diff --git a/goe_rrg_bulk_universality_v2.txt b/goe_rrg_bulk_universality_v2.txt
index ed994f9..7506b09 100644
--- a/goe_rrg_bulk_universality_v2.txt
+++ b/goe_rrg_bulk_universality_v2.txt
@@ -1497,5 +1497,1221 @@ whose required derivatives are bounded by polynomials in
 $\|y\|,\|b_0\|$ times $C\exp(Cs\|b_0\|\|y\|)$; since
 $\|b_0\|\le K_n=n^{o(1)}$ and $u=n^{-\tau+o(1)}$, their
 $L^3(\varphi_D)$ norms are $n^{o(1)}$, uniformly over the source
-values.
+values. Consequently
+\[
+  \int_D|f_u-1|^3\,\dd\varphi_D\le Cu^3n^{o(1)} ,
+\]
+and, since $|(1+h)\log(1+h)-h-h^2/2|\le C|h|^3$ for $|h|\le\tfrac12$,
+while $\int(f_u-1)\dd\varphi_D=0$, we get
+\begin{equation}\label{eq:ent-chi-square}
+  \Ent(\Law(Z_u)\mid\varphi_D)
+  =\frac12\int_D(f_u-1)^2\dd\varphi_D+O(u^3n^{o(1)}).
+\end{equation}
+
+\emph{Extracting the $j$-th Hermite coefficient.} Orthogonality of the
+Hermite tensors gives, for the squared norm above,
+\[
+  \int_D(f_u-1)^2\dd\varphi_D
+  =u^{2j}\frac{\|H_j\|^2}{j!}
+   +\sum_{k\ne j}u^{2k}\frac{\|H_k\|^2}{k!} .
+\]
+The terms with $k<j$ are $n^{-2k\tau}O(n^{-2c_k})$, hence polynomially
+small. For the whole sum we have a cruder bound that suffices: the
+left side is a $\chi^2$ divergence, and conditionally on $B_n=b_0$ the
+density ratio is
+$\rho_{b_0,u}(y)=(1-u^2)^{-\dim D/2}
+ e^{\langle y,\mu\rangle-\|\mu\|^2/2}
+ e^{-u^2\|y\|^2/(2(1-u^2))}$
+with $\mu=ub_0/(1-u^2)$; a Gaussian integral then gives
+$\chi^2=O\bigl(u^2(\dim D+\|b_0\|^2)\bigr)$ for small $u$, whose mean is
+$O(u^2)$, because $\EE\|B_n\|^2=\dim D+o(1)$. Therefore
+\begin{equation}\label{eq:ent-relative-expansion}
+  \Ent(\Law(Z_u)\mid\varphi_D)
+  =\frac{u^{2j}}{2}\frac{\|H_j\|^2}{j!}+O(n^{-2\tau+o(1)}),
+\end{equation}
+and the same identity holds on $U_i$ with $H_j$ replaced by
+$P_i^{\otimes j}H_j$.
+
+\emph{The quadratic-form inequality.} Substitute
+\eqref{eq:ent-relative-expansion} and its projected analogue into
+\eqref{eq:ent-relative-deficit}, multiply by $2j!/u^{2j}$, and use
+$u^{-2j}=n^{2j\tau+o(1)}$. The result is
+\begin{equation}\label{eq:ent-quadratic-small}
+  \|H_j\|^2-\frac12\sum_{i=1}^d\bigl\|P_i^{\otimes j}H_j\bigr\|^2
+  \le n^{-c_*+2j\tau+o(1)}\le n^{-\delta}
+\end{equation}
+for some $\delta>0$, because $2j\tau<c_*/2$.
+
+\emph{Returning to the kernel.} Let $K$ be the common kernel, i.e.\ the
+closed subspace of symmetric $j$-tensors satisfying
+$\sigma_i^{\otimes j}P_i^{\otimes j}H=P_i^{\otimes j}H$ for every $i$.
+Write $L_i=(\sigma_i^{\otimes j}-\Id)P_i^{\otimes j}$, so that
+$L_iH_j$ has norm $O(n^{-c_*})$ by hypothesis, and note that
+$K=\bigcap_i\ker L_i$. The equality analysis of
+Lemma~\ref{ent:tensor} shows precisely that no nonzero tensor in
+$K^{\perp}$ is annihilated by all the $L_i$; since $D^{\otimes j}$ is
+finite-dimensional and the $L_i$ are fixed, there is $c=c(d,E,j)>0$
+with
+\begin{equation}\label{eq:ent-kernel-distance}
+  \sum_{i=1}^d\|L_iH\|^2\ \ge\ c\;\operatorname{dist}(H,K)^2
+\end{equation}
+for every symmetric $H$. Hence $\operatorname{dist}(H_j,K)=O(n^{-c_*})$.
+Let $H$ be the orthogonal projection of $H_j$ onto $K$ and $H'$ the
+complement. The quadratic form
+$Q(X)=\|X\|^2-\frac12\sum_i\|P_i^{\otimes j}X\|^2$ is bounded on
+$D^{\otimes j}$, so
+\[
+  |Q(H_j)-Q(H)|\le C\|H_j+H\|\,\|H'\|\le n^{-c_*+o(1)},
+\]
+where $\|H_j\|=n^{o(1)}$ was used. By
+\eqref{eq:ent-quadratic-small}, $Q(H)\le n^{-\delta'}$ for some fixed
+$\delta'>0$, and by Lemma~\ref{ent:tensor},
+$Q(H)\ge c_j\|H\|^2$. Therefore
+\[
+  \|H_j\|\le\|H\|+\|H'\|=O\bigl(n^{-\delta'/2}\bigr)+O(n^{-c_*})
+  =O(n^{-c_j})
+\]
+with a new fixed $c_j>0$. This closes the induction.
+
+\emph{Polynomial moments.} Every monomial of degree at most $q$ equals
+$\ip{T}{y^{\otimes j}}$ for some fixed symmetric tensor $T$ and some
+$j\le q$, and $y^{\otimes j}$ differs from
+$\operatorname{He}_j(y)$ by lower-order polynomial terms that are
+already controlled. Hence
+$\EE\ip{T}{B_n^{\otimes j}}=\ip{T}{H_j}+o(n^{-c})$ equals its Gaussian
+value in every fixed degree. The constants in the bounds above depend
+only on $d,E,j,c_*$ and the fixed error constants; the fact that the
+choice of $\epsilon$ depends on $j$ is harmless because only finitely
+many $j\le q$ occur. This proves the lemma.
+\end{proof}
+
+\begin{proof}[Proof of Proposition~\ref{prop:edge-moments}]
+Condition on a graph in the common event of Lemma~\ref{ent:type-count}
+and Proposition~\ref{prop:spectral-input}, and let $B_n=B_V$, the
+projected star field of Section~\ref{sec:entropy-projection}. Complete
+delocalization gives the deterministic bound $\|B_n\|\le n^{o(1)}$.
+Lemma~\ref{ent:projected-entropy} supplies the entropy hypothesis, the
+first two moments $\EE B_n=0$ and
+$\EE B_nB_n^{\mathsf T}=\Id_D+O(n^{-1+o(1)})$, and the moment
+symmetries $(\sigma_i^{\otimes j}-\Id)\EE(P_iB_n)^{\otimes j}=O(n^{-1+o(1)})$
+for every fixed $j$, all uniformly over the selected eigenindices and
+in $0\le\epsilon\le1$. Averaging \eqref{eq:ent-projected-entropy} over
+$i$ gives the criterion's entropy hypothesis, with the harmless factor
+$d$.
+Lemma~\ref{ent:gaussian-criterion}, applied with $q$ replaced by
+$\max(q,2)$, now gives that every moment of $B_n$ of degree at most $q$
+differs from the corresponding standard Gaussian moment in $D$ by
+$O(n^{-c})$, uniformly over the choices.
+\end{proof}
+
+\emph{From the projected star to the edge values.} The observations of
+the standard Gaussian $G$ along $w$ and $v_i$ form a $2\ell$-vector
+with covariance $C_\ell$, because $G$ has covariance $\Id_D$ in the
+$d\ell$ dimensions of $D$ and $\ip{w}{v_i}=p$. It remains to compare
+these coordinates with the actual edge values
+$\bigl(V(v),V(v')\bigr)\in\R^{\ell}\oplus\R^{\ell}$ of Section
+\ref{sec:entropy-count}. Write $B_V=(Z,\frac1{\sqrt a}PX)$ as in
+Section~\ref{sec:entropy-projection}. Then
+\[
+  \ip{v_i}{B_V}
+  =p\ip{w}{B_V}+\ip{Pe_i}{PX}
+  =pZ+X_i-\frac1d\sum_{j=1}^dX_j ,
+\]
+and $\sum_jX_j=R_{\rm res}+EZ$ by definition of the residual, so
+\[
+  \ip{v_i}{B_V}
+  =X_i+\Bigl(p-\frac Ed\Bigr)Z-\frac1dR_{\rm res} .
+\]
+Here $p=E/d$ \emph{exactly}, and $|R_{\rm res}|=O(n^{-1+o(1)})$
+because $|\lambda-E|\le S^2/n$ and $|V(v)|=n^{o(1)}$. Hence the edge
+value vector equals the $(w,v_i)$-observations of $B_V$ up to
+$O(n^{-1+o(1)})$ in every coordinate, and since all coordinates are at
+most $n^{o(1)}$ in absolute value, every fixed coordinate monomial
+changes by $O(n^{-1+o(1)})$. This is the only place where the exact
+identity $p=E/d$ enters, and it is
+essential.
+
+\emph{Several edges.} For $k$ conditionally independent uniform
+oriented edges, a coordinate monomial factors over the edges, because
+the edges are chosen independently and the eigenvectors are fixed once
+the graph is. The one-edge estimate, applied with $\ell$ channels,
+therefore gives the product estimate with an error of order
+$k\,n^{-c+o(1)}$, and $k$ is fixed. This concludes the proof of
+Proposition~\ref{prop:edge-moments}.
+
+\begin{remark}\label{rem:entropy-cost}
+The section has produced Gaussian moments at \emph{one} spectral
+energy $E$, for a fixed number of eigenvectors, fixed number of edges
+and fixed moment order, with a uniform error $n^{-c}$ over graph
+dependent choices of eigenvectors. Nothing here is uniform in the
+moment order or the number of edges, and nothing here applies to a
+window of energies; both limitations are consistent with our later use
+and are the reason the proof strategy lets $n\to\infty$ before any of
+those parameters grow.
+\end{remark}
+
+\section{From smoothed counts to one energy}\label{sec:limits}
+
+We now have Gaussian moments at sampled edges. Two obstacles remain.
+First, all of Section~\ref{sec:entropy} concerns a graph on which we
+already know the local law and rigidity; those are statements about
+\emph{smoothed} eigenvalue counts, and at the end we must pass to an
+unsmoothed statement at the single energy $E$. Second, moments of
+eigenvector entries are not yet a point process. This section solves
+both: it constructs the microscopic limit at $E$ and identifies its
+unmarked marginal with the Kesten--McKay density in the correct
+microscopic coordinates.
+
+The only tool is an exact symmetry of the uniform graph ensemble, the
+two-edge switch, which we turn into a self-consistency identity for the
+resolvent at a fixed height. It is worth saying what that identity says
+in one line:
+\begin{equation}\label{eq:switch-slogan}
+  \text{average over edges of } G_{vw}
+  \ =\ \frac{1+\zeta m_n(\zeta)}{d} + \text{small},
+\end{equation}
+where $m_n(\zeta)=\frac1n\Tr(A-\zeta)^{-1}$. The right side is the
+Kesten--McKay (tree) formula with the \emph{actual} $m_n$ inserted
+rather than the deterministic $m_d$. Such an identity, valid at every
+fixed height, is exactly what a stochastic Airy-type argument normally
+needs and normally lacks; it is available here because the switch is an
+exact graph operation, not an approximation.
+
+\subsection{The switch}
+
+Fix four distinct vertices $v,w,x,y$ such that $(v,x)$ and $(w,y)$ are
+edges and $(v,w)$, $(x,y)$ are not. The \emph{switch} deletes the two
+edges and inserts $(v,w)$ and $(x,y)$. The result is again a simple
+$d$-regular graph on the same vertex set: degrees are preserved, the new
+edges join previously non-adjacent pairs, and the deleted edges are
+distinct. Moreover the switch is invertible, the inverse switch being
+the same operation applied to the new edge pair; hence the switch is an
+involution of the finite set of simple $d$-regular graphs, and a uniform
+graph pushed forward by the switch is again uniform. In particular, for
+every function $F$ of the graph alone,
+\begin{equation}\label{eq:switch-invariance}
+  \EE F(A)=\EE F(A+Z),
+\end{equation}
+where $Z$ is the matrix of the switch. If the switch is not defined at a
+sampled quadruple, both sides are interpreted by applying the identity
+to the identity operation; the undefined event has probability bounded
+away from one and its contribution is controlled by the same error
+estimates as below. Exactness holds on the event that the sampled pair
+really is switchable.
+
+The defining convenience of the switch is that $Z$ has rank two in the
+following explicit sense. Put
+\[
+  u=e_v-e_y,\qquad v=e_w-e_x,\qquad
+  Z=uv^{\mathsf T}+vu^{\mathsf T}.
+\]
+Expanding,
+\[
+  Z=e_ve_w^{\mathsf T}+e_we_v^{\mathsf T}+e_xe_y^{\mathsf T}
+    +e_ye_x^{\mathsf T}
+    -e_ve_x^{\mathsf T}-e_xe_v^{\mathsf T}
+    -e_we_y^{\mathsf T}-e_ye_w^{\mathsf T},
+\]
+so $Z$ deletes $(v,x)$ and $(w,y)$ and inserts $(v,w)$ and $(x,y)$,
+as required. Because $Z$ has rank at most two, the resolvent equation
+has an explicit elementary solution.
+
+\begin{lemma}[Rank-two resolvent identity]\label{lem:switch-identity}
+Let $W=A-\zeta$ with $\Im\zeta>0$, $G=W^{-1}$, $G'=(W+Z)^{-1}$, and
+\[
+  A_{uv}=v^{\mathsf T}Gu,\qquad A_{uu}=u^{\mathsf T}Gu,\qquad
+  A_{vv}=v^{\mathsf T}Gv,\qquad
+  D=(1+A_{uv})^2-A_{uu}A_{vv}.
+\]
+Then $D\neq0$ and
+\begin{align}\label{eq:switch-entries}
+  G'_{xy}
+  &=G_{xy}-D^{-1}\Bigl[(G_{xv}-G_{xy})
+      \bigl[(1+A_{uv})(G_{yw}-G_{yx})-A_{vv}(G_{yv}-G_{yy})\bigr]
+      \notag\\
+  &\hspace{22mm}+(G_{xw}-G_{xx})
+      \bigl[(1+A_{uv})(G_{yv}-G_{yy})-A_{uu}(G_{yw}-G_{yx})\bigr]
+      \Bigr],
+\end{align}
+and the same formula with $(x,y)$ replaced by $(v,w)$ throughout gives
+$G'_{vw}$.
+\end{lemma}
+
+\begin{proof}
+From $(W+uv^{\mathsf T}+vu^{\mathsf T})G'=I$ we get
+\[
+  G'=G-Guv^{\mathsf T}G'-Gvu^{\mathsf T}G' .
+\]
+Set $X=v^{\mathsf T}G'$ and $Y=u^{\mathsf T}G'$, row vectors. Multiplying
+the displayed identity on the left by $v^{\mathsf T}$ and by
+$u^{\mathsf T}$ gives the linear system
+\[
+  \begin{pmatrix}1+A_{uv}&A_{vv}\\ A_{uu}&1+A_{uv}\end{pmatrix}
+  \begin{pmatrix}X\\ Y\end{pmatrix}
+  =\begin{pmatrix}v^{\mathsf T}G\\ u^{\mathsf T}G\end{pmatrix},
+\]
+whose determinant is $D$. Since $\Im G$ is negative definite and $Z$ is
+symmetric with zero exponential type, $W+Z$ has its spectrum in the open
+upper half-plane, so $G'$ exists; consistency of the system gives
+$D\ne0$ (otherwise $W+Z$ would have a null vector with real
+eigenvalue). Cramer's rule gives
+\[
+  X=D^{-1}\bigl[(1+A_{uv})v^{\mathsf T}G-A_{vv}u^{\mathsf T}G\bigr],\quad
+  Y=D^{-1}\bigl[(1+A_{uv})u^{\mathsf T}G-A_{uu}v^{\mathsf T}G\bigr].
+\]
+Now take the $(x,y)$ entry of $G'=G-Guv^{\mathsf T}G'-Gvu^{\mathsf T}G'$:
+\[
+  G'_{xy}=G_{xy}-(Gu)_x(e_y^{\mathsf T}v^{\mathsf T}G')
+                  -(Gv)_x(e_y^{\mathsf T}u^{\mathsf T}G'),
+\]
+and $e_y^{\mathsf T}v^{\mathsf T}G'=(G'v)_y=(X)_y$,
+$e_y^{\mathsf T}u^{\mathsf T}G'=(Y)_y$, while
+$(Gu)_x=G_{xv}-G_{xy}$ and $(Gv)_x=G_{xw}-G_{xx}$ because
+$u=e_v-e_y$ 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}
+
+\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}
+
+\subsection{Resolvents at sampled edge endpoints}\label{sec:limits-endpoints}
+
+Conditionally on the graph, sample an infinite sequence of independent
+uniform oriented edges. For fixed $k\ge1$, let $W_{n,k}$ be the $n\times2k$
+matrix whose columns are the coordinate vectors of the first $k$
+starting vertices followed by their $k$ ending vertices. Define
+\begin{equation}\label{eq:lim-endpoint-resolvent}
+ M_{n,k}(z)=W_{n,k}^{\mathsf T}\Bigl(A-E-\frac zn\Bigr)^{-1}W_{n,k}
+ =\sum_i\frac{b_ib_i^{\mathsf T}}{x_i-z},\qquad
+ b_i=W_{n,k}^{\mathsf T}u_i,\quad \Im z>0 .
+\end{equation}
+The normalization $\|u_i\|^2=n$ accounts for the missing factor $n^{-1}$.
+We show that, at any fixed microscopic height, the scalar $m_n$ and the
+matrix $M_{n,k}$ concentrate on their tree values $m_*$ and $K_k(m_*)$.
+
+\begin{proposition}[Endpoint control]\label{prop:endpoint-control}
+There is a deterministic $\xi>0$, depending only on $d,E$, such that for
+every fixed $k\ge1$ and $D>0$ there are constants $C_{k,D}<\infty$ and
+$y_{k,D}<\infty$ with the following property. For every fixed $s\in\R$
+and every fixed $y\ge y_{k,D}$,
+\begin{equation}\label{eq:lim-endpoint-control}
+ \limsup_{n\to\infty}\PP\Bigl(
+ |m_n(s+\mathrm iy)-m_*|
+ +\|M_{n,k}(s+\mathrm iy)-K_k(m_*)\|>y^{-\xi}\Bigr)\le C_{k,D}\,y^{-D}.
+\end{equation}
+The constants and the threshold do not depend on $s$, and $s+\mathrm iy$
+is fixed before the limit.
+\end{proposition}
+
+\begin{proof}
+It suffices to treat $k\ge2$, since smaller matrices are compressions;
+throughout $k,s,y$ are fixed and error exponents may decrease from one
+occurrence to the next.
+
+\emph{Initial estimates.} At $z_0=s+\mathrm iS$,
+Proposition~\ref{prop:spectral-input} and Lemma~\ref{lem:tree-identities}
+give, for some $c>0$,
+\begin{equation}\label{eq:lim-initial-estimates}
+ m_n(z_0)=m_*+O(S^{-c}),\quad M_{n,k}(z_0)=K_k(m_*)+O(S^{-c}),\quad
+ \Im m_n(\mathrm iS^2)+\|\Im M_{n,k}(\mathrm iS^2)\|=O(1).
+\end{equation}
+For the off-block entries, independent uniform vertices $v,w$ satisfy
+$\EE_{v,w}|R_{vw}|^2=n^{-2}\Tr R^*R=\Im m_n(z_0)/S$, so Markov bounds
+them by $O(S^{-c})$. Endpoints of distinct sampled edges are independent
+uniform vertices, and within one edge the typical-star estimate gives the
+adjacent entry $(1+Em_*)/d$. Also $\#\{|x_i|\le S^2\}=O(S^2)=n^{o(1)}$.
+
+\emph{Concentration under a trace bound.} Descend from height $S$ to $y$
+by halving the imaginary part at fixed real part $s$; let $\mathcal Z_n$
+be the $O(\log S)$ dyadic arguments. For every fixed $C<\infty$, outside
+an event of limiting probability $O_{k,D,C}(y^{-D})$, simultaneously for
+$z\in\mathcal Z_n$,
+\begin{equation}\label{eq:lim-concentration-implication}
+ |m_n(z)|\le C\ \Longrightarrow\ M_{n,k}(z)=K_k(m_n(z))+O\bigl((\Im z)^{-c_2}\bigr),
+\end{equation}
+with $c_2>0$ independent of $D,k,s,C$. To see this, compare with $z_0$:
+for $|x|>S^2$ and large $S$,
+$|(x-z)^{-1}-(x-z_0)^{-1}|\le CSx^{-2}\le CS^{-1}\Im(x-\mathrm iS^2)^{-1}$,
+so the outside-band parts are $O(S^{-1})$ (positive residues make the
+matrix bound immediate). Inside the band, with
+$w_i(z)=(x_i-z)^{-1}-(x_i-z_0)^{-1}$, a bounded trace gives
+$\sum_{|x_i|\le S^2}|w_i(z)|^2\le 2\Im m_n(z)/\Im z+2\Im m_n(z_0)/S\le C'/\Im z$.
+Replacing the marks by independent $N(0,C_k)$ vectors, each entry of
+$\sum(b_ib_i^{\mathsf T}-C_k)w_i(z)$ has $2j$-th absolute moment
+$O_{j,k,C}((\Im z)^{-j})$ (any spectral index occurring once has zero
+expectation; the remaining partitions are bounded by Gaussian moments and
+the weight squares). Conditionally on a graph satisfying the fixed-order
+moment estimates, the same holds up to polynomial error, with only
+$n^{o(1)}$ terms. Pick $0<c_2<\min(c,1/4)$; Markov at threshold
+$(\Im z)^{-c_2}$ 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
+function by a bounded amount, so
+\begin{equation}\label{eq:lim-rank-trace}
+ |\widehat m_n(z)-m_n(z)|\le C/\Im z .
+\end{equation}
+Write the first two pairs as $(v_1,w_1),(v_2,w_2)$ and set
+$f_1=(e_{v_1}-e_{v_2})/\sqrt2$, $f_2=(e_{w_1}-e_{w_2})/\sqrt2$,
+$F=(f_1,f_2)$, $B=\begin{pmatrix}0&-2\\-2&0\end{pmatrix}$; the switch adds
+$FBF^{\mathsf T}$, and the resolvent identity reads
+$(I+UB)\widehat U=U$ with $U=F^{\mathsf T}(A-E-z/n)^{-1}F$. Assuming
+$|m_n(z)|\le C$ and applying \eqref{eq:lim-concentration-implication} and
+\eqref{eq:lim-rank-trace} to both data sets,
+\begin{equation}\label{eq:lim-switch-compressions}
+ U=\begin{pmatrix}m&q\\q&m\end{pmatrix}+O\bigl((\Im z)^{-c_2}\bigr),\quad
+ \widehat U=\begin{pmatrix}m&-q\\-q&m\end{pmatrix}+O\bigl((\Im z)^{-c_2}\bigr),
+ \quad q=\frac{1+Em}{d},\quad m=m_n(z),
+\end{equation}
+the sign change in $\widehat U$ coming from expressing the updated pairs
+in the old columns. The off-diagonal of $(I+UB)\widehat U=U$ gives
+\begin{equation}\label{eq:lim-scalar-equation}
+ q^2-q-m^2=O\bigl((\Im z)^{-c_2}\bigr).
+\end{equation}
+As a quadratic in $m$ this has simple conjugate roots in the strict bulk;
+by Lemma~\ref{lem:tree-identities} the upper root is $m_*$ and the lower
+stays a fixed distance from $\{\Im m\ge0\}$. Hence, in the bounded range,
+\begin{equation}\label{eq:lim-scalar-stability}
+ m_n(z)=m_*+O\bigl((\Im z)^{-c_2}\bigr).
+\end{equation}
+
+\emph{Closing the descent.} No trace bound is assumed at lower steps.
+Instead, for $\eta/2\le\eta'\le\eta$, term by term
+$|x-s-\mathrm i\eta'|\ge(\eta'/\eta)|x-s-\mathrm i\eta|$, so
+\begin{equation}\label{eq:lim-halving-bound}
+ |m_n(s+\mathrm i\eta')-m_n(s+\mathrm i\eta)|
+ \le\frac{\eta-\eta'}{\eta'}\Im m_n(s+\mathrm i\eta)\le\Im m_n(s+\mathrm i\eta).
+\end{equation}
+Choose a fixed trace range containing the initial value and every
+next-step value of \eqref{eq:lim-halving-bound} whenever the preceding
+value is within $1$ of $m_*$, enlarged once more for the switched trace
+via \eqref{eq:lim-rank-trace}; take $y$ large enough that
+\eqref{eq:lim-scalar-stability} returns within $1$ of $m_*$. Induction
+down the heights proves the trace bounds and the endpoint estimates at
+every step. Absorbing fixed constants into a smaller $\xi$ proves
+\eqref{eq:lim-endpoint-control}; the only $s$-dependence is the harmless
+requirement $S\ge C(|s|+y+1)$, which holds before the limit.
+\end{proof}
+
+\subsection{The limiting marked process}\label{sec:limits-microscopic}
+
+For a locally finite point measure $\nu=\sum_j\delta_{x_j}$, write $\PV$
+for the symmetric truncation $\lim_{R\to\infty}\sum_{|x_j|\le R}$.
+
+\begin{proposition}[Microscopic limits]\label{prop:microscopic-limit}
+From every sequence of admissible sizes tending to infinity one can
+extract a subsequence along which
+\[
+ (\nu_n,m_n,(M_{n,k})_{k\ge1})\Longrightarrow(\nu,m,(M_k)_{k\ge1})
+\]
+jointly, in the local vague topology for point measures and the
+compact-open topology for analytic functions. The limit $\nu=\sum_j\delta_{x_j}$
+is locally finite, with multiplicities permitted, and there are
+deterministic $\xi,\chi>0$ such that the following hold.
+
+For every fixed $0<\delta<1$, almost surely for all sufficiently large $R$,
+\begin{equation}\label{eq:mesoscopic-limit}
+ m(e+\mathrm iu)=m_*+O(u^{-\xi}),\qquad |e|\le R,\ R^\delta\le u\le R^2,
+\end{equation}
+and
+\begin{equation}\label{eq:discrepancy}
+ \nu([v,w])=\rho(w-v)+O(R^{1-\chi}),\qquad -R\le v\le w\le R,\ R\to\infty .
+\end{equation}
+On a possible extension of the probability space, for every fixed $k$
+there are standard Gaussian vectors $g_j\in\R^{2k}$, independent over
+$j$ conditionally on $\nu$, such that
+\begin{equation}\label{eq:marked-representation}
+ m(z)=m_{\rm r}+\PV\sum_j\frac1{x_j-z},\qquad
+ M_k(z)=K_k(m_{\rm r})+\PV\sum_j\frac{C_k^{1/2}g_jg_j^{\mathsf T}C_k^{1/2}}{x_j-z}.
+\end{equation}
+These representations are consistent as $k$ grows by keeping the first
+$k$ channels, a separate mark is assigned to each occurrence of a
+repeated point, the principal values converge locally uniformly on $\C_+$,
+and $\Im M_k(z)$ is positive definite there, almost surely simultaneously
+for all finite $k$ and all $z\in\C_+$.
+\end{proposition}
+
+\begin{proof}
+\emph{Tightness.} For fixed $L,y>0$, positivity gives
+$\nu_n([-L,L])\le(L^2+y^2)y^{-1}\Im m_n(\mathrm iy)$; using
+Proposition~\ref{prop:endpoint-control} with large fixed $y$ (chosen per
+tolerance) gives tightness of the local counts. For compact-open
+tightness, fix a compact $\mathcal K\subset\C_+$ and $y>0$. For a resolvent
+sum $f$ with nonnegative residues, comparing kernels on the real line
+gives $\Im f(z)\le C_{\mathcal K,y}\Im f(\mathrm iy)$ and
+$|f(z)-f(\mathrm iy)|\le C_{\mathcal K,y}\Im f(\mathrm iy)$ for
+$z\in\mathcal K$; applying these to $m_n$ and, by polarization, to the
+real quadratic forms of $M_{n,k}$ bounds each fixed compact, and Cauchy's
+estimates give equicontinuity on smaller compacts. Extracting diagonally
+over growing compacts and channel dimensions gives the joint limit;
+local vague limits with locally bounded counts are point measures with
+integer multiplicities, and the analytic limits have nonnegative
+imaginary parts.
+
+\emph{Conditional Gaussian marks on compact intervals.} Choose nested
+compact position intervals exhausting $\R$ with deterministic boundaries
+almost surely uncharged by $\nu$ (possible since only countably many
+points have positive probability of being atoms). On each interval
+truncate the point count at a fixed integer, order the prelimit indices
+using graph data alone, and extract positions and marks. By
+Proposition~\ref{prop:edge-moments} every fixed joint moment of the marks
+is polynomially close to that of independent $N(0,C_k)$ vectors, uniformly
+over the selected indices. Diagonal selection of the graph exceptional
+events (choose $J_n\to\infty$ slowly enough that all orders at most $J_n$
+match to $J_n^{-1}$ on a common event of probability $\ge1-J_n^{-1}$; only
+finitely many fixed-order assertions enter at each stage) upgrades
+fixed-moment matching to conditional weak convergence to the product
+Gaussian law, uniformly over bounded lists: a list failing to converge
+would admit a deterministic sequence of graphs and index lists failing
+while all fixed moments converged, contradicting tightness and Gaussian
+moment determinacy. Passing to the truncation limit identifies the marking
+law given $\nu$, with covariance $C_k$ over the $2k$ endpoints; on an
+extension of the probability space one assigns an independent mark
+$b_j=C_k^{1/2}g_j$ to each occurrence of a repeated point.
+
+\emph{Removing imaginary tails.} For fixed $z\in\C_+$ and large $L$, the
+scalar kernels satisfy, for $|x|>L$,
+$\Im(x-z)^{-1}+|(x-z)^{-1}-(x-\mathrm i)^{-1}|\le C_zL^{-1}\Im(x-\mathrm iL)^{-1}$;
+for matrix sums use positivity of the residues and bound the norm by the
+trace of the positive imaginary part. By Proposition~\ref{prop:endpoint-control}
+these traces at $\mathrm iL$ are tight as $L\to\infty$ after the upper
+limit in $n$; the factor $L^{-1}$ removes the tails in probability, and
+compact marked sums converge, so
+\begin{equation}\label{eq:lim-difference-identities}
+ \begin{aligned}
+ \Im m(z)&=\sum_j\Im\frac1{x_j-z},&
+ m(z)-m(\mathrm i)&=\sum_j\Bigl(\frac1{x_j-z}-\frac1{x_j-\mathrm i}\Bigr),\\
+ \Im M_k(z)&=\sum_jb_jb_j^{\mathsf T}\Im\frac1{x_j-z},&
+ M_k(z)-M_k(\mathrm i)&=\sum_jb_jb_j^{\mathsf T}\Bigl(\frac1{x_j-z}-\frac1{x_j-\mathrm i}\Bigr).
+ \end{aligned}
+\end{equation}
+No imaginary spectral mass is lost at infinity; real additive constants
+are not yet excluded and are fixed below.
+
+\emph{Uniform bounds after the limit.} Pass
+Proposition~\ref{prop:endpoint-control} to the limit by the portmanteau
+inequality on the open event of large error (its constants are independent
+of $s$), and slightly decrease $\xi<1$. Fix $0<\delta<1$ and take a unit
+grid on the padded region $|e|\le2R$, $\tfrac12R^\delta\le u\le2R^2$,
+giving $O(R^3)$ arguments. For fixed dyadic $R$ this is a finite
+collection in the exact limit; the failure probability is at most
+$CR^{3-\delta D}$, and Borel--Cantelli over dyadic $R$ with large $D$
+controls it. The kernel identities give $|m'(z)|\le\Im m(z)/\Im z$, and
+imaginary parts at comparable height and bounded distance are comparable,
+so filling the grid costs $O(1/u)$; padding reaches all large $R$. A
+countable set of $\delta$'s gives every fixed-$\delta$ conclusion. The
+limit in $n$ preceded this growing-grid argument.
+
+\emph{From smoothing to counts.} Equation~\eqref{eq:mesoscopic-limit} gives
+$\nu([-R,R])=O(R)$. Integrating $\pi^{-1}\Im m(e+\mathrm i\sqrt R)$ over
+$e\in[v,w]$ yields $\rho(w-v)+O(R^{1-\xi/2})$. Compare with the sharp
+count using $P_{v,w}(x)=\pi^{-1}\int_v^w\frac{\sqrt R}{(x-e)^2+R}\dd e$:
+points within $R^{3/4}$ of an endpoint number $O(R^{3/4})$; on
+$[-2R,2R]$ off the two strips, $|P_{v,w}(x)-\mathbf 1_{[v,w]}(x)|=O(R^{-1/4})$,
+for total error $O(R^{3/4})$; for $|x|>2R$ the bound
+$P_{v,w}\le CR^{3/2}/x^2$ with the linear count contributes
+$O(\sqrt R)$. This gives \eqref{eq:discrepancy} for any
+$0<\chi<\min(1/4,\xi/2)$.
+
+\emph{Scalar principal values.} Partial summation in \eqref{eq:discrepancy}
+gives $\PV\sum_{|x_j|>R}x_j^{-1}=O(R^{-\chi})$ and
+$\sum_{|x_j|>R}x_j^{-2}=O(R^{-1})$ (the leading density terms cancel in
+the symmetric sum on the two half-lines). Hence
+$S_\nu(z)=\PV\sum_j(x_j-z)^{-1}$ is analytic and locally uniformly
+convergent, and comparing with the constant density gives
+$S_\nu(\mathrm iR)=\mathrm i\pi\rho+o(1)$. By
+\eqref{eq:lim-difference-identities}, $m-S_\nu$ is constant with zero
+imaginary part; the two displays identify it as $m_{\rm r}$.
+
+\emph{Marked principal values.} Fix $k$, condition on $\nu$, and set
+$Z_j=b_jb_j^{\mathsf T}-C_k$, independent centered with bounded second
+moments. Ordered by increasing $|x_j|$, $\sum_jZ_j/(x_j-\mathrm i)$
+converges almost surely entrywise, and
+$\sum_{|x_j|>1}\|Z_j\|x_j^{-2}<\infty$ conditionally almost surely, so
+the difference series between $z$ and $\mathrm i$ converges absolutely
+and locally uniformly; with the scalar PV this gives local uniform
+convergence of $S_{k,\nu}(z)=\PV\sum_jb_jb_j^{\mathsf T}(x_j-z)^{-1}$.
+The linear count gives $\sum_j|x_j-\mathrm iR|^{-2}=O(R^{-1})$, so each
+entry of the centered marked series at $\mathrm iR$ has conditional
+second moment $O(R^{-1})$, whence
+$S_{k,\nu}(\mathrm iR)\to\mathrm i\pi\rho\,C_k$ in probability. Since
+$M_k-S_{k,\nu}$ is almost surely constant in $z$ and
+$M_k(\mathrm iR)\to K_k(m_*)$ in probability, that constant equals
+$K_k(m_*)-\mathrm i\pi\rho\,C_k=K_k(m_{\rm r})$.
+
+Finally, \eqref{eq:discrepancy} supplies infinitely many points and $C_k$
+is positive definite; $2k$ independent Gaussian marks span $\R^{2k}$
+almost surely, so the positive contributions to $\Im M_k(z)$ make it
+positive definite for every $z\in\C_+$, simultaneously over a countable
+set of $k$.
+\end{proof}
+
+\section{Continuous switches and Gaussian insertion}\label{sec:switching}
+
+We work with a microscopic subsequential law $\nu$ supplied by
+Proposition~\ref{prop:microscopic-limit}. Its endpoint marks are exactly
+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.
+
+\subsection{From permutations to orthogonal switches}
+
+Fix $k\ge2$, write $M=M_k$, and for $O\in O(k)$ set
+$D_O=\begin{pmatrix}I_k&0\\0&O\end{pmatrix}$ and
+$B_O=D_OJ_kD_O^{\mathsf T}-J_k$, which is symmetric. For an analytic
+symmetric matrix function $M$ with positive definite imaginary part, put
+\begin{equation}\label{eq:sw-transformation}
+ \mathcal T_O(m,M)=\Bigl(m-\partial_z\log\det(I_{2k}+B_OM),\,
+ D_O^{\mathsf T}(M^{-1}+B_O)^{-1}D_O\Bigr).
+\end{equation}
+This is well defined: $M$ is invertible and
+$\Im(M^{-1})=-M^{-*}(\Im M)M^{-1}<0$, so $M^{-1}+B_O$ is invertible and
+its inverse has positive definite imaginary part. Only the logarithmic
+derivative is used, so no branch of the logarithm is chosen.
+
+\begin{proposition}[Continuous switching symmetry]\label{prop:continuous-switching}
+For every fixed $k\ge2$ and every $O\in O(k)$, the joint law of $(m,M_k)$
+under $\nu$ is invariant under $\mathcal T_O$.
+\end{proposition}
+
+\begin{proof}
+Let $O$ be a permutation matrix. On the event, of probability
+$1-O_k(n^{-1})$, that the $2k$ sampled endpoints are distinct and induce
+exactly the sampled matching, the map $A\mapsto A+WB_OW^{\mathsf T}$,
+$W\mapsto WD_O$ permutes the pairing between the two endpoint lists, is a
+bijection of the restricted graph/list space preserving all weights, and
+is inverted by the inverse permutation. The determinant and inverse
+identities for finite-rank perturbations give
+$\widehat m_n=m_n-\partial_z\log\det(I_{2k}+B_OM_{n,k})$ and
+$\widehat M_{n,k}=D_O^{\mathsf T}(M_{n,k}^{-1}+B_O)^{-1}D_O$ (the factor
+$n^{-1}$ in $m_n=n^{-1}\Tr R$ supplies the derivative). Compact-open
+convergence of analytic functions gives convergence of derivatives, and
+positive-definite imaginary parts make the inverse regular, so passing to
+the limit proves the claim for permutation matrices.
+
+For $U\in O(k)$ put $Q_U=\diag(U,U)$. As $C_k$ and $K_k(m_{\rm r})$
+commute with $Q_U$, the Gaussian representation
+\eqref{eq:marked-representation} shows that
+$\mathcal R_U(m,M)=(m,Q_U^{\mathsf T}MQ_U)$ preserves the joint law. Since
+$Q_UD_OQ_U^{\mathsf T}=D_{UOU^{\mathsf T}}$ and
+$Q_UB_OQ_U^{\mathsf T}=B_{UOU^{\mathsf T}}$, we get
+$\mathcal R_U^{-1}\mathcal T_O\mathcal R_U=\mathcal T_{UOU^{\mathsf T}}$;
+so the law is invariant under every orthogonal conjugate of a permutation
+switch. The maps satisfy $B_{OO'}=B_O+D_OB_{O'}D_O^{\mathsf T}$ and
+$D_{OO'}=D_OD_{O'}$, so $\mathcal T_O\mathcal T_{O'}=\mathcal T_{OO'}$
+(the inverse updates compose and the determinant ratios multiply before
+differentiation). A transposition is a hyperplane reflection; its
+orthogonal conjugates are all hyperplane reflections, and finite products
+generate $O(k)$.
+\end{proof}
+
+\subsection{The finite Gaussian model}
+
+Fix a discrepancy exponent $0<\chi<1/4$ from \eqref{eq:discrepancy}. Use
+$r$ planar rotations through a small angle $\theta$, with aggregate
+variance of order $T=r\theta^2$. A cutoff $K$ marks the finite block that
+will receive GOE noise, and a much larger temporary cutoff $L$ first turns
+the principal-value resolvent into a finite matrix. Choose once and for all
+\begin{equation}\label{eq:sw-parameters}
+ 0<\tau<\frac{\chi}{100000},\quad k=2r,\quad K=r^{1/100},\quad T=r^{\tau},
+\quad \theta=\sqrt{T/r},\quad L=r^{100/\chi},\quad r\in\N .
+\end{equation}
+For each $r$ let $O=O_r$ be the direct sum of $r$ planar rotations of angle
+$\theta$, write $B=B_{O_r}$ and $M^0=K_k(m_{\rm r})$. The real symmetric
+matrix
+\begin{equation}\label{eq:sw-coefficient}
+ \Gamma_r=C_k^{1/2}(B^{-1}+M^0)^{-1}C_k^{1/2}
+\end{equation}
+is well defined for large $r$: the singular values of $B$ are all
+$2|\sin(\theta/2)|$ and $\|M^0\|$ is bounded independently of $r$. Let
+$\beta$ be a diagonal matrix of eigenvalues of $\Gamma_r$ and set
+\begin{equation}\label{eq:sw-moments}
+ s_1=\Tr\beta,\qquad s_2=\Tr\beta^2 .
+\end{equation}
+Index the limiting positions with multiplicity, put
+$I=\{j:|x_j|\le K\}$, $q_*=|I|$ and $D_I=\diag(x_j:j\in I)$. Conditionally
+on $\nu$, let $\mathcal W$ be a raw GOE matrix of size $q_*$ (variances $1$
+off the diagonal and $2$ on it), independent of $\nu$ apart from its
+dimension, and set
+\begin{equation}\label{eq:sw-inserted-matrix}
+ H_r=D_I+s_1\Id_{q_*}+\sqrt{s_2}\,\mathcal W .
+\end{equation}
+
+\begin{proposition}[Gaussian insertion]\label{prop:gaussian-insertion}
+The coefficients satisfy
+\begin{equation}\label{eq:sw-coefficient-bounds}
+ \|\beta\|\le C\theta,\qquad cT\le s_2\le CT,\qquad |s_1|\le CT
+\end{equation}
+with deterministic positive constants. As $r\to\infty$,
+\begin{equation}\label{eq:sw-trace-limit}
+ m_{\rm r}+\Tr(H_r-z)^{-1}\ \Longrightarrow\ m(z)
+\end{equation}
+jointly at every fixed finite list of $z\in\C_+$, and the unscaled
+eigenvalue point measure of $H_r$ converges locally vaguely in law to
+$\nu$. More precisely, generating the Gaussian marks of $M_k$
+conditionally on $\nu$ separately for each $r$, and writing
+$\mathcal S_r(z)=m(z)-\partial_z\log\det(I_{2k}+BM_k(z))$, expectations of
+every bounded smooth function with bounded derivatives of finitely many
+real and imaginary trace values differ by $o(1)$ when
+$m_{\rm r}+\Tr(H_r-z)^{-1}$ is replaced by $\mathcal S_r(z)$.
+\end{proposition}
+
+\emph{First cut at $L$.} Elementary consequences of \eqref{eq:discrepancy}
+are
+\begin{equation}\label{eq:sw-tail-sums}
+ \nu([-R,R])=2\rho R+O(R^{1-\chi}),\quad
+ \Bigl|\PV\sum_{|x_j|>R}\frac1{x_j}\Bigr|=O(R^{-\chi}),\quad
+ \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
+$M_L(z)=M^0+\sum_{|x_j|\le L}C_k^{1/2}g_jg_j^\top C_k^{1/2}(x_j-z)^{-1}$.
+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),
+\end{equation}
+since the mean tail is $O(L^{-\chi})C_k$ (and $O(L^{-1})C_k$ for the
+derivative), each centered entry of the Gaussian tail has variance
+$O(L^{-1})$, and there are $(4r)^2$ entries. Keeping only $|x_j|\le r^4$
+(of which there are $\sim2\rho r^4$) and using Gaussian fourth moments
+gives
+\begin{equation}\label{eq:sw-imaginary-lower}
+ \Im M(z)\ge r^{-5}\Id_{4r},\qquad \Im M_L(z)\ge r^{-5}\Id_{4r}
+\end{equation}
+with probability tending to one (using $L\gg r^4$). As $B^{-1}$ is real
+symmetric, this bounds $\|(B^{-1}+M)^{-1}\|$ and $\|(B^{-1}+M_L)^{-1}\|$ by
+$r^5$, so cutting both scalar and marked sums in
+$\mathcal S_r(z)=m(z)-\Tr(B^{-1}+M(z))^{-1}M'(z)$ costs $o_{\PP}(1)$: the
+marked error is $O_{\PP}(r\cdot r^5\cdot r^{-98})+O_{\PP}(r\cdot r^{10}
+\cdot r^{-98}\cdot r^2)=O_{\PP}(r^{-85})$, and the scalar tail is
+$O(L^{-\chi})$. Diagonalizing $\Gamma_r$ and rotating the independent
+Gaussian row vectors in channel space gives a matrix $\mathcal G$ of
+independent standard Gaussian entries with $4r$ columns, and the
+finite-rank inverse identity (Woodbury) yields
+\begin{equation}\label{eq:sw-finite-matrix}
+ \mathcal S_r(z)=m_{\rm r}+\Tr(D+A'-z)^{-1}+o_{\PP}(1),\qquad
+ A'=\mathcal G\beta\mathcal G^\top,\quad D=\diag(x_j:|x_j|\le L).
+\end{equation}
+The coefficient bounds follow from
+$(B^{-1}+M^0)^{-1}=(\Id+BM^0)^{-1}B=B+O(\theta^2)$: every singular value
+is comparable to $\theta$, so $\|\beta\|=O(\theta)$ and $s_2\asymp
+r\theta^2=T$; and $s_1=\Tr(C_kB)+O(r\theta^2)$ with
+$\Tr(C_kB)=2p\,\Tr(O-\Id_k)=O(r\theta^2)$.
+
+\emph{Removing the outer positions.} Let $F=\{j:K<|x_j|\le L\}$ and put
+$h=T^2$, $U_I=(|D_I|+h)^{-1/2}$, $U_F=|D_F|^{-1/2}$. Partial summation
+using \eqref{eq:sw-tail-sums} gives
+\begin{equation}\label{eq:sw-weight-bounds}
+ \Tr U_I^2+\Tr U_F^2=O(\log r),\qquad \|U_I^2\|\le h^{-1},\quad
+ \|U_F^2\|\le K^{-1}.
+\end{equation}
+In the estimates below, $\operatorname{polylog}(r)$ denotes a fixed power
+of $\log r$ whose exponent may rise from one occurrence to the next.
+
+\begin{lemma}[Weighted block estimates]\label{sw:weighted-blocks}
+For $\mathcal A\in\{I,F\}$ put $a_I=h$, $a_F=K$. With probability tending
+to one,
+\begin{equation}\label{eq:sw-bernstein}
+ \bigl\|U_{\mathcal A}(A'_{\mathcal A\mathcal A}-s_1\Id)U_{\mathcal A}\bigr\|
+ \le\operatorname{polylog}(r)\bigl(\sqrt{T/a_{\mathcal A}}+\theta\bigr),
+\end{equation}
+and, for $P_0=U_IA'_{IF}U_F$,
+\begin{equation}\label{eq:sw-cross-block}
+ \|P_0\|_{\HS}^2\le T\operatorname{polylog}(r).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The centered matrix in \eqref{eq:sw-bernstein} is a sum of independent
+self-adjoint $\sum_{\alpha=1}^{4r}\beta_{\alpha\alpha}(v_\alpha
+v_\alpha^\top-P)$ with $P=U_{\mathcal A}^2$ and $v_\alpha$ Gaussian of
+covariance $P$. Gaussian fourth moments give
+$\EE[(v_\alpha v_\alpha^\top)^2]=(\Tr P)P+2P^2$. Truncating each Gaussian
+column at $|\text{entry}|\le\log r$ (a superpolynomially small event, with
+superpolynomially small centering correction) leaves summands of norm at
+most $C\|\beta\|(\log r)^2\Tr P\le\theta\operatorname{polylog}(r)$, and
+bounds the matrix variance parameter by $Cs_2(\Tr P)\|P\|\le
+(T/a_{\mathcal A})\operatorname{polylog}(r)$. The bounded matrix Bernstein
+inequality gives a two-sided tail $2|\mathcal A|\exp\bigl(-ct^2/
+(\sigma_{\rm mat}^2+bt)\bigr)$, and a large power of $\log r$ in $t$
+proves \eqref{eq:sw-bernstein}. For the cross block, independence of
+distinct Gaussian rows gives $\EE\|P_0\|_{\HS}^2=s_2(\Tr U_I^2)(\Tr U_F^2)$,
+so Markov with \eqref{eq:sw-weight-bounds} gives \eqref{eq:sw-cross-block}.
 \end{proof}
+
+Fix $z\in\C_+$ and write $G_F=(D_F+A'_{FF}-z)^{-1}$,
+$G_I=(D_I+A'_{II}-z)^{-1}$.
+
+\begin{lemma}[Far block and Schur correction]\label{sw:schur}
+Let $\delta_F=\operatorname{polylog}(r)\bigl((1+T)/K+\sqrt{T/K}+\theta\bigr)$.
+Then $\delta_F=o(1)$ and, with probability tending to one,
+\begin{equation}\label{eq:sw-far-resolvent}
+ \|U_F^{-1}(G_F-D_F^{-1})U_F^{-1}\|\le\delta_F=o(1),\qquad
+ \Tr G_F=o_{\PP}(1),
+\end{equation}
+and, for $\Sigma=A'_{IF}G_FA'_{FI}$,
+\begin{equation}\label{eq:sw-schur-small}
+ \|U_I\Sigma U_I\|=o_{\PP}(T^{-5}).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Conjugating by $U_F$,
+$U_F(D_F+A'_{FF}-z)U_F=\operatorname{sign}(D_F)+U_F(A'_{FF}-z)U_F$, and the
+second term has norm at most $\delta_F=o(1)$ by
+Lemma~\ref{sw:weighted-blocks}, $|s_1|=O(T)$ and \eqref{eq:sw-weight-bounds};
+a Neumann expansion proves \eqref{eq:sw-far-resolvent}, so
+$Q_F=U_F^{-1}G_FU_F^{-1}$ has bounded norm. Also
+$|\Tr(G_F-D_F^{-1})|\le\delta_F\Tr U_F^2=o(1)$, so $\Tr G_F=o_{\PP}(1)$.
+
+For $\Sigma$, replacing $G_F$ by $D_F^{-1}$ costs at most
+$\|P_0\|_{\HS}^2\delta_F\le T\operatorname{polylog}(r)\delta_F$ in the
+$U_I$-weighted norm. For $A'_{IF}D_F^{-1}A'_{FI}$, condition on the inner
+block and set $Q=\mathcal G_I\beta^2\mathcal G_I^\top$; the columns $a_j$
+of $A'_{IF}$, $j\in F$, are independent centered Gaussian with covariance
+$Q$, so $\EE[\sum_{j\in F}a_ja_j^\top/x_j\mid\cdot]=Q\sum_{j\in F}x_j^{-1}$
+and the entrywise variance is $(Q_{aa}Q_{bb}+Q_{ab}^2)\sum_{j\in F}x_j^{-2}$.
+With $\EE[Q_{aa}^2]=s_2^2+2\Tr\beta^4=O(T^2)$ and $|Q_{ab}|^2\le Q_{aa}Q_{bb}$,
+\eqref{eq:sw-tail-sums} and \eqref{eq:sw-weight-bounds} give
+$\EE\|U_IA'_{IF}D_F^{-1}A'_{FI}U_I\|_{\HS}^2\le C T^2(\log r)^2
+(K^{-2\chi}+K^{-1})$, so by Markov this norm is
+$T\operatorname{polylog}(r)(K^{-\chi}+K^{-1/2})$ with probability tending to
+one. Combining and multiplying by $T^5$, the exponents
+$6\tau-\chi/100$, $\tfrac{13}2\tau-\tfrac1{200}$ and
+$\tfrac{13}2\tau-\tfrac12$ are all negative under \eqref{eq:sw-parameters},
+giving \eqref{eq:sw-schur-small}.
+\end{proof}
+
+\begin{lemma}[Inner resolvent and trace cut]\label{sw:trace-cut}
+For every fixed $z\in\C_+$,
+\begin{equation}\label{eq:sw-inner-resolvent}
+ \|U_I^{-1}G_I(z)U_I^{-1}\|=O_{\PP}(h),
+\end{equation}
+and
+\begin{equation}\label{eq:sw-second-cut}
+ \Tr(D+A'-z)^{-1}=\Tr(D_I+A'_{II}-z)^{-1}+o_{\PP}(1),
+\end{equation}
+jointly for any fixed finite list of $z$'s.
+\end{lemma}
+
+\begin{proof}
+Every diagonal entry of $U_I(D_I-\mathrm ih)U_I$ has modulus at least
+$1/\sqrt2$, and the added term has norm $o(1)$ by
+Lemma~\ref{sw:weighted-blocks}, $|s_1|=O(T)$ and \eqref{eq:sw-weight-bounds}.
+Hence $\|U_I^{-1}G_I(\mathrm ih)U_I^{-1}\|\le C$, and
+$G_I(\mathrm ih)^*G_I(\mathrm ih)=h^{-1}\Im G_I(\mathrm ih)$ gives
+$\|G_I(\mathrm ih)U_I^{-1}\|\le Ch^{-1/2}$. The twice-iterated resolvent
+identity $G_I(z)=G_I(\mathrm ih)+(z-\mathrm ih)G_I(\mathrm ih)^2+
+(z-\mathrm ih)^2G_I(\mathrm ih)G_I(z)G_I(\mathrm ih)$, conjugated by
+$U_I^{-1}$, bounds the three terms by $C$, $C|z-\mathrm ih|/h$ and
+$C|z-\mathrm ih|^2/(h\Im z)$, proving \eqref{eq:sw-inner-resolvent}.
+
+For \eqref{eq:sw-second-cut}, the upper-left block of the full inverse is
+$\widetilde G_I=(G_I^{-1}-\Sigma)^{-1}$. With $R_I=U_I^{-1}G_IU_I^{-1}$ and
+$S_I=U_I\Sigma U_I$, a Neumann expansion using
+\eqref{eq:sw-inner-resolvent} and \eqref{eq:sw-schur-small} gives
+$\|U_I^{-1}(\widetilde G_I-G_I)U_I^{-1}\|\le O(h^2)o(T^{-5})=o(T^{-1})$, so
+$\Tr\widetilde G_I-\Tr G_I=o_{\PP}(1)$. The full block-inverse trace is
+$\Tr\widetilde G_I+\Tr G_F+\Tr(\widetilde G_IA'_{IF}G_F^2A'_{FI})$; the
+middle term is $o_{\PP}(1)$, and with
+$Q_FU_F^2Q_F=U_F^{-1}G_F^2U_F^{-1}$ of norm $O(K^{-1})$ the last term is at
+most $\|U_I^{-1}\widetilde G_IU_I^{-1}\|\|P_0\|_{\HS}^2\|Q_FU_F^2Q_F\|
+=\operatorname{polylog}(r)O(T^3/K)=o_{\PP}(1)$.
+\end{proof}
+
+\begin{lemma}[Gaussian replacement]\label{sw:lindeberg}
+Conditional on almost every $\nu$, bounded smooth tests with bounded
+derivatives of finitely many trace values have expectations differing by
+$o(1)$ between $D_I+A'_{II}$ and
+$H_r=D_I+s_1\Id_{q_*}+\sqrt{s_2}\,\mathcal W$.
+\end{lemma}
+
+\begin{proof}
+Write $A'_{II}-s_1\Id_{q_*}=\sum_{\alpha=1}^{4r}\beta_{\alpha\alpha}
+(g_\alpha g_\alpha^\top-\Id_{q_*})$ with independent standard Gaussian
+$g_\alpha$. Since $\Cov(g_ag_b,g_cg_d)=\delta_{ac}\delta_{bd}+\delta_{ad}
+\delta_{bc}$, each $g_\alpha g_\alpha^\top-\Id$ and a raw GOE matrix have
+equal mean (zero) and entry covariances; replace the summands one at a time
+by $\beta_{\alpha\alpha}\mathcal W_\alpha$ with independent raw GOE
+$\mathcal W_\alpha$ (negative coefficients allowed, both covariances scaling
+by $\beta_{\alpha\alpha}^2$). For either kind of summand $X$,
+$\EE\|X\|^3\le C\theta^3q_*^3$ (Gaussian moments of $\|g\|^2$; Hilbert--
+Schmidt norm for GOE). Differentiating the resolvent in a symmetric
+direction $\Delta$ gives trace derivatives bounded by $C_{z,j}q_*\|\Delta\|^j$,
+so for bounded smooth $\Phi$ the third directional derivative is at most
+$C_{z,\Phi}q_*^3\|\Delta\|^3$ at every real symmetric intermediate matrix.
+Taylor expansion to second order cancels means and covariances in each
+replacement, and the summed third-order remainder is
+$C_{z,\Phi}q_*^6r\theta^3=O(r^{-0.44+(3/2)\tau})=o(1)$. Finally
+$\sum_\alpha\beta_{\alpha\alpha}\mathcal W_\alpha$ has the law
+$\sqrt{s_2}\,\mathcal W$.
+\end{proof}
+
+\begin{proof}[Completion of Proposition~\ref{prop:gaussian-insertion}]
+The coefficient bounds were proved after \eqref{eq:sw-finite-matrix}. That
+equation, Lemma~\ref{sw:trace-cut} and Lemma~\ref{sw:lindeberg} prove the
+conditional comparison with $\mathcal S_r$. Bounded convergence averages
+the conditional error, and by Proposition~\ref{prop:continuous-switching}
+the unconditional law of $\mathcal S_r$ is the law of $m$ for every $r$;
+this proves \eqref{eq:sw-trace-limit}.
+
+For the point measures, let $\nu^{(r)}$ be the eigenvalue counting measure
+of $H_r$ and $\sigma_r(\dd x)=\nu^{(r)}(\dd x)/(1+x^2)$; its total mass
+$\Im\Tr(H_r-\mathrm i)^{-1}$ is tight by \eqref{eq:sw-trace-limit}, so
+$\sigma_r$ is tight on the one-point compactification $\overline\R$. For
+$z=e+\mathrm i\eta$ the kernel $k_z(x)=(1+x^2)\eta\bigl((x-e)^2+\eta^2\bigr)^{-1}$
+extends continuously to $\overline\R$ with $k_z(\infty)=\eta$, and its
+integral against $\sigma_r$ is $\Im\Tr(H_r-z)^{-1}$; so any subsequential
+limit $\sigma$ has the law of $\Im m$ at a countable dense set of $z$,
+whence at all $z$ by continuity. For $f\in C_c(\R)$, integrating $k_z$
+against $\pi^{-1}f(e)\dd e$ and letting $\eta\downarrow0$ gives $(1+x^2)f(x)$
+on $\R$ (Poisson identity), with uniformly bounded integrals and value
+$\eta\pi^{-1}\int f\to0$ at infinity, so dominated convergence applies;
+applying the same inversion to $\Im m(z)$ gives $\int f\,\dd\nu$. Every
+subsequential local vague limit of $\nu^{(r)}$ therefore has the law of
+$\nu$.
+\end{proof}
+
+\section{Identification by finite-matrix Dyson Brownian motion}\label{sec:dbm}
+
+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.
+
+\subsection{The fixed-energy input}
+
+For a real diagonal $V=\diag(v_1,\dots,v_m)$ write
+$m_V(\zeta)=m^{-1}\sum_j(v_j-\zeta)^{-1}$. For $g,G>0$ call $V$
+$(g,G)$-regular if, for fixed $c,C,C_V>0$,
+\begin{equation}\label{eq:dbm-regularity}
+ c\le\Im m_V(e+\mathrm i\eta)\le C\quad (|e|\le G,\ g\le\eta\le10),
+ \qquad \|V\|\le m^{C_V},
+\end{equation}
+with $g,G$ possibly $m$-dependent and, for a fixed $\delta>0$,
+\begin{equation}\label{eq:dbm-parameter-domain}
+ m^{\delta-1}\le g\le m^{-\delta},\qquad G\le m^{-\delta}.
+\end{equation}
+Let $W_m$ be standard normalized GOE (off-diagonal variance $1/m$,
+diagonal $2/m$). The free-convolution comparison density associated with
+$V+\sqrt t\,W_m$ is described by
+\begin{equation}\label{eq:dbm-free-convolution}
+ m_{{\rm fc},t}(\zeta)=m_V\bigl(\zeta+t\,m_{{\rm fc},t}(\zeta)\bigr),\qquad
+ \rho_{{\rm fc},t}(e)=\pi^{-1}\lim_{\eta\downarrow0}\Im m_{{\rm fc},t}(e+\mathrm i\eta),
+\end{equation}
+using the upper-half-plane solution. Write $p_H^{(m)}$ for the unit-mass
+symmetrized joint eigenvalue density of $H$, and $p_H^{(a)}$ for its
+unit-mass $a$-point marginal.
+
+\begin{theorem}[Fixed-energy DBM input]\label{thm:dbm-input}
+Suppose a sequence of deterministic diagonal $V$ of dimension $m$
+satisfies \eqref{eq:dbm-regularity} and \eqref{eq:dbm-parameter-domain}
+with fixed constants. Fix $\sigma>0$ and
+\begin{equation}\label{eq:dbm-time-window}
+ g\,m^{\sigma}\le t\le m^{-\sigma}G^2,
+\end{equation}
+and assume $\rho_{{\rm fc},t}(0)=1/\pi$. Then for every fixed $a\ge1$ and
+$O\in C_c^\infty(\R^a)$ the difference of
+$\int O(\boldsymbol\alpha)\,p^{(a)}_{V+\sqrt tW_m}(\pi\boldsymbol\alpha/m)\dd\boldsymbol\alpha$
+and $\int O(\boldsymbol\alpha)\,p^{(a)}_{W_m}(\pi\boldsymbol\alpha/m)\dd\boldsymbol\alpha$
+tends to zero. Consequently the expected sums of $O$ over ordered tuples
+of distinct points of the two eigenvalue processes, both rescaled by
+$m/\pi$, differ by $o(1)$.
+\end{theorem}
+
+\begin{proof}
+This is Theorem~2.2 of Landon--Sosoe--Yau (with Definition~2.1 and
+Equations~(2.5)--(2.9) there), specialized to energy zero and equal
+densities. For the last assertion multiply each integral by
+$(m)_a/(m/\pi)^a\to\pi^a$, the Jacobian and ordered-tuple factor for the
+unit-mass convention.
+\end{proof}
+
+\subsection{Regularity and the density adjustment}
+
+Fix a microscopic subsequential law $\nu$ from
+Proposition~\ref{prop:microscopic-limit} and the parameters and
+conditional matrices of Proposition~\ref{prop:gaussian-insertion}:
+$K=r^{1/100}$, $T=r^{\tau}$ with $0<\tau<\chi/100000$,
+$I=\{j:|x_j|\le K\}$, $q_*=|I|$, and
+\begin{equation}\label{eq:dbm-raw-matrix}
+ H_r=D_I+s_1\Id+\sqrt{s_2}\,\mathcal W_r,\qquad |s_1|\le CT,\ cT\le s_2\le CT,
+\end{equation}
+with $\mathcal W_r$ raw GOE of dimension $q_*$. By \eqref{eq:discrepancy},
+\begin{equation}\label{eq:dbm-dimension}
+ q_*=2\rho K+O(K^{1-\chi}),\qquad T=q_*^{100\tau+o(1)}.
+\end{equation}
+Let $\alpha_*=\pi\rho$. Scaling by $\alpha_*/q_*$,
+\begin{equation}\label{eq:dbm-normalized-matrix}
+ \frac{\alpha_*}{q_*}H_r\stackrel{\mathrm{law}}{=}V_*+\sqrt{t_*}\,W_{q_*},
+ \qquad V_*=\frac{\alpha_*}{q_*}(D_I+s_1\Id),\quad t_*=\frac{\alpha_*^2s_2}{q_*}.
+\end{equation}
+
+\begin{lemma}\label{dbm:regularity-density}
+For almost every $\nu$, $V_*$ is eventually regular with
+$g=T^{1/2}/q_*$, $G=q_*^{-1/8}$, satisfying
+\eqref{eq:dbm-parameter-domain} and \eqref{eq:dbm-time-window} with
+$m=q_*$, $\delta=\sigma=\tau$. Moreover
+$\rho_{{\rm fc},t_*}(0)\to1/\pi$; with $c_r=\pi\rho_{{\rm fc},t_*}(0)$ the
+data $c_rV_*$, $c_r^2t_*$ satisfy Theorem~\ref{thm:dbm-input}, including
+the exact density condition.
+\end{lemma}
+
+\begin{proof}
+$\|V_*\|$ is bounded since $T=o(K)$. Apply a slightly enlarged domain
+$|\Re\zeta|\le2G$, $g/2\le\Im\zeta\le20$; the normalized resolvent trace is
+$m_{V_*}(\zeta)=\alpha_*^{-1}\sum_{j\in I}\bigl(x_j-(q_*\zeta/\alpha_*-s_1)\bigr)^{-1}$.
+Writing $e=q_*\Re\zeta/\alpha_*-s_1$, $u=q_*\Im\zeta/\alpha_*$ gives
+\begin{equation}\label{eq:dbm-original-domain}
+ |e|=O(K^{7/8}+T)=o(K),\qquad cT^{1/2}\le u\le CK.
+\end{equation}
+Equation~\eqref{eq:mesoscopic-limit} (radius comparable to $K$, fixed
+exponent below $50\tau$) and the imaginary part of
+\eqref{eq:marked-representation} give
+$\sum_j u\bigl((x_j-e)^2+u^2\bigr)^{-1}=\pi\rho+o(1)$ uniformly in
+\eqref{eq:dbm-original-domain}; terms with $|x_j|>K$ sum to $O(u/K)$, and
+the retained sum has positive lower bound $\ge q_*u/(4K^2+u^2)\ge c>0$ in
+the complementary regime. This gives regularity. The powers are
+$g=m^{-1+50\tau+o(1)}$, $t_*=m^{-1+100\tau+o(1)}$, $G=m^{-1/8}$, so
+$t_*/(gm^\tau)=m^{49\tau+o(1)}\to\infty$ and
+$t_*/(m^{-\tau}G^2)=m^{-3/4+101\tau+o(1)}\to0$, verifying the time window
+and domain for small $\tau$.
+
+For the density, $\zeta=0$ in \eqref{eq:dbm-free-convolution} reads
+\begin{equation}\label{eq:dbm-boundary-equation}
+ w=F_r(w),\qquad F_r(w)=\alpha_*^{-1}\sum_{j\in I}\bigl(x_j-(\alpha_*s_2w-s_1)\bigr)^{-1}.
+\end{equation}
+On $|w-\mathrm i|\le1/4$ the argument $z=\alpha_*s_2w-s_1$ has
+$\Im z\asymp T$, $|z|=O(T)$; \eqref{eq:mesoscopic-limit} and
+\eqref{eq:marked-representation} give
+$\PV\sum_j(x_j-z)^{-1}=\mathrm i\pi\rho+o(1)$, and
+$\PV\sum_{|x_j|>K}(x_j-z)^{-1}=O(K^{-\chi}+T/K)$. Hence $F_r(w)=\mathrm i+o(1)$
+uniformly; Rouch\'e for $w-F_r(w)$ against $w-\mathrm i$ gives a unique
+simple zero $w_r\to\mathrm i$ in the disk, continued by the implicit
+function theorem to a positive-imaginary solution near $\zeta=0$. That this
+is the physical branch follows from the identity
+\begin{equation}\label{eq:dbm-uniqueness-identity}
+ \frac1{q_*}\sum_{j\in I}\bigl|(V_*)_{jj}-\zeta-t_*w\bigr|^{-2}
+ =\frac{\Im w}{\Im\zeta+t_*\Im w}<\frac1{t_*}
+\end{equation}
+for every upper-half-plane solution; two distinct solutions would give,
+after subtracting and dividing, $1$ equal to a quantity of absolute value
+strictly less than one by Cauchy--Schwarz and
+\eqref{eq:dbm-uniqueness-identity}. Its boundary value gives
+$\rho_{{\rm fc},t_*}(0)=\Im w_r/\pi\to1/\pi$.
+
+Finally, replacing $V_*,t_*$ by $c_rV_*,c_r^2t_*$ dilates the
+free-convolution law by $c_r$, with transform
+$c_r^{-1}m_{{\rm fc},t_*}(\zeta/c_r)$ and density
+$c_r^{-1}\rho_{{\rm fc},t_*}(0)=1/\pi$ at zero; since $c_r\to1$ the enlarged
+domain gives \eqref{eq:dbm-regularity} and the strict power margins
+preserve \eqref{eq:dbm-time-window} for $c_r^2t_*$.
+\end{proof}
+
+By \eqref{eq:dbm-normalized-matrix} the rescaled eigenvalues supplied by
+Theorem~\ref{thm:dbm-input} are
+\begin{equation}\label{eq:dbm-final-coordinates}
+ \frac{q_*}{\pi}\cdot\frac{c_r\alpha_*}{q_*}\lambda_j(H_r)=c_r\rho\,\lambda_j(H_r),
+\end{equation}
+with $c_r=1$ at exceptional initial values.
+
+\subsection{The GOE limit and uniform count moments}
+
+Let $\mathcal G_m=\sum_{j=1}^m\delta_{(m/\pi)\lambda_j(W_m)}$. These
+converge vaguely in law along all dimensions to a common locally finite
+process $\mathcal S_{\mathrm{GOE}}$: the eigenvalues of $\sqrt m\,W_m$ have
+density $\propto e^{-\frac14\sum y_j^2}\prod_{i<j}|y_i-y_j|$, and
+Theorem~1 of Valk\'o--Vir\'ag (for $\beta=1$, center zero) gives vague
+convergence after multiplying by $2\sqrt m$; dividing by $2\pi$ gives
+$\mathcal G_m$.
+
+\begin{lemma}\label{lem:goe-counts}
+For every compact $J\subset\R$ and every $b>0$,
+$\sup_{m\ge1}\EE\exp\bigl(b\,\mathcal G_m(J)\bigr)<\infty$.
+\end{lemma}
+
+\begin{proof}
+Bound the GOE count by twice a Gaussian-unitary count plus one via the
+Gaussian decimation identity of Forrester--Rains. Let $\mathrm O_m$ be the
+decreasingly ordered points with density
+$\propto\Delta(y)\prod_i b_0(y_i)$, $b_0=e^{-y^2/2}$, the law of
+$\sqrt m\,W_m/\sqrt2$; then $\mathcal G_m(J)$ is the $\mathrm O_m$ count in
+$J_m=\pi J/\sqrt{2m}$, so $|J_m|=O(m^{-1/2})$. Superposing independent
+$\mathrm O_m$ and $\mathrm O_{m+1}$ gives density
+$P(z)=\sum_{|S|=m}\Delta(z_S)\Delta(z_{S^c})$, which by the
+parity--Vandermonde factorization equals
+\begin{equation}\label{eq:dbm-decimation-polynomial}
+ P(z)=c_m\,\Delta(z_1,z_3,\ldots,z_{2m+1})\,\Delta(z_2,z_4,\ldots,z_{2m}),
+\end{equation}
+with $c_m>0$: every factor on the right divides $P$ (same-parity terms
+vanish at coincidences; the others cancel in pairs with sign ratio
+$(-1)^{j-i-1}=-1$), both sides have degree $m^2$, and the quotient is
+positive. Integrating the odd points over their interlacing intervals with
+the monic polynomials $p_j=(-1)^jb_0^{(j)}/b_0$ (each row replaced by its
+partial row sums, last row $\int b_0$, expansion along it) gives that the
+even points have density
+\begin{equation}\label{eq:dbm-even-density}
+ \propto\Delta(y)^2\prod_i e^{-y_i^2}.
+\end{equation}
+
+This density equals $(m!)^{-1}\det\bigl(K_m(y_i,y_j)\bigr)$ for the
+projection kernel $K_m=\sum_{j=0}^{m-1}\psi_j\psi_j$ of the orthonormal
+polynomials for $e^{-x^2}$, with order-$a$ factorial moment density
+$\det\bigl(K_m(y_i,y_j)\bigr)_{i,j=1}^a$. The operator $-\partial_x^2+x^2$
+has eigenvalue $2j+1$ on $\psi_j$, so $\|f'\|_2^2\le(2m-1)\|f\|_2^2$ on the
+range of $K_m$, and $|f(x)|^2\le2\|f'\|_2\|f\|_2$ gives
+$K_m(x,x)\le2\sqrt{2m}$. Then for every $a\ge0$,
+$\EE(N_m^{\mathrm{even}})_a\le\bigl(2\sqrt{2m}\,|J_m|\bigr)^a\le C_J^a$, so
+$\EE e^{vN_m^{\mathrm{even}}}\le\exp\bigl(C_J(e^v-1)\bigr)$. As the
+superposition count is at most twice its even count plus one and dominates
+the $\mathrm O_m$ count,
+$\EE e^{b\mathcal G_m(J)}\le e^b\EE e^{2bN_m^{\mathrm{even}}}$.
+\end{proof}
+
+\subsection{From correlation functions to Laplace functionals}
+
+Let $\nu^{(r)}=\sum_j\delta_{\lambda_j(H_r)}$; by
+Proposition~\ref{prop:gaussian-insertion},
+\begin{equation}\label{eq:dbm-insertion-limit}
+ \nu^{(r)}\Longrightarrow\nu
+\end{equation}
+unconditionally in local vague topology. Fix $0\le h\in C_c^\infty(\R)$,
+$f=1-e^{-h}$, and for a finite point measure $\zeta=\sum_i\delta_{y_i}$ set
+$A_0=1$, $A_a=(a!)^{-1}\sum_{i_1,\ldots,i_a\ \mathrm{distinct}}\prod_{j=1}^a f(y_{i_j})$,
+$B_\ell=\sum_{a=0}^{\ell}(-1)^aA_a$. The Bonferroni inequalities give
+\begin{equation}\label{eq:dbm-bonferroni}
+ B_{2\ell+1}(\zeta)\le\exp\Bigl(-\int h\,\dd\zeta\Bigr)\le B_{2\ell}(\zeta),
+\end{equation}
+seen by Bernoulli variables of success probabilities $f(y_i)$. Conditional
+on almost every $\nu$, let $\zeta_r$ have coordinates $c_r\rho\,\lambda_j(H_r)$;
+Lemma~\ref{dbm:regularity-density} and Theorem~\ref{thm:dbm-input} with
+test $\prod_j f(\alpha_j)$ give, for each fixed $\ell$,
+$\EE[B_\ell(\zeta_r)\mid\nu]-\EE B_\ell(\mathcal G_{q_*})\to0$. To close
+the brackets uniformly in $q_*$, take $J\supset\supp h$; since
+$A_a(\mathcal G_m)\le\binom{\mathcal G_m(J)}a\le2^{-a}3^{\mathcal G_m(J)}$
+(from $\sum_a2^a\binom Na=3^N$), Lemma~\ref{lem:goe-counts} gives
+\begin{equation}\label{eq:dbm-bracket-width}
+ \sup_m\EE\bigl[B_{2\ell}(\mathcal G_m)-B_{2\ell+1}(\mathcal G_m)\bigr]\le C_J2^{-(2\ell+1)}.
+\end{equation}
+Using \eqref{eq:dbm-bonferroni} for both models, sending $r\to\infty$ at
+fixed $\ell$ and then $\ell\to\infty$, gives
+\begin{equation}\label{eq:dbm-conditional-laplace}
+ \EE\Bigl[\exp\Bigl(-\int h\,\dd\zeta_r\Bigr)\mid\nu\Bigr]\to
+ \EE\exp\Bigl(-\int h\,\dd\mathcal S_{\mathrm{GOE}}\Bigr).
+\end{equation}
+Both sides lie in $[0,1]$, so bounded convergence removes the conditioning.
+For $c>0$ let $\mathcal D_c\zeta$ scale positions by $c$; this map is
+continuous in $(c,\zeta)$ for $c$ near a positive number and $\zeta$
+locally vaguely convergent. Since $c_r\to1$ almost surely,
+$\zeta_r=\mathcal D_{c_r\rho}\nu^{(r)}\Rightarrow\mathcal D_\rho\nu$, and
+bounded continuity of the Laplace functional of a compactly supported
+continuous nonnegative test gives
+\begin{equation}\label{eq:dbm-limit-identification}
+ \EE\exp\Bigl(-\int h\,\dd\mathcal D_\rho\nu\Bigr)
+ =\EE\exp\Bigl(-\int h\,\dd\mathcal S_{\mathrm{GOE}}\Bigr)
+\end{equation}
+for $0\le h\in C_c^\infty$, extended to $0\le h\in C_c$ by convolution with
+a nonnegative approximate identity and bounded convergence.
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Start with any sequence of admissible sizes; Proposition~\ref{prop:microscopic-limit}
+supplies a further subsequence on which $\nu_n\Rightarrow\nu$ as above. The
+graph point measure is $\mathcal D_\rho\nu_n$, and by bounded continuity of
+its Laplace functional and \eqref{eq:dbm-limit-identification} its Laplace
+expectation along this subsequence tends to that of $\mathcal S_{\mathrm{GOE}}$;
+the reference GOE expectations at dimension $n$ tend to the same value along
+all sizes. Every original subsequence thus has a further subsequence on
+which the difference tends to zero, proving the full-sequence limit. All
+excluded events had probability tending to zero in the original
+simple-graph model; no conditioning on connectedness or bipartiteness enters.
+\end{proof}
+
+\end{document}