\chapter{Random-matrix inputs and the equilibrium Lagrange multiplier}\label{sec:spectral}\label{sec-spectral} This section collects the properties of the GOE spectrum used in the paper, and it determines the Lagrange multiplier of \Cref{def:model-multiplier} in each temperature regime. \Cref{sec:spec-inputs} states the published random-matrix results that we use, in our normalization. These are rigidity and convergence at the spectral edge (\Cref{sp-goe-rigidity}), simultaneous control of the eigenvalue locations (\Cref{sp-location-control}), a localized estimate for the Stieltjes transform (\Cref{sp-published-counts}), and the regularized inverse-gap statistic (\Cref{sp-raw-gap-resolvent}). The proofs below give the precise published estimates used to derive the stated bounds. \Cref{sec:spec-geometry} proves two-sided bounds on all eigenvalue gaps below the top eigenvalue at every fixed confidence (\Cref{sp-fixed-confidence-geometry}), and derives from them the spectral hypotheses used at low temperature (\Cref{sp-condensed-geometry}). \Cref{sec:spec-counts} proves a law of large numbers for the eigenvalues near the edge at every scale between $N^{-2/3}$ and $1$ (\Cref{sp-mesoscopic-counts}), together with its version for the gaps below the top eigenvalue (\Cref{mc-raw-edge-measure}). \Cref{sgl-scalar-section} determines the Lagrange multiplier at fixed high temperature and along every approach $c_N=1-\delta_N$ with $N\delta_N^3\to\infty$ (\Cref{sgl-canonical-quantitative,sp-canonical-saddle}). Finally, \Cref{sec:spec-critical} treats the critical window. It proves almost-sure properties of the reflected GOE Airy point process and of the limiting equation for the Lagrange multiplier (\Cref{sp-airy-counts}), and the convergence of the scaled half-precisions together with uniform bounds on them (\Cref{cg-canonical-weyl}). This section uses the definitions of \Cref{sec:model}. \Cref{sp-canonical-saddle,sp-mesoscopic-counts} are used at high temperature, in \Cref{sec:hot-fixed,sec:hot-meso,sec:infinite}, and \Cref{mc-raw-edge-measure} is used in \Cref{sec:hot-meso}. \Cref{sp-condensed-geometry} supplies the spectral hypotheses in the approach to the critical temperature from below (\Cref{sec:cold-meso}). At a fixed low temperature, the proof of the main result of \Cref{sec:cold-fixed} uses instead \Cref{sp-fixed-confidence-geometry,sp-goe-rigidity}, through the law of large numbers for the inverse gaps of \Cref{lem:coldf-inverse-gap-lln}. The critical window uses \Cref{cg-canonical-weyl} and the conditions \eqref{cg-weyl-hypotheses}, in \Cref{sec:inequalities,sec:density,sec:crit-process,sec:crit-profile}. \Cref{sp-airy-counts} is used for the limits of the critical profile far from the critical temperature (\Cref{sec:outer-hot,sec:outer-cold}). \paragraph{Notation.} Throughout this section, $\lambda_1\ge\cdots\ge\lambda_N$ are the eigenvalues of $W_N$. For $E>\lambda_1$ and $E>2$, respectively, we write \begin{equation}\label{eq:spec-stieltjes} \mathfrak m_N(E)=\frac1N\sum_{i=1}^N\frac1{E-\lambda_i},\qquad \mathfrak m_{\mathrm{sc}}(E)=\int\frac{\rho_{\mathrm{sc}}(x)}{E-x}\,dx=\frac{E-\sqrt{E^2-4}}2 \end{equation} for the Stieltjes transforms of the empirical eigenvalue distribution and of the semicircle law. With this sign convention both are positive. The papers \cite{ErdosYauYin2012,Landon2022,LandonSosoe2022} use the opposite sign, which does not affect absolute differences. We let $\gamma_1\ge\gamma_2\ge\cdots\ge\gamma_N$ be the semicircle quantiles, defined by $\int_{\gamma_i}^2\rho_{\mathrm{sc}}(x)\,dx=i/N$. We write \[ C_N(t)=\#\{i:\lambda_i\ge2-tN^{-2/3}\},\qquad \Delta_{i,N}=N^{2/3}(\lambda_1-\lambda_i)\quad(1\le i\le N), \] for the number of eigenvalues within distance $tN^{-2/3}$ of the edge and for the scaled gaps below the top eigenvalue. The following elementary bounds on the quantiles are used several times. \begin{lemma}[Semicircle quantiles]\label{lem:spec-quantiles} For $1\le i\le N$, we have \[ \Big(\frac{3\pi i}{2N}\Big)^{2/3}\le2-\gamma_i\le7\Big(\frac iN\Big)^{2/3}. \] Moreover, $2-\gamma_i=(3\pi i/(2N))^{2/3}\big(1+O((i/N)^{2/3})\big)$ uniformly for $1\le i\le N/2$. \end{lemma} \begin{proof} For $0\le v\le4$, we have $\rho_{\mathrm{sc}}(2-v)=(2\pi)^{-1}\sqrt{v(4-v)}$. Using $\sqrt{4-v}\le2$, we find that $i/N=\int_0^{2-\gamma_i}\rho_{\mathrm{sc}}(2-v)\,dv\le2(2-\gamma_i)^{3/2}/(3\pi)$, which is the lower bound. For $i\le N/2$, we have $\gamma_i\ge0$ by symmetry of $\rho_{\mathrm{sc}}$. Since $\sqrt{4-v}\ge\sqrt2$ for $v\le2$, we then have $i/N\ge\sqrt2\,(2-\gamma_i)^{3/2}/(3\pi)$ and $2-\gamma_i\le(3\pi/\sqrt2)^{2/3}(i/N)^{2/3}\le7(i/N)^{2/3}$. For $i>N/2$, we use $2-\gamma_i\le4\le7(i/N)^{2/3}$. Finally, since $\sqrt{4-v}=2(1+O(v))$, we have $\int_0^u\rho_{\mathrm{sc}}(2-v)\,dv=2u^{3/2}(1+O(u))/(3\pi)$ for $u\le2$. Setting $u=2-\gamma_i\le7(i/N)^{2/3}$ and solving for $u$, we obtain the last claim. \end{proof} \section{Published random-matrix inputs}\label{sec:spec-inputs} \paragraph{Imported results and their normalization.} Apart from standard tools of analysis and probability, the following are the only random-matrix inputs to the proofs of Theorems~1.1\nobreakdash--1.4: eigenvalue rigidity; convergence at the edge to the reflected GOE Airy point process, including the GOE Tracy--Widom law; published counting estimates, used with rigidity to obtain simultaneous control of the eigenvalue locations; the localized Stieltjes-transform estimate; and the regularized inverse-gap statistic, including Du and Huang's joint inverse-gap convergence in $\R\times\ell^2$. The versions collected in this subsection suffice for the downstream profile estimates; proofs also cite the sharper published estimates from which they are obtained. Du and Huang's limiting equation $\mathscr S$ and their critical half-precision result at $\sigma=0$ are identified where they occur in the critical-window statements below; the quantitative bounds and the extensions in $\sigma$ are proved here. All statements use the GOE variances $1/N$ off the diagonal and $2/N$ on the diagonal, the reflected edge coordinates $N^{2/3}(2-\lambda_i)$, and the Stieltjes-transform sign convention fixed above. We record the published results that we use, in the normalization of \Cref{sec:model}. The random-matrix ensemble parameter of Ram\'irez, Rider, and Vir\'ag \cite{RamirezRiderVirag2011} equals $1$ for the GOE and is unrelated to the inverse temperature $\beta$. \begin{lemma}[{Rigidity and convergence at the edge \cite[Theorem~2.1, (2.16)]{Landon2022}, \cite[Theorem~2.2, (2.25)]{ErdosYauYin2012}, \cite[Theorem~1.1, (1.7), and Proposition~3.5]{RamirezRiderVirag2011}}]\label{sp-goe-rigidity}\label{sp-goe-edge} \leavevmode \begin{enumerate} \item For every $\eta\in(0,1)$ there exist $C>0$ and $N_0$ such that, for $N\ge N_0$, the event \begin{equation}\label{sgl-rigidity-input} F_C=\Big\{\max_{1\le i\le N}|\lambda_i-\gamma_i|\le C N^{-5/12}\Big\} \end{equation} has probability at least $1-\eta$. \item There exists a random sequence $\chi_1<\chi_2<\cdots$, the reflected GOE Airy point process, which almost surely is strictly increasing, has no finite accumulation point, and satisfies $\chi_i\to\infty$, such that for every fixed $k$, \[ \big(N^{2/3}(2-\lambda_i)\big)_{i=1}^k\Longrightarrow(\chi_i)_{i=1}^k . \] The law of $-\chi_1$ is the GOE Tracy--Widom law $\mathrm{TW}_1$. In particular, $N^{2/3}(\lambda_1-2)$ is tight, and for every fixed $k$, $(\Delta_{i,N})_{i=2}^k\Rightarrow(\Delta_i)_{i=2}^k$, where $\Delta_i=\chi_i-\chi_1\in(0,\infty)$ almost surely. \item Since $-W_N$ has the law of $W_N$, the same statements hold at the lower edge. In particular, $\lambda_N\to-2$ in probability, and $\max_i|\lambda_i|\le2+C N^{-5/12}$ on $F_C$. \end{enumerate} \end{lemma} \usedin{The \hyperref[sec:hot-fixed]{high-temperature}, \hyperref[sec:cold-fixed]{low-temperature}, and \hyperref[sec:crit-process]{critical-window} analyses, through the consequences below.} \begin{proof} The first part is a consequence of the sharper published rigidity estimate \cite[Theorem~2.1]{Landon2022}, based on \cite[Theorem~2.2]{ErdosYauYin2012}. Choosing its exponent smaller than $1/4$ gives the stated fixed-confidence bound. The variance normalization in the latter reference applies to $\sqrt{N/(N+1)}\,W_N$; undoing this scaling changes eigenvalues by $O(N^{-1})$ on a bounded-spectrum event and is absorbed by $N^{-5/12}$. We use the sharper published estimates in a few proofs below, with explicit citations, but state only the bounds needed here. For the second part, the eigenvalues of $\sqrt N\,W_N$ have the law of the Hermite $\beta$-ensemble with $\beta=1$, whose density is proportional to $\prod_{i0$, an integer $N_0$, and measurable events $\mathcal H_N^\eta$ with $\PP(\mathcal H_N^\eta)\ge1-\eta$ for $N\ge N_0$, on which \begin{equation}\label{eq:spec-location-control} N^{2/3}|\lambda_i-\gamma_i|\le C_\eta(1+i)^{1/4}\qquad(1\le i\le N). \end{equation} \end{lemma} \usedin{The fixed-confidence gap bounds, mesoscopic edge counts, and almost-sure Airy geometry below.} \begin{proof} Write $q_{i,N}=N^{2/3}(2-\gamma_i)$ and $a=2/(3\pi)$. Choose a fixed $b>0$ so small that $7b^{2/3}<1/2$. By \Cref{lem:spec-quantiles}, uniformly for $i\le bN^{2/5}$, \[ q_{i,N}\asymp i^{2/3},\qquad aq_{i,N}^{3/2}=i+O(i^{5/3}N^{-2/3})=i+O(1). \] Consequently, for all sufficiently large fixed $K$ and all large $N$, the thresholds $s_\pm=q_{i,N}\pm i^{1/4}$, $K\le i\le bN^{2/5}$, lie in $[t_0,N^{4/15}]$ and satisfy \[ as_+^{3/2}-i\ge c i^{7/12},\qquad i-as_-^{3/2}\ge c i^{7/12}, \] where $t_0$ is any fixed sufficiently large constant. Here the exponent $7/12$ is the sum of $1/3$, from the derivative of $s^{3/2}$ at $s\asymp i^{2/3}$, and $1/4$. The published bounds \cite[Proposition~6.5, (6.11)--(6.12)]{LandonSosoe2022} give $|\E C_N(s)-as^{3/2}|\le C$ and $\operatorname{Var}(C_N(s))\le C\log(2+s)$ throughout this range. These counting estimates follow there from the Airy estimates of Soshnikov \cite{Soshnikov2000}, the GUE estimates of Gustavsson \cite{Gustavsson2005}, and the superposition identity of Forrester and Rains \cite{ForresterRains2001}. If $N^{2/3}(2-\lambda_i)s_+$, then $C_N(s_+)i^{1/4}\big) \le C\log(2+i)i^{-7/6}\qquad(K\le i\le bN^{2/5}). \] The series on the right is summable. Choose $K$, depending only on $\eta$, so that its sum over $i\ge K$ is at most $\eta/4$. For the remaining indices $i>bN^{2/5}$, use the sharper rigidity estimate \cite[Theorem~2.1]{Landon2022}, with exponent $1/10$, on an event whose probability tends to one. If $i\le N/2$, it gives \[ N^{2/3}|\lambda_i-\gamma_i|\le N^{1/10}i^{-1/3}\le C_b i^{1/4}, \] since $i>bN^{2/5}$; if $i>N/2$, it gives the upper bound $N^{1/10}\le C i^{1/4}$. Finally, the first $K-1$ scaled eigenvalues are tight by \Cref{sp-goe-rigidity}, and $q_{i,N}\to(3\pi i/2)^{2/3}$ for each fixed $i$. We can therefore choose $C_\eta$ so that \eqref{eq:spec-location-control} holds simultaneously for these first indices with limiting failure probability at most $\eta/4$. Enlarge $C_\eta$ to dominate the constants for the other indices. Intersecting the three events gives failure probability at most $\eta$ for all sufficiently large $N$. \end{proof} \begin{lemma}[Localized Stieltjes transform estimate]\label{sp-published-counts} For every $\eta\in(0,1)$ there are $U_\eta,C_\eta>0$, an integer $N_0$, and measurable events $\mathcal R_N^\eta$ of probability at least $1-\eta$ for $N\ge N_0$, such that $\lambda_1<2+U_\eta N^{-2/3}$ on these events and, for every deterministic $u\ge U_\eta$, with $E=2+uN^{-2/3}$, \begin{equation}\label{sp-local-resolvent} \E\Big[\ind_{\mathcal R_N^\eta}\,\big|\mathfrak m_N(E)-\mathfrak m_{\mathrm{sc}}(E)\big|\Big] \le C_\eta N^{-1/3}\Big(u^{-1/8}+N^{-1/3}u^{-1/2}+N^{-1/6}+u^{-1}\Big). \end{equation} The random variable inside the expectation is defined to be zero off $\mathcal R_N^\eta$; on this event $E>\lambda_1$, so the resolvent is defined. \end{lemma} \usedin{The high-temperature multiplier estimates below.} \aliaslabel{sgl-localized-resolvent-input}{sp-local-resolvent} \begin{proof} Fix $\xi=1/200$. The localized estimate of \cite[Lemmas~2.2 and~2.6]{Landon2022} supplies, on a common event of confidence $1-\eta$, the bound $C_\eta\{N^{-1/3}(1+\log^2u)/u+N^{-1+\xi}\}$ for $U_\eta\le u0$, \[ \frac1N\sum_{i=2}^N\frac1{c(\lambda_1-\lambda_i)}=\frac1c+\frac{\Xi_N}{cN^{1/3}} . \] \item If $c_N=1-\delta_N$ with $\delta_N\to0$ and $M_N=N\delta_N^3\to\infty$, then \[ \frac1{\delta_N}\Big(\frac1N\sum_{i=2}^N\frac1{c_N(\lambda_1-\lambda_i)}-1\Big)\longrightarrow1\quad\text{in probability}. \] \end{enumerate} \end{lemma} \usedin{The \hyperref[sec:hot-meso]{high-temperature multiplier estimates}, the \hyperref[sec:cold-fixed]{low-temperature} and \hyperref[sec:cold-meso]{varying low-temperature} analyses, the \hyperref[sec:crit-process]{critical process} and \hyperref[sec:crit-profile]{profile}, through the consequences below.} \begin{proof} The first part, and the convergence in distribution of $\Xi_N$ to $\Xi$, are due to Landon and Sosoe \cite[Theorem~6.1, (6.3)--(6.4)]{LandonSosoe2022}. Note that $2\sqrt{t_k}/\pi=\pi^{-1}\int_0^{t_k}x^{-1/2}\,dx$ and that the random variable in (6.3) there equals $\Xi_N$. The joint convergence in $\R\times\ell^2$ is \cite[Proposition~4.1, (4.1)]{DuHuang2026}, whose inverse-gap vector has an additional first coordinate equal to $0$. See also \cite[Proposition~1.3]{DuHuang2026}, which restates the result of Landon and Sosoe. For the third part, recall that $\sum_{i\ge2}\Delta_{i,N}^{-1}=\Xi_N+N^{1/3}$ by the definition of $\Xi_N$. Then \[ \frac1N\sum_{i=2}^N\frac1{c(\lambda_1-\lambda_i)}=\frac{N^{2/3}}{cN}\big(\Xi_N+N^{1/3}\big)=\frac1c+\frac{\Xi_N}{cN^{1/3}} . \] For the fourth part, the third part implies that \[ \frac1{\delta_N}\Big(\frac1N\sum_{i=2}^N\frac1{c_N(\lambda_1-\lambda_i)}-1\Big) =\frac1{c_N}+\frac{\Xi_N}{c_N\delta_NN^{1/3}} =\frac1{c_N}+\frac{\Xi_N}{c_NM_N^{1/3}}, \] where we used $1/c_N-1=\delta_N/c_N$. The right side tends to $1$ in probability, because $c_N\to1$, $M_N\to\infty$, and $\Xi_N$ is tight. The error is $O(\delta_N)+O_{\PP}(M_N^{-1/3})$. \end{proof} \section{Eigenvalue gaps at fixed confidence}\label{sec:spec-geometry} The published results determine the locations of any fixed number of edge eigenvalues and give quantitative control away from the edge. The next lemma provides two-sided bounds on all gaps $\lambda_1-\lambda_i$ with constants independent of $N$, at every fixed confidence in the sense of \Cref{sec:model-quantifiers}. The proof uses the simultaneous location bound of \Cref{sp-location-control}, and treats the first few gaps through edge convergence. \begin{lemma}[Eigenvalue gaps at fixed confidence]\label{sp-fixed-confidence-geometry} For every $\eta\in(0,1)$, there exist constants $A,K,C,c_\eta,C_\eta>0$, depending only on $\eta$, and events $\mathcal G_N^\eta$ with $\liminf_N\PP(\mathcal G_N^\eta)\ge1-\eta$, on which, for all sufficiently large $N$, \begin{equation}\label{sp-count-envelope} \begin{gathered} 2-\lambda_i\ge-AN^{-2/3}\quad(1\le i\le N),\qquad \#\{i:\lambda_i>2\}\le K,\\ \#\{i:2-\lambda_i\le u\}\le C(1+Nu^{3/2})\quad(u\ge0), \end{gathered} \end{equation} and \begin{equation}\label{sp-indexed-raw-gaps} c_\eta(i-1)^{2/3}\le\Delta_{i,N}=N^{2/3}(\lambda_1-\lambda_i)\le C_\eta(i-1)^{2/3}\qquad(2\le i\le N). \end{equation} \end{lemma} \begin{proof} Fix $\eta\in(0,1)$ and use \Cref{sp-location-control} with confidence $\eta/2$. Write $d_{i,N}=N^{2/3}(2-\lambda_i)$ and $q_{i,N}=N^{2/3}(2-\gamma_i)$. On its event, $|d_{i,N}-q_{i,N}|\le C_{\eta/2}(1+i)^{1/4}$ simultaneously for every $i$. Since \Cref{lem:spec-quantiles} gives $(3\pi i/2)^{2/3}\le q_{i,N}\le7i^{2/3}$, there exist $K\ge2$ and $02\}\le K. \] This proves \eqref{sp-count-envelope} after increasing its constant $C$. Choose $i_0\ge K$ so large that $c_2i_0^{2/3}\ge2A$ and $C_2i_0^{2/3}\ge A$. For $i\ge i_0$, subtracting $d_{1,N}$ from \eqref{eq:spec-edge-distances} gives \[ \frac{c_2}2 i^{2/3}\le\Delta_{i,N}=d_{i,N}-d_{1,N}\le2C_2i^{2/3}. \] For the first few gaps, \Cref{sp-goe-rigidity} gives $(\Delta_{2,N},\Delta_{i_0,N})\Rightarrow(\Delta_2,\Delta_{i_0})$, where $0<\Delta_2\le\Delta_{i_0}<\infty$ almost surely. Choose $0<\delta_*1$ and an approach $c_N=1+\epsilon_N$ to the critical temperature from below. The low-temperature sections use its conclusions as follows. The bounds of the second part are the deterministic spectral hypotheses there, and the last part describes the Gaussian vector of \Cref{sec:model-equilibrium}. \begin{lemma}[Eigenvalue gaps and the top fraction at low temperature]\label{sp-condensed-geometry}\label{lem:spec-cold-geometry} We consider the following two cases, with $c_\infty=\lim_Nc_N$: \begin{itemize} \item[(F)] $c_N=c>1$ is fixed; \item[(M)] $c_N=1+\epsilon_N$ with $\epsilon_N>0$, $\epsilon_N\to0$, and $N\epsilon_N^3\to\infty$. \end{itemize} Let $a_i=c_N(\lambda_1-\lambda_i)/2$, and let $\theta_N$, $R_N$, $m_N$ be as in \eqref{eq:model-cold-parameters} with $c=c_N$. \begin{enumerate} \item We have the identity \begin{equation}\label{eq:spec-top-fraction} \theta_N=1-\frac1{c_N}-\frac{\Xi_N}{c_NN^{1/3}} . \end{equation} In case (F), $\theta_N\to\theta=1-1/c$ in probability. In case (M), $\theta_N/\epsilon_N\to1$ in probability, and in particular $\theta_N>0$ with probability tending to one. In both cases, $N^{2/3}a_2\Rightarrow c_\infty\Delta_2/2$, and $m_N\to\infty$ in probability. In case (F), moreover, $\log R_N/\log N\to1/3$ and $R_N/\log^2N\to\infty$ in probability. \item For every $\eta\in(0,1)$, there exist constants $B,c_*,C_*,\varepsilon_*>0$, depending only on $\eta$ (and on $c$ in case (F)), and events with limiting probability at least $1-\eta$, on which for all sufficiently large $N$: \begin{equation}\label{eq:spec-cold-geometry} \begin{gathered} c_*(i-1)^{2/3}N^{-2/3}\le a_i\le C_*(i-1)^{2/3}N^{-2/3}\quad(2\le i\le N),\\ a_N\le B,\qquad \frac{a_3}{a_2}\ge1+\varepsilon_* . \end{gathered} \end{equation} On these events, for every integer $2\le k\le N$ and every integer $m'\ge2$, \begin{equation}\label{eq:spec-cold-consequences} \frac{a_k}{a_2}\le\frac{C_*}{c_*}(k-1)^{2/3},\qquad a_2^2\sum_{i=2}^Na_i^{-2}\le4\Big(\frac{C_*}{c_*}\Big)^2,\qquad \sum_{i>m'}\Big(\frac{a_2}{a_i}\Big)^2\le4\Big(\frac{C_*}{c_*}\Big)^2m'^{-1/3}, \end{equation} and $c_*N^{1/3}\le R_N\le C_*N^{1/3}$. In case (F), for any fixed $\theta_0\in(0,1-1/c)$, these events may also be chosen so that $\theta_N\ge\theta_0$. \item Conditional on $W_N$, let $(Z_i)_{i\ge2}$ be independent centered Gaussian random variables with variances $(2a_i)^{-1}$. Then $\E[\sum_{i\ge2}Z_i^2\mid W_N]=N(1-\theta_N)$, and for every $\varepsilon>0$, \[ \PP\Big(\Big|\sum_{i=2}^NZ_i^2-N(1-\theta_N)\Big|>\varepsilon N\Bigm| W_N\Big)\longrightarrow0\quad\text{in probability}. \] More precisely, on the events of the second part, this conditional probability is at most $C\varepsilon^{-2}N^{-2/3}$, with $C$ depending only on $c_*$. \end{enumerate} \end{lemma} \begin{proof} The identity \eqref{eq:spec-top-fraction} is the third part of \Cref{sp-raw-gap-resolvent}, since $(2a_i)^{-1}=(c_N(\lambda_1-\lambda_i))^{-1}$. In case (F), we have $\theta_N\to1-1/c$, because $\Xi_N$ is tight. In case (M), using $1-1/c_N=\epsilon_N/(1+\epsilon_N)$, we find \[ \frac{\theta_N}{\epsilon_N}=\frac1{1+\epsilon_N}-\frac{\Xi_N}{(1+\epsilon_N)\,\epsilon_NN^{1/3}}\longrightarrow1 \] in probability, because $\epsilon_NN^{1/3}=(N\epsilon_N^3)^{1/3}\to\infty$. Next, $N^{2/3}a_2=c_N\Delta_{2,N}/2\Rightarrow c_\infty\Delta_2/2$ by \Cref{sp-goe-rigidity}, and the limit is almost surely positive and finite. We have $m_N=\theta_NN^{1/3}\cdot N^{2/3}a_2$. In both cases $\theta_NN^{1/3}$ tends to infinity in probability, and therefore $m_N\to\infty$ in probability. In case (F), $\log R_N=(\log N)/3+\log(N^{2/3}a_2)$ with $\log(N^{2/3}a_2)=O_{\PP}(1)$, which implies the remaining two claims of the first part. For the second part, fix $\eta$ and work on the events $\mathcal G^{\eta/3}_N$ of \Cref{sp-fixed-confidence-geometry}. Using \eqref{sp-indexed-raw-gaps} and $1\le c_N\le C$, we obtain the two-sided bound on $a_i$ in \eqref{eq:spec-cold-geometry}, with $c_*=c_{\eta/3}/2$ and $C_*=CC_{\eta/3}/2$. Taking $i=N$, we also have $a_N\le C_*$. For the third gap, $a_3/a_2=\Delta_{3,N}/\Delta_{2,N}\Rightarrow\Delta_3/\Delta_2$, which is almost surely larger than $1$ because the GOE Airy point process is simple (\Cref{sp-goe-rigidity}). Let $\varepsilon_*>0$ be such that $\PP(\Delta_3/\Delta_2\le1+\varepsilon_*)\le\eta/3$. The portmanteau theorem gives $\limsup_N\PP(a_3/a_2\le1+\varepsilon_*)\le\eta/3$. In case (F), the event $\{\theta_N\ge\theta_0\}$ has probability tending to one by the first part. The intersection of these events has the properties stated in the second part. The bounds \eqref{eq:spec-cold-consequences} follow from \eqref{eq:spec-cold-geometry}, the bounds $\sum_{j\ge1}j^{-4/3}\le4$ and $\sum_{j\ge m'}j^{-4/3}\le m'^{-4/3}+\int_{m'}^\infty x^{-4/3}\,dx\le4m'^{-1/3}$, and $R_N=N^{1/3}\cdot N^{2/3}a_2$. For the third part, the conditional mean is $\sum_{i\ge2}(2a_i)^{-1}=N(1-\theta_N)$ by the definition of $\theta_N$, and the conditional variance is $2\sum_{i\ge2}(2a_i)^{-2}$. On the events of the second part, it is at most $c_*^{-2}N^{4/3}\sum_{j\ge1}j^{-4/3}/2\le2c_*^{-2}N^{4/3}$. Chebyshev's inequality bounds the conditional probability by $2c_*^{-2}\varepsilon^{-2}N^{-2/3}$ on these events, and the convergence in probability follows from \Cref{lem:model-localization}. \end{proof} \section{Edge counts at a varying scale}\label{sec:spec-counts} We now prove a law of large numbers for the eigenvalues within distance $y/h_N$ of the edge, at each scale $h_N^{-1}$ between $N^{-2/3}$ and $1$. The theorem is used at the scale $h_N\asymp\delta_N^{-2}$ in \Cref{sp-canonical-saddle}, and at the smaller scales $h_N=t$ of the times $t$ in the high-temperature profile windows. To cover the second use, we allow a random shift of the edge and a factor $\alpha_N\to1$. For counts, the restriction $h_N=o(N^{2/3})$ is what allows the number of eigenvalues at the scale $h_N^{-1}$ to grow arbitrarily slowly. \begin{theorem}[Edge counts at a varying scale]\label{sp-mesoscopic-counts} Let $h_N>0$ be deterministic with $h_N\to\infty$ and $h_N=o(N^{2/3})$, and set $K_N=Nh_N^{-3/2}$. Let $\alpha_N>0$ be deterministic with $\alpha_N\to1$, and let $\xi_N$ be real random variables with $h_N\xi_N\to0$ in probability. We define the measure \begin{equation}\label{eq:spec-edge-measure} \omega_N=\frac1{K_N}\sum_{i=1}^N\delta_{\alpha_Nh_N(2-\lambda_i-\xi_N)} . \end{equation} \begin{enumerate} \item For every compact $Y\subset\R$, $\sup_{y\in Y}\big|\omega_N((-\infty,y])-2y_+^{3/2}/(3\pi)\big|\to0$ in probability. In particular, for $\alpha_N=1$ and $\xi_N=0$, the first limit in \eqref{sp-heat-counts} below holds. \item With probability tending to one, every atom of $\omega_N$ lies in $(-1/2,\infty)$. For every $\eta\in(0,1)$, there exist $C_\eta$ and events with limiting probability at least $1-\eta$ on which $\omega_N((-\infty,y])\le C_\eta(K_N^{-1}+y^{3/2})$ for all $y\ge1$. \item Let $k>3/2$, and let $\varphi:[-1/2,\infty)\to\R$ be continuous with $\sup_{y\ge0}(1+y)^k|\varphi(y)|<\infty$. Then \begin{equation}\label{eq:spec-weighted-tests} \int\varphi\,d\omega_N\longrightarrow\frac1\pi\int_0^\infty\varphi(y)\sqrt y\,dy\quad\text{in probability}. \end{equation} The same holds for deterministic continuous $\varphi_N$ that converge to $\varphi$ uniformly on compact subsets of $[-1/2,\infty)$ and satisfy $\sup_N\big(\sup_{[-1/2,0]}|\varphi_N|+\sup_{y\ge0}(1+y)^k|\varphi_N(y)|\big)<\infty$. In particular, $\omega_N$ converges vaguely in probability to $\pi^{-1}\sqrt y\,\ind_{\{y\ge0\}}\,dy$. \item For $\alpha_N=1$ and $\xi_N=0$, we have, for each $y>0$ and each fixed $k>3/2$, \begin{equation}\label{sp-heat-counts} \begin{gathered} \frac1{K_N}\#\Big\{i:2-\lambda_i\le\frac y{h_N}\Big\}\longrightarrow\frac2{3\pi}y^{3/2}\quad\text{in probability},\\ \sum_{i=1}^Ne^{-h_N(2-\lambda_i)}=\frac N{2\sqrt\pi\,h_N^{3/2}}\big(1+o_{\PP}(1)\big),\\ \sum_{i=1}^N\big(1+h_N(2-\lambda_i)\big)^{-k}=K_NC_k\big(1+o_{\PP}(1)\big), \end{gathered} \end{equation} where $C_k=\pi^{-1}B(3/2,k-3/2)$. In the last sum, the finitely many terms with $1+h_N(2-\lambda_i)\le1/2$ are excluded, and with probability tending to one there are none. \end{enumerate} In the third part, the integral is taken on the event of the second part that all atoms lie in $(-1/2,\infty)$; on the complementary event, we define it to be $0$. Any other measurable finite choice gives the same convergence in probability, and the same convention applies to $\varphi_N$. \end{theorem} \begin{proof} The proof has four steps. We first prove the count asymptotic for $\alpha_N=1$, $\xi_N=0$, at a fixed $y>0$, using the simultaneous location bound. We then use the uniform upper bound \eqref{sp-count-envelope} on the counting function to bound the counts uniformly in $y$, add the shift and the factor $\alpha_N$, and integrate. \paragraph{Step 1: counts without shift.} Let $F^0_N(y)=K_N^{-1}\#\{i:h_N(2-\lambda_i)\le y\}$, and fix $y>0$. We have $K_N\to\infty$ and $K_N=o(N)$. Fix $r\in(0,2y^{3/2}/(3\pi))$ and let $i_\pm=\lfloor(2y^{3/2}/(3\pi)\pm r)K_N\rfloor$. On the events of \Cref{sp-location-control}, for every fixed confidence $\eta>0$, \[ h_N|\lambda_{i_\pm}-\gamma_{i_\pm}|\le C_\eta h_NN^{-2/3}(1+i_\pm)^{1/4} \le C_{\eta,y,r}K_N^{-5/12}\longrightarrow0, \] because $h_NN^{-2/3}=K_N^{-2/3}$. By \Cref{lem:spec-quantiles}, $h_N(2-\gamma_{i_\pm})\to(y^{3/2}\pm3\pi r/2)^{2/3}$. Thus, for all large $N$ on these events, ordering brackets $K_NF_N^0(y)$ between $i_-$ and $i_+$. Letting $r\downarrow0$ and then $\eta\downarrow0$ proves $F_N^0(y)\to2y^{3/2}/(3\pi)$ in probability, however slowly $K_N$ tends to infinity. For $y\le0$, we have $F^0_N(y)\le K_N^{-1}\#\{i:\lambda_i\ge2\}$, which tends to zero in probability by \eqref{sp-count-envelope} and \Cref{lem:model-localization}. Altogether, $F^0_N(y)\to F(y)=2y_+^{3/2}/(3\pi)$ for every $y\in\R$. Each $F^0_N$ is nondecreasing and $F$ is continuous. Convergence at the points of a finite mesh with spacing $\varrho$ of a compact interval $Y'$, together with monotonicity, then gives $\sup_{Y'}|F^0_N-F|\le o_{\PP}(1)+\sup\{|F(y)-F(y')|:y,y'\in Y',|y-y'|\le\varrho\}$. Letting $\varrho\to0$, we obtain $\sup_{Y'}|F^0_N-F|\to0$ in probability. \paragraph{Step 2: a uniform bound on the counts.} On the events of \Cref{sp-fixed-confidence-geometry} with confidence $\eta/2$, the last bound of \eqref{sp-count-envelope} with $u=y/h_N$ shows that $F_N^0(y)\le C(K_N^{-1}+y^{3/2})$ for all $y\ge0$, and by the first bound we have $h_N(2-\lambda_i)\ge-Ah_NN^{-2/3}$, which tends to zero. Let $F_N(y)=\omega_N((-\infty,y])$. On the further event $\{|h_N\xi_N|\le1/8,\ \alpha_N\in[1/2,2]\}$, whose probability tends to one, every atom $\alpha_N(h_N(2-\lambda_i)-h_N\xi_N)$ of $\omega_N$ is larger than $-1/2$ for $N$ large, and for $y\ge1$, \[ F_N(y)=F_N^0\big(y/\alpha_N+h_N\xi_N\big)\le F_N^0(2y+1)\le C\big(K_N^{-1}+(3y)^{3/2}\big). \] This proves the second part. \paragraph{Step 3: shift and factor.} Fix a compact interval $Y$ and $\varrho\in(0,1)$. With probability tending to one, $|y/\alpha_N+h_N\xi_N-y|\le\varrho$ for all $y\in Y$, and then $F^0_N(y-\varrho)\le F_N(y)\le F^0_N(y+\varrho)$ by monotonicity. By Step 1 on the interval $Y'=Y+[-1,1]$ and the uniform continuity of $F$, we obtain $\sup_Y|F_N-F|\le o_{\PP}(1)+\sup_{y\in Y}|F(y\pm\varrho)-F(y)|$. Letting $\varrho\to0$ proves the first part. \paragraph{Step 4: integration.} Let $\varphi$ be as in the third part, with $\sup_{[-1/2,0]}|\varphi|+\sup_{y\ge0}(1+y)^k|\varphi(y)|\le\Lambda$. Fix $\eta,\varepsilon>0$ and work on the events of the second part. For $R\ge1$, we decompose $(R,\infty)$ into the intervals $(2^jR,2^{j+1}R]$ and find \[ \int_{(R,\infty)}|\varphi|\,d\omega_N\le\Lambda\sum_{j\ge0}(2^jR)^{-k}F_N(2^{j+1}R) \le C_{\eta,k}\Lambda\big(R^{3/2-k}+K_N^{-1}R^{-k}\big), \] and also $\pi^{-1}\int_R^\infty|\varphi(y)|\sqrt y\,dy\le C_k\Lambda R^{3/2-k}$. Since $k>3/2$, we may choose $R$, depending on $\eta,\varepsilon,k,\Lambda$, such that both tails are at most $\varepsilon$ for all large $N$. On $[-1/2,R]$, we fix a mesh $-1/2=y_00$ be deterministic with \begin{equation}\label{mc-time-hypotheses} \delta_N^2t_N\to\infty,\qquad \delta_N^2t_N=o\big(M_N^{2/3}\big). \end{equation} The second condition is equivalent to $t_N=o(N^{2/3})$, because $M_N^{2/3}=N^{2/3}\delta_N^2$. Note that $c_N(\lambda_1-\lambda_i)=p_i-d_N$ in terms of the precisions \eqref{sp-canonical-definition}. \begin{corollary}[Gaps below the top eigenvalue at a smaller scale]\label{mc-raw-edge-measure} Assume $\delta_N\to0$, $M_N\to\infty$, and \eqref{mc-time-hypotheses}. Let \[ \omega_N^{\mathrm{gap}}=\frac{t_N^{3/2}}N\sum_{i=1}^N\delta_{t_Nc_N(\lambda_1-\lambda_i)} . \] Then $\omega_N^{\mathrm{gap}}$ converges vaguely in probability to $\pi^{-1}\sqrt y\,dy$ on $[0,\infty)$. If $\varphi:[0,\infty)\to\R$ is continuous with $\sup_{y\ge0}(1+y)^2|\varphi(y)|<\infty$, then \[ \int\varphi\,d\omega_N^{\mathrm{gap}}\longrightarrow\frac1\pi\int_0^\infty\varphi(y)\sqrt y\,dy\quad\text{in probability}. \] The same conclusion holds for deterministic continuous $\varphi_N$ that converge to $\varphi$ uniformly on compact subsets of $[0,\infty)$ and satisfy $\sup_N\sup_{y\ge0}(1+y)^2|\varphi_N(y)|<\infty$. The atom of $\omega^{\mathrm{gap}}_N$ at $0$ has mass $t_N^{3/2}/N\to0$. \end{corollary} \begin{proof} We use \Cref{sp-mesoscopic-counts} with $h_N=t_N$, $\alpha_N=c_N$, and $\xi_N=2-\lambda_1$. Then $K_N=Nt_N^{-3/2}$ and $\omega_N=\omega_N^{\mathrm{gap}}$. Its hypotheses hold. Indeed, $t_N\to\infty$ because $\delta_N\to0$ and $\delta_N^2t_N\to\infty$, we have $t_N=o(N^{2/3})$ by \eqref{mc-time-hypotheses}, and $h_N\xi_N=t_NN^{-2/3}\cdot N^{2/3}(2-\lambda_1)\to0$ in probability, since $N^{2/3}(2-\lambda_1)$ is tight by \Cref{sp-goe-rigidity}. All atoms are nonnegative. We may therefore extend $\varphi$ and $\varphi_N$ to $[-1/2,0)$ by their values at $0$, and the claims follow from the third part of \Cref{sp-mesoscopic-counts} with $k=2$. \end{proof} Only $t_N\to\infty$, $t_N=o(N^{2/3})$ and $c_N\to1$ were used in this proof. \section{The Lagrange multiplier at high temperature}\label{sgl-scalar-section}\label{sec:spec-hot} At high temperature, $0\le c<1$, the Lagrange multiplier is determined by the Stieltjes transform of the empirical eigenvalue distribution. Indeed, for $c>0$, set $E_N=z_N/c$. Then $E_N>\lambda_1$, and \eqref{sp-canonical-definition} is equivalent to \begin{equation}\label{eq:spec-root-equation} \mathfrak m_N(E_N)=c . \end{equation} For $00$, and let $f(x)=\sum_{i=1}^nw_i(x-\lambda_i)^{-1}$ for $x>\lambda_1$. Let $x_*\in\R$ and $h>0$ with $x_*-h>\lambda_1$, let $I=[x_*-h,x_*+h]$, and let $\bar f\in C^1(I)$ with $\Lambda=-\bar f'(x_*)>0$ and \[ (1-\alpha)\Lambda\le-\bar f'(x)\le(1+\alpha)\Lambda\qquad(x\in I) \] for some $\alpha\in[0,1)$. \begin{enumerate} \item If $|f(x_*\pm h/2)-\bar f(x_*\pm h/2)|\le e$ for both signs, where $e<(1-\alpha)\Lambda h/2$, then $\bar f(x_*)>0$, the equation $f(x)=\bar f(x_*)$ has a unique solution $x_f\in(\lambda_1,\infty)$, and $|x_f-x_*|\bar f(x_*),\qquad \bar f(x_*)>\bar f(x_*+a)+e_a\ge f(x_*+a)>0 . \] The function $f$ is continuous and strictly decreasing on $(\lambda_1,\infty)$, with limits $+\infty$ and $0$. As the level $\bar f(x_*)$ is positive, it is attained exactly once, at a point of $(x_*-a,x_*+a)$. The first part is the case $a=h/2$. For the second part, the function $q=-f'=\sum_iw_i(x-\lambda_i)^{-2}$ is positive and decreasing on $(\lambda_1,\infty)$. We have $x_*-h0$ there exist $L,N_0$ such that for all $N\ge N_0$ and all deterministic $E$ with $|E-E_*|\le\upsilon/2$, \begin{equation}\label{sgl-one-resolvent-error} \PP\Big(E\le\lambda_1\ \text{ or }\ \big[E>\lambda_1\text{ and }|\mathfrak m_N(E)-\mathfrak m_{\mathrm{sc}}(E)|>L\,\delta e\big]\Big)\le\varepsilon . \end{equation} Here $\mathfrak m_N(E)$ is considered only on the event $E>\lambda_1$, where it is defined. This is a bound for each deterministic point, uniform in its choice, and not a bound for the supremum over a continuum of points. We apply it below at four points. Let $u=N^{2/3}(E-2)$, so that $u\asymp M^{2/3}$ uniformly over these deterministic points. For large $N$ it exceeds the fixed threshold in \Cref{sp-published-counts}. On its common confidence event $\mathcal R_N$, $E>\lambda_1$, and \eqref{sp-local-resolvent}, divided by $\delta=M^{1/3}N^{-1/3}$, gives \[ \begin{aligned} \frac1\delta\E\big[\ind_{\mathcal R_N}|\mathfrak m_N(E)-\mathfrak m_{\mathrm{sc}}(E)|\big] &\le C_\varepsilon M^{-1/3}\Big(M^{-1/12}+N^{-1/3}M^{-1/3}+N^{-1/6}+M^{-2/3}\Big)\\ &=C_\varepsilon e. \end{aligned} \] Choose the event with failure probability at most $\varepsilon/2$, then apply Markov's inequality with a sufficiently large $L$. This proves \eqref{sgl-one-resolvent-error}; the constants are uniform in the deterministic choice of $E$. \paragraph{Step 3: the root.} Fix $\varepsilon>0$, let $L$ be as in \eqref{sgl-one-resolvent-error}, and set $a=16L\upsilon e$, which is at most $h_0\upsilon$ for $N$ large. By \eqref{sgl-one-resolvent-error} at the two points $E_*\pm a$, with probability at least $1-2\varepsilon-o(1)$ both errors are at most $L\delta e$, and $I_0\subset(\lambda_1,\infty)$. The threshold in the third part of \Cref{sgl-five-point-bracket} with $\alpha=1/2$ is $\Lambda_Na/2\ge2L\delta e$, by Step 1. Recall that $\mathfrak m_N(E_N)=c=\mathfrak m_{\mathrm{sc}}(E_*)$. The lemma implies that $|E_N-E_*|<16L\upsilon e$, and since $\varepsilon$ is arbitrary, this proves \eqref{sgl-quantitative-root}. \paragraph{Step 4: the derivative and the consequences.} Fix $\varepsilon>0$ and $L$ as before, and apply the first two parts of \Cref{sgl-five-point-bracket} with $h=\upsilon\sqrt e\le h_0\upsilon$ and $\alpha=C_0\sqrt e$. With probability at least $1-4\varepsilon-o(1)$, the errors at the four points $E_*\pm h/2$, $E_*\pm h$ are at most $L\delta e$. The condition $L\delta e<(1-\alpha)\Lambda_Nh/2$ holds for $N$ large, since $\Lambda_Nh\ge\delta\sqrt e/4$ and $e\to0$. By \eqref{sgl-five-point-derivative}, \[ \Big|\frac{-\mathfrak m_N'(E_N)}{\Lambda_N}-1\Big|\le C_0\sqrt e+\frac{4L\delta e}{\Lambda_N\upsilon\sqrt e}\le(C_0+16L)\sqrt e . \] We have $N^{-1}\sum_iq_i^2=c^{-2}N^{-1}\sum_i(E_N-\lambda_i)^{-2}=c^{-2}(-\mathfrak m_N'(E_N))$ and $c^{-2}\Lambda_N=(1-c^2)^{-1}$, which gives \eqref{sgl-quantitative-trace}. For the consequences, $1\le M\le N$ eventually, so $e\le4M^{-5/12}$. Thus $\ell^2e\to0$ and $\ell\sqrt e\to0$. Since $z_N-(1+c^2)=c(E_N-E_*)=O_{\PP}(\delta^2e)$, the first limit in \eqref{sgl-log-precise-consequences} follows. The second follows from \eqref{sgl-quantitative-trace}. Finally, since $1+c^2-\delta^2=2c$, \begin{equation}\label{eq:spec-d-identity} d_N-\delta^2=\big(z_N-(1+c^2)\big)+c\,(2-\lambda_1). \end{equation} The last term is $O_{\PP}(N^{-2/3})$ by \Cref{sp-goe-rigidity}, and $N^{-2/3}/\delta^2=M^{-2/3}$. Dividing \eqref{eq:spec-d-identity} by $\delta^2$, we find $\ell|d_N/\delta^2-1|=O_{\PP}(\ell^2e+\ell M^{-2/3})\to0$. \end{proof} At a fixed temperature the same comparison is simpler, because the edge stays at a fixed distance from $E_*$. \begin{lemma}[The Lagrange multiplier at fixed high temperature]\label{lem:spec-fixed-inversion} Fix $00$. On $F_C$ of \Cref{sp-goe-rigidity}, all denominators on $I$ stay uniformly positive for large $N$. Comparing eigenvalues with quantiles, and then using monotone semicircle quadrature, gives \[ \sup_{E\in I}|\mathfrak m_N(E)-\mathfrak m_{\mathrm{sc}}(E)|\le C_c(CN^{-5/12}+N^{-1}). \] The root bracket \Cref{sgl-five-point-bracket} gives $|E_N-E_*|\le C_{c,C}N^{-5/12}$ on $F_C$. Taking confidence arbitrarily close to one proves the stated multiplier rate. For the derivative bracket take $h=N^{-5/24}$; the relative derivative error is bounded by $C_{c,C}(h+N^{-5/12}/h)\to0$ on $F_C$. Since $N^{-1}\sum_iq_i^2=c^{-2}(-\mathfrak m_N'(E_N))$ and $c^{-2}\Lambda=(1-c^2)^{-1}$, this gives the trace limit. \end{proof} We collect the consequences used in the high-temperature sections. \begin{theorem}[The Lagrange multiplier above the critical temperature]\label{sp-canonical-saddle} \leavevmode \begin{enumerate} \item For fixed $03/2$, \begin{equation}\label{sp-inverse-traces} \begin{gathered} \sum_{i=1}^Nq_i^k=Nc_N^{-3/2}\delta_N^{3-2k}C_k\big(1+o_{\PP}(1)\big),\qquad C_k=\frac1\pi B\Big(\frac32,k-\frac32\Big),\\ \frac{\max_iq_i}{\sqrt{2\sum_iq_i^2}}=M_N^{-1/2}\big(1+o_{\PP}(1)\big). \end{gathered} \end{equation} \item In the setting of the second part, fix $A>0$, and let $J_N$ be the number of indices $i$ with $2-\lambda_i\le A\delta_N^2/\ell_N$. Then \begin{equation}\label{sp-growing-head} J_N=\frac{2A^{3/2}}{3\pi}\,\frac{M_N}{\ell_N^{3/2}}\big(1+o_{\PP}(1)\big),\qquad \frac{\sum_{i\le J_N}q_i^2}{\sum_{i=1}^Nq_i^2}=\frac{4A^{3/2}}{3\pi}\,\ell_N^{-3/2}\big(1+o_{\PP}(1)\big). \end{equation} \end{enumerate} \end{theorem} \begin{proof} In the first part, the statements on $z_N$ and on $\sum_iq_i^2$ are contained in \Cref{lem:spec-fixed-inversion}. The limits of $d_N$ and $s_N$ follow from the identities $d_N=(z_N-(1+c^2))+(1-c)^2+c(2-\lambda_1)$ and $s_N=(z_N-(1+c^2))+(1+c)^2-c(2+\lambda_N)$, together with $\lambda_1\to2$ and $\lambda_N\to-2$ in probability (\Cref{sp-goe-rigidity}). In the second part, all statements except $s_N\to4$ are \eqref{sgl-log-precise-consequences} and \eqref{sgl-edge-precise-consequences}, and $s_N\to4$ follows from the same identity for $s_N$. For the third part, write $\upsilon_N=\delta_N^2/c_N$ and $E_N=z_N/c_N$ as in \Cref{sgl-canonical-quantitative}, and set $y_i=(2-\lambda_i)/\upsilon_N$ and $\vartheta_N=(E_N-2)/\upsilon_N$. Then $q_i=c_N^{-1}(E_N-\lambda_i)^{-1}=c_N^{-1}\upsilon_N^{-1}(\vartheta_N+y_i)^{-1}$, and $\vartheta_N=1+O_{\PP}(e_N)\to1$ in probability by \eqref{sgl-quantitative-root}. We apply \Cref{sp-mesoscopic-counts} with $h_N=\upsilon_N^{-1}$, $\alpha_N=1$, and $\xi_N=0$; this is admissible because $h_N\to\infty$ and $h_N/N^{2/3}=c_N/M_N^{2/3}\to0$, and then $K_N=N\upsilon_N^{3/2}$. With probability tending to one, $\min_iy_i\ge-1/4$ and $|\vartheta_N-1|\le1/4$, and then, by the mean value theorem, \[ \big|(\vartheta_N+y_i)^{-k}-(1+y_i)^{-k}\big|\le k\,2^{k+2}\,|\vartheta_N-1|\,(1+y_i)^{-k}, \] since every point between $\vartheta_N+y_i$ and $1+y_i$ is at least $(1+y_i)/2$, and $1+y_i\ge3/4$. The fourth part of \Cref{sp-mesoscopic-counts} gives $K_N^{-1}\sum_i(1+y_i)^{-k}\to C_k$ in probability. We conclude that $\sum_i(\vartheta_N+y_i)^{-k}=K_NC_k(1+o_{\PP}(1))$, and \[ \sum_{i=1}^Nq_i^k=c_N^{-k}\upsilon_N^{-k}K_NC_k\big(1+o_{\PP}(1)\big)=Nc_N^{-k}\upsilon_N^{3/2-k}C_k\big(1+o_{\PP}(1)\big), \] and $c_N^{-k}\upsilon_N^{3/2-k}=c_N^{-3/2}\delta_N^{3-2k}$. For the ratio, we have $\max_iq_i=q_1=d_N^{-1}=\delta_N^{-2}(1+o_{\PP}(1))$ by the second part, and $\sum_iq_i^2=(N/(2\delta_N))(1+o_{\PP}(1))$ by the case $k=2$, where $C_2=1/2$. These two asymptotics give $q_1/\sqrt{2\sum_iq_i^2}=\delta_N^{-2}(N/\delta_N)^{-1/2}(1+o_{\PP}(1))=M_N^{-1/2}(1+o_{\PP}(1))$. For the fourth part, apply \Cref{sp-mesoscopic-counts} with $h_N=\ell_N/(A\delta_N^2)$, $\alpha_N=1$, and $\xi_N=0$. This is admissible because $h_N\to\infty$ and $h_N/N^{2/3}=\ell_N/(AM_N^{2/3})\to0$. Here $K_N=A^{3/2}M_N/\ell_N^{3/2}$, and the asymptotic for $J_N$ is the count at $y=1$. The eigenvalues are ordered, and the selected indices are $1,\dots,J_N$. For these indices, by \eqref{eq:spec-d-identity}, \[ p_i=z_N-c_N\lambda_i=\delta_N^2+\big(z_N-(1+c_N^2)\big)+c_N(2-\lambda_i), \] where $z_N-(1+c_N^2)=o_{\PP}(\delta_N^2)$, $0\le c_N(2-\lambda_i)\le A\delta_N^2/\ell_N=o(\delta_N^2)$ when $\lambda_i\le2$, and $c_N(2-\lambda_i)\ge-c_N(\lambda_1-2)=O_{\PP}(N^{-2/3})=o_{\PP}(\delta_N^2)$ otherwise. Thus $q_i^2=\delta_N^{-4}(1+o_{\PP}(1))$ uniformly for $i\le J_N$. Dividing $J_N\delta_N^{-4}(1+o_{\PP}(1))$ by $\sum_iq_i^2=(N/(2\delta_N))(1+o_{\PP}(1))$ we arrive at the second claim, since $2J_N/(N\delta_N^3)=2J_N/M_N$. \end{proof} \begin{remark}[Scales of the precisions and counts when $t_N=o(N^{2/3})$]\label{rem:spec-small-time-scales} In the setting of the second part of \Cref{sp-canonical-saddle}, assume also \eqref{mc-time-hypotheses}. Then, in probability, \begin{equation}\label{mc-scales} \frac{d_Nt_N}{\delta_N^2t_N}\to1,\qquad \frac{Nd_N^{3/2}}{M_N}\to1,\qquad t_N=o(N^{2/3}),\qquad Nt_N^{-3/2}\to\infty,\qquad \frac{t_N}{N\delta_N}=\frac{\delta_N^2t_N}{M_N}\to0 . \end{equation} \end{remark} \section{The critical window}\label{sec:spec-critical} In the critical window, $c_N=1+\sigma N^{-1/3}$, the Lagrange multiplier is determined by the gaps at the top of the spectrum. This section first studies the GOE Airy point process and the limiting equation for the multiplier, and then the half-precisions at finite $N$. \paragraph{The limiting equation.} Recall the reflected GOE Airy point process $\chi_1<\chi_2<\cdots$ of \Cref{sp-goe-rigidity}, the limit of $N^{2/3}(2-\lambda_i)$, its gaps $\Delta_i=\chi_i-\chi_1$, the regularized inverse-gap statistic $\Xi$ of \Cref{sp-raw-gap-resolvent}, and $t_k=(3\pi k/2)^{2/3}$. By \cite[Proposition~1.4]{DuHuang2026}, almost surely $\chi_k/t_k\to1$, and the function \begin{equation}\label{sp-airy-root-function} \mathscr S(x)=\Xi+\frac1x+\sum_{i=2}^\infty\Big(\frac1{x+\Delta_i}-\frac1{\Delta_i}\Big) =\lim_{k\to\infty}\Big(\sum_{i=1}^k\frac1{x+\Delta_i}-\frac2\pi\sqrt{t_k}\Big)\qquad(x>0) \end{equation} is well defined, strictly decreasing on $(0,\infty)$, and tends to $+\infty$ as $x\downarrow0$ and to $-\infty$ as $x\to\infty$. We prove these facts, with quantitative bounds, in \Cref{sp-airy-counts} below. For $\sigma\in\R$, we let $\varrho_\sigma>0$ be the unique solution of $\mathscr S(\varrho_\sigma)=\sigma$. We set \begin{equation}\label{eq:spec-limiting-half-precisions} R_1^\sigma=\frac{\varrho_\sigma}2,\qquad R_i^\sigma=\frac{\Delta_i+\varrho_\sigma}2\quad(i\ge2). \end{equation} We call the $R_i^\sigma$ the limiting scaled half-precisions. In the third part of \Cref{cg-canonical-weyl} below, we prove that the scaled half-precisions $N^{2/3}r_i$ converge to them in distribution, jointly for finitely many $i$. The map $\sigma\mapsto\varrho_\sigma$ is a continuous strictly decreasing bijection of $\R$ onto $(0,\infty)$. At $\sigma=0$, the root $\varrho_0$ is the quantity $\Delta(\chi)$ of \cite[Proposition~1.4(b)]{DuHuang2026}, which describes the Gibbs measure at the critical temperature. \begin{theorem}[{The GOE Airy point process and the limiting equation (cf.\ \cite[Proposition~1.4]{DuHuang2026})}]\label{sp-airy-counts}\label{sp-airy-saddle} Let $C_A(T)=\#\{i:\chi_i\le T\}$ and $\mathcal E(T)=C_A(T)-2T^{3/2}/(3\pi)$. There exists an event of probability one on which the following hold, with random constants $C_\omega$ (and $C_{\omega,k}$, depending also on $k$). \begin{enumerate} \item $|\mathcal E(T)|\le C_\omega(1+T)^{7/8}$ for all $T\ge0$, and $|\chi_k-t_k|\le C_\omega t_k^{3/8}$ for all $k\ge1$. In particular, $C_\omega^{-1}(i-1)^{2/3}\le\Delta_i\le C_\omega(i-1)^{2/3}$ for $i\ge2$, and $\sum_{i\ge2}\Delta_i^{-2}<\infty$. \item The series in \eqref{sp-airy-root-function} converges locally uniformly on $(0,\infty)$, together with its derivative, both expressions in \eqref{sp-airy-root-function} agree, and $\mathscr S$ is continuously differentiable and strictly decreasing, with $\mathscr S(x)\to+\infty$ as $x\downarrow0$ and $\mathscr S(x)\to-\infty$ as $x\to\infty$. For $x>\max(0,\chi_1)$ and $t=x-\chi_1$, \begin{equation}\label{sp-airy-stieltjes} \mathscr S(x)=-\sqrt t+\int_0^\infty\frac{\mathcal E(u)}{(t+u)^2}\,du+B_\omega(t),\qquad B_\omega(t)=\sum_{i:\chi_i<0}\Big(\frac1{t+\chi_i}-\frac1t\Big). \end{equation} \item As $h\downarrow0$ and $x\to\infty$, respectively, and for each fixed $k>3/2$, \begin{equation}\label{sp-airy-traces} \begin{gathered} \sum_{i\ge1}e^{-h\Delta_i}=\frac{1+O_\omega(h^{5/8})}{2\sqrt\pi\,h^{3/2}},\qquad -\mathscr S'(x)=\frac{1+O_\omega(x^{-5/8})}{2\sqrt x},\\ \sum_{i\ge1}\frac1{(x+\Delta_i)^k}=C_kx^{3/2-k}\big(1+O_{\omega,k}(x^{-5/8})\big), \end{gathered} \end{equation} where $C_k=\pi^{-1}B(3/2,k-3/2)$. \item As $S\to\infty$, we have $\varrho_{-S}=S^2+O_\omega(S^{3/4})$ on this event. Moreover, with probability over the GOE Airy point process, \begin{equation}\label{sp-outer-airy-saddle} \varrho_{-S}=S^2+\chi_1+O_{\PP}\Big(S^{3/4}\Big)\qquad(S\to\infty). \end{equation} \end{enumerate} \end{theorem} \usedin{The \hyperref[sec:outer-hot]{high-temperature} and \hyperref[sec:outer-cold]{low-temperature} limits of the critical profile.} \begin{proof} Let $a=2/(3\pi)$. We first transfer the simultaneous location bound to the Airy limit and derive its counting asymptotics. We then use Stieltjes integration to study $\mathscr S$, the traces, and the root. \paragraph{Step 1: the points and the counting function.} Fix $\eta\in(0,1)$ and use \Cref{sp-location-control}. For each fixed $L$, the vector $(N^{2/3}(2-\lambda_i))_{i\le L}$ converges in law to $(\chi_i)_{i\le L}$, and $q_{i,N}=N^{2/3}(2-\gamma_i)\to t_i$. Hence, for every $\varepsilon>0$ and all large $N$, \[ \PP\big(|N^{2/3}(2-\lambda_i)-t_i|\le C_\eta(1+i)^{1/4}+\varepsilon\text{ for }1\le i\le L\big)\ge1-\eta. \] The set defined by these inequalities is closed. The portmanteau theorem, followed by $\varepsilon\downarrow0$ and then $L\to\infty$, gives \[ \PP\big(|\chi_i-t_i|\le C_\eta(1+i)^{1/4}\text{ for all }i\ge1\big)\ge1-\eta. \] Taking the union of these events for $\eta=2^{-j}$, $j\ge1$, gives an event of probability one on which $|\chi_i-t_i|\le C_\omega t_i^{3/8}$ for all $i$. In particular, $\chi_i/t_i\to1$. Intersect this event with the probability-one events of \Cref{sp-goe-rigidity,sp-raw-gap-resolvent} on which the $\chi_i$ are distinct, tend to infinity, and $\Xi$ exists. For large $T$, write $k=C_A(T)$, so that $\chi_k\le T<\chi_{k+1}$. The point asymptotics first imply $t_k\asymp T\asymp t_{k+1}$, and the point bounds then give \[ t_k\le T+C_\omega T^{3/8},\qquad t_{k+1}\ge T-C_\omega T^{3/8}. \] Using $k=at_k^{3/2}$ and $k+1=at_{k+1}^{3/2}$ in these inequalities proves $|k-aT^{3/2}|\le C_\omega T^{7/8}$. The counting function is finite on bounded intervals, so enlarging $C_\omega$ proves $|\mathcal E(T)|\le C_\omega(1+T)^{7/8}$ for every $T\ge0$. \paragraph{Step 2: the gaps.} The point bound gives $\Delta_i=\chi_i-\chi_1\asymp_\omega i^{2/3}$ for large $i$. The finitely many remaining $\Delta_i$, $i\ge2$, are positive, so the two-sided bound in the first part follows, as does $\sum_{i\ge2}\Delta_i^{-2}<\infty$. For all sufficiently large $k$, we also have \begin{equation}\label{eq:spec-sqrt-difference} \sqrt{\chi_k}-\sqrt{t_k}=\frac{\chi_k-t_k}{\sqrt{\chi_k}+\sqrt{t_k}}=O_\omega(t_k^{-1/8})\longrightarrow0 . \end{equation} \paragraph{Step 3: the function $\mathscr S$.} For $x>0$ and $i\ge2$, we have $|(x+\Delta_i)^{-1}-\Delta_i^{-1}|\le x\Delta_i^{-2}$ and $|\partial_x(x+\Delta_i)^{-1}|\le\Delta_i^{-2}$. By the first part, the series in \eqref{sp-airy-root-function} and its termwise derivative converge uniformly on compact subsets of $(0,\infty)$, and $\mathscr S$ is continuously differentiable with \[ -\mathscr S'(x)=\sum_{i\ge1}(x+\Delta_i)^{-2}>0 . \] The two expressions in \eqref{sp-airy-root-function} agree because the partial sums of the second expression equal $x^{-1}+\sum_{i=2}^k((x+\Delta_i)^{-1}-\Delta_i^{-1})+(\sum_{i=2}^k\Delta_i^{-1}-2\sqrt{t_k}/\pi)$, and the last bracket tends to $\Xi$ by \eqref{sp-regularized-edge}. As $x\downarrow0$, we have $\mathscr S(x)\to+\infty$, because the term $x^{-1}$ tends to infinity and the series stays bounded. Now fix $x>\max(0,\chi_1)$ and let $t=x-\chi_1>0$, so that $x+\Delta_i=t+\chi_i>0$ for all $i$. For a real cutoff $T>0$, we have the exact identity \begin{equation}\label{eq:spec-finite-stieltjes} \sum_{i:\chi_i\le T}\frac1{t+\chi_i}=\frac{C_A(T)}{t+T}+\int_0^T\frac{C_A(u)}{(t+u)^2}\,du+B_\omega(t). \end{equation} Indeed, a point $\chi_i\in[0,T]$ contributes $(t+T)^{-1}+\int_{\chi_i}^T(t+u)^{-2}\,du=(t+\chi_i)^{-1}$ to the first two terms on the right, and a point $\chi_i<0$ contributes $(t+T)^{-1}+\int_0^T(t+u)^{-2}\,du=t^{-1}$, which $B_\omega(t)$ corrects to $(t+\chi_i)^{-1}$. Inserting $C_A(u)=au^{3/2}+\mathcal E(u)$ and integrating by parts in the continuum term, we obtain \[ \frac{aT^{3/2}}{t+T}+\int_0^T\frac{au^{3/2}}{(t+u)^2}\,du=\frac1\pi\int_0^T\frac{\sqrt u}{t+u}\,du=\frac2\pi\Big(\sqrt T-\sqrt t\arctan\sqrt{T/t}\Big), \] where we used $3a/2=1/\pi$. We obtain \[ \sum_{i:\chi_i\le T}\frac1{t+\chi_i}-\frac2\pi\sqrt T =-\frac2\pi\sqrt t\arctan\sqrt{T/t}+\frac{\mathcal E(T)}{t+T}+\int_0^T\frac{\mathcal E(u)}{(t+u)^2}\,du+B_\omega(t). \] As $T\to\infty$, the right side tends to the right side of \eqref{sp-airy-stieltjes}, since $\mathcal E(T)/(t+T)=O_\omega(T^{-1/8})$ and the integral converges absolutely by Step 1. On the left side, for $\chi_k\le T<\chi_{k+1}$, the sum equals $\sum_{i\le k}(x+\Delta_i)^{-1}$, and $2(\sqrt T-\sqrt{t_k})/\pi\to0$ by \eqref{eq:spec-sqrt-difference} applied at $k$ and $k+1$, together with $\sqrt{t_{k+1}}-\sqrt{t_k}\to0$. By the second expression in \eqref{sp-airy-root-function}, the left side tends to $\mathscr S(x)$, and \eqref{sp-airy-stieltjes} follows. For $t\ge1$, Step 1 gives \[ \Big|\int_0^\infty\frac{\mathcal E(u)}{(t+u)^2}\,du\Big|\le C_\omega\int_0^\infty\frac{(1+u)^{7/8}}{(t+u)^2}\,du\le C_\omega t^{-1/8}, \] and $|B_\omega(t)|\le C_\omega t^{-2}$ for $t\ge2|\chi_1|$, since only finitely many points are negative and all are at least $\chi_1$. Combining these bounds with \eqref{sp-airy-stieltjes}, we find $\mathscr S(x)=-\sqrt{x-\chi_1}+O_\omega((x-\chi_1)^{-1/8})\to-\infty$. This completes the proof of the second part. \paragraph{Step 4: traces.} For $h\in(0,1]$, the same bookkeeping as in \eqref{eq:spec-finite-stieltjes} shows that \[ \sum_{i\ge1}e^{-h\chi_i}=h\int_0^\infty e^{-hu}C_A(u)\,du+\sum_{i:\chi_i<0}\big(e^{-h\chi_i}-1\big). \] The leading part is $ah\int_0^\infty e^{-hu}u^{3/2}\,du=a\Gamma(5/2)h^{-3/2}=(2\sqrt\pi)^{-1}h^{-3/2}$, the part coming from $\mathcal E$ is at most $C_\omega h\int_0^\infty e^{-hu}(1+u)^{7/8}\,du\le C_\omega h^{-7/8}$, and the finite sum is $O_\omega(h)$. Multiplying by $e^{h\chi_1}=1+O_\omega(h)$, we obtain the first formula in \eqref{sp-airy-traces}. For $k>3/2$ and $t=x-\chi_1\ge1$, similarly, \[ \sum_{i\ge1}\frac1{(t+\chi_i)^k}=\int_0^\infty\frac{k\,C_A(u)}{(t+u)^{k+1}}\,du+\sum_{i:\chi_i<0}\Big(\frac1{(t+\chi_i)^k}-\frac1{t^k}\Big). \] The leading part is $\pi^{-1}\int_0^\infty\sqrt u\,(t+u)^{-k}\,du=\pi^{-1}t^{3/2-k}B(3/2,k-3/2)=C_kt^{3/2-k}$, the part coming from $\mathcal E$ is at most $C_{\omega,k}t^{7/8-k}$, and the finite sum is $O_{\omega,k}(t^{-k-1})$. The third formula follows, because $x+\Delta_i=t+\chi_i$ and $t^{3/2-k}=x^{3/2-k}(1+O_{\omega,k}(x^{-1}))$. The second formula is the case $k=2$, since $-\mathscr S'(x)=\sum_i(x+\Delta_i)^{-2}$ and $C_2=\pi^{-1}B(3/2,1/2)=1/2$. \paragraph{Step 5: the root.} Let $x=\varrho_{-S}$ and $t=x-\chi_1$. We have $x\to\infty$ as $S\to\infty$, because $\mathscr S$ decreases from $+\infty$ to $-\infty$. From the end of Step 3, we have $\sqrt t=S+O_\omega(t^{-1/8})$. Then $t/S^2\to1$ and $\sqrt t-S=O_\omega(S^{-1/4})$, and multiplying by $\sqrt t+S\le3S$, we obtain $t-S^2=O_\omega(S^{3/4})$, that is, $\varrho_{-S}=S^2+\chi_1+O_\omega(S^{3/4})=S^2+O_\omega(S^{3/4})$. For \eqref{sp-outer-airy-saddle}, the pathwise bound just proved and the finiteness of $\chi_1$ imply that $S^{-3/4}(\varrho_{-S}-S^2-\chi_1)$ is bounded eventually on each sample path. The root is continuous in $S$, so its eventual bounds can be tested on rational $S$ and then localized by measurable events at any fixed confidence. This gives the asserted $O_{\PP}(S^{3/4})$ estimate. \end{proof} \paragraph{The half-precisions at finite $N$.} The critical sections use the following deterministic conditions on the half-precisions. Recall from \eqref{eq:model-precision-form} that the Gibbs measure is $\pi\propto\exp(-\sum_ir_ix_i^2)\nu_N$, where $r_i=p_i/2$. \begin{definition}[Critical spectral conditions]\label{def:spec-critical-conditions} Let $00$. A sequence $00$, $N>(2M)^3$, and $|\sigma|\le M$, let $c_{N,\sigma}=1+\sigma N^{-1/3}\in[1/2,3/2]$. Let $z_{N,\sigma}$ be the Lagrange multiplier of $c_{N,\sigma}W_N$, let $d_{N,\sigma}=z_{N,\sigma}-c_{N,\sigma}\lambda_1$ be the smallest precision, and let \[ r_{i,N,\sigma}=\frac{z_{N,\sigma}-c_{N,\sigma}\lambda_i}2,\qquad \varrho_{N,\sigma}=\frac{N^{2/3}d_{N,\sigma}}{c_{N,\sigma}} . \] Then $r_{1,N,\sigma}=d_{N,\sigma}/2$ and $N^{2/3}r_{i,N,\sigma}=c_{N,\sigma}(\varrho_{N,\sigma}+\Delta_{i,N})/2$, where $\Delta_{1,N}=0$. The condition $N>(2M)^3$ ensures $c_{N,\sigma}>0$. Without such a condition the analysis below fails at small $N$: if $c_{N,\sigma}<0$, then $c_{N,\sigma}\lambda_1$ is the smallest of the numbers $c_{N,\sigma}\lambda_i$, and the half-precisions are no longer ordered as above. \begin{lemma}[{Half-precisions in the critical window (cf.\ \cite[Proposition~4.1, (4.2)]{DuHuang2026})}]\label{cg-canonical-weyl} Fix $M>0$, and let $N>(2M)^3$. \begin{enumerate} \item For $|\sigma|\le M$, the root $\varrho_{N,\sigma}$ is the unique positive solution of $\mathscr S_N(\varrho_{N,\sigma})=\sigma$, where \begin{equation}\label{eq:spec-finite-root} \mathscr S_N(x)=\Xi_N+\frac1x+\sum_{i=2}^N\Big(\frac1{x+\Delta_{i,N}}-\frac1{\Delta_{i,N}}\Big), \end{equation} and $\Xi_N$ is as in \Cref{sp-raw-gap-resolvent}. The map $\sigma\mapsto\varrho_{N,\sigma}$ is strictly decreasing. \item For every $\eta\in(0,1)$, there exist $00$, depending only on $\eta$ and $M$, and events $\mathcal A_N$ with $\liminf_N\PP(\mathcal A_N)\ge1-\eta$, on which, for every $\sigma\in[-M,M]$ simultaneously, the sequence $(r_{i,N,\sigma})_{i=1}^N$ satisfies the critical spectral conditions \eqref{cg-weyl-hypotheses} with constants $(c_0,C_0,\delta)$. \item Let $\sigma_1,\dots,\sigma_m\in[-M,M]$. Let $\mathbf v_{N,\sigma}=((N^{2/3}r_{i,N,\sigma})^{-1})_{i\ge1}$, padded with zeros for $i>N$, and $\mathbf v_\sigma=((R_i^\sigma)^{-1})_{i\ge1}$, with $R^\sigma_i$ as in \eqref{eq:spec-limiting-half-precisions}, where $\varrho_\sigma$ solves \begin{equation}\label{cg-airy-root} \Xi+\frac1x-\sum_{i\ge2}\frac{x\,\Delta_i^{-2}}{1+x\Delta_i^{-1}}=\sigma . \end{equation} Then, jointly in distribution, \[ \big(\Xi_N,\mathbf b_N,(\varrho_{N,\sigma_j})_{j\le m},(\mathbf v_{N,\sigma_j})_{j\le m}\big)\Longrightarrow\big(\Xi,(\Delta_i^{-1})_{i\ge2},(\varrho_{\sigma_j})_{j\le m},(\mathbf v_{\sigma_j})_{j\le m}\big) \] in $\R\times\ell^2\times\R^m\times(\ell^2)^m$, with $\mathbf b_N$ as in \Cref{sp-raw-gap-resolvent}. All limits are functions of the same GOE Airy point process. \end{enumerate} \end{lemma} \usedin{The \hyperref[sec:inequalities]{functional inequalities}, \hyperref[sec:density]{density estimates}, \hyperref[sec:crit-process]{critical process}, \hyperref[sec:crit-profile]{critical profile}.} The left side of \eqref{cg-airy-root} is $\mathscr S(x)$, since $(x+\Delta_i)^{-1}-\Delta_i^{-1}=-x\Delta_i^{-2}/(1+x\Delta_i^{-1})$; thus $\varrho_\sigma$ is the root defined before \Cref{sp-airy-counts}. Similarly, $\mathscr S_N$ is $\mathscr S$ with the GOE Airy point process replaced by the scaled eigenvalues. \begin{proof} We write $c=c_{N,\sigma}$ and $\varrho_N=\varrho_{N,\sigma}$ when $\sigma$ is fixed. The proof has three steps: the finite equation, convergence through a deterministic continuous map, and the bounds at fixed confidence. \paragraph{Step 1: the finite equation.} We have $z_{N,\sigma}-c\lambda_i=cN^{-2/3}(\varrho_N+\Delta_{i,N})$, and the equation \eqref{sp-canonical-definition} reads $\sum_{i=1}^N(\varrho_N+\Delta_{i,N})^{-1}=cN^{1/3}=N^{1/3}+\sigma$. Subtracting $\sum_{i\ge2}\Delta_{i,N}^{-1}=\Xi_N+N^{1/3}$, which is finite because the eigenvalues are almost surely distinct, we obtain $\mathscr S_N(\varrho_N)=\sigma$. The function $\mathscr S_N$ is continuous and strictly decreasing on $(0,\infty)$, with limit $+\infty$ at $0$ and limit $\Xi_N-\sum_{i\ge2}\Delta_{i,N}^{-1}=-N^{1/3}$ at $\infty$. Since $\sigma\ge-M>-N^{1/3}$, the level $\sigma$ is attained exactly once, and the root is decreasing in $\sigma$. The condition $z_{N,\sigma}>c\lambda_1$ of \Cref{def:model-multiplier} is equivalent to $\varrho_N>0$. \paragraph{Step 2: convergence.} For $\xi\in\R$, a sequence $u=(u_i)_{i\ge2}\in\ell^2$ with $u_i\ge0$, and $x>0$, let \[ \Theta(\xi,u;x)=\xi+\frac1x-\sum_{i\ge2}\frac{xu_i^2}{1+xu_i}, \] so that $\mathscr S_N(x)=\Theta(\Xi_N,\mathbf b_N;x)$ and $\mathscr S(x)=\Theta(\Xi,(\Delta_i^{-1})_{i\ge2};x)$. For $0M$ and $\mathscr S(x_+)<-M$. For $N$ large, we have $\mathscr S_N(x_-)>M$ and $\mathscr S_N(x_+)<-M$, and by monotonicity all roots $\varrho_{N,\sigma}$, $|\sigma|\le M$, lie in $(x_-,x_+)$. For $\varepsilon\in(0,x_-/2)$, the quantity \[ \gamma_\varepsilon=\min_{|\sigma|\le M}\min\big(\mathscr S(\varrho_\sigma-\varepsilon)-\sigma,\ \sigma-\mathscr S(\varrho_\sigma+\varepsilon)\big) \] is positive, by continuity and compactness. Once $\sup_{[x_-/2,2x_+]}|\mathscr S_N-\mathscr S|<\gamma_\varepsilon$, we have $|\varrho_{N,\sigma}-\varrho_\sigma|<\varepsilon$ for all $|\sigma|\le M$, by monotonicity. We conclude that $\varrho_{N,\sigma}\to\varrho_\sigma$ almost surely, uniformly for $|\sigma|\le M$. Finally, \[ \big(N^{2/3}r_{1,N,\sigma}\big)^{-1}=\frac2{c\,\varrho_{N,\sigma}},\qquad \big(N^{2/3}r_{i,N,\sigma}\big)^{-1}=\frac2c\,\frac{\Delta_{i,N}^{-1}}{1+\varrho_{N,\sigma}\Delta_{i,N}^{-1}}\quad(i\ge2). \] The map $v\mapsto v/(1+xv)$ is $1$-Lipschitz on $[0,\infty)$, and $|v/(1+xv)-v/(1+x'v)|\le v^2|x-x'|$. These two bounds show that the map $(x,u)\mapsto(u_i/(1+xu_i))_{i\ge2}$ from $(0,\infty)\times\ell^2_+$ to $\ell^2$ is continuous, with modulus bounded by $\|u-w\|_2+\|u\|_4^2|x-x'|$. Since $c_{N,\sigma}\to1$, we conclude that $\mathbf v_{N,\sigma}\to\mathbf v_\sigma$ in $\ell^2$ almost surely, for each $\sigma$. This proves the third part. \paragraph{Step 3: bounds at fixed confidence.} Fix $\eta\in(0,1)$. By \Cref{sp-fixed-confidence-geometry} with confidence $\eta/2$, there exist $0