diff --git a/uniform_marked_polygon_bridge_v2.txt b/uniform_marked_polygon_bridge_v2.txt
index 8bb3573..49d4925 100644
--- a/uniform_marked_polygon_bridge_v2.txt
+++ b/uniform_marked_polygon_bridge_v2.txt
@@ -378,9 +378,6 @@ 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.
-
-\end{document}
-
 % ============================================================
 \section{A finite formula for strip crossing mass}
 \label{c6:strip-transfer}
@@ -2495,3 +2492,1349 @@ $12iL$ over a period, compensated by the edge ratio's $-12iL$.
 \end{proof}
 
 
+
+\subsection{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.
+
+\subsection{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.
+
+\subsection{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.
+
+\subsection{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.
+
+\subsection{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}
+
+\subsection{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 $d$.
+
+\section{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}
+
+\subsection{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}
+
+\subsection{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.
+
+\subsection{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.
+
+\subsection{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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