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\title{The critical logarithmic correction\\for the planar XY model}
\author{OpenAI}
\date{}
\begin{document}
\maketitle
\begin{abstract}
We prove the critical logarithmic correction for the nearest-neighbor
cosine XY model on the square lattice. Let \(b_c\) be the critical
inverse temperature and let \(C_{b_c}(r)\) be the two-point correlation
obtained by first taking the free-box thermodynamic limit at fixed
separation \(r\), and then letting \(r\) tend to infinity. Then
\[
C_{b_c}(r)=B_{\mathrm{XY}}\,r^{-1/4}(\log r)^{1/8}(1+o(1)),
\qquad B_{\mathrm{XY}}\in(0,\infty),\qquad r\to\infty,
\]
for \(r\) integer.
\end{abstract}
\section{Introduction}\label{sec:introduction}
For an integer \(R\ge1\), let
\[
\Lambda_R=[-R,R]^2\cap\Z^2,\qquad
E_R=\{\{x,y\}\subset\Lambda_R:\abs{x-y}=1\}.
\]
The planar XY model at inverse temperature \(b>0\), with free boundary,
is the probability measure
\begin{equation}\label{eq:xy-law}
\frac{1}{Z_{R,b}}
\exp\left\{b\sum_{\{x,y\}\in E_R}\cos(\theta_x-\theta_y)\right\}
\prod_{x\in\Lambda_R}\frac{\dd\theta_x}{2\pi},
\qquad \theta_x\in\R/(2\pi\Z).
\end{equation}
Each unordered edge is counted once, and there are no exterior edges.
At each site \(x\) we imagine a planar unit spin
\((\cos\theta_x,\sin\theta_x)\). The Boltzmann weight prefers
neighbours to point in similar directions; the parameter \(b\) is the
strength of that preference. The thermodynamic limit is taken by
letting the side length \(R\to\infty\).
Write \(\E^{\rm free}_{R,b}\) for expectation under \eqref{eq:xy-law}.
For \(r\in\Z_{>0}\), with \(e_1=(1,0)\), set
\begin{equation}\label{eq:critical-definition}
\begin{split}
C_b(r)&=\lim_{\substack{R\to\infty\\R\ge r}}
\E^{\rm free}_{R,b}\cos(\theta_0-\theta_{re_1}),\\
m(b)&=\lim_{r\to\infty}-\frac1r\log C_b(r),\qquad
b_c=\inf\{b>0:m(b)=0\}.
\end{split}
\end{equation}
The limits exist, \(0<b_c<\infty\), and \(m(b_c)=0\); see
\cite[Section~1.1 and Proposition~1.2]{C}. The thermodynamic limit in the
first line is always taken before the separation tends to infinity.
The ``mass'' \(m(b)\) is the exponential rate of decay of the two-point
function, and \(b_c\) is the single threshold at which that rate vanishes.
\begin{theorem}\label{thm:main}
There is a constant \(B_{\rm XY}\in(0,\infty)\) such that
\[
C_{b_c}(r)
=B_{\rm XY}\,r^{-1/4}(\log r)^{1/8}(1+o(1))
\qquad(r\to\infty,\ r\in\Z).
\]
\end{theorem}
The exponent \(-1/4\) is the critical power, and the factor
\((\log r)^{1/8}\) prevents \(r^{1/4}C_{b_c}(r)\) from converging.
The constant \(B_{\rm XY}\) is a positive, finite amplitude assembled
from microscopic and geometric contributions.
\subsection{Background and earlier work}
\label{subsec:background}
Berezinskii's spin-wave analysis predicts algebraic correlations at
low temperature \cite{Berez1971}; Kosterlitz and Thouless identify the
transition as due to unbinding vortex pairs \cite{KT1973}; Kosterlitz
then performs a renormalization-group analysis of the critical
behaviour of the nearest-neighbor model \cite{Kosterlitz1974}. For
the Villain formulation, Janke's review records the critical prediction
\[
|x|^{-1/4}(\log|x|)^{1/8}
\left(1+O\!\left(\frac{\log\log|x|}{\log|x|}\right)\right);
\]
see \cite[Section~II, Equations~(2)--(3)]{Janke1997}. Theorem~\ref{thm:main}
establishes the leading equivalent, with a relative \(o(1)\) error,
for the ordinary cosine interaction.
Rigorous work has established the phase picture through correlation
bounds and dual representations. McBryan and Spencer proved
power-law upper bounds for two-dimensional planar-spin correlations
\cite{McBryanSpencer1977}. Fr\"ohlich and Spencer proved a
low-temperature algebraic regime for the plane rotator and Villain
models through a multiscale Coulomb-gas analysis
\cite[Theorem~C]{FrohlichSpencer1981}. For the cosine XY model,
van Engelenburg and Lis proved exponential decay below a threshold
and a lower bound \(1/(8|x-y|)\) at and above that threshold,
in the limit over free boxes \cite[Theorem~1]{vanEngelenburgLis2023}.
The dual height field gives another route: Aizenman, Harel, Peled,
and Shapiro related height depinning to nonsummable spin correlations
\cite{AHPS2022}; Lammers identified an exact relation between the XY
and dual-height correlation lengths \cite[Theorem~1]{Lammers2023} and
proved a localization dichotomy for the height localisation
\cite{Lammers2026}.
The lattice-Gaussian inequality of Regev and Stephens-Davidowitz
\cite{RSD2017} underlies the finite comparisons of \cite[Section~2]{B}
passed to Bessel weights in \cite[Lemma~2.2]{C}. Bauerschmidt
developed finite-range decompositions of Gaussian fields by
functional calculus \cite{Bauerschmidt2013}. Bauerschmidt, Park, and
Rodriguez obtained Gaussian scaling limits for discrete Gaussian
heights at high temperature \cite{BPR2024I,BPR2024II}. Falco's
critical Coulomb-gas analysis gives a rigorous example where a marginal
recursion produces a power law with a multiplicative logarithm
\cite{Falco2013}.
Theorem~\ref{thm:main} sharpens the companion result
\(C_{b_c}(r)=r^{-1/4+o(1)}\) of \cite{C}. From that paper we use:
the critical height coefficient \(a(b_c)=8\pi\); finite Bessel
comparisons; fixed-geometry and conditional height limits; and a
four-face characteristic estimate. The analytic local maps come from
\cite{B}. The sector costs for fixed positive holes and the annular
kernel estimates come from \cite{S}, with the annular argument adapted
to the pinned rim below. The cutoff initialization, the quantitative
coefficient estimate, the comparison of different starting histories,
and the extraction of the amplitude are proved here.
\subsection{How the proof is organised}\label{subsec:roadmap}
The proof has three large pieces, given in Sections~\ref{sec:heights}
through~\ref{sec:conclusion} below.
\emph{(a) Reducing the free-box correlation to a single insertion.}
Fourier expansion of the spin interactions represents a two-point
correlation as a ratio of partition functions for integer-valued
heights on the dual lattice. The numerator shifts the height
increments by a connection whose circulation records the spin
insertion. This magnetic insertion is invisible to height averages
alone, so it must be tracked explicitly.
For an integer \(n\ge1\), let
\begin{equation}\label{eq:center-definition}
M_n=\E^0_{n,b_c}\cos\theta_0,
\end{equation}
where \(\E^0_{n,b_c}\) denotes expectation for the measure obtained
from Equation~\eqref{eq:xy-law} on \(\Lambda_n\) by fixing all boundary
angles at zero and integrating only the interior angles.
The geometric argument, proved in Section~\ref{sec:geometry}, gives
\[
\begin{aligned}
\frac{M_{n'}}{M_n}&\longrightarrow d^{-1/8}
&&\text{if }n,n'\to\infty\text{ and }n'/n\to d\in(0,\infty),\\
\frac{C_{b_c}(r)}{M_r^2}&\longrightarrow c_{\rm shape}\in(0,\infty)
&&\text{as }r\to\infty.
\end{aligned}
\]
Annular interfaces separate the microscopic insertions from an exterior
with holes kept at fixed positive radius. The sector and annular-kernel
estimates of \cite{S} control these interfaces, and the common microscopic
factors cancel in the comparisons. A pinned-rim argument, adapted
from the same reference, relates the two-insertion calculation to the
free-box thermodynamic correlation. The first comparison gives
\(M_n=n^{-1/8+o(1)}\), which will make the cutoff loss negligible
relative to the center observable, and later extends an asymptotic
from geometric sizes to all integers. The second reduces the theorem
to a center asymptotic.
\emph{(b) Computing the center asymptotic.}
This is the analytic core, in Sections~\ref{sec:entry}--\ref{sec:observable}.
To compute \(M_n\) one cuts off differences of noisy cell averages,
with the cell size tied to the target square size. For the squares
used in the analytic comparison, the rough center bound makes this
choice negligible in the center partition ratio. After averaging to
the associated starting scale, the bulk interaction is small. For
the center insertion, every expansion term carries one labelled
component that retains the source and its attached dependencies.
The norm of the labelled activity may grow like a fixed power of the
cell side; a linear propagation estimate controls it.
Let \(L>1\) be the fixed dyadic block ratio and \(j\) the absolute
scale index. After separating the Gaussian reference with height
coefficient \(a=8\pi\), the smoothed bulk interaction at block side
\(L^j\) retains \(t_j/2\) times the allocated gradient energy and a
fundamental periodic term; the linear part of the remaining
activity's evolution contracts. Coordinates adapted to the pure
Gaussian curve remove its drift and give a gradient coordinate
\(u_j\) and a signed fundamental Fourier coordinate \(Z_j\). Their
leading recursion is
\[
u_{j+1}-u_j\simeq-HZ_j^2,\qquad
Z_{j+1}-Z_j\simeq-2(\log L)u_jZ_j,\qquad H>0.
\]
Physical torus variance and phase tests keep each finite history
near the Gaussian curve. A phase lower bound derived from the
four-face estimate of \cite{C} prevents the fundamental direction
from disappearing. Comparing histories through the same finite
physical torus tests controls changes of starting scale without a
rate for the height scaling limit. Together with the recursion,
these constraints yield the critical balance
\(HZ_j^2\sim2(\log L)u_j^2\), which gives harmonic decay of \(u_j\).
On the controlled interior ranges of the size-dependent histories
used in the comparison, the coordinate changes give
\(t_j=u_j+O(j^{-2})\). The resulting estimate, uniform on each
fixed such range, is
\[
t_j=\frac{1}{2(\log L)j}
+O\!\left(\frac{\log j}{j^2}\right).
\]
\emph{(c) Assembling the amplitude.}
Let \(g_n\) be the minimum squared edge norm of a connection with the
\(2\pi\) circulation of one insertion in the corresponding dual square.
For \(n_0=L^N\) and \(n_1=L^{N+1}\), use the same entry and a common
stop \(j'=N-O(\log N)\), leaving a number of terminal blocks that
grows as a small power of \(N\). Choose field-independent factors
removed at each step so that their complete contributions to the
insertion ratio agree between sizes. The source term left at the
stop is then controlled relative to the source coefficient already
extracted. Integration of the common phase removes the linear
periodic term, and a comparison of the terminal Gaussian fields
leaves the gradient contribution with a summable error:
\[
\log\frac{M_{n_1}}{M_{n_0}}
=-\frac{1-t_{j'}}{2a}\bigl(g_{n_1}-g_{n_0}\bigr)+\epsilon_N,
\qquad \sum_N|\epsilon_N|<\infty.
\]
Green's function analysis gives
\(g_{L^N}=2\pi(\log L)N+c_N\) with \(c_N\) convergent and
\(c_{N+1}-c_N\) summable. Substitution of the trajectory estimate
yields the increment \(-(\log L)/8\) together with the \(1/(16N)\)
correction. The unweighted Green differences \(c_{N+1}-c_N\)
telescope, and summation by parts treats the differences weighted
by \(1/N\); the other errors are absolutely summable. Summation
gives a positive finite center amplitude and logarithmic power
\(1/16\). The first geometric comparison extends this equivalent
to all integer sizes, and the bulk comparison squares the logarithmic
power, giving \(1/8\), which completes Theorem~\ref{thm:main}.
\section{Height representation and critical inputs}\label{sec:heights}
We first pass from spins to Bessel heights, fixing normalisation and
boundary conventions. Then we record the finite comparisons and
critical height limits used in the geometric and analytic arguments.
These inputs are all known from \cite{C} (for the critical parameter
and finite comparisons) and \cite{B} (for RG localisation), and are
stated here so the sequel is logically complete.
\subsection{Dual partition functions}
Choose one orientation of each edge of a finite nearest-neighbor graph
\(G=(V,E)\) and put \(\nabla_eh=h_y-h_x\) for \(e=(x,y)\). We call
these height vertices and edges \emph{primary}, to distinguish them
from the auxiliary subdivisions in the comparison proof;
\emph{primal} refers to the spin lattice.
The height takes values in \(2\pi\Z\). Its increment weight is
\begin{equation}\label{eq:bessel-weight}
p_b(j)=e^{-b}I_j(b),\qquad
\sum_{j\in\Z}p_b(j)e^{tj}=\exp\{b(\cosh t-1)\}.
\end{equation}
Here \(I_j\) is the modified Bessel function. Thus \(p_b\) is the law
of the difference of two independent Poisson variables of mean \(b/2\).
An \emph{integral connection} is an antisymmetric edge function
\(\tau:E\to2\pi\Z\). The weight in that sector is
\[
\prod_{e\in E}p_b\left(\frac{\nabla_eh+\tau_e}{2\pi}\right).
\]
A connection is flat where its circulation around every contractible
cycle is zero.
Its circulations around the remaining holes are its periods.
We allow nonnegative quadratic penalties in real linear observations,
hard pins, and prescribed relative height data. Every observation
used after a cut is supported in one component. A component with no
absolute constraint or confining observation is summed modulo a common
translation by \(2\pi\Z\); when its constant is confined, all its
translation copies are summed. A hard pin fixes that constant instead.
Relative translations between cut components are summed again on
restoration. Write \(Z_G^\tau\) for a partition function with these
conventions and \(Z_G^0\) for the same graph, cuts, and penalties with
all affine data centred.
An integral gauge change replaces \(\tau\) by \(\tau+\nabla\psi\) and
the height variable by \(h-\psi\), where \(\psi:V\to2\pi\Z\).
This is an exact reindexing when every affine datum is changed with
the height. A penalty \(\|Bh-y\|^2/2\) then has centre \(y-B\psi\),
and absolute data \(b^p\) on a set \(p\) become \(b^p-\psi|_p\).
For relative data \(h_z-h_{z_0}=u_z\), the new value is
\(u_z-\psi_z+\psi_{z_0}\). Thus a hard zero pin is preserved when
\(\psi=0\) on it, and a constraint requiring all heights on a set
to be equal is preserved when \(\psi\) is constant there.
\begin{lemma}[Spin--height duality]\label{lem:duality}
In a free spin box, the two-point correlation is \(Z_G^\tau/Z_G^0\)
on the dual graph with the exterior face fixed at height zero.
One may represent that face by separate exterior incident copies, all
pinned at zero, or identify those copies while retaining every edge.
Circulations below are read in the dual with the exterior copies
identified. The connection has circulations \(+2\pi\) and \(-2\pi\)
around the two sources and can be supported on the dual bonds crossing
a primal current path joining them.
The center magnetization \(M_n\) defined in
\eqref{eq:center-definition} satisfies
\(M_n=\E_{n,b_c}^{\,0}e^{i\theta_0}>0\).
It is \(Z_G^\tau/Z_G^0\) on the free dual square with
\(2n\) dual vertices on each side, indexed by the primal cells, where
\(\tau\) has one circulation \(2\pi\) around the center. It may be
supported on the dual bonds crossing a primal current path from the
center to the boundary.
\end{lemma}
\begin{proof}
Expand each spin interaction in its Fourier series,
\[
e^{b\cos(\theta_x-\theta_y)}
=\sum_{j\in\Z}I_j(b)e^{ij(\theta_x-\theta_y)}.
\]
In the fixed-boundary square, first cancel the interactions whose two
endpoints are fixed boundary spins. Integrating each remaining unfixed
spin imposes its integer divergence condition. Subtract a fixed primal
current carrying the inserted charges along the stated path. The
remaining current is divergence free at every integrated vertex: at all
vertices in the free spin box and at the interior vertices in the
fixed-boundary square. Rotation onto dual edges then gives an integral
gradient. In the free spin box its exterior height is fixed; in the
fixed-boundary square it is determined modulo a common translation.
The factors \(e^b\) in \eqref{eq:bessel-weight} cancel in the ratio.
The numerator and denominator are sums of strictly positive Bessel
weights, so the ratios are positive. Current reversal identifies each
insertion with its conjugate, giving the stated real cosine
expectations. This is the duality of \cite[Lemma~2.1]{C} and
\cite[Lemma~2.1]{S}, with their respective boundary conventions.
For the classical current and Bessel-height representations, see also
\cite[Appendix~A]{FrohlichSpencer1981} and
\cite[Section~2]{vanEngelenburgLis2023}.
\end{proof}
The reference form on a retained primary graph is the unit-conductance
form
\begin{equation}\label{eq:reference-energy}
\ip{g}{A_Gg}=\sum_{\{x,y\}\in E}(g_x-g_y)^2.
\end{equation}
It converges under lattice scaling to \(\int|\nabla g|^2\).
For the four nearest steps this is exactly the operator \(A_0\) in
\cite{B}. We write \(A\) when the graph is understood.
\subsection{The critical height inputs}
\begin{proposition}[Finite affine comparisons]\label{prop:comparisons}
On every fixed primary graph, the Bessel height law has the following
properties.
\begin{enumerate}
\item A shifted-to-centred partition ratio is at most one and
decreases when the same nonnegative quadratic precision is added to
the affine and centred coordinates. Affine data may include an
integral connection, observation centers, and compatible nonzero
equality values. The monotonicity remains valid with a fixed real
linear statistic in the numerator exponent, although the tilted
ratio need not be at most one.
\item For zero affine data, centred covariances and centred real moment
generating functions decrease under added centred precision. Centred
characteristic coefficients allowed by the constant convention in both
laws are nonnegative and increase under added precision. They are
strictly positive for the finite-graph phases
used below, whose coefficient total on every unconfined component is
an integer.
\item If \(X=\sum_e q_e\nabla_eh\) in a centred law, then
\[
\Var X\le C_b\sum_eq_e^2,\qquad
\E|X|^k\le C_{k,b}\left(\sum_eq_e^2\right)^{k/2}.
\]
More generally, after releasing endpoint, curl, and period constraints
while retaining independent primary increments, a tilt with
\(\max_e|q_e|\le K\) costs at most
\(\exp(C_{b,K}\sum_eq_e^2)\).
\end{enumerate}
The statements include independent Gaussian observation noises.
Released constants are taken modulo translations or confined; a real
statistic used in releasing a constant must annihilate it.
\end{proposition}
\begin{proof}
The comparison assertions are \cite[Lemma~2.2]{C}. That lemma replaces
each primary bond by a chain of lattice-Gaussian bonds, applies the
finite affine comparison on the enlarged lattice, and passes to the
limit at the fixed primary graph. Equality constraints are limits
of positive quadratic penalties. Thus the same affine statistic is
kept when observations or constraints are released.
Strict positivity for the phases used here follows from the
phase-transfer estimate \cite[Equation~(2.2) and its proof]{C}: on an
unconfined component subtract an integer point charge and interpolate
the remaining zero-total finite-variance test from zero; on a confined
component interpolate directly using its finite fixed-graph variance.
The finite lattice-Gaussian comparison underlying this argument is
developed in \cite[Lemma~2.1]{B} from the inequality of
\cite[Theorem~2.1 in the arXiv version]{RSD2017}.
For the last assertion, perform these releases before estimating the
tilt. The independent primary transform is the product of the
transforms in \eqref{eq:bessel-weight}, with \(t=2\pi q_e\).
For \(|q_e|\le K\),
\(\cosh(2\pi q_e)-1\le C_Kq_e^2\). This gives the last bound.
Normalise \(q\) by \((\sum q_e^2)^{1/2}\) and differentiate bounded
exponential moments to obtain the stated moment bounds. These are
the primary reference-cost bounds of \cite[Lemma~2.3]{C}; no bound
on the microscopic Gaussian precision of the subdivided chains is
used.
\end{proof}
The next proposition uses absolute pin data: \(h_H=0\) fixes every
primary height in \(H\), and \(b^p\) specifies a compatible absolute
height at every primary vertex of \(p\). Its approximation by finite
guard observations is understood in probability, at each prescribed
tolerance, under the sampled law stated there. The finite list of
smooth guard probes is fixed before the lattice limit and enlarged
afterward to reduce the tolerance.
\begin{proposition}[Critical coefficient and mixed limits]\label{prop:height-inputs}
For the critical parameter in \eqref{eq:critical-definition}, the
Bessel height coefficient of \cite{C} is
\begin{equation}\label{eq:critical-coefficient}
a=a(b_c)=8\pi.
\end{equation}
At this parameter the following fixed-geometry statements hold.
\begin{enumerate}
\item Let the fixed geometry belong to the polygonal Lipschitz class
of \cite[Theorem~3.1]{C}: it is a finite union of bounded rectangular
pieces with nondegenerate faces and widths. Pieces in one component
are joined along full interfaces, which may be subdivided into segments;
point contacts do not join components. All passages have positive width.
Allow free walls and
hard pins on the primary vertices in aligned unions of positive-area
boxes. On each component
\(U\), fixed bounded, piecewise continuous density smears of the height
have Gaussian Laplace and moment limits
with covariance the inverse of
\[
a^{-1}\int_U|\nabla v|^2
\quad\text{on}\quad
\{v\in H^1(U):v=0\text{ on the pin}\}.
\]
For an unpinned component this is a statement about mean-free smears;
its fractional mean modulo \(2\pi\) becomes independent and uniform.
Fixed finite-rank nonnegative penalties in box averages give the
corresponding Gaussian laws, with constants integrated when the
penalties confine them. The statements hold along every admissible
mesh sequence, including bounded rounding of the fixed geometry.
They include energy-approximable shifts in the sense of
\cite[Theorem~3.1]{C}.
\item Start with no hard pins and use the centred law whose
component-local box-average penalties confine every pre-pin component
constant. Let \(H,p\) be positively separated unions of boxes. Put
\[
D(H,p)=\log\frac{\Prob(h_H=0\mid h_p=0)}{\Prob(h_H=0)},\qquad
r_H(b^p)=\frac{\Prob(h_H=0\mid h_p=b^p)}
{\Prob(h_H=0\mid h_p=0)}.
\]
Then \(D(H,p)\ge0\) converges to the Gaussian pin interaction,
\(0<r_H\le1\), and
\(\E_{\mathrm{cen}} r_H=e^{-D(H,p)}\), where the expectation samples
the guard data in the centred law.
For every fixed finite list of these density smears and box averages,
the conditional Laplace transforms and \(r_H\) admit approximations
by fixed finite lists of smooth observations of the guard data, whose
laws and functions converge to their Gaussian counterparts. The
list is enlarged after the lattice limit. These convergence assertions
also hold when the data are sampled under a fixed finite real linear
tilt; \(D\) and \(r_H\) retain their centred definitions above.
\end{enumerate}
\end{proposition}
\begin{proof}
Equation~\eqref{eq:critical-coefficient} is the conclusion of
\cite[Section~6]{C}. The first and second parts are respectively
\cite[Theorems~3.1 and~3.2]{C}. In the second part, observations are
component-local before pins are imposed. The finite-observation
formulation is part of the theorem; it does not assert uniform
convergence at every microscopic guard value.
\end{proof}
We also use the restoration consequence \cite[Lemma~3.9]{C}.
Suppose every endpoint of a restored bond, and the support of every
restored form, lies in a core \(H\) separated from a guard \(p\).
If \(r_\Delta\) is the conditional restoration factor divided by
its value at zero guard data, then \(r_H\le r_\Delta\le1\).
The guard laws in the two topologies transfer events whose
probabilities tend to zero in either direction, also under a fixed
real linear tilt. The finite-observation approximations above concern
conditional Laplace transforms and \(r_H\); no additional limit for
\(r_\Delta\) is asserted. In each use, the graph, pin regions, and
observation cells are fixed before the lattice limit.
\section{Fixed-geometry comparison}\label{sec:geometry}
The center magnetization isolates the microscopic cost of one spin
insertion. We compare boxes by placing the same finite nest of annuli
around each corresponding source. An attachment calculation separates
the ratio of a nest from that of its exterior; the nest ratios then
cancel, leaving exterior energies at fixed geometry. The first
comparison is between square sizes, and the second concerns the
free-box thermodynamic correlation of two insertions. All continuum
limits used here keep the holes and every separating region at a
fixed positive size.
\begin{proposition}[Comparison with center magnetization]
\label{geo:comparison}
For integer sequences \(n,n'\to\infty\) with \(n'/n\to d\in(0,\infty)\),
\begin{equation}\label{geo:center-ratio}
\frac{M_{n'}}{M_n}\longrightarrow d^{-1/8}.
\end{equation}
There is a constant \(c_{\rm shape}\in(0,\infty)\) such that
\begin{equation}\label{geo:bulk-ratio}
\frac{C_{b_c}(r)}{M_r^2}\longrightarrow c_{\rm shape}
\qquad(r\to\infty,\ r\in\Z).
\end{equation}
Here \(C_{b_c}\) is the thermodynamic limit in
\eqref{eq:critical-definition}; that limit precedes \(r\to\infty\).
In particular, \(M_n=n^{-1/8+o(1)}\).
\end{proposition}
The proposition has two parts. \eqref{geo:center-ratio} says that
\(M_n\) obeys a clean power law, so later we may compare any two
geometric sizes. \eqref{geo:bulk-ratio} says that the free-box
correlation is \(M_r^2\) up to a fixed positive geometric factor; the
log-power \(1/8\) in \(C_{b_c}\) is then the square of the \(1/16\)
in the center asymptotic of Section~\ref{sec:conclusion}.
\subsection{Separating the insertion from the exterior}
For a graph with the boundary and constant conventions of
Section~\ref{sec:heights}, write
\(\Phi_G^\tau=Z_G^\tau/Z_G^0\) for its bare sector ratio.
The fixed-domain sector input evaluates this ratio on a free exterior.
Let \(U\) be a bounded connected rectilinear domain with a simple
outer boundary and finitely many disjoint simple holes, all with
dyadic vertices, positive side lengths, and positive passages.
Give the height graph free walls. If \(\vartheta_0\) is a smooth
closed one-form near \(\overline U\) whose periods about the holes are
prescribed multiples of \(2\pi\), set
\[
\mathcal M_U(\vartheta_0)
=\inf_{g\in H^1(U)}\int_U|\vartheta_0+\nabla g|^2.
\]
Let \(\mathcal G_\nu(U)\) be its nearest-neighbor primary graph at mesh
\(\nu^{-1}\). For integral lattice connections that are flat on the
retained graph away from the holes and have these periods,
the sector theorem of \cite[Theorem~3.1]{S} states that
\begin{equation}\label{geo:free-sector}
\Phi_{\mathcal G_\nu(U)}^\tau
\longrightarrow \exp\{-\mathcal M_U(\vartheta_0)/(2a)\}
\qquad(\nu\to\infty),
\end{equation}
where \(a=8\pi\). The limit holds along
every coherent bounded-step rounding of this fixed geometry. The
energy depends only on the periods. This statement concerns free
height walls; a pinned exterior will be treated below.
To connect \eqref{geo:free-sector} to a microscopic source, use the
annular construction of \cite[Sections~4 and~5]{S}. Fix a small
\(\varepsilon>0\) and a rectilinear disk \(P\) with dyadic vertices
such that
\begin{equation}\label{geo:polygon}
B(0,1-\varepsilon)\subset P\subset B(0,1+\varepsilon).
\end{equation}
Choose a sufficiently large power of four \(\lambda\), and an integer
\(\varrho_0\) aligned so that all relevant array divisions lie between
rows of dual vertices. Put \(\varrho_i=\varrho_0\lambda^i\).
The \(i\)-th annulus has walls \(\varrho_{i-1}P\) and
\(\varrho_iP\). Write \(Q_s=(-s,s)^2\) and
\(A_\lambda=\lambda P\setminus P^\circ\). In the reference annulus
\(A_\lambda\), the guard is
\(\overline{Q_{2\sqrt\lambda}}\setminus Q_{\sqrt\lambda}\), and the
inner and outer end regions are \(A_\lambda\cap\overline{Q_4}\) and
\(A_\lambda\setminus Q_{\lambda/4}\), respectively. Scale these
three regions by \(\varrho_{i-1}\), and call the guard \(\mathsf P_i\).
It is a full separating band, and each end contains every endpoint
in that annulus of a bond to be restored at its wall. These prescribed
bands, with positive gaps, are the geometry used in the flat-pin
estimate below.
Choose one vertex in \(\mathsf P_i\). A guard datum records every
height difference from that vertex. In the annular quotient choose
its height to be zero, so the datum specifies every guard height.
The nearest-neighbor weight then factors between the two sides
conditionally on that datum. Write \(\mu_i^\tau\) for the guard law
in the cut annulus, with its one common constant quotiented out.
A core is a connected finite block filling the first hole; it may
carry the source and has no guard data.
At an interface in this construction, let \(T^\tau(u)\) be the restored
partition sum divided by the cut partition sum at its incident guard
data \(u\),
and put \(d_T=T^0(0)\). The restored sum includes the relative
integer translations of the joined components. The same definition
applies to one joint interface joining several last annuli to a
connected exterior, with no direct bonds between distinct incident
annuli. Its data are those in the incident annuli, while all exterior
heights are summed together inside that one kernel.
For a vertex region \(H\), let \([H]\) denote the event that all its
heights are equal, without prescribing their common value. In a
centred cut annulus with guard \(\mathsf P_i\), put
\[
\overline D_i(H,\mathsf P_i)
=\log\frac{\Prob([H]\mid[\mathsf P_i])}{\Prob([H])}.
\]
The upper kernel bound of \cite[Lemma~5.1]{S}, its Equation~(8), holds
for all these interfaces, including a core interface. For an
annulus--annulus or joint exterior interface, its Equation~(9) also
gives the centred average bound:
\begin{equation}\label{geo:kernel-bounds}
0<\frac{T^\tau(u)}{d_T}\le1,
\qquad
\int\frac{T^0}{d_T}\prod_i\dd\mu_i^0
\ge \exp\left\{-\sum_i\overline D_i(H_i,\mathsf P_i)\right\}.
\end{equation}
The sum and product use the incident annuli and the appropriate end
in each. The first bound is pointwise in the data, and the second
is an average under independent centred cut annuli. The centred
divisor \(d_T\) therefore works for the shifted kernel as well.
The flat-pin estimate of \cite[Proposition~4.1]{S}
makes each interaction on the right arbitrarily small in the mesh
limit by taking \(\lambda\) large at fixed \(P\).
Two consequences of \cite[Proposition~5.3 and Corollary~5.5]{S}
control how the averaged loss passes through a whole nest.
For any \(\xi\in(0,1/64)\), one can first take
\(\varepsilon\) small, then \(\lambda\) large, and finally the fixed
aligned threshold \(\varrho_0\) large, so that every interface between
successive annuli of these radii satisfies
\begin{equation}\label{geo:regular-loss}
\int\left(1-\frac{T_i^\tau}{d_{T_i}}\right)
\dd(\mu_i^\tau\otimes\mu_{i+1}^\tau)\le\xi.
\end{equation}
This holds in the centred sector and in either unit-period sector,
when restoration introduces no source. It follows from the exact
sector identity (10) in that proposition and
\eqref{geo:free-sector}; it is the normalised estimate (11) there.
All late annuli obey the same bound because they refine the same
fixed geometry. Fix one such core, containing the source in a unit
sector, and fix its connection once on the increasing family. In a nest
of \(J\) annuli, the law of its last guard then has density at most \(2\)
with respect to
\(\mu_J^\tau\), for all sufficiently large \(J\). This holds in
all three sectors and for identical lattice translates. The only
lower assumption on the first core kernel is its strict positivity;
no uniform positive lower bound on that kernel is required.
This density bound compares the actual last-guard law, after all inner
blocks have been joined, with the cut law used in the averaged estimate.
It remains to attach the whole nests to their exterior.
Let \(\mathcal N_J\) be the bare sector ratio of one whole
free nest of depth \(J\), with its core and a unit source. Integral
gauge changes and reversal of all heights show that the same
\(\mathcal N_J\) is obtained from any lattice translate and either
sign. This is the common factor that will cancel in both comparisons.
Suppose \(k\) such nests, with disjoint closures, attach to a
connected exterior. The graph of cut blocks, with an edge for each
interface, is a tree rooted at the exterior; there are no other block
adjacencies, in particular no direct bonds between distinct incident
last annuli. Write \(\mu_{J,i}^\tau\) for the cut guard law in
the \(i\)-th incident last annulus, with its restricted connection.
In the following ratio, ``joined'' means the
exterior after joining only the \(k\) last annuli; the holes then have
the inner radii of those annuli. Define
\begin{equation}\label{geo:joining-ratio}
\mathfrak q^\tau_{\rm ext}
=\frac{\Phi_{\rm joined}^\tau}
{\Phi_{\rm ext}^\tau\prod_{i=1}^k\Phi_{{\rm last},i}^\tau}.
\end{equation}
Releasing those interfaces gives \(0<\mathfrak q^\tau_{\rm ext}\le1\).
Exact summation of the interface kernel, which is Equation~(10) of
\cite{S}, gives
\begin{equation}\label{geo:kernel-sector-identity}
\int\frac{T^\tau_{\rm ext}}{d_{\rm ext}}
\prod_i\dd\mu_{J,i}^\tau
=\left(\int\frac{T^0_{\rm ext}}{d_{\rm ext}}
\prod_i\dd\mu_{J,i}^0\right)\mathfrak q^\tau_{\rm ext}.
\end{equation}
Thus a sufficient attachment condition, along the lattice sequence
in question, is
\begin{equation}\label{geo:attachment-condition}
\sum_i\overline D_i(H_i,\mathsf P_i)\le\delta+o(1),
\qquad \log\mathfrak q^\tau_{\rm ext}\ge-\delta-o(1),
\end{equation}
with \(\delta>0\) arbitrarily small. Indeed,
\eqref{geo:kernel-bounds} and \eqref{geo:kernel-sector-identity}
bound the shifted average loss by
\(1-e^{-2\delta-o(1)}\), and the centred one by
\(1-e^{-\delta-o(1)}\). This estimate is averaged over the independent
cut-annulus laws, without any uniform lower bound at individual data.
The lower thermodynamic comparison below retains a hard pin in the
exterior. We therefore check that the same reasoning applies when
that pin fixes the exterior constant and is kept in both the restored
and cut graphs. The entire hard constraint remains in the exterior
partition sum and joint kernel; for the rim below, this includes every
equality \(h_x=0\) on the rim. Accordingly,
\(d_{\rm ext}=T_{\rm ext}^0(0)\) is evaluated with that pin retained.
Prescribed relative guard data are affine equalities, so
Proposition~\ref{prop:comparisons}, with the independent increments
on cut bonds, still gives \(0<T^\tau/d_T\le1\). To prove the centred
bound in \eqref{geo:kernel-bounds}, impose the flat end event \([H_i]\)
in each incident annulus and choose its representative to be zero on
that end. The restored-bond factor now depends on the relative integer
translations and the exterior heights, and is independent of the
guard data. Adding these end constraints decreases the data-to-zero
ratio, giving the product of the annular flat-pin ratios in
\eqref{geo:kernel-bounds}. This proof uses no unpinned exterior law.
For multiplication, place the exterior, with its full pin constraint,
at the root of the tree of cut blocks. Its configurations are summed
subject to every retained pin equality. Once its additive constant is
fixed, the remaining block constants are in bijection with the integer
differences along the tree edges. Conditioning on the full annular guards
separates their two sides, so the tree multiplication of
\cite[Lemma~5.2]{S} and the identity
\eqref{geo:kernel-sector-identity} follow with this pin retained.
To finish the attachment calculation, let \(e_\tau\) be the expectation
of \(T^\tau_{\rm ext}/d_{\rm ext}\) under the product of the propagated
last-guard laws of the whole nests, and define \(e_0\) in the same way.
Their densities relative to the independent cut laws are at most
\(2^k\), so \eqref{geo:attachment-condition} implies
\(1-e_\tau,1-e_0\le 2^k(1-e^{-2\delta-o(1)})\).
Exact multiplication and cancellation of \(d_{\rm ext}\) give
\begin{equation}\label{geo:attachment-factorization}
\Phi_{\rm full}^\tau
=\mathcal N_J^k\Phi_{\rm ext}^\tau\frac{e_\tau}{e_0},
\qquad
\log\Phi_{\rm full}^\tau
=k\log\mathcal N_J+\log\Phi_{\rm ext}^\tau
+O_k(\delta)+o(1).
\end{equation}
Here \(\delta\) is small at fixed \(k\), and \(J\to\infty\) with
the core, \(P,\lambda,\varrho_0\) fixed. The first identity is exact;
the second is the attachment consequence we shall use.
\subsection{A pinned comparison for the thermodynamic correlation}
We first place the desired thermodynamic correlation between two finite
height ratios. For a large dyadic \(R_{\rm out}\ge8\), let
\[
\mathsf F_{R_{\rm out}}
=\overline{Q_{7R_{\rm out}/8}}\setminus Q_{3R_{\rm out}/4}.
\]
The region \(\mathsf F_{R_{\rm out}}\) is a positive-width rim,
expressible as an aligned union of positive-area boxes with dyadic
vertices. In lattice units let
\(G_{R_{\rm out},r}\) be the dual primary array in \(rQ_{R_{\rm out}}\),
with coherent rounding, and let \(\mathsf F_{R_{\rm out},r}\) be the
filled primary boxes representing \(r\mathsf F_{R_{\rm out}}\). Put
\[
\Phi_{R_{\rm out},r}^{\rm fr}=\Phi_{G_{R_{\rm out},r}}^\tau,
\qquad
\Phi_{R_{\rm out},r}^{\rm pin}
=\Phi_{G_{R_{\rm out},r},\,h_{\mathsf F_{R_{\rm out},r}}=0}^\tau,
\]
where the sources are \(0,re_1\), with periods \(+2\pi,-2\pi\).
There is one connection suitable for both comparisons. In the primal
lattice take the actual nearest-neighbor path
\[
(0,0),(1,0),\ldots,(r,0)
\]
and give its edges the unit current directed from one source to the
other. The dual connection is supported on the dual bonds crossing
these path edges. They lie inside \(rQ_{R_{\rm out}/2}\), for all
large \(r\), so the connection vanishes on every bond meeting the
rim or lying outside it. Use this connection in every larger free
spin box.
For a box \(\Lambda_R\) containing \(rQ_{R_{\rm out}}\) in its interior,
release the matching equalities for all bonds crossing the boundary of
\(G_{R_{\rm out},r}\), retaining their independent Bessel increments as
in the finite comparison. The resulting inner height component has
a free wall; the connection on the exterior component is zero and its
ratio cancels. Proposition~\ref{prop:comparisons} therefore gives the
upper bound by \(\Phi_{R_{\rm out},r}^{\rm fr}\). For the lower bound,
add the common hard pin \(h_{\mathsf F_{R_{\rm out},r}}=0\) before making
any cut. Added precision decreases the sector ratio. This filled rim
separates the interior from the entire exterior. Because the
connection is supported on the specified path inside the rim, the
partition factor for all exterior variables, including the original
fixed exterior face, is identical in the numerator and denominator.
It cancels. The same cancellation of the outer collar occurs in
\(G_{R_{\rm out},r}\) with this rim pinned, so the remaining ratio
is exactly \(\Phi_{R_{\rm out},r}^{\rm pin}\). Hence, for every
sufficiently large \(R\) at each fixed \(R_{\rm out},r\),
\[
\Phi_{R_{\rm out},r}^{\rm pin}\le
\E^{\rm free}_{R,b_c}\cos(\theta_0-\theta_{re_1})
\le \Phi_{R_{\rm out},r}^{\rm fr}.
\]
Taking \(R\to\infty\) at this point proves
\begin{equation}\label{geo:thermodynamic-bracket}
\Phi_{R_{\rm out},r}^{\rm pin}\le C_{b_c}(r)
\le \Phi_{R_{\rm out},r}^{\rm fr}.
\end{equation}
The finite comparisons just used are the affine and matching-constraint
comparisons of \cite[Lemma~2.2]{C}; the exterior cancellation is why
the location of the connection matters.
To estimate the lower comparator, we need a sector lower bound with
the rim pinned. The next lemma supplies exactly that bound.
\begin{lemma}[Trial energy with a pinned rim]\label{geo:pinned-sector}
Let \(U\) be a bounded connected polygonal Lipschitz domain in the
fixed-geometry class of Proposition~\ref{prop:height-inputs}. Its outer
boundary and the boundaries of its two holes are simple and rectilinear,
with dyadic vertices and positive side lengths; the hole closures are
disjoint and strictly inside the outer boundary, leaving positive-width
passages. Let \(\mathsf F\) be an aligned union of positive-area boxes
with dyadic vertices in \(U\), permitted as a pin in that fixed geometry.
Suppose a simple dyadic rectilinear arc joins
interiors of sides of the two holes transversely and has a neighborhood
at positive distance from \(\mathsf F\) and all nonincident boundaries.
Give the holes periods \(+2\pi,-2\pi\).
Let \(\vartheta=dv\) be a smooth closed one-form near \(\overline U\)
with these periods, with a multivalued primitive whose only jump is
across that arc and whose branch is zero on a neighborhood of
\(\mathsf F\). Choose the canonical integral connections obtained
from the jumps of this branch along a coherent rounded discretisation
of the arc, or connections
related to them by integral gauges whose potentials vanish on every
component of \(\mathsf F\). Then, along every admissible rounded mesh sequence,
\begin{equation}\label{geo:pinned-sector-bound}
\liminf_{\nu\to\infty}
\Phi_{\mathcal G_\nu(U),\,h_{\mathsf F}=0}^\tau
\ge \exp\left\{-\frac1{2a}\int_U|\vartheta|^2\right\}.
\end{equation}
\end{lemma}
\begin{proof}
We adapt the confined-sector argument in the proof of
\cite[Proposition~3.4]{S}, retaining the rim pin. A permitted gauge
preserves the pin and the ratio, so we use the canonical connection.
Round the arc within a bounded number of lattice steps and extend its
transverse ends to the rounded hole walls. Transport the local branches
across this rounding, which remains away from \(\mathsf F\), and sample
them at the lattice vertices,
using a coherent one-sided assignment when the rounded cut meets a
vertex. Choose the integral jump \(\tau_e\) on each edge with its
matching sign. The resulting real lattice values \(v_\nu\) satisfy
\[
\nabla_e v_\nu+\tau_e=\int_e\vartheta,
\qquad v_\nu=0\text{ on the rounded pin }\mathsf F.
\]
The identity follows by integrating \(dv\) along each edge, including
its integral jump when the edge crosses or meets the rounded cut.
The branches differ by integral multiples of \(2\pi\), so
\(\tau_e\in2\pi\Z\); supported bonds lie within one lattice step of
the rounded arc. The chosen branch is
zero near every component of the pin. These branches are fixed before
refining the lattice.
Isolate a narrow rectilinear strip following the arc between the holes
by cutting the bonds across both of its sides. Choose a wider strip
patch \(W\) and full guard bands \(\mathsf P\) on the two sides of the
narrow strip, with positive buffers from the restored bonds and from
the artificial boundary of \(W\). The strip and patch meet the hole
walls in interiors of sides. Choose all their widths and coordinates
dyadically and keep \(W\) at positive distance from \(\mathsf F\).
For every sufficiently fine mesh, the narrow strip contains all bonds
supporting \(\tau\). After this cut the strip admits a single-valued
lift. The remaining domain excludes that entire support, so its lift
is the original branch, already zero near every component of
\(\mathsf F\). The patch \(W\), which is disjoint from the pin, admits
a lift both before and after restoring the strip.
Fix an endpoint region \(H\) inside \(W\) containing both primary
endpoints of every restored strip bond, and a region \(H'\) containing
both endpoints of every bond across the artificial boundary of \(W\).
Choose these as aligned unions of positive-area boxes with dyadic
vertices, separated from \(\mathsf P\). At the transverse exits, their
intervals on the hole sides follow the same nested order as the strip,
guard, and patch boundaries, with positive gaps from each other and
the side corners. The guard bands continue to the original hole walls
between the intervals for \(H\) and \(H'\). Hence every path in the
retained domain from a restored strip interface to the artificial
boundary of \(W\) meets a guard, including paths near the exits.
Use separate boxes on the two sides of every pre-pin cut, so each box
lies in one pre-pin component. The clearance from \(W\) to the rim
also permits \(H'\) to be disjoint from \(\mathsf F\).
Fix one dyadic cell side \(\ell>0\) and one cell partition aligned with
the holes, the rim, the strip, \(W\), and all guard and endpoint bands.
At each topology determine its components before imposing pins. The
restriction of this same partition has every whole observation cell in
one such component. Using one box average on every cell confines the
constant of every pre-pin component, including those later fixed by
\(\mathsf F\). On every topology, with \(\mathsf F\)
already pinned, add \(\frac12\sum_Q(h_Q-(v_\nu)_Q)^2\) in the numerator
and \(\frac12\sum_Qh_Q^2\) in the denominator. Denote the resulting
full graph ratio by \(\Phi^{\tau,\mathrm{conf}}_{\nu,\ell}\); below,
a topology subscript specifies its cut version. The new observation
residuals have the same positive precision, and the pin is compatible
since \(v_\nu=0\) there. Thus
\begin{equation}\label{geo:confinement-below-bare}
0<\Phi^{\tau,\mathrm{conf}}_{\nu,\ell}
\le \Phi_{\mathcal G_\nu(U),\,h_{\mathsf F}=0}^\tau.
\end{equation}
On a cut component \(V\), choose its single-valued lift \(v_V\) and
sample it at the lattice vertices with \(v_V-v_\nu\in2\pi\Z\)
pointwise. Since \(\nabla_e v_V=\int_e\vartheta\), the integral
potential \(\psi=v_\nu-v_V\) satisfies \(\tau+\nabla\psi=0\).
If \(V\) meets \(\mathsf F\), both lifts use the zero branch there,
so \(\psi=0\) on the pin. Writing \(B\) for the cell-average map,
the gauge rule of Section~\ref{sec:heights} changes the observation
center \(Bv_\nu\) to \(Bv_\nu-B\psi=Bv_V\). The mixed height limit
in Proposition~\ref{prop:height-inputs}, at this fixed \(\ell\),
and Gaussian completion give for its confined ratio the limit
\(\exp\{-I_{\ell,V}(v_V)/2\}\), where
\[
I_{\ell,V}(v_V)=
\inf_{\substack{u\in H^1(V)\\u=0\ \mathrm{on}\ V\cap\mathsf F}}
\left\{a^{-1}\int_V|\nabla u|^2+
\sum_{Q\subset V}(u_Q-(v_V)_Q)^2\right\}.
\]
An empty pin in this display imposes no constraint. The unpinned
constant is integrated. The converging lattice cell centers can be used
in the Laplace limit: transforms at nearby fixed arguments give local
equicontinuity by H\"older's inequality. Trying \(u=v_V\) is allowed,
and cell Poincar\'e gives, for bounded values of the displayed functional,
\[
\|u-v_V\|_{L^2(V)}^2
\le C\ell^2\left(\|\nabla u\|_2^2+\|\nabla v_V\|_2^2+
\sum_Q(u_Q-(v_V)_Q)^2\right).
\]
Weak compactness and lower semicontinuity therefore show that
\(I_{\ell,V}(v_V)\to a^{-1}\int_V|\nabla v_V|^2\) as
\(\ell\downarrow0\). This is the single-lift argument using the
hard-pin energy space of \cite[Theorem~3.1]{C}.
We verify that restoring the strip has a vanishing averaged loss.
For full absolute data \(u\) on \(\mathsf P\), let
\[
K^\tau(u)=\frac{Z_{\rm open,\mathsf F}^\tau(u)}
{Z_{\rm cut,\mathsf F}^\tau(u)},
\qquad d_0=K^0(0),
\]
with the confinement retained. Here \(\mathrm{open}\) restores the bonds
cut across the two sides of the narrow strip, while \(\mathrm{cut}\)
retains those cuts. The guard separates the restored bonds
from the artificial side boundary of \(W\) and from \(\mathsf F\).
No observation cell crosses a guard division. Conditioning on the full
guard hence makes
\(K^\tau(u)\) exactly the kernel computed on \(W\) alone, so it is
unchanged by the exterior pin. Finite affine comparison gives
\(0<K^\tau(u)\le d_0\).
Adding the pin on the endpoint region \(H\) to the centred
data-to-zero comparison makes the restored factor a scalar.
Bayes' formula then gives
\[
1\ge\E_{{\rm cut}\,W,0}\frac{K^0}{d_0}
\ge e^{-D_{\nu,\ell}^{W}(H,\mathsf P)}.
\]
Here \(D_{\nu,\ell}^{W}\) is the absolute pin interaction in the centred
cut patch with its component-local cell observations, before either
hard pin is imposed. The localisation estimate of
\cite[Lemma~3.2]{S} applies to this pre-pin law with no existing hard
pins: for every fixed dyadic cut mixed geometry and positively separated
aligned pin regions that are unions of positive-area boxes with dyadic vertices,
\begin{equation}\label{geo:cell-pin-localization}
\lim_{\ell\downarrow0}\limsup_{\nu\to\infty}
D_{\nu,\ell}(H,\mathsf P)=0.
\end{equation}
It uses the Gaussian pin limit of \cite[Theorem~3.2]{C} for precision
\(a^{-1}\int|\nabla u|^2+\sum_Q u_Q^2\); \(\ell\) decreases through
aligned dyadic values after the mesh limit. Both topologies of \(W\)
have the single-lift limit just proved, with the same energy since the
deleted interfaces have zero area. Consequently the nonnegative patch
loss satisfies
\begin{align*}
L_W^\tau
&=\sum_u\frac{Z_{{\rm cut}\,W}^\tau(u)}{Z_{{\rm cut}\,W}^0}
\left(1-\frac{K^\tau(u)}{d_0}\right)\\
&=\Phi_{{\rm cut}\,W}^{\tau,\mathrm{conf}}-
\Phi_{{\rm open}\,W}^{\tau,\mathrm{conf}}\,
\E_{{\rm cut}\,W,0}\frac{K^0}{d_0}
\longrightarrow0
\qquad(\nu\to\infty,\text{ then }\ell\downarrow0).
\end{align*}
The centred loss has the same conclusion.
It remains to pass this loss from \(W\) to the whole pinned graph.
Let \(G\) be the current graph with the narrow strip cut, and let
\(G'\) further cut the bonds across the artificial boundary of \(W\).
At prescribed guard data, finite comparison, with the rim pin retained,
gives
\begin{equation}\label{geo:global-data-comparison}
\frac{Z_{G,\mathsf F}^\tau(u)}{Z_{G,\mathsf F}^0}
\le
\frac{\Prob_{G,\mathsf F,0}(h_{\mathsf P}=0)}
{\Prob_{G',\mathsf F,0}(h_{\mathsf P}=0)}
\frac{Z_{G',\mathsf F}^\tau(u)}{Z_{G',\mathsf F}^0}.
\end{equation}
Use the endpoint region \(H'\) already chosen around that artificial
boundary, separated from \(\mathsf P\).
If \(R(u)\) denotes the centred restoration factor from \(G'\) to
\(G\), summation gives the probability ratio in
\eqref{geo:global-data-comparison} as \(R(0)/\E_{G',\mathsf F,0}R\).
The finite comparison argument in the proof of \cite[Lemma~3.9]{C}
can be applied with the common rim constraint retained: adding the
pin on \(H'\) fixes both endpoints of each restored bond, so its
restoration factor becomes independent of the guard data. Dividing
the data-to-zero ratios and averaging by Bayes' formula yields
\[
\frac{\E_{G',\mathsf F,0}R}{R(0)}
\ge e^{-D^{\mathsf F}_{\nu,\ell}(H',\mathsf P)}.
\]
To control this conditional interaction by the available unpinned
theorem, use the centred confined law on the pre-pin graph \(G'\) and
write all pin events with zero values. Bayes' formula gives exactly
\begin{equation}\label{geo:conditional-pin-identity}
\begin{split}
0\le D^{\mathsf F}_{\nu,\ell}(H',\mathsf P)
&:=\log\frac{\Prob(h_{H'}=0\mid h_{\mathsf P}=0,h_{\mathsf F}=0)}
{\Prob(h_{H'}=0\mid h_{\mathsf F}=0)}\\
&=D_{\nu,\ell}(H'\cup\mathsf F,\mathsf P)
-D_{\nu,\ell}(\mathsf F,\mathsf P)\\
&\le D_{\nu,\ell}(H'\cup\mathsf F,\mathsf P).
\end{split}
\end{equation}
The first inequality is centred pin monotonicity in the law already
pinned on \(\mathsf F\); the last uses the nonnegativity of the
unconditional interaction. The union \(H'\cup\mathsf F\) remains
at positive distance from \(\mathsf P\). Its boxes and the observation
cells are component-local on \(G'\) before pins are imposed.
Thus \eqref{geo:cell-pin-localization}, applied to this union in that
pre-pin geometry, makes the last expression tend to zero in the stated
order. This derives the needed conditional estimate from the
unconditional pin theorem.
Multiply \eqref{geo:global-data-comparison} by
\(1-K^\tau(u)/d_0\ge0\) and sum over \(u\). On \(G'\), the part outside
\(W\) factors off as a confined shifted-to-centred ratio, at most one
by Proposition~\ref{prop:comparisons}. The global loss is therefore
at most \(e^{D_{\nu,\ell}(H'\cup\mathsf F,\mathsf P)}L_W^\tau\), and
tends to zero. The same argument applies in the centred sector. If
these two losses, normalised by the centred cut sum, are \(L_\tau,L_0\),
exact summation of the restored sums gives
\[
\Phi_{\rm open}^{\tau,\mathrm{conf}}
=\frac{\Phi_{\rm cut}^{\tau,\mathrm{conf}}-L_\tau}
{1-L_0}.
\]
The positive scalar \(d_0\) has canceled. The losses vanish, and the
cut components' single-lift energies add to \(\int_U|\vartheta|^2\).
This proves the ordered estimate
\[
\lim_{\ell\downarrow0}\limsup_{\nu\to\infty}
\left|\Phi^{\tau,\mathrm{conf}}_{\nu,\ell}
-\exp\left\{-\frac1{2a}\int_U|\vartheta|^2\right\}\right|=0.
\]
For each fixed aligned \(\ell\),
\eqref{geo:confinement-below-bare} bounds the bare pinned ratio below
by this confined ratio. Taking its lattice liminf and then letting
\(\ell\downarrow0\) proves \eqref{geo:pinned-sector-bound}.
The holes, rim, strip, patch, and bands were all fixed before these
limits.
\end{proof}
\subsection{The energies outside the nests}
For the small source in a fixed domain, isoperimetric arguments
provide trial forms that estimate the energies that survive in the
two comparisons. All these estimates are for fixed geometries; they
enter the final answers only through their logarithmic parts.
\emph{The centred insertion.} For a small circular hole of radius
\(t\) around a single unit source in the reference square \(D\),
Green's identity on the exterior with a logarithmic trial gives
\begin{equation}\label{geo:single-energy}
\mathcal M_{{\rm cen},D}(t)
=2\pi\log(1/t)+4\pi^2R_D(0,0)+O(t),
\end{equation}
where \(R_D\) is the Green function of \(-\Delta\) on \(D\) with
Dirichlet data zero on the outer boundary and a mean-zero convention
on the inner potential of the hole. The \(O(t)\) error is
independent of the mesh parameters. Replacing the circular hole by
the polygonal \(tP\) of \eqref{geo:polygon} changes the right side by
\(O(\varepsilon)\).
\emph{The pair.} For the pair \(0,e_1\) in \(Q_{R_{\rm out}}=R_{\rm out}D\),
put
\[
\mathfrak h_{R_{\rm out}}
=R_{Q_{R_{\rm out}}}(0,0)+R_{Q_{R_{\rm out}}}(e_1,e_1)
-2G_{Q_{R_{\rm out}}}(0,e_1).
\]
Green scaling gives
\(R_{Q_{R_{\rm out}}}(z,w)=(2\pi)^{-1}\log R_{\rm out}
+R_D(z/R_{\rm out},w/R_{\rm out})\).
The singular logarithm between \(0\) and \(e_1\) is zero, so
\[
\mathfrak h_{R_{\rm out}}
=R_D(0,0)+R_D(e_1/R_{\rm out},e_1/R_{\rm out})
-2R_D(0,e_1/R_{\rm out})
\longrightarrow0.
\]
The \(\log R_{\rm out}\) terms have canceled by neutrality. The same
exterior formula therefore reads
\begin{equation}\label{geo:free-pair-energy}
\mathcal M_{{\rm pair},R_{\rm out}}(t)
=4\pi\log(1/t)+4\pi^2\mathfrak h_{R_{\rm out}}
+O_{R_{\rm out}}(t).
\end{equation}
In both \eqref{geo:single-energy} and \eqref{geo:free-pair-energy},
replacing circular holes by \(tP\) changes the right side by
\(O(\varepsilon)\): restrict admissible forms between the exteriors
of the circles of radii \(t(1-\varepsilon)\) and
\(t(1+\varepsilon)\), and use their logarithmic terms. The constant
in this error is independent of small \(t\). The same restriction
argument on a circular annulus proves, for either unit period,
\begin{equation}\label{geo:annular-energy}
\mathcal M_{\lambda P\setminus P^\circ}(\pm1)
=2\pi\log\lambda+O(\varepsilon),
\end{equation}
with an error independent of large fixed \(\lambda\). This is also
the annular energy estimate in the proof of
\cite[Proposition~5.3]{S}.
For the pinned lower bound we exhibit a trial attaining the same pair
asymptotics. Identify the plane with \(\mathbb C\), and let
\[
\Theta(z)=\arg z-\arg(z-1).
\]
Use the branch on \(\mathbb C\setminus[0,1]\) that tends to zero at
infinity. Its gradient has periods \(+2\pi,-2\pi\), and
\(\Theta(z)=O(|z|^{-1})\), \(\nabla\Theta(z)=O(|z|^{-2})\) at
infinity. Rotation identifies \(\nabla\Theta\), up to orientation,
with the gradient of \(\log|z|-\log|z-1|\). Green's identity on the
plane outside the two radius-\(t\) disks gives
\begin{equation}\label{geo:plane-pair-energy}
\int_{\R^2\setminus(B(0,t)\cup B(e_1,t))}|\nabla\Theta|^2
=4\pi\log(1/t)+O(t).
\end{equation}
Indeed each small circle contributes \(2\pi\log(1/t)+O(t)\).
The cross logarithm is \(\log|e_1|=0\), and the boundary term at
infinity vanishes by the displayed decay. Harmonicity of each regular
part removes a logarithmically amplified error, as in the preceding
Green calculation.
\emph{The exterior constants.} For the center computation the exterior
in the reference square \(D\) is the same for the two sizes. The
closed-cell discrete Green function \(R_g^{(s)}\) on the fixed
reference domain, with its mean-zero convention, converges to \(R_D\).
Choosing dyadic aligned discrete holes gives the displayed estimates
with \(o(1)\), uniformly in the mesh. The same exterior formula
restricts the pair exterior to the fixed domain \(D\) in the larger
box \(Q_{R_{\rm out}}\); the \(\log R_{\rm out}\) term is fixed and
cancels in the ratio. Thus the exterior contributes the same
constant to \(\log\Phi^{\rm ext}\) in \eqref{geo:center-exterior}
and \eqref{geo:pair-exterior}, once \(R_{\rm out}\) is fixed.
We record carefully how these energies apply when the last radius moves
with the lattice. Suppose \(t_\nu\to t>0\). Around each limiting source,
choose fixed dyadic stair holes lying just inside the disk of radius
\((1-3\varepsilon)t\) and just outside the disk of radius
\((1+3\varepsilon)t\). Small further perturbations of their dyadic
centers and sides give strict eventual inclusions of the rounded moving
holes \(t_\nu P\). For the pair, choose exit sides whose interiors
meet the fixed source-to-source path transversely.
All these holes contain the source, remain separated,
and leave positive passages. Cut comparison brackets the three sector
ratios, using restrictions of the one connection chosen before the cuts.
For the pinned ratios keep exactly the same rim in both brackets;
the rim is connected, so the integral potential relating two flat
representatives with the same periods and zero connection near the rim
is constant there and can be normalized to zero. Apply
\eqref{geo:free-sector} to the fixed free exteriors and
Lemma~\ref{geo:pinned-sector} to the fixed pinned inner bracket.
The changes of radius cost \(O(\varepsilon)\) in the energy formulas.
This is the moving-hole argument of \cite[Lemma~6.2]{S}, with the pin
retained in the lower comparison. Bounded rounding of the outer arrays
and filled rim boxes is included in the fixed mixed limits. In
particular, every sector limit just invoked has holes of positive radius.
Let \(\Phi_{{\rm cen},\nu}^{\rm ext}(t_\nu)\) denote the free exterior in
the reference square \(D\), with one moving polygonal hole at the center.
The preceding argument gives
\begin{equation}\label{geo:center-exterior}
\limsup_{\nu\to\infty}\left|
\log\Phi_{{\rm cen},\nu}^{\rm ext}(t_\nu)+
\frac{2\pi\log(1/t)+4\pi^2R_D(0,0)}{2a}\right|
\le C(\varepsilon+t).
\end{equation}
For the pair exterior in \(Q_{R_{\rm out}}\), write
\(\Phi_{{\rm pair},R_{\rm out},\nu}^{\rm ext,fr}(t_\nu)\) for the free-wall ratio and
\(\Phi_{{\rm pair},R_{\rm out},\nu}^{\rm ext,pin}(t_\nu)\) for the ratio with
\(\mathsf F_{R_{\rm out}}\) pinned. Releasing that pin bounds the latter by the
former. Combining \eqref{geo:free-pair-energy}, the
plane energy trial \eqref{geo:plane-pair-energy} (which gives a
pin-admissible test at the same leading energy), and
Lemma~\ref{geo:pinned-sector}, gives the following useful uniform
statement. Given \(\delta>0\),
first choose \(R_{\rm out}\) sufficiently large, then choose a bound
\(\eta>0\) and \(\varepsilon>0\) sufficiently small. For either
boundary choice \(\star\in\{\mathrm{fr},\mathrm{pin}\}\) and every
sequence \(t_\nu\to t\in(0,\eta]\),
\begin{equation}\label{geo:pair-exterior}
\limsup_{\nu\to\infty}\left|
\log\Phi_{{\rm pair},R_{\rm out},\nu}^{\mathrm{ext},\star}(t_\nu)
+\frac{4\pi\log(1/t)}{2a}\right|\le\delta.
\end{equation}
Here one uses \(\mathfrak h_{R_{\rm out}}\to0\) and \(R_{\rm out}^{-2}\to0\) first, then the
bound \(\eta\) in Lemma~\ref{geo:pinned-sector}.
\subsection{Cancellation of the microscopic costs}
We combine the attachment formula \eqref{geo:attachment-factorization}
with the energy formula \eqref{geo:center-exterior} and its pair
analogue. Fix the entire nest: its depth \(J\), the polygonal hole \(P\),
the annulus ratios \(\lambda,\varrho_0\), and every guard and
separation band with positive widths. All these are fixed before any
lattice or continuum limit. In both sizes we use the same nest and
the same connection. The bare nest ratio \(\mathcal N_J\) and the
fixed exterior constant in each formula therefore appear only in
combinations that are unchanged by passing from one size to another.
We verify the attachment condition for either square. After joining
the last annulus to its exterior, the hole radius changes from \(t\)
to \(t/\lambda\). By \eqref{geo:free-sector} and
\eqref{geo:annular-energy}, its annular log ratio is
\(-2\pi\log\lambda/(2a)+O(\varepsilon)+o(1)\). In
\eqref{geo:center-exterior} the two exterior constant terms agree,
and the difference of their radius energies is exactly
\(2\pi\log\lambda\). Thus the leading terms in
\(\log\mathfrak q^\tau_{\rm ext}\) cancel and its limiting absolute error is
\(O(\varepsilon+\eta)\). In the other square the error is
\(O_d(\varepsilon+\eta)\), using \(t/d\) and \(t/(d\lambda)\).
These bounds hold uniformly for the possible \(t\) at large fixed
\(\lambda\). The flat-pin interactions can also be made arbitrarily
small by choosing \(\lambda\) large. More precisely, for any desired
attachment accuracy, choose \(\varepsilon,\eta\) small, then
\(\lambda\) large for the flat-pin estimate, and finally the fixed
\(\varrho_0\) large for \eqref{geo:regular-loss}. The inequalities
\eqref{geo:attachment-condition} then hold with arbitrarily small
\(\delta\).
\begin{proof}[Proof of Proposition~\ref{geo:comparison}]
\emph{Center ratio.} Fix \(R_{\rm out}\) large and then take the
nest parameters as above, all fixed. Take a \(t\in(0,\eta]\) with
\(tR_{\rm out}<1\). At the start, the current hole radius is
\(\varrho_0Pt\). Use this same nest of annuli and the same
connection placed identically around each of the two sources. The
annular log ratios telescope through the depth \(J\), and every
exterior error is bounded by the same small constant.
Equation~\eqref{geo:attachment-factorization} then gives, for two
sizes \(n/n'\to d\) as in the statement,
\[
\log\frac{M_{n'}}{M_n}
=\log\frac{\Phi_{{\rm cen},n'}^{\rm ext}(t)}{\Phi_{{\rm cen},n}^{\rm ext}(t)}
+o(1).
\]
By \eqref{geo:center-exterior} the right side converges to
\(-(\log d)/(8)\), giving \eqref{geo:center-ratio}. The
\(o(1)\) absorbs the finitely many inner annuli and the mesh errors.
\emph{Bulk ratio.} Repeat the same construction with the pair of
sources in \(Q_{R_{\rm out}}\) instead of a single source. The
exterior ratios in \eqref{geo:pair-exterior} now contribute the
common factor \(R_{\rm out}^{-2}\) times
\(\exp(-4\pi^2\mathfrak h_{R_{\rm out}}/(2a))\). The microscopic nests
and attachment cancellations are identical to the center case, and
the two sources' contributions combine to the same microscopic
factor. Thus
\(\log(\Phi_{R_{\rm out},r}^{\rm fr}/\Phi_{R_{\rm out},r}^{\rm pin})
\to0\), and, keeping the rims fixed before taking the mesh and
thermodynamic limits,
\[
\log C_{b_c}(r)=-\frac{4\pi}{2a}\log r+O(1)
+\frac{2\pi^2}{a}\mathfrak h_{R_{\rm out}}
\;\longrightarrow\;-\frac14\log r+O(1).
\]
More precisely, the same nest construction applied to the center
\(M_r\) doubles to the pair contribution, and the interior of
\(Q_{R_{\rm out}}\) contributes the constant
\(\exp(-2\pi^2\mathfrak h_{R_{\rm out}}/a)\), which tends to one
as \(R_{\rm out}\to\infty\) because \(\mathfrak h_{R_{\rm out}}\to0\).
Here \(a=8\pi\) makes $4\pi/(2a)=1/4$, giving the stated
\(r^{-1/4}\) decay, and the remaining constant is uniform in \(r\).
The ratio \(C_{b_c}(r)/M_r^2\) therefore converges to a finite
positive limit \(c_{\rm shape}\). This is \eqref{geo:bulk-ratio}.
\end{proof}
\section{A scale-dependent cutoff and entry}\label{sec:entry}
We build a small local interaction for the critical Bessel heights at
a starting side that may grow with the size of the square. Away from
the magnetic source the construction gives the small activities
required by the height map of Section~\ref{sec:trajectory}; at the
source it gives one labelled activity, whose norm may grow like a
fixed power of the observation-cell side. The cutoff that makes
these bounds possible has negligible cost in the finite physical
partition functions.
The analytic machinery is the local subset map of
\cite[Proposition~4.8 and Sections~4, 8, 9]{B}. We state the
resulting identities precisely and give the estimates that the
later sections use; the deep shell, localization, and covariance
estimates of that reference apply verbatim, with the two additions
we verify here (the \emph{observation-regulator} of
\cite[Lemma~9.2]{B} and the wall/box verification).
\subsection{The averaged interaction and its subset targets}
From now on \(p=p_{b_c}\), \(\alpha=\sqrt a\), and
\(\omega=2\pi/\alpha\). With \(a=8\pi\) this gives
\(\alpha=\sqrt{8\pi}\) and \(\omega=\sqrt{\pi/2}\); only the
\(\omega\)-periodicity of the interaction enters below.
Let \(G\) be a finite retained graph whose primary vertices are
partitioned into whole square cells of side \(s\). Let \(B_s\)
be the cell average. Introduce independent standard Gaussian noise
\(e\) and observations
\[
y=B_sh+e.
\]
For prescribed \(y\), let \(Z_G^\tau(y)\) include the penalty
\(\exp(-\|B_sh-y\|^2/2)\), with every retained observation row.
The corresponding unpinned real Gaussian energy is
\begin{equation}\label{eq:observation-energy}
E_G^\tau(y)=\min_u
\left\{a^{-1}\norm{\nabla u+\tau}_G^2+\norm{B_su-y}^2\right\}.
\end{equation}
On a cut graph the minimum varies every free component constant.
At fixed \(y\), the observation penalty confines those constants
in the height sum as well; all component translation copies are
summed.
We use the reference representation on the connected full graphs:
a compatible torus in the centered sector, or an aligned free square
in its centered or center-unit sector. On such a graph choose \(v\)
so that \(\eta=\nabla v+\tau\) is co-closed, including at a free
boundary. Orthogonality to gradients gives
\begin{equation}\label{eq:sector-completion}
E_G^\tau(y)=a^{-1}\norm{\eta}_G^2+E_G^0(y-B_sv).
\end{equation}
Let \(N_{\rm c}\) be the number of observation cells and
\(\mathcal Y_G=\R^{N_{\rm c}}/(2\pi\Z\mathbf1)\). Integrals on
this quotient use ambient Lebesgue measure on a fundamental domain.
Write \(Z_G^\tau\) without an observation argument for the bare
partition sum, modulo one common height translation. Summing all
height translation copies in \(Z_G^\tau(y)\) and integrating their
Gaussian penalties gives exactly
\[
\int_{\mathcal Y_G}Z_G^\tau(y)\,\dd y
=(2\pi)^{N_{\rm c}/2}Z_G^\tau .
\]
\subsection{The Gaussian reference and the local map}
Fix a dyadic ratio \(L\), and for an integral \(u\ge1\) put
\[
P_u(\cos t)=
\left(\frac{\sin(ut/2)}{u\sin(t/2)}\right)^{64},
\qquad \mathsf P_u=P_u(1-A/64).
\]
The removable values are understood. The split of
\cite[Equation~(9.11)]{B}, with no independent first piece, is
\begin{equation}\label{ent:reference-split}
\begin{split}
C_{<u}&=A^{-1}(1-\mathsf P_u),\\
\Gamma_u&=A^{-1}(\mathsf P_u-\mathsf P_{Lu}),\\
C_{\ge u}&=A^+\mathsf P_u\quad\hbox{on nonconstant modes}.
\end{split}
\end{equation}
The first two quotients use their polynomial continuations at zero.
They are positive, have ranges \(O(u)\) and \(O(Lu)\), and telescope
on nonconstant modes. Their continued constant modes may be
integrated along with the other modes: the final common mean is
uniform modulo \(\omega\), so adding an independent Gaussian constant
does not change a periodic integral. On a free square these
operators are the even restrictions of the operators on the doubled
torus.
The scaled measurement of distance and derivatives uses the point
norm
\begin{equation}\label{eq:point-norm}
\begin{split}
\norm{f}_{u,D}
&=\max_{x\in D}\bigl\{|f(x)|,u|\nabla f(x)|,
u^2|\nabla^2f(x)|\bigr\},
\end{split}
\end{equation}
on every dilated stencil. For a field polynomial
$F$ on a finite activity support \(X\) and background \(\phi\),
the analytic activity norm is the sum of directional derivatives
at a fixed radius \(h_0\):
\begin{equation}\label{eq:activity-norm}
|F|_{u,X,\phi}
=\sum_{k\ge0}\frac{h_0^k}{k!}
\sup_{\norm{f_i}_{u,D_u(X)}\le1}
|D^kF(\phi)[f_1,\ldots,f_k]|.
\end{equation}
The padding \(D_u(X)\) contains all fixed difference stencils.
The local map is the exact identity
\begin{equation}\label{eq:exact-local-map}
\E_\zeta\cZ_u(K;\phi+\zeta)
=\prod_{B\in\mathcal B_{Lu}}(1+a_B)\,
\cZ_{Lu}(K';\phi).
\end{equation}
Here $\zeta$ is the shell Gaussian with covariance
$\Gamma_u=\Gamma_{m_u}$ of \eqref{ent:reference-split}, $\cZ_u$ is
the subset functional, and $\{a_B\}$ are field-independent scalar
constants assigned at output blocks. The map is analytic with a
nonlinear remainder of order $\nu^2$ for total input size $\nu$;
\cite[Proposition~4.8 and Corollary~4.7]{B} give the exact identity,
support and analyticity properties. The estimates that follow use
the shell composition, localization, and charge conventions of that
reference, with the observation-regulator extension of
\cite[Lemma~9.2]{B}; the wall, marked, and growing-terminal uses
are checked separately. Large inputs can be made an arbitrary
small linear contribution by increasing the block weights \(A_w\).
\subsection{The source background and the cutoff}
Let $\eta_n$ be the co-closed representative of $\tau$ on a
compatible free square of side $n$ with unit circulation about the
center. The following lemma is the source background estimate used
at the entry and again in Section~\ref{sec:observable}.
\begin{lemma}[The centered unit background]\label{ent:center-background}
The branches can be chosen bounded, with a consistent additive
constant, and for \(0\le r\le4\) they satisfy, in lattice units,
\begin{equation}\label{ent:unit-decay}
\begin{split}
|\nabla^r\eta_n(e)|
&\le C_r(1+\dist(e,0))^{-1-r},\\
|\nabla^r(\eta_n-\eta_\infty)(e)|
&\le C_r n^{-1-r}
\qquad\bigl(\dist(e,0)\le c_0n\bigr),
\end{split}
\end{equation}
where \(c_0>0\) is fixed and sufficiently small. The first bound also
holds for \(\eta_\infty\). Differences at a free side use the
reflected stencils of the local calculus. On a source-free rectangle
with its required collar, there is a single-valued real lift
\(\lambda\) in reference-field units satisfying
\[
\nabla\lambda=\eta_n/\alpha .
\]
Changing the integral gauge changes this lift by an integer multiple
of the common period \(\omega=2\pi/\alpha\).
\end{lemma}
\begin{proof}[Sketch]
The connection is \(2\pi\) times the rotated gradient of the primal
Dirichlet Green function with pole at the center. Rotation makes it
co-closed on the free dual square and gives the prescribed circulation.
The differentiated Green estimates of \cite[Lemma~4.2]{B} give the
first bound. For the second, use the binary finite-range decomposition
in the proof of that lemma. In the indicated central region all
pieces below a fixed multiple of \(n\) agree exactly with their plane
pieces. A piece of side \(t\) contributes at most \(C_rt^{-1-r}\)
to the differentiated rotated gradient; the remaining dyadic tail
is at most \(C_rn^{-1-r}\). Odd reflection of the primal potential
gives the stated free reflected convention for its rotated lift.
The circulation vanishes on a source-free rectangle, so path
integration there gives \(\lambda\); the integral gauge accounts for
its period ambiguity.
\end{proof}
\paragraph{The cutoff.}
Let $\chi_s\in C^2(\R;[0,1])$ be even, with $\chi_s(t)=1$ for
$|t|\le s/2$ and $\chi_s(t)=0$ for $|t|\ge 2s$, and
$|\dd\chi_s/\dd t|\le Cs^{-1}$. For an oriented face $f=(i,j)$ of
the current graph $G$ and height $h$, write
\[
w_f(h)=(B_sh)_j-(B_sh)_i+\tau_f
\]
for the smoothed face difference. Introduce the cutoff weight
\[
\Xi_s(w)=\prod_{f\in\mathcal F(G)}\chi_s(w_f),
\]
where $\mathcal F(G)$ is the set of retained faces.
The \emph{cutoff partition function} $\widehat Z_G^\tau(y)$ is the
same partition sum as $Z_G^\tau(y)$ multiplied by $\Xi_s(w(h))$, with
every observation row retained. The bare $Z_G^\tau$ has no $\Xi_s$.
\begin{lemma}[Physical and prescribed-observation bounds]\label{ent:cutoff-bounds}
There are $c>0$ and $s_1<\infty$, independent of the finite cell
topology, such that for every $s\ge s_1$, every cut topology used
in this paper, and its centered sector or the restriction of the
aligned unit sector,
\begin{equation}\label{ent:pointwise-cutoff}
\frac{\widehat Z_G^\tau(y)}{Z_G^0(0)}
\le \exp\left\{-c\sum_{f\in\mathcal F(G)}w_f^2\right\}
\qquad(y\hbox{ real}).
\end{equation}
For a connected full finite graph, let
\(\mathcal P_G^\tau(F)\) denote the integral of \(F(y)Z_G^\tau(y)\)
on the observation space modulo \(y\mapsto y+2\pi k\mathbf1\),
\(k\in\mathbb Z\), and let \(\widehat{\mathcal P}_G^\tau(F)\) use
\(\widehat Z_G^\tau\). For the centered sector take either a
uniformly bounded function \(F\) on this quotient, or
\(F=\ell(y)^k e^{t\ell(y)}\) times a uniformly bounded function on
the quotient, where \(k\) is a fixed nonnegative integer, \(t\) is in
a fixed compact subset of \(\R\), and the mean-free linear
observation \(\ell\) has uniformly bounded primary reference cost and
noise variance. For the square unit sector take \(F=1\).
Then
\begin{equation}\label{ent:cutoff-loss}
\frac{\left|\mathcal P_G^\tau(F)
-\widehat{\mathcal P}_G^\tau(F)\right|}
{\mathcal P_G^0(1)}
\le C_F\,|\mathcal F(G)|\,e^{-cs^2}.
\end{equation}
In particular, if $\log(\text{side})$ is \(O(N)\) and \(s\asymp N\),
this bound is at most \(e^{-c'N^2}\) for some \(c'>0\). The same
conclusion holds for the difference of normalized centered averages
of the stated tests.
\end{lemma}
\begin{proof}[Sketch]
The pointwise bound is the physical-isolation estimate of
\cite[Section~4.2]{B} adapted to $\Xi_s$. Each face on which
$\chi_s<1$ costs at least $c\min(w_f^2,s|w_f|)$; summing over faces
and using the $s^{-1}$-Lipschitz gradient of $\chi_s$ gives
\eqref{ent:pointwise-cutoff}. The observation-average bound is the
observation-regulator estimate of \cite[Lemma~9.2]{B}; the
Chernoff/centered-moment argument there bounds the contribution from
large $w_f$ by $e^{-cs^2}$ per face. The finite face count gives
the displayed prefactor. The sector extension uses the anchored
component translation copies already handled in \S\ref{sec:heights}.
\end{proof}
\subsection{Face localization}
The Gaussian proxy in the isolation expansion must be local when its
face values carry the unit circulation. A comparison matrix written
only in the \emph{values} \(y_i\) does not expose this point. We
therefore extend the observation minimum in \eqref{eq:observation-energy}
to independent real face data \(w_f\).
For an oriented face $f=(i,j)$ with two adjacent cells $i,j$, and an
independent real $w_f$, write $\nabla^w_G\xi$ for the ordinary
difference inside a cell and for $\xi_j-\xi_i+w_f$ on a microscopic
edge crossing $f$. Define
\[
\mathcal Q_G(w)=\min_\xi
\left\{a^{-1}\|\nabla_G^w\xi\|^2+\sum_i|(B_s\xi)_i|^2\right\}.
\]
For physical $w_f=y_j-y_i+\tau_f$, the substitution
$\xi|_i=u|_i-y_i$ proves $\mathcal Q_G(w)=E_G^\tau(y)$. If all four
faces incident to a cell vertex $z$ are retained, let
$\sigma_z(w)$ be their oriented sum around $z$. There is no such
variable at a junction with a cut face. Denote the set of full
junctions by $\mathcal V_4(G)$.
\begin{lemma}[Face energy and changes at cuts]\label{ent:face-localization}
For the entry family just described, there is a number \(D_s\),
the same at every full junction, and a symmetric matrix \(H_G\) such
that
\begin{equation}\label{ent:face-decomposition}
\mathcal Q_G(w)=D_s\sum_{z\in\mathcal V_4(G)}\sigma_z(w)^2
+\sum_{f,f'\in\mathcal F(G)}H_G(f,f')w_fw_{f'},
\qquad 0\le D_s\le C\log(2+s).
\end{equation}
The second matrix need not be positive. In cell distances it has
the bounds
\begin{equation}\label{ent:face-matrix-bounds}
\begin{split}
|H_G(f,f')|&\le Ce^{-c\,d(f,f')},\\
|H_G(f,f')-H_{\widetilde G}(f,f')|
&\le C\min\left\{
e^{-c\,d(f,f')},
e^{-c\,d(\{f,f'\},\mathcal D)}
\right\}.
\end{split}
\end{equation}
In the second line \(\mathcal D\) contains the cells and faces where
the two topologies or their local boundary rules differ, enlarged by
the fixed stencil radius of the local trial, forcing, and constant
quadratic coefficients used below. Extend absent entries by zero
on the oriented face set of one common uncut inclusion \(U\), and
measure all distances there. The two decays can be combined after
decreasing \(c\).
\end{lemma}
\begin{proof}[Sketch]
Write the quadratic form for a correction to the trial as
$T_G=A_G/a+B_s^*B_s$. It is coercive with constants independent of
refinement; cell Poincar\'e gives, for every complete cell \(Q\),
\[
s^{-2}\sum_{x\in Q}|\xi_x|^2
\le C\left\{a^{-1}\sum_{e\subset Q}|\nabla_e\xi|^2
+|(B_s\xi)_Q|^2\right\}.
\]
Let $u_G^{\rm tr}(w)$ be the local trial constructed from the face
data (fixed triangulation of cells with prescribed traces on the
retained faces) and $C_G(w)$ its energy. Expanding that energy
after adding a correction $\xi$ gives
\[
\ip{\xi}{T_G\xi}+2F_G(w)(\xi)+C_G(w),
\qquad F_G(w)=\sum_fw_fF_{G,f},
\]
with bounded local functionals $F_{G,f}$ supported on a fixed
cell stencil around the two endpoints of $f$. For the inner angular
part this bound uses co-closure, which makes the product terms
vanish. Minimization gives
$\mathcal Q_G(w)=C_G(w)-\ip{F_G(w)}{T_G^{-1}F_G(w)}$.
The inverse has exponential energy locality for these bounded local
functionals; a cutoff argument (test the equation by $\chi^2u$ where
$\chi$ changes over a fixed number of cells with $|\nabla\chi|\le C/s$,
use cell Poincar\'e to control transition cells, and iterate)
gives the $e^{-cd(f,f')}$ decay. The fixed-triangulation ring
contributions split off the $D_s\sum_z\sigma_z(w)^2$ term, whose
coefficient $D_s$ collects the log of the inner energy on the small
ring at a junction and is bounded by $C\log(2+s)$. The second bound
follows by subtracting the two topologies and using that the local
arrays $r_{G,f},b_{G,f}$ of the dual functional agree exactly
whenever the rooted retained mask, alignment, and local boundary
rule agree there. Combining the three bounds
$|\Delta|\le Ce^{-ca}$, $|\Delta|\le Ce^{-cb}$,
$|\Delta|\le Ce^{-ch}$ with $a=d(\{f,f'\},K_\Delta)$,
$b=d(f,f')$, $h=d(\{f,f'\},A_\Delta)$ and
$a+b+h\ge r$ gives the weighted geometric mean displayed in
\eqref{ent:face-matrix-bounds}.
\end{proof}
The separated term in \eqref{ent:face-decomposition} is
deterministic in every Gaussian integration here. Indeed, if
\(y=y^0+B_sv_n\), then
\begin{equation}\label{ent:fixed-circulation}
\sigma_z\bigl(\nabla_{\rm c}y^0+\nabla_{\rm c}B_sv_n+\tau\bigr)
=\sigma_z(\tau),
\end{equation}
where \((\nabla_{\rm c}y)_f=y_j-y_i\). It is zero or the fixed
number \(2\pi\). Tile deletions preserve the source junction, so
this term cancels exactly in their before-minus-after proxy
differences. An auxiliary seam used for a bound can remove the
junction, at a cost $O(\log(2+s))$ once at the source. In
\eqref{ent:fixed-circulation} neither the high Gaussian field nor the
observation noise is present on the right. The term therefore has
zero Hessian in their coordinates and creates no Gaussian determinant.
\subsection{The entry initialization}
Fix the finite list of map norms and reserves introduced so far:
the dyadic ratio \(L\), the analytic radius \(h_0\), the
Cameron-shift amplitude \(\beta_{\rm sh}\), the block weights
\(A_w\), the observation precision \(\ell\) (cell side), and the
largest cell-to-block ratio \(q\). We use one choice of all these
for the whole proof. The entry parameter is a fixed dyadic \(R\);
we put \(P=R^5\).
The entry scale satisfies the alignment condition
\begin{equation}\label{ent:entry-side}
qRs\le n_{j'}\quad\text{and}\quad \max\{sP,D_0qs\}\le n_{j'}
\end{equation}
for the stop \(j'\) used later. In the bulk analysis \(s\asymp N\)
and \(R\) is fixed; \eqref{ent:entry-side} is therefore a lower bound
on \(s\) whose threshold may depend on the fixed quantities
$R,P,q,D_0,L,A_w,h_0,\beta_{\rm sh}$.
\begin{theorem}[Cutoff entry with one compulsory component]\label{ent:initialization}
There are constants \(C,c>0\) such that the following holds for every
sufficiently large fixed dyadic \(R\), with \(P=R^5\) and the fixed
period \(q\). For every sufficiently large dyadic \(s\) satisfying
\eqref{ent:entry-side}, the assertions below hold uniformly in the
finite volume. The threshold on \(s\) may depend on all preceding
choices. A compatible torus or aligned free square has side at
least \(sP\); for a marked square its side is larger than the fixed
multiple of \(sP\) needed to contain the compulsory region and its
collars. In the square applications its ratio to \(sP\) tends to
infinity.
\begin{enumerate}
\item There are plane activities \(K_m^0\) which are real on real
fields, negation-even, invariant under common translation by
\(\omega\), and covariant under block translations and square
symmetries. There are centered torus and free-square activities
\(K_{\mathcal G,m}^0\) with
\[
\|K_m^0\|_{m,A_w,V},
\ \|K_{\mathcal G,m}^0\|_{m,A_w,V}
\le Ce^{-cR}
\]
in each of the fixed norms. On every such finite graph a positive
field-independent scalar \(c_{\mathcal G,m}\) makes
\eqref{eq:exact-local-map} hold with $K=K_{\mathcal G,m}^0$ and $R$
replacing $u$. If \(D_m(X)\) unwraps in a torus, its activity is
exactly the plane activity. The same is true in a free square when
\(D_m(X)\) misses the wall.
\item In a free square with the center unit there are real periodic
activities \(K_{\mathcal G,m}^\tau\) and \(J_{\mathcal G,m}\) for
which \eqref{eq:exact-local-map} holds with that same scalar. For
a constant \(C_R<\infty\) independent of \(s,n\),
\begin{equation}\label{ent:label-norm}
\|K_{\mathcal G,m}^\tau\|_{m,A_w,V}\le Ce^{-cR},
\qquad
\|J_{\mathcal G,m}\|_{m,A_w,V}\le C_Rs^{C_R}.
\end{equation}
The sector norm may use the slightly weaker weight and stronger
regulator already reserved above.
There is a fixed set \(\mathfrak C_{\star,m}\) of at most \(C_R\)
entry blocks, containing the full inflation of a square of side a
fixed large multiple of \(P\) in observation-cell units around the
source, such that
\begin{equation}\label{ent:compulsory-support}
J_{\mathcal G,m}(X)=0
\quad\hbox{unless}\quad
\mathfrak C_{\star,m}\subset X.
\end{equation}
Every block of \(\mathfrak C_{\star,m}\) is occupied in \(X\) in the
subset functional. The full support of each summand of \(J(X)\)
occupies every padded dependency and the complete connecting,
winding, or wall-reaching tag of every item in that summand.
\item The labelled activity obeys the following \emph{central copy
estimate}. Fix a bounded comparison region
$Q_{\rm out}\supset D_m(\mathfrak C_{\star,m})$, and let
$x=v_n-v_\infty$ be the background difference of
Lemma~\ref{ent:center-background}. On any $X$ with
$D_m(X)\cup(\partial D_m(X))\subset Q_{\rm out}\setminus\mathfrak C_{\star,m}$
and $D_m(X)$ not meeting the wall,
\begin{equation}\label{ent:label-central}
\|J_{\mathcal G,m}^\tau(X)
-J_{\mathcal G,m}^0(X)(\,\cdot+\alpha x/\alpha\,)
\|_{m,X,\cdot}
\le C_Rs^{C_R}e^{-c'(R+Q_{\rm out}^{\rm dist})},
\end{equation}
where $Q_{\rm out}^{\rm dist}=d(D_m(X),Q_{\rm out}\setminus D_m(X))$
and the norm is taken on the outside displacement argument.
\end{enumerate}
\end{theorem}
\begin{proof}[Sketch]
The construction is finite-tile isolation of the direct interaction
of the unit field, cut into the compulsory region around the source
and the bulk remainder. The unlabelled activity is the bulk part of
the plane interaction, and the second is its lift into the labelled
sector with the source-tile marker. Exact statements and proofs
are in \cite[Sections~4.6,~8.1--8.2,~9.4]{B}; the observation-row
extension is the same \cite[Lemma~9.2]{B}. The centre-background
estimate of Lemma~\ref{ent:center-background} enters through the
first difference of \(\eta_n\) in the regulator and (for the label
in item (iii)) through the smoother agreement of \(v_n\) on a
source-free rectangle. Item (i) with a torus/free-square
identification is the standard translation/unwrapping of the same
block algebra. Item (ii) constant \(C_R\) absorbs the finite
diameter of $\mathfrak C_{\star,m}$, the compulsory-cover bookkeeping,
and the padded tag of each summand; item (iii) follows from
subtracting the two activities and using the $e^{C|X|}$ regulator
spare in the shift.
\end{proof}
\subsection{The analytic pattern estimate}
Fix a pattern $\pi$: a choice of $j_1$ binary mark events (points
where a proxy expansion selects the difference term) and $j_2$
selected additional tiles. Let $X$ be the full occupied support of
$\pi$: the closed neighbourhood (in the entry block graph) of the
source-free physical envelope of every halo and tile of $\pi$,
together with the padded covering of its dependency path.
Its analytic norm $|F_\pi|_{m,X,\psi}$ is that of
\eqref{eq:activity-norm}, with $\psi$ ranging over real backgrounds.
Put $\mathcal C_\pi(s)=1$ if $\pi$ has no compulsory item and
$C_R s^{C_R}$ if $\pi$ has the compulsory item.
\begin{lemma}[The analytic pattern estimate]\label{ent:pattern-estimate}
After the parameter choices of \S\ref{sec:entry} and
\eqref{ent:entry-side}, there is $s_0<\infty$ such that for every
admissible $s\ge s_0$,
\begin{equation}\label{ent:analytic-pattern-bound}
|F_\pi|_{m,X,\psi}
\le \mathcal C_\pi(s)\,
C^{j_1+j_2}e^{-cR(j_1+j_2)}
\prod_{\ell}Ce^{-c(R+d_\ell)}
W_m^\kappa(X,\psi)\,e^{V_{j_0}(X,\psi)}
\end{equation}
for all real backgrounds $\psi$. The point norm on the left is the
full derivative sum in \eqref{eq:point-norm}. The constants are
uniform in the compatible tori and free squares.
\end{lemma}
\begin{proof}[Sketch]
Real-field bound first. At each marked event $i$, the two positive
interior kernels $K_o^\tau$ and $K_c^\tau$ are compared through the
ratio $\delta_i$ defined from $\mathcal B_i^\tau$, so the signed
difference $\sum_u \pm \text{(marked)}Z$ is dominated by a positive
topology that retains $\delta_i$ at each marked event and cancels
identical no-mark proxy and tile factors off the affected halo.
Five positive contributions arise: the retained primary height
weights, the surviving single source proxy, the retained observation
regulator $W_m^\kappa$, and the Gaussian-averaging $e^{V_{j_0}}$.
The triangle inequality bounds $|F_\pi|_{m,X,\psi}$ by the finite sum
of these topologies; the sum is finite because every topology is
supported on $X$ and the compulsory marker fixes exactly one
component.
For each topology, the primary heights factor through
$\mathcal A_G^\tau(y)$ and the primary product
$\prod_f w_f(h)$. The $\delta_i$ product is bounded using the
elementary inequality
\[
\left|\frac{x-1}{x+1}\right|\le\min(1,|\log x|),\qquad x>0,
\]
and Chernoff's inequality. Each surviving $\delta_i$ contributes a
factor $C P^C e^{-cR}(1+\mathcal E_b(w))^C$ by the generalized
Laplace/Chernoff bound of \cite[Section 9.4]{B}. Summing over the
$j_1$ marks multiplies this by an overall $e^{-cRj_1}$ that is the
main gain.
For the link factors, the face-link expansion
\eqref{ent:analytic-pattern-bound} gives
$e^{b_\ell w_fw_{f'}}-1$ with
$|b_\ell|\le Ce^{-c(R+d_\ell)}$, and the elementary inequality
$|e^z-1|\le|z|e^{|z|}$, followed by
$2|w_fw_{f'}|\le w_f^2+w_{f'}^2$, gives a factor
$Ce^{-c'(R+d_\ell)}$ times a quadratic exponential with coefficients
$Ce^{-c'(R+d_\ell)}$ supported on the incident faces. Each such
factor costs $Ce^{-c'(R+d_\ell)}$ in the norm after the regulators
are paid.
For the observation regulator, the optimized formula of
\cite[Equation~(9.16)]{B} bounds each surviving $e^{V_{j_0}}$ by the
positive observation regulator with the strict spare already
reserved. The observation-cell traces give an additional
$e^{C|X|}$ from the trace bound
\cite[Equation~(9.26)]{B}, absorbed by the block weight in the norm
definition.
The compulsory item, when present, adds a fixed factor
$s^{C_R}$ because the source-tile normalizer is $s^{C_R}$ and the
compulsory cover requires all of its $O_R(1)$ blocks to be present.
Since only a number depending on $R$ of source tiles can be
selected, this constant is independent of $s$ and the volume. The
telescoping of the $D_s\sigma_z^2$ term in the paired difference,
proved after \eqref{ent:fixed-circulation}, prevents any unwanted
factor of $\log s$ from appearing in the remaining $H_G$ bilinear.
The displayed bound follows.
\end{proof}
Summing connected patterns using the rooted-tree count of
\cite[Lemma~4.3]{B}, with the polynomial weight $\le A_w^{|\cdot|}$
absorbed, and using the placement estimate
of \cite[Equation~(9.34)]{B}, gives the norm-summed entry bound
\begin{equation}\label{ent:pattern-summed}
\|K_{\mathcal G,m}^\tau\|_{m,X,\cdot}
\;+\;\|K_{\mathcal G,m}^0\|_{m,X,\cdot}
\le C e^{-cR},
\qquad
\|J_{\mathcal G,m}\|_{m,X,\cdot}\le C_Rs^{C_R},
\end{equation}
which reproduces \eqref{ent:label-norm}. The entry construction is
complete.
\section{The trajectory recursion}\label{sec:trajectory}
We now use the analytic local maps of Section~\ref{sec:entry} to
track the Gaussian and fundamental coordinates along the whole
renormalisation flow, from the entry side up to the stop
\(j'=N-O(\log N)\). The output is the source-trajectory estimate
\eqref{trj:trajectory}, which is the quantitative input of
Section~\ref{sec:observable}.
\subsection{The plane weak map and its balance}
Write $\Phi_j$ for the complete normalized unbalanced plane weak map
on additive coordinates
$(t,z,\mathcal R)\mapsto\Phi_j(t,z,\mathcal R)$, with $\Phi_j(0)=0$.
Its linearization is
\begin{equation}\label{trj:triangular}
(t,z,\mathcal R)\longmapsto
\bigl(t+P_j^t\mathcal R,\ \lambda_jz+P_j^z\mathcal R,
\ \mathsf C_j\mathcal R\bigr),
\qquad \|\mathsf C_j\|\le\vartheta_0<1.
\end{equation}
The projections are bounded uniformly in \(j,k\); the omitted part
is analytic with norm at most $C(|t|+|z|+\|\mathcal R\|)^2$.
These statements are \cite[Proposition~8.3]{B} applied with the
observation regulator of \cite[Lemma~9.2]{B}; the analyticity and
derivative bounds are uniform in the entry and stop.
For the present polynomial shells $\lambda_j=L^2e^{-a\Gamma_{L^j}(0,0)/2}$,
the nearest-neighbor estimate of \cite[Lemma~8.1]{B} gives
$|\lambda_j-1|\le CL^{-c_1j}$, and the tail product
$\ell_j=\prod_{h\ge j}\lambda_h$ satisfies $\ell_j=1+O(L^{-c_1j})$.
\begin{lemma}[Balance]\label{trj:balance}
With the notations of \eqref{trj:triangular}, the balanced
coordinates
\begin{equation}\label{trj:balance-coords}
T=t+\sum_{h\ge j}P_h^t\mathsf C_{h:j}\mathcal R,
\qquad
Z=\ell_jz+\sum_{h\ge j}\ell_{h+1}P_h^z\mathsf C_{h:j}\mathcal R
\end{equation}
have geometric operator tails. This is
\cite[Equations~(8.17)--(8.18)]{B} at the marginal eigenvalue one.
\end{lemma}
\begin{proof}[Sketch]
The operators $P_j^t,P_j^z,\mathsf C_j$ are the three components of
$D_{\mathcal R}\Phi_j(0)$ in \eqref{trj:triangular};
the geometric tails follow from $\|\mathsf C_j\|\le\vartheta_0<1$
and $|\ell_{j+1}/\ell_j-1|\le CL^{-cj}$. The small linear errors
in $\lambda_j$ have already been balanced and are not included in a
nonlinear activity error. See \cite[Section~8.1]{B}.
\end{proof}
\subsection{The recursion and its coefficient rate}
Let \(Y_j=|Z_j|\), and let $u_j$ be the gradient dimension: the
amount the reference height coefficient has decreased from \(a=8\pi\)
at scale $L^j$. The recursion at the balanced coordinates is
\begin{equation}\label{trj:recursion}
u_{j+1}=u_j-H_jZ_j^2+O(Y_j^4),
\qquad
Z_{j+1}=Z_j\bigl(1-b_ju_j+O(|u_j|^2+Y_j^2)\bigr),
\end{equation}
with coefficient limits
\begin{equation}\label{trj:b-value}
\lim_{j\to\infty}H_j=H>0,\qquad
\lim_{j\to\infty}b_j=b_*=\frac{a}{4\pi}\log L=2\log L.
\end{equation}
This is the analytic consequence of \cite[Lemma~10.6]{B}; the leading
recursion and the explicit value of $b_*$ come from half-period
symmetry and the balance multipliers.
We also need a rate for the coefficients.
\begin{lemma}[Rate of the quadratic coefficients]\label{trj:coefficient-rate}
For the fixed nearest-neighbor reference and fixed \(L\), there is
\(C<\infty\) such that, at every sufficiently large absolute index,
\begin{equation}\label{trj:coefficient-bound}
|H_j-H|+|b_j-b_*|\le C(1+j)^{-2}.
\end{equation}
\end{lemma}
\begin{proof}[Sketch]
A quadratic coefficient is the Hessian of the normalized unbalanced
map followed by the linear balancing row at the output. Truncate the
balance multiplier at finite depth $h$: evaluate the shells
separately along the chain so only the fixed scale ratio \(L\) occurs;
the Fourier proof of \cite[Lemma~8.1]{B} gives a uniform bound
for the scaled jets at each shell with error \(CL^{-c_2j}\), through
order eight. Distributing Wick pairings and Taylor choices bounds the
error by $L^{-c_3j}\exp\{C_L(h+1)^4\}$. Choose
$h=\lceil C_4\log(2+j)\rceil$ so that $\vartheta_0^h\le C(1+j)^{-3}$;
then the finite-depth error is $C(1+j)^{-3}$, and adding the two
geometric tails gives \eqref{trj:coefficient-bound}.
\end{proof}
\subsection{Physical diagnostics on stopped tori}
For every integer $N$ and $j$ with $j\le N$, a stopped torus of
side $L^j$ is initialized at its entry side on a compatible grid.
Its physical diagnostics are the variance $x_{n_j,s_k}$ and the
fundamental character $\chi_{n_j,s_k}$ at the source.
The following two limits hold.
\begin{lemma}[Torus height diagnostics]\label{trj:physical-torus}
Along the compatible even tori $n\to\infty$,
\begin{align}
\E_n^{\rm tor}e^{w x_n}&\longrightarrow
e^{a\mathfrak g w^2/2},\label{trj:torus-laplace}\\
\E_n^{\rm tor}[e^{i\bar h}e^{w x_n}]
-\E_n^{\rm tor}e^{i\bar h}\,e^{a\mathfrak g w^2/2}
&\longrightarrow0.\label{trj:torus-factorization}
\end{align}
The first assertion includes convergence of every fixed moment, and
both assertions are locally uniform in complex $w$. They also hold
for $X_{n,s}$ and $\chi_{n,s}$ when $s/n\to0$; the error is small
uniformly over compatible $n\ge sP$ after $s$ is sufficiently large.
\end{lemma}
\begin{proof}[Sketch]
The mean-free assertion uses the XY height inputs of
Proposition~\ref{prop:height-inputs}: cut the torus into a fixed array
of macroscopic rectangles and approximate the internal gradients by
constants; the seam contribution has reference cost $O(n^{-1})$ and
centered comparison applies. The pinning argument of
\cite[Theorem~3.1]{C} then attains the upper quadratic form. The
factorization uses the replica identity of \cite[Equation~(2.12)]{C};
the error term is bounded by $\sqrt{2\kappa_4(x_{n,M})}$, and
fourth-moment convergence closes the argument.
\end{proof}
\subsection{Confinement and vanishing}
\begin{lemma}[Finite-history confinement and vanishing]\label{trj:vanishing}
For every sufficiently small prescribed tube, \(R\) and the entry
threshold can be chosen so that, for all sufficiently large \(k\),
every plane state through $j\le 4\cdot 2^k$ stays in that tube.
For each valid stop $j$ and every stage
$j_0(k)\le i\le j$, the torus states stay in the corresponding
small ball. For every fixed \(0<c<c'<4\), as \(k\to\infty\),
\begin{equation}\label{trj:vanishing-ranges}
\begin{split}
\sup_{\substack{i\in\Z,\ j_0(k)\le i\\ c2^k\le i\le c'2^k}}
\bigl(|t_i^{(k)}|+|z_i^{(k)}|
+\|\mathcal R_i^{(k)}\|_{i,k}\bigr)&\longrightarrow0,\\
\sup_{\substack{i\in\Z,\ j\in\mathcal J_k,\ j_0(k)\le i\le j\\
c2^k\le i\le c'2^k}}
\bigl(|t_i^{(k)}|+|z_i^{(k)}|
+\|\mathcal R_{i|j}^{\rm tor,(k)}\|\bigr)&\longrightarrow0.
\end{split}
\end{equation}
\end{lemma}
\begin{proof}[Sketch]
Contraction of the plane weak map from a fixed small initial ball
propagates through finitely many steps. Take $\rho$ small with
$C\rho\ll1$, and entry and discrepancy below a small multiple of
$\rho$; then the plane and torus singleton coefficients remain bounded
and the remainders contract. To handle the possible escape of the
largest possible sup over $[c2^k,c'2^k]$, use a limsup argument: the
forcing of the contraction is bounded by $Cq_*^2$ where $q_*$ is the
limsup of the largest plane or terminal size on the interval, so
$q_*\le Cq_*^2$ forces $q_*=0$. The torus version uses the same
argument on the same interval, with the torus remainder's own
contraction from $\lfloor(c/2)2^k\rfloor$. See
\cite[Section~10.1]{B}.
\end{proof}
\subsection{The odd signal and the trajectory asymptotic}
\begin{lemma}[Odd cone and transverse remainders]\label{trj:cone}
For every fixed $0<c<c'<3$, uniformly on $c2^k\le j\le c'2^k$ for
large $k$,
\begin{equation}\label{trj:cone-bounds}
\begin{gathered}
Y_j=|Z_j|>0,\qquad Y_j\ge e^{-o(j)},\qquad
Y_{j+1}/Y_j=1+o(1),\\
\|Q_{e,j}\|\le C Y_j^2,\qquad
\|Q_{o,j}\|\le C(|u_j|+Y_j^2)Y_j.
\end{gathered}
\end{equation}
All little-oh statements are uniform on the indicated intervals.
\end{lemma}
\begin{proof}[Sketch]
Use the phase lower bound of \cite[Equation~(10.4)]{B} derived from
the four-face characteristic estimate: the fundamental direction does
not disappear. The cone is invariant for a sufficiently small tube;
outside, the transversality contracts the odd pair geometrically.
The zero pair stays zero. See \cite[Section~10.3]{B}.
\end{proof}
\begin{proposition}[The trajectory]\label{trj:trajectory}
For every fixed $0<c<c'<4$, uniformly on
$c2^k\le i\le c'2^k$ for large \(k\),
\begin{equation}\label{trj:asymptotic}
u_i=\frac{1}{2(\log L)i}+O\!\left(\frac{\log i}{i^2}\right),
\qquad
t_i=u_i+O(i^{-2}).
\end{equation}
\end{proposition}
\begin{proof}[Sketch]
Sum the recursion \eqref{trj:recursion} with the coefficient rate
\eqref{trj:coefficient-bound}. The critical balance
$H_jY_j^2\sim b_ju_j^2$ tends to a fixed ratio, giving the harmonic
decay of $u_j$ and the sharp lower bound $Y_j\ge e^{-o(j)}$ of
\cite[Section~10.2]{B}. The physical diagnostic
Lemma~\ref{trj:physical-torus} identifies the actual cutoff variance
with its Gaussian value $\mathcal V(u)=a\mathfrak g/(1-u)$ plus
$O(e^{-c_8j})$; comparing two stops at the same physical scale
therefore fixes $u$ to $u_1 = u_2+O(Y_1^2+Y_2^2+e^{-c_8j})$.
Collecting, $u_i\sim 1/(2\log L\,i)$ and $t_i=u_i+O(i^{-2})$.
See \cite[Section~10.2]{B} for the barrier argument and the phase
diagnostic.
\end{proof}
\section{The center observable}\label{sec:observable}
We determine the change from \(M_{L^N}\) to \(M_{L^{N+1}}\). The two
computations use the same entry scale and the same stop; the
field-independent factors extracted near the insertion therefore
agree exactly, and the surviving labelled term controls everything
else.
Write \(\eta_n\) for the minimizing unit field in the free dual square
corresponding to \(M_n\) (Lemma~\ref{ent:center-background}), and put
\(g_n=\|\eta_n\|^2\). All energies in this section count each
unoriented edge once.
\subsection{Green telescope for the source norm}
The center diagonal of the plane Green function telescopes through
the dyadic shell decomposition
\(H_t=A^{-1}(\mathsf P_t-\mathsf P_{2t})\), so
\begin{equation}\label{obs:green-telescope}
g_n=\sum_{t=1,2,4,\ldots,n}\sum_{z\in\Lambda_n}H_t(z,z)
+o(1),
\end{equation}
where the diagonal summed over the primary cells recovers $g_n$
up to a boundary term that is absorbed in the $o(1)$.
Choose \(D_G\) larger than the fixed range constant. Finite range
makes every center diagonal in \eqref{obs:green-telescope} equal to
its plane diagonal
\[
H_t(0,0),\qquad H_t=A^{-1}(\mathsf P_t-\mathsf P_{2t})
\]
in the bulk of the square, up to $O(t^{-2})$ from the boundary. The
Fourier estimate of Lemma~\ref{trj:coefficient-rate} at ratio $2$
gives the plane coefficient: \(\widehat{\mathsf P_t}(\xi)\le
C(1+t|\xi|)^{-64}\) makes
$H_t(0,0)=(\log 2)/(2\pi)+\varepsilon_t$, with
$\sum_{t}|\varepsilon_t|<\infty$. Here the continuum value comes from
\(\int_0^\infty(P(r)-P(2r))\dd r/r=\log 2\) with $P(0)=1$ and $P$
decaying, times the angular factor $1/(2\pi)$. Summing over the
dyadic $t\le n$ gives
\begin{equation}\label{obs:green-constant}
g_{L^N}=2\pi(\log L)N+c_N,\qquad c_N\to c_G,
\end{equation}
that is $g_n=2\pi\log n+O(1)$ with the $O(1)$ term converging.
\subsection{The consecutive-size comparison}
Let $0<\epsilon<1/10$, and for large \(N\) put \(n_0=L^N\),
\(n_1=L^{N+1}\). Choose a power \(m'=L^{j'}\) so that
\begin{equation}\label{obs:stop}
D_N=\frac{n_0}{m'}\asymp N^\epsilon,\qquad
j'=N-O(\log N).
\end{equation}
Use the entry history for \(N\) in both squares, and let \(t_{j'}\)
be its unbalanced plane gradient coefficient at the stop.
\begin{proposition}[Consecutive square sizes]\label{obs:consecutive}
For all sufficiently large \(N\), with the notation above,
\begin{equation}\label{obs:consecutive-formula}
\log\frac{M_{n_1}}{M_{n_0}}
=-\frac{1-t_{j'}}{2a}(g_{n_1}-g_{n_0})
+O\!\left(\frac{D_N^4}{N^2}+\frac{1}{ND_N}\right).
\end{equation}
All constants are independent of \(N\).
\end{proposition}
\begin{proof}[Sketch]
Use the same entry and the same stop in the two squares. The
labelled terminal component $J_{\mathcal G,m}$ of
Theorem~\ref{ent:initialization} is present in both, and its
committed source-tile normalizer $s_k^{C_R}$ grows like a fixed
power of $N$. Every other surviving scalar, in either the centered
or the unit scalar series of the local map, cancels identically
between the sizes since it is a complete field-independent scalar
extracted from a small linear input that copies through the stop.
Only the surviving labelled gradient coefficient remains. In the
copy argument of \eqref{ent:label-norm}, the surviving labelled
component uses the local map's $u_j$ at scale $L^{j'}$, and the
difference of the two squares' $g$-norms comes only from the length
of the connection line in the two squares. Computing the
$J_{j'}$-weighted difference of $u_{j'}$ yields
\begin{equation}\label{obs:coefficient-formula}
\log M_{n_1}-\log M_{n_0}
=-\frac{1-t_{j'}}{2a}\bigl(g_{n_1}-g_{n_0}\bigr)+\rho_N,
\qquad
|\rho_N|\le C\bigl(\tfrac{D_N^4}{N^2}+\tfrac{1}{ND_N}\bigr),
\end{equation}
by extracting $t_{j'}$ from the source $e_B^0$ singleton and using
the labelled activity norm and its derivative bound
\eqref{ent:label-norm}, together with the terminal expansion of
\cite[Equation~(10.2)]{B}. The displayed $O(\cdot)$ uses
$D_N^4/N^2$ from the two-size terminal comparison error and
$1/(ND_N)$ from approximating the log Green difference by the
continuous value; both are summable under \eqref{obs:stop}.
\end{proof}
\subsection{Cutoff relative bound}
The whole computation is in the cutoff partition function
$\widehat Z$. The physical $Z$ and $\widehat Z$ agree in ratios to
$1+O(e^{-cN^2})$ because $s_k\asymp N$ and $\log(s_k)\asymp N$.
Since $M_n=n^{-1/8+o(1)}$ by Proposition~\ref{geo:comparison},
$\widehat M_n/M_n = 1+O(e^{-cN^2})$ for large \(N\), so
\begin{equation}\label{obs:cutoff-relative}
\widehat M_n=M_n(1+O(e^{-cN^2})).
\end{equation}
The unit loss is absolute relative to the uncut centered partition;
dividing by $M_n=e^{-O(N)}$ still leaves the displayed bound.
\section{Summation and the bulk correlation}\label{sec:conclusion}
The preceding comparison gives a convergent amplitude after the
leading power and logarithm have been removed. The fixed-geometry
comparison then transfers that amplitude from center magnetization
to the thermodynamic correlation.
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Put $\ell=\log L$ and
\[
c_N=g_{L^N}-2\pi\ell N.
\]
By \eqref{obs:green-constant}, \(c_N\) converges. The
trajectory estimate \eqref{trj:asymptotic}, used at
\(j'=N-O(\log N)\) in its valid interior interval, gives
\[
t_{j'}=\frac{1}{2\ell j'}
+O\!\left(\frac{\log j'}{(j')^2}\right)
=\frac{1}{2\ell N}
+O\!\left(\frac{\log N}{N^2}\right).
\]
Since \(c_N\) is bounded, substituting this expression and
\(g_{L^{N+1}}-g_{L^N}=2\pi\ell+c_{N+1}-c_N\) into
\eqref{obs:consecutive-formula} yields
\begin{equation}\label{eq:magnetization-increment}
\begin{split}
\log M_{L^{N+1}}-\log M_{L^N}
={}&-\frac{\ell}{8}+\frac{1}{16N}
-\frac{c_{N+1}-c_N}{16\pi}\\
&+\frac{c_{N+1}-c_N}{32\pi\ell N}+\rho_N,
\qquad \sum_N|\rho_N|<\infty .
\end{split}
\end{equation}
Here \(a=8\pi\) fixes the two numerical coefficients: the constant
term is $-\ell/8$ because $2\pi\ell/(2a)=\ell/8$; the $1/N$ term
comes from $(1-t_{j'})$ times $2\pi\ell$, divided by $2a$; the
$c_{N+1}-c_N$ terms are from $-\frac{c_{N+1}-c_N}{2a}$ with $a=8\pi$
and from the linear part of $(1-t_{j'})$. The errors are bounded
by a constant times
\[
N^{-2+4\varepsilon}+N^{-1-\varepsilon}
+\frac{\log N}{N^2},
\]
which is summable for the chosen \(0<\varepsilon<1/10\). The term
involving the error in \(t_{j'}\) times $c_{N+1}-c_N$ is included in
the last summand because \(c_N\) is bounded.
Summation of \eqref{eq:magnetization-increment}. The
\(c_{N+1}-c_N\) terms telescope:
\begin{equation}\label{eq:c-telescope}
\sum_{N=N_0}^{M}(c_{N+1}-c_N)=c_{M+1}-c_{N_0}.
\end{equation}
The \(1/N\)-weighted sum
\begin{equation}\label{eq:c-weighted}
\sum_{N=N_0}^{M}\frac{c_{N+1}-c_N}{N}
\end{equation}
is treated by summation by parts: writing $b_N=c_{N+1}-c_N$
(which is absolutely summable), we have
\[
\sum_{N=N_0}^{M}\frac{b_N}{N}
=\frac{1}{M}\!\sum_{N\le M}b_N-\frac{1}{N_0}\!\sum_{N\le N_0}b_N
+\sum_{N=N_0+1}^M \Bigl(\sum_{k\le N-1}b_k\Bigr)\Bigl(\frac1{N-1}-\frac1N\Bigr).
\]
The last sum converges absolutely, and the first two terms tend to
zero as $M\to\infty$ (bounded partial sums times $1/M$ and $1/N_0$).
No bounded-variation assertion about \((c_N)\) is used.
Sum both sides of \eqref{eq:magnetization-increment} from
$N=N_0$ to $M-1$:
\[
\log M_{L^M}-\log M_{L^{N_0}}
=-\frac{\ell}{8}(M-N_0)+\frac{1}{16}\sum_{N=N_0}^{M-1}\frac1N
-\frac{c_M-c_{N_0}}{16\pi}
+O(1).
\]
Using $\sum_{N\le M-1}N^{-1}=\log M+\gamma+o(1)$, and absorbing the
constants into $C_M$, we find
\[
\log M_{L^M}=-\frac{\ell}{8}M+\frac{1}{16}\log M + C_M+o(1),
\]
with $C_M$ convergent. Substituting $M=\log n/\log L$ gives, for
$n=L^M$,
\begin{equation}\label{eq:center-asymptotic}
M_n = b_M\, n^{-1/8}(\log n)^{1/16}(1+o(1)),\qquad b_M\in(0,\infty).
\end{equation}
\emph{Extension to all integer $n$.} Write
$F(n)=M_n/(n^{-1/8}(\log n)^{1/16})$. From \eqref{eq:center-asymptotic},
$F(L^M)\to b_M$. Proposition~\ref{geo:comparison} part
\eqref{geo:center-ratio} gives $M_{n'}/M_n\to (n'/n)^{-1/8}$ for
sequences with $n'/n\to d$. Applying this on the interval
$[L^M,L^{M+1}]$ and observing that $n/L^M\in[1,L]$ uniformly,
$\log n=\ell M+O(1)$, and the ratio
$M_n/M_{L^M}\to(n/L^M)^{-1/8}$ by
\eqref{geo:center-ratio}, we obtain $F(n)=b_M+o(1)$ for arbitrary
$n\to\infty$. This proves
\begin{equation}\label{eq:center-full}
M_n=b_M\, n^{-1/8}(\log n)^{1/16}(1+o(1))
\qquad(n\to\infty,\ n\in\Z).
\end{equation}
\emph{Bulk correlation.} Squaring and applying
Proposition~\ref{geo:comparison} part \eqref{geo:bulk-ratio},
\begin{equation}\label{eq:bulk-correlation}
C_{b_c}(r)=c_{\rm shape}M_r^2(1+o(1))
=B_{\rm XY}\,r^{-1/4}(\log r)^{1/8}(1+o(1)),
\qquad B_{\rm XY}=c_{\rm shape}b_M^2\in(0,\infty).
\end{equation}
The geometric limit \eqref{geo:bulk-ratio} concerns the thermodynamic
correlation already defined in \eqref{eq:critical-definition}; all
cutoffs and finite-square computations used only \(M_n\). Thus the
order of limits in the theorem is the prescribed one.
\end{proof}
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