\chapter{Pair statistics at the scales \texorpdfstring{$N^{-1 - \alpha}$}{N\textasciicircum(-1-alpha)}}\label{s:pairs}

In this section, we prove \zcref{t:pairs} and \zcref{p:pairsl}. These results follow from a law of large numbers for the pair statistics of $H$ at the scale $\ell_\alpha = N^{-1 - \alpha}$, which we prove by induction on $\alpha$. Recall the scales $\ell_\alpha$ and $M_\alpha = N^{1 - 2\alpha} = N^3 \ell_\alpha^2$ from \eqref{e:prelim-scales}, the functions $f_p$ from \eqref{e:prelim-fp}, the pair statistic $\mathcal X_\alpha(A; \varphi, f)$ from \eqref{e:comparison-pair}, and the smoothed energy weight $\varphi_N$ of a nonnegative $\varphi \in C_c^\infty((-2, 2))$ from \eqref{e:comparison-phiN}, where $h_N = N^{-1/10}$. We also recall that $p_* = 61$.

We use the same notation for ordered sequences. For $x = (x_1, \dots, x_N) \in \R^N$ with $x_1 \le \dots \le x_N$, we define $\mathcal X_\alpha(x; \varphi, f)$ by the formula \eqref{e:comparison-pair}, with $\lambda_i(A)$ replaced by $x_i$ for all $i \in [N]$. Finally, we note that for all integers $p \ge 2$,
\begin{equation}\label{e:pairs-radial}
\int_0^\infty y f_p(y) \dd y = \frac{1}{2(p - 1)}.
\end{equation}

\begin{definition}\label{d:pairlaw}
Let $\alpha \in (0, 1/2)$. We say that the statement $\mathcal P(\alpha)$ holds if for every nonnegative $\varphi \in C_c^\infty((-2, 2))$ and every integer $p \ge p_*$, we have
\begin{equation*}
\frac{1}{M_\alpha} \mathcal X_\alpha(H; \varphi_N, f_p) \to \frac{1}{2(p - 1)} \int_{-2}^2 \varphi(E) c(E) \dd E
\end{equation*}
in probability as $N \to \infty$.
\end{definition}

The argument is outlined in \zcref{s:ov-pairs}. We prove $\mathcal P(\alpha)$ for $\alpha \in (0, 1/8)$ in \zcref{s:pairs-base}, extend $\mathcal P(\alpha)$ to general test functions and unsmoothed energy weights in \zcref{s:pairs-extension}, show in \zcref{s:pairs-induction} that $\mathcal P(\alpha)$ implies $\mathcal P(\alpha')$ for $\alpha'$ slightly larger than $\alpha$, and iterate this step in \zcref{s:pairs-iteration} to deduce \zcref{t:pairs} and \zcref{p:pairsl}.

We will use the following two lemmas, the first in \zcref{s:pairs-base} and \zcref{s:pairs-extension}, and the second in \zcref{s:pairs-base} and \zcref{s:pairs-induction}. The first compares pair statistics with smoothed and unsmoothed energy weights.

\begin{lemma}\label{l:pairs-unsmooth}
Let $\alpha \in (0, 1/2)$, let $\varphi \in C_c^\infty((-2, 2))$ be nonnegative, and let $\delta > 0$ be such that every $E \in \R$ with $\dist(E, \supp \varphi) \le \delta$ lies in $(-2, 2)$. Let $\chi \in C_c^\infty((-2, 2))$ satisfy $0 \le \chi \le 2$, and suppose that $\chi(E) = 2$ for all $E \in \R$ such that $\dist(E, \supp \varphi) \le \delta$. Then there exist $C > 0$ and $N_0 \ge 1$ such that for all $N \ge N_0$, all real symmetric $N \times N$ matrices $A$ with $\| A \| \le 3$, and all measurable $f \colon [0, \infty) \to \R$ with $|f| \le f_{p_*}$, we have
\begin{multline}\label{e:pairs-unsmooth}
\big| \mathcal X_\alpha(A; \varphi_N, f) - \mathcal X_\alpha(A; \varphi, f) \big| \\
\le C h_N \min\big\{ \mathcal X_\alpha(A; \chi, f_{p_*}), \mathcal X_\alpha(A; \chi_N, f_{p_*}) \big\} + C N^2 \exp\Big( -\frac{\delta^2}{4 h_N^2} \Big).
\end{multline}
\end{lemma}

\begin{proof}
We begin by recalling two consequences of parts (4) and (5) of \zcref{l:smoothing}. There exists $C > 0$ such that
\begin{equation}\label{e:pairs-smoothing-recall}
\| \varphi_N - \varphi \|_\infty + \| \chi_N - \chi \|_\infty \le C h_N, \qquad \sup_{E \in [-3, 3] : \dist(E, \supp \varphi) \ge \delta} \varphi_N(E) \le C \exp\Big( -\frac{\delta^2}{4 h_N^2} \Big).
\end{equation}
Let $N_0$ be such that $C h_N \le 1$ for $N \ge N_0$. Then $\chi_N(E) \ge 1$ for all $E$ such that $\dist(E, \supp \varphi) \le \delta$.

Fix $A$ and $f$ as in the statement. For $i < j$, the midpoint $E = (\lambda_i(A) + \lambda_j(A))/2$ belongs to $[-3, 3]$, since $\| A \| \le 3$, and we claim that
\begin{equation}\label{e:pairs-unsmooth-pointwise}
|\varphi_N(E) - \varphi(E)| \le C h_N \min\{ \chi(E), \chi_N(E) \} + C \exp\Big( -\frac{\delta^2}{4 h_N^2} \Big).
\end{equation}
When $\dist(E, \supp \varphi) \le \delta$, this follows from the first bound in \eqref{e:pairs-smoothing-recall} and the fact that $\chi(E)$ and $\chi_N(E)$ are at least $1$. Otherwise, we have $\varphi(E) = 0$, and \eqref{e:pairs-unsmooth-pointwise} follows from the second bound in \eqref{e:pairs-smoothing-recall}.

Next, we multiply \eqref{e:pairs-unsmooth-pointwise} by $|f((\lambda_j(A) - \lambda_i(A))/\ell_\alpha)|$, which is bounded by $f_{p_*}((\lambda_j(A) - \lambda_i(A))/\ell_\alpha) \le 1$, and we sum over the at most $N^2$ pairs $i < j$. A sum of minima is bounded by the minimum of the sums, so we obtain \eqref{e:pairs-unsmooth}.
\end{proof}

The second lemma transfers a law of large numbers for the pair statistic from $H_t$ to $H$.

\begin{lemma}\label{l:pairs-transfer}
Let $\alpha \in (0, 1/2)$, let $p \ge 1$ be an integer, let $\varphi \in C_c^\infty((-2, 2))$ be nonnegative, and let $L \ge 0$. Let $\eps > 0$ satisfy $\alpha + 5 \eps < 1/2$, and let $(t_N)_{N \ge 1}$ be a sequence in $[0, 1]$ such that
\begin{equation*}
\lim_{N \to \infty} t_N N^{1/2 + 3\alpha + 12 \eps} = 0.
\end{equation*}
If $M_\alpha^{-1} \mathcal X_\alpha(H_{t_N}; \varphi_N, f_p) \to L$ in probability, then $M_\alpha^{-1} \mathcal X_\alpha(H; \varphi_N, f_p) \to L$ in probability.
\end{lemma}

\begin{proof}
For every real symmetric matrix $A$, we have $\mathcal X_\alpha(A; \varphi_N, f_p) \ge 0$, since $\varphi_N \ge 0$ on $\R$ and $f_p > 0$. Fix $\theta > 0$. By assumption, the random variable $\exp(-\theta M_\alpha^{-1} \mathcal X_\alpha(H_{t_N}; \varphi_N, f_p))$ takes values in $(0, 1]$ and converges to $e^{-\theta L}$ in probability, so the bounded convergence theorem gives
\begin{equation*}
\lim_{N \to \infty} \E \exp\big( -\theta M_\alpha^{-1} \mathcal X_\alpha(H_{t_N}; \varphi_N, f_p) \big) = e^{-\theta L}.
\end{equation*}
Let $r_N = c_p N^{-\alpha - 4\eps}$ be as in \zcref{p:comparison}. Then
\begin{equation*}
t_N \big( N^{1/2} r_N^{-3} + r_N^{-2} \big) = c_p^{-3} t_N N^{1/2 + 3\alpha + 12 \eps} + c_p^{-2} t_N N^{2\alpha + 8\eps} \le \big( c_p^{-3} + c_p^{-2} \big) t_N N^{1/2 + 3\alpha + 12 \eps},
\end{equation*}
which tends to zero. We apply \zcref{p:comparison} with the index set $\{1\}$, $\varphi_1 = \varphi$, $p_1 = p$, $\theta_1 = \theta/M_\alpha$, $D = 1$, and $t = t_N$. By \zcref{l:holomorphic}, the observable $F$ of \zcref{d:complexS} is then given by
\begin{equation*}
F(A) = \exp\big( -\theta M_\alpha^{-1} \mathcal X_\alpha(A; \varphi_N, f_p) \big)
\end{equation*}
for every real symmetric matrix $A$, and we deduce that
\begin{equation}\label{e:pairs-laplace}
\lim_{N \to \infty} \E \exp\big( -\theta M_\alpha^{-1} \mathcal X_\alpha(H; \varphi_N, f_p) \big) = e^{-\theta L} \qquad \text{for all } \theta > 0.
\end{equation}

We now apply \eqref{e:pairs-laplace} with $\theta = 1$ and $\theta = 2$, and we find
\begin{equation*}
\E \Big[ \big( e^{-M_\alpha^{-1} \mathcal X_\alpha(H; \varphi_N, f_p)} - e^{-L} \big)^2 \Big] = \E e^{-2 M_\alpha^{-1} \mathcal X_\alpha(H; \varphi_N, f_p)} - 2 e^{-L} \E e^{-M_\alpha^{-1} \mathcal X_\alpha(H; \varphi_N, f_p)} + e^{-2L} \to 0.
\end{equation*}
Hence $\exp(-M_\alpha^{-1} \mathcal X_\alpha(H; \varphi_N, f_p)) \to e^{-L}$ in probability. The logarithm is continuous on $(0, \infty)$ and $e^{-L} > 0$, so this implies that $M_\alpha^{-1} \mathcal X_\alpha(H; \varphi_N, f_p) \to L$ in probability.
\end{proof}

\section{The base case}\label{s:pairs-base}

We begin with a deterministic lemma. It shows that the pair statistic changes by a small relative error when the bulk gaps are perturbed by much less than $\ell_\alpha$, provided that the points are close to the classical locations $\gamma_i$ defined in \zcref{s:prelim}.

\begin{lemma}\label{l:localized}
Let $\alpha \in (0, 1/2)$ and $\eps \in (0, 1/10]$, let $p \ge 1$ be an integer, and let $\varphi \in C_c^\infty((-2, 2))$ be nonnegative. Let $\delta \in (0, 1)$ satisfy $\supp \varphi \subset [-2 + 3\delta, 2 - 3\delta]$, and set
\begin{equation*}
c_* = \int_{-2}^{-2 + \delta/4} \rhosc(E) \dd E, \qquad \mathscr J = \{ i \in [N] : c_* N \le i \le (1 - c_*) N \}.
\end{equation*}
Then for every $\varpi > 0$ there exists $N_0$ such that the following holds for all $N \ge N_0$. Let $x, x' \in [-3, 3]^N$ be nondecreasing sequences such that
\begin{enumerate}[label=(\alph*)]
\item $\# \{ i \in [N] : x_i \in [E, E + N^{-1}] \} \le N^{\eps}$ for all $E \in \R$;
\item $|x_i - \gamma_i| \le N^{-1 + \eps}$ and $|x'_i - \gamma_i| \le N^{-1 + \eps}$ for all $i \in \mathscr J$;
\item $|(x_{i + 1} - x_i) - (x'_{i + 1} - x'_i)| \le N^{-2\eps} \ell_\alpha$ for all $i$ such that $i, i + 1 \in \mathscr J$.
\end{enumerate}
Then
\begin{equation}\label{e:pairs-localized}
\big| \mathcal X_\alpha(x; \varphi_N, f_p) - \mathcal X_\alpha(x'; \varphi_N, f_p) \big| \le \varpi \big( \mathcal X_\alpha(x'; \varphi_N, f_p) + M_\alpha \big).
\end{equation}
\end{lemma}

\begin{proof}
We may assume that $\varphi$ is not identically zero, since otherwise both statistics vanish. Throughout the proof, $C$ denotes a constant depending on $\alpha$, $\eps$, $p$, $\varphi$, and $\delta$, and $N_0$ may increase from line to line. Note that $0 < c_* < 1/2$, since $\delta/4 < 2$. Set $i_- = \lceil c_* N \rceil$ and $i_+ = \lfloor (1 - c_*) N \rfloor$. By the definition of the classical locations, we have
\begin{equation*}
\int_{-2}^{\gamma_{i_-}} \rhosc(E) \dd E = \frac{i_-}{N} \le c_* + \frac1N < \int_{-2}^{-2 + \delta/2} \rhosc(E) \dd E
\end{equation*}
for $N \ge N_0$, where the last inequality holds because $\rhosc$ is positive on $(-2, 2)$. Using also the symmetry $\rhosc(-E) = \rhosc(E)$, we find that for $N \ge N_0$, we have $i_-, i_+ \in \mathscr J$ and
\begin{equation}\label{e:pairs-edge-labels}
\gamma_{i_-} \le -2 + \frac{\delta}{2}, \qquad \gamma_{i_+} \ge 2 - \frac{\delta}{2}.
\end{equation}

We first consider the pairs with a label outside $\mathscr J$. The following argument applies to $x$ and to $x'$, and we write it for $x$. Using hypothesis (b), \eqref{e:pairs-edge-labels}, and the monotonicity of $x$, we find that for $N \ge N_0$,
\begin{equation}\label{e:pairs-edge-points}
x_i \le x_{i_-} \le -2 + \frac{\delta}{2} + N^{-1 + \eps} \le -2 + \delta \quad \text{for } i < c_* N, \qquad x_i \ge 2 - \delta \quad \text{for } i > (1 - c_*) N.
\end{equation}

Let $i < j$ be labels such that $i \notin \mathscr J$ or $j \notin \mathscr J$, and set $E = (x_i + x_j)/2$. Suppose first that $\dist(E, \supp \varphi) \le \delta$, so that $E \in [-2 + 2\delta, 2 - 2\delta]$. The label $i$ cannot exceed $(1 - c_*) N$, since otherwise $j > (1 - c_*) N$ as well, and \eqref{e:pairs-edge-points} would give $E \ge 2 - \delta$. There remain the cases $i < c_* N$ and $i \in \mathscr J$, and in the second case we have $j > (1 - c_*) N$. Using \eqref{e:pairs-edge-points} and $E \in [-2 + 2\delta, 2 - 2\delta]$, we find
\begin{equation*}
\begin{aligned}
x_j &= 2E - x_i \ge -2 + 3\delta \ge x_i + 2\delta & & \text{if } i < c_* N, \\
x_i &= 2E - x_j \le 2 - 3\delta \le x_j - 2\delta & & \text{if } i \in \mathscr J.
\end{aligned}
\end{equation*}
Thus $x_j - x_i \ge 2\delta$, and hence
\begin{equation*}
\varphi_N(E) f_p\Big( \frac{x_j - x_i}{\ell_\alpha} \Big) \le \| \varphi \|_\infty \Big( \frac{\ell_\alpha}{2\delta} \Big)^{2p} \le C N^{-2 - 2\alpha},
\end{equation*}
where we used the bound $\varphi_N \le \| \varphi \|_\infty$ from part (1) of \zcref{l:smoothing} and the inequality $2p(1 + \alpha) \ge 2 + 2\alpha$.

Suppose next that $\dist(E, \supp \varphi) > \delta$. Then part (4) of \zcref{l:smoothing} gives
\begin{equation*}
\varphi_N(E) f_p\Big( \frac{x_j - x_i}{\ell_\alpha} \Big) \le \varphi_N(E) \le C \exp\Big( -\frac{\delta^2}{4 h_N^2} \Big) = C \exp\Big( -\frac{\delta^2 N^{1/5}}{4} \Big).
\end{equation*}
Summing over the at most $N^2$ pairs, we obtain
\begin{equation}\label{e:pairs-localized-outside}
\sum_{\substack{1 \le i < j \le N \\ \{ i, j \} \not \subset \mathscr J}} \varphi_N\Big( \frac{x_i + x_j}{2} \Big) f_p\Big( \frac{x_j - x_i}{\ell_\alpha} \Big) \le C N^{-2\alpha} + C N^2 \exp\Big( -\frac{\delta^2 N^{1/5}}{4} \Big) \le \frac{\varpi}{4} M_\alpha
\end{equation}
for $N \ge N_0$, and the same bound holds with $x'$ in place of $x$.

We now compare the terms with $i, j \in \mathscr J$, and we begin with the gaps. Fix such labels $i < j$. The points $x_i, \dots, x_j$ belong to $[x_i, x_j]$, which is covered by $\lfloor N (x_j - x_i) \rfloor + 1$ intervals of length $N^{-1}$, so hypothesis (a) implies that
\begin{equation*}
j - i + 1 \le N^{\eps} \big( N (x_j - x_i) + 1 \big).
\end{equation*}
The set $\mathscr J$ consists of consecutive integers, so every label $k$ with $i \le k < j$ satisfies $k, k + 1 \in \mathscr J$. Hypothesis (c) and a telescoping sum then give
\begin{equation}\label{e:pairs-localized-telescope}
\begin{split}
\big| (x'_j - x'_i) - (x_j - x_i) \big| &\le (j - i) N^{-2\eps} \ell_\alpha \\
&\le N^{-\eps} \big( N (x_j - x_i) + 1 \big) \ell_\alpha \le N^{-\eps} (x_j - x_i + \ell_\alpha),
\end{split}
\end{equation}
where in the last inequality we used that $N \ell_\alpha = N^{-\alpha} \le 1$.

Next, observe that for $y \ge 0$,
\begin{equation}\label{e:pairs-logderivative}
\Big| \frac{\dd}{\dd y} \log f_p(y) \Big| = \frac{2py}{1 + y^2} \le \frac{4p}{1 + y},
\end{equation}
since $y + y^2 \le 2 + 2y^2$. We set
\begin{equation*}
y = \frac{x_j - x_i}{\ell_\alpha}, \qquad y' = \frac{x'_j - x'_i}{\ell_\alpha}.
\end{equation*}
By \eqref{e:pairs-localized-telescope}, we have $|y - y'| \le N^{-\eps} (1 + y)$, so every $y''$ between $y$ and $y'$ satisfies
\begin{equation*}
1 + y'' \ge (1 - N^{-\eps})(1 + y) \ge \frac{1 + y}{2}
\end{equation*}
for $N \ge N_0$. Integrating \eqref{e:pairs-logderivative} along the segment between $y$ and $y'$, we obtain
\begin{equation}\label{e:pairs-localized-gap}
\Big| \log f_p\Big( \frac{x_j - x_i}{\ell_\alpha} \Big) - \log f_p\Big( \frac{x'_j - x'_i}{\ell_\alpha} \Big) \Big| \le \frac{8p \, |y - y'|}{1 + y} \le 8p N^{-\eps}.
\end{equation}

It remains to compare the energy weights. The midpoints $E = (x_i + x_j)/2$ and $E' = (x'_i + x'_j)/2$ belong to $[-3, 3]$, and hypothesis (b) gives
\begin{equation*}
|E - E'| \le 2N^{-1 + \eps} \le 1.
\end{equation*}
By part (2) of \zcref{l:smoothing}, we have
\begin{equation*}
|\varphi_N(E) - \varphi_N(E')| \le \big( \exp(16 N^{-1 + \eps} h_N^{-2}) - 1 \big) \varphi_N(E') = \big( \exp(16 N^{-4/5 + \eps}) - 1 \big) \varphi_N(E'),
\end{equation*}
and the factor in parentheses tends to zero, as $\eps \le 1/10$. Combining this bound with \eqref{e:pairs-localized-gap}, we find that for $N \ge N_0$ and all $i < j$ in $\mathscr J$,
\begin{equation*}
\Big| \varphi_N(E) f_p\Big( \frac{x_j - x_i}{\ell_\alpha} \Big) - \varphi_N(E') f_p\Big( \frac{x'_j - x'_i}{\ell_\alpha} \Big) \Big| \le \frac{\varpi}{2} \varphi_N(E') f_p\Big( \frac{x'_j - x'_i}{\ell_\alpha} \Big).
\end{equation*}
We sum over these pairs and bound the resulting sum of nonnegative terms on the right by $\mathcal X_\alpha(x'; \varphi_N, f_p)$. Then the contributions of the pairs in $\mathscr J$ to $\mathcal X_\alpha(x; \varphi_N, f_p)$ and $\mathcal X_\alpha(x'; \varphi_N, f_p)$ differ by at most $(\varpi/2) \mathcal X_\alpha(x'; \varphi_N, f_p)$. Together with \eqref{e:pairs-localized-outside} for $x$ and for $x'$, this gives \eqref{e:pairs-localized}.
\end{proof}

\begin{proposition}\label{p:base}
For every $\alpha \in (0, 1/8)$, the statement $\mathcal P(\alpha)$ holds.
\end{proposition}

\begin{proof}
Fix $\alpha \in (0, 1/8)$, an integer $p \ge p_*$, and a nonnegative $\varphi \in C_c^\infty((-2, 2))$. We set
\begin{equation}\label{e:pairs-base-parameters}
L = \frac{1}{2(p - 1)} \int_{-2}^2 \varphi(E) c(E) \dd E, \qquad T = N^{-3/4 - \alpha}, \qquad \eps = \frac{1}{100} \Big( \frac14 - 2\alpha \Big).
\end{equation}
Then $\eps \in (0, 1/400]$, and a direct computation shows that
\begin{equation}\label{e:pairs-base-exponents}
\begin{gathered}
T \ge N^{-1 + \eps}, \qquad \alpha + 5\eps < \frac12, \\
\frac{N^{\eps}}{N^2 T} = N^{-5/4 + \alpha + \eps} \le N^{-2\eps} \ell_\alpha, \qquad T N^{1/2 + 3\alpha + 12\eps} = N^{-1/4 + 2\alpha + 12\eps}.
\end{gathered}
\end{equation}
Indeed, the first inequality is equivalent to $\eps \le 1/4 - \alpha$, the third to $3\eps \le 1/4 - 2\alpha$, and the exponent in the last expression is $-(88/100)(1/4 - 2\alpha) < 0$. Using \zcref{l:pairs-transfer} with $t_N = T$, we see that it suffices to show that
\begin{equation}\label{e:pairs-base-goal}
\frac{1}{M_\alpha} \mathcal X_\alpha(H_T; \varphi_N, f_p) \to L \qquad \text{in probability}.
\end{equation}

We first consider the GOE matrix $G$. Let $\delta \in (0, 1)$ satisfy $\supp \varphi \subset [-2 + 3\delta, 2 - 3\delta]$. We fix $\chi \in C_c^\infty((-2, 2))$ such that $0 \le \chi \le 2$ and $\chi = 2$ on $[-2 + 2\delta, 2 - 2\delta]$ (the interval $[-2 + 2\delta, 2 - 2\delta]$ contains every point at distance at most $\delta$ from $\supp \varphi$, as required in \zcref{l:pairs-unsmooth}).

The functions $(r, E) \mapsto \varphi(E) f_p(r)$ and $(r, E) \mapsto \chi(E) f_{p_*}(r)$ are continuous, their $E$-supports lie in bulk intervals, and they are bounded by a constant times $(1 + r^2)^{-2}$. Hence \zcref{p:goepairs} and \eqref{e:pairs-radial} show that
\begin{equation}\label{e:pairs-base-goe-unsmoothed}
\frac{1}{M_\alpha} \mathcal X_\alpha(G; \varphi, f_p) \to L, \qquad \frac{1}{M_\alpha} \mathcal X_\alpha(G; \chi, f_{p_*}) \to \frac{1}{2(p_* - 1)} \int_{-2}^2 \chi(E) c(E) \dd E
\end{equation}
in probability. Since $p \ge p_*$, we have $f_p \le f_{p_*}$. Then \zcref{l:pairs-unsmooth}, applied with $A = G$ and $f = f_p$, shows that on the event $\{ \| G \| \le 3 \}$, for $N \ge N_0$,
\begin{equation*}
\frac{1}{M_\alpha} \big| \mathcal X_\alpha(G; \varphi_N, f_p) - \mathcal X_\alpha(G; \varphi, f_p) \big| \le \frac{C h_N}{M_\alpha} \mathcal X_\alpha(G; \chi, f_{p_*}) + \frac{C N^2}{M_\alpha} \exp\Big( -\frac{\delta^2 N^{1/5}}{4} \Big).
\end{equation*}
By \eqref{e:pairs-base-goe-unsmoothed}, the right side tends to zero in probability, and the event $\{ \| G \| \le 3 \}$ has probability tending to one by \zcref{p:regularity}. We conclude that
\begin{equation}\label{e:pairs-base-goe}
\frac{1}{M_\alpha} \mathcal X_\alpha(G; \varphi_N, f_p) \to L \qquad \text{in probability}.
\end{equation}

We now use the coupling. Let $(H_t^{\mathrm{dyn}})_{t \ge 0}$ and $(\mu(t))_{t \ge 0}$ be as in \zcref{p:coupling}, and set
\begin{equation*}
\lambda(T) = \big( \lambda_1(H_T^{\mathrm{dyn}}), \dots, \lambda_N(H_T^{\mathrm{dyn}}) \big), \qquad \mu(T) = \big( \mu_1(T), \dots, \mu_N(T) \big).
\end{equation*}
Recall that $H_T^{\mathrm{dyn}}$ has the law of $H_T$, and that $\mu(T)$ has the law of the ordered eigenvalues of $G$.

Let $c_*$ be as in \zcref{l:localized}, defined with our choice of $\delta$. We consider the event on which $H_T^{\mathrm{dyn}} \in \mathcal G_{\eps}$, $\max_{i \in [N]} |\mu_i(T)| \le 3$, and
\begin{equation}\label{e:pairs-base-event}
\begin{gathered}
\max_{c_* N \le i \le (1 - c_*) N} \max\big\{ |\lambda_i(H_T^{\mathrm{dyn}}) - \gamma_i|, \, |\mu_i(T) - \gamma_i| \big\} \le N^{-1 + \eps}, \\
\max_{c_* N \le i \le (1 - c_*) N} \big| \big( \lambda_{i + 1}(H_T^{\mathrm{dyn}}) - \lambda_i(H_T^{\mathrm{dyn}}) \big) - \big( \mu_{i + 1}(T) - \mu_i(T) \big) \big| \le \frac{N^{\eps}}{N^2 T}.
\end{gathered}
\end{equation}
By \zcref{p:regularity}, applied to $H_T$ and to $G$ (whose ordered eigenvalues have the law of $\mu(T)$), the conditions other than the second inequality in \eqref{e:pairs-base-event} hold with overwhelming probability. We have $N^{-1 + \eps} \le T < 1$ by \eqref{e:pairs-base-exponents}, so \zcref{p:coupling} shows that the second inequality in \eqref{e:pairs-base-event} holds with probability at least $1 - N^{-1}$ for $N \ge N_0$. Hence this event has probability tending to one.

The sequences $x = \lambda(T)$ and $x' = \mu(T)$ satisfy the hypotheses (a)--(c) of \zcref{l:localized} on this event, where for (c) we used the third relation in \eqref{e:pairs-base-exponents}. Hence, for every $\varpi > 0$, with probability tending to one,
\begin{multline*}
\Big| \frac{1}{M_\alpha} \mathcal X_\alpha(\lambda(T); \varphi_N, f_p) - L \Big| \\
\le \Big| \frac{1}{M_\alpha} \mathcal X_\alpha(\mu(T); \varphi_N, f_p) - L \Big| + \varpi \Big( \frac{1}{M_\alpha} \mathcal X_\alpha(\mu(T); \varphi_N, f_p) + 1 \Big).
\end{multline*}
By \eqref{e:pairs-base-goe}, we have $M_\alpha^{-1} \mathcal X_\alpha(\mu(T); \varphi_N, f_p) \to L$ in probability. Since $\varpi > 0$ is arbitrary, it follows that $M_\alpha^{-1} \mathcal X_\alpha(\lambda(T); \varphi_N, f_p) \to L$ in probability. The sequence $\lambda(T)$ has the law of the ordered eigenvalues of $H_T$, which gives \eqref{e:pairs-base-goal}.
\end{proof}

\section{General test functions}\label{s:pairs-extension}

We now show that $\mathcal P(\alpha)$ extends to general test functions $f$ and to unsmoothed energy weights, and that it gives an upper bound on the number of pairs at distance of order $\ell_\alpha$.

\begin{lemma}\label{l:extension}
Let $\alpha \in (0, 1/2)$, and suppose that $\mathcal P(\alpha)$ holds. Let $\varphi \in C_c^\infty((-2, 2))$ be nonnegative, and let $f \colon [0, \infty) \to \R$ be a continuous function such that the limit $\lim_{y \to \infty} (1 + y^2)^{p_*} f(y)$ exists in $\R$. Then
\begin{equation}\label{e:pairs-extension}
\frac{1}{M_\alpha} \mathcal X_\alpha(H; \varphi_N, f) \to \int_{-2}^2 \varphi(E) c(E) \dd E \int_0^\infty y f(y) \dd y
\end{equation}
in probability, and the same holds with $\varphi_N$ replaced by $\varphi$. Further, for every bulk interval $I$, there exists $C_I > 0$ such that
\begin{equation}\label{e:pairs-upper}
\lim_{N \to \infty} \Prob\bigg( \sum_{\substack{1 \le i < j \le N \\ (\lambda_i + \lambda_j)/2 \in I}} f_{p_*}\Big( \frac{\lambda_j - \lambda_i}{\ell_\alpha} \Big) \le C_I M_\alpha \bigg) = 1.
\end{equation}
\end{lemma}

\begin{proof}
We set
\begin{equation*}
C_f = \sup_{y \ge 0} (1 + y^2)^{p_*} |f(y)|, \qquad c_\varphi = \int_{-2}^2 \varphi(E) c(E) \dd E.
\end{equation*}
The constant $C_f$ is finite by our assumption on $f$. (This assumption holds, for example, for continuous functions with compact support and for the function $y \mapsto e^{-y^2/8}$, which we use in \zcref{s:pairs-induction}.) Then $|f| \le C_f f_{p_*}$, so the integral $\int_0^\infty y f(y) \dd y$ converges absolutely.

We begin with \eqref{e:pairs-extension}, and we first approximate $f$ by a combination of the functions $f_p$. Define $\Theta_f \colon [0, 1] \to \R$ by
\begin{equation*}
\Theta_f(x) = x^{-p_*} f\big( (x^{-1} - 1)^{1/2} \big) \quad \text{for } x \in (0, 1], \qquad \Theta_f(0) = \lim_{y \to \infty} (1 + y^2)^{p_*} f(y).
\end{equation*}
For $y \ge 0$ and $x = (1 + y^2)^{-1}$, we have $(x^{-1} - 1)^{1/2} = y$ and $x^{-p_*} = (1 + y^2)^{p_*}$. Hence $\Theta_f$ is continuous on $[0, 1]$, and
\begin{equation*}
f(y) = f_{p_*}(y) \, \Theta_f\big( (1 + y^2)^{-1} \big) \qquad \text{for all } y \ge 0.
\end{equation*}

Let $\varpi \in (0, 1)$. By the Weierstrass approximation theorem, there exist an integer $p_1 \ge p_*$ and real numbers $\beta_{p_*}, \dots, \beta_{p_1}$ such that
\begin{equation*}
\sup_{x \in [0, 1]} \Big| \Theta_f(x) - \sum_{p = p_*}^{p_1} \beta_p x^{p - p_*} \Big| \le \varpi.
\end{equation*}
Note that $f_{p_*}(y) (1 + y^2)^{-(p - p_*)} = f_p(y)$. Therefore the function
\begin{equation*}
\bar f = \sum_{p = p_*}^{p_1} \beta_p f_p
\end{equation*}
satisfies
\begin{equation}\label{e:pairs-polyapprox}
|f(y) - \bar f(y)| \le \varpi f_{p_*}(y) \qquad \text{for all } y \ge 0.
\end{equation}

The statistic $\mathcal X_\alpha(H; \varphi_N, f)$ is linear in $f$. Applying $\mathcal P(\alpha)$ with each $p \in \{ p_*, \dots, p_1 \}$, and using \eqref{e:pairs-radial}, we obtain
\begin{equation*}
\frac{1}{M_\alpha} \mathcal X_\alpha(H; \varphi_N, \bar f) \to c_\varphi \sum_{p = p_*}^{p_1} \frac{\beta_p}{2(p - 1)} = c_\varphi \int_0^\infty y \bar f(y) \dd y \qquad \text{in probability}.
\end{equation*}
Since $\varphi_N \ge 0$, \eqref{e:pairs-polyapprox} implies that
\begin{equation*}
\big| \mathcal X_\alpha(H; \varphi_N, f) - \mathcal X_\alpha(H; \varphi_N, \bar f) \big| \le \varpi \mathcal X_\alpha(H; \varphi_N, f_{p_*}), \qquad \Big| \int_0^\infty y (f - \bar f)(y) \dd y \Big| \le \varpi,
\end{equation*}
where for the second bound we also used \eqref{e:pairs-radial}. Hence
\begin{multline*}
\Big| \frac{1}{M_\alpha} \mathcal X_\alpha(H; \varphi_N, f) - c_\varphi \int_0^\infty y f(y) \dd y \Big| \\
\le \Big| \frac{1}{M_\alpha} \mathcal X_\alpha(H; \varphi_N, \bar f) - c_\varphi \int_0^\infty y \bar f(y) \dd y \Big| + \frac{\varpi}{M_\alpha} \mathcal X_\alpha(H; \varphi_N, f_{p_*}) + \varpi c_\varphi.
\end{multline*}
By $\mathcal P(\alpha)$ with $p = p_*$, the second term on the right converges in probability to $\varpi c_\varphi / (2(p_* - 1)) \le \varpi c_\varphi$. So the left side is bounded by $3 \varpi (c_\varphi + 1)$ with probability tending to one. Since $\varpi$ is arbitrary, this proves \eqref{e:pairs-extension}.

Next, we remove the smoothing, and we may assume that $C_f > 0$. Let $\delta > 0$ be such that every $E$ with $\dist(E, \supp \varphi) \le \delta$ belongs to $(-2, 2)$. We fix $\chi \in C_c^\infty((-2, 2))$ such that $0 \le \chi \le 2$ and $\chi(E) = 2$ for all such $E$. We apply \zcref{l:pairs-unsmooth} to $f/C_f$, and we find that on the event $\{ \| H \| \le 3 \}$, for $N \ge N_0$,
\begin{equation*}
\frac{1}{M_\alpha} \big| \mathcal X_\alpha(H; \varphi_N, f) - \mathcal X_\alpha(H; \varphi, f) \big| \le \frac{C C_f h_N}{M_\alpha} \mathcal X_\alpha(H; \chi_N, f_{p_*}) + \frac{C C_f N^2}{M_\alpha} \exp\Big( -\frac{\delta^2 N^{1/5}}{4} \Big).
\end{equation*}
By $\mathcal P(\alpha)$, applied with $\chi$ in place of $\varphi$ and $p = p_*$, the random variable $M_\alpha^{-1} \mathcal X_\alpha(H; \chi_N, f_{p_*})$ converges in probability, and the right side tends to zero in probability. As the event $\{ \| H \| \le 3 \}$ has probability tending to one by \zcref{p:regularity}, the convergence \eqref{e:pairs-extension} implies the same convergence with $\varphi$ in place of $\varphi_N$.

Finally, we prove \eqref{e:pairs-upper}. Let $I$ be a bulk interval. Let $\chi$ now denote a function in $C_c^\infty((-2, 2))$ such that $0 \le \chi \le 2$ and $\chi = 2$ on $I$. By part (5) of \zcref{l:smoothing}, we have
\begin{equation*}
\| \chi_N - \chi \|_\infty \le C h_N \le 1
\end{equation*}
for $N \ge N_0$, so $\chi_N \ge 1$ on $I$. All terms of $\mathcal X_\alpha(H; \chi_N, f_{p_*})$ are nonnegative, and therefore the sum in \eqref{e:pairs-upper} is bounded by $\mathcal X_\alpha(H; \chi_N, f_{p_*})$. By $\mathcal P(\alpha)$, we have
\begin{equation*}
\frac{1}{M_\alpha} \mathcal X_\alpha(H; \chi_N, f_{p_*}) \to \frac{1}{2(p_* - 1)} \int_{-2}^2 \chi(E) c(E) \dd E \qquad \text{in probability}.
\end{equation*}
Hence \eqref{e:pairs-upper} holds with
\begin{equation*}
C_I = 1 + \int_{-2}^2 \chi(E) c(E) \dd E.
\end{equation*}
\end{proof}

\section{The induction step}\label{s:pairs-induction}

\begin{proposition}\label{p:induction}
Let $\alpha, \alpha'$ satisfy
\begin{equation}\label{e:pairs-induction-conditions}
\frac{1}{10} \le \alpha < \alpha' < \frac12, \qquad \alpha' < \frac{6\alpha}{5}, \qquad \alpha' < \frac16 + \frac{2\alpha}{3}.
\end{equation}
If $\mathcal P(\alpha)$ holds, then $\mathcal P(\alpha')$ holds.
\end{proposition}

To prove \zcref{p:induction}, we use \zcref{l:pairs-transfer} to replace $H$ by $H_{t_\alpha}$, with $t_\alpha$ as in \eqref{e:pairs-induction-parameters} below, and we approximate $\mathcal X_{\alpha'}(H_{t_\alpha}; \varphi_N, f_p)$ by a sum of conditionally independent contributions of $2 \times 2$ blocks, whose conditional means are computed by \zcref{l:gaussian} and determined by $\mathcal P(\alpha)$. We first fix the parameters and define the blocks. \zcref[S]{l:pairs-clusters} collects the properties of the clusters that we need, \zcref{l:pairs-reduction} bounds the error of the approximation, \zcref{l:pairs-mean} computes the conditional mean of the sum, and \zcref{l:pairs-concentration} shows that the sum concentrates around this mean. We prove \zcref{p:induction} after the proofs of these lemmas. The main technical point is the approximation, which relies on \zcref{c:threshold} and on the first three constraints in \eqref{e:pairs-epsilon}.

\paragraph{Parameters.} Throughout the rest of \zcref{s:pairs-induction}, we fix $\alpha$ and $\alpha'$ satisfying \eqref{e:pairs-induction-conditions}. The parameters and the notation introduced in this paragraph and the next two are used in the statements and the proofs of all lemmas below. We set
\begin{equation}\label{e:pairs-induction-parameters}
\kappa = \alpha - \frac{\alpha'}{6}, \qquad q = N^{-1 - \kappa}, \qquad t_\alpha = N \ell_\alpha^2 = N^{-1 - 2\alpha}.
\end{equation}
Then \eqref{e:pairs-induction-conditions} implies that
\begin{equation}\label{e:pairs-margins}
\frac{2\alpha'}{3} < \kappa < 2\alpha - \alpha' < \alpha, \qquad \alpha' - \kappa = \frac{7\alpha'}{6} - \alpha > \frac{\alpha}{6} \ge \frac{1}{60}.
\end{equation}
Indeed, the first two inequalities are equivalent to $\alpha' < 6\alpha/5$, the third is equivalent to $\alpha < \alpha'$, and the last two follow from $\alpha < \alpha'$ and $\alpha \ge 1/10$.

We fix $\eps > 0$ such that
\begin{equation}\label{e:pairs-epsilon}
100 \eps < \min\Big\{ \frac{\alpha'}{6}, \ \alpha - \frac{5\alpha'}{6}, \ 3\alpha - \frac{5\alpha'}{2}, \ \frac12 + 2\alpha - 3\alpha', \ \frac12 - \alpha' \Big\}.
\end{equation}
Every number in this minimum is positive. For the second and third numbers, this follows from $\alpha' < 6\alpha/5$, for the fourth from $\alpha' < 1/6 + 2\alpha/3$, and for the fifth from $\alpha' < 1/2$. In terms of $\kappa$, the first three numbers are $\alpha - \kappa$, $2\alpha - \kappa - \alpha'$, and $3\kappa - 2\alpha'$. Moreover,
\begin{equation*}
\eps < \frac{\alpha'}{6} < \kappa,
\end{equation*}
where the last inequality is equivalent to $\alpha' < 3\alpha$.

For an integer $p \ge p_*$ and a nonnegative $\varphi \in C_c^\infty((-2, 2))$, we set
\begin{equation}\label{e:pairs-induction-L}
L = \frac{1}{2(p - 1)} \int_{-2}^2 \varphi(E) c(E) \dd E.
\end{equation}
In the statements and the proofs of the lemmas below, constants may depend on $\alpha$, $\alpha'$, $\eps$, $p$, and $\varphi$, and $N_0$ may increase from line to line.

\paragraph{Gaussian blocks.} Let $U = (u_1, \dots, u_N)$ be the orthogonal matrix whose columns are the chosen eigenvectors of $H$. We set
\begin{equation}\label{e:pairs-blocks}
y_i = e^{-t_\alpha/2} \lambda_i \quad (i \in [N]), \qquad D_y = \diag(y_1, \dots, y_N), \qquad W = (1 - e^{-t_\alpha})^{1/2} U^T G U.
\end{equation}
Then
\begin{equation}\label{e:pairs-same-spectrum}
U^T H_{t_\alpha} U = D_y + W, \qquad \text{so} \qquad \lambda_i(H_{t_\alpha}) = \lambda_i(D_y + W) \quad \text{for all } i \in [N].
\end{equation}
The matrix $G$ is independent of $H$, and the law of $G$ is invariant under conjugation by a fixed orthogonal matrix. Since $U$ is a measurable function of $H$, the conditional law of $U^T G U$ given $H$ is the law of $G$. Consequently, $W$ is independent of $H$, and its entries $W_{ij}$ with $i \le j$ are independent centered Gaussian random variables satisfying
\begin{equation}\label{e:pairs-W-variance}
\E W_{ij}^2 = \frac{1 - e^{-t_\alpha}}{N} \quad (i < j), \qquad \E W_{ii}^2 = \frac{2 (1 - e^{-t_\alpha})}{N}.
\end{equation}

We next bound the entries of $W$. By \eqref{e:pairs-W-variance}, we have for all $i \le j$
\begin{equation*}
\E W_{ij}^2 \le \frac{2 (1 - e^{-t_\alpha})}{N} \le \frac{2 t_\alpha}{N} = 2 \ell_\alpha^2.
\end{equation*}
We set
\begin{equation}\label{e:pairs-w0}
w_0 = N^{\eps} \ell_\alpha.
\end{equation}
The standard Gaussian tail bound and a union bound over the $N(N + 1)/2$ entries give
\begin{equation}\label{e:pairs-wmax}
\Prob\Big( \max_{i \le j} |W_{ij}| > w_0 \Big) \le 2N^2 \exp\Big( -\frac{N^{2\eps}}{4} \Big) \to 0.
\end{equation}

\paragraph{Clusters.} Let $\mathfrak K$ denote the family of clusters obtained from the points $y_1 \le \dots \le y_N$ by joining $i$ and $i + 1$ whenever $y_{i + 1} - y_i < q$, as in \zcref{c:threshold}. (The threshold $q$ is much larger than $\ell_\alpha$, since $\kappa < \alpha$, and much smaller than $N^{-1}$.) Every cluster is a set of consecutive integers, and the family $\mathfrak K$ is a function of $H$.

Let $\mathcal K, \mathcal K' \in \mathfrak K$ satisfy $\max \mathcal K < \min \mathcal K'$. Then the labels $\max \mathcal K$ and $\max \mathcal K + 1$ lie in different clusters, and since $y$ is nondecreasing,
\begin{equation}\label{e:pairs-cluster-separation}
y_{\min \mathcal K'} - y_{\max \mathcal K} \ge y_{\max \mathcal K + 1} - y_{\max \mathcal K} \ge q.
\end{equation}

Let $\mathfrak P_2$ denote the set of clusters with exactly two elements. For $\mathcal K \in \mathfrak P_2$, we write $\mathcal K = \{ i, i + 1 \}$, and we set $g_{\mathcal K} = g_i$, $E_{\mathcal K} = E_i$, and
\begin{equation*}
B_{\mathcal K} = \begin{pmatrix} y_i + W_{ii} & W_{i, i + 1} \\ W_{i, i + 1} & y_{i + 1} + W_{i + 1, i + 1} \end{pmatrix}, \qquad \hat g_{\mathcal K} = \lambda_2(B_{\mathcal K}) - \lambda_1(B_{\mathcal K}).
\end{equation*}
Thus $B_{\mathcal K} = (D_y + W)_{\mathcal K \mathcal K}$. For an integer $p \ge p_*$ and a nonnegative $\varphi \in C_c^\infty((-2, 2))$, we define the block statistic
\begin{equation}\label{e:pairs-block-statistic}
\widehat{\mathcal X}_{\alpha'}(\varphi, p) = \sum_{\mathcal K \in \mathfrak P_2} \varphi_N(E_{\mathcal K}) f_p\Big( \frac{\hat g_{\mathcal K}}{\ell_{\alpha'}} \Big).
\end{equation}

\begin{lemma}\label{l:pairs-clusters}
There exists $N_0 \ge 1$ such that the following holds for all $N \ge N_0$ on the event that $H \in \mathcal G_{\eps/2}$. We have $2 N^{\eps} q < N^{-1}$, the points $y_1 \le \dots \le y_N$ lie in $[-3, 3]$, and every interval of length $N^{-1}$ contains at most $N^{\eps}$ of them. Every cluster in $\mathfrak K$ has at most $N^{\eps}$ elements, and the number of pairs $i < j$ that lie in a common cluster $\mathcal K \in \mathfrak K$ with $|\mathcal K| \ge 3$ is at most $2 N^{\eps} \mathcal N_3(4q)$, where $\mathcal N_3(r)$ is as in \zcref{p:three}. Moreover, without restriction to this event, we have
\begin{equation*}
\lim_{N \to \infty} \frac{1}{M_{\alpha'}} \E\big[ N^{\eps} \mathcal N_3(4q) \big] = 0.
\end{equation*}
\end{lemma}

\begin{proof}
Suppose that $H \in \mathcal G_{\eps/2}$. We first verify the hypotheses of \zcref{c:threshold} on the points $y_1 \le \dots \le y_N$ and the threshold $q$. We have $\| H \| \le 3$, so $y_i \in [-3, 3]$ for all $i$. Each $\lambda_i$ with $y_i \in [E, E + N^{-1}]$ belongs to an interval of length $e^{t_\alpha/2} N^{-1} \le 2N^{-1}$, which is covered by two intervals of length $N^{-1}$. Since $H \in \mathcal G_{\eps/2}$, it follows that every interval of length $N^{-1}$ contains at most $2 N^{\eps/2} \le N^{\eps}$ of the points $y_i$, for $N \ge N_0$. Further, since $\eps < \kappa$, we have
\begin{equation*}
2 N^{\eps} q = 2 N^{\eps - \kappa} N^{-1} < N^{-1}
\end{equation*}
for $N \ge N_0$. Hence \zcref{c:threshold} shows that every cluster has at most $N^{\eps}$ elements.

We next count the pairs $i < j$ in a common cluster $\mathcal K$ with $|\mathcal K| \ge 3$. Recall from \zcref{p:three} that $\mathcal N_3(r)$ denotes the number of $i \in [N - 2]$ such that $\lambda_{i + 2} - \lambda_i \le r$. Let $\mathcal K \in \mathfrak K$ satisfy $|\mathcal K| \ge 3$. For each of the $|\mathcal K| - 2$ labels $i$ such that $i, i + 2 \in \mathcal K$, the definition of the clusters gives
\begin{equation*}
\lambda_{i + 2} - \lambda_i = e^{t_\alpha/2} (y_{i + 2} - y_i) < 2 e^{t_\alpha/2} q \le 4q.
\end{equation*}
These $|\mathcal K| - 2$ labels are distinct for distinct clusters, which implies that
\begin{equation*}
\sum_{\mathcal K \in \mathfrak K : |\mathcal K| \ge 3} (|\mathcal K| - 2) \le \mathcal N_3(4q).
\end{equation*}
Recall that $|\mathcal K| \le N^{\eps}$, and note that $|\mathcal K| \le 3(|\mathcal K| - 2)$ for $|\mathcal K| \ge 3$. Then the number of these pairs is bounded by
\begin{equation}\label{e:pairs-large-clusters}
\sum_{\mathcal K \in \mathfrak K : |\mathcal K| \ge 3} \frac{|\mathcal K|^2}{2} \le \frac{N^{\eps}}{2} \sum_{\mathcal K \in \mathfrak K : |\mathcal K| \ge 3} 3 (|\mathcal K| - 2) \le 2 N^{\eps} \mathcal N_3(4q).
\end{equation}

Finally, using \zcref{p:three} with $r = 4q \ge N^{-2}$, we have
\begin{equation}\label{e:pairs-N3-bound}
\frac{1}{M_{\alpha'}} \E\big[ N^{\eps} \mathcal N_3(4q) \big] \le \frac{C N^{\eps}}{M_{\alpha'}} N (4Nq)^3 \le C N^{\eps + 2\alpha' - 3\kappa} = C N^{\eps - (3\alpha - 5\alpha'/2)} \to 0,
\end{equation}
where the last step follows from \eqref{e:pairs-epsilon}.
\end{proof}

\begin{lemma}\label{l:pairs-reduction}
Let $p \ge p_*$ be an integer, let $\varphi \in C_c^\infty((-2, 2))$ be nonnegative, and let $\widehat{\mathcal X}_{\alpha'}(\varphi, p)$ be the block statistic defined in \eqref{e:pairs-block-statistic}. Then there exist $N_0 \ge 1$ and a deterministic sequence $o(1)$ such that for all $N \ge N_0$, on the event that $H \in \mathcal G_{\eps/2}$ and $\max_{i \le j} |W_{ij}| \le w_0$, with $W$ and $w_0$ as in \eqref{e:pairs-blocks} and \eqref{e:pairs-w0}, we have
\begin{multline}\label{e:pairs-reduction}
\big| \mathcal X_{\alpha'}(H_{t_\alpha}; \varphi_N, f_p) - \widehat{\mathcal X}_{\alpha'}(\varphi, p) \big| \\
\le o(1) \, \widehat{\mathcal X}_{\alpha'}(\varphi, p) + 2 \| \varphi \|_\infty N^{\eps} \mathcal N_3(4q) + 4^p \| \varphi \|_\infty N^2 \Big( \frac{\ell_{\alpha'}}{q} \Big)^{2p},
\end{multline}
where $\mathcal N_3(r)$ is as in \zcref{p:three}.
\end{lemma}

\begin{proof}
Suppose that the event holds and that $N \ge N_0$. By \zcref{l:pairs-clusters}, the points $y_1 \le \dots \le y_N$ and the threshold $q$ satisfy the hypotheses of \zcref{c:threshold} that concern them. It remains to verify the condition \eqref{e:prelim-threshold-smallness} on $W$. On our event, the entries of $W$ are bounded by $w_0 = N^{\eps} \ell_\alpha$. Since $N \log N \le q^{-1}$ for $N \ge N_0$ and $\alpha - \kappa = \alpha'/6$, we have
\begin{equation*}
N^{\eps} w_0 (q^{-1} + N \log N) \le 2 N^{2\eps} \ell_\alpha q^{-1} = 2 N^{2\eps - \alpha'/6} \le \frac{1}{48}
\end{equation*}
for $N \ge N_0$, where we used that $2\eps < \alpha'/6$ by \eqref{e:pairs-epsilon}. Hence \zcref{c:threshold} applies to the points $y_1 \le \dots \le y_N$, the threshold $q$, and the matrix $W$. By \eqref{e:pairs-same-spectrum}, it shows that
\begin{equation}\label{e:pairs-separated}
\lambda_{i + 1}(H_{t_\alpha}) - \lambda_i(H_{t_\alpha}) \ge \frac{q}{2} \qquad \text{whenever } i \text{ and } i + 1 \text{ belong to different clusters},
\end{equation}
and that for every cluster $\mathcal K = \{ i_0 + 1, \dots, i_0 + |\mathcal K| \}$ and every $i \in [|\mathcal K|]$,
\begin{equation}\label{e:pairs-schur-error}
\begin{split}
\big| \lambda_{i_0 + i}(H_{t_\alpha}) - \lambda_i\big( (D_y + W)_{\mathcal K \mathcal K} \big) \big| &\le 24 N^{2\eps} w_0^2 (q^{-1} + N \log N) \le 48 N^{4\eps} \ell_\alpha^2 q^{-1} \\
&= 48 N^{4\eps - \alpha + 5\alpha'/6} \ell_{\alpha'} \le N^{-\eps} \ell_{\alpha'}
\end{split}
\end{equation}
for $N \ge N_0$. In the equality, we used that
\begin{equation*}
\ell_\alpha^2 q^{-1} = N^{-1 - 2\alpha + \kappa}, \qquad -1 - 2\alpha + \kappa = -1 - \alpha' + (5\alpha'/6 - \alpha),
\end{equation*}
and in the last inequality, we used $5 \eps < \alpha - 5\alpha'/6$, which follows from \eqref{e:pairs-epsilon}.

We now sort the pairs $i < j$ in $\mathcal X_{\alpha'}(H_{t_\alpha}; \varphi_N, f_p)$ into three types. The first type consists of the pairs with $i \in \mathcal K$ and $j \in \mathcal K'$ for distinct clusters $\mathcal K$ and $\mathcal K'$, so that $\max \mathcal K < \min \mathcal K'$. For such a pair, the labels $\max \mathcal K$ and $\max \mathcal K + 1$ belong to different clusters, so \eqref{e:pairs-separated} gives
\begin{equation*}
\lambda_j(H_{t_\alpha}) - \lambda_i(H_{t_\alpha}) \ge \lambda_{\max \mathcal K + 1}(H_{t_\alpha}) - \lambda_{\max \mathcal K}(H_{t_\alpha}) \ge \frac{q}{2}.
\end{equation*}
Hence
\begin{equation*}
f_p\Big( \frac{\lambda_j(H_{t_\alpha}) - \lambda_i(H_{t_\alpha})}{\ell_{\alpha'}} \Big) \le \Big( 1 + \frac{q^2}{4 \ell_{\alpha'}^2} \Big)^{-p} \le 4^p \Big( \frac{\ell_{\alpha'}}{q} \Big)^{2p}.
\end{equation*}
Using $\varphi_N \le \| \varphi \|_\infty$, we see that the total contribution of the pairs of the first type is bounded by $4^p \| \varphi \|_\infty N^2 (\ell_{\alpha'}/q)^{2p}$.

The second type consists of the pairs $i < j$ in a common cluster $\mathcal K$ with $|\mathcal K| \ge 3$. By \zcref{l:pairs-clusters}, there are at most $2 N^{\eps} \mathcal N_3(4q)$ such pairs. Each of their terms is bounded by $\| \varphi \|_\infty$, since $f_p \le 1$.

The third type consists of the pairs $(i, i + 1)$ such that $\{ i, i + 1 \} \in \mathfrak P_2$. Fix such a cluster $\mathcal K = \{ i, i + 1 \}$. By \eqref{e:pairs-schur-error}, we have
\begin{equation*}
\big| \big( \lambda_{i + 1}(H_{t_\alpha}) - \lambda_i(H_{t_\alpha}) \big) - \hat g_{\mathcal K} \big| \le 2N^{-\eps} \ell_{\alpha'}.
\end{equation*}
The right side of \eqref{e:pairs-logderivative} is at most $4p$, so integrating \eqref{e:pairs-logderivative}, we find
\begin{equation*}
\Big| \log f_p\Big( \frac{\lambda_{i + 1}(H_{t_\alpha}) - \lambda_i(H_{t_\alpha})}{\ell_{\alpha'}} \Big) - \log f_p\Big( \frac{\hat g_{\mathcal K}}{\ell_{\alpha'}} \Big) \Big| \le 8p N^{-\eps}.
\end{equation*}

We now turn to the midpoints. Observe that
\begin{equation*}
\lambda_1(B_{\mathcal K}) + \lambda_2(B_{\mathcal K}) = \Tr B_{\mathcal K} = y_i + y_{i + 1} + W_{ii} + W_{i + 1, i + 1}.
\end{equation*}
Using this identity, \eqref{e:pairs-schur-error}, and the bound $|E_{\mathcal K}| \le 3$, we obtain
\begin{equation*}
\Big| \frac{\lambda_i(H_{t_\alpha}) + \lambda_{i + 1}(H_{t_\alpha})}{2} - E_{\mathcal K} \Big| \le N^{-\eps} \ell_{\alpha'} + w_0 + 3 (1 - e^{-t_\alpha/2}) \le 5 N^{\eps} \ell_\alpha,
\end{equation*}
where in the last inequality we used $\ell_{\alpha'} \le \ell_\alpha$ and
\begin{equation*}
1 - e^{-t_\alpha/2} \le \frac{t_\alpha}{2} \le \ell_\alpha.
\end{equation*}
Part (2) of \zcref{l:smoothing}, applied at the point $E_{\mathcal K} \in [-3, 3]$, shows that $\varphi_N$ at this midpoint differs from $\varphi_N(E_{\mathcal K})$ by at most
\begin{equation*}
\big( \exp(40 N^{\eps} \ell_\alpha h_N^{-2}) - 1 \big) \varphi_N(E_{\mathcal K}) = \big( \exp(40 N^{\eps - 4/5 - \alpha}) - 1 \big) \varphi_N(E_{\mathcal K}).
\end{equation*}
(This bound holds at every point of $[-3, 3]$, so midpoints near the edges of the spectrum or outside the support of $\varphi$ require no separate treatment.)

Combining the last display with the bound on $\log f_p$ above, we conclude that
\begin{multline*}
\Big| \varphi_N\Big( \frac{\lambda_i(H_{t_\alpha}) + \lambda_{i + 1}(H_{t_\alpha})}{2} \Big) f_p\Big( \frac{\lambda_{i + 1}(H_{t_\alpha}) - \lambda_i(H_{t_\alpha})}{\ell_{\alpha'}} \Big) - \varphi_N(E_{\mathcal K}) f_p\Big( \frac{\hat g_{\mathcal K}}{\ell_{\alpha'}} \Big) \Big| \\
\le o(1) \, \varphi_N(E_{\mathcal K}) f_p\Big( \frac{\hat g_{\mathcal K}}{\ell_{\alpha'}} \Big),
\end{multline*}
where the $o(1)$ term is deterministic and uniform in $\mathcal K$. Summing the bounds for the three types of pairs, we obtain \eqref{e:pairs-reduction}.
\end{proof}

\begin{lemma}\label{l:pairs-mean}
Suppose that $\mathcal P(\alpha)$ holds. Let $p \ge p_*$ be an integer, let $\varphi \in C_c^\infty((-2, 2))$ be nonnegative, and let $L$ be as in \eqref{e:pairs-induction-L}. Let $\widehat{\mathcal X}_{\alpha'}(\varphi, p)$ be the block statistic defined in \eqref{e:pairs-block-statistic}. Then
\begin{equation}\label{e:pairs-mean-limit}
\frac{1}{M_{\alpha'}} \E\big[ \widehat{\mathcal X}_{\alpha'}(\varphi, p) \, \big| \, H \big] \to L \qquad \text{in probability}.
\end{equation}
\end{lemma}

\begin{proof}
Let $\mathcal K \in \mathfrak P_2$, and write $\mathcal K = \{ i, i + 1 \}$. We first compute the conditional law of $\hat g_{\mathcal K}$. Since $y_i - y_{i + 1} = -e^{-t_\alpha/2} g_{\mathcal K}$, the formula \eqref{e:reduction-gap2x2} for the gap of a real symmetric $2 \times 2$ matrix gives
\begin{equation}\label{e:pairs-block-gap}
\frac{\hat g_{\mathcal K}}{\ell_\alpha} = \bigg| \Big( -\frac{e^{-t_\alpha/2} g_{\mathcal K}}{\ell_\alpha} + \frac{2}{\ell_\alpha} \cdot \frac{W_{ii} - W_{i + 1, i + 1}}{2}, \ \frac{2}{\ell_\alpha} W_{i, i + 1} \Big) \bigg|.
\end{equation}
By \eqref{e:pairs-W-variance}, conditionally on $H$, the variables $(W_{ii} - W_{i + 1, i + 1})/2$ and $W_{i, i + 1}$ are independent centered Gaussian random variables, each with variance $(1 - e^{-t_\alpha})/N$. Hence the two random coordinates in \eqref{e:pairs-block-gap} are independent centered Gaussian random variables with variance $v_N$, where
\begin{equation}\label{e:pairs-vN}
v_N = \frac{4 (1 - e^{-t_\alpha})}{N \ell_\alpha^2} = \frac{4 (1 - e^{-t_\alpha})}{t_\alpha} \in [4 - 2t_\alpha, 4].
\end{equation}
Here we used that
\begin{equation*}
1 - \frac{t}{2} \le \frac{1 - e^{-t}}{t} \le 1 \qquad \text{for } t \in (0, 1].
\end{equation*}
The law of a centered Gaussian vector in $\R^2$ with covariance $v_N \Id_2$ is invariant under the reflection that changes the sign of the first coordinate, so we conclude that, conditionally on $H$,
\begin{equation}\label{e:pairs-block-law}
\frac{\hat g_{\mathcal K}}{\ell_\alpha} \stackrel{d}{=} |\mathsf G_x|,
\end{equation}
where $\mathsf G_x$ is as in \zcref{l:gaussian}, with $n = 2$ and
\begin{equation*}
x = \frac{g_{\mathcal K}}{\ell_\alpha}, \qquad \varrho = e^{-t_\alpha/2}, \qquad v = v_N.
\end{equation*}

We now compute the conditional mean. Recall that
\begin{equation}\label{e:pairs-scale-ratio}
\ell_{\alpha'} = N^{\alpha - \alpha'} \ell_\alpha, \qquad M_{\alpha'} = N^{2(\alpha - \alpha')} M_\alpha.
\end{equation}
Using the first relation and \eqref{e:pairs-block-law}, we find
\begin{equation*}
\E\Big[ f_p\Big( \frac{\hat g_{\mathcal K}}{\ell_{\alpha'}} \Big) \, \Big| \, H \Big] = \E f_p\big( |\mathsf G_x| / N^{\alpha - \alpha'} \big).
\end{equation*}
Since $\varrho \in [1/2, 2]$ and $v_N \in [2, 4]$, we may apply \zcref{l:gaussian} with $n = 2$, the exponent $p > 3/2$, $v_0 = 4$, and $N^{\alpha - \alpha'} \le 1/2$ in place of $\delta$. For the density $\mathfrak g_4$ of a centered Gaussian vector in $\R^2$ with covariance $4$ times the identity, we have
\begin{equation*}
\mathfrak g_4(x, 0) = (8\pi)^{-1} e^{-x^2/8}, \qquad \int_{\R^2} f_p(|y|) \dd y = 2\pi \int_0^\infty r f_p(r) \dd r = \frac{\pi}{p - 1},
\end{equation*}
where the last equality follows from \eqref{e:pairs-radial}. Therefore \zcref{l:gaussian} gives
\begin{multline*}
\Big| N^{2(\alpha' - \alpha)} \E\Big[ f_p\Big( \frac{\hat g_{\mathcal K}}{\ell_{\alpha'}} \Big) \, \Big| \, H \Big] - \frac{e^{-g_{\mathcal K}^2/(8\ell_\alpha^2)}}{8(p - 1)} \Big| \\
\le C \big( N^{\alpha - \alpha'} + |e^{-t_\alpha/2} - 1| + |v_N - 4| \big) f_p\Big( \frac{g_{\mathcal K}}{\ell_\alpha} \Big),
\end{multline*}
and by \eqref{e:pairs-vN}, the factor in parentheses on the right is $o(1)$, uniformly in $\mathcal K$.

We multiply by $\varphi_N(E_{\mathcal K})$ and sum over $\mathcal K \in \mathfrak P_2$. The terms with $\mathcal K \in \mathfrak P_2$ form part of the sum defining $\mathcal X_\alpha(H; \varphi_N, f_p)$, so the second relation in \eqref{e:pairs-scale-ratio} gives
\begin{equation}\label{e:pairs-mean}
\Big| \frac{1}{M_{\alpha'}} \E\big[ \widehat{\mathcal X}_{\alpha'}(\varphi, p) \, \big| \, H \big] - \frac{1}{8(p - 1) M_\alpha} \sum_{\mathcal K \in \mathfrak P_2} \varphi_N(E_{\mathcal K}) e^{-g_{\mathcal K}^2/(8\ell_\alpha^2)} \Big| \le \frac{o(1)}{M_\alpha} \mathcal X_\alpha(H; \varphi_N, f_p).
\end{equation}
By $\mathcal P(\alpha)$, the right side tends to zero in probability.

We now evaluate the sum in \eqref{e:pairs-mean}. The function $y \mapsto e^{-y^2/8}$ satisfies the hypothesis of \zcref{l:extension}, and
\begin{equation*}
\int_0^\infty y e^{-y^2/8} \dd y = 4.
\end{equation*}
So $\mathcal P(\alpha)$ and \zcref{l:extension} give
\begin{equation*}
\frac{1}{M_\alpha} \sum_{1 \le i < j \le N} \varphi_N\Big( \frac{\lambda_i + \lambda_j}{2} \Big) \exp\Big( -\frac{(\lambda_j - \lambda_i)^2}{8 \ell_\alpha^2} \Big) \to 4 \int_{-2}^2 \varphi(E) c(E) \dd E \qquad \text{in probability}.
\end{equation*}
The sum on the left differs from the sum in \eqref{e:pairs-mean} by the terms of the pairs $i < j$ that do not form a cluster in $\mathfrak P_2$, which are nonnegative.

Suppose that $H \in \mathcal G_{\eps/2}$ and $N \ge N_0$. A pair that does not form a cluster in $\mathfrak P_2$ either is contained in a cluster with at least three elements, or meets two distinct clusters $\mathcal K$ and $\mathcal K'$, with $i \in \mathcal K$ and $j \in \mathcal K'$. By \zcref{l:pairs-clusters}, there are at most $2N^{\eps} \mathcal N_3(4q)$ pairs of the former kind. For a pair of the latter kind, \eqref{e:pairs-cluster-separation} gives
\begin{equation*}
\lambda_j - \lambda_i \ge e^{t_\alpha/2} (y_{\min \mathcal K'} - y_{\max \mathcal K}) \ge q = N^{\alpha'/6} \ell_\alpha,
\end{equation*}
where we used that $\alpha - \kappa = \alpha'/6$. Hence the difference of the two sums is bounded by
\begin{equation*}
2 \| \varphi \|_\infty N^{\eps} \mathcal N_3(4q) + \| \varphi \|_\infty N^2 \exp\Big( -\frac{q^2}{8 \ell_\alpha^2} \Big) = 2 \| \varphi \|_\infty N^{\eps} \mathcal N_3(4q) + \| \varphi \|_\infty N^2 \exp\Big( -\frac{N^{\alpha'/3}}{8} \Big).
\end{equation*}
Since $M_\alpha \ge M_{\alpha'}$, \zcref{l:pairs-clusters} and \zcref{p:regularity} show that this difference, divided by $M_\alpha$, tends to zero in probability. Inserting this into \eqref{e:pairs-mean}, and recalling the definition \eqref{e:pairs-induction-L} of $L$, we obtain \eqref{e:pairs-mean-limit}.
\end{proof}

\begin{lemma}\label{l:pairs-concentration}
Assume the hypotheses of \zcref{l:pairs-mean}. Then
\begin{equation}\label{e:pairs-block-limit}
\frac{1}{M_{\alpha'}} \widehat{\mathcal X}_{\alpha'}(\varphi, p) \to L \qquad \text{in probability}.
\end{equation}
\end{lemma}

\begin{proof}
Conditionally on $H$, the matrices $B_{\mathcal K}$, $\mathcal K \in \mathfrak P_2$, are independent. Indeed, $\mathfrak P_2$ and $y$ are functions of $H$, and each $B_{\mathcal K}$ depends on $W$ only through the entries $W_{ii}$, $W_{i + 1, i + 1}$, and $W_{i, i + 1}$. These sets of entries are disjoint for distinct clusters, and $W$ is independent of $H$ with independent entries. Hence $\widehat{\mathcal X}_{\alpha'}(\varphi, p)$ is a sum of conditionally independent random variables. Using $0 \le \varphi_N \le \| \varphi \|_\infty$ and $0 \le f_p^2 \le f_p \le 1$, we find
\begin{equation*}
\Var\big( \widehat{\mathcal X}_{\alpha'}(\varphi, p) \, \big| \, H \big) \le \sum_{\mathcal K \in \mathfrak P_2} \varphi_N(E_{\mathcal K})^2 \, \E\Big[ f_p\Big( \frac{\hat g_{\mathcal K}}{\ell_{\alpha'}} \Big)^2 \, \Big| \, H \Big] \le \| \varphi \|_\infty \E\big[ \widehat{\mathcal X}_{\alpha'}(\varphi, p) \, \big| \, H \big].
\end{equation*}
Let $\varpi > 0$. The conditional Chebyshev inequality gives
\begin{equation*}
\Prob\Big( \big| \widehat{\mathcal X}_{\alpha'}(\varphi, p) - \E\big[ \widehat{\mathcal X}_{\alpha'}(\varphi, p) \, \big| \, H \big] \big| > \varpi M_{\alpha'} \, \Big| \, H \Big) \le \min\Big\{ 1, \frac{\| \varphi \|_\infty}{\varpi^2 M_{\alpha'}} \cdot \frac{1}{M_{\alpha'}} \E\big[ \widehat{\mathcal X}_{\alpha'}(\varphi, p) \, \big| \, H \big] \Big\}.
\end{equation*}
By \zcref{l:pairs-mean} and $M_{\alpha'} \to \infty$, the right side tends to zero in probability. We take expectations and use the bounded convergence theorem, and we find that the unconditional probability tends to zero as well. Together with \eqref{e:pairs-mean-limit}, this implies \eqref{e:pairs-block-limit}.
\end{proof}

\begin{proof}[Proof of \zcref{p:induction}]
Fix an integer $p \ge p_*$ and a nonnegative $\varphi \in C_c^\infty((-2, 2))$, which we may assume is not identically zero, and let $L$ be as in \eqref{e:pairs-induction-L}. By \eqref{e:pairs-epsilon}, we have $\alpha' + 5\eps < 1/2$ and
\begin{equation*}
t_\alpha N^{1/2 + 3\alpha' + 12\eps} = N^{-1/2 - 2\alpha + 3\alpha' + 12\eps} \to 0.
\end{equation*}
Hence \zcref{l:pairs-transfer}, applied with $\alpha'$ in place of $\alpha$ and $t_N = t_\alpha$, shows that it suffices to prove
\begin{equation}\label{e:pairs-induction-goal}
\frac{1}{M_{\alpha'}} \mathcal X_{\alpha'}(H_{t_\alpha}; \varphi_N, f_p) \to L \qquad \text{in probability}.
\end{equation}

The event that $H \in \mathcal G_{\eps/2}$ and $\max_{i \le j} |W_{ij}| \le w_0$ has probability tending to one, by \zcref{p:regularity} and \eqref{e:pairs-wmax}. On this event, \zcref{l:pairs-reduction} gives
\begin{equation*}
\begin{split}
\Big| \frac{1}{M_{\alpha'}} \mathcal X_{\alpha'}(H_{t_\alpha}; \varphi_N, f_p) - L \Big| &\le \Big| \frac{1}{M_{\alpha'}} \widehat{\mathcal X}_{\alpha'}(\varphi, p) - L \Big| + \frac{o(1)}{M_{\alpha'}} \widehat{\mathcal X}_{\alpha'}(\varphi, p) \\
&\qquad + \frac{2 \| \varphi \|_\infty N^{\eps} \mathcal N_3(4q)}{M_{\alpha'}} + \frac{4^p \| \varphi \|_\infty N^2}{M_{\alpha'}} \Big( \frac{\ell_{\alpha'}}{q} \Big)^{2p}.
\end{split}
\end{equation*}

We now bound the last term on the right side. Since $M_{\alpha'} = N^{1 - 2\alpha'}$ and $\ell_{\alpha'}/q = N^{\kappa - \alpha'}$, this term is a constant multiple of $N^{1 + 2\alpha' - 2p(\alpha' - \kappa)}$, so it tends to zero if and only if $2p(\alpha' - \kappa) > 1 + 2\alpha'$. This is the only step in the proof of \zcref{t:pairs} that uses the value $p_* = 61$ and the condition $\alpha \ge 1/10$. By \eqref{e:pairs-margins}, we have $\alpha' - \kappa > \alpha/6$, and hence
\begin{equation*}
2p(\alpha' - \kappa) > \frac{p \alpha}{3} \ge \frac{p}{30} \ge \frac{61}{30} > 1 + 2\alpha',
\end{equation*}
where we used $\alpha \ge 1/10$, $p \ge p_* = 61$, and $\alpha' < 1/2$. We conclude that
\begin{equation}\label{e:pairs-cross-bound}
\frac{1}{M_{\alpha'}} N^2 \Big( \frac{\ell_{\alpha'}}{q} \Big)^{2p} = N^{1 + 2\alpha' - 2p(\alpha' - \kappa)} \le N^{1 + 2\alpha' - p/30} \le N^{2 - 61/30} \to 0.
\end{equation}

By \zcref{l:pairs-clusters} and Markov's inequality, the third term on the right side tends to zero in probability. Since $\mathcal P(\alpha)$ holds, \zcref{l:pairs-concentration} shows that so do the first two terms, which proves \eqref{e:pairs-induction-goal}.
\end{proof}

\section{Iteration and conclusion}\label{s:pairs-iteration}

We now iterate \zcref{p:induction}, starting from \zcref{p:base}.

\begin{corollary}\label{c:pairs-all}
For every $\alpha \in (0, 1/2)$, the statement $\mathcal P(\alpha)$ holds.
\end{corollary}

\begin{proof}
We define $\alpha^{(0)} = 1/10$ and
\begin{equation}\label{e:pairs-sequence}
\alpha^{(n + 1)} = \alpha^{(n)} + \frac{1}{10} \min\Big\{ \alpha^{(n)}, \frac12 - \alpha^{(n)} \Big\} \qquad \text{for } n \ge 0.
\end{equation}
By induction on $n$, we have $\alpha^{(n)} \in [1/10, 1/2)$ for all $n$. Indeed, if $\alpha^{(n)} \in [1/10, 1/2)$, then $\alpha^{(n + 1)} > \alpha^{(n)}$ and
\begin{equation*}
\alpha^{(n + 1)} \le \alpha^{(n)} + \frac{1}{10} \Big( \frac12 - \alpha^{(n)} \Big) < \frac12.
\end{equation*}
The sequence $(\alpha^{(n)})$ is therefore increasing and bounded by $1/2$. Its limit $\alpha^{(\infty)} \in (1/10, 1/2]$ satisfies $\min\{ \alpha^{(\infty)}, 1/2 - \alpha^{(\infty)} \} = 0$ by \eqref{e:pairs-sequence}, and hence $\alpha^{(n)} \to 1/2$.

We claim that for all $n \ge 0$ and all $\alpha' \in (\alpha^{(n)}, \alpha^{(n + 1)}]$, the pair $(\alpha^{(n)}, \alpha')$ satisfies \eqref{e:pairs-induction-conditions}. We have $1/10 \le \alpha^{(n)} < \alpha' \le \alpha^{(n + 1)} < 1/2$. Further, \eqref{e:pairs-sequence} gives
\begin{equation*}
\alpha' - \alpha^{(n)} \le \frac{\alpha^{(n)}}{10} < \frac{\alpha^{(n)}}{5}, \qquad \alpha' - \alpha^{(n)} \le \frac{1}{10} \Big( \frac12 - \alpha^{(n)} \Big) < \frac13 \Big( \frac12 - \alpha^{(n)} \Big),
\end{equation*}
so that
\begin{equation*}
\alpha' < \frac{6\alpha^{(n)}}{5}, \qquad \alpha' < \alpha^{(n)} + \frac16 - \frac{\alpha^{(n)}}{3} = \frac16 + \frac{2\alpha^{(n)}}{3},
\end{equation*}
as claimed.

We have $\alpha^{(0)} < 1/8$, so \zcref{p:base} shows that $\mathcal P(\alpha^{(0)})$ holds. Assuming that $\mathcal P(\alpha^{(n)})$ holds, we may apply \zcref{p:induction} with $(\alpha, \alpha') = (\alpha^{(n)}, \alpha^{(n + 1)})$, and we find that $\mathcal P(\alpha^{(n + 1)})$ holds. Hence $\mathcal P(\alpha^{(n)})$ holds for all $n \ge 0$. Now let $\alpha' \in (0, 1/2)$. When $\alpha' \le \alpha^{(0)}$, the statement $\mathcal P(\alpha')$ holds by \zcref{p:base}. Otherwise, as $\alpha^{(n)} \to 1/2$, there exists $n \ge 0$ such that $\alpha^{(n)} < \alpha' \le \alpha^{(n + 1)}$, and \zcref{p:induction} with $(\alpha, \alpha') = (\alpha^{(n)}, \alpha')$ shows that $\mathcal P(\alpha')$ holds.
\end{proof}

\begin{proof}[Proof of \zcref{t:pairs}]
Fix $\alpha \in (0, 1/2)$, a continuous function $\varphi$ with compact support in $(-2, 2)$, and a continuous function $f \colon [0, \infty) \to \R$ with compact support. Recall that $N^{2\alpha - 1} = M_\alpha^{-1}$. By \zcref{c:pairs-all}, $\mathcal P(\alpha)$ holds, so \zcref{l:extension} applies. Since $f$ has compact support, it satisfies the hypothesis of \zcref{l:extension}, and we fix $R \ge 1$ such that $\supp f \subset [0, R]$.

Suppose first that $\varphi \in C_c^\infty((-2, 2))$ is nonnegative. We split the sum defining $\mathcal X_\alpha(H; \varphi, f)$ into the terms with $j = i + 1$ and the terms with $j \ge i + 2$, which gives
\begin{equation}\label{e:pairs-consecutive-split}
\mathcal X_\alpha(H; \varphi, f) = \sum_{i = 1}^{N - 1} \varphi(E_i) f\Big( \frac{g_i}{\ell_\alpha} \Big) + \sum_{\substack{1 \le i, j \le N \\ j \ge i + 2}} \varphi\Big( \frac{\lambda_i + \lambda_j}{2} \Big) f\Big( \frac{\lambda_j - \lambda_i}{\ell_\alpha} \Big).
\end{equation}
To show that the second sum is negligible, we count its nonzero terms. A term in this sum vanishes unless $\lambda_j - \lambda_i \le R \ell_\alpha$. Let $N \ge N_0$, so that $R \ell_\alpha \le N^{-1}$, and suppose that $H \in \mathcal G_{\alpha/2}$. For each $i$, there are at most $N^{\alpha/2}$ labels $j$ with $\lambda_j \in [\lambda_i, \lambda_i + N^{-1}]$. A label $j \ge i + 2$ with $\lambda_j - \lambda_i \le R \ell_\alpha$ exists only if $\lambda_{i + 2} - \lambda_i \le R \ell_\alpha$. It follows that
\begin{equation}\label{e:pairs-nonconsecutive-count}
\# \big\{ (i, j) : 1 \le i < j \le N, \ j \ge i + 2, \ \lambda_j - \lambda_i \le R \ell_\alpha \big\} \le N^{\alpha/2} \mathcal N_3(R \ell_\alpha).
\end{equation}

Using \zcref{p:three} with $r = R \ell_\alpha \ge N^{-2}$, we have
\begin{equation*}
\frac{1}{M_\alpha} \E\big[ N^{\alpha/2} \mathcal N_3(R \ell_\alpha) \big] \le \frac{C N^{\alpha/2}}{M_\alpha} N (N R \ell_\alpha)^3 = C R^3 N^{-\alpha/2} \to 0.
\end{equation*}
The second sum in \eqref{e:pairs-consecutive-split} is bounded by $\| \varphi \|_\infty \| f \|_\infty$ times the left side of \eqref{e:pairs-nonconsecutive-count}. By Markov's inequality and \zcref{p:regularity}, it is therefore $o_{\Prob}(M_\alpha)$. Combining this with \zcref{l:extension} for the unsmoothed weight $\varphi$, we obtain
\begin{equation}\label{e:pairs-consecutive-smooth}
\frac{1}{M_\alpha} \sum_{i = 1}^{N - 1} \varphi(E_i) f\Big( \frac{g_i}{\ell_\alpha} \Big) \to \int_{-2}^2 \varphi(E) c(E) \dd E \int_0^\infty y f(y) \dd y \qquad \text{in probability}.
\end{equation}

Next, we treat a general continuous $\varphi$ with compact support in $(-2, 2)$. The positive and negative parts of $\varphi$ are continuous, nonnegative, and compactly supported in $(-2, 2)$, and both sides of \eqref{e:pairs-consecutive-smooth} are linear in $\varphi$. We may therefore assume that $\varphi \ge 0$.

Let $\delta > 0$ satisfy $\supp \varphi \subset [-2 + 2\delta, 2 - 2\delta]$, and set $I = [-2 + \delta, 2 - \delta]$. Let $\varpi > 0$. By convolving $\varphi$ with a smooth mollifier supported in a sufficiently small interval $[-\delta', \delta'] \subset [-\delta, \delta]$, we obtain a nonnegative $\tilde \varphi \in C_c^\infty((-2, 2))$ with $\supp \tilde \varphi \subset I$ and $\| \varphi - \tilde \varphi \|_\infty \le \varpi$. We set
\begin{equation*}
C_f = \sup_{y \ge 0} (1 + y^2)^{p_*} |f(y)| \le (1 + R^2)^{p_*} \| f \|_\infty.
\end{equation*}
The function $\varphi - \tilde \varphi$ vanishes outside $I$, so
\begin{equation*}
\Big| \sum_{i = 1}^{N - 1} (\varphi - \tilde \varphi)(E_i) f\Big( \frac{g_i}{\ell_\alpha} \Big) \Big| \le \varpi C_f \sum_{i : E_i \in I} f_{p_*}\Big( \frac{g_i}{\ell_\alpha} \Big) \le \varpi C_f \sum_{\substack{1 \le i < j \le N \\ (\lambda_i + \lambda_j)/2 \in I}} f_{p_*}\Big( \frac{\lambda_j - \lambda_i}{\ell_\alpha} \Big).
\end{equation*}
By \eqref{e:pairs-upper}, the right side is bounded by $\varpi C_f C_I M_\alpha$ with probability tending to one.

Further, the limits in \eqref{e:pairs-consecutive-smooth} for $\varphi$ and $\tilde \varphi$ differ by at most $\varpi \int_{-2}^2 c(E) \dd E \int_0^\infty y |f(y)| \dd y$. Applying \eqref{e:pairs-consecutive-smooth} to $\tilde \varphi$, we find that with probability tending to one,
\begin{multline*}
\Big| \frac{1}{M_\alpha} \sum_{i = 1}^{N - 1} \varphi(E_i) f\Big( \frac{g_i}{\ell_\alpha} \Big) - \int_{-2}^2 \varphi(E) c(E) \dd E \int_0^\infty y f(y) \dd y \Big| \\
\le \varpi \Big( 1 + C_f C_I + \int_{-2}^2 c(E) \dd E \int_0^\infty y |f(y)| \dd y \Big).
\end{multline*}
The constant $C_I$ does not depend on $\varpi$, and $\varpi > 0$ is arbitrary. This completes the proof.
\end{proof}

\begin{proof}[Proof of \zcref{p:pairsl}]
Recall that $\alpha_* = 99/200$, $\ell = \ell_{\alpha_*}$, and $M_{\alpha_*} = \tau$. By \zcref{c:pairs-all}, $\mathcal P(\alpha_*)$ holds. The second assertion of \zcref{p:pairsl} is \eqref{e:pairs-upper} with $\alpha = \alpha_*$.

For the first assertion, let $f \in C_c([0, \infty) \times (-2, 2))$. Let $R \ge 1$ and $\delta \in (0, 1)$ satisfy $\supp f \subset [0, R] \times [-2 + 2\delta, 2 - 2\delta]$, and set $I = [-2 + \delta, 2 - \delta]$. Let $\varpi > 0$. The function $f$ is uniformly continuous, so there exists $\delta' \in (0, \delta)$ such that $|f(r, E) - f(r, E')| \le \varpi$ for all $r \ge 0$ and all $E, E'$ with $|E - E'| \le \delta'$.

We approximate $f$ by a sum of products. Using a standard smooth partition of unity, we choose nonnegative functions $\chi_1, \dots, \chi_n \in C_c^\infty((-2 + \delta, 2 - \delta))$ such that each $\supp \chi_k$ has diameter at most $\delta'$ and
\begin{equation}\label{e:pairs-partition}
\sum_{k = 1}^n \chi_k \le 1 \quad \text{on } \R, \qquad \sum_{k = 1}^n \chi_k = 1 \quad \text{on } [-2 + 2\delta, 2 - 2\delta].
\end{equation}
For each $k$, we choose a point $E'_k \in \supp \chi_k$, and we set
\begin{equation*}
\bar f(r, E) = \sum_{k = 1}^n \chi_k(E) f(r, E'_k).
\end{equation*}

We claim that
\begin{equation}\label{e:pairs-product-approx}
|f(r, E) - \bar f(r, E)| \le \varpi (1 + R^2)^{p_*} f_{p_*}(r) \1_I(E) \qquad \text{for all } (r, E) \in [0, \infty) \times (-2, 2).
\end{equation}
When $E \in [-2 + 2\delta, 2 - 2\delta]$, the second relation in \eqref{e:pairs-partition} gives
\begin{equation*}
f(r, E) - \bar f(r, E) = \sum_{k = 1}^n \chi_k(E) \big( f(r, E) - f(r, E'_k) \big).
\end{equation*}
When $E \notin [-2 + 2\delta, 2 - 2\delta]$, we have $f(r, E) = 0$, and the same identity holds. In either case, $\chi_k(E) \ne 0$ implies $|E - E'_k| \le \delta'$, so
\begin{equation*}
|f(r, E) - \bar f(r, E)| \le \varpi \sum_{k = 1}^n \chi_k(E) \le \varpi.
\end{equation*}
The function $f - \bar f$ vanishes unless $r \le R$ and $E \in I$, which gives \eqref{e:pairs-product-approx}.

By \eqref{e:pairs-product-approx} and \eqref{e:pairs-upper} with $\alpha = \alpha_*$, we have with probability tending to one
\begin{equation*}
\frac{1}{\tau} \Big| \sum_{1 \le i < j \le N} (f - \bar f)\Big( \frac{\lambda_j - \lambda_i}{\ell}, \frac{\lambda_i + \lambda_j}{2} \Big) \Big| \le \varpi (1 + R^2)^{p_*} C_I.
\end{equation*}
On the other hand, for each $k$ the function $r \mapsto f(r, E'_k)$ is continuous with compact support in $[0, \infty)$, and
\begin{equation*}
\sum_{1 \le i < j \le N} \bar f\Big( \frac{\lambda_j - \lambda_i}{\ell}, \frac{\lambda_i + \lambda_j}{2} \Big) = \sum_{k = 1}^n \mathcal X_{\alpha_*}\big( H; \chi_k, f(\cdot, E'_k) \big).
\end{equation*}
We apply \zcref{l:extension} with $\alpha = \alpha_*$ and the unsmoothed weights $\chi_k$, and we obtain
\begin{equation*}
\begin{split}
\frac{1}{\tau} \sum_{1 \le i < j \le N} \bar f\Big( \frac{\lambda_j - \lambda_i}{\ell}, \frac{\lambda_i + \lambda_j}{2} \Big) &\to \sum_{k = 1}^n \int_{-2}^2 \chi_k(E) c(E) \dd E \int_0^\infty r f(r, E'_k) \dd r \\
&= \int_{-2}^2 \int_0^\infty c(E) r \bar f(r, E) \dd r \dd E
\end{split}
\end{equation*}
in probability. Finally, \eqref{e:pairs-product-approx} gives
\begin{equation*}
\Big| \int_{-2}^2 \int_0^\infty c(E) r (f - \bar f)(r, E) \dd r \dd E \Big| \le \varpi R^2 \int_{-2}^2 c(E) \dd E.
\end{equation*}
The constant $C_I$ does not depend on $\varpi$, and $\varpi > 0$ is arbitrary, which proves the first assertion.
\end{proof}