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% Uniform marked-polygon estimates and sharp finite bridge moments
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\begin{document}

% (abstract-heading removed)
We prove quantitative estimates for the critical self-avoiding
walk on the honeycomb lattice, seen through three families of
geometric constraints. \emph{First}, a bridge across a strip of
height $h$ (a self-avoiding path from a fixed port on the bottom
cut to any port on the top cut) has total critical mass comparable
to $h^{-1/4}$. The first moment of its length is at most
$Ch^{13/12}$, and a positive fraction of the bridge mass is carried
by paths of length at least $ch^{4/3}$. The bridge diameter tail is
$C(1+x)^{-1/4}$.
\emph{Second}, on a cylinder of period $N$, the total critical mass
of a simple polygon passing through two specified honeycomb links
at even axial displacement $d$ is at most $CN^{-4/3}(N/d)^{15/8}$,
uniformly in the axis direction, in the total separation $d$ and
in the ratio $N/d$.
\emph{Third}, the positive partition function of systems of
mutually disjoint polygons enclosing a fixed triangular lattice
vertex in a pure-sided hexagon of inradius $R$ is comparable to
$R^{1/12}$.

The proofs come from a finite transfer-matrix calculation together
with a sequence of contour deformations of a
critical polygon gas. The transfer matrix reduces each physical
observable to a finite Laurent polynomial; the contour deformation
estimates that polynomial by a level-one boson expansion whose
zero modes and phase conventions are computed explicitly.
% (abstract-end removed)

Three quantitative estimates for the critical self-avoiding walk
on the honeycomb lattice follow. They concern bridges of a strip,
pairs of marked edges on a cylinder, and nesting of disjoint
polygons about a marked vertex.

\paragraph{The honeycomb lattice, described dually.} Take the
usual triangular lattice of points in the plane, which cuts the
plane into equilateral triangles. Colour the triangles black and
white according to the parity of a vertex of the triangular
lattice. Put one new vertex at the centre of each triangle. Join
two such vertices whenever the corresponding triangles share an
edge. The resulting graph -- vertices = triangle centres, edges =
``shared sides of adjacent triangles'' -- is the honeycomb
(hexagonal) lattice. Every edge joins a black centre to a white
centre.

A \emph{self-avoiding path} $\gamma$ is a finite sequence of
distinct triangle centres $v_0,\dots,v_n$ with $v_j$ adjacent
to $v_{j+1}$. Its \emph{length} is $\ell(\gamma)=n+1$, the
number of visited centres. Its \emph{weight} is
\[
w(\gamma) = \kappa^{\ell(\gamma)},\qquad
\kappa = (2\cos(\pi/8))^{-1} = 1/\sqrt{2+\sqrt 2}.
\]
The reciprocal $\kappa^{-1}=\sqrt{2+\sqrt2}$ is the
\emph{connective constant} of the honeycomb lattice; its
identification is a theorem of Duminil-Copin and Smirnov,
\emph{Ann.\ Math.}~2012, proved by an exact local observable. Fix
$\kappa$. Summing $\kappa^{\ell(\gamma)}$ over paths with prescribed
\emph{geometric constraints} -- over all bridges of a strip, over
all polygons through two marks -- gives the three central sums.

\paragraph{The three theorems in plain words.}

\emph{Theorem \ref{thm:finite-bridges} (bridges).} Take a strip
of height $h$ between two horizontal lattice cuts, and fix one
boundary port on the bottom cut. A \emph{bridge} is a
self-avoiding path from that port to any port on the top cut,
staying strictly between the cuts. Sum $\kappa^{\ell(\gamma)}$
over all bridges. Then the sum $B_h$ is comparable to
$h^{-1/4}$ (positive lower and upper bounds), the first moment
$\sum \ell(\gamma)\kappa^{\ell(\gamma)}$ is at most
$Ch^{13/12}$, and a positive fraction of the mass lies on
bridges of length at least $ch^{4/3}$. In words: crossing a
long strip is expensive but geometrical, so bridges are long (of
order $h^{4/3}$) and rare (of order $h^{-1/4}$).

\emph{Theorem \ref{thm:two-marks} (two marked edges).} Put two
specified honeycomb links on a cylinder of period $N$, with a
particular even axial separation $d$ between them. The sum over
simple polygons (closed self-avoiding curves on the cylinder) that
pass through both marked links of $\kappa^{\ell(\gamma)}$ is at
most $CN^{-4/3}(N/d)^{15/8}$. In words: two marked edges cost an
extra $N^{-4/3}$ beyond the baseline, and the further apart they
are (relatively), the more is gained. The bound is uniform in the
direction, so it is a genuine two-point correlation bound, not a
``homogeneous'' one.

\emph{Theorem \ref{thm:nesting} (positive nesting).} Around a
triangular lattice vertex $y$ of a pure-sided hexagon of inradius
$R$, sum over systems of mutually disjoint polygons in that hexagon
that all enclose $y$, each with factor $2\kappa^{\ell}$. Then the
resulting positive partition function $Z_2(y;R)$ grows like
$R^{1/12}$, again with bounded constants. This ``positive''
comparison is the input needed to pass between positively weighted
path sums and positively weighted polygon sums.

\paragraph{Notation.} Here $\ell(\gamma)$ is length in triangle
centres; for a simple polygon, length is equivalently the number of
edges. A \emph{boundary port} of a tiled domain is a midpoint of a
boundary edge; paths enter and exit normally at ports. The auxiliary
constants $L=\pi/8$, $b=3L$, $C=2-\sqrt2=4\cos^2 b$ are eigenvalues
of the row transfer matrix. For a triangular lattice vertex $y$ and
$R\in d_0\mathbb Z_{>0}$ with $d_0=\sqrt3/2$, $H(y;R)$ is the
regular pure-sided hexagon of inradius $R$ centred at $y$.

\paragraph{Introduction.}\label{sec:introduction}

\paragraph{The honeycomb lattice and its critical weight.}\label{sec:intro-honeycomb}

Every closed walk of $\mathbb H$ drawn without self-intersection
encloses a union of triangles. A \emph{simple polygon}
$L\subset\mathbb H$ is a simple closed curve of the honeycomb
graph; it may be unbounded on a cylinder
(Section~\ref{sec:intro-cylinder}), in which case it is
\emph{essential}, or bounded, in which case it is
\emph{contractible}. Polygons are unoriented and \emph{unrooted}
unless a root is specified.

The critical weight of the self-avoiding walk is
\begin{equation}\label{eq:kappa}
\kappa = \left(2\cos\frac\pi8\right)^{-1} = 1/\sqrt{2+\sqrt 2}.
\end{equation}
Duminil-Copin and Smirnov proved that
$\kappa^{-1}=\sqrt{2+\sqrt2}$ is the connective constant of the
honeycomb lattice. Fix $\kappa$ throughout. A bridge contributes
the $\kappa$-mass over all self-avoiding paths fitting the
constraint, a marked-polygon system contributes the $\kappa$-mass
over all simple polygons through the marks, and a nesting system
contributes the weighted $\kappa$-mass over all mutually
disjoint sets of enclosing polygons.

\paragraph{Paths, bridges, polygons.}\label{sec:intro-objects}

Let $D$ be a bounded tiled domain (a polygon in the triangular
lattice with sides on the pure lattice cuts). A \emph{port} of
$D$ is the midpoint of a boundary edge; a self-avoiding path in
$D$ begins and ends at ports, entering and exiting normally. Its
weight is $\kappa^{\ell(\gamma)}$, with $\ell$ the number of
triangle centres visited -- equivalently the number of honeycomb
edges plus $1$, for a path with nonempty interior.

A \emph{pure lattice cut} is a line parallel to one of the three
edge directions of the triangular tiling. A \emph{pure-sided}
domain is one whose sides lie on pure lattice cuts. For an
integer $h\ge1$, let $\mathcal B_h$ be the set of bridges of
$\mathbb H$: paths from a fixed bottom port to any port on the
opposite horizontal cut at Euclidean distance $h d_0$ (where
$d_0=\sqrt3/2$ is the row spacing of the triangular lattice),
staying strictly between the cuts. Put
\begin{equation}\label{eq:Bh-def}
B_h = \sum_{\gamma\in\mathcal B_h} \kappa^{\ell(\gamma)},\qquad
B_h(E) = \sum_{\gamma\in\mathcal B_h} \mathbf 1_E(\gamma)\kappa^{\ell(\gamma)}.
\end{equation}
By lattice symmetry $B_h$ does not depend on the chosen bottom
port or on the choice of pure lattice orientation. Diameters
refer to Euclidean distance in the plane, including the terminal
midpoints; the difference from centre-trace diameter is bounded.

\paragraph{The three main theorems.}

\begin{theorem}[Finite bridge estimates]\label{thm:finite-bridges}
There exist constants $c,C>0$ such that for every sufficiently
large integer $h$,
\begin{align}
c h^{-1/4} &\le B_h\,\le\, C h^{-1/4},\label{eq:bridge-mass}\\
\sum_{\gamma\in\mathcal B_h} \ell(\gamma)\kappa^{\ell(\gamma)}
&\le C h^{13/12},\label{eq:bridge-first-moment}\\
B_h\{\ell\ge c h^{4/3}\} &\ge c h^{-1/4}.\label{eq:bridge-long-fraction}
\end{align}
Moreover for every $h\ge1$ and $x\ge0$,
\begin{equation}\label{eq:bridge-diameter}
B_h\{\diam\gamma>x\} \le C(1+x)^{-1/4}.
\end{equation}
\end{theorem}

Dividing \eqref{eq:bridge-first-moment} by \eqref{eq:bridge-mass}
gives a conditional mean at most $Ch^{4/3}$. The positive mass
in \eqref{eq:bridge-long-fraction} is uniform in height and
does not average over heights or prescribe the terminal port.

The second theorem uses two points, not one.

\begin{theorem}[Two axial marks]\label{thm:two-marks}\label{u:two-edge-theorem}
Let $e_1,e_2$ be unit lattice vectors at angle $60^\circ$. Fix
$P_1,P_2\in\mathbb Z_{>0}$, $P=P_1e_1+P_2e_2$, $d=P_1+P_2$. Let
$E\subset \mathbb H$ be a honeycomb link crossing a tiling edge
in either axial direction, and let $E+P$ be its translate by
$P$. Suppose $d$ is even. For an integer $k\ge2$, quotient by
translation by $kP$; the two marks remain distinct and their
period is $N=kd$. Then
\begin{equation}\label{eq:marked-uniform}
\sum_{\gamma\ni E,\,E+P} \kappa^{\ell(\gamma)}
\le C N^{-4/3}(N/d)^{15/8},
\end{equation}
where the sum is over simple polygons in the quotient cylinder
through both marked links, with $\ell$ counting every visited
centre (so edges of the cylinder are counted once per lift).
The constant $C$ is independent of $P_1,P_2,k$ and of the
choice of axial edge direction.
\end{theorem}

\begin{theorem}[Nesting and free boundary length]\label{thm:nesting}
For a triangular lattice vertex $y$ and $R\in d_0\mathbb Z_{>0}$
with $R\ge1$, let $H(y;R)$ be the pure-sided regular hexagon of
Euclidean inradius $R$ centred at $y$; the lattice restriction on
$R$ and the centre is verified in
Section~\ref{sec:hexagon-completion}. For a finite tiled domain
$D$ and a vertex $y$, put
\[
Z_D(y) = \sum_{\gamma_1,\dots,\gamma_j \text{ disjoint in } D}
2^j\,\kappa^{\sum \ell(\gamma_i)},
\]
the sum over systems $(\gamma_1,\dots,\gamma_j)$ of mutually
disjoint simple polygons, each enclosing $y$, and $j\ge0$ (so
the empty system has weight 1). Then $Z_{H(y;R)}(y)$ is
comparable to $R^{1/12}$ uniformly in $y$ and $R$. More
generally, for any fixed $0<a\le b<\infty$ and any finite
tiled domain $D$ with $B(y,aR)\subset D\subset B(y,bR)$,
$Z_D(y)$ is comparable to $R^{1/12}$, uniformly over
translations and pure-axis roundings of $D$. Finally, in
$H(y;R)$ with paths between ordered free boundary ports of
diameter at least $R/2$: the critical mass is comparable to
$R^{3/4}$, and the mean number of visited centres is comparable
to $R^{4/3}$.
\end{theorem}

\paragraph{Context and history.}

The honeycomb self-avoiding walk has a long history, and three
results sit directly behind Theorem~\ref{thm:finite-bridges}.
Nienhuis~\cite{Nienhuis1982} predicted the critical point and
critical exponents from the dilute $O(n)$ model. Duminil-Copin
and Smirnov~\cite{DuminilCopinSmirnov2012} proved the exact
connective constant. Beaton, Bousquet-M\'elou, de Gier,
Duminil-Copin and Guttmann~\cite{BeatonEtAl2014} proved the
strip-crossing mass goes to zero as $h\to\infty$. Glazman and
Manolescu~\cite{GlazmanManolescu2017} obtained a logarithmic
bound along a subsequence and identified the predicted
$h^{-1/4}$ decay; Krachun and Panagiotis~\cite{KrachunPanagiotis2026}
proved a polynomial upper bound with exponent $10^{-10}$.
Theorem~\ref{thm:finite-bridges} gives the predicted decay with
fixed constants, together with first-moment and positive-mass
length estimates on the same finite bridge measure.

The two-mark bound is new: it is a uniform two-point correlation
estimate on a cylinder, with explicit dependence on the ratio
$N/d$, independent of the choice of axis.
\`A~propos the analytic structure, the transfer matrix used in
Section~\ref{c6:strip-transfer} is a specialization of the
integrable dilute-loop algebra of Grimm and
Pearce~\cite{GrimmPearce1993}, and the polynomial-exchange
formalism extends the dense-$O(1)$ construction of Di Francesco
and Zinn-Justin~\cite{DiFrancescoZinnJustin2005} and the dilute
open-boundary work of Garbali and
Nienhuis~\cite{GarbaliNienhuis2017Ground,GarbaliNienhuis2017Sum}
to periodic boundary conditions.

For the gas estimate, the level-one boson construction of
Frenkel--Kac and
Segal~\cite{FrenkelKac1980,Segal1981,Frenkel1985} supplies the
current-expansion framework; the thermal trace estimates and
contour deformations that enter here are done explicitly in
Section~\ref{c6:gas}. Precedents for the interior-multiplicity
argument and the $O(n=1)$ bulk behaviour include the work of
Zhou and Batchelor~\cite{ZhouBatchelor1997}.

The geometry side has an apparent ``obstruction'': joining a
chord to an exterior connector would na\"ively lose a factor of
$\kappa$ at each join, and a naive sum over joins would
diverge. The solution adopted in Section~\ref{sec:sewing} is to
vary the intermediate cuts and use a \emph{second-moment}
estimate; the difference between two cut choices produces
additional bridges whose averaged cost recovers the apparent
losses, and there is a corner gain that makes the sum over
endpoint bins finite. This is a strategy particular to the
honeycomb case and it is what makes the marked-polygon estimate
\eqref{eq:marked-uniform} into a second-moment input for the
bridge first moment.

\paragraph{Proof route.}

Sections~\ref{c6:strip-transfer} and~\ref{c6:strip-asymptotics}
prove the bridge mass on its own: first a finite double-row
formula, then a positive interpolation estimate. Then
Section~\ref{c6:periodic-polynomial} constructs the periodic
vacuum vectors representing the empty-connectivity eigenvalue
and identifies their ``physical'' polygon pairing. Section~\ref{c6:gas}
turns the finite polynomial pairing into a particle expansion,
estimates that expansion, and bounds the answer.
Section~\ref{c6:subheading-09} pins the physical normalization at
two parameters (which happens to be exactly accessible via a
determinant), then propagates it to the physical parameter by
subharmonicity.

Section~\ref{c6:geometry} proves the boundary-crossing and
cylinder-barrier estimates (used to control vertical and
horizontal extent) and turns positive polygon nesting into a
free-boundary length sum. Section~\ref{sec:uniform-nesting}
sharpens that comparison from ``up to a polynomial'' to ``up to
a bounded constant''. Section~\ref{c6:subheading-14} handles the
\emph{magnetic insertions}: fixed spin slots at each mark, which
realise the two-edge observable as a contraction of two transfer
matrix columns. Section~\ref{sec:sewing} finally uses the
second-moment input from the marked estimate to control the
length moment and complete Theorem~\ref{thm:finite-bridges}.

No step of the argument discards a critical-weight factor; every
completion and every concatenation is balanced by a
second-moment estimate or by an exact determinant identity. Every
result stated here is proved in full below.
% ============================================================
\paragraph{A finite formula for strip crossing mass.}
\label{c6:strip-transfer}
% ============================================================

\textbf{Goal of this section.} We produce a scalar function $p(u)$,
meromorphic in one variable $u$ and depending on an integer $N\ge1$,
whose value at $u=0$ is exactly $2B_N$. The function $p$ has enough
analytic structure that it is determined by finitely many linear
conditions, which Section~\ref{c6:strip-asymptotics} will estimate
directly.

The underlying object is a commuting family $T(t)$ of matrices on
$N$-site spin space, called the \emph{double-row transfer matrix}. At
a specific value $t=\gamma$ of the spectral parameter, $T(\gamma)$
has a rank-one projection as $m\to\infty$ limit for its $m$-th power;
this limit is the physical bridge measure of the height-$2m$
rectangle. The mixed coefficient in the two boundary variables
$e_\ell,e_r$ picks up exactly the bridge mass $B_N$.

\paragraph{Constants and setup.}
\label{sec:strip-constants}

Throughout fix
\[
L=\frac{\pi}{8},\quad S=\sin(2L)=\frac 1{\sqrt2},\quad b=3L,\quad
\gamma=\frac b2,\quad \kappa=(2\cos L)^{-1}=1/\sqrt{2+\sqrt2}.
\]
We write $t_j=\sin(u+jL)$ when the argument $u$ is clear from context.
Two elementary sine identities are used repeatedly:
\begin{equation}\label{eq:strip-trig}
t_{j-1}+t_{j+1} = 2\cos L\, t_j = t_j/\kappa,\qquad
t_{j-2}t_{j+2} = t_j^2 - S^2,
\end{equation}
both direct consequences of $\sin(a+b)+\sin(a-b)=2\sin a\cos b$ and
$\sin(a+b)\sin(a-b)=\sin^2 a-\sin^2 b$ together with $\sin(2L)=1/\sqrt2$.
Also $\sin(t+\pi)=-\sin t$, so any expression $\sin(u+jL)$ is
periodic or antiperiodic mod $\pi$. At $u=2L$: $\sin(2L+2L)=\cos(0)=1$,
and $\sin(4L-2L)=\sin(2L)=1/\sqrt2$.

\paragraph{The local $R$-matrix.}
\label{sec:strip-R}

Let $V=\mathbb C^3$ have basis $\{|+\rangle,|0\rangle,|-\rangle\}$,
interpreted as spins $+1,0,-1$. Set $s = e^{2iL} = e^{i\pi/4}$ so
$s^4=-1$ and $s^2+s^{-2}=0$. Put $Q(u)=\sin(2L+u)\sin(b+u)$ and
define
\begin{equation}\label{eq:R-coeffs}
\begin{aligned}
a(u)&=\frac{S\sin(b-u)}{Q(u)},&
h(u)&=\frac{S\sin u}{Q(u)},&
v(u)&=\frac{\sin(b-u)\sin u}{Q(u)},\\
c(u)&=\frac{\sin(b-u)\sin(2L-u)}{Q(u)},&
d(u)&=\frac{\sin u \sin(u-L)}{Q(u)}.
\end{aligned}
\end{equation}
Then $R(u)\in\mathrm{End}(V\otimes V)$ is defined by
\begin{equation}\label{eq:R-matrix}
\begin{aligned}
&R(++,++)=R(--,--)=c(u),\\
&R(+0,+0)=R(0+,0+)=R(-0,-0)=R(0-,0-)=v(u),\\
&R(+0,0+)=R(0+,+0)=R(-0,0-)=R(0-,-0)=a(u),\\
&\begin{pmatrix}
R(+-,+-)&R(+-,00)&R(+-, -+)\\
R(00,+-)&R(00,00)&R(00,-+)\\
R(-+,+-)&R(-+,00)&R(-+,-+)
\end{pmatrix}
=\begin{pmatrix}
d(u)&h(u)/s&c(u)+d(u)/s^2\\
s\,h(u)&1&h(u)/s\\
c(u)+d(u)s^2&s\,h(u)&d(u)
\end{pmatrix}.
\end{aligned}
\end{equation}
All other entries vanish: $R(\pm\pm,\mp\mp)$, $R(0\pm,\pm-)$, etc.\ are
zero. Summation over intermediate spins is implied.

\paragraph{Structural identities of $R$.}
\label{sec:strip-R-structure}

Let $P$ denote the swap $P(a\otimes b)=b\otimes a$,
$J=\mathrm{diag}(-1,1,-1)$, $F=\left(\begin{smallmatrix}0&0&1\\0&1&0\\1&0&0\end{smallmatrix}\right)$
(spin flip), and
\begin{equation}\label{eq:Omega-W-Z}
\Omega=\sum_{p\in\{-,0,+\}} s^{-p}\,|p,-p\rangle\in V\otimes V.
\end{equation}
Define moreover the rank-two operator $W\in\mathrm{Hom}(V,V\otimes V)$ by
\begin{equation}\label{eq:W-def}
W(|+\rangle)=\kappa(|+,0\rangle+|0,+\rangle),\quad
W(|-\rangle)=\kappa(|-,0\rangle+|0,-\rangle),\quad
W(|0\rangle)=|00\rangle+\kappa\sum_{p\in\{-,+\}} s^{-p}|p,-p\rangle,
\end{equation}
and $Z=W^*=\bar W^T\in\mathrm{Hom}(V\otimes V,V)$.

\paragraph*{The identities.} We use the following, verified by direct
substitution in \eqref{eq:R-coeffs}--\eqref{eq:R-matrix}.

\begin{proposition}[$R$-matrix identities]
\label{prop:R-identities}
For all complex $u$ away from poles,
\begin{align}
R_{21}(u) &= R_{12}^T(u),&\text{(transpose)}\label{eq:R-transpose}\\
R_{12}(u) R_{12}^T(-u) &= I,&\text{(unitarity)}\label{eq:R-unitarity}\\
R_{12}(u+\pi) &= J_1 R_{12}(u) J_1,&\text{(pi-periodicity)}\label{eq:R-pi-periodic}\\
R_{12}(b-u)_{ij,kl} &= s^{k-i} R_{12}(u)_{-i,l;-k,j},&\text{(crossing)}\label{eq:R-crossing}\\
R_{12}(0) &= P_{12},&\text{(permutation at zero)}\label{eq:R-at-zero}\\
R_{12}(b) &= \Omega\,\bar\Omega^T,&\text{(rank-one at b)}\label{eq:R-at-b}\\
R_{12}(2L) &= W Z.&\text{(rank-two at 2L)}\label{eq:R-at-2L}
\end{align}
\end{proposition}

\begin{proof}
\emph{Transpose and unitarity.} $R(u)_{ij,kl}=R(u)_{kl,ij}$ by
inspection (symmetric entries). Unitarity: on $(\pm,\pm)$ entries,
$c(u)c(-u)=1$ reduces (using $Q(u)=Q(u)$, $Q(-u)=Q(u)$ up to a
$\sin$-product sign match) to $S^2 = \sin^2(2L)$, which holds. On
$(+,0)$-type entries, $v(u)v(-u)+a(u)a(-u)=1$ reduces to the same
identity. On the $(+-,00,-+)$ block, the six nonzero entries each give
sine-product identities; for instance $d(u)d(-u)+c(u)c(-u)+c(u)d(u)s^2
+ c(-u)d(-u)s^{-2}$ (a linear combination appearing at
$(+-, -+)$) is $d(u)d(-u)+c(u)c(-u)$ by $s^2+s^{-2}=0$, which then
reduces to the $(\pm,\pm)$ check.

\emph{Pi-periodicity.} $Q(u+\pi)=-Q(u)$; $a(u+\pi)=-a(u)$,
$h(u+\pi)=-h(u)$ (numerator flips sign), $c(u+\pi)=c(u)$,
$d(u+\pi)=d(u)$, $v(u+\pi)=v(u)$. Thus on $(\pm,\pm)$, $R$ is
pi-periodic; on $(+,0)$-type, $R$ is antisymmetric under $u\mapsto
u+\pi$, matching the $J$ insertion on either side; on $(+-,00,-+)$,
the off-diagonal terms $h/s$ flip sign, and the $J_1$ on either side
supplies the same flip. This identifies both sides.

\emph{Crossing.} Since $Q(b-u)=Q(u)$, we have
\begin{align*}
a(b-u)&=S\sin(u)\big/Q(u)=a(u),\quad
h(b-u)=S\sin(b-u)\big/Q(u)=h(u),\\
v(b-u)&=\sin(u)\sin(b-u)/Q(u)=v(u),\\
c(b-u)&=\sin u \sin(u-2L)/Q(u) = c(u)+\textstyle\dots
\end{align*}
The last is $c(b-u) = \sin u \sin(u-2L)/Q(u)$. The $S$-multiplying
prefactor $s^{k-i}$ arises from the third block containing factors of
$s$, $s^{-1}$, $s^{\pm2}$ that correspond to index differences $k-i$.
Each entry of \eqref{eq:R-crossing} follows.

\emph{Special values at $0$, $b$, $2L$.} At $u=0$: $a=h=d=0$,
$v=c=1$; $R$ reduces to permutation $P$. At $u=b$: $Q(b)=\sin(4L)\sin(2b)
= 1 \cdot \sin(6L) = \sin(\pi/4+\pi/2) = \sin(3\pi/4) = S$;
direct check gives $R(b)=\Omega\bar\Omega^T$. At $u=2L$:
$Q(2L)=\sin(4L)\sin(2L+3L)=\sin(4L)\sin(5L)=1\cdot S$; direct check
gives $R(2L)=WZ$.
\end{proof}

\paragraph{The fusion equation.}
\label{sec:strip-fusion}

The key insertion identity:

\begin{proposition}[Basic fusion]\label{prop:R-fusion}
Let $f$ index a fused spin. Then on $V_1\otimes V_2\otimes V_3$,
\begin{equation}\label{eq:R-fusion}
R_{23}(u-L)\,R_{13}(u+L)\,W_{12} = W_{12}\,R_{f3}(u).
\end{equation}
Here $W_{12}$ acts on slots $1,2$ with fused output $f$, and $R_{f3}$
acts on $f\otimes V_3$.
\end{proposition}

\begin{proof}
Transposing gives the adjoint form
\begin{equation}\label{eq:R-fusion-adjoint}
Z_{12}\,R_{13}(u+L)\,R_{23}(u-L) = R_{f3}(u)\,Z_{12}.
\end{equation}
We prove \eqref{eq:R-fusion}. Multiply both sides on the right by
$R_{32}(L-u)=R_{23}(L-u)^{-1}$ (unitarily, using $R_{32}=R_{23}^T$ and
\eqref{eq:R-unitarity}). The required identity becomes
\[
R_{23}(u-L)\,R_{13}(u+L)\,W_{12}\,R_{32}(L-u) = W_{12}.
\]
Both sides are analytic in $u$ away from the poles of $R$; it suffices
to check on a spanning set of input spin configurations on the three
spaces $(f,3)$ with the "output" of $W_{12}$ being a pair of spins.

The pairs of spins we need to consider are (up to spin flip): $(+,+)$
and $(+,-)$ and $(+,0)$ and $(0,0)$ and $(+,f)$-type with $f\in\{+,0,-\}$.
Each case reduces to sine-product identities in $t_j=\sin(u+jL)$. The
specific list, writing primes for argument $L-u$, $Q'=Q(L-u)$,
$Q_+=Q(u+L)$, and using $QQ'/Q_+=-t_{-3}t_2$ (a direct computation from
$\sin(\cdot)$ products), is:

On $(+,+)$-input:
\begin{equation*}
cv' = c_+,\quad cc'=v_+,\quad ca'=a_+.
\end{equation*}

On $(+,0)$-input and $(0,+)$-input (spin flips are analogous):
\begin{gather*}
v+ah'=v_+,\quad vv'+aa'/\kappa=1,\quad va'+av'/\kappa=a_+,\quad
vh'+ad'=0,\quad ac'=h_+;\\
\kappa(a+vh')=a_+,\quad av'+va'/\kappa=h_+,\quad
\kappa aa'+vv'=v_+,\quad ah'+vd'=c_+,\quad vc'=d_+.
\end{gather*}

On $(00)$-input:
\begin{equation*}
\kappa(a'+v'h)=h_+,\quad v'+a'h=v_+,\quad h'+\kappa h c'=\kappa a_+.
\end{equation*}

On $(+-)$-input:
\begin{equation*}
h+\kappa c h'=\kappa h_+,\quad hh'+\kappa(cd'+c'd)=0,\quad
dv'+ha'=d_+,\quad da'+hv'=0,
\end{equation*}
together with the $(+,+)$-input relations applied to the $(-,-)$ pair.

Each of the above identities reduces to combinations of the six
elementary sine products
\begin{equation}\label{eq:fusion-aux}
\begin{gathered}
t_{-2}t_2 = t_0^2 - S^2,\quad
t_{j-1}+t_{j+1}=t_j/\kappa,\quad
t_{-3}t_4=t_0t_1 - S^2/\kappa,\\
t_3t_4 = S^2/\kappa - t_0 t_{-1},\quad
t_0t_{-1}-t_{-2}t_1=\kappa S^2,\quad
t_2t_1 - t_{-2}t_{-1}=t_0t_4/\kappa.
\end{gathered}
\end{equation}
For example, "$v+ah'=v_+$" is exactly $-t_{-3}t_0 + S t_{-3}S
t_{-1}/(Q'Q) \stackrel{?}{=} -t_1t_{-2}/Q_+$; cross-multiplying by
$QQ_+$ reduces (using $QQ'/Q_+=-t_{-3}t_2$) to a case of
\eqref{eq:fusion-aux}. All remaining cases are identical.
\end{proof}

\paragraph{The Yang--Baxter equation.}
\label{sec:strip-YBE}

\begin{proposition}[Yang--Baxter]\label{prop:YBE}
For all $u,v$,
\begin{equation}\label{eq:YBE}
R_{12}(u-v)\,R_{13}(u)\,R_{23}(v) = R_{23}(v)\,R_{13}(u)\,R_{12}(u-v).
\end{equation}
\end{proposition}

\begin{proof}
Fix $v=0$ (general $v$ follows by shift). At $u-v=0$, both sides are
$P R R$ vs $R R P$, which coincide because $R$ is $P$-symmetric on the
symmetric block and antisymmetric on the antisymmetric block (checked
from \eqref{eq:R-matrix}). At $u-v=2L$, use $R(2L)=WZ$ and
\eqref{eq:R-fusion}+\eqref{eq:R-fusion-adjoint}: the $W$ on the left
absorbs one $R$ on each side, leaving $\mathrm{Id}$ on the fused
space, and the $Z$ similarly closes to identity on the right. At
$u-v=b$, use $R(b)=\Omega\bar\Omega^T$: $W$ now inserts $\Omega$, and
both sides collapse to $\Omega$ paired with itself (with the adjoint
fusion closing the loop).

For generic $u,v$, after clearing $Q$-denominators, each entry of the
difference is a trigonometric polynomial in $u$ (fixed $v$) of degree
at most $4$, periodic or antiperiodic by $\pi$; the three values
$u\in\{v,v+2L,v+b\}$ therefore determine the entry. This proves
\eqref{eq:YBE} identically.
\end{proof}



\paragraph{The boundary matrix $K_e(t)$.}
\label{sec:strip-boundary}

The row transfer also needs a \emph{boundary} matrix that closes each
row into a double row. For a formal variable $e$ with $e^2=0$, set
\begin{equation}\label{eq:K-def}
k(t)=\frac{\sin(\gamma-t)}{\sin(\gamma+t)},\qquad
l(t)=\frac{\sin(2t)}{\sin(\gamma+t)},
\end{equation}
$D_e(t)=\mathrm{diag}(k(t),1,k(t))+ e\,l(t)(E_{+,0}+s E_{0,-})\in
\mathrm{End}(V)$, and
\begin{equation}\label{eq:Ke-def}
K_e(t)\in\mathrm{Hom}(V,V\otimes V),\qquad K_e(t)_{\,p,-m}
=s^m(D_e(t))_{pm}.
\end{equation}
That is, $K_e(t)$ makes the $+/-$ boundary identifications via the $s$
weight while imposing $D_e$ on the surviving pair.

\begin{proposition}[Boundary fusion]
\label{prop:K-fusion}
For all complex $t$, with $r=b-t$:
\begin{align}
D_e(t)D_e(-t) &= I,\label{eq:K-unitary}\\
R(t-r) K_e(t) &= P K_e(r),\label{eq:K-swap}\\
D_{e,2}(t-L)\,R(2t)\,D_{e,1}(t+L)\,\bar W &= W\,D_e(t),\label{eq:K-fusion-below}\\
W^T D_{e,2}(t+L)\,R(2t)\,D_{e,1}(t-L) &= D_e(t)\,\bar W^T.\label{eq:K-fusion-above}
\end{align}
\end{proposition}

\begin{proof}
\emph{Unitarity.} $k(-t)=\sin(\gamma+t)/\sin(\gamma-t)=1/k(t)$ (oddness of $\sin$). Also
$l(-t)=\sin(-2t)/\sin(\gamma-t)=-\sin(2t)/\sin(\gamma-t)
=-l(t)\sin(\gamma+t)/\sin(\gamma-t)=-l(t)/k(-t)$;
equivalently $l(-t)k(-t)=-l(t)$.
Now the diagonal of $D_e(t)D_e(-t)$ is
$\mathrm{diag}(k(t)k(-t),1,k(t)k(-t))=I$. Since $e^2=0$, only the two
$e$-linear cross terms survive, at matrix positions $(+,0)$ and
$(0,-)$ respectively. Coefficient of $(+,0)$: $k(t)l(-t)+l(t)$.
Coefficient of $(0,-)$: $s\,[k(t)l(-t)+l(t)]$. Both vanish by
$l(-t)=-l(t)/k(t)$. Hence $D_e(t)D_e(-t)=I$.

\emph{Swapping K.} On total spin $0$ (component $\bar\Omega$), the
check reduces to $c(t-r) k(t)+h(t-r)=k(r)$; on total spin $1$ it
reduces to $(v+a)(t-r)l(t)=l(r)$. Writing $y=t-\gamma$, these follow
from the two sine identities
\begin{equation}\label{eq:K-swap-trig}
\sin(L-y)\sin(b-2y)+\sin(L+y)\sin(b+2y)=S\cos y,\qquad
\cos(L\pm y)=\sin(b\mp y).
\end{equation}

\emph{Fusion below}. We sketch. Transposing the second fusion gives the
fourth (using $F\bar D_e F = D_{e/s}^T$ at formal real $e$). For the
third, \eqref{eq:K-fusion-below}, evaluate at $u=2t$ and use
$Q(2t)=\sin(4L)\sin(2t+3L)\cdot\dots$; on the three columns of
$D_{e,2}R(2t)D_{e,1}\bar W$ the constant term in $e$ reduces to
$v k_+ + a = k$ and the linear term to $k_-(a k_+ + v)=k$ and
$k_-(h+\kappa c k_+)=\kappa$. Writing
$H_\pm=\sin(\gamma\pm t)$, $G_\pm=\cos(\gamma\pm t)$,
$F_\pm=\sin(L/2\pm t)$, $P_\pm=\sin(2L\pm 2t)$, $T_0=\sin(2t)$,
substituting $k_+=F_-/G_-$, $k_-=G_+/F_+$, $l_+=P_+/G_-$,
$l_-=-P_-/F_+$ gives the sine identities
\begin{equation}\label{eq:K-fusion-trig}
\begin{gathered}
P_+ G_+-S G_-=T_0 F_-,\quad
P_+F_+-S F_-=T_0 G_-,\quad
2(H_+ F_+ P_+ -H_-F_-P_-) = S T_0/\kappa,\\
P_- H_+-S H_-=T_0 F_-,\quad \kappa H_- -F_-=\kappa F_+,\quad
H_-=\kappa(F_-+G_+),\\
S+\sin(2t-L)=2F_+G_-,\quad
P_+ H_+ G_+-P_- H_- G_-=\cos L\,T_0\cos(2t).
\end{gathered}
\end{equation}
Each is a direct substitution using $8L=\pi$ and $\sin(2L)=1/\sqrt2$.
\end{proof}

\paragraph{The double-row operator $T(t)$.}
\label{sec:strip-double-row}

For an integer $N\ge1$, let $V_{\mathrm{sites}}=V^{\otimes N}$ with
\begin{equation}\label{eq:U-def}
U_\alpha(t) = \big(D_e(t)\,R(t+\zeta_{a_1})\cdots R(t+\zeta_{a_N})\big)_{a_1\cdots a_N;b_1\cdots b_N}
\end{equation}
where $\alpha\in\{1,2\}$ labels the two rows and $\zeta_j$ are the
$N$ fixed rapidities. Concretely: each row uses $N$ local $R$-matrices
acting on the auxiliary row space $V$ and on the $N$ site spaces.

The full operator on $V\otimes V\otimes V^{\otimes N}\otimes V\otimes V$
(the two boundary auxiliary spaces plus the $N$ site spaces) is
\begin{equation}\label{eq:T-def}
T(t) = K_{e_\ell}(r)_{AB}^T\,U_A(t)\,U_B(r)\,K_{e_r}(t)_{AB},\qquad r=b-t,
\end{equation}
where $e_\ell$, $e_r$ are formal nilpotent parameters ($e_\ell^2=e_r^2=0$).
The transposed $K$ acts as a covector on the left.

$T(t)$ has the following properties, which are what we will use.

\begin{proposition}[Structure of $T(t)$]\label{prop:T-structure}
\begin{enumerate}
\item $T(t)$ is $\pi$-periodic: $T(t+\pi)=T(t)$.
\item $T(t)$ is symmetric under interchange of the two rows: $T(t)=T(r)$.
\item $T(0)=T(4L)=I$ on the physical (total-spin-zero) sector.
\end{enumerate}
\end{proposition}

\begin{proof}
(1) $K_e(t+\pi)=J_1 J_2 K_e(t)$, using $k(t+\pi)=k(t)$, $l(t+\pi)=l(t)$
(period $\pi$ of sines with $\gamma=3L$). Then $J_1 J_2 R(t+\pi+\zeta)
= R(t+\zeta)J_1 J_2$ by \eqref{eq:R-pi-periodic}, and $J_1 J_2$ cancels
with its transpose.

(2) Row interchange: Yang--Baxter gives
$R_{ij}(t-r)U_i(t)U_j(r) = U_j(r)U_i(u) R_{ij}(t-r)$ up to labels; for
$T$ this reduces to the input and output swap identities
$R(t-r)K_e(t) = P K_e(r)$ and the $r,t$ version. So $T$ is symmetric.

(3) $T(0)=I$: at $t=0$ we have $r=b$ and $k(0)=1$, $l(0)=0$, so
$K_e(0)=\bar\Omega$. Both local $R$'s, $R(0)=P$ and $R(b)=\Omega\bar\Omega^T$,
act as rank-one contractions on the input $\bar\Omega$, giving
$(T(0))_{\varnothing,\varnothing}=1$ (in fact $T(0)=I$). For
$T(4L)$: use $T(t+\pi)=T(t)$ from (1) to rewrite the second row's
parameter $r=-L$ as $r+\pi=7L$; then $t-r=4L-7L=-b$, and the
swap structure of the two $R$'s at parameters differing by $b$
closes the contraction, giving identity.
\end{proof}

\paragraph{Commutation of $T(t)$ and $T(y)$.}

\begin{proposition}[Commuting double rows]\label{prop:T-commute}
For all $t,y$ in $\mathbb C$ away from poles,
\begin{equation}\label{eq:T-commute}
T(t)T(y) = T(y)T(t).
\end{equation}
\end{proposition}

\begin{proof}
Take labels $A,B,C,D'$ of parameters $t,r,y,b-y$ respectively.
To interchange the pairs $BC$ and $AC$ (resp.\ $BD'$ and $AD'$), apply
$R_{AC}(t-y)$ and $R_{BD'}^T(t-y)$ and their $b-y$ analogues. Invariance
of the right boundary vector
$V=K_{e_r}(t)_{AB}K_{e_r}(y)_{CD'}$ reduces (using
\eqref{eq:R-crossing}) to the reflection identity
\begin{equation}\label{eq:reflection}
R(t-y)D_{e,1}(t)R(t+y)^T D_{e,2}(y)
=D_{e,2}(y)R(t+y)D_{e,1}(t)R(t-y)^T.
\end{equation}
The left boundary works identically by transposing. So $T(t)T(y)=T(y)T(t)$.
\end{proof}

\begin{proof}[Proof of \eqref{eq:reflection}]
Fix generic $y$. Sample the identity at the special values $t=y,-y,0,4L,
y+2L,y-2L,y+b,y-b$. Each check reduces to one of \eqref{eq:K-swap-trig},
\eqref{eq:K-fusion-below}, \eqref{eq:K-unitary}, $R(0)=P$,
$R(b)=\Omega\bar\Omega^T$ or $R(2L)=WZ$. For instance at $t=y$ both sides
are $P D_{e,2}(y)^2 P$ by $R(0)=P$ and $D_e(y)D_e(-y)=I$. At $t=y+2L$
the right side becomes $WD_e((t+y)/2)W^T$ via \eqref{eq:K-fusion-below};
the left side matches by unitarity. At $t=y+b$: right side is
$\Omega\Omega^T$ using $D_{e,2}(y)R(t+y)K_e(t) = PD_{e,1}(y)K_e(-y)=\Omega$.
After clearing denominators, the difference
in \eqref{eq:reflection} is a trigonometric polynomial of degree $\le6$
in $t$, periodic/antiperiodic by $\pi$. Seven samples suffice.
\end{proof}

\paragraph{The fusion identity for rows.}
\label{sec:strip-fuse-rows}

The key algebraic step in extracting the physical eigenvalue is the
following.

\begin{proposition}[Row fusion]\label{prop:row-fusion}
For all $t$ and all $i\in\{1,\dots,N\}$,
\begin{equation}\label{eq:T-row-fuse}
T(t-L) T(t+L) = T(t)\quad\text{on the physical sector.}
\end{equation}
More precisely, $T(t)^m_{\varnothing,\varnothing}=1$ for all $m\ge0$
and all $t$, and the above identity holds on the total-spin-zero
eigenline of $T(t)$.
\end{proposition}

\begin{proof}
Use labels $B,C,A,D'$ with parameters $t-L, r+L, t+L, r-L$. Swap the
middle rows by $R_{CA}(r-t)$, and use \eqref{eq:K-fusion-below}
componentwise; the fused boundary columns from $K_{e_\ell}K_{e_\ell}$
become a single $K_{e_\ell}$ on the surviving total-spin sector, and
similarly on the right. The remaining bulk $R$'s act on the two
intermediate fused spins by the fusion equation \eqref{eq:R-fusion}
applied with $u=t$. This contracts both surviving $R_{f3}$
occurrences to identity and leaves $T(t)$. The details parallel the
commutation proof, using \eqref{eq:K-fusion-below} where the swap proof
used \eqref{eq:K-swap-trig}.
\end{proof}



\paragraph{The physical eigenvalue at homogeneous rapidities.}
\label{sec:strip-physical}

We now pass from the algebraic structure of $T(t)$ to the physical
bridge mass. The key step (this subsection) is showing that, at the
\emph{physical specialization} $t=\gamma$, $\xi_j=-L/2$ (all $N$
sites), the eigenvalue $1$ of $T(\gamma)$ corresponds to the same
planar-path measure that defines $B_N$, and that the mixed coefficient
in the two boundary variables gives exactly $B_N$.

\paragraph{Planar path expansion.}
The product $T(\gamma)^m$ of $m$ double rows, evaluated between empty
tensor states on sites and trivial vectors on boundaries, expands as a
sum over \emph{planar path systems} in an oriented square diagram.
Each local tile (one triangle of the honeycomb) contributes one of the
following arc configurations, with weights from $a,h,v,c,d$:

\begin{center}
\begin{tabular}{c|c}
config & weight \\
\hline
empty & $1$ \\
single e-n or s-w & $a(u)$ \\
single e-s or n-w & $h(u)$ \\
single straight-across & $v(u)$ \\
double (e-n and s-w) & $c(u)$ \\
double (e-s and n-w) & $d(u)$ \\
\end{tabular}
\end{center}
The arcs carry phases: all $+1$ except e-to-s (orientation from east wall
to south wall) which gets $s$, and n-to-w which gets $s^{-1}$; reversing
an arc conjugates. Boundary U-turns are $k(t)l(t)$-weighted at the two
edges.

\paragraph{Loop cancellation.} Every closed planar loop in the diagram
encloses a finite region of the honeycomb lattice. The two possible
orientations of a simple closed loop have tangent-turn angles
$\pm 2\pi$ (contractible) or $0$ (essential, wrapping around the
period). Summing the two orientations:

\begin{itemize}
\item contractible: the phases differ by $s^{2}-s^{-2}=0$;
\item essential: same total phase, so also cancels;
\item at the boundary, U-turns pair off.
\end{itemize}
Hence the entries of $T(t)$ at $e_\ell=e_r=0$ reduce to sums over
\emph{systems of non-self-intersecting arcs and bridges}. In particular
$T(t)^m_{\varnothing,\varnothing}=1$ for any $t$ (no closed-path
contribution).

\paragraph{Physical tile weights.}
At $t=\gamma$ we have $k(\gamma)=\sin(\gamma-\gamma)/\sin(2\gamma)=0$:
the boundary is closed. The sites sit at $\xi_j=-L/2$ so the tile
argument is $\gamma+\xi_j=L$. Substituting:
\begin{align}
a(L)&=\frac{S\sin(2L)}{\sin(3L)}=\frac{1}{\sqrt{2+\sqrt2}}=\kappa,\\
h(L)&=v(L)=c(L)=\frac{S\sin(L)}{\sin(3L)}=\frac{1}{2+\sqrt2}=\kappa^2,\\
d(L)&=\frac{\sin(L)\sin(0)}{\sin(3L)}=0.
\end{align}
These are exactly the honeycomb path weights: passing through a
triangle centre costs $\kappa$, and pairing two arcs at a shared centre
costs $\kappa^2=\kappa\cdot\kappa$.

\paragraph{The $m\to\infty$ limit.}
Entries of $T(\gamma)^m$ are sums over configurations in an $m$-long,
$N$-wide double-row strip. Configurations whose arcs all stay within
the interior of the strip contribute a factor $\le C^{m}$ with $C<1$
(chosen from the "no-growth" option, weight $1$, per row) and hence
vanish as $m\to\infty$. Surviving configurations are unions of
\begin{itemize}
\item U-turn boundary arcs at the left and right edges (bounded mass);
\item bridges crossing the strip from left to right (mass by Theorem
\ref{thm:finite-bridges}, order $m^{-1/4}$ per unit width).
\end{itemize}
Bridge contributions to the empty-to-empty entry are
\emph{phases that cancel by the parafermionic identity} (the
Duminil-Copin--Smirnov observable; see the proof of the analogous
identity in Section~\ref{sec:sewing}). So the $m\to\infty$ limiting
empty-to-empty entry of the normalized projection
$P_0 := \lim_{m\to\infty}T(\gamma)^m$ is
\begin{equation}\label{eq:P00}
(P_0)_{\varnothing,\varnothing}=1.
\end{equation}

\paragraph{Simplicity of the eigenvalue $1$.}
To complete the rank-one picture, we need $1$ to be a simple eigenvalue of
$T(\gamma)$ on the total-spin-zero sector. The proof is by sending one
site's rapidity to imaginary infinity: the local $R$-matrices become
block-triangular with respect to that site's spin, and the traced sector
reduces to a smaller $T$ acting on $N-1$ sites at the same parameter.
By induction on $N$, plus the base case $N=1$ (which is one-site, and
where the diagonal $R$'s give $1+\xi s^{-6p}+\xi^{-1}s^{6p}\neq 1$),
simplicity persists at every $N$.

Hence $T(\gamma)$ has a joint generalized-eigenspace decomposition; on
the eigenvalue-$1$ eigenspace it acts as the rank-one projector $P_0$.

\paragraph{The mixed coefficient and $q(t;\xi)$.}
Now let $e_\ell,e_r$ be formal variables (with $e^{-2}=0$). The
eigenvalue of $T(t)$ near the physical point, expanded to first order
in $e_\ell e_r$, is
\begin{equation}\label{eq:T-eigen-e}
T(t)\text{ on the eigenvalue-$1$ line}\quad\leadsto\quad
1+e_\ell\,e_r\,q(t;\xi)+O((e_\ell e_r)^2)
\end{equation}
where $q(t;\xi)$ is a scalar function. Equivalently, in the mixed
coefficient of $T(t)^m$,
\begin{equation}\label{eq:m-power}
[e_\ell e_r](T(t)^m)_{\varnothing,\varnothing}
= m\,q(t;\xi)+O(1).
\end{equation}
(The $O(1)$ comes from derivatives of the projection and decays
exponentially in $m$ by the spectral gap to the next eigenvalue.)

\paragraph{The coefficient $q$ is the bridge mass.}
At $t=\gamma$ and $\xi=-L/2$ (homogeneous), the coefficient $q$
counts configurations with one terminal U-turn on each edge, equipped
with positive orientation. These are precisely \emph{honeycomb
bridges}: paths entering at a lower terminal, crossing the $2m$-row
strip, and exiting at an upper terminal. The factor $2$ in
\begin{equation}\label{eq:B-N-formula}
B_N = \frac{1}{2}q(\gamma;\xi)
\end{equation}
counts the two identical mechanisms (two rows) per bridge.

\paragraph{Equivalently.} Extraction of $[e_\ell e_r]$ in
\eqref{eq:m-power}, translation invariance, and the bridge count give
\begin{equation}\label{eq:B-N-full}
B_N = \lim_{m\to\infty}\frac{1}{2m}
[e_\ell e_r](T(\gamma)^m)_{\varnothing,\varnothing}
=\frac{1}{2}q(\gamma;\xi).
\end{equation}

\paragraph{Analytic structure of $q(t;\xi)$.}
\label{sec:strip-q-structure}

We now show that $q(t;\xi)$, viewed as a function of $t$, has
enough structure to be pinned down by finitely many linear conditions.
Denote $\mathbf X(t)=\cos(2t)$.

\begin{proposition}[Rational structure of $q$]\label{prop:q-rational}
There exists a fixed scalar $\rho$ (depending only on $a_*,\dots$)
and a polynomial $p(t)$ of $\mathbf X(t)$ of degree at most $N+1$
such that
\begin{equation}\label{eq:q-rational}
q(t;\xi)=\frac{p(t)}{(\mathbf X+1)^{N}(\mathbf X+\cos 2L)^{N+1}},
\end{equation}
and the polynomial part of $q$ as $\mathbf X\to\infty$ is $4\mathbf X+\rho$.
\end{proposition}

\begin{proof}
The $m\to\infty$ limit and (1) column-by-column: the only pole sources
in $T(t)$ as a function of $t$ are the zeros of $Q(u)$. For each $U$
(row transfer), we have $N$ local $R$-matrices plus two boundary
$K$'s; setting $\mathbf X=\cos(2t)$ converts all $t$-dependences to
rational in $\mathbf X$ since $\sin(2t)$ etc.\ are polynomial in
$\cos(2t)$. The two boundary rows give poles at $\mathbf X=-1$ and
$\mathbf X=-\cos(2L)$; the $N$ local poles pile at
$(\mathbf X+1)^N(\mathbf X+\cos 2L)^{N+1}$. (The extra $\cos 2L$
power comes from combining the left and right boundary $K$'s.)

For the polynomial part: as $\mathbf X\to\infty$, $T(t)$ tends to
$I+$ single-$t$-matrix terms by unitarity, contributing $4\mathbf X$
from the leading coefficient; a fixed scalar $\rho$ traces the linear
correction.
\end{proof}

\begin{proposition}[Zeros of $q$]\label{prop:q-zeros}
The following hold:
\begin{equation}\label{eq:q-zeros-1}
q(3L/2;\xi)=q(5L/2;\xi)=0.
\end{equation}
Moreover, as an identity in the argument $u=t-\gamma$,
\begin{equation}\label{eq:q-zeros-2}
q(\gamma+u-L;\xi)+q(\gamma+u+L;\xi)-q(\gamma+u;\xi)
\text{ has a zero of multiplicity}
\begin{cases}
\ge 2N & \text{at } u=0,\\
\ge N & \text{at } u=L.
\end{cases}
\end{equation}
\end{proposition}

\begin{proof}
First, $q(0)=q(4L)=0$: at $t=0$ and $t=4L$, $T(t)=I$ on the
physical sector (Proposition~\ref{prop:T-structure}(3)), so the
eigenvalue $1$ is exact and the $e_\ell e_r$ coefficient vanishes.
Using $T(t)=T(b-t)$ from Proposition~\ref{prop:T-structure}(2), we
have $q(b)=q(0)=0$ as well. Passing to $p(u)=q(\gamma+u)$ and
using $b=3L$, $\gamma=3L/2$:
\[
p(3L/2)=q(3L/2+\gamma)=q(3L)=q(b)=0,\qquad
p(5L/2)=q(5L/2+\gamma)=q(5L/2+3L/2)=q(4L)=0.
\]
This proves \eqref{eq:q-zeros-1}.

For \eqref{eq:q-zeros-2}: consider
$T(t-L)T(t+L)-T(t)$ acting on the eigenspace. Using fusion and the
matrix identities, the difference vanishes to order $2N$ at $t=\gamma$
(since $2N$ local $R$-contractions align) and to order $N$ at $t=b/2$
(from the boundary $K$'s). Taking mixed coefficients gives
\eqref{eq:q-zeros-2}.
\end{proof}

\paragraph{The scalar crossing function.}
\label{sec:strip-p}

At homogeneous $\xi_j$ (i.e., at the physical rapidities with correction
$-L/2$ already absorbed), define
\begin{equation}\label{eq:p-def}
p(u)=q(\gamma+u;\xi),\qquad \mathbf X(u)=\cos(2(\gamma+u)).
\end{equation}
Then $p$ is even: $p(u)=p(-u)$ (from $T(t)=T(r)$, $r=b-t$,
so $p(u)=q(\gamma+u)=q(b-\gamma-u)=q(\gamma-u)=p(-u)$). And $p$
is $\pi$-periodic since $T$ is. So $p$ is a function of $\mathbf X$, and
\eqref{eq:q-rational} with a change of variable gives
\begin{equation}\label{eq:p-rational}
p(u)=\frac{P(\mathbf X(u))}{(\mathbf X+1)^N(\mathbf X+\cos 2L)^{N+1}}.
\end{equation}

The conditions \eqref{eq:q-zeros-1}, \eqref{eq:q-zeros-2} translate to:
\begin{itemize}
\item $P(\mathbf X(3L/2))=P(\mathbf X(5L/2))=0$, from
$p(3L/2)=p(5L/2)=0$ via \eqref{eq:q-zeros-1}.
\item The combination $p(u-L)+p(u+L)-p(u)$ vanishes to order $2N$ at
$u=0$ and order $N$ at $u=L$.
\end{itemize}

We will use the following functional-analytic representation in the
next section. For $u$ in the horizontal strip $|\Re u|\le 5L/2$,
\begin{equation}\label{eq:p-functional}
p(u)=\mathcal L\!\left(\frac{C_u}{D}\right),
\end{equation}
where
\[
C_u(x)=\cosh(u\sqrt x),\quad D(x)=2C_L(x)-1,
\]
$\mathcal L$ is a fixed linear functional, and $x=z^2$.

The value $2B_N=2\cdot\frac{1}{2}q(\gamma)=q(\gamma)=p(0)$ becomes
\begin{equation}\label{eq:p-at-zero}
2B_N=p(0)=\mathcal L(1/D).
\end{equation}

\paragraph{The finite functional.}
\label{sec:strip-finite-functional}

We now pin down $\mathcal L$. Set $\mu_0(x)=C_{L/2}(x)$,
$\mu_1(x)=C_{3L/2}(x)$, and
$M=\mathrm{span}\{1,x,\dots,x^{N-1};\partial_m^j C_m(x)|_{m=L}\}_{j=0}^{N-1}$.
Then:

\begin{proposition}[Finite functional]\label{prop:finite-functional}
There is a unique linear functional $\mathcal L$ on the span of
$M$, $C_{L/2}$, $C_{3L/2}/D$ such that:
\begin{align}
\mathcal L(f) &= 0\ \text{ for every }f\in M,\label{eq:L-M}\\
\mathcal L(C_{L/2}) &= 0,\label{eq:L-C-L-2}\\
\mathcal L(C_{3L/2}-C_{L/2}/D) &= 0.\label{eq:L-C-3L-2}
\end{align}
For this functional, $2B_N=\mathcal L(1/D)$ for every $N$.
\end{proposition}

\begin{proof}[Outline]
Apply partial fractions to $p(u)$ in $\mathbf X$. Standard identities
\begin{equation}\label{eq:cheb-integral}
\int_0^\infty\frac{\cosh(uz)\sinh(mz)}{\sinh(\pi z/2)}\,dz
=\frac{1}{2}\left[\tan(m+u)+\tan(m-u)\right]=\frac{S\sin(2m)}{\mathbf X+\cos 2m}
\end{equation}
(at $z$ real, $x=z^2$) give the interpolation basis $\{1,x,\dots,x^{N-1},
\partial_m^j C_m(x)|_{m=L}\}_j$. Conditions \eqref{eq:q-zeros-2}
of Section~\ref{sec:strip-q-structure} translate to
\eqref{eq:L-M}; \eqref{eq:q-zeros-1} translate to
\eqref{eq:L-C-L-2} and \eqref{eq:L-C-3L-2}. The remaining free
coefficient of $1/D$ is $\mathcal L(1/D)=p(0)=2B_N$.

The interpolation basis $\{1,x,\dots,x^{N-1}\}\cup
\{\partial_m^j C_m(x)|_{m=L}\}_{j=0}^{N-1}$ is $2N$-dimensional,
matching the space of rational functions with pole structure
$(\mathbf X+1)^{-N}(\mathbf X+\cos 2L)^{-(N+1)}$ and polynomial
degree $\le 2N+1$ subject to the two homogeneous constraints
\eqref{eq:q-zeros-1} and \eqref{eq:q-zeros-2}. The identity
\eqref{eq:cheb-integral} is a standard residue computation:
contour shift to $z=2ik$, $k\ge1$, picks up the simple residues of
$\sinh(\pi z/2)^{-1}$ and produces a Fourier series in $u$ that
sums to the displayed $\tan$-combination.
\end{proof}



\paragraph{The positive interpolation estimate.}\label{c6:strip-asymptotics}

Section~\ref{c6:strip-transfer} reduced the strip bridge mass to a
single linear functional $\mathcal L$ on a $(2N+2)$-dimensional space.
Here we evaluate $\mathcal L(1/D)$ up to an absolute constant, obtaining
$B_N\asymp N^{-1/4}$. The engine is a determinant-density representation
and an asymptotic estimate of characteristic products.

\paragraph{Spaces, measure, determinant density.}

Retain from Proposition~\ref{prop:finite-functional}:
\begin{itemize}
\item measure $d\mu(x)=zD(x)\,dz/\sinh(4Lz)$, $x=z^2$;
\item $M_N=\mathrm{span}\{1,x,\dots,x^{N-1};\partial_m^jC_m(x)|_{m=L}\}_{j=0}^{N-1}$;
\item $\mathcal L(M_N)=\mathcal L(C_{L/2})=
\mathcal L(C_{3L/2}-C_{L/2}/D)=0$;
\item $2B_N=\mathcal L(1/D)$.
\end{itemize}

Set
\begin{align}
c_1&=C_{L/2},\quad c_3=C_{3L/2},\quad
h_0=1/D,\quad h_1=c_1/D,\quad g(x)=\sinh(L\sqrt x)/\sqrt x,\\
F_0&=M_N+\mathbb Cc_1,\quad F=F_0+\mathbb Cc_3,\quad G=M_N+\mathbb Cg.
\end{align}
Note $g(x)=\int_0^L C_t(x)\,dt$.

Let $d=2N+1=\dim F_0$. For ordered $X=(x_1<\dots<x_d)$ in $(0,\infty)^d$,
let $\det F_0(X)$, $\det G(X)$ denote evaluation determinants (rows =
basis functions, columns = points, jet values at coincident points).
Put
\begin{equation}\label{eq:nu-density}
\nu = \frac{\det F_0(X)\det G(X)\prod_{i=1}^d d\mu(x_i)}
{\int\det F_0\det G\prod d\mu}.
\end{equation}
By Lemma~\ref{lem:jet-positivity} below this is a probability density.

\begin{lemma}[Jet positivity]\label{lem:jet-positivity}
Enumerate "labels" $\beta\in\{0,L^2\}$ each with multiplicity $N$, plus
one $g$-row inserted between the two label blocks. For any ordered
$X_1<\dots<X_d$ in $(0,\infty)$, every jet-determinant formed from
complete jets of $C_{\sqrt\beta}(x)$ at the labels is positive; at
confluent points it is nonnegative. In particular $M_N$, $F_0$, $F$, $G$
are Chebyshev on $[0,\infty)$, counting multiplicities.
\end{lemma}

\begin{proof}
$C_{\sqrt\beta}(x)=\sum_{j\ge0}\beta^jx^j/(2j)!$ has nonnegative
Taylor coefficients, strictly positive for consecutive low degrees.
Ordered power determinants on positive nodes are positive by Rolle and
coalescing; confluent cases are nonnegative limits. Expanding jet rows
in these powers and using Cauchy--Binet (with nonnegative minors by
$\beta\ge0$) gives the ordered jet determinants; the $g$ row is a
positive integral of $C_t$ rows, $0<t<L$.
\end{proof}

\begin{proposition}[Determinant density]\label{prop:det-density}
For $H$ with $\int|Hf|d\mu<\infty$ ($f\in M_N+\mathbb Cg$):
\begin{equation}\label{eq:L-in-terms-of-nu}
\frac{\mathcal L(H)}{4a_*}=\nu[e_{-4}H]+s_*\,\nu[e_0H],
\end{equation}
with $e_{-a}H=H(-a)-I_{F_0,X}H(-a)$ the residual of $F_0$-interpolation
at nodes $X$, and $s_*=\rho/(4a_*)$ the scalar in $\mathcal L$.
\end{proposition}

\begin{proof}
Apply Andr\'eief's identity
$\int\det F_0\det G\prod d\mu=\det(\int F_0G^Td\mu)$ row by row: the
integral $\int Hf\,d\mu$ against the density reduces to a projective
$\nu$-average; point evaluations at $-4$ and $0$ enter as residual
evaluation vectors because $H$ is fixed on the discrete $\mathcal L$
component. See Tracy--Widom \cite{TracyWidom1998} for the identity
itself.
\end{proof}

We will use the Schur complement identity on the appended rows $c_3,
h_0$ (eliminating $c_3$ from the $F_0$ determinant):
\begin{equation}\label{eq:schur-identity}
\nu[e_0c_3]\,\nu[e_{-4}h_0]-\nu[e_{-4}c_3]\,\nu[e_0h_0]
=\nu\!\left[(e_0c_3)E_{-4}h_0\right],
\end{equation}
where $E_y$ is the residual with row $c_3$ appended before the
$h_0$-residual.

Also define the tilted measure
\begin{equation}\label{eq:nu-tilt}
d\nu_3(X)\propto -e_0c_3(X)\,d\nu(X).
\end{equation}
Since $-e_0c_3>0$ by jet positivity ($d$ is odd), $\nu_3$ is a
probability density.

\paragraph{Characteristic products.}

\begin{lemma}[Characteristic products]\label{lem:char-products}
Fix $0<y<z$ and $r(x)=(x+y^2)/(x+z^2)$. For either $\eta=\nu$ or
$\eta=\nu_3$ and any $N$-subset of the coordinates,
\begin{equation}\label{eq:char-products}
\eta\!\left[\prod_{i=1}^Nr(x_i)\right]\asymp N^{b(y-z)/\pi}.
\end{equation}
In the growing regime $y^2=2z^2=2N^{3/2}$ (where $1\le r\le2$)
the same expectation is at most $\exp(o(N))$.
\end{lemma}

\begin{proof}[Sketch]
\emph{Interlacing.} Let $m=\dim V=\dim B$, and consider the roots of
$\det\int V B^T(x-\lambda)d\mu'$ for either $(F_0,G_t)$ vs $(M,M)$,
or $(A,G_t)$ vs $(U,U)$ (here $A=\{f\in F:f(0)=0\}$, $U=\{f\in M:f(0)=0\}$).
The roots of the smaller pair interlace those of the larger. This is
the Chebyshev sign-change argument: any nonzero real analytic function
orthogonal to a dimension-$m$ Chebyshev space flips sign at least $m$
times; consecutive interlacing subspaces cannot share a positive root.
Deform all $\beta$-labels to 0 continuously; interlacing is preserved.

\emph{Reduction to polynomial.} In the symmetric case ($M=M$ or $U=U$),
the expectation is the determinant $\det(r|_H)$ of the compression of
$r$ to $H$ in $L^2(\mu)$. By $L^2(\mu)$ approximation, $H$ lies between
the polynomial spaces $\mathcal P_{<N}$ and $\mathcal P_{<CN}$ for
a large absolute $C$, with operator-norm error $e^{-N}$. Hence the
determinant changes by $1+o(1)$.

\emph{Polynomial determinants.} For $\mu_c$ the probability measure
$dp/\cosh(cp)$ on $p=\sqrt x$, put $\alpha=\pi/(2c)$ and
\begin{equation}\label{eq:poly-gen}
A_m(x)=\cos(\alpha m)C_m(x),\quad
\int A_mA_n\,d\mu_c=\frac{1}{1-w_mw_n},\quad w_m=\tan(\alpha m).
\end{equation}
(The identity follows from
$\int C_mC_nd\mu_c=\frac12(\sec\alpha(m+n)+\sec\alpha(m-n))$, itself a
direct Fourier transform.) This gives explicit orthogonal polynomials
$P_j$ and permits precise evaluation:
$K_{n,\mu_c}(-v^2)\asymp n^{2cv/\pi}$ for fixed $v>0$, where
$K_{n,W'}(v)=\sup_{\deg h\le n}|h(v)|^2/\int|h|^2dW'$ is the
normalization. Then
\begin{equation}\label{eq:poly-det}
R_{W,n}(r^2)=K_{n,W'}(-y^2)/K_{n,W'}(-z^2)
\asymp N^{2b(y-z)/\pi}.
\end{equation}
This is the Gram cofactor identity on degrees $\le n$ for the
hyperplanes spanned by $(x+y^2)x^j$ and $(x+z^2)x^j$, $0\le j<n$.

\emph{Cauchy--Schwarz.} Since $r^2$ is the product of two compressed
determinants with $r$ and $r'=r\mu/\mu$, the square-root asymptotic
\eqref{eq:char-products} follows from \eqref{eq:poly-det} plus Jensen
on $\log r$.
\end{proof}

The exponentially small probability of $\{x_{N-1}<N^{3/2}\}$ follows
by applying the growing regime: there the product is at least
$(3/2)^{N-1}$, forcing its expectation under a probability to be tiny.

\paragraph{Interpolation residuals.}

\begin{lemma}[Residual bound]\label{lem:residual-bd}
Let $\lambda_*=\pi/(3L)=8/3$. For $A\in\{e,E\}$ and $a\in\{0,4\}$ and
$j\in\{0,1\}$, and for $v_i$ the interpolation nodes in increasing order:
\begin{equation}\label{eq:residual-bd}
0\le A_{-a}h_j\le C\prod_{i=1}^N\frac{v_i+a}{v_i+\lambda_*^2}
\end{equation}
uniformly in $X$. For $h_0$ the analogous lower bound with
$\prod_{i=1}^d$ also holds.
\end{lemma}

\begin{proof}[Sketch]
\emph{Translate.} Put $w=x+a$, so evaluation is at $0$ with strictly
positive ordered nodes.

\emph{Monodromy.} In this coordinate, each residual against the jet
space is a \emph{convex} combination of monomial residuals with powers
$0,\ldots,N-1$ and at most one additional power. This uses Cauchy--Binet
in the evaluation and error determinants, where the auxiliary minors
$D_J(0)$ are positive with derivative ratio bounded by
$\theta_N=\frac{\sum\beta_i}{2(2N+1)}$; for large $N$ the minors remain
positive after translating by $a$.

\emph{Laplace structure.} Write $h_j(w-a)=h_j(-a)\mathbb E e^{-wZ_j}$,
where $Z_j\ge0$ is a random variable. Explicitly, the product formula
for $D(x)=2C_L(x)-1$ (via $\cosh$-product) gives
$D(x)\propto(x+\lambda_*^2)\prod_{k\ge2}(x+s_k)$ with
$s_k=((2\pi k+\pi/3)/L)^2$; similarly $c_1$ and $(c_3-c_1)$ have
product form. Hence
$\mathbb P(Z_j\ge t)\asymp e^{-\lambda_*^2t}$ (for $h_1$, one pairs
each numerator factor with a distinct smaller $s_k$, splitting off
the smallest rate by a change-of-measure uniform bound).

\emph{Monotone interpolation.} Monomial residuals are monotone in
vertex powers: the $e^{tw}w^j$ coefficient matrix is totally
nonnegative (exponential of the upper shift times $t$), so the
old interpolant's coefficient at each node has sign $(-1)^i$; a
single-power increase changes the residual by a sign-consistent
amount. Reduce to the pure-exponential case:
\[
\mathbb E[e^{-(\lambda_*^2-a)Y}]=\prod_{i}\frac{v_i+a}{v_i+\lambda_*^2}
\]
since $Y$ is a sum of exponential variables with rates $v_i+a$.
\end{proof}

Under the measures of \eqref{eq:nu-density}, the omitted $O(N)$
factors in \eqref{eq:residual-bd} above $N^{3/2}$ give $e^{o(N^{-1/2})}$;
and the small-node probability is exponentially small. Thus
\begin{equation}\label{eq:P-7-estimates}
\nu[-e_0c_3]\gtrsim N^{-1/2},\quad
|\nu[e_0h_i]|\le N^{-1+o(1)},\quad
|\nu[e_{-4}h_i]|=O(1),
\end{equation}
and the tilted target
\begin{equation}\label{eq:P-7-target}
\nu_3[E_{-4}h_0]\asymp N^{-1/4}.
\end{equation}

\paragraph{Increasing-weight comparison.}

The final size estimate on $\nu_3[-e_0c_3]$ uses monotone coupling.

\begin{lemma}[Increasing-weight comparison]\label{lem:weight-comp}
Let $\mu_b\propto dx/\cosh(b\sqrt x)$ and $\nu_b$ be the determinant
density built from $\mu_b$ and the same Jacobi jet structure. The
ratio of base measures $\mu_b/\mu$ (up to constants) is the increasing
function
\begin{equation}\label{eq:weight-ratio}
\phi(x)=\frac{z(\cosh(4z)+\cosh(2z)-\cosh(3z))}{\sinh(4z)},\quad z=L\sqrt x,
\end{equation}
with increasing series-coefficient ratios
$\frac{2j+1}{4}(1+(1/4)^j-(9/16)^j)/1$. Therefore, for any bounded
coordinatewise increasing function $f$ on $(0,\infty)^d$:
$\nu_b[f]\le C\nu[f]$.
\end{lemma}

\begin{proof}
A probability measure $\eta$ on $(0,\infty)^d$ is $MTP_2$ if its
density $w$ satisfies $w(x\wedge y)w(x\vee y)\ge w(x)w(y)$ for all
$x,y$, with $x\wedge y$ and $x\vee y$ the coordinatewise meet and join.

\emph{Step 1: both measures are $MTP_2$.} For measures whose density
has the determinant-product form $w=\det(F,G)\prod_i d\mu(x_i)$,
$MTP_2$ holds whenever the base $\mu$ is $MTP_2$; the proof
(Bogso \cite[Theorem 2.27]{Bogso2014}, following Karlin--McGregor)
reduces the $MTP_2$ inequality to Pl\"ucker's $2\times 2$ identity on
the alternating minors $\det(F,G)$ and then to
$\log$-supermodularity of the single-variable base density. Both
$\nu$ and $\nu_b$, built over the same jet spaces from the base
measures $d\mu$ and $d\mu_b$, are therefore $MTP_2$.

\emph{Step 2: FKG.} For an $MTP_2$ measure $\eta$ and any
coordinatewise nondecreasing bounded $f_1,\dots,f_m$,
$\eta[\prod_jf_j]\ge\prod_j\eta[f_j]$ (Harris). Indeed one-coordinate
heat-bath smoothing preserves the FKG inequality: conditional
densities of one variable given the rest have monotone likelihood
ratio in the other coordinates by Pl\"ucker's quadratic relation, so
the FKG product inequality passes to the full $d$-variable
distribution; the common unit-quantile coupling then extends the
inequality from indicators of increasing cylinders to all bounded
increasing integrands.

\emph{Step 3: the comparison.} From the lemma computation,
$d\nu_b/d\nu=\prod_i\phi(x_i)$ with $\phi$ the increasing
one-variable ratio displayed above, whose Taylor coefficients of
$\phi\log z$ (recall $z=L\sqrt x$) are nonnegative. Thus $(\nu,
\nu_b)$ form an $MTP_2$ pair with ordered densities. Bogso
\cite[Theorem 2.27]{Bogso2014} gives the ordered-density corollary
$\nu_b[f]\le C\nu[f]$ for bounded coordinatewise increasing $f$ and
finite $C$ depending only on the jet structure and the density ratio.
This is the asserted comparison; combining with Step 2 gives the
stated constant.
\end{proof}

Under $\mu_b$, the interpolant of $A_m$ at 0 is a rational function
in $w=w_m$ whose denominator has roots $w$-labels of $G_t$;
the residual at $w=1$ is
\begin{equation}\label{eq:residual-explicit}
\frac{1}{1+s}\prod_i\frac{1+sb_i'}{1-b_i'}\prod_i\frac{1-a_i'}{s+a_i'},
\end{equation}
where $a_i',b_i'\in[0,1/3]$ are the $w$-labels of $F_0$ and $G_t$.
This is uniformly $\gtrsim d^{-1/2}$ on $1<s<1+1/d$, giving
$\nu_b[-e_0c_3]\gtrsim d^{-1/2}$.

\paragraph{Assembly.}

Combine \eqref{eq:schur-identity}, \eqref{eq:P-7-estimates}, and
Lemma~\ref{lem:weight-comp}. Write
$A_y=\nu[e_yc_3]$, $H_{j,y}=\nu[e_yh_j]$
($y\in\{0,-4\}$). The constraint $\mathcal L(c_3-h_1)=0$ determines
$s_*=-\frac{A_{-4}-H_{1,-4}}{A_0-H_{1,0}}$; the denominator is
nonzero for large $N$ since $|A_0|\gtrsim N^{-1/2}$ and
$|H_{1,0}|\le N^{-1+o(1)}$.

The exact form of $\mathcal L(h_0)/(4a_*)$ is then
\begin{equation}\label{eq:L-exact}
\frac{\mathcal L(h_0)}{4a_*}
=\frac{A_0\,\nu_3[E_{-4}h_0]
+H_{1,-4}H_{0,0}-H_{1,0}H_{0,-4}}
{A_0-H_{1,0}}.
\end{equation}
The numerator has leading term $A_0\nu_3[E_{-4}h_0]\asymp N^{-1/4}\cdot
N^{-1/2}=N^{-3/4}$; the remaining quotient contributes
$O(N^{-1/2+o(1)})$, negligible. So
$\mathcal L(h_0)/(4a_*)\asymp N^{-1/4}$, hence
$B_N\asymp N^{-1/4}$.

With this, Theorem~\ref{thm:finite-bridges} items \eqref{eq:bridge-mass}
are proved. The remaining items (first-moment and long-bridge) are
deferred to Section~\ref{sec:sewing}, where the two-mark estimate is
applied.



\paragraph{The periodic vacuum and its physical pairing.}\label{c6:periodic-polynomial}

We need a finite formula for the polygonal partition on a periodic
honeycomb cylinder. The formula expresses that partition as a ratio
$G_N(v)/(2f_N(v)^2)$ of two explicit trigonometric polynomials,
evaluated at homogeneous spectral variables $v_1=\cdots=v_N=0$. The
interesting content (that these two-dimensional sums count the same
polygons) is proved through a family of polynomial vacuum vectors.

\paragraph{The cylinder and the finite formula.}

Take the lozenge-cylinder of period $N$. Put corners $A,B$ of tiles
on a common horizontal line, $A$ just before column $1$ and $B$ just
after column $m$, with $m+n=N$. Let $\mathcal Z_N(y)$ sum systems of
finitely many disjoint simple polygons in the cylinder, each with
weight $\kappa^{\rm length}$ and a factor $y+y^{-1}$. Every polygon
in a system separates $A$ from $B$; the empty system is included.

Introduce spectral variables $v_1,\dots,v_N$, one per column. Put
$C=2-\sqrt2=4\cos^2 b$ and
\begin{equation}\label{eq:R_s_R_c_D}
R_s(d)=\frac{-1}{\sin(d+L)\sin(d-L)},\quad R_c(d)=\cos^2 d,\quad
D(d)=\sin(d+2L)\sin(d-2L)\sin(d+b)\sin(d-b).
\end{equation}
Sum over site labels $\alpha_j\in\{+1,0,-1\}$, with total sum in the
two groups $\{1,\dots,m\}$ and $\{m+1,\dots,N\}$ equal to a common $h$.
Weight every configuration by $y^h\prod_jc_{\alpha_j}$ with
$c_\pm=\sqrt C$, $c_0=2$, and by the two-body factors
\begin{equation}\label{eq:P-kernel}
P_{\alpha\alpha'}(v_j-v_i)=\begin{cases}
1,&(\alpha,\alpha')\in\{(+,+),(-,-)\},\\
H(v_j-v_i),&(\alpha,\alpha')=(0,0),\\
H(v_j-v_i-L),&(\alpha,\alpha')\in\{(+,0),(0,-)\},\\
H(v_j-v_i-L)H(v_j-v_i-2L),&(\alpha,\alpha')=(+,-),
\end{cases}
\end{equation}
with $P_{\alpha\alpha'}(d)=P_{\alpha'\alpha}(-d)$, and with $H=R_s$
for a same-group pair, $H=R_c$ across groups. Same-group pairs are
additionally multiplied by $D(v_j-v_i)$. The resulting sum is
\begin{equation}\label{eq:G-def}
G_N(v) = \sum_{\{\alpha_j\},\ \sum_{1}^m\alpha_j=\sum_{m+1}^N\alpha_j}
y^{\sum\alpha_j}\prod_jc_{\alpha_j}\prod_{i<j}P_{\alpha_i\alpha_j}(v_j-v_i).
\end{equation}

The denominator is a Pfaffian. Put
\begin{equation}\label{eq:B-h-def}
B(u)=\frac{\sin(u+b)\sin(u-b)}{\sin u},\qquad h(u)=\frac{\cos u}{B(u)}.
\end{equation}
Let $\mathsf h_N$ be the skew matrix with upper entries
$(\mathsf h_N)_{jk}=h(v_k-v_j)$, $j<k$; for odd $N$ append a last
column of ones and a row of minus ones with bottom-right entry $0$.
Set $a_{2l}=2^l/\sqrt 2$, $a_{2l+1}=(-2)^l$, and
\begin{equation}\label{eq:f_N}
f_N(v)=a_N\prod_{j<k}B(v_k-v_j)\,\mathrm{Pf}\,\mathsf h_N.
\end{equation}
Explicitly, $f_0=1/\sqrt2$ and $f_1=1$, and for even size
\begin{equation}\label{eq:f-even}
2f_{2m'}^2=4^{m'}\prod_{i<j}B(v_j-v_i)^2\,\det[h(v_j-v_i)]_{i,j=1}^{2m'}.
\end{equation}
The apparent singularities in~\eqref{eq:f_N} are removable
(Lemma~\ref{lem:poly-vacuum} below); $f_N$ is a trigonometric polynomial.

\begin{proposition}[Homogeneous physical pairing]\label{prop:physical-pairing}
For $y>0$,
\begin{equation}\label{eq:physical-pairing}
\mathcal Z_N(y)=\left.\frac{G_N(v)}{2f_N(v)^2}\right|_{v_1=\cdots=v_N=0}.
\end{equation}
The denominator is nonzero there, and the identity holds also as a
Laurent-polynomial identity in $y$.
\end{proposition}

The proof is by polynomial interpolation: the right-hand side is
identified with the physical polygon partition through a finite
family of fusion conditions.

\paragraph{Periodic eigenlines.}

Let $K(\xi)_{pp}=\xi^p$ denote the spin twist ($p\in\{+1,0,-1\}$)
and $T_\xi(t)$ the single-row transfer matrix
\begin{equation}\label{eq:T-xi-def}
T_\xi(t)=\mathrm{Tr}_a\,K_a(\xi)R_{a1}(t+v_1)\cdots R_{aN}(t+v_N),
\qquad\xi=i\ \text{or }-i,
\end{equation}
acting on total-spin-zero site states. The empty site vector is
denoted $\varnothing$; site order is site 1 first even though operator
action is from the right.

\begin{lemma}[Polynomial vacuum]\label{lem:poly-vacuum}
For $N\ge1$ and generic $(v_j)$, $T_i(t)$ has a simple common eigenline
with eigenvalue $1$. Its normalized eigenvector $w^i$ has
$w^i_\varnothing=1$, and $\Psi_N:=f_Nw^i$ (with $\Psi_0=1/\sqrt 2$)
is a trigonometric polynomial of degree $\le N-1$ in each $v_j$,
with the following properties:
\begin{enumerate}
\item\emph{Parity.} On a component with site spin $p_j$, $\Psi_N$ is
$(-1)^{N-1+p_j}$-parity in $v_j\mapsto v_j+\pi$.
\item\emph{Fusion.} With remaining list $U$,
\begin{equation}\label{eq:psi-fusion-2L}
\Psi_N(u,u+2L,U)=\prod_{z\in U}\cos(z-u-L)\cdot W_{12}\Psi_{N-1}(u+L,U),
\end{equation}
\begin{equation}\label{eq:psi-fusion-b}
\Psi_N(u,u+b,U)=2\cos b\prod_{z\in U}\cos(z-u-L)\cos(z-u-2L)\cdot\Omega_{12}\Psi_{N-2}(U).
\end{equation}
\item\emph{Infinity.} As $|v_N|\to i\infty$,
$\Psi_N(v)/\prod_{j<N}\cos(v_N-v_j)\to\sqrt2\Psi_{N-1}(v_1,\dots,v_{N-1})\otimes|0\rangle_N$.
\item\emph{Wheel zero.} $\Psi_N$ vanishes at the wheel
$(v_1,v_2,v_3)=(u,u+2L,u+5L)$ (order $\ge1$; order $\ge2$ for twist $-i$).
\end{enumerate}
\end{lemma}

\begin{proof}[Proof sketch]
\emph{Eigenvalue and simplicity.} Let $T_j=T_\xi(-v_p+jL)$ at any site $p$.
Yang--Baxter and spin conservation give
\begin{equation}\label{eq:fusion-T}
T_0T_2=T_1,\quad T_0T_3=I,\quad T_1T_3=T_2.
\end{equation}
(Check by applying $R_{2p}(0)R_{1p}(2L)=I$ on the image of $W$.)
Any finite product at common twist has $\varnothing,\varnothing$ entry
$1$ (loop cancellation, as in Section~\ref{sec:strip-physical}).
Judicious imaginary-limit sampling reduces the simplicity on total spin
zero to a base case $N=1$.

\emph{Vector normalization.} We set $w^\xi=P^\xi|\varnothing\rangle$,
$\ell^\xi=\langle\varnothing|P^\xi$, both rational with empty entry $1$,
and related by crossing $\ell^\xi(v)=w^{\xi^{-1}}(-v)^T$. Exchange and
fusion of $w$ follow as in Section~\ref{sec:strip-physical}: at
$v_q-v_p\in\{0,2L,b\}$, the local $R_{pq}$ and its fusions act as stated.

\emph{Degree bound.} For each row $T_\xi(t)$ on the vacuum line,
Fourier-Laurent expansion gives
\[
T_\xi(t)=I+\sum_k[A_k\cot(t+v_k+2L)+B'_k\cot(t+v_k+3L)],
\qquad\sum_k(A_k+B'_k)=0.
\]
Applying the fusion equations to $T_j=T_\xi(-v_p+jL)$ gives commuting-operator
annihilator rows $\mathcal A_{p,0}=c_0-c_1+c_2+X_0c_2$ and
$\mathcal A_{p,1}=c_1-c_2+c_3+X_1c_3$. Collecting into operator
matrix $\mathcal A$, one has
$\det\mathcal A=(\det H_{\mathcal A})P^\xi$, where $H_{\mathcal A}$
is the cotangent kernel $r(u)=\cot(u+3L)-\cot(u+4L)+\cot(u+5L)$.
The scalar Pfaffian $J_N=\mathrm{Pf}(H_{\mathcal A})\prod_{j<k}p(v_k-v_j)$,
\begin{equation}\label{eq:p-pfaffian}
p(u)=\frac{\sin(u+2L)\sin(u-2L)\cos u\ [\sin(u+b)\sin(u-b)]^2}
{\sin^2u\,\sin(u+L)\sin(u-L)},
\end{equation}
is a polynomial of trigonometric degree $\le 3(N-1)$ per variable.
Next, the two-site computation
\begin{equation}\label{eq:J-2-sites}
J_2=\cos u\,[S t_*-(t_*^2/2)\cos 2u],\qquad t_*=-\frac{1}{1+S},
\end{equation}
identifies three simple roots modulo $\pi$: one real at $u\equiv\pi/2$,
and two complex from $\cos 2u=-1-\sqrt2$. This gives generic simplicity
at $v_j-v_k\in\{L,2L\}$. Combining with
fusion via $W$ and $\Omega$, the total pole degree of $P^\xi$ is
bounded by $3(N-1)$, hence $d_N^i+d_N^{-i}\le 3(N-1)$.

\emph{Wheel zeros.} Consider $\Psi$ at $(v_1,v_2,v_3)=(u,u+2L,u+5L)$
in the order $1,2,3$ with remaining list $Y$. Perturbing
$v_2=u+2L+z$, $v_3=u+5L+z'$ and using~\eqref{eq:psi-fusion-2L}:
at $z=0$, $\Psi\in\mathrm{im}(W_{12})$; on $z'=z$ and $z'=0$ we get
specific columns $b_0,c_0$ proportional to $\Omega_{23}$ and to
$K_3 J^{\rm spin}_1 \Omega_{31}$ (composed with site-3 factors). The
column is in $\mathrm{im}(W_{12})$ only at specific spin coefficient
identities; matching at spin $\pm1$ then at spin $0$ forces both
$\gamma,\alpha=0$ (for $\xi=-i$), giving vanishing of order $\ge2$.

\emph{The Pfaffian normalization.} The candidate $f_N$ of
\eqref{eq:f_N} also satisfies the fusions~\eqref{eq:psi-fusion-2L},
\eqref{eq:psi-fusion-b} (using $B(d)B(d-b)=-\cos(d-L)\cos(d-2L)$ and
induction on $N$) and has the required degree, parity and symmetry.
The base cases $f_0,f_1,f_2$ match by direct computation. A polynomial
of degree $\le N-1$ with fixed parity and $2(N-1)$ prescribed values
modulo $\pi$ is unique, giving $f_N=\Psi_N^\varnothing$.

The remaining claims on $\Psi_N$ (degree control, parity) follow from
the fusion and normalized-vector structure.
\end{proof}

The equality of the exact degree to $3(N-1)$ is then obtained from
divergence at $v_2-v_1\in\{b\}$ in the double fusion~\eqref{eq:psi-fusion-b}:
the wheel factor there forces $d_N^i\ge N-1$ and $d_N^{-i}\ge 2(N-1)$.
Matching \eqref{eq:f_N} gives the exact scalar fusion identities
\begin{align}
\Psi_N(u,u+2L,U) &= c_1\prod_{z\in U}\cos(z-u-L)\cdot W_{12}\Psi_{N-1}(u+L,U),\label{eq:exact-fusion-2L}\\
\Psi_N(u,u+b,U) &= c_2\prod_{z\in U}\cos(z-u-L)\cos(z-u-2L)\cdot\Omega_{12}\Psi_{N-2}(U),\label{eq:exact-fusion-b}
\end{align}
with $c_1=1$ (after normalization) and $c_2=2\cos b$
(cross-checked at $N=2,3$ and by induction).

\paragraph{The physical contraction.}

\begin{proof}[Proof of Proposition~\ref{prop:physical-pairing}]
Define
$Z(v)=\ell^{-i}(v)(\prod_{j\le m}K_j(\delta))w^i(v)$,
with $\delta=i y$. Then $U=2f_N^2Z$ is a trigonometric polynomial
of degrees $\le 2(N-1)$ in each variable, $\pi$-periodic by spin-sign
covariance, and separately symmetric in the two groups (spin
conservation plus exchange).

Both $U$ and $G_N$ (from~\eqref{eq:G-def}) share:
\begin{itemize}
\item separate symmetry in the two groups,
\item $\pi$-periodicity in each $v_j$,
\item exact fusion to the smaller list at fused rapidities
$(v_{j+1}-v_j)\in\{0,\pm2L,\pm b\}$ (with normalizations from $f_N$),
\item growth behavior at imaginary infinity by~\eqref{eq:psi-fusion-b},
\item initial conditions $G_{N=0}=1$, $G_{N=1}=1$ (identical to $U$).
\end{itemize}
By induction on $N$, $G_N=U$. Indeed, the difference $G_N-U$ has zero
degree after dividing by the fusion-denominator product and vanishes
at each step. The base cases $N=0,1$ match.

Evaluating at $v_1=\cdots=v_N=0$ with the sandwiched $K_j(\delta)$'s:
\begin{enumerate}
\item $\mathcal Z_N$ at homogeneous $v$ has a natural expansion as an
oriented sum over polygon systems; the contractible and essential
sectors get phases $i$ and $\pm i$ (reversing conjugates), and the
`left-side switch' factor $(-1)^{1_{A\in D'}}$ plus
$\delta^{1_{A\in D'}-1_{B\in D'}}$ from the sandwich. Reversing
orientation inverts the factor, so a polygon contributes $0$ unless
$A,B$ are separated, giving weight $y+y^{-1}$ exactly.
\item The tile weights in $G_N$ without phases are honeycomb path
weights.
\item The empty system gives $1$, matching $(\varnothing,\varnothing)$
entry.
\end{enumerate}
For positive $y$, exhaustion proves $\mathcal Z_N=G_N/(2f_N^2)|_{v=0}$,
and the equality extends to a Laurent polynomial identity.
\end{proof}



\paragraph{The two-line gas expansion.}\label{c6:gas}

Section~\ref{c6:periodic-polynomial} expressed the physical cylinder
partition as a finite trigonometric ratio. We now expand that ratio as
a grand-canonical sum over \emph{particles}, and control the sum by a
coercivity bound on the kernel. The output is a rigorous upper bound
on $\log|P_m|$ (Proposition~\ref{prop:gas-upper}) which
Section~\ref{c6:subheading-09} saturates.

\paragraph{Normalization and the target.}

Put
\begin{equation}\label{eq:P-m-normalized}
P_m(e^r)=\frac{G(e^r)}{C^mD(0)^{m(m-1)}}.
\end{equation}
With $L=\pi/8$, $b=3L$, $C=2-\sqrt2$, $\widehat\rho(p)=1/(2\cosh Lp-1)$,
and $V(s)$ the real log of $R_s(is)R_c(is)$ for
$R_s(z)=-1/[\sin(z-L)\sin(z+L)]$, $R_c(z)=\cos^2z$, put
$I=\int_{\mathbb R}V(s)\rho(s)\,ds$.

\begin{proposition}[Homogeneous gas upper bound]\label{prop:gas-upper}
As $m\to\infty$, locally uniformly in $|\Im r|<.54\pi$:
\begin{equation}\label{eq:gas-upper}
\log|P_m(e^r)|\le m^2I+\left(\frac38+\frac{2}{3\pi^2}\Re r^2\right)\log m+o(\log m).
\end{equation}
\end{proposition}

\paragraph{Residue particles.}

Each lowering of a site label in the finite double-pole formula
produces one or two residue particles. A particle $d$ carries a color
sign $q_d\in\{\pm1\}$, a species, a contour, and a real position $s_d$.

The elementary mode potential is $e^{mV(w)}$, and the pair weights are
\begin{equation}\label{eq:pair-weight}
t_{gh}(w)=\frac{R_{gh}(iw)}{R_{gh}(iw+L)R_{gh}(iw-L)},
\end{equation}
with measure $\tau dz/(2\pi i)$ and $\tau=2\sin(2L)/\sqrt C$.

After periodization (replacing each $\sinh(w+ijL)$ by
$\frac12S_T(w+ijL)$ with $S_T(w)=2\sinh w\prod_{n\ge1}(1-e^{-2nT+2w})
(1-e^{-2nT-2w})$, and multiplying the integrand by
$\sum_le^{-2Tl^2+4l\sum q_jw_j}$) we obtain a $T$-periodic
$P_{m,T}(e^r)$, agreeing with $P_m(e^r)$ at $e^{-2T}=0$. We work on
$P_{m,T}$ and return by maximum modulus in the nome $e^{-T}$.

\paragraph{Species and offsets.}

After contour manipulation and fusion, five species remain in each
color. Their weights $w_i$ (exponent of $y$), offsets $\delta_i$
(multiples of $L$), and chemical activities $\sigma_i$ are:
\begin{center}
\begin{tabular}{c|ccccc}
$i$ & $0$ & $1$ & $2$ & $3$ & $4$ \\
\hline
label & $S$ & $D$ & $I_+'$ & $I_-'$ & $F$ \\
$w_i$ & $1$ & $1$ & $-1$ & $1$ & $2$ \\
$12\delta_i$ & $6$ & $1$ & $5$ & $-3$ & $-1$ \\
$\sigma_i$ & $0$ & $1$ & $1$ & $-1$ & $0$
\end{tabular}
\end{center}
The neutral-winding constraint on total particle counts is
\begin{equation}\label{eq:winding-neutrality}
\sum_{g\in\{c,-\}}\sum_{i=1}^5w_i\nu_{g,i}=0.
\end{equation}

\paragraph{The interaction kernel.}

The particle kernel for two particles at real frequency $p$ is
\begin{equation}\label{eq:Z-def}
Z^{(c)}_{ij}(p)=\frac{2\pi}{p}e^{(\delta_i-\delta_j)Lp}F^{(c)}_{ij}(e^{Lp}),
\end{equation}
with $F^{(c)}_{ij}(z)$ expressed through an auxiliary function $H(a;\sigma)$
that sums residues on the unit circle:
\begin{equation}\label{eq:H-aux}
H(a;\sigma)(z)=\sum_{j=0}^7h_j\frac{z^{[a+j]_8}}{z^8-1}+\frac{1-\sigma}{2}h_{[-a]_8},
\quad h=(2,-1,1,-2c,2c,-2c,1,-1).
\end{equation}
Let $H_*=H(0;0)$, $A_0=\sum_{a\in(2)}H(a;1)$, $A_1=\sum_{a\in(2,3)}H(a;-1)$,
$A_2=1$, $A_3=-1$, $A_4=A_1+A_3$. For $0\le i,j<4$, set
\begin{equation}\label{eq:F-ij}
F^{(c)}_{ij}(z)=P_{ij}(z)+\frac{A_i(z)A_j(z^{-1})}{H_*(z)}
+(-1+2c)\operatorname{sgn}(i-j){\mathbf 1}_{\{i,j\}=\{1,3\}},
\end{equation}
$F_{i4}=F_{i1}+F_{i3}+{\mathbf 1}_{i=3}-{\mathbf 1}_{i=1}$,
$F_{4i}(z)=-F_{i4}(z^{-1})$, $F_{44}=F_{11}+F_{13}+F_{31}+F_{33}$,
where $P_{ij}$ collects lower-derivative residue rules
$P_{ij}(z)=\sum_{a\in b_i,k\in b_j}H(a-k;\operatorname{sgn}(j-i))$ for $i,j<2$
and $\operatorname{sgn}(j-i)$ for $i,j\ge2$, and $0$ otherwise.

On the two $R$-lines (before fusion), the matrix symbol has a clean form.
Let $K=K_{RR}$ and $\widehat\rho(p)=1/(2\cosh Lp-1)$; then
\begin{equation}\label{eq:K-R-est}
K=(1-e(p))K_0,\quad
e(p)=\frac{\sinh(3Lp)-\sinh(2Lp)}{\sinh(4Lp)}\in(0,1),
\end{equation}
\begin{equation}\label{eq:K-0}
K_0=\frac{2\pi}{p}\begin{pmatrix}\coth(bp)&-\operatorname{csch}(bp)\\
-\operatorname{csch}(bp)&\coth(bp)\end{pmatrix},\quad V_R=K{\mathbf 1}\widehat\rho.
\end{equation}
(Here $V_R$ denotes the potential contribution.)

\paragraph{Coercivity.}

The key analytic input is a positive-definite bound on $Z$.

\begin{proposition}[Coercivity]\label{prop:coercivity}
Put $h_*=3\pi b/4$ and $\alpha=2\pi^2/h_*$, and let $W=(w,w)^t$ be the
weight column. Then for every real $p\ne0$ there is $\eta>0$ with
\begin{equation}\label{eq:Z-coercive}
Z_{\rm s}(p)\succcurlyeq\frac{\eta\alpha}{p^2}WW^t+\epsilon_0(1+p^2)^{-1}{\bf I},
\end{equation}
for some $\epsilon_0>0$ independent of $T$ and $m$, where $Z_s=(Z+Z^t)/2$.
\end{proposition}

\begin{proof}[Sketch]
On each $R$-line, $K=(1-e(p))K_0$ with $e(p)\in(0,1)$ and $K_0$
positive definite for $p\ne0$. The leading behavior at small $p$ is
$K_0\sim2\pi/(p)(\coth bp{\bf I}-\operatorname{csch}bp\,P)$ where
$P$ is the color-swap permutation matrix; the eigen-decomposition is
common and gives the $WW^t/p^2$ leading term. Away from $p=0$, the
residual is bounded below by $\epsilon_0(1+p^2)^{-1}{\bf I}$ after
partial-fraction expansion of the explicit $F$-rules.
\end{proof}

\begin{equation}\label{eq:Z-small-p}
Z(p)=\alpha WW^t/p^2+M_*/p+O(1)\quad(p\to0),
\quad M_*\text{ real skew}.
\end{equation}
\begin{equation}\label{eq:Z-large-p}
Z(p)=\frac{2\pi}{|p|}{\mathbf I}+2ff^t+O(e^{-\epsilon|p|})\quad(|p|\to\infty),\quad f={\mathbf e}_4.
\end{equation}

\paragraph{The grand sum.}

\begin{theorem}[Grand sum]\label{thm:grand-sum}
For every integer $m\ge1$ and every sufficiently large real $T$,
$P_{m,T}(e^r)$ equals
\begin{equation}\label{eq:grand-sum}
\sum_{\{\nu_{g,i}:\,(\ref{eq:winding-neutrality})\}}
\frac{1}{\prod_{g,i}\nu_{g,i}!}
\,e^{c\cdot 2\pi iTr\,\sigma^t\nu/(4h_*)-S_{\rm bulk}(T)-S_{\rm gas}[\nu;r]},
\end{equation}
with:
\begin{itemize}
\item $S_{\rm bulk}(T)=\frac{1}{2T}\sum_{p\ne0}\widehat J^tK(p)\widehat J\,\widehat\rho(p)^2
-\frac{1}{2}(T/3-2\log2)(M_A-M_B)^2$,
a real quadratic form;
\item $S_{\rm gas}[\nu;r]=-\frac{1}{2T}\sum_{p\ne0}\sum_{d,e}Z_{g_de_d}(p)e^{-ip(s_d-s_e)}-\frac{1}{T}\sum_{\text{linear}}\dots$,
the pairwise and single-particle interactions against real positions
$s_d\in(-T/2,T/2)$;
\item $\varepsilon_\nu\in\{\pm1\}$ is a fixed phase depending only on the multiplicity pattern;
\item $c$ is a fixed scalar.
\end{itemize}
The sum and integral converge absolutely, and the sum localizes on
\begin{equation}\label{eq:gas-localization}
\max_d|\Re s_d|\lesssim\log m+\text{const}.
\end{equation}
\end{theorem}

\paragraph{Convergence and localization.}

\begin{proof}[Sketch]
Proposition~\ref{prop:coercivity} bounds the off-diagonal $Z_s$ terms
from below by a positive quadratic form. After regularizing the diagonal,
the sum over $\nu_{g,i}$ is bounded by a geometric series in
$\sum_{g,i}\nu_{g,i}$ weighted by a Gaussian-decaying kernel in the
positions $s_d$ and a bounded activity from the chemical potential. For any configuration with a particle at $|\Re s_d|>\log m+\text{const}$,
the one-particle suppression from $S_{\rm bulk}$ dominates the local
gain, and the sum over such configurations is at most $e^{-\epsilon m}$.
Consequently the whole sum is dominated by the localized region
\eqref{eq:gas-localization}.
\end{proof}



\paragraph{Saturation at the two physical parameters.}\label{c6:saturation}

We have the upper bound $\log|P_m(e^r)|\le m^2I+(3/8+2\Re(r^2)/(3\pi^2))
\log m+o(\log m)$ for $|\Im r|<.54\pi$ (Proposition~\ref{prop:gas-upper}),
plus an exact evaluation of $P_m(i)$ via a determinant at the "empty
polygon" value. We now show both bounds are sharp and take their ratio.

\paragraph{The saturation statement.}

\begin{auxproposition}[Saturation at $y=1$ and $y=i$]\label{prop:saturation}
\begin{equation}\label{eq:saturation-aux}
\log P_m(1)=Im^2+\frac38\log m+o(\log m),\qquad
\log P_m(i)=Im^2+\frac5{24}\log m+o(\log m).
\end{equation}
Equivalently, the nesting ratio satisfies
\begin{equation}\label{eq:nesting-ratio}
\mathcal Z_{m,m}(1)=\frac{P_m(1)}{P_m(i)}=m^{1/6+o(1)}.
\end{equation}
\end{auxproposition}

\paragraph{The reference determinant at $y=i$.}

At $y=i$ the finite polynomial $G$ becomes $2f_N^2$: only the empty
polygon system contributes. In its determinantal expression replace
the kernel $h(d)=(\cot(d+b)+\cot(d-b))/2$ by
$h_0(d)=-\lambda\tan(\lambda d)/2$, $\lambda=\pi/(2b)=4/3$. The
normalized homogeneous value of this modified partition evaluates
exactly, by Schur's Pfaffian identity, to
\begin{equation}\label{eq:reference-det}
P_0^{(\lambda)}=\big[\lambda^4\sin^8(b)/D(0)\big]^{m^2}=e^{Im^2}.
\end{equation}
Indeed, Schur's Pfaffian of $G_{jk}=(X_j-X_k)/(X_j+X_k)$ with
$X_j=e^{2i\lambda v_j}$ equals $\prod_{i<j}(X_i-X_j)^2/
\prod(X_i+X_j)^2$ (clearing denominators, both sides alternate with
the same Vandermonde degree; the coefficient is fixed by a base case),
and $B(u)\tan(\lambda u)\to-\lambda\sin^2b$ at coincidence while
$D(0)=C\sin^4b$.

We are reduced to bounding the ratio $P_m(i)/P_0^{(\lambda)}$.

\paragraph{Bounding the ratio at $y=i$.}

Write $P_m(i)/P_0^{(\lambda)}$ as the product over $m$ "duplicate
pairs" $v_p,v_p+L$ of the linear-perturbation ratios
\begin{equation}\label{eq:duplicate-ratio}
\prod_{k=1}^m\frac{\det(1+\Delta_k)}{\det(1)},
\end{equation}
where $\Delta_k$ is the rank-one modification between the $h$ and $h_0$
kernels restricted to the $k$-th pair.

\emph{Weyl chamber decomposition.} Decompose the integration domain by
the partition of $\{1,\dots,m\}$ into the subsets $\Omega^A$, where $A$
is the unique subset containing the up-to-relabeling largest $|z_j|$.
On each $\Omega^A$, moving the rapidities with $j\in A$ to imaginary
infinity is pole-free within the chamber, and the two kernels (for $h$
and $h_0$) differ by a bounded factor with a Sobolev/$L_p$ estimate
from the linear-differential operator computation.

For each $k$, one gets uniformly
\begin{equation}\label{eq:weyl-bound}
\big|\det(1+\Delta_k)\big|\le e^{-\epsilon\,\mathrm{dist}(A_k,
\partial\mathrm{chamber})}.
\end{equation}
Summing chambers and applying maximum modulus:

\begin{equation}\label{eq:log-modulus-total}
\log\big|P_m(i)/P_0^{(\lambda)}\big|\le\frac5{24}\log m+O(1).
\end{equation}

Matching lower bound: restrict to configurations with all duplicates
within bounded distance of center, on which $\det(1+\Delta_k)=1-o(1)$.
This gives $|P_m(i)/P_0^{(\lambda)}|\ge m^{5/24-o(1)}$, hence
\begin{equation}\label{eq:P-m-i-final}
\log P_m(i)=Im^2+\frac5{24}\log m+o(\log m).
\end{equation}

\paragraph{Propagation to $y=1$.}

Set $r=\log y$ and put
\begin{equation}\label{eq:f-m}
f_m(r)=\frac{\log|P_m(e^r)|-Im^2}{\log m}
-\left(\frac38+\frac{2\Re(r^2)}{3\pi^2}\right).
\end{equation}
From Proposition~\ref{prop:gas-upper}, $f_m\le o(1)$ locally
uniformly in $|\Im r|<.54\pi$. From \eqref{eq:P-m-i-final} at
$r=i\pi/2$ (i.e. $y=i$), $f_m(i\pi/2)\to0$.

The function $f_m$ is subharmonic. Apply the subharmonic area
mean inequality on a finite chain of disks starting at $r=i\pi/2$ and
reaching any compact subset of $|\Im r|<.54\pi$: the two ends of the
chain give $f_m\to0$ in local $L^1$ throughout the strip. In
particular at $r=0$:
\begin{equation}\label{eq:saturation-at-1}
\log P_m(1)=Im^2+\frac38\log m+o(\log m).
\end{equation}

Combining \eqref{eq:P-m-i-final} and \eqref{eq:saturation-at-1}:
\begin{equation}\label{eq:nesting-ratio-concl}
\mathcal Z_{m,m}(1)=P_m(1)/P_m(i)=m^{1/6+o(1)},
\end{equation}
as claimed.
\end{proof}



\paragraph{Crossings, cylinder barriers, and positive nesting.}\label{c6:geometry}

Section~\ref{c6:strip-asymptotics} gave the strip-mass estimate
$B_k\asymp k^{-1/4}$. Here we extract (a) escape and confinement
estimates, (b) essential-polygon formation on a cylinder, (c) positive
polygon nesting, and (d) a boundary-occupancy identity. These are the
geometric inputs for Section~\ref{sec:sewing}.

All constants are uniform in $N$, $h$, and $H$.

\paragraph{The critical weight at the port convention.}

Adapting the Hammersley--Welsh argument to boundary-port convention
confirms that $\kappa$ is the critical self-avoiding-walk weight at
fixed starting triangle. The route uses strip bridges: for $x<\kappa$
the mass of weak bridges of height $k$ is at most $Ce^{-ck}$, and
decomposition over weak half-space walks gives the susceptibility
bounds. Constants depend only on $x$.

\paragraph{Boundary crossings.}

\begin{proposition}[Boundary crossings]\label{prop:crossings}
Fix a convex pure-sided tiled domain $D$ and a port $p$ on its
boundary (including strips and half-planes by exhaustion). Then:
\begin{enumerate}
\item The critical mass of completed paths of diameter $\ge r$ is at
most $C(1+r)^{-1/4}$.
\item The half-plane mass $K_l$ from a base port to another at lattice
distance $l$ is decreasing in $l$ and satisfies $K_l\asymp l^{-5/4}$.
\end{enumerate}
\end{proposition}

\begin{proof}[Sketch]
\emph{Contour comparison.} The signed exit identity (Section~\ref{c6:strip-flux}
with $\beta=3/8$) gives
\begin{equation}\label{eq:positive-exit}
\sum_{\gamma:p\to\partial D}\kappa^{\ell(\gamma)}
\cos(\beta W(\gamma))=1,\quad c\le\cos(\beta W(\gamma))\le1.
\end{equation}
Restricting to a convex subdomain and using $c=\cos(\beta\pi)>0$ gives
the first assertion.

\emph{Triangle escape.} For a centered triangle $T(p,n_j,t)$ of width
$t$, the exit mass $S(t)$ to each sloping side is comparable to
$(1+t)^{-1/4}$. The upper bound comes from the strip embedding; the
lower bound from the base-return identity
$2\cos(\beta\pi/3)S=1-cA_T$ and the diameter bound.

\emph{Monotonicity.} A non-crossing identity for $K$ and an intermediate
port between sources $a$ and $Q=P+1$ gives
\begin{equation}\label{eq:K-monotone}
(K_{P-a}-K_{Q-a})\sin(\beta\pi)=D_1\sin(2\beta\pi)\ge0,
\end{equation}
using the flux rule with an artificial arc below the base line. Combined
with the diameter bound over a window of length $\asymp l$: $K_l\lesssim l^{-5/4}$.

\emph{Deep exits.} For an exit from $T(a,n_0,t)$ through the side with
outward normal $n_1$, at depth $q=n_0\cdot(p_1-a)$,
\begin{equation}\label{eq:deep-exit}
\mathop{\mathrm{average}}_{t\in[\alpha R,AR]}\sum_{q\ge\delta R}k(t,q)
\ge c_{\alpha,A,\delta}R^{-1/4},
\end{equation}
for $\delta$ small, by translating along the base to align the exit
ports and cutting strictly before all starts.

\emph{Three exits and matching lower bound.} Iterating the deep-exit
construction at $p_1,p_2$ builds a three-exit path. Averaging over $t$
and $R$ gives mass $\gtrsim R^{-3/4}$ of paths with $q\ge\delta R$ and
$X\ge\eta R$, whence a second-moment bound
\begin{equation}\label{eq:three-exit-second-moment}
\mu[Y^2]\le CR^{-5/4},
\end{equation}
using the difference bridges with $B_{|t-t'|/d_0}$. Cauchy--Schwarz gives
$\mu[Y>0]\gtrsim R^{-1/4}$ and the matching lower bound for $K_l$ by
monotonicity.
\end{proof}

\paragraph{Narrow-channel bridges.}

\begin{lemma}[Narrow-channel bridges]\label{lem:narrow-channel-bridges}
For every sufficiently large absolute constant $A$, there are
constants $C,R_0>0$ such that, for every strip of height $H$, every
starting port on one cap, and every integer $M\ge1$ with $H/M\ge R_0$,
the critical mass of bridges whose transverse displacement from the
starting port stays at most $4AH/M$ throughout is at least
\begin{equation}\label{eq:channel-bridge-mass}
e^{-CM}(H/M)^{-1/4}.
\end{equation}
\end{lemma}

\begin{proof}[Sketch]
Put $s=H/M$ and choose $M-1$ internal levels independently in windows
of length $\asymp s$ around equally spaced cuts. Each piece has height
between $s/2$ and $2s$ and transverse excursion $\le As$, with
endpoint shift directed towards the original axis. Reflection retains
at least half the mass after the excursion restriction; the strip
bound on each piece gives $cs^{-1/4}$; Cauchy--Schwarz over the lists
of internal levels gives \eqref{eq:channel-bridge-mass}.
\end{proof}

\paragraph{Essential polygons in a slab.}

\begin{proposition}[Slab essential polygons]\label{prop:slab-essential-polygons}
There are absolute constants $c,C>0$ such that, for every sufficiently
large integer $N$ and every horizontal slab $S$ of height $CN$ on the
cylinder of period $N$:
\begin{equation}\label{eq:slab-essential-mass}
\sum_{\gamma\subset S,\ \gamma\text{ essential simple}}
\kappa^{\ell(\gamma)}\ge c.
\end{equation}
\end{proposition}

\begin{proof}[Sketch]
Build confined arches via collar decompositions at two ports
$p,q$ separated by $D\in[N/2-\epsilon N,N/2+\epsilon N]$. Each collar
uses \eqref{eq:deep-exit}, \eqref{eq:channel-bridge-mass}, and one
$S(t)$ factor, so a collar pair has mass $\gtrsim N^{-3/2}$, and the
connector adds another $N^{-5/4}$ factor by the $K_l$ estimate.
Averaging over the collar-cut lists and Cauchy--Schwarz gives a
slab-mass bound independent of the period. Joining upper and lower
confined arches and averaging over their shared base level gives
\eqref{eq:slab-essential-mass}.
\end{proof}

\paragraph{Positivity of the negative polymer polynomial.}

Let $D$ be the finite cylinder between horizontal row-boundary caps,
with height $H=h d_0$ and quotient by $N$ steps. Let $\mathcal P$ be
its set of simple polygons. For $0\le\beta'\le3/8$, put
$\alpha=\pi\beta'/3$, $x=(2\cos\alpha)^{-1}$, and
\begin{equation}\label{eq:omega-weight}
\omega(L)=x^{|L|}\cdot\begin{cases}
-2\cos(4\alpha),&L\text{ contractible},\\
-2\cos(2\alpha),&L\text{ essential}.
\end{cases}
\end{equation}
For $\mathcal A\subseteq\mathcal P$:
\begin{equation}\label{eq:Q-def}
Q(\mathcal A)=\sum_{\substack{\mathcal F\subseteq\mathcal A\\
\text{polygons in }\mathcal F\text{ disjoint}}}
\prod_{L\in\mathcal F}\omega(L).
\end{equation}

\begin{auxproposition}[Positivity]\label{prop:Q-positive}
For every $N\ge1$, $h\ge1$, $0\le\beta'\le3/8$, $\mathcal A\subseteq\mathcal P$:
$Q(\mathcal A)>0$.
\end{auxproposition}

\begin{proof}[Sketch]
Short periods $N=1$ or $h=1$ follow from the explicit row-cycle
formulas. For $N,h\ge2$: elementary polygons (six-centre loops around
interior lattice points) form a connected overlap family meeting every
triangle centre, so the incompatibility graph is connected. The signed
exit identity gives
\begin{equation}\label{eq:Q-expression}
Q(\mathcal P)=\sum_{\gamma:p\to\partial D}x^{|\gamma|}
e^{i\beta'W(\gamma)}Q(\{L:L\cap\gamma=\varnothing\}).
\end{equation}
Continue from $\beta'=0$, $x=1/2$ where defect terms are nonnegative,
to $\beta'=3/8$. A minimal-zero subset $\mathcal A$ must be all of
$\mathcal P$ by connectedness; but a fixed bridge through every level
gives a positive residual polynomial, contradiction.
\end{proof}

\paragraph{Deletion and free-energy bounds.}

At the endpoint parameters replace $\omega(L)$ by the essential-only
weight $-t\kappa^{|L|}$ with $t\in[0,\sqrt2]$. Positivity holds by
multiaffinity. The deletion ratio is
\begin{equation}\label{eq:cylinder-deletion-ratio}
H_H(\gamma;t)\ge\exp\Big(t\!\sum_{\ell\ \text{essential},\ \ell\cap\gamma\ne\varnothing}\!\kappa^{|\ell|}\Big).
\end{equation}
Combining \eqref{eq:cylinder-deletion-ratio} with
\eqref{eq:slab-essential-mass} and \eqref{eq:strip-upper} gives
\begin{equation}\label{eq:cylinder-free-energy}
-\log Z_H(\sqrt2)\le CH/N\qquad(H\ge N),
\end{equation}
and hence
\begin{equation}\label{eq:cylinder-integrated-ratio}
\int_1^{\sqrt2}\sum_{\ell\text{ essential}}\kappa^{|\ell|}
H_H(\ell;t)\,dt\le CH/N.
\end{equation}

\paragraph{Cylinder barriers and height tails.}

\begin{auxproposition}[Barriers and tails]\label{prop:cylinder-barriers}
On the infinite cylinder of period $N$:
\begin{enumerate}
\item The individual critical weights of essential polygons meeting a
fixed horizontal level have bounded sum.
\item From a fixed bottom port in a semi-infinite cylinder, arches
reaching height $s$ have mass at most $Ce^{-cs/N}$.
\item For $s\ge1$, all polygons meeting a fixed level with height span
in $[s,2s]$ have total mass at most $\operatorname{poly}(N,s)
e^{-cs/N}$.
\end{enumerate}
\end{auxproposition}

\begin{proof}[Sketch]
Combine \eqref{eq:cylinder-deletion-ratio} with
\eqref{eq:slab-essential-mass}. A polygon of span $s$ meets order
$s/(CN)$ disjoint slabs, each forcing positive-size cost:
$H_H(\gamma;t)\gtrsim e^{c's/N}$. Sum by spans between $(jN,(j+1)N)$.
The case $N=2$ uses an explicit two-strip surgery on the two choices of
upward-centre; $N=1$ is a row-cycle count.
\end{proof}

\paragraph{Local nesting and boundary occupancy.}

\begin{proposition}[Local nesting]\label{prop:local-nesting}
Let $Z(R)$ be the positive partition sum of pairwise disjoint simple
polygons in the plane, factor $2$, critical weight, supported within
radius $R$ of a fixed triangular-lattice point, including the empty
system. Then
\begin{equation}\label{eq:Z-R}
Z(R)=R^{1/12+o(1)}.
\end{equation}
\end{proposition}

\begin{proof}[Sketch]
On a cylinder, disregard essential separators at bounded multiplicative
cost by \eqref{eq:cylinder-deletion-ratio}. Contractible separators
lift to plane polygons around the marked point; the barrier bounds
\eqref{eq:cylinder-barriers} discard those with excessive vertical or
horizontal span. Systems inject. Hence
$\mathcal Z\lesssim Z(CN^{1+2\xi})^2$ for any $\xi>0$, and combining
with \eqref{eq:nesting-ratio} and taking $\xi\downarrow0$ gives
\eqref{eq:Z-R}.
\end{proof}

\begin{proposition}[Boundary occupancy]\label{prop:occupancy}
Let $D$ be a finite convex tiled domain and let $e$ be an internal
honeycomb link crossing a tiling edge $[y,z]$. If $U_D(e)$ is the
critical mass of completed external paths using $e$ with ordered free
boundary ports, then
\begin{equation}\label{eq:occupancy}
U_D(e)\asymp Z_D(y)+Z_D(z)
\end{equation}
with absolute constants.
\end{proposition}

\begin{proof}[Sketch]
The cut-link identity gives a signed exit mass
$S_e=Z_e+(M_y+M_z)/\sqrt2$, where $Z_e$ is the enclosing-polygon
partition in the cut domain and $M_y,M_z$ are contributions of a
polygon that uses $e$ with $y$ or $z$ inside ($Z_D(y)=Z_e+2M_y$,
$Z_D(z)=Z_e+2M_z$). Reversing path prefixes and summing over sources
yields \eqref{eq:occupancy}.
\end{proof}



\paragraph{Bounded-factor nesting estimates.}\label{sec:uniform-nesting}

The saturation bound of Section~\ref{c6:saturation} identifies a nonzero
normalization at exponent level. We now sharpen two inputs to bounded
factors: the geometric comparison between plane and cylinder, and the
analytic estimate of the normalized edge contributions. Both are used
in the sewing argument (\S\ref{sec:sewing}).

\paragraph{Deletion ratio and the first-hit comparison.}

Let $D$ be either a finite cylinder between flat caps, or a finite
convex tiled polygon in the plane. Write
\begin{equation}\label{eq:Q-D}
Q_D=\begin{cases}
Z_H(\sqrt2),&D\text{ cylinder},\\
1,&D\text{ plane},
\end{cases}
\end{equation}
the positive essential-polygon partition. For a set $K$ of occupied
centres:
\begin{equation}\label{eq:theta-D-K}
\Theta_D(K)=Q_{D\setminus K}/Q_D.
\end{equation}

\begin{lemma}[Deletion monotonicity]\label{lem:deletion-monotone}
$\Theta_D(K)$ is increasing in both the set of available polygons and
the deleted set $K$, for $0\le t\le\sqrt2$.
\end{lemma}

\begin{proof}
For a single deleted polygon $v$:
\begin{equation}\label{eq:single-deletion-ratio}
Q_P/Q_{P\setminus v}=1-t\kappa^{|v|}\cdot Q_{P\setminus N[v]}/Q_{P\setminus v},
\end{equation}
where $N[v]$ is the closed conflict neighbourhood. By positivity
\eqref{eq:Q-def} the ratio is $\ge1$. Induction on $|P|$ (deleting
neighbours successively in the smaller set) gives the multi-deletion
inequality; extra neighbours only increase the ratio.
\end{proof}

For a simple polygon $L\subset D$ and a boundary source $a$, let
$P(a,L)$ be the real part of the normalized signed mass of stems that
first hit $L$. Contour identity:
\begin{equation}\label{eq:first-hit-comparison}
1\ge P(a,L)\ge
\sum_{\substack{\gamma:a\to\partial D\\ \gamma\cap L\ne\varnothing}}
\kappa^{|\gamma|}\cos(\beta W(\gamma))H_D(\gamma)\ge0.
\end{equation}

\paragraph{Boundary-flux primitive.}

\begin{auxlemma}[Primitive of boundary flux]\label{lem:flux-primitive}
Choose a dual-edge chain $\Gamma$ from a boundary lattice vertex $y_0$
to an interior vertex $y$. Define $\operatorname{cr}_\Gamma(\gamma)$ to
be the signed number of $\Gamma$-crossings of a completed path and put
\begin{equation}\label{eq:current-I}
I=\Im\sum_{a\in\partial D}\sum_{\gamma:a\to\partial D}
\operatorname{cr}_\Gamma(\gamma)\,\kappa^{|\gamma|}e^{i\beta W(\gamma)}H_D(\gamma).
\end{equation}
Then
\begin{equation}\label{eq:flux-primitive-bound}
2\!\!\sum_{L\text{ contractible enclosing }y}\!\!\kappa^{|L|}H_D(L)\sum_aP(a,L)
\le|I|+\sqrt2\!\!\sum_{L\text{ essential}}\!\!\kappa^{|L|}H_D(L)\sum_aP(a,L).
\end{equation}
The essential sum is absent in the plane.
\end{auxlemma}

\begin{proof}[Sketch]
Evaluate the same prefix current $J$ two ways.
\emph{(i)} Telescoping along $\Gamma$ via the cut-link identity of
Prop.~\ref{prop:occupancy}: $J=-(\mathcal Y(y)/Q_D-1)/\sqrt2$.
\emph{(ii)} Continuing each prefix to its first exit gives
$J=I+2A+E$, where $A$ is the contractible enclosing sum and $|E|\le
\sqrt2\sum_{\text{essential}}\kappa^{|L|}H_D(L)\sum_aP(a,L)$ (turn
increments $\pm4\pi/3$ versus $\pm2\pi/3$). Comparing gives the
inequality.
\end{proof}

\paragraph{The annular bound.}

\begin{proposition}[Planar annular bound]\label{prop:annular-bound}
For every $s\ge1$ and every interior lattice point $y$:
\begin{equation}\label{eq:annular-mass}
\sum_{\substack{L\text{ contractible enclosing }y\\
\operatorname{diam}(L)\in[s,2s]}}\kappa^{|L|}\le C.
\end{equation}
\end{proposition}

\begin{proof}[Sketch]
Take a regular lattice hexagon of inradius $\asymp s$ around $y$. Only
chords that separate $y$ from $y_0$ contribute to $I$, with bounded net
crossing; summing the boundary diameter bound over the sources gives
$|I|\le Cs^{3/4}$.

A polygon enclosing $y$ with diameter in $[s,2s]$ fits inside a
doubled hexagon and has transverse span $\ge cs$ in some pure direction.
For order $s$ sources on the corresponding cap, transverse tubes of
width $\ll s$ force intersection of every confined path with $L$, so
\eqref{eq:channel-bridge-mass} and \eqref{eq:first-hit-comparison} give
$\sum_aP(a,L)\ge cs^{3/4}$. The essential sum vanishes in the plane; the
auxlemma gives the bound.
\end{proof}

\paragraph{Bounded cost of large specified lifts.}

\begin{auxproposition}[Bounded cost of large lifts]\label{prop:large-lifts}
Fix $c_2>0$. On a cylinder of period $N$, the total individual critical
weight of contractible polygons enclosing a fixed mark $y$ whose
specified lifts have diameter greater than $c_2N$ is at most $C_{c_2}$,
independently of $N$.
\end{auxproposition}

\begin{proof}[Sketch]
\emph{Good heights.} Put $F_H(t)=-\log Z_H(t)$. Superadditivity of
$F_H$ in $H$ plus the free-energy bound \eqref{eq:cylinder-free-energy}
make
\begin{equation}\label{eq:good-height}
a_j=\frac{NF_{H_j}(\sqrt2)}{H_j},\qquad H_j=d_0N2^j,
\end{equation}
a nondecreasing sequence in $[0,C]$. For any $u>0$, at most $4rC/u$
indices (for $k=2^r$ a fixed dyadic integer depending on $u$) violate
$F_{H_j}(\sqrt2)-kF_{H_j/k}(\sqrt2)\le uH_j/(4N)$, and then
\begin{equation}\label{eq:good-height-conclusion}
\sum_{L\text{ essential in }H_j\text{-slab}}\kappa^{|L|}H_{\text{slab}}(L)
\le e^{uH_j/N}.
\end{equation}

\emph{Good-height contractible sums.} For a good height $H$,
\begin{equation}\label{eq:good-contractible}
\sum_{\substack{L\text{ contractible enclosing }y\\
\operatorname{span}_{\text{vert}}\in[H/4,H/2]}}\kappa^{|L|}\le Ce^{-cH/N},
\end{equation}
by binning their minimum levels into windows of width $\delta N$ (bounded
number of bins by good-height hypothesis), applying
\eqref{eq:flux-primitive-bound} in an encompassing slab, and using
\eqref{eq:cap-first-hit} and the slab-bound \auxref{prop:cylinder-barriers}
to control both sides.

\emph{Tilted cylinder projection.} Dyadic bands with vertical span
$\ge cN$ project injectively onto a cylinder tilted by $60^\circ$, of
period $\asymp$ span. The tilted cylinder has its own dyadic-height
decomposition; by the above, all but boundedly many heights are good, so
\begin{equation}\label{eq:large-lifts-bound}
\sum_{\substack{L\text{ contractible enclosing }y\\
\text{lift-diam}>c_2N}}\kappa^{|L|}\le C_{c_2}.
\end{equation}
The remaining finite collection of exceptional bands is bounded by the
planar annular bound \eqref{eq:annular-mass} applied to the corresponding
diameter intervals.
\end{proof}

\paragraph{Classwise convergence of the normalization.}

\begin{auxproposition}[Classwise convergence]\label{prop:classwise-convergence}
For $r\in\{0,i\pi/2\}$ put $\sigma(r)=3/8+2r^2/(3\pi^2)$ and
$\mathcal A_m(r)=e^{-Im^2}m^{-\sigma(r)}P_m(e^r)$. There exist a fixed
integer $Q\ge1$, a $\delta>0$, and positive constants
$\mathfrak a_b(r)$, $0\le b<Q$, such that
\begin{equation}\label{eq:classwise-convergence}
\mathcal A_m(r)=\mathfrak a_{m\bmod Q}(r)+O(m^{-\delta}).
\end{equation}
Consequently
\begin{equation}\label{eq:uniform-cylinder-ratio}
\mathcal Z_{m,m}(1)=P_m(1)/P_m(i)\asymp m^{1/6}.
\end{equation}
\end{auxproposition}

\begin{proof}[Sketch]
Work with the periodic grand sum at period
$T=2a_0^{-1}\log m+g+i\theta$, $|\theta|\le\pi/2$, for a suitably large
fixed $g$. The stability estimate \eqref{eq:Z-coercive} and
localization \eqref{eq:gas-localization} continue to hold with the
coercivity error $O(1/g)\mathcal Q$ absorbed by $g$ large; the period
graph is fixed-mod-$g$ independent of $m$ on half intervals.

Multiplying the lifted polynomial by
$e^{-2Am^2e^{-2T}}e^{-Im^2}m^{-\sigma(r)}$ produces an
$F_m(\theta,r)$ whose circle mean is exactly $\mathcal A_m(r)$. We show
$F_m$ is Cauchy at exponential speed along each class modulo $Q$.

\emph{Scalar factor.} The quadratic bulk contribution gives
$m^{3/8}$; the logarithmic determinant gives
$\pi T/(16L)=\frac{5}{24}\log m$ plus constant; combining with the
edge integrals gives $m^{\sigma(r)}$ exactly.

\emph{Edge integrals.} Retain only particles within $\epsilon T'$ of an
edge, at most $T'$ particles, charge $j$ matching the right edge. The
omitted mass is $e^{-cT'}$, and standard small-slope deformation gives
agreement of the two retained integrands up to $e^{-cT'}$.

\emph{Discrete phases.} The $m$-dependent phase factors are roots of
unity from fixed logarithmic branch multiplies; the entire phase
depends only on $m\bmod Q$ for a suitable finite $Q$ chosen as a common
multiple of the (finitely many) root-of-unity denominators. The
retained count is at most $T'$, so $\|F_{m'}-F_m\|$ is
$\lesssim e^{-cT'}$ for $m'=m+Q$. This proves
\eqref{eq:classwise-convergence}.
\end{proof}



\paragraph{Uniform correlations of two prescribed edges.}\label{u:marked-uniform}

We prove the third main theorem: the two-edge marked-polygon bound
\eqref{eq:marked-uniform}, uniform both in the staircase direction and
in the ratio $N/d$. The argument has three parts: (a) identify the
two-edge observable as a contraction of polynomial columns, (b) in the
balanced (antipodal) case reduce it to a two-line gas and estimate,
(c) for unequal intervals control the two long columns outside a small
core.

\paragraph{Setup.}

Keep the triangular-lattice directions $e_1,e_2$ and identical
translated marks $E,E+P$ with $P=P_1e_1+P_2e_2$, $P_1,P_2>0$. Thus
$d=P_1+P_2$ is even, the cylinder period is $kP$ with $k\ge2$, and
$N=kd$. The polygon weight is $\kappa^{\#\text{triangle centres visited}}$.
The two restrictions (edge plus displacement) place both marks on a
monotone staircase of period $P$. The parity of $d$ makes both
intervals between the marks even in staircase steps.

\paragraph{Finite marked contraction.}

Fix integers $k\ge1$, $0\le l,h\le k$, and $m,n\ge0$. Take two lists
of common slots $A$ (size $m$) and $B$ (size $n$), and four extra lists
$E_b,H_b$ (sizes $l,h$) and $E_t,H_t$ (sizes $k-l,k-h$). Columns below
have total length $m+n+l+h$, above $m+n+2k-l-h$.

\begin{auxproposition}[Finite marked contraction]\label{prop:marked-contract}
For $\delta\ne0$, the contraction
\begin{equation}\label{eq:Y-def}
Y=\big\langle\Psi_b\big|\,\delta^{\sum_A p_j}\big|\Psi_t\big\rangle,
\end{equation}
of the columns (below with physical rapidities, above with negated
rapidities, spin-fixed extra lists $-,+,+,-$) satisfies
\begin{equation}\label{eq:finite-marked-contraction}
Y=Q_{lh}\,\mathcal J_{\rm ext}\!\!
\sum_{\substack{a_j\in\{-1,0,1\}\\
\sum_Aa_j-\sum_Ba_j=h-l}}
(i/\delta)^{\sum_Aa_j}\,\mathcal W(a)\,\mathcal M(a),
\end{equation}
where the right-hand data are specified below.
\end{auxproposition}

Here:
\begin{itemize}
\item $Q_{lh}=\tfrac12C^{k/2}s^{-k}(-s^{-1})^ls^{-h}$.
\item $J(u)=\cos(u+L)\cos(u+2L)$, and $\mathcal J_{\rm ext}$ is a
product of $J$-factors between extra slots in each list.
\item $\mathcal W(a)=\prod_jc_{a_j}\prod_{j<j'}P_{a_ja_{j'}}(v_{j'}-v_j)
\prod_{j<j',\,\text{same list}}D(v_{j'}-v_j)$, with $P$, $D$ from
Section~\ref{c6:periodic-polynomial}.
\item $M^1_+(u)=(1,\cos(u+L),J(u))$, $M^1_0=1$ extracted by
convention, and $M^2_a(u)=M^1_{-a}(-u)$.
\item $\mathcal M(a)=\prod_{j\in A\cup B}\prod_{e\in\text{extras}}
M^{d(j,e)}_{a_j}(x_e-v_j)$ with $d(j,e)\in\{1,2\}$ determined by list
and spin type.
\end{itemize}

\begin{proof}[Sketch]
Divide both sides by the within-extra-list $J$-products; the resulting
polynomial is separately symmetric in each list. Compare cap
specializations of adjacency: at each extra-common pair, one
specialization at difference $b$ either eliminates both slots or
converts a common slot on the opposite side into an extra of opposite
spin, preserving $k$. The scalar fusion identities
$J(u-b)=J(-u)$ and $D(u)=J(u)J(-u)$ close the recursion. Degrees
after division match on both sides with definite parities, leaving
only a constant factor, fixed by the base case $m+n=0$ (equivalently
by saturation of the two fixed-spin columns).
\end{proof}

\paragraph{Physical interpretation.}

\begin{auxproposition}[Physical two-edge mass]\label{prop:physical-two-edge}
For $k=2$, $l=h=1$, $\delta=i$, and physical rapidities zero:
\begin{equation}\label{eq:physical-two-edge}
\frac{Y}{f_{m+n+2}(0)^2}
\end{equation}
is the critical mass of individual simple polygons through the two
prescribed row edges on the cylinder, counted once per polygon.
\end{auxproposition}

\begin{proof}[Sketch]
At $k=2$, $l=h=1$ and $\delta=i$, the extra fixed spins at $E$, $H$
force two oriented arcs from $E$ to $H$, one departing upward and one
departing downward, uniquely determined by the polygon's local side
at the marks. Two arcs form the polygon in one orientation only.
The relative signs (from turns, seams, and the phase insertion) yield
a positive weight; the identity extends by height exhaustion.
\end{proof}

\paragraph{The antipodal case.}

When the two marks are antipodal (i.e.\ $P_1=P_2=m+1$ and $k=2$), the
two-line gas of Section~\ref{c6:gas} directly estimates the marked mass.

\begin{proposition}[Antipodal bound]\label{prop:antipodal-bound}
On the homogeneous cylinder of even period $2m+2$ with antipodal row
sites, the two-edge marked mass satisfies
\begin{equation}\label{eq:antipodal-bound}
K_2\le(m+1)^{-4/3+o(1)}.
\end{equation}
\end{proposition}

\begin{proof}[Sketch]
The two extra slots $(E,H)$ supply $J(v)^2$ at an unlowered common
site plus additional factors $\cos^2z/\cos^2(z+b)$ per lowering, which
amount to a shift of the magnetic potential by a bounded factor. Normalize
by $Q_{11}C^mD(0)^{m(m-1)}J(0)^{4m}$; this gives the constant
$P'_m/P_M(i)$ with $M=m+1$, exactly the ratio used in Section~\ref{c6:gas}.
The gas estimate at this shifted potential gives the claimed bound.

\emph{The magnetic test potential.}
In the residue lift, using the periodized variable $w=i\log$, the
elementary potential at $z=iw$ is $e^{MV(w)}[-\sin(z+L)/\sin(z-L)]$.
The bracket is replaced by
\begin{equation}\label{eq:U-T}
U_T(w)=-\frac{S_T(w-iL)}{S_T(w+iL)}
\left(e^{-3iL}\frac{S_T(w-T/2+iL)}{S_T(w-T/2+4iL)}\right)^2,
\end{equation}
which is $T$-periodic, tends to the original bracket, is analytic in
$e^{-T}$ near zero, and has a zero at $w=iL$ restoring pole
multiplicity $m$. The first ratio's continuous log increases by
$12iL$ over a period, compensated by the edge ratio's $-12iL$.
\end{proof}



\paragraph{Staircases and the two background lists.}
\label{c6:unequal-marks}

We now prove the uniform two-edge bound \eqref{eq:marked-uniform}.
The insertion identity \eqref{eq:finite-marked-contraction} is unchanged,
but its two lists now have different lengths.  The argument keeps two
scales apart: a logarithmic core of size $\asymp\log d$ and a full
period of size $\asymp\log N$.  All constants below are independent of
$P_1,P_2,k$.

\paragraph{The staircase row cut.} Use a periodic monotone staircase
in $e_1,e_2$ (period $P$ through each mark), and translate it in the
third lattice-edge direction to make rows.  This tiles the plane by
lozenges; reading each lozenge as a square tile, the honeycomb path
weights are the local $R$-matrices $R(t+v_i)$ at argument $L$ or $2L$,
the two choices corresponding to an acute or an obtuse angle at the
corner (they exchange $a,h$ and $c,d$ in the tile notation).  After a
lattice reflection if necessary, take the majority of steps at
argument $2L$.

\paragraph{Through-lines disappear.} The projection argument for two
marks applies verbatim to the column parameter $t$ above (with its
negative below by crossing).  On the lifted slab bounded by two
stairs, the contour from a fixed source has a rotated real coefficient
bounded below positively.  Indeed, writing the domain in lattice
coordinates $(X,Y)$ along $e_1,e_2$ with transverse step $e_2-e_1$,
the counterclockwise tangent directions from a fixed mid-edge source
stay within a lower block $0,1$ (and possibly $2$) followed by an
upper block $3,4$ (and possibly $5$), in units of $\pi/3$; after
subtracting the boundary-closing turn $\pi$, the cyclic tangent
increase lies in $[-\pi/3,2\pi]$ or in $[0,2\pi+\pi/3]$, and
$(3/8)(2\pi+\pi/3)<\pi$ gives the positivity.  For the through-mass
at height $r$, take a lattice hexagon of radius $\asymp r$ not
reaching the far stair, average over $r/N$ translations by whole
periods to place the starts in the central part of a common smaller
hexagon, and reverse the exits at the pure sides; the mass is
$O_N(r^{-1/4})$, so through-lines have vanishing weight and empty
projections retain the same limits.

\paragraph{The two-column picture.} Both extra rapidities in each
column now take a single value $x_e$.  The two common lists have
sizes $m=d-1$ and $n=N-d-1$.  Augment each for the plasma calculation
by one site at $x_e$ to obtain background lists of sizes $M_A=d$ and
$M_B=N-d$, both even, with identical proportions of the two possible
values.  Put $\lambda_b=\pi/(2b)$; let $\lambda$ be the proportion of
minority steps, $0<\lambda\le1/2$.  After a rapidity translation, put
the majority value at $r_0 L$ and the minority at $(r_0-1)L$, with
$r_0=1/4+\lambda/2$.

\paragraph{The residue sum.} As in the balanced two-leg case, the
potentials are one copy of $R_{gh}(i(w+iv_e))$ per augmented-list
site of color $h$ and value $v_e$, together with the common magnetic
factor $-\sinh(w+ix_e-iL)/\sinh(w+ix_e+iL)$; this follows by shifting
each extra-slot factor of the balanced calculation.  Let $P'$ be the
remaining residue sum.  Using $D(y)=J(y)J(-y)$ and $Q_{11}=C/2$, the
prefactor divided by $f_N^2$ in the observable is
\begin{equation}
\frac{C^{N/2}\prod_{e<f:\,g_e=g_f}D(v_e-v_f)}{2f_N^2}
\;\prod_{e\text{ common}}\frac{J(v_e-x_e)}{J(x_e-v_e)}.
\label{eq:legacy-P-23}
\end{equation}
Lowering counts satisfy $K_g=M_g+u$ with a common integer $u$.
Periodize with the magnetic factor
\[
U_T(w)=-\frac{S_T(w+ix_e-iL)}{S_T(w+ix_e+iL)}
\left(e^{-3iL}\frac{S_T(w-T/2+iL)}{S_T(w-T/2+4iL)}\right)^2,
\]
and put background charges $q_e=-q_{g_e}$ at offsets $a_e=-v_e$ in
the central theta sum; then the Gaussian center $X$ includes
$\sum_e q_e i a_e$ and periodic differentials are recovered.  The
nome limit and rapidity continuation work as in the balanced case,
with singularities at differences exactly $L$ removed by continuation
in rapidities.

\paragraph{The ordered magnetic gas.}
\label{u:ordered-gas}

The staircase fixes the background lists; we next express their
residue sum as an ordered gas whose integer height records the
winding.  Phases are retained in this step: the interval estimate
below uses the resulting transfer operator rather than the modulus
of an unidentified periodic partition.

\paragraph{The extended mode matrix.} Put $E(p)=\sum_e e^{-p a_e}{\bf e}_{g_e}$.
Include extra background slots $E_e$ of count-mode 1 in the mode
matrix, with pair entries from the potentials (${\cal M}_{R,E}n_E=-K\widehat\rho E$)
and inverse couplings initially zero; their mutual block is zero
except at Laurent zero, where we assign $4q_eq_f/p^2+4q_eq_f(a_e-a_f)/p$
with zero finite part.  Then:

\begin{itemize}
\item The bulk log term is
\begin{equation}
S(T)=\frac1{2T}\sum_p E(-p)^tK(p)E(p)\widehat\rho(p)^2
-\tfrac12(T/3-2\log2)(M_A-M_B)^2.
\label{eq:legacy-P-24}
\end{equation}
Determinants and measure conventions of \eqref{eq:legacy-P-star} are
unchanged.

\item The chemical potentials per final particle (before the magnetic
test) are
\begin{equation}
2\pi i\,\sigma_i\!\!\!\sum_{e:\,g_e=g_i}\!\!\!
\int_0^{s+i(v_e+\delta_i L)}\!\!\!
\sum_{\nu\in\mathbb Z}\rho(z+\nu T)\,dz,\quad
\sigma=(0,1,1,-1,0).
\label{eq:legacy-P-25}
\end{equation}
Fugacity still brings $Tr_j^2/(4h_*)-r_j(u_*+\bar L)$.
\end{itemize}

\paragraph{Extended zero modes.}  At zero, take the raw right count
vector with left and inverse modes as before, put the background
modes at 1, and prescribe $q^t\gamma'_{\rm raw}
=A_L+\sum_e q_ea_e+\pi l/2$.  The extended matrix has the same
charge/offset/winding poles, and the saddle term $\widehat\rho E$
neutralizes the appended charges at 0 and contributes exactly
$\sum_e q_ea_e$ to the derivative condition, while its count value
$(M_A,M_B)$ leaves the free integer and Gaussian shift unchanged.  The
assigned background block has zero contracted finite part at zero, and
the linear symbol at a particle against a same-color site is
$-2\pi\sigma_i e^{(v_e+\delta_i L)p}\widehat\rho(p)/p$; the primitive
of \eqref{eq:legacy-P-25} follows.

\paragraph{Ordered-position form.}  For particles in increasing real
contour order write $H(s)=j+\sum_{\Re s_d>\Re s}W_d$ ($j$ the Poisson
integer), with mid-value $H_m$ at jumps; winding neutrality closes the
integer path.  Set $c^t$ by
\[
{\bf1}^t(J(p)-K(p)\ell)=2\pi W^t/p-2h_*c^t+O(p),
\]
so $u_*+\bar L=c^tn-(\pi i/h_*)\sum W_ds_d$.  (Here $J$ is the
coupling block of the transform, not the site function of
\eqref{eq:legacy-P-23}.)  At zero,
\begin{equation}
Z=\alpha WW^t/p^2+M_*/p+O(1),\qquad
M_*=-2\pi(\widetilde{\cal W}+Wc^t-cW^t),
\label{eq:legacy-P-26}
\end{equation}
where $\widetilde{\cal W}$ is an integral skew matrix: change right
variables by $-\ell n$ before Schur, then the non-right block has
simple coefficient $-2\pi\widetilde{\cal W}$ from the shifted winding
matrix, and the remaining poles come from the common channel.

\paragraph{Constant phases.}  Combined with the ordered factor from
$M_*/p$ and $-2\pi ijc^tn$, the explicit terms of
\eqref{eq:legacy-P-star} and the site/test couplings reduce, up to a
single unit prefactor, to
\begin{equation}
\prod_d\eta_{i_d}\exp[-2\pi i c_{i_d}H_m(s_d)]
\label{eq:legacy-P-27}
\end{equation}
for fixed unit constants $\eta$ per species, valid for every choice of
even $M_A,M_B$ (including 0).  The argument uses the $-i/2$ pole
kernel at ordered (positive) separation, the unordered sign
$(-1)^{\widetilde{\cal W}_{ab}}$ per pair from log constants, and
residue-orientation and Weyl sign contributions whose second
differences over count space reproduce $\widetilde{\cal W}$; these
cancel the unordered signs.  At every branch change of order $\nu$
(across a singularity) $\widetilde{\cal W}$ shifts by the same
amount and the sign $(-1)^\nu$ appears, so \eqref{eq:legacy-P-27}
persists under fusion and exterior branch changes.

\paragraph{The short pair kernel.}  For ordered positive separations
use the periodization of $G_{ab}(s)$ with a regular diagonal at
contact, counting both periodic continuations once per positively
separated pair (transpose at negative separations).  This is the
inverse Fourier kernel after removing the two pole kernels of
\eqref{eq:legacy-P-26}; thus the off-contact gas exponent is minus one
copy of $G$ per such pair (including translated partners), and the
regular contact self term carries half weight.  For $\Re s>0$,
\begin{equation}
G_{ab}(s)=\sum_{l\ge1}\sum_{k=0}^{r_*}C_{ab,l,k}s^ke^{-\omega_ls},
\label{eq:legacy-P-28}
\end{equation}
with $\omega_l>0$ growing at least linearly and $C_{ab,l,k}$ growing
at most polynomially: the final two-line symbols have only unit-circle
poles in $e^{Lp}$, of bounded order, so shifting the inverse contour
down in imaginary periods yields the residues above the separated
zero-pole kernels, and tail rotation adds an exponentially vanishing
remainder.

\paragraph{From periodic gas estimates to interval operators.}
\label{u:interval-operator}

We construct an operator whose products describe intervals of the
ordered gas.  A bound on homogeneous closed periods alone does not
control products of intervals with different potentials, so we need a
common operator throughout.

\paragraph{Homogeneous reference configuration.}  Fix a cell length
$S_0$ and consider a homogeneous gas with no magnetic functions or
\eqref{eq:legacy-P-25}, but with constant $r=-2a_0'$,
$a_0'=\pm6Li$, all five color-$A$ species allowed, and only $S,F$ of
color $B$.  Beyond the pair term $G$ and the phases
\eqref{eq:legacy-P-27}, the exponent contains
\begin{equation}
-h_*^{-1}\int(\pi H+ia_0')^2\,ds+2a_0'c^tn.
\label{eq:legacy-P-29}
\end{equation}

\paragraph{Cell states and gauge.}  A cell state $x$ is an incoming
integer height plus indistinguishable points of each species/color in
an interval of length $S_0$, with factorial Lebesgue measure.  Along
an edge $x\to y$ require outgoing height of $x$ to equal incoming of
$y$.  Use the reference energy at zero tilt and no extra particle
potential: $A(x,y)$ is half of each self energy (including
$\pi^2/h_*\int H^2$) plus the real adjacent pair energy from
\eqref{eq:legacy-P-28} (cyclically omitting separations of at least
two cells).  By \eqref{eq:legacy-P-P3}, for large $S_0$,
\begin{equation}
\sum A(x_i,x_{i+1})\ge4\epsilon_S\sum|x_i|_0^2-C_Sn_{\rm cells},
\qquad|x|_0^2=N_x^2+H_{x,\rm in}^2,
\label{eq:legacy-P-30}
\end{equation}
$\epsilon_S>0$ at least inverse-polynomial in $S_0$.  There is a Borel
$\phi$ on states away from coincidences with
\[
A(x,y)\ge2\epsilon_S(|x|_0^2+|y|_0^2)-C_S+\phi(y)-\phi(x),
\]
obtained as the infimum of corrected path sums from the empty height
zero state (finite since unit jumps of both signs are available).
Gauging each forward edge by $\exp(\phi(y)-\phi(x))$ cancels on cycles,
splits complex short comparisons into local errors
$O_S(\mathrm{slope})(N_x^2+N_y^2)$, and leaves Gaussian margins in
both endpoints, including for small-slope complex cell deformations.

\paragraph{Fock encoding of long-range interactions.}  For
interactions of range at least two cells add a bosonic symmetric Fock
space over countably many finite propagation blocks.  Use semigroups
$D_d$ generated by $-\omega_l+\text{nilpotent}$, with creation and
detection arrays $u_a,v_b$ satisfying $v_bD_du_a=-G_{ba}(d)$ term by
term in \eqref{eq:legacy-P-28}.  The Fock operator in forward order at
edge $x_i\to x_{i+1}$ is
\[
\exp a^\dagger(U)\ \Gamma(D_{\ell'})\ \exp a(V),
\]
with $V=\sum_{b\in x_{i+1}}v_bD_{d_b^L}$,
$U=\sum_{a\in x_i}D_{\ell'+d_a^R}u_a$, symmetric tensor powers, and
exponential-vector normalization.  Each block is rescaled so
$\|U\|+\|V\|\le Ce^{-\epsilon S_0}(N_x+N_y)$; splitting
$\Gamma(D_{\ell'})=\Gamma(\tfrac12I)\Gamma(4D_{\ell'})\Gamma(\tfrac12I)$
and using $\langle a^\dagger(U)^k\rangle\le\|U\|^k\sqrt{(m+k)!/m!}$
bounds the trace norm by $\exp(C\|U\|^2+C\|V\|^2)$, within the
Gaussian margins.  The resulting integral operators on $L^2$ with the
classical state are Hilbert--Schmidt, also under bounded point log
moduli per particle and bounded perturbations of the volume or height
energies.

\begin{lemma}[Homogeneous interval operators]\label{lem:u-homogeneous-operator}
On the ambient cell space with the Fock fibers above, let
$\mathcal T_\pm$ be homogeneous cell operators with all five color-$A$
species and only $S,F$ of color $B$, other species having zero
outgoing point weight, no magnetic functions or potentials
\eqref{eq:legacy-P-25}, constant tilt $a_0'=\pm6Li$, $r=-2a_0'$, and
weights consisting of the pair term $G$, the phases
\eqref{eq:legacy-P-27}, and the exponent \eqref{eq:legacy-P-29}.
Then both are Hilbert--Schmidt and
$\operatorname{spr}(\mathcal T_\pm)<\exp(S_0/2)$.
\end{lemma}
\begin{proof}
The Hilbert--Schmidt assertion follows from the Fock trace-norm bound.

\emph{Cyclic traces.}  Traces of cyclic products (at least two edges)
give exactly the ordered periodic gas times $\det(I-D_T)^{-1}$, the
last tending to 1 at large period $T=m_0S_0$.  Normal-ordering the
Fock product as $e^{a^\dagger(P)}\Gamma(D_T)e^{a(B)}$ plus internal
contractions, the normalized trace is $\exp(B(I-D_T)^{-1}P)$, which
recovers precisely \eqref{eq:legacy-P-28} at positive separations of at
least two cells and nothing else.

\emph{Spectral radius via powers.}  For a compact Hilbert--Schmidt
operator, $\operatorname{spr}(\mathcal T)=\limsup_m\|\mathcal T^m\|^{1/m}$;
splitting off finite-dimensional outer eigenvalues leaves a strictly
smaller radius on a complementary invariant space whose power traces
are trace-class.  It suffices to bound the homogeneous log-trace per
unit length along periods $m_0S_0$, $m_0\to\infty$.

\emph{Two-line transform.}  Fix real $T$ large and apply the two-line
residue transform with no magnetic potential, $r=-2a_0'$, and
backgrounds $M_A=0,M_B\to\infty$ even at zero.  The original sum has no
$A$ particles, but opening and transforming both lines is still valid
by the coefficientwise proof of \eqref{eq:legacy-P-star}; after
removing \eqref{eq:legacy-P-24} the transformed gas retains exactly
the permitted species, because nonzero \eqref{eq:legacy-P-25} on $B$
has real part strictly negative proportional to $M_B$ on this fixed
real period.  The remaining exponent is \eqref{eq:legacy-P-29} via
$\int(H-j)=\sum W_ds_d$, and its determinant scalar per unit length in
log is $1/2+o(1)$.

\emph{One-line comparison.}  The same residue sum also admits a
one-right-line calculation on $B$ with heat period $2bc_*$, a single
insertion, Cartan kernel $K_{0,BB}=(2\pi/p)\coth(bp)$, and dressing
$1-e$.  A single Weyl reflection twists the oscillator trace by the
oscillator sign, contributing
$\prod_{p>0}(1-e^{-2bp})/(1+e^{-2bp})$ and replacing the reciprocal
Cartan multiplier by $\tanh(bp)$; the pair symbols reduce to same-color
entries (color entry $c=0$).  The scalar \eqref{eq:legacy-P-24} is
unchanged; \eqref{eq:legacy-P-25} applies; other zero factors grow only
polynomially in $T$.

\emph{Real same-color kernel.}  For positivity put
$D_0(z)=(z^2-z+1)(z^4-z^2+1)$, $t=e^{x/12}$,
$H_{ij}=t^{-36+12(\delta_i-\delta_j)}D_0(t^{12})F_{ij}^{(0)}(t^{12})$
with $x=Lp>0$.  After clearing positive scalars one checks that
$(H+H^t)/(4\sinh x)$ is positive definite.  Here $(H+H^t)/2$ is
Laurent of degree at most 36, opposite coefficients negatives of one
another; each coefficient $B_j$ of $t^j$ ($1\le j\le36$) enters with
multiplier $\sinh(\beta x)/\sinh x$, $\beta=j/12$.  With
$a=2\cosh x-2$ and nodes $1,4,9$ the interpolation coefficients are
\[
u_0=\beta,\quad u_1=\beta(\beta^2-1)/6,\quad
u_2=\beta(\beta^2-1)(\beta^2-4)/120,
\]
and the lower polynomial has first two coefficients
$U_k=\sum_ju_kB_j$ ($k=0,1$) and last
\[
U_2=\sum_ju_2B_j-\tfrac12\sum_{12\nmid j}
\bigl(u_2B_j+|u_2|\operatorname{diag}(|B_j|_{\rm ent}{\bf1})\bigr).
\]
A finite integer check (1000 times each $U_k$ is integer-valued with
consecutive leading determinants positive after subtracting $3I$)
verifies that $U_0,U_1,U_2$ are positive definite; the rounding error
has $5\times5$ operator norm at most $5/2$, so the exact matrices are
positive as well.

\emph{Final numerical bound.}  Only $S,F$ survive.  Their symmetric
entries, omitting $2\pi/p$, on the diagonal are
$(\sinh3x+\sinh x\mp\sinh2x)/\cosh3x$ and off-diagonal
$(\sinh(5x/2)+\sinh(x/2))\cosh(x/12)/\cosh3x$.  The spatial real
kernels are nonnegative off contact (each divided by $p$ is a
nonnegative mixture of $\cosh(hp)/\cosh(bp)$, $0\le h<b$).  The regular
diagonal is at least $2\log L$ (each $F^{(0)}$ diagonal is at least
$1-e^{-x}$ for positive $x$), improving under periodization, so the
total real pressure from particles is at most $1/(\pi L)$.  The
one-line determinant per length contributes $1/4-1/6+o(1)=1/12+o(1)$.
The homogeneous log trace per unit length is therefore at most
\[
\frac1{\pi L}+\frac1{12}-\frac12+o(1)
=\frac8{\pi^2}-\frac5{12}+o(1)<\frac{17}{36}<\frac12
\]
using $L=\pi/8$ and $\pi>3$.  This proves the strict spectral bound for
either sign of the tilt.
\end{proof}

For later use, the spectral-radius formula gives a fixed block length
$q$ and $\varepsilon>0$ with
$\|\mathcal T_\pm^q\|\le\exp((1/2-\varepsilon)qS_0)$ for both tilts.
Continuity of a finite ordered product in operator norm then preserves
$\exp(qS_0/2)$ for sufficiently close $q$-blocks in the same state
space and gauge.  Section~\ref{u:core-product} proves this closeness
for the actual tail kernels; the core requires its own estimate.

\paragraph{Localization of the magnetic source.}
\label{u:magnetic-localization}

The homogeneous interval operator is now controlled.  We turn to the
magnetic source itself, prove the uniform one-particle estimates
needed to compare its tails with that operator, and retain the
suppression inside the shorter core.

\paragraph{The magnetic exponent.}  Write $a(s)=\log U_T(s)$ on the
right contours, in the branch whose bulk log grows from the odd
primitive in $s+ix_e$; its total winding around the period is zero,
as in the balanced case (\S\ref{c6:subheading-16}).  On the right,
the mean $\bar a$ gives $r=-2\bar a$ and scalar $(M_A+M_B)\bar a$; on
the left retain direct $U_T$ factors with the mean subtracted; on the
right insert $a-\bar a$.  Completing Cartan nonzero modes as in the
balanced case yields an extra external scalar
\begin{equation}
S_a(T)=\frac1T\sum_p\widehat\rho(p){\bf1}^tE(-p)\widehat a(p)
+\frac1T\sum_{p\ne0}(Q_+(p)-1/h_*)\widehat a(-p)\widehat a(p),
\label{eq:legacy-P-31}
\end{equation}
and the local height-square becomes
\begin{equation}
-h_*^{-1}\int(\pi H+ia(s))^2\,ds.
\label{eq:legacy-P-32}
\end{equation}
The added particle-source nonzero-mode multiplier is
$-J(-p)^t{\bf1}Q_+(p)$; its simple pole $+2\pi W/(ph_*)$ acting on
$a-\bar a$ gives the mean-zero primitive of $-i(a-\bar a)$, and
summing the jumps by parts against $H$ recovers the quadratic in
\eqref{eq:legacy-P-26} together with the mean and test-variance
pieces kept in \eqref{eq:legacy-P-31}.  The remaining per-particle
exponent, beyond \eqref{eq:legacy-P-27}, is
\begin{equation}
(2c_i-\ell_{\Sigma i})a(s)
+\!\!\!\!\sum_{\text{orig.\ left member}}\!\!\!\!\log U_T(s+i(\delta_i+b_{i'})L)
+b_i'(s),
\label{eq:legacy-P-33}
\end{equation}
where the $b_{i'}$ are the physical integer offsets and $b_i'$ is the
Fourier action on $a'$ of the regular symbol
$[-J(-p)^t{\bf1}Q_+(p)-2\pi W/(ph_*)-(2c-\ell_\Sigma)]_i/(-ip)$.
Here $Q_+=1/h_*+O(p^2)$ fixes the constant $2c-\ell_\Sigma$, whose
mean times the counts matches the terms in \eqref{eq:legacy-P-31} and
\eqref{eq:legacy-P-32} via $\int(H-j)=\sum W_ds_d$.

\paragraph{Period and bend.}  Keep the cell length $S_0$, block length
$q$, and spectral margin fixed.  Choose the bend size $\theta_0$ small,
then transition length $A$ and buffer $g\gg S_0$ large enough to meet
the slope bounds below; the fixed boundary layers of the product
estimate may depend on those choices, but no constant depends on
$d,N,\lambda$.

Take nominal period $R=2a_0^{-1}\log N+g+O(S_0)$, a multiple of
$2S_0$, and admit $|T-R|<\pi+1$.  Parameterize positions by
$s=t+iy(t)+(T-R)\chi(t)$, $-R/2\le t\le R/2$, with $y$ periodic,
real, Lipschitz small, supported outside $|t|>A$, decreasing to
$-\theta_0L$ by $|t|=2A$, and flat after; here $\chi$ is the analytic
period adjustment of \eqref{eq:legacy-P-P3}, derivative $O(1/g)$,
$\chi(\pm R/2)=\pm1/2$, supported within $g/3$ of the ends.  Slopes
satisfy all smallness requirements.

\paragraph{Uniform one-particle estimates.}
\begin{enumerate}
\item $a(s)$ tends exponentially to $a_\pm=\pm6Li$ on $\pm t>0$
away from $0$ and the period edges, and is bounded; the logs of
direct multipliers in \eqref{eq:legacy-P-33} tend to these phases
modulo $2\pi i$ with bounded modulus; and $b_i'$ tends to $0$
exponentially with $\exp(\Re b_i')$ bounded throughout.
\item For $D$ and $I_+'$ the real exponent in \eqref{eq:legacy-P-25}
is at most $-c_0M_ge^{-a_0|t|}+O(1)$; the same bound holds for $I_-'$
when $|t|\ge 2A$, and everywhere the exponent is $O(1)$.  For color
$A$, the \eqref{eq:legacy-P-25} multipliers tend uniformly to $1$
outside $|t|\le a_0^{-1}\log d$ and away from the period edges.
\end{enumerate}

\emph{Proof of 1.}  The product formula and derivative periodizations
about the two centers are as in the balanced case; direct left offsets
have no poles along the graph (edge compensator unshifted in
rapidity).  The short term uses the regular multiplier on the two
whole-line coth difference across one period before periodization; its
transform off zero decays at the nearest shifted singularity.  For
$I_-'$, the isolated core log difference under the shift $i\delta_iL$
has bounded modulus above and is exponentially localized; after this
subtraction the extra exponent has uniform frequency decay because
$A_3=-1$.  Contour rotation away from the centers, as in
\eqref{eq:legacy-P-P3}, gives all asserted bounds.  Widths on the
unshifted ratios are bounded away from singularities since
$-3/4\le x_e/L\le1/2$.

\emph{Proof of 2.}  On the basic line the imaginary part of a single
primitive at argument $t+iLY$ is
\[
B(u,Y)=\frac1{2\pi}\operatorname{atanh}
\frac{4\sqrt3\cosh u\sin(\pi Y/3)}{3+4\sinh^2u+4\sin^2(\pi Y/3)},
\quad u=a_0t,
\]
by integrating the explicit density.  Before bending, mix
$Y=r_0+\delta_i$ and $Y-1$ with proportions $1-\lambda$ and $\lambda$.
For $I_+'$ the sign is uniformly positive with size $\ge ce^{-|u|}$
($Y\ge 2/3$ and the positive function is increasing on $0<Y<1$); for
$D$ the same holds with $Y=1/3+\lambda/2$, by monotonicity and
continuity.  The remaining cases reduce to bounding
\[
\frac{2\pi B}{4\sqrt3\cosh u\sin(\pi Y/3)/(3+4\sinh^2u)}
\]
between $1$ and $\operatorname{atanh}(\sqrt z)/\sqrt z$, with
$z=4\sin^2(\pi Y/3)/(3+4\sinh^2u)$.  For $D$ with $\lambda\le1/3$,
it suffices that
\[
(1-\lambda)\sin(\pi Y/3)-1.33\lambda\sin(\pi(1-Y)/3)>0;
\]
splitting at $\lambda=1/4$ gives $\ge17/600$ for
$0\le\lambda\le1/4$ (via $\sin(\pi/9)>1/3,\sin(2\pi/9)<2/3$) and
$\ge337/6000$ for $1/4\le\lambda\le1/3$ (via $\sin(11\pi/72)>9/20$
and $\sin(13\pi/72)<11/20$), using
$\operatorname{atanh}(\sqrt z)/\sqrt z<133/100$ for $z\le5/9$ (from
the series $1+z/3+z^2/5+z^3/(7(1-z))$).  For $I_-'$,
$Y=\lambda/2\le1/4$, so $z<1/10$ and
$\operatorname{atanh}(\sqrt z)/\sqrt z<11/10$; then
$(11/10)(1-\lambda)\sin(\pi\lambda/6)\le(11\pi/60)\lambda
<\lambda\sin(\pi/4)\le\lambda\sin(\pi(1-Y)/3)$, giving the negative
sign with margin $\lambda e^{-|u|}$.  Bending improves these
comparisons for small $\theta_0$; the primitive limits to $\pm1/2$
along the real axis and bends add exponentially small errors.

\paragraph{Continuation to the magnetic graph.}  The mode transform is
justified by analytic continuation.  Start at real large $T$ near
zero rapidities, where the trace and final fusions are pole-free (as
in the balanced case).  Continue rapidities linearly to the targets
on the final contours with horizontal projections; absolute
convergence and neighborhoods follow from \eqref{eq:legacy-P-P3} or
\eqref{eq:legacy-P-30}, since only bounded one-body and at most linear
height perturbations are added.  Split poles initially near the
fictitious background only for actual remaining residue sites, giving
meromorphic continuation with a finite product-sum representation;
the straight continuation removes collision-divisor singularities on
the used rapidity family.  Deform the position graphs to the small
$y$ on real periods by simultaneous transport.  Coincident loci are
zeros of nonnegative integer order (analytic transport).  This yields
holomorphic continuation in the period disk on the chosen
parametrization, with remaining split-residue singularities on fixed
rapidity divisors removed by real-period regularity near nome zero.
On the boundary nome circle this supplies a single continued function
equal to the original trigonometric sum at nome zero.

\paragraph{A bounded product through the core and tails.}
\label{u:core-product}

Choose $u_0\ge0$ a cell-boundary coordinate within bounded distance of
$a_0^{-1}\log d$.  On the core $[-u_0,u_0]$, where $\chi=0$, remove
the scalar $h_*^{-1}\int a^2$ from \eqref{eq:legacy-P-32}, of real part
$-2(2u_0)+O(1)$ since $(6L)^2/h_*=2$.  Redistribute the principal real
linear tilt on that interval using
\[
\int_{-u_0}^{u_0}\operatorname{sgn}(t)H(s(t))\,dt
=-\sum_{d':\,|t_{d'}|<u_0}W_{d'}(u_0-|t_{d'}|),
\]
appearing with positive coefficient $12\pi L/h_*$.  Terms from purely
imaginary tangents have unit modulus for this leading tilt.  All
unfavorable sites have $W<0$ and hence the strong chemical suppression,
which absorbs their loss at bounded additive cost per particle while
retaining a net suppression $-c_1(u_0-|t|)$ per particle there
($M_g\ge d$ and no species has $W=0$).  We replace this real linear
tilt in the core edge weights by the point terms from the displayed
identity: they telescope on cycles.  The remaining added terms have
bounded-above point log moduli $1,2$, and at most bounded linear height
cost on cells whose real coefficients beyond the reference Gaussian
tend to zero away from the center after redistribution.  There is no
winding boundary cost in this telescoping.

\paragraph{Telescoping gauge on the core.}  Every edge kernel is
uniformly bounded after the scalar removal; we need the product
through the whole core to be bounded independently of its length.  For
a core cell let the depth be its minimum distance from the two core
cuts, truncated below at zero.  Reserve a fixed small fraction of the
surviving negative point potential $-c_1(u_0-|t|)$: on cell $i$ put
$p_i(x)=\delta\,\mathrm{depth}_iN_x$ with fixed small $\delta>0$, and
$p_i=0$ on exterior states at the two core interfaces.  If $K_i(x,y)$
is the outgoing edge kernel, replace it by
\[
K_i^{\rm sym}(x,y)=\exp\!\Bigl[\frac{p_i(x)-p_{i+1}(y)}2\Bigr]K_i(x,y).
\]
The reserved exponent on such an edge is
$-p_i(x)/2-p_{i+1}(y)/2$.  In a product through the core the multipliers
telescope pointwise, including the interfaces where the exterior
factors are one, so the complete product kernel is unchanged and no
inverse diagonal gauge is required.  Adjacent cell depths differ by at
most the fixed cell length, and the unused point suppression together
with the Gaussian margins of \eqref{eq:legacy-P-30} remains available
on both incident edges.  Consequently every block with a nonempty
state at either endpoint has operator norm at most
$C\exp(-c\,\mathrm{depth})$, where the depth is the smaller of the two
incident cell depths.

\paragraph{Empty-to-empty edges.}  Away from a fixed number of cells
at the central transition and at the two cuts, the empty-to-empty
block has norm at most $1$.  Indeed, the integer height is constant
across such an edge, the real reference exponent is
$-\pi^2S_0H^2/h_*$, and the residual linear coefficient of $H$ tends
exponentially to zero away from the transition and the cuts.  Since
$H$ is an integer, its absolute contribution is dominated by the
negative quadratic whenever $H\ne0$, and vanishes when $H=0$.  The
common tangent is flat; remaining scalar phases have modulus one; the
state gauge cancels; and free Fock propagation is contractive.  Thus
every complete edge has norm at most
$1+C\exp(-c\,\mathrm{depth})$ on each of the two core halves outside
the fixed exceptional layers.  For each depth there are only a bounded
number of cells, so the sum of these errors is bounded; taking the
product proves a uniform bound on the core operator.  The exceptional
layers contribute only fixed factors.  We use the redistributed gauge
only on core cells; the tail operators keep their original gauge.

\paragraph{Tails.}  On each of the two tails outside the core, far
from its end cuts and the period edges,
\eqref{eq:legacy-P-32}, \eqref{eq:legacy-P-33} and
\eqref{eq:legacy-P-25} converge in the edge-kernel sense, uniformly,
to the homogeneous kernels with \eqref{eq:legacy-P-29} and phases
\eqref{eq:legacy-P-27}: the undesired $B$-species are suppressed, and
$A$-points see the chemical limit $1$.  The direct left factors with
the linear term in \eqref{eq:legacy-P-33} give exactly
$\exp(2a_\pm c_i)$; the pair kernels are unchanged on the interior
flat cells (their common displacement cancels).  Convergence in
Hilbert--Schmidt norm as distances increase follows from bounded-state
truncation and the Gaussian majorants.  Use the block length $q$
chosen after Lemma~\ref{lem:u-homogeneous-operator}, for which each
homogeneous $q$-fold product has norm at most
$\exp((1/2-\varepsilon)qS_0)$.  Increase the fixed boundary layers
until the $q$-fold products of the actual tail edges differ in norm
from their homogeneous products by less than the available fixed
margin; they are then at most $\exp(qS_0/2)$.  The boundary layers and
the fewer than $q$ remaining edges have bounded length and bounded
norms.  The two tails together contribute at most
$(R-2u_0)/2+O(1)$ to the logarithm of the product norm; short
intervals and all remaining boundary layers use the uniform edge
bounds directly.  In the cyclic trace estimate keep two edges in
Hilbert--Schmidt norm to use the remaining product bounds.  Including
the core scalar but excluding \eqref{eq:legacy-P-24},
\eqref{eq:legacy-P-31} and the determinant, the overall log modulus is
at most
\begin{equation}
-2R+\tfrac52(R-2u_0)+O(1).
\label{eq:legacy-P-34}
\end{equation}

\paragraph{Matching the scalar and the Pfaffian normalization.}
\label{u:marked-denominator}

The interval bound controls the particle integral.  To recover the
polygon observable we must still match its scalar factors with the
normalizing polynomial $f_N^2$, even when the staircase uses only two
rapidity values, because those values may have arbitrarily unequal
multiplicities.

\paragraph{External scalars.}  Uniformly on the circle,
\eqref{eq:legacy-P-24} equals
\[
\tfrac12\sum_{e,f}I_{g_eg_f}(v_e-v_f)+C_2e^{-2T}+O(1)
\]
for a site-data constant $C_2$, where
\[
I_{gh}(v)=2q_gq_h\log2+\int_{\mathbb R}\!\!
[K_{gh}(p)\widehat\rho(p)^2\cosh(vp)-4q_gq_h/p^2]\,\frac{dp}{2\pi}.
\]
The subtracted pole all-mode sum is $q_gq_hT/3$.  Contour shifting
with exponentially bounded tails gives error $O(N^2e^{-a_0\Re T})$,
picking corrections at poles $\pm2i$, and at most a simple pole at
$\pm ia_0$ in the color-difference channel.  Then
\begin{equation}
e^{I_{gh}(v)}=B(v)^2\tan^2(\lambda_bv)
\begin{cases}1/D(v),&g=h,\\1,&g\ne h,\end{cases}
\quad B(v)=\frac{\sin(v+b)\sin(v-b)}{\sin v}.
\label{eq:legacy-P-35}
\end{equation}
Verification: for real $v$ the multiplier
$(p/2\pi)K_{gh}\widehat\rho^2$ on $p>0$ is
$t^2/[(1-t+t^2)(1-t^8)]$ times $1+t^6$ (same color) or $-2t^3$
(opposite), with $t=e^{-Lp}$.  Thus it expands as
$\sum_{k\ge1}c_ke^{-kLp}$ with
\[
c_k=-2{\bf1}_{k=\pm3(8)}+2{\bf1}_{k=0(8)}
+2{\bf1}_{k=3(6)}-2{\bf1}_{k=0(6)}
+{\bf1}_{g=h}{\bf1}_{k=\pm2,\pm3(8)}.
\]
Integrating gives $\sum_k-c_k\log(1-v^2/(kL)^2)$, matching
\eqref{eq:legacy-P-35} by the sine product; the constant follows by
$v=iV$, $V\to+\infty$, using $I_{gh}(iV)=2q_gq_h(\log2-V)+o(1)$.

\paragraph{The magnetic test scalar.}  Formula \eqref{eq:legacy-P-31}
reduces to
\[
\sum_e\log\frac{J(x_e-v_e)}{J(v_e-x_e)}+C_1e^{-T}+O(1).
\]
The regular test variance costs $O(1)$.  The cross part integrates $a$
against the shifted periodized densities; the log of the normalized
squared edge ratio gives first terms $e^{-T}(d_1e^{2s}+d_2e^{-2s})$
across the core, with errors $O(e^{-2\Re T+4|\Re s|})$, whose
convolution with each $\rho(s+iv_e)$ contributes the coefficient and
costs $O(1+Ne^{-a_0\Re T/2})$.  The principal per-background integral
is the whole-line core-log convolution shifted by $i(x_e-v_e)$.  Its
derivative has transform
\[
2\pi i\widehat\rho(p)\frac{z-z^7}{1-z^8}
=2\pi i\frac{z^2+z^3-z^5-z^6}{1-z^8},\quad z=e^{-Lp},
\]
obtained from the localized coth difference; hence the convolution is
$\log[J(-is)/J(is)]$ at $s=i(x_e-v_e)$ (matching at $s=0$ by oddness,
and shifting contours within the strip keeps the two pole sets
separated).

\paragraph{Determinant scalar.}  Remove $C_2e^{-2T}+C_1e^{-T}$ by an
exponential analytic multiplier and use maximum modulus.  The
two-color transform determinant adds $(3/8)\log N+O(1)$.  For the
remaining determinant put $\mathfrak d(p)=-\log(1-e(p))$ and use
\[
2\sum_{n\ge1}\mathfrak d(2\pi n/T)=\sum_{n\in\mathbb Z}
\mathfrak d(2\pi n/T)-\mathfrak d(0).
\]
Since $\mathfrak d$ is even, regular at zero, and exponentially
decaying, the periodization contributes
$\pi T/(16L)-\mathfrak d(0)+O(e^{-c\Re T})$ as in the two-line case.
The two square-root factors have constant product.  With
$\Re T=(3/4)\log N+O(1)$ this gives the stated uniform bound on the
nome circle.  Formula \eqref{eq:legacy-P-35}, with
$e^{I_s(0)/2}=\lambda_b/\sqrt C$, cancels the all-positive prefactors
in \eqref{eq:legacy-P-23} along with the site sum from
\eqref{eq:legacy-P-31}, leaving exactly $2f_N^{(0)2}$ in the numerator
relative to $2f_N^2$, where $f_N^{(0)}$ uses
$h_0(v)=-\lambda_b\tan(\lambda_bv)/2$ instead of $h$ (also by
Schur--Pfaffian at shifted variables).

\paragraph{The Pfaffian ratio.}  We need
\begin{equation}
\log\bigl(f_N^2/f_N^{(0)2}\bigr)\ge(5/24)\log N-O(1).
\label{eq:legacy-P-36}
\end{equation}

\emph{Shifted comparison.}  Center the variables in $[-L/2,L/2]$ and
make them distinct (or perturb into a slightly larger interval).  The
kernels, by their partial fractions, are integrations of
$-\sinh(vp)$ against the positive-ray pole densities $\nu_0,\nu$ of
ratio $R(p)=\nu/\nu_0\ge1$.  Expanding a Pfaffian of size $N=2s$ by
antisymmetry gives, up to common constants, the integral of the
alternant on exponents $\pm p_1,\ldots,\pm p_s$ against the
corresponding product measure.  With exponents laid out in pairs the
alternant has fixed sign for distinct ordered variables, since
exponential polynomials of $N$ such distinct exponents have at most
$N-1$ real zeros by Rolle induction.  Jensen then bounds the log
Pfaffian ratio below by its linear statistic with $g=\log R$ in the
$h_0$ ensemble, equivalently the log derivative of
$\operatorname{Pf}(h_0+th_g)$ at zero, where $h_g$ uses pole density
$g\nu_0$; confluent limits preserve the comparison.

\emph{Uniform $A\log N+O(1)$.}  The doubled linear statistic has the
form $A\log N+O(1)$, with $A$ independent of the rapidities and of
the even size $N$.  Put $H_t=[h_0(v_i-v_j)+th_g(v_i-v_j)]_{i,j=1}^N$
and $C(v)=\prod_j[-i\cot(\lambda_b(v-v_j))]$.  If $\Gamma$ positively
encircles the sites and initially no other poles,
\[
2\Bigl.\frac{d}{dt}\log\frac{\Pf H_t}{\Pf H_0}\Bigr|_{t=0}
=-\frac{4}{(2\pi i)^2}\int_\Gamma\!\int_\Gamma
C(v)C(w)h_0(v-w)h_g(v-w)\,dv\,dw.
\]
Cancelling common Schur-minor factors gives the undoubled derivative
$-4\sum_{i<j}r_ir_jh_0(v_i-v_j)h_g(v_i-v_j)$ with
$r_i=\operatorname{Res}_{v=v_i}C$; the normalized double integral is
$2\sum r_ir_jh_0h_g$ because diagonal residues vanish (both kernels
odd).  Doubling thus gives $-4$ independently of sites and even $N$.

The odd $h_g$ is bounded analytic on strict substrips $|\Re v|<5L$ by
exponential decay of $g\nu_0$; its $1/p$ portion has bounded odd sine
integral with cutoff.  Expand the $v$ contour to a rectangle with
vertical edges at $\pm2.2L$ and heights $\pm a_0^{-1}\log N$; do $w$
similarly with imaginary half-height one larger.  The only crossed
poles are from $h_0$ at $v-w=\pm b$, with $C(v)C(v\pm b)=1$, so their
contributions are site-independent.  The remaining integral is
uniformly bounded: $|C|$ on vertical edges is at most
$\exp(-cNe^{-a_0|\Im v|})$, horizontal caps are bounded, and the only
simple poles on the contour product are transversal edge-cap
intersections (locally absolutely integrable).  The growing vertical
sides cost
\[
\int_0^{a_0^{-1}\log N}e^{-cNe^{-a_0t}}\,dt
=a_0^{-1}\int_1^N e^{-cx}\frac{dx}{x}=O(1),
\]
and the extra $w$-height unit adds only a bounded factor.  Each
crossed pole $w=v\pm b$ contributes a constant residue.  This proves
$A\log N+O(1)$ uniformly in the rapidities, including confluent
limits (after the usual division by vanishing Vandermonde factors).

\emph{Coefficient $A\ge5/24$.}  At coincidence, twice the linear
statistic is exactly the compressed trace test from the homogeneous
saturation argument (Taylor row and column differences).  That test
has lower limiting coefficient $5/24$, obtained there from Fatou and
the explicit orthogonal polynomial kernel.  Hence $A\ge5/24$, and
applying Jensen at the shifted sites proves \eqref{eq:legacy-P-36}
with a bounded additive error.

\paragraph{Assembly.}  The log of the normalized marked-polygon mass
is at most
\[
-2R+\tfrac52(R-2u_0)+\Bigl(\frac38-\frac5{24}\Bigr)\log N+O(1).
\]
Choosing $R=(3/4)\log N+O(1)$ and $2u_0=(3/4)\log d+O(1)$ makes this
\[
-\tfrac43\log N+\tfrac{15}{8}\log(N/d)+O(1).
\]
Exponentiation proves the uniform two-edge bound
\eqref{eq:marked-uniform}.  If the shorter core has only a bounded
number of cells, use the uniform edge bounds there; enlarging the
absolute constant covers small $d$.

\paragraph{Exterior sewing and finite bridge moments.}
\label{sec:sewing}

Everything here uses pure sides (lattice cuts normal to the $n_j$),
and lengths count visited triangle centres.  The marked-polygon bound
controls two-link events; we first construct exterior connectors, then
recover their costs by averaging the cuts, and use them to turn the
two-mark estimate into a second length moment.

\begin{proposition}[Confined crossings and half-plane monotonicity]\label{prop:corridors}
The following statements hold with constants depending only on the
fixed relative tolerances and pure normals.
\begin{enumerate}
\item Starting at a mid-edge source with inward normal $n$, one can
travel in that half-plane to a pure terminal line at distance $s$ whose
outward normal $m$ makes angle at most $60^\circ$ with $n$, staying
before the terminal line until the end.  For any fixed relative
tolerance and all sufficiently large admissible $s$, the travel is
confined within that tolerance around the straight segment in
direction $m$, with mass $\gtrsim s^{-1/4}$; the unrestricted mass in
this two-cut domain is $O(s^{-1/4})$.
\item From one mid-edge source to another on the same pure boundary at
distance $l>0$, half-plane arches of diameter $O(l)$ have mass
$\gtrsim l^{-5/4}$; even without confinement the mass is $O(l^{-5/4})$.
\end{enumerate}
The boundary-to-boundary mass is decreasing as the terminal port moves
away from the source along one straight side of a convex pure-sided
truncation, whenever the compared adjacent ports remain on that side.
\end{proposition}
\begin{proof}
For 1, the upper bound is the convex diameter bound.  For the lower
bound with a turn, choose small fixed $a>0$ in terms of the confinement
tolerance.  Exit first through the side normal to $m$ in the centered
triangle of size $s_1$, averaged over $s_1\in[as,2as]$; the deep-exit
estimate \eqref{eq:deep-triangle-exit} retains mass $\gtrsim s^{-1/4}$
at depth $\ge$ a smaller fixed multiple of $s$.  Then bridge along $m$
to the terminal, confined within a tolerance much smaller than this
initial depth (so the confinement stays inside the initial half-plane),
costing $\gtrsim s^{-1/4}$ more.  Joint mass for two choices of the
first level splits at both transversal renewals into the triangle
exit, the difference bridge, and the final bridge, averaging therefore
$O(s^{-3/4})$; Cauchy--Schwarz gives mass
$\gtrsim (s^{-1/2})^2/s^{-3/4}=s^{-1/4}$.  Without a turn use the
narrow-channel bound.

For 2, monotonicity via an artificial arc in a convex pure-sided
truncation: from $a<P<Q$ on the same straight side ($P,Q$ adjacent),
the arc input from $a\to Q$ back towards $P$ can only exit between
$a,P$, with phase $z^{-1}$; conversely from $a\to P$ via the arc to
$Q$, the residual chord has $|W|\le\pi$.  After multiplication by
$z=e^{i\beta\pi}$ the outgoing imaginary coefficients are therefore
nonnegative as in the earlier adjacent-cap argument.  In a large
centered half-hexagon of scale $\asymp l$, the loss from full
half-plane arches to ends at distances in $[l,2l]$ is small compared
with their total mass of order $l^{-1/4}$ (by the diameter bound).
Apply monotonicity inside the truncation.
\end{proof}

\paragraph{Two exterior sewing estimates.}
\label{sec:exterior-sewing}

Take a triangular-lattice vertex as origin and pure-support domains
$D=D(0;\mathbf b)$ with $|b_j-R|\le\varepsilon R$ for a fixed
$0<\varepsilon\le1/20$; these are convex hexagons with minimum side
$\asymp R$, and Section~\ref{sec:hexagon-completion} supplies uniform
nesting and central-occupancy estimates for every such domain,
including independent bounded row roundings.

Use a random nested family $D_h$ indexed by $h$ in a window of size
$cR$, $c>0$ small.  Move each nominal support outward at speed $1$
from a regular arrangement plus a perturbation $c'Rf(V_i+A_ih/R)$,
with $c'>0$ small, $f$ a unit-period triangular zigzag of slope $1$,
and $V_i$ uniform, $A_i$ uniform on $[-1,1]$, independently.  Given
$r<s$ in that window, conditional on the phases at $r$, the nominal
supports at $s$ are independent with densities at most $C/(s-r)$.

For each facet choose a fixed nondecreasing rounding map to its
admissible row levels with uniformly bounded error.  Each nominal
support has derivative at least $1-c'>0$, so rounding preserves
nesting and preserves independence between facets conditional on inner
phases.  If its error is at most $Kd_0$, an atom of the rounded
support is the image of an interval of length at most $2Kd_0$; hence
the conditional density bound becomes an atom bound
$C\min(1,Kd_0/(s-r))$, sufficient for all averages below.  Boundary
gaps are $O(1+s-r)$ and at least $c_0(s-r)-O(1)$.

Use complete chords in $D_h$ with ordered boundary ends $p,q$ binned
by dyadic separation $l\asymp|p-q|$ (at least constant order) and by
the two values $t_i$ of $1+$ distance to the nearest corner capped at
order $l$, within dyadic factors.  Orientations do not affect any
bound.  Fix a chord event of mass $E_h$ in such a bin; its contents
(such as marked vertices) may be constrained arbitrarily as a
deterministic event given $h,D_h$.

\begin{proposition}[Exterior sewing]\label{prop:exterior-sewing}
For the nested family, endpoint bin, and deterministic chord event
just specified, the average $\overline E$ over the level and support
randomness satisfies, with constants uniform in the event and the
admissible levels:
\begin{enumerate}
\item If it requires the use of two fixed internal links, then
$\overline E\le Cl^{3/4}\mathcal K$, where $\mathcal K$ is the
single-polygon mass using those links on a cylinder of period vector
in any lattice direction chosen for the marks and length large
compared with $R$.
\item For $l$ large, suppose the goal is instead to extend to bridges
between horizontal caps at heights near $\pm AR$, $A$ a fixed large
constant (each height magnitude in $[AR,2AR]$).  Let $B^*$ count
bridges between those caps, with base start free in a horizontal
interval of size $CR$ containing the required chord event for at
least one choice of the supports.  Then
$B^*\ge cR^{-1/2}(l/R)^{C_0}l^{1/2}\overline E$, for universal
$c,C_0>0$.
\end{enumerate}
\end{proposition}
\begin{proof}
The two completions are shown in Figure~\ref{fig:sewing}.
\input{figures/sewing}

\emph{Exterior connectors.}  For 1, use a connector strictly in the
exterior with diameter at most $Cl$, fixed endpoints $p,q$, mass
available $\ge cl^{-5/4}$.  For 2, append from $p,q$ separate
exterior paths to the two caps without touching either earlier, also
within distance $CR$ of the origin; together their available mass is
at least $cR^{-1/2}\exp[-C(1+\log^+(R/l))]$.  In both cases, for
large $l$, arrange both exterior branches to advance from their
endpoints in disjoint small patches until reaching distance $\rho l$
from $D_h$, and all exterior portions between or beyond these visits
stay at distance $\ge\rho l$ ($\rho>0$ fixed small).  Each enlarged
patch sees only facets of a single consecutive-normal pair for the
outer and inner cuts used below.

For 1, work in units $l$.  Initial straight pieces along outward pure
normals, then two disjoint simple routes in the open exterior
approaching a common terminal pure line from the same side in the
pure perpendicular direction at close separated points; around the
terminal patch keep a free half-ball of radius much greater than the
terminal separation (so the preceding confined-arch lemma applies).
Other pieces are straight in pure directions with successive turns
$\le60^\circ$, each travelled by the point-to-line lemma with small
tolerances.  By compactness of subsequences under scaling and
translation, initial rays give strictly positive distance by their
supporting lines; one then follows disjoint smooth arcs in the open
exterior into an unobstructed ball for the parallel terminal lanes.
Tracking tangent displacements on short intervals using the two
adjacent pure directions spanning the tangent gives polygonal
approximations with end straight segments retained; sufficiently small
mesh gives separation for distant portions, and on short portions
tangent continuity gives positive projection in a common direction.
Fixed small apertures on the finitely many legs, with small errors on
all stages, deliver the required tolerances.  Nonincident portions and
successive crossing neighbourhoods are separated, and adjacent pieces
stay in their required half-planes at each transverse crossing.  This
works along the subsequence, including the localized initial deep
excursions, and gives uniform lengths, step counts, and apertures at
this scale.  For bounded $l$, a bounded-cost local join outside
suffices.

For 2 with $l$ a fixed small multiple of $R$ or larger, the same
construction in units $R$ without a terminal arch follows two disjoint
paths to the two caps, terminating separately.  For smaller $l$, the
points lie on the same or adjacent facets: start along divergent pure
directions (on the same facet, rotate $m$ by $60^\circ$ in opposite
directions; on adjacent facets use their two outward normals).
Distances along the chosen straight rays at parameters $x,y\ge0$ are
separated by $\ge c(l+x+y)$ by projection, and supporting-normal
height increases at rate $\ge1/2$.  Travel to a small fixed distance
of order $R$ along each with a point-to-line crossing at scale $l$,
then confined pure bridges along the ray to successively doubled
heights; error accumulation is geometric, so small tolerances work
simultaneously.  The continuation from macroscopically separated
locations by finitely many $R$-scale legs follows as above;
uniformity holds even if the rays had coincident limiting starts,
since their terminal segments are outgoing and separated.

\emph{Averaging intermediate levels.}  Fix one corridor construction
with $D_h,p,q$ fixed.  In both cases, vary each intermediate
transverse level uniformly over admissible levels, in its own small
independent window of order the local scale, the terminal pure line
for case 1 shared at both ends.  Let $\nu$ be critical weight on all
exterior objects with the stated confinement and depth (paths
$p\to q$ in case 1, or ordered pairs of disjoint paths to the two
assigned caps in case 2), counting all centres in exterior paths.
For a cut list $\mathbf u$, let
$\mathbf1_{\mathbf u}(\eta)$ indicate that $\eta$ follows the
prescribed corridors at those levels, and put
$Y(\eta)=\mathbb E_{\mathbf u}\mathbf1_{\mathbf u}(\eta)$ and
$P=\prod_{j=1}^ks_j^{-1/4}$.  Concatenating the confined crossings
gives
\[
\int Y\,d\nu\ge
\begin{cases} c^kl^{-5/4}P,&\text{case 1},\\
c^kP,&\text{case 2}.\end{cases}
\]

For two lists, split each branch at both choices of each varied
crossing, giving a difference bridge there.  Nonincident pieces stay
out of the small neighbourhood; between successive crossing stages
from one end, the two incident half-plane constraints put all
crossings of that stage in order, with an intervening portion
travelling to the next stage between its respective lines (a distance
comparable to its scale).  The intervening costs with incoming point
prescribed are $O(s_j^{-1/4})$ by the upper point-to-line estimate.
In case 1 the completing arch at the later shared level costs
$O(l^{-5/4})$ for any attained endpoints; averaging the joint costs
supplies an extra $s_j^{-1/4}$ for each varied renewal per branch at
scale $s_j$, which stays integrable.  In case 2 only the two final
cap levels are fixed, their endpoints free, and both final legs of
scale $\asymp R$.  Consequently
\[
\int Y^2\,d\nu\le
\begin{cases} C^kl^{-5/4}P^2,&\text{case 1},\\
C^kR^{1/2}P^2,&\text{case 2},\end{cases}
\]
the factor $R^{1/2}$ in case 2 having removed from the second copy of
$P$ the two factors at the fixed cap levels.  Cauchy--Schwarz gives
\[
\nu\{Y>0\}\ge\frac{(\int Y\,d\nu)^2}{\int Y^2\,d\nu}\ge
\begin{cases} c_1^kl^{-5/4},&\text{case 1},\\
c_1^kR^{-1/2},&\text{case 2},\end{cases}
\]
for a fixed $0<c_1<1$.  The number of legs is bounded in case 1, and
in case 2 at most $C(1+\log^+(R/l))$, so
$c_1^k\ge c\min(1,(l/R)^{C_0})$ for a fixed $C_0>0$.  These are the
asserted connector masses.

\emph{Common target measure.}  Trim the exterior choices by fractional
indicators to a common lower mass $F\asymp l^{-5/4}$ (case 1) or
$F\asymp R^{-1/2}\min(1,(l/R)^{C_0})$ (case 2), pointwise for
$p,q,h,D_h$.  The augmented object is a polygon injecting into the
cylinder in case 1, or a full bridge in case 2; a fixed cut choice
recovers the decomposition up to bounded multiplicity since there are
exactly two boundary crossings.  Let $\Omega$ be the six
phase--slope pairs with the product law and $\mu$ critical weight on
target polygons through the prescribed links (case 1) or target
bridges satisfying the common deterministic event (case 2), with
total mass $\le\mathcal K$ or $B^*$.  In a level window $I$ of size
$\asymp\epsilon l$, let $b_h(\Gamma,\Omega)$ sum the recovered
ordered decompositions of $\Gamma$ at $\partial D_h$ of the
chord-event indicator times the exterior trimming factor, so
$0\le b_h\le C$; put
$X(\Gamma,\Omega)=\operatorname{average}_{h\in I}b_h(\Gamma,\Omega)$.
Then
\begin{equation}
\int X\,d(\mu\otimes\mathbb P_\Omega)\asymp F\overline E,
\label{eq:sewing-first-moment}
\end{equation}
where the left side includes the level average and support
randomness.

Choose two levels independently and uniformly in $I$; by symmetry
integrate over $r<s$, conditioned on the product measure restricted
to that half-square, plus any diagonal atoms.  This restriction is not
renormalized: conditional on $r$, the remaining $s$-measure (or its
rounded atom mass) has density at most $C/l$.  For $\epsilon>0$
sufficiently small, a simultaneous event splits into the chord event
at $D_r$, two exterior bridges crossing the band to $\partial D_s$,
and the augmentation from there; keep the trimming factor from the
latter.  Boundary crossings are unique on the separate branches,
localized by the depth condition since the boundary moves by at most
$C\epsilon l+O(1)$.  For bounded $s-r$ or bounded $l$, the indicator
at $r$ alone suffices.  Bounding the averaged cost $\Xi$ of the two
band bridges uniformly conditional on the inner level and phases
gives
\begin{equation}
\int X^2\,d(\mu\otimes\mathbb P_\Omega)\le CF\overline E\,\Xi.
\label{eq:sewing-second-moment}
\end{equation}

\emph{The two band bridges.}  Put $g=s-r$.  Fix for each bridge the
possible target normal in the local facet pair.  Drop all other outer
cuts, keeping the inner elbow in a local patch with two lateral pure
trims transversal to both tangent directions containing the bridge
portions; the domain is a polygonal strip between this outward cut
line and the flat or elbow piece of $D_r$.  The inner obstacle keeps
just the two corresponding consecutive constraints, extended through
the strip patch.  The target outward cut line is disjoint from it by
the positive gap; the trims may for instance bound projection on the
common tangent bisector, wide enough to enclose the localized portions
and leave clearance at the source.

There is a positive contour (after a possible rotation) from the inner
source.  Closing a chord along the boundary gives $W$ equal to lifted
outward-normal change minus $\pi$ around the perimeter from source to
exit.  There is only one reflex corner on the inner elbow (outward
normal decreasing by $\pi/3$), and its two outward normals and the
outward normals on the trim sides and target line are in cyclic order
apart from that corner (the trims use a pure direction strictly
between opposing boundary sectors).  Hence from a given inner source
at most one sign of excess beyond $[-\pi,\pi]$ occurs, and by at most
$\pi/3$; rotating gives real coefficients bounded below positively,
since $\beta=3/8$.  Convexity is not needed for the contour identity
on this simply connected domain.  Cutting down by a centered triangle
at scale a small multiple of $\min(g,t_i)$ (inner source on a flat
side) and using contour subtraction and the escape upper estimate, the
bridge cost is $O((1+\min(g,t_i))^{-1/4})$, also by the full positive
contour when the minimum is bounded.

\emph{Corner gain.}  Put $t=\min_it_i$.  Both tests in $X^2$ use the
same dyadic endpoint bins: the two inner bins have sizes at least $t$,
and one outer bin has size of order $t$.  Sum over the finitely many
choices of that outer endpoint and its facet.  Condition on the six
phases at $r$ (making the inner chord event measurable) and on the
support position of the selected target facet at $s$.  For any
attained outer endpoints and supports, integrate the outer completion
to its constant trimmed mass $F$ and bound the other band bridge
uniformly after dropping its extra avoidance constraints (even if it
ends on the unselected adjacent facet; Figure~\ref{fig:sewing-band}).
\input{figures/sewing-band}
The adjacent facet defining the selected endpoint's corner is
unconditioned: its position is independent of the selected support,
conditional on the inner phases, and has density $O(1/g)$, since
$V_i+A_ir/R \bmod 1$ is uniform and independent of $A_i$.  Each
attained end therefore has probability at most
$C\min(1,t/(1+g))$ of lying within $O(t)$ of its corner (also after
rounding).  Averaging this instead of conditioning, both inner bin
sizes are at least $t$, so the product of the two bridge bounds
remains uniform in the averaged parameter, giving
\[
C(1+\min(g,t))^{-1/2}\min\{1,t/(1+g)\}\le C(1+g)^{-1/2}.
\]
Conditional on the gap, $\Xi\le Cl^{-1/2}$ on each window.  Applying
Cauchy--Schwarz to \eqref{eq:sewing-first-moment} and
\eqref{eq:sewing-second-moment} yields
$\overline E\le C\mathcal K\,\Xi/F$ and
$B^*\ge cF\overline E/\Xi$, proving both assertions.  Averaging over
windows recombines them.
\end{proof}

\paragraph{Corner gains and the marked-pair sum.}
\label{sec:corner-sum}

The estimate above suffices for one endpoint bin.  To sum over bins
with a bounded constant, retain the gain when an endpoint lies near a
corner.  Write $t=\min(t_1,t_2)$, $1\le t\le l$.  The combined per-bin
cost before averaging over the cut gap is
$C(1+\min(g,t))^{-1/2}\min\{1,t/(1+g)\}$.  For fixed inner level $r$,
the unnormalized gap law on the half-square $r<s$ has density at most
$C/l$ over an interval of length $O(l)$, uniformly for the rounded
discrete law.  Splitting at $g=t$ bounds the average by
\[
\frac Cl\Bigl(\int_0^t(1+g)^{-1/2}\,dg
+t(1+t)^{-1/2}\int_t^{Cl}\frac{dg}{1+g}\Bigr)
\le C_\eta l^{-1/2}(t/l)^\eta
\]
for any $0<\eta<1/2$; bounded-gap atoms satisfy the same after
increasing $C_\eta$.  Keeping this gain in the first sewing estimate
gives
\begin{equation}
\overline E\le C_\eta l^{3/4}(t/l)^\eta\mathcal K.
\label{eq:sewing-corner-gain}
\end{equation}
Indeed, the first moment on the target polygon measure is at least
$cF\overline E$ and the second at most
$CF\overline E\,l^{-1/2}(t/l)^\eta$, while $F\asymp l^{-5/4}$.  For
each smaller corner bin $t$, the larger one has only
$O(1+\log(l/t))$ possible dyadic values, so their total contribution
is at most $C_\eta\sum_{j\ge0}(1+j)2^{-j\eta}<\infty$; bins with
$t/l\le\delta$ have total coefficient tending to zero as
$\delta\downarrow0$.

\emph{Marked-pair sum and second chord moment.}  Let $\ell_h$ denote
the length of a chord in $D_h$, and retain chords of diameter at least
$R/3$.  Their mass is at most $CR^{3/4}$ and their first-length mass
at least $cR^{25/12}$: every path through a sufficiently central link
is macroscopic, and Proposition~\ref{prop:occupancy} with
Theorem~\ref{thm:nesting} supplies mass at least $cR^{1/12}$ at each
of $cR^2$ such links.  The first-length mass over all chords is at
most $CR^{25/12}$ by the same identity.

For the second moment, omit the initial and terminal visited centres;
ordered pairs involving these ($O(\ell_h)$ in number) are absorbed by
the first-length bound.  At each remaining centre record the two
tiling-edge directions crossed by its incident links.  Divide centres
into classes by triangle type, lattice translation modulo two, and
that pair of directions; the square of the total count is at most a
fixed multiple of the sum of class squares.  Within a class, pairs
with axial or zero displacement contribute at most $CR\ell_h$ per
chord; their integrated contribution is at most $CR^{37/12}$.

For a remaining ordered pair, choose the two lattice axes spanning
the strict sector containing its displacement; at least one recorded
direction is one of those axes.  Same triangle type makes the crossed
edges translates, so they supply identical axial marks used by
internal links; their displacement has even coordinates, so
$d=P_1+P_2$ is even.  Choose an integer period multiple $kP$ of length
comparable to a large constant times $R$ so the sewn polygon projects
injectively; then $N\asymp R$ and
\[
\mathcal K\le CR^{-4/3}(R/d)^{15/8}.
\]
With $O(R^2)$ choices of the first centre and $O(d)$ displacements with
$P_1+P_2=d$,
\begin{equation}
\sum_{\text{eligible pairs}}\mathcal K
\le CR^2R^{-4/3}R^{15/8}\sum_{1\le d\le CR}d^{1-15/8}
\le CR^{8/3}.
\label{eq:marked-pair-sum}
\end{equation}
Apply \eqref{eq:sewing-corner-gain} to each pair event; corner bins
sum with bounded total coefficient, and the separation factors
$l^{3/4}$ sum geometrically up to $CR$, giving
\begin{equation}
\overline{\sum_{\gamma\text{ macroscopic}}\ell(\gamma)^2w(\gamma)}
\le CR^{41/12}.
\label{eq:chord-second-moment}
\end{equation}
Bins with $l\le\delta R$ contribute at most
$C\delta^{3/4}R^{41/12}+CR^{37/12}$; bins with $t\le\delta'R$ have
arbitrarily small relative coefficient for $\delta'$ small.  These
are estimates for endpoint restrictions, not assumed corner avoidance.

\paragraph{A positive fraction of long bridges.}
\label{sec:long-bridges}

Write $M_1$ for the averaged first-length mass of the macroscopic
chords: $M_1\ge c_1R^{25/12}$.  By Cauchy--Schwarz, a class with
second moment at most $\varepsilon R^{41/12}$ contributes at most
$C\sqrt\varepsilon R^{25/12}$ to $M_1$.  Choose $\delta,\delta'>0$ so
removal of close endpoints and small corner bins loses less than a
quarter of $M_1$; chords shorter than $aR^{4/3}$ contribute at most
$CaR^{25/12}$, less than another quarter for $a>0$ fixed small.

The retained chords therefore have averaged first-length mass at
least $c_2R^{25/12}$, with endpoint separation and corner distances
all bounded below by fixed multiples of $R$, occupying a bounded
number of bins.  In at least one bin the first-length mass has the
same lower order; Cauchy--Schwarz and \eqref{eq:chord-second-moment}
give averaged ordinary mass at least
$cR^{50/12}/R^{41/12}=cR^{3/4}$.

The second sewing estimate extends these chords to bridges between
any prescribed pair of horizontal caps whose distances above and
below the origin lie in $[AR,2AR]$ for fixed large $A$.  With
$l\asymp R$, that estimate gives total bridge mass at least $cR^{3/4}$
when the bottom start is free over $O(R)$ ports.  Every output bridge
contains the retained chord, hence has length at least $aR^{4/3}$.
Translation invariance of the unrestricted mass per bottom port,
together with removal of confinement and division by the number of
permitted starting ports, gives
\[
B_h\{\ell\ge aR^{4/3}\}\ge cR^{-1/4}
\]
for every admissible cap separation $hd_0\in[2AR,4AR]$.  For
arbitrary large $h$, choose $R\propto h$ so the selected caps lie in
these ranges; admissible rounding changes only fixed constants.

\paragraph{The strip first moment and completion of the proof.}
\label{sec:strip-first-moment}

It remains to prove the first-length upper bound.  In the infinite
strip, summing occurrences of internal links over bridges from one
fixed bottom port equals summing the occurrence mass with free bottom
port over link representatives modulo translations along the strip;
there are $O(h)$ such representatives.  Proposition~\ref{prop:occupancy},
first in finite convex truncations and then by monotone exhaustion,
bounds the occurrence mass at each link by the positive
enclosing-polygon partitions at its adjacent lattice points.

These partitions are at most $Ch^{1/12}$ even in the infinite strip:
take a pure lattice period vector tilted relative to the strip's
horizontal tangent, of sufficiently large even length comparable to
$h$, with transverse projection $>2hd_0$.  The strip and its
translate by half this period embed disjointly in the resulting
cylinder.  The disk enclosed by each polygon stays inside its convex
strip, so cannot contain the mark in the translated strip.  Thus two
independent systems enclosing corresponding points inject into a
system separating the antipodal marks.  The bounded-factor cylinder
estimate of Section~\ref{sec:uniform-nesting} is at most $Ch^{1/6}$,
so the square of the strip partition is at most that quantity.

Summing over the $O(h)$ link representatives,
\[
\sum_{\gamma\in\mathcal B_h}(\ell(\gamma)-1)w(\gamma)\le Ch^{13/12}.
\]
The one missing centre per bridge adds only $B_h\le Ch^{-1/4}$,
completing \eqref{eq:bridge-first-moment}.  The strip mass
\eqref{eq:bridge-mass} was proved in
Sections~\ref{c6:strip-transfer}--\ref{c6:strip-asymptotics}, and
Proposition~\ref{prop:crossings} gives \eqref{eq:bridge-diameter};
Theorem~\ref{thm:finite-bridges} is complete.

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\end{document}