==== TABLE OF CONTENTS ====
Cylinder-loop-weights-and-planar-nesting//00_introduction.tex
Cylinder-loop-weights-and-planar-nesting//02_cylinder.tex
Cylinder-loop-weights-and-planar-nesting//03_gas_mean.tex
Cylinder-loop-weights-and-planar-nesting//03_reference.tex
Cylinder-loop-weights-and-planar-nesting//04_gas_fluctuations.tex
Cylinder-loop-weights-and-planar-nesting//04b_interpolation.tex
Cylinder-loop-weights-and-planar-nesting//04c_exponent.tex
Cylinder-loop-weights-and-planar-nesting//05_pressure.tex
Cylinder-loop-weights-and-planar-nesting//06_all_count.tex
Cylinder-loop-weights-and-planar-nesting//main.tex
Cylinder-loop-weights-and-planar-nesting//references.tex
===========================

==== BEGIN Cylinder-loop-weights-and-planar-nesting//00_introduction.tex ====
\section{Introduction}\label{sec:E-introduction}

The critical activity fixes the weight of each honeycomb polygon, but
the sum over disjoint polygons surrounding a point requires control
on every spatial scale. We determine
the growth of this nesting partition function by first calculating a
partition function for polygons separating two prescribed points on a
cylinder. Two independent pressure estimates then relate the cylinder
to planar nests. The same analysis also gives bounds for large polygon
counts and for the length of critical self-avoiding paths in a strip.

\subsection{The model and its exponents}

We realize the honeycomb lattice as the dual of the equilateral
triangular tiling with side length one and a horizontal edge direction.
A triangular band has height $\sqrt3/2$. A polygon is an unoriented
simple cycle of the dual graph. Its length $|P|$ is its number of
vertices, and its critical weight is $\rho^{|P|}$, where
\[
 \rho=\frac1{\sqrt{2+\sqrt2}}.
\]
Nienhuis predicted this activity through the dilute $O(n)$ model
\cite{Nienhuis1982}; Duminil-Copin and Smirnov proved that its
reciprocal is the honeycomb connective constant \cite{DCS2012}.
Their parafermionic observable supplies the local boundary cancellation
used here. We also use the finite strip transfer construction and the
critical strip-crossing estimate of \cite{StripCompanion}.

Identify the plane with $\mathbb C$, with the horizontal lattice
direction real and a face center at zero. For even $N=2m$, quotient by
the translation $iN\sqrt3/2$ and mark the images of $0$ and
$m e^{2\pi i/3}$.
Their vertical separation is $m$ bands. This is the balanced cylinder
geometry used throughout: a lattice cut follows $m$ edges in direction
$e^{2\pi i/3}$ and then $m$ in direction $e^{\pi i/3}$, closing after
one period. Compactifying the cylinder's two infinite ends gives a
sphere. A polygon \emph{separates the markers} if they lie in distinct
components of its complement on that sphere. Such a polygon may wind
around the cylinder, or may be contractible and enclose exactly one
marker in its disk.

Give every separating polygon an additional factor $\chi$, called its
loop fugacity, and define
\[
 Z_N(\chi)=\sum_{\mathcal P}\prod_{P\in\mathcal P}
                                  \chi\rho^{|P|}.
\]
The sum runs over finite families of pairwise vertex-disjoint separating
polygons, including the empty family with weight one. The transfer
construction in Section~\ref{sec:02-cylinder} proves convergence of
these masses. Every separating polygon meets the finite arc of the
prescribed cut between the markers, so the number of disjoint polygons
is bounded at fixed $N$. Thus $Z_N$ is a polynomial with nonnegative
coefficients and $Z_N(0)=1$.

\begin{theorem}[Cylinder fugacity exponent]
\label{thm:04-cylinder-exponent}\label{E:cylinder-exponent}
For every fixed $\chi>0$, along even $N$,
\[
 Z_N(\chi)=N^{\tau(\chi)+o(1)},\qquad
 \tau(\chi)=
 \begin{cases}
 \displaystyle\frac16-\frac{2}{3\pi^2}
              \bigl(\arccos(\chi/2)\bigr)^2,&0<\chi\le2,\\[4pt]
 \displaystyle\frac16+\frac{2}{3\pi^2}
              \bigl(\operatorname{arcosh}(\chi/2)\bigr)^2,&\chi\ge2.
 \end{cases}
\]
\end{theorem}

For the planar counterpart, let $P_\chi(r)$ be the sum of
$\prod_{P\in\mathcal P}\chi\rho^{|P|}$ over disjoint nests around
one fixed face center, with every polygon of diameter at most $r$;
again include the empty nest. Theorem~\ref{thm:05-nest-normalization}
proves the bounded-factor comparison
\[
 P_\chi(c_0N)^2\le Z_N(\chi)
       \le C_{\chi,c_0}P_\chi(c_0N)^2
\]
for every fixed $\chi\ge0$ and sufficiently small fixed $c_0>0$.
Consequently, for every fixed $\chi>0$,
\[
 P_\chi(r)=r^{\tau(\chi)/2+o(1)}.
\]
The square has a geometric origin. Removing essential polygons and
large contractible polygons costs a bounded factor; the remaining
polygons form two independent small planar nests, one near each marker.
Figure~\ref{fig:E-planar-comparison} shows the two planar neighborhoods.
The two pressure calculations below provide the estimates needed for
this removal.

In particular, $Z_N(2)=N^{1/6+o(1)}$ and
$P_2(r)=r^{1/12+o(1)}$. Fugacity two arises in the boundary winding
identity because an enclosing polygon has two orientations. If $S_k$
is the same fugacity-two nesting mass around a middle face of a strip
of $2k$ bands, with no diameter restriction, then
Corollary~\ref{E:strip-nesting} gives
\[
 S_k=k^{1/12+o(1)}.
\]

\subsection{Uniform counts and path length}

The fixed-fugacity theorem concerns one parameter at a time. To control
polygon counts that grow with $N$, we prove a separate estimate uniform
over an unbounded fugacity range. Define the critical coefficient mass
$w_N(a)$ by
\[
 Z_N(\chi)=\sum_{a\ge0}w_N(a)\chi^a.
\]
Thus $w_N(a)$ sums $\prod_P\rho^{|P|}$ over families with exactly
$a$ separating polygons. Section~\ref{sec:06-all-count} proves
\[
 Z_N(2\cosh v)\le\exp\!\bigl(C\log N(1+v^2)\bigr)
       \quad(v\ge0),
\]
with $C$ independent of $v$, and hence
\[
 w_N(a)\le\exp\!\left(C\log N-c\frac{a^2}{\log N}\right)
       \quad(a\ge0).
\]
These are unnormalized coefficient masses. The uniform fugacity bound
uses the coarse real-sector estimate before any counts are truncated;
the fixed-parameter asymptotic alone would not give it.

The nesting law also controls a path-length observable. A \emph{port}
is the midpoint of a triangular-tile edge. In a strip of $h$ bands,
consider self-avoiding paths from a fixed bottom boundary port to
either boundary, with no other boundary visit. They include their
initial and final half-edges, and $n(\gamma)$ counts the visited dual
vertices. Proposition~\ref{E:raw-first-moment} proves
\[
 \sum_\gamma\rho^{n(\gamma)}n(\gamma)
       \le h^{13/12+o(1)}.
\]
Under the normalized critical weights, their expected length conditional
on reaching the top boundary of the bottom $\lfloor h/4\rfloor$ bands is
$h^{4/3+o(1)}$ (Proposition~\ref{prop:02-length-reduction}).
The difference between these exponents comes from the conditioning
event, whose unnormalized mass is comparable to $h^{-1/4}$ by the
strip-crossing theorem of \cite[Theorem~1.1]{StripCompanion}.
A positive identity relating visits to an interior edge to nests
around its two adjacent faces supplies the required length comparison.

\subsection{Proof strategy}

The proof has an algebraic and analytic route to the cylinder exponent,
followed by a geometric route to nesting. We describe the two routes
separately because the pressure estimates do not follow from the
cylinder asymptotic.

\paragraph{A finite representation with its normalization.}
Section~\ref{sec:02-cylinder} constructs the cylinder vacuum from a
polynomial open-strip state. A vacuum state records the pairings of
occupied ports made by paths on one side of a cut, with no external
source. Contracting the two half-cylinder states with the separation
rule gives $Z_N$. Degree bounds and reductions at special parameters
determine this contraction exactly.

Writing $\chi=2\cos\eta$, a finite contour deformation expresses
its numerator as a sum of particle integrals with factor $e^{i\eta M}$.
The integer $M$ fixes the two groups' elementary particle totals as
$m-M$ and $m+M$. Each group contains main particles and
residue-generated far single particles and fused pairs, the latter
counted twice in the elementary degree.
Only terms of elementary degree $N$ enter the identity.

\paragraph{The logarithmic contribution of the particle integrals.}
Section~\ref{sec:03-gas-mean} centers each main group at the mean
density $m\varrho(u)$, where $\varrho$ is a fixed probability density
with exponential tails. Walls at distance proportional to $\log N$
leave only $O(1)$ mean mass outside; this is the spatial scale of the
particle calculation. The real quadratic energy of the remaining fluctuations
controls their imbalance, their far-particle counts, and excursions
beyond that interval. A Legendre orthogonal-polynomial ensemble gives
a positive reference law for the two large interior groups. Subtracting
its logarithmic interaction removes the diagonal singularity, so the
remaining energy is meaningful on empirical particle measures.

The coefficient of $\log N$ must be kept throughout this comparison.
The relative fluctuation determinant contributes $7/8$, the two
reference partition masses contribute $-1/2$, and the scalar vacuum
denominator contributes $-5/24$. Their sum is
\[
 \frac78-\frac12-\frac5{24}=\frac16.
\]
To obtain the quadratic term in $\eta$, the phase $e^{i\eta M}$
must be retained during integration.
Section~\ref{sec:E-reference} proves the reference integration
bounds. Section~\ref{sec:04-gas-fluctuations} realizes the needed
complex displacement by moving actual particles and computing its
Jacobian. Because the interior particle numbers also change with $M$,
Section~\ref{sec:E-interpolation} constructs a holomorphic approximation
to the integer-dimensional integrals before moving the imbalance sum.
The combined particle and imbalance-contour deformations convert the
phase into a quadratic contribution to the real action.
These steps yield, locally uniformly for $|\operatorname{Re}\eta|<\pi$,
\[
 \frac{\log|Z_N(2\cos\eta)|}{\log N}
 \le \frac16-\frac{2}{3\pi^2}\operatorname{Re}(\eta^2)+o(1).
\]
Section~\ref{sec:E-cylinder-exponent} obtains the matching lower
exponent from this harmonic upper envelope. The anchor $\eta=\pi/2$
has the exact value $Z_N(0)=1$; subharmonicity propagates near equality,
and coefficient positivity transfers it to each fixed positive
fugacity. Thus no lower estimate of a single complex sector is needed.

\paragraph{From the cylinder to planar polygons and strip paths.}
Section~\ref{sec:05-pressure} varies the weight of essential loops
in an annular transfer. Its first two pressure derivatives at zero fugacity control
both their total mass and their first horizontal-span moment. These
bounds make the cost of removing separating essential polygons uniform
in $N$.
Section~\ref{sec:05-open-pressure} instead varies the weight of all
loops in an open strip. Its first pressure derivative at zero fugacity
is affine in $N$ up to an error $O(N^{-1})$. Counting each shape's
vertical placements and comparing two strip widths then gives
\[
 \sum_{[P]:\,\diam P>r}\rho^{|P|}\le C(1+r)^{-2}.
\]
Here $[P]$ is an unrooted, unoriented polygon modulo triangular-lattice
translations. This estimate controls large contractible polygons.
Section~\ref{sec:05-nesting} uses both removal bounds to separate
the two planar nests, then proves the length--nest identity and the
raw and conditioned strip-length estimates.

\subsection{Earlier work}

The electric exponent has a direct Coulomb-gas predecessor. Gamsa and
Cardy study the change in weight of loops separating marked points
\cite[Section~3, equation~(6)]{GamsaCardy2006}. In the dilute
zero-loop specialization, their scaling dimension for
$\chi=2\cos\eta$, $0<\chi\le2$, is
$x=\eta^2/(3\pi^2)-1/12$, giving the predicted two-point power
$-2x=\tau(\chi)$. Their argument assumes the Gaussian-field scaling
description. Cardy's annulus calculation likewise permits an independent
weight for winding loops and gives the corresponding radial power
\cite{Cardy2006Annulus}. The present argument establishes the finite
lattice normalization and the estimates required for the cylinder and
planar observables above; it also proves the stated positive-fugacity
branch for $\chi>2$.

Periodic dilute-loop transfer matrices with seams that change wrapping
loop weights were analyzed by Zhou and Batchelor
\cite{ZhouBatchelor1997}. Their spectral calculation provides related
electric-parameter formulas. The observable here is a contraction with
two specified markers, and includes contractible polygons surrounding
one marker. The finite contraction and its nonvanishing scalar
normalization must therefore be identified in this geometry.
Ikhlef and Cardy related local parafermionic equations to the
Yang--Baxter weights of dilute-loop models \cite{IkhlefCardy2009}.
Glazman and Manolescu proved invariance of the boundary two-point
function for columnwise rhombic half-planes with angles in
$[\pi/3,2\pi/3]$ \cite{GM2019}. We use their local-weight convention,
specialized to the regular honeycomb lattice in the final statements.

Polynomial exchange equations and special-rapidity reductions have
earlier uses in integrable loop constructions. Di Francesco and
Zinn-Justin use them to prove a multiparameter sum rule for the dense
$O(1)$ model \cite{DiFrancescoZinnJustin2005}. Garbali and Nienhuis
develop polynomial states and fusion recurrences for the dilute
$O(1)$ open strip, and obtain determinant expressions for its
normalization \cite{GarbaliNienhuis2017Ground,GarbaliNienhuis2017Sum}.
They also note that the polynomial-state method applies at zero loop
weight \cite[Section~4]{GarbaliNienhuis2017Ground}. Here the
open-strip polynomial state is supplied by \cite{StripCompanion};
we construct its periodic limit, the normalized vacuum quotient,
and the two-marker contraction with their physical interpretation.

The analytic reference measure is a classical orthogonal-polynomial
ensemble. Andr\'eief's identity gives its Gram normalization and
tilted factorial determinants \cite{TracyWidom1998}. The circle
comparison is the finite upper bound associated with the strong
Szeg\H{o} theorem \cite{Simon2005Szego}; its Schur-polynomial proof
is related to the unitary trace calculations of Diaconis--Shahshahani
and Diaconis--Evans \cite{DiaconisShahshahani1994,DiaconisEvans2001}.
We include these arguments in the required
normalization. To retain the logarithmic exponent, the present
application also needs uniform source bounds for growing physical
tests, a particle realization of the complex quadratic change of
variables, and continuation of the integer dimensions. These are
the roles of Sections~\ref{sec:E-reference}--\ref{sec:E-interpolation}.
==== END Cylinder-loop-weights-and-planar-nesting//00_introduction.tex ====

==== BEGIN Cylinder-loop-weights-and-planar-nesting//02_cylinder.tex ====
\section{Cylinder partition functions and contour sectors}
\label{sec:02-cylinder}

Our first task is to express the positive cylinder observable as a
finite sum to which asymptotic analysis can be applied. We prove two
exact representations: a sum over three labels at each site, and an
integral over finitely many contour sectors. The normalization is part
of each identity. We then estimate that normalization on the balanced
physical cylinder. The construction begins with the strip estimate needed to control
paths in a fixed-width periodic slab. The positive length--nest
comparison will be proved with its applications in
Section~\ref{sec:05-nesting}.

We use the honeycomb lattice dual to the equilateral triangular lattice
of side length one. A port is the midpoint of a tile edge. A path between
ports visits each honeycomb vertex at most once, includes its endpoint
half-edges, and has weight $\rho^{n(\gamma)}$, where $n(\gamma)$ counts
visited vertices. A polygon of length $r$ has $r$ vertices. Put
\[
 \lambda=\frac\pi8,\qquad
 d=\cos(2\lambda)=\frac1{\sqrt2},\qquad c=\cos(3\lambda),\qquad
 \rho=\frac1{\sqrt{2+\sqrt2}}.
\]
For the strip of $H$ triangular bands, let $\mathcal A_H$ be the mass
of paths from one fixed bottom port to the other bottom ports and let
$\mathcal B_H$ be their mass to the top boundary. We use the following
results from \cite[Theorem 1.1 and Lemmas 2.1, 2.3]{StripCompanion}:
\begin{equation}\label{eq:E-strip-input}
 c\mathcal A_H+\mathcal B_H=1,\qquad
 0<\mathcal B_H\asymp H^{-1/4},\qquad
 \mathcal B_H\text{ is nonincreasing in }H.
\end{equation}
The same source proves exponential decay of a one-defect transfer at
each fixed height, and boundedness of the corresponding vacuum
half-state. Here a defect means one strand attached to a specified
source and continuing to the current cut. We will name each use of
these finite-height facts.

\subsection{Strip input and the critical walk bound}

The critical activity is the reciprocal of the connective constant,
as proved by Duminil-Copin and Smirnov~\cite{DCS2012}. For later use
we give the short deduction from the strip estimates; it also supplies
a subexponential bound without a limit interchange.

\begin{proposition}[Critical weight]\label{prop:02-critical}
If $c_n$ is the number of $n$-edge self-avoiding walks from a specified
honeycomb vertex and $\mu=\lim_{n\to\infty}c_n^{1/n}$, then
$\rho=\mu^{-1}$. Moreover, $c_n\rho^n=\exp(o(n))$ as an upper bound.
\end{proposition}
\begin{proof}
We use the bridge-decomposition method of Hammersley and
Welsh~\cite{HammersleyWelsh1962}; see also
Madras and Slade~\cite{MadrasSlade1993} and
Duminil-Copin and Hammond~\cite[Proposition~2.1]{DuminilCopinHammond2013}.
We give the decomposition with the present honeycomb endpoint conventions,
using the positive strip bound to control each weak bridge.

Submultiplicativity gives existence of $\mu$. Split a walk at a vertex
of minimum height, choosing the first such vertex in a fixed traversal
order when necessary. Its two tails start at that vertex and stay above
its height. Translate this vertex to one of two fixed representatives,
according to its triangular-band type. Recording the two tails and the
split time recovers the original walk, because its original root is fixed.

Decompose each nonempty tail by stopping at its last global maximum,
then at the last minimum of the remainder, and continuing alternately.
Every piece is a weak bridge: all its vertices lie between the heights
of its endpoints. There are no horizontal edges. Its positive spans,
measured in thirds of a triangular band, are integers, are nonincreasing,
and are strictly decreasing after a possible equality of the first two.
Their sum is at most $2n$. Consequently there are $O(\sqrt n)$ pieces
and at most $\exp(O(\sqrt n\log(n+1)))$ possible span lists, even after
recording the split time and the two initial types.

For a fixed initial vertex and a specified directed span, the total
$\rho^{\#\text{edges}}$-weight of weak bridges is bounded by an absolute
constant. To obtain this from the strip bound $\mathcal B_H\le1$,
choose horizontal band boundaries immediately beyond the two endpoint
heights. At a lower endpoint of lower-center type append the vertical
half-edge to the lower boundary. At the other type first append the
fixed descending edge, then the vertical half-edge. Make the reflected
construction at the upper endpoint. All added vertices are outside the
old height range, so the extension preserves self-avoidance and has
bounded multiplicity. The starting boundary port and strip height are
determined by the starting vertex and span. The extension adds at most
three vertex-weight factors relative to the old edge count. The same
argument applies to downward bridges by reflection.

For each span list, ignore avoidance between successive pieces and
sum their weights successively. This gives a constant to the power
$O(\sqrt n)$, and hence
\[
 c_n\rho^n\le \exp(O(\sqrt n\log(n+1))).
\]
Taking $n$th roots yields $\rho\le\mu^{-1}$. If the inequality were
strict, choose $r$ with $\rhoj+2N$ and take the south-to-east turn in row $i+kN$.
The three kinds of one-arc tile used here have strictly positive
weights at both physical parameters $\lambda,2\lambda$; only one arc
is placed in each tile. Since different rhombi have disjoint interiors,
the lifted path is self-avoiding and meets the column's west and east
cuts only at its endpoints. After projection, however, its intermediate
south-to-north route repeats vertices in a row visited one period
earlier. This is a width-one connector from type $j$ to type $i$ with
some positive weight $a$, depending only on the fixed cylinder.

\begin{figure}[tbp]
\centering
\begin{tikzpicture}[scale=.86,every node/.style={font=\small}]
 \draw (-.8,-.8) rectangle (.8,.8);
 \node at (0,0) {$R(v)$};
 \foreach \x/\y in {0/-.8,-.8/0,.8/0,0/.8}
    \fill (\x,\y) circle (2pt);
 \node[below,align=center] at (0,-.9) {south\\input 1};
 \node[left,align=center] at (-.9,0) {west\\input 2};
 \node[right,align=center] at (.9,0) {east\\output 1};
 \node[above,align=center] at (0,.9) {north\\output 2};
 \node at (0,-2.05) {local port order};
\end{tikzpicture}
\hspace{1cm}
\begin{tikzpicture}[scale=.77,every node/.style={font=\small}]
 \foreach \y in {0,1.1,2.9,4.7}
    \draw (-.65,\y-.4) rectangle (.65,\y+.4);
 \draw[very thick,blue!60!black] (-.65,0) -- (-.2,0)
     to[out=0,in=-90] (0,.4) -- (0,1.5);
 \draw[very thick,blue!60!black,densely dotted] (0,1.5) -- (0,2.5);
 \draw[very thick,blue!60!black] (0,2.5) -- (0,3.3);
 \draw[very thick,blue!60!black,densely dotted] (0,3.3) -- (0,4.3);
 \draw[very thick,blue!60!black] (0,4.3)
     to[out=90,in=180] (.25,4.7) -- (.65,4.7);
 \node[left] at (-.8,0) {row $j$};
 \node[right] at (.8,1.1) {row $j+1$};
 \node[right] at (.8,2.9) {row $j+N+1$};
 \node[right] at (.8,4.7) {row $i+kN$};
 \draw[<->] (-1.65,1.1) -- (-1.65,2.9)
     node[midway,left] {one period};
 \node[below,align=center] at (0,-.65) {one column in the lift};
\end{tikzpicture}
\caption{The boxes show cyclic port order, rather than physical angles.
On the right, dotted portions contain the omitted intermediate rhombi.
The two displayed south--north passages are one period apart, so they
visit the same vertices after projection onto the cylinder. In the
planar lift all rhombi used are distinct.}
\label{fig:E-local-connector}
\end{figure}

Concatenate lifts of bridges counted by the divergent entry with
copies of this connector, always in successive disjoint column slabs.
Their interiors are disjoint, so the resulting planar path is
self-avoiding. To recover the concatenation, list all column cuts
that the path meets exactly once. Between two adjacent column cuts
in this list, a width-one subbridge inside any original valid slab
is still projection-valid. The inserted connectors are precisely the
projection-invalid width-one subbridges: they occupy one entire
column and their two cuts are crossed only at their endpoints.
Thus the connectors, the intervening slabs, and their original paths
are uniquely recovered from the concatenated path.

Choose finitely many valid slab widths whose total mass is $B$ with
$aB>1$. The width generating function of one valid-slab--connector
block is $g(z)=a\sum_M b_M(i,j)z^{M+1}$, with the sum restricted to
those widths. It satisfies $g(0)=0$ and $g(1)>1$. Hence the generating
function $\sum_{r\ge0}g(z)^r$ of recoverable concatenations has radius
strictly less than one. Each slab has at most a fixed constant times
its width in vertices, because its projection is self-avoiding on a
fixed cylinder; the connector has bounded length. Consequently all
these planar paths of total width $w$ have length at most $C_Nw$.
Their total critical weight is bounded above by the sum of
$c_n\rho^n$ for $n\le C_Nw$, up to fixed endpoint factors. This is
$\exp(o(w))$ by Proposition~\ref{prop:02-critical}, contradicting the
radius just obtained.

\end{proof}

\begin{proposition}[Physical cylinder vacuum]
At physical row parameters $u_i\in\{\lambda,2\lambda\}$,
\label{prop:02-cylinder-vacuum}
the column $T(3\lambda)$ restricted to vectors with zero empty
component has spectral radius less than one. Its unique stationary
state with empty component one is the regular continuation of $U_N$.
It equals the absolutely convergent sum of half-cylinder configurations
with vacant far boundary. The right-half state is
$U_N(3\lambda-u_1,\ldots,3\lambda-u_N)$.
\end{proposition}
\begin{proof}
A nonzero diagram propagated from a nonempty input pairing either
has a west-to-east spanning path or closes a discarded cycle. The
number of path segments is bounded at fixed $N$. Discarding their
mutual avoidance bounds their weights by a product of an exponentially
decaying spanning sum from Lemma~\ref{lem:02-periodic-slab} and
subexponential same-side sums. Therefore
the transfer on the subspace of zero empty component has spectral
radius strictly below one. The empty component itself is invariant.
Writing the transfer in these two blocks proves uniqueness and local
analyticity of its normalized fixed vector, and convergence from a
vacant far boundary proves the half-cylinder interpretation.
The rational stationary vector $V_N/h_N$ agrees with it wherever
defined nearby, hence extends to the physical parameters. Horizontal
reflection gives the stated right-half arguments. Vertical reflection
also recovers the covariance proved algebraically above.
\end{proof}

\subsection{The finite separating-loop formula}

Split the cyclic list into consecutive groups $X,Y$ of sizes $s,b$,
where $N=s+b$, and mark the two seams. Define $Z_{X,Y}(\chi)$ by
contracting the left and right vacuum states, assigning a closed loop
weight $\chi$ if it crosses the $X$ portion of the cut an odd number
of times, and weight zero otherwise. The empty configuration contributes
one. If a group is empty all loops have weight zero, so $Z=1$.
At physical parameters the markers are face centers. Compactifying
the cylinder ends gives a sphere, on which the odd-intersection rule
is precisely separation of the markers. Therefore $Z_{X,Y}(\chi)$
counts disjoint polygon families all separating the markers, with
weight $\prod_\ell\chi\rho^{|\ell|}$. Every such loop meets the cut;
disk connectivity on its two sides suffices to test separation.
The half-cylinder sums converge absolutely, and a family has a
bounded number of loops at fixed $N$, so this interpretation is valid
for every fixed $\chi$.

For the balanced observable $Z_N$ of the introduction, $N=2m$ and
the prescribed row list is $m$ copies of $2\lambda$ followed by $m$
copies of $\lambda$. Identify the triangular lattice with the complex
plane and start its column cut at the face center $f_1=0$. Its successive
edges have these row vectors, so the second marker and period are
\[
 f_2=m e^{2\pi i/3}=-\frac N4+i\frac{N\sqrt3}{4},\qquad
 V=m(e^{2\pi i/3}+e^{\pi i/3})=i\frac{N\sqrt3}{2}.
\]
The two groups are the two portions of this prescribed cut from $0$
to $f_2$ and from $f_2$ to $V$. Translation or reflection gives the
same observable. In particular the markers are half a period apart in
height; their longitudinal displacement is fixed by this construction.

\begin{lemma}[Characterization of the contraction]\label{lem:02-characterization}
The rational functions $Z_{X,Y}(\chi)$ are uniquely determined by the
following properties, with $\chi$ fixed.
\begin{enumerate}
\item $H_{X,Y}=h_N(X,Y)^2Z_{X,Y}$ is polynomial of degree at most
$2N-2$ in each argument, and $Z$ is symmetric within each group.
\item An ascending merge or deletion within either group leaves $Z$
unchanged after making the corresponding replacement or deletion.
\item Sending one argument to zero or infinity omits that site.
\item Sending $x\in X$ and $y\in Y$ to infinity at fixed generic
ratio gives
\[
 Z_{X,Y}\longrightarrow Z_{\widehat X,\widehat Y}Z_{x,y},
 \qquad Z_{x,y}=1+\chi(2c)^2\frac{xy}{(x+y)^2}.
\]
\item At size zero, $Z=1$.
\end{enumerate}
All limits are rational limits at generic values of the remaining
arguments. Hats mean omission of the indicated sites.
\end{lemma}
\begin{proof}
The right-half arguments are $q^3/x_i$. Equation~\eqref{eq:02-reciprocity},
homogeneity, and the per-slot degree bounds for $V_N$ give the
polynomial bound for $H$: occupied half-powers occur in pairs in the
contraction. To exchange sites within a group, move a symmetric braid
box from one side of the contraction to the other by transposition.
The two states braid in opposite directions because the starred
arguments reverse differences. Loop parity is preserved: following
the arcs in the box layer shows that each resulting loop has an even
total number of $X$-hits on the two interfaces of that layer. Thus
any cycle discarded on one side has zero gluing weight as well.
This proves symmetry and also permits a same-group splitting layer
to be moved through the contraction. At the descending specialization
on the starred side, apply $R(j\lambda)=S_jS_j^{\mathsf t}$ and the
ascending splitting identity. To cancel $S_j$, project its output onto diagrams whose first new
slot is vacant. For $S_2$, omitting that slot leaves the diagonal map
with coefficients $1$ on a vacancy and $\rho$ on an occupied strand.
For $S_3$, both new slots are then vacant; omitting them gives the
identity. Thus $S_j$ is injective, and
$S_jU_{\rm red}=S_jS_j^{\mathsf t}U_{\rm desc}$ implies
$S_j^{\mathsf t}U_{\rm desc}=U_{\rm red}$. This proves
the reduction of $Z$.

Cyclic covariance and the last-slot leading coefficient show that
sending one argument to infinity inserts a vacant site in the left
vacuum. Reflection gives the same statement in the right vacuum.
Horizontal reflection gives the limit at zero.

For completeness, the simultaneous two-site limit requires more than
the individual limits. Reorder within groups so that $x,y$ are adjacent
across a marker, and rotate them to the last two positions. Write
$P$ for the other $N-2$ arguments. We claim
\begin{equation}\label{eq:02-pair-limit}
 \frac{V_N(P,x,y)}{(xy)^{N-2}}
 =V_{N-2}(P)\otimes V_2(x,y)+o(|x|),
 \qquad
 V_2(x,y)=(x+y)\varnothing+2c\sqrt{xy}\,\mathrm{cup}.
\end{equation}
Fix square-root branches on the scaling ray. In the interpolation in
$y$, the deletion nodes are $y_z=q^3z$ for $z\in(P,x)$. For $N\ge3$,
nodes belonging to $P$ have data $O(|x|^{N-1})$, cardinal factors
$O(|x|^{N-3})$, and additional occupied square-root ratios at most
$O(|x|^{1/2})$. After division by $(xy)^{N-2}$ they are $o(|x|)$.
The leading vacant interpolation term becomes
$(y-y_x)V_{N-2}\otimes\varnothing+o(|x|)$. At $y_x=q^3x$ the data are
\[
 (x+y_x)\prod_{p\in P}(p-q^{-3}x)(p-q^6x)\,S_3V_{N-2}(P).
\]
The cardinal factor is asymptotic to $(y/y_x)^{N-2}$, with an extra
$\sqrt{y/y_x}$ on occupied output. Since $q^{-3}q^6=q^3$, these
terms give exactly the remaining terms in Equation~\eqref{eq:02-pair-limit}.
The expression for $V_2$ follows directly from the size-two
interpolation. Division by the empty components gives factorization
of the normalized left vacuum, and reflection gives that of the
right vacuum. The intersection-parity rule factors on the two
subsystems, proving both the pair limit and the displayed $Z_{x,y}$.

To prove uniqueness, the zero-site value and omission already settle
$N=1$. At $N=2$ with one site in each group, omission at zero and
infinity leaves only a possible difference $Cxy$ of the cleared
numerators. Its quotient by $h_2^2=(x+y)^2$ has a nonzero paired
scaling limit unless $C=0$, so the specified pair limit settles this
case. For all other group sizes, induct on $N$ and subtract two
candidate cleared polynomials. The known
scalar reductions of $h_N$ turn the reductions of $Z$ into identical
values of these polynomials at each split. Omission at zero kills
their constant coefficient, and
omission at infinity kills their leading coefficient. In a variable
of a group of size $s$, symmetry and the two reductions supply
$4(s-1)$ roots, namely $q^{\pm2}$ and $q^{\pm3}$ times every other
same-group argument. If $s>b$, these roots, together with the zero
constant coefficient, exceed the remaining degree $2N-3$.
If $s=b\ge2$, the difference must be a constant multiple of
\begin{equation}\label{eq:02-uniqueness-factor}
 \left(\prod_i x_i\right)
 \prod_{\substack{i0$.
Thus $I+PRP$ is strictly accretive and invertible. Since the removable
factors $H_*(0)$ and $H_*(i\beta)$ are nonzero, the compression identity
already proves $h_N\ne0$ at these balanced confluent sites. The rest
of the proof determines its logarithmic size.

The projection has a useful exact localization formula:
\begin{equation}\label{eq:02-projection-complement}
 \|(I-P)k_z\|^2=\|k_z\|^2
 \prod_{\pm}
 \left|\frac{e^{\gamma z}-e^{\pm i\gamma\beta/2}}
 {e^{\gamma z}+\overline{e^{\pm i\gamma\beta/2}}}\right|^{2m}.
\end{equation}
To see this, augment the reference Gram matrix by $k_z$ and take its
Schur complement. Under $w=e^{\gamma z}$, its kernel is
$1/(w+\overline{w'})$ up to separate one-point factors. The Cauchy
determinant formula follows by clearing denominators and using
antisymmetry, with the constant fixed by residues. Its ratio after
augmentation is exactly Equation~\eqref{eq:02-projection-complement}.
Divided differences give its confluent version and prove the
nondegeneracy of the derivative Gram matrix.

For each fixed horizontal line in the evaluation strip, this implies
\begin{equation}\label{eq:02-projection-bound}
 \|Pk_z\|^2\le C\min\{1,Ne^{-\gamma|\Re z|}\}.
\end{equation}
Indeed, for $a=e^{\pm i\gamma\beta/2}$ and $w=e^{\gamma z}$,
\[
 1-\frac{|w-a|^2}{|w+\overline a|^2}
 =\frac{4\Re w\Re a}{|w+\overline a|^2}
 \asymp e^{-\gamma|\Re z|}.
\]
The product in Equation~\eqref{eq:02-projection-complement} is
therefore within $CNe^{-\gamma|\Re z|}$ of one, and is at most
$\exp(-cNe^{-\gamma|\Re z|})$. In particular, translated to the
frame of any real $x$ with $|x|\le(1-\delta)L$, the projections
$P_N$ converge strongly to the identity, uniformly for fixed
$\delta>0$. The complement tends uniformly to zero on every fixed
evaluation vector, and these vectors span a dense subspace by Fourier
uniqueness; contraction of the projections extends this to all of
$\mathcal H$.

Integrating Equation~\eqref{eq:02-projection-bound} on horizontal
lines near the two edges of the evaluation strip gives, for every
fixed $\epsilon>0$,
\begin{equation}\label{eq:02-trace-envelope}
 \operatorname{Tr}(Pe^{-\epsilon|k|}P)\le C_\epsilon L.
\end{equation}
Here multipliers are written as functions of $k$. More explicitly,
Plancherel on the line of height $y$ gives the multiplier
$v^{-1}e^{-2ky}$, and its trace against $P$ is
$\int_\R\|Pk_{x+iy}\|^2\,dx$. Choose $y$ near each strip edge.
The sum of the two resulting positive multipliers dominates a
constant multiple of $e^{-\epsilon|k|}$, while each integral is
$O(L)$ by Equation~\eqref{eq:02-projection-bound}.

We claim, for each integer $p\ge1$,
\begin{equation}\label{eq:02-trace-limit}
 \lim_{N\to\infty}\frac{\operatorname{Tr}((PRP)^p)}{2L}
       =\int_\R R(k)^p\,\frac{dk}{2\pi}.
\end{equation}
First truncate the exponentially decaying multiplier $R$ to a bounded
frequency interval. For any multiplier $H$,
\[
 \|PHP\|_1\le\operatorname{Tr}(P|H|P),
\]
by factoring through $|H|^{1/2}P$ and using the Hilbert--Schmidt
inequality. Equation~\eqref{eq:02-trace-envelope} makes the normalized
trace error tend to zero uniformly as the frequency interval grows;
a telescoping product gives the same statement for each fixed power.

For a bounded compactly supported symbol $S$, cyclicity of the trace
writes the corresponding trace as
\[
 \operatorname{Tr}\bigl(v^{-1}(Sv)P(SP)^{p-1}\bigr).
\]
The resolution
$\int_\R|k_x\rangle\langle k_x|\,dx=v^{-1}$ turns this trace into
$\int_\R F_N(x)\,dx$, where
\[
 F_N(x)=\left\langle(Sv)P(SP)^{p-1}k_x,k_x\right\rangle_{\mathcal H}.
\]
The real-line evaluation vectors have a fixed norm. The multiplier
$Sv$ is bounded on the compact frequency support, and the rightmost
operator acting on $k_x$ contains $P$. Hence
$|F_N(x)|\le C\|Pk_x\|$.
Translate $k_x$ to $k_0$ and conjugate every $P$ by the same real
translation. The multipliers are unchanged. In the bulk
$|x|\le(1-\delta)L$, strong convergence of these translated
projections therefore gives, uniformly,
\[
 F_N(x)\longrightarrow
 \left\langle S^pv k_0,k_0\right\rangle_{\mathcal H}
 =\int_\R S(k)^p\,\frac{dk}{2\pi}.
\]
The transition region has length $O(\delta L)$; beyond $(1+\delta)L$,
Equation~\eqref{eq:02-projection-bound} has an integrable exponential
tail of size $o(L)$. Divide by $2L$, then let $\delta\downarrow0$.
This proves the compact-frequency limit and hence
Equation~\eqref{eq:02-trace-limit}.

On the real frequency axis, $|R(k)|\le1$ and $\Re R(k)\ge0$:
here $2\beta\Re\zeta=\pi/3$. Moreover
$\|PRP\|_1=O(L)$ by Equation~\eqref{eq:02-trace-envelope}.
For $00$. The logarithm becomes
$\log[(1-y^3)/((1-y)(1+y^2))]$. Termwise integration of its
convergent logarithmic series, or monotone limits of their integrals,
gives
\[
 \frac1\beta\int_0^1
 \frac{\log(1-y^3)-\log(1-y)-\log(1+y^2)}{y}\,dy
 =\frac1\beta\left(-\frac{\pi^2}{18}
                         +\frac{\pi^2}{6}-\frac{\pi^2}{24}\right)
 =\frac{5\pi^2}{72\beta}.
\]
Combining this with $2L/(2\pi)$ and
$\beta\gamma=\pi/3$ gives the coefficient
$5\pi/(72\beta\gamma)=5/24$ in
Equation~\eqref{eq:02-denominator}.
\end{proof}
==== END Cylinder-loop-weights-and-planar-nesting//02_cylinder.tex ====

==== BEGIN Cylinder-loop-weights-and-planar-nesting//03_gas_mean.tex ====
\section{Centering, coercivity, and the fluctuation spectrum}\label{sec:03-gas-mean}

The finite contour formula expresses the cylinder partition function as
an oscillatory sum of particle integrals. We first identify its exact
mean density. Subtracting this density leaves an interaction whose real
part is positive and therefore controls both the particle counts and
excursions beyond a logarithmic wall. We then compare the interior
interaction with a reference ensemble whose normalization and linear
statistics can be evaluated. The energy estimates have two uses. The untruncated real energy estimate controls every valid particle count; the sharper imaginary trial fields identify the tilt cost in the fixed-fugacity exponent. Both use the same exact compensated-sector identity, Lemma~\ref{lem:03-compensated-sector}. The determinant asymptotic in Theorem~\ref{thm:03-determinant} supplies the remaining fluctuation contribution. Lemmas~\ref{lem:03-core} and~\ref{lem:03-schur} then give the finite-dimensional approximation needed to realize that determinant by moving actual particle coordinates.

\subsection{The exact mean density}

Throughout this section we specialize Theorem~\ref{thm:02-contour-sectors} to \(N=2m,\ s=b=m,\ \varepsilon=1\), all real site offsets zero. Fourier transforms in the real coordinate use \(e^{-iku}\) and inverse measure \(dk/(2\pi)\). A particle is specified by its real coordinate \(u\) and its ray label; the imaginary part of its logarithmic coordinate is determined by that label. An interaction matrix kernel acts by convolution; symmetric means \(H^{JI}(-u)=H^{IJ}(u)\), and forms with complex arguments are continued \emph{bilinearly}. Write
\[
 \gamma=\pi/(3\beta),\quad L=(\log N)/\gamma,\quad
 p=2\beta,\quad \kappa=(3\beta)^2 .
\]
Labels are \(I=(\sigma,J)\), where \(\sigma\in\{+1,-1\}\) and \(J=0,F,P\) denote the main, far-single and far-pair species. Their elementary shifts are, respectively, \(0\), \(11/4\), and \((5/2,7/2)\), as in Theorem~\ref{thm:02-contour-sectors}. Their multiplicities \(w_I\) are \(1,1,2\); a pair still has only one real integration variable. Write \(H\) for minus the sign-normalized particle-particle log interaction, \(V\) for the log (without negation) of the sign-normalized site-particle interaction (particle row, site column). Differences in these kernels take the first real coordinate minus the second, e.g. the exponentiated factors use \(H^{I I'}(u-u')\) with minus sign for labels \(I,I'\). More explicitly use linear combinations of
\[
 K_\theta(u)=-\log[2(\sin(\theta/2)\cosh(u/2)-i\cos(\theta/2)\sinh(u/2))],\qquad
 K_0(u)=-\log|2\sinh(u/2)|
\]
reducing angles to \([0,2\pi)\). Thus \(H\) uses the exponents of the log-sinh factors in \(E\) (fused for pair labels), and \(V\) the negative exponents in the site-particle product; cross terms in reverse order transpose and reflect the kernel. The exponents at angle zero in \(H\) are \(2\delta_{IJ}\) by the shift lists; there are no angle-zero terms of \(V\). These conventions agree with the sign normalization of the contour-sector identity.

Modulo distributions at zero frequency the transforms of \(K_\theta\) are
\[
 \pi e^{(\pi-\theta)k}/(k\sinh(\pi k)),\qquad
 \widehat K_0=\pi\cosh(\pi k)/(k\sinh(\pi k)).
\]
Indeed \(K_\theta''=\tfrac14/\sinh^2((u+i\theta)/2)\); shifting by a period and taking the double pole residue gives \(-\pi k e^{(\pi-\theta)k}/\sinh(\pi k)\). The second formula follows by even averaging as \(\theta\downarrow0\). Affine terms in comparisons below are fixed by tails: real parts \(-|u|/2+O(e^{-|u|})\), imaginary parts odd with limits given by \(\tfrac12(\pi-\theta)\operatorname{sign}u\) for \(0<\theta<2\pi\). For the main (near) labels,
\[
 \widehat H_{nn}(k)=\frac{2\pi c_*(k)}{k\sinh(\pi k)}
 \begin{pmatrix}\cosh(3\beta k)& e^{pk}\\ e^{-pk}&\cosh(3\beta k)\end{pmatrix},
 \qquad c_*(k)=2\cosh(\beta k)-1.
\]
Define \(\widehat\varrho=1/c_*\). Shifting by a period and taking simple residues gives
\[
 \varrho(u)=\frac{e^{-\gamma|u|}-e^{-5\gamma|u|}}{\sqrt3\,\beta(1-e^{-6\gamma|u|})}.
\]
The value at zero is its removable continuation. This is a positive,
even, analytic probability density, and \(\varrho(u)\asymp
e^{-\gamma|u|}\). The lattice activity is denoted by
\(\rho\).

Let \(\mu\) have density \(m\varrho\) on each main label and vanish
on the four far labels. We use \((f,Hg)\) for the sum of the
corresponding double integrals over labels and real coordinates.

\begin{lemma}[Exact centering]\label{lem:03-centering}\label{E:centering}
The particle--site kernel satisfies \(V=H_{\cdot n}*\varrho\).
Combining the site--site factors with
\(\exp((\mu,H\mu)/2)\) gives
\[
 C_*^N\prod_{i0\) small; it stays fixed through \(N\to\infty\). Take a positive convolution probability \(\psi=\psi^{(h)}\) with transform the average of
\(\cosh((\pi-ht)k)/\cosh(\pi k)\) for \(00\) zero-extended, by Cauchy estimates on circles of radius \(c x\). Subtract from \(H_{ne}\), only on same-label entries (thus involving an exterior main particle), the real nonnegative kernel
\[
 R_h(u)=\chi(u)(S_0-S_0*\psi)(u).
\]
Call the resulting interaction \(\widetilde H_{ne}\), extending \(R_h\) by zero on other entries. The interaction \(\widetilde H_{ne}\) is smooth Gevrey (some finite order, for fixed \(h\)) with at most linear growth and bounded higher derivatives of each positive order. Here and below Gevrey bounds follow equivalently for bounded decaying terms from stretched exponential Fourier estimates; for example differentiating under a bound \(C\exp(-c|k|^\tau)\) gives \(C^{l+1}(l!)^{1/\tau}\) bounds, and multiplication of Gevrey functions preserves a finite order by Leibniz. For \(S_0*\psi\) one can first take two derivatives in the Fourier estimates.

For smooth trial densities define the complex energy
\[
 \mathcal E(\alpha,\zeta)=\tfrac12(\alpha-\mu,H_{nn}(\alpha-\mu))
 +(\alpha-\mu,\widetilde H_{ne}\zeta)+\tfrac12(\zeta,H\zeta)_{\rm off}.
\]
The last term omits all individual explicit self terms. One may fix generic distinct explicit real positions (null sets of coincidences never needed). Trial bounds will apply for example to smooth densities, or integrable densities of finite log energy with sufficient decay, by approximation. Point configurations have logarithmic self singularities, so this
formula must be compensated before it is used in a particle integral.
The following reference kernel makes that compensation explicit. Let \(u\mapsto t\) be an increasing bijection \([-L,L]\to[0,\pi]\) smooth with strictly positive derivative in the open interval; set \(p_L=dt/\pi\) as a probability there in \(u\) (we will specify the map). Use
\[
 H_{\rm ref}(t,t')=-2\log|2(\cos t-\cos t')|
\]
separately on each main group, zero cross. Its \(p_L\)-mean vanishes by circle reflection (mean of \(\log|2\sin(t/2)|\) is zero by its Fourier series). Use this kernel for independent reference integrals with measures
\[
 \frac{1}{\nu_\sigma!}\prod_{i=1}^{\nu_\sigma}\frac{2\sin t_i\,dt_i}{2\pi}
 \prod_{i0\), with no residual distributions at zero. Indeed nonsingular terms have transform \(O(e^{-c|k|}(1+k^{-2}))\), multiplied by \(1-\widehat\psi\). For the singular \(S_0\) term the difference in real space is \((1-\chi)(S_0-S_0*\psi)\) with derivatives in \(L^1\) bounded by \(O(h^2)C^{l+1}(l!)^2\) by even averaging of imaginary shifts off the singularity, and the cutoff bounds. This gives \(O(h^2)\exp(-c\sqrt{|k|})\) in frequency, sufficient by the lower bound for \(\widehat\psi\). Thus the mixed-kernel error is bounded by
\(C h^{c_0}\|\alpha-\mu\|_{-1/2}\|\psi*\zeta\|_{-1/2}\).
For explicit-explicit pairs only the real kernel matters. Every smooth real summand (apart from diagonal \(S_0\)) changes under \(\psi*\psi\) averaging by \(O(h)e^{-c|u|}\), using exponentially decaying second derivatives and the tail and second moment of \(\psi\). The diagonal singularity can only decrease. And the missing same-variable diagonals in the smoothed model cost \(O_h(n_e)\).

To absorb these errors we need a quadratic count bound that is valid even when the interior trial densities are signed. We obtain it by testing far labels separately and testing main labels only outside the wall, where those trial densities vanish. Partition the explicit coordinates into unit-scale cells and let \(n_j\) denote their variable counts. We claim
\begin{equation}
 \sum n_j^2\le C(1+|\log h|)(B(x)+1).
 \label{eq:G-2}
\end{equation}
For far labels use translates of a nonnegative fixed bump covering a unit interval, to test separately each \(x_I=\psi*\zeta_I\) positive. For outer main counts on either side of the wall use bumps supported \emph{outside} the wall, adapted to cells starting from the wall (the grid may change by bounded overlaps). For the first cell use a bump rising from zero at the wall to 1 within distance \(h\), plateauing to distance 2, then cutting off. This captures at least a constant smeared mass from each unsmeared point of the first cell, by the displayed real kernels. On all these outer tests \(\operatorname{Re}\alpha\) does not intervene and \(\mu\) has cell masses summably bounded. The \(H^{1/2}\) norms squared of the bumps are \(O(1+|\log h|)\) even at the wall, by the kernel seminorm \(\int|b(u)-b(v)|^2|u-v|^{-2}du\,dv\) (equivalence by Plancherel); for the transition of length \(h\), the translation differences in \(L^2\) squared cost \(O(\min(l,l^2/h))\) for translations by \(0L\), the total count beyond \(t\) is at most twice its positive smeared cumulative there, hence its square integrates on \([L,\infty)\) to \(O(B(x)+1)\) by \(\|D\|^2\) and the \(\mu\)-tail. Same on the other side. The cumulative counts are nonnegative integers, so their squares dominate the counts themselves. Integrating gives the distance sum bound. It also bounds exterior counts away from the wall; \eqref{eq:G-2} controls the remaining cells near the wall and the far-particle cells inside it.

Total \emph{smeared} mass outside the wall is at most \(C_h(1+\sqrt{B(x)})\) also, by the cumulative test outside distance one from the wall and \eqref{eq:G-2} convolved by \(\psi\) for the first cell. Finally \(\int r=-\operatorname{Re}M\); test on the wall region with a smooth plateau extending outside by distance one, cost \(O(\sqrt L)\) in \(H^{1/2}\), and bound the remainder by the positive tail masses. In particular, retaining a fraction of \(B(x)\) in a trial lower bound gives exponential damping \(\exp(-c_h\ell_e)\), sufficient for convergence and tail truncation of the explicit absolute integrals. Summing their factorial measures with factors \(\exp(C_hn_e)\) costs at most \(\exp(O_h(L))\): far variables have effective volume \(O(L)\), and exterior main variables have effective volume \(O(1)\). This is the coarse cost; part 4 of Proposition~\ref{prop:03-trial-bounds} will improve it for the configurations needed by the exponent.

\end{proof}

\subsection{The energy gain from an imaginary tilt}

The cylinder parameter enters a sector through
\(e^{i(\eta+\pi f)M}\). We seek trial imaginary densities that
convert this oscillation into a positive real cost while preserving
the prescribed totals. The estimates here are variational statements;
the particle contour deformation implementing them is proved in
Section~\ref{sec:04-gas-fluctuations}.

Let \(a=\operatorname{Re}\eta,\ b_1=\operatorname{Im}\eta\) (here \(b_1\) is not a size); suppose \(|a|\le a_0<\pi,\ |b_1|\le b_0\) fixed. Allow Fourier aliases \(j\in\mathbb Z\). In the real negative-log action \(\mathcal A\), add to \(\operatorname{Re}\mathcal E\) the negative log modulus of \(e^{i(\eta+\pi f+2\pi j)M}\). Use \(D_z=\partial^{-1}\sum z_\sigma\), \(z_r=z_+\), so \(\int z_r=-\operatorname{Im}M\). Let \(f_R^\psi\) be the cumulative positive \emph{weighted} smeared far count to the right, and \(f_R^0\) the unsmeared version. The sign tails and \eqref{eq:G-1} give
\begin{equation}
\begin{split}
 \mathcal A\ \ge\
 B(x)&-B(z)-2p\int(D z_r+r D_z)-2\sum_{I:\rm far}g_I\int x_I D_z
 -(x,J_o z)\\
 &-\int \Theta z_r-b_1\int r-s(h)(B(x)+B(z)+1)-C_h n_e,
\end{split}
 \label{eq:G-4}
\end{equation}
Here \(\Theta=a+2\pi(j+f_R^\psi)\). Indeed the baseline \(+\) versus \(-\) sign tails give the \(2p\) terms by total neutrality and antisymmetry. The additional far rows give sign against \(z_++z_-\) with coefficient \(g_I\), and against \(z_r\) with coefficient \(\pi w_I\); the constant from \(2 f_R^\psi-f\) thus cancels the sign-bias \(\pi f\).

The next proposition starts with the unshifted real action and then gives
three uses of an imaginary trial field.  Part 2 permits every Fourier
alias.  Part 3 suppresses aliases whose effective phase is large near an
end of the wall.  Part 4 gives the quadratic dependence on the tilt when
those end phases are bounded.

We make the common assumptions and the end phases precise.  An admissible
near density \(\alpha=(\alpha_+,\alpha_-)\) is supported on \([-L,L]\),
has the prescribed totals \(m'_\sigma(M)\), and has finite trial energy.
We allow signed real parts.  Put
\[
 x=\operatorname{Re}\alpha-\mu+\psi*\zeta,
 \qquad z=\operatorname{Im}\alpha,
 \qquad
 \mathcal A=\operatorname{Re}\mathcal E(\alpha,\zeta)
             -\log\left|e^{i(\eta+\pi f+2\pi j)M}\right|.
\]
Thus \(\int z_\sigma=-\sigma\operatorname{Im}M\).
For parts 2--4 below assume that \(p_L\) has uniformly bounded
\(H^{-1/2}\) norm.  For each sufficiently small fixed \(\delta>0\),
assume that it has a normalized smooth truncation \(p_L^\circ\), supported
in \(|u|\le(1-\delta/2)L\), whose difference from \(p_L\) is exponentially
small in total variation and \(H^{-1/2}\).  The derivatives of each fixed
order of \(p_L^\circ\) are to be polynomially bounded in \(L\).
The hard-wall coordinates constructed below supply these properties.

For a fixed \(\delta\), choose nonnegative smooth probes
\(\varphi_\pm\le1\), supported away from the ends of
\(\pm[(1-2\delta)L,(1-\delta)L]\), obtained by rescaling fixed bumps,
with \(\int\varphi_\pm\asymp\delta L\).  For
\(a=\operatorname{Re}\eta\), define
\[
 \Theta(u)=a+2\pi\bigl(j+f_R^\psi(u)\bigr),\qquad
 \theta_\pm=\frac{\int\Theta\varphi_\pm}{\int\varphi_\pm},
 \qquad U=\max\{|\theta_+|,|\theta_-|\}.
\]
Here \(f_R^\psi\) is the weighted smeared far count to the right,
as in \eqref{eq:G-4}.  A polynomial bound throughout the proposition
means a fixed power of \(L\) with a fixed multiplicative constant.

\begin{proposition}[Trial bounds with tilt]\label{prop:03-trial-bounds}
Fix \(a_0<\pi\) and \(b_0<\infty\), and let
\(\eta=a+ib_1\) satisfy \(|a|\le a_0\), \(|b_1|\le b_0\).
The following bounds hold uniformly for admissible real parts of
\(\alpha\), with the imaginary fields specified below.  In parts 1--3,
\(h>0\) is fixed and sufficiently small.
\begin{enumerate}
\item If \(M\) is real and \(z=0\), then for sufficiently small
fixed \(h>0\), with no truncation of the explicit counts,
\[
 \mathcal A\ge cB(x)-C_hL-C_hn_e.
\]

\item Suppose \(n_e\) is polynomially bounded, and that \(p_L\)
has the properties stated above.  For any
\(v=\operatorname{Im}M\in\mathbb R\) and any \(j\in\mathbb Z\),
the choice \(z_\sigma=-\sigma vp_L\) gives
\[
 \mathcal A\ge cB(x)-C_hL-C_hn_e
       +2\pi jv-C_h\bigl(v^2+(1+f)|v|\bigr).
\]
There is no polynomial restriction on \(j\) in this estimate.

\item Suppose in addition that \(|j|\) is polynomially bounded.
There are a value \(v=\operatorname{Im}M\), independent of
\(\operatorname{Re}M\), and an imaginary field \(z\) with the
required totals such that
\[
 \mathcal A\ge cB(x)+c_\delta U^2L-C_hL-C_hn_e.
\]
The value \(v\) is polynomially bounded.  Each
\(z_\sigma+\sigma vp_L\) is the derivative of a smooth primitive
supported in \(|u|\le(1-\delta/2)L\), with polynomial bounds on
all required fixed-order derivatives.  Moreover, the number
of far particles in
\([-(1-2\delta)L,(1-2\delta)L]\) is at most \(CU\) for large \(L\).

\item Under the same polynomial hypotheses, suppose \(U\le U_0\).
Choose a desired error \(\varepsilon>0\), then \(h\) sufficiently
small, then \(\delta\) sufficiently small.  The constant \(U_0\)
may be fixed arbitrarily after these choices.  There are a value
\(v=\operatorname{Im}M\), independent of \(\operatorname{Re}M\),
and an imaginary field \(z\), with the same support and polynomial
regularity properties as in part 3, such that
\[
 \mathcal A\ge c'_\varepsilon B(x)
       +2L\frac{\operatorname{Re}(\eta^2)}{4\kappa}
       -C\varepsilon L-o(L)-C_hn_e.
\]
The selected \(v\) is polynomially bounded.
The retained coefficient \(c'_\varepsilon>0\) can be taken
independent of \(\delta,U_0\).  On these bounded-\(U\) configurations,
integrating the explicit factorial measures with the factor
\(\exp(C_hn_e)\), while using a fixed portion of the retained
\(B(x)\)-damping, costs at most
\[
 \exp\bigl(C_{h,\varepsilon}\delta L+o(L)\bigr).
\]
The same conclusion allows further fixed exponential factors per
explicit particle after reducing \(\delta\) as needed.
\end{enumerate}
The constants are uniform once the polynomial bounds and the previously
fixed parameters are fixed.  In part 4 the asymptotic limit is taken
only after \(\varepsilon,h,\delta,U_0\) have been chosen in that order;
constants depending on those fixed choices are allowed in \(o(L)\).
\end{proposition}
\begin{proof}
For part 1, set \(z=0\) in \eqref{eq:G-4}. The remaining
linear term satisfies
\[
 |b_1\textstyle\int r|\le C_h\sqrt{L(B(x)+1)}
                    \le \tfrac14B(x)+C_hL
\]
by Corollary~\ref{cor:03-counts}; choose \(h\) so that the smoothing
loss consumes at most another fixed fraction of \(B(x)\).
This proves the first bound without any count truncation.

For part 2 suppose that \(n_e\) is polynomially bounded and use the
stated norm and truncation properties of \(p_L\). For \(z_\sigma=-\sigma v p_L,\ v=\operatorname{Im}M\), \eqref{eq:G-4} gives
\begin{equation}
 \mathcal A\ge c B(x)-C_h L-C_h n_e+2\pi jv-C_h(v^2+(1+f)|v|).
 \label{eq:G-5}
\end{equation}
Here \(D_z=0\), and the mixed baseline and odd-remainder errors cost \(C|v|\sqrt{B(x)}\). This bound uses no polynomial restriction on \(j\).

For part 3 suppose also \(|j|\le{\rm poly}(L)\), with the polynomial fixed. We select \(v=\operatorname{Im}M\), independent of \(\operatorname{Re}M\), together with \(z\). In the choices below we first specify smooth \(z_r,D_z\) supported on \(|u|\le(1-\delta)L\) (and \(v=-\int z_r\)); then we actually add \(v(p_L^\circ-p_L)\) to \(z_r\), leaving \(D_z\) unchanged. Thus each difference \(z_\sigma+\sigma v p_L\) is a smooth density derivative with primitive compactly supported in \(|u|\le(1-\delta/2)L\); all its required sizes and fixed-order derivatives are polynomially bounded. In evaluating \eqref{eq:G-4} we can neglect the exponentially small correction, at cost \(o(1)(1+\sqrt{B(x)})\) by the norm bounds, total variation, and the polynomial count bounds.

Take smooth nonnegative probes \(\varphi_\pm(u)\le1\), supported on \(\pm[(1-2\delta)L,(1-\delta)L]\), rescaled fixed bumps supported away from their interval edges with mass comparable to \(\delta L\). Set \(\theta_\pm\) equal to the corresponding weighted averages of \(\Theta\); let \(U=\max|\theta_\pm|\). A first, coarse tilt uses
\[
 D_z=0,\qquad z_r=-t_0\sum_\pm\theta_\pm\varphi_\pm
\]
with small fixed \(t_0>0\). Then \(-\int\Theta z_r=t_0\sum\theta_\pm^2\int\varphi_\pm\). The \(B(z)\) cost is at most \(C t_0^2\sum\theta_\pm^2\int\varphi_\pm\); baseline and odd mixed terms are absorbed similarly with a fraction of \(B(x)\). The term with \(b_1\) is absorbed as in the real bound. Thus
\begin{equation}
 \mathcal A\ge cB(x)+c_\delta U^2 L-C_h L-C_h n_e .
 \label{eq:G-6}
\end{equation}
By positivity, the number of far particles on \([-(1-2\delta)L,(1-2\delta)L]\) is \(\le C U\) for large \(L\): their smeared masses contribute at least a constant each to the drop between the two probe averages.

For part 4 suppose \(U\le U_0\), where \(U_0\) may be arbitrarily large but fixed. Given a desired small error, choose a small parameter \(\varepsilon>0\), then \(h\) sufficiently small (so \(s(h)\ll\varepsilon\)), then \(\delta>0\) sufficiently small relative to these. \(U_0\) can be chosen after these parameters. Use a smooth mask \(0\le w(u)\le1\) supported on \([-(1-3\delta)L,(1-3\delta)L]\), vanishing at distances \(\le\sqrt L\) of far positions, equal to 1 except in boundary buffers of total width \(O(\delta L)\) inside \([-L,L]\) and holes of total width \(O_{U_0}(\sqrt L)\). Take fixed-order positive derivatives \(O_{U_0,\delta,l}(L^{-l/2})\). This is possible by the count conclusion. Put \(s(u)=\operatorname{sign}(a+2\pi(j+f_R^0(u)))\) and \(A_0=|a|/(2\kappa),\ B_0=-p b_1/(2\kappa)\). Use
\[
 z_r=-A_0\,s(u)w(u),\qquad D_z=B_0 w(u),
\]
with the correction convention above. These are smooth before correction even at jumps. Since \(|a|<\pi\), \(|a+2\pi(j+f_R^0)|\ge |a|\). On the mask support all far smearing tails from points outside their own respective side of \(u\) are superpolynomially small. Consequently \(-\int\Theta z_r\ge |a| A_0\int w-o(1)\), and \(\sum g_I\int x_I D_z=o(1)\). Moreover
\[
 B(z)\le (B_0^2+(\kappa-p^2)A_0^2)\int w+o(L),\qquad
 |(x,J_o z)|\le o(\sqrt L)\sqrt{B(x)} .
\]
Indeed before correction the scaled \(L^2\) masses of \(z_r,D_z\) (squared divided by \(L\)) concentrate at low frequency since their positive-order derivatives have norms \(o(\sqrt L)\). Use the symbol expansion and the upper norm bound, and the odd-remainder multiplier bound.

We give details for the real-field optimization so that no slow-variation assumption on \(x\) is needed. Let subscript \(\ell\) mean a smooth compact low-pass Fourier cutoff equal to 1 near zero, even real with values in \([0,1]\), supported on sufficiently small fixed frequencies depending on \(\varepsilon\). By the positive small-frequency expansion, reserving a fraction on all frequencies,
\[
 B(x)\ \ge\ c_\varepsilon B(x)
 +(1-C\varepsilon)\int \big(D_\ell^2+Q(r_\ell,x_{{\rm far},\ell})\big).
\]
On the mask \(x_{{\rm far},\ell}\) is superpolynomially small (Schwartz kernel tails, count bound). Thus in this lower bound we can use \((1-C'\varepsilon)\int w(D_\ell^2+(\kappa-p^2)r_\ell^2)-o(1)\) besides the reserve. The pairings against \(ws,w\) can replace \(D,r\) by \(D_\ell,r_\ell\) at \(o(\sqrt L)\sqrt{B(x)}\) cost by the derivative bounds and Sobolev norms. Also
\[
 |\int(1-w)r|\ \le\
 (C\sqrt{\delta L}+o(\sqrt L))\sqrt{B(x)}+C_h .
\]
Use the smooth wall plateau of \eqref{eq:G-3} minus \(w\), whose \(L^2\) norm has the stated bound by the support estimate and whose derivative norm is \(o(\sqrt L)\), then positive outer tail bounds for the rest. Constants depending on \(U_0,\delta,h\) in these estimates are allowed inside little-oh terms.

Absorb errors with a portion of the reserve, including \eqref{eq:G-1}, losing \(C\varepsilon L+o(L)\) with \(\delta\) sufficiently small. It remains on measure \(w\,du\) to complete squares for \(D_\ell,r_\ell\), whose linear coefficients in \eqref{eq:G-4} are \(2p A_0 s,\ -(b_1+2p B_0)\). The lower cost from these optimizations per unit mass is at worst
\[
 -p^2 A_0^2-\frac{(b_1+2p B_0)^2}{4(\kappa-p^2)}-O(\varepsilon).
\]
Adding \(-B_0^2-(\kappa-p^2)A_0^2+|a| A_0\) leaves \((a^2-b_1^2)/(4\kappa)\) before errors. Thus
\begin{equation}
 \mathcal A\ \ge\ c'_\varepsilon B(x)+2L\,\frac{\operatorname{Re}(\eta^2)}{4\kappa}
 -C\varepsilon L-o(L)-C_h n_e.
 \label{eq:G-7}
\end{equation}
The retained constant can be taken independent of \(\delta,U_0\). For these bounded-\(U\) estimates the explicit factorial measures with \(\exp(C_h n_e)\) and a portion of the retained damping cost at most \(\exp(C_{h,\varepsilon}\delta L+o(L))\): outer main points cost bounded grand mass by tail damping, only a bounded number of far points can be on the interior interval just counted, and the remaining far points use an effective volume \(O(\delta L)+O_{h,\varepsilon}(1)\) per variable, with factorial divisors. This also works with more fixed exponential factors per particle, by further reducing \(\delta\). Bounds \eqref{eq:G-5}–\eqref{eq:G-7} are deterministic bounds on trial densities. Implementing them for the complex integrals, including fluctuation determinants and the discreteness of \(M\), takes further work below.


\end{proof}

\subsection{Coordinates that remove the diagonal singularity}

The trial estimates require a probability coordinate concentrated near
\(\varrho\), while the reference ensemble uses a fixed compact
interval. We choose a map that has both properties and makes the
true-minus-reference interaction smooth through the endpoints.

Use a reference parameter \(s\in[-A,A]\), \(A=L+d_0\), and \(u=W(s)\). This is odd and \(W(A-v)=L-\omega(v)\) at the positive end, where \(\omega\) is quadratic near 0, Gevrey, increasing with derivative comparable to \(\min(v,1)\), and \(\omega(v)=v-d_0\) for \(v\ge1\). Construct it by integrating a cutoff interpolation of the derivative between \(v\) and 1 (fixes \(d_0\)). Thus \(W=s\) except within bounded endpoint distance and \(W\) reflects evenly at the ends. Set
\[
 \lambda_L(s)=C_L\varrho(W(s)),\qquad
 t=\theta(s)=\pi\int_{-A}^s\lambda_L,\qquad
 C_L^{-1}=\int_{-A}^{A}\varrho(W(v))\,dv=1+O(e^{-\gamma L}).
\]
So \(p_L\) is \(\lambda_L ds\) pushed to physical \(u\). Its stated probability and truncation norm requirements hold: in the core \(s=u\), and near each wall its density in \(u\) is bounded by \(C e^{-\gamma L}(L-|u|)^{-1/2}\); the log-kernel estimate for the \(H^{-1/2}\) norm thus suffices. Cut smoothly in the core to form \(p_L^\circ\). Derivatives of \(\lambda_L\) are bounded by \(C^{l+1}(l!)^{P}\lambda_L\) for a fixed \(P\), and \(\lambda_L\asymp e^{-\gamma|s|}\). Constants in Gevrey bounds below can change (including order \(P\)), but not with \(N\). These elementary closure estimates under smooth Gevrey multiplication, composition on a bounded scale, and reciprocal of a function bounded away from zero follow by multiplying formal Taylor series with coefficients bounded by \(C^l(l!)^{P}\); in composition of positive degrees totaling \(l\), products of factorials are bounded by \(l!\), and the number of compositions is at most exponential.

Let
\[
 K(s,v)=H_{nn}(W(s)-W(v))-\mathbf 1\,H_{\rm ref}(\theta(s),\theta(v)),
 \qquad \mathscr K=\partial_s\partial_v K ,
\]
where \(\mathbf 1\) denotes the identity matrix.

\begin{lemma}[Regularity of the compensated kernel]\label{lem:03-kernel-regularity}
The kernel \(K\) is smooth, extending evenly by reflection in each endpoint variable. \(K\) and all derivatives of total order \(l\) satisfy bounds \(C^{l+1}(l!)^P(1+L)\), and
\(\mathscr K\) and its derivatives satisfy \(C^{l+1}(l!)^P e^{-c|s-v|}\). For the latter estimates one only extends locally across endpoints. Here and below \(\operatorname{poly}(L)\) prefactors may replace \(1+L\) without harm.

\end{lemma}
\begin{proof}
First suppose \(|s-v|>c_0>0\). The sinh terms have the required estimates directly. Split the cos-difference logarithm into its two sine logarithms. Then
\(\sin(|\theta(s)-\theta(v)|/2)\) is bounded below by \(c\max_{[s,v]}\lambda_L\), reversing the interval if needed, and \(\sin((\theta(s)+\theta(v))/2)\) is at least as large. One derivative in each variable supplies the decaying ratio
\[
 \frac{\lambda_L(s)\lambda_L(v)}{(\max_{[s,v]}\lambda_L)^2}.
\]
Higher derivatives preserve this bound, proving the off-diagonal estimate for \(\mathscr K\).

Near the diagonal and away from the endpoints, divide both \(2\sinh((W(s)-W(v))/2)\) and \(\cos\theta(s)-\cos\theta(v)\) by \(s-v\). Their natural local scales are respectively \(1\) and \(\lambda_L^2\), since \(\sin\theta\asymp\lambda_L\) there. Averaging derivatives along the joining interval gives uniform Gevrey bounds and lower bounds on those scales. Their logarithmic ratio is therefore smooth.

Near a common endpoint, let \(d_s,d_v\) be the distances from that endpoint. Both differences have the additional factor \(d_s+d_v\): even reflection makes their zeros occur at both \(s=v\) and the reflected diagonal. Divide by
\[
 (s-v)(d_s+d_v).
\]
The division is smooth Gevrey, obtained by averaging a derivative of the first divided difference from its reflected zero. The quadratic fold \(\omega'(d)\asymp d\) and the sine factors give positive lower bounds for both quotients on their respective scales. The same log-ratio argument now proves regularity through the corner. Even reflection in either variable follows from the construction of \(W\) and \(\theta\).

Finally, in variables at distance at least \(\delta L\) from the endpoints, the compensated kernels continue analytically for uniformly small complex displacements, with the same fixed-order derivative control in the remaining variables. Here \(W(s)=s\) and \(\theta'=C_L\pi\varrho\) is analytic with relative bounds on uniformly small disks, so the divided-difference arguments apply there as well.
\end{proof}

The next statement supplies the linear and constant terms needed together with this kernel in the particle deformation and dimension interpolation. It explains why the exponentially large mean particle number does not make the compensated coefficients exponentially large.

\begin{lemma}[Regularity of the compensated source]\label{lem:03-source-regularity}
Suppose \(|M|,n_e,\ell_e\) have fixed polynomial bounds in \(L\). In the \(s\)-coordinate, the linear source
\[
 b=H_{nn}(m'p_L-\mu)+\widetilde H_{ne}\zeta
\]
in \(\Phi_M\) is smooth and even under endpoint reflection. Its derivatives of order \(r\) are bounded by
\(\operatorname{poly}(L)C^{r+1}(r!)^P\), for a fixed Gevrey order \(P\), and it has the core analytic continuation just described. Writing
\[
 E_e=\tfrac12(\zeta,H\zeta)_{\rm off},
\]
the difference \(\mathcal E(m'p_L,\zeta)-E_e\) is polynomially bounded in absolute value. The unchanged explicit-pair term \(E_e\), independent of \(M\), has real part bounded below by a negative polynomial; no upper bound is required.
\end{lemma}
\begin{proof}
For each main component, decompose the centered mean as
\[
 m'_\sigma p_L-m\varrho
   =(m'_\sigma C_L-m)\varrho
       +m'_\sigma(p_L-C_L\varrho).
\]
The coefficient in the first term is polynomially bounded, since \(m'_\sigma-m\) is polynomial and \(C_L-1=O(e^{-\gamma L})\). Its convolution is controlled by \(H_{nn}*\varrho=V\).

The two measures in the second term agree in the core. Near or beyond the ends their densities, written in \(ds\) for \(p_L\) and \(du\) for \(\varrho\), have derivative bounds with a factor \(O(e^{-\gamma L})\) and exponentially bounded tails. This factor cancels \(m'_\sigma=O(e^{\gamma L})\). Insert a fixed-width physical Gevrey cutoff there. For convolution of the first density with the diagonal logarithm, subtract \(-2\log|s-v|\) and its reflected-image term using the divided differences of \(W\). The remaining kernel is smooth. Transfer derivatives of the ordinary translated logarithm to the compactly cut off, evenly reflected density. For convolution with the second density, transfer derivatives in physical \(u\) and then compose with \(W\). Both terms are analytic in the deep core by separation from their supports.

The source \(\widetilde H_{ne}\zeta\) has the same bounds for fixed \(h\). Its regularization concerns only exterior main interactions and leaves an analytic kernel in the core. These arguments also give polynomial total variation and moments for \(m'p_L-\mu\), hence the stated constant-term bound by log integrability. Finally, real logarithmic singularities of explicit pairs are repulsive; away from them the kernels have at most linear growth. The polynomial count and excursion bounds therefore give \(\operatorname{Re}E_e\ge-\operatorname{poly}(L)\), including when distinct explicit particles approach each other.
\end{proof}

\subsection{The relative interaction operator}

The compensated kernel is now smooth. It remains to compare its
quadratic fluctuations with those of the reference law, uniformly as
the interval grows. Let \(Q=(Q_+,Q_-)\) be a primitive of a change
of zero mass on each main label in the \(s\)-coordinate, initially
smooth and zero at both endpoints. Define
\[
 h_0(Q,Q)=(Q',H_{\rm ref}Q'),\qquad
 H_{\rm form}(Q,Q)=h_0(Q,Q)+
   \iint Q(s)^{\mathsf t}\mathscr K(s,v)Q(v)\,ds\,dv
\]
where \(H_{\rm form}\) is the true near interaction form. The second expression is complex symmetric; on complex fields both energy forms are extended bilinearly. The space \(\mathcal X\) has squared norm \(h_0\); in the \(t\)-coordinate the fields extend as odd functions at both ends, completed from smooth ones. Per label
\[
h_0=\pi^2\sum_{n\ge1} n |b_n|^2,\qquad Q=\sum b_n\sin(nt)
\]
for the real form/norm, since \(H_{\rm ref}(t,t')=4\sum_{n\ge1}\cos(nt)\cos(nt')/n\) as an integrable-kernel Fourier expansion and \(\int\cos(nt)\,dQ=n\pi b_n/2\) on Dirichlet fields. For symmetric energy calculations on complex fields \(h_0(\cdot,\cdot)\) extends bilinearly; norms and operator adjoints use the ordinary complex Hilbert convention. The inverse kernel using \(ds\) pairing is
\[
 G(s,v)=\frac{1}{2\pi^2}\log\left(\frac{\sin((\theta(s)+\theta(v))/2)}
                  {\sin(|\theta(s)-\theta(v)|/2)}\right).
\]
In particular \(j:\mathcal X\to L^2(ds)\) the inclusion has bounded norm uniformly, \(jj^*=G\) separately per label: the log singularity is square integrably bounded locally by \(C(1+|\log|s-v||)\) using \(\sin\theta\le C\lambda_L\); off diagonal the kernel and derivatives decay exponentially, by writing the squared ratio as \(1+\sin\theta(s)\sin\theta(v)/\sin^2((\theta(s)-\theta(v))/2)\). Schur's integral bound applies.

The next estimate controls the number of nearly degenerate directions,
not just the smallest eigenvalue. This distinction is needed when
passing from trace estimates to logarithmic determinants.

\begin{lemma}[Trace bound and small eigenvalues]\label{lem:03-spectral}\label{E:spectral-bounds}
On the complexification of \(\mathcal X\), let
\[
 T=I+j^*\mathscr K j,
 \qquad A_T=\tfrac12(T+T^*).
\]
There are constants independent of \(L\) such that
\(\|T-I\|_1\le CL\), \(\|T\|\le C\), and
\begin{equation}
 A_T\ge cL^{-2}I,
 \qquad N_{A_T}(t):=\dim\mathbf1_{(-\infty,t)}(A_T)
       \le CL\sqrt t\quad(02d(s)\), using \(|Q(s)-Q(v)|^2\le2|Q(s)|^2+2|Q(v)|^2\).
A fixed enlargement of the endpoint region causes no change in these
bounds. This proves the second comparison.

For completeness, divide any interval of length comparable to
\(b\ge1\) into cells of lengths in \([1/2,1]\), and let \(q_a\)
be the mean of \(Q\) on cell \(a\). The local seminorm controls
\[
 \sum_a\int_{I_a}|Q-q_a|^2
       +\sum_a|q_{a+1}-q_a|^2.
\]
The first bound is the elementary variance identity on a cell; the
second follows by averaging the difference over two adjacent cells,
whose mutual distances are at most \(2\). If the mean on the whole
block vanishes, the discrete Poincare inequality gives
\(\sum_a|q_a|^2\le Cb^2\sum_a|q_{a+1}-q_a|^2\).
Thus \(\|Q\|_2^2\le Cb^2\mathfrak e(Q)\) for fields with zero
mean on each such block. On the whole interval, the wall term controls
the first and last cell means; summing the successive differences then
gives the same estimate with \(b\asymp L\), without any constraints.
The two comparisons now give the gap \(cL^{-2}\).

For \(t\) above a fixed multiple of \(L^{-2}\), choose
\(b=c_1t^{-1/2}\) with \(c_1\) sufficiently small. Vanishing block
means impose at most \(CL/b\) linear constraints, and on their common
kernel the Rayleigh quotient of \(A_T\) is at least \(t\).
The min--max principle proves the counting bound. Below the global
gap the count is zero; the intervening range is handled by changing
the constants. Compressing a positive Hermitian form cannot increase
this count. Finally, if a subspace satisfies
\(\|TQ\|0\) and all sufficiently
large \(L\), there are \(\delta_*>0\), independent of \(L\), and a
real orthogonal projection \(P\) in \(\mathcal X\), of rank \(O_\xi(L)\),
whose range consists of smooth fields supported in
\(|s|\le(1-\delta_*)L\), such that
\begin{equation}
 \|T-I-P(T-I)P\|_1\le\xi L.
 \label{eq:G-10}
\end{equation}
An orthonormal basis of its range can be chosen with every required
fixed derivative bounded by a polynomial in \(L\). The images of these
basis fields under \(h_0\) and \(H_{\rm form}\), regarded as test
densities for the \(ds\) pairing, have the same bounds and smooth odd
reflections at the endpoints. Constants may depend on \(\xi\) and on
the derivative order.
\end{lemma}
\begin{proof}
We first approximate \(\mathscr K\) on \(L^2(ds)\). Truncate the cosine
basis in each unit-scale cell at frequency \(J\). The coefficient
estimate in Lemma~\ref{lem:03-spectral} gives trace-norm error at most
\(CL/J\), since \(\sum_{n>J}(1+n^2)^{-1}\le C/J\) and the sum over
cell separations converges. Approximate the finitely many retained
cosines in each cell by smooth functions compactly supported inside
that cell, with \(L^2\)-error at most \(\eta\), and orthonormalize.
For fixed \(J\) and sufficiently small \(\eta\), their Gram matrices
are uniformly invertible. Replacing a rank-one term \(u\otimes v\)
by its smoothed counterpart changes its trace norm by at most
\(\|u-\widetilde u\|\|v\|+
\|\widetilde u\|\|v-\widetilde v\|\).
Summing the absolutely convergent cell expansions therefore adds at
most \(C_J\eta L\).

Discard cells meeting \(|s|>(1-2\delta_*)L\). There are
\(O(\delta_*L+1)\) such cells, and the total trace-norm contribution
of terms with either index in these cells is of that same order.
Denote the resulting real bump projection by \(Q_b\). Then
\[
 \|\mathscr K-Q_b\mathscr KQ_b\|_1
 \le C\bigl(J^{-1}+C_J\eta+\delta_*\bigr)L+O(1).
\]
Choose \(J\), then \(\eta\), then \(\delta_*\), so the coefficient
on the right is as small as required. There are \(O_\xi(L)\) bumps;
each has uniformly bounded derivatives of every fixed order, with
constants allowed to depend on these fixed choices.

Let \(b_1,\ldots,b_d\) be the orthonormal bump basis and put
\(v_i=j^*b_i\). As a function in \(s\), \(v_i=Gb_i\).
Their Gram matrix in \(\mathcal X\) is uniformly conditioned. The
upper bound follows from \(\|j\|\le C\). For the lower bound, every
\(b\) in the bump span is a permissible smooth field and satisfies
\(\|b\|_{\mathcal X}\le C_\xi\|b\|_2\), by the local upper norm
estimate and its uniformly bounded \(H^1/L^2\) ratio. Duality gives
\[
 \|j^*b\|_{\mathcal X}
 \ge\frac{|\langle j^*b,b\rangle_{\mathcal X}|}
          {\|b\|_{\mathcal X}}
 =\frac{\|b\|_2^2}{\|b\|_{\mathcal X}}
 \ge c_\xi\|b\|_2.
\]
Thus the map \(b\mapsto Gb\) loses no linear independence or uniform
conditioning.

Choose a smooth cutoff equal to one on a neighborhood of the bump
support and zero outside \(|s|\le(1-\delta_*)L\), leaving a separation
at least \(c\delta_*L\) between the bumps and its nonconstant region.
Multiply each \(v_i\) by this cutoff, obtaining \(\widetilde v_i\).
The off-diagonal derivative bounds for \(G\) imply
\[
 \|v_i-\widetilde v_i\|_{\mathcal X}
       \le C_\xi L^C e^{-c\delta_*L}.
\]
To check the endpoint term, the explicit sine ratio gives
\[
 G(s,v)=O\bigl(d(s)e^{-c|s-v|}\bigr)
\]
when \(s\) is close to an endpoint and \(v\) lies in a retained cell.
Every fixed positive-order derivative has the same exponential bound,
without necessarily retaining the factor \(d(s)\). The difference
field vanishes at the endpoint. In particular, the elementary estimate
\[
 \int_0^2\frac{|q(d)|^2}{d}\,dd
 \le 2\int_0^2|q'(d)|^2\,dd,\qquad q(0)=0,
\]
controls its wall contribution.
The upper local norm estimate, or its consequence
\(h_0(q,q)\le C\|q\|_{H^1(ds)}^2\) for such fields, then proves the
claim. The Gram matrix remains uniformly invertible because
\(d=O_\xi(L)\). If \(P_0,P\) are the orthogonal projections onto the
spans of \(v_i,\widetilde v_i\), respectively, the formula
\(V(V^*V)^{-1}V^*\) for a span projection yields
\[
 \|P-P_0\|\le C_\xi L^C e^{-c\delta_*L}.
\]
The operator \(B=j^*Q_b\mathscr KQ_bj\) has both its range and its
adjoint range in \(\operatorname{ran}P_0\); hence \(B=P_0BP_0\).
Writing \(A_1=T-I\), we obtain
\[
 \|A_1-PA_1P\|_1
 \le2\|A_1-B\|_1+2\|P-P_0\|\,\|B\|_1
 \le C\bigl(J^{-1}+C_J\eta+\delta_*\bigr)L+o(1).
\]
This proves \eqref{eq:G-10} after the parameter choices above.

Finally, \(Gb_i\) has controlled derivatives of any fixed order in
the core. Locally subtract the ordinary diagonal logarithm from \(G\);
the remaining divided differences are smooth with uniform derivative
bounds. For the logarithm itself, transfer derivatives to the smooth
bump, and use the exponentially decaying off-diagonal estimates on the
rest. Multiplication by the cutoff preserves these bounds. Passing to
an orthonormal basis uses the inverse square root of a uniformly
conditioned \(d\)-dimensional Gram matrix; its operator norm is bounded,
so pointwise derivative bounds increase by at most \(O(\sqrt d)\).
For the action of \(h_0\), write the test density as the derivative of
\(H_{\rm ref}\) applied to the derivative of the core field. Subtract
its local logarithm and transfer derivatives to that smooth input.
The even endpoint reflections of the kernel give odd reflections of
the resulting test density. Adding the smooth operator \(\mathscr K\)
gives the same assertions for \(H_{\rm form}\).
\end{proof}

The finite core carries almost all of the determinant. Eliminating
its coordinates also leaves a positive interaction on its orthogonal
complement. The following statement quantifies both assertions using
the small-value count, rather than the gap alone.

\begin{lemma}[Determinants and Schur complements]\label{lem:03-schur}\label{E:schur-comparison}
Let \(S=\operatorname{ran}P\), \(R=S^\perp\), and
\[
 T_e=T_{RR}-T_{RS}T_{SS}^{-1}T_{SR}.
\]
The Hermitian part \(\operatorname{Re}T_e=(T_e+T_e^*)/2\) has the gap
and small-eigenvalue count in \eqref{eq:G-8}. Moreover,
\begin{equation}
 \left|\log|\det T_{SS}|-\log|\det T|\right|=o_\xi(L),
 \qquad
 \sum_{\substack{\lambda<1\\
        \lambda\in\operatorname{spec}(\operatorname{Re}T_e)}}
       -\log\lambda=o_\xi(L).
 \label{eq:G-11}
\end{equation}
Eigenvalues are counted with multiplicity. Here the notation means
\[
 \lim_{\xi\downarrow0}\limsup_{L\to\infty}
       \frac{|o_\xi(L)|}{L}=0.
\]
It does not assert an \(o(L)\) error for a fixed \(\xi\). One may take errors bounded respectively by
\(C\xi^{1/5}L\) and
\(C\xi^{1/3}(1+|\log\xi|)L\), after the core approximation is made
with tolerance \(\xi\).
\end{lemma}
\begin{proof}
The gap for \(T_{SS}\) makes it invertible. Set
\[
 Jy=(-T_{SS}^{-1}T_{SR}y,y),\qquad y\in R.
\]
Then \(TJy=(0,T_ey)\), so
\[
 J^*(\operatorname{Re}T)J=\operatorname{Re}T_e,
 \qquad \|Jy\|\ge\|y\|.
\]
The gap follows immediately. If \(E\subset R\) is a subspace on which
the Rayleigh quotient of \(\operatorname{Re}T_e\) is less than \(t\),
the quotient of \(\operatorname{Re}T\) on \(JE\) is also less than
\(t\). Since \(J\) is injective, min--max proves the count. These
identities use Hermitian adjoints; the matrices of the bilinear energy
are, separately, complex symmetric in a real basis.

We make explicit how the counting bound controls logarithms. If the
positive eigenvalues or singular values of any operator under
consideration have counting function \(N(t)\le CL\sqrt t\), then
\[
 \sum_{s_i<\tau}\log\frac{\tau}{s_i}
     =\int_0^\tau\frac{N(t)}t\,dt\le2CL\sqrt\tau.
\]
The lower gap justifies the identity even for the finite-size operators.
Put \(B=T_{SS}\oplus I_R\). Equation~\eqref{eq:G-10} gives
\(\|T-B\|_1\le\xi L\), and \(\|T\|,\|B\|\le C\), whence
\(\|T^*T-B^*B\|_1\le C\xi L\).
The function \(y\mapsto\log\max(y,\tau^2)\) has Lipschitz constant
\(\tau^{-2}\). The trace difference of this function on two positive
operators is bounded by that constant times their trace-norm difference;
this follows first for finite matrices by the eigenvalue variation
inequality and then by finite compressions. Removing the clipping costs
\(CL\sqrt\tau\) by the preceding integral identity. Thus
\[
 \left|\log|\det T|-\log|\det B|\right|
       \le CL\bigl(\sqrt\tau+\xi\tau^{-2}\bigr).
\]
Take \(\tau=\xi^{2/5}\) to obtain the first assertion.

For the second assertion choose \(0<\alpha0\), interlacing
and integration of its counting function below one imply
\[
 \operatorname{Tr}(I-\operatorname{Re}T_e)_+
       \le\operatorname{rank}F+\|E\|_1.
\]
Indeed a rank-\(r\) perturbation changes a count by at most \(r\), and
integration over \((0,1)\) then bounds the contribution of \(F\) by
\(r\); the contribution of \(E\) is bounded by its trace norm.
For \(0<\tau0\), and let \(F-I\) be trace class. Let \(l\in\mathcal X\) be real. Suppose that there are a smooth real vector test \(b(s)\), odd under endpoint reflection, and a smooth real matrix kernel \(\mathscr C(s,v)\) satisfying \(\mathscr C(s,v)=\mathscr C(v,s)^{\mathsf t}\), odd under reflection in either variable, such that
\[
 h_0(l,Q)=\int b(s)^{\mathsf t}Q(s)\,ds,\qquad
 h_0(Q,(F-I)Q)=\iint Q(s)^{\mathsf t}\mathscr C(s,v)Q(v)\,ds\,dv.
\]
Let \(c\in\R\). For the compensated cost
\[
 V(Q)=c+h_0(l,Q)+\tfrac12h_0(Q,(F-I)Q),\qquad
 m_F=c-\tfrac12h_0(l,F^{-1}l),
\]
one has
\begin{equation}
 \log\E_0e^{-V(Q)}
 \le -m_F+CL+\tfrac12\sum_{\lambda<1,\ \lambda\in\operatorname{spec}(F)}
                    -\log\lambda .
 \label{eq:G-18}
\end{equation}
Here and below smooth compensated expressions are evaluated on empirical
primitives by distributional pairing; their individual singular
\(h_0\)-energies are not defined. For the sharp form used below, assume \(\nu_\sigma\asymp m\),
\(g^{-1}+\|F\|+\|l\|_{\mathcal X}\le CL^a\) for fixed \(C,a\), and polynomial physical derivative bounds for \(b\) and \(\mathscr C\) in the convention above. Then for every fixed \(D\), \eqref{eq:G-18} holds with \(CL\) replaced by \(o(L)\) and expectation restricted to \(\|Q\|_{-2,A}\le L^D\). The error is uniform once these bounds and \(D\) are fixed. The constant term \(c\) is unrestricted: its factor \(e^{-c}\) occurs on both sides of the estimate.
\end{lemma}

\begin{proof}
First suppose that all data are generated by finitely many physical sine
tests. Put \(Q_*=-F^{-1}l\), and complete the square before discarding
any positive quadratic terms:
\[
 -V(Q)=-m_F+h_0(Q_*,Q)-\tfrac12\|Q_*\|^2
       -\tfrac12h_0(Q-Q_*,(F-I)(Q-Q_*)).
\]
With \(J=(I-F)_+\), the last term is at most
\(\tfrac12h_0(Q-Q_*,J(Q-Q_*))\). Let \(g_J\) be a real Gaussian
vector of covariance \(J\). Gaussian linearization and
\eqref{eq:G-13} give
\begin{align*}
 \E_0e^{-V(Q)}
 &\le e^{-m_F-\|Q_*\|^2/2}
   \E_{g_J}\!\left[e^{-h_0(g_J,Q_*)}
                      \E_0 e^{h_0(Q_*+g_J,Q)}\right]\\
 &\le e^{-m_F+CL}\E_{g_J}e^{\|g_J\|^2/2}
   =e^{-m_F+CL}\det(I-J)^{-1/2}.
\end{align*}
The cancellation of the mixed and \(Q_*\)-terms in the second line is
exact. The nontrivial eigenvalues of \(I-J\) are precisely those
eigenvalues of \(F\) below one, which proves the formula.

For general smooth data, truncate in the physical \(s\)-sine basis.
The resulting kernels converge with enough derivatives to converge on
every fixed empirical primitive. Their operators converge in trace norm,
and the linear Riesz vectors converge in \(\mathcal X\); these statements
follow by summing the Fourier coefficient tails and using the bounded
embedding \(\mathcal X\hookrightarrow L^2(ds)\). For sufficiently large
cutoffs the gap is at least \(g/2\), so the infima converge. To control
the determinant term, on any finite-dimensional negative-excess subspace
compare a truncated operator to the corresponding compression of \(F\).
For positive operators bounded below by \(g/2\), the integral identity
for \(\log\), or differentiation along their line segment, bounds the
change of log determinant by \(2g^{-1}\) times the trace-norm error.
Interlacing bounds the negative log sum of a compression by that of
\(F\). Fatou's lemma now proves \eqref{eq:G-18}.

The coarse argument also allows the additional cost
\(-t\|Q\|_{-2,A}^2\). Indeed its operator \(B_A\), defined by
\(h_0(Q,B_AQ)=\|Q\|_{-2,A}^2\), has bounded norm and trace
\(O(L)\): factor it through the bounded \(L^2(ds)\) embedding and
sum \((1+(k/A)^2)^{-2}\). Its physical sine truncations also converge
on empirical fields. Choosing \(2t\|B_A\|\le g/2\) preserves a gap.

For the sharpened estimate, take a physical sine cutoff \(K=L^B\).
Smoothness gives errors smaller than any prescribed inverse power of
\(L\), by choosing a sufficiently high fixed derivative order and then
\(B\). Dual Sobolev estimates make the cost error \(o(1)\) on
\(\|Q\|_{-2,A}\le L^D\); trace-norm error divided by the gap and
norm error in the completed square are also \(o(1)\).
The Riesz vectors of the first \(K\) physical sine tests have a Gram
matrix with inverse-polynomial lower bound. To verify this, test the
linear combination of their functionals on the same sine combination
as a primitive. Its \(L^2\)-pairing is the Euclidean squared norm,
whereas the local derivative and boundary estimate for \(h_0\) bounds
its \(\mathcal X\)-norm by a polynomial times that norm. Duality gives
the claimed lower bound. Consequently every polynomially bounded vector
in this finite-dimensional space has a smooth dual test with polynomial
derivative bounds.

Apply \eqref{eq:G-14} in the displayed Gaussian argument when
\(\|g_J\|\le L^{B'}\). Outside this ball use \eqref{eq:G-13}.
After the factor \(e^{\|g_J\|^2/2}\), the normalized tilted Gaussian
has covariance \(J(I-J)^{-1}\), of polynomial norm, in polynomial
dimension. For covariance norm at most \(L^c\) and dimension at most
\(L^c\), its tail outside radius \(L^{B'}\) is bounded, for large
\(B'\), by \(\exp(-L^{2B'-c}/4)\) times an exponential of a fixed
polynomial. Choose \(B'\) to absorb the factor \(e^{CL}\).
This gives the stated \(o(L)\) loss.
\end{proof}

==== END Cylinder-loop-weights-and-planar-nesting//03_reference.tex ====

==== BEGIN Cylinder-loop-weights-and-planar-nesting//04_gas_fluctuations.tex ====
\section{The electric contour integral}\label{sec:04-gas-fluctuations}

The quadratic integration estimate is real. Electric sectors also carry
an oscillatory imbalance factor, and obtaining its sharp contribution
requires a complex displacement. We first implement the deterministic
trial fields by moving the actual particles, while keeping their integer
dimensions fixed. The next section then varies the dimensions through
an explicitly constructed holomorphic approximation.


\subsection{Reference dimensions and compensated quadratic costs}

Freeze the positions, labels, and counts of the explicit particles
\(\zeta\) from Section~\ref{sec:03-gas-mean}. For a pair of
nonnegative integers \(\nu=(\nu_+,\nu_-)\) and an independent
parameter \(M\in\C\), define
\[
 F_\nu(M)=\int A(X_\nu,\zeta)\exp(-\Phi_M(q)),
 \qquad q_\sigma=X_{\nu,\sigma}-\nu_\sigma p_L,
\]
with the two unnormalized reference measures from
Lemma~\ref{lem:03-compensated-sector}. Thus \(F_\nu\)
includes their masses \(J_{\nu_+}(w_0)J_{\nu_-}(w_0)\), but not
the extracted site product, the explicit-particle prefactor, or the
imbalance phase. The energy \(\Phi_M\) retains the prescribed
trial totals \(m'(M)\); the empirical field is centered at the
actual integer dimensions \(\nu\). These two quantities need not
agree. Their independence will be needed for interpolation in the
next section.

Write
\[
 d_e=\ell_e+\frac{(\operatorname{Re}M)^2+n_e^2}{L},
 \qquad H_0=H_{\rm ref}.
\]
We use \(H_0\) in the \(s\)-coordinate on both groups.
Derivatives of primitives are densities or distributions in that
coordinate and are pushed forward by \(W\) when inserted in
\(\mathcal E\).

\begin{proposition}[Bounds at integer reference dimensions]\label{prop:04-integer-bounds}
Fix \( |\operatorname{Re}\eta|\le a_0<\pi\) and
\( |\operatorname{Im}\eta|\le b_0<\infty\), with the parameter
conventions of Proposition~\ref{prop:03-trial-bounds}. For
nonnegative integer dimensions \(\nu_\sigma\), put
\[
 \mathcal L_{\nu,j}(M)
 =\log|e^{i(\eta+\pi f+2\pi j)M}F_\nu(M)|,
 \qquad v=\operatorname{Im}M.
\]
The following estimates hold.
\begin{enumerate}
\item If \(M\) is real and \(0\le\nu_\sigma\le C e^{\gamma L}\),
then, without any restriction on the explicit counts,
\[
 \mathcal L_{\nu,j}(M)\le C_hL+C_hn_e-c_hd_e.
\]
This bound is independent of the integer alias \(j\).
\item Suppose \(n_e\) is polynomially bounded, as in part 2 of
Proposition~\ref{prop:03-trial-bounds}. In the same dimension range,
for every real \(v\) and integer \(j\),
\begin{equation}
 \mathcal L_{\nu,j}(M)\le C_h L+C_h n_e-c_h d_e-2\pi jv
                 +C_h(v^2+(1+f)|v|).
 \label{eq:G-15}
\end{equation}
\item Assume the common hypotheses of parts 3 and 4 of
Proposition~\ref{prop:03-trial-bounds}, polynomial bounds on
\(n_e,|j|,|M|,\ell_e\), and \(\nu_\sigma\asymp m\).
Choose the imaginary imbalance and trial density from part 3 of
that proposition. Then
\begin{equation}
 \mathcal L_{\nu,j}(M)\le C_h L+C_h n_e-c_h d_e-c_\delta U^2 L+o(L).
 \label{eq:G-16}
\end{equation}
If \(U\le U_0\), the choice from part 4 instead gives
\begin{equation}
 \mathcal L_{\nu,j}(M)\le\frac{3\gamma L}{8}
 -2L\frac{\operatorname{Re}(\eta^2)}{4\kappa}
 +C\varepsilon L+o(L)+C_h n_e-c_{h,\varepsilon}d_e.
 \label{eq:G-17}
\end{equation}
Here \(\delta\) can be taken sufficiently small without degrading
the displayed damping and per-particle constants, and \(U_0\) may
then be fixed arbitrarily, as in part 4 of that proposition.
\end{enumerate}
The estimates are uniform once the parameters in the trial bounds
and the polynomial size powers are fixed.
\end{proposition}

We prove Proposition~\ref{prop:04-integer-bounds} using the real integration bound of Lemma~\ref{lem:04-quadratic} and the particle deformation constructed below.

Throughout this section, a polynomial bound means a bound by a fixed power of \(L\), whose exponent may depend on previously fixed parameters. It never refers to the exponentially large reference dimensions. Energy expressions use the complex bilinear extension of \(h_0\), whereas norms and adjoints use the Hermitian convention. Angle brackets against physical test functions denote distributional pairing.

For smooth real Hilbert fields \(Y\), and a real smooth primitive \(Z_0\) compactly supported in the core \(|s|\le(1-\delta_c)L\) for some fixed \(\delta_c>0\), use the functional
\[
 C_z(Y)=\mathcal E\big(m'p_L+(Y+iZ_0)',\zeta\big)-i(\eta+\pi f+2\pi j)M
       =c_z+h_0(u_z,Y)+h_0(Y,TY)/2
\]
with complex bilinear convention; write \(C_0\) for \(Z_0=0\), so \(C_z(Y)=C_0(Y+iZ_0)\). In \eqref{eq:G-16},\eqref{eq:G-17}, the pushforward of \(Z_0'\)
by \(W\) is the density \(z_\sigma+\sigma v p_L\) chosen in
parts 3 and 4 of Proposition~\ref{prop:03-trial-bounds}. Its required fixed-order derivative bounds are polynomial. Everywhere in the polynomial regimes \(u_z\) is a bounded Hilbert vector of polynomial norm, obtained by duality from a smooth odd-reflected test in \(s\) with polynomial fixed-order derivatives. Indeed use Lemma~\ref{lem:03-source-regularity}, and the action of \(H_{\rm form}\) on smooth core fields (localized diagonal log convolved by their smooth derivatives, as in the core-subspace proof). Also \(c_z\) is polynomially bounded except for the unchanged explicit-explicit constant \((\zeta,H\zeta)_{\rm off}/2\), whose real part is at least \(-{\rm poly}(L)\); the equality in polynomial size for \(c_z\) here uses \(|j|\) polynomial as well. Without polynomial truncation we will only use \(Z_0=0\); the forms are still continuous at fixed sizes and generic frozen positions. The infimum
\[
 m_z=\inf_{Y\ {\rm real\ in}\ \mathcal X}\operatorname{Re} C_z(Y)
\]
satisfies the indicated trial lower bounds (with \(B(x)\) replaced using \eqref{eq:G-3} by positive damping in \(d_e\), minus a constant). Indeed apply them first on dense smooth Dirichlet fields, then use continuity. Every such variation preserves the two specified totals. Primitives correspond to \(s\)-densities here, so integrable square-root singularities at physical endpoints do not impede the energy bounds.

The real reference bound already handles parts 1 and 2. For the sharp
estimates in part 3, the imaginary trial density and the relative
determinant must both be realized inside the original particle integral.
The following deformation does this while leaving its integer dimensions
unchanged.

\begin{lemma}[Particle contour deformation]\label{lem:04-contour}\label{E:particle-transport}
Under the hypotheses of part 3 of
Proposition~\ref{prop:04-integer-bounds},
choose the core projection \(P\) of \eqref{eq:G-10}, with
\(S=P\mathcal X\), \(R=S^\perp\), and Schur complement \(T_e\).
For every prescribed fixed tail exponent, a sufficiently large fixed
\(D\) gives an exact contour deformation of the reference integral.
On \(\|Q\|_{-2,A}\le L^D\), its compensated cost is
\(C_0(Q+a(Q))-h_0(Q,Q)/2+o(1)\), where
\[
 Q+a(Q)=iZ_0+Q_R+T_{SS}^{-1/2}Q_S
             -T_{SS}^{-1}Pu_z-T_{SS}^{-1}T_{SR}Q_R.
\]
The extra Jacobian, after the one-coordinate reference-density
cancellation, has modulus
\((1+o(1))|\det T_{SS}|^{-1/2}\). The full cost \(\operatorname{Re}C_0(Q+a(Q))\) has Hessian
\(I\) on \(S\) and \(\operatorname{Re}T_e\) on \(R\), and its
infimum is at least \(m_z\). The complementary contribution is smaller
than the prescribed stretched exponential.
\end{lemma}

\begin{proof}
We first construct an affine path of smooth fields with a positive real
Hessian. We then implement this path by particle displacements, compute
the compensated cost and the remaining Jacobian, and estimate the region
where a cutoff switches off the deformation.

\paragraph{The affine path and its real cost.}
Choose \(P,S,R,T_e\) as in \eqref{eq:G-10},\eqref{eq:G-11} with \(\xi\) sufficiently small (here \(R=S^\perp\)). After reducing \(\delta_c>0\), all core basis and shift functions are supported in \(|s|\le(1-\delta_c)L\). The frozen affine shifts \(Q\mapsto Q+a(Q)\) along the successive stages are:
\begin{enumerate}
\item Add \(i\tau Z_0\), \(0\le\tau\le1\).
\item Keeping \(i Z_0\) added, replace \(Q_S=P Q\) by \(Q_S-\tau(T_{SS}^{-1}P u_z+B_s Q_R)\), where \(B_s=T_{SS}^{-1}T_{SR}\) here denotes a block map, \(Q_R=(I-P)Q\).
\item In the fully shifted coordinates of stage 2 replace the free input \(Q_S\) of that expression by \(D_\tau Q_S\), with
\[
 D_\tau=((1-\tau)I+\tau T_{SS})^{-1/2}.
\]
\end{enumerate}
Thus the output primitive in the last stage equals \(i Z_0+Q_R+D_\tau Q_S-T_{SS}^{-1}P u_z-B_s Q_R\).
Use a single smooth path parameter in \([0,1]\) by flattening at stage boundaries. Matrix functions and their required derivatives here have polynomial norms by \eqref{eq:G-8}. For instance the symmetric inverse square root is given by the positive-resolvent integral \(\pi^{-1}\int_0^\infty y^{-1/2}(yI+(1-\tau)I+\tau T_{SS})^{-1}dy\), using accretivity (its identity follows from the scalar formula and derivatives on Jordan blocks). All nontrivial input uses of \(Q\) in \(a\) are through smooth dual tests with polynomial bounds.

The real Hessian of the full cost \(\operatorname{Re}C_0(Q+a(Q))\)
has an inverse-polynomial gap throughout this path. To check this
without treating complex changes as real orthogonal maps, write
\(A_S=T_{SS}\), \(C_S=T_{SR}\), and \(B_s=A_S^{-1}C_S\).
The linear part of stage 2 sends \((s,r)\) to \((s-\tau B_sr,r)\).
Direct expansion of the complex bilinear quadratic form gives
\[
 T[(s-\tau B_sr,r),(s-\tau B_sr,r)]
 =T[(s,(1-\tau)r),(s,(1-\tau)r)]
       +(2\tau-\tau^2)T_e[r,r].
\]
For real \(s,r\), both terms have positive real part, and the
coefficients on \(r\) add to one. Stage 1 leaves the Hessian unchanged.
At stage 3 it is block diagonal, with blocks \(T_e\) and
\(A_SH_\tau^{-1}\), where \(H_\tau=(1-\tau)I+\tau A_S\).
The identity
\[
 H_\tau^*\operatorname{Re}(A_SH_\tau^{-1})H_\tau
       =(1-\tau)\operatorname{Re}A_S+\tau A_S^*A_S
\]
and the polynomial norms and inverse norms of these matrices give the
required gap. Here \(D_\tau\) commutes with \(A_S\) and is symmetric,
so \(D_\tau^{\mathsf t}A_SD_\tau=A_SH_\tau^{-1}\).

All residual kernels, linear dual tests, and their path derivatives
have polynomial physical derivative bounds, by the core-subspace
construction. Their trace norms are polynomial as well, since the
change from \(T-I\) has polynomial norm and rank \(O(L)\). At the
endpoint the Hessian is \(I\oplus\operatorname{Re}T_e\).
For the infimum comparison, fix a real \(r\). The stationary point of
\(C_z(s+r)\) in the complex \(S\)-variable is
\(s_*=s_1+is_2=-A_S^{-1}(Pu_z+C_Sr)\). Completing the complex square
at \(s_*\) yields
\[
 \operatorname{Re}C_z(s_*+r)
 =\operatorname{Re}C_z(s_1+r)
       +\tfrac12h_0(s_2,\operatorname{Re}A_Ss_2)
 \ge m_z.
\]
The remaining endpoint input contributes \(\|s\|^2/2\). This proves
the infimum claim even though the final contour need not have the
initial imaginary trial profile.

\paragraph{Realizing the path as an exact contour deformation.}
Choose a \(C^\infty\) function \(\chi:[0,\infty)\to[0,1]\), equal
to one on \([0,1]\) and zero on \([4,\infty)\). If the flattened
path just constructed is \(a_\tau\), use the particle-dependent
parameter
\[
 \tau(Q)=\chi(\|Q\|_{-2,A}^2/L^{2D}),
 \qquad a(Q)=a_{\tau(Q)}(Q),
\]
and homotope from zero by replacing \(\tau(Q)\) with
\(u\tau(Q)\), \(0\le u\le1\). The empirical primitive, as a function
of a particle at \(s_i\), has first derivative \(-\delta_{s_i}\)
and second derivative \(\delta'_{s_i}\). Both belong continuously
to the physical \(H^{-2}\) space: in the sine basis their squared
norms are bounded respectively by constant multiples of
\(A^{-1}\sum_k(1+(k/A)^2)^{-2}\) and
\(A^{-1}\sum_k(k/A)^2(1+(k/A)^2)^{-2}\).
Thus the deformation is \(C^2\) in the real particle variables,
which is sufficient for Stokes' theorem.

Move a particle of label \(\sigma\) at \(s\) to
\[
 \Psi_\sigma(s;Q)=s+\Delta_\sigma(s;Q),\qquad
 \Delta_\sigma(s;Q)=-\frac{a_\sigma(Q)(s)}
                              {\nu_\sigma\lambda_L(s)}.
\]
For any fixed derivative order, on the support of this motion,
\[
 \|\partial_s^k\Delta\|_\infty
 \le L^{C_{D,k}}e^{-\gamma\delta_c L}.
\]
Indeed \(a\) and its required derivatives are polynomial on
\(\|Q\|_{-2,A}\le2L^D\), and
\(\nu_\sigma\lambda_L(s)\ge c e^{\gamma\delta_cL}\) in its fixed
core. This estimate concerns spatial differentiation at frozen \(Q\).

For completeness, the exact change of contour is a change of an
integration chain, not an assertion that the map \(\Psi\) is
holomorphic or globally injective. Partition every original real
coordinate into a slightly enlarged core interval and its complement,
and fix the complementary coordinates. The factor \(A\) is independent
of all active core coordinates. The remaining density times the
coordinate volume form is a holomorphic top form \(\Omega\) in a
neighborhood of the active intervals and their images. Its reference
part is a squared Vandermonde times the holomorphic one-body density,
so coincidences are zeros rather than poles. The regular exponent is
holomorphic there by the core regularity estimates of the preceding
section. Hence \(d\Omega=0\). On a side face of the homotopy prism,
the corresponding coordinate lies at an interval endpoint and its
displacement vanishes in a neighborhood; its differential is zero
on that face, so the pullback of \(\Omega\) vanishes. Stokes' theorem
therefore equates the two integrals. Apply this argument for almost
every frozen configuration and sum the finitely many partitions.
The estimates below give absolute integrability; configurations with
coincident explicit particles may be omitted as null sets.

\paragraph{The compensated cost after displacement.}
The contour identity is exact. We now estimate its pullback uniformly on \(\|Q\|_{-2,A}\le 2L^D\). Write \(X=X_\nu\), \(\Psi={\rm Id}+\Delta\). Separate the fixed-field single-coordinate Jacobians \(1+\Delta_\sigma'\). The diagonal omission in the Vandermonde log change contributes half the diagonal of
\[
 H_0(\Psi(s),\Psi(v))-H_0(s,v)
\]
per particle (kernels here in \(s\), with appropriate labels). Its exponential cancels the transformed \(2\sin\theta\,\theta'\) ratio and these single-coordinate Jacobians exactly. The difference kernel is defined by the log of the cos-difference ratio, with analytic branch near 1, and its required derivatives are exponentially small with odd/even reflections as appropriate. Indeed near a moved diagonal use the divided-difference bounds in the analytic core and small derivatives of \(\Delta\); off diagonal use the sine separation bounds with displacements relative to \(\theta'\). Thus the resulting base negative-log change is half the difference kernel on \(X,X\). Combining with \(\Phi_M\) evaluated after movement and including the phase in the cost, one obtains uniformly up to \(o(1)\)
\begin{equation}
 C_0(Q+a(Q))-\tfrac12 h_0(Q,Q).
 \label{eq:G-19}
\end{equation}
This notation always means the compensated smooth expansion, not the two singular forms separately on an empirical field.

We justify the error in \eqref{eq:G-19} by a single scalar Taylor
expansion, which avoids any dimension-dependent multivariable
remainder. Freeze \(Q\) and \(a\), and replace \(\Delta\) by
\(z\Delta\). On the disk
\( |z|\le R_L=\exp(\gamma\delta_cL/4)\), all required spatial
derivatives of \(z\Delta\) remain exponentially small. Core
analyticity and the divided-difference estimates keep every logarithmic
ratio in its branch near one, including at coincident particle
positions. After removing the unchanged explicit--explicit constant,
the logarithmic cost on this disk is bounded by
\(L^C e^{2\gamma L}\): every term is linear or quadratic in measures
of total variation \(O(e^{\gamma L})\), with polynomially bounded
kernels on the disk. Cauchy's estimate bounds the remainder after
order \(K\), at \(z=1\), by
\[
 C L^C\exp\!\left(\left[2\gamma-
              (K+1)\gamma\delta_c/4\right]L\right).
\]
Choose the fixed integer \(K\) so that \(K+1>8/\delta_c\).

Each retained coefficient is a polynomial in the components of the
frozen displacement in the real span of the core basis and \(Z_0\).
It is therefore enough to compute it for real displacements and extend
the resulting polynomial identity to complex coefficients. For real
displacements, the pushforward of the smooth mean has coefficients
\[
 d_k=\frac{(-\partial_s)^k}{k!}
               (\nu_\sigma\lambda_L\Delta_\sigma^k).
\]
Here \(d_1=a_\sigma'\), while all fixed-order derivatives of \(d_k\),
\(k\ge2\), are exponentially small. The useful cancellation is
\(\nu_\sigma\lambda_L\Delta_\sigma^k=-a_\sigma\Delta_\sigma^{k-1}\);
using the total variation of the mean alone would not prove this
assertion near the edge of the core. All these densities are supported
in the core. For the centered distribution \(q=Q'\), every
positive-order push coefficient, tested against a smooth \(f\), is
\(\langle q,\Delta^k f^{(k)}\rangle/k!\). Integration by parts and
duality with \(\|Q\|_{-2,A}\) make it exponentially small, with all
fixed test derivatives needed below.

These rules give the substitution \(q\mapsto q+a'\) directly for the
smooth kernel \(K\) and the smooth linear term in \(\Phi_M\).
For the base-kernel difference, its \(q,q\) term is exponentially
small by the ratio-kernel bounds and Sobolev duality in both variables.
Terms containing means use \(H_0(\lambda_Lds)=0\). The one-mean
coefficients reduce to a pushed \(q\) paired with \(H_0d_k\),
and the two-mean coefficients pair positive-order \(d_k,d_l\).
These identities first hold for real diffeomorphisms. Differentiation
is legitimate because the changed mean has smooth core density;
local convolution with the diagonal logarithm transfers derivatives
to that density. It gives polynomial smooth norms for \(H_0d_k\).
Consequently the base half-difference is
\[
 (q,H_0a')+\tfrac12(a',H_0a')+o(1).
\]
Polynomial continuation of the finite jets, followed by the scalar
remainder estimate, proves the same identity for complex displacements.
No pushforward density on the real line is asserted for a complex
contour.

Explicitly \(\Phi_M(q)\) is a constant plus its smooth linear test on \(q\) plus \((q,K q)/2\). Hence replacing \(q\) by \(q+a'\) there (and subtracting the phase log) gives exactly \eqref{eq:G-19} minus \((q,H_0 a')+(a',H_0 a')/2\), without ever needing a bare self energy of \(q\). All the Taylor estimates are uniform even close to particle coincidences: on the growing scalar disk the shifted cos-difference over the original one in a same-label base ratio remains near 1, by the simultaneous divided difference (the same frozen shift function is used on the two arguments and has tiny spatial derivatives). The huge constant from a possible very close explicit-explicit pair never participates in these jets.

\paragraph{The Jacobian from the dependence on the configuration.}
The preceding cancellation includes each single-coordinate Jacobian at
fixed \(Q\). It remains to estimate the part arising from
the dependence of \(a\) on the whole configuration. On the full-shift
region write
\[
 D_Qa=\sum_{b=1}^{r}e_b\otimes\phi_b,\qquad r=\dim S=O(L),
\]
where \(\phi_b\) are smooth physical dual tests, and put
\(J_i=1+\partial_s\Delta_{\sigma_i}(s_i;Q)\). The sign
\(\partial_{s_i}Q=-\delta_{s_i}\) cancels the minus sign in
\(\Delta=-a/(\nu\lambda_L)\). The matrix determinant lemma gives
exactly, after factoring \(\prod_iJ_i\), the reduced determinant
\[
 \det(I+V),\qquad
 V_{bb'}=\sum_i
 \frac{\phi_{b,\sigma_i}(s_i)e_{b',\sigma_i}(s_i)}
      {J_i\nu_{\sigma_i}\lambda_L(s_i)}.
\]
For the summand test \(g=\phi_b e_{b'}/(J\nu\lambda_L)\), centered
quadrature is quantified by
\[
 \left|\sum_{i:\sigma_i=\sigma}g(s_i)
                     -\nu_\sigma\int\lambda_Lg\,ds\right|
 =|\langle Q_\sigma,g'\rangle|
 \le \|Q_\sigma\|_{-2,A}\|g'\|_{H^2}
 \le L^C e^{-cL}.
\]
The support of \(e_{b'}\) is inside the core, so all norms on the
right have the required exponential factor. Replacing \(J\) by one
in the mean term has the same error. Thus
\(V_{bb'}=\int\phi_b e_{b'}\,ds+O(L^Ce^{-cL})\), with the
labels summed. On \(S\), the derivative of the endpoint affine map
is \(D_1\); hence \(I+V=D_1+E_L\),
\(\|E_L\|\le L^Ce^{-cL}\). Since \(\|D_1^{-1}\|\) is polynomial,
\[
 \big|\log\det(I+D_1^{-1}E_L)\big|
 \le 2r\|D_1^{-1}E_L\|=o(1).
\]
The principal matrix square root satisfies
\(|\det D_1|=|\det T_{SS}|^{-1/2}\), giving the asserted modulus.

\paragraph{The transition region and the outer tail.}
In the transition region, differentiation of the cutoff adds a single
rank-one term with output \(\partial_\tau a_\tau\in S+\C Z_0\).
The total rank is at most \(r+1\). The cutoff differentials on point
deltas have polynomial bounds on this region, and every entry of the
reduced matrix is at most \(e^{C_DL}\), even if bounded without
centering. Hadamard's inequality therefore bounds its modulus by
\(\exp(O_D(L^2))\). This estimate uses the reduced matrix of size
\(O(L)\), never the exponentially large particle matrix. Outside
the cutoff region the extra determinant is one.

Finally fix a path parameter \(\tau\). Its compensated real cost has
inverse-polynomial gap and polynomial linear and trace-norm bounds,
with constant bounded below by minus a polynomial. These bounds are
independent of the cutoff exponent \(D\). Choose \(E\) once so large
that subtracting \(L^{-E}\|Q\|_{-2,A}^2\) preserves half the gap.
Lemma~\ref{lem:04-quadratic} and the elementary inequality
\(\mathbf1_{\{\|Q\|>L^D\}}\le
 \exp(L^{-E}\|Q\|^2-L^{2D-E})\) yield
\[
 \E_0[\mathbf1_{\{\|Q\|_{-2,A}>L^D\}}e^{-V_\tau(Q)}]
 \le \exp(-L^{2D-E}+L^{E_1}),
\]
with \(E_1\) fixed independently of \(D\). For example, the
negative log sum is at most a polynomial because
\(-\log\lambda\le g^{-1}(1-\lambda)\) on \([g,1]\), while the
trace norm and inverse gap are polynomial.

The parameter in the transition region depends on \(Q\), so a
fixed-parameter estimate alone is insufficient. On
\(\|Q\|_{-2,A}\le2L^D\), the physical kernel bounds give
\(|\partial_\tau V_\tau(Q)|\le L^{C+2D}\), after factoring out
any unchanged explicit-pair constant. A parameter grid of mesh
\(L^{-C-2D-1}\) approximates every such cost within one. Bound its
exponential by \(e\) times the sum over grid costs and apply the
preceding tail estimate at every grid point. The additional logarithmic
cost is only \(O_D(\log L)\). The undeformed outer tail already
has fixed parameter zero. Combining this estimate with the transition
Jacobian bound and bounded reference masses gives
\[
 \exp\{-L^{2D-E}+L^{E_1}+O_D(L^2)\}.
\]
Increasing \(D\) makes this smaller than any prescribed
\(\exp(-L^{E_2})\). The factor \(A\), which depends only on fixed
coordinates, remains in \([0,1]\) in all these absolute estimates.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:04-integer-bounds}]
For the estimates without a shift, take absolute values, use \(0\le A\le1\),
and apply Lemma~\ref{lem:04-quadratic} with \(F=\operatorname{Re}T\)
and infimum \(m_0\). The negative log sum is \(O(L)\) by
\eqref{eq:G-8} and the trace bound; the reference masses are bounded
above. The real trial estimate and \eqref{eq:G-5}, with
\eqref{eq:G-3}, prove the first two assertions.

On the full-shift range use the sharp version of \eqref{eq:G-18}, \eqref{eq:G-11}, and the infimum comparison to \(m_z\). Including the full Jacobian and both \(J_{\nu_\sigma}(w_0)\) gives the log bound there
\[
 -m_z+(7\gamma/8)L-\gamma L/2+o_\xi(L)+o(L)
\]
by \eqref{eq:G-9}. Taking \(\xi\) small, then applying the deterministic lower bounds on \(m_z\), and the separate negligible bad-region estimate (choose its power after the polynomial restrictions), proves \eqref{eq:G-16} and \eqref{eq:G-17}.
\end{proof}

==== END Cylinder-loop-weights-and-planar-nesting//04_gas_fluctuations.tex ====

==== BEGIN Cylinder-loop-weights-and-planar-nesting//04b_interpolation.tex ====
\section{Continuing the imbalance sum}\label{sec:E-interpolation}

The number of reference particles must next be varied with \(M\).
There is no integral of noninteger dimension to which Cauchy's theorem
could be applied directly. We construct a holomorphic approximation
from polynomial moments of the integer-dimensional integrals. The
construction has two outputs: arbitrarily small absolute error on the
integer grid, and a coarse growth bound on a complex dimension polydisk.
The grid-transfer lemma at the end of the section uses both outputs to
carry the sharp integer-dimension estimates to the complex dimensions
needed for the imbalance contour shift.

\begin{proposition}[Interpolation of the reference dimension]\label{prop:04-interpolation}
Fix polynomial bounds on \(n_e,\ell_e,|M|\). For every fixed \(b>0\)
there is a jointly holomorphic function \(\widehat F(\nu,M)\), with
\(\nu=(\nu_+,\nu_-)\in\mathbb C^2\), on a neighborhood of the indicated \(M\)-disk and the polydisk
\(|\nu_\sigma-m|\le\sqrt m\), such that
\begin{equation}
 |\widehat F(\nu,M)|\le \exp(L^C),\qquad
 |\widehat F(\nu,M)-F_\nu(M)|\le e^{-L^b}
       \quad\text{for }\nu\in\mathbb Z_{\ge0}^2\text{ in the polydisk}.
 \label{eq:G-20}
\end{equation}
Here \(C\) is fixed after the truncation bounds and \(b\) are fixed.
The bounds are uniform in the real explicit-particle positions on their
labeled lines. The integrals include their reference masses
\(J_{\nu_+}(w_0)J_{\nu_-}(w_0)\), but no external phase factor.
\end{proposition}

The next three subsections prepare the proof of
Proposition~\ref{prop:04-interpolation}: polynomial approximation,
a finite endpoint-subset expansion, and continuation of the retained
factorial moments. The proof is assembled after these constructions.

\subsection{Polynomial approximation under the integer laws}

We first make all errors that will be estimated under a positive
integer-dimensional probability law. In \(\Phi_M\), truncate the
physical sine expansions of the linear dual test and of the primitive
kernel \(\mathscr K\). The degrees can be polynomial in \(L\), with
error \(e^{-L^{B_0}}\) in any prescribed fixed smooth norm. Indeed a
Gevrey bound \(A^k(k!)^P\) and \(k\) integrations by parts bound the
\(n\)-th Fourier coefficient by \(C A^k(k!)^P n^{-k}\);
optimizing \(k\) gives \(C\exp[-c(n/A)^{1/P}]\). The Gevrey scale
here is polynomial in \(L\). Fixed Fourier cutoffs preserve
holomorphy in \(M\).

The total variation of each particle or mean measure is
\(O(e^{\gamma L})\), so the exponent error is bounded by
\(e^{-L^{B_0}+O(L)}\) uniformly on configurations. Independently of
this Fourier cutoff, Lemma~\ref{lem:04-quadratic} bounds the absolute
expectation of the original exponential by \(\exp(L^{C_0})\), using
the gap, polynomial norm of the linear term, and polynomial lower
bound on the real constant. Since
\(|e^{-z}-1|\le |z|e^{|z|}\), choosing \(B_0\) larger than the
required precision and \(C_0\) makes the error in expectation
negligible. Multiplication by \(A\in[0,1]\) does not alter this bound.
Thus this first precision choice precedes all moment constructions.

Orthogonalize the Riesz vectors of the retained physical sine tests.
The subspace and its real orthogonalization are independent of
\(M,\nu\). Let \(Y\in\R^d\) be the vector of resulting tests on
\(Q\). The truncated exponential is
\[
 g(Y)=\exp(-c_1-l^{\mathsf t}Y-\tfrac12Y^{\mathsf t}DY),
 \qquad I+\operatorname{Re}D\succeq\delta I,
 \qquad \delta=L^{-r_0},
\]
with \(D\) complex symmetric, \(d,\|D\|,\|l\|\) polynomial, and
\(\operatorname{Re}c_1\ge-L^C\). Operator-norm convergence of the
truncated kernel preserves this gap. Each coordinate of \(Y\) is a
centered sum of a physical cosine polynomial \(f(s)\), with
polynomial degree and coefficients. The inverse-polynomial Gram
bound established in Lemma~\ref{lem:04-quadratic} justifies this last
assertion after orthogonalization.

\begin{lemma}[Weighted polynomial approximation]\label{lem:04-hermite}
For the preceding Gaussian exponential, choose \(a=1-\delta/2\).
Given any fixed precision exponent, there is a polynomial \(P(Y)\)
of polynomial total degree \(K_1\) such that
\[
 \sup_{y\in\R^d}e^{-a|y|^2/2}|g(y)-P(y)|\le e^{-L^{B_1}}.
\]
Its coefficients are holomorphic in \(M\) and bounded by an
exponential of a fixed polynomial. The degree and coefficient bounds
are fixed before any endpoint expansion.
\end{lemma}

\begin{proof}
Use the Gaussian probability measure
\(d\mu_a=(a/\pi)^{d/2}e^{-a|y|^2}dy\), and its orthonormal Hermite
polynomials defined by
\[
 e^{\sqrt{2a}z^{\mathsf t}y-z^{\mathsf t}z/2}
       =\sum_\alpha H_\alpha(y)\frac{z^\alpha}{\sqrt{\alpha!}}.
\]
Gaussian integration proves orthogonality. For completeness,
polynomials are dense: an orthogonal complement would have an entire
Gaussian-weighted Laplace transform with every derivative at zero
equal to zero; Fourier uniqueness then makes the complement zero.
The integrability needed here follows from Gaussian decay.

Let \(c_\alpha=\int gH_\alpha\,d\mu_a\). Completing the square gives
\begin{align*}
 \sum_\alpha c_\alpha\frac{z^\alpha}{\sqrt{\alpha!}}
 &=e^{-c_1}\det(I+D/(2a))^{-1/2}\\
 &\quad\times\exp\!\left\{
 \tfrac12l^{\mathsf t}(2aI+D)^{-1}l
 -\sqrt{2a}z^{\mathsf t}(2aI+D)^{-1}l
 -\tfrac12z^{\mathsf t}D(2aI+D)^{-1}z\right\}.
\end{align*}
The square root is the analytic branch specified by the Gaussian
integral. Accretivity ensures its existence and polynomial inverse
norms. Set \(B=D(2aI+D)^{-1}\). The identity
\[
 (2aI+D)^*(2aI+D)-D^*D
       =4a(aI+\operatorname{Re}D)\succeq2a\delta I
\]
shows \(\|B\|\le1-L^{-C}\). The other factors in the generating
function have size at most \(\exp(L^C)\). Therefore, for some
\(r=1+L^{-C_1}>1\), integrating the squared generating function at
\(rz\) against \(\pi^{-d}e^{-|z|^2}\,dz\) on \(\C^d\) gives
\[
 \sum_\alpha r^{2|\alpha|}|c_\alpha|^2\le e^{L^C}.
\]
This follows from monomial orthogonality; convergence follows from
\(r^2\|B\|<1\), with inverse-polynomial margin.

The Hermite generating identity also gives Mehler's diagonal formula
\cite[equation~18.18.28]{DLMF18}
\[
 \sum_\alpha t^{|\alpha|}|H_\alpha(y)|^2
 =(1-t^2)^{-d/2}
       \exp\!\left(\frac{2at}{1+t}|y|^2\right),\quad 0K_e}\frac{(2\beta_L)^k}{k!}.
\]
If \(K_e+1\ge4e\beta_L\), this tail is at most
\(2e^{L^{C_2'}}(2e\beta_L/(K_e+1))^{K_e+1}\).
Choose the polynomial \(K_e\) large enough to attain the desired
precision. Crucially, this does not increase \(K_1\): the endpoint
subset size and the global polynomial degree are separate orders.

\subsection{Continuation of the retained factorial moments}

The determinant formulas here are the standard consequences of
Andr\'eief's identity for an orthogonal-polynomial ensemble
\cite[equations (1.3), (3.1), and Section 5]{TracyWidom1998}.
We give the expansion because the subsequent
continuation uses its precise finite degrees.

We describe one label, with integer dimension \(n\), later continued
to \(\nu\). Introduce formal source variables \(\mathbf t=(t_j)\),
and \(G(x)=\sum_jt_jg_j(x)\), where the \(g_j\) are the global
polynomial tests. The order-\(k\) factorial density, tilted by
\(e^{\sum_iG(x_i)}\) and without the divisor \(k!\), is
\[
 \mathcal D_n(\mathbf t)
       \det[\mathcal C_n(x_h,x_{h'};\mathbf t)]_{h,h'\le k},
\]
where
\[
 \mathcal D_n=\det_{\Pi_n}(\Pi_ne^G\Pi_n),\qquad
 \mathcal C_n=\Pi_n(\Pi_ne^G\Pi_n|_{\Pi_n})^{-1}\Pi_ne^G.
\]
These identities are formal at \(\mathbf t=0\), where the inverse
exists. To prove them, insert an additional factor
\(\prod_i(1+h(x_i))\) in the tilted expectation. Integration of the
two polynomial evaluation determinants gives the Gram determinant
with weight \(e^G(1+h)\). Factor out its value at \(h=0\), and
expand the remaining determinant in its minors. The coefficient of
\(h(x_1)\cdots h(x_k)\) is exactly the formula above.

For a labeled endpoint set write
\[
 I(B')=\sum_{|\alpha|\le K_1}
        c_\alpha(B';M,\nu)S_1^\alpha.
\]
The coefficients are holomorphic in \(M,\nu\): they are obtained
from \(P\), centerings linear in \(\nu\), and sums or products of
bounded real endpoint functions evaluated at the specified points.
Their magnitudes are at most \(e^{L^C}\). For each split of
\(|B'|=k\) into \(k_+,k_-\), integrate these coefficients against
\(\partial_{\mathbf t}^{\alpha}|_{\mathbf t=0}\) of the product
of the two tilted factorial densities, and divide by \(k_+!k_-!\).
This is precisely the expected subset sum. Only source degrees at
most \(K_1\) are required, regardless of \(k\le K_e\).

In the polynomial-index basis, multiplication by \(x\) is the
tridiagonal Jacobi matrix \(J\), with off-diagonal coefficients
\[
 a_i=\frac{i}{\sqrt{(2i-1)(2i+1)}}
       =(4-i^{-2})^{-1/2}.
\]
At source degree \(r\), multiplication by \(e^G\) has bandwidth at
most \(rd_g\). In
\[
 \log\mathcal D_n
   =\operatorname{Tr}_{\Pi_n}
         \log(I+\Pi_n(e^G-1)\Pi_n),
\]
a closed index path of degree \(r\) has total displacement at most
\(rd_g\). If its starting index lies farther than \(rd_g\) below
\(n\), it never meets the upper projection boundary; every inserted
\(\Pi_n\) can be removed. The complete coefficient then equals
that of \(\log e^G=G\), and vanishes for \(r\ge2\). This argument
is an identity of formal series, including the cancellations among
the finitely many logarithm terms. All higher cumulants therefore
involve only indices within \(2K_1d_g+O(1)\) of \(n\).

The kernel correction has the same property. Exactly,
\(\mathcal C_n\Pi_n=\Pi_n\) and
\((I-\Pi_n)\mathcal C_n=0\), so
\(\mathcal C_n-\Pi_n\) has rows below \(n\) and columns at least
\(n\). Its degree-\(r\) inverse-series coefficient has bandwidth
\(rd_g\), forcing both indices within \(rd_g\) of \(n\).
All these assertions concern polynomial kernels coefficient by
coefficient; pointwise evaluation at endpoints is legitimate.

Continue the required finite blocks by replacing \(a_{n+j}\) with
\[
 a_{\nu+j}=(4-(\nu+j)^{-2})^{-1/2},
\]
using the branch near \(1/2\), on \(|\nu-m|\le cm\) for a small
fixed \(c\). Choose a fixed offset block, for example
\(|j|\le m/8\), and let the projection test the sign of the offset.
Our choice \(K_1d_g=o(m)\) ensures that no retained coefficient
reaches its artificial endpoints. The block differs in norm by
\(O(m^{-2})\) from the real constant tridiagonal matrix with entries
\(1/2\). The latter has spectrum in \([-1,1]\). On a contour at
a sufficiently large constant times \(m^{-2}\) from this interval,
a resolvent identity bounds the continued resolvent by \(O(m^2)\).
For the Chebyshev polynomial \(T_p(x)=\cos(p\arccos x)\), the
quadratic-root formula gives
\( |T_p(x)|\le C e^{Cp/m}\) on this contour. Expanding each
\(g_j\) in its cosine coefficients and applying the resolvent
formula thus bounds \(\|g_j(J)\|\) by \(e^{CL}L^C\).
There are at most exponentially many indices but only polynomially
many factors in any retained term; all products, traces, and formal
coefficients therefore have bounds \(e^{L^C}\). This resolvent
estimate is essential because an absolute enumeration of paths of
length \(d_g\) would give a much larger bound.

The degree-one trace
\(T_j(n)=\operatorname{Tr}(\Pi_ng_j\Pi_n)\) requires a separate
construction. Its forward difference
\(T_j(n+1)-T_j(n)=(g_j(J))_{nn}\) is a near-index entry, hence has
an analytic continuation \(h_j(\nu)\) on \(|\nu-m|\le cm\)
with bound \(e^{CL}L^C\). If
\(h_j(m+w)=\sum_{p\ge0}h_{jp}w^p\), Cauchy's estimate gives
\(|h_{jp}|\le e^{CL}L^C(C/m)^p\). Let \(S_p\) be the polynomial
with \(S_p(0)=0\) and
\(S_p(w+1)-S_p(w)=w^p\). Define
\[
 \widehat T_j(m+w)=T_j(m)+\sum_{p=0}^{P_3}h_{jp}S_p(w).
\]
This agrees, at integer \(w\), with summing the truncated forward
differences in either direction from zero. The generating function
\[
 \sum_{p\ge0}S_p(w)\frac{t^p}{p!}
          =\frac{e^{tw}-1}{e^t-1}
\]
and Cauchy's estimate on a circle of radius comparable to
\((p+1)/(p+1+|w|)\) give
\( |S_p(w)|\le[C(1+|w|+p)]^{p+1}\).
For \(|w|\le\sqrt m\) and any fixed polynomial \(P_3\), the
continued trace is consequently bounded by \(e^{CL}L^C\), uniformly
in the chosen truncation power for sufficiently large \(L\).
Its integer-grid error is at most
\(e^{CL}L^C\sqrt m(C/\sqrt m)^{P_3+1}\).
Increasing the polynomial order gives any required
\(e^{-L^{B_3}}\) precision.

The same construction continues the reference masses. The exact norm
product gives
\[
 \Delta^2\log J_n(w_0)
   =\log\!\left(1+\frac1{(2n+1)(2n+3)}\right).
\]
Keep the exact \(\log J_m\) and \(\Delta\log J_m=O(m^{-1})\),
Taylor-expand this analytic second difference at \(m\), and sum
twice by polynomial antidifferences. Its Taylor coefficients are
\(O(m^{-2})(C/m)^p\). The polynomial twice-antidifference with
values zero at \(w=0,1\) has generating function
\[
 \frac{e^{tw}-1-w(e^t-1)}{(e^t-1)^2}
\]
and bound \([C(1+|w|+p)]^{p+2}\). The resulting analytic logarithm
stays \(O(L)\) on \(|w|\le\sqrt m\). Integer errors acquire at
most a factor \(m\) when summed twice; after exponentiation they
still attain arbitrary \(e^{-L^{B_3}}\) accuracy. These uniform
size bounds do not grow with the final polynomial Taylor order.

Endpoint evaluations are continued directly. The constant kernel is
represented by Christoffel--Darboux~\cite[equation~18.2.12]{DLMF18}:
\[
 \Pi_n(x,y)=a_n
   \frac{p_n(x)p_{n-1}(y)-p_{n-1}(x)p_n(y)}{x-y},
\]
with the removable diagonal filled by its limit. For \(x\) near
\(+1\), continue a wavefunction with index \(z=\nu+j\) by
\[
 p_z(x)=\sqrt{(2z+1)/2}
 \sum_{r=0}^{\infty}
 \frac{\prod_{l=0}^{r-1}(z-l)(z+l+1)}{(r!)^2}
                  \left(\frac{x-1}{2}\right)^r.
\]
This is the Rodrigues expansion at integer \(z\). For
\(|j|\le m/8\), \(|\nu-m|\le\sqrt m\), and
\(|x-1|\le C L^{2D_E}/m^2\), the terms with \(r\le4m\) are
bounded by \((C L^{D_E})^{2r}/(r!)^2\), apart from the square-root
prefactor. For \(r>4m\), the ratio of successive absolute
majorants is at most \(C|x-1|<1/2\). Thus the function is bounded
by \(e^{L^C}\), also on slightly larger complex endpoint disks.
Cauchy estimates give the same type of bound for its first derivative,
including the factor \(m^2\) from the disk radius. This controls
same-endpoint diagonal quotients; opposite-endpoint denominators
are bounded away from zero.

Near \(-1\), use the value at \(-x\) and multiply by
\((-1)^j\), omitting any putative \((-1)^\nu\). At integer \(n\)
this prescription gives
\(\widetilde p_{n+j}(x)=d_n(x)p_{n+j}(x)\), where
\(d_n(x)=1\) near \(+1\) and \(d_n(x)=(-1)^n\) near \(-1\).
Consequently both the Christoffel--Darboux kernel (with its original
\(x-y\) denominator) and every retained correction kernel are
changed by \(K(x,y)\mapsto d_n(x)K(x,y)d_n(y)\). For any endpoint
set their determinant is unchanged, because the row and column
multipliers contribute \(\prod_h d_n(x_h)^2=1\). This proves
agreement at integer dimensions even for configurations using both
endpoints, while providing a holomorphic continuation in \(\nu\).

\begin{proof}[Proof of Proposition~\ref{prop:04-interpolation}]
Use the polynomial and endpoint-subset approximations just constructed.
In each retained subset integral, replace the higher cumulants and
kernel corrections by their continued near-index formulas, the
constant kernel by the continued Christoffel--Darboux expression,
and the linear traces and masses by their polynomial-antidifference
approximants. Reconstruct \(\mathcal D_n\) by formal exponentiation
through source degree \(K_1\), and take the indicated source
derivatives. The two labels are combined by independence.

Every expression before endpoint integration is holomorphic in
\(\nu,M\), with a uniform bound \(e^{L^C}\). Indeed source types,
source orders, and determinant sizes are polynomial. Near-index
blocks have length \(O(m)\), so an expression with polynomially
many index sums has at most \(\exp(\operatorname{poly}(L))\)
terms. Permutation expansions, source coefficient extractions, and
factorials of polynomial orders satisfy the same bound. Coefficients
\(c_\alpha\) include only the fixed polynomial \(P\), centerings,
and bounded endpoint data. Integration over fixed real endpoint
regions preserves these bounds and holomorphy by dominated
convergence (or Morera's theorem). No analyticity of the endpoint
factors themselves as functions of their real positions is needed.

We record the order of the error choices to exclude a circular
precision requirement. First fix \(B_0\) using the original
integer-law expectation bound. This fixes the dimension and gap in
the Gaussian approximation. Next fix \(B_1\) and \(K_1\); then
\(D_1,B_2,D_E\), in that order; finally choose \(K_e\) for the
factorial tail. All errors to this point have already been bounded
on the actual integer laws by \(e^{-L^b}\), with a freely reserved
margin. They are not passed through an analytic continuation formula.
The remaining formulas, through degree \(K_1\), are polynomial in
the linear trace coefficients and in the kernel entries. Their
sensitivity to the final trace and normalization errors is at most
\(e^{L^{C_*}}\), for an exponent \(C_*\) depending only on the
choices already made. The preceding size bounds for trace and mass
approximants are independent of their final polynomial Taylor order.
Choose \(B_3>\max(b,C_*)+1\), and the corresponding orders, last.
The resulting errors are negligible after this final amplification.
Taking the initially reserved margins, for example replacing each
requested error by \(e^{-2L^b}\), makes their finite sum at most
\(e^{-L^b}\) for large \(L\). This proves both assertions in
\eqref{eq:G-20}.
\end{proof}

\begin{lemma}[Transfer from the integer grid]\label{lem:04-grid}
Suppose \(M\) is in the fixed polynomial box and
\(|F_\nu(M)|\le B\) at all integer grid points of
\(|\nu_\sigma-m|\le\sqrt m\), with
\(B\ge e^{-L^{b'}}\), \(b'b'\) then give \(S\le(1+o(1))B\).

The tensor Chebyshev coefficients are at most \(4S\), and there
are at most \((K+1)^2\) of them. At the target, each rescaled
coordinate has magnitude \(\operatorname{poly}(L)/R\). Its
complex inverse-cosine angle differs from \(\pi/2\) by the same
order, so every Chebyshev basis value up to degree \(K\) is
bounded by \(\exp(CK\operatorname{poly}(L)/R)=1+o(1)\).
Thus \(|P(\nu^*)|\le C(K+1)^2B\), which is
\(B\exp(o(L))\). The Taylor error is negligible relative to \(B\).
\end{proof}

For fixed explicit data, the prescribed totals \(m'_\sigma(M)\)
are affine in \(M\). Under the retained polynomial restrictions,
\(m'(M)-m\) has polynomial size and lies in the dimension polydisk.
Consequently
\[
 H(M)=\widehat F(m'(M),M)
\]
is holomorphic throughout the chosen polynomial \(M\)-disk. At every
valid integer \(M\) in that disk, it approximates the original
integer-dimensional integral to the precision in \eqref{eq:G-20}.
The grid-transfer lemma supplies the sharper bounds on \(H\) away
from those integer values; they do not follow from the coarse
holomorphic bound alone.

==== END Cylinder-loop-weights-and-planar-nesting//04b_interpolation.tex ====

==== BEGIN Cylinder-loop-weights-and-planar-nesting//04c_exponent.tex ====
\section{The cylinder exponent}\label{sec:E-cylinder-exponent}

We can now sum the contour sectors. The interpolation just proved is
needed only during the summation over imbalance: it gives a holomorphic
function approximating the fixed-dimension integrals to arbitrarily high
exponential precision. After moving that sum, we restore the scalar
normalization and obtain the physical lower bound from positivity.


Take the balanced physical cylinder with \(N=2m\). Its row
parameters are \(2\lambda\) on the first group and \(\lambda\)
on the second. Their multiplicative site values \(q^2,q\) become
\(1,q^{-1}\) under the common rescaling by \(q^{-2}\), which leaves
the contraction unchanged. Thus the logarithmic sites used below are
\(0,-i\beta\), as in Section~\ref{sec:02-cylinder}.

Define the scalar
\begin{equation}\label{eq:04-physical-normalization}
 \Delta_N=
 \frac{\widetilde h_N^2}
 { (\gamma/2)^N\prod_{iU_0\), or \eqref{eq:G-7} if \(U\le U_0\). The choice depends only on alias, bias, and explicit data, and has a polynomial bound independent of \(T\). The connecting verticals and the original end integrals are all negligible by \eqref{eq:G-15}: there the loss \(c_h T^2/L\) can dominate all other powers, including possible growth from \(|j v|\), by taking the power of \(T\) sufficiently large after \(C_J\). These choices only require a fixed polynomial box in \eqref{eq:G-20}; the precision there is now chosen as just described. All movements of \(H\) times its phases are thus within its holomorphy domain.

\paragraph{Summing the shifted bounds.}
On the shifted lines \eqref{eq:G-16},\eqref{eq:G-17} apply, up to \(o(L)\) by the transfer. In the large-\(U\) case integrate the bound \eqref{eq:G-16} using exponential \(\ell_e\)-damping and factorials, and absorb its \(O_h(L)\) costs by taking \(U_0\) sufficiently large (after \(\delta\) and \(h\)). In the bounded-\(U\) case the bound \eqref{eq:G-17} costs after prefactors and integration at most an additional \(C_{h,\varepsilon}\delta L+o(L)\) in the log, as following \eqref{eq:G-7}. More explicitly this case is supported on explicit configurations with at most \(C_{U_0}\) far particles on \(|u|\le(1-2\delta)L\), no restriction needed for near outer particles other than being exterior. Thus this same integrated bound works for each alias and there are only polynomially many; likewise horizontal length costs at most polynomially. This pointwise split does not require any contour movement of explicit variables. All constants allow \(\delta\) small after \(\varepsilon,h\), then \(U_0\) arbitrarily large fixed, little-oh errors then for these fixed choices. Sending the small errors to zero proves \eqref{eq:G-23}.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{E:cylinder-exponent}]
By \eqref{eq:04-physical-normalization}, the physical exponent is the
contour numerator exponent minus the scalar exponent. The deformation
determinant contributes \(7/8\), the two reference masses contribute
\(-1/2\), and the scalar normalization contributes \(-5/24\), all
as coefficients of \(\log N\). Thus the leading constants are
\[
 \frac78-\frac12-\frac5{24}=\frac16,
 \qquad \frac1{2\gamma\kappa}=\frac2{3\pi^2},
\]
since \(\gamma=4/3\) and \(\kappa=(3\pi/4)^2\). Thus
Proposition~\ref{prop:04-sector-sum} gives, locally uniformly in
\(|\operatorname{Re}\eta|<\pi\),
\[
 u_N(\eta):=\frac{\log|Z_N(2\cos\eta)|}{\log N}
 \le\mathcal H(\eta)+o(1),\qquad
 \mathcal H(\eta)=\frac16-\frac{2}{3\pi^2}
                                  \operatorname{Re}(\eta^2).
\]
\paragraph{Propagation from the empty-configuration value.}
The function \(\mathcal H\) is harmonic. Since \(Z_N\) has
nonnegative coefficients and \(Z_N(0)=1\), the interior point
\(\eta=\pi/2\) satisfies \(u_N(\pi/2)=\mathcal H(\pi/2)=0\).
We show that every fixed disk inside the strip contains points
\(\eta_N\) with
\(u_N(\eta_N)\ge\mathcal H(\eta_N)-o(1)\).

Join the anchor to the disk by a compact path in the strip and
choose a connected neighborhood with positive clearance from its
boundary. The locally uniform upper bound permits a constant
\(\epsilon_N\downarrow0\) such that
\(v_N=\mathcal H+\epsilon_N-u_N\ge0\) on this neighborhood.
It is superharmonic and locally integrable; zeros of \(Z_N(2\cos\eta)\)
give only logarithmic singularities, and the anchor ensures this
holomorphic function is not identically zero. At the anchor,
\(v_N=\epsilon_N\).

Choose a finite chain of disks of radius \(r\), with successive
centers at distance at most \(r\), leading into the target disk.
Take \(r\) small enough that, from any point in one disk, the
radius-\(3r\) disk stays in the chosen neighborhood and covers the
next disk. The area mean inequality for a nonnegative superharmonic
function implies that if \(v_N(z)\le a_N\) at the chosen point,
then its integral over the next radius-\(r\) disk is at most
\(9\pi r^2a_N\). Some point there therefore has value at most
\(9a_N\). Starting at the anchor and iterating the fixed finite
chain yields \(v_N(\eta_N)\le9^k\epsilon_N=o(1)\) in the target.

\paragraph{Comparison with positive real fugacities.}
The disk argument supplies a near-saturating point, not a pointwise
lower bound throughout the disk. Positivity of the physical polynomial
converts this into the desired lower bound at a positive fugacity.
For \(\chi=2\), take target disks about small positive real values
of \(\eta\), sufficiently narrow that \(|2\cos\eta|<2\)
throughout. Positivity of coefficients gives
\(Z_N(2)\ge|Z_N(2\cos\eta_N)|\). Thus the lower limit of
\(\log Z_N(2)/\log N\) is at least the infimum of
\(\mathcal H\) on each such disk. Letting these disks approach
zero gives \(1/6\); the upper bound at zero matches it.

If \(0<\chi<2\), put \(\eta_\chi=\arccos(\chi/2)\).
Choose real disk centers decreasing to \(\eta_\chi\) from above,
and radii sufficiently small that \(|2\cos\eta|\le\chi\)
throughout. The same positivity argument gives the lower exponent
\(\mathcal H(\eta_\chi)\), matching the upper bound there.
If \(\chi>2\), use
\(\eta_\chi=i\operatorname{arcosh}(\chi/2)\) and disk centers
approaching it along the imaginary axis from smaller imaginary
parts. Again their radii can be chosen so that
\(|2\cos\eta|\le\chi\). This proves the remaining formula.
The argument obtains a lower bound from the polynomial's positivity
and the interior anchor; it requires no lower estimate on a single
complex contour sector.
\end{proof}
==== END Cylinder-loop-weights-and-planar-nesting//04c_exponent.tex ====

==== BEGIN Cylinder-loop-weights-and-planar-nesting//05_pressure.tex ====
\section{Winding pressure and essential polygons}
\label{sec:05-pressure}

Throughout the pressure and nesting arguments, except during the explicitly
indicated variation of the loop fugacity, put
\[
 \lambda=\frac{\pi}{8},\qquad \beta=2\lambda=\frac{\pi}{4},
 \qquad \gamma=\frac{\pi}{3\beta}=\frac43,
 \qquad \rho=\frac1{\sqrt{2+\sqrt2}}.
\]
A geometric honeycomb polygon has weight \(\rho^{|P|}\), where
\(|P|\) counts vertices. Polygons are unrooted and unoriented; when
translation classes are used, the translation group will always be specified.

The cylinder exponent concerns polygons separating two marked points.
To obtain a nesting law around one planar point, we must remove two
kinds of polygon with a bounded multiplicative cost: essential polygons
that wind around the cylinder, and contractible polygons too large to fit
near either marker. We treat them separately. The first two derivatives
of an annular pressure control the total essential-polygon weight and
its horizontal-span moment. Section~\ref{sec:05-open-pressure} obtains the planar diameter tail
needed for contractible polygons from a separate open-strip pressure.
Section~\ref{sec:05-nesting} then compares the surviving configurations
with two independent planar nests and derives the strip-length estimates.
The present section supplies the essential-polygon removal bound.

\subsection{The annular transfer and its interpolation data}

Let \(N=2m\), and assign the additive site parameters
\[
 b_1=\cdots=b_m=\lambda/2,
 \qquad b_{m+1}=\cdots=b_{2m}=-\lambda/2.
\]
The auxiliary argument \(z_0=3\lambda/2\) in the column of boxes \(R(z-b_i)\) gives the physical box arguments \(\lambda\) and \(2\lambda\), respectively. In the transfer considered here an essential closed curve receives fugacity \(\chi\), whereas a contractible closed curve receives zero. This convention is specific to the winding pressure.

A state records a cyclic noncrossing partial matching together with a marked complementary region, namely the region containing the past end of the cylinder. Equivalently, draw the matching in a disk with a marked puncture and no strand ending at the puncture. There are finitely many states: an ordinary cyclic noncrossing matching has finitely many regions in which to place the puncture. This information determines composition. Indeed two realizations of the same state decompose the disk into disks and one punctured disk; homeomorphisms of these pieces fixing their boundary arcs identify their sewing rules and distinguish essential from contractible cycles.

Let \(T_\chi(z)\) carry the auxiliary slot around all sites, joining its final output to its initial input. The local identities of the strip calculation apply, since every cycle internal to one of those identities is contractible. At \((\chi,z)=(0,z_0)\), the empty-coordinate functional is invariant and the restriction to vectors with zero empty coordinate has spectral radius less than one. To see the latter assertion in the refined state space, a nonzero diagram with nonempty initial connectivity either closes a killed loop or contains a path joining its two longitudinal cuts. The periodic-slab spanning estimate in Lemma~\ref{lem:02-periodic-slab} therefore gives decay of its powers, exactly as for the unmarked connectivity space. Consequently the eigenvalue \(1\) is simple and strictly dominant, with analytic continuation under small changes of \(\chi\) and of the site parameters.

The matrices \(T_\chi(z)\) commute as \(z\) varies. Insert an invertible crossing between two auxiliary lines, move it through the sites by the three-box identity, and cancel it at the other end; the resulting diagram reverses the two columns. This proves commutation at generic parameters and hence as a rational identity. The simple eigenspace at \(z_0\) is therefore invariant under every \(T_\chi(z)\). Denote its eigenvalue by \(\Lambda(\chi;z)\), normalizing an eigenvector to have empty coordinate one. At \(\chi=0\), the empty-coordinate rule gives \(\Lambda(0;z)=1\).

\begin{lemma}[Annular pinch and analytic data]
\label{lem:05-winding-data}
The function \(\Lambda(\chi;z)\) is rational in \(e^{2iz}\), with denominator dividing \(\prod_i\mathscr D(z-b_i)\), where \(\mathscr D\) is the local denominator in \eqref{eq:E-local-weights}, and tends to \(1+\chi\) at both ends of this multiplicative coordinate. At the balanced specialization it is invariant under \(z\mapsto3\lambda-z\). For \(u=b_i\) and generic remaining sites,
\begin{equation}
 \Lambda(\chi;u+j\lambda)\Lambda(\chi;u)
 =\begin{cases}
   \Lambda(\chi;u+\lambda),&j=2,\\
   1,&j=3.
  \end{cases}
 \label{eq:05-annular-pinch}
\end{equation}
Set \(w=2i(z-z_0)\) and, with derivatives evaluated at \(\chi=0\), define
\begin{equation}
 \begin{gathered}
 f(w)=\partial_\chi\Lambda(\chi;z),\qquad
 h_2(w)=\partial_\chi^2\Lambda(\chi;z),\qquad
 g(w)=h_2(w)-f(w)^2,\\
 (\mathscr L v)(w)=v(w+i\beta)+v(w-i\beta)-v(w).
 \end{gathered}
 \label{eq:05-pressure-derivatives}
\end{equation}
Then \(\mathscr L f\) and \(\mathscr L g\) vanish to orders \(2m,m,m\) at \(0,i\beta,-i\beta\), respectively. The functions \(f-1\) and \(h_2\) are even rational functions of \(e^w\), vanish at both real infinities, and have poles only at the translates of these three knots by \(i\pi\), modulo \(2\pi i\), with the same respective order bounds.
\end{lemma}

\begin{proof}
The common eigenvector is independent of \(z\). Computing the empty output of the transfer on that vector proves the rationality and denominator bound. A shift \(z\mapsto z+\pi\) inserts occupancy signs that cancel around the closed auxiliary. At either imaginary infinity, \(A,U\to0\), \(B\to1\), and all entries remain bounded. Only empty input reaches empty output; it contributes the empty diagram and one essential auxiliary circle, with weights \(1\) and \(\chi\). Reflection reverses the row order and negates all \(b_i\); the balanced list is thereby preserved, and simplicity identifies the reflected eigenline with the original one.

For the pinch, pass the two auxiliaries in order \([u+j\lambda,u]\). At a host with \(x=u-b_k\) their crossing is
\[
 M_x=R_1(x+j\lambda)R_2(x),\qquad
 (I\otimes S_j^{\mathsf t})M_x
   =\widehat R(S_j^{\mathsf t}\otimes I).
\]
At the marked host, \(x=0\) and
\(M_0=(S_j\otimes I)(S_j^{\mathsf t}\otimes I)\).
Close the paired auxiliary trace and move the transpose splitter backwards through all other hosts. At the marked host the remaining contraction is
\[
 (I\otimes S_j^{\mathsf t})(S_j\otimes I)
   =\widehat R\big|_{x=0}.
\]
For completeness, this identity follows by multiplying on the right by \(S_j^{\mathsf t}\otimes I\), using the transposed splitting identity at \(x=0\), and cancelling that factor. Cancellation is valid because \(S_j^{\mathsf t}\) is surjective: insert vacancy, or one through strand into a fixed output and divide by its nonzero coefficient. The fused auxiliary is the one of argument \(u+\lambda\) for \(j=2\), and disappears for \(j=3\). These are identities of annular diagrams, so they prove \eqref{eq:05-annular-pinch} with the essential-loop fugacity retained.

The first two derivatives of \(\log\Lambda\) are \(f\) and \(g\). Taking their derivatives in \eqref{eq:05-annular-pinch} makes \(\mathscr L f\) and \(\mathscr L g\) vanish at the points corresponding to both \(b_i+\lambda\) and \(b_i+2\lambda\). At generic sites the factors giving these zeros are distinct. Analyticity of the normalized eigenvector near the balanced physical sites, and regularity of the boxes at all the evaluation points, permit confluence. The two families give the stated multiplicities. The two sine factors in each \(\mathscr D(z-b_i)\) give the pole list. Since the denominator is independent of \(\chi\), taking two \(\chi\)-derivatives does not increase those orders.
\end{proof}

\subsection{Uniform Hardy-space interpolation}

The interpolation conditions just obtained will estimate the pressure
at the physical point \(w=0\). Their number grows with the cylinder
width, so invertibility alone is insufficient: the inverse must remain
bounded as the knots coalesce and their multiplicities grow. The
following Hardy-space realization separates that issue from the
particular pressure derivatives.

We establish the analytic estimates with constants independent of \(N\). Let \(H\) be the Hardy space on \(|\operatorname{Im}w|0
 \quad(k\in\mathbb R),\qquad
 \|(P_ND_a|_{P_NH})^{-1}\|\le c_a^{-1}.
 \label{eq:05-coercivity}
\end{equation}
Moreover, writing \(\|\cdot\|_{\partial,p}\) for the sum of the two boundary \(L^p\) norms, one has
\begin{equation}
 \|e^{aw}(1-f)\|_{L^p(\operatorname{Im}w=y)}
       \le C_a e^{aA_N},
 \qquad p=2,4,\quad |y|\le h+\beta.
 \label{eq:05-first-derivative-norms}
\end{equation}
For fixed \(00\). This is sufficient since the absolute ratio of the imaginary and real parts of the denominator is at most \(\cot(\pi a)\). Finally
\[
 D_a(+\infty)=e^{i(\pi/2-3\beta a)},\qquad
 D_a(-\infty)=e^{-i(\pi/2-3\beta a)}.
\]
These limits have positive real part. Continuity proves the uniform lower bound and boundedness. Compressing the real-part inequality to the finite-dimensional jet space proves the inverse bound.

The product defining \(B_N\) gives, throughout the closed strip,
\begin{equation}
 |1-B_N(w)|\le C\min\{1,e^{-\gamma(\operatorname{Re}w-A_N)}\}.
 \label{eq:05-blaschke-tail}
\end{equation}
For large positive real part, sum the deviations of individual factors from one; for the remaining points use \(|B_N|\le1\). It follows by direct integration, for \(p=2,4\), that
\[
 \|e^{aw}(1-B_N)\|_{\partial,p}\le C_a e^{aA_N}.
\]
The jet conditions give the exact interpolation equation
\begin{equation}
 P_ND_ax_N=P_N[e^{aw}(1-B_N)].
 \label{eq:05-first-interpolation}
\end{equation}
Thus \(\|x_N\|_H+\|D_ax_N\|_H\le C_ae^{aA_N}\).

\paragraph{The product estimate.}
The second pressure derivative involves \(f^2\). We therefore need a
fourth-norm estimate in addition to the Hilbert-space bound just proved;
it will control this product without losing a power of \(N\).
Here are the additional bounds needed for products of analytic functions. Embed \(H\) in the two-component boundary space, and let \(\Pi\) be the orthogonal projection onto its traces. In each Fourier fiber it projects onto \((e^{-hk},e^{hk})\). Its diagonal entries differ from the appropriate positive- or negative-frequency projections by bounded square-integrable symbols; its off-diagonal entries decay exponentially. Such remainders map \(L^2\) into \(L^2\cap L^\infty\), hence into \(L^4\). The half-frequency projections are bounded on \(L^4\), the \(p=4\) case of the classical M.~Riesz conjugate-function boundedness theorem; for a real-line reference see~\cite{Grafakos1997,GrafakosSavage2015}. Only boundedness, not a sharp norm, is used here. An elementary proof for real Schwartz data writes its positive-frequency part as \(u+iv\), with \(u\) half the original data. Fourier support gives \(\int(u+iv)^4=0\), whence
\[
 \int v^4\le6\left(\int u^4\int v^4\right)^{1/2}.
\]
The identity is justified first for frequency cutoffs, whose inverse transforms are bounded with \(O(|x|^{-1})\) tails; density and complex linearity then prove the assertion. Therefore \(\Pi\) is bounded from boundary \(L^2\cap L^4\) to itself, with the sum norm.

On the full boundary space,
\[
 P_N=\Pi-B_N\Pi\overline{B_N}.
\]
Boundary multiplication by \(B_N\) preserves every \(L^p\) norm. Hence these projection bounds are uniform in \(N\). Moreover, the commutator of \(P_N\) with any fixed constant diagonal matrix maps \(L^2\) to \(L^4\) uniformly: only the off-diagonal entries of \(\Pi\) occur in its commutator, and multiplication by \(B_N\) commutes with the diagonal matrix.

Let \(D_{\rm edge}\) be the diagonal matrix whose two entries are the limiting values of \(D_a\) in the dominant frequency direction of the corresponding boundary trace. On \(H\), \(D_a-D_{\rm edge}\) maps boundary \(L^2\) to \(L^4\). On each dominant half-axis this follows from exponential convergence of the symbol; on the other half-axis the trace has exponential decay relative to the weighted norm \eqref{eq:05-hardy-norm}. Since \(x_N=P_Nx_N\), the preceding commutator estimate yields
\[
 P_ND_ax_N=D_{\rm edge}x_N+E_N,
 \qquad \|E_N\|_{\partial,4}\le C_a\|x_N\|_H.
\]
The right side of \eqref{eq:05-first-interpolation} has boundary fourth norm at most \(C_ae^{aA_N}\); the diagonal entries of \(D_{\rm edge}\) are nonzero. This proves the same fourth-norm bound on \(x_N\) and on \(D_ax_N\).

Invert the weighted shift multiplier in \eqref{eq:05-partial-fractions}. To move a trace from height \(h\) to height \(h+\beta\), or from \(-h\) to \(-h-\beta\), the multiplier is, up to a constant phase,
\[
 \frac{e^{\pm\beta k}}{2\cosh(\beta(k+ia))-1}.
\]
It has finite constant limits on the two half-axes and exponentially decreasing remainders, so the preceding \(L^2\) and \(L^4\) argument applies. Interior lines follow by the two-boundary Fourier representation. This proves \eqref{eq:05-first-derivative-norms}.

\paragraph{The second derivative and the residual.}
Now fix \(a<1/2\), and write
\(Y_2=e^{2aw}\mathscr Lh_2\in D_{2a}P_NH\).
Since \(\mathscr Lg\) has the prescribed jets,
\begin{equation}
 P_NY_2=P_N[e^{2aw}(\mathscr L(f^2)-B_N)].
 \label{eq:05-second-interpolation}
\end{equation}
If \(v=1-f\), then \(\mathscr L(f^2)=1-2\mathscr Lv+\mathscr L(v^2)\). The second-norm bound at weight \(2a\), the fourth-norm bound at weight \(a\) on the shifted lines, and \eqref{eq:05-blaschke-tail} show that the interpolation input has \(H\)-norm at most \(C_ae^{2aA_N}\). Coercivity gives this bound for \(Y_2\) as well.

Subtract the right interpolation input from each left input in \eqref{eq:05-first-interpolation} and \eqref{eq:05-second-interpolation}. The differences lie in \(B_NH\). Division by \(B_N\) is isometric, and evaluation at a real point has uniformly bounded norm in \(H\). In the first equation the difference is
\(e^{aw}(B_N-\mathscr Lf)\); in the second it is
\(e^{2aw}(\mathscr Lg+B_N)\).
This proves \eqref{eq:05-jet-residual} and its first-derivative variant.
\end{proof}

\subsection{The size and first spatial moment of winding pressure}

We now turn the uniform interpolation estimate into two bounds at the
physical point: the first derivative has order \(N^{-1}\), and the
second logarithmic derivative has absolute value \(O(N^{-1})\).
Their polygon interpretation will then bound the total essential weight
and its first horizontal-span moment.

We will use the inverse of \(\mathscr L\). Its convolution kernel on the real line is
\begin{equation}
 \rho_*(s)=\frac1{\beta\sqrt3}
 \frac{\sinh(2\pi s/(3\beta))}{\sinh(\pi s/\beta)},
 \qquad
 \widehat{\rho_*}(k)=\frac1{2\cosh(\beta k)-1},
 \label{eq:05-resolvent}
\end{equation}
where the quotient at zero is interpreted continuously. This is exactly
the mean density \(\varrho\) from Section~\ref{sec:03-gas-mean}:
it canceled the particle--site sources there, and here it inverts
the shift operator \(\mathscr L\). The Fourier identity follows by a residue computation, or by integrating after the substitution \(u=e^{\beta k}\). In particular
\begin{equation}
 \rho_*(s)>0,\qquad \int\rho_*=1,\qquad
 \rho_*(s)\le Ce^{-\gamma|s|},\qquad
 \lim_{s\to+\infty}e^{\gamma s}\rho_*(s)=\frac1{\beta\sqrt3}.
 \label{eq:05-resolvent-tail}
\end{equation}

\begin{lemma}[Winding pressure bounds]
\label{lem:05-pressure-bounds}
There are constants \(00\),
\[
 d_+(s):=\int e^{-sv}K_+(dv)
        =\exp\!\left(\int_0^\infty e^{-sv}q(v)\,dv\right)>0.
\]
Since \(K_-*l\) is supported on \(v\le0\), applying the positive-support inverse of \(K_+\) to \eqref{eq:05-half-line-equation} at \(v<0\) gives
\[
 (K_-*l)(v)=d_+(p)^{-1}e^{pv}\mathbf1_{v<0}.
\]
Consequently the residual on the entire line is
\begin{equation}
 e^{pv}-\mathcal Dl
   =d_+(p)^{-1}K_+*(e^{pv}\mathbf1_{v>0}).
 \label{eq:05-positive-residual}
\end{equation}
The half-line identities use ordinary \(L^2\) convolution on the left; on the right they are valid with the displayed exponential weights. Total variation of the \(L^1\) kernels justifies all exchanges below. Integrating \eqref{eq:05-laplace-transform} against \(e^{-(\gamma+a)t}\,dt\), then substituting \(e^{v-\gamma t}\), yields
\begin{align}
 \int_{\mathbb R}e^{-\gamma t}b_\infty(t)\,dt
 &=\frac{\Gamma(3/2+p)}{\gamma}
       \int_{\mathbb R}e^{-(1+p)v}(e^{pv}-\mathcal Dl)(v)\,dv\notag\\
 &=\frac{\Gamma(3/2+p)}{\gamma}
       \frac{d_+(1+p)}{d_+(p)}>0.
 \label{eq:05-positive-moment}
\end{align}
Combining \eqref{eq:05-pressure-limit}, \eqref{eq:05-positive-moment}, and \(e^{\gamma A_N}=3N\) completes the proof.
\end{proof}

\begin{proposition}[Suppression of essential polygons]
\label{prop:05-winding-suppression}
On the balanced cylinder with period \(V=iN\sqrt3/2\), let \([\ell]\) range over simple essential polygons modulo integer horizontal translations, and set \(W(\ell)=\rho^{|\ell|}\). If \(d_x(\ell)\) is their horizontal span, then
\begin{equation}
 \sum_{[\ell]}W(\ell)\asymp N^{-1},\qquad
 \sum_{[\ell]}W(\ell)d_x(\ell)\le C.
 \label{eq:05-winding-moment}
\end{equation}
The total weight of essential polygons separating the two marked points in the balanced two-marker geometry is bounded uniformly in \(N\). At each fixed nonnegative fugacity, deleting all these polygons changes its partition function by at most a bounded multiplicative factor.
\end{proposition}

\begin{proof}
In a slab of \(M\) columns with vacant outer cuts, differentiate the logarithm of the loop partition at \(\chi=0\). Its first derivative counts single essential polygons. Its second derivative is minus the weight of ordered incompatible pairs, including a polygon paired with itself. After division by \(M\), strict spectral dominance gives convergence of the two derivatives to those of \(\log\Lambda(\chi;z_0)\): the spectral projection has nonzero vacuum overlap, equal to one at zero fugacity, and the remaining spectrum decays exponentially, locally uniformly in the analytic parameter. Combinatorially each finite polygon or intersecting pair occupies a finite consecutive interval of columns; its number of placements divided by \(M\) increases to one. Monotone convergence gives
\begin{equation}
 \begin{split}
 f(0)&=\sum_{[\ell]}W(\ell),\\
 -g(0)&=\sum_{[\ell],[\ell']}W(\ell)W(\ell')
        \sum_{j\in\mathbb Z}
          \mathbf1_{\{\ell\cap(\ell'+j)\ne\varnothing\}}.
 \end{split}
 \label{eq:05-cluster-derivatives}
\end{equation}
The physical tiles encode precisely the honeycomb polygons with their vertex weights, and on this embedded trivalent graph intersection is precisely incompatibility.

Let \(\Gamma\) be the triangular translation lattice modulo \(V\). Its integer horizontal subgroup has index \(N\). Fix vertices \(P,Q\) of \(\ell\) of the same bipartite type, with \(\Delta_x=x(Q)-x(P)>0\), and put \(\Delta=Q-P\in\Gamma\). For a translating vector \(a\in\Gamma\), let \(I_P(a)\) indicate that \(P\) lies strictly in the right-hand component of the complement of \(\ell'+a\). Along each horizontal coset it is one on the negative tail and zero on the positive tail. Moreover \(I_Q(a)=I_P(a-\Delta)\). The difference is finitely supported, and
\[
 \sum_{a\in\Gamma}(I_Q(a)-I_P(a))=N\Delta_x.
\]
For an integer horizontal \(\Delta\), this is ordinary telescoping in each of the \(N\) cosets. In general multiply \(\Delta\) by \(N\), which is an integer horizontal translation modulo \(V\), and telescope the sum of the \(N\) translates of the original difference. If the indicators disagree, the path in \(\ell\) from \(P\) to \(Q\) meets \(\ell'+a\); a point on its boundary also gives intersection. Thus at least \(N\Delta_x\) translating vectors give an intersection. Choosing the vertices near the two horizontal extremes, and changing one to a neighbor when necessary to match bipartite types, gives \(\Delta_x\ge d_x(\ell)-C\).

Average the inner intersection count in \eqref{eq:05-cluster-derivatives} over representatives of \(\Gamma/\mathbb Z\). Translation by each representative permutes the horizontal polygon classes and preserves their weights. The resulting inequality is
\[
 f(0)\sum_{[\ell]}W(\ell)(d_x(\ell)-C)\le-g(0).
\]
Now apply Lemma~\ref{lem:05-pressure-bounds}; in particular \(-g(0)\ge0\) by \eqref{eq:05-cluster-derivatives}. This proves \eqref{eq:05-winding-moment}.

Every separating essential polygon meets a fixed arc joining the two markers. That arc has horizontal span \(O(N)\). A polygon class can therefore have at most \(CN+d_x(\ell)+C\) translates meeting it. The two estimates in \eqref{eq:05-winding-moment} bound their total weight. If this bound is \(C\), summing all sets of removed polygons without any compatibility restrictions costs at most \(\sum_{k\ge0}(\chi C)^k/k!=e^{\chi C}\). Positivity provides the opposite inequality by retaining only configurations with no removed polygon.
\end{proof}

\section{Open-strip pressure and the planar polygon tail}
\label{sec:05-open-pressure}

Essential polygons are now controlled. For large contractible polygons
we need a planar estimate, and the next transfer counts all polygons
inside a strip. Let \([P]\) range over unoriented planar polygons modulo all
triangular-lattice translations. Index the triangular bands by integers
and let \(H(P)\) be the difference between the highest and lowest
occupied band indices. Thus \(P\) occupies \(H(P)+1\) bands between
its extremes. The observable to be recovered from the derivative at
zero loop fugacity is
\[
 Q_N=\sum_{[P]}\rho^{|P|}(N-H(P))_+.
\]
The decisive precision is
\(Q_N=c_1N+c_2+O(N^{-1})\): positivity and comparison at two widths
will convert that error into a
\(N^{-2}\) height tail. Here translation classes are taken modulo the
full triangular lattice; this differs from the horizontal classes in
Proposition~\ref{prop:05-winding-suppression}.

\subsection{The variable-fugacity transfer and its boundary terms}

We first derive the transfer identities, including the boundary terms
created by varying the fugacity.

For the transfer construction, vary \(\lambda\) near \(\lambda_0=\pi/8\), and put
\(n=-2\cos(4\lambda)\), \(d=\sin(2\lambda)\).
Thus \(n\) is a local analytic coordinate, since \(dn/d\lambda=8\) at \(\lambda_0\). A closed cycle now receives weight \(n\). Define the raw two-slot box \(r(y)\) by
\begin{equation}
 \begin{aligned}
 e_y&=d\sin(3\lambda)+\sin y\sin(3\lambda-y),
 & A_y&=d\sin(3\lambda-y),\\
 U_y&=d\sin y,
 & B_y&=\sin y\sin(3\lambda-y),\\
 E_y&=\sin(2\lambda-y)\sin(3\lambda-y),
 & F_y&=\sin(y-\lambda)\sin y.
 \end{aligned}
 \label{eq:05-variable-box}
\end{equation}
These are the dilute \(O(n)\) weights listed in
\cite[equation~(1.2)]{ZhouBatchelor1997}, with the common factor
\(\sin(2\lambda)\sin(3\lambda)\) restored.
They are, respectively, the coefficients of vacancy, a single vertical strand, a cup or cap alone, a single diagonal strand, two vertical strands, and a cap together with a cup. Put
\[
 L_y=\sin(2\lambda+y)\sin(3\lambda+y),\qquad
 R_y=r(y)/L_y,\qquad P_y=r(y)/e_y.
\]
At \(\lambda_0\), \(e_y=L_y\), so both normalized boxes coincide with the original box. Use the same splitter diagrams, now with cup coefficients
\(q_2=(2\cos\lambda)^{-1}\), \(q_3=1\), and with occupied-leg coefficient \(q_2\) for \(S_2\).

\begin{lemma}[Local identities away from zero fugacity]
\label{lem:05-variable-algebra}
With \(\widehat P=P_{x+\lambda}\) for \(j=2\), and the identity for \(j=3\),
\begin{equation}
 \begin{gathered}
 R_yR_{-y}=I,\qquad P_{j\lambda}=S_jS_j^{\mathsf t},\\
 (P_x)_2(P_{x+j\lambda})_1(I\otimes S_j)
   =d_j(x)(S_j\otimes I)\widehat P,\qquad
 d_j(x)=1+nq_j\frac{A_{x+j\lambda}U_x}
                          {e_{x+j\lambda}e_x}.
 \end{gathered}
 \label{eq:05-variable-splitting}
\end{equation}
The raw boxes also satisfy the three-box Yang--Baxter identity.
\end{lemma}

\begin{proof}
For unitarity, the one-occupancy sector is the two-by-two matrix with diagonal \(A\) and off-diagonal \(B\). Direct substitution in \eqref{eq:05-variable-box} gives its inverse after normalization by \(L_yL_{-y}\). In the even-occupancy sectors the required identities are
\[
 \begin{gathered}
 e_ye_{-y}+nU_yU_{-y}=E_yE_{-y}=L_yL_{-y},\\
 e_{-y}U_y+(E_y+nF_y)U_{-y}=0,\\
 E_yF_{-y}+F_yE_{-y}+nF_yF_{-y}+U_yU_{-y}=0.
 \end{gathered}
\]
They prove \(R_yR_{-y}=I\). The factorization at \(j\lambda\) follows by substituting \(y=2\lambda,3\lambda\).

We record the coefficient check for splitting to retain the terms arising from the nonzero cycle weight. In the following tables, subscript \(1\) means argument \(x+j\lambda\), subscript \(2\) means argument \(x\), and every letter has been divided by the corresponding \(e\). Unsubscripted letters belong to \(\widehat P\); for deletion take \(B=1\) and the other nonempty two-slot coefficients zero. Write \(q=q_j\) and \(k=q_2\) for \(j=2\), \(k=0\) for \(j=3\). Outputs are numbered \(1,2,3\), with \(1,2\) the split slots; a parenthesized pair denotes an additional cup. In the last block a parenthesized output pairs the two input strands, whereas an unparenthesized pair denotes through strands. The coefficient equalities are
\[
\begin{array}{c|c|c|c}
\text{input}&\text{output}&\text{two boxes}&\text{fused value}\\\hline
00&\varnothing&d_j&d_j\\
&12&U_1A_2+qB_1B_2&d_jq\\
&13&U_1B_2+qB_1A_2&d_jkU\\
&23&U_2+qA_1(E_2+nF_2)&d_jkU\\\hline
10&1&A_1+q(F_1+nE_1)U_2&d_jkA\\
&2&B_1A_2+qU_1B_2&d_jkA\\
&3&B_1B_2+qU_1A_2&d_jB\\
&3,(12)&qF_1E_2&d_jqB\\
&1,(23)&A_1U_2+q(E_1E_2+F_1F_2+nE_1F_2)&0
\end{array}
\]
and, for \(j=2\),
\[
\begin{array}{c|c|c|c}
01&1&k(B_1+U_1U_2)&d_2kB\\
&2&k(A_1A_2+B_2)&d_2kB\\
&3&k(A_1B_2+A_2)&d_2A\\
&1,(23)&k(B_1U_2+U_1F_2)&0\\
&3,(12)&kU_1E_2&d_2qA\\\hline
11&\varnothing&k(U_1+B_1U_2)&d_2U\\
&12&k(E_1A_2+A_1B_2)&0\\
&(12)&kF_1A_2&d_2qU\\
&13&k(E_1B_2+A_1A_2)&d_2kE\\
&(13)&kF_1B_2&d_2kF\\
&23&kB_1E_2&d_2kE\\
&(23)&k(U_1U_2+B_1F_2)&d_2kF
\end{array}
\]
The terms proportional to \(n\) are exactly the closed cups in the contraction. These tables can be verified without diagrammatic simplifications: put \([s]=T^s-T^{-s}\), \(T=e^{i\lambda}\), and \(p=x/\lambda\). At argument \(x+l\lambda\), the normalized tuple \((A,B,U,E,F)\) is
\[
 \frac{([2][3-p-l],\,[p+l][3-p-l],\,[2][p+l],\,
         [3-p-l][2-p-l],\,[p+l-1][p+l])}
      {[2][3]+[p+l][3-p-l]}.
\]
Substitute \(q_2=[1]/[2]\), \(n=-(T^4+T^{-4})\), and clear denominators. Multiplication gives the displayed right columns, proving all occupancy sectors of \eqref{eq:05-variable-splitting}.

For the three-box identity, specialize its second argument to \(y=j\lambda\), \(j=2,3\). Splitting, transposition, and reflection reduce both sides to the same fused diagram with scalar \(d_j(x)\). Unitarity gives the further specializations \(y=j\lambda-x\); \(y=0\) follows from \(R_0=I\). After clearing scalar denominators, each occupancy component has exponents between \(-4\) and \(4\) in \(e^{iy}\), all of fixed parity. The five values modulo \(\pi\) just listed therefore determine the polynomial at generic \(x\), and rational continuation proves the identity everywhere it is defined.
\end{proof}

Let \(x_i\) be host arguments near \(\lambda/2\), and use the notation
\[
 \bar z=3\lambda-z,
 \qquad \mathfrak m(z)=\frac{\sin(3\lambda/2-z)}{\sin(3\lambda/2+z)},
 \qquad K(a)=\text{vacancy}+a\,\text{cup}.
\]
The bar here denotes this affine reflection, not complex conjugation. Define an open transfer \(t(z)\) on ordinary disk states by passing auxiliaries \([\bar z,z]\), the rightmost crossing each host first, with normalized box \(P_{\text{auxiliary}-\text{host}}\). Insert \(K(\mathfrak m(\bar z))\) at the bottom and contract with \(K(\mathfrak m(z))^{\mathsf t}\) at the top.
Figure~\ref{fig:05-open-transfer} records the auxiliary order and the
boundary tensors. The caps are part of the transfer: they contribute
the two scalar factors when auxiliaries are fused.

\begin{figure}[!hbp]
\centering
\begin{tikzpicture}[scale=1,every node/.style={font=\small}]
 \foreach \x/\lab in {-1/{\bar z},1/{z}} {
   \draw[blue!60!black,thick,->] (\x,.15)--(\x,4.3);
   \node[above] at (\x,4.3) {$\lab$};
 }
 \foreach \y/\lab in {1/{x_1},2/{x_k},3/{x_N}} {
   \draw[->] (2.6,\y)--(-2.5,\y);
   \node[right] at (2.6,\y) {$\lab$};
   \foreach \x in {-1,1} {
      \fill[white] (\x-.16,\y-.16) rectangle (\x+.16,\y+.16);
      \draw (\x-.16,\y-.16) rectangle (\x+.16,\y+.16);
   }
 }
 \node at (2.2,2.5) {$\vdots$};
 \node at (2.2,1.5) {$\vdots$};
 \draw[dashed,thick] (-1,.15) to[out=-90,in=-90] (1,.15);
 \draw[dashed,thick] (-1,3.8) to[out=90,in=90] (1,3.8);
 \node[below] at (0,-.4) {$K(\mathfrak m(\bar z))$};
 \node[above] at (0,4.5) {$K(\mathfrak m(z))^{\mathsf t}$};
 \node[left,align=right] at (-2.5,2) {host lines};
\end{tikzpicture}
\caption{The open transfer, drawn as an operator diagram rather than a
physical embedding. The ordered auxiliaries are \([\bar z,z]\), so the
rightmost line crosses each host first; each box has argument auxiliary
minus host. A dashed boundary arc denotes the tensor
\(K(a)=\text{vacancy}+a\,\text{cup}\), not a compulsory occupied arc.
At \(z=\bar z=3\lambda/2\) both cup coefficients vanish, leaving the
two physical columns with vacant auxiliary boundaries.}
\label{fig:05-open-transfer}
\end{figure}


\begin{lemma}[Open transfer and its pinches]
\label{lem:05-open-pinch}
The matrices \(t(z)\) commute and are invariant under \(z\leftrightarrow\bar z\). At \(\lambda=\lambda_0\), \(x_i=\lambda/2\), and \(z=3\lambda/2\), this is the transfer of two physical columns with vacant auxiliary boundaries. Its simple eigenvalue continuing \(1\) will be denoted by \(\Lambda_o(n;z)\). It is rational and \(\pi\)-periodic in \(z\), equals one at \(n=0\), and tends to \(1+n\) at both imaginary infinities. For \(u=x_i\), \(v=u+j\lambda\), its pinch rule is
\begin{equation}
 \Lambda_o(n;v)\Lambda_o(n;u)
 =c_bc_t\prod_kd_j(u-x_k)d_j(\bar v-x_k)
 \begin{cases}
 \Lambda_o(n;u+\lambda),&j=2,\\
 1,&j=3,
 \end{cases}
 \label{eq:05-open-pinch}
\end{equation}
where \(w_0=u+v-3\lambda\) and
\begin{equation}
 c_b=\frac{e_{w_0}+nq_j\mathfrak m(\bar u)A_{w_0}}{L_{w_0}},
 \qquad
 c_t=\frac{e_{-w_0}+nq_j\mathfrak m(v)A_{-w_0}}{L_{-w_0}}.
 \label{eq:05-boundary-scalars}
\end{equation}
\end{lemma}

\begin{proof}
Exchanging adjacent auxiliaries \(v,u\) uses the crossing \(R_{v-u}\) at the bottom, moved to the top by the three-box identity. For exchanging the two lines of one transfer, the boundary identity is
\[
 R_{\bar z-z}K(\mathfrak m(\bar z))
   \ \text{is proportional to}\ K(\mathfrak m(z)).
\]
The proportionality follows from, at \(y=\bar z-z\),
\[
 U_y+\mathfrak m(\bar z)(E_y+nF_y)
 =\mathfrak m(z)(e_y+n\mathfrak m(\bar z)U_y).
\]
The transposed inverse crossing contributes the reciprocal scalar, proving reflection symmetry.

For two pairs, the exchanges taking
\([\bar v,v,\bar u,u]\) to \([\bar u,u,\bar v,v]\) have operator product
\[
 (R_{-w_0})_2(R_h)_3(R_{-h})_1(R_{w_0})_2,
 \qquad h=v-u,\quad w_0=v+u-3\lambda,
\]
with the rightmost factor acting first. They take the product of bottom caps exactly to its reversal. By unitarity, this reduces to left-right symmetry of
\((r(h))_3(r(w_0))_2[K(a)\otimes K(b)]\),
where \(a=\mathfrak m(\bar v)\), \(b=\mathfrak m(\bar u)\). Only the paired output types \(12,34\) and \(13,24\) need comparison; their equalities are
\begin{equation}
 \begin{aligned}
 aA_{w_0}e_h+ab(nE_{w_0}+F_{w_0})U_h
   &=e_{w_0}U_h+bA_{w_0}(E_h+nF_h),\\
 (a-b)B_{w_0}A_h&=(1-ab)U_{w_0}B_h.
 \end{aligned}
 \label{eq:05-cap-exchange}
\end{equation}
Substitution from \eqref{eq:05-variable-box} and the sine formula for \(\mathfrak m\) proves them. Replacing \(v,u\) by \(\bar u,\bar v\) and transposing gives the top exchange, hence commutation.

At the physical center both cap coefficients vanish. Spectral simplicity of the open-strip transfer gives an analytic simple eigenspace nearby, invariant under all \(t(z)\) by commutation. The empty-coordinate rule at \(n=0\) makes its eigenvalue identically one. At imaginary infinity, \(A_y/e_y,U_y/e_y\to0\), \(B_y/e_y\to1\), and \(\mathfrak m(z)\mathfrak m(\bar z)\to1\). Thus the empty-output computation is \(1+n\). Occupancy signs prove periodicity, and computing on the fixed common eigenvector proves rationality.

For the pinch, exchange the middle lines using \(R_{w_0}\) on the bottom caps and \(R_{-w_0}\) on the top, reaching order \([\bar v,\bar u,v,u]\). The descending pair \(v,u\) crosses first. At host \(x_i=u\), the factorization of \(P_{j\lambda}\), together with \(P_0=I\), inserts a transpose splitter at input. Move it backwards through the other hosts by the transposed version of \eqref{eq:05-variable-splitting}. Its contraction against the exchanged bottom caps splits the first pair by \(S_j\), with scalar \(c_b\). In raw coefficients the necessary identities are
\begin{equation}
 \begin{aligned}
 aA_{w_0}+q_jab(nE_{w_0}+F_{w_0})
   &=q_j(e_{w_0}+nq_jbA_{w_0}),\\
 aB_{w_0}+abU_{w_0}
   &=U_{w_0}+bB_{w_0}\\
   &=\mathfrak m(\bar u-\lambda)
                  (e_{w_0}+nq_jbA_{w_0})\quad(j=2).
 \end{aligned}
 \label{eq:05-cap-fusion}
\end{equation}
Here \(a=\sin(v-3\lambda/2)/\sin(9\lambda/2-v)\), and likewise for \(b\); multiplying out proves \eqref{eq:05-cap-fusion}, just as for \eqref{eq:05-cap-exchange}. Move the first pair's splitter through the hosts by the reflected splitting identity. The reflected top calculation, with \(\bar u,\bar v\), gives \(c_t\). The remaining descending pair has both an input and output transpose splitter; transposed intertwining reduces it to the other fused transfer. Cancelling the surjective input splitter is valid as in the annular proof. Every crossed host contributes the two factors displayed in \eqref{eq:05-open-pinch}. This proves that identity for generic other sites, hence by continuation.
\end{proof}

\subsection{The forced interpolation problem}

We have obtained the open transfer and its exact boundary scalars.
We next differentiate its pinches to identify the forced interpolation
problem whose value at the physical center is the planar polygon
pressure. Define \(F_o(z)=\partial_n\Lambda_o(n;z)|_{n=0}\). In all formulas that follow, raw coefficients and auxiliary arguments are evaluated at \(\lambda_0\). Movement of evaluation arguments while differentiating costs nothing, because \(\Lambda_o(0;z)\) is constant. Introduce
\begin{equation}
 \begin{aligned}
 \phi_j(x)&=q_j\frac{U_xA_{x+j\lambda}}{e_xe_{x+j\lambda}},\\
 b_j(u)&=-\frac{U_{w_0}U_{-w_0}}{e_{w_0}e_{-w_0}}
  +q_j\left[
      \mathfrak m(\bar u)\frac{A_{w_0}}{e_{w_0}}
      +\mathfrak m(u+j\lambda)\frac{A_{-w_0}}{e_{-w_0}}
       \right],\qquad w_0=2u+(j-3)\lambda.
 \end{aligned}
 \label{eq:05-open-forcing}
\end{equation}
The first term of \(b_j\) comes from differentiating scalar unitarity on the vacuum sector. Formula~\eqref{eq:05-open-pinch} gives the remaining terms.

There is one additional evaluation. At \(z=0\), the top cap has \(\mathfrak m(0)=1\) and folds the two columns into one another. A quarter-turn of a raw box changes its argument to \(3\lambda-y\), and \(e_y=e_{3\lambda-y}\). Each folded pair therefore cancels by unitarity at \(x_k,-x_k\), contributing
\(1+nU_{x_k}U_{-x_k}/(e_{x_k}e_{-x_k})\); the remaining bottom factor is \(1+n\mathfrak m(3\lambda)\). Hence
\begin{equation}
 F_o(0)=\mathfrak m(3\lambda)
            +\sum_k\frac{U_{x_k}U_{-x_k}}{e_{x_k}e_{-x_k}}.
 \label{eq:05-folded-value}
\end{equation}
At \(8\lambda=\pi\), direct sine identities give
\begin{equation}
 \begin{gathered}
 \phi_3(x)=\phi_2(x)+\phi_2(x+\lambda),\qquad
 b_3(u)=b_2(u)+b_2(u+\lambda),\\
 b_2(\lambda)=\mathfrak m(3\lambda),\qquad
 \phi_2(\lambda-x)+\phi_2(-x)
       =\frac{U_xU_{-x}}{e_xe_{-x}}.
 \end{gathered}
 \label{eq:05-forcing-identities}
\end{equation}
Combining the two differentiated pinches using \eqref{eq:05-forcing-identities} shows that the even residual
\[
 F_o(z+\lambda)+F_o(z-\lambda)-F_o(z)
 -\sum_k[\phi_2(z-\lambda-x_k)+\phi_2(2\lambda-z-x_k)]
 -b_2(z-\lambda)
\]
vanishes at \(z=x_i+\lambda,x_i+2\lambda\) and their reflections about \(3\lambda/2\). Formula~\eqref{eq:05-folded-value} gives zeros at \(z=\lambda,2\lambda\) as well.

Now set every \(x_i=\lambda/2\) and use \(s=2i(z-3\lambda/2)\). To distinguish the physical center from the folded evaluation
\eqref{eq:05-folded-value}, write
\[
 \mathcal F_o(s)=F_o(3\lambda/2+s/(2i)).
\]
Thus \(\mathcal F_o(0)\) is the pressure derivative at the physical
center \(z=3\lambda/2\), while \(F_o(0)\) in
\eqref{eq:05-folded-value} is the folded value at \(z=0\). Set
\begin{equation}
 Y(s)=\mathscr L\mathcal F_o(s)-N\phi(s)-b(s),\qquad
 \phi(s)=\phi_2(y)+\phi_2(-y),\quad
 b(s)=b_2(\lambda/2+y),\quad y=s/(2i).
 \label{eq:05-open-residual}
\end{equation}
Confluence yields zeros of orders \(2N,N,N\) at \(0,i\beta,-i\beta\), and simple zeros at \(\pm i\beta/2\). The function \(\mathcal F_o\) is even, rational in \(e^s\), tends to one at both real infinities, and has poles of orders at most \(N+1\) at \(\pm3i\beta\), and \(2N\) at \(4i\beta\), modulo \(2\pi i\). These bounds count the two box denominators per host and the two boundary denominators. Differentiation does not increase the cleared denominator at zero fugacity: if \(A(n,s)/D(n,s)\) equals one at \(n=0\), its derivative is \((A_n-D_n)/D\) there.

Both forcing terms are rational in \(e^s\) and are \(O(e^{-|\operatorname{Re}s|})\) at real infinity. The function \(\phi\) is analytic in a neighborhood of \(|\operatorname{Im}s|\le h\), and \(b\) has at most simple poles at the two boundary points \(\pm ih\). To check the possibly closer poles of \(b\), use the raw coefficients at \(y=\lambda\):
\[
 \frac{U_{-2\lambda}}{A_{-2\lambda}}
    =\frac{\sin(-2\lambda)}{\sin(5\lambda)}
    =\mathfrak m(7\lambda/2),\qquad
 \frac{U_{2\lambda}}{e_{2\lambda}}=q_2.
\]
The two singular terms at \(y=\lambda\) cancel; reflection gives the cancellation at \(-\lambda\). This cancellation is independent of the hosts. Thus all preceding evaluations and coalescences, including the shifted \(\mathcal F_o\) terms, occur at regular points.

\begin{lemma}[Open-strip interpolation estimate]
\label{lem:05-open-interpolation}
For some fixed \(a\in(\gamma/2,1)\), and uniformly in \(N\),
\begin{equation}
 |Y(s)|\le C e^{-cNe^{-\gamma s}}
               (1+N^{a/\gamma}e^{-as}),\qquad s\ge0.
 \label{eq:05-open-error}
\end{equation}
Consequently
\begin{equation}
 \mathcal F_o(0)=(\rho_* *(N\phi+b))(0)+O(N^{-1}).
 \label{eq:05-open-center}
\end{equation}
\end{lemma}

\begin{proof}
The poles of \(\mathcal F_o-1\) determine the space of unknowns, whereas
zeros of \(Y\) determine the interpolation equations. These spaces have
the same dimension but different two-dimensional additions to their
common base. We first show that the resulting interpolation remains
uniformly invertible. We then remove the bulk forcing \(N\phi\), treat
the two boundary poles of \(b\), and estimate what remains.

\paragraph{The unknown and equation spaces.}
Use the Hardy space \(H\) on \( |\operatorname{Im}s|<3\beta/2\),
with norm \eqref{eq:05-hardy-norm}, and the multiplier \(D_a\) in
\eqref{eq:05-multiplier}. Let \(P\) project
onto the base jets of orders \(2N,N,N\) at \(0,i\beta,-i\beta\).
Let \(P'\) increase the two outer orders by one, and let \(P''\) instead
add the simple knots \(i\beta/2,-i\beta/2\). Their complementary inner
functions are \(B,B',B''\), respectively. The extra orders of \(P'\)
come from the two boundary denominators of the transfer; the extra knots
of \(P''\) come from the folded evaluation \eqref{eq:05-folded-value}.
Partial fractions, as in \eqref{eq:05-partial-fractions}, give
\begin{equation}
 e^{as}\mathscr L(\mathcal F_o-1)=D_ax,\qquad x\in P'H.
 \label{eq:05-open-partial-fractions}
\end{equation}
We first prove that
\begin{equation}
 P''D_a:P'H\longrightarrow P''H
 \quad\text{has inverse norm bounded independently of }N
 \label{eq:05-open-inverse}
\end{equation}
for all sufficiently large \(N\) and all \(a\) in a fixed neighborhood of \(\gamma/2\).

At \(a=\gamma/2\), the compression to \(PH\) is uniformly invertible by \eqref{eq:05-coercivity}, and \(D_a(k)\to d_0=1\) at both frequency infinities. The orthogonal decompositions of the two enlarged spaces are
\[
 P'H=PH\oplus B E',\qquad P''H=PH\oplus B E'',
\]
where \(E'\) and \(E''\) are the fixed two-dimensional jet spaces at \(\pm i\beta\) and \(\pm i\beta/2\), respectively. For each fixed vector \(e\) in either space,
\[
 \|(D_a-d_0I)Be\|_H+
 \|(D_a^*-d_0I)Be\|_H\longrightarrow0.
\]
Indeed \(B(s)\to0\) boundedly at every real point, so \(Be\to0\) in real-line \(L^2\) by domination. Its weighted Fourier mass on a fixed compact interval consequently tends to zero. Outside that interval \(D_a-d_0\) is uniformly small, while \(\|Be\|_H=\|e\|_H\). The same argument applies to the adjoint; finite dimensionality makes convergence uniform on unit balls. The off-diagonal blocks of \(P''D_a|_{P'H}\) therefore tend to zero, and its extra block tends to \(d_0\) times the overlap projection from \(E'\) to \(E''\). The cross-Gram matrix of the kernel-vector bases is, up to the fixed positive kernel constant,
\[
 \begin{pmatrix}
 \sec(\pi/4)&\sec(\pi/12)\\
 \sec(\pi/12)&\sec(\pi/4)
 \end{pmatrix},
 \qquad \det=2-\sec^2(\pi/12)>0.
\]
Thus the limiting extra block is invertible. The block inverse proves \eqref{eq:05-open-inverse} at \(a=\gamma/2\). Operator-norm continuity of \(D_a\), uniform for \(a\) in compact subsets of \((0,1)\), extends the bound to a fixed neighborhood. Choose \(a\) slightly larger than \(\gamma/2\) within it.

\paragraph{Removing the bulk forcing.}
Set \(\mu=\gamma-a\) and
\[
 X=D_a^{-1}(e^{as}\phi(s)).
\]
The unrestricted vector \(NX\) solves the bulk part of the equation
exactly. Its projection \(NP'X\) is an admissible unknown; the following
estimate ensures that projecting it introduces an error of order
\(N^{a/\gamma}\), rather than order \(N\). We claim
\begin{equation}
 \|(I-P')X\|_H\le C N^{-\mu/\gamma}.
 \label{eq:05-forcing-approximation}
\end{equation}
First, \(X\) extends to a slightly wider spatial strip and obeys
\begin{equation}
 |X(s)|\le C e^{-\mu|\operatorname{Re}s|}
 \quad\text{on that wider strip}.
 \label{eq:05-forcing-decay}
\end{equation}
Here is the required Fourier-contour justification. Since \(\phi\) is analytic on a wider strip, the transform of \(e^{as}\phi\) decays exponentially in \(|\operatorname{Re}k|\) at a rate greater than \(h\). Choose \(\epsilon>0\) such that
\[
 \mu+\epsilon<\min(\gamma/2,a).
\]
Its continuation between the horizontal frequency lines at heights \(\pm(\mu+\epsilon)\) has only a possible simple pole at \(i(1-a)\), originating in its leading positive-real tail. Explicitly, subtract a suitable multiple of \(e^{as}/(1+e^s)\). This removes that tail; the remaining negative-real tail is \(O(e^{as})\), sufficient because \(a>\mu+\epsilon\). Shifting the spatial contour inside the wider analytic strip proves the same large-frequency decay on all these continued frequency lines. Multiplication by \(D_a^{-1}\) cancels the pole at \(i(1-a)\) through its sine numerator. Its shift denominator introduces at most a simple pole at \(i\mu\); the poles arising from \(1/\cosh(3\beta k)\) begin at \(\pm i\gamma/2\), outside the chosen region. On horizontal frequency tails the bounds remain unchanged up to constants. Fourier inversion on the displaced contours, including the residue at \(i\mu\), proves \eqref{eq:05-forcing-decay}.

To estimate the complementary projection, pair \(X\) with \(B'G\), \(\|G\|_H=1\), using the two boundary integrals. On each boundary \(\overline{B'}=1/B'\). Move the upper integral a small fixed distance upward and the lower integral downward, extending the conjugate of \(G\) by reflection across the original boundary. Choose the distance so that the new lines remain in the wider strip of \eqref{eq:05-forcing-decay} and before the nearest poles of the finite Blaschke product. The reflected points are interior to \(H\), so the test functions have uniformly bounded \(L^2\) norms there. The contour move is justified first for finite linear combinations of reproducing kernels, which are dense and have exponential tails, and then by norm convergence. On the displaced lines, the coordinate \(e^{\gamma s}\) has negative real part. The explicit factors give
\[
 |1/B'(s)|\le C\exp(-cN e^{-\gamma|\operatorname{Re}s|}).
\]
Cauchy--Schwarz and \eqref{eq:05-forcing-decay} now bound the pairing by the square root of
\[
 C\int_{\mathbb R}e^{-2\mu|u|}
                  e^{-cN e^{-\gamma|u|}}\,du
 \le C N^{-2\mu/\gamma}.
\]
This proves \eqref{eq:05-forcing-approximation}.

\paragraph{Removing the boundary forcing.}
It remains to interpolate \(e^{as}(b-1)\), whose boundary poles prevent it from belonging directly to \(H\). Choose \(b_p\) with simple poles at \(\pm ih\), having the residues of \(b\) divided by the values of \(B''\) there. These values have modulus one. One explicit choice is a sum of the corresponding polar parts multiplied by Gaussians equal to one at their own pole; this gives uniform decay on every fixed horizontal line and a uniform bound on the real line. Put
\begin{equation}
 U(s)=e^{as}[b(s)-1+B''(s)-B''(s)b_p(s)].
 \label{eq:05-boundary-interpolant}
\end{equation}
The poles cancel, and \(U\) has the same prescribed jets as \(e^{as}(b-1)\). Furthermore
\begin{equation}
 \|U\|_H\le C N^{a/\gamma},\qquad
 |b(s)-1-e^{-as}U(s)|\le C|B''(s)|\quad(s\in\mathbb R).
 \label{eq:05-boundary-interpolant-bound}
\end{equation}
For the first assertion, \(e^{as}(B''-1)\) has norm \(O(N^{a/\gamma})\) by \eqref{eq:05-blaschke-tail}. Near either boundary pole, the canceled polar parts are bounded by
\(C\min(N,|u|^{-1})\), since the tangential derivative of \(B''\) is \(O(N)\) on the boundary. Their squared integral is \(O(N)\). Away from the poles, the rational tails and the Gaussian terms have bounded weighted norm. Since \(a/\gamma>1/2\), the pole contribution is absorbed. Removability of the poles and the boundary bounds imply \(U\in H\). The second assertion follows directly from \eqref{eq:05-boundary-interpolant} and the real-line bound on \(b_p\).

\paragraph{The residual and the center value.}
The exact jet equation for \eqref{eq:05-open-residual} is
\[
 P''(D_ax-ND_aX-U)=0.
\]
Compare \(x\) with \(NP'X\), using \eqref{eq:05-open-inverse}. Since
\(N\|(I-P')X\|_H\le C N^{a/\gamma}\),
we obtain
\[
 \|D_ax-ND_aX-U\|_H\le C N^{a/\gamma}.
\]
This difference is in \(B''H\), so divide by \(B''\) and use bounded real-line evaluation. Equations~\eqref{eq:05-open-partial-fractions} and \eqref{eq:05-boundary-interpolant-bound} give
\[
 |Y(s)|\le C|B''(s)|(1+N^{a/\gamma}e^{-as}).
\]
The product estimate \eqref{eq:05-inner-decay} proves \eqref{eq:05-open-error}. Evenness gives the corresponding estimate on the negative half-line.

Finally, the real-line functions are regular, and \(\mathcal F_o\) tends exponentially to one. Thus the inverse shift formula gives
\(\mathcal F_o=\rho_* *(N\phi+b+Y)\).
The error at zero is at most
\[
 C\int_0^\infty e^{-\gamma s}e^{-cNe^{-\gamma s}}
                    (1+N^{a/\gamma}e^{-as})\,ds
 =\frac{C}{\gamma N}\int_0^N e^{-cu}(1+u^{a/\gamma})\,du
 =O(N^{-1}),
\]
proving \eqref{eq:05-open-center}.
\end{proof}

\subsection{Polygon density and the diameter tail}

We now identify the analytic derivative with a positive polygon sum.
There are two normalization factors to keep track of: one transfer step
advances two columns, and a polygon shape has several vertical placements
in the strip.

\begin{proposition}[Linear open-strip pressure]
\label{prop:05-open-pressure}
For the polygon sum \(Q_N\) defined at the start of this section, there
are constants \(c_1,c_2\) such that
\begin{equation}
 Q_N:=\sum_{[P]}\rho^{|P|}(N-H(P))_+
       =c_1N+c_2+O(N^{-1}).
 \label{eq:05-strip-polygon-pressure}
\end{equation}
\end{proposition}

\begin{proof}
At the moving center \(z=3\lambda/2\), and with the hosts also taken at \(\lambda/2\), both auxiliary caps are vacancies for all nearby \(\lambda\). The local vacuum weights for \(P_y\) are identically one. Differentiating the finite-strip partition at \(n=0\) therefore counts exactly one polygon, with its original critical honeycomb weight: derivatives of local occupied weights have an additional factor \(n\) and disappear at zero. Changing the arguments with \(\lambda\) contributes nothing to the eigenvalue derivative, since the base eigenvalue is constant.

The transfer \(t\) advances two horizontal columns. Spectral dominance, as in \eqref{eq:05-cluster-derivatives}, identifies \(\mathcal F_o(0)\) with twice the polygon density per single column. Normalize a representative of a full translation class so that its lowest
occupied band is zero. Its lowest band can then be placed at precisely
\(0,1,\ldots,N-H(P)-1\) in the strip. Horizontal half-unit offsets
caused by a one-band translation do not affect density: for each of these
\((N-H(P))_+\) placements, integer horizontal translates have density one
per column. Monotone counting of finite longitudinal placements thus gives \(Q_N=\mathcal F_o(0)/2\). All passages from finite length use nonnegative polygon sums and the analytic finite-dimensional transfer; they require no interchange of an uncontrolled infinite derivative. Equation~\eqref{eq:05-open-center} proves \eqref{eq:05-strip-polygon-pressure}, with
\[
 c_1=\tfrac12(\rho_* *\phi)(0),\qquad
 c_2=\tfrac12(\rho_* *b)(0).
\]
\end{proof}

\begin{theorem}[Planar polygon tail]
\label{thm:05-planar-tail}\label{E:polygon-tail}
Uniformly for \(R\ge0\),
\begin{equation}
 \sum_{[P]:\,\operatorname{diam}(P)>R}\rho^{|P|}
       \le\frac{C}{(1+R)^2}.
 \label{eq:05-planar-tail}
\end{equation}
The sum is over unrooted, unoriented polygons modulo triangular-lattice translations.
\end{theorem}

\begin{proof}
Divide \eqref{eq:05-strip-polygon-pressure} by \(N\). Since
\((1-H(P)/N)_+\) increases to one, monotone convergence gives
\(\sum_{[P]}\rho^{|P|}=c_1<\infty\).
Define
\[
 D_N=c_1N-Q_N=\sum_{[P]}\rho^{|P|}\min(N,H(P)).
\]
This sequence is increasing, and
\(D_N=-c_2+O(N^{-1})\).
Choose an admissible integer \(M\asymp N/2\). Every polygon with \(H(P)>N\) contributes \(N-M\) to
\(\min(N,H(P))-\min(M,H(P))\), so
\[
 (N-M)\sum_{[P]:H(P)>N}\rho^{|P|}
 \le D_N-D_M=O(N^{-1}).
\]
Thus the height tail is \(O(N^{-2})\). This argument also works if the pressure formula is used only along even widths, by taking \(M,N\) even. Rotate the lattice to obtain the same bound in the three band directions. The Euclidean diameter is bounded by a constant times the largest of these three band spans, plus a fixed lattice constant. The union bound proves \eqref{eq:05-planar-tail} for large \(R\); total summability covers bounded \(R\).
\end{proof}

\section{Planar nests and strip-length consequences}
\label{sec:05-nesting}

The two pressure calculations have supplied the needed geometric inputs:
essential polygons have bounded total weight in the two-marker geometry,
and planar polygon shapes have a quadratic diameter tail. We first turn
the latter into a bounded removal cost for large contractible polygons.
The cylinder exponent then gives the planar and strip nesting exponents.
The final step converts these nest sums into length-weighted path sums.

\subsection{Removing large polygons and comparing two nests}

\begin{lemma}[Macroscopic removal]
\label{lem:05-macroscopic-removal}
Fix \(c_0>0\). In an \(N\)-band strip, the total weight of polygons enclosing a specified face center and having diameter greater than \(c_0N\) is bounded by a constant depending only on \(c_0\). The same is true of contractible polygons enclosing a specified marker on the balanced cylinder, with diameter measured on a planar lift. At each fixed nonnegative fugacity, deleting these polygons changes a nesting partition function by at most a bounded multiplicative factor.
\end{lemma}

\begin{proof}
For a planar shape of diameter comparable to \(R\), the number of translations fitting in the strip and enclosing the specified point is at most \(CN(1+R)\): there are \(O(N)\) possible vertical placements, and enclosure forces its horizontal range to contain the point. For dyadic \(R\ge c_0N\), Theorem~\ref{thm:05-planar-tail} bounds the weight of all such shapes by \(CR^{-2}\). Thus their total weight is bounded by
\[
 C\sum_{j\ge0}N(1+2^jc_0N)(1+2^jc_0N)^{-2}
 \le C_{c_0}.
\]
A simple contractible cylinder polygon lifts to translates of a simple planar polygon with disjoint interior disks. The boundary lifts are disjoint; if two of their bounded disks overlapped, planar separation would force nesting, which is impossible for distinct translates of equal-area bounded disks. Thus its disk projects injectively to the cylinder. For a fixed planar translation class, the number of placements modulo the period containing the marked point is again at most \(CN(1+R)\): there are \(N\) vertical translation classes modulo period and its horizontal range must straddle the marker. Counting all these placements, including those not valid under projection, is an upper bound. The same dyadic estimate applies.

For a fixed remaining nest, sum the removed polygons while ignoring disjointness both among themselves and with the nest. If their total single-polygon weight is at most \(C_{c_0}\), this contributes at most \(e^{\chi C_{c_0}}\). Positivity gives the reverse inequality by taking the empty removed set.
\end{proof}

Let \(P_\chi(r)\) be the planar partition over finite disjoint nests around one face center, with every polygon of diameter at most \(r\), each weighted by \(\chi\rho^{|P|}\); include the empty nest. Recall that \(Z_N(\chi)\) is the balanced-cylinder partition in which every polygon separates the two prescribed markers.

\begin{theorem}[Normalization by planar nests]
\label{thm:05-nest-normalization}\label{E:planar-nesting}
For every fixed \(\chi\ge0\), and every sufficiently small fixed
\(c_0>0\),
\begin{equation}
 P_\chi(c_0N)^2
 \le Z_N(\chi)
 \le C_{\chi,c_0}P_\chi(c_0N)^2
 \label{eq:05-cylinder-planar-comparison}
\end{equation}
for all sufficiently large even \(N\). Consequently, for every fixed \(\chi>0\),
\begin{equation}
 P_\chi(r)=r^{\tau(\chi)/2+o(1)}\qquad(r\longrightarrow\infty),
 \label{eq:05-planar-all-fugacity}
\end{equation}
where \(\tau\) is the exponent in Theorem~\ref{thm:04-cylinder-exponent}.
In particular,
\begin{equation}
 P_2(r)=r^{1/12+o(1)}\qquad(r\longrightarrow\infty).
 \label{eq:05-planar-nest-scaling}
\end{equation}
Here \(P_\chi(r)\) includes the empty nest, and the diameter cutoff
applies separately to every polygon in a nest.
\end{theorem}


\begin{figure}[!hbp]
\centering
\begin{tikzpicture}[scale=1.06,every node/.style={font=\small}]
 \draw[gray] (-3,0)--(3,0) (-3,4)--(3,4);
 \draw[gray,densely dotted] (-3,0)--(-3,4) (3,0)--(3,4);
 \node at (-3.3,2.6) {$\cdots$};
 \node at (3.3,2.6) {$\cdots$};
 \draw[gray,dashed] (-.577,0)--(0,1)--(-1.155,3)--(-.577,4);
 \foreach \xx/\yy in {0/1,-1.155/3} {
   \fill[blue!7] (\xx,\yy) circle (.55);
   \draw[blue!45!black] (\xx,\yy) circle (.55);
   \draw[blue!65!black] (\xx,\yy) ellipse (.25 and .20);
   \draw[blue!65!black] (\xx,\yy) ellipse (.15 and .12);
   \fill (\xx,\yy) circle (1.6pt);
 }
 \node[left] at (-.65,1) {$f_1$};
 \node[left] at (-1.805,3) {$f_2$};
 \draw[<->] (1.15,1)--(1.15,3)
   node[midway,right] {$N/2$ bands};
 \draw[<->] (-3.8,0)--(-3.8,4)
   node[midway,left,align=center] {period\\$N$ bands};
 \draw[->,gray] (2.6,-.15)--(2.6,.15);
 \draw[->,gray] (2.6,3.85)--(2.6,4.15);
 \node[below] at (1.1,-.1) {identified boundaries};
 \draw[->] (.5,3.65)--(-.8,3.4);
 \node[right] at (.55,3.65) {radius $c_0N$};
\end{tikzpicture}
\caption{A schematic fundamental band of the cylinder, with the lattice
suppressed. The dashed column cut follows the two balanced row blocks;
its marked seams have displacement $-N/4+iN\sqrt3/4$ and are separated
vertically by half the period. For small fixed $c_0$,
their radius-$c_0N$ neighborhoods are disjoint and embed isometrically
in the plane. After essential and larger polygons are deleted at bounded
cost, each remaining polygon belongs to a nest in one of these two
neighborhoods. The independent choices give the square of the planar
partition in \eqref{eq:05-cylinder-planar-comparison}.}
\label{fig:E-planar-comparison}
\end{figure}

\begin{proof}
The two cylinder markers have disjoint neighborhoods of radius
\(c_0N\), each isometric to its planar neighborhood, if \(c_0\) is
small enough: along the standard balanced column cut their displacement
is \(-N/4+iN\sqrt3/4\), and their vertical separation is half the
balanced period.
Figure~\ref{fig:E-planar-comparison} shows the two embedded neighborhoods.
Delete all separating essential polygons, using
Proposition~\ref{prop:05-winding-suppression}, and all contractible
polygons of lifted diameter greater than \(c_0N\), using
Lemma~\ref{lem:05-macroscopic-removal} at both markers. The product of
the deletion costs is uniformly bounded.

Every remaining polygon is contractible and encloses exactly one marker
in its disk. A planar simple curve of diameter at most \(r\) enclosing a
point lies in the radius-\(r\) neighborhood of that point: the point is
in the curve's convex hull, so its distance to each curve point is at
most the curve's diameter. Hence each remaining polygon lies in its
marker's embedded neighborhood. Conversely, a planar nest counted by
\(P_\chi(c_0N)\) projects into that neighborhood and separates the
cylinder markers. The neighborhoods are disjoint, so the two nests can
be chosen independently. Retaining just these configurations gives the
lower bound, and the preceding deletion gives the upper bound in
\eqref{eq:05-cylinder-planar-comparison}.

For each fixed \(\chi>0\), Theorem~\ref{thm:04-cylinder-exponent} gives
\(Z_N(\chi)=N^{\tau(\chi)+o(1)}\) along even \(N\). Taking logarithms
in the comparison yields
\(P_\chi(c_0N)=N^{\tau(\chi)/2+o(1)}\). Monotonicity in the radius and
bracketing an arbitrary radius between consecutive admissible values
prove \eqref{eq:05-planar-all-fugacity}. At \(\chi=2\),
\(\tau(2)=1/6\), giving \eqref{eq:05-planar-nest-scaling}.
\end{proof}

\subsection{Nests around the middle of a strip}

Let \(S_k\) be the partition sum of nests around a face center \(f\)
on the middle line of the strip \(\mathcal S_{2k}\) of \(2k\) triangular
bands. Thus \(S_k\) sums the finite vertex-disjoint nests enclosing \(f\),
with weight \(\prod_P2\rho^{|P|}\), including the empty nest.

\begin{corollary}[Middle-face nesting]\label{E:strip-nesting}
As \(k\longrightarrow\infty\),
\begin{equation}
 S_k=k^{1/12+o(1)}.
 \label{eq:C-nest}
\end{equation}
\end{corollary}
\begin{proof}
Choose a fixed \(c_0>0\) so small that every nest enclosing \(f\),
with each polygon of diameter at most \(c_0k\), fits inside the strip. By
Lemma~\ref{lem:05-macroscopic-removal}, deleting all larger polygons
has a bounded cost. Therefore
\[
 P_2(c_0k)\le S_k\le C P_2(c_0k).
\]
Apply Theorem~\ref{thm:05-nest-normalization}.
\end{proof}

\subsection{A positive comparison between length and nests}

We use the boundary winding identity to count visits to one internal
edge by summing over all ordered boundary sources and endpoints. This
local comparison will convert the nesting exponent into a length
exponent. Recall \(t=3/8\), \(c=\cos(3\pi/8)\), and \(d=1/\sqrt2\).

For a finite simply connected union $D$ of triangular tiles with simple
polygonal boundary and a boundary port $a$, the boundary identity of \cite[Lemma~2.1]{StripCompanion} is
\begin{equation}\label{eq:E-boundary-input}
 \sum_{b\in\partial D}\sum_{\gamma:a\to b}
       \rho^{n(\gamma)}e^{itW(\gamma)}=1,
\end{equation}
where $W$ measures signed turning from the inward initial direction to
the outward final direction. Its local rule sends the current weight
into the two new continuations with multipliers
$\rho e^{i\lambda}$ and $\rho e^{-i\lambda}$; their sum is one.
At a first self-collision, reversing the closing cycle gives the
canceling winding increments $\pm4\pi/3$. In a convex domain every
boundary-to-boundary path satisfies $|W|\le\pi$, and therefore
$c\le\cos(tW)\le1$. These conventions also specify the modified
interior-source calculation below.


For a finite convex union $D$ of triangular tiles and an internal
honeycomb mid-edge $p$, define
\[
 L_p(D)=\sum_{a,b\in\partial D}\ \sum_{\gamma:a\to b}
 \rho^{n(\gamma)}\cos(tW(\gamma))\,
 \mathbf1_{\{p\in\gamma\}}.
\]
The boundary ports are ordered, and $W(\gamma)$ is the total signed
turning from the initial inward direction to the final outward direction.
Convexity gives $c\le\cos(tW(\gamma))\le1$.
For a point $r$ not on a polygon, let $\mathcal Z_r(D)$ be the sum over
finite sets of pairwise vertex-disjoint simple polygons, all strictly
enclosing $r$, with weight $\prod_\ell 2\rho^{|\ell|}$; the empty set
has weight one. The same definition applies when $r$ is a mid-edge,
excluding polygons through that mid-edge.

\begin{lemma}[Length and polygon nests]\label{lem:02-length-nests}\label{E:length-nest}
Let $f_1,f_2$ be the two honeycomb face centers adjacent to $p$. Then
\[
 d\bigl(\mathcal Z_{f_1}(D)+\mathcal Z_{f_2}(D)\bigr)
 \le L_p(D)\le
 \mathcal Z_{f_1}(D)+\mathcal Z_{f_2}(D).
\]
The inequalities also hold, with finite quantities, in every homogeneous
infinite strip of fixed positive height.
\end{lemma}
\begin{proof}
Cut the edge at $p$ into two stopping outlets. For a boundary source
$a$, the winding splitting identity still has total one: every collision
stem approaches its closing cycle from outside, so the two orientations
cancel. Subtract this identity from the one without the cut. The total
winding weight of paths visiting $p$ equals that of paths from $a$
stopping at the cut. Reversing these paths and taking real parts gives
$L_p(D)=F_+(D)+F_-(D)$, where $F_\sigma$ is the real winding-weighted
sum of paths from the specified side $p^\sigma$ to the exterior. The
initial direction points into the neighboring vertex on that side.

Apply the same splitting procedure from this interior source, retaining
all stopping possibilities. A return to the other side of the cut traces
a polygon $\ell'$ through $p$. For a fixed initial side it is counted
once and has turning $\pm2\pi$, hence real factor
$\cos(2\pi t)=-d$. A collision with a polygon not enclosing $p$
cancels with its reversed orientation. If its polygon $\ell$ strictly
encloses $p$, the stem approaches from inside. The closing turn of the
counterclockwise polygon at the collision vertex is $-\pi/3$, whereas
the turn from the stem into that orientation is $+\pi/3$. The added
winding is therefore $8\pi/3$, or $-8\pi/3$ for the other orientation.
Both have factor $\exp(\pm it8\pi/3)=-1$.

For a fixed enclosing $\ell$, sum the stems in its interior, excluding
its vertices. This is exactly the corresponding $F_\sigma$ for the
smaller domain: the stem stops at the mid-edge before its aimed-at
vertex, whose weight belongs to $\ell$. If its initial direction already
points at an excluded vertex, this stem sum is one. No further inner
polygon is then possible. Write \(D_\ell\) for the domain strictly inside an enclosing polygon
\(\ell\), with its vertices excluded and with stopping outlets on the
incident half-edges. Conservation of the initial mass, followed by the
collision classification above, gives the recurrence
\[
 F_\sigma(D)=1+d\sum_{\ell'\ni p}\rho^{|\ell'|}
       +2\sum_{\ell\text{ enclosing }p}\rho^{|\ell|}F_\sigma(D_\ell).
\]
The factor two in the last sum comes from the two orientations of the
enclosing polygon, each with real factor \(-1\) on the stopped side
of the conservation identity. The initial-source convention just
specified gives \(F_\sigma(D_\ell)=1\) if its first vertex is excluded.

Iterate this recurrence, choosing an outermost enclosing polygon first
and then polygons strictly inside it. Every nested family is obtained
once in this order. The iteration terminates either with the constant
term one or with a polygon through \(p\); such a polygon must be the
innermost selected cycle: \(p\) lies on its boundary, so no disjoint
cycle strictly inside its bounded disk can enclose \(p\). Every enclosing cycle removes vertices,
so the iteration is finite. It gives, for either sign,
\[
 F_\sigma(D)=\mathcal Z_p(D)+d\sum_{\ell'\ni p}
       \rho^{|\ell'|}\mathcal Z_{\mathrm{around}}(\ell'),
\]
where $\mathcal Z_{\mathrm{around}}(\ell')$ sums the disjoint outer
polygons surrounding $\ell'$, with fugacity two per polygon. This
recursion remains valid if the opposite neighbor of $p$ is excluded;
in that case there is no return to the cut.

A polygon enclosing exactly one of $f_1,f_2$ uses their separating
edge. There can be at most one such polygon in a disjoint nest, and it
is the innermost polygon of that nest: any smaller polygon around the
same face would also have to use that edge. Consequently
\[
 \mathcal Z_{f_1}(D)+\mathcal Z_{f_2}(D)
 =2\mathcal Z_p(D)+2\sum_{\ell'\ni p}
       \rho^{|\ell'|}\mathcal Z_{\mathrm{around}}(\ell').
\]
Together with the preceding expression for $F_++F_-$ and $0k\)
bands from either boundary. Its centered \(2k\)-band strip therefore
fits inside \(\mathcal S_N\), so its nesting mass is at least \(S_k\).
There are at least \(c_1N\) such internal edge orbits, and
Lemma~\ref{lem:02-length-nests} gives a lower bound \(c_2NS_k\) for
their total winding-weighted visit mass. Any bottom-source path counted
there reaches the top boundary of the bottom \(k\) bands; the rotated statement holds
for a top source. Since the winding cosine is at most one and the
number of counted visits is at most the path length, the unnormalized
first-length mass of the conditioning event is at least \(c_3NS_k\).
Its upper bound is \(\mathcal M_N(a)\). By
Corollary~\ref{E:strip-nesting} and
Proposition~\ref{E:raw-first-moment}, these bounds both have exponent
\(13/12\). Division by \(q_N\) gives
\(N^{13/12+1/4+o(1)}=N^{4/3+o(1)}\).
\end{proof}
==== END Cylinder-loop-weights-and-planar-nesting//05_pressure.tex ====

==== BEGIN Cylinder-loop-weights-and-planar-nesting//06_all_count.tex ====
\section{A uniform bound for the number of separating polygons}
\label{sec:06-all-count}

The fixed-fugacity cylinder asymptotic determines the nesting exponent.
A different consequence of the same finite sector identity controls the
number of separating polygons, including counts that grow with the
width. For that purpose the fugacity must be allowed to grow without a
fixed upper bound. The real, untruncated sector estimate provides this
uniformity directly.

Use the physical balanced two-marker cylinder with even width
\(N=2m\), band parameters \(2\lambda\) on the first half and
\(\lambda\) on the second, and \(\lambda=\pi/8\). For each integer
\(a\ge0\), let \(w_N(a)\) be the critical mass of families of exactly
\(a\) pairwise vertex-disjoint polygons, each separating the two
markers. The weight of a family is \(\prod_P\rho^{|P|}\), with
\(\rho=(2+\sqrt2)^{-1/2}\) per polygon vertex. In particular the
factor two for a polygon is not included in \(w_N(a)\). The physical
partition is
\begin{equation}
 Z_N(\chi)=\sum_{a\ge0}w_N(a)\chi^a,
 \qquad w_N(a)\ge0,
 \qquad w_N(0)=1.
 \label{eq:06-positive-coefficients}
\end{equation}
This is the separating-loop contraction defined in
Section~\ref{sec:02-cylinder}. At fixed \(N\) it is a polynomial with
finite coefficients; a cut between the two markers bounds the number
of disjoint separating polygons.

\begin{theorem}[Uniform fugacity bound]\label{E:all-fugacity-bound}
There are constants \(C<\infty\) and \(N_0\) such that, for every
even \(N\ge N_0\) and every real \(v\ge0\),
\begin{equation}
 Z_N(2\cosh v)
 \le \exp\!\bigl(C\log N\,(1+v^2)\bigr).
 \label{eq:06-all-fugacity}
\end{equation}
The constants are independent of \(v\).
\end{theorem}
\begin{proof}
Set \(L=(\log N)/\gamma\), where \(\gamma=4/3\), and fix the
smoothing parameter used in the real-sector bound. It is fixed
independently of \(N\) and \(v\). We use the exact sector identity
\eqref{eq:G-22} at \(\eta=iv\), retaining all valid integer sectors
and dimensions. The imbalance \(M\) remains a real integer throughout
this proof. Thus \(v=\operatorname{Im}\eta\), whereas
\(\operatorname{Im}M=0\).

Use the normalization \eqref{eq:04-physical-normalization},
\[
 G_N(\eta)=\Delta_N Z_N(2\cos\eta).
\]
The scalar \(\Delta_N\) is nonzero and independent of \(\eta\).
We first bound \(G_N(iv)\), and then divide by \(|\Delta_N|\).

Recall the quantities in that identity. The explicit configuration
\(\zeta\) consists of exterior main particles and all far particles;
\(n_e\) counts its integration variables, with a fused double counted
as one variable, and
\[
 \ell_e=\sum_{z\in\zeta}(|\operatorname{Re}z|-L)_+.
\]
The interior dimensions \(m'_\sigma(M)\) are nonnegative integers
at most \(N\). Each explicit species retains its factorial divisor;
its elementary multiplicity is one for a singleton and two for a
fused double. The frozen prefactor has modulus at most
\(\exp(C_hn_e)\). It is independent of \(\eta\) and \(M\).

At \(\eta=0\), part 1 of
Proposition~\ref{prop:04-integer-bounds} gives
\begin{equation}
 |F_{m'(M)}(M)|
 \le\exp\!\left(C_hL+C_hn_e
       -c_h\left(\frac{M^2+n_e^2}{L}+\ell_e\right)\right).
 \label{eq:06-real-sector}
\end{equation}
Its dimension range \(0\le m'_\sigma\le C e^{\gamma L}\) includes
every valid sector because \(e^{\gamma L}=N\). In particular,
\eqref{eq:06-real-sector} requires no polynomial restriction on
\(M\), \(n_e\), or \(\ell_e\). That assertion uses the coarse real
reference-source inequality and compensated quadratic integration;
the later sharp source estimate and interpolation of dimensions are
not needed here.

The phase of the exact sector identity is
\(\exp(i(\eta+\pi f)M)\), where the far-particle integer \(f\) is
fixed by \(\zeta\). Replacing \(\eta=0\) by \(iv\) therefore
multiplies its modulus by \(e^{-vM}\le e^{v|M|}\). Every other factor
in \eqref{eq:06-real-sector} is unchanged. Combining the prefactor
and this estimate bounds the divided numerator by
\[
 e^{C_hL}
 \sum_{M\in\mathbb Z}e^{-c_hM^2/L+v|M|}
 \sum_{\zeta}\!\int
       e^{C'_hn_e-c_h\ell_e}.
\]
We have discarded the nonpositive \(n_e^2\) term and relaxed the
valid-count constraints, which only enlarges the positive majorant.
All integrals in the last sum retain their species factorials.

Here is the explicit-configuration estimate. For a far species its
real parameter ranges over \(\mathbb R\), and
\[
 \int_{\mathbb R}e^{-c_h(|u|-L)_+}\,du=2L+2/c_h.
\]
For an exterior main species it ranges over \(|u|>L\), giving
\(2/c_h\). Fixed contour-measure and activity constants can be
absorbed into \(e^{C'_h}\). For each of the finitely many species,
with parameter domain \(D_I\), the factorial sum is at most
\[
 \sum_{n=0}^{\infty}\frac{e^{C'_hn}}{n!}
 \left(\int_{D_I}e^{-c_h(|u|-L)_+}\,du\right)^n
 \le e^{C''_h(L+1)}.
\]
Taking the product gives \(e^{C_hL}\), for \(L\ge1\).
The elementary multiplicity two of a fused double affects the
admissibility constraint already dropped; it does not remove its
factorial divisor or introduce a second integration variable.

Only the imbalance sum remains. Young's inequality gives
\[
 -c_hM^2/L+v|M|
 \le-\frac{c_hM^2}{2L}+\frac{Lv^2}{2c_h},
\]
so
\[
 \sum_{M\in\mathbb Z}e^{-c_hM^2/L+v|M|}
 \le C_h\sqrt L\,e^{C_hLv^2}.
\]
Absorbing \(\sqrt L\) into \(e^{C_hL}\) proves
\( |G_N(iv)|\le e^{C_hL(1+v^2)}\) for the divided numerator
\(G_N\) in \eqref{eq:G-22}, uniformly for all \(v\ge0\).

Finally divide by the \(\eta\)-independent denominator \(\Delta_N\).
The balanced denominator theorem, Theorem~\ref{thm:02-denominator},
gives \(\log|\Delta_N|=(5/24)\log N+o(\log N)\). In particular
it is nonzero and \(|\Delta_N|^{-1}\le e^{C\log N}\) for all
sufficiently large even \(N\). This bound is independent of \(v\).
The physical value \(Z_N(2\cosh v)\) is nonnegative by
\eqref{eq:06-positive-coefficients}. Dividing the numerator estimate
and using \(L=(\log N)/\gamma\) proves
\eqref{eq:06-all-fugacity}.
\end{proof}

\begin{corollary}[Tail in the polygon count]\label{E:all-count-tail}
\label{lem:11-polygon-tail}
There are constants \(c,C>0\) such that for every sufficiently large
even \(N\) and every integer \(a\ge0\),
\begin{equation}
 w_N(a)\le
 \exp\!\left(C\log N-\frac{ca^2}{\log N}\right).
 \label{eq:L-40}
\end{equation}
\end{corollary}
\begin{proof}
For every \(v\ge0\), positivity in
\eqref{eq:06-positive-coefficients} gives
\[
 w_N(a)(2\cosh v)^a\le Z_N(2\cosh v).
\]
Since \(\log(2\cosh v)\ge v\),
Theorem~\ref{E:all-fugacity-bound} yields
\[
 w_N(a)\le\exp\!\left(C\log N(1+v^2)-av\right).
\]
Take \(v=a/(2C\log N)\). The exponent becomes
\(C\log N-a^2/(4C\log N)\), as required. This includes \(a=0\);
indices beyond the polynomial degree have zero mass.
\end{proof}
==== END Cylinder-loop-weights-and-planar-nesting//06_all_count.tex ====

==== BEGIN Cylinder-loop-weights-and-planar-nesting//main.tex ====
\documentclass[11pt,oneside]{article}
\ifdefined\pdftrailerid\pdftrailerid{}\fi
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\input{glyphtounicode.tex}
\input{glyphtounicode-cmex.tex}
\pdfglyphtounicode{mapsto}{007C}
\pdfglyphtounicode{vextendsingle}{23D0}
\pdfglyphtounicode{mu}{03BC}
\pdfglyphtounicode{Delta}{0394}
\pdfglyphtounicode{Omega}{03A9}
\pdfgentounicode=1

\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
\usepackage{booktabs,array,longtable}
\usepackage{microtype}
\usepackage{xcolor}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,positioning,calc}
\usepackage{hyperref}
\hypersetup{hypertexnames=false,colorlinks=true,linkcolor=blue!45!black,citecolor=green!35!black,urlcolor=blue!55!black,pdftitle={Cylinder loop weights and planar nesting},pdfauthor={OpenAI}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\C}{\mathbb C}
\newcommand{\R}{\mathbb R}
\newcommand{\Z}{\mathbb Z}
\newcommand{\N}{\mathbb N}
\newcommand{\E}{\mathbb E}
\newcommand{\Prob}{\mathbb P}
\DeclareMathOperator{\diam}{diam}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\dist}{dist}
\DeclareMathOperator{\Var}{Var}
\allowdisplaybreaks[2]
\setlength{\emergencystretch}{2em}
\title{Cylinder loop weights and planar nesting}
\author{OpenAI}
\date{September 26, 2026}
\usepackage{accsupp}
\newcommand{\MathActualText}[2]{%
  \BeginAccSupp{method=hex,unicode,ActualText=#1}%
  #2%
  \EndAccSupp{}%
}
\let\OriginalMapsto\mapsto
\renewcommand{\mapsto}{\mathrel{\MathActualText{21A6}{\OriginalMapsto}}}
\let\OriginalHookrightarrow\hookrightarrow
\renewcommand{\hookrightarrow}{\mathrel{\MathActualText{21AA}{\OriginalHookrightarrow}}}
\let\OriginalLongrightarrow\longrightarrow
\renewcommand{\longrightarrow}{\mathrel{\MathActualText{27F6}{\OriginalLongrightarrow}}}

\begin{document}
\maketitle
\begin{abstract}
We determine the growth exponent, at every fixed positive loop fugacity, of the critical honeycomb partition function for disjoint polygons separating two prescribed markers on a balanced cylinder. At fugacity two the cylinder exponent is $1/6$; the corresponding planar nesting exponent and middle-strip nesting exponent are $1/12$.
\end{abstract}
\tableofcontents
\input{sections/00_introduction}
\input{sections/02_cylinder}
\input{sections/03_gas_mean}
\input{sections/03_reference}
\input{sections/04_gas_fluctuations}
\input{sections/04b_interpolation}
\input{sections/04c_exponent}
\input{sections/06_all_count}
\input{sections/05_pressure}
\clearpage
\phantomsection
\addcontentsline{toc}{section}{References}
\input{references}
\end{document}
==== END Cylinder-loop-weights-and-planar-nesting//main.tex ====

==== BEGIN Cylinder-loop-weights-and-planar-nesting//references.tex ====
\begin{thebibliography}{99}
\bibitem{Cardy2006Annulus}
J.~Cardy. The $O(n)$ model on the annulus.
\emph{Journal of Statistical Physics} \textbf{125} (2006), 1--21.
\href{https://doi.org/10.1007/s10955-006-9186-8}{doi:10.1007/s10955-006-9186-8}.
\href{https://arxiv.org/abs/math-ph/0604043}{arXiv:math-ph/0604043}.

\bibitem{DiFrancescoZinnJustin2005}
P.~Di Francesco and P.~Zinn-Justin.
Around the Razumov--Stroganov conjecture: proof of a multi-parameter sum rule.
\emph{Electronic Journal of Combinatorics} \textbf{12} (2005), Research Paper~6.
\href{https://doi.org/10.37236/1903}{doi:10.37236/1903}.
\href{https://arxiv.org/abs/math-ph/0410061}{arXiv:math-ph/0410061}.

\bibitem{DiaconisEvans2001}
P.~Diaconis and S.~N.~Evans.
Linear functionals of eigenvalues of random matrices.
\emph{Transactions of the American Mathematical Society}
\textbf{353} (2001), 2615--2633.
\mbox{\href{https://doi.org/10.1090/S0002-9947-01-02800-8}{doi:10.1090/S0002-9947-01-02800-8}}.

\bibitem{DiaconisShahshahani1994}
P.~Diaconis and M.~Shahshahani.
On the eigenvalues of random matrices.
\emph{Journal of Applied Probability} \textbf{31A} (1994), 49--62.
\href{https://doi.org/10.1017/S0021900200106989}{doi:10.1017/S0021900200106989}.

\bibitem{DCS2012}
H.~Duminil-Copin and S.~Smirnov.
The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt2}$.
\emph{Annals of Mathematics} \textbf{175} (2012), 1653--1665.
\href{https://doi.org/10.4007/annals.2012.175.3.14}{doi:10.4007/annals.2012.175.3.14}.

\bibitem{DuminilCopinHammond2013}
H.~Duminil-Copin and A.~Hammond. Self-avoiding walk is sub-ballistic.
\emph{Communications in Mathematical Physics} \textbf{324} (2013), 401--423.
\href{https://doi.org/10.1007/s00220-013-1811-1}{doi:10.1007/s00220-013-1811-1}.
\href{https://arxiv.org/abs/1205.0401}{arXiv:1205.0401}.

\bibitem{GamsaCardy2006}
A.~Gamsa and J.~Cardy.
Correlation functions of twist operators applied to single self-avoiding loops.
\emph{Journal of Physics A: Mathematical and General}
\textbf{39} (2006), 12983--13003.
\href{https://doi.org/10.1088/0305-4470/39/41/S12}{doi:10.1088/0305-4470/39/41/S12}.
\href{https://arxiv.org/abs/math-ph/0606065}{arXiv:math-ph/0606065}.

\bibitem{GarbaliNienhuis2017Ground}
A.~Garbali and B.~Nienhuis.
The dilute Temperley--Lieb $O(n=1)$ loop model on a semi infinite strip:
the ground state.
\emph{Journal of Statistical Mechanics: Theory and Experiment} (2017), 043108.
\href{https://doi.org/10.1088/1742-5468/aa6a30}{doi:10.1088/1742-5468/aa6a30}.
\href{https://arxiv.org/abs/1411.7020}{arXiv:1411.7020}.

\bibitem{GarbaliNienhuis2017Sum}
A.~Garbali and B.~Nienhuis.
The dilute Temperley--Lieb $O(n=1)$ loop model on a semi infinite strip:
the sum rule.
\emph{Journal of Statistical Mechanics: Theory and Experiment} (2017), 053102.
\href{https://doi.org/10.1088/1742-5468/aa6bc3}{doi:10.1088/1742-5468/aa6bc3}.
\href{https://arxiv.org/abs/1411.7160}{arXiv:1411.7160}.

\bibitem{GM2019}
A.~Glazman and I.~Manolescu.
Self-avoiding walk on $\mathbb Z^2$ with Yang--Baxter weights: universality of critical fugacity and 2-point function.
\emph{Annales de l'Institut Henri Poincar\'e, Probabilit\'es et Statistiques}
\textbf{56} (2020), 2281--2300.
\href{https://doi.org/10.1214/19-AIHP1024}{doi:10.1214/19-AIHP1024}.

\bibitem{Grafakos1997}
L.~Grafakos.
Best bounds for the Hilbert transform on $L^p(\mathbb R^1)$.
\emph{Mathematical Research Letters} \textbf{4} (1997), no.~4, 469--471.
\href{https://grafakos.missouri.edu/preprints/pichorides.pdf}{Author's text}.

\bibitem{GrafakosSavage2015}
L.~Grafakos and T.~Savage.
Best bounds for the Hilbert transform on $L^p(\mathbb R^1)$: A corrigendum.
\emph{Mathematical Research Letters} \textbf{22} (2015), no.~5, 1333--1335.
\href{https://grafakos.missouri.edu/preprints/grafakos-savage2.pdf}{Author's text}.

\bibitem{HammersleyWelsh1962}
J.~M.~Hammersley and D.~J.~A.~Welsh.
Further results on the rate of convergence to the connective constant
of the hypercubical lattice.
\emph{The Quarterly Journal of Mathematics} \textbf{13} (1962), 108--110.
\href{https://doi.org/10.1093/qmath/13.1.108}{doi:10.1093/qmath/13.1.108}.

\bibitem{IkhlefCardy2009}
Y.~Ikhlef and J.~Cardy.
Discretely holomorphic parafermions and integrable loop models.
\emph{Journal of Physics A: Mathematical and Theoretical}
\textbf{42} (2009), 102001.
\href{https://doi.org/10.1088/1751-8113/42/10/102001}{doi:10.1088/1751-8113/42/10/102001}.
\href{https://arxiv.org/abs/0810.5037}{arXiv:0810.5037}.

\bibitem{IshikawaOkadaTagawaZeng2006}
M.~Ishikawa, S.~Okada, H.~Tagawa and J.~Zeng.
Generalizations of Cauchy's determinant and Schur's Pfaffian.
\emph{Advances in Applied Mathematics} \textbf{36} (2006), 251--287.
\href{https://doi.org/10.1016/j.aam.2005.07.001}{doi:10.1016/j.aam.2005.07.001}.
\href{https://arxiv.org/abs/math/0411280}{arXiv:math/0411280}.

\bibitem{MadrasSlade1993}
N.~Madras and G.~Slade.
\emph{The Self-Avoiding Walk}.
Birkh\"auser, Boston, 1993.

\bibitem{DLMF18}
National Institute of Standards and Technology.
\emph{Digital Library of Mathematical Functions}, Chapter~18:
Orthogonal Polynomials.
\href{https://dlmf.nist.gov/18.18.E28}{Equation~18.18.28 (Mehler)};
\href{https://dlmf.nist.gov/18.2.E12}{equation~18.2.12 (Christoffel--Darboux)}.

\bibitem{Nienhuis1982}
B.~Nienhuis.
Exact critical point and critical exponents of $O(n)$ models in two dimensions.
\emph{Physical Review Letters} \textbf{49} (1982), 1062--1065.
\href{https://doi.org/10.1103/PhysRevLett.49.1062}{doi:10.1103/PhysRevLett.49.1062}.

\bibitem{StripCompanion}
OpenAI.
Critical strip-crossing mass on the honeycomb lattice.
OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026/main.pdf}{OAI:Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026},
2026.

\bibitem{Sarason2007}
D.~Sarason. Algebraic properties of truncated Toeplitz operators.
\emph{Operators and Matrices} \textbf{1} (2007), 491--526.
\href{https://doi.org/10.7153/oam-01-29}{doi:10.7153/oam-01-29}.

\bibitem{Simon2005Szego}
B.~Simon. The sharp form of the strong Szeg\H{o} theorem.
In \emph{Geometry, Spectral Theory, Groups, and Dynamics},
Contemporary Mathematics \textbf{387}, American Mathematical Society,
2005, 253--275.
\href{https://arxiv.org/abs/math/0402110}{arXiv:math/0402110}.

\bibitem{TracyWidom1998}
C.~A.~Tracy and H.~Widom.
Correlation functions, cluster functions and spacing distributions
for random matrices.
\emph{Journal of Statistical Physics} \textbf{92} (1998), 809--835.
\href{https://doi.org/10.1023/A:1023084324803}{doi:10.1023/A:1023084324803}.
\href{https://arxiv.org/abs/solv-int/9804004}{arXiv:solv-int/9804004}.

\bibitem{WienerHopf1931}
N.~Wiener and E.~Hopf.
\"Uber eine Klasse singul\"arer Integralgleichungen.
\emph{Sitzungsberichte der Preussischen Akademie der Wissenschaften,
Physikalisch-Mathematische Klasse} (1931), 696--706.

\bibitem{ZhouBatchelor1997}
Y.~K.~Zhou and M.~T.~Batchelor.
Critical behaviour of the dilute $O(n)$, Izergin--Korepin and dilute
$A_L$ face models: Bulk properties.
\emph{Nuclear Physics B} \textbf{485} (1997), 646--664.
\href{https://doi.org/10.1016/S0550-3213(96)00654-2}{doi:10.1016/S0550-3213(96)00654-2}.
\href{https://arxiv.org/abs/cond-mat/9611156}{arXiv:cond-mat/9611156}.
\end{thebibliography}
==== END Cylinder-loop-weights-and-planar-nesting//references.tex ====