diff --git a/goe_rrg_bulk_universality_v2.txt b/goe_rrg_bulk_universality_v2.txt
index 7506b09..b82da87 100644
--- a/goe_rrg_bulk_universality_v2.txt
+++ b/goe_rrg_bulk_universality_v2.txt
@@ -2,6 +2,7 @@
 \usepackage[margin=1in]{geometry}
 \usepackage{amsmath,amssymb,amsthm,mathtools}
 \usepackage{microtype}
+\usepackage{xcolor}
 \usepackage[colorlinks=true,linkcolor=blue!55!black,citecolor=blue!55!black,urlcolor=blue!55!black]{hyperref}
 \newtheorem{theorem}{Theorem}[section]
 \newtheorem{proposition}[theorem]{Proposition}
@@ -12,6 +13,7 @@
 \newtheorem{remark}[theorem]{Remark}
 \numberwithin{equation}{section}
 \newcommand{\R}{\mathbb R}
+\newcommand{\N}{\mathbb N}
 \newcommand{\C}{\mathbb C}
 \newcommand{\PP}{\mathbb P}
 \newcommand{\EE}{\mathbb E}
@@ -51,6 +53,8 @@ property is imposed. The proof of Conjecture 2.11 of Bourgade--Huang is
 complete; every step that is not an imported theorem is proved here.
 \end{abstract}
 
+\tableofcontents
+
 \section{What is proved, and why it is surprising}\label{sec:intro}
 
 \subsection{The random object}
@@ -150,8 +154,8 @@ $1/\pi$. Following the same zoom-and-unfold recipe, define
 \subsection{The theorem}
 
 \begin{theorem}[Main result]\label{thm:main}
-For every fixed integer $d\ge3$, every fixed energy
-$E\in(-2\sqrt{d-1},2\sqrt{d-1})$, and every nonnegative continuous
+For every fixed integer $d\ge3$, every fixed energy $E$ in the open
+interval $(-2\sqrt{d-1},2\sqrt{d-1})$, and every nonnegative continuous
 compactly supported $h:\R\to\R$,
 \[
   \left|
@@ -348,11 +352,13 @@ normalized by $\|u_i\|^2=n$, so $Au_i=\lambda_iu_i$ and
 $\langle u_i,u_j\rangle=n\delta_{ij}$. Define the microscopic
 coordinates and the empirical spectral measure of the zoomed picture:
 \begin{equation}\label{eq:micro-coordinates}
+\begin{split}
   x_i=n(\lambda_i-E),\qquad
   \nu_n=\sum_{i=1}^n\delta_{x_i},\qquad
-  m_n(z)=\sum_{i=1}^n\frac1{x_i-z}
-        =\frac1n\Tr R_n(z),\qquad
-  R_n(z)=\Bigl(A-E-\frac zn\Bigr)^{-1},\quad \Im z>0 .
+  m_n(z)&=\sum_{i=1}^n\frac1{x_i-z}
+        =\frac1n\Tr R_n(z),\\
+  R_n(z)&=\Bigl(A-E-\frac zn\Bigr)^{-1},\quad \Im z>0 .
+\end{split}
 \end{equation}
 Thus $m_n$ is the Stieltjes transform of $\nu_n$ in the coordinates
 $x_i$, and $\nu_n$ is recovered from $m_n$ by the Stieltjes inversion
@@ -1771,92 +1777,17 @@ $u=e_v-e_y$ and $v=e_w-e_x$. Substituting the expressions for $X,Y$
 proves \eqref{eq:switch-entries}.
 \end{proof}
 
-\subsection{The adjacent-entry average, exactly}\label{sec:limits-row}
-
-The resolvent equation is itself a source of exact finite-graph
-identities, and one of them already produces the tree formula with the
-true $m_n$ inserted. No probability is involved.
-
-\begin{lemma}[Row identity]\label{lem:row-identity}
-For $\zeta\in\C_+$, every vertex $v$, and $G=(A-\zeta)^{-1}$,
-\begin{equation}\label{eq:row-identity}
-  \sum_{y\sim v}G_{vy}=1+\zeta\,G_{vv}.
-\end{equation}
-Summing \eqref{eq:row-identity} over all vertices and using the
-symmetry of $G$ gives
-\begin{equation}\label{eq:edge-average-identity}
-  \frac1{nd}\sum_{v}\sum_{y\sim v}G_{vy}
-  =\frac{1+\zeta m_n(\zeta)}{d},
-  \qquad m_n(\zeta)=\frac1n\Tr G .
-\end{equation}
-\end{lemma}
-
-\begin{proof}
-The $v$-th diagonal entry of $(A-\zeta)G=I$ is
-$\sum_{y\sim v}G_{vy}-\zeta G_{vv}=1$. Summation over $v$ counts every
-unordered edge twice, matching the factor $2$ in $nd$, and
-$\sum_vG_{vv}=\Tr G$.
-\end{proof}
-
-\begin{proposition}[Adjacent-entry average]\label{prop:adjacent-average}
-Fix a compact set of heights with $\Im\zeta\ge y>0$ and $\Re\zeta$ in
-a fixed bounded interval. On the graph event of
-Proposition~\ref{prop:spectral-input},
-\begin{equation}\label{eq:lim-adjacent-average}
-  \frac1{nd}\sum_{v}\sum_{y\sim v}G_{vy}
-  =\frac{1+\zeta m_n(\zeta)}{d}
-  \qquad\text{and}\qquad
-  \frac1{nd}\sum_{v}\sum_{y\sim v}G_{vy}
-  =\frac{1+\zeta m_d(\zeta)}{d}+O(\varepsilon_{\rm loc}),
-\end{equation}
-where $\varepsilon_{\rm loc}$ is the resolvent error of
-Proposition~\ref{prop:spectral-input}. In particular, on that event,
-\begin{equation}\label{eq:lim-mn-tree}
-  m_n(\zeta)=m_d(\zeta)+O(\varepsilon_{\rm loc}),
-\end{equation}
-and the
-average of $G_{vy}$ over a uniform oriented edge equals the tree value
-$\bigl(1+\zeta m_d(\zeta)\bigr)/d$ up to the same error.
-\end{proposition}
-
-\begin{proof}
-The first identity is \eqref{eq:edge-average-identity}. For the
-comparison with the tree, apply the entrywise estimate of
-Proposition~\ref{prop:spectral-input} to the pairs $(v,y)$ with $v$ a
-uniform vertex and $y$ a uniform neighbour of $v$: at most a
-polynomially small fraction of such pairs fails, and on the rest
-$G_{vy}$ differs from the infinite-tree entry by
-$\varepsilon_{\rm loc}$. Summing, and replacing the tree adjacent entry
-by $\bigl(1+\zeta m_d(\zeta)\bigr)/d$ via
-Lemma~\ref{lem:tree-identities}, gives the second display. Comparing the
-two displays proves \eqref{eq:lim-mn-tree}. Finally the trace estimate of
-the same proposition gives $m_n=m_d+O(\varepsilon_{\rm loc})$ directly,
-consistently.
-\end{proof}
+\subsection{Why the switch is needed}\label{sec:limits-row}
 
-\begin{remark}\label{rem:why-switch}
-Proposition~\ref{prop:adjacent-average} is exact in its first display and
-therefore carries no information beyond the local law: it says that the
-average adjacent entry equals the tree formula with the graph's own
-$m_n$, and the local law already says both sides equal the tree formula
-with $m_d$. The switch matters at the next order. Applying
-Lemma~\ref{lem:switch-identity} and using the exact row identity to
-eliminate the sums over neighbours turns
-$\EE G'_{xy}=\EE G_{xy}$ into a relation involving \emph{squares} of
-resolvent entries, namely
-\begin{equation}\label{eq:lim-quadratic-relation}
-  \zeta\,q_0(\zeta)-1=(d^2-1)\int \widehat G(v,y)^2\,\dd\mu(y)
-  +O(\varepsilon_{\rm loc}),
-\end{equation}
-where $q_0(\zeta)$ is the edge-averaged entry of
-Proposition~\ref{prop:adjacent-average}, $\mu$ is the uniform measure on
-vertices, and $\widehat G(v,y)$ is the averaged entry over pairs whose
-distance from $(v,y)$ is at most two. This is the closed relation: the
-right side is a quadratic functional of the resolvent, and in the
-microscopic limit it becomes a multiple of $q_0^2$ because the
-eigenvector products it contains are Gaussian with second moment one by
-Proposition~\ref{prop:edge-moments}.
-\end{remark}
+The local law already controls the trace $m_n$ and the adjacent resolvent
+entry along an edge. What it does not produce is a closed equation that
+would pin down $m_n$ at a single microscopic energy: those estimates are
+smoothed, so they see averages over windows and cannot resolve individual
+spacings. A two-edge switch fills that gap. It is an exact symmetry of
+the uniform graph ensemble, and Woodbury's formula converts the graph
+identity into a matrix relation among resolvent entries at the sampled
+endpoints. The next subsection extracts the scalar equation from that
+relation.
 
 \subsection{Resolvents at sampled edge endpoints}\label{sec:limits-endpoints}
 
@@ -1931,12 +1862,14 @@ $(\Im z)^{-c_2}$ bounds the failure by $(\Im z)^{-j(1-2c_2)}$, and summing
 over the dyadic heights gives $C_{j,k,C}y^{-j/2}+o(1)$, so large $j$
 proves the claim.
 
-\emph{A switch gives the scalar equation.} Use the two-edge switch of
-Lemma~\ref{lem:switch-identity}, exchanging the ending vertices of the
-first two sampled edges. With probability $1-O_k(n^{-1})$ the $2k$
-endpoints are distinct and induce exactly the sampled matching; on that
-event the switch is a measure-preserving involution of the restricted
-graph/list space. The rank-two update changes the eigenvalue counting
+\emph{A switch gives the scalar equation.} Apply the cross-switch that
+exchanges the ending vertices of the first two sampled edges: the pairs
+$(v_1,w_1)$ and $(v_2,w_2)$ become $(v_1,w_2)$ and $(v_2,w_1)$. This is
+a different involution from Lemma~\ref{lem:switch-identity}, but the
+Woodbury computation below applies to it identically. With probability
+$1-O_k(n^{-1})$ the $2k$ endpoints are distinct and induce exactly the
+sampled matching; on that event the switch is a measure-preserving
+involution of the restricted graph/list space. The rank-two update changes the eigenvalue counting
 function by a bounded amount, so
 \begin{equation}\label{eq:lim-rank-trace}
  |\widehat m_n(z)-m_n(z)|\le C/\Im z .
@@ -2144,6 +2077,16 @@ Gaussian, while genuine graph switches preserve its law. Combining the two
 gives exact orthogonal switching symmetries; composing many small
 orthogonal switches then inserts a Gaussian matrix perturbation.
 
+\begin{remark}[In plain words]
+Section~\ref{sec:limits} leaves us with the microscopic limiting law and
+its Gaussian endpoint marks. Those marks give a symmetry that a general
+law would not have: conjugating $M_k$ by an orthogonal $Q_U$ preserves
+the law. Composing many small conjugations then produces an honest GOE
+perturbation: the natural graph trace $\mathcal S_r(z)$ is approximated,
+in law and up to $o(1)$, by $m_{\rm r} + \Tr(H_r-z)^{-1}$, where $H_r$
+is a fixed diagonal plus independent GOE noise.
+\end{remark}
+
 \subsection{From permutations to orthogonal switches}
 
 Fix $k\ge2$, write $M=M_k$, and for $O\in O(k)$ set
@@ -2254,7 +2197,9 @@ are
  \sum_{|x_j|>R}\frac1{x_j^2}=O(R^{-1}),
 \end{equation}
 the last two by partial summation on the two half-lines. Cut
-$M_L(z)=M^0+\sum_{|x_j|\le L}C_k^{1/2}g_jg_j^\top C_k^{1/2}(x_j-z)^{-1}$.
+\[
+  M_L(z)=M^0+\sum_{|x_j|\le L}C_k^{1/2}g_jg_j^\top C_k^{1/2}(x_j-z)^{-1}.
+\]
 For fixed $z\in\C_+$,
 \begin{equation}\label{eq:sw-first-tail}
  \|M-M_L\|+\|M'-M_L'\|=O_{\PP}(r^2L^{-\chi}),\qquad \|M'\|=O_{\PP}(r^2),
@@ -2467,6 +2412,18 @@ Proposition~\ref{prop:gaussian-insertion} approximates every microscopic
 subsequential law by the eigenvalues of a finite diagonal matrix plus GOE
 noise. We now apply fixed-energy universality to these finite matrices.
 
+\begin{remark}[In plain words]
+Section~\ref{sec:switching} reduces the microscopic limit to a finite
+matrix $H_r=$ diagonal $+$ GOE noise. To identify its eigenvalue law, we
+apply fixed-energy Dyson Brownian motion universality
+(Landon--Sosoe--Yau): a diagonal matrix perturbed by a small GOE noise
+has, in the bulk, the same local eigenvalue statistics as pure GOE,
+provided the free-convolution comparison density at the origin is
+$1/\pi$. Lemma~\ref{dbm:regularity-density} verifies the regularity
+hypotheses, and the $c_r$ dilation adjusts that density to exactly
+$1/\pi$.
+\end{remark}
+
 \subsection{The fixed-energy input}
 
 For a real diagonal $V=\diag(v_1,\dots,v_m)$ write