small gates exclude swallowed excursions]
\label{chordal:gates}
In a joint subsequential limit of the physical strand and its
Loewner driving function, the physical curve cannot enter an open
component of a Loewner hull that is disjoint from the Loewner trace
and the original boundary.
\end{lemma}
\begin{proof}
Use the continuous Brownian SLE trace
\cite{RohdeSchramm2005}; here $4<\kappa\le6$.  Suppose that the
claimed excursion occurs.  By a countable choice, there is a small
closed ball $J$ with a larger buffered neighborhood disjoint from
the SLE trace and original boundary, approached deeply inside a
swallowed component at finite limiting capacity.  Fix bounds for
this capacity and the driver, and a high basepoint corresponding
to $iM$ in the initial half-plane coordinate.  Take the first discrete
approach to an intermediate buffer around $J$, before the alleged
hit.  Along a subsequence of these approach times the prefix keeps
a fixed spare distance from $J$, whereas its kernel viewed from the
basepoint excludes $J$.  This follows from driving convergence and
equality of the accessible Loewner domains; the whole buffered ball,
being disjoint from the trace, lies in one excluded component.

Nevertheless $J$ is still internal in every discrete simple slit
disk.  Normalize its inverse disk map $\Phi_n$ at the basepoint and
write $z_n=\Phi_n^{-1}(\operatorname{center}J)$.  Kernel convergence
forces $|z_n|\to1$.  Put $\epsilon_n=1-|z_n|$, and take circular
disk crosscuts centered at $z_n/|z_n|$, with radii between
$2\epsilon_n$ and $\sqrt{\epsilon_n}$.  If $L_n(t)$ is their image
length, Cauchy--Schwarz and the area formula give
\begin{equation}\label{chordal:gate-area}
 \int_{2\epsilon_n}^{\sqrt{\epsilon_n}}
            L_n(t)^2\,\frac{dt}{t}
       \le2\pi\operatorname{area}(U_n).
\end{equation}
The right side is uniformly bounded and the logarithmic integration
range diverges, so one image crosscut has diameter tending to zero.
Its disk cap contains $z_n$, and its endpoints access the slit
boundary.  Since that boundary stays a fixed distance from $J$, the
crosscut separates all of $J$ from the basepoint for sufficiently
large $n$.

The target is on the basepoint side.  Indeed the driver and capacity
bounds give a fixed route high above the half-plane hulls from the
basepoint to the target, with positive physical separation from the
prefix.  A small gate touching the prefix misses this route.  A gate
with both endpoints on the original outer wall cannot separate two
fixed interior balls when its diameter tends to zero, by uniform
Jordan convergence.  Adding a boundary-to-interior simple slit
disjoint from that gate does not change its separation.  Gates with
an endpoint at the initial slit attachment are included among gates
touching the prefix.

We must use a stopping rule, not a gate selected from the future
excursion.  Fix a small $\ell>0$ and stop at the first discrete step
at which there exists a crosscut of diameter at most $\ell$
separating $J$ from both basepoint and target, while the prefix still
keeps the chosen spare distance from $J$ and the basepoint is safe.
The preceding construction shows that this stop occurs before the
offending buffered hit for fine meshes.  Choices can be fixed
measurably from each finite history.  At first occurrence a witnessing
gate has an endpoint on the latest microscopic step, including its
attachment point.  Otherwise it was a crosscut of the previous slit
domain with the same separation: adding a disjoint simple slit
cannot change that separation.  This contradicts first occurrence.
The tip therefore reaches the gate and its target-side bank within
a ball of radius $\ell+O(\delta_n)$.

Choose a fixed outer radius smaller than the spare distance to $J$.
In the annulus from $4\ell$ to this radius about the new gate
endpoint, components on the trapped side are avoidable: a
tip-to-target route crosses the gate inside the inner ball and stays
on its other side.  Any later hit of $J$ requires a radial traversal
on the trapped side, since every return through the gate is inside
the inner rim.  Lemma~\ref{chordal:avoidability}, applied at this
first-gate stop, makes its conditional probability vanish as
$\ell\downarrow0$ after the mesh limit.  There is no sum over possible
gate locations.  This contradicts the alleged excursion.
\end{proof}

\begin{theorem}[Dobrushin convergence for $1\le q<4$]
\label{chordal:trace}
For every bounded marked Jordan domain and every marked uniform
polygon approximation satisfying Equation~\eqref{intro:marked-jordan}
with the boundary conventions of Subsection~\ref{intro:model}, the Dobrushin
strand converges in law, in the uniform Euclidean oriented-curve
topology modulo increasing reparametrization, to chordal
$\mathrm{SLE}_{\kappa(q)}$.
\end{theorem}
\begin{proof}
By Proposition~\ref{chordal:driver} and
Lemma~\ref{chordal:compactness}, take a joint subsequence realized
with compact driving convergence and uniform physical curve
convergence up to reparametrization.  Write $Y(t)$ for the physical
Brownian SLE trace, and $\gamma(s)$ for the full limiting physical
curve in an arbitrary forward parametrization.  Before its first
target visit, let $c(s)$ be the half-plane capacity of the hull of
its initial segment.  Kernel convergence makes $c$ continuous and
nondecreasing and identifies it as the limit of the corresponding
discrete capacities.  These mapped segments are bounded locally
away from the target.  As the target is approached, $c(s)\to\infty$:
a driver bounded on every finite capacity interval has its finite-time
hulls in bounded portions of the initial half-plane.

The domain accessible toward the target after time $s$ is the
Loewner remaining domain of $Y$ at time $c(s)$, by kernel convergence
and the Loewner ODE.  On every interval where $c$ changes, the newly
added trace segments of $\gamma$ and $Y$ have a point in common:
a boundary point of the smaller shared remaining domain inside the
older domain belongs to both increments.  Continuity, with shrinking
such intervals, implies
\[
 \gamma(s)=Y(c(s))
\]
at every time outside the interiors of plateaus of $c$, including
their active endpoints.  A nonconstant plateau is therefore an extra
excursion based at $Y(t)$, with $t$ its constant capacity, contained
in the old closed hull or original boundary.

Lemma~\ref{chordal:gates} excludes motion inside an open swallowed
component.  Hence an extra plateau can move only on $Y([0,t])$ or
the original boundary.  By Lemma~\ref{chordal:compactness}, it has
no varying subinterval entirely on that boundary.  If the plateau
were nonconstant, choose a compact varying subinterval in its
interior whose image avoids both the boundary and its basepoint.
Every image point has a unique old SLE time: two old visits together
with this genuine extra visit would give three interior visits
separated by nonconstant excursions, forbidden by
Lemma~\ref{chordal:compactness}.  This old-time inverse is continuous.
Indeed any sequence of old times has a convergent subsequence in
$[0,t]$, and trace continuity and uniqueness identify its limit.
Composing with the extra subcurve yields a continuous nonconstant
real function, whose image contains a nonempty capacity interval.
Lemma~\ref{chordal:dense-doubles} provides a point in that interval
with two old visits, a contradiction.  Thus every plateau is
constant.  The same reasoning at capacity zero excludes an initial
extra motion.

Corollary~\ref{chordal:target} excludes further motion after the first
limit target visit.  The capacity change of time therefore identifies
the entire ordered curve with $Y$, up to the permitted increasing
reparametrizations and constant waiting intervals.  Finally
Lemma~\ref{moving:drawing} transfers this conclusion from the rounded
simple drawing to the specified medial strand at distance
$O(\delta_n)$, including repeated midpoint visits with fixed pairings.
All subsequential limits agree, proving the claimed full-sequence
convergence.
\end{proof}
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//chordal.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//endpoint-field.tex ====
\section{The endpoint field and a four-arm estimate}
\label{sec:q4tools}

Throughout this section and the next, $q=4$ and $p=p_c=2/3$.
We use the critical isotropic square-lattice law, with mesh $\delta$.
The medial loop weight is $2$.  Resolve each contact inside a disk of
radius less than $\delta/100$, preserving the prescribed pairing, so
that the lattice loops are disjoint simple curves.  These resolutions
change uniform curve distance by $O(\delta)$ and do not cross an actual
open primal or dual edge.  Orient each strand by the convention that
primal vertices lie on its left.  Independently for each loop, choose
with equal probabilities this orientation or its reverse.  The resulting
height has jumps of size one.  In the plane it is defined modulo a
constant; its pairing with a smooth compactly supported density of
integral zero is well defined.

We fix the normalization
\begin{equation}\label{q4field:normalization}
 G_{\mathbb C}(x,y)=-\frac{1}{2\pi}\log|x-y|,
 \qquad \sigma^2=\frac{2}{\pi},
 \qquad \lambda_0=\sqrt{\frac{\pi}{8}},
 \qquad 2\sigma\lambda_0=1.
\end{equation}
Thus $\Phi_V$ will denote the Dirichlet free field whose covariance is
the Green operator for $-\Delta$ on $V$.

\begin{lemma}[Comparison of height conventions]\label{q4field:height-comparison}
Couple the cellwise lattice height $h_\delta^{\mathrm{cell}}$ and
the height $h_\delta^{\mathrm{geom}}$ defined by the smoothed loop
interiors using the same independent loop signs.  For every smooth
compactly supported neutral density $\rho$,
\begin{equation}\label{q4field:height-comparison-L2}
 \mathbb E\left|
   \langle h_\delta^{\mathrm{geom}}
          -h_\delta^{\mathrm{cell}},\rho\rangle\right|^2
 \longrightarrow0.
\end{equation}
This conclusion uses only bulk crossing and passage-count estimates.
\end{lemma}

\begin{proof}
Decompose $\rho$ into a finite linear combination of differences of
smooth unit masses.  For a single such difference, sample its two
endpoints independently at each of the two occurrences in the second
moment.  For a loop $L$, let $d_i^{\mathrm{geom}}(L)$ and
$d_i^{\mathrm{cell}}(L)$ be its winding-indicator differences for
the two height conventions at the $i$-th endpoint pair.  Averaging
the common fair signs gives the mixed second-moment integrand
\[
 \mathbb E\sum_L
   \bigl(d_1^{\mathrm{geom}}(L)-d_1^{\mathrm{cell}}(L)\bigr)
   \bigl(d_2^{\mathrm{geom}}(L)-d_2^{\mathrm{cell}}(L)\bigr).
\]
At a fixed tuple of four distinct endpoints, put $d_*>0$ for their
minimum separation, truncated at one.  A nonzero term requires the
same loop to enter an $O(\delta)$ neighborhood of an endpoint of
each pair.  For small enough $\delta$ that loop has diameter at
least $d_*/2$.  The bulk one-arm estimate makes such a macroscopic
loop's approach to a mesh-sized endpoint neighborhood negligible.
Higher passage-count moments give uniform integrability on each
fixed size range; the arm bound for a loop visiting the bounded
support and reaching a large radius removes the remaining tail.
Thus this integrand tends to zero at every distinct endpoint tuple.

The octave passage-count bounds also give the integrable majorant
$C(1+|\log(d_*\vee\delta)|)^2$.  Indeed every contributing loop has
size at least the endpoint separation up to a mesh error, the count
has bounded moments per octave, and the outer octaves have a
power-decaying tail.  The same estimate holds for each of the four
mixed terms in the displayed product.  Dominated convergence in
the independently sampled endpoints proves
\eqref{q4field:height-comparison-L2}; tuples with coincidences have
zero integration measure.  A finite linear decomposition and
Cauchy--Schwarz give the assertion for $\rho$.
\end{proof}

\begin{proposition}[Whole-plane height convergence]\label{q4field:wholeplane-input}
For every real $\rho\in C_c^\infty(\mathbb C)$ with $\int\rho=0$,
the independent fair unit-jump height of critical square-lattice
$\FK_4$ satisfies
\begin{equation}\label{q4field:plane-characteristic}
 \lim_{\delta\downarrow0}
 \mathbb E\exp\bigl(i\langle h_\delta,\rho\rangle\bigr)
 =\exp\left\{-\frac{\sigma^2}{2}
  \iint\rho(x)G_{\mathbb C}(x,y)\rho(y)\,dx\,dy\right\}.
\end{equation}
The same assertion holds jointly for finitely many such tests.
\end{proposition}

\begin{proof}
Corollary~\ref{cor:rr-gff-input} gives this statement for the cellwise
height.  Lemma~\ref{q4field:height-comparison} transfers it to the
smoothed geometric height, since the difference of the characteristic
functions is bounded by
$\mathbb E|\langle h_\delta^{\mathrm{geom}}
-h_\delta^{\mathrm{cell}},\rho\rangle|$.  Apply the same argument
to every real linear combination of finitely many tests to obtain
joint convergence.
\end{proof}

The parameters in \citet[Theorem~2.8 and Corollary~3.2]{SixVertexGFF2026}
are $c=2$, $\Delta=-1$, and $\mu=0$.  The loop factor is then an
ordinary cosine, exactly the characteristic function of a fair sign
multiplying a unit jump.  The proposition concerns only the
whole-plane field; finite-domain and conditional-field statements
are proved below.

\subsection{Moments and binary transport}

We first record why characteristic-function convergence supplies the
moments used below.  This argument uses only the bulk crossing theorem,
which includes $q=4$ \citep{DCLM2019Isoradial}, and the actual-component
counting argument of Section~\ref{sec:preliminaries}.

\begin{lemma}[Neutral moment bounds]\label{q4field:neutral-moments}
For finitely many smooth compactly supported neutral densities, their
height pairings have uniformly bounded moments of every fixed order.
All their joint moments converge to the corresponding Gaussian
moments.  The bounds and limits
are unchanged by the $O(\delta)$ drawing modifications just described.
\end{lemma}

\begin{proof}
It suffices, by linear decomposition, to use differences of two smooth
unit masses.  Expand a product of $m$ such pairings, integrating the two
endpoints of each occurrence separately.  For fixed sampled endpoints
$a_i,b_i$, put
\[
 d_i(L)=\mathbf1_{\{a_i\in\operatorname{int}L\}}
       -\mathbf1_{\{b_i\in\operatorname{int}L\}}.
\]
Conditional on the loops, averaging the fair signs gives exactly
\begin{equation}\label{q4field:even-assignment}
 \mathbb E_{\epsilon}\prod_{i=1}^m
       \sum_L\epsilon_Ld_i(L)
 =\sum_{f:\{1,\ldots,m\}\to\{\text{loops}\}}
       \prod_i d_i(f(i))
       \prod_L\mathbf1_{\{|f^{-1}(L)|\text{ is even}\}}.
\end{equation}
In a nonzero term every used loop separates the endpoints in each slot
assigned to it and is used in at least two slots.  If $d_*$ is the
smallest distance between the $2m$ sampled endpoints, truncated at $1$,
such a loop has diameter at least $d_*$.  Notice that even repeated
copies of the same smeared charge have separately sampled endpoints;
coincidences have zero integration measure.

For each endpoint and each size octave, the number of loops of that
size which wind around it has moments of every order bounded uniformly
in the mesh.  Indeed such a loop supplies a passage in one of a bounded
number of buffered annuli of comparable size.  Ordering their medial
passages produces distinct actual bank components, with bounded
multiplicity.  Successive actual-component exploration and bulk RSW
give a geometric tail for this count.  Between $d_*$ and the diameter
of the fixed support there are $O(1+|\log d_*|)$ octaves.  For larger
loops, separation of two support points forces a visit to a fixed
bounded slit joining them.  Reaching the octave of radius $R$ then
requires an actual arm from that bounded region to scale $R$, whose
probability decays as a power of $R$.  Cauchy--Schwarz and higher
passage-count moments make the sum of these outer octaves finite.
Consequently the absolute value of the integrand in
\eqref{q4field:even-assignment}, after expectation, is bounded by
\[
 C_m\bigl(1+|\log(d_*\vee\delta)|\bigr)^m.
\]
The same reasoning bounds any fixed higher absolute moment of the
geometric majorant.  Logarithmic singularities are locally integrable
in the independently sampled endpoints.  This proves uniform
integrability of every fixed moment.  Proposition~\ref{q4field:wholeplane-input}
and uniform integrability now give joint moment convergence.

If a drawing modification changes a contributing indicator at one of
a fixed collection of distinct sampled endpoints, a loop of positive
diameter enters an $O(\delta)$ neighborhood of that endpoint.  The
bulk one-arm estimate makes this event negligible; the preceding
higher moment bounds allow its removal under the integrals.
\end{proof}

The binary index propagates by equality along every directed strand.
At a tile, sum the two possible pairings with equal coefficient.
An inserted matrix acts on this binary index, and a closed loop with
ordered insertions $M_1,\ldots,M_k$ has conditional response
$\tfrac12\operatorname{tr}(M_1\cdots M_k)$.  Thus the unperturbed loop
weight is $\operatorname{tr}I=2$.

\begin{lemma}[The reflected vacuum]\label{q4field:reflection}
The plane binary-index network is reflection positive across the
translated lattice symmetry lines parallel to either axis or to a
diagonal.  Reflection reverses directed transport and takes inserted
matrices to their adjoints.  Simultaneous conjugation by $SU(2)$ is
unitary on the reflected Hilbert space.
\end{lemma}

\begin{proof}
For a vertical cut through a primal column, a bisected tile contributes
the seam operator
\[
 I+|\mathrm{Bell}\rangle\langle\mathrm{Bell}|,
 \qquad \mathrm{Bell}=e_1\otimes e_1+e_2\otimes e_2,
\]
which is positive.  The horizontal case is identical.  A diagonal cut
along tile edges contracts equal indices, and is positive as well.
These statements give a positive Gram form for arbitrary finite
network replacements strictly on one side of the cut.  Each switch
intertwines the transports on its reference-directed pairs, so
simultaneous conjugation preserves all contractions; reflection
therefore implements it unitarily.

One may first use large reflection-periodic tori, with two reflection
seams.  For a diagonal seam take periods along and perpendicular to the
diagonal.  The all-loop torus measure differs from periodic FK by a
bounded genus factor.  A fixed patch is, with probability tending to
one, screened by contractible actual circuits of both colors.  On that
event the genus data are exterior to the patch.  Bulk uniqueness and
the circuit tails identify every local subsequential limit with the
plane FK state.  Equivalently, finite closure of all strands meeting
the patch gives its plane domain Markov law directly.  Independent
binary indices pass to this limit.  The finite-volume positive forms
therefore give the asserted plane form.
\end{proof}

\subsection{From the spin-two sector to four arms}

Write $\pi_4^\delta(r,R)$ for the plane probability of four alternating
actual primal and dual arms across a square annulus.  Fixed changes of
its rims change this probability by at most constant factors.  We use
also separated prescribed landing sectors and their buffered
extensions, at a bounded multiplicative cost.  These are bulk
statements at $p_c$; the published alternating-arm result includes
$q=4$ \citep[Proposition~2.3 and the arm-separation discussion]{DM2022Scaling}.
No boundary RSW theorem at $q=4$ is used.

\begin{proposition}[A bulk four-arm bound]\label{q4field:four-arm}
There is a constant $C$ such
that, for $0<r<R$,
\begin{equation}\label{q4field:four-arm-bound}
 \limsup_{\delta\downarrow0}\pi_4^\delta(r,R)
 \le C(r/R)^2.
\end{equation}
The same bound, with a constant depending on the fixed bulk collar,
holds inside a domain with arbitrary exterior boundary data.
\end{proposition}

\begin{proof}
Scale the outer radius to a fixed small constant.  Choose mirror boxes
of radius $s$ at horizontal distance one, centered at $x$ and $y$.
In the box at $x$, take a positive unit mass in a disk of radius
$\varepsilon s$ at $x+(s/2,0)$ and a negative unit mass near
$x-(s/2,0)$; use their reflected placement at $y$.  Fix smooth mass
profiles and connecting slits inside the boxes.  Their height
differences are denoted $h_x,h_y$.

For a traceless complex $2\times2$ matrix $V$, insert
$\exp(taV)$ along each successive directed flux piece, where $a$ is
its contribution to this smeared dipole.  Equivalently, if $\rho_x$
is the dipole density, choose a compactly supported one-form
$\alpha_x$ in its box with $d\alpha_x=\rho_x\,dx\wedge dy$,
and integrate $\alpha_x$ along the directed pieces.  Such a form
exists because $\int\rho_x=0$.  Let $A(V)$ be the
second-order coefficient as a vector in the reflected space.  It is
an equivariant quadratic map on the complexified conjugation
representation of $SU(2)$.  Its symmetric square decomposes into a
scalar representation and one irreducible spin-two representation.
Set
\[
 E=e_1e_2^*,\qquad F=E^*,\qquad H=\operatorname{diag}(1,-1).
\]
The tensor $E\otimes E$ lies entirely in spin two, whereas the
spin-two projection of $H\otimes H$ is nonzero.  Unitary equivariance
and irreducibility therefore give, with an absolute constant,
\begin{equation}\label{q4field:spin-two}
 \|A(E)\|^2
 \le C\left\|A(H)-\frac12\mathbb E(h_x^2)\mathbf1\right\|^2
 =\frac C4\operatorname{Cov}(h_x^2,h_y^2).
\end{equation}
Here $\mathbf1$ is the vacuum vector.  This does not assert that the
entire scalar sector consists of multiples of $\mathbf1$: projection
onto spin two annihilates $\mathbf1$, which is all the inequality
requires.  For diagonal $H$, the second coefficient is $h_x^2/2$,
giving the last equality with the reflected orientation convention.

By Lemma~\ref{q4field:neutral-moments}, the covariance in
\eqref{q4field:spin-two} converges to its Gaussian value.  The covariance
of the two neutral charges is $O(s^2)$, since the logarithmic kernel
is smooth between the boxes and each charge has total mass zero and
first absolute moment $O(s)$.  The Gaussian identity
$\operatorname{Cov}(X^2,Y^2)=2\operatorname{Cov}(X,Y)^2$ gives
\begin{equation}\label{q4field:spin-upper}
 \limsup_{\delta\downarrow0}\|A(E)\|^2\le C_\varepsilon s^4.
\end{equation}

We obtain a matching lower bound from two passages at each box.
For a loop visiting both boxes, compress each maximal run at one box
before its next visit to the other into one factor.  Nilpotence gives
alternating factors
\[
 I+ta_1E,\ I+ub_1F,\ I+ta_2E,\ I+ub_2F,\ldots.
\]
Close an excursion by an arc through the omitted other box,
where the relevant one-form vanishes.  The closing arc can be chosen
disjoint from the excursion, giving a simple curve.  Its flux is an
averaged winding difference, so $|a_i|,|b_i|
\le1$.  The fluxes are integers if the excursions avoid the endpoint
supports.  Let $M$ be the total number of alternating periods, over
all loops visiting both boxes.  There are $2M$ disjoint connecting
passages.  The coefficient $K$ of $t^2u^2$ in the product of normalized
traces satisfies
\begin{equation}\label{q4field:coefficient-bound}
 |K|\le C M^4,\qquad K=0\quad\hbox{if }M<2.
\end{equation}
Loops visiting only one box contribute $1$ and can be omitted.

If $M=2$ and the endpoint supports are missed, then $K\ge0$.
For a single two-period loop its only possible nonzero contribution is
$a_1a_2b_1b_2/2$.  The full winding difference belongs to
$\{-1,0,1\}$, so nonzero integer fluxes satisfy
$a_1=-a_2$ and $b_1=-b_2$; the coefficient is $1/2$.
For two one-period loops the coefficient is
$a_1b_1a_2b_2/4$.  If nonzero, both simple loops separate the two
endpoint pairs.  Their interiors are disjoint or nested, so their
choices of the contained endpoints either agree in both boxes or are
complementary in both.  Hence the products $a_1b_1$ and $a_2b_2$ have
the same sign, and this coefficient is $1/4$.

For completeness, the errors in this sign calculation must be
\emph{relative} to two four-arm probabilities.  Let
$\pi_4=\pi_4^\delta(c_1s,c_2)$, for fixed suitable constants.  Bulk
component searches give
\begin{align}
 \mathbb E\bigl[M^4\mathbf1_{\{M\ge3\}}\bigr]
   &\le o_s(1)\pi_4^2,\label{q4field:relative-tail}\\
 \mathbb P(M=2,\text{ an endpoint support is hit})
   &\le C\varepsilon^\alpha\pi_4^2,\label{q4field:relative-hit}
\end{align}
for some $\alpha>0$, with the mesh taken sufficiently small at each
fixed $s,\varepsilon$.  Here is the exploration underlying these
estimates.  Around either box the $2M$ radial separators, cropped with
end margins, give $2M$ alternating bank arms and at least $M$ distinct
actual primal crossing components of a cropped annulus.  Conversely,
two distinct actual primal crossing components force two intervening
dual separators, hence four alternating arms in a slightly narrower
annulus.  Thus the first two successful discoveries already cost
$C\pi_4$.  Expose each complete actual component, including all its
exiting edges revealed closed, and do not count wire identifications
as open paths.  The remaining graph is dominated by
the original annular lattice with maximal identifications at its
original rims.  In separated dyadic rings, a buffered dual circuit
blocks every actual radial crossing with uniformly positive
conditional probability.  Each further discovered crossing therefore
costs at most $Cs^\alpha$.  Iterating this bound gives a summable tail
even after multiplication by $M^4$.  The two box annuli have disjoint
bulk collars, so their first-four-arm costs multiply up to a constant;
this proves \eqref{q4field:relative-tail}.  An endpoint hit additionally
requires a bank arm from radius $\varepsilon s$ to a fixed small
multiple of $s$, in a buffered region inside the hole of its
corresponding four-arm annulus.  Both four-arm costs are retained.
The same conditional blocking estimate proves
\eqref{q4field:relative-hit}.  In particular an absolute $o(1)$ bound
would not suffice in either estimate.

Finally there is a compatible positive event of probability at least
$c\pi_4^2$, with $c$ independent of sufficiently small $\varepsilon$.
Prescribe primal east/west and dual north/south landings
at each box annulus, with buffered extensions.  Attach the two primal
inner landings to small actual circuits around the positive and
negative endpoint disks, respectively, at radii $\eta s$,
where the small constant $\eta>0$ is fixed before choosing
$\varepsilon\ll\eta$.  At the large scale join the inward primal arms
directly between the boxes, and the outward pair around their
exterior.  Join the upper dual arms to each other and the lower dual
arms to each other between these routes.  At each inner rim join its
two dual arms in a central corridor between the terminals.  Together
these dual connections form a closed oval separating the two primal
groups.  Retain one inner-to-outer landing pairing of comparable
probability at the first annulus and its mirror at the second; the
gluing works for either pairing.  One of the two pairing classes
has at least half the separated-arm probability, so no additional
theorem concerning a prescribed pairing is needed.

\begin{figure}[tbp]
\centering
\begin{tikzpicture}[scale=.82]
 \draw[gray,dashed] (-2,0) circle (.64);
 \draw[gray,dashed] (-2,0) circle (.91);
 \draw[gray,dashed] (2,0) circle (.64);
 \draw[gray,dashed] (2,0) circle (.91);
 \draw[very thick] (-1.65,0)--(1.65,0);
 \draw[very thick,rounded corners=8pt]
   (-2.35,0)--(-3.15,0)--(-3.15,-2)--(3.15,-2)--(3.15,0)--(2.35,0);
 \draw[blue!70!black,thick]
   (-2,0) .. controls (-2,1.55) and (2,1.55) .. (2,0)
          .. controls (2,-1.55) and (-2,-1.55) .. cycle;
 \foreach \x in {-2.35,-1.65,1.65,2.35}
   {\draw[thick,fill=white] (\x,0) circle (.11);}
 \node at (-1.65,.28) {$+$};
 \node at (1.65,.28) {$+$};
 \node at (-2.35,.28) {$-$};
 \node at (2.35,.28) {$-$};
 \node at (-2,-.3) {$x$};
 \node at (2,-.3) {$y$};
 \node[blue!70!black] at (0,1.42) {dual oval};
\end{tikzpicture}
\caption{The bulk gluing used in the lower bound.  Black connections
join the positive terminals and the negative terminals separately;
the dual oval keeps these two actual primal clusters distinct.
Each terminal is protected by a small primal circuit.  Only
connectivity is depicted; the four-arm annuli are schematic.}
\label{q4field:gluing-figure}
\end{figure}

All these joins can be made in finitely many disjoint tubes of the
requested colors, outside the core annuli except for their landing
extensions.  Use transverse crossings in overlap rectangles to join
the extensions to the tubes.  Condition on the complement of one
color's tubes and apply FKG and buffered RSW there, then do the other
color.  Since only that color's extensions enter its tubes, the
requirements are increasing in the appropriate conditional model.
The two core annulus probabilities multiply by the bulk collar
comparison.  The resulting positive and negative terminal circuits
belong to two distinct actual primal clusters, the first connecting
the two positive terminals and the second the two negative ones.

There are at least two distinct closed interfaces separating these
two connected primal groups: color parity makes their number even,
and at least one is necessary.  Each such interface separates both
endpoint pairs and misses their support disks.  On the part of this
event with $M=2$, the two-loop coefficient is $1/4$.  Subtracting
\eqref{q4field:relative-tail} and \eqref{q4field:relative-hit}, first
choose $\varepsilon>0$ small and fixed, and then $s$ small.  Positivity
on all the remaining $M=2$ configurations and
\eqref{q4field:coefficient-bound} show
\[
 \|A(E)\|^2=\mathbb E K\ge c'\pi_4^2.
\]
Combining this with \eqref{q4field:spin-upper} proves
$\limsup_{\delta\downarrow0}\pi_4\le Cs^2$.
Quasi-multiplicativity and fixed-factor rim changes give
\eqref{q4field:four-arm-bound}.  Bulk collars give its domain version.
\end{proof}

\begin{corollary}[Macroscopic contacts]\label{q4field:contact-dimension}
In any tight subsequential
limit of actual loops or Dobrushin strands, the interior set of
genuinely distinct visits by macroscopic curves has Hausdorff dimension
at most zero, almost surely.  Constant parametrization pauses are
discarded.  No assertion about boundary contacts is made here.
\end{corollary}

\begin{proof}
Fix an interior compact set and an outer displacement cutoff $R>0$.
A point with two genuinely distinct visits, each having displacement
at least $R$, gives four medial passages across every sufficiently
small surrounding annulus.  With rim margins, their actual banks
give four alternating arms.  A grid cover at scale $r$ has
$O(r^{-2})$ cells, each with limiting probability at most $C(r/R)^2$.
Its expected number of required cells is therefore bounded uniformly
in $r$, after the mesh limit.  For every $\eta>0$ the expected
$\eta$-dimensional covering content is $O(r^\eta)$; along dyadic
scales it tends to zero almost surely.  Countable compact sets,
displacement cutoffs, and positive $\eta$ prove the claim.
\end{proof}

\section{The field in regular domains}
\label{sec:q4regular}

We now derive the finite-domain statement required for the loop
identification.  It is not part of
Proposition~\ref{q4field:wholeplane-input}.

\subsection{Actual walls and elementary bounds}

Write $C\in\{\mathsf P,\mathsf D\}$ for a color and $C'$ for the other.
An actual homogeneous wall is a simple polygon of $C$-diagonals,
whose switches are forced open in color $C$.  A mixed wall consists
of an actual primal-diagonal arc, an actual dual-diagonal arc, and two
tile sides joining their ends.  Each tile side is a one-port
transition gate.  All interior switches not on a wall are sampled,
including a switch whose diagonal endpoints happen both to lie on
the wall.

\begin{lemma}[Vacuum realization of a wall]\label{q4field:wall-law}
Conditioning the plane vacuum on exactly the stated wall switches
gives the FK law inside with the indicated actual connections.
For a closed $C$-wall, the $C'$ model on the strictly internal sites
and its internal edges has free boundary condition.
In a mixed wall, fixing the binary index at one transition port fixes
the index of the distinguished strand; this condition has probability
$1/2$ conditional on every switch configuration.
\end{lemma}

\begin{proof}
An exterior connection of either color can enter only through the
same-color wall arc, already connected by actual open edges.  Hence
it creates no additional interior partition.  For a closed wall no
$C'$ vertex is wired through that wall, giving the stated free model
(equivalently, apply planar duality).  There are exactly two transition
ports in the mixed experiment.  The loop through one must leave the
interior through the other; its one independent fair index is fixed
by the condition at the first port.
\end{proof}

Set the height to zero on a homogeneous wall.  In a mixed wall, take
height zero on its primal arc and choose the distinguished index so
that its dual arc has height one.  Conditional on the geometry,
\begin{equation}\label{q4field:domain-height}
 h_\delta(z)=u_{*,\delta}(z)
      +\sum_{L\text{ internal}}\epsilon_L
                    \mathbf1_{\{z\in\operatorname{int}L\}},
\end{equation}
where the signs are independent and fair.  In the homogeneous case
$u_{*,\delta}=0$; in the mixed case it is the $0/1$ side function of
the distinguished strand.  Values on the drawn curves are immaterial
in integrals.  For $f\in C_c^\infty$ in the interior, the pairing
$\langle h_\delta,-\Delta f\rangle$ depends only on switches and
binary indices near the support of $f$: integration by parts writes
it as oriented flux against a compactly supported smooth one-form.
It consequently does not depend on the relative height constant.

\begin{lemma}[Free-bank estimates]\label{q4field:free-bank}
Consider a patch in which the bank color $C'$ has no forced open edge
and no boundary identification on any domain wall meeting the patch.
The graph may omit sites or edges.  Counts of disjoint medial
passages between buffered annular rims have moments of every order,
uniformly in the mesh and in the remaining exterior data.  The
probability of an actual $C'$ arm across a ratio $R/r$ is at most
$C_0(r/R)^\beta$ for some $\beta>0$.  These statements hold as well
for a product of free $C'$ models on disjoint subgraphs.  In a bulk
patch either color may be used.
\end{lemma}

\begin{proof}
Order the medial radial separators and crop their actual $C'$ bank
paths with margins at both ends.  Different strips cannot communicate
across a separator, and each strip is charged at most twice.  Thus
many passages force a proportional number of distinct actual
crossing components of the cropped graph.  Reveal these components
successively from one rim, ignoring partition jumps.  A deleted
component does not wire surviving interior sites through its former
neighbors.  Since the wall is free for this color, the remaining
graph is dominated by the full annular lattice with maximal ties
only at its original rims.  A buffered circuit of the other color
blocks actual crossing, with conditional probability bounded below
by bulk RSW.  Iteration gives a geometric tail for the component
count and hence all its polynomial moments.  Applying these blockers
in separated dyadic rings proves the power bound for one arm.
Missing edges or sites only lower actual connectivity; the same
comparison applies to a product on disjoint subgraphs.  Enlarge
inner radii by a fixed mesh multiple whenever needed.
\end{proof}

Take now a fixed simple rectilinear polygon $V$, with two distinct
marked transition points on interiors of straight sides in the mixed
case.  Approximate it by the corresponding actual diagonal walls
within $O(\delta)$, in marked boundary order.  Such approximations
exist: round each color arc on its own lattice and adjust its ends
by $O(\delta)$ to join the two colors by a single tile side at each
transition.  This class of regular domains will suffice for the
later comparison argument.

\begin{lemma}[Moment kernels and compactness]\label{q4field:moment-kernels}
For the regular polygons just specified, the height moment kernels at
distinct interior points $z_1,\ldots,z_m$ satisfy
\begin{equation}\label{q4field:log-kernel}
 \left|\mathbb E\prod_{i=1}^m h_\delta(z_i)\right|
 \le C_m\bigl(1+|\log(d_*\vee\delta)|\bigr)^m,
 \qquad d_*=1\wedge\min_{i<j}|z_i-z_j|,
\end{equation}
with $d_*=1$ when $m=1$.  Here $C_m$ depends on $V$ and $m$,
uniformly over the interior tuples, including those approaching
the boundary.  Geometric majorants for the conditional
signed expansions have higher polynomial moments with corresponding
logarithmic bounds.  Integrated moments are uniformly integrable;
the height laws are tight as local distributions, and their moment
kernels are asymptotically equicontinuous on compact sets of distinct
interior tuples.  The count bounds hold also for arbitrarily shaped
homogeneous walls in a fixed bounded box and for products of disks
of one wall color.
\end{lemma}

\begin{proof}
A loop winding around $z$ and of diameter in $[r,2r]$ lies within
$O(r)$ of $z$ and supplies a macroscopic passage in one of a bounded
number of annuli of radius comparable with $r$.  For a homogeneous
wall use its free bank color.  In a fixed mixed polygon, choose a
subpassage away from the transition points by a fixed fraction of
$r$.  The two wall arcs separate linearly away from a transition,
so the chosen patch avoids one of them and has a free bank color.
Lemma~\ref{q4field:free-bank} gives bounded moments per octave.
At mesh scale a deterministic local bound suffices.  The same free
bank argument works for arbitrary homogeneous wall shapes and for
the stated products.

Expand \eqref{q4field:domain-height} and average the signs.  A loop
which survives must be assigned to at least two of the distinct
points and therefore has diameter at least $d_*$.  The remaining
offset factors have absolute value at most one.  Minkowski's
inequality over $O(1+|\log(d_*\vee\delta)|)$ octaves gives
\eqref{q4field:log-kernel} and the asserted higher moment majorants.
Integrating the logarithmic singularities gives uniform
integrability.  Uniform second moments for compactly supported
Fourier tests give tightness in a sufficiently weak local negative
Sobolev space, hence tightness as distributions.

To move one point a small distance while the tuple stays distinct,
any changed contributing loop indicator must belong to a loop of
diameter bounded below which approaches that point.  In the mixed
case a changed offset requires the distinguished strand to approach
it.  The bulk one-arm estimate makes these events uniformly rare;
the higher count moments control their contribution.  This proves
equicontinuity, allowing the $O(\delta)$ ambiguity of the drawing.
The same annular passage counts give tight macroscopic curve counts
and tightness modulo reparametrization by the displacement-arc
subdivision argument of Section~\ref{sec:loops}.
\end{proof}

\subsection{Neutral tests through a conditioned wall}

\begin{theorem}[The regular-domain field]\label{q4field:regular-domain}
Let $V$ be a fixed
polygon and let its actual wall approximations be as above.  Then
\begin{equation}\label{q4field:Q}
 h_\delta\ \Longrightarrow\ u+\sigma\Phi_V
 \qquad\text{as distributions in }V,
\end{equation}
where $u=0$ for a homogeneous wall and $u$ is the harmonic extension
of the boundary values $0,1$ for a mixed wall.  All joint integrated
moments converge.  At every tuple of distinct fixed interior
positions, the moment kernels converge to those of the same field.
\end{theorem}

The proof has three parts.  First we transfer local Laplacian tests
through separated pieces of wall.  Next we close the gaps between
those pieces.  Finally we determine the harmonic ambiguity left by
the Laplacian tests.

\begin{lemma}[Analytic displacement across a vacant slab]
\label{q4field:analytic-displacement}
Consider two groups of vacuum insertions supported in finitely many
pairwise separated rectangles, with a vacant projection slab between
the two groups.  Its normal is parallel to a lattice axis or a
diagonal.  Normalize each wall constraint by the probability of its
isolated switch constraints; allow bounded charge and index factors.
Translate one whole group farther across the slab.  Every
subsequential continuum correlation is real analytic in the
translation distance on a neighborhood of $[0,\infty)$.
This assertion also holds with other, previously fixed translations.
\end{lemma}

\begin{proof}
First note the needed uniform norm bound.  For a wall factor let
$W_j$ denote its switch constraints only, and let $p_j$ be their
isolated vacuum probability.  Allowed lattice translations and
reflections preserve $p_j$ exactly.
The arbitrary-event comparison across vacant bulk collars gives
\begin{equation}\label{q4field:normalized-wall}
 c\le\mathbb E\prod_j\frac{\mathbf1_{W_j}}{p_j}\le C.
\end{equation}
The constants may depend on the fixed gaps and rectangles.  A single
marked-port projection contributes exactly $1/2$ to the expectation
of the positive wall constraints.  Bounded charge or index factors
can be omitted in absolute value.  The same upper bound applies to
separated subfamilies, their lattice translates, and their reflected
copies.  In particular the reflected norms needed below are bounded
uniformly in the mesh.  No lower bound on an individual, possibly
very rare, wall probability $p_j$ is asserted.

On the algebraic reflected space let $S_\delta$ translate an insertion
deeper into the half-plane by an admissible lattice period in the
normal direction.  Reflection and translation invariance make this
operator symmetric.  Cauchy--Schwarz gives
\[
 \|S_\delta^n v\|^2
       \le \|v\|\,\|S_\delta^{2n}v\|.
\]
At fixed mesh every finite insertion is bounded, including its
normalizing constants, so the norms on the right are bounded as the
displacement varies.  Iterating this inequality proves
$\|S_\delta^n v\|\le\|v\|$.  It also shows that null vectors remain
null.  Thus $S_\delta$ extends to a self-adjoint contraction on the
reflected Hilbert space.  Use $T_\delta=S_\delta^2$ to obtain a
positive self-adjoint contraction.  Write $p_\delta=O(\delta)$ for
its physical translation step.

Reserve a small fixed part of the vacant slab.  After absorbing the
rest into the positions of the two base vectors, the correlations
have the spectral interpolation
\begin{equation}\label{q4field:spectral-interpolation}
 F_\delta(z)=
  \left\langle a_\delta,
    T_\delta^{N_\delta+z/p_\delta}b_\delta\right\rangle,
 \qquad N_\delta p_\delta\longrightarrow a>0.
\end{equation}
The vector norms are bounded by \eqref{q4field:normalized-wall}.
Spectral calculus therefore makes these a bounded holomorphic
family on every closed half-plane strictly inside
$\operatorname{Re}z>-a$.  A bounded integer rounding of the exponent
changes a correlation by $o(1)$ locally there: for $n$ of order
$\delta^{-1}$ and bounded real $\theta$,
\[
 \sup_{0\le t\le1}|t^{n+\theta}-t^n|=O(n^{-1}).
\]
Thus rounding a physical shift by a bounded number of admissible
normal translation periods gives the same limit.  Take a diagonal
subsequence on a countable array of
rational shifts and apply normal-family compactness to
\eqref{q4field:spectral-interpolation}.  Every resulting limit is
holomorphic on a neighborhood of the entire nonnegative real ray,
including zero.  The assertion for one cut only uses the other
translations as fixed parameters.
\end{proof}

\begin{lemma}[Removal of a local neutral factor]
\label{q4field:neutral-removal}
Let finitely many cells of one finite rectangular grid, separated
by positive-width grid gaps, contain normalized
wall factors, bounded index factors, and charge factors.  Let one
cell contain only
\[
 X_\delta(f)=\exp\bigl(i\langle h_\delta,-\Delta f\rangle\bigr),
 \qquad f\in C_c^\infty(\text{cell};\mathbb R).
\]
In the expectation of
the product, this factor may be replaced with error $o(1)$ by
\begin{equation}\label{q4field:neutral-mean}
 \mu(f)=\exp\left(-\frac{\sigma^2}{2}
                            \int|\nabla f|^2\right).
\end{equation}
The geometry and all gaps are fixed before $\delta\downarrow0$.
\end{lemma}

\begin{proof}
Put $Y_\delta=X_\delta(f)-\mu(f)$.  If an allowed reflection line
separates this cell from all the others with a strict projection
margin, reflection Cauchy--Schwarz proves the claim immediately.
Indeed the squared norm of $Y_\delta$ tends to zero: the two reflected
Laplacian tests have disjoint supports and their whole-plane Gaussian
covariance is
\[
 \sigma^2\int\nabla f\cdot\nabla(f\circ\theta)=0.
\]
Proposition~\ref{q4field:wholeplane-input} consequently factors their
characteristic functions.  Reflection conjugations and orientation
signs do not change this conclusion.  The other reflected vector has
bounded norm by \eqref{q4field:normalized-wall}.

For general cell placements choose the original cells from a finite
rectangular grid with positive gaps, and call the row and column of
$Y_\delta$ its central row and column.  The following finite
displacement procedure creates the required separating line.
\begin{enumerate}[label=\arabic*.]
 \item Translate all rows above the central row upward, and all rows
 below it downward, each as a single group.
 \item Translate the columns to the right of the central column
 rightward, and those to the left leftward, by amounts larger than
 the preceding vertical spread.
 \item The central cell is now isolated in its $x+y$ projection.
 Partition all other cells into the groups with larger and smaller
 $x+y$ projections.  Move these groups far in the directions
 $(1,1)$ and $(-1,-1)$, respectively, until the former are above
 and to the right of all remaining cells and the latter below and
 to their left.
 \item Move the upper group farther right and the lower group
 farther downward.  All other cells then have larger $x-y$
 projection than the central cell.
\end{enumerate}
Each movement begins with a vacant projection slab and increases its
width.  Individual rectangular collars stay vacant: each translated
group moves rigidly, and the two groups at a cut only become farther
apart.  Each named directional movement is a separate one-parameter
translation.  Empty groups need no movement.  The amounts are chosen
successively sufficiently large, with each threshold allowed to
depend on all earlier choices.

In the final configuration a diagonal reflection line separates
$Y_\delta$, so every limiting correlation is zero.  Apply
Lemma~\ref{q4field:analytic-displacement} to the last displacement.
Analytic uniqueness extends the zero correlation from all
sufficiently large values to zero.  Continue in reverse order.
At each stage the earlier parameters are still sufficiently large
for the relevant vacant slab, and the later ones have been returned
to zero.  A countable set of rational choices suffices for a common
subsequence.  The triangular dependence of the thresholds therefore
causes no loss of parameters.  Eventually the correlation at the
original placements is zero.  Every subsequence has the same
conclusion, proving the $o(1)$ assertion.
\end{proof}

We apply this lemma by cutting small positive-width gaps in the wall.
Fix a tuple of interior positions and group its coincident positions.
Choose a finite horizontal and vertical grid so that every group
lies alone in an interior cell with strict margins.  The grid lines
miss all corners and transitions, and meet the relevant wall sides
transversely.  Remove the wall constraints in a thin strip at each
such line.  The remaining wall pieces have separated rectangular
supports.  The one marked-port projection, if present, belongs to
one of those wall cells.  In each interior cell combine all tests
assigned to its group into one exponential.  Formula
\eqref{q4field:normalized-wall} bounds the normalizing expectation
away from zero and infinity for this fixed gapped geometry.
Successively applying Lemma~\ref{q4field:neutral-removal} identifies
the joint Laplacian-test law under this gapped conditioning.

\begin{lemma}[Closing homogeneous wall gaps]\label{q4field:close-gaps}
For fixed interior charge supports and, when present, a marked port,
the local normalized charge laws with gapped and complete walls
become arbitrarily close as follows: first take the mesh limit at
fixed positive gap widths, and then let those widths tend to zero.
\end{lemma}

\begin{proof}
Consider one gap of width $\eta$ at a homogeneous wall point.  The
switches still to be fixed lie in a ball of radius $O(\eta)$, and are
to be open in the wall color $C$.  First omit the marked-index
condition.  In the opposite color $C'$, the gapped model is the
larger of the two models.  Couple by exploring all actual $C'$
components incident to the fixing edges in this larger sample,
starting from both endpoints and revealing also all exiting edges.
Use the ordered FK conditional probabilities for common queried
edges.  Explore actual open edges only, without following boundary
partition identifications.

The only differing fixed data lie on the fixing edges.  On the event
that the discovered components stay inside a fixed small homogeneous
neighborhood, their outside edges are closed in both samples.  This
screens the changes, so the unqueried graphs have identical induced
laws and may be sampled identically.  This is also a direct
whole-plane exploration with only specified edge states; it requires
no boundary wire jump.  The probability that the exploration leaves
the neighborhood is at most $C(\eta/R)^\beta$ by
Lemma~\ref{q4field:free-bank}, where $R>0$ is its fixed radius.

Every discrepant switch is open for $C'$ in the larger sample, so
both its $C'$ vertices belong to the discovered components.  A loop
or strand touching a discrepancy in either model has its actual
$C'$ bank in those components.  On the screening event it therefore
cannot reach any charge support or the marked port, all outside the
neighborhood.  All remote loops and their binary indices can be
matched exactly.  Restoring the marked-index condition changes the
exceptional probability by at most a factor two, since its
conditional probability is exactly $1/2$.  Repeat at the finitely
many gaps and let their widths tend to zero after the mesh limit.
\end{proof}

\subsection{Identification of the harmonic part}

\begin{proof}[Proof of Theorem~\ref{q4field:regular-domain}]
Take a subsequential distributional limit $h$, using
Lemma~\ref{q4field:moment-kernels}.  The preceding gap construction,
Lemmas~\ref{q4field:neutral-removal} and \ref{q4field:close-gaps}
show that, in a neighborhood of every tuple of interior positions,
the joint Laplacian tests have the Gaussian law with covariance
\begin{equation}\label{q4field:laplacian-covariance}
 \mathbb E\bigl[L(f)L(g)\bigr]
       =\sigma^2\int_V\nabla f\cdot\nabla g,
 \qquad L(f)=\langle h,-\Delta f\rangle.
\end{equation}
For coincident positions all the tests are combined in their single
vacant cell before applying the characteristic-function argument.
This detail includes the diagonals of $V^m$ for every $m$.

More explicitly, the integrated moment bounds give moment
distributions of every order for $h$, and hence for its Laplacian.
On product neighborhoods of every position tuple they coincide with
the Gaussian moment distributions just computed.  Tensor test
functions, partitions of unity, and their density in smooth compactly
supported tests show equality on all of $V^m$.  This excludes any
unseen distribution supported on a coincidence diagonal.  Every
linear combination of finitely many $L(f)$ has the Gaussian moments;
Gaussian moment determinacy identifies its law.  Thus $L$ is a
centered Gaussian isonormal field on $C_c^\infty(V)$ with
\eqref{q4field:laplacian-covariance}.

Extend $L$ by mean-square density to $H_0^1(V)$.  For a smooth
compactly supported density $g$, let $v_g\in H_0^1(V)$ solve
$-\Delta v_g=g$ weakly, and set
\[
 h^0(g)=L(v_g).
\]
Then $h^0$ has the law of $\sigma\Phi_V$.  For
$f\in C_c^\infty(V)$, $v_{-\Delta f}=f$, so
$\langle h-h^0,-\Delta f\rangle=0$.  Consequently
\begin{equation}\label{q4field:harmonic-ambiguity}
 H=h-h^0-u
\end{equation}
is distributionally harmonic, and therefore represented almost
surely by a smooth harmonic function.  No independence between $H$
and $h^0$ has been assumed or is needed.

It remains to show that $H=0$.  We use moment kernels, not pointwise
values of the limiting random distribution.  Let $x,y$ approach
distinct boundary points which are neither corners nor transitions.
The uncentered second kernel of $h-u$ tends to zero.  To see this
before passing to the limit, use
\eqref{q4field:domain-height}.  In the mixed case the wrong
$0/1$ side at a point near its prescribed wall requires the marked
strand to cross a short segment from that point to the wall.  It
must then leave a fixed local patch, since both marked ports are
elsewhere.  Likewise, a loop containing both $x$ and $y$ has diameter
bounded below; a segment from $x$ to the nearby wall must cross it,
so the loop approaches that wall point while leaving the patch.
The free bank one-arm estimate makes both events rare.  The uniform
higher count moments in Lemma~\ref{q4field:moment-kernels} absorb
their contributions.  Finally the deterministic harmonic function
$u$ tends to the prescribed boundary value.  The homogeneous case
has no marked term and is the same argument.  Thus
\begin{equation}\label{q4field:boundary-kernel}
 K_{h-u}(x,y)\longrightarrow0
 \quad\text{at distinct ordinary boundary points}.
\end{equation}
Kernel convergence and equicontinuity on interior compact sets
justify taking the mesh limit before this boundary limit.

Let $\psi:\mathbb D\to V$ be conformal.  Polygon corner behavior
implies that $\psi$ and its inverse extend H\"older continuously to
the boundary.  In particular, for some $a,C>0$,
\[
 |\psi(z)-\psi(w)|\ge C^{-1}|z-w|^a.
\]
If $r,s\ge1/2$, then
$|re^{i\theta}-se^{i\varphi}|\ge c|e^{i\theta}-e^{i\varphi}|$.
The logarithmic second-kernel bound consequently gives an integrable
majorant
\begin{equation}\label{q4field:angular-majorant}
 C\left(1+
    \bigl|\log|e^{i\theta}-e^{i\varphi}|\bigr|\right)^2
\end{equation}
uniformly over radii approaching one.  The same type of bound holds
for the Dirichlet Green kernel, whose value tends to zero at two
distinct boundary points.

Choose nonnegative smooth functions $\rho_t$ of integral one, each
supported on a compact radial interval in $(0,1)$ whose two endpoints
tend to one.  For every integer $n$, consider the smooth distribution
test which, for an ordinary function $v$, is
\[
 B_{n,t}(v)=\int_0^1\rho_t(r)
       \frac1{2\pi}\int_0^{2\pi}
       v(\psi(re^{i\theta}))e^{-in\theta}\,d\theta\,dr.
\]
For distributions this is the pullback test; its density in $V$
includes the conformal change-of-variables Jacobian.  It is smooth
and compactly supported for each fixed $t$.  Equations
\eqref{q4field:boundary-kernel} and
\eqref{q4field:angular-majorant}, followed by dominated convergence
in the two independent angles and radii, give
\[
 B_{n,t}(h-u)\longrightarrow0\quad\text{in }L^2.
\]
The Dirichlet kernel gives the same conclusion for $B_{n,t}(h^0)$.
Hence $B_{n,t}(H)\to0$ in $L^2$.

The harmonic function $H\circ\psi$ has Fourier expansion with
$n$-th circle coefficient $c_n r^{|n|}$.  Therefore
\[
 B_{n,t}(H)=c_n\int_0^1\rho_t(r)r^{|n|}\,dr
       \longrightarrow c_n.
\]
All coefficients, including $c_0$, vanish almost surely.  Thus $H=0$,
and \eqref{q4field:Q} follows for every subsequential limit.
Uniform integrability gives convergence of all integrated moments.
The equicontinuity of the kernels at distinct interior tuples then
upgrades their distributional convergence to pointwise convergence
there, proving the last assertion of the theorem.
\end{proof}
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//endpoint-field.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//endpoint-loops.tex ====
\section{The endpoint in rough domains}
\label{sec:q4rough}

We work at $q=4$ throughout this section.  We use the whole-plane
height convergence of Proposition~\ref{q4field:wholeplane-input}
and its regular-domain form, Theorem~\ref{q4field:regular-domain}.
Write
\[
 \sigma^2=\frac{2}{\pi},\qquad \lambda_0=\sqrt{\frac{\pi}{8}},
 \qquad \sigma\lambda_0=\frac12.
\]
The normalization of the Dirichlet Gaussian free field is
$G_D=(-\Delta_D)^{-1}$, so its Green function has singularity
$(2\pi)^{-1}\log(1/|z-w|)$.  A signed height has an independent fair
unit jump across each closed contour.  Its offset is specified separately.
Our purpose is to identify the conditional field in domains cut out by
the contours themselves.  Their boundaries need not be Jordan curves at
this stage.

\subsection{Pockets and their conditional laws}

A \emph{wall of color $C$} means an actual connected open wall of that
color, rather than an identification of separated vertices.  Denote the
other color by $C'$.  All the following discrete drawings may be separated
within a distance $O(\delta)$, where $\delta$ is the mesh, so that their
contours are simple and have the prescribed traversal order.

\begin{lemma}[Exterior-readable pockets]
\label{q4pocket:discrete-pockets}
In a homogeneous $C$-wall disk, each outer contour is the external
perimeter of an outermost actual $C'$-cluster.  Behind it lie pockets
bounded by the maximal simple $C'$-open cycles, ordered by inclusion of
their interiors.  Their interiors are disjoint; several pockets can
belong to one outer contour.  The contours, these cycles, their
assignment to contours, and all switches outside the pocket interiors
can be determined without querying strictly interior switches.
Conditional on these data, the pocket interiors have independent
homogeneous $C'$-wall laws.

In a two-arc disk, first reveal the distinguished path.  On either side,
its bank together with that side's wall arc determines analogous
maximal-cycle pockets of the wall color.  Conditional on the path and
the exterior pocket data, all these pockets have independent
homogeneous laws.  A point wound by a containing contour or side
closure, and farther than $O(\delta)$ from its bank walk, lies in one of
the corresponding pockets.  Every further contour winding that point
is contained in that pocket.
\end{lemma}

\begin{proof}
Draw each cluster as a small regular neighborhood of its actual graph.
Its boundary follows the paired medial pieces within $O(\delta)$.
Every perimeter behind its external perimeter lies in a bounded face
of that graph.  Such a face is enclosed by a simple cycle.  Equivalently,
the bank of the external perimeter is a closed graph walk; decomposing
this walk into simple cycles shows that every wound point with the
specified clearance belongs to a maximal cycle interior.

Two maximal cycle interiors cannot overlap without one containing the
other.  Otherwise their boundaries meet at two or more vertices, their
union is contained in one biconnected block, and the simple outer boundary
of that block encloses both.  This contradicts maximality.  Shared
vertices or edges cause no difficulty, since the open interiors remain
disjoint.

We justify the assertion about revealing information.  Fix a possible
list of cycles and expose its wall edges and everything outside its
open disks.  Any connection between vertices of one cycle that uses
its interior already has a connection along that cycle.  More strongly,
adding arbitrary interior edges cannot join two distinct exterior
biconnected blocks.  After removal of an exterior cut vertex, the
attachments belonging to any one wall cycle remain connected along
the rest of that cycle.  Thus an interior addition cannot repair an
exterior cut vertex.  The exterior block containing a maximal cycle
already has all its vertices in that cycle's closed disk: otherwise
the simple external boundary of the block would be a larger cycle.
A new cycle that both enters and leaves a listed disk would have two
distinct wall attachments and would consequently belong to its wall
block.  It cannot enlarge that block outside the disk.  The same
argument applies if several listed disks are traversed, since they
would then have to belong to the same exterior block.  Nor can internal
changes alter an external cluster perimeter.  Consequently the event
specifying the proposed list and its exterior is measurable from the
revealed edges.  Each wall is actually connected, so the domain Markov
property gives the asserted product of homogeneous laws.

For the marked path, close each side using its wall arc and the
appropriate half-sides of the terminal gates.  The same-color bank and
wall form a closed walk of actual open edges.  The path lies outside
every cycle made from this walk, so its exploration queries no switch
strictly inside such a cycle.  The preceding argument therefore applies
on both sides.  Finally, a further contour leaving a pocket and winding
a point inside it would have to surround its wall cycle.  This is
impossible because that cycle belongs to an actual cluster connected
to the original wall.
\end{proof}

For a family $\mathcal D$ of disjoint homogeneous disks, let
$H_{\mathcal D}$ be the signed height with zero offset and independent
laws in the disks.  Its value at a point refers to the disk containing
that point.

\begin{lemma}[Comparison of zero-offset moments]
\label{q4pocket:moment-comparison}
Let $\mathcal D_-$ and $\mathcal D_+$ be two families of homogeneous
discrete disks of the same wall color.  Suppose that each disk of
$\mathcal D_-$ is contained in a disk of $\mathcal D_+$.  Assume all
disks lie in a fixed bounded region.  For distinct points
$z_1,\ldots,z_m$ in the smaller interiors,
\begin{equation}
 \mathbb E\prod_{j=1}^m H_{\mathcal D_-}(z_j)
 \ \leq\
 \mathbb E\prod_{j=1}^m H_{\mathcal D_+}(z_j)+o(1).
 \label{q4pocket:moment-order}
\end{equation}
The error tends to zero with the mesh, uniformly when the points have
fixed positive pairwise separation.  The statement compares moments, not signed
heights in an ordered coupling.
\end{lemma}

\begin{proof}
On graphs of the opposite color, the smaller laws are independent free
random-cluster laws.  Their product is stochastically dominated by the
restriction of the larger laws: reveal the remaining edges and use
the induced boundary partitions, which dominate free boundary
conditions.  Couple these opposite-color configurations by inclusion.

Consider a smaller outer contour that is relevant to the test points.
Its cluster is included in a larger-model cluster, but the latter may
itself be surrounded by a different cluster.  Select the \emph{outermost
larger contour containing the filled hull} of the smaller cluster.
One must make this selection even when the two clusters do not merge.
Using nested regular neighborhoods of included graphs, or the
cycle decomposition in Lemma~\ref{q4pocket:discrete-pockets}, shows
that this selected contour contains all the wound test points with
clearance.  Every relevant smaller maximal cycle lies in a maximal
cycle behind the selected larger contour.  In the surrounding case,
the smaller cycle lies in a bounded face of the outer cluster and is
therefore enclosed by one of its maximal cycles.

Retain only the two sets of exterior and pocket data from this
coupling.  Now resample their interiors using the conditional laws
of Lemma~\ref{q4pocket:discrete-pockets}.  Their wall color has reversed,
so the same free opposite-color comparison applies at the next stage.
Iterate.  This is a sequence of couplings of conditional marginals;
it does not claim monotonicity of nesting depth in the original
edgewise coupling.  At every fixed number of stages it gives inclusion
at the same numerical nesting levels.  A smaller contour's group of
test points can merge with another such group, but cannot split.
There is no extra contour of that level between a selected outer
contour and a pocket behind it: the pocket wall is part of that very
contour's cluster.

To translate this construction into moments, expand each height as
the sum of signs of its surrounding contours and average the signs.
A term survives exactly when every assigned contour receives an even
number of the indices $1,\ldots,m$.  Encode that term by the vector of
nesting levels $(\ell_1,\ldots,\ell_m)$ assigned at the respective
points.  The same vector is admissible in the larger model: mergers
add even multiplicities and hence preserve their parity.  Distinct
vectors remain distinct assignments.  This is an injection between
the terms counted by the two sign-averaged moments.

For a surviving term, each assigned contour winds at least two distinct
test points and has diameter bounded below by their separation.  All
its ancestors have that lower bound as well.  Truncate the number of
such contours and ancestors.  The free-bank arm bound makes the event
that any of these contours passes within $O(\delta)$ of a test point
negligible.  On its complement all the required pocket containments
hold.  Lemma~\ref{q4field:free-bank} and the higher count moments
in Lemma~\ref{q4field:moment-kernels} remove the exceptional event
and then the truncation by uniform integrability.  This proves
\eqref{q4pocket:moment-order}.
\end{proof}

\subsection{Geometry before identification}

The arm and passage estimates give three properties of every
subsequential contour limit that we shall use.  We spell out their
endpoint content to avoid importing a crossing estimate at a true
$q=4$ boundary.

\begin{lemma}[Status disks and boundary traversal]
\label{q4pocket:geometry}
For every positive-diameter limiting contour, and for the limiting
distinguished path, the set of distinct multiple visits in the
original-domain interior has Hausdorff dimension at most zero.
Deterministic interior points are almost surely avoided.  At every
interior trace point there are disks of each winding status arbitrarily
nearby.  In the marked case the statuses are read after closing the
path on either wall arc.  No nonconstant subtraversal of a limit lies
entirely on the original Jordan boundary.

These statements and curve tightness hold both for the regular wall
experiments and for the marked Jordan approximations in the theorem.
\end{lemma}

\begin{proof}
The first two assertions follow from
Proposition~\ref{q4field:four-arm} and
Corollary~\ref{q4field:contact-dimension}, together with the one-arm
estimate.  For the two-status assertion use the attached-circuit
argument of Section~\ref{sec:loops}, with both banks retained.  Here
is the only modification required at $q=4$.  Enlarge the first
crossing band a little laterally, and give a label only to an actual
component containing a crossing of the original core.  After a
component has been searched and removed, comparison with full wiring
on the enlarged band's boundary is still valid.  A buffered
opposite-color transversal extending past both lateral ends of the
core blocks any further core crossing.  Bulk RSW gives this blocker
a uniformly positive probability.  The number of labels is therefore
tight, and each label represents an actual connection in the revealed
past.  The leftmost crossings from a label's bottom sources in
successive core slabs can now be used exactly as in the attached-circuit
argument; the fresh-side circuits lie in interior balls.  For a
distinguished path, close it along the remote wall when reading the
two statuses.  The argument also works with exposed bulk data outside
the test patch.  Only buffered bulk RSW is used
\citep{DCLM2019Isoradial}.

For the boundary assertion, fix a deterministic boundary point away
from the marks.  A macroscopic passage approaching it supplies an
actual arm of the bank color that is not wired there.  In local
annuli this arm is bounded by the free-bank comparison.  Along a
primal wired arc the dual bank uses internal square faces and has no
dual exit through that arc.  Along the complementary arc the primal
bank uses actual sampled edges and has no primal ties.  Exterior
dual identifications occur only beside the complementary arc.
Turns can be followed by adjacent actual bank edges within
$O(\delta)$.  Any other induced ties in a buffered annulus away from
the other arc attach the surviving graph only at its rims.  Thus the
same arm bound applies in the specified Jordan approximations.

A nonconstant boundary subtraversal would cover an arc and hit a
point of a fixed countable dense subset away from the marks, by a
macroscopic passage.  The preceding bound excludes this.  For
tightness, at each displacement scale use a finite cover of
subpassage tests away from the two marks.  Every displacement contains
a smaller displacement a fixed fraction of that scale away from
both marks.  Marked uniform Jordan convergence and separation of
the two boundary arcs allow its annulus to be cropped in a
neighborhood missing at least one arc.  The same passage estimates
apply there.  Interior no-spur and multiple-visit tests always retain
positive clearance from the original boundary.
\end{proof}

\subsection{Protected cuts and pointed kernels}

We next give the deterministic approximation needed for the conditional
field.  A single limit contour may have several open winding components;
filling in its hull would incorrectly join some of them.

\begin{lemma}[Outer comparisons with protected cuts]
\label{q4pocket:protected-cuts}
Let $L_n$ be simple discrete contour closures in a bounded Jordan
domain $V$, converging uniformly modulo reparametrization to $L$.
The closures may include one original wall arc.  Assume the geometric
properties of Lemma~\ref{q4pocket:geometry}.  Let $T$ be the trace
of $L$, and let $U$ and $E$ be, respectively, the open sets of winding
status one and zero off $T$, with infinity in $E$.  Then the
components of $U$ are simply connected pointed kernels of the
interiors of $L_n$.

For every $\beta>0$ there exist simple regular comparison polygons
$B_j$, and protecting balls centered on $T\cap V$ or $\partial V$,
such that:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the pointed kernels of $B_j$ at points of $U$ are precisely
the separate components of $U$;
\item every compact subset of $U$ is eventually contained in $B_j$;
\item at each fixed $j$, deleting the part of a sufficiently fine
discrete pocket outside $B_j$ changes edges only in finitely many
protecting balls, of radii $r_i$, with
$\sum_i r_i^\beta\longrightarrow0$ as $j\to\infty$;
\item $B_j$ is connected to a region outside $\overline V$.
\end{enumerate}
The finite protecting selection is made after constructing the cuts.
\end{lemma}

\begin{proof}
Every point of $T\cap V$ is approached by $E$, by the two-status
property.  Points of $T\cap\partial V$ are approached by the exterior
of $\overline V$.  Hence
\begin{equation}
 \overline E=\mathbb C\setminus U.
 \label{q4pocket:exterior-closure}
\end{equation}
Compact subsets of $U$ persist in the approximating interiors by
winding stability.  Conversely, every excluded point is arbitrarily
close to persistent exterior points, by
\eqref{q4pocket:exterior-closure}.  Thus the component containing a
fixed point of $U$ is exactly its pointed kernel.  Kernels of simply
connected domains are simply connected.

Let $M\subset T\cap V$ be the distinct multiple-visit set, after
constant pauses have been erased.  Given $\eta>0$, cover \emph{all}
of $M$ by a countable collection of open balls $D_i$, centered on
$M$, with
\[
 \sum_i\operatorname{rad}(D_i)^\beta<\eta,
 \qquad \sup_i\operatorname{rad}(D_i)<\eta.
\]
This is possible because $\dim_{\mathrm H}M\leq0$.  Set
$G=E\cup\bigcup_iD_i$.

We first show that any $x\in E\cap V$ can be joined to the exterior
through $G$, after adding at most one arbitrarily small ball centered
on $\partial V$.  If the closure of the path component of $x$ in
$G$ reaches $\partial V$, such a ball connects it to the Jordan
exterior.  Otherwise that component has a compact frontier
$S\subset V\setminus G$ separating $x$ from infinity.  The set
$S$ meets $T$ only at unique-preimage points; its other points lie
in $U$.

There are polygonal closed walks, arbitrarily close to $S$, of
nonzero winding about $x$.  To see this, grid the bounded component
of $\mathbb C\setminus S$ containing $x$, retain its edge-adjacent
squares reachable from $x$ without meeting $S$, and trace their
external perimeter.  We can lift these walks, with an error tending
uniformly to zero, into the closed disks bounded by $L_n$.  Near a
point of $S\cap U$, a fixed interior ball persists.  Near a point
$z\in S\cap T$, all boundary parameters approaching $z$ tend to
its unique preimage under $L$.  Nearby points of the approximating
closed disk can therefore be connected near $z$: first go by a
closest segment to the boundary, and then use the short boundary
arc between the resulting nearby parameters.  The closest segments
stay in the closed disk.  The errors are uniform along the walk;
otherwise compactness would give one of the two local situations
just considered and contradict failure of uniformity.  The lifted
closed walk still winds about $x$, whereas $x$ lies outside the
approximating disk.  This contradicts contractibility of that disk
and proves the escape assertion.

Choose a large square $Q$ containing $\overline V$ in its interior.
Take finite increasingly fine nets of $\overline E\cap Q$ whose
vertices belong to $E$.  Join each net vertex to an exterior collar
of $Q$ using the escape assertion.  The finitely many additional
boundary-centered balls can have arbitrarily small total
$\beta$-content.  Select a finite polygonal tree joining these
paths to one root on $\partial Q$, otherwise lying in the open
allowed region; gridding the paths and deleting cycles gives such
a tree with axis-parallel edges.  Remove a sufficiently thin
polygonal neighborhood of this tree, attached to $\partial Q$.
Using narrow rectangular links and square junctions leaves one
simple rectilinear polygon $B$.  Keep an open part of the collar
outside $\overline V$ in $B$.

As the nets become finer, the removed tree approaches every point
of $\overline E\cap Q$.  Any part of it entering $U$ lies in the
protecting balls, whose radii tend to zero and whose centers lie
on $T$ or $\partial V$.  Hence compact subsets of $U$ persist,
while the pointed kernels cannot extend through an excluded point.
This proves the first two assertions.

For the finite selection, fix the tree and its closed thin
neighborhood.  It is compact and lies in the union of $E$ and
the countable protecting balls.  A finite subcover uses compact
subsets of $E$ and finitely many of those balls, with a positive
margin.  The compact subsets of $E$ are disjoint from the closed
disks bounded by $L_n$ for all large $n$.  Thus only the finitely
many selected balls can contain cuts affecting the discrete
pockets.  Their total content is no larger than that of the full
cover.  This is why the countable cover is used before selecting
the tree.  A further $O(\delta)$ adjustment makes the comparison
polygon a regular wall polygon on the appropriate color lattice,
without losing the protecting margins.
\end{proof}

\subsection{Transfer of the conditional field}

\begin{proposition}[Rough-interior moment transfer]
\label{q4pocket:rough-transfer}
Condition on exterior pocket data converging to a containing contour
as in Lemma~\ref{q4pocket:protected-cuts}.  At every fixed distinct
tuple of points in $U$, the joint moments of the residual zero-offset
height converge to the Wick moments of independent fields
$\sigma\Phi^0_\Omega$ in the components $\Omega$ of $U$.
This holds along every such convergent sequence of exterior data.
Several containing contours at one generation may be treated
simultaneously, taking the product over them.
\end{proposition}

\begin{proof}
Write $\beta>0$ for the power in the free-bank arm bound.  Fix a
comparison polygon $B$ and the finite protecting balls supplied
by Lemma~\ref{q4pocket:protected-cuts}.  In every pocket delete
opposite-color edges outside the lattice approximation of $B$.
Equivalently, intersect its set of elementary wall-color cells
with the corresponding cell set of $B$, retaining only
opposite-color edges between surviving adjacent cell centers.
All changes are in the protecting balls.

The original opposite-color laws are product free laws.  Couple
them with the modified laws by exploring, in the larger law, the
actual clusters incident to the deleted edges.  Once those
clusters are exposed, all discrepancies are shielded and the
unqueried graphs have the same conditional law.  The argument
is the same as the gap-closing coupling in
Section~\ref{sec:q4regular}.  If the ball radii are $r_i$ and $e>0$
is fixed, the probability that a discrepancy reaches farther than
$e$ from their centers is at most
\begin{equation}
 C\sum_i(r_i/e)^\beta.
 \label{q4pocket:cut-error}
\end{equation}
Indeed every such discrepancy supplies an actual opposite-color
arm in the original product free law, dominated by the full-lattice
comparison.  Contours through changed switches have banks in
these discovered clusters and obey the same locality bound.
Choose $e$ smaller than the tuple's separation and its clearance
from $T\cup\partial V$.  Every sign-averaged residual moment term
uses a contour of at least the pair-separation diameter.
The macroscopic count moments and H\"older's inequality therefore
make the moment error tend to zero with the total protecting
content in \eqref{q4pocket:cut-error}.

We verify the laws and offsets of the retained regions.  Each
side-connected component of the retained wall-color cells is
again a simple polygon disk.  Every globally missing cell has
a path of missing cells to the exterior: it is outside at least
one of the two simple grid polygons being intersected.  Thus a
retained component has no hole.  Its boundary cannot pinch at
a vertex with two opposite retained cells in that component
and two opposite missing cells.  The retained pair would be
connected by a side-adjacent path, while the missing pair is
connected through the exterior; these two disjoint connections
between alternating sectors contradict planar separation.
Apply this argument separately to each old pocket.  The
retained free opposite-color graph hence factors into
independent homogeneous disk laws.

Each retained component also reaches an \emph{old} pocket wall.
Start from one of its cells and take a side-adjacent path in
$B$ to the part of $B$ outside $\overline V$.  The first exit
from the old pocket crosses an old wall edge; the path before
that exit remains in the same retained component.  The component
therefore has an actual wall-color connection to the old wall.
No exterior contour can surround it and add an unrecorded
height jump.  Its residual offset is zero.  Finally, a connected
compact neighborhood of any test point began in one old pocket
for large $n$: the bank walk tracking $L_n$ winds around that
compact, so Lemma~\ref{q4pocket:discrete-pockets} places it in
a pocket.

For the upper bound, apply
Lemma~\ref{q4pocket:moment-comparison} from the retained disks
into $B$, and then apply
Theorem~\ref{q4field:regular-domain}.  For the lower bound,
make no deletions.  Choose one regular polygon disk compactly
inside each involved component $\Omega$, containing its test
points, and compare their product laws into the original
pockets.  Such inner disks can exhaust each simply connected
$\Omega$ in the pointed-kernel sense.

Dirichlet Green functions converge along simply connected
pointed kernels.  Between points whose limiting kernels are
different, they tend to zero; this follows by conformal mapping
convergence, or by the vanishing probability of crossing the
separating bottleneck before exit.  Therefore the Wick kernels
of both the inner polygons and the protected outer polygons
converge to those of independent Dirichlet fields in the
components $\Omega$.  Take the mesh limit first at fixed
polygons and protectors; next improve the inner and outer
approximations and let their total protecting content vanish.
The two bounds coincide.  Already assigned ancestor signs, or
the marked path's side constants, multiply lower-order residual
moments and are restored afterwards.  They do not enter the
moment comparison.
\end{proof}

\section{Identification at the endpoint and the boundary convention}
\label{sec:q4identification}

\subsection{A global conditional decomposition}

Take a joint subsequential limit of the height and the distinguished
path in a regular two-arc polygon.  For a homogeneous initial square,
take a joint limit of the height and the full contour collection,
including nesting levels, parent relations, and independent fair signs.
These marked geometric limits can be obtained at countably many
positive-diameter cutoffs decreasing to zero, moving a cutoff when
necessary to avoid atoms.  At each cutoff there are only a tight
number of relevant contours and ancestors.  Thus finite marked
relations stabilize, and a diagonal extraction retains every positive
limiting contour.  The signs remain conditionally independent and
fair given the geometry, which was sampled without using them.

Fix a nesting level $k$ in the square.  Every deterministic interior
point lies almost surely in a positive winding region at that level.
Indeed alternating actual circuits on successively separated bulk
scales force arbitrarily many surrounding contours.  The domain
Markov bounds and bulk RSW make the failure probability tend to zero
as the number of scales grows, with the mesh limit taken first at
each finite selection of scales.  The same statement holds for
Lebesgue-almost every point by Fubini.  Winding regions of different
contours at a fixed level are disjoint, as their discrete predecessors
are.  Positive coincident traversal arcs would also contradict the
multiple-visit dimension bound.  In the marked case the two side
closures likewise cover almost every point away from their traces.

Let $\mathcal F_k$ denote the limiting exterior geometry and signs
through level $k$, or let $\mathcal F$ denote the limiting marked
path in the two-arc experiment.  Let $\mathcal O$ be the collection
of components of the corresponding open winding regions.  Define
$v$ on each component to be the sum of its containing signs through
level $k$, or its marked side value $0$ or $1$.  Its value on the
remaining area-zero set is immaterial.

\begin{proposition}[Conditional field in all complementary components]
\label{q4pocket:conditional-field}
Conditional on the stated limiting exterior data, the height has the
distributional decomposition
\begin{equation}
 h\ \stackrel{\mathrm{law}}=\
 v+\sigma\sum_{\Omega\in\mathcal O}\Phi^0_\Omega,
 \label{q4pocket:conditional-decomposition}
\end{equation}
where the Dirichlet fields are independent and extended by zero.
The identity holds as an identity of distributions on the whole
initial-domain interior, not merely after restriction to individual
components.
\end{proposition}

\begin{proof}
For a test function $f$, the independent sum on the right exists
in $L^2$, since Green-function monotonicity gives
\begin{equation}
 \sum_{\Omega\in\mathcal O}
 \iint_{\Omega^2}|f(z)|G_\Omega(z,w)|f(w)|\,dz\,dw
 \leq
 \iint_{V^2}|f(z)|G_V(z,w)|f(w)|\,dz\,dw.
 \label{q4pocket:covariance-domination}
\end{equation}
The same estimate, with a fixed negative Sobolev norm, defines the
sum as a random distribution.

Use a representation on which the extracted geometric data converge.
A fixed distinct tuple of deterministic interior points is almost
surely eventually covered, with positive limiting clearance, by the
relevant winding components.  Its offsets converge.  Conditional
on the discrete exterior pocket data, including the actual cycle
walls, its residual moments converge by
Proposition~\ref{q4pocket:rough-transfer}.  That proposition applies
along every realized convergent data sequence; it requires no
regularity of the unexplored cycle walls beyond the proved pocket
properties.

To pass to conditioning only on the stated limiting data, multiply
the discrete moment identities by a bounded continuous function of
those data.  The necessary uniform integrability follows from
Lemma~\ref{q4field:moment-kernels}.  At level $k$, terms already
assigned to ancestors have offsets bounded by $k$.  The remaining
sign-averaged terms require shared contours of at least the relevant
pair-separation diameter, and their counts have all polynomial
moments under the initial law.  The marked offset is bounded by
one.  Thus, for each $m$, a bound of the form
\[
 C_{m,k}\bigl(1+|\log d_*|\bigr)^m,
 \qquad
 d_* = \min\bigl\{1,|z_i-z_j|:i\ne j\bigr\},
\]
and its higher-moment versions apply also before coverage by the
limiting components has stabilized.  Such bounds are integrable
on tuples in compact subsets of $V$, including near diagonals.

Smear the identities against tensor products of smooth test
densities.  Fubini and uniform integrability pass every joint
moment to the limit, with every bounded continuous geometric
multiplier retained.  The almost-everywhere coverage removes the
remaining set from these integrals.  Higher even moments give
uniform integrability of the smeared variables themselves.
A countable determining family of test densities then yields the
conditional moment identities for distributions.  For each fixed
conditional geometry, the Gaussian law with the bounded offset
on the right of \eqref{q4pocket:conditional-decomposition} is
moment determining; equivalently, apply the one-dimensional
Gaussian moment criterion to every rational linear combination
of finitely many tests.  Hence the moment identities identify
the conditional law.  Because the tests range over the entire
initial domain, this argument also excludes a residual
distribution supported on the area-zero complement of the
components.
\end{proof}

\subsection{Local-set uniqueness with the exact height normalization}

We use two continuum identifications.  For the zero-boundary GFF,
the two-valued bounded-type thin local set with values $-a,b$ is
unique and field measurable when $a,b>0$ and
$a+b\geq2\lambda_0$.  At $a=b=2\lambda_0$ it is the non-nested
$\mathrm{CLE}_4$ gasket, with independent fair labels on its
holes \citep[Propositions~1 and~2 and Sections~4 and~6.3]
{AruSepulvedaWerner2019}.

For boundary data we use Theorem~3.21 of
\citet{AruLupuSepulveda2020}, with its immediately preceding
boundary criterion.  In the specialization needed here, the
initial domain is a disk, $u$ is bounded harmonic with finitely
piecewise constant boundary data in $[-a,b]$, and
$a+b\geq2\lambda_0$.  Uniqueness holds for bounded-type thin
local sets whose total conditional harmonic function is
constantly $-a$ or $b$ on each connected boundary component
of each complementary domain.  The interval here is closed:
boundary values equal to $-a$ or $b$ are permitted.  The
standard completed chordal $\mathrm{SLE}_4$ level line, for
boundary data $-\lambda_0,\lambda_0$ on the two arcs, has
exactly this conditional field decomposition
\citep{SchrammSheffield2013}.

\begin{lemma}[The limiting sets are bounded-type thin local sets]
\label{q4pocket:local-set}
For the first contour level in the square, let
\[
 A=\overline V\setminus\bigcup_{\Omega\in\mathcal O}\Omega.
\]
In the marked experiment let $A$ be the path trace together with
$\partial V$.  After the normalizations below, these are
bounded-type thin local sets.  In the homogeneous case their
harmonic values are $\pm2\lambda_0$.  In the marked case their
\emph{total} harmonic values are $\pm\lambda_0$.
\end{lemma}

\begin{proof}
In the homogeneous case, write $\Phi=h/\sigma$ and $m=v/\sigma$.
Theorem~\ref{q4field:regular-domain} says that $\Phi$ is a
zero-boundary GFF.  At the first level $v\in\{-1,1\}$, so
$m\in\{-2\lambda_0,2\lambda_0\}$.  In the marked case put
\[
 X=\frac{h-1/2}{\sigma}=\Phi+u,
 \qquad
 m=\frac{v-1/2}{\sigma}-u,
\]
where $\Phi$ is a zero-boundary GFF and $u$ is the bounded
harmonic function with boundary data $-\lambda_0,\lambda_0$
on the respective arcs.  Here $|m|\leq2\lambda_0$ and
$m+u\in\{-\lambda_0,\lambda_0\}$ on each component.

In either case Proposition~\ref{q4pocket:conditional-field}
gives the full conditional Dirichlet law of $\Phi-m$ in
$V\setminus A$.  This law depends on the exterior data only
through $A$.  Conditioning down to $(A,m)$ proves locality.
Moreover $m$ is an ordinary bounded function, integrated over
$V\setminus A$, and the distributional identity has no
additional part supported on $A$.  This is thinness.  The
uniqueness of a local set's harmonic distribution from the
field and set is also part of the local-set formalism
\citep[Lemma~2.3(1)]{AruLupuSepulveda2020}; no extra choice
of component labels is being added at this step.

For completeness, $A$ is connected and contains the entire
Jordan boundary.  Its complementary components are simply
connected by Lemma~\ref{q4pocket:protected-cuts}.  If $A$
were disconnected, split it into two nonempty disjoint compact
sets, putting the connected $\partial V$ in the first.  A fine
polygonal boundary around part of the second, at positive
distance from the first and from $\partial V$, would be a
closed walk in a component of $V\setminus A$ with nonzero
winding about a point of $A$.  This contradicts that
component's simple connectivity.  Thus $A$ is a nontrivial
connected compact set.  It has no isolated points, and no
component disjoint from $\partial V$.  These facts verify
the topological condition in the definition of a bounded-type
thin local set.  Conformal transport to a disk preserves all
these assertions.
\end{proof}

\begin{proposition}[Regular marked endpoint]
\label{q4pocket:regular-chordal}
The distinguished curves in the regular two-arc experiments
converge as ordered curves, up to increasing changes of parametrization,
to chordal $\mathrm{SLE}_4$ between the prescribed marks.
\end{proposition}

\begin{proof}
Apply the boundary-data uniqueness theorem with
$a=b=\lambda_0$.  The initial domain, after conformal transport,
has one complementary component, which is not a singleton.
The boundary function has two changes and takes values in the
closed allowed interval.  Its interior range meets
$(-\lambda_0,\lambda_0)$ by the strong maximum principle.
The height-gap condition holds with equality.  In every
component of $V\setminus A$, the total conditional mean is
constant $-\lambda_0$ or $\lambda_0$, so the stated boundary
criterion holds on every boundary component.  Finally
Lemma~\ref{q4pocket:local-set} supplies thinness, boundedness,
and the required topology.  All hypotheses of
\citet[Theorem~3.21]{AruLupuSepulveda2020} are therefore met.

The completed Schramm--Sheffield level line coupled to the
same field satisfies these same conditions.  Uniqueness
identifies the limiting closed set, together with its harmonic
labels, with that level line plus the original boundary.
In particular its interior trace is the simple chordal
$\mathrm{SLE}_4$ arc, which has no hits on the interiors of
the two boundary arcs.

This closed-set statement identifies the original ordered
curve as well.  Any extra boundary point off the identified
compact arc has a neighborhood disjoint from that arc; reaching
it would force a nonconstant boundary subtraversal, excluded
by Lemma~\ref{q4pocket:geometry}.  On a simple arc, any
genuine backtracking repeats a nondegenerate subarc.  Such a
subarc has positive Hausdorff dimension, contradicting the
dimension-zero multiple-visit bound.  After suppressing constant
pauses, the curve therefore traverses the arc once, in the
prescribed endpoint order.
\end{proof}

\subsection{Contours, labels, and all nesting levels}

\begin{proposition}[Endpoint contours in a square]
\label{q4pocket:square-cle}
In a homogeneous square, the entire signed contour hierarchy
converges to nested $\mathrm{CLE}_4$, with independent fair
signs at each generation.  The identification includes each
individual contour as a simple-loop element, its parent, and
its traversal order up to the loop convention.
\end{proposition}

\begin{proof}
The first-level set $A$ in Lemma~\ref{q4pocket:local-set}
has harmonic values $\pm2\lambda_0$.  Ordinary two-valued-set
uniqueness identifies its coupling with the field as the
labelled non-nested $\mathrm{CLE}_4$ gasket.

We must still identify the discrete contour elements, since
one such element could a priori have several winding
components.  Every positive contour has an interior-status
disk, by the two-status property and the exclusion of
boundary-only traversal.  Thus it winds at least one hole
of the identified gasket.  Select two distinct holes by
the first points they contain in a fixed countable dense
set.  Conditional on the gasket $A$, the two unit offsets
in our construction satisfy
\begin{equation}
 \mathbb E[v(\Omega_1)v(\Omega_2)\mid A]
 =\mathbb P\{\Omega_1\text{ and }\Omega_2
        \text{ belong to one contour}\mid A\}.
 \label{q4pocket:label-grouping}
\end{equation}
Indeed the signs were independent and fair given the entire
unlabelled contour geometry; different contours contribute
zero covariance, whereas the same contour contributes one.
In the unique two-valued-set coupling, different
$\mathrm{CLE}_4$ holes have independent fair labels, so the
left side of \eqref{q4pocket:label-grouping} is zero.
Countability excludes all grouping of distinct holes.

Consequently each positive contour winds exactly one simple
CLE hole.  Its full interior trace equals that hole's Jordan
boundary: the two-status property puts every trace point
on the boundary of its winding region, and winding stability
puts that boundary in the trace.  Additional boundary pieces
are excluded as in Proposition~\ref{q4pocket:regular-chordal}.
The limiting winding is one.  Any extra covering or
backtracking repeats a nondegenerate subarc and violates the
dimension-zero bound.  Thus the original contour is exactly
the corresponding simple-loop element, not just a set with
the same filled interior.

Proceed conditionally through the hierarchy.  Given the
geometry and labels through an identified level, the
residual fields in its holes are independent zero-boundary
GFFs by Proposition~\ref{q4pocket:conditional-field}.  The
next-level decomposition, with the parent constant subtracted,
makes its missing set in each hole a thin local set with
harmonic values $\pm2\lambda_0$.  Child winding components
lie in the parent by discrete winding stability at points
with clearance.  They are simply connected and cover almost
every point, by the arguments already given.  Every positive
child has an interior-status disk and is supported on the
closure of its winding regions by the two-status property.
Thus the same topology and thinness verification applies
inside each parent hole.

Conditional uniqueness, including measurability from the
respective residual field, gives independent fresh CLEs
in the parent holes.  Equation~\eqref{q4pocket:label-grouping}
and the trace argument again identify every individual child
and its sign.  Its parent can be read at any interior disk
point off the traces.  Induction identifies all finite
generations.  Finally every positive-diameter contour and
all its ancestors occur in the retained finite-cutoff
hierarchies, whose counts are tight.  Sending the cutoff
to zero identifies the full collection in the stated
matching topology.
\end{proof}

\begin{corollary}[Endpoint whole-plane ensemble]
\label{q4pocket:whole-plane}
The infinite-volume free critical FK loop ensemble at $q=4$
converges to the nested whole-plane $\mathrm{CLE}_4$ ensemble,
in the spherical matching metric for all loops.
\end{corollary}

\begin{proof}
Use the expanding-square coupling of Section~\ref{sec:loops}.
Its endpoint ingredients are bulk estimates: monotone
boundary coupling and exploration of actual boundary
connections couple the infinite-volume free switches with
the square switches deep inside a large square, except
when a boundary connection reaches the inner region.
Actual dual circuits on successive annuli block that
event, uniformly even under the maximal boundary law.
Actual-color annular blockers likewise make a loop from
a fixed disk reaching the coupling boundary unlikely.
The finite-square limit is given by
Proposition~\ref{q4pocket:square-cle}.

Nested simple CLEs in domains of conformal radius tending
to infinity converge locally to nested whole-plane CLE;
their spherical whole-plane law is M\"obius invariant
\citep[Theorem~1, Proposition~4, and Section~3.2]
{KemppainenWerner2016}.  Its range
$8/3<\kappa\leq4$ includes the present endpoint, and
centered expanding squares meet its conformal-radius
condition.

To pass from the local comparison to the spherical metric,
first fix a spherical diameter cutoff.  Loops lying
entirely outside a sufficiently large fixed disk are below
that cutoff.  With high probability, every other loop
above the cutoff remains inside the switch-coupling region,
by the preceding escape bound.  The number of such loops
is tight, and the finite-square matching identifies them
bijectively with their traversal order.  Take the mesh
limit, then enlarge the square and the fixed disk, and
finally send the diameter cutoff to zero.  This is the
required global matching statement, including all nesting
levels.
\end{proof}

\subsection{The prescribed marked Jordan approximations}

It remains to transfer Proposition~\ref{q4pocket:regular-chordal}
to the precise wired/free edge convention of the theorem.
This transfer uses actual color walls, so that a boundary
identification is never mistaken for a path in the lattice.

\begin{proposition}[Endpoint marked Jordan convergence]
\label{q4pocket:jordan-chordal}
For every marked uniform Jordan approximation specified in
the theorem, its critical $q=4$ distinguished curve converges
to chordal $\mathrm{SLE}_4$ in the limiting marked domain.
\end{proposition}

\begin{proof}
Use the two-change disk drawing from
Lemma~\ref{moving:drawing}: place tiles on every polygon-graph
edge, give boundary edges distinct exterior tabs, and put
local caps along the wired primal and complementary dual
arcs.  Flattening the tabs and separating the drawing in
a thin collar changes the ordered curve by only
$O(\delta_n)$ in uniform distance.  Boundary edges on the
wired primal arc may be fixed open, while edges on the
free arc retain their sampled law.

The primal bank of the distinguished path, closed along
the primal wall, is a walk of actual open edges in the
wired cluster.  On the other side, the actual internal
dual bank, its exits to exterior copies across the free
arc, and the local complementary-wire links give the
dual closing walk.  All artificial dual links and copies
stay within $O(\delta_n)$ of that free arc.  No dual
exit through the primal wired arc is used.  At fixed
bulk points away from limiting traces these walks
therefore have stable complementary winding statuses
$1$ and $0$.  Lemma~\ref{q4pocket:geometry} gives tightness,
the two-status property, and the restrictions on multiple
visits and boundary traversal for these drawings.

Construct a regular comparison polygon $V^+$ close to the
limiting marked domain $V$ with the following two strict
separations: the original primal wired arc, including
its endpoints, lies outside $\overline{V^+}$; and the
comparison dual wall lies outside $\overline V$.
All margins are fixed before taking the mesh limit.
One construction uses plane Schoenflies coordinates for
$\partial V$.  Move the primal portion slightly inward,
extending it a little beyond both marks, then rise
outside the unit circle near the two ends and put
both comparison transitions there.  Keep the whole
comparison dual portion outside the unit circle.
Approximate the resulting simple curve by a simple
rectilinear polygon, preserving all strict margins,
and place the transitions inside straight sides.
Its actual primal-open and dual-open wall approximations
are regular experiments.  Reverse the colors to construct
$V^-$: the original complementary arc is outside
$\overline{V^-}$, and the comparison primal wall is
outside $\overline V$.  Both comparisons approach the
same marked Jordan domain as the offsets vanish.

Couple the original law with that in $V^+$ so that
every original primal open edge in the common region
is open in the comparison.  To check the boundary
partition needed for this ordering, restrict the
original configuration to the common edges.  Any
connection departing from the comparison disk can
induce ties only at its actual primal wall vertices:
its dual wall and transitions lie outside the entire
original closure, while the original wired vertices
lie outside the comparison.  The comparison connects
all those primal wall vertices by fixed open wall
edges.  Its induced partition therefore dominates
the original one, and random-cluster monotonicity
gives the coupling.  The cut is along a primal
lattice wall, so no unaligned edge crossing is
being treated as a wall vertex.

Every piece inside $V^+$ of the original primal
closing bank walk belongs to the comparison primal
wall cluster: follow its original connection to
the outer wire and take its first exit from $V^+$.
Suppose a test point has original primal winding
status but comparison dual status, with positive
clearance.  On the comparison dual side it can be
joined to the comparison dual wall without crossing
the comparison primal wall cluster, and hence
without crossing the original closing walk inside
$V^+$.  From that wall, which is outside the entire
original polygon, it can escape to infinity through
the original exterior.  This contradicts the
original winding.  Thus original primal status
implies comparison primal status.

For $V^-$ apply exactly the dual ordering.  Original
dual identifications and artificial bank links lie
beside the complementary arc, outside $V^-$.  In
the common region the dual walk uses only internal
dual steps, and departing connections induce ties
only at the aligned comparison dual wall vertices.
An originally wired primal edge admits no dual
crossing step.  Original dual status therefore
implies comparison dual status.  The free-bank arm
bound removes near-hits and tile ambiguities at
any prescribed finite tuple of bulk points.

Let $I_n(z)$ be the original primal-side indicator,
and let $I_n^-(z),I_n^+(z)$ denote those in the
two comparisons.  The separate monotone couplings
give, for every fixed finite tuple $\boldsymbol z$,
the stochastic bounds
\begin{equation}
 (I_n^-(z))_{z\in\boldsymbol z}
 \ \preceq\
 (I_n(z))_{z\in\boldsymbol z}
 \ \preceq\
 (I_n^+(z))_{z\in\boldsymbol z},
 \label{q4pocket:side-sandwich}
\end{equation}
up to errors vanishing with the mesh.  At fixed
comparison polygons the limiting laws are the
regular chordal $\mathrm{SLE}_4$ laws by
Proposition~\ref{q4pocket:regular-chordal}.  Now let
the offsets vanish.  Conformal maps and their
boundary extensions converge under marked uniform
Jordan convergence, using uniform local connectivity
of these boundary parametrizations.  The two
comparison SLE laws therefore have the same limit.
Deterministic interior points avoid SLE, so their
side-status arrays converge as well.  Equation
\eqref{q4pocket:side-sandwich} identifies every
finite-dimensional status law of an original
subsequential limit.

On a fixed countable dense set, the statuses
determine the interior trace: an interior trace
point has both statuses in every neighborhood,
whereas a point off the trace has a small disk
of constant status.  This follows from
Lemma~\ref{q4pocket:geometry}.  Thus the limiting
interior trace is the SLE trace.  No boundary
traversal and the dimension-zero multiple-visit
bound identify the complete ordered curve exactly
as in Proposition~\ref{q4pocket:regular-chordal}.
Every tight subsequence has the same law, proving
the full-sequence assertion.
\end{proof}
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//endpoint-loops.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//introduction.tex ====
\section{Introduction and precise statements}\label{sec:introduction}

Planar critical models are expected to have conformally invariant scaling
limits.  For random-cluster models this prediction identifies the law of an
interface, the joint law of all surrounding loops, and crossing
probabilities in arbitrary shapes.  These are distinct conclusions: a
limit for one interface does not by itself identify its coupling with
other interfaces, and convergence of loop traces on compact sets does not
by itself retain multiplicities, nesting, or complete traversals.

Schramm's characterization of conformally invariant interfaces led to
Schramm--Loewner evolution \citep{Schramm2000}.  Cardy's prediction for
critical percolation crossings \citep{Cardy1992} was proved by Smirnov for
site percolation on the triangular lattice \citep{Smirnov2001}.  The
random-cluster Ising model has a discrete holomorphic observable
\citep{Smirnov2010}, and its interface convergence is established in
\citet{ChelkakDCHonglerKemppainenSmirnov2014}.  Conformal loop ensembles
provide the corresponding continuum laws for complete families of
interfaces \citep{SheffieldWerner2012,Sheffield2009Exploration}.
The results below concern the nearest-neighbor square lattice, with its
ordinary Euclidean embedding and its original random-cluster probability
law.

\subsection{The model and its boundary conventions}
\label{intro:model}

Fix a real number $q\in[1,4]$ and put
\begin{equation}\label{intro:parameters}
 p_q=\frac{\sqrt q}{1+\sqrt q},
 \qquad
 \kappa(q)=\frac{4\pi}{\arccos(-\sqrt q/2)}.
\end{equation}
The inverse cosine has values in $[0,\pi]$.  Thus $4\le\kappa(q)\le6$,
with $\kappa(1)=6$ and $\kappa(4)=4$.  On a finite graph $G=(V,E)$,
let $\xi$ be a partition of designated boundary vertices.  For
$A\subseteq E$, let $k_\xi(A)$ be the number of components of $(V,A)$
after identification within each block of $\xi$, including isolated
vertices.  The random-cluster law is
\begin{equation}\label{intro:finite-law}
 \phi_{G,q}^{\xi}(A)
 =\frac{1}{Z_{G,q}^{\xi}}
 p_q^{|A|}(1-p_q)^{|E|-|A|}q^{k_\xi(A)}.
\end{equation}
An \emph{actual open path} always consists of open edges of $G$.
Identification in $\xi$ is used to compute the probability weight and
does not supply an additional edge in such a path.

Let $D$ be a bounded Jordan domain.  A lattice approximation at mesh
$\delta_n\downarrow0$ is the bounded interior $D_n$ of a simple closed
nearest-neighbor polygon on $\delta_n\Z^2$.  Its graph $G_n$ has every
lattice vertex in $\overline{D_n}$ and every nearest-neighbor edge whose
closed segment is contained in $\overline{D_n}$.  In particular, boundary
edges are part of $G_n$.

For two marked boundary points $a,b$, we require counterclockwise
parametrizations $\beta_n,\beta:[0,1]\to\C$, injective on $[0,1)$,
such that
\begin{gather}\label{intro:marked-jordan}
 \beta_n(0)=\beta_n(1)=a_n,\qquad \beta_n(1/2)=b_n,\qquad
 \beta(0)=\beta(1)=a,\qquad \beta(1/2)=b,\\
 \sup_{t\in[0,1]}|\beta_n(t)-\beta(t)|\longrightarrow0.\notag
\end{gather}
The free arc $F_n$ goes counterclockwise from $a_n$ to $b_n$; the other
arc $W_n$ goes counterclockwise from $b_n$ to $a_n$.  Wire together all
vertices of $W_n$, including the two marked endpoints.  All other
boundary vertices remain singleton blocks.  Remove from the sampled edge
set the boundary edges of $W_n$ and declare those edges open for the
interface drawing.  This is equivalent on all remaining edges to
\eqref{intro:finite-law} with the same wired partition: the endpoints of
each removed edge were already identified.  Edges on $F_n$ remain random.

The medial graph has vertices at the midpoints of primal edges, with
medial edges joining the midpoints of consecutive sides of a lattice
square.  At an open primal edge the four incident medial half-edges are
paired in quarter turns around the adjacent dual faces; at a closed
primal edge they are paired around the primal endpoints.  At the
boundary one uses the turns between the primal wired arc and its
complementary dual wired arc.  There is a unique medial strand between
the two changes of boundary condition.  We orient it from $a_n$ to $b_n$
and denote it by $\eta_n$.  This is the usual Dobrushin construction
\citep[Section~5.2.2]{DC2017Lectures}.  Occurrence order at a twice-visited
medial vertex is part of the curve.  The tile drawings used later preserve
this order and differ from the straight medial drawing by at most a
constant times $\delta_n$ in uniform curve distance.

For continuous oriented curves $\gamma_1,\gamma_2:[0,1]\to\C$, write
\begin{equation}\label{intro:curve-metric}
 d_{\mathrm{curv}}(\gamma_1,\gamma_2)
 =\inf_{\varphi_1,\varphi_2}
   \sup_{0\le t\le1}
   |\gamma_1(\varphi_1(t))-\gamma_2(\varphi_2(t))|,
\end{equation}
where the infimum is over increasing homeomorphisms of $[0,1]$.
Zero-distance curves are identified.  Chordal $\SLE_\kappa$ in $(D;a,b)$
is the conformal image of the upper-half-plane Loewner trace driven by
$\sqrt\kappa B_t$ from $0$ to $\infty$, with $B$ a standard Brownian
motion.  The remaining positive dilation of the conformal coordinate
does not change this law.

\subsection{Complete loop collections}
\label{intro:collections}

Let $\phi_q^0$ be the weak limit of \eqref{intro:finite-law} in increasing
square boxes of $\Z^2$ with free boundary conditions.  Form its complete
medial interface collection, using the same local pairings as above, and
then scale the embedding by $\delta$.  Denote the resulting collection
by $\Gamma_\delta$.  It contains every loop at every nesting level.
The elementary circuit argument in Section~\ref{sec:preliminaries}
shows that its discrete interfaces are almost surely finite, also at
$q=4$.

Individual loops are unrooted and unoriented continuous maps of the
circle into the Riemann sphere $\widehat\C$.  Their distance is the
infimum of the uniform spherical distance over circular
orientation-preserving homeomorphisms and reversal, with zero-distance
representatives identified.  We retain distinct loops even when their
images coincide.  For two collections, an $\eps$-matching is a partial
bijection that covers all loops of spherical diameter greater than
$\eps$ in both collections, has matched-loop distance at most $\eps$,
and leaves only loops of diameter at most $\eps$ unmatched.  Convergence
of collections means the existence of such matchings with
$\eps\downarrow0$.  The continuum law in the next statement is the
standard nested whole-plane $\CLE_\kappa$ in this topology.

\begin{theorem}[Interfaces and all nested loops]
\label{intro:main}
For every fixed real $1\le q\le4$, the following assertions hold.
\begin{enumerate}[label=\textup{(\roman*)}]
\item For every bounded Jordan domain $D$ with distinct boundary marks
$a,b$, and every approximation satisfying
\eqref{intro:marked-jordan}, the Dobrushin strand $\eta_n$ converges in
law, in \eqref{intro:curve-metric}, to chordal $\SLE_{\kappa(q)}$ from
$a$ to $b$ in $D$.
\item Under the specified free infinite-volume law,
$\Gamma_\delta$ converges in law as $\delta\downarrow0$ to the nested
whole-plane $\CLE_{\kappa(q)}$, in the spherical collection topology
just defined.  The limit is M\"obius invariant.
\end{enumerate}
Both conclusions hold along every mesh sequence.  No uniformity in $q$
is asserted.
\end{theorem}

Theorem~\ref{intro:main} establishes the SLE/CLE conformal-invariance
prediction for critical random-cluster models on the square lattice
throughout $q\in[1,4]$.  The word ``critical'' refers to the edge
parameter $p_q$.  At $q=4$ the limits are chordal $\SLE_4$ and the
standard simple nested whole-plane $\CLE_4$.

The endpoint proof uses a whole-plane height-field theorem, with its
precise law and symmetry input verified in
Section~\ref{sec:q4-rotation-repair}.  That theorem supplies no
finite-domain field convergence, conditional convergence after an
exploration, or interface convergence.  Sections~\ref{sec:q4regular}--
\ref{sec:q4identification} prove these further implications.  The
strict angular and six-arm estimates used for $q<4$ are not extended to
$q=4$ by a limiting argument.

\subsection{Square-lattice bond crossings}
\label{intro:crossings}

For four distinct counterclockwise boundary marks $a,b,c,d$ of $D$,
require the same uniform convergence as in
\eqref{intro:marked-jordan}, with the respective four marks at
parameters $0,1/4,1/2,3/4$.  Sample \emph{all} edges of $G_n$
independently with probability $1/2$, including every boundary edge, and
make no identifications or external connections.  Let $C_n$ be the
event of an actual open path from the closed counterclockwise arc
$[a_n,b_n]$ to the closed counterclockwise arc $[c_n,d_n]$.

\begin{theorem}[Cardy's formula for square-lattice bond percolation]
\label{intro:cardy}
Let $\psi:D\to\H$ send $a,c,d$ to $0,1,\infty$, and put
$x=\psi(b)\in(0,1)$.  Under the boundary and approximation conventions
above,
\begin{equation}\label{intro:cardy-formula}
 \lim_{n\to\infty}\mathbb P(C_n)
 =\frac{\displaystyle\int_0^x[t(1-t)]^{-2/3}\,\dd t}
        {\displaystyle\int_0^1[t(1-t)]^{-2/3}\,\dd t}.
\end{equation}
\end{theorem}

Theorem~\ref{intro:cardy} proves the square-lattice bond-percolation form
of Cardy's prediction with the free-boundary convention specified here.
We obtain it directly from the alternating four-change calculation and
a deterministic comparison on a common lattice graph.  In particular,
its proof supplies the boundary transfer for the actual crossing event;
it does not infer continuity of that event from a curve limit.

\subsection{Proof structure and technical contributions}

For $1\le q<4$, use the parameters
\begin{equation}\label{intro:lambda-rho}
 d=\sqrt q=2\cos\lambda,\qquad
 \rho=\lambda/\pi,\qquad 0<\lambda\le\pi/3.
\end{equation}
The first part of the proof is a four-change boundary calculation.  In
the upper half-plane, put the boundary changes at $0,\chi,1,\infty$.
For the open cap normalization, in which the four unmatched ends carry
no additional pairing weights, the connection probability is
\begin{equation}\label{intro:fourchange-formula}
 f(\chi)=\frac{I_C(\chi)}{I_B(\chi)+I_C(\chi)},
 \quad
 \begin{cases}
 I_B(\chi)=\displaystyle\int_\chi^1
 u^{\rho-1}|u-\chi|^{\rho-1}|u-1|^{1-3\rho}\,\dd u,\\[3pt]
 I_C(\chi)=\displaystyle\int_1^\infty
 u^{\rho-1}|u-\chi|^{\rho-1}|u-1|^{1-3\rho}\,\dd u.
 \end{cases}
\end{equation}
Separate deterministic cap wires on the two designated intervals change
this to $f/(f+d(1-f))$.  Keeping these two normalizations distinct is
essential when $d\ne1$.

\paragraph{Positive comparisons before complex observables.}
Section~\ref{sec:preliminaries} identifies the capped tile law exactly
with the random-cluster law and establishes the collar and passage-count
comparisons used later.  It keeps actual components distinct from
boundary partition blocks.  Bulk box crossing at $q\le4$ and the stronger
up-to-boundary crossing estimates for $q<4$ have different roles
\citep{DCLM2019Isoradial,DMT2021Fractal}.  Section~\ref{sec:exponents}
derives the local exponent inequalities needed to control corner, tip,
and source factors.  All complex currents are subsequently estimated in
these positive laws.

\paragraph{Angular transfer and boundary holomorphicity.}
Sections~\ref{sec:angular} and \ref{sec:current} construct the transfer
operators, their reflected norms, and the path-current primitive.
The argument distinguishes a common complex continuation radius from
norm constants that may depend on a vector in the cyclic core.  A
scalar spectral factorization then suppresses the angular modes that
would obstruct holomorphicity.  The boundary calculation in
Section~\ref{sec:fourchange} is initially conditional on this
holomorphicity.  Sections~\ref{sec:pairedwall}--\ref{sec:holomorphic}
discharge the condition by a distant-probe comparison, closing finitely
many wall gaps, and a folded mixed-stencil estimate.  The folds retain
the external states in their reflected norms; local exponent savings
pay for extraction and normalization costs.  This produces
\eqref{intro:fourchange-formula} without assuming a lattice conformal
invariance theorem.

\paragraph{Moving domains and the chordal driver.}
The fixed-shape formula must remain useful after a random exploration.
Section~\ref{sec:movingtests} proves this transfer using a finite chart
approximation and an averaged six-arm estimate under the original
switch law.  It does not require a uniform six-arm theorem in every
possible explored domain.  Section~\ref{sec:chordal} first normalizes a
shrinking test into a bounded martingale and then transfers it to the
true Dobrushin law.  Two tests identify the driver and its bracket.
Tightness and a separate traversal argument give the Euclidean uniform
curve conclusion of Theorem~\ref{intro:main}(i) for $q<4$.

\paragraph{Joint branches and loop identities.}
Sections~\ref{sec:peeling}--\ref{sec:loops} construct a target tree by
uniform-walled peeling.  They establish the reflected angular dynamics,
exclude forcepoint variation supported on endpoint times, and identify
independent continuation noises after separation, conditional on the
common causal history at separation.  The resulting branching SLE description is then matched
to the original discrete loops: winding information determines loop
labels, while additional estimates determine full traversal order.
An escape bound upgrades convergence in bounded windows to the
spherical matching topology for every nesting generation.

\paragraph{The endpoint field argument.}
At $q=4$ the loop height uses independent fair signs and unit jumps.
Section~\ref{sec:q4-rotation-repair} extracts a bijection of selected
loops from the public FK homotopy coupling
\citep{DCKKMO2026Rotation}.  Even sign moments depend only on loops
that separate at least two distinct pairs of endpoints.  These loops
are macroscopic, and the bijection preserves their winding differences
up to one harmless sign per loop.  This supplies the endpoint symmetry
input for the whole-plane six-vertex field theorem
\citep{SixVertexGFF2026}, including the multiplicities that a loop-set
distance alone would not determine.
Section~\ref{sec:q4tools} combines binary transport and reflection
positivity with the whole-plane height input to control double
contacts.  The error estimates are relative to the square of the
four-arm probability; an absolute error would be insufficient.
Section~\ref{sec:q4regular} derives the Dirichlet field in regular
domains by translating separated charge groups and determining the
harmonic boundary term.  Section~\ref{sec:q4rough} then treats rough
explored pockets by a conditional moment comparison, with contour
injections at each numerical nesting level and a regular-domain
sandwich.  The final identification verifies locality, thinness,
bounded harmonic values, and the relevant topology before invoking
two-valued local-set uniqueness
\citep{AruSepulvedaWerner2019,AruLupuSepulveda2020}.

\subsection{Conventions on constants and limits}

Every theorem is for a fixed $q$.  Constants may depend on $q$,
macroscopic shapes, positive separations, and an explicitly fixed
finite set of tests.  Mesh limits are taken with those data fixed;
auxiliary separations, gaps, cutoffs, and localization parameters are
removed in the stated order.  A claim uniform over random exploration
histories is made only where the corresponding comparison has been
proved.  In particular, convergence in probability over histories is
not replaced by a deterministic estimate for every history.

The proof uses established public results at their stated hypotheses.
It uses no theorem about an annealed random planar map, no random
embedding, and no transfer from a random-map metric limit to the square
lattice.  The finite Dobrushin experiments and the free infinite-volume
experiment are different probability spaces; the theorems assert their
separate convergence and require no coupling between their samples.
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//introduction.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//main.tex ====
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\title{Natural Occupation Measures for\\[0.15em]
Critical Square-Lattice FK Interfaces}
\author{OpenAI}
\date{October 5, 2026}
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==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//main.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//observable.tex ====
\section{The four-change observable and its boundary values}
\label{sec:fourchange}

Throughout Sections~\ref{sec:fourchange}--\ref{sec:holomorphic}, let
$1\le q<4$, $d=\sqrt q=2\cos\lambda$, and $\rho=\lambda/\pi$.
The construction has two distinct parts.  We first determine the boundary
values of a disk observable by an exact calculation on its boundary-arc
tree.  We then prove its holomorphicity by placing two copies on opposite
banks of a wall.  The boundary identification in
Proposition~\ref{four:conditional-map} is explicitly conditional until
Theorem~\ref{hol:disk} supplies this last premise.

\subsection{Boundary data and the open-cap law}

Let $\Omega$ be a fixed simple orthogonal polygon in tile coordinates,
with four distinct boundary points $p,b,c,d_0$ in counterclockwise order,
each in the relative interior of a side.  Write $A,B,C,D$ for the successive
intervals beginning at $p$.  Approximate $\Omega$ by whole-tile disks of
mesh $n^{-1}$, with side positions and marks changed by $O(n^{-1})$,
using the balanced corner parities of Corollary~\ref{exp:polygon-budget}.
Give $A,C$ one designated wire color $\mathfrak A$, and $B,D$ the other,
$\mathfrak B$.  At each mesh choose the global color assignment for which
the deterministic-cap partition, completed around $A,C$, is at least
that for the reversed assignment.  Changing just the pairing of the four
unmatched ports changes a positive partition by a bounded factor.

Under the \emph{open-cap law}, the four unmatched ports remain open;
every closed loop, including those closed by local caps, receives weight
$d$, in addition to the common tile coefficients.  This law and either
deterministic completion have uniformly bounded density ratios.  Let
$E_{\mathfrak A}$ be the event of an actual $\mathfrak A$-path in the disk
joining $A$ to $C$, and define $E_{\mathfrak B}$ similarly for $B,D$.
Eligible endpoints of intervals count as boundary vertices.  Set
$P_{\mathfrak A}=\mathbb P_{\rm open}(E_{\mathfrak A})$ and
$P_{\mathfrak B}=\mathbb P_{\rm open}(E_{\mathfrak B})$.

Use the boundary factors and inward-normal lift of
Section~\ref{sec:exponents}, with bases
\begin{equation}
 (H_A,H_B,H_C,H_D)=(s+1,1,s,0),\qquad
 h_i=H_R+\Delta\mathbf1_{\{i\text{ is high}\}},\quad i\in R.
 \label{four:bases}
\end{equation}
The boundary order starts at $p$.  On continuing to a second lap the
base decreases by $1$, to account for the curvature lift, so the locally
forward raw drop at $p$ is $-s$.  Introduce
\begin{equation}
 x=e^{-i\lambda},\quad y=e^{i\lambda},\quad
 P_0=x^{s+1},\quad G=1-y^2,\quad A_R=x^{2H_R+\Delta}.
 \label{four:normalization}
\end{equation}
In particular, $d=y+y^{-1}$ and $x^{2\Delta}=-y^3$.

Take the primitive of Section~\ref{sec:current} from $p$, initially
directed inward, to a lattice vertex $z$.  Removing the common source
factor $c_*=e^{-i\lambda}/(2\sin\lambda)$, and dividing each diagram by
its open-cap weight, gives the random variable $H_{V(z)}$ below.  The
normalized observable is
\begin{equation}
 h_n(z)=\frac{\mathbb E_{\rm open}H_{V(z)}}{P_0G}.
 \label{four:observable}
\end{equation}

\subsection{The boundary-arc tree}

Cut the disk along its raw boundary arcs, leaving its closed internal
loops in place only for their ordinary weights.  The adjacency graph of
the resulting regions is a tree, rooted at the region incident to $p$.
Its vertices have the common color of their boundary-gap visits; colors
alternate across every edge.  Let $V(z)$ be the region containing $z$
when the internal loops are ignored.  Call a tree vertex \emph{wired}
if it visits a gap of its own color on an interval wiring that color.

\begin{lemma}[Tree normalization and connection alternatives]
\label{four:tree}
The number of loops closed by the local boundary caps is
\begin{equation}
 \mathcal I=\#\{\text{unwired tree vertices}\}.
 \label{four:loop-count}
\end{equation}
Exactly one of $E_{\mathfrak A},E_{\mathfrak B}$ occurs.  Completing the
four open ports around $A,C$ adds respectively one and two loops on these
events.

For an edge $e$ represented by an arc $i\to j$, $i<j$, put
\begin{equation}
 m_e=h_i-h_j,\qquad
 w_e=\frac{\cos((m_e-\tfrac12)\lambda)}{\sin\lambda},\qquad
 k_e=-\frac{\cos((m_e+\tfrac12)\lambda)}{\sin\lambda},\qquad
 a_e=x^{h_i+h_j}.
 \label{four:edge-weights}
\end{equation}
Let $\mathcal P(V)$ be the root-to-$V$ edge path.  Define
\begin{align}
 D_V&=d^{-\mathcal I}\prod_{e\in\mathcal P(V)}k_e
                         \prod_{e\notin\mathcal P(V)}w_e,\label{four:D}\\
 H_V&=d^{-\mathcal I}\sum_{e\in\mathcal P(V)}a_e
       \prod_{f\text{ before }e\text{ in }\mathcal P(V)}k_f
       \prod_{f\text{ not before }e,\ f\ne e}w_f.
       \label{four:H}
\end{align}
Thus $H_{\rm root}=0$.  If $e$ joins its parent $u$ to its child $v$, the
common product $D_e^\circ$, with the factor on $e$ omitted, satisfies
\begin{equation}
 H_v-H_u=a_eD_e^\circ,\qquad
 D_u=w_eD_e^\circ,\qquad D_v=k_eD_e^\circ.
 \label{four:recursion}
\end{equation}
\end{lemma}

\begin{proof}
Attach a thin exterior annulus containing the local caps, with the four
unmatched ports continued to its outer boundary.  Each cap cuts off a
sliver at the intervening gap.  The remaining annular regions connect
the wire-color visits of each interval and lead to the four outer
intervals.  Consequently precisely the wired classes remain accessible
from that boundary.  Two noncrossing open strands cut a disk into three
boundary regions, with one further region for each boundary loop.  The
other tree regions give exactly the loops in \eqref{four:loop-count}.
This also gives the opposite-pair alternative.

These alternatives concern actual connections.  Indeed the switch arcs
are perimeters separating the two thickened diagonal graphs.  Visits of
one color in a single boundary region belong to the same actual component;
reinstating internal loops cannot divide that boundary class.  There
are only two designated intervals of each color.  The two possible
pairings of four ports agree or disagree with the deterministic exterior
pairing, adding two or one loops, respectively, as asserted.  Finally,
the path rule of Lemma~\ref{cur:tree-weights} gives
\eqref{four:D}--\eqref{four:H}; all ordinary internal-loop weights cancel.
Separating the term marked on $e$ gives \eqref{four:recursion}.
\end{proof}

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[scale=1.05, every node/.style={font=\small}]
\foreach \dx/\ev in {0/{E_{\mathfrak A}},4.2/{E_{\mathfrak B}}}{
  \begin{scope}[xshift=\dx cm]
  \draw[thick] (0,1.15) arc[start angle=90,end angle=450,radius=1.15];
  \foreach \pos/\lab in {(0,1.15)/p,(-1.15,0)/b,(0,-1.15)/c,(1.15,0)/d_0}
    \fill \pos circle[radius=1.4pt];
  \node[above] at (0,1.15) {$p$};
  \node[left] at (-1.15,0) {$b$};
  \node[below] at (0,-1.15) {$c$};
  \node[right] at (1.15,0) {$d_0$};
  \node at (-1.02,1.02) {$A$};
  \node at (-1.02,-1.02) {$B$};
  \node at (1.02,-1.02) {$C$};
  \node at (1.02,1.02) {$D$};
  \node at (0,-1.75) {$\ev$};
  \end{scope}
}
\draw[very thick] (0,1.15) .. controls (0,.35) and (.35,0) .. (1.15,0);
\draw[very thick] (-1.15,0) .. controls (-.35,0) and (0,-.35) .. (0,-1.15);
\begin{scope}[xshift=4.2cm]
\draw[very thick] (0,1.15) .. controls (0,.35) and (-.35,0) .. (-1.15,0);
\draw[very thick] (0,-1.15) .. controls (0,-.35) and (.35,0) .. (1.15,0);
\end{scope}
\end{tikzpicture}
\caption{The two four-port pairings after local caps are attached.
The boundary is drawn as a circle only to show cyclic order; $A,C$
wire $\mathfrak A$ and $B,D$ wire $\mathfrak B$.  Completing around
$A,C$ adds one loop on the left and two on the right.  The thick curves
are open medial strands, not cluster paths.}
\label{four:pairing-figure}
\end{figure}

For later use we make the finite tree computation explicit.  Let $L(e)$
be the number of marks among $b,c,d_0$ in the child boundary interval of
$e$.  Those marks form a consecutive subset.  Their drops are $s,t,s$;
the endpoint high statuses agree when $L$ is odd and differ when it is
even.  Substitution in \eqref{four:edge-weights} gives
\begin{equation}
\begin{array}{c|cc}
 L&w_e&k_e\\ \hline
 0&d,\ 1&-1,\ -d\\
 1\ (b\text{ or }d_0)&1&0\\
 1\ (c)&1&-d\\
 2&1,\ d&0,\ -(d^2-1)\\
 3&0&1
\end{array}
\label{four:table}
\end{equation}
Two entries refer, in order, to a high and a low first port.

The $L>0$ subtree has four terminal legs, possibly of zero length.
The $p$-leg has $L=3$, and the other legs have $L=1$.  Between the
branches is either one shared node or a chain of $m\ge1$ edges with
$L=2$.  A chain of type $X$ encloses $c,d_0$ and has ports on $B,D$;
write $\mathfrak C=\mathfrak B$ in this case.  A chain of type $Y$
encloses $b,c$, has ports on $A,C$, and has $\mathfrak C=\mathfrak A$.
These are the only possibilities because descendant marked sets are
nested.  Number the chain nodes $0,\ldots,m$ away from the $p$-leg,
and write $c_j$ for their colors.  For a shared branch use $m=0$.

Leg and branch nodes are wired: each meets intervals of both designated
colors, also when a leg collapses at a mark.  An internal chain node
meets only intervals wiring $\mathfrak C$, and cannot meet a mark because
its descendant marked set is unchanged along the chain.  The child of
an $L=0$ edge is unwired exactly when its first port is high.  Hence
\begin{equation}
 \mathcal I=\mathcal I_0+\mathcal I_{\rm mid},\qquad
 \mathcal I_{\rm mid}=\#\{0<j<m:c_j\ne\mathfrak C\},
 \label{four:count-split}
\end{equation}
where $\mathcal I_0$ counts high-starting $L=0$ edges.  If $m>0$,
the endpoint regions of each chain edge both visit the two
$\mathfrak C$-intervals, so $E_{\mathfrak C}$ holds.  If $m=0$,
the shared node visits all four intervals, giving the connection of
color $c_0$.  Since a chain edge has $w_e=d$ precisely when its child
has color $\mathfrak C$,
\begin{equation}
 F:=d^{-\mathcal I_{\rm mid}}\prod_{L(e)=2}w_e
 =\begin{cases}
 d^{\mathbf1_{\{c_0\ne\mathfrak C\}}},&m>0,\\
 1,&m=0.
 \end{cases}
 \label{four:F}
\end{equation}
Every off-path $L=0$ weight cancels its contribution to $d^{\mathcal I}$.
Inside an $L=0$ branch contained in $R$, define
\begin{equation}
 g_R(\text{wire color})=\tfrac12,\qquad
 g_R(\text{other color})=-\tfrac12.
 \label{four:g}
\end{equation}
Then
\begin{equation}
 \frac{k_e}{w_e}
 =-d^{g_R(\operatorname{col}(v))-g_R(\operatorname{col}(u))},
 \qquad a_e=A_R.
 \label{four:tail}
\end{equation}
In particular, two successive increments of $H$ in such a branch cancel.

\begin{lemma}[Marked values and corrected observable]
\label{four:marked}
Call the tree \emph{typical} when all four terminal legs have positive
length.  On a typical tree,
\begin{equation}
 D_p=D_b=D_{d_0}=0,\qquad
 \frac{H_b}{P_0G}=\begin{cases}1&E_{\mathfrak A},\\d&E_{\mathfrak B},\end{cases}
 \quad
 \frac{H_{d_0}}{P_0G}=y\begin{cases}d&E_{\mathfrak A},\\1&E_{\mathfrak B}.
 \end{cases}
 \label{four:marked-values}
\end{equation}
On the gap visits of any interval $R$,
\begin{equation}
 \begin{gathered}
 \Re(H_V/A_R)-c_R(\operatorname{col}(V))D_V
 \quad\text{is constant},\\
 c_R(\text{wire color})=-\cos\lambda,\qquad
 c_R(\text{other color})=\cos(2\lambda).
 \end{gathered}
 \label{four:line-invariant}
\end{equation}
Set
\begin{equation}
 \Gamma(\mathfrak A)=P_0y^2,\qquad
 \Gamma(\mathfrak B)=-P_0y,\qquad
 H_V^\circ=H_V-\Gamma(\operatorname{col}(V))D_V.
 \label{four:corrected}
\end{equation}
The variables $H_V^\circ$ are bounded uniformly over all trees and vertices.
On a typical tree they have value $P_0GT$ along the $c$-leg and all its
descendant branches, where
\begin{equation}
 T=\mathbf1_{E_{\mathfrak A}}+y\mathbf1_{E_{\mathfrak B}}.
 \label{four:T}
\end{equation}
On gap visits along $B$, their only possible values are $H_b,P_0GT$;
along $C$, they are $H_{d_0},P_0GT$.
\end{lemma}

\begin{proof}
An unpassed $L=3$ edge kills every term of $H$, so only its last edge
can mark.  Once the first $b$- or $d_0$-leg edge has been passed,
$k_e=0$ kills all later marked terms.  At node $0$,
\begin{equation}
 D=F,\qquad H/P_0=
 \begin{cases}F,&c_0=\mathfrak A,\\-y^3F,&c_0=\mathfrak B.
 \end{cases}
 \label{four:branch-values}
\end{equation}
The first $b$-leg coefficient $a_e/P_0$ is $-y^2$ or $y^{-1}$,
according as the parent color is $\mathfrak A$ or $\mathfrak B$;
the corresponding $d_0$-leg coefficients are $-y^4$ or $y$.
This gives \eqref{four:marked-values} at the near branch or shared node.

On the middle chain, $a_e/P_0=y$ in type $X$ and $-y^2$ in type $Y$.
If the first chain edge has $k_e=0$, its weight is $w_e=1$ and only
that chain mark contributes.  Otherwise $F=d$; the chain terms and
the target mark contribute $(2-d^2)$ times the chain amplitude.
Indeed the second chain edge already has $k=0,w=1$, or, when $m=1$,
the target edge has the same amplitude.  The identities
\begin{equation}
 d+(2-d^2)y=yG,\qquad
 -dy^3-(2-d^2)y^2=G
 \label{four:far-algebra}
\end{equation}
give the far-branch values; the first-$k=0$ cases give the same values.

For \eqref{four:line-invariant}, consider first the earlier port of an
arc, and let $r=1$ for a high port and $r=-1$ otherwise.  With
``before'' and ``after'' referring to boundary-gap visits,
\begin{equation}
 \Re(a_e/A_R)=\cos((m_e-r\Delta)\lambda)
       =c_{\rm after}k_e-c_{\rm before}w_e.
 \label{four:early-port}
\end{equation}
At the later port, using its own high status,
\begin{equation}
 -\Re(a_e/A_R)=-\cos((m_e+r\Delta)\lambda)
       =c_{\rm after}w_e-c_{\rm before}k_e.
 \label{four:late-port}
\end{equation}
These product-to-sum identities and \eqref{four:recursion} prove the
invariant, including across a corner in the same interval.

For $R=B,C$, direct substitution gives
\begin{equation}
 \Re(\Gamma/A_R)=c_R,\qquad
 d\Gamma(\text{wire color})+\Gamma(\text{other color})=-A_R.
 \label{four:Gamma-identities}
\end{equation}
Thus $H^\circ$ is unchanged on $L=0$ edges in $B,C$.  It is unchanged
on $c$-leg edges as well: those have $w=1,k=-d$, and $a=A_C$ for
a high first port, whose parent color is $\mathfrak B$, and $a=A_B$
otherwise.  The table bounds all other parts of the tree: an unpassed
$p$-leg kills terms; the middle chain kills a complete prefix in at most
two steps; and an $L=0$ tail has the two-step rule
\eqref{four:tail}.  The $c$-leg is the only place where the uncorrected
variables can grow, and the correction is constant there.  This proves
the uniform bound for $H^\circ$, including atypical trees.

At a node incident to a $c$-leg edge, the $B,C$ line invariants,
anchored by \eqref{four:marked-values}, intersect at $P_0GT$ when $d>1$.
The four line directions after division by $P_0G$ are
\begin{equation}
 1,\quad -y^{-1},\quad y^2,\quad -y
 \qquad\text{on }A,B,C,D,
 \label{four:directions}
\end{equation}
as follows from $iA_R/(P_0G)$ up to nonzero real factors.  At $d=1$
the same identity follows by continuity for each fixed finite tree.
The descendants beyond a $c$-leg edge lie in its $B,C$ boundary disk;
its $L=0$ side branches therefore preserve this value.

The gap walk along $B$ goes from the end of the $b$-leg to the end
of the $c$-leg.  Its $L=0$ excursions preserve $H^\circ$, so only
type-$X$ middle nodes need checking.  Similarly only type-$Y$ nodes
need checking for $C$.  At node $0$,
\begin{equation}
 H^\circ/(P_0G)=F\quad(c_0=\mathfrak A),\qquad
 H^\circ/(P_0G)=yF\quad(c_0=\mathfrak B).
 \label{four:corrected-branch}
\end{equation}
After the first zero $k$, the value stays at the far-branch value.
If the first $k$ is nonzero, the first jump is $-y^{-1}$ in type $X$
with $c_0=\mathfrak A$, and $-y^2$ in type $Y$ with
$c_0=\mathfrak B$.  Thus node $1$ already has value $T$ in normalized
units.  These are precisely the alternatives in the statement.
\end{proof}

\subsection{Moments, shielding, and disappearance of the diagonal term}

\begin{lemma}[Compactness and boundary shielding]
\label{four:compactness}
For every fixed positive power, the moments of $|H_{V(z)}|+|D_{V(z)}|$
are bounded by a power of $\min(1,|z-c|)^{-1}$, uniformly when $z$
is farther than a mesh-scale distance from $c$.  The expectations in
\eqref{four:observable} are asymptotically equicontinuous on interior
compact sets.  Every lattice parity therefore has the same locally
continuous subsequential limits.

Typicality has probability tending to one.  As a macroscopic interior
point approaches an ordinary boundary point, including a polygon corner,
its tree vertex visits that point's interval with probability tending to
one after the mesh limit.  Near $b,c,d_0$ it lies beyond an edge of the
corresponding terminal leg with high probability; near $p$ an unpassed
$L=3$ edge remains with high probability.
\end{lemma}

\begin{proof}
The table and the two-step cancellation bound $|H_V|+|D_V|$ by
$C\exp(C\ell)$, where $\ell$ is the number of $c$-leg edges passed
on the root-to-$V$ path.  A passed $c$-leg arc has a child disk containing
$z$.  It cannot be wholly inside a box at $c$ of radius a small fixed
multiple of $\min(1,|z-c|)$, because the flat wall makes such a child
disk local.  Fixed buffers and the dyadic bands between that radius and
macroscopic scale count these passages.  The conditional passage
estimates of Lemmas~\ref{prelim:passages} and~\ref{prelim:tilts}, grouped into disjoint
buffer colors, give the asserted moments.  Bounded open-cap reweighting
allows their use under a deterministic-cap FK law.  If two nearby
interior vertices have different boundary-tree regions, a raw boundary
arc travels from their vicinity to the wall.  Bulk RSW circuits make
this probability uniformly small; H\"older's inequality and the moment
bound give equicontinuity, including between lattice parities.

For the boundary claims, work on separated scales in the relevant wall
sector.  In disjoint buffers impose two actual wall-to-wall crosscuts,
of opposite colors, the first inside the second.  Up-to-wall RSW gives
a uniform conditional lower bound.  A raw medial separator lies between
the two thickened paths and joins the two banks.  One direct verification
is to trace the perimeters: on either bank the gaps at the two crosscut
ends have opposite colors, so an odd number of ports lies between them;
the paths prohibit radial escape, and at least one of these ports must
be paired to the other bank.  Such an arc separates the deeper box
from the exterior.  The same argument works in a corner sector.
Iteration gives probability tending to one at diverging scale ratio.
It produces the claimed local raw arch at an ordinary point and the
appropriate terminal-leg separator at each mark.
\end{proof}

\begin{lemma}[Diagonal residual cancellation]
\label{four:residual}
For every bounded function $\gamma$ of the two colors and every boundary
point $u$ distinct from the four marks,
\begin{equation}
 \lim_{z\to u}\limsup_{n\to\infty}
 \left|\mathbb E_{\rm open}
       [\gamma(\operatorname{col}(V(z)))D_{V(z)}]\right|=0.
 \label{four:boundary-residual}
\end{equation}
Here $z$ in the outer limit is a macroscopic interior location, and any
lattice vertices tending to it may be used in the inner limit.
\end{lemma}

\begin{proof}
We first cancel a positively weighted residual at a fixed interior $z$.
Replace the marked primitive by the full $K(Q)$ string from $p$ to $z$.
If $N$ is the number of internal closed loops surrounding $z$, their
weight ratio is $-2/d$.  Let $\sigma$ take opposite signs on the two
colors.  Nesting parity gives
$(-1)^N=\sigma(\operatorname{col}(z))\sigma(\operatorname{col}(V))$.
Equality of the two winding expansions implies
\begin{equation}
 \mathbb E_{\rm open}
       [\sigma(\operatorname{col}(V))D_V(2/d)^N]=o(1).
 \label{four:weighted-residual}
\end{equation}
Here is the required absolute estimate in the wrong expansion.
Relative to the complementary cap partition it is at most
\begin{equation}
 K_*^{1+X_*}
 \prod_{\nu=p,b,d_0}\mathbf1_{\{N_\nu=0\}}\,
 d^{N_c}\prod_{v\text{ a corner}}f(g_v)^{N_v/2},
 \qquad g_v\in\{-\tfrac12,\tfrac12\}.
 \label{four:wrong-bound}
\end{equation}
The counts use disjoint small macroscopic cutoffs.  The variable $X_*$
counts macroscopic passages and has every fixed exponential moment;
its constants may depend on $z$.  Away from singularities, ordinary
small arcs have the complementary zero-drop cap weights.  At $b,d_0,c$
the effective wrong drops are $-s,-s,-t$, with weights $0,0,d$.
At $p$, use a locally forward boundary interval.  Its arc meets the
exiting string with net intersection $-1$; after changing the winding
convention the effective drop is $s+1$, again giving weight zero.
This depends on the net intersection, not the placement of the string.
Corner alternation and cap elimination give the remaining factors in
\eqref{four:wrong-bound}.  Internal loops around $z$ have absolute
weight $2|\cos(2\lambda)|\le d$, and each nonlocal arc costs at most
a bounded factor per passage.

Without the $K_*^{1+X_*}$ factor, the logarithm of the expectation is
bounded above by
\begin{equation}
 3P+D_*+\sum_{v\text{ convex}}C_{i(v)}
          +\sum_{v\text{ concave}}L_{i(v)}+O(1).
 \label{four:residual-budget}
\end{equation}
Its second moment is polynomial in $n$.  H\"older with exponent slightly
above one absorbs $X_*$ at an arbitrarily small loss in the coefficient
of $\log n$, by Lemma~\ref{prelim:absorb}.  The balanced-corner budget
\eqref{exp:polygon-saving} is strictly negative by
\eqref{exp:strict-arm}.  Our global color choice makes the complementary
cap partition at most a constant times the correct open-cap partition.
This proves \eqref{four:weighted-residual}.

We next remove the positive weight, retaining relative error estimates.
Let $\mu$ be the correct deterministic-cap FK law and
$R_*=d^{\ell_{\rm open}-\ell_{\rm det}}$; open expectation is
$\mathbb E_\mu[R_*\cdot]/\mathbb E_\mu R_*$.  For the interval $R$
containing $u$, put
\begin{equation}
 B_*=R_*\sigma(\operatorname{col}(V))
             d^{-g_R(\operatorname{col}(V))}D_V,
 \qquad
 T_*=d^{g_R(\operatorname{col}(V))}(2/d)^N.
 \label{four:BT}
\end{equation}
Thus $\mathbb E_\mu[B_*T_*]=o(1)$ at every fixed $z$.
Choose scales in the regular sector at $u$ with
\begin{equation}
 |z-u|\ll r_1\ll r_2\ll r_3\ll1,
 \label{four:residual-scales}
\end{equation}
all ratios diverging as $z\to u$, and take the mesh limit first.

Except on an event of vanishing probability, $B_*$ is determined outside
$r_2$.  To see this, use actual screens to ensure that no raw strand
from the $r_2$ rim reaches $r_3$.  Every changing boundary arc then lies
in one small part of $R$, so has $L=0$.  All $L>0$ data, the marked
subtree and $R_*$ are fixed.  Regard arcs touching the connected patch
as affected and arcs wholly outside it as stable.  Stable separators
either separate the entire patch from $p$ or do not; noncrossing puts
the affected on-path arcs after them.  Off-path factors cancel, while
the on-path $k/w$ suffix telescopes by \eqref{four:tail}.  The color
factor in $B_*$ cancels that suffix exactly.  Set $B_{\rm out}=B_*$
on this exterior good event and zero otherwise.

Truncate $T_*$ to $T_{\rm in}$ by counting only enclosing loops wholly
inside $r_1$ and reconstructing the tree color from their parity and the
fixed lattice color of $z$.  This is exact unless a raw path runs from
$O(|z-u|)$ to the $r_1$ scale: every enclosing loop meets the segment
from $z$ to its nearest wall point.  To measure both truncation errors,
let $N_0$ count the enclosing loops wholly inside the box of radius
$\operatorname{dist}(z,\partial\Omega)/8$ at $z$, and put
$U_z=(2/d)^{N_0}$.  Work under the positive tilt
$\nu_{U_z}\propto U_z\mu$.  The rare events remain rare under this
tilt by conditional RSW outside its support.  The variables $B_*$,
$T_*/U_z$, and $T_{\rm in}/U_z$ have uniformly bounded fixed moments
as $z\to u$.  For $B_*$, every relevant $c$-leg arc gives a macroscopic
passage away from $u$; an arc wholly local at $u$ would have $L=0$.
Every extra loop in $T_*$ gives a passage in a band at distance comparable
to $\operatorname{dist}(z,\partial\Omega)$, by the segment observation
and its escape from the cutoff.  Finitely many bulk and sector buffers
cover this band uniformly.  Near a corner, use a corner buffer and small
straight-bank buffers, or only the latter if the corner is farther away
than this scale.  The tilt-edge passage estimates apply in each buffer.
H\"older therefore makes the product truncation errors
$o(\mathbb E_\mu U_z)$, and
\begin{equation}
 \mathbb E_\mu T_{\rm in}\asymp\mathbb E_\mu U_z\ge1.
 \label{four:T-comparison}
\end{equation}

The inward relative comparison of Lemma~\ref{prelim:relative-collar}
between $r_1$ and $r_2$, applied
to this positive inner factor and the absolute parts of $B_{\rm out}$,
gives
\begin{equation}
 \frac{\mathbb E_\mu[B_{\rm out}T_{\rm in}]
 -\mathbb E_\mu B_{\rm out}\,\mathbb E_\mu T_{\rm in}}
 {\mathbb E_\mu U_z}=o(1).
 \label{four:relative-decoupling}
\end{equation}
First use \eqref{four:weighted-residual} at fixed $z$, then let $z\to u$.
Equations~\eqref{four:T-comparison}--\eqref{four:relative-decoupling}
show that $\mathbb E_\mu B_*$ vanishes in this order of limits.
Finally apply the same separation without a tilt to $B_*$ times any
bounded function of the tree color, reconstructing that color from the
inner-loop parity.  Its expectation factors up to $o(1)$ into
$\mathbb E_\mu B_{\rm out}$ and a bounded inner expectation.
Taking the bounded color factor that reverses the extra factors in
\eqref{four:BT}, and dividing by $\mathbb E_\mu R_*$, proves
\eqref{four:boundary-residual}.
\end{proof}

\subsection{The boundary problem, conditional on holomorphicity}

\begin{proposition}[Identification of a holomorphic subsequential limit]
\label{four:conditional-map}
Suppose an interior subsequential limit $h$ of $h_n$ is holomorphic in
the ordinary complex coordinate, and pass simultaneously to a limit
$r$ of $P_{\mathfrak A}$.  Then $0<r<1$.  Put
\begin{equation}
 U_0=r+d(1-r),\qquad V_0=dr+1-r,\qquad T_0=r+y(1-r).
 \label{four:target-vertices}
\end{equation}
The function $h$ maps $\Omega$ conformally onto the interior of the
simple polygon with successive vertices $0,U_0,T_0,yV_0$, taking
$p,b,c,d_0$ to those vertices.  If a conformal half-plane coordinate
takes the marks to $0,\chi,1,\infty$, respectively, then
\begin{equation}
 r=\frac{\displaystyle\int_1^\infty
        u^{\rho-1}|u-\chi|^{\rho-1}|u-1|^{1-3\rho}\,du}
       {\displaystyle\int_\chi^\infty
        u^{\rho-1}|u-\chi|^{\rho-1}|u-1|^{1-3\rho}\,du}.
 \label{four:crossing-formula}
\end{equation}
Under deterministic separate $\mathfrak A$ caps the corresponding
crossing probability is $r/(r+d(1-r))$.
\end{proposition}

\begin{proof}
Fixed up-to-wall RSW corridors give both actual crossings positive
probability under deterministic caps, and hence under open caps;
thus $0<r<1$.  Shielding and the moment bounds give the interior
limits $0,U_0,yV_0$ at $p,b,d_0$: near $p$ an unpassed $L=3$ edge
kills $H$, and near $b,d_0$ the first $k=0$ edge fixes it at its marked
value.  At every other boundary point the line invariant, shielding,
and Lemma~\ref{four:residual} put $h$ on the corresponding line of
direction \eqref{four:directions}, with offsets fixed by these values.
In a conformal half-plane coordinate, whose boundary extension is
homeomorphic, the line-normal harmonic component vanishes continuously
on each unmarked interval.  Harmonic Schwarz reflection, followed by
extension of its harmonic conjugate, extends $h$ holomorphically across
that interval.

On $B,C$ the traces lie on the actual segments
$[U_0,T_0]$ and $[T_0,yV_0]$, respectively.  Indeed $H^\circ$ has,
for each connection event, only the two values in
Lemma~\ref{four:marked}; the segment direction is fixed, so averaging
keeps it between the averaged endpoints.  Its correction has zero
limiting expectation at every fixed boundary point.  As that point
approaches $c$, shielding gives a $c$-leg separator enclosing it and
nearby interior points.  The uniform bound for $H^\circ$ therefore
makes the boundary trace tend to $T_0$.

There is also continuity at $c$ from the interior.  In a half-plane
coordinate $\zeta$ with $c=0$, the flat wall and the moment bounds
give polynomial growth of $h-T_0$.  Its two boundary directions are
$-y^{-1}$ on the negative and $y^2$ on the positive real axis.
Consequently
\begin{equation}
 e^{-2i\lambda}\zeta^{3\rho-1}(h(\zeta)-T_0)
 \label{four:corner-reflection}
\end{equation}
has real boundary values on both sides, with the branch in the upper
half-plane.  Schwarz reflection makes it holomorphic on a punctured
disk; polynomial growth makes its singularity at worst a pole.
Its leading Laurent power, after undoing the multiplier, must have
positive exponent because the two boundary traces tend to zero.
Thus $h\to T_0$ throughout the interior approach as well.

The target polygon is counterclockwise and simple.  For $d>1$ its
vertex $T_0$ is reflex: in the basis $1,y$ it lies strictly inside the
triangle with vertices $0,U_0,yV_0$.  When $d=1$, it lies on the
third side of that triangle.  For any point off the four boundary
lines, straighten each continuous boundary trace in its own line.
Its winding number is that of the target polygon.  The argument
principle therefore gives one preimage of every generic interior
point and none of every generic exterior point.  The open mapping
theorem puts the open image inside the polygon, and then the traces
on $A,D$ also lie on their actual segments.  The segment control on
$B,C$ already established is needed at the reflex corner.  No interior
point lies on the trace, and the preimage count proves conformal
bijectivity.

The four target angles are
$\rho\pi,\rho\pi,(2-3\rho)\pi,\rho\pi$.  The
Schwarz--Christoffel formula therefore has derivative proportional to
$\zeta^{\rho-1}(\zeta-\chi)^{\rho-1}(\zeta-1)^{1-3\rho}$;
see, for example, \cite{Pommerenke1992}.  The $B$ and $C$ side vectors
are $-(1-r)y^{-1}$ and $ry^2$, so their lengths are $1-r$ and $r$.
Their ratio gives \eqref{four:crossing-formula}.  At $d=1$ the straight
marked point has exponent zero, as required.  Lemma~\ref{four:tree}
reweights the two open-cap alternatives by $d$ and $d^2$, giving the
last assertion.
\end{proof}

For $q=1$, one has $d=1$, $\rho=1/3$, and the substitution $v=\chi/u$
turns \eqref{four:crossing-formula} into Cardy's integral
\cite{Cardy1992}.  No boundary-law reweighting of actual paths is then
needed.  Holomorphicity and the boundary approximation transfers are
still required before this conditional computation becomes a scaling
limit statement.

\section{Paired walls and distant charges}
\label{sec:pairedwall}

We now put a source primitive on each bank of the polygon wall.  The
purpose of the exterior source is to leave a nonconstant second factor
after applying a mixed bulk estimate; no conformal invariance of the
exterior probability law is used.

\subsection{The paired-wall tensor and its normalization}

Let $O=\mathbb C\setminus\overline\Omega$.  At every used wall port,
replace the ordinary spin contraction by the product of two boundary
factors.  Use the same factor $b_i$ on the interior and exterior banks,
each evaluated on the arrow sign relative to that bank's inward normal.
The exterior designated color is opposite to the interior one.
Starting at $p$, the exterior order is $D,C,B,A$, and the high statuses
on the two banks agree.  Its physical normal lift is
$\theta_O=\theta_\Omega+\pi$, has total turn $-2\pi$, and has raw
bases $H_R-\tfrac12$.

Take the two source primitives from $p$ to $z\in\Omega$ and from $p$
to $u$ in an exterior collar strip along $D$.  The strip is connected
and simply connected and approaches both $p$ and $d_0$; initially
we restrict $u$ to its interior compact sets.  Use the respective
inward initial lifts.  A \emph{pure string of charge $a$} along a
directed tile path contributes $e^{i\lambda ac}$ at a crossed spin,
where $c=1$ denotes an arrow in the path's left-normal direction.
Add the following strings:
\begin{enumerate}[label=(\roman*),leftmargin=*]
\item In exterior boundary order put
\begin{equation}
 P_D=0,\quad P_C=-2s,\quad P_B=-2(s+t),\quad P_A=-2(2s+t).
 \label{wall:short-charges}
\end{equation}
For each nonzero $P_R$, run a charge-$P_R$ string forward, parallel
to its interval in boundary homotopy, with its endpoints at the interval
ends and its interior off the wall.  Sliding this path out through a
complete wall adds $P_R$ to that interval's raw base.
\item Choose two finite sets of positive numbers $a_j$, each with sum
$1$, satisfying $a_j<1$ and $(1+a_j)\lambda<\pi/2$.
Run charge-$(-a_j)$ strings from $p$ to exterior bulk vertices $w_j$,
at successively separated distances
\begin{equation}
 r_j\asymp R_0L^{j-1}
 \label{wall:distant-scales}
\end{equation}
from a fixed pole $z_*\in\Omega$.  Their initial portions follow a
common simple cut from $p$ out to distant scales before splitting;
they never return near the polygon.  Continue this cut notionally to
infinity, and back through $\Omega$ to $z_*$.
\end{enumerate}
First fix a sufficiently large $L$, then let $R_0$ become large after
the mesh limit.  Exterior probe paths stay on the starting $D$ side
of the cut and avoid the short strings and their sliding homotopies.
To arrange this, choose thin topological boundary strips separated
at the marks by sector curves, reserve the $D$ strip for probes, and
send the cut out at $p$.  Each fixed geometry admits piecewise smooth
paths with clear corridors and then tile-edge approximations.

At shared initial tile edges use ordered parallel lanes, with pure
strings on the non-probe side when they separate.  This specifies the
matrix order: the lanes cross the same smooth raw strand near a port
midpoint, and diagonal phases act on the specified side of $L_-$.
Near tile vertices they can be joined without extra strand crossings.
Pure lanes need no mutual ordering.  Source paths may be chosen in
fixed homotopy with continued lift throughout their domains, also when
small wall gaps are present away from the marks and these corridors.

For a positive reference law, cap each designated primal interval
separately on each wall component.  At a fragment tip the designated
color changes between banks.  Write $\mu$ for this deterministic-cap
FK law, and $Z^{\rm cap}$ for its all-loop partition relative to the
uncut plane vacuum, including common tile coefficients.  If
$\Xi(z,u)$ denotes the vacuum spin expectation of the boundary factors,
sources, and pure strings, the winding expansion gives
\begin{equation}
 \frac{\Xi(z,u)}{Z^{\rm cap}}
       =\mathbb E_\mu\mathcal O(z,u),
 \label{wall:diagram-normalization}
\end{equation}
where $\mathcal O$ is the oriented diagram sum divided, configuration
by configuration, by the deterministic-cap loop weight.  All changes
are finite at each mesh.  The almost-sure finiteness of vacuum loops,
or the finite-cylinder construction of Section~\ref{sec:angular},
justifies this identity even in the plane; unchanged remote loops have
ratio one after orientation summation.

For each $j$, let $N_j$ count closed raw loops surrounding $w_j$ wholly
inside a box of a fixed small relative radius at scale $r_j$.  Set
\begin{equation}
 U=\prod_j u_+(a_j)^{N_j},\qquad
 u_+(a)=\frac{\cos((1+a)\lambda)}{\cos\lambda}>0.
 \label{wall:U}
\end{equation}
The patches are disjoint and buffered.  For gapped walls a further
positive tip factor $T_g$ will be introduced in
Section~\ref{sec:closinggaps}; it equals $1$ for a complete wall.
The normalized paired-wall function is
\begin{equation}
 \mathcal H_g(z,u)=
 \frac{\Xi_g(z,u)}{Z_g^{\rm cap}\mathbb E_{\mu_g}[UT_g]}.
 \label{wall:H-normalized}
\end{equation}

\subsection{The ideal exterior observable and its phases}

With a complete wall, the capped interior and exterior are independent
by the loop weights and domain Markov property.  The interior factor
is $h_n(z)$ times $c_*P_0G$ and the bounded ratio of its open and
deterministic-cap partitions.  The ideal exterior object is the same
four-change disk-tree observable on $O\cup\{\infty\}$, viewed after
inversion about $z_*$: its boundary order is $D,C,B,A$, its bases in
that order are $(s+1,1,s,0)$, its launch is $p$, and it uses the exterior
high designation.  Raw boundary arcs form a finite disk tree, since
none escapes to infinity almost surely.  This definition needs no
tiles at infinity and does not change the exterior positive law.
Let $J_n(u)$ be its open-law expectation normalized by $c_*P_0G$
as in \eqref{four:observable}.

The tree table gives local boundedness and asymptotic equicontinuity of
$J_n$ in the probe strip.  A relevant nonlocal arc has a macroscopic
passage near the bounded wall, and separating nearby probe points
requires a boundary-reaching arm.  Shielding at the flat marks gives
limit zero at $p$ and a value bounded away from zero at $d_0$.
Consequently every locally uniform subsequential limit $J$ is continuous
and nonconstant on this connected strip.

The physical phases agree with the ideal ones for bounded open arcs.
First slide the short strings onto the complete wall, avoiding the
source path.  The new raw profile $H_R-\tfrac12+P_R$ differs from the
desired exterior profile by the global constant $-s-\tfrac32$.
Since there is one outward source, this contributes only a fixed phase.
Let
\begin{equation}
 \psi=\arg\left(\left(\frac1{z-z_*}\right)'\right)
 \label{wall:inversion-angle}
\end{equation}
with the cut and continuous lift just chosen.  The inverted normal is
$\theta_O+\psi$ and has total turn $2\pi$: on the clockwise exterior
lap, $\psi$ gains $4\pi$ before the cut is reset.  An oriented arc's
image turn is its physical turn plus the endpoint $\psi$ difference,
minus $4\pi$ times its signed intersections with the outward cut.
Boundary normal factors cancel the endpoint terms.  For a marked arc,
the two departing pieces have additional start terms, canceled exactly
by the change of the gauge $e^{i\rho\alpha}$ at the source.
Prefix intersections are unchanged topologically.  Before the distant
strings split, their total charge is $-2$, which supplies precisely the
remaining image multiplier.  Thus the physical and ideal open-arc
factors agree up to one common phase whenever those arcs stay before
the splitting scales.  This is a calculation with turn lifts of smooth
disjoint drawn arcs, independent of any invariance of their law.

\subsection{A nonzero distant factor}

For a closed exterior loop $\gamma$, let
$A_\gamma=\sum_{w_j\text{ inside }\gamma}a_j$, using its physically
bounded inside.  Such a loop surrounds $p$ exactly when it surrounds
the whole wall.  Its ratio to the ordinary weight is
\begin{equation}
 \begin{cases}
 v(A_\gamma)=\cos((A_\gamma-1)\lambda)/\cos\lambda,
       &\gamma\text{ surrounds the wall},\\
 u(A_\gamma)=\cos((1+A_\gamma)\lambda)/\cos\lambda,
       &\text{otherwise}.
 \end{cases}
 \label{wall:closed-ratios}
\end{equation}
These formulas follow by counterclockwise intersection with the strings.
A closed loop cannot separate the source of a nonzero open-arc term
from the connected wall.  Thus \eqref{wall:closed-ratios} lists all
changes to its weight, including when some open arcs go far.
Let $Y$ be the product of these ratios; only finitely many are nontrivial.

\begin{lemma}[Distant normalization]
\label{wall:distant}
For sufficiently large fixed $L$, under the exterior positive tilt
$\nu_U\propto U\mu$,
\begin{equation}
 \mathbb E_{\nu_U}(Y/U)\ge c_L>0,\qquad
 \mathbb E_{\nu_U}|Y/U|^p\le C_{p,L}
 \label{wall:distant-moments}
\end{equation}
for every fixed $p>0$, uniformly for large $R_0$ and fine mesh.
\end{lemma}

\begin{proof}
Between consecutive endpoint scales use corridor annuli with radii
$Kr_j$ and $r_{j+1}/K$, for a large fixed geometric constant $K$.
Each winding loop wholly in corridor $j$ has ratio
$v(\sum_{i\le j}a_i)\ge1$, because the total charge magnitude is $2$.
Let $M_j$ be the product of these ratios and $M=\prod_jM_j$.
Besides the $U$ nests and these corridor loops, all nontrivial loops
are counted by a variable $X$ having every fixed exponential moment
uniformly under $\nu_U$, with constants independent also of sufficiently
large $L$.  Indeed a loop excluding $z_*$ but surrounding an endpoint
outside its local cutoff has a passage at that endpoint scale, since it
separates the endpoint from $z_*$.  A loop winding about $z_*$ and
enclosing some but not all endpoints either lies in its corridor or
passes a bordering scale.  Fixed box and shell covers, disjoint from
other endpoint buffers, and the tilt-edge passage estimates give the
claim.  Therefore, for a fixed $C$,
\begin{equation}
 |Y/U|\le MC^X,\qquad
 Y/U\ge MC^{-X}\quad\text{off }\mathcal B,
 \label{wall:MY}
\end{equation}
where $\mathcal B$ is the event that some loop excluding $z_*$ encloses
at least two endpoints.  Off $\mathcal B$, all remaining ratios belong
to a fixed finite positive set.  Moreover
\begin{equation}
 \mathbb E_{\nu_U}M^b\le C_bL^{C_b}\qquad(b>0\text{ fixed}).
 \label{wall:M-moments}
\end{equation}
To prove this, use $O(\log L)$ dyadic passage bands per corridor,
group disjoint buffers by colors, and apply H\"older; those buffers
avoid the supports of the endpoint tilts.

If a bad loop has smallest enclosed endpoint index $j$, it crosses
corridor $j$ radially, and hence excludes every winding loop counted
by $M_j$.  Write $\mathcal B_j$ for this event.  Successive actual
circuit blockers give
$\mathbb P(\mathcal B_j\mid U\text{ patches},(M_i)_{i\ne j})
\le CL^{-\alpha}$, uniformly.  On $\mathcal B_j$ one has $M_j=1$.
First apply this conditional estimate to
$\prod_{i\ne j}M_i$, then H\"older with exponent $1+\delta$ close
to one to absorb $C^X$, interpolating against the polynomial moment
bound \eqref{wall:M-moments}.  For some $\alpha'>0$ independent of $L$,
\begin{equation}
 \mathbb E_{\nu_U}[MC^X\mathbf1_{\mathcal B}]
       \le C'L^{-\alpha'}\mathbb E_{\nu_U}M.
 \label{wall:bad-relative}
\end{equation}
The same interpolation, using arbitrary fixed exponential rates for
$X$, gives for every $\varepsilon>0$
\begin{equation}
 \mathbb E_{\nu_U}[MC^{-X}]
       \ge c_\varepsilon L^{-\varepsilon}\mathbb E_{\nu_U}M.
 \label{wall:positive-relative}
\end{equation}
Equivalently, Jensen under the $M$ tilt and the exponential bounds
give an arbitrarily small coefficient of $\log L$ in its mean of $X$.
Take $\varepsilon<\alpha'$ and then $L$ large.  Equations
\eqref{wall:MY}--\eqref{wall:positive-relative}, and $M\ge1$, prove
the positive lower bound in \eqref{wall:distant-moments}.  Its moment
bound follows from \eqref{wall:MY}, \eqref{wall:M-moments}, and H\"older.
\end{proof}

\subsection{Absolute source bounds and factorization}

We record the source-path placement used here and in gap closure.
Both paths can be taken in regular simple tubes, on their nominal
sides of the wall except at the flat launch.  For an absolute estimate,
temporarily slide the exterior pure-string endpoints at $p$ a small
macroscopic distance toward the non-probe, $A$ side of the same solid
wall.  Keep all gaps and other marks farther away, and include the
charge along the traversed bank interval as diagonal port phases.
This is an equality of the summed pairing expression: ordered lanes
allow a homotopy onto the bank without crossing a source path, and
ice conservation gives the port phases.  Pure lanes may cross one
another.  Charges still count from their original endpoints, so no
extra charge is created at a temporary slide-end.  The paired factors
and the conventions used for gluing do not change.

After this temporary slide each tube has $p$ in the relative interior
of a flat entrance face, with a clear portion on both sides; its target
is deep inside, and its other sides avoid the wall, strings, and the
other tube.  Finitely many such choices cover compact probe sets.
For each pairing, choose a path that crosses minimally the disjoint
raw crosscuts of its tube.  A sufficiently small separating crosscut
cuts off $p$ locally on the entrance face: the small disk on one side
cannot contain the deep target, so must contain the launch.  Because
the entrance is solid wall, it is an entire launch-straddling raw arc,
not a short return of a larger strand.  Every other traversed crosscut
gives a macroscopic subpassage, even if one full strand visits the tube
repeatedly.  This construction also avoids entering tiny ordinary
boundary-arc disks elsewhere.

The local launch weights are decisive.  After the short strings and
distant total charge are counted, the effective unpassed raw drop at
$p$ is $-s$.  Passing the exiting source prefix makes it $-s-1$, with
the $Q^{-1}$ minus sign.  Thus small nested launch arches have weight
zero while unpassed and unit absolute weight while passed; only the
last can mark a nonzero term.  The other marked drops are $s,t,s$,
with local unit weights.  Ordinary same-bank arcs cancel local caps,
and a physical corner without a wire change has zero-drop staggering.
All remaining factors cost at most a constant per macroscopic passage.
Consequently the source sum, divided by $U$ and the deterministic loop
weights, is bounded by
\begin{equation}
 |Y/U|\,C^{1+X_0},
 \label{wall:absolute-source}
\end{equation}
where $X_0$ counts passages near the bounded wall and probe corridors
and has uniform fixed exponential moments.

\begin{proposition}[Complete-wall factorization]
\label{wall:factorization}
For the complete wall, there are probe-independent complex numbers
$a_n(R_0,L)$, bounded above and away from zero in modulus for the fixed
large $L$, such that
\begin{equation}
 \mathcal H_{\rm full}(z,u)
       =a_n(R_0,L)h_n(z)J_n(u)+o(1)
 \label{wall:factorization-eq}
\end{equation}
uniformly on interior compact probe sets.  The error tends to zero
with mesh first and then $R_0\to\infty$.  No limit of $a_n$ is asserted
or needed.
\end{proposition}

\begin{proof}
Under $\nu_U$, bulk RSW circuits in the empty scales between the wall
and the first endpoints make the probability that a raw open arc reaches
the splitting scales tend to zero.  On its complement the phase comparison
above is exact.  Lemma~\ref{wall:distant} and
\eqref{wall:absolute-source} make the error negligible even on the
exceptional event, by H\"older.

Choose intermediate radii
$1\ll R_1\ll R_2\ll R_0$.  The ideal tree integrand, including its
open-to-deterministic reweighting, is readable inside $R_1$ when its
boundary arcs stay there; set it to zero otherwise.  Similarly $Y/U$
is readable outside $R_2$ unless a raw path joins that scale to the first
endpoint bands: new loops at smaller scales have trivial ratios.
All product truncation errors vanish under $\nu_U$ by the moment
estimates and RSW.  The inward relative comparison across the intervening
bulk annuli then factors their expectation up to $o(1)$.  Applied to
absolute parts, that comparison also replaces the inner tilted
expectation by the unweighted one.  The exterior coefficient is the
fixed phase times $\mathbb E_{\nu_U}(Y/U)$ and the bounded tree/cap
normalizations.  Interior-exterior independence completes
\eqref{wall:factorization-eq}; Lemma~\ref{wall:distant} bounds the
coefficient away from zero.
\end{proof}

\section{Closing finitely many wall gaps}
\label{sec:closinggaps}

Reflection cuts will require vacancies in the wall.  The following
comparison removes them without estimating a signed observable by an
additive error in its partition function.

Choose finitely many separated gap locations away from the marks and
source or string corridors.  Remove intervals of lengths comparable
to $h$; at a genuine corner remove such intervals on both adjacent
sides.  Their endpoints are lattice vertices.  At each tip, let
$N_{\rm tip}$ count raw arcs joining the two banks of that ray around
its tip, wholly inside a sufficiently small cutoff box of scale $h$.
Use disjoint buffered boxes, and set
\begin{equation}
 T_g=\prod_{\text{tips}}(\cot\lambda)^{N_{\rm tip}}.
 \label{gap:tip-tilt}
\end{equation}
All these factors are positive.  Other gaps may already be present at
fixed sizes and separated locations; include their analogous factors.

\begin{proposition}[Gap closure]
\label{gap:closure}
Keep the distant charges and their locations fixed.  Let $g$ denote
the gapped wall and $f$ the wall obtained by closing the specified
$h$-gaps, retaining any other gaps and tip factors.  There are positive
probe-independent numbers $b_n$, bounded above and below, such that
\begin{equation}
 \mathcal H_g(z,u)=b_n\mathcal H_f(z,u)+o(1)
 \label{gap:comparison}
\end{equation}
uniformly on interior compact probe sets.  The error vanishes in the
mesh limit followed by $h\to0$.  Constants may depend on the fixed
remaining features.  In particular, a second, smaller collection of
gaps may be closed before the first collection.
\end{proposition}

\begin{proof}
Use the same tile pairings in the cut conventions $g,f$.  A port
interaction or raw arc is \emph{affected} if it uses a newly missing
port; the gapped diagram joins the filled paths terminating on the two
sides there.  Write
\begin{equation}
 S=UT_f,\qquad T=T_g/T_f,\qquad
 \nu_g\propto ST\mu_g,\qquad \nu_f\propto S\mu_f.
 \label{gap:tilted-laws}
\end{equation}
The supports of $S$ are away from the closing gaps.

\paragraph{Confinement and the local multiplier.}
In the common collars at radii $h\ll r_1\ll1$, require a primal
screen joining the wall rays in the sector carrying primal wire
portions, and a dual screen in the other sector, which has no primal
through wire.  With diverging scale ratio, conditional RSW makes this
inner good event tend to probability one under $\nu_g$, even given
the remote tilts and deeper tip data.  It confines all affected paths
to these neighborhoods.  Moreover the outer primal connectivity
induced by the inner fillings is the same for $g,f$: every connection
using the deeper patch must enter the already connected primal screen,
including bypasses through its joined wire portions.  Represent the
wires by deterministic connections along their designated intervals.
The FK Euler formula now makes $\ell_f-\ell_g$ local, equal to a
difference of internal component counts plus deterministic constants.
This can first be checked with all distant edges fixed.

On this good event there is an exact local multiplier $B$ such that
\begin{equation}
 \mathcal O_g=B\mathcal O_f.
 \label{gap:pointwise}
\end{equation}
It is the ratio of the affected spin weights, multiplied by
$d^{\ell_f-\ell_g}$.  It is positive.  To check the latter assertion,
view the local patch as a disk with two radial notches ending at the
separated tips.  A same-bank arc, even one threading the gap, has its
usual staggered $d,1$ weights: simple-arc isotopy fixes its turn with
the prescribed normal ends, and its endpoint high statuses are
opposite by color parity.  Orient an opposite-bank arc from its
interior to its exterior port.  Its endpoint directions are now the
two interior normals; the high statuses agree, since the color to
the left is the preceding interior gap at both ends.  The matched
boundary factors give amplitude
\begin{equation}
 \frac{\cos(\rho(W+\theta_i-\theta_j)/2)}{\sin\lambda}.
 \label{gap:cross-bank-amplitude}
\end{equation}
For ports on the same ray, turn minus normal difference is $\pm2\pi$,
giving $\cot\lambda$.  For different rays it is zero, giving
$1/\sin\lambda$.  Start with collinear rays, drawing a path around
the finite tip or across the slot, respectively.  Bending one ray
through the intended right angle rotates its endpoint normal by the
same amount, so the conclusion persists at a corner gap.  All simple
crosscuts with the same ends have the same lifted turn by isotopy.
New closed internal loops have ordinary weight $d$, while the filled
arcs have homogeneous-interval weights.  Every local factor is positive.

We need a comparison uniform as $h\to0$:
\begin{equation}
 K^{-(1+X_h)}T\le B\le K^{1+X_h}T,
 \label{gap:B-bound}
\end{equation}
where $X_h$ is a finite sum of passage counts at scale $h$ and has
uniform fixed exponential moments under $\nu_g$.  The following
cap elimination avoids counting every straddle at intermediate scales.
In both conventions remove affected boundary arcs of diameter below
a small fixed fraction of $h$, together with their disk descendants.
In $f$ these lie on homogeneous continuous intervals, including across
the missing positions.  In $g$, a short affected arc with solid ends
is near a single tip, cutting off a disk against one continuous bank
or across that tip change.  Its disk is also short by the straight or
single-corner geometry.  No such disk spans both separated tips; at
a corner both removed side portions have lengths comparable to $h$.
Also remove, in both conventions, any common unaffected descendants
needed for either elimination.  Such descendants have both ends on a
common bank piece and the same neutral disk and intervening arcs;
their union is therefore an allowed common removal.

Eliminate innermost arcs first.  Zero-change cap increments cancel the
$d,1$ factors.  Across a single tip change the two statuses are both
high or both low; one unmatched port is transported and no loop is
created, leaving exactly the $\cot\lambda$ factor.  No whole interval
is erased by these local removals.  Remaining caps still pair adjacent
high-low ports with inherited status, apart from boundedly many end
ties.  The number of removed tip arcs differs from the tilt count by
$O(X_h)$.  Every surviving affected boundary arc has a non-short
passage at the slot scale, so their number is also $O(X_h)$, using
bulk, straight-wall, and tip buffers.  For this count, filled paths
are disjoint segments of gapped paths.  The tilt-edge moment bounds
apply also near the cutoff boxes.

Remaining common raw arcs have the same ends.  Their residual cap
pairings differ at only $O(1+X_h)$ ports: on every common solid
interval, both pair consecutive high-low ports, interrupted only by
interval ends and noncommon survivors.  Thus their boundary-loop
counts differ by at most this number, irrespective of long-range
ties.  New local closed loops cancel their ordinary spin weights.
This proves \eqref{gap:B-bound} and retains the exact positivity.

\paragraph{Absolute moments, including exceptional configurations.}
For every fixed power, both $\mathcal O_g/(ST)$ and $\mathcal O_f/S$
have bounded moments under $\nu_g$, uniformly as the indicated gaps
shrink; $\mathcal O_f/S$ has the same bound under $\nu_f$.
Choose the two simple source corridors as in
Section~\ref{sec:pairedwall}, with the ordered lanes and temporary
start-clearance slide common to both cut conventions.  On each
pairing deform the paths to minimal tube separators.  Both markers
are outward sources and lie on distinct open arcs.  Closed loops
cannot mark.  Apart from entire local launch arches, possible marked
crosscut components are counted by macroscopic passages, also for
repeated visits of a full strand.  Local launch arches are zero while
unpassed and have unit magnitude while passed, so only the last nested
candidate can mark.  This remains true when gaps open elsewhere.
Other local flips have unit weights and can be avoided by the paths.
Pure strings may meet remote arcs; at a wall mark their effects are
exactly their specified bank charges and they do not alter ordinary
small disks away from their endpoints.

The short eliminations and the residual cap comparison just used
remain valid as count estimates without confinement of long affected
paths.  Short affected arcs and loops are away from all insertions and
are neutral except for their tip factors.  Every non-short affected
boundary arc costs a bounded factor per $X_h$.  For the other boundary
weights, use the residual count of $f$ and eliminate its local disks
up to a fixed small macroscopic threshold, chosen relative to retained
gaps and feature separations.  This includes straddles at the newly
filled homogeneous locations.  Ordinary common disks cancel caps;
flip and launch disks obey the preceding rules; a retained tip gives
its designated positive factor, up to cutoff-scale passage errors.
Any affected long survivor already costs $O(X_h)$.  Thus the residual
comparison has no loss per scale between $h$ and macroscopic distance.

Internal closed ratios outside the local bulk tilt patches are trivial
unless the loop has a macroscopic passage.  For example, a loop
enclosing a marker without its start also encloses an end of the
marker's boundary arc on a solid of length bounded below, and hence
cannot be tiny.  The same observation applies to loops cutting off
starts on solids.  The other strings have only their designated local
endpoint nests.  Very large loops enclosing all finite data have ratio
one.  All remote scales, including the distant charges, stay fixed
while these gaps close.  Consequently the absolute source sums,
after their respective positive tilt factors are removed, are bounded
by
\begin{equation}
 K^{1+X_0+X_h}.
 \label{gap:absolute}
\end{equation}
Here $X_0$ is a fixed-scale passage count with every fixed exponential
moment.  Any trajectory of genuinely fixed diameter near a shrinking
slot has a fixed-size subpassage outside a small neighborhood of its
core, where the ordinary bulk or sector buffers apply.  This also
covers passages involving the newly opened wall; in $f$ use the
disjoint raw segments already noted.  Equation~\eqref{gap:absolute}
proves all the stated moment bounds.

\paragraph{Relative decoupling.}
Choose further scales
$r_1\ll r_2\ll r_3\ll1$, all ratios diverging.  There is an
exterior-readable approximation $F_{\rm out}$ to $\mathcal O_f/S$
from outside the $r_2$ patches, exact unless raw passages traverse
the common collar to $r_3$; set it to zero otherwise.  Indeed, on this
exterior good event, changing the fills inside $r_2$ affects only
local arcs against homogeneous continuous bank intervals.  Eliminate
them and their intervening arcs with no net weight and the same
remaining caps.  To make this elimination common to all fills,
remove the union of ports affected, or lying in an intervening
interval, for some fill.  Stable arcs from such an interval pair
inside it and have that same neutral disk with its descendants.
The untouched endpoints therefore see precisely the common
adjacent-pair reconnection of the deleted blocks.  Modified internal
loops are ordinary and all remaining source data are unchanged.
The moment bounds control its error under both relevant tilted laws.

Let
\begin{equation}
 D_{\rm in}=(B/T)\mathbf1_{\{\text{inner good}\}}.
 \label{gap:D-inner}
\end{equation}
It is nonnegative, measurable inside $r_1$, and has bounded fixed
moments and expectation in $[c,C]$, by \eqref{gap:B-bound} and the
high probability of the good event.  The outward two-sector case of
Lemma~\ref{prelim:relative-collar} says that the conditional law of edges outside
$r_2$, given an inner fill under $\nu_g$, differs by relative $o(1)$
from its law under $\nu_f$.  The graph and local wire representatives
in the intervening collars and farther out are common; only the inner
partition can differ.  For several patches, iterate this uniform
comparison patch by patch.  Conditioning the other interiors adds
only local ties, with their common external wires represented along
the sides.  The factor $T$ is held inside and $S$ is remote, so the
relative comparison persists after both tilts.  It follows that
\begin{align}
 \mathbb E_{\nu_g}[\mathcal O_g/(ST)]
 &=\mathbb E_{\nu_g}[D_{\rm in}F_{\rm out}]+o(1)\notag\\
 &=\mathbb E_{\nu_g}D_{\rm in}\,
       \mathbb E_{\nu_f}[\mathcal O_f/S]+o(1).
 \label{gap:decoupling}
\end{align}
The errors follow from the preceding rare-event and moment bounds.
The first factor is positive, bounded above and below, and independent
of the probes.  By \eqref{wall:H-normalized} and
\eqref{gap:tilted-laws}, the two outer expectations in
\eqref{gap:decoupling} are exactly $\mathcal H_g$ and $\mathcal H_f$.
This proves the proposition.
\end{proof}

\section{Holomorphicity from reflection and extraction}
\label{sec:holomorphic}

Assume now that the coordinates of $\Omega$ are rational in the tile
frame.  Thus its true vertical wall levels are congruent to some $X_*$
modulo a fixed $G_*>0$.  All geometric constructions in this section
are fixed before the mesh tends to zero.  We first use small compact
probe neighborhoods with the following strict margins: the two probe
heights are distinct and avoid horizontal walls and feature heights;
their abscissas avoid vertical wall levels; and the two probes are not
on a common axis diagonal.  Distant endpoints are placed at heights
outside the probe bands.  We remove these temporary exclusions at
the end.

\subsection{The mixed stencil and the gaps}

For $v=z,u$ and a tile-axis unit vector $e_k$, set
\begin{equation}
 \delta_{k,v}^{(2)}F=F(v+2n^{-1}e_k)-F(v).
 \label{hol:two-step}
\end{equation}
The relevant stencil is
\begin{equation}
 (\delta_{1,z}^{(2)}+i\delta_{2,z}^{(2)})\delta_{j,u}^{(2)},
 \qquad j\in\{1,2\}.
 \label{hol:mixed-stencil}
\end{equation}
Simple two-step extensions with a common arriving prefix express it
as a vacuum tensor expectation containing one high and one low local
insertion from Section~\ref{sec:current}.  The first is the high
combination because the two positive axis directions have relative
coefficient $i$ in \eqref{hol:mixed-stencil}.  All dependence on the
stencil terms is localized in the two small probe boxes; all other
paths are common diagonal strings.  Fix the starts, homotopies, and
lifts as in Section~\ref{sec:pairedwall}, with separate corridors and
a final incoming path compatible with both positive extensions.
These are valid tensor identities with gaps present.

The \emph{primary gaps} have widths comparable to $h$.  They include
all true corners, and openings on every vertical wall in each probe
height band.  One anchor height $y_{*,i}$ per probe box suffices.
Choose $H_i>0$, with $2H_i$ dividing $G_*$, sufficiently small that
for every probe height $y_i$ in that box,
\begin{equation}
 l_i=\frac{y_i+y_{*,i}-H_i}{2},\qquad
 y_i\in(l_i,l_i+H_i)
 \label{hol:marker-cell}
\end{equation}
with uniform margins.  The two cell intervals are disjoint and lie in
clear horizontal feature bands.  Put a primary gap at height $y_{*,i}$
on every vertical wall crossing the band.  Round its ends to lattice
vertices.  We will also use \emph{secondary gaps}, of much smaller
width $\sigma$, at finitely many points of the remaining straight
solids.  Thus every solid component used in the estimate is a separated
straight fragment.

\subsection{Reflection inequalities and angular extraction}

An axis-parallel cut through tile vertices, vacant of solids,
endpoints, and stencil boxes, gives a Hilbert-space Cauchy--Schwarz
inequality for the plane vacuum.  Each child keeps one half and adds
its conjugate, arrow-reversed mirror, with exponent $1/2$ in the
multiplicative inequality.  This is the square-row Gram construction
of Section~\ref{sec:angular}, or its finite-cylinder limit.  Diagonal
strings can cross the cut transversely at its vertices: ice
conservation reroutes any runs along the cut within a few vacant rows.
Conjugation inverts the reflected charge, while reversing the joined
path direction inverts it again.  The flux therefore continues across
the join, with no new endpoint or phase charge.

Call a stencil copy \emph{favorable} if the slope-$+1$ line through
it lies in a clear slab of fixed positive macroscopic thickness,
containing no solids, endpoints, other features, or other stencils.
Its diagonal-time projection must differ from the others'.  Parallel
diagonal cuts inside this slab, with many positive-time rows on both
sides, allow Proposition~\ref{cur:slab-estimate}, based on
Theorem~\ref{bulk:angular-estimate}, to extract the insertion with cost
\begin{equation}
 O(n^{-1-\rho})\quad\text{for a high insertion},\qquad
 O(n^{-1+\rho})\quad\text{for a low insertion}.
 \label{hol:angular-costs}
\end{equation}
This extraction retains the two external state norms.  We spell out
which path identities make those norms independent of the choice
within the stencil.  After an odd number of reflections the local
insertion is the $L_+,Q^{-1}$ alternative; its angular parameter is
still $\tau=+$, with the reflected high coefficient.  Rewrite any
backward increment locally as the negative of its forward counterpart.
The common start and end and the compensation pairs from
Section~\ref{sec:current} put both high terms on positive-step paths.
The arriving charge belongs to the background even if the original
prefix came from the future side.  In the vacant slab, funnel that
background to common seams and one skirt avoiding the entire marker
rectangle.  The difference of the original and new routings is a
closed chain whose face coefficient is the same at every prospective
source edge.  Ice conservation therefore gives one common scalar
phase, rather than term-dependent phases.  These changes stay inside
the intact stencil box and vacant band, so all other stencil sums
remain in the external states unchanged.

The two state norms are themselves reflected diagrams, now omitting
the extracted middle band and using the positive intervening
square-layer metric at the two cuts.  Their diagonal chains join at
shared level vertices with continuous flux.  Thus their exponents
are again $1/2$.  No bound on arbitrary boundary-state norms is being
assumed.  Other favorable copies to either side, and their mirrors,
remain favorable if the bands are sufficiently narrow.

Suppose a node has $k$ favorable copies.  Extract one at a median
projection.  If $k_-,k_+$ copies are strictly before and after it, its
two norm children have $2k_-,2k_+$ favorable copies.  Each is at most
$k$, and their half-average is $k-1$.  Under the successive half-weights
the count reaches zero with expected time at most the initial count.
Truncating at a sufficiently large finite depth therefore leaves an
arbitrarily small expected residual count.  Every extracted insertion
is counted once at its node; otherwise total counts are preserved in
half-average.  All cuts and extraction bands avoid the other intact
local data.

\subsection{The complete cost of the norm diagrams}

At a final leaf, bound the remaining tensor sum by an absolute winding
expansion.  Every retained straight segment, its features, and every
remote endpoint is an intact reflected copy.  In one common $+$
expansion, even-reflection copies use the original local weights,
whereas odd-reflection copies use the absolute wrong-expansion
weights.  Indeed spatial reflection and travel reversal preserve the
signed strand turn together, while conjugation reverses its phase.
For positive comparison use direct cap wires on even copies and
reflected complementary wires on odd copies, completing the disjoint
straight slits by any deterministic planar positive caps.  The local
tables can then be read in their original bank coordinates.

Ordinary local arcs have exactly their high-low cap weights.  Each
flat mark contributes both banks.  In the correct expansion the local
magnitudes at all four flips are $1$; in the wrong expansion they are
zero per local straddle at $p,b,d_0$ and $d$ at $c$, on each bank.
At $p$ use the effective raw drop $-s$ and the exiting diagonal prefix;
at the other marks use $s,t,s$.  Negation of the effective shifts on
a flat wall gives the wrong weights.  Reflection cuts preserve these
outgoing local charge data even if the strings subsequently continue
to mirrors.  Tip-cross factors on a straight fragment are $\cot\lambda$
in either absolute convention.  Endpoints of charge magnitude $a$
have local closed-loop factors $u_+(a)$ on even copies and $u_-(a)$
on odd copies.  There are no true corners on the solids.

Choose all local cutoffs smaller than the fixed cut margins.  At any
unextracted marker, a nonzero term requires its strand to reach a
solid or another marker: a closed component cannot change orientation
only once.  It is therefore macroscopic, and no tiny surrounding loop
can isolate the marker.  Using vertex instead of source-edge endpoints
changes positions by boundedly many mesh steps and costs only bounded
passage factors.  Every other nonordinary arc or loop has a macroscopic
passage.  There is no flux to infinity: each diagonal chain has total
divergence zero after its source terminations are included, and the
mirror joins create no new endpoints.  Thus a closed loop enclosing
all finite data has ordinary weight; one returning to the data while
going far is counted among the macroscopic passages.

Cap elimination therefore bounds the absolute leaf expression,
relative to its deterministic caps, by $C^{1+X}$ times the product
of the local factors just listed, where $X$ has every fixed
exponential moment.  A zero local factor means the corresponding
absence indicator.  H\"older and the positive local-tilt comparisons
absorb $C^X$ at arbitrarily small loss in the coefficient of $\log n$.
The two banks at a mark count separately: conditional on their exterior
data they are two half-sector patches separated by the solid wall,
with only outer cap ties.  Tip expectations agree up to constants with
the original positive tip normalizations.  For a complementary copy,
reflect across the ray to exchange banks and wire assignments, then
use radius and collar comparisons.  Surface partition costs of the
disjoint straight fragments compare to the products of their original
single-fragment costs; the one-step polarity shift supplies the same
comparison for complementary copies.  These are the separated-patch
and surface comparisons of Lemmas~\ref{prelim:collar} and~\ref{prelim:surface}.

Accordingly each retained feature carries its original baseline:
the vacuum-relative segment partition, the tip expectation, or
$\log F_+(a)$ for a bulk endpoint.  The extra logarithmic costs on
wrong copies are
\begin{equation}
 2P\text{ per colocated }s\text{-flip pair},\qquad
 2D_*\text{ per }t\text{-flip pair},\qquad
 \log\frac{F_-(a)}{F_+(a)}\text{ per bulk endpoint}.
 \label{hol:wrong-costs}
\end{equation}
The errors are one-sided $o(\log n)$ in the sense of
Section~\ref{sec:exponents}.  No decay from an unextracted insertion
is used; its finitely many local terms cost only a constant.

Interpret the exponents $1/2$ on the reflection tree as probability
weights.  An original segment or other feature has expected copy count
one: it is never cut or removed in an extracted middle band.  Ensure
at least one preliminary split.  In each individual child of every
split, every retained feature is paired with its opposite-parity
mirror.  Hence the final expected even and odd counts are both $1/2$
per original feature.  The baseline costs average back to
\begin{equation}
 \log\bigl(Z_g^{\rm cap}\mathbb E_{\mu_g}[UT_g]\bigr)+O(1).
 \label{hol:baseline}
\end{equation}
There are three $s$ marks and one $t$ mark, with both banks at each,
and two endpoint-charge sets of total magnitude one.  Thus the extra
averaged cost is at most
\begin{equation}
 3P+D_*+\frac12\sum_j\log\frac{F_-(a_j)}{F_+(a_j)}
 \le 2P+D_*+o(\log n)
 \le P+o(\log n).
 \label{hol:strict-budget}
\end{equation}
The first inequality uses \eqref{exp:charge-budget} twice, and the
second follows from \eqref{exp:rectangle-budget} and $C_i\ge0$.  Since
$P\le-c_0\log n+O(1)$ for some $c_0>0$, this is a strict power saving
available to pay for incomplete extraction and all small H\"older
losses.  Keeping the norm diagrams, the baseline, and the parity costs
is essential to this saving.

\subsection{Making almost all stencil copies favorable}

\begin{lemma}[Finite reflection templates]
\label{hol:templates}
For fixed primary gap widths and every $\varepsilon>0$, one can choose
finitely many reflection and extraction steps and finitely many
secondary gaps so that the expected extracted count of each original
marker is at least $1-\varepsilon$.  The cuts have strict macroscopic
clearances.  On any compact set of eligible probe pairs, finitely many
such templates suffice, and one may use the union of their secondary
gaps.  All choices precede the mesh limit.
\end{lemma}

\begin{proof}
We first work at one probe pair with ideal real cuts, before a small
generic perturbation.  Divide a bounded vertical interval containing
all data, with empty end margins, into finitely many distinctly labeled
cells, including the two marker cells \eqref{hol:marker-cell}.
Horizontal folds act on a word of cell labels.  Splitting a word of
length $m$ after position $j$ makes the two words
\begin{equation}
 WW^{\rm rev}\quad\text{and}\quad V^{\rm rev}V,
 \label{hol:word-split}
\end{equation}
where $W$ is the prefix of length $j$ and $V$ the suffix of length
$m-j$.  Their lengths are $2j$ and $2(m-j)$.

Under the size-biased weights $j/m$ and $1-j/m$, the vector of label
frequencies is a bounded martingale.  The expected terminal label
frequency multiplied by the initial word length equals the expected
terminal label count under the ordinary half-weights.  Choose $j$
within a bounded distance of $m/2$ to maximize the conditional frequency
variance.  These frequencies converge almost surely to a pure label.
Here are the details.  If the frequency vector $\pi$ stays away from
the vertices of its simplex, then some admissible near-middle prefix
has
\begin{equation}
 \|\operatorname{counts}_{\le j}-j\pi\|\ge c>0.
 \label{hol:frequency-deviation}
\end{equation}
Indeed two consecutive choices differ by a single-label vector minus
$\pi$; they cannot both have small norm.  The mixed length-two case
is immediate.  The conditional frequency variance is then at least
$c'/m^2$.

Along the size-biased chain, $\sum m^{-2}=\infty$ almost surely.
Lengths change by bounded amounts; the probabilities of its symmetric
increments are $1/2+O(1/m)$.  Both the absolute conditional drift and
conditional variance of $\log m$ are $O(m^{-2})$.  If the sum were
finite, localization and martingale convergence would make $\log m$
converge to a finite limit.  Then $m$ would be bounded, contradicting
the same assumed summability.  A bounded frequency martingale cannot
accumulate the divergent conditional variance in
\eqref{hol:frequency-deviation} while tending to a mixed limit.
Length one, if reached, is already pure.  This proves purity and hence
arbitrarily small expected impurity at a sufficiently large finite depth.

Outer-end folds retaining the whole word and its reflection can also
make its length arbitrarily large; the empty child has zero size weight.
In a branch, concentrate on its majority label if that label is a
marker.  For any prescribed finite run depth, only a proportion bounded
by that depth times the minority frequency, plus the end contribution,
fails to lie deep in a pure run of the majority.  Their expected count
can therefore be made arbitrarily small.  Adjacent equal-label cells
in the folded word alternate orientation.  In a pure run of marker
cell $i$, the vertical projection to the original cell is a triangular
wave of period $2H_i$.  It sends every marker ordinate $b'$ to $y_i$
and $b'+H_i$ to $y_{*,i}$, by \eqref{hol:marker-cell}.

We next choose vertical cuts to align the marker abscissas with the
primary gaps.  All copies of a majority marker start from the same
original abscissa $x_0$.  Its farther clear side up to a neighboring
vertical wall has width at least a fixed fraction of $G_*$; suppose
this is the left side, reflecting the following construction otherwise.
Take $H_i$ much smaller than this width.  Cut just to the right of
the current rightmost copy $a_{\max}$ and retain the left child,
including all marker columns there.  The discarded child's marker
count is zero, so this loses no expected majority count.  The left
tail below $x_0$ still projects identically.  If the cut is $c'$, the
new rightmost copy and right-tail projection are
\begin{equation}
 a'_{\max}=2c'-x_0,\qquad
 s'\longmapsto a'_{\max}+x_0-s'.
 \label{hol:tail-fold}
\end{equation}
Keep the central projection in the wall-free abscissa interval.
The first cut is close to $x_0$ on the right; subsequent distances
$c'-a_{\max}>0$ can lie in the ample leftward clearance sampled
by the old right tail.

Drift the copy abscissas modulo $2H_i$ toward
$A_*=X_*-H_i$.  Lift their distribution initially to mean $x_0$
and variance zero.  At each step, with lifted mean $\mu$, choose
$c'\equiv\mu+v\pmod{H_i}$; reflected lifts are
$2(\mu+v)-a$.  The new mean is $\mu+v$ and the variance increases
by exactly $v^2$.  Choose $Nv$ to realize the required residue
displacement to $A_*$ in the rightward direction, with bounded positive
total displacement, and then choose $N$ large.  The first cut fits
its small right clearance and the later cuts their larger available
clearance.  Chebyshev's inequality makes all but an arbitrarily small
fraction of copies have residue within a small fraction of the primary
gap width of $A_*$.  At the last step,
\begin{equation}
 a_{\max}+x_0\equiv2A_*\pmod{2H_i}.
 \label{hol:right-tail-residue}
\end{equation}

For such a successful copy $(a,b')$, consider its slope-$+1$ line.
Choose the pure-run depth larger than the entire folded horizontal
support span divided by the cell width; this finite bound is known
from the abscissa schedule before the word depth is chosen.  Throughout
the horizontal support of all data, the line then projects vertically
into the clear marker band, so meets no horizontal wall or other
feature height.  In the central portion its abscissas avoid vertical
walls.  In the left tail, every projected vertical wall lies at
$X_*$ modulo $2H_i$, and hence at horizontal offset $H_i$ from $a$
up to the small residue error.  The same is true in the right tail by
\eqref{hol:tail-fold}--\eqref{hol:right-tail-residue}.
The triangular-wave identity therefore sends the ordinate at each
such encounter into the primary gap at $y_{*,i}$.  The line has a
wall-free slab of positive thickness.  By designing first the finite
abscissa schedule and its required run depths for the fixed $h$, then
the word depth, the expected non-targeted or unsuccessful marker
counts are arbitrarily small for each of the two original markers.

Finally perturb every cut generically by a sufficiently small fixed
amount.  The finite successful inventory and strict clearances persist.
Distinct stencil copies acquire distinct time projections: for copies
with different reflection histories, their difference depends
nontrivially on the last cut where the histories differ; distinct
original markers with identical histories were separated by the
generic-coordinate assumption.  Keep cuts away from endpoints,
marked features, tips, and coincident runs along solids.  At every
transverse solid intersection insert a secondary gap at its preimage
on the original wall.  There are finitely many such preimages, away
from marks and corners after the perturbation.  Thus all intended
axis cuts become vacant.

Each template works on a neighborhood of its initial probe pair.
A finite cover of a compact support gives finitely many templates;
take the union of all their secondary gaps, then choose $\sigma$
sufficiently small.  Additional removed portions only improve the
clearances.  Round cuts to vertex lines.  These roundings and the
$O(1)$-mesh shifts of the polygon preserve all fixed margins for fine
mesh.  Apply the finite median-extraction procedure above, with a
further arbitrarily small residual count.  This proves the lemma.
\end{proof}

\begin{proposition}[Mixed derivative saving]
\label{hol:mixed-bound}
For fixed primary gaps and sufficiently small fixed secondary gaps
chosen as in Lemma~\ref{hol:templates}, there is $c>0$ such that
\begin{equation}
 \left| (\delta_{1,z}^{(2)}+i\delta_{2,z}^{(2)})
               \delta_{j,u}^{(2)}\mathcal H_g(z,u)\right|
       =O(n^{-2-c})
 \label{hol:mixed-decay}
\end{equation}
uniformly on the eligible compact probe sets.  The constants and
reflection-tree size may depend on these fixed gaps.
\end{proposition}

\begin{proof}
Choose the expected extraction deficit $\varepsilon$ in
Lemma~\ref{hol:templates} small relative to $c_0$ in
\eqref{hol:strict-budget}.  The angular costs contribute at most
\begin{equation}
 -(1+\rho)(1-\varepsilon)\log n
 -(1-\rho)(1-\varepsilon)\log n
       =(-2+2\varepsilon)\log n.
 \label{hol:extracted-cost}
\end{equation}
The baseline \eqref{hol:baseline} cancels the precise normalization
in \eqref{wall:H-normalized}.  The remaining feature cost is at most
$-c_0\log n+o(\log n)$ by \eqref{hol:strict-budget}.
There are finitely many nodes, so all H\"older allowances may be chosen
with total smaller than, say, $c_0/4$.  Taking
$2\varepsilon<c_0/4$ leaves a strictly negative extra exponent.
This proves \eqref{hol:mixed-decay}.  It uses finite templates fixed
before the mesh, not any uniformity as their gaps shrink.
\end{proof}

\subsection{Removing the gaps and the exterior factor}

\begin{theorem}[Holomorphic disk limit]
\label{hol:disk}
For every rational orthogonal polygon with the four flat marks and
balanced approximations specified above, every locally continuous
subsequential limit of $h_n$ is holomorphic.  Consequently the boundary
identification and crossing formula of
Proposition~\ref{four:conditional-map} hold without its holomorphicity
premise for these polygons.
\end{theorem}

\begin{proof}
Take a simultaneous locally convergent subsequence $h_n\to h$,
$J_n\to J$.  The ideal compactness statements do not use holomorphicity.
On a product lattice coset for the step-two stencil, multiply
\eqref{hol:mixed-decay} by a compactly supported smooth test function
$\varphi(z,u)$ in an eligible product neighborhood and sum with
normalized mesh weights.  Twice summing by parts and dividing by
$(2/n)^2$ gives
\begin{equation}
 \int \mathcal H_g(z,u)
       (\partial_1+i\partial_2)_z\partial_{j,u}\varphi(z,u)
       \,d^2z\,d^2u=o(1),
 \label{hol:gapped-distribution}
\end{equation}
where the integral denotes the corresponding lattice Riemann sum
before passage to a limit.  The difference-to-derivative error vanishes
because $\mathcal H_g$ is locally bounded at fixed gaps by the absolute
moment bounds.  Every coset gives the same conclusion.

First let the mesh tend to zero.  Proposition~\ref{gap:closure}, with
its positive multiplier bounded above and below, transfers the zero
limit from the secondary gaps to the primary-gapped wall as
$\sigma\to0$.  Apply it again as the primary widths $h\to0$ to reach
the complete wall.  Finally use Proposition~\ref{wall:factorization},
whose scalar is bounded away from zero, and then let $R_0\to\infty$.
The ideal functions have compact convergence, so their lattice sums
are ordinary Riemann sums.  We obtain
\begin{equation}
 \int h(z)J(u)(\partial_1+i\partial_2)_z
             \partial_{j,u}\varphi(z,u)\,d^2z\,d^2u=0.
 \label{hol:product-distribution}
\end{equation}
No convergence of either comparison scalar is required: for each fixed
geometric stage its inverse is bounded, and that stage's error tends
to zero in the stated order.

Fix an interior $z$ off the finitely many feature projection lines.
There is a generic exterior probe box eligible for this $z$ on which
some distributional derivative $\partial_jJ$ is nonzero.  Otherwise
both derivatives would vanish off the finitely many excluded lines
in the connected strip; continuity would make $J$ constant on the
whole strip, contradicting its different limiting values at $p,d_0$.
Shrink the $z$ neighborhood and this probe box to retain all strict
margins.  Product tests in \eqref{hol:product-distribution}, with a
probe test pairing nontrivially with $\partial_jJ$, give
$(\partial_1+i\partial_2)h=0$ distributionally near $z$.
Thus $h$ is holomorphic there.  Its continuity extends holomorphicity
across the finitely many exceptional straight lines by Morera's
theorem, splitting triangles along those lines and passing to the
limit.  Proposition~\ref{four:conditional-map} now applies.
\end{proof}
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//observable.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//peeling.tex ====
\section{Peeling along monochromatic walls}
\label{sec:peeling}

Throughout Sections~\ref{sec:peeling}--\ref{sec:loops}, fix
$1\le q<4$ and write $4<\kappa=\kappa(q)\le6$.
We now construct a joint exploration toward interior targets.  The boundary
conditions in this construction are part of the proof: its initial law is
the wired FK law in a square, and its final application will be to the
free infinite-volume law of Subsection~\ref{intro:collections}.

\subsection{Actual walls, caps, and the exact exploration law}
\label{peel:construction}

Begin with a nearest-neighbor square whose boundary primal edges are
deterministically open.  Attach to the inward side of each boundary edge
the corresponding half-tile.  Its two ports are paired around its dual
corner; this triangle contains no random switch.  More generally, a
\emph{homogeneous wall} of color $C\in\{{\sf P},{\sf D}\}$ consists of
actual open $C$-edges and these deterministic triangles on its inward
side.  Actual paths along the wall are allowed.  This distinction from
an abstract boundary identification will be used in every crossing
argument below.

All strands are oriented with primal on the left.  To start an
exploration in a primal-walled component, remove one wall triangle.
Enter at its earlier port in counterclockwise boundary order; the other
port is the force port.  On a dual wall reverse the boundary order.  The
boundary of an active component, in counterclockwise order, is
\begin{equation}\label{peel:boundary}
 P_WD_W,\qquad [D_W,D_O]_{\sf D},\qquad
 D_OP_O,\qquad [P_O,P_W]_{\sf P}.
\end{equation}
The first and third pieces are tile sides, while the second and fourth
are actual open dual and primal paths with their deterministic inward
triangles.  Either path may have length zero.  Boundary occurrences,
rather than just geometric vertices, specify the accesses in this
description.

Here is the update at the tip side $P_WD_W$.
\begin{enumerate}[label=\textup{(\roman*)}]
\item If the next tile is fresh, reveal its switch.  In the primal-open
case, cut along its open diagonal $P_WP'$, remove the triangle traversed
by the strand, and continue from $P'D_W$ on the two-change side.  The
other half-tile becomes a deterministic wall triangle.
\item If $P'$ is an interior vertex, there is only one active component.
If it is a boundary occurrence, the new diagonal is a proper crosscut
joining the primal arc to itself.  It cuts off a homogeneous primal
pocket.  Retain the component containing the target.  In the pocket,
remove the triangle on the \emph{next counterclockwise old-wall edge}
starting from the hitting occurrence $P'$ and restart there.  For targets
on the two-change side, continue the old strand.
\item For a dual-open switch use the same rule with the colors exchanged
and clockwise order.  If the tip meets a known triangle, remove it and
follow its cap to the other leg.  This shortens the wall instead of
adding an edge.
\end{enumerate}
The rule is understood in the sector seen from the tile.  The opposite
corner of a newly retained wall triangle lies on its indicated side of
the crosscut.  A known triangle has two consecutive boundary sides in
that sector, so clipping it leaves just the other leg.  In particular
one never glues the two banks of a zero-width slit.  Starting from the
square, nonempty completed components are physically Jordan domains:
an update either indents the boundary through a new interior corner,
cuts it by a proper crosscut, or clips consecutive boundary edges.
The same observations apply when a homogeneous wall is opened.  The
only degenerate terminal configurations lie within a bounded number of
tiles of the target.  They cannot occur on a bounded radial horizon
from a macroscopic starting domain.

\begin{lemma}[Exact pocket law]\label{peel:factorization}
Conditionally on any finite exploration history, the unqueried switches
in its remaining components have the positive cap laws described by
their boundaries.  Distinct components are conditionally independent.
Before two targets split into distinct components their explorations
are identical.  A strand between successive pocket switches is a
portion of one original closed loop.
\end{lemma}

\begin{proof}
At a cut, every matching connection is either the consumed switch pair
or a local cap on one of its sides.  Consequently the factors
$q^{\ell/2}$ for unstarted loops, and the factor for eventually closing
the current strand, have no remaining dependence involving both sides
of the cut.  Their product therefore factors over the components.  A
two-change component has one strand between its ports; closing these
ports outside gives its marginal cap law.  The deterministic triangles
implement exactly those identifications.  No random switch is hidden
in such a triangle.

Until a switch into a homogeneous pocket, every traversed pair belongs
to the loop through the originally opened cap.  At a pocket switch
that loop remains on the other side of the actual open diagonal.
The new cap is on a loop not yet traced in the retained pocket.  This
also proves the last assertion.  For finitely many targets keep the
cut-off components, including their boundary states, and process active
branches in a fixed priority order.  The factorization gives the claimed
conditional independence before taking any limit.
\end{proof}


\subsection{Radial clocks and forcepoint resets}
\label{peel:radial}

For an interior target $z$ let
\[
 t=\log\frac{\operatorname{crad}(U_0,z)}
                    {\operatorname{crad}(U_t,z)},
 \qquad \Phi_t:U_t\longrightarrow\mathbb D,
 \quad \Phi_t(z)=0,
\]
where the argument of $\Phi_t'(z)$ is fixed.  On $t\le T$, Koebe's
theorem supplies a ball about $z$ of radius bounded below in terms of
$T$ and the initial conformal radius.  The chart modulus of
Section~\ref{sec:movingtests} therefore applies uniformly to connected
short paths and to their boundary accesses.

Let $e^{iW_t}$ be the tip and $e^{iO_t}$ the force access.  Choose lifts
so that
\begin{equation}\label{peel:angle}
 \theta_t=O_t-W_t\in[0,2\pi]
\end{equation}
is the angle of the counterclockwise dual boundary arc.  The force
access is at the wall-color endpoint of the force side chosen at the
last opening: after a primal opening this is $P_O$, approached beside
the interior of the side.

We use the following interpolation, which fixes the behavior at
collisions.  Move along the small tip side to the near endpoint of the
new diagonal, grow the diagonal as a slit, and, on the retained
two-change side, grow the other leg of the traversed triangle to clip
it.  Move along the resulting boundary leg to the new port.  For a
known triangle grow its other leg across to clip it.  Boundary motion
itself has zero growing time.  When the slit makes a pocket crosscut,
the force access collapses to the tip: $\theta=0$ in a primal pocket
and $\theta=2\pi$ in a dual pocket.  At a primal restart, move $O$
counterclockwise \emph{ahead}, to the far endpoint of the next edge,
and remove that edge's triangle by growing its two legs in wall order.
Then continue with $W$ at the new start.  Reverse both directions at
a dual restart.  Triangle removals on separated sides of a common
crosscut may subsequently be processed separately.

These updates have disjoint growing-time interiors and produce
exactly the designated target components.  They differ from the medial
strand, including its reset connectors, only inside a bounded number
of adjacent tiles in the accessed sectors.  The chart modulus implies
that all boundary-angle jumps vanish uniformly on bounded horizons.
The capacities of individual updates also vanish: their cuts are
chart-small, and kernel convergence preserves the basepoint component.
The total angular oscillation within one update vanishes as well.
Indeed its cut is in a vanishing sector of its starting chart; the
normalized new maps converge to the old map on compact interior sets
and, by reflection, on closed boundary arcs away from that sector.
The new tip is in that sector, while a force access is either regular
or belongs to the same collapsing sector.

Between the designated resets, the radial boundary equation is
\begin{equation}\label{peel:force-ode}
 dO_t=\cot(\theta_t/2)\,dt.
\end{equation}
If its endpoint is hit but the force side is retained, continue using
the access beside that side.  Convergence in the retained sector,
which can be checked by truncating the access with short circular
gates, gives the same one-sided equation on the next slit.  Its
improper integral at a collision is finite: near the collision the
motion has one sign and the force access has a finite angular limit.
No additional jump of $O$ is introduced.  These conventions also apply
when the end of the newly opened triangle is reached.  Stops at
completed updates suffice in what follows; allowing the switch of a
just-started update changes nothing in the limit.

\subsection{Crossings in the moving charts}
\label{peel:crossings}

We need actual paths reaching the explored wall, including where that
wall is irregular.  The following form specifies both the geometry and
the information retained after a crossing is found.

\begin{lemma}[Wall crossings and screening]\label{peel:wall-rsw}
Fix a buffered corridor of bounded aspect in a normalized disk chart.
Its terminal wall intervals lie strictly inside designated homogeneous
arcs; one terminal interval may instead be an interior bar.  After the
mesh limit at this fixed scale, an actual crossing of either prescribed
color $X$ has conditional probability bounded below, uniformly in the
exploration history and its boundary partitions.  At an $X$-wall it
reaches an actual wall vertex.  At an opposite-color wall it reaches
the $X$-corner of an inward wall triangle.  In the latter case a segment
inside that triangle, ending at the midpoint of its base, completes
the crossing to a screen and meets an actual $X$-path only at its
starting corner.

The two-sided bounded-density collar comparison of
Section~\ref{sec:preliminaries} holds for such buffered chart collars
on a homogeneous wall, with a bounded number of through-wire blocks.
\end{lemma}

\begin{proof}
We explain the reduction to a simple square-lattice quad, so that the
up-to-boundary crossing theorem applies with its exact hypotheses.
For the color $X$, retain the edge-connected component around the
basepoint of square faces whose centers have $\ell^\infty$ distance
more than $k\delta$ from the planar complement; a fixed $k=20$
suffices.  The failing faces are edge-connected, as one sees by
rasterizing a thickened path in the connected complement, including
slit banks.  Other components of passing faces are assigned to the
exterior.  Both face families are then edge-connected.  Alternating
interior and exterior faces at a self-touch vertex would separate one
of these families, so the retained polygon has a simple boundary.
It lies in unqueried territory and eventually contains every compact
chart core.

An incident failing face on its perimeter can be followed, through
$O(k)$ adjacent failing faces, to the first wall contact.  Thus every
perimeter point has a short exterior access to the actual wall.  The
chart modulus implies that the perimeter approaches the unit circle
uniformly and that its separated accesses have the correct cyclic
order.  The winding about the basepoint fixes that order globally.

Approximate the two longitudinal sides of the desired corridor by
simple $X$-lattice crosscuts inside this polygon.  First approximate
their compact interior portions with positive clearance; then append
terminal paths in disjoint chart neighborhoods, stopping at the first
perimeter contact.  Grid shadowing and loop erasure preserve the chart
margins, since each short connected error is chart-small.  Together
with the appropriate perimeter intervals these crosscuts bound a
simple lattice quad.  The transverse margins and angular order bound
its extremal distance by a constant depending only on the buffered
corridor.

This last bound can be made directly in discrete extremal length.
Thicken vertex weights over balls of a few mesh steps, with density
scaled by $\delta^{-1}$.  Each member of a fixed transverse family of
longitudinal chart paths shadows an edge path in the quad, cropped to
the two terminal sides.  Grouping successive connected pieces of
displacement $\delta$ charges only boundedly many nearby vertices.
The shadowed length is therefore bounded by a constant times its
density integral; the squared area integral is bounded by a constant
times the sum of squared vertex weights.  The fixed chart family gives
the asserted extremal-distance bound.

Theorem~1.2 of \citet{DMT2021Fractal} now gives a positive crossing
probability for the simple quad at $p_q$, for $1\le q<4$, with arbitrary
boundary partition.  Equivalently, use the free law as the lower
comparison.  Do not query edges running along its terminal perimeter
sides.  From a reached vertex, an outward failing-face chain gives
$O(k)$ available $X$-edges toward the intended wall.  The extension
uses only those unqueried terminal-side edges and edges outside the
quad.  Finite energy gives a uniform positive conditional cost.  Stop
at the first wall contact; if an opposite-color wall blocks the next
step, the current vertex is already the $X$-corner of its triangle.
This proves the crossing assertion, including its endpoint convention.
Circuits and interior connections with strict margins follow by
ordinary box-crossing and gluing.

For the collar statement, find the same-color screen nearest the
unknown-facing side of a deterministic buffered subcollar.  Paths may
be restricted to the strip between the prescribed wall intervals,
pruned at terminal returns, and augmented by ordered spokes to common
endpoints.  Minimizing the region on the unknown side selects a
simple outermost crossing.  Any competing path entering strictly
nearer that side has an open excursion that can replace part of the
selected path and decrease this region.  Hence the selection tests
only on and on that side of the screen.  Its other side retains an
FK law with a boundary partition.

Actual paths of the screen's color must attach to it in order to cross.
An opposite-color path cannot pass it except through the bounded
through-wire blocks; the added segment in a wall triangle creates no
other bypass.  Splitting or merging those blocks changes a weight by
a bounded factor.  The primal and dual screening searches used in the
collar proof of Section~\ref{sec:preliminaries} thus apply without
change to arbitrary-event densities.  All chart margins are fixed
before the mesh limit, as required.
\end{proof}

\subsection{A bound on generations}
\label{peel:generations}

\begin{lemma}[Disjoint passages and reference walls]
\label{peel:passages}
The number of edge-disjoint actual monochromatic cycles surrounding a
fixed macroscopic ball in the initial square, and using no deterministic
initial wall edge, is tight as the mesh tends to zero.  Along finitely
many target branches with bounded radial horizons, the number of
reference-wall generations defined below is jointly tight.
\end{lemma}

\begin{proof}
Such cycles provide edge-disjoint vertical passages in a fixed-height
middle horizontal band.  By bounded degree and Menger's theorem,
their number is bounded by a fixed multiple of the maximal number of
vertex-disjoint passages.  For primal paths one may add random switches
on the initial wired boundary edges, which does not change the interior
marginal under consideration.  For dual paths use the rectangle of
interior faces.  Thus it suffices to work in an ordinary fixed-aspect
color-lattice rectangle, with its induced positive FK law.

Prune bottom-to-top paths so they do not return to their terminal rows.
Select the leftmost open path, then the leftmost path to its right that
is vertex-disjoint from it, and repeat.  To justify the information
claim, attach noncrossing spokes just below and above the terminal rows
to common endpoints, pad the rectangle sides, and minimize left area.
Another eligible path entering strictly left gives an excursion with
distinct endpoints on the augmented path and permits replacement in
either traversal direction, decreasing left area.  End excursions
through a spoke use only terminal contacts.  The selected path is
therefore determined by its own edges and the edges on its left.  If
$m$ vertex-disjoint passages exist, at least $m-1$ remain to its right,
and the argument iterates.

After a selection $p$, there is a uniformly positive conditional
probability of blocking another disjoint passage.  If $p$ comes within
ten mesh steps of the right wall on some horizontal half-row, force an
opposite-color connection along that row from outside the right wall
to first adjacency to $p$; its finite-energy cost is bounded.  If $p$
already blocks, no forcing is necessary.  Otherwise follow $p$ on its
right through opposite-lattice face centers, joining consecutive ones
inside the right wedges at its vertices.  Erase loops and trim terminal
returns to obtain a simple path $p_*$ between the extreme interior
half-rows.  Those rows are single intervals on the right side because
$p$ was pruned.  With them and the rightmost interior column, $p_*$
bounds a simple opposite-lattice quad of fresh edges.

Horizontal rows give a bounded-extremal-distance family across this
quad: there are order-height disjoint rows, each of length at most
order-width.  Lemma~\ref{peel:wall-rsw}, or directly
\citet[Theorem~1.2]{DMT2021Fractal}, supplies an opposite-open crossing
with uniformly positive probability.  Extend it one step beyond the
right boundary at finite-energy cost.  At its other endpoint, append
a segment in the incident square to a vertex of $p$.  This screen
blocks all actual vertical paths on the right avoiding vertices of
$p$.  Its edges are fresh under the leftmost-path conditioning.
Consequently the successive selections have a geometric tail; blocker
tests need not themselves be exposed when continuing the search.
In particular, if $N$ is their number, then for constants depending
only on $q$ and the fixed rectangle shape,
\begin{equation}\label{peel:passage-tail}
 \mathbb P(N\ge m)\le(1-c)^{m-1},\qquad
 \sup_{\delta}\mathbb E[e^{aN}]<\infty
 \quad\text{for }0<a<-\log(1-c).
\end{equation}
This proves the first assertion, and also gives the corresponding
bound for a strictly bulk rectangle.

A generation begins in a homogeneous $C$-pocket, whose present wall is
its \emph{reference wall}.  Read the $C$-arc from $O$ to $W$ in
counterclockwise order for primal and clockwise order for dual.  It
consists of a prefix of the remaining reference interval followed by
a suffix of actual open edges revealed during this generation.  A
triangle step shortens this path; a fresh diagonal extends it or
returns to an earlier vertex.  At a same-color switch retaining part
of the prefix, the next gap is precisely its first retained reference
edge.  The description is thus preserved, even when the prefix becomes
empty.  These switches take place on the reference wall.

Start a new generation at the first homogeneous pocket whose retained
arc has no reference edge; every opposite-color switch has this
property.  Its wall consists entirely of actual diagonals discovered
since the previous start.  Consumed edges are not reintroduced by
known triangles.  The successive new reference cycles are consequently
edge-disjoint, separately for each color.  They surround the basepoint
ball present up to the fixed horizon.  If a wall walk has repeated
vertices, its winding extracts a surrounding cycle.  The first
assertion bounds the number of generations.  For finitely many
branches sum the bounds, allowing an ancestral generation to be
counted along each of its descendants.
\end{proof}

\subsection{Local chordal tests in a peeled domain}
\label{peel:localtests}

Put $h=1-2\rho$, with $\rho$ as in Section~\ref{sec:exponents}.
At a causal completed state with $\theta$ in a compact subinterval of
$(0,2\pi)$, map the active domain to $\mathbb H$ so that the tip is
$0$, the force access is $\infty$, and the target has imaginary part
one.  The positive real arc is dual-designated.  Stop a short subsequent
segment at fixed driver and chordal-capacity bounds, chosen so the
target remains safely inside and connected to the force access.  There
are no resets on this segment.  Let $g_t$ and $W_t^{\mathbb H}$ be its
chordal map and driver.

\begin{proposition}[Tests away from coincidence]\label{peel:martingales}
For $x>0$ beyond these localization bounds, the quantities
\begin{equation}\label{peel:test}
 M_t(x)=\left(\frac{xg_t'(x)}{g_t(x)-W_t^{\mathbb H}}\right)^h
\end{equation}
satisfy the approximate martingale identities between arbitrary causal
stops on the segment.  Tests may include the entire earlier exploration
history of finitely many targets.  The errors tend to zero after the
mesh limit.  The assertion concerns the averaged identities for this
history law, not a uniform assertion over all deterministic mazes.
\end{proposition}

\begin{proof}
We supply the modifications needed to apply the moving-domain
comparison of Section~\ref{sec:movingtests}.

\paragraph{The test notch.}
Fix first a small chart scale $s>0$.  Cut a notch around $x$, of size
between $s$ and $2s$, with two transverse ends on the old dual wall.
Put a new separate primal wire only on the middle portion of its new
wall, a positive chart distance from those ends, and keep dual
designations on both flanks.  Approximate a regular semicircular or
polygonal notch by openable dual diagonals near its ends and openable
primal diagonals in its strictly interior middle, with transition tile
sides.  The artificial diagonals are deterministic in the modified
experiment.  The result has four changes with the same triangular-cap
convention as before.

This construction is possible in unqueried territory.  Compact
interior parts have ordinary positive-clearance approximations; end
parts shadow dual-grid paths in the prescribed accessed corridors up
to first actual dual-wall contact.  Loop erasure preserves their order
by the chart margins.  We also choose the notch to have controlled
physical terminal tails.  Length-area, applied to a positive-width
family of chart curves, bounds the integral of the inverse-curve tail
lengths below height fraction $\epsilon$ by $O(\sqrt\epsilon)$, since
the physical area is bounded.  Thus a curve can be chosen with tails
vanishing at shrinking height cutoffs.  One may make the height cutoff
decrease slowly with the mesh.  Its retained interior is approximated
with positive one-sided clearance, and the new primal interval stays
at positive physical distance from all previous and locally stopped
trajectory pieces for fixed $s$.

\paragraph{Exact identities and comparison of the histories.}
Sample the notched law conditionally at the start of the segment.
Follow its strand until the localized stop, measuring the stop in the
unnotched geometry.  The strand cannot reach or swallow the notch
data: real and complex Loewner ODE bounds keep its growth away from
that chart neighborhood.  Every locally discarded pocket is
homogeneous and contains neither notch nor force data.

The pairing probability with the two primal wires separate is an
exact martingale.  Identifying the two primal intervals gives the
complementary dual comparison; its law is a bounded tilt depending
only on the pairing.  These identities are taken at completed states.
For additional targets keep reporting geometry and scheduling from
the unnotched cuts until the local stop or an active horizon, without
changing a target assignment near the notch.  Still-shared targets
see this same growth, and separated components have their ordinary
independent laws by Lemma~\ref{peel:factorization}.  If a target clock
freezes, truncate the identity there.

Lemma~\ref{peel:wall-rsw} compares the exterior history laws, for fixed
$s$, by bounded density constants.  To obtain a coupling error tending
to zero as $s\downarrow0$, bracket the exterior primal partitions at
the notch by the extremal partitions, expose actual attachment
clusters in the stronger law, and block them by dual wall-to-wall
paths in successive dyadic chart half-collars.  These lie away from
the old primal wire, so there is no primal wire bypass.  Conditional
success probabilities at each fixed-shape scale are uniform after
the mesh limit.  Only the separate-primal-wire law is needed in this
last coupling, not its pairing tilt.

\paragraph{Rough-domain obstacles.}
For a later stopped notched domain, its four-change rectangle is
nondegenerate for fixed $s$.  The central possible-attachment vertices
and their optimistic color paths are defined exactly as in the
moving-domain argument.  Accessible identifications and deterministic
open adjacencies have short accesses to their assigned wires.
Away from those wires a central attachment begins with an actual
uncertain-edge path to macroscopic distance.  Protected-ball deletion
still has only rim attachments in the relevant components; splitting
a physical vertex weakens these connections.  No notch cap supplies
a shortcut of a color away from that color's electrodes.

Use the initial square, the relevant ancestral and current exploration
pieces, and the notch as obstacles.  First truncate the numbers of
branches and generations, using Lemma~\ref{peel:passages}.  Within
generation $j$, retain excursions reaching depth $r_j$ from the initial
boundary and the earlier-generation obstacles, starting and ending at
much smaller depth $h_j$, or at an active endpoint.  The earlier
obstacles contain its reference wall up to a step-sized error.  Away
from them, each retained leg is jump-free and has actual open paths
of both colors beside one loop portion.  Bulk passage bounds, on a
finite grid of annuli, make the number of these legs and of their
short internal blocks tight.  A trajectory does not retrace a loop
portion: at a switch the old loop remains in the other component.

Choose accuracies backwards through the generations, approximating
each earlier obstacle much better than the descendant depth $h_j$.
Excursion endpoints then attach to earlier routes in protected small
balls, and omitted shallow pieces contribute only $r_j$ to the
Hausdorff error.  Each retained block has a known actual path $\pi$ of
the requested color in a short bulk patch.  The routing and flag
construction of Section~\ref{sec:movingtests} therefore applies on a
finer deterministic grid.  Route the notch on its removed side, with
clearance on compact interior portions, and attach its short terminal
tails in protected balls.  Specifically, fix the two protection radii,
choose still smaller physical tails as above, then approximate the
retained interior by a polygon with one-sided clearance.  At fixed
height cutoffs, normal-family bounds allow all perturbations and
snappings to be chosen from finite lists.  Make earlier obstacle
accuracies smaller afterward, and take mesh errors last.  The resulting
rounded Jordan comparison domain contains every required internal
compact path and has all the limiting complement contacts needed for
kernel convergence; its marked prime data agree in interior charts.

\paragraph{Why the flag cost is the original six-arm cost.}
Use original past switches and actual wire edges for the known legs.
In testing the absence of a short optimistic bridge to $\pi$, allow
even the original unnotched ambient completion.  An actual path leaving
the old domain first hits its designated wall; opposite-color wall
diagonals block it.  The same is true on first contact with a previously
queried edge of its color.  Near the current old wire portions, the
chart modulus excludes a shortcut from a central vertex in a
sufficiently small physical disk.

There are two different locality checks near the notch.  A primal
central path that locally crosses it stays near the old positive
dual-wall patch in the starting chart, far from any old primal contact
or locally grown trajectory; it cannot reach a known primal history
edge there.  For a dual central vertex near a flagged historical
detail, the entire notch is far away in the old chart.  Indeed a
central dual abscissa near the notch must avoid both dual flanks and
their ends, and hence lie in a compact interior chart patch beside the
new primal interval.  That patch has positive physical distance from
the old history and the locally stopped pieces, first by the regular
notch margins and then by the ODE bounds.  Kernel convergence and
analytic continuation away from the local hull give the same assertion
on every shape subsequence.

All these arguments follow connected paths in the old domain until
their first old-wall contact.  Boundary-sector identifications cannot
undo the actual diagonal barriers.  The sector-duality argument thus
produces the original known flags from the two witness arms, and an
actual extra arm produces six alternating arms in the original
ambient switch lattice.

For dual extra arms near flags, the required narrow arm pieces lie
outside the notch and its narrower protecting collar.  The arbitrary-event
collar density comparison transfers their probability, including
the local history, to the original law; the pairing tilt costs another
bounded factor.  For primal extra arms, first compare the stopped
history through the outer collar.  Conditionally on that history the
extra-arm event is increasing in primal edges.  Restore the deleted
notch and wire only the new middle primal vertices locally.  This
dominates the modified law: the dual flanks had imposed no primal
wiring, and their known dual edges had only removed primal choices.
The flagged local arm pieces are physically separated from that new
interior wiring by a vacant bulk collar, containing no earlier history.
The bulk density comparison then transfers their probability to the
original conditional law.

For either color, summing over fine-grid flagged cells consequently
costs a bounded multiple of the original annealed six-arm sum.  Its
decay follows from \citet[Corollary~6.7]{DMT2021Fractal}, after the
collar transfer of Section~\ref{sec:preliminaries}.  Take the grid
scale small after all collar margins, trims, and block cutoffs.
Protected-ball errors, in contrast, are estimated directly
conditionally in the modified law.  The chart and notch may depend on
the segment-start history: at fixed scales their margins and comparison
constants are uniform under the stated normalizations.

\paragraph{The alternating formula and the shrinking notch.}
These checks are the additional inputs required by the upper barrier
comparison of Section~\ref{sec:movingtests}.  Its directed routes have
clearance from central candidates except for the controlled deletions.
The level fibers of the remaining kernel and comparison template agree
outside protected end balls, giving the same shared-graph partition
domination for the abscissa-trimmed quad.  Fix generation cutoffs,
block cutoffs, grids, and all intermediate scales before the mesh
limit.  The preceding averaged bounds make adverse conditional costs
negligible in history probability.  The two color upper bounds and the
pairing tilt therefore identify the stopped conditional pairing
probability with the separate-wire value $g(\chi)$.

Divide its exact martingale identity by the initial formula value,
which is bounded below for fixed $s$.  As $s\downarrow0$, the ratio
tends locally uniformly to \eqref{peel:test}.  To see the conformal
normalization, remove the same small boundary hull near $x$ in the
starting and later chordal charts.  The map between them is analytic
near that hull, with derivative $g_t'(x)$ bounded above and below and
bounded second derivative.  Rescaling the notch gives uniform Jordan
shape convergence, so the flattened prepoint interval widths have
ratio $g_t'(x)(1+o(1))$, while their base positions have ratio
$(g_t(x)-W_t^{\mathbb H})/x$.  The asymptotic
$g(\chi)\sim c\chi^h$ gives \eqref{peel:test}.  Finally use the
vanishing separate-law coupling error to return bounded tests to the
original law.  This order of limits proves the proposition.
\end{proof}

\section{Identification of the joint angular tree}
\label{sec:tree}

\subsection{Interior evolution and an oriented boundary estimate}
\label{tree:tightness}

All assertions in this section are first made for finitely many
fixed deterministic interior targets and bounded radial horizons.
The horizons may subsequently increase.  The small-update estimates
of Subsection~\ref{peel:radial} absorb port ambiguities and the difference
between an interpolated stop and a completed-state stop.

\begin{lemma}[Interior characteristics]\label{tree:interior}
Away from $\theta\in\{0,2\pi\}$, every limiting branch has the local
characteristics
\begin{equation}\label{tree:sde}
 \begin{aligned}
 dW_t&=\sqrt\kappa\,dB_t+
          \frac{6-\kappa}{2}\cot(\theta_t/2)\,dt,\\
 dO_t&=\cot(\theta_t/2)\,dt,\qquad \theta_t=O_t-W_t.
 \end{aligned}
\end{equation}
Here $t$ is that branch's radial clock; the martingale identities hold
relative to the full causal exploration history.
\end{lemma}

\begin{proof}
Use two ordered test points $x$ in Proposition~\ref{peel:martingales}.
The relative-convexity inequality \eqref{chordal:two-test} and the inversion
argument of Proposition~\ref{chordal:driver} identify the stopped
chordal driver and its bracket.  They also give
stopped second-moment estimates.  On a sufficiently short segment
started with $\theta\in[e,2\pi-e]$, the chordal and radial clock rates
and coordinate changes are bounded above and below.  Thus crossing a
fixed interior angular interval in vanishing radial time has
probability tending to zero, uniformly at causal starting stops.
A macroscopic angular increment started near an endpoint must cross
such an interior interval.  Aldous' criterion and the vanishing
within-update oscillations give continuous tightness of $\theta$;
in particular its numbers of fixed-size alternating swings are tight.

After the tightness of $W,O$ proved below, localize on rational start
ranges and inner continuity levels to pass the re-centered identities
to a joint subsequential limit, retaining tests of the past.  The
ordinary chordal-to-radial target change, before any loss of the force
access, gives \eqref{tree:sde}.  This is precisely the stopped
coordinate change in \citet[Proposition~3.12]{Sheffield2009Exploration};
no continuation through collisions is inferred from that proposition.
\end{proof}

The missing tightness, and later the absence of an endpoint push, come
from an estimate whose direction depends on the restart rule.

\begin{lemma}[Fast motion near coincidence]\label{tree:bar}
There are positive constants $c,C$ with the following property.  Start
at a causal stop with $\theta<r/16$, where $r>0$ is small.  For
$0<u\ll r^2$, the probability that $W$ advances counterclockwise by
$r$ within additional radial time $u$, while $\theta<r/16$ until that
advance, is at most
\begin{equation}\label{tree:bar-bound}
 C\exp(-cr/\sqrt u)+o(1).
\end{equation}
The error vanishes in the mesh limit for fixed $r,u$.  The constants
in the limiting estimate are uniform in these parameters.  Near
$2\pi$ reverse the direction of motion and exchange the colors.
\end{lemma}

\begin{proof}
Work in the disk chart at the starting stop.  The radial Loewner ODE
confines the updates during time $u$ to a strip of depth $C\sqrt u$
from its boundary.  A driver advance of $r$ forces a positive angular
traversal through a buffered part of the corridor from angle
$W_0+r/5$ to $W_0+r/2$.

Here is the boundary-coordinate justification of this implication.
Stop at the first $|W-W_0|=r$, so the lift cannot wrap around.
If the advancing growth never passes, for example,
$W_0+2r/3$ in the starting chart, a neighborhood of radius comparable
to $r$ at $W_0+5r/6$ remains untouched.  Its prepoint image stays
$O(\sqrt u)$ from its original angle.  First check the ODE at depth
a large constant times $\sqrt u$.  Near the untouched arc the map
reflects analytically; radial monotonicity bounds the log-radius
harmonic part by the absolute original log radius.  In the stated
neighborhood this is $O(r)$, so the interior gradient estimate bounds
the angular derivative.  The boundary estimate follows by reflection.
The untouched prime access is distinct from the tip, and the small
boundary relocations also miss it.  The driver therefore cannot pass
its lifted image, contrary to the stipulated advance.  This proves the
required physical traversal in the starting chart.  All within-step
and strand-surrogate errors vanish for fixed $r,u$.

Place order $r/\sqrt u$ disjoint buffered bars across that angular
corridor.  Each has width comparable to $\sqrt u$ and extends to a
larger fixed multiple of the allowed depth.  By
Lemma~\ref{peel:wall-rsw}, each has a uniformly positive conditional
chance of an actual dual-open path from the dual corner of a triangle
on the primal wall to its inner end.  Complete this path across that
triangle to the midpoint of its primal base.

One successful bar prevents the required advance before a dual reset.
For completeness, follow the actual traversed switch pairs and add a
microscopic connector at each primal restart.  At that restart the
connector passes from the encountered pair through the neighborhood
of the open primal diagonal and hitting vertex to the new cap's
\emph{earlier} port, approaching along its near base endpoint.  It
stays in the retained sectors and avoids all actual dual paths.
A positive angular passage within the shallow strip would have to
cross the completed bar positively.  It cannot cross its actual dual
part.  Traversal of the cap pair in the fixed triangle crosses the
added segment only negatively, because primal is on the left.  If
this triangle itself is selected for the restart, the connector reaches
the earlier port without crossing the segment.  Every other reset
connector stays beside its own primal vertices and avoids that fixed
segment.  An excluded triangle is never used again.  Thus the positive
crossing is impossible.  This argument is in the prime closure of the
stop domain, where all short accessed moves are uniformly chart-small.

While $\theta<r/16$ there is no dual reset.  Multiplying the conditional
failure bounds for the disjoint bars proves \eqref{tree:bar-bound}.
Also, a negative motion of order $r$ is impossible in this zone:
$O$ is nondecreasing, including its primal resets, and $W=O-\theta$.
The reflected argument proves the other endpoint statement.
\end{proof}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=1.05]
 \draw[thick] (0,0) circle (1.55);
 \draw[very thick,dashed,->] (15:1.55) arc (15:132:1.55);
 \draw[very thick,->] (140:1.55) arc (140:365:1.55);
 \fill (0,0) circle (1.5pt) node[below] {$z$};
 \fill (15:1.55) circle (2pt) node[right] {$W$};
 \fill (140:1.55) circle (2pt) node[above left] {$O$};
 \draw[->] (15:.62) arc (15:140:.62);
 \node at (77:.92) {$\theta$};
 \node at (70:1.95) {dual arc};
 \node at (255:1.94) {primal arc};
 \node[align=center,text width=4.8cm] at (5.05,1.02)
 {$\theta=0$: primal pocket\\
  advance $O$ to the next counterclockwise wall edge};
 \node[align=center,text width=4.8cm] at (5.05,-1.02)
 {$\theta=2\pi$: dual pocket\\
  reverse the wall order};
\end{tikzpicture}
\caption{Normalized chart of a two-change component.  The angle
$\theta=O-W$ is measured counterclockwise along the dual arc.
The arrows on the circle indicate boundary order, not strand
orientation.  The forcepoint reset uses the next old-wall edge on the
retained pocket.  A full turn changes the endpoint of the reflected
angular interval.  The drawing is schematic; microscopic tip and
force sides are suppressed.}
\label{tree:figure}
\end{figure}

Away from coincidences use Lemma~\ref{tree:interior}.  In an endpoint
zone, a fast exit toward the interior is controlled by the tightness
of $\theta$, and fast motion remaining in that zone is controlled by
Lemma~\ref{tree:bar} and the sign of $O$.  Thus $W$ and $O$ are
continuously tight, also at variable stops.  Moreover, on a sufficiently
small fixed endpoint zone, both the continuous part of $O$ and every
reset have the same sign.  Its variation on each visit is bounded by
its angular oscillation there.  Count visits by alternations between
nested endpoint zones, whose number is tight; outside these zones the
cotangent rate is bounded.  We obtain tightness of
\begin{equation}\label{tree:variation}
 \operatorname{Var}_{[0,T]}(O)
 \quad\hbox{and}\quad
 \int_0^T|\cot(\theta_t/2)|\,dt.
\end{equation}
In any limit the endpoint sets have zero radial Lebesgue measure,
$O$ has locally finite variation, and \eqref{peel:force-ode} holds off
those sets.  A remaining singular part could only be nonnegative at
$0$ and nonpositive at $2\pi$.  We next rule it out.

\subsection{Instantaneous reflection and removal of a singular push}
\label{tree:endpoints}

\begin{proposition}[The endpoint convention is determined]
\label{tree:reflection}
In its own radial clock, the limiting gap is the unique instantaneous
reflected diffusion on $[0,2\pi]$ with interior generator
\begin{equation}\label{tree:generator}
 \mathcal A f(\theta)=\frac\kappa2 f''(\theta)
       +\frac{\kappa-4}{2}\cot(\theta/2)f'(\theta).
\end{equation}
The forcepoint has no singular push:
\begin{equation}\label{tree:no-push}
 O_t=O_0+\int_0^t\cot(\theta_s/2)\,ds.
\end{equation}
Equation~\eqref{tree:sde} consequently holds globally, including
collisions, with locally integrable drift.
\end{proposition}

\begin{proof}
For a smooth function constant near both endpoints, the localized
interior identities glue to the martingale problem for
$\mathcal A$.  To admit a smooth function with
$f'(0)=f'(2\pi)=0$, approximate it by such cutoffs with uniformly
bounded second derivatives.  Its cotangent term is bounded near the
ends, since $|f'(\theta)|=O(\min(\theta,2\pi-\theta))$.
The endpoint time is zero, so dominated convergence gives the limiting
martingale identities and, by applying the same argument to $f^2$,
their brackets.

In particular $X=\cos(\theta/2)$ solves
\begin{equation}\label{tree:jacobi}
 dX_t=\frac{\sqrt\kappa}{2}\sqrt{1-X_t^2}\,dB_t
           -\frac{3\kappa-8}{8}X_t\,dt,
 \qquad -1\le X_t\le1.
\end{equation}
The diffusion coefficient is one-half H\"older and the drift is
Lipschitz; the one-dimensional square-root uniqueness criterion gives
pathwise uniqueness.  Hence this equation fixes the instantaneous
reflected law of $\theta$; a sticky alternative is also excluded by
zero endpoint time.

Near either endpoint the scaled distance $R$ has Bessel dimension
\begin{equation}\label{tree:bessel-dimension}
 \mathfrak d=3-\frac8\kappa\in(1,2).
\end{equation}
Its drift differs from $(\mathfrak d-1)/(2R)$ by a bounded smooth
term of order $R$.  Stopped Girsanov, equivalently in the squared
coordinate where the excess drift divided by the diffusion coefficient
extends boundedly, transfers the usual Bessel path properties.  Thus
$\theta$ is locally H\"older of every exponent below $1/2$, and each
endpoint set has Hausdorff dimension at most
\begin{equation}\label{tree:zero-dimension}
 d_0=1-\mathfrak d/2<1/2.
\end{equation}
The needed Bessel H\"older statement is, for example,
\citet[Theorem~1.2 and Corollary~2.14]{HutzenthalerJentzenNoll2014}.
The zero-set statement follows from the inverse-local-time description
in \citet[Exercise~XI.1.25]{RevuzYor1999} and the stable-range dimension
in \citet[Theorem~III.15]{Bertoin1996}.  Only a bounded stopped
perturbation at one endpoint is being used here.

Write the possible remaining term as the signed measure
\[
 d\nu=dO-\cot(\theta/2)\,dt.
\]
It is supported on the two endpoint sets and has the sign specified
after \eqref{tree:variation}.  We give the time-block argument
explicitly, so that a Hausdorff-dimension bound, rather than an unstated
Minkowski bound, suffices.  Choose $\eta>0$ so small that
\[
 p=\tfrac12-\eta>d_0,
 \qquad u_m=2^{-m},\qquad r_m=u_m^p.
\]
By the H\"older property, every sufficiently fine dyadic block of
length $u_m$ meeting an endpoint set has its end-gap at most $r_m/32$
throughout.  Indeed choose a H\"older exponent strictly between $p$
and $1/2$.  Apply Lemma~\ref{tree:bar} with harmless fixed buffer
changes to the finitely many blocks at level $m$, taking the mesh
limit first.  The probability that some level-$m$ block stays in the
stated endpoint zone and has one-signed $O$-variation exceeding $Cr_m$
is at most
\begin{equation}\label{tree:block-sum}
 C(1+T/u_m)\exp(-c u_m^{-\eta}).
\end{equation}
This is summable in $m$.  The estimate for $O$ follows from that for
$W$ because their difference is the end-gap, of size at most $r_m/32$
on such a block.  Borel--Cantelli gives, almost surely, the variation
bound $Cu_m^p$ on every sufficiently fine endpoint-meeting block,
separately for the two endpoints and on each fixed horizon.

On an endpoint neighborhood, $dO$ is one-signed, and so is the
cotangent time measure.  Its restriction to the endpoint set is zero,
whereas the restriction of $dO$ there is precisely $d\nu$.
Consequently $|\nu|(I)\le\operatorname{Var}_I(O)$ for every small
interval in that neighborhood.  Cover the endpoint set by intervals
of variable lengths whose sum of $p$th powers is arbitrarily small,
using \eqref{tree:zero-dimension}.  Replace each by a bounded number
of comparable dyadic intervals that meet the set.  The preceding
variation bound yields arbitrarily small $|\nu|$-mass.  Thus
$\nu=0$, proving \eqref{tree:no-push}.

Finally, a Bessel process of dimension greater than one is a
semimartingale with integrable inverse-distance drift and no local-time
term in its ordinary coordinate.  Undoing \eqref{tree:jacobi} in the
endpoint neighborhoods, and using the interior identities elsewhere,
gives
$d\theta=-\sqrt\kappa\,dB+((\kappa-4)/2)\cot(\theta/2)dt$.
Together with \eqref{tree:no-push} this is \eqref{tree:sde} globally.
There is therefore no unspecified boundary pushing rate.
\end{proof}

\subsection{Common clocks and independence after separation}
\label{tree:joint}

The single-branch conclusion does not by itself identify the joint
tree.  In particular a narrow discrete corridor may survive after its
two limiting target kernels have separated.  We address this before
using any continuum tree theorem.

\begin{proposition}[Joint target tree]\label{tree:joint-law}
For every finite family of deterministic interior targets, the peeling
converges in its joint radial driving data to the target-invariant
branching radial $\mathrm{SLE}_\kappa(\kappa-6)$ tree.  Branches agree
until their target kernels separate and then have independent
continuation noises.  At successive full turns the force side is
reversed, equivalently the angular gap remains instantaneously
reflected on $[0,2\pi]$.
\end{proposition}

\begin{proof}
Use the sum of currently accrued target clocks as a common parameter
$v$, freezing each target at its prescribed horizon.  Every clock
$t_i(v)$ is nondecreasing and $1$-Lipschitz.  Several run together while
their targets share an active component; separate components are
processed in serial priority order.  A flat clock has a constant state.
All tightness bounds and the interior identities hold between adapted
common-clock stops, including tests of the whole past.  Work in other
components during an interrupted segment is conditionally independent
of its unqueried switches by Lemma~\ref{peel:factorization}; neither
the schedule nor the stop gives advance information.

Extract limits jointly with the clocks.  The preceding endpoint
arguments can be applied in each marginal's own radial clock, using
tightness and inverse-clock extraction across flat stretches.  Thus
the martingale part $M_i$ of $W_i$ is a continuous local martingale
relative to the common past, with
\begin{equation}\label{tree:brackets}
 d\langle M_i\rangle=\kappa\,dt_i.
\end{equation}
The Loewner equations determine its target kernels and normalized
maps.

While targets $z_i,z_j$ remain in one component, put
$\xi=\Phi_i(z_j)$.  Radial Loewner evolution and the conformal-radius
change give
\begin{equation}\label{tree:clock-change}
 dt_j=P^2\,dt_i,\qquad
 d\log|\xi|=P\,dt_i,\qquad
 P=\frac{1-|\xi|^2}{|e^{iW_i}-\xi|^2}.
\end{equation}
Their marked boundary angles correspond by the normalized disk
M\"obius map.  On a compact interval with $|\xi|<1$ these formulas
are regular and pass to the limit.  The targets are still in the same
kernel exactly while they retain an interior connecting path: kernel
convergence preserves paths with positive clearance.  Thus they have
the same growth up to kernel separation.

Suppose now their limiting kernels have separated but their discrete
components have not yet split.  Along such a portion $|\xi|\to1$ by
monotonicity.  On a subinterval where both limiting angle gaps stay in
the interior, the two marked prepoints remain separated in both
charts.  The disk M\"obius formula forces one of
$e^{iW_i},e^{iO_i}$ to be within $C(1-|\xi|)$ of $\xi$ in the
prelimit.  Since they are separated in the $i$-chart, the choice cannot
change on a connected subinterval.  If it is the tip, $P\to\infty$;
if it is the forcepoint, $P\to0$.  Equation~\eqref{tree:clock-change}
then forces one limiting clock to be flat.  This remains true along
subsequences of exact discrete split times.  After an exact split,
serial growth gives disjoint time interiors for the two clocks.
There are only boundedly many changes of component choice for a finite
target list and finite horizons.  Endpoint angle sets have zero mass
for the corresponding own clock by Proposition~\ref{tree:reflection}.
The covariance inequality for continuous local martingales and
\eqref{tree:brackets} therefore show that
\begin{equation}\label{tree:orthogonal}
 d\langle M_i,M_j\rangle=0
 \quad\hbox{after their kernel separation}.
\end{equation}
Frozen targets are omitted from this comparison.

We spell out why these characteristics determine independent
continuations, without asserting that the clocks are independent.
In each still-accessible active group choose its first indexed target
as representative.  Until a separation or freeze, the representative's
motion determines all other states and clocks by
\eqref{tree:clock-change}.  Separation and freeze times are intrinsic
stopping stages: mutual accessibility is read from the radial ODE up
to first approach to unit modulus.  While two other members remain
regular with the representative, they cannot lose their shared access
before either is lost from that representative.  A nonrepresentative's
horizon is likewise read by its clock correspondence.  At a multiple
separation, take the continuous state limits in all own clocks; the
subgroups are exactly those retaining internal mutual paths in their
kernels.  Serial speeds of unrelated groups have no effect on these
stages.

For each target accumulate its martingale coordinate only over the
suffix during which it is an active representative, keeping it flat
otherwise.  More precisely, with the predictable choice at the stage
endpoints, set
\[
 N_i(v)=\int_0^v
 \mathbf 1_{\{i\text{ is an active representative at }s\}}\,dM_i(s).
\]
Once it becomes representative it remains representative of its
descendant group until it freezes.  These representative martingales
$N_i$ are pairwise orthogonal by \eqref{tree:orthogonal}.
Their terminal brackets can be finite, so make the completion explicit.
Let $T_i$ be the prescribed horizons and take the deterministic common
terminal parameter $V=\sum_iT_i$, extending all stopped states constantly
if necessary.  Divide each $N_i$ by $\sqrt\kappa$, and enlarge the
common filtration after $V$ with
independent Brownian motions $\widehat B_i$, independent also of the
entire original terminal sigma-field.  Set
\[
 \widehat N_i(V+s)=N_i(V)/\sqrt\kappa+\widehat B_i(s),\qquad s\ge0,
\]
and use $\widehat N_i(v)=N_i(v)/\sqrt\kappa$ for $v\le V$.
The extended processes are continuous local martingales in this
enlarged common filtration, remain pairwise orthogonal, and have
infinite terminal brackets.  Knight's orthogonal-martingale time-change
theorem now gives independent standard Brownian motions in their
respective bracket clocks
\citep[Theorem~V.1.9]{RevuzYor1999}; see also
\citet[Theorem~2]{Meyer1971Knight}.
Restricting to each original terminal bracket supplies precisely its
original representative martingale.  The auxiliary continuation changes
neither the past filtration nor any stage of the exploration.

For the conditional assertion, let $\mathcal G_v$ be the common causal
filtration and let $S$ be a separation or freeze stage.  Restart the
active representative martingales at $S$, stop at the next common stage,
and pad constantly to a deterministic horizon.  After localizing to
bounded brackets, these restarted processes remain orthogonal
martingales under $\mathbb P(\,\cdot\mid E)$ for every $E\in\mathcal G_S$ of
positive probability, with the same bracket identities.  Completing
them as above and applying Knight's theorem under this measure gives
the same product Wiener law for every $E$.  Thus the continuation
noises are independent of $\mathcal G_S$.  A continuing representative
uses its original martingale increments, and a newly selected one
starts its representative suffix.  No conditioning on terminal
bracket values or future separation data is involved.

The pathwise uniqueness of \eqref{tree:jacobi}, followed by
\eqref{tree:no-push}, determines the angles and maps through the next
intrinsic stage.  A newly assigned representative uses its assigned
independent noise; a continuing one uses the continuation of the same
Brownian solution.  A change caused solely by a representative's
horizon gives a fresh continuation at that stopping stage.  Induction
over the finite target groups identifies the usual common-branch,
independent-after-separation construction.

The resulting equations and angular convention are the target-invariant
radial tree of
\citet[Propositions~3.13--3.14 and Section~4.2]{Sheffield2009Exploration}:
a one-sided force weight $\kappa-6$, with the side reversed in an
enclosed component after a full turn.  Our sign convention is
$\theta=O-W$; exchanging the sign of its Brownian motion translates
to the opposite angular convention.  Increasing the finite horizons
proves the assertion at all finite clocks.
\end{proof}

\section{Recovery of the nested loop collection}
\label{sec:loops}

Continue first in a fixed axis-parallel square with the primal-open
wall and wired law of Subsection~\ref{peel:construction}.  Draw the two
pairs at each tile contact disjointly in a midpoint disk of radius
less than $\delta/100$, taking each turn on its prescribed side.
The resulting actual loops are simple, pairwise disjoint, and avoid
actual open diagonals of both colors.  Use the analogous drawing for
wall triangles and orient every loop primal-left.  Matching successive
traversal pieces inside those disks changes each loop by at most
$2\delta/100$ in Euclidean uniform curve distance, and by a constant
multiple of this in spherical distance, simultaneously for all loops.
Thus the drawing convention has no effect on the desired topology.

\subsection{Compactness and two sides of every limiting loop}
\label{loop:compactness}

\begin{lemma}[Positive-law loop geometry]\label{loop:geometry}
The closed loop collections in the fixed square are tight for matching
uniform curves modulo reparametrization and orientation.  Constant
limit curves may be removed.  Their positive-diameter subsequential
limits have the following properties almost surely.
\begin{enumerate}[label=\textup{(\roman*)}]
\item Each deterministic interior point lies on no loop.  Its absolute
winding number about any loop is $0$ or $1$, and is stable along a
converging loop sequence.
\item No interior point has three distinct visits in total among the
loop elements.  Visits include multiple traversals after constant
pauses are removed and also multiplicity of different elements.
\item At every interior point of a positive loop, every neighborhood
contains open disks of both absolute winding statuses for that loop.
\end{enumerate}
\end{lemma}

\begin{proof}
\textit{Passage counts and compactness.}
Consider a vertical band of fixed positive width, with its vertical
sides interior to the square and its height equal to that of the
square.  Each disjoint medial passage between its sides has an actual
dual path beside it: consecutive adjacent dual vertices are identical
or joined by an open dual diagonal, including at wall caps.  Crop to
a slightly narrower band.  The medial separators continue beyond both
cropped sides and are disjoint from dual-open edges, so they divide
the band into strips.  Each accompanying dual path lies in the strip
on its separator's dual side, with at most two assignments to a strip.
Paths in different such strips cannot communicate in the actual dual
graph there.  Hence many passages force many distinct actual dual
components crossing the cropped band.

The graph of internal dual faces is a full lattice rectangle with
arbitrary induced boundary data; it has no dual crossing of the
initial deterministic primal wall.  The disjoint-passage proof of
Lemma~\ref{peel:passages} bounds its crossing-component count.
Even with all boundary vertices wired, an \emph{actual} left-right
crossing is bounded away from probability one by an up-to-boundary
primal blocker.  End that blocker just beyond the extreme dual rows
by bounded finite-energy extensions to neighboring internal primal
vertices.  The rectangle's dual boundary edges themselves are random,
so the crossing theorem applies.  The same argument holds for
horizontal bands.

A finite family of trimmed bands at each fixed scale bounds both the
number of loops with diameter above that scale and the number of their
disjoint subarcs with a given positive displacement.  These bounds
yield compactness modulo reparametrization.  To make the extraction
explicit, subdivide each positive loop by successive first-displacement
times into arcs of diameter at most $2^{-j+O(1)}$, retaining marks from
previous subdivisions.  The number of marks at each fixed $j$ is
tight.  Pass to subsequences on which their positions and cyclic orders
converge.  Assign nested parameter intervals with maximal length
tending to zero and positive lengths at every fixed subdivision,
splitting an interval further in two when necessary.  The corresponding
polygonal curves approximate uniformly.  Diagonalize over diameter
cutoffs tending to zero, retaining multiplicities of matched elements.
This proves collection tightness.

\paragraph{Avoidance and the visit bound.}
For a deterministic interior point, RSW circuits in successive
concentric annuli obstruct an actual arm to a fixed positive distance.
The probability that a macroscopic loop approaches that point tends
to zero with the inner radius.  A countable union over diameter
cutoffs gives (i).  Winding is stable under uniform convergence away
from the point, and each simple discrete loop has absolute winding
$0$ or $1$.

Three distinct positive visits near an interior point would yield six
medial passages through arbitrarily small annuli out to a fixed
positive scale.  This includes visits belonging to different collection
elements.  In a trimmed intermediate annulus, order those radial
separators and assign their actual color arms to the intervening
strips.  Arms in distinct strips stay separated.  Where a strip
receives two assignments retain both if their colors differ, and one
if their colors agree.  Each separator marks a color transition, so
the resulting cyclic sequence has an alternating subfamily of at
least six arms.  Radial buffers justify all croppings.  The six-arm
bound of \citet[Corollary~6.7]{DMT2021Fractal}, transferred to this
positive square law through a bulk collar, has exponent $2+c$ for
some $c>0$.  A grid union over possible centers therefore vanishes
like a positive power of the inner scale.  Taking mesh limits first
proves (ii).

\paragraph{A circuit attached to each macroscopic bank.}
Fix a bulk core rectangle, with buffer space around it, and a color
$X$.  We claim that, with probability tending to one as $m\to\infty$
after the mesh limit, the actual cluster of every bottom-to-top core
crossing contains an $X$-open circuit in the buffered patch enclosing
a disk of radius $c/m$.

In a first subrectangle ending at a middle row, label the actual
connected components joining its two terminal rows.  Their number is
tight by the passage bound.  Reveal the past up to that middle row.
Above it use $m$ fresh transverse slabs of height $h\asymp1/m$,
with all levels rounded on the $X$-grid.  For a given label, its
sources are its actually connected bottom vertices.  At each new row
ask whether these sources connect to that row in the cumulative core
rectangle.  A full crossing survives all these tests under one of
the initial labels.

When a label survives to a row, choose the leftmost simple open path
from its sources to that row.  The edges strictly to its right are
fresh with domain Markov data.  Indeed attach auxiliary spokes from
one point below to the sources and from the new top row to a point
above, in a disk with these auxiliary endpoints on its boundary.
Simple paths use spokes only at their ends.  Minimizing left area
selects a path determined by its own edges and those on its left:
any path entering strictly left supplies an open excursion between
two distinct selected-path vertices that decreases the left region.
The ordinary lattice part $p$ is actually connected to the label.

At the slab's middle height, approach horizontally from the right
exterior of the core and choose $x$ on the right at distance $h/16$
from $p$.  The ball $B(x,h/3)$ lies in the wider fresh patch, far
from the past, spokes, and slab ends.  The fresh graph on the right
of $p$ inside a slightly buffered ball dominates the restriction of a
free comparison on the full ball graph.  All attachments from omitted
left-side or path edges go through vertices of $p$, which are wired
by actual connectivity in the conditioned law.  This argument applies
to either color, using the dual FK law for dual $X$.

By interior RSW and FKG in the free ball comparison, with probability
at least a fixed $c'>0$ there is an open circuit between radii
$h/64,h/24$, connected there to one between radii $h/8,h/4$.
The outer circuit meets the deterministic geometric path $p$, since
$p$ comes within $h/16$ of $x$ and exits the ball.  Both use the same
color graph.  The inner circuit is wholly on the right of $p$, as is
its connection up to first contact with $p$.  Therefore this event
supplies an actual attached disk-circuit using only fresh right-side
edges in the slab.  Conditionally on survival, failure to acquire such
an attachment costs a factor at most $1-c'$ at each stage.  Truncate
the initial number of labels and iterate.  This proves the claim.

\paragraph{Visibility of both winding statuses.}
Use countably many core tests with rational positions and buffered
sizes.  A varying portion of a limiting loop in an interior ball
crosses two horizontal or vertical levels of one such test with
clearance.  In its discrete approximants the two actual color banks
supply crossings.  Their clusters lie on opposite sides of the simple
loop and do not cross it.  Choose the buffered test much smaller than
the loop's diameter, so neither of the attached disk-circuits can
contain the entire loop.  The disks strictly inside the two circuits
then have the two corresponding winding statuses.  Shrink them to
retain positive clearance; subsequential limits preserve disks of
positive radius in the prescribed ball.  The preceding estimate makes
failure arbitrarily unlikely.  Every neighborhood of a point on a
nonconstant loop contains a varying portion, so the countable tests
prove (iii) simultaneously.
\end{proof}

\subsection{Canonical loops and their continuation in the tree}
\label{loop:canonical}

Take joint subsequential couplings of the loop collections and of the
angular trees for a countable dense set $\mathcal Z$ of deterministic
interior targets, with arbitrarily large finite clock horizons.
Lemma~\ref{loop:geometry}(i) ensures that all these targets avoid all
positive limiting loops almost surely.  Incidental lattice-edge
positions do not require an extra convention: each active finite
horizon retains a ball around its target.

We specify the established continuum input before making the lattice
identification.  For $4<\kappa<8$, the target-invariant tree of
\citet[Sections~4.2--4.3]{Sheffield2009Exploration} constructs the
surrounding loops by full-turn angular excursions and completes each
partially traced loop through the other branches.  At the time of
that paper, its stated continuity and invariance conclusion in
Proposition~5.1 was conditional.  Here the needed hypotheses follow
from \citet[Theorem~1.3]{MillerSheffield2016IG1} and
\citet[Theorem~1.2 and the ensuing CLE discussion]{MillerSheffield2016IG3}.
For our single one-sided force weight, $\kappa-6>-2$, so force
collisions do not reach the continuation threshold.  For reversal,
the two one-sided weights $\kappa-6$ and $0$ are both at least
$\kappa/2-4$.  The starting square is a Jordan domain, and all
specified targets are distinct interior points off the loops.  These
results supply continuous non-simple CLE loops and their conformal
and anticonformal invariance under exactly the parameters used here.
They assert no lattice convergence or traversal identification.

The tree in Proposition~\ref{tree:joint-law} therefore has a coupled
canonical nested $\mathrm{CLE}_\kappa$ collection in the square.
Starting from a primal wall, let $b_0=0$, and let $b_1,b_2,\ldots$
be the successive hits of the alternating opposite endpoints:
first $2\pi$, then $0$, and so on.  Let $a_k$ be the last visit to
the starting endpoint before $b_k$.  The canonical surrounding loop
is denoted $\widetilde L_z^k$.  It follows the branch toward $z$ on
$(a_k,b_k)$ and is then completed to its starting point by the tree's
continuation of the current strand.  These are the successive loops
with nonzero winding about $z$, and every canonical loop is obtained
from a target in $\mathcal Z$.

The endpoint reversal amounts to using reflected exploration chirality
in the new enclosed component.  Nesting supplies independent CLEs in
those components, and anticonformal invariance preserves their
unoriented laws.  Thus the alternating endpoint convention gives the
same nested ensemble.  The scalar full-turn description is also the
one in \citet[Proposition~2]{SchrammSheffieldWilson2009}; that result
identifies the conformal-radius decrement, while the joint branch
coupling and loop completion used here come from the exploration-tree
construction.  Endpoint accessibility of the diffusion with
$1<\mathfrak d<2$ makes every $b_k$ finite, and $b_k\to\infty$:
otherwise its continuous angle would make infinitely many full
$2\pi$ swings on a compact clock interval.

We will need the following concrete interpretation of the completion.

\begin{lemma}[Default continuation and local double visits]
\label{loop:continuation}
At an interior-angle time $s\in(a_k,b_k)$, the remaining traversal
of $\widetilde L_z^k$ is the default strand from $W_s$ to $O_s$ in
the current domain.  In a chart sending those accesses to $0,\infty$,
every finite portion of this continuation is ordinary chordal
$\mathrm{SLE}_\kappa$ and is recorded by branches to targets from
$\mathcal Z$ that remain accessible high above that portion.
Every open nonconstant traversal interval of $\widetilde L_z^k$
contains two visits to one square-interior point.
\end{lemma}

\begin{proof}
The stopped chordal equivalence is only valid while its force access
is available.  To continue after the original target is cut off, use
the branch that retains this access, as prescribed by the tree
construction.  A bounded finite chordal-capacity portion has bounded
hull and driver.  An open set of targets mapped to sufficiently large
heights remains accessible above it, with tip and force angles strictly
separated on that portion.  The countable dense set supplies such a
target.  Its branch records the default continuation even if it has
already separated from $z$.  Target invariance gives consistency when
the target or the finite portion is enlarged.  This realizes all
finite portions and hence the complete strand; it does not extend
Proposition~3.12 of \citet{Sheffield2009Exploration} beyond its stopping
condition.

All these curves are continuous.  Over a finite trace time the
remaining domains have locally connected boundaries, by the
locally-connected-continuum boundary theorem applied to the continuous
trace and the starting boundary.  Their prime-access conformal images
can thus be regarded as continuous physical curves.

For ordinary chordal SLE, \citet[Theorem~1.1 and Corollary~1.7]{MillerWu2017}
give double points in the interior when $4<\kappa<8$.  Indeed their
double-point dimension exceeds the boundary-hit dimension by
$(\kappa-4)^2/(8\kappa)>0$.  Both visits occur at finite times.
Scaling makes the probability of such a double visit before time $t$
independent of $t>0$; since its union over finite times has probability
one, this probability is one for every $t>0$.  The domain Markov
property then gives an interior double visit in every rational open
capacity interval.  Conformal maps preserve an interior double visit.
Every open traversal interval of the canonical loop contains an
interior-time portion either of its full-turn excursion or of its
finite default continuation.  On a sufficiently short localized
portion its law is ordinary chordal SLE.  The preceding arbitrary-short-time
statement applies before any positive localization time,
proving the assertion.
\end{proof}

\subsection{Episodes and full-turn labels}
\label{loop:episodes}

A discrete \emph{episode} starts when a cap is opened and ends at the
next switch retaining a homogeneous pocket.  Its color is the
homogeneous wall color just before the opening.  Call it successful
when its loop winds around the target.

\begin{lemma}[Discrete winding order]\label{loop:episode-order}
As long as the target is macroscopically separated from a consumed
tile or star, an episode is successful exactly when its terminal
switch changes wall polarity.  These polarity changes enumerate all
loops surrounding the target, in their order from outside inward.
A positive loop with positive clearance from the target is reached
before a finite radial horizon depending on that clearance.
Conversely a successful switch by a bounded horizon has macroscopic
clearance from the target.
\end{lemma}

\begin{proof}
The loop through the newly opened cap lies inside its homogeneous
starting pocket.  At the cap, the side toward the base, of the
starting color, is exterior to this loop; the other side is interior.
During the episode the actual color banks preserve connections to
these sides.  If a bank hits its same-color wall, the retained
homogeneous target pocket is connected to the cap along that bank.
The traversed loop is outside the pocket: its past pairs have been
removed, the new open diagonal is a barrier, and the continuation to
the old force side stays in the main component.  The target can reach
that pocket's wall color without crossing the loop.  Comparing this
color with the starting color proves that the target is inside the
loop precisely at a polarity change.  The same argument works with
separate boundary occurrences.

Every other unstarted candidate surrounding the target stays intact
in the retained component.  Actual diagonals and walls cannot cross
it, and a loop wholly inside a simply connected pocket has its
interior there too.  Its deterministic pairs in any wall triangles
are also intact, because cuts by other loops never traverse them.
If the loop entered at a new cap surrounds the target, it is outermost
among the remaining candidates: its access to the homogeneous wall
at that cap crosses none of them.  Thus successful episodes enumerate
the surrounding loops, with possible unsuccessful same-color episodes
between them.

Before such a loop's episode, growth remains outside its enclosed
region; during the episode it follows that loop up to step-sized
errors.  A ball centered at the target inside its region therefore
survives until the switch.  If its radius is $r>0$, then
$\operatorname{crad}(U_t,z)\ge r$ up to that time, bounding the
radial clock by $\log(\operatorname{crad}(U_0,z)/r)$.
Conversely, Koebe's bound on a fixed horizon gives a positive-radius
ball in the newly retained pocket, and its surrounding successful
loop lies outside that ball, up to vanishing step errors.
\end{proof}

An endpoint approach could otherwise be mistaken for a reset in the
limit.  The next estimate excludes this possibility.

\begin{lemma}[Endpoint resets cannot be skipped]
\label{loop:no-skip}
At a bounded-clock completed state with $\theta$ approaching $0$,
the probability of returning to a fixed positive angle before a
primal-pocket reset tends to zero, with the mesh limit first and the
endpoint tolerance second.  The analogous assertion near $2\pi$
uses a dual-pocket reset.  Consequently the true successive discrete
polarity-change times $b_k^n$ converge in probability to $b_k$ on
sufficiently large finite horizons.

Moreover, in an episode that succeeds by a bounded horizon, after a
fixed interior-angle advance the probability of retreating arbitrarily
near its starting endpoint before its successful switch tends to zero
in the same order of limits.
\end{lemma}

\begin{proof}
Near $0$ the entire dual wall arc and both end sides are in a tiny
boundary neighborhood of the stopping chart.  Until a primal reset,
the growing dual bank remains actually connected to that arc; stop
also before a fixed interior-angle threshold, so no opposite-end
switch has yet occurred.  In successive buffered dyadic chart
half-annuli around this neighborhood, place actual primal transversal
paths from primal wall to primal wall.  Lemma~\ref{peel:wall-rsw}
gives a uniformly positive conditional success chance at each
separated scale, after the mesh limit at the chosen cutoffs.  A
successful path confines the connected dual bank, and hence the slit
pieces, inside its small wall sector.

A hull confined to such a sector has a small new angular footprint:
its basepoint-side maps converge to the identity at the origin and,
by reflection, on each closed boundary arc away from that sector.
The trapped dual arc remains in its short footprint.  Therefore it
cannot produce a fixed positive angular advance.  Letting the number
of half-annuli grow proves the first assertion.  Exchange colors and
reverse directions at $2\pi$.

A true polarity change approaches the specified opposite endpoint,
up to vanishing update errors.  Uniform angle convergence and induction
therefore preclude a premature macroscopic limit for $b_k^n$.
Conversely, after the limiting first endpoint hit, instantaneous
reflection produces a positive excursion within any prescribed
additional time almost surely.  Truncate the clocks and choose an
exit size so that failure has arbitrarily small probability.  Start
at the first completed-state approach within a small tolerance after
the previous true polarity change.  The just-proved estimate excludes
an excursion of this size before the corresponding reset.  Let the
tolerance tend to zero.  An earlier matching reset already suffices.
Thus no endpoint hit is skipped, and $b_k^n\to b_k$.

For the final assertion, stop at successive approaches to the starting
endpoint after a fixed-size advance.  A successful episode that makes
such an approach must then advance macroscopically without the
corresponding same-color reset, since that reset would end the episode.
The first assertion bounds each such event.  The number of fixed-size
swing tests is tight by angular tightness, so truncation and iteration
give the stated retreat estimate.
\end{proof}

It follows in particular that on every compact subinterval of
$(a_k,b_k)$, the prelimit target branch is, with probability tending
to one, in the same episode as at its $k$th successful switch.  The
angle stays away from the starting endpoint on that compact interval,
and the preceding polarity change occurs before it.

Include the true winding-loop labels $L_z^{n,k}$ in the coupling and
extract their limits $L_z^k$, increasing clock and diameter cutoffs
as needed.  They have positive size at bounded $b_k^n$ by
Lemma~\ref{loop:episode-order}.  They exhaust the positive limiting
loops winding about $z$: any such loop has positive clearance from
$z$ and is reached before a bounded horizon, while the number of full
swings there is tight.  Winding tests are stable.  For finite lists of
labels stabilize also the indicators of equality as collection
elements and then diagonalize.  Distinct elements in this bookkeeping
have distinct discrete representatives, even if their geometric limits
might initially appear identical.

\subsection{Ordered containment and equality of the labels}
\label{loop:matching}

\begin{lemma}[The canonical traversal is contained in order]
\label{loop:ordered}
For each $z\in\mathcal Z$ and $k\ge1$, the entire traversal of
$\widetilde L_z^k$ occurs in cyclic order in $L_z^k$, possibly with
extra traversal portions inserted.  Its order occupies at most one
turn of $L_z^k$.  If two canonical labels give the same element, their
lattice-limit labels give the same collection element.
\end{lemma}

\begin{proof}
Fix a rational $s\in(a_k,b_k)$.  In any open subinterval of
$(a_k,s)$ choose $t<t'$ strictly inside it.  The target kernel strictly
shrinks between these times.  There is a new boundary point inside
the earlier open kernel; it belongs to the continuous canonical trace
over $[t,t']$.  Every small ball about this point with closure in the
earlier kernel must encounter new prelimit growth between the
corresponding times.  Otherwise the ball stays clear of new boundary.
It also meets a compact internal patch of the later kernel.  Connectedness
then puts that whole ball in the retained prelimit component, extending
the limiting kernel through its alleged boundary point, a contradiction.
These prelimit boundary pieces are slit updates tracing the successful
episode, up to vanishing physical step errors.

Apply this test simultaneously to any finite ordered list of such
intervals.  Completed-update stops can be moved slightly inside them
because capacities and angular within-step errors vanish.  We obtain
points of the discrete winding-loop traversal, in chronological order,
converging to canonical trace points in those intervals.

Now take any finite list of finite chordal intervals on the default
continuation after $s$.  Choose, realization by realization, a target
$w\in\mathcal Z$ in the current kernel whose chart image is safely
above the entire bounded portion in question.  It was still accessible
with $z$ up to $s$.  Their time correspondence there is regular and
passes to the mesh; they cannot have split earlier because the limiting
kernel contains internal connecting paths with clearance.  The branch
toward $w$ follows the default continuation with tip and force angles
separated by a positive margin on the selected portion, by
Lemma~\ref{loop:continuation}.  Therefore its discrete branch has no
reset there and continues the same original winding loop, in order,
after the common prefix.  The new-boundary test applies to these
intervals in the $w$-clock.  The target $w$ need not have a successful
episode for this loop.  Its post hoc choice costs no uniformity
assumption: all targets, clocks, and regular sharing correspondences
from the countable set have already been included in the coupling.

By continuity, shrinking the testing intervals and increasing the
finite continuation range shows that every finite increasing list of
canonical traversal points has representatives in order on $L_z^k$.
On a countable dense set of canonical parameter values, take a
single nondecreasing cyclic choice of representative parameters by
compactness and a diagonal subsequence.  Omitted endpoint values are
included by continuity.  The order uses at most one turn: the discrete
episode and its no-reset continuation traverse a simple closed loop
from its released cap without passing that cap a second time.
Two separated canonical visits cannot collapse to one representative
parameter, because every intervening open traversal interval has
actual variation.

If $\widetilde L_z^k=\widetilde L_w^j$ but the two lattice-limit
elements were distinct, each would contain the same canonical
interior double visit from Lemma~\ref{loop:continuation}.  This would
give more than two total limiting visits at that point, contrary to
Lemma~\ref{loop:geometry}(ii).  Thus equal canonical labels have equal
lattice-limit labels, with multiplicity understood as stipulated above.
\end{proof}

The converse is essential: ordered containment alone would permit
several canonical labels inside one discrete limiting element.

\begin{lemma}[The converse label identification]
\label{loop:label-equivalence}
Almost surely, simultaneously for all $z,w\in\mathcal Z$ and
$k,j\ge1$,
\begin{equation}\label{loop:label-iff}
 L_z^k=L_w^j
 \quad\Longleftrightarrow\quad
 \widetilde L_z^k=\widetilde L_w^j
\end{equation}
as collection elements.
\end{lemma}

\begin{proof}
Only the forward implication remains.  On the event that the discrete
labels are shared, the two branches cannot split before entering this
common loop: before its episode the loop and its interior remain
intact across every cut by other loops.  Choose clock horizons large
enough to include both successful switches except on an arbitrarily
small event.

While they share a domain, write $\xi=\Phi_z(w)$.  For a fixed small
$e>0$, stop at first approach to $|\xi|\ge1-e$, if it occurs before
the relevant winding switches.  For fine mesh this can be a completed
state also satisfying $|\xi|\le1-e/2$.  An exact split cannot precede
this threshold inside the comparison range: the ODE and clock change
are regular for $|\xi|\le1-e/2$, and elementary capacities vanish
there.

We first show that an \emph{unstarted} loop surrounding both targets
at this stop has probability tending to zero as $e\downarrow0$.
Place separated dyadic chart half-annuli around the boundary direction
of $\xi$, between radius a multiple of $e$ and a fixed small radius.
Choose their buffered wall intervals to avoid the two marked accesses
and sides; deleting the bounded number of affected scales leaves a
number tending to infinity.  Each bar separates a sector containing
the radial access from $w$ to the wall from the sector containing $z$
and a remote wall portion.

Use an actual-color transversal path in each bar, choosing the color
to match one wall end.  At that end it reaches an actual wall vertex.
At the other it does so as well if the wall has the same color.  If
that wall has the other color, complete the path from its specified-color
triangle corner to the base by the segment of
Lemma~\ref{peel:wall-rsw}.  After trimming at first wall contacts this
is a simple crosscut in the prime closure.  Its added segment meets
only the one local cap pair, once.

An unstarted simple loop inside the domain surrounding both targets
must cross this screen at least twice: it must meet both of the
separated sectors, and its interior contains both targets.  It cannot
cross the actual open path, and cannot traverse that one cap pair
twice.  Thus one successful screen precludes the event, even with
mixed wall colors.  The conditional success probability of each
fixed-aspect buffered bar is bounded below by
Lemma~\ref{peel:wall-rsw}.  Iteration, with the mesh limit first,
proves the asserted small probability.

We next obtain a shared interior precursor for a shared successful
loop.  If the common loop is processed before the near-separation
threshold is needed, its episode visibly passes through interior
angles, comparable in the two charts while $|\xi|\le1-e/2$.
If the loop is still current at the threshold, its $z$-angle cannot
be arbitrarily close to that episode's starting endpoint and later
succeed without a same-color reset, by Lemma~\ref{loop:no-skip}.
Otherwise, its progression from the start to the current state has
already passed a fixed interior margin.  The $w$-angle at that shared
state has a comparable interior margin, with constants allowed to
depend on $e$.  The unstarted case has just been made unlikely.
Thus, except on an arbitrarily small event, the two successful
episodes have a common state before kernel separation, with both
angles interior by fixed positive margins.

From this precursor to the respective winding switches, neither angle
returns arbitrarily near its episode's starting endpoint, by the
successful-episode retreat estimate in Lemma~\ref{loop:no-skip}.
Choose in order the clock cutoff, the near-separation threshold, the
interior margins, and the retreat tolerance, making their error
probabilities successively small.  With strict buffers a subsequential
precursor remains in a common kernel with $|\xi|\le1-e/2$, lies after
both previous polarity changes, and lies in the interiors of both
limiting successful excursions.  Their endpoint histories and starting
excursion time agree along the common branch by the regular chart
correspondence.  They share the successful front to this interior
state and use the same default continuation to the same force access
for its rest.  Therefore their canonical loop elements coincide.
Let all the preceding errors tend to zero.  Countability of the
labels proves the simultaneous assertion.
\end{proof}

\subsection{Trace and traversal identification}
\label{loop:traversals}

\begin{proposition}[Identification away from the square wall]
\label{loop:identification}
Every positive lattice-limit loop with an interior point has a
canonical label.  Matched elements have the same winding indicators
on $\mathcal Z$.  If either matched element stays away from the square
wall, the two elements are equal as curves modulo reparametrization
and orientation.  Without that restriction, every canonical element
is still contained in order in its lattice-limit partner.
\end{proposition}

\begin{proof}
The two-winding-disk property of Lemma~\ref{loop:geometry}(iii)
provides a disk of winding one for every positive limiting loop with
an interior point.  It contains a target from $\mathcal Z$, so the
loop is one of its exhausted winding labels.  For a labeled loop,
Lemma~\ref{loop:label-equivalence} makes its dense-target winding
indicators identical to those of its canonical partner: winding one
at a target is equivalent to equality with one of that target's
successive surrounding-loop labels, in both collections.

Suppose an interior trace point of the lattice-limit loop were outside
its canonical partner trace.  A small ball about it avoids that
canonical trace and has constant canonical winding.  The two-winding-disk
property supplies disks of both lattice winding statuses in the
same ball, each containing a dense target.  This contradicts the
indicator equality.  Hence the entire interior lattice trace lies
on its partner.  The reverse trace inclusion follows from ordered
containment.  If the canonical partner is internal, the connected
lattice trace containing it cannot reach the wall without adding an
interior point outside that partner.  If the lattice trace is internal,
so is the contained partner.  Thus in either wall-away case the full
traces are equal.

It remains to exclude extra traversal of this trace.  Invert the
nondecreasing cyclic representative selection in
Lemma~\ref{loop:ordered} by a generalized inverse.  Because every
canonical open traversal interval has actual variation, this inverse
is continuous, nondecreasing, and of degree one.  The traces agree at
all its non-plateau-interior times.  A plateau would insert an extra
portion of the lattice traversal anchored at one canonical point.

Assume such a portion varies.  By trace equality it lies on the
canonical trace.  At a visited value different from its anchor there
can be only one canonical parameter: two canonical visits plus this
inserted visit would violate the total-visit bound of
Lemma~\ref{loop:geometry}(ii).  On a varying subarc away from the
anchor, the assignment to that unique canonical parameter is
continuous, by compactness.  It therefore sweeps a nontrivial canonical
traversal interval all of whose values have unique canonical visits.
This is impossible by the interior-double-visit property of
Lemma~\ref{loop:continuation}.  Hence every inserted portion is
constant.  Equality modulo reparametrization follows, including the
preserved multiplicity of collection elements.
\end{proof}

\subsection{From squares to the free plane law and the sphere}
\label{loop:wholeplane}

\begin{theorem}[Nested full-plane convergence for $q<4$]
\label{loop:plane}
For every fixed $1\le q<4$, the loop collection of the free
infinite-volume critical square-lattice FK law converges, through the
full mesh sequence, to the nested whole-plane
$\mathrm{CLE}_{\kappa(q)}$ in the spherical loop-matching topology
of Subsection~\ref{intro:collections}.  The limit is M\"obius invariant and includes every
nested generation.
\end{theorem}

\begin{proof}
We first give the uniform comparison estimates that allow an upgrade
from a local limit.  For a fixed compact disk $K$, the probability
that any discrete loop visits $K$ and reaches a distant surrounding
circle tends to zero as that circle tends to infinity, uniformly for
sufficiently small meshes and for larger initial wired squares.
Actual monochromatic circuits in buffered annuli block such a loop;
conditional RSW gives a fixed success chance at each separated scale.
Use a slightly enlarged $K$ and strict radial margins when passing
to limits.

The free infinite-volume law and the homogeneous-square law can be
coupled identically on a growing intermediate disk with probability
tending to one as the square grows.  Bracket the exterior partition
by the minimal and maximal partitions, use monotone FK coupling, and
expose actual boundary-attachment clusters in the maximal sample.
Dual RSW screens make their probability of reaching the deep interior
tend to zero.  Once exterior influence is separated, the remaining
conditional laws agree and one uses identical switches.  Combining
this with the preceding escape estimate makes the \emph{entire}
loops visiting $K$ coincide, since those loops stay in the intermediate
disk.  This argument uses the specified free plane law and FK domain
Markov coupling; it assumes no interface limit.

In a fixed-square subsequential coupling,
Proposition~\ref{loop:identification} identifies all wall-away
macroscopic loop elements, with their traversals.  Its remaining
possible discrepancies involve the wall.  The escape estimate makes
it unlikely that a lattice-limit loop both hits $K$ and reaches that
wall.  The same holds for a canonical CLE loop, because that loop is
contained in its lattice-limit partner.  Thus, when the square is
large, the square CLE and the free plane lattice law have arbitrarily
close matching configurations on loops visiting $K$, in the
successive mesh and square limits.

The large-domain local limit of nested CLE is established in
\citet[Appendix~A, Theorem~A.1]{MillerWatsonWilson2015}.  Its comparison
uses conformal identifications of the relevant random-region loop
configurations, with maps approaching the identity on compact sets;
it does not assert equality at unchanged Euclidean coordinates.
The work cited here is \emph{The conformal loop ensemble nesting
field}, arXiv:1401.0218, whose Appendix~A contains that theorem.
For $4<\kappa<8$ this construction agrees with the whole-plane
branching construction by
\citet[Remark~2.6]{GwynneMillerQian2021}, and its law is invariant
under inversion, hence under M\"obius maps, by their Theorem~1.1.
Centered squares and disks give the same local limit: normalize their
conformal radii, and the comparison maps tend to the identity on
compact subsets.  All these parameter restrictions hold for our
$4<\kappa\le6$.

To pass from this local law to the required spherical topology, fix
$\epsilon>0$ and choose $K$ so large that its exterior has spherical
diameter less than $\epsilon$.  Loops entirely outside $K$ are then
irrelevant at that matching accuracy.  The uniform escape bounds
control the loops that visit $K$ but leave a much larger disk.  They
hold for square CLEs uniformly as the squares grow, by their
subsequential square-law couplings and the containment argument above.
Consequently the laws of the square CLEs are Cauchy for spherical
matchings: compare two large squares on a large fixed intermediate
disk to the same free-plane lattice law, let the mesh tend to zero,
and use the escape tails and the exterior diameter bound.

Any resulting spherical limit has the established local whole-plane
CLE law.  That local law, together with these tails, determines the
full spherical collection: at each accuracy all relevant loops are
seen in a bounded window except on an arbitrarily small event.
Conversely, the same comparisons force every subsequence of the
free-plane lattice laws to this limit.  Hence convergence holds for
the full mesh sequence.  The target labels ranged over all $k\ge1$
and the clock cutoffs increased without bound, so no nested generations
have been discarded.  This proves Theorem~\ref{intro:main}(ii) for $1\le q<4$.
\end{proof}
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//peeling.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//preliminaries.tex ====
\section{Tiles, positive laws, and localization}\label{sec:preliminaries}

Except where an endpoint statement is explicitly made, fix $1\le q<4$ and write
\begin{equation}\label{prelim:parameters}
 d=\sqrt q=2\cos\lambda,\qquad \rho=\lambda/\pi,
 \qquad 0<\lambda\le\pi/3.
\end{equation}
All constants may depend on this fixed parameter and on the fixed geometric
shapes under consideration. In particular, none of the estimates below is
asserted uniformly as $q\uparrow4$.

\subsection{Tiles and boundary caps}

We use a square tile array whose corner sites alternate between primal
($\mathsf P$) and dual ($\mathsf D$) sites. The primal lattice consists of the
primal diagonals, so its mesh is $\sqrt2$ times the tile-edge length. Rotating
the tile coordinates gives the prescribed orientation of the primal square
lattice. Each tile carries one of the two noncrossing pairings of its four
side midpoints, called \emph{ports}. Pairing around the dual corners declares
the primal diagonal open; the other pairing declares the dual diagonal open.
An \emph{actual color path} uses these open diagonals and never uses boundary
identifications as edges. A strand has an actual primal path on one side and
an actual dual path on the other.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[scale=1.05]
 \begin{scope}
  \draw[gray] (0,0) rectangle (2,2);
  \draw[very thick] (0,0)--(2,2);
  \draw[thick] (0,1) arc[start angle=-90,end angle=0,radius=1];
  \draw[thick] (1,0) arc[start angle=180,end angle=90,radius=1];
  \node[below left] at (0,0) {$\mathsf P$};
  \node[above right] at (2,2) {$\mathsf P$};
  \node[above left] at (0,2) {$\mathsf D$};
  \node[below right] at (2,0) {$\mathsf D$};
  \foreach \x/\y in {0/1,1/0,2/1,1/2}
    \fill (\x,\y) circle (1.5pt);
  \node at (1,-.65) {primal open};
 \end{scope}
 \begin{scope}[xshift=5cm]
  \draw[gray] (0,0) rectangle (2,2);
  \draw[dashed,gray] (0,0)--(2,2);
  \draw[thick] (1,0) arc[start angle=0,end angle=90,radius=1];
  \draw[thick] (2,1) arc[start angle=-90,end angle=-180,radius=1];
  \node[below left] at (0,0) {$\mathsf P$};
  \node[above right] at (2,2) {$\mathsf P$};
  \node[above left] at (0,2) {$\mathsf D$};
  \node[below right] at (2,0) {$\mathsf D$};
  \foreach \x/\y in {0/1,1/0,2/1,1/2}
    \fill (\x,\y) circle (1.5pt);
  \node at (1,-.65) {primal closed};
 \end{scope}
\end{tikzpicture}
\caption{The two tile switches. The solid diagonal is an open primal
edge; the dashed diagonal records the absent primal edge. The curved
arcs specify the pairing of ports and hence the order of strand
occurrences when a medial vertex is visited twice.}
\label{prelim:switch-figure}
\end{figure}

For topological arguments, draw the two arcs of a switch disjointly within
its tile, normally to the sides at their ports. This drawing and the straight
medial drawing can be parametrized piece by corresponding piece, preserving
the order of occurrences at a double visit. Their uniform distance is at
most twice the tile diameter. Thus these local changes of drawing disappear
in the curve topology of the theorem.

A wall cuts tile connections at ports. Distinct sectors at a wall corner
have distinct copies of that corner, as on a surface cut open. The end of a
finite slit has one sector of angle $2\pi$. All shapes in the tile arguments
are unions of whole tiles; separating repeated boundary incidences by a
small thickening leaves their combinatorics unchanged.

Partition each boundary component into intervals of alternating designated
wire colors. Within an interval, pair the two intervening ports between
successive occurrences of the designated color, including eligible end
vertices. These are the \emph{local caps}. A genuine change leaves one
unmatched port, on the side whose color differs from the change vertex.
Use the analogous cyclic caps on a boundary component without changes.
Intervals will always be much longer than one mesh.

A \emph{positive cap completion} joins the unmatched ports around the primal
intervals in the added boundary disk. Equivalently, wire each primal
interval separately, with a dual sea between these wires. This adds only
boundedly many connections beyond the local caps when the number of changes
is bounded. It adds no primal identification on a dual interval.

\begin{lemma}[The positive loop law]\label{prelim:loop-law}
On a genus-zero tile surface with positive cap completions, the switch
weight $d^\ell$, where $\ell$ counts completed medial loops, is proportional
to the critical FK weight with the corresponding boundary partition.
Conditional laws on common subdomains are therefore FK laws with induced
partitions. The same assertion holds after interchanging the colors.
On a torus without caps the two weights differ by a factor bounded in terms
of $q$ alone.
\end{lemma}

\begin{proof}
Draw noncrossing exterior primal trees realizing the designated
identifications and thicken the resulting open primal graph. Its perimeter
is the medial loop collection, apart from a fixed number of exterior
components. Euler's formula gives the configuration-dependent part of the
perimeter count as
\[
 \ell=2k_\xi(A)+|A|+\text{constant}.
\]
Since $d^2=q$ and $p_q/(1-p_q)=d$, the weights agree up to a constant
independent of $A$. The domain Markov property gives the conditional-law
statement. On a torus the thickened graph has Euler characteristic
$|V|-|A|=2k-\ell-2g$, with $g\le1$, so the additional factor is bounded.
\end{proof}
\subsection{Crossing estimates and finite plane loops}

We use two distinct crossing inputs. The buffered bulk box-crossing theorem
for canonical weights on doubly periodic isoradial graphs applies for
$1\le q\le4$ \citep[Theorem~1.1]{DCLM2019Isoradial}. Square rhombi have
canonical odds $\sqrt q$, so this includes our square lattice. For example,
for fixed $a>0$, horizontal crossing of
$[-an,an]\times[-n,n]$ inside
$[-(a+1)n,(a+1)n]\times[-2n,2n]$ has probability bounded away from zero
and one, uniformly in the boundary partition on the larger rectangle.
The same theorem gives equality of the free and wired infinite-volume
laws. Hence planar duality at these self-dual odds preserves the plane
loop law, including when a tile translation exchanges the two colors.

When a crossing must reach a domain wall, we use the stronger theorem for
critical square FK with $1\le q<4$: a discrete simple-polygon quad of
bounded extremal distance has an actual open crossing with probability
bounded away from zero, uniformly in its boundary partition
\citep[Theorem~1.2]{DMT2021Fractal}. We do not apply this statement at $q=4$.
FKG and the domain Markov property apply because $q\ge1$; duality supplies
the corresponding statements for the other color.

Here is the concrete wall application needed below. A tile-coordinate
rectangle contains a nearest-neighbor quad of either color graph with
staircase sides within boundedly many mesh layers of its boundary.
For bounded aspect ratio its discrete extremal distance is bounded:
there are order its width in mesh units disjoint zigzag paths, each of
length at most a constant times its length; distributing a unit flow among
them bounds the resistance. The geometric staircase quad likewise has
bounded modulus. At an electrode on a wall, the last crossing vertex can
be connected to a wall vertex of the same color using boundedly many fresh
edges. Conditional finite energy bounds the extra cost below uniformly.
Keep the electrode away from the ends of its designated wall interval.

Crossings in a buffered piecewise rectangular corridor follow by using
strips of width a fixed fraction of its clearance and overlapping end
squares. Transversal crossings in these squares join successive
longitudinal crossings; FKG combines the finitely many requirements of one
color. This constructs wall-to-wall paths in annular sectors with openings
$\pi/2,\pi,3\pi/2,2\pi$, including between the two banks of a slit, as
well as circuits in a full annulus. Different colors in tile-disjoint
sectors can be imposed successively by their conditional lower bounds.
All paths here are actual paths. Radii may use tile sup-norm rims, and all
buffers have fixed relative width and sufficiently large fixed mesh width.

\begin{lemma}[Finiteness in the plane]\label{prelim:finite-loops}
For every $1\le q\le4$, all medial strands of the specified free
infinite-volume square FK law are almost surely finite loops.
Finite deterministic surgeries admit positive normalized loop laws with
the FK conditional laws of Lemma~\ref{prelim:loop-law}.
\end{lemma}

\begin{proof}
Buffered bulk crossing gives a conditional probability bounded below for
an actual primal circuit around any fixed box in each of successively
separated annuli. The probability of missing the first $m$ such screens
is at most $(1-c)^m$, regardless of the exterior partition. The same
argument gives arbitrarily large actual dual circuits. These two colors
of screens prevent a medial strand meeting the box from escaping to
infinity. Taking the countable union over boxes proves finiteness.

A surgery at finitely many tile connections changes only finitely many
loops at a fixed mesh. Relative to the unmodified plane law, its weight
ratio is $d^{\Delta\ell}$, bounded above and below by constants that may
depend on this finite set of connections. Normalization defines its
positive law. Equivalently, exhaust by planar domains; the same circuit
argument makes the local completion independent of remote boundaries.
Lemma~\ref{prelim:loop-law} then gives the conditional law.
\end{proof}

\subsection{Comparison through a collar}

\begin{lemma}[Bounded densities across a collar]\label{prelim:collar}
Suppose two positive laws have the same graph in a collar between radii
$r$ and $16r$. In this collar the graph is a full annulus, or a bounded
number of sectors separated by straight coordinate walls; coincident slit
banks are allowed. Their deterministic wire portions agree there. Only
boundedly many wire blocks traverse the collar, and no other
identifications jump it. One radial side faces an arbitrary unknown
partition, whereas the graph and deterministic wires toward the other
side are fixed. The marginal densities on edges beyond that other side
are mutually bounded by a constant independent of scale and of the far
graph. Equivalently, their expectations of every nonnegative remote
function are comparable with this same constant.

The conclusion also holds for a fixed-shape rectangular bulk collar
shielding a long box, with a constant depending on its shape.
\end{lemma}

\begin{proof}
First work on a finite graph and a finite remote edge set. Split each
through wire transversally, preserving its connections on each side and
retaining just one representative connection across the cut. The unknown
data induce a partition $\xi$ at their section. Include all common
deterministic wires for every choice of $\xi$.

For two ordered partitions, the likelihood ratio of the coarser law to
the finer law is increasing: the rank of the additional identifications
decreases as open components merge. Its marginal likelihood ratio on
the remote set remains increasing, by the domain Markov property and
conditional FKG. Consequently its maximum is attained at the all-open
configuration and its minimum at the all-closed configuration. It is
enough to compare those two events for the maximal and minimal partitions;
every other partition lies between them.

An actual primal crosscut in every sector, or a primal circuit in the
annulus, screens the remote graph with uniformly positive probability.
Such screens can be discovered from the unknown side without examining
edges strictly beyond them. In a sector, explore all actual dual paths
from its unknown-facing arc and query edges incident to the reached dual
vertices. If a primal wall-to-wall crossing exists, the search does not
reach the remote arc. The elementary complementary crossing alternative
then places a blocking primal path among the queried edges: otherwise,
setting the unqueried primal edges closed would produce a dual crossing.
The alternative follows by tracing the perimeter of the actual component
reachable from the starting side; corner gap vertices belong to the arcs
when their color permits. The annular version is the alternative between
a primal circuit and a radial dual connection. Perform the searches in a
subcollar leaving space before both sections. Reversing colors gives the
dual searches as well.

Condition on a successful primal search under the minimal law. Each
screen is already connected. Joining all screens and the bounded number
of through-wire representatives requires only boundedly many additional
block identifications. Each has an unnormalized weight ratio in
$[q^{-1},1]$, hence changes expectations by a bounded density factor.
The resulting partition dominates every partition transmitted from the
unknown side. In particular it dominates the maximal law restricted at
this same screen geometry, without requiring that law to contain the
screen. The positive probability of the search proves the desired
comparison of all-open probabilities.

For all-closed probabilities, make the actual dual searches under the
maximal law. No actual primal path crosses these screens. Only the
boundedly many through-wire representatives can still transmit ties;
remove these ties at bounded cost. The resulting free cut is
stochastically smaller on the remote graph than any other imposed cut
partition, including the restriction of the minimal law. This proves
the decreasing-event comparison. The extremal likelihood-ratio argument
now gives the assertion for every remote event and every nonnegative
remote function. Exhaustion proves the infinite-exterior version.
For a rectangular bulk collar use the corresponding enclosing-circuit
searches.
\end{proof}

\begin{lemma}[Relative comparison]\label{prelim:relative-collar}
Under the hypotheses of Lemma~\ref{prelim:collar}, suppose successive
collars have one of the following geometries:
\begin{enumerate}[label=(\roman*)]
\item a full annulus, or one sector whose through wires are precisely
continuous primal wire portions along its side rays;
\item two sectors, with primal wires along both side rays of one sector
and no through wires in the other, and no further fixed contacts
between the sectors.
\end{enumerate}
After $m$ further collars of a sufficiently large fixed radius ratio,
the remote expectations of every fixed nonnegative function under two
possible unknown-side laws have ratio
$1+O((1-c)^m)$, or both expectations vanish. The conclusion holds in
either radial direction, also after positive modifications or tilts
supported on the unknown side.
\end{lemma}

\begin{proof}
Use a radius ratio such as $256$ to leave two subcollars of the preceding
type. Lemma~\ref{prelim:collar}, applied nearer the unknown side, gives a
common marginal part of uniformly positive mass on the other subcollar.
Couple those configurations identically on a fixed fraction of a
reference marginal. In case~(i), additionally impose a primal screen
meeting and joining every through wire. In case~(ii), impose a primal
screen on the wired side and a dual screen on the other side. These
requirements have uniformly positive conditional probability. On their
occurrence the partition transmitted to the next section depends only
on the shared configuration: every route through the unknown region
must enter and leave through a primal screen already connected to its
through wires, and the other sector transmits no primal connection.

For a nonnegative remote function, take the largest and smallest
expectations over partitions at each successive section. Conditioning
nests these intervals of possible expectations. The common part just
constructed contracts their difference by a factor $1-c$. Their ratio
is bounded initially by Lemma~\ref{prelim:collar}; iterating therefore
gives the stated relative estimate. A positive unknown-side tilt is a
mixture over the same partitions. In particular, when propagating away
from a gap, wire connections made or lost inside that gap belong to the
unknown partition, which explains the need for both colors of screen
in case~(ii).
\end{proof}

\subsection{Passages and positive local tilts}

Raw strands use only tile switches and stop when they reach a wall port.
A passage count in a buffered annulus counts disjoint strand passages
through its prescribed inner and outer rims; the outer buffer allows
cropping at both rims.

\begin{lemma}[Exponential passage moments]\label{prelim:passages}
In any fixed buffered annulus or sector of the preceding types, a raw
passage count $X$ satisfies
\begin{equation}\label{prelim:passage-moment}
 \mathbb E[\exp(BX)\mid\text{exterior}]\le C_B
 \qquad(0\le B<\infty),
\end{equation}
uniformly in scale and in exterior states inducing the specified graph
and arbitrary boundary partition. The same holds for a fixed finite
family of macroscopic passage counts in a fixed geometry.
\end{lemma}

\begin{proof}
First obtain one positive exponential rate. Disjoint simple radial
strand subarcs split a slightly cropped annulus into ordered strips.
Along each passage is an actual primal crossing path. Paths in distinct
strips cannot join in the cropped graph, because open primal edges cannot
cross the intervening medial separators. Choose the endpoint rim bands
strictly inside the region traversed by the separators. Thus $X$ passages
force at least $X/2-O(1)$ different crossing components in the actual
cropped graph.

Explore complete actual components successively from one rim, ignoring
all external identifications when deciding membership in a component.
After a component is removed, its incident unexplored boundary edges
are closed. The surviving FK law is dominated by full wiring at the
original rims and walls; removing vertices can only decrease actual
crossings. Even with this full wiring an actual dual blocker across the
cropped annulus has probability at least $c>0$. Conditional on every
successful discovery, the chance of another crossing is therefore at
most $1-c$. The number of components has a geometric tail, proving
\eqref{prelim:passage-moment} for one $B>0$.

To obtain any fixed rate, divide the radial distance into $J$ separated
slices of thickness comparable to $r/J$. Cover a middle rim in each slice
by $O(J)$ small boxes whose buffered annuli remain within the slice.
Use wall-centered boxes near a straight wall and smaller bulk boxes
elsewhere, counting the two sides of a wall separately. The overlap graph
has a bounded coloring independent of $J$. Every original passage crosses
one small annulus in each slice, so
\[
 JX\le\sum_i X_i,
\]
with $O(J^2)$ counts having the first exponential bound. Conditional
iteration within each color class, followed by H\"older across the
bounded number of colors, gives a rate $c'J$. Choose $J$ arbitrarily
large but fixed. Before these boxes fit in the mesh, a deterministic
finite-size bound suffices. A finite covering by bulk, straight-wall,
and corner buffers gives the macroscopic version: a strand segment of
the designated diameter must leave one box of an inner cover whose
outer buffer is much smaller than that diameter.
\end{proof}

Let $N_j$ count raw nests wholly inside radius $r_j$, where the radii are
dyadic and the local sector geometry extends beyond the largest radius
by a buffer. A nest is a closed loop surrounding a bulk site, or an arc
joining the opposite side rays around a boundary gap, corner, change, or
slit tip. Then
\begin{equation}\label{prelim:nest-increments}
 0\le N_j-N_{j-1}\le X_j,
\end{equation}
where $X_j$ is a finite family of passage counts in a band near $r_j$.
Its buffer is disjoint from $B_{r_{j-k}}$ for a fixed $k$. Indeed a newly
included nest either travels a fixed fraction of the radius or goes
around the point within the band. At a slit tip, reaching the other bank
without approaching the tip requires traversing the full-angle sector.
Small-box coverings away from the center give the asserted $X_j$.
Start the levels at a fixed mesh radius so the initial count is bounded.

\begin{lemma}[Local tilt bounds]\label{prelim:tilts}
For fixed $b>0$, put
\[
 F_j=\mathbb E b^{N_j},\qquad
 d\nu_j=F_j^{-1}b^{N_j}\,d\mathbb P.
\]
The ratios $F_j/F_{j-1}$ are bounded above and below uniformly.
Under $\nu_j$, the shell counts $X_i$ at and beyond the outer tilt
level, $i\ge j$, have all fixed exponential moments uniformly bounded.
The assertions remain conditional on exterior data of the specified
geometry and allow adjoining a fixed family of passage counts with
buffers at or beyond that level.
\end{lemma}

\begin{proof}
The case $b=1$ is Lemma~\ref{prelim:passages}. For $0<b<1$, set
$D_i=\log(F_{i-1}/F_i)\ge0$. Jensen's inequality and the exponential
bound give, for any sufficiently small fixed $\varepsilon>0$,
\begin{align*}
 D_i
 &\le |\log b|\,\mathbb E_{\nu_{i-1}}X_i\\
 &\le\varepsilon\log
       \mathbb E_{\nu_{i-1}}
          \exp\bigl(|\log b|X_i/\varepsilon\bigr)\\
 &\le C_\varepsilon+
       \varepsilon\sum_{i-k<l<i}D_l.
\end{align*}
In the last inequality, drop decreasing factors back to level $i-k$.
The remaining tilt is supported outside the buffer of $X_i$, so the
conditional passage bound applies. Choose $\varepsilon(k-1)<1$;
induction bounds every $D_i$. Dropping the same bounded number of levels
then proves every asserted exponential moment.

For $b>1$, choose $B>k\log b$ as large as desired and induct on $j$ that
$\mathbb E_{\nu_j}e^{BX_i}\le C_*$ for $i\ge j$. When $i\ge j+k$,
the tilt and the buffer are disjoint. Otherwise let $j_0=i-k<j$.
The normalizing increment from $j_0$ to $j$ is at least one, so
\[
 \mathbb E_{\nu_j}e^{BX_i}
 \le\mathbb E_{\nu_{j_0}}
 \exp\left(BX_i+(\log b)\sum_{j_0<l\le j}X_l\right).
\]
Give each summand in the sum H\"older weight $\log b/B$ and use the
inductive bound at the earlier tilt $j_0$. The remaining H\"older
factor is an exponential moment of $X_i$ at a larger fixed rate,
whose buffer is disjoint from the tilt at $j_0$. The right side is at
most $C C_*^{k\log b/B}$, which closes the induction for sufficiently
large $C_*$. Negative levels mean no tilt. The moment bounds give the
ratio bounds using \eqref{prelim:nest-increments}. The argument is
unchanged after adjoining the indicated passage counts.
\end{proof}

Consequently the expectations of these local weights and their fixed
powers have logarithms of size $O(\log n)$ up to radius $O(n^C)$ in mesh
units. Changing a cutoff
radius by a fixed factor costs a bounded ratio. Spatially separated local
tilts can be treated successively by conditioning on their disjoint
patches, or by Lemma~\ref{prelim:collar}. Finally, log convexity between
a given power and twice that power shows that increasing the power by
$\varepsilon$ costs only $n^{O(\varepsilon)}$.

\begin{lemma}[Absorbing macroscopic errors]\label{prelim:absorb}
Suppose $Y_n\ge0$, $\mathbb E Y_n\ge n^{-A}/C$, and
$\mathbb E Y_n^2\le Cn^A$, whereas $X_n\ge0$ has uniformly bounded
exponential moments at every fixed rate. For fixed $C_0\ge1$ and every
$\varepsilon>0$,
\[
 C_\varepsilon^{-1}n^{-\varepsilon}
 \le
 \frac{\mathbb E[Y_n C_0^{\pm X_n}]}{\mathbb E Y_n}
 \le C_\varepsilon n^{\varepsilon}.
\]
\end{lemma}

\begin{proof}
Choose $\eta>0$ so that $\eta\log C_0<\varepsilon$ and split at
$X_n=\eta\log n$. On the first event the multiplicative factor lies
between $n^{-\varepsilon}$ and $n^{\varepsilon}$. Cauchy--Schwarz
and the exponential moment bound at an arbitrarily large fixed rate make
both $\mathbb E[Y_n;X_n>\eta\log n]$ and
$\mathbb E[Y_n C_0^{X_n};X_n>\eta\log n]$ smaller than any prescribed
inverse power after accounting for the polynomial second moment.
Choose that power larger than $A+1$. Division by the stated lower bound
for $\mathbb E Y_n$ proves both estimates, enlarging constants for
bounded $n$.
\end{proof}

\subsection{Comparing slit partition functions}

\begin{lemma}[Surface partition comparison]\label{prelim:surface}
Let finitely many straight finite wall segments have fixed macroscopic
shapes and positive mutual separation. Complete each as a positive
isolated boundary slit. Let $Z_s$ be its partition function relative to
the unmodified plane law, and $Z_{\rm all}$ the corresponding partition
function with all slits. Uniformly in the mesh,
\begin{equation}\label{prelim:surface-factorization}
 Z_{\rm all}\asymp\prod_s Z_s.
\end{equation}
Reversing every cap-wire polarity on both banks of a straight isolated
segment with boundedly many changes alters $Z_s$ by a bounded factor.
The same comparison holds for a segment and its image, with transported
cap data, under a symmetry of the homogeneous tile lattice, with the two
color names exchanged when the symmetry exchanges the color lattices.
\end{lemma}

\begin{proof}
Enclose each segment in its own patch with a vacant bulk tile collar and
room for a further disjoint buffer. Let $X_s$ be a passage count through
that collar. Compare the incremental loop change $\Delta\ell_s$ of
the slit surgery with $\Delta\ell_s^\circ$ computed after closing the
patch ports by the same arbitrary pairing before and after the surgery.
Their difference is $O(X_s)$. To see this, cut the segment ports. The
background pairing is determined within the patch except for ends whose
strand first traverses the collar. There are $O(X_s)$ such ends, and
changing their pairing changes either completed cycle count by at most
their number, also when other surgeries have been made outside.

Put $G_s=d^{\Delta\ell_s^\circ}\mathbf1_{\{X_s\le K\}}$.
For fixed sufficiently large $K$, simultaneous truncation has probability
bounded below under the normalized law with all completed slits, by
Lemma~\ref{prelim:passages}. On the truncated event the surgery weights
and their local versions are comparable. Therefore
\[
 Z_{\rm all}\asymp\mathbb E_0\prod_sG_s,
 \qquad Z_s\asymp\mathbb E_0G_s,
\]
where $\mathbb E_0$ is the unmodified plane expectation. Apply
Lemma~\ref{prelim:collar} successively to the separated patch factors
to obtain \eqref{prelim:surface-factorization}.

Translate an isolated segment by one tile step in its tangent direction.
This is a symmetry of the homogeneous plane loop law, with duality
interchanging the colors when needed. Its shifted local caps agree with
the reversed caps except at boundedly many ports near tips and changes.
Their remaining pairings differ on only boundedly many ends, which has
bounded cycle-count cost. Applying such a lattice symmetry to the isolated
slit and its cap data gives the last assertion.
This is a comparison of partition functions; it does not assert bounded
pointwise density for changing all pairings along a wall.
\end{proof}

\section{Logarithmic short-distance comparisons}\label{sec:exponents}

Only the strict range $1\le q<4$ is considered in this section. We require
inequalities between logarithmic costs, not the existence of limiting
critical exponents. Put
\begin{equation}\label{exp:parameters}
 \Delta=\frac{\pi}{2\lambda}-\frac32,\qquad
 s=\Delta+1,\qquad t=-\Delta,\qquad
 f(g)=\frac{\sin((2-g)\lambda)\sin((1+g)\lambda)}
                 {\sin(2\lambda)\sin\lambda}.
\end{equation}
In the local experiments below, the cutoff radius is comparable to $n$
in mesh units, the specified local geometry extends beyond it by a fixed
buffer, and the exterior has any fixed compatible positive completion.

\subsection{Local costs and their conventions}

At a flat change of wire color, let $P(n)$ be the logarithm of the
probability of an actual arm of the gap vertex's color reaching the
cutoff radius. Let $N$ count raw arcs straddling the change, from one
side ray to the other, wholly inside that radius, and set
\[
 D_*(n)=\log\mathbb E[d^N].
\]
At a corner without a wire flip, write $i=0$ if the gap color agrees
with the designated wire color and $i=1$ otherwise. For a convex
right-angle corner and a reentrant right-angle turn respectively, define
\[
 C_i(n)=\log\mathbb E[f(1/2)^{N/2}],\qquad
 L_i(n)=\log\mathbb E[f(-1/2)^{N/2}],
\]
where again $N$ counts local straddling raw arcs. Finally, at a bulk
vertex, let $N$ count closed raw nests surrounding it and set
\begin{equation}\label{exp:bulk-weights}
 F_\pm(a;n)=\mathbb E[u_\pm(a)^N],\qquad
 u_\pm(a)=\frac{\cos((1\pm a)\lambda)}{\cos\lambda},
\end{equation}
for fixed
\begin{equation}\label{exp:charge-range}
 0<a<1,\qquad (1+a)\lambda<\pi/2.
\end{equation}
We usually suppress $n$. In particular,
$D_*\ge0$, $C_i\ge0$, and $L_i\le0$: comparing the products of
sines at their common mean gives $f(1/2)>1$ and
$0<f(-1/2)<1$.

\begin{lemma}[Local conventions and strict arm decay]\label{exp:local-costs}
Changing a cutoff radius by a fixed factor, changing the fixed compatible
exterior completion, applying a tile-lattice symmetry to the experiment, or
interchanging both color names and wire designations changes each of
these logarithms by $O(1)$. The expectations of the local weights,
their fixed powers, and the same arm indicator have logarithms of size
$O(\log n)$. For some $c>0$,
\begin{equation}\label{exp:strict-arm}
 P(n)\le-c\log n+O(1).
\end{equation}
\end{lemma}

\begin{proof}
For nest weights these claims, apart from the signs already noted, are
Lemmas~\ref{prelim:collar} and~\ref{prelim:tilts}. Symmetries of the
local loop weights and caps give the geometric and color symmetries.
At a flat change reflection exchanges the two assignments relative to
the gap color. Bending or erasing distant cap connections changes only
the possible exterior partition.

We give the arm argument because it concerns actual connectivity. In a
tile half-box, boundary regions separated by raw boundary-to-boundary
arcs are precisely the actual color components touching boundary gaps.
Closed interior loops cannot split these boundary classes. If the region
at the marked gap does not reach the outer rim, some raw arc with both
ends on the flat wall straddles the gap. One way to see this is to form
the tree of regions separated by the disjoint arcs. The wall and rim
walks both traverse the tree path between their common end regions.
A wall visit to a region absent from the rim excursion is entered and
left across the same separating edge within the wall portion. Conversely,
a straddling arc cuts off the gap and its actual color paths. Thus absence
of all straddling arcs inside radius $r$ forces an arm to a comparable
radius, and an arm to a larger fixed multiple excludes all such arcs.

Arm probabilities at constant multiples of a radius are comparable.
Given the shorter arm, impose a same-color wall-to-wall screen in an
intermediate annulus and a crossing from its inner side to the desired
outer radius. The arm meets the screen and the added crossing meets it;
FKG and the wall crossing estimates give a uniform extension cost.
The construction is valid under arbitrary distant boundary data.
Starting from a bounded-mesh event and iterating gives a polynomial
lower bound. Conversely, actual opposite-color screens in separated
annuli obstruct the arm and have uniformly positive conditional
probability. Their iteration gives \eqref{exp:strict-arm}. Dual-color
requirements use FKG for decreasing primal events. No wire identification
has been used as an edge of an arm.
\end{proof}

For clarity, a one-sided error denoted $o(\log n)$ in an upper bound
means an allowance
\begin{equation}\label{exp:error-convention}
 \varepsilon\log n+O_\varepsilon(1)
 \quad\text{for every fixed }\varepsilon>0.
\end{equation}
Its constants may depend on all fixed charges and geometric choices.

\begin{proposition}[Three logarithmic comparisons]\label{exp:comparisons}
With the definitions above,
\begin{align}
 2P+2D_*+4(C_0+C_1)&\le o(\log n),\label{exp:rectangle-budget}\\
 C_0+C_1+L_0+L_1&\le o(\log n),\label{exp:corner-budget}\\
 P+\sum_j\log\frac{F_-(a_j)}{F_+(a_j)}
   &\le o(\log n),\label{exp:charge-budget}
\end{align}
where the finite list of fixed charges in the last inequality satisfies
\eqref{exp:charge-range} and $\sum_j a_j=1$.
\end{proposition}

We prove the proposition in three finite auxiliary geometries. Their
common algebra and cap bookkeeping are recorded first.

\subsection{The two winding expansions}

Orient every switch strand and, for either global sign, assign the turn
weight $\exp(\pm i\rho W/2)$, where $W$ is its signed turn. Give both
switches the common coefficient $\sin(\lambda/2)/\sin\lambda$ and sum
consistent orientations. The resulting real arrow tensor is identical
for the two signs. Indeed it requires two incoming arrows. For adjacent
incoming positions only one pairing works and its two turns cancel;
for opposite positions the two pairing contributions sum to
$2\cos(\lambda/2)$. Thus every finite contraction, with any prescribed
boundary spin factors and inserted edge matrices, has the same value
in these two expansions. Common tile coefficients will always cancel.

Traverse a boundary with the domain on its left. Let $\theta_j$ be a
lift of its inward normal angle at port $j$, continued by the boundary
turn at a corner. For spin $c=+1$ or $-1$ according as the arrow points
inward or outward, use
\begin{equation}\label{exp:boundary-spin}
 b_j(c)=(2\sin\lambda)^{-1/2}
          \exp\{ic(\rho\theta_j/2+\lambda h_j)\}.
\end{equation}
A raw arc from port $i$ to port $j$ bounding a disk against the
intervening forward boundary interval has turn
$\theta_j-\theta_i-\pi$: close it against that interval, using the
normal endpoint tangents. Summing its two orientations cancels curvature
in the positive-sign expansion and gives
\begin{equation}\label{exp:arc-weight}
 w(h_i-h_j),\qquad
 w(m)=\frac{\cos((m-1/2)\lambda)}{\sin\lambda}.
\end{equation}
For the negative-sign expansion set $h'_j=-h_j-\theta_j/\pi$;
its weight is $w(h'_i-h'_j)$.

Take $h_j=H_j+\Delta\mathbf1_{\{j\text{ high}\}}$, where $H$ is
piecewise constant. A port is high if its immediately preceding gap
vertex in the interval has the designated wire color. Across a wire
flip the adjacent ports are both high or both low. Since the ports
strictly inside a disk arc's boundary interval pair among themselves,
its endpoint high statuses are opposite for an even number of flips
and equal for an odd number. Thus
$h_i-h_j$ is the sum of the drops of $H$ plus $r\Delta$, where
$r=\pm1$ for an even number of flips (positive when the first port is
high) and $r=0$ otherwise. In the negative-sign expansion, high ports
refer to the complementary wires; base drops are negated and augmented
by the boundary turn divided by $\pi$. The identities
\begin{equation}\label{exp:staggered-weights}
 w(g+\Delta)=\frac{\sin((2-g)\lambda)}{\sin\lambda},
 \qquad
 w(g-\Delta)=\frac{\sin((1+g)\lambda)}{\sin\lambda}
\end{equation}
give weights $d,1$ at zero drop.

\begin{lemma}[Local cap elimination]\label{exp:cap-elimination}
For disk arcs whose boundary intervals contain at most one wire flip,
their positive cap-loop factors are $d$ for a zero-flip high starting
port and $1$ for a zero-flip low starting port or a one-flip arc, up to
a bounded overall loop-count error and an error $d^{O(X)}$ for remaining
macroscopic arcs. A nest of $N$ straddling arcs at a corner of turn
$g\pi$ without a wire flip has negative-expansion weight relative to
its caps comparable to $f(g)^{N/2}$.
\end{lemma}

\begin{proof}
Take a family of disk arcs closed under the arcs inside their disks,
with no holes, and remove them from the inside outward. Removing two
adjacent remaining endpoints creates a loop exactly when their exterior
partners already pair them; otherwise reconnect those partners.
The caps still pair successive high-low ports on each remaining
interval, with arbitrary pairings of its unmatched ends. A high-low arc
within an interval creates one loop. A low-high arc merely transports
the pattern. Across a flip, equal high statuses transport an unmatched
end without creating a loop. If two transported unmatched ends were
already tied, the extra loop erases an entire remaining interval. There
are boundedly many such exceptions; also allow the final bounded error
for a no-flip circle. The unremoved boundary loops are at most the number
of remaining arcs.

Include all sufficiently small arcs, together with arcs inside their
disks. Any others contribute a macroscopic passage count $X$ in boundary
bands. Along a local straddling nest the starting parities alternate,
because ports between consecutive starts pair internally on that side.
The two relative weights in \eqref{exp:staggered-weights} multiply to
$f(g)$ after division by the corresponding cap weights $d$ and $1$.
One unpaired factor is bounded. This proves the claimed $f(g)^{N/2}$
comparison. Cutoff patches are disjoint with wider buffers, and a
straddling arc omitted because it exits its patch is charged to $X$.
\end{proof}

\subsection{A rectangle: the flat-change and convex-corner costs}

\begin{proof}[Proof of \eqref{exp:rectangle-budget}]
Take a convex tile rectangle with both side lengths comparable to $n$.
Put a change in the middle of the bottom side and another in the middle
of the top side. Give $H$ drops $s,t$ at these changes and no others.
Since $s+t=1$ and the total boundary turn is $2\pi$, the factors
\eqref{exp:boundary-spin} are periodic around the contour.

In the positive-sign expansion, an arc not separating the changes uses
an interval containing neither change and has its ordinary cap weight.
An arc enclosing one change has $w(s)=w(t)=1$. Closed interior loops
have weight $d$. Lemma~\ref{exp:cap-elimination} and
Lemma~\ref{prelim:passages} therefore bound the spin partition function
above by a constant times the positive cap partition $Z_+^{\rm cap}$.
The latter includes the common tile coefficients.

In the negative-sign expansion the change drops are $-s,-t$ and each
convex corner contributes $1/2$. An arc containing neither change has
an interval containing at most two corners and has positive weight for
both high statuses. An arc separating the changes can use the interval
containing the $s$ mark and $k$ corners; its weight is
$\sin(k\lambda/2)/\sin\lambda$. These weights are nonnegative and
vanish exactly for $k=0$, namely a straddling arc on the bottom side.
Let $E$ be the event that the $s$-gap vertex has an actual path of its
color to another side of the rectangle. Every weight is strictly
positive on $E$, and the local straddles at $t$ have weight $d$.
Relative to the cap law with the complementary wires, the spin
partition is consequently at least
\begin{equation}\label{exp:rectangle-lower}
 \mathbb E_-
 \left[\mathbf1_E d^{N_t}
    \prod_{v\text{ corner}} f(1/2)^{N_v/2}
    C^{-(1+X)}\right].
\end{equation}
Here $X$ counts macroscopic boundary passages. For instance, choose all
small arcs below a fixed diameter fraction so their disks contain at
most one feature. The cap elimination accounts for those arcs, and
remaining factors on $E$ lie in a fixed finite set of positive values.

Without $C^{-(1+X)}$, the expectation in
\eqref{exp:rectangle-lower} is bounded below by a constant times
\[
 \exp\left(P+D_*+\sum_v C_{i(v)}\right).
\]
Indeed the separated positive local factors compare conditionally by
Lemma~\ref{prelim:collar}. After conditioning on their small patches,
extend the local $s$ arm through a fixed corridor avoiding those patches
to another side, by the screens and crossing extensions in
Lemma~\ref{exp:local-costs}. The remaining law is FK with an induced
partition. Lemma~\ref{prelim:absorb} loses only
$n^\varepsilon$ upon reinstating the passage factor, because all local
first and second moments have polynomial cost by
Lemma~\ref{prelim:tilts}.

Choose odd horizontal tile length and even vertical tile length. The
four corners then have the same relative gap-versus-wire type, since
the left and right sides belong to opposite wire intervals. Compare the
two tensor expansions and repeat on this very same rectangle with every
wire polarity reversed. The ratios
$Z_+^{\rm cap}/Z_-^{\rm cap}$ cancel upon multiplication. One run has
four type-$0$ corners and the other four type-$1$ corners; both have the
same $P,D_*$ up to bounded changes of local convention. Taking logarithms
proves \eqref{exp:rectangle-budget} with the error convention
\eqref{exp:error-convention}.
\end{proof}

\subsection{A long cylinder: balancing the two corner turns}

\begin{proof}[Proof of \eqref{exp:corner-budget}]
Take a tile cylinder whose spatial period has two positive coordinate
components, each comparable to $n$, with even sum. In a lift, its lower
boundary repeats all the east steps followed by all the north steps.
The upper boundary is a parallel translated copy at transverse distance
of order $Kn\log n$, where $K$ will be a sufficiently large fixed
constant. Each boundary has one convex and one concave corner. There
are no wire flips and $H$ is constant. Use positive caps for the
reference laws and free periodic spin contraction, with no flux
projection, for the tensor. The normal-angle lift is periodic.

Every same-boundary raw arc bounds a disk against the boundary interval
it cuts off. Its lifted endpoint interval spans less than one period:
otherwise its endpoints and those of a translate would interleave,
forcing intersecting arcs. Its positive-expansion weights are ordinary
cap weights, and its negative-expansion drops are $0$ or $\pm1/2$,
so those weights are positive. A contractible interior loop has weight
$d$. An essential loop has total turn zero and hence weight $2$; the
zero-turn assertion follows by isotopy of a simple generator in the
flat cylinder.

Reserve bottom and top bands of width $O(n)$ containing all boundary
features and their enlarged buffers. In the empty middle strip let
$k_0$ count essential closed medial loops wholly within that strip, and
put
\begin{equation}\label{exp:middle-tilt}
 M=(2/d)^{k_0}.
\end{equation}
It depends only on middle-strip tiles and is at least one. Under either
positive cap law,
\begin{equation}\label{exp:middle-moments}
 \mathbb E M^p\le n^{O_p(K)}\qquad(p<\infty).
\end{equation}
To prove this, cover the middle strip by $O(K\log n)$ buffered bulk
boxes of size a small fraction of the period. Each essential loop
supplies a passage, and conditional iteration of
Lemma~\ref{prelim:passages} with a bounded coloring proves the bound.
Essential loops omitted from $M$, macroscopic boundary arcs, and other
macroscopic loops approaching features are charged to a count $X$ in
the enlarged boundary bands. This $X$ has all fixed exponential moments
uniformly, independently of $K$.

Let $T$ be the event of a raw arc joining the two boundaries. Then
$M=1$ on $T$, and
\begin{equation}\label{exp:through-rarity}
 \mathbb P_\pm(T)\le C n^{-cK}.
\end{equation}
Indeed, in each successive transverse band of width comparable to the
period, actual crossings can form a noncontractible cycle of one color.
Use a necklace of overlapping strip crossings with transversal joins;
its buffers are injective on the cylinder. In the lift the connected
crossings continue through every translate, and their projection
contains a noncontractible cycle. FKG and crossing estimates give a
uniform conditional success probability. Such a cycle blocks $T$, so
iteration through order $K\log n$ bands proves
\eqref{exp:through-rarity}.

The positive expansion has absolute partition value at most
\[
 Z_+^{\rm cap}\,\mathbb E_+[M C^{1+X}].
\]
Off $T$, the negative expansion is positive and its integrand relative
to $Z_-^{\rm cap}$ is at least
\[
 M\prod_v f(g_v)^{N_v/2}C^{-(1+X)},
 \qquad g_v\in\{1/2,-1/2\}.
\]
On $T$, its absolute integrand is bounded by the same local factors
times $C^{1+X}$, without $M$. These statements are
Lemma~\ref{exp:cap-elimination}; through arcs are macroscopic, so their
bounded weights are included in $X$. H\"older, the polynomial local
tilt bounds, and \eqref{exp:through-rarity} bound the possible signed
through contribution by $n^{-c'K+O(1)}$ relative to its caps.

Condition on the middle-strip tiles and use the separated feature
patches to compare the positive main expectation, without its $X$
factor, to
\[
 (\mathbb E_-M)
 \exp\left(\sum_{v:g_v=1/2}C_{i(v)}+
           \sum_{v:g_v=-1/2}L_{i(v)}\right).
\]
It is bounded below by an inverse power of $n$ whose exponent is
independent of $K$, because $M\ge1$. Choose $K$ large enough that the
through error is negligible. Now keep $K$ fixed and apply
Lemma~\ref{prelim:absorb}, using \eqref{exp:middle-moments}, to absorb
all $C^{\pm X}$ factors at an arbitrarily small power loss.

Run the comparison for both choices of the two boundary wire colors,
then reverse both colors on exactly the same cylinder, cutoffs and
middle strip. Tensor equality cancels the two ratios of
$Z_\pm^{\rm cap}\mathbb E_\pm M$ when the inequalities are multiplied.
The two runs together count twice each of $C_0,C_1,L_0,L_1$.
Taking logarithms and dividing by two proves
\eqref{exp:corner-budget}.
\end{proof}

\subsection{A charged cylinder: the bulk costs}

\begin{proof}[Proof of \eqref{exp:charge-budget}]
Use a cylinder with straight horizontal boundaries, with even horizontal
period comparable to $n$, and lower boundary traversed eastward. Put the
two flat changes on the bottom, separated by order $n$, and no change
on top. Place the finitely many bulk points $z_j$ at mutually separated
distances of order $n$ in the bottom band. Their buffered cutoff patches
are disjoint; make the band wide enough to leave clear corridors nearer
the bottom. The height is again of order $Kn\log n$.

Draw tile-edge strings from the $s$-gap to each $z_j$ and multiply the
spin crossing its string by $\exp(-i\lambda a_j c)$, where $c=+1$
denotes an arrow along the left normal of the oriented string step.
Use raw drops $s-1,t$ for $H$ on the lower boundary, with total zero,
and constant $H$ on top. Oriented intersections give the following
sign convention: in the positive expansion, a string of charge $-a_j$
shifts a boundary disk arc's cosine argument, in units of $\lambda$,
by $+a_j$ for a launch in its interval and by $-a_j$ for an endpoint
in its disk. The boundary child lies to the arc's right when the arc
is followed from the earlier to the later port. Thus the local effective
bottom drops are $s,t$; in the negative expansion they are $-s,-t$,
with $a=\sum_{z_j\text{ in disk}}a_j$ added for the endpoints inside.

For a boundary disk arc the negative-expansion numerators, over the
common denominator $\sin\lambda$, are
\begin{equation}\label{exp:charged-table}
\begin{array}{c|l}
 \text{number of enclosed flips}&\text{numerators}\\ \hline
 0&\sin((2-a)\lambda),\quad\sin((1+a)\lambda)\\
 1&\sin(a\lambda)\quad[s],\qquad
      \sin((2-a)\lambda)\quad[t]\\
 2&\sin((3-a)\lambda),\quad\sin(a\lambda).
\end{array}
\end{equation}
The interval spans less than one period, so each change is counted at
most once. Top boundary disks use the zero-flip row. Since $0\le a\le1$
and $0<\lambda\le\pi/3$, every entry is nonnegative. A vanishing
entry always belongs to a child interval containing $s$; this includes
the additional zero $\sin(3\lambda)=0$ at $q=1$ in the two-flip row.
Contractible closed loops in the negative expansion have relative weight
$\cos((1-a)\lambda)/\cos\lambda>0$; their interior is on the left
for counterclockwise orientation. The two expansions therefore give
the prescribed local weights $u_\pm(a_j)$. Essential loops have
positive weight $2\cos(a\lambda)$, where $a$ is the charge separated
from the launches, and loops in the empty middle strip have weight $2$.
Keep $M$ as in \eqref{exp:middle-tilt}.

Let $E$ be the event that the $s$-gap vertex connects by actual paths
of its color to a noncontractible open cycle. Construct that cycle near
the bottom, before the conditioned endpoint patches and middle strip.
Conditionally on those patches and the middle,
\[
 \mathbb P_-(E\mid\text{patches and middle})\gtrsim e^P.
\]
This follows from the local arm comparison, same-color FKG and crossing
extension, and the periodic necklace construction used for
\eqref{exp:through-rarity}. On $E$ no child disk arc can cut off $s$,
since the noncontractible cycle cannot fit behind it, and there is no
through arc.

Set $U_\pm=\prod_j u_\pm(a_j)^{N_j}$ and let $Y_\pm$ be the spin
integrands relative to their respective positive cap laws. Local cap
and loop counting gives
\begin{align*}
 |Y_+|&\le M U_+ C^{1+X},\\
 Y_-&\ge M U_- C^{-(1+X)}\quad\text{on }E,\\
 Y_-&\ge0\quad\text{off }T.
\end{align*}
The local $t$-straddles in the negative expansion contribute $d$ each,
which may be discarded in this lower bound. A loop whose endpoint
enclosure differs from a local nest has a macroscopic passage near a
feature: a contractible disk containing $z_j$ must separate it from
the bottom. Thus it is charged to $X$, with all cutoffs small compared
with the feature separations. On the through event $T$ one still has
\[
 |Y_-|\le U_-d^{N_t}C^{1+X},\qquad M=1.
\]
All these local costs are polynomial and the through probability is
$O(n^{-cK})$, so choose $K$ fixed and large to dominate the signed
through error.

Conditioning and Lemma~\ref{prelim:collar} compare the separated local
factors to $\prod_jF_\pm(a_j)$, the middle factor to
$\mathbb E_\pm M$, and the event $E$ to its lower arm cost $e^P$.
Lemma~\ref{prelim:absorb} removes the passage errors at $n^\varepsilon$
cost. Repeat with every wire designation reversed, using the same
domain, strings, drops, cutoffs and $M$. Equality of the two tensor
expansions cancels both cap and middle-factor ratios upon multiplying
the two comparisons. The resulting inequality is
\[
 2P+2\sum_j\log\frac{F_-(a_j)}{F_+(a_j)}
 \le o(\log n),
\]
which gives \eqref{exp:charge-budget}.
\end{proof}

\subsection{The polygon budget}

\begin{corollary}[Balanced corner parities]\label{exp:polygon-budget}
Let a fixed simple rectilinear polygon have $m$ concave and $m+4$
convex true corners, with wire changes in side interiors. It has tile
approximants differing by $O(1)$ mesh units such that at least
$\lfloor m/2\rfloor$ corners of each relative wire type occur in each
of the convex and concave sets, also after global wire reversal.
For these approximants,
\begin{equation}\label{exp:polygon-saving}
 3P+D_*+\sum_{v\,\mathrm{convex}}C_{i(v)}
       +\sum_{v\,\mathrm{concave}}L_{i(v)}
 \le\frac12P-\frac32D_*+o(\log n).
\end{equation}
Each fixed family of negative endpoint charges with magnitudes summing
to one has total wrong-to-correct local logarithmic cost at most
$-P+o(\log n)$.
\end{corollary}

\begin{proof}
First prescribe the desired relative corner types for one designation.
There remain at least four spare convex corners, one of which adjusts
the parity of the sum of the absolute corner parities. If $b_i$ denotes
the integer coordinate parity of side $i$, a corner between sides
$i-1$ and $i$ has parity $p_i=b_{i-1}+b_i$ modulo two. This cyclic
system is solvable exactly when $\sum_i p_i=0$ modulo two. Choose the
side coordinates with those parities within boundedly many mesh units
of the prescribed ones. For the fixed simple polygon, sufficiently
small independent shifts preserve simplicity. Global wire reversal
just exchanges the relative types.

Pair the balanced convex and concave sets and apply
\eqref{exp:corner-budget}. If $m$ is even, four convex terms remain;
if $m$ is odd, five convex terms and one nonpositive concave term remain.
Since $C_i\ge0$, Equation~\eqref{exp:rectangle-budget} implies
\[
 C_i\le-\tfrac12(P+D_*)+o(\log n).
\]
Paying at most five such terms yields
\eqref{exp:polygon-saving}. With no corner factors,
\eqref{exp:rectangle-budget} alone gives
$P+D_*\le o(\log n)$, which is the corresponding flat budget.
The endpoint-charge assertion is exactly
\eqref{exp:charge-budget}.
\end{proof}
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//preliminaries.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//rotation-repair.tex ====
\section{Rotation of the fair-signed height at \texorpdfstring{$q=4$}{q=4}}
\label{sec:q4-rotation-repair}

The endpoint argument needs rotational invariance of height correlations,
with the loops counted with their multiplicities.  The loop-set distance
in the statement of the FK rotation theorem does not by itself retain
those multiplicities.  We obtain the required assertion directly from
the homotopy coupling used in its proof
\cite[Sections~3.5--3.6 and~4.2]{DCKKMO2026Rotation}.
Only qualitative bulk RSW estimates are used in this transfer.

For $k\ge2$, let
\[
 \mathcal D_k=
 \left\{\boldsymbol u=(u_1,u'_1,\ldots,u_k,u'_k):
 \{u_i,u'_i\}\cap\{u_j,u'_j\}=\varnothing\quad(i\ne j)\right\}.
\]
Coincidence within one pair is allowed.  Give the critical $q=4$ FK
loops independent fair signs and let $h_\delta$ be the resulting height,
with jumps of size one and an immaterial additive constant.  At a
continuum point use any fixed face representative within $O(\delta)$.
Put
\begin{equation}
 \Phi_{k,\delta}(\boldsymbol u)
 =\mathbb E\prod_{i=1}^k
       \bigl(h_\delta(u'_i)-h_\delta(u_i)\bigr).
 \label{eq:rr-Phi}
\end{equation}

\begin{theorem}[Isometry invariance of the endpoint correlations]
\label{thm:rr-rotation}
For every even $k\ge2$ and every compact $K\subset\mathcal D_k$,
\begin{equation}
 \lim_{\delta\downarrow0}
 \sup_{\boldsymbol u\in K}\sup_{I\in\operatorname{Isom}(\R^2)}
 \left|\Phi_{k,\delta}(\boldsymbol u)
             -\Phi_{k,\delta}(I\boldsymbol u)\right|=0.
 \label{eq:rr-target}
\end{equation}
\end{theorem}

The exact plane correspondence is recorded first, including the
passage from torus loops to the plane.  This avoids any need for an
unpublished general-parameter correspondence.

\begin{lemma}[The endpoint plane correspondence]
\label{lem:rr-bkw}
After the fixed similarity identifying the medial square grid with
$\Z^2$, the fair unit-jump height of critical plane $\FK_4$ has the
same gradient law as the zero-slope six-vertex height with
$\mathbf a=\mathbf b=1$ and $\mathbf c=2$.
\end{lemma}

\begin{proof}
On an even square torus, give equal weight to each pair consisting
of noncrossing local switch pairings and an orientation of each
resulting closed loop.  Erasing orientations gives weight $2^\ell$
to a configuration with $\ell$ loops.  Erasing pairings gives an
ice-rule arrow configuration: each of the four $a$- and $b$-type
vertices admits one compatible pairing, and each of the two
$c$-type vertices admits two.  Thus the arrow marginal has weights
$(1,1,2)$, and the loop orientations, conditionally on the
unoriented loops, are independent and fair.  This is the endpoint
specialization of the local calculation in
\cite[Section~2, Equations~(2.1)--(2.3)]{Lis2021}.

The torus weight $2^\ell$ need not equal the torus FK weight.
Precisely the needed plane passage is supplied by
\cite[Lemma~3.1]{Lis2021}: its unoriented loop law converges locally
to critical plane $\FK_4$.  Fix a finite set of observed medial
edges.  Bulk annular RSW implies that every plane FK loop meeting
this set is finite almost surely.  For any $\varepsilon>0$, choose
a box containing all these loops with probability at least
$1-\varepsilon$.  The event that the observed loops close inside
this box is determined by its finitely many pairings.  On that
event, local convergence transfers their full loop identities,
and hence their independent fair orientations.  Letting the box
grow proves convergence of the local arrow law to independently
oriented plane FK loops.

On the other hand, the unrestricted six-vertex torus laws have the
zero-slope plane limit used in
\cite[Theorem~2.2 and the paragraph following it]{SixVertexGFF2026}.
That paragraph identifies the unrestricted torus limit with the
slope-zero law defined through balanced cylinders.  This is a
lattice Gibbs-law fact preceding the GFF
argument.  The two descriptions of the arrow limit agree, and
arrows determine unit height differences.  Additive constants do
not affect the gradient law.  This proves the lemma.
\end{proof}

Consequently Theorem~\ref{thm:rr-rotation} supplies exactly the
$\mathbf c=2$ case of the symmetry input
\cite[Theorem~4.1]{SixVertexGFF2026}.  Its proof below does not use
any GFF limit or any sharp four-arm estimate.

\subsection{Even moments retain only macroscopic loops}

Smooth the discrete contours locally into disjoint Jordan loops.  This
changes their location by $O(\delta)$.  For such a loop $L$ set
\[
 D_i(L)=\mathbf1_{\operatorname{int}L}(u'_i)
                   -\mathbf1_{\operatorname{int}L}(u_i).
\]
A reference orientation may multiply all the $D_i(L)$ by one sign
depending on $L$; the fair sign absorbs that choice.  Let
$\Pi_{\mathrm{even}}(k)$ denote the partitions of $\{1,\ldots,k\}$
whose blocks all have even size.  Conditional expectation over the
signs gives the exact identity
\begin{equation}
 P_k(\mathcal L;\boldsymbol u)
 :=\mathbb E\left[\prod_i
       \bigl(h_\delta(u'_i)-h_\delta(u_i)\bigr)
                         \,\middle|\,\mathcal L\right]
 =\sum_{\pi\in\Pi_{\mathrm{even}}(k)}
   \ \sum_{\substack{f:\pi\to\mathcal L\\f\ \mathrm{injective}}}
          \prod_{B\in\pi}\prod_{i\in B}D_i(f(B)).
 \label{eq:rr-even-polynomial}
\end{equation}
At positive mesh only finitely many loops separate any prescribed
pair: each such loop crosses a fixed finite lattice path between its
endpoints.  Hence the expansion is a finite identity.  In particular,
there is no limiting interchange in its derivation.

Call a loop relevant if at least two coordinates of
$(D_1(L),\ldots,D_k(L))$ are nonzero.  Every loop in a nonzero summand
of \eqref{eq:rr-even-polynomial} is relevant.  If endpoints belonging
to distinct slots are separated by $d>0$, a relevant loop surrounds
one endpoint from each of two distinct slots.  Therefore
\begin{equation}
 \diam L\ge d.
 \label{eq:rr-macro}
\end{equation}
Here we used that the closure of a Jordan domain is contained in the
convex hull of its boundary.  Also, a separating loop intersects the
segment joining the separated endpoints.  Thus, if the endpoints lie
in $B(0,M)$, every relevant loop meets that disk.  For rounded
continuum endpoints, replace $d$ by $d-O(\delta)$.

We record the quantitative estimates needed below.  They hold also
for the two-angle mixed isoradial lattices in the rotation
construction.  For any fixed choice of the two angles, their mesh
geometry has bounded distortion and their bulk conditional RSW
bounds are available from
\cite[Theorem~2.2, Remark~2.3 and Proposition~2.12]{DCKKMO2026Rotation}.

\begin{lemma}[Counts, escape, and deterministic-point avoidance]
\label{lem:rr-estimates}
Let $N(a,M)$ be the number of loops of diameter at least $a$ meeting
$B(0,M)$.  For $\delta$ sufficiently smaller than $a$, and every
fixed $p<\infty$,
\begin{equation}
 \|N(a,M)\|_{L^p}\le C_p(1+M/a)^2.
 \label{eq:rr-count}
\end{equation}
There are $C,\gamma>0$ such that, with harmless fixed changes of the
radii,
\begin{align}
 \mathbb P\{\exists L:L\cap B(0,M)\ne\varnothing,
                            L\not\subset B(0,R)\}
      &\le C(M/R)^\gamma, && R\ge10M,
       \label{eq:rr-escape}\\
 \mathbb P\{\exists L:\diam L\ge a,
                       \dist(z,L)\le r\}
      &\le C\bigl((r+C\delta)/a\bigr)^\gamma,
          &&0<r<a/100.
       \label{eq:rr-point}
\end{align}
The constants are independent of the deterministic center $z$.
\end{lemma}

\begin{proof}
Bulk RSW in a sequence of buffered annuli gives a polynomial bound
for an actual primal or dual arm between their extreme rims: each
annulus has a uniformly positive conditional probability of a
blocking circuit of the opposite color.

We also need the standard elementary count consequence of this
argument.  The number of distinct actual clusters of one color
crossing a fixed-aspect annulus has a geometric tail.  Explore
complete actual components from the inner rim, revealing their
closed exiting edges.  Remove every explored component, including
noncrossing ones, until the next crossing component is found.
At this stopping time, extend the surviving configuration by closed
edges on the deleted graph.  Its FK law is dominated by the law in
the original annulus with all vertices on both original rims
wired together.  Removed bulk
components do not wire surviving bulk vertices.  A dual circuit in
a buffered middle annulus of this auxiliary residual configuration
blocks another actual primal crossing, and has probability bounded
below even under the maximal primal boundary condition.  Repeating the exploration bounds the chance
of each additional crossing by a fixed number below one.  Apply
the same argument with the colors exchanged.

Each loop is the unique exterior contour of its interior-side
actual cluster, primal or dual.  Thus distinct loops give distinct
colored clusters.  Cover $B(0,M)$ by $O((1+M/a)^2)$ balls of radius
$a/100$.  A loop of diameter at least $a$ meeting one of these balls
has an interior-side bank crossing an annulus about its center
whose two radii are fixed multiples of $a$.  Different global
clusters give different actual components in that annulus.
The preceding geometric tail and Minkowski's inequality prove
\eqref{eq:rr-count}.

For \eqref{eq:rr-escape}, a bank of an escaping loop gives an
actual arm from scale $M$ to scale $R$.  For \eqref{eq:rr-point}, a
bank of the indicated loop gives an arm from scale $r+C\delta$
about $z$ to a fixed fraction of $a$.  The polynomial arm bound
proves both assertions.
\end{proof}

Fix a compact $K\subset\mathcal D_k$, write $d_K>0$ for its minimum
cross-slot endpoint distance, and choose $M$ containing every
endpoint in $K$.  If $k=2m$, an even partition has at most $m$
blocks, so \eqref{eq:rr-even-polynomial} implies
\begin{equation}
 |P_{2m}(\mathcal L;\boldsymbol u)|
       \le C_m N(d_K/2,M+1)^m,
 \qquad \boldsymbol u\in K,
 \label{eq:rr-UI}
\end{equation}
for sufficiently small mesh.  All fixed moments of the right-hand
side are uniformly bounded by Lemma~\ref{lem:rr-estimates}.  These
are bounds for the \emph{sign-averaged polynomial}; no uniform
absolute-moment assertion about a product of pointwise height
differences is being made.

Two consequences will be useful.  First, removing loops not
contained in $B(0,R)$ changes the expectation of the polynomial by
at most $C_K(M/R)^{\gamma/2}$, by
\eqref{eq:rr-escape}, \eqref{eq:rr-UI}, and Cauchy--Schwarz.
Second, if corresponding endpoints of two tuples in a compact
subset of $\mathcal D_k$ move by at most $r$, evaluate both
polynomials in the same loop sample.  Off the events in
\eqref{eq:rr-point} at those endpoints, every relevant loop has
unchanged endpoint indicators.  Hence
\begin{equation}
 |\Phi_{k,\delta}(\boldsymbol u)
                   -\Phi_{k,\delta}(\boldsymbol v)|
       \le C_K\bigl((r+C\delta)/d_K\bigr)^{\gamma/2}.
 \label{eq:rr-continuity}
\end{equation}
This estimate is uniform in the absolute location of the tuple.
It covers within-slot coincidences and changes of the rounding
convention.  The supremum over $K$ is a supremum of deterministic
tuple estimates; we do not require one avoidance event to hold
simultaneously at all points occurring in $K$.

\subsection{What the published coupling preserves}

Let $\mathbb L(\alpha)$ be the rectangular isoradial lattice in
\cite[Section~1.4]{DCKKMO2026Rotation};
$\mathbb L(\pi/2)$ is a fixed rotation and dilation of the square
lattice.  The next lemma extracts the precise part of its coupling
that we use.  Its assertion is about loop multiplicities, not merely
about proximity of loop sets.

\begin{lemma}[Homotopy coupling with dense marked clusters]
\label{lem:rr-nails}
Fix $\alpha\in(0,\pi)$.  As the mesh tends to zero, the construction
of \cite[Section~3.6]{DCKKMO2026Rotation} gives a coupling of two
auxiliary loop samples.  Their restrictions to a deterministic
window exhausting the plane have, unconditionally, the respective
homogeneous laws on $\delta\mathbb L(\pi/2)$ and
$\delta\mathbb L(\alpha)$.

Outside an event of probability tending to zero, there are
correspondingly indexed, pairwise disjoint marked
clusters, called nails, whose mesh tends to zero and whose window
exhausts the plane.  Corresponding nails have diameter tending to
zero and lie at distance tending to zero from the same grid point.
There is a bijection $\psi$ between all loops surrounding at least
two but not all nails in the two samples.  It preserves their
oriented reduced crossing words relative to the nail grid.
\end{lemma}

\begin{proof}
This is the construction preceding the metric conclusion, with its
parameters retained.  In the notation of
\cite[Definition~3.8 and Section~3.6, pp.~49--50]{DCKKMO2026Rotation},
choose $\eta=N^{-b}$ and $t=N^{-b^2}$, for $b>0$ sufficiently small.
On an event of probability $1-O(N^{-c})$, the selected nail
clusters survive the whole extrema-resampling and track-exchange
chain.  The same construction preserves the bijection of
non-trivial loops and their reduced words at each step.
This covers both colors of contour.

Initially, the nail grid spacing is $t^{1/2}N$, nail diameters
are $O(tN)$, and the nail window has size comparable to
$t^{1/4}N$.  The terminal nail positions differ by these errors
from their images under $M_{\pi/2,\alpha}$.  The independent FK
argument in \cite[Section~4.2, proof of Theorem~1.7]{DCKKMO2026Rotation}
proves $M_{\pi/2,\alpha}=\mathrm{id}$.  We may therefore use the
already constructed coupling with corresponding nails at the same
limiting positions.

For clarity, these scales can be arranged for every small physical
mesh $\delta$.  Set $a=b^2$ and choose
$N=2\lfloor\tfrac12\delta^{-1/(1-3a/8)}\rfloor$, so $N$
is even as required by the track construction.  Then the physical grid
spacing, nail diameter, and nail-window size are respectively of
orders
\[
 N^{-a/8},\qquad N^{-5a/8},\qquad N^{a/8}.
\]
The homogeneous observation window has scale $\delta N$, which
also diverges.  Center the nail window by translating each
endpoint sample by a period vector of its own homogeneous
lattice nearest the same centering vector.  The two chosen
translations differ by $O(\delta)$; this only adds a vanishing
error to corresponding nail positions.  The endpoint restrictions
have exactly the homogeneous laws, by
\cite[Proposition~2.11]{DCKKMO2026Rotation}.  After centering, both
the nail window and the homogeneous observation windows exhaust
the plane.  The probability $O(N^{-c})$ tends to zero.  This proves
the stated extraction; no conclusion about curve-set convergence
is used to recover the bijection.
\end{proof}

Here is the topological information in a crossing word that will
matter.  Orient each loop by the reference convention and let
$s_L\in\{-1,1\}$ be its winding number at an interior point.
For any two nails, join chosen points in them by a path of nail-grid
edges, with transverse perturbations if necessary.  The algebraic
intersection number of $L$ with this path equals the difference of
its winding numbers at the two nail points.  It is read from the
oriented crossing word.  Cyclic permutation and deletion of an
adjacent inverse pair do not change it.  Consequently, if $n,n'$
and $\widetilde n,\widetilde n'$ are corresponding nail points,
\begin{equation}
 s_L\bigl(\mathbf1_{\operatorname{int}L}(n')
                  -\mathbf1_{\operatorname{int}L}(n)\bigr)
 =s_{\psi(L)}
       \bigl(\mathbf1_{\operatorname{int}\psi(L)}(\widetilde n')
                  -\mathbf1_{\operatorname{int}\psi(L)}(\widetilde n)\bigr).
 \label{eq:rr-winding-difference}
\end{equation}
We need only these differences.  In particular, no inference of an
absolute inside set from a finite grid word is required.

\begin{figure}[t]
\centering
\begin{tikzpicture}[scale=.83,>=Stealth]
 \begin{scope}
  \draw[thick,blue!55!black] (0.8,0) ellipse (1.22 and 1.48);
  \node[blue!55!black] at (1.8,1.5) {$L$};
  \foreach \y/\j in {.7/1,-.7/2}{
   \fill (-1.25,\y) circle (1.6pt);
   \fill (.8,\y) circle (1.6pt);
   \node[anchor=east] at (-1.34,\y) {$u_{\j}$};
   \node[anchor=west] at (.88,\y) {$u'_{\j}$};
   \draw[gray,densely dashed] (-1.25,\y)--(-1.1,\y+.15);
   \draw[gray,densely dashed] (.8,\y)--(.65,\y+.15);
   \draw[red!70!black,fill=white,semithick] (-1.1,\y+.15) rectangle ++(.12,.12);
   \draw[red!70!black,fill=white,semithick] (.65,\y+.15) rectangle ++(.12,.12);
  }
  \node at (.35,-1.95) {two separated slots};
 \end{scope}
 \draw[->,thick] (2.4,0)--(4.1,0)
        node[midway,above] {$\psi$};
 \begin{scope}[xshift=6cm]
  \draw[thick,blue!55!black]
      (-.35,-1.1) .. controls (-.9,-.1) and (-.25,1.5) .. (.9,1.4)
      .. controls (2,1.2) and (2,-1.4) .. (.75,-1.5)
      .. controls (.1,-1.7) and (-.15,-1.4) .. (-.35,-1.1);
  \node[blue!55!black] at (1.8,1.65) {$\psi(L)$};
  \foreach \y/\j in {.7/1,-.7/2}{
   \fill (-1.25,\y) circle (1.6pt);
   \fill (.8,\y) circle (1.6pt);
   \draw[gray,densely dashed] (-1.25,\y)--(-1.1,\y+.15);
   \draw[gray,densely dashed] (.8,\y)--(.65,\y+.15);
   \draw[red!70!black,fill=white,semithick] (-1.1,\y+.15) rectangle ++(.12,.12);
   \draw[red!70!black,fill=white,semithick] (.65,\y+.15) rectangle ++(.12,.12);
  }
  \node at (.35,-1.95) {$D(\psi(L))=\pm D(L)$};
 \end{scope}
\end{tikzpicture}
\caption{Nearby nails (squares) detect the separations of endpoint
pairs (dots).  The homotopy bijection preserves oriented winding
differences.  A single sign change of a loop's whole separation
vector cancels when that loop occurs an even number of times.}
\label{fig:rr-nails}
\end{figure}

\subsection{Transfer of the correlation polynomial}

Fix $\alpha\in(0,\pi)$ and compact $K\subset\mathcal D_k$.
Choose $0<r<d_K/1000$.  Apply Lemma~\ref{lem:rr-nails} and, for
each endpoint, select an indexed nail close to it.  For all small
enough meshes both corresponding nails lie within $r/2$ of that
endpoint.  Nails selected for endpoints in distinct slots are
distinct.  In both samples discard the event that a loop of
diameter at least $d_K/4$ comes within $r$ of an endpoint.
Lemma~\ref{lem:rr-estimates} bounds the probability of these
exceptions by $C_K(r/d_K)^\gamma+o(1)$, uniformly for the
individual tuples in $K$.

On the remaining event, a relevant loop $L$ surrounds the nails
near one endpoint of each of two distinct slots, and excludes a
nail near the other endpoint of a separated slot.  It is therefore
non-trivial in the exact sense of Lemma~\ref{lem:rr-nails}, so
$\psi(L)$ is defined.  For those two slots the winding differences
at the corresponding nails are nonzero by
\eqref{eq:rr-winding-difference}.  Thus $\psi(L)$ separates each
of the two nail pairs.  In particular, it encloses one nail from
each of two distinct slots, whose distance is at least
$d_K-O(r+\delta)$.  This proves that $\psi(L)$ itself has diameter
at least $d_K/4$.  Its endpoint-avoidance event is consequently
applicable; smallness of the partner has not been assumed away.

Endpoint indicators and nearby-nail indicators agree for both
loops.  Equation~\eqref{eq:rr-winding-difference} now gives,
for \emph{every} slot,
\begin{equation}
 D_i(\psi(L))=\frac{s_L}{s_{\psi(L)}}D_i(L).
 \label{eq:rr-signature}
\end{equation}
The inverse bijection gives the same statement starting with a
relevant loop in the second sample.  Hence it is a bijection
between all relevant loops, preserving their entire separation
vectors up to one sign per loop.  For a block $B$ in an even
partition,
\[
 \prod_{i\in B}D_i(\psi(L))
    =\left(\frac{s_L}{s_{\psi(L)}}\right)^{|B|}
                       \prod_{i\in B}D_i(L)
    =\prod_{i\in B}D_i(L).
\]
Transporting the injective labels in
\eqref{eq:rr-even-polynomial} therefore makes the two polynomials
\emph{exactly equal} on the good event.  This uses the actual
bijection, so distinct loops are never identified with one another.
Cauchy--Schwarz and \eqref{eq:rr-UI} bound the expectation error by
$C_K(r/d_K)^{\gamma/2}+o(1)$.

It remains to pass from the auxiliary mixed samples to the intended
homogeneous ones.  Fix $R>10M$ and couple their restrictions
identically in $B(0,R)$ using the local-law assertion of
Lemma~\ref{lem:rr-nails}; for small enough mesh this disk lies in
both homogeneous windows.  Off the escaping-loop event in either
sample, all relevant complete loops are identical.  Equations
\eqref{eq:rr-escape} and \eqref{eq:rr-UI} bound the expectation
error by $C_K(M/R)^{\gamma/2}$.  Let the mesh tend to zero,
then $r\downarrow0$ and $R\uparrow\infty$.  Writing
$\Phi^{\alpha}_{k,\delta}$ for the fair-signed correlation on
$\delta\mathbb L(\alpha)$, we obtain
\begin{equation}
 \lim_{\delta\downarrow0}\sup_{\boldsymbol u\in K}
 \left|\Phi^{\pi/2}_{k,\delta}(\boldsymbol u)
                   -\Phi^{\alpha}_{k,\delta}(\boldsymbol u)\right|=0.
 \label{eq:rr-isoradial-transfer}
\end{equation}
No diameter threshold needs to be a continuity point: all small
loops were removed by the even-moment identity itself.

\begin{proof}[Proof of Theorem~\ref{thm:rr-rotation}]
The rectangular model on $\mathbb L(\alpha)$ is invariant under
reflection $S_{\alpha/2}$ about $e^{i\alpha/2}\R$
\cite[Remark~1.8]{DCKKMO2026Rotation}.  Its fair-sign height
correlations inherit that symmetry.  If the chosen endpoint
rounding is not reflection equivariant, the reflected
representatives differ by $O(\delta)$, which is harmless by
\eqref{eq:rr-continuity}.  Apply
\eqref{eq:rr-isoradial-transfer} at the original and reflected
tuples.  The square model is consequently asymptotically invariant
under $S_{\alpha/2}$, uniformly on every compact subset of
$\mathcal D_k$.  Composing with its exact horizontal-reflection
symmetry yields invariance under rotation through $\alpha$.
Square symmetries cover the endpoints and the other half of the
circle.  This proves the assertion for each fixed rotation.

The estimate \eqref{eq:rr-continuity} upgrades this to uniformity
in the rotation angle.  Indeed, the orthogonal images of $K$ form
a compact subset of $\mathcal D_k$, and a finite angular net
approximates every rotation with endpoint displacement at most
$C_K$ times its angular mesh.  First use convergence at the
finitely many net angles, and then let the angular mesh decrease.
Reflections follow by composition with an exact square symmetry.
This argument requires no estimate uniform in a degenerating
isoradial angle.

Finally, any translation differs from a period translation of the
mesh-$\delta$ square lattice by a vector of length $O(\delta)$.
Exact period invariance and the location-uniform estimate
\eqref{eq:rr-continuity} handle this residual displacement,
uniformly over \emph{all} translation vectors.  Every plane
isometry is a translation followed by an orthogonal map.
The fixed rotation and dilation identifying
$\mathbb L(\pi/2)$ with $\Z^2$ merely change the mesh convention
and compact set.  These observations prove
\eqref{eq:rr-target} for the original square lattice.
\end{proof}


\subsection{The whole-plane field input}

\begin{corollary}[The endpoint whole-plane field limit]
\label{cor:rr-gff-input}
The fair unit-jump plane height of critical square-lattice $\FK_4$
converges on neutral smooth compactly supported tests to
$\sqrt{2/\pi}$ times the whole-plane GFF with Green kernel
$G(x,y)=-(2\pi)^{-1}\log|x-y|$.  In particular,
Proposition~\ref{q4field:wholeplane-input} holds.
\end{corollary}

\begin{proof}
Lemma~\ref{lem:rr-bkw} identifies the discrete gradient law with
the isotropic six-vertex law at $(\mathbf a,\mathbf b,\mathbf c)
=(1,1,2)$.  Theorem~\ref{thm:rr-rotation} gives precisely the
compact-$\mathcal D_k$, all-isometries symmetry statement used as
Theorem~4.1 in \cite{SixVertexGFF2026}, at this parameter.
We may therefore apply its isotropic Theorem~2.8, using the
remaining spectral, regularity, and free-energy results cited and
proved there.  The passage from correlations to smooth field
tests is part of its Section~8.  No anisotropic extension or
general-parameter loop-functional assertion is needed here.

At $\mathbf c=2$ one has $\Delta=-1$, so the variance multiplier
in that theorem is $2/\arccos(-1)=2/\pi$.  A fixed rotation and
dilation of the lattice does not change the neutral whole-plane
Green quadratic form: the dilation adds a constant to the
logarithmic kernel and each test has integral zero.  For every
real $\rho\in C_c^\infty(\R^2)$ with $\int\rho=0$, the convergence
therefore gives
\[
 \mathbb E e^{i\langle h_\delta,\rho\rangle}
 \longrightarrow
 \exp\left\{-\frac1\pi
       \iint \rho(x)G(x,y)\rho(y)\,dx\,dy\right\}.
\]
Apply the same assertion to every real linear combination of a
finite family of neutral tests to obtain joint convergence.
This is exactly Proposition~\ref{q4field:wholeplane-input}.
\end{proof}
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//rotation-repair.tex ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//sources/companions/results/gaussian-free-field-balanced-six-vertex-model/build/sections/01-introduction.tex ====
\section{Introduction}\label{sec:introduction}

The six-vertex model assigns an arrow to each edge of the square lattice,
subject to the rule that two arrows enter and two arrows leave each vertex.
Its height function is a discrete random surface. We prove the Gaussian
free field limit of this surface throughout the critical range
\(a=b=1\), \(0<c\le2\), for the plane state specified below.

\subsection{The plane law and the height convention}

The two vertices having both horizontal arrows directed inward, or both
directed outward, have weight \(c\); the other four vertices have weight
one. A finite configuration has the product of its vertex weights.
On
\[
 T_{M,L}=(\Z/M\Z)\times(\Z/L\Z),\qquad M,L\ge4,\quad L\ \text{even},
\]
condition every vertical column of horizontal edges to contain \(L/2\)
left arrows and \(L/2\) right arrows. This is a conserved sector of the row
transfer matrix. We first take \(M\to\infty\), and then even \(L\to\infty\),
in the local weak topology of arrow configurations. We denote the resulting
plane law by \(\P_c\). Existence in the previously untreated range is part
of the theorem.

Give the face on the left of an arrow height one more than the face on its
right. The ice rule makes this a well-defined height \(h\), modulo an
additive constant, on the faces of the plane. For \(x=(x_1,x_2)\in\R^2\),
let \(h(x)\) be the height of the face with bottom-left corner
\((\lfloor x_1\rfloor,\lfloor x_2\rfloor)\), and set
\[
 h_\delta(x)=h(x/\delta),\qquad \delta>0.
\]
This convention also fixes the value on grid lines.

Let \(\Gamma\) be the real plane Gaussian free field modulo constants,
normalized by the Green kernel
\begin{equation}\label{eq:green}
 G(x,y)=-\frac1{2\pi}\log|x-y|.
\end{equation}
Thus its covariance on smooth compactly supported tests of total integral
zero is the integral of \(G\) against their product.
For compactly supported finite signed measures \(\mu,\nu\) of total mass
zero, write
\[
 \mathcal E(\mu,\nu)=\iint G(x,y)\,\mu(\dd x)\nu(\dd y).
\]
In saying that \(\mu\) has finite energy, we require this to be an ordinary
finite signed-measure integral for \(\nu=\mu\). We do not use cancellation
between infinite positive and negative logarithmic energies.

For the negative regularity assertion we retain, in addition to the usual
distributional restriction spaces, the literal seminorm in
\cite[Definition~2.6(iii)]{DKLM2026}. Let \(\mathcal T_1(Q)\), with
\(Q=(-1,1)^2\), consist of compactly supported signed densities in \(Q\),
of integral zero, whose zero extensions are \(1\)-Lipschitz. Set
\begin{equation}\label{eq:source-holder}
 \|f\|_{\mathcal C^\alpha(U)}
 =\sup_{\substack{0<\varepsilon\le1,\ \varphi\in\mathcal T_1(Q)\\
                   \supp\varphi\subset U/\varepsilon}}
   \varepsilon^{-\alpha}
   \left|\int f(x/\varepsilon)\varphi(x)\,\dd x\right|.
\end{equation}
This is a seminorm of the global plane field: \(U\) indexes the allowed
tests, and the sampled points \(x/\varepsilon\) need not lie in \(U\).
It is distinct from the ordinary restriction topology on \(U\).
The space is the completion modulo the kernel of this seminorm.
The absolute value does not change the source convention because the test
family is symmetric. We prove convergence for this convention as well as
for the usual local restriction Hölder, Besov and Sobolev spaces.

\begin{theorem}\label{thm:main}
For every fixed \(c\in(0,2]\), the prescribed iterated balanced-torus limit
\(\P_c\) exists. Define
\begin{equation}\label{eq:multiplier}
 \Delta=1-\frac{c^2}{2},\qquad
 \sigma(c)^2=\frac2{\arccos\Delta}=\frac1{\arcsin(c/2)}.
\end{equation}
As \(\delta\downarrow0\) through arbitrary positive real values,
the fields \(h_\delta\) converge to
\(\sigma(c)\Gamma\) in each of the following senses.
\begin{enumerate}
\item For every finite family \(\mu_1,\ldots,\mu_m\) of compactly supported
finite signed measures of total mass zero and finite energy,
\[
 \bigl(\langle h_\delta,\mu_j\rangle\bigr)_{j=1}^m
 \ \Longrightarrow\
 \mathcal N\!\left(0,
       \bigl(\sigma(c)^2\mathcal E(\mu_i,\mu_j)\bigr)_{i,j=1}^m\right).
\]
\item For every bounded open \(U\subset\R^2\) and \(-1<\alpha<0\), there
is convergence in law, with tightness, in \(\mathcal C^\alpha(U)\) modulo
constants, and in the corresponding spaces
\[
 B^\alpha_{p,q}(U),\qquad W^{\alpha,p}(U),
 \qquad 1\le p<\infty,\quad 1\le q\le\infty.
\]
The usual local restriction Hölder topology is also included, and no
regularity of \(\partial U\) is required. One may fix a
smooth compactly supported \(\rho\) of integral one and replace each field
\(X\) by \(X-\langle X,\rho\rangle\), using the same pinning for the limit.
\item For every \(k\ge1\) and every tuple
\((u_i,u_i')_{i=1}^k\) with
\(\{u_i,u_i'\}\cap\{u_j,u_j'\}=\varnothing\) for \(i\ne j\),
\begin{equation}\label{eq:increment-main}
 \E_c\prod_{i=1}^k
    \bigl(h_\delta(u_i')-h_\delta(u_i)\bigr)
 \longrightarrow
 \sigma(c)^k\sum_{\pi}\prod_{\{i,j\}\in\pi}C_{ij},
\end{equation}
where the sum is over pairings of \(\{1,\ldots,k\}\), is zero for odd \(k\),
and
\[
 C_{ij}=G(u_i',u_j')-G(u_i',u_j)-G(u_i,u_j')+G(u_i,u_j).
\]
The convergence is uniform on compact sets of such tuples.
\end{enumerate}
\end{theorem}

The theorem concerns this particular zero-slope plane state. It makes no
uniqueness assertion among all Gibbs states or all slopes, and no estimate
uniform as \(c\downarrow0\).

\subsection{Prior work and the new ingredients}

The prediction that critical six-vertex height fluctuations have a Gaussian
free field limit connects this interacting lattice surface with conformal
universality. Earlier rigorous approaches include the dimer correspondence
at the free-fermion point \(c=\sqrt2\) and constructive renormalization in
a neighborhood of that point; see Kenyon~\cite{Kenyon2000},
Giuliani--Mastropietro--Toninelli~\cite{GMT2017}, and the account in
\cite[Section~2.4]{DKLM2026}.
Duminil-Copin, Kozlowski, Lammers and Manolescu proved the stated convergence
for \(\sqrt3\le c\le2\) and identified the multiplier
\eqref{eq:multiplier}~\cite[Theorem~2.8]{DKLM2026}.
Their work also provides the balanced plane state for \(1\le c\le2\),
its equilibrium characterization, and a conditional identification of the
multiplier from the curvature of the free energy
\cite[Theorem~2.2, the paragraph after (6), and Theorems~4.3--4.4]{DKLM2026}.
Theorem~\ref{thm:main} resolves the Gaussian free field scaling-limit
conjecture for this prescribed balanced plane law throughout \(0<c\le2\),
retaining these established parameter values.

The finite algebra in our argument is the algebraic Bethe ansatz.
We use scalar-product and norm determinants in their trigonometric and
diagonally twisted forms~\cite{Slavnov1989,KMT1999,KMST2005}.
Those finite identities are separate from the uniform thermodynamic
estimates proved here. We also use the local row algebra and folded angular
continuation mechanism in~\cite[Section on angular transfer]{AngularRepository2026}.
Its original parameter range does not cover the whole theorem.
Lemma~\ref{lem:scalar-continuation} checks the finite local continuation
identities for the present range, and Sections~\ref{sec:thermodynamic}
and~\ref{sec:angular} prove the limiting kernel estimates used here.

Three parts of the argument are useful beyond this particular organization
of the proof. First, a root-tail estimate and an edge maximum principle
give uniform inverses for Gaudin matrices and for mixed discrete--continuous
quadrature operators. Combined with finite-dimensional residue conditions,
these bounds control simultaneous escapes of several rapidities in
determinant formulas. Second, angular spectra split into two widely
separated scales. Bounds on all zeros outside the two scale windows
justify the operator limits that yield holomorphic and antiholomorphic
Gaussian currents. Third, the normalization for \(c<1\) follows from
matching a homogeneous stiffness lower bound with a staggered
vacuum-overlap upper bound. The latter turns an exact finite determinant
into a spectral packing inequality. Ordinary moment estimates and analytic
desmoothing then recover point-increment correlations and the full test
classes in Theorem~\ref{thm:main}.

\subsection{Proof overview}\label{sec:overview}

For the new range \(0<c<\sqrt3\), introduce
\[
 \lambda=\arccos(-\Delta)\in(\pi/3,\pi),
 \qquad \eta=\lambda/\pi.
\]
Section~\ref{sec:finite} defines commuting row operators and their diagonal
marks. Section~\ref{sec:thermodynamic} constructs their limiting matrix
elements, proves the necessary uniform tail bounds, and shows that words
with widely separated angular parameters decorrelate.

These matrix elements define a Hilbert space with commuting angular rows
in Section~\ref{sec:angular}. Their joint spectral values are inner functions
of a complex row parameter. A mark acts as a current. The finite mass
\(\kappa\) of the spectrum transverse to common angular dilation is the
coefficient of the logarithmic height covariance. Section~\ref{sec:normalization}
computes \(\kappa\) directly when \(c<1\); for \(c\ge1\) the cited conditional
normalization theorem applies once convergence has been established.

Section~\ref{sec:chiral} separates the angular spectrum into two scales and
proves the Wick rules for the limiting currents. Section~\ref{sec:probability}
passes from these currents to smooth tests, finite-energy measures,
negative regularity and off-diagonal point increments. The final section
assembles these conclusions with the established interval. Each
thermodynamic passage below specifies which size tends to infinity first;
none relies on exchanging the auxiliary limits.
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//sources/companions/results/gaussian-free-field-balanced-six-vertex-model/build/sections/01-introduction.tex ====

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  year = {2014},
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  note = {Theorem numbering follows the author-hosted text}
}
==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//references.bib ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//sources/companions/results/critical-square-lattice-random-cluster-interfaces/build/references.bib ====

@article{DCLM2019Isoradial,
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@article{DMT2021Fractal,
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@article{DM2022Scaling,
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@misc{DC2017Lectures,
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@misc{SixVertexGFF2026,
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@misc{DCKKMO2026Rotation,
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@article{Smirnov2010,
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@article{ChelkakDCHonglerKemppainenSmirnov2014,
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@article{SheffieldWerner2012,
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@article{MillerSheffield2016IG3,
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@article{MillerWu2017,
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@article{GwynneMillerQian2021,
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==== END Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//sources/companions/results/critical-square-lattice-random-cluster-interfaces/build/references.bib ====

==== BEGIN Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces//sources/companions/results/gaussian-free-field-balanced-six-vertex-model/build/references.bib ====
@misc{DKLM2026,
  author       = {Duminil-Copin, Hugo and Kozlowski, Karol Kajetan and
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  title        = {Gaussian free field convergence of the six-vertex model
                  with {$-1\leq\Delta\leq-\frac12$}},
  year         = {2026},
  eprint       = {2603.06268},
  archivePrefix = {arXiv},
  primaryClass = {math-ph},
  doi          = {10.48550/arXiv.2603.06268},
  url          = {https://arxiv.org/abs/2603.06268v1},
  note         = {\href{https://arxiv.org/abs/2603.06268v1}{arXiv:2603.06268v1}, 6 March 2026}
}

@article{Kenyon2000,
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@article{GMT2017,
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}

@book{Sheffield2005,
  author = {Sheffield, Scott},
  title = {Random Surfaces},
  series = {Ast{\'e}risque},
  number = {304},
  publisher = {Soci{\'e}t{\'e} Math{\'e}matique de France},
  year = {2005},
  url = {https://www.numdam.org/item/AST_2005__304__R1_0/},
  note = {\href{https://www.numdam.org/item/AST_2005__304__R1_0/}{Numdam edition}}
}

@misc{AngularRepository2026,
  author = {{OpenAI}},
  title = {Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces},
  year = {2026},
  month = oct,
  howpublished = {OpenAI Math Release preprint,
                  OAI:Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026},
  note = {October 5, 2026; Section ``Angular transfer and its two leading modes''}
}

@article{LammersTassy2024,
  author = {Lammers, Piet and Tassy, Martin},
  title = {Macroscopic behavior of {Lipschitz} random surfaces},
  journal = {Probability and Mathematical Physics},
  volume = {5},
  year = {2024},
  pages = {177--267},
  doi = {10.2140/pmp.2024.5.177},
  url = {https://doi.org/10.2140/pmp.2024.5.177},
  eprint = {2004.15025},
  archivePrefix = {arXiv},
  primaryClass = {math.PR},
  note = {Cited locators refer to
          \href{https://arxiv.org/abs/2004.15025v1}{arXiv:2004.15025v1}, 30 April 2020}
}
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