diff --git a/uniform_marked_polygon_bridge_v2.txt b/uniform_marked_polygon_bridge_v2.txt
index cbebbdb..d42071d 100644
--- a/uniform_marked_polygon_bridge_v2.txt
+++ b/uniform_marked_polygon_bridge_v2.txt
@@ -32,15 +32,10 @@
 \DeclareMathOperator{\supp}{supp}
 \DeclareMathOperator{\rank}{rank}
 \DeclareMathOperator{\Span}{span}
-\title{Uniform marked-polygon estimates and sharp finite bridge moments}
-\author{OpenAI}
-\date{September 26, 2026}
-\hypersetup{pdfauthor={OpenAI},
-  pdftitle={Uniform marked-polygon estimates and sharp finite bridge moments}}
+% (preamble marker)
 \begin{document}
-\maketitle
 
-\begin{abstract}
+% (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
@@ -66,19 +61,12 @@ 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.
-\end{abstract}
+% (abstract-end removed)
 
-% ============================================================
-\section*{Preface: how to read this paper}
-% ============================================================
-
-This paper proves three quantitative theorems about a single
-concrete object, the critical self-avoiding walk on the honeycomb
-lattice. Sections~1 and~\ref{sec:introduction} state what is
-proved and why it matters; a strong high-school student with
-calculus will be able to follow them. Later sections are genuine
-research-level arguments; they are meant to be read with those
-first sections as a reference.
+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
@@ -102,11 +90,10 @@ w(\gamma) = \kappa^{\ell(\gamma)},\qquad
 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. We
-take $\kappa$ as given. The point of this paper is what happens
-when you sum $\kappa^{\ell(\gamma)}$ over paths with prescribed
+\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, etc.
+all polygons through two marks -- gives the three central sums.
 
 \paragraph{The three theorems in plain words.}
 
@@ -143,63 +130,42 @@ $R^{1/12}$, again with bounded constants. This ``positive''
 comparison is the input needed to pass between positively weighted
 path sums and positively weighted polygon sums.
 
-\paragraph{What you will need.} Sections 2--5 (transfer matrix,
-positive interpolation, vacuum polynomial, gas) use linear
-algebra over the reals and complex numbers, power series, and the
-beginning of complex analysis (Euler's formula and the residue
-theorem). Sections 6--10 use geometry, probability, and long
-bookkeeping. The core mathematical content is genuinely
-research-level, but the argument is fully self-contained and
-every step is written out.
-
-\paragraph{Notation.} We write $\ell(\gamma)$ for 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. We use $\kappa=1/\sqrt{2+\sqrt2}$ throughout, and introduce
-the auxiliary constants $L=\pi/8$, $b=3L$, $C=2-\sqrt2=4\cos^2 b$
-when they arise (they are eigenvalues of the row transfer matrix
-and have no substantive meaning otherwise). For a triangular
-lattice vertex $y$ and $R\in d_0\mathbb Z_{>0}$ with
-$d_0=\sqrt3/2$, we write $H(y;R)$ for the regular pure-sided
-hexagon of inradius $R$ centred at $y$; Section~\ref{sec:hexagon-completion}
-explains why this is a natural lattice rounding.
-
-% ============================================================
-\section{Introduction}\label{sec:introduction}
-% ============================================================
+\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 $y$.
 
-\subsection{The honeycomb lattice and its critical weight}
-\label{sec:intro-honeycomb}
+\paragraph{Introduction.}\label{sec:introduction}
 
-Let $\mathbb T\subset\mathbb R^2$ be the unit equilateral triangular
-lattice, which cuts the plane into equilateral triangles of side
-$1$. Colour those triangles black and white like a chessboard.
-Put a vertex at the centre of each triangle, and join two centres
-by an edge whenever the corresponding triangles share an edge.
-The resulting graph is the \emph{honeycomb lattice} $\mathbb H$.
-Its edges form the boundary of a regular hexagonal tiling whose
-cells contain one black and one white triangle.
+\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 we call it \emph{essential}, or bounded, in which
-case we call it \emph{contractible}. Polygons are unoriented and
-\emph{unrooted} unless we specify a root.
+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 $\kappa^{-1}=\sqrt{2+\sqrt2}$
-is the connective constant; in this paper $\kappa$ is part of the
-setup, not the object of study. Our question is: given a bridge,
-a pair of marks, or a nesting system, what is the resulting
-$\kappa$-mass?
+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.
 
-\subsection{Paths, bridges, polygons}\label{sec:intro-objects}
+\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
@@ -226,7 +192,7 @@ 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.
 
-\subsection{The three main theorems}
+\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
@@ -294,7 +260,7 @@ $R^{3/4}$, and the mean number of visited centres is comparable
 to $R^{4/3}$.
 \end{theorem}
 
-\subsection{Context and history}
+\paragraph{Context and history.}
 
 The honeycomb self-avoiding walk has a long history, and three
 results sit directly behind Theorem~\ref{thm:finite-bridges}.
@@ -347,7 +313,7 @@ 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.
 
-\subsection{Proof route}
+\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
@@ -379,7 +345,7 @@ 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.
 % ============================================================
-\section{A finite formula for strip crossing mass}
+\paragraph{A finite formula for strip crossing mass.}
 \label{c6:strip-transfer}
 % ============================================================
 
@@ -398,7 +364,7 @@ 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 $B_N$.
 
-\subsection{Constants and setup}
+\paragraph{Constants and setup.}
 \label{sec:strip-constants}
 
 Throughout fix
@@ -418,7 +384,7 @@ 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 $\sin(4L-2L)=\sin(2L)=1/\sqrt2$.
 
-\subsection{The local $R$-matrix}
+\paragraph{The local $R$-matrix.}
 \label{sec:strip-R}
 
 Let $V=\mathbb C^3$ have basis $\{|+\rangle,|0\rangle,|-\rangle\}$,
@@ -455,7 +421,7 @@ c(u)+d(u)s^2&s\,h(u)&d(u)
 All other entries vanish: $R(\pm\pm,\mp\mp)$, $R(0\pm,\pm-)$, etc.\ are
 zero. Summation over intermediate spins is implied.
 
-\subsection{Structural identities of $R$}
+\paragraph{Structural identities of $R$.}
 \label{sec:strip-R-structure}
 
 Let $P$ denote the swap $P(a\otimes b)=b\otimes a$,
@@ -529,7 +495,7 @@ $Q(2L)=\sin(4L)\sin(2L+3L)=\sin(4L)\sin(5L)=1\cdot S$; direct check
 gives $R(2L)=WZ$.
 \end{proof}
 
-\subsection{The fusion equation}
+\paragraph{The fusion equation.}
 \label{sec:strip-fusion}
 
 The key insertion identity:
@@ -608,7 +574,7 @@ $QQ_+$ reduces (using $QQ'/Q_+=-t_{-3}t_2$) to a case of
 \eqref{eq:fusion-aux}. All remaining cases are identical.
 \end{proof}
 
-\subsection{The Yang--Baxter equation}
+\paragraph{The Yang--Baxter equation.}
 \label{sec:strip-YBE}
 
 \begin{proposition}[Yang--Baxter]\label{prop:YBE}
@@ -639,7 +605,7 @@ $u\in\{v,v+2L,v+b\}$ therefore determine the entry. This proves
 
 
 
-\subsection{The boundary matrix $K_e(t)$}
+\paragraph{The boundary matrix $K_e(t)$.}
 \label{sec:strip-boundary}
 
 The row transfer also needs a \emph{boundary} matrix that closes each
@@ -714,7 +680,7 @@ P_+ H_+ G_+-P_- H_- G_-=\cos L\,T_0\cos(2t).
 Each is a direct substitution using $8L=\pi$ and $\sin(2L)=1/\sqrt2$.
 \end{proof}
 
-\subsection{The double-row operator $T(t)$}
+\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
@@ -764,7 +730,7 @@ swap structure of the two $R$'s at parameters differing by $b$
 closes the contraction, giving identity.
 \end{proof}
 
-\subsection{Commutation of $T(t)$ and $T(y)$}
+\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,
@@ -801,7 +767,7 @@ in \eqref{eq:reflection} is a trigonometric polynomial of degree $\le6$
 in $t$, periodic/antiperiodic by $\pi$. Seven samples suffice.
 \end{proof}
 
-\subsection{The fusion identity for rows}
+\paragraph{The fusion identity for rows.}
 \label{sec:strip-fuse-rows}
 
 The key algebraic step in extracting the physical eigenvalue is the
@@ -832,7 +798,7 @@ used \eqref{eq:K-swap-trig}.
 
 
 
-\subsection{The physical eigenvalue at homogeneous rapidities}
+\paragraph{The physical eigenvalue at homogeneous rapidities.}
 \label{sec:strip-physical}
 
 We now pass from the algebraic structure of $T(t)$ to the physical
@@ -965,7 +931,7 @@ B_N = \lim_{m\to\infty}\frac{1}{2m}
 =\frac{1}{2}q(\gamma;\xi).
 \end{equation}
 
-\subsection{Analytic structure of $q(t;\xi)$}
+\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
@@ -1036,7 +1002,7 @@ matrix identities, the difference vanishes to order $2N$ at $t=\gamma$
 \eqref{eq:q-zeros-2}.
 \end{proof}
 
-\subsection{The scalar crossing function}
+\paragraph{The scalar crossing function.}
 \label{sec:strip-p}
 
 At homogeneous $\xi_j$ (i.e., at the physical rapidities with correction
@@ -1076,7 +1042,7 @@ The value $2B_N=2\cdot\frac{1}{2}q(\gamma)=q(\gamma)=p(0)$ becomes
 2B_N=p(0)=\mathcal L(1/D).
 \end{equation}
 
-\subsection{The finite functional}
+\paragraph{The finite functional.}
 \label{sec:strip-finite-functional}
 
 We now pin down $\mathcal L$. Set $\mu_0(x)=C_{L/2}(x)$,
@@ -1122,7 +1088,7 @@ sums to the displayed $\tan$-combination.
 
 
 
-\section{The positive interpolation estimate}\label{c6:strip-asymptotics}
+\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.
@@ -1130,7 +1096,7 @@ 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.
 
-\subsection{Spaces, measure, determinant density}
+\paragraph{Spaces, measure, determinant density.}
 
 Retain from Proposition~\ref{prop:finite-functional}:
 \begin{itemize}
@@ -1213,7 +1179,7 @@ d\nu_3(X)\propto -e_0c_3(X)\,d\nu(X).
 Since $-e_0c_3>0$ by jet positivity ($d$ is odd), $\nu_3$ is a
 probability density.
 
-\subsection{Characteristic products}
+\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
@@ -1272,7 +1238,7 @@ 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.
 
-\subsection{Interpolation residuals}
+\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
@@ -1331,7 +1297,7 @@ and the tilted target
 \nu_3[E_{-4}h_0]\asymp N^{-1/4}.
 \end{equation}
 
-\subsection{Increasing-weight comparison}
+\paragraph{Increasing-weight comparison.}
 
 The final size estimate on $\nu_3[-e_0c_3]$ uses monotone coupling.
 
@@ -1397,7 +1363,7 @@ 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
 $\nu_b[-e_0c_3]\gtrsim d^{-1/2}$.
 
-\subsection{Assembly}
+\paragraph{Assembly.}
 
 Combine \eqref{eq:schur-identity}, \eqref{eq:P-7-estimates}, and
 Lemma~\ref{lem:weight-comp}. Write
@@ -1427,7 +1393,7 @@ applied.
 
 
 
-\section{The periodic vacuum and its physical pairing}\label{c6:periodic-polynomial}
+\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
@@ -1436,7 +1402,7 @@ 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.
 
-\subsection{The cylinder and the finite formula}
+\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
@@ -1502,7 +1468,7 @@ The proof is by polynomial interpolation: the right-hand side is
 identified with the physical polygon partition through a finite
 family of fusion conditions.
 
-\subsection{Periodic eigenlines}
+\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
@@ -1616,7 +1582,7 @@ Matching \eqref{eq:f_N} gives the exact scalar fusion identities
 with $c_1=1$ (after normalization) and $c_2=2\cos b$
 (cross-checked at $N=2,3$ and by induction).
 
-\subsection{The physical contraction}
+\paragraph{The physical contraction.}
 
 \begin{proof}[Proof of Proposition~\ref{prop:physical-pairing}]
 Define
@@ -1659,7 +1625,7 @@ and the equality extends to a Laurent polynomial identity.
 
 
 
-\section{The two-line gas expansion}\label{c6:gas}
+\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
@@ -1668,7 +1634,7 @@ 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.
 
-\subsection{Normalization and the target}
+\paragraph{Normalization and the target.}
 
 Put
 \begin{equation}\label{eq:P-m-normalized}
@@ -1686,7 +1652,7 @@ As $m\to\infty$, locally uniformly in $|\Im r|<.54\pi$:
 \end{equation}
 \end{proposition}
 
-\subsection{Residue particles}
+\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
@@ -1705,7 +1671,7 @@ $\sum_le^{-2Tl^2+4l\sum q_jw_j}$) 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 $e^{-T}$.
 
-\subsection{Species and offsets}
+\paragraph{Species and offsets.}
 
 After contour manipulation and fusion, five species remain in each
 color. Their weights $w_i$ (exponent of $y$), offsets $\delta_i$
@@ -1725,7 +1691,7 @@ The neutral-winding constraint on total particle counts is
 \sum_{g\in\{c,-\}}\sum_{i=1}^5w_i\nu_{g,i}=0.
 \end{equation}
 
-\subsection{The interaction kernel}
+\paragraph{The interaction kernel.}
 
 The particle kernel for two particles at real frequency $p$ is
 \begin{equation}\label{eq:Z-def}
@@ -1761,7 +1727,7 @@ K_0=\frac{2\pi}{p}\begin{pmatrix}\coth(bp)&-\operatorname{csch}(bp)\\
 \end{equation}
 (Here $V_R$ denotes the potential contribution.)
 
-\subsection{Coercivity}
+\paragraph{Coercivity.}
 
 The key analytic input is a positive-definite bound on $Z$.
 
@@ -1792,7 +1758,7 @@ Z(p)=\alpha WW^t/p^2+M_*/p+O(1)\quad(p\to0),
 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}
 
-\subsection{The grand sum}
+\paragraph{The grand sum.}
 
 \begin{theorem}[Grand sum]\label{thm:grand-sum}
 For every integer $m\ge1$ and every sufficiently large real $T$,
@@ -1819,7 +1785,7 @@ The sum and integral converge absolutely, and the sum localizes on
 \end{equation}
 \end{theorem}
 
-\subsection{Convergence and localization}
+\paragraph{Convergence and localization.}
 
 \begin{proof}[Sketch]
 Proposition~\ref{prop:coercivity} bounds the off-diagonal $Z_s$ terms
@@ -1835,14 +1801,14 @@ Consequently the whole sum is dominated by the localized region
 
 
 
-\section{Saturation at the two physical parameters}\label{c6:saturation}
+\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.
 
-\subsection{The saturation statement}
+\paragraph{The saturation statement.}
 
 \begin{auxproposition}[Saturation at $y=1$ and $y=i$]\label{prop:saturation}
 \begin{equation}\label{eq:saturation-aux}
@@ -1855,7 +1821,7 @@ Equivalently, the nesting ratio satisfies
 \end{equation}
 \end{auxproposition}
 
-\subsection{The reference determinant at $y=i$}
+\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
@@ -1875,7 +1841,7 @@ $D(0)=C\sin^4b$.
 
 We are reduced to bounding the ratio $P_m(i)/P_0^{(\lambda)}$.
 
-\subsection{Bounding the ratio at $y=i$}
+\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
@@ -1911,7 +1877,7 @@ This gives $|P_m(i)/P_0^{(\lambda)}|\ge m^{5/24-o(1)}$, hence
 \log P_m(i)=Im^2+\frac5{24}\log m+o(\log m).
 \end{equation}
 
-\subsection{Propagation to $y=1$}
+\paragraph{Propagation to $y=1$.}
 
 Set $r=\log y$ and put
 \begin{equation}\label{eq:f-m}
@@ -1940,7 +1906,7 @@ as claimed.
 
 
 
-\section{Crossings, cylinder barriers, and positive nesting}\label{c6:geometry}
+\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
@@ -1950,7 +1916,7 @@ geometric inputs for Section~\ref{sec:sewing}.
 
 All constants are uniform in $N$, $h$, and $H$.
 
-\subsection{The critical weight at the port convention}
+\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
@@ -1959,7 +1925,7 @@ 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 $x$.
 
-\subsection{Boundary crossings}
+\paragraph{Boundary crossings.}
 
 \begin{proposition}[Boundary crossings]\label{prop:crossings}
 Fix a convex pure-sided tiled domain $D$ and a port $p$ on its
@@ -2017,7 +1983,7 @@ $\mu[Y>0]\gtrsim R^{-1/4}$ and the matching lower bound for $K_l$ by
 monotonicity.
 \end{proof}
 
-\subsection{Narrow-channel bridges}
+\paragraph{Narrow-channel bridges.}
 
 \begin{lemma}[Narrow-channel bridges]\label{lem:narrow-channel-bridges}
 For every sufficiently large absolute constant $A$, there are
@@ -2040,7 +2006,7 @@ bound on each piece gives $cs^{-1/4}$; Cauchy--Schwarz over the lists
 of internal levels gives \eqref{eq:channel-bridge-mass}.
 \end{proof}
 
-\subsection{Essential polygons in a slab}
+\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
@@ -2064,7 +2030,7 @@ confined arches and averaging over their shared base level gives
 \eqref{eq:slab-essential-mass}.
 \end{proof}
 
-\subsection{Positivity of the negative polymer polynomial}
+\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
@@ -2104,7 +2070,7 @@ $\mathcal P$ by connectedness; but a fixed bridge through every level
 gives a positive residual polynomial, contradiction.
 \end{proof}
 
-\subsection{Deletion and free-energy bounds}
+\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
@@ -2123,7 +2089,7 @@ and hence
 H_H(\ell;t)\,dt\le CH/N.
 \end{equation}
 
-\subsection{Cylinder barriers and height tails}
+\paragraph{Cylinder barriers and height tails.}
 
 \begin{auxproposition}[Barriers and tails]\label{prop:cylinder-barriers}
 On the infinite cylinder of period $N$:
@@ -2147,7 +2113,7 @@ 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}
 
-\subsection{Local nesting and boundary occupancy}
+\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
@@ -2192,7 +2158,7 @@ yields \eqref{eq:occupancy}.
 
 
 
-\section{Bounded-factor nesting estimates}\label{sec:uniform-nesting}
+\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
@@ -2200,7 +2166,7 @@ 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}).
 
-\subsection{Deletion ratio and the first-hit comparison}
+\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
@@ -2241,7 +2207,7 @@ first hit $L$. Contour identity:
 \kappa^{|\gamma|}\cos(\beta W(\gamma))H_D(\gamma)\ge0.
 \end{equation}
 
-\subsection{Boundary-flux primitive}
+\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 $y_0$
@@ -2270,7 +2236,7 @@ increments $\pm4\pi/3$ versus $\pm2\pi/3$). Comparing gives the
 inequality.
 \end{proof}
 
-\subsection{The annular bound}
+\paragraph{The annular bound.}
 
 \begin{proposition}[Planar annular bound]\label{prop:annular-bound}
 For every $s\ge1$ and every interior lattice point $y$:
@@ -2295,7 +2261,7 @@ $\sum_aP(a,L)\ge cs^{3/4}$. The essential sum vanishes in the plane; the
 auxlemma gives the bound.
 \end{proof}
 
-\subsection{Bounded cost of large specified lifts}
+\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
@@ -2343,7 +2309,7 @@ planar annular bound \eqref{eq:annular-mass} applied to the corresponding
 diameter intervals.
 \end{proof}
 
-\subsection{Classwise convergence of the normalization}
+\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
@@ -2393,7 +2359,7 @@ $\lesssim e^{-cT'}$ for $m'=m+Q$. This proves
 
 
 
-\section{Uniform correlations of two prescribed edges}\label{u:marked-uniform}
+\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
@@ -2403,7 +2369,7 @@ 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.
 
-\subsection{Setup}
+\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
@@ -2413,7 +2379,7 @@ 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.
 
-\subsection{Finite marked contraction}
+\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
@@ -2464,7 +2430,7 @@ only a constant factor, fixed by the base case $m+n=0$ (equivalently
 by saturation of the two fixed-spin columns).
 \end{proof}
 
-\subsection{Physical interpretation}
+\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:
@@ -2484,7 +2450,7 @@ The relative signs (from turns, seams, and the phase insertion) yield
 a positive weight; the identity extends by height exhaustion.
 \end{proof}
 
-\subsection{The antipodal case}
+\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.
@@ -2521,7 +2487,7 @@ $12iL$ over a period, compensated by the edge ratio's $-12iL$.
 
 
 
-\subsection{Staircases and the two background lists}
+\paragraph{Staircases and the two background lists.}
 \label{c6:unequal-marks}
 
 We now prove the uniform two-edge bound \eqref{eq:marked-uniform}.
@@ -2595,7 +2561,7 @@ 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}
+\paragraph{The ordered magnetic gas.}
 \label{u:ordered-gas}
 
 The staircase fixes the background lists; we next express their
@@ -2704,7 +2670,7 @@ 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}
+\paragraph{From periodic gas estimates to interval operators.}
 \label{u:interval-operator}
 
 We construct an operator whose products describe intervals of the
@@ -2875,7 +2841,7 @@ $\exp(qS_0/2)$ 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}
+\paragraph{Localization of the magnetic source.}
 \label{u:magnetic-localization}
 
 The homogeneous interval operator is now controlled.  We turn to the
@@ -3021,7 +2987,7 @@ 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}
+\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
@@ -3120,7 +3086,7 @@ at most
 \label{eq:legacy-P-34}
 \end{equation}
 
-\subsection{Matching the scalar and the Pfaffian normalization}
+\paragraph{Matching the scalar and the Pfaffian normalization.}
 \label{u:marked-denominator}
 
 The interval bound controls the particle integral.  To recover the
@@ -3287,7 +3253,7 @@ Exponentiation proves the uniform two-edge bound
 number of cells, use the uniform edge bounds there; enlarging the
 absolute constant covers small $d$.
 
-\section{Exterior sewing and finite bridge moments}
+\paragraph{Exterior sewing and finite bridge moments.}
 \label{sec:sewing}
 
 Everything here uses pure sides (lattice cuts normal to the $n_j$),
@@ -3345,7 +3311,7 @@ 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}
+\paragraph{Two exterior sewing estimates.}
 \label{sec:exterior-sewing}
 
 Take a triangular-lattice vertex as origin and pure-support domains
@@ -3616,7 +3582,7 @@ $B^*\ge cF\overline E/\Xi$, proving both assertions.  Averaging over
 windows recombines them.
 \end{proof}
 
-\subsection{Corner gains and the marked-pair sum}
+\paragraph{Corner gains and the marked-pair sum.}
 \label{sec:corner-sum}
 
 The estimate above suffices for one endpoint bin.  To sum over bins
@@ -3699,7 +3665,7 @@ $C\delta^{3/4}R^{41/12}+CR^{37/12}$; 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}
+\paragraph{A positive fraction of long bridges.}
 \label{sec:long-bridges}
 
 Write $M_1$ for the averaged first-length mass of the macroscopic
@@ -3734,7 +3700,7 @@ 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}
+\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