\chapter{A comparison estimate for pair statistics}\label{s:comparison}

In this section, we compare Laplace transforms of pair statistics of $H$ with those of the Ornstein--Uhlenbeck interpolation $H_t$ defined in \eqref{e:prelim-Ht}. In \zcref{s:pairs}, pair statistics of $H_t$ are computed for suitable small times $t$ using the Gaussian component of $H_t$, and \zcref{p:comparison} below transfers these computations to $H$. The pair statistics count pairs of eigenvalues at distance of order $\ell_\alpha = N^{-1 - \alpha}$, weighted by a smooth function of their midpoint, and the observables we compare are exponentials of nonpositive multiples of such statistics.

Our argument follows the continuity estimate of Bourgade and Yau \cite[Lemma~A.1]{bourgade2017eigenvector} (see also \cite[Chapter~15]{erdos2017dynamical}), which is a continuous version of the Lindeberg replacement strategy \cite{chatterjee2006generalization,tao2011random} along the Ornstein--Uhlenbeck interpolation. It requires bounds on the derivatives of the observable in one matrix entry, up to order three, or four when $\E X^3 = 0$. Obtaining these bounds is the main difficulty. Derivatives of eigenvalues of order two and higher contain inverse gaps $(\lambda_i - \lambda_j)^{-1}$, and bounding the resulting sums would require control of pairs of nearby eigenvalues, which is what the comparison is meant to establish. We therefore avoid eigenvalue perturbation theory. Instead, we show that the pair statistic extends to a holomorphic function of the matrix entries whose real part is nonnegative as long as the spectrum stays in a thin rectangle of height of order $\ell_\alpha$ around $[-4, 4]$. The observable then has modulus at most one on a complex disk in each entry, whose radius is slightly smaller than $N^{-\alpha}$, and Cauchy's estimates bound its derivatives by powers of the inverse radius, uniformly in the normalization of the statistic. These bounds suffice because the radius is much larger than the typical size $N^{-1/2}$ of an entry.

\zcref[S]{s:comparison-smoothing} introduces the smoothed energy weights. In \zcref{s:comparison-strip}, we show that a complex perturbation of one entry of a regular matrix keeps its spectrum in the thin rectangle; this is the only place in this section where we use the conditions of \zcref{d:regular}. \zcref[S]{s:comparison-extension} constructs the holomorphic extension, and \zcref{s:comparison-proof} proves the comparison estimate. The proof uses the observable only through the two properties listed in \zcref{r:comparison-inputs}.

\section{Smoothed energy weights}\label{s:comparison-smoothing}

Recall the scale $\ell_\alpha = N^{-1 - \alpha}$ from \eqref{e:prelim-scales}. For $\alpha \in (0, 1/2)$, a real symmetric $N \times N$ matrix $A$, and functions $\varphi \colon \R \to \R$ and $f \colon [0, \infty) \to \R$, we define the pair statistic
\begin{equation}\label{e:comparison-pair}
\mathcal X_\alpha(A; \varphi, f) = \sum_{1 \le i < j \le N} \varphi \Big( \frac{\lambda_i(A) + \lambda_j(A)}{2} \Big) f \Big( \frac{\lambda_j(A) - \lambda_i(A)}{\ell_\alpha} \Big).
\end{equation}
Throughout the rest of this section, we fix $\alpha \in (0, 1/2)$. Recall the functions $f_p$ from \eqref{e:prelim-fp}. For an integer $p \ge 1$, we extend $f_p$ to $\C \setminus \{ \iu, -\iu \}$ by the same formula, $f_p(z) = (1 + z^2)^{-p}$.

We set $h_N = N^{-1/10}$, and for a nonnegative $\varphi \in C_c^\infty((-2, 2))$, we define the smoothed energy weight
\begin{equation}\label{e:comparison-phiN}
\varphi_N(w) = \frac{1}{\sqrt{2 \pi} \, h_N} \int_\R \varphi(y) \exp \Big( - \frac{(w - y)^2}{2 h_N^2} \Big) \dd y, \qquad w \in \C.
\end{equation}
We use this notation for every nonnegative function in $C_c^\infty((-2, 2))$; for example, $\varphi_{j, N}$ in \zcref{d:complexS} below denotes the function defined by \eqref{e:comparison-phiN} with $\varphi_j$ in place of $\varphi$. Differentiating under the integral sign, we see that $\varphi_N$ is an entire function. For real $x$, the number $\varphi_N(x)$ is the average of $\varphi$ with respect to the Gaussian density with mean $x$ and variance $h_N^2$. For $z \in \C \setminus \{ 0 \}$, we let $\arg z \in (-\pi, \pi]$ denote the principal argument of $z$.

Part (3) of the following lemma controls the argument of $\varphi_N$ near the real axis. We will use it, together with part (1), in \zcref{s:comparison-extension}, and we will use parts (1), (2), (4), and (5) in \zcref{s:pairs} and \zcref{s:hermitian}.

\begin{lemma}\label{l:smoothing}
Let $\varphi \in C_c^\infty((-2, 2))$ be nonnegative. Then the following statements hold for all $N \ge 1$.
\begin{enumerate}[label=(\arabic*)]
\item For all $x \in \R$, we have $0 \le \varphi_N(x) \le \| \varphi \|_\infty$. If $\varphi$ is not identically zero, then $\varphi_N(x) > 0$ for all $x \in \R$.
\item For all $x \in \R$ with $|x| \le 5$ and all $w \in \C$ with $|w - x| \le 1$, we have
\begin{equation}\label{e:comparison-relative}
| \varphi_N(w) - \varphi_N(x) | \le \big( \exp(8 |w - x| h_N^{-2}) - 1 \big) \varphi_N(x).
\end{equation}
If moreover $w \in \R$, then
\begin{equation}\label{e:comparison-relative-real}
\exp(- 8 |w - x| h_N^{-2}) \varphi_N(x) \le \varphi_N(w) \le \exp(8 |w - x| h_N^{-2}) \varphi_N(x).
\end{equation}
\item For all $w \in \C$ such that $(|\Re w| + 2) |\Im w| \le h_N^2$, either $\varphi_N(w) = 0$ or
\begin{equation}\label{e:comparison-phase}
| \arg \varphi_N(w) | \le (|\Re w| + 2) |\Im w| h_N^{-2}.
\end{equation}
\item For all $x \in \R$, we have
\begin{equation}\label{e:comparison-tailweight}
\varphi_N(x) \le 2 \| \varphi \|_\infty \exp \big( - \dist(x, \supp \varphi)^2 / (2 h_N^2) \big).
\end{equation}
\item We have $\sup_{x \in \R} | \varphi_N(x) - \varphi(x) | \le \| \varphi' \|_\infty h_N$.
\end{enumerate}
\end{lemma}

\begin{proof}
The first claim in part (1) holds since $\varphi_N(x)$ is the average of $\varphi$ against a probability density. The second holds since this density is positive everywhere and $\varphi$ is continuous, nonnegative, and not identically zero.

We now prove part (2). Fix $x \in \R$ with $|x| \le 5$ and $w \in \C$ with $|w - x| \le 1$. For $y \in \supp \varphi$, we have $|x - y| \le 7$, and expanding $(w - y)^2$ around $x - y$ gives
\begin{equation}\label{e:comparison-split}
\exp \Big( - \frac{(w - y)^2}{2 h_N^2} \Big) = \exp \Big( - \frac{(x - y)^2}{2 h_N^2} \Big) \exp \Big( - \frac{2 (x - y)(w - x) + (w - x)^2}{2 h_N^2} \Big).
\end{equation}
The exponent in the second factor satisfies
\begin{equation}\label{e:comparison-exponent}
\frac{| 2 (x - y)(w - x) + (w - x)^2 |}{2 h_N^2} \le \frac{14 |w - x| + |w - x|^2}{2 h_N^2} \le 8 |w - x| h_N^{-2}.
\end{equation}
Using $|e^z - 1| \le e^{|z|} - 1$ for $z \in \C$ and integrating \eqref{e:comparison-split} against $\varphi(y) \dd y$, we find
\begin{align*}
| \varphi_N(w) - \varphi_N(x) | & \le \frac{1}{\sqrt{2 \pi} \, h_N} \int_\R \varphi(y) \exp \Big( - \frac{(x - y)^2}{2 h_N^2} \Big) \big( e^{8 |w - x| h_N^{-2}} - 1 \big) \dd y \\
& = \big( e^{8 |w - x| h_N^{-2}} - 1 \big) \varphi_N(x),
\end{align*}
which is \eqref{e:comparison-relative}.

When $w$ is real, the exponent in the second factor of \eqref{e:comparison-split} is real, so by \eqref{e:comparison-exponent}, we have
\begin{equation*}
e^{-8 |w - x| h_N^{-2}} \le \exp \Big( - \frac{2 (x - y)(w - x) + (w - x)^2}{2 h_N^2} \Big) \le e^{8 |w - x| h_N^{-2}}, \qquad y \in \supp \varphi.
\end{equation*}
Integrating \eqref{e:comparison-split} against $\varphi(y) \dd y$ gives \eqref{e:comparison-relative-real}.

To prove part (3), we expand
\begin{equation*}
(w - y)^2 = (\Re w - y)^2 - (\Im w)^2 + 2 \iu \Im w \, (\Re w - y),
\end{equation*}
which gives
\begin{multline*}
\varphi_N(w) = \frac{1}{\sqrt{2 \pi} \, h_N} \int_\R \varphi(y) \exp \Big( - \frac{(\Re w - y)^2 - (\Im w)^2}{2 h_N^2} \Big) \\
\times \exp \Big( - \frac{\iu \Im w \, (\Re w - y)}{h_N^2} \Big) \dd y.
\end{multline*}

The first two factors in the integrand are nonnegative. For $y \in \supp \varphi$, the phase of the last factor has absolute value at most
\begin{equation*}
\beta = (|\Re w| + 2) |\Im w| h_N^{-2} \le 1.
\end{equation*}
Since $\beta < \pi/2$, the closed sector of half-angle $\beta$ is an intersection of two closed half-planes,
\begin{equation*}
\{ 0 \} \cup \{ z \neq 0 : |\arg z| \le \beta \} = \big\{ z : \Re \big( e^{\iu (\pi/2 - \beta)} z \big) \ge 0 \big\} \cap \big\{ z : \Re \big( e^{- \iu (\pi/2 - \beta)} z \big) \ge 0 \big\}.
\end{equation*}
Each of these half-planes is preserved by integration against a nonnegative weight, so $\varphi_N(w)$ belongs to the sector, as claimed in \eqref{e:comparison-phase}.

For parts (4) and (5), we change variables in \eqref{e:comparison-phiN} and write, for $x \in \R$,
\begin{equation*}
\varphi_N(x) = \frac{1}{\sqrt{2 \pi}} \int_\R \varphi(x + h_N y) e^{-y^2/2} \dd y.
\end{equation*}
The integrand vanishes unless $h_N |y| \ge \dist(x, \supp \varphi)$, so \eqref{e:comparison-tailweight} follows from the Gaussian tail bound
\begin{equation*}
\int_{|y| \ge y_0} e^{-y^2/2} \dd y \le 2 \sqrt{2 \pi} \, e^{-y_0^2/2}, \qquad y_0 \ge 0.
\end{equation*}
Further, the mean value theorem gives
\begin{equation*}
| \varphi_N(x) - \varphi(x) | \le \frac{1}{\sqrt{2 \pi}} \int_\R | \varphi(x + h_N y) - \varphi(x) | e^{-y^2/2} \dd y \le \Big( \frac{2}{\pi} \Big)^{1/2} \| \varphi' \|_\infty h_N \le \| \varphi' \|_\infty h_N,
\end{equation*}
which is part (5).
\end{proof}

\section{Spectra under a perturbation of one entry}\label{s:comparison-strip}

For an integer $p \ge 1$, we set
\begin{equation}\label{e:comparison-omega}
\omega_p = \frac{1}{1000 (p + 1)}, \qquad c_p = \frac{\omega_p}{100},
\end{equation}
and we define the open rectangle
\begin{equation}\label{e:comparison-rectangle}
\mathfrak{R}_{\alpha, p} = \{ w \in \C : |\Re w| < 4, \ |\Im w| < \omega_p \ell_\alpha \}.
\end{equation}
Recall the set $\mathcal{G}_\eps$ of regular matrices from \zcref{d:regular} and the matrices $V^{ab}$ from \eqref{e:prelim-V}. In the next lemma, the size $8 c_p N^{-\alpha - 4 \eps}$ of the perturbation is much larger than the typical size $N^{-1/2}$ of an entry of $H_t$ when $\alpha + 4 \eps < 1/2$.

\begin{lemma}\label{l:strip}
Let $\eps > 0$, and let $p \ge 1$ be an integer. There exists $N_0 = N_0(\alpha, p)$ such that the following holds for all $N \ge N_0$. Let $A \in \mathcal{G}_\eps$, let $a, b \in [N]$ with $a \le b$, and let $\zeta \in \C$ with $|\zeta| \le 8 c_p N^{-\alpha - 4 \eps}$. Then $\Spec(A + \zeta V^{ab}) \subset \mathfrak{R}_{\alpha, p}$.
\end{lemma}

\begin{proof}
We fix a spectral decomposition
\begin{equation*}
A = \sum_{i = 1}^N \lambda_i(A) v_i v_i^T,
\end{equation*}
where $(v_i)_{i \in [N]}$ is an orthonormal basis of $\R^N$. Recall from the definition of $\mathcal{G}_\eps$ that $\| A \| \le 3$ and $\| v_i \|_\infty \le N^{-1/2 + \eps}$ for all $i \in [N]$.

We write $\partial \mathfrak{R}_{\alpha, p}$ for the boundary of $\mathfrak{R}_{\alpha, p}$. It consists of two horizontal sides on the lines $\Im w = \pm \omega_p \ell_\alpha$ and two vertical sides on the lines $\Re w = \pm 4$.

We begin by bounding the entries of the resolvent $(A - w)^{-1}$ for $w \in \partial \mathfrak{R}_{\alpha, p}$. Let $w$ lie on a horizontal side, so that $|\Re w| \le 4$ and $|\Im w| = \omega_p \ell_\alpha$. For $a', b' \in \{ a, b \}$, the spectral decomposition gives
\begin{equation}\label{e:comparison-resolvent}
\big| \big( (A - w)^{-1} \big)_{a' b'} \big| \le \sum_{i = 1}^N \frac{|v_i(a')| \, |v_i(b')|}{|\lambda_i(A) - w|} \le N^{-1 + 2 \eps} \sum_{i = 1}^N \frac{1}{|\lambda_i(A) - w|}.
\end{equation}

To bound the last sum, we partition the real line into the intervals
\begin{equation*}
\mathcal{I}_n = \big[ \Re w + n/N, \Re w + (n + 1)/N \big), \qquad n \in \Z.
\end{equation*}
Every eigenvalue of $A$ belongs to some $\mathcal{I}_n$ with $|n| \le 7 N$, since $\lambda_i(A) \in [-3, 3]$ and $|\Re w| \le 4$. Moreover, the definition of $\mathcal{G}_\eps$ implies that each $\mathcal{I}_n$ contains at most $N^\eps$ eigenvalues of $A$, counted with multiplicity. For $\lambda_i(A) \in \mathcal{I}_n$, we have
\begin{equation}\label{e:comparison-cases}
|\lambda_i(A) - w| \ge
\begin{cases}
\omega_p \ell_\alpha & \text{if } n \in \{ -1, 0 \}, \\
n / N & \text{if } n \ge 1, \\
(|n| - 1) / N & \text{if } n \le -2,
\end{cases}
\end{equation}
where the first bound uses $|\Im w| = \omega_p \ell_\alpha$, and the other two use $|\lambda_i(A) - w| \ge |\lambda_i(A) - \Re w|$. Summing over $n$, we obtain
\begin{equation*}
\sum_{i = 1}^N \frac{1}{|\lambda_i(A) - w|} \le \frac{2 N^\eps}{\omega_p \ell_\alpha} + 2 N^\eps \sum_{n = 1}^{7 N} \frac{N}{n} \le \frac{2 N^{1 + \alpha + \eps}}{\omega_p} + 2 N^{1 + \eps} \big( 1 + \log(7 N) \big).
\end{equation*}

We now choose $N_0 = N_0(\alpha, p)$ such that
\begin{equation*}
2 \omega_p \big( 1 + \log(7 N) \big) \le N^\alpha \qquad \text{for all } N \ge N_0.
\end{equation*}
For such $N$, the right side is bounded by $3 N^{1 + \alpha + \eps} / \omega_p$, and inserting this bound into \eqref{e:comparison-resolvent}, we obtain
\begin{equation}\label{e:comparison-resolvent-bound}
\big| \big( (A - w)^{-1} \big)_{a' b'} \big| \le \frac{3 N^{\alpha + 3 \eps}}{\omega_p}, \qquad a', b' \in \{ a, b \}.
\end{equation}
It remains to treat the vertical sides. There we have $|\Re w| = 4$, so $\dist(w, \Spec A) \ge 1$ since $\| A \| \le 3$. Hence $\| (A - w)^{-1} \| \le 1$ there, and \eqref{e:comparison-resolvent-bound} holds on all of $\partial \mathfrak{R}_{\alpha, p}$.

Next, we reduce the characteristic polynomial of $A + \zeta V^{ab}$ to a determinant of size at most two. Recall Sylvester's determinant identity
\begin{equation*}
\det(\Id_N + Q_1 Q_2) = \det(\Id_2 + Q_2 Q_1), \qquad Q_1 \in \C^{N \times 2}, \quad Q_2 \in \C^{2 \times N}.
\end{equation*}
For $a < b$, we have
\begin{equation*}
\zeta V^{ab} = Q_1 Q_2, \qquad Q_1 = (e_a, e_b), \qquad Q_2 = \zeta (e_b, e_a)^T.
\end{equation*}
Factoring out $A - w$ and applying the identity, we find, for $w \notin \Spec A$ and $x \in [0, 1]$,
\begin{equation}\label{e:comparison-det}
\det(A + x \zeta V^{ab} - w) = \det(A - w) \det \bigg( \Id_2 + x \zeta \begin{pmatrix} ((A - w)^{-1})_{ba} & ((A - w)^{-1})_{bb} \\ ((A - w)^{-1})_{aa} & ((A - w)^{-1})_{ab} \end{pmatrix} \bigg).
\end{equation}
For $a = b$, the same reasoning with $Q_1 = e_a$ and $Q_2 = \zeta e_a^T$ gives
\begin{equation}\label{e:comparison-det-diagonal}
\det(A + x \zeta V^{aa} - w) = \det(A - w) \big( 1 + x \zeta ((A - w)^{-1})_{aa} \big).
\end{equation}

Now let $w \in \partial \mathfrak{R}_{\alpha, p}$ and $x \in [0, 1]$. Observe that $\partial \mathfrak{R}_{\alpha, p}$ does not meet $[-3, 3]$, so $w \notin \Spec A$.
Using \eqref{e:comparison-resolvent-bound} and our assumption on $\zeta$, we see that every entry of the $2 \times 2$ matrix in \eqref{e:comparison-det}, multiplied by $x \zeta$, has modulus bounded by
\begin{equation*}
8 c_p N^{-\alpha - 4 \eps} \cdot \frac{3 N^{\alpha + 3 \eps}}{\omega_p} = \frac{24 c_p}{\omega_p} N^{-\eps} \le \frac{1}{4},
\end{equation*}
where we used $c_p = \omega_p / 100$. For a $2 \times 2$ matrix $Q'$ whose entries have modulus at most $1/4$, we have
\begin{equation*}
|\det(\Id_2 + Q')| = |1 + \Tr Q' + \det Q'| \ge 1 - \frac{1}{2} - \frac{1}{8} > 0.
\end{equation*}
In the case $a = b$, the last factor in \eqref{e:comparison-det-diagonal} has modulus at least $3/4$. We conclude that
\begin{equation}\label{e:comparison-nonvanishing}
\det(A + x \zeta V^{ab} - w) \neq 0 \quad \text{for all } w \in \partial \mathfrak{R}_{\alpha, p} \text{ and } x \in [0, 1].
\end{equation}

Finally, we use the argument principle. For $x \in [0, 1]$, we set
\begin{equation*}
\chi_x(w) = \det(w \Id_N - A - x \zeta V^{ab}), \qquad w \in \C.
\end{equation*}
This is a monic polynomial of degree $N$ in $w$, whose coefficients are polynomials in $x$. Combining \eqref{e:comparison-nonvanishing} with the compactness of $[0, 1] \times \partial \mathfrak{R}_{\alpha, p}$, we see that $|\chi_x(w)|$ is bounded below by a positive constant for $(x, w)$ in this set. Hence, with $\partial \mathfrak{R}_{\alpha, p}$ oriented counterclockwise, the integral
\begin{equation*}
\frac{1}{2 \pi \iu} \oint_{\partial \mathfrak{R}_{\alpha, p}} \frac{\partial_w \chi_x(w)}{\chi_x(w)} \dd w,
\end{equation*}
which counts the zeros of $\chi_x$ in $\mathfrak{R}_{\alpha, p}$ with multiplicity, is a continuous function of $x \in [0, 1]$. It takes integer values, so it is constant.

For $x = 0$, the zeros of $\chi_0$ are the eigenvalues of $A$, which belong to $[-3, 3] \subset \mathfrak{R}_{\alpha, p}$, so the integral is $N$ for all $x$. Therefore all $N$ zeros of $\chi_1$, which are the eigenvalues of $A + \zeta V^{ab}$ counted with algebraic multiplicity, lie in $\mathfrak{R}_{\alpha, p}$. This completes the proof.
\end{proof}

\section{A holomorphic extension of the pair statistic}\label{s:comparison-extension}

We will need the following elementary fact.

\begin{lemma}\label{l:comparison-open}
Let $\mathcal{O} \subset \C$ be open. Then the set of matrices $A \in \C^{N \times N}$ such that $\Spec A \subset \mathcal{O}$ is open in $\C^{N \times N}$.
\end{lemma}

\begin{proof}
Let $(A_n)_{n \ge 1}$ be a sequence of matrices converging to $A$ such that each $A_n$ has an eigenvalue $w_n \notin \mathcal{O}$. The sequence $(w_n)$ is bounded, since $|w_n| \le \| A_n \|$, and after passing to a subsequence, we may assume that $w_n \to w$. Then $w \notin \mathcal{O}$, since $\C \setminus \mathcal{O}$ is closed, and
\begin{equation*}
\det(A - w) = \lim_{n \to \infty} \det(A_n - w_n) = 0.
\end{equation*}
Hence the set of matrices with an eigenvalue outside $\mathcal{O}$ is closed.
\end{proof}

We now introduce the tensor products used to define the extension. We identify $\C^N \otimes \C^N$ with $\C^{N^2}$, and we order the basis $(e_i \otimes e_j)_{i, j \in [N]}$ lexicographically in $(i, j)$. For $A \in \C^{N \times N}$, we define
\begin{equation}\label{e:comparison-tensor}
\mathfrak{D}(A) = A \otimes \Id - \Id \otimes A, \qquad \mathfrak{M}(A) = \frac{1}{2} (A \otimes \Id + \Id \otimes A).
\end{equation}
By Schur's theorem, there exist a unitary matrix $O$ and an upper triangular matrix $T$ such that $A = O T O^*$, where the diagonal entries $T_{ii} = w_i$ are the eigenvalues of $A$, listed with algebraic multiplicity (in any prescribed order). Then
\begin{equation}\label{e:comparison-triangular}
\begin{gathered}
(O \otimes O)^* \mathfrak{D}(A) (O \otimes O) = T \otimes \Id - \Id \otimes T, \\
(O \otimes O)^* \mathfrak{M}(A) (O \otimes O) = \frac{1}{2} (T \otimes \Id + \Id \otimes T).
\end{gathered}
\end{equation}
The matrices $T \otimes \Id$ and $\Id \otimes T$ are upper triangular with respect to the lexicographic order, since
\begin{equation*}
(T \otimes \Id)(e_i \otimes e_j) = \sum_{i' \le i} T_{i' i} \, e_{i'} \otimes e_j, \qquad (\Id \otimes T)(e_i \otimes e_j) = \sum_{j' \le j} T_{j' j} \, e_i \otimes e_{j'},
\end{equation*}
and their diagonal entries at $e_i \otimes e_j$ are $w_i$ and $w_j$, respectively. Hence the right sides of \eqref{e:comparison-triangular} are upper triangular, with diagonal entries $w_i - w_j$ and $(w_i + w_j)/2$ at $e_i \otimes e_j$.

For a square matrix $M$, we define
\begin{equation}\label{e:comparison-matrixfunction}
\varphi_N(M) = \sum_{n = 0}^\infty \frac{\varphi_N^{(n)}(0)}{n!} M^n,
\end{equation}
where the series converges absolutely since $\varphi_N$ is entire. Every power of an upper triangular matrix is upper triangular, with the corresponding powers of the diagonal entries on its diagonal. Hence, for an upper triangular matrix $M$, the matrix $\varphi_N(M)$ is upper triangular with diagonal entries $\varphi_N(M_{ii})$. Similarly, the inverse of an invertible upper triangular matrix is upper triangular, with the reciprocal diagonal entries. Moreover, for every square matrix $M$ and every unitary matrix $Q$ of the same size, we have
\begin{equation}\label{e:comparison-conjugation}
\varphi_N(Q^* M Q) = Q^* \varphi_N(M) Q.
\end{equation}

Suppose now that every eigenvalue $w_i$ of $A$ satisfies $|\Im w_i| < \omega_p \ell_\alpha$. By \eqref{e:comparison-triangular}, the matrix $\Id + \ell_\alpha^{-2} \mathfrak{D}(A)^2$ is similar to an upper triangular matrix with diagonal entries $1 + ((w_i - w_j) / \ell_\alpha)^2$. For $z = (w_i - w_j)/\ell_\alpha$, we have $|\Im z| < 2 \omega_p$, and
\begin{equation*}
\Re(1 + z^2) = 1 + (\Re z)^2 - (\Im z)^2 \ge 1 - 4 \omega_p^2 > 0.
\end{equation*}
Therefore $\Id + \ell_\alpha^{-2} \mathfrak{D}(A)^2$ is invertible, which justifies the following definition.

\begin{definition}\label{d:complexS}
For an integer $p \ge 1$, we define
\begin{equation}\label{e:comparison-domain}
\mathcal{U}_{\alpha, p} = \{ A \in \C^{N \times N} : |\Im w| < \omega_p \ell_\alpha \text{ for every eigenvalue } w \text{ of } A \}.
\end{equation}
For a nonnegative $\varphi \in C_c^\infty((-2, 2))$, an integer $p \ge 1$, and $A \in \mathcal{U}_{\alpha, p}$, we define
\begin{equation}\label{e:comparison-complexS}
\mathcal X_\alpha(A; \varphi_N, f_p) = \frac{1}{2} \Tr \Big[ \varphi_N \big( \mathfrak{M}(A) \big) \big( \Id + \ell_\alpha^{-2} \mathfrak{D}(A)^2 \big)^{-p} \Big] - \frac{1}{2} \Tr \varphi_N(A).
\end{equation}
Let $\mathcal{J}$ be a finite nonempty set. For each $j \in \mathcal{J}$, let $\varphi_j \in C_c^\infty((-2, 2))$ be nonnegative, let $p_j \ge 1$ be an integer, and let $\theta_j \ge 0$. Set $p = \max_{j \in \mathcal{J}} p_j$. We define the observable
\begin{equation}\label{e:comparison-F}
F(A) = \exp \Big( - \sum_{j \in \mathcal{J}} \theta_j \mathcal X_\alpha(A; \varphi_{j, N}, f_{p_j}) \Big), \qquad A \in \mathcal{U}_{\alpha, p}.
\end{equation}
\end{definition}

Since $\omega_p \le \omega_{p_j}$, we have $\mathcal{U}_{\alpha, p} \subset \mathcal{U}_{\alpha, p_j}$ for all $j \in \mathcal{J}$, so \eqref{e:comparison-F} is well defined. By \zcref{l:comparison-open}, the set $\mathcal{U}_{\alpha, p}$ is open. It contains every real symmetric matrix and every matrix with spectrum in $\mathfrak{R}_{\alpha, p}$. The observable $F$ depends on $N$ and on the data $(\varphi_j, p_j, \theta_j)_{j \in \mathcal{J}}$, which we suppress from the notation. The parameters $\theta_j$ may depend on $N$, and all bounds below are uniform in them.

\begin{lemma}\label{l:holomorphic}
Let $p \ge 1$ be an integer, and let $\varphi \in C_c^\infty((-2, 2))$ be nonnegative. Then the following statements hold for all $N \ge 1$.
\begin{enumerate}[label=(\arabic*)]
\item The function $A \mapsto \mathcal X_\alpha(A; \varphi_N, f_p)$ is holomorphic in the entries of $A$ on $\mathcal{U}_{\alpha, p}$. If $A \in \mathcal{U}_{\alpha, p}$ has eigenvalues $w_1, \dots, w_N$, listed with algebraic multiplicity in any order, then
\begin{equation}\label{e:comparison-eigen}
\mathcal X_\alpha(A; \varphi_N, f_p) = \sum_{1 \le i < j \le N} \varphi_N \Big( \frac{w_i + w_j}{2} \Big) f_p \Big( \frac{w_j - w_i}{\ell_\alpha} \Big).
\end{equation}
In particular, \eqref{e:comparison-complexS} agrees with \eqref{e:comparison-pair} for every real symmetric matrix $A$.
\item Let $A \in \mathcal{U}_{\alpha, p}$, and suppose that $(|\Re w| + 2) |\Im w'| \le h_N^2 / 200$ for all eigenvalues $w$ and $w'$ of $A$. Then every summand on the right side of \eqref{e:comparison-eigen} is either zero or has argument of absolute value at most $1/100$. In particular, $\Re \mathcal X_\alpha(A; \varphi_N, f_p) \ge 0$.
\item If $\Spec A \subset \mathfrak{R}_{\alpha, p}$, then $A \in \mathcal{U}_{\alpha, p}$, and $A$ satisfies the hypothesis of part (2).
\item Let $F$ be as in \zcref{d:complexS}, and suppose that $p = \max_{j \in \mathcal{J}} p_j$. Then $F$ is holomorphic in the entries of $A$ on $\mathcal{U}_{\alpha, p}$, and $|F(A)| \le 1$ for every $A \in \mathcal{U}_{\alpha, p}$ satisfying the hypothesis of part (2). In particular, $|F(A)| \le 1$ if $\Spec A \subset \mathfrak{R}_{\alpha, p}$ or if $A$ is real symmetric.
\end{enumerate}
\end{lemma}

\begin{proof}
We begin with holomorphy. The entries of $\mathfrak{M}(A)^n$ are polynomials in the entries of $A$, and $\| \mathfrak{M}(A)^n \| \le \| A \|^n$. Since $\varphi_N$ is entire, the series defining $\varphi_N(\mathfrak{M}(A))$ therefore converges uniformly on bounded subsets of $\C^{N \times N}$, and the entries of $\varphi_N(\mathfrak{M}(A))$ are holomorphic functions of the entries of $A$, as locally uniform limits of polynomials. The same holds for $\varphi_N(A)$. By Cramer's rule, the entries of $(\Id + \ell_\alpha^{-2} \mathfrak{D}(A)^2)^{-1}$ are quotients of polynomials in the entries of $A$, whose common denominator $\det(\Id + \ell_\alpha^{-2} \mathfrak{D}(A)^2)$ does not vanish on $\mathcal{U}_{\alpha, p}$. Hence the right side of \eqref{e:comparison-complexS} is holomorphic on $\mathcal{U}_{\alpha, p}$.

Next, we prove \eqref{e:comparison-eigen}. Let $A = O T O^*$ as above, with $T_{ii} = w_i$. Using \eqref{e:comparison-triangular} and \eqref{e:comparison-conjugation} with $Q = O \otimes O$, we find that
\begin{multline*}
(O \otimes O)^* \varphi_N \big( \mathfrak{M}(A) \big) \big( \Id + \ell_\alpha^{-2} \mathfrak{D}(A)^2 \big)^{-p} (O \otimes O) \\
= \varphi_N \Big( \frac{1}{2} (T \otimes \Id + \Id \otimes T) \Big) \Big( \Id + \ell_\alpha^{-2} (T \otimes \Id - \Id \otimes T)^2 \Big)^{-p}.
\end{multline*}
By the facts on triangular matrices recalled after \eqref{e:comparison-matrixfunction}, the right side is a product of upper triangular matrices. The diagonal entry of this product at $e_i \otimes e_j$ is
\begin{equation*}
\varphi_N \Big( \frac{w_i + w_j}{2} \Big) f_p \Big( \frac{w_i - w_j}{\ell_\alpha} \Big).
\end{equation*}
Similarly, $O^* \varphi_N(A) O = \varphi_N(T)$ is upper triangular with diagonal entries $\varphi_N(w_i)$. Taking traces, we obtain
\begin{align*}
\mathcal X_\alpha(A; \varphi_N, f_p) & = \frac{1}{2} \sum_{i, j = 1}^N \varphi_N \Big( \frac{w_i + w_j}{2} \Big) f_p \Big( \frac{w_i - w_j}{\ell_\alpha} \Big) - \frac{1}{2} \sum_{i = 1}^N \varphi_N(w_i) \\
& = \sum_{1 \le i < j \le N} \varphi_N \Big( \frac{w_i + w_j}{2} \Big) f_p \Big( \frac{w_j - w_i}{\ell_\alpha} \Big),
\end{align*}
where we used that $f_p(0) = 1$, so that the terms with $i = j$ cancel the second sum, and that the summand is symmetric in $(i, j)$, since $f_p(-z) = f_p(z)$. For a real symmetric $A$, we may take $w_i = \lambda_i(A)$, and \eqref{e:comparison-eigen} becomes \eqref{e:comparison-pair}.

To prove part (2), we first bound the argument of $f_p$. Let $z \in \C$ with $|\Im z| \le 1/2$. Then
\begin{equation}\label{e:comparison-realpart}
\begin{gathered}
\Re(1 + z^2) = 1 + (\Re z)^2 - (\Im z)^2 \ge \frac{3}{4} \big( 1 + (\Re z)^2 \big) > 0, \\
|\Im(1 + z^2)| = 2 |\Re z| \, |\Im z| \le \big( 1 + (\Re z)^2 \big) |\Im z|.
\end{gathered}
\end{equation}
The real part of $1 + z^2$ is positive, so \eqref{e:comparison-realpart} and the inequality $|\arctan y| \le |y|$ give
\begin{equation}\label{e:comparison-argrational}
| \arg(1 + z^2) | \le \frac{|\Im(1 + z^2)|}{\Re(1 + z^2)} \le \frac{4}{3} |\Im z|.
\end{equation}
For such $z$ with the additional property $(4/3) p |\Im z| < \pi$, it follows that $\arg f_p(z) = - p \arg(1 + z^2)$, and therefore
\begin{equation}\label{e:comparison-argfp}
| \arg f_p(z) | \le \frac{4}{3} p |\Im z|.
\end{equation}

Now let $A$ be as in part (2), fix $i < j$, and set $z = (w_j - w_i)/\ell_\alpha$. The assumption $A \in \mathcal{U}_{\alpha, p}$ gives
\begin{equation*}
|\Im z| < 2 \omega_p \le \frac{1}{1000},
\end{equation*}
and in particular $(4/3) p |\Im z| < \pi$. By \eqref{e:comparison-argfp}, we have
\begin{equation*}
| \arg f_p(z) | \le \frac{8 p \, \omega_p}{3} = \frac{8 p}{3000 (p + 1)} < \frac{1}{375}.
\end{equation*}
For the midpoint $(w_i + w_j)/2$, we have
\begin{equation*}
\Big| \Re \frac{w_i + w_j}{2} \Big| \le \max(|\Re w_i|, |\Re w_j|), \qquad \Big| \Im \frac{w_i + w_j}{2} \Big| \le \max(|\Im w_i|, |\Im w_j|),
\end{equation*}
so that our assumption gives
\begin{equation*}
\Big( \Big| \Re \frac{w_i + w_j}{2} \Big| + 2 \Big) \Big| \Im \frac{w_i + w_j}{2} \Big| \le \max_{w, w' \in \{ w_i, w_j \}} (|\Re w| + 2) |\Im w'| \le \frac{h_N^2}{200}.
\end{equation*}
Applying part (3) of \zcref{l:smoothing}, we find that either $\varphi_N((w_i + w_j)/2) = 0$ or $|\arg \varphi_N((w_i + w_j)/2)| \le 1/200$. In the latter case, we have
\begin{equation*}
\Big| \arg \Big( \varphi_N \Big( \frac{w_i + w_j}{2} \Big) f_p(z) \Big) \Big| \le \Big| \arg \varphi_N \Big( \frac{w_i + w_j}{2} \Big) \Big| + | \arg f_p(z) | \le \frac{1}{200} + \frac{1}{375} < \frac{1}{100},
\end{equation*}
where the first inequality holds because the middle expression is smaller than $\pi$. Every nonzero complex number whose argument has absolute value at most $1/100$ has positive real part, so the real part of the sum in \eqref{e:comparison-eigen} is nonnegative.

For part (3), let $\Spec A \subset \mathfrak{R}_{\alpha, p}$. Then $A \in \mathcal{U}_{\alpha, p}$, and all eigenvalues $w$ and $w'$ of $A$ satisfy
\begin{equation*}
(|\Re w| + 2) |\Im w'| < 6 \omega_p \ell_\alpha \le \frac{6}{2000} N^{-1} \le \frac{N^{-1/5}}{200} = \frac{h_N^2}{200}.
\end{equation*}

It remains to prove part (4). The function $F$ is holomorphic on $\mathcal{U}_{\alpha, p}$ by part (1), since $\mathcal{U}_{\alpha, p} \subset \mathcal{U}_{\alpha, p_j}$ for all $j \in \mathcal{J}$. Let $A \in \mathcal{U}_{\alpha, p}$ satisfy the hypothesis of part (2). Applying part (2) with $p_j$ and $\varphi_j$ in place of $p$ and $\varphi$, we find that $\Re \mathcal X_\alpha(A; \varphi_{j, N}, f_{p_j}) \ge 0$ for every $j \in \mathcal{J}$. Using $\theta_j \ge 0$, we obtain
\begin{equation*}
|F(A)| = \exp \Big( - \sum_{j \in \mathcal{J}} \theta_j \Re \mathcal X_\alpha(A; \varphi_{j, N}, f_{p_j}) \Big) \le 1.
\end{equation*}
The last claim follows from part (3) and, for real symmetric $A$, from the fact that $\mathcal X_\alpha(A; \varphi_{j, N}, f_{p_j}) \ge 0$, which holds by \eqref{e:comparison-pair}, part (1) of \zcref{l:smoothing}, and the positivity of $f_{p_j}$ on $\R$.
\end{proof}

For a function $\mathcal Q$ defined on an open set of $N \times N$ matrices containing the real symmetric matrices, a real symmetric matrix $B$, indices $a \le b$, and an integer $n \ge 1$, we write
\begin{equation}\label{e:comparison-partial}
\partial_{ab}^n \mathcal Q(B) = \frac{\dd^n}{\dd x^n} \mathcal Q(B + x V^{ab}) \Big|_{x = 0}
\end{equation}
whenever this derivative exists, and $\partial_{ab} = \partial_{ab}^1$. In the real coordinates $x_{ab}$ obtained by writing a real symmetric matrix as $\sum_{a \le b} x_{ab} V^{ab}$, the operator $\partial_{ab}$ is the partial derivative with respect to $x_{ab}$. The next lemma gives a crude bound on the derivatives of $F$ that holds at every real symmetric matrix, without any regularity assumption. We will use it to verify the hypotheses of the derivative formula in \zcref{l:comparison-generator} below and to control contributions of events of very small probability.

\begin{lemma}\label{l:globalderivative}
Let $F$ be as in \zcref{d:complexS}, and let $p = \max_{j \in \mathcal{J}} p_j$. For every real symmetric matrix $B$ and all $a \le b$, the function $x \mapsto F(B + x V^{ab})$ is smooth on $\R$. Moreover, for every integer $n \ge 1$, we have
\begin{equation}\label{e:comparison-global}
| \partial_{ab}^n F(B) | \le n! \Big( \frac{2}{\omega_p} \Big)^n \Big( \frac{1 + \| B \|}{\ell_\alpha h_N^2} \Big)^n.
\end{equation}
\end{lemma}

\begin{proof}
Fix $B$ and $a \le b$, and set
\begin{equation*}
r_B = \frac{\omega_p \ell_\alpha h_N^2}{2 (1 + \| B \|)}.
\end{equation*}

We first locate the eigenvalues of $B + \zeta V^{ab}$ for $\zeta \in \C$. Let $w \in \C$ satisfy $\dist(w, \Spec B) > |\zeta|$. Since $B$ is real symmetric and $\| V^{ab} \| = 1$, we have
\begin{equation*}
\| \zeta (B - w)^{-1} V^{ab} \| \le |\zeta| \dist(w, \Spec B)^{-1} < 1,
\end{equation*}
so the matrix
\begin{equation*}
B + \zeta V^{ab} - w = (B - w) \big( \Id + \zeta (B - w)^{-1} V^{ab} \big)
\end{equation*}
is invertible. Hence every eigenvalue $w$ of $B + \zeta V^{ab}$ satisfies $\dist(w, \Spec B) \le |\zeta|$, and in particular
\begin{equation}\label{e:comparison-eigenlocation}
|\Im w| \le |\zeta|, \qquad |\Re w| \le \| B \| + |\zeta|.
\end{equation}

The bound \eqref{e:comparison-eigenlocation} shows that $B + \zeta V^{ab} \in \mathcal{U}_{\alpha, p}$ whenever $|\zeta| < \omega_p \ell_\alpha$. Hence, by part (4) of \zcref{l:holomorphic}, the function $\zeta \mapsto F(B + \zeta V^{ab})$ is holomorphic on the disk $\{ |\zeta| < \omega_p \ell_\alpha \}$. Applying this with $B + x_0 V^{ab}$ in place of $B$ for each $x_0 \in \R$, we see that $x \mapsto F(B + x V^{ab})$ is real analytic, and in particular smooth, on $\R$.

Now let $|\zeta| \le r_B$, and let $w$ and $w'$ be eigenvalues of $B + \zeta V^{ab}$. Using \eqref{e:comparison-eigenlocation}, $r_B \le 1$, and $r_B < \omega_p \ell_\alpha$, we find that $B + \zeta V^{ab} \in \mathcal{U}_{\alpha, p}$ and that
\begin{equation*}
(|\Re w| + 2) |\Im w'| \le (\| B \| + 3) r_B \le 3 (1 + \| B \|) r_B = \frac{3}{2} \omega_p \ell_\alpha h_N^2 \le \frac{h_N^2}{200}.
\end{equation*}
Hence $B + \zeta V^{ab}$ satisfies the hypothesis of part (2) of \zcref{l:holomorphic}, and part (4) of that lemma gives $|F(B + \zeta V^{ab})| \le 1$ for $|\zeta| \le r_B$. Cauchy's estimate on the circle of radius $r_B$ about the origin gives
\begin{equation*}
|\partial_{ab}^n F(B)| \le n! \, r_B^{-n},
\end{equation*}
which is \eqref{e:comparison-global}.
\end{proof}

\section{Comparison along the Ornstein--Uhlenbeck interpolation}\label{s:comparison-proof}

For $a \le b$ and $t \in [0, 1]$, we let $x_{ab}(t) = (H_t)_{ab}$ denote the $(a, b)$ coordinate of $H_t$, so that
\begin{equation}\label{e:comparison-coordinates}
H_t = \sum_{a \le b} x_{ab}(t) V^{ab}, \qquad x_{ab}(t) = e^{-t/2} H_{ab} + (1 - e^{-t})^{1/2} G_{ab},
\end{equation}
and we let $H_t^{[ab]} = H_t - x_{ab}(t) V^{ab}$ denote the matrix obtained from $H_t$ by setting its $(a, b)$ and $(b, a)$ entries to zero. We also set $v^G_{ab} = \E G_{ab}^2$, so that $v^G_{ab} = 1/N$ for $a < b$ and $v^G_{aa} = 2/N$. The variables $(H_{ab})_{a \le b}$ and $(G_{ab})_{a \le b}$ are independent, so for each $t$, the coordinates $(x_{ab}(t))_{a \le b}$ are independent. In particular, $x_{ab}(t)$ is independent of $H_t^{[ab]}$.

We record some properties of the coordinates. Recall from \zcref{d:wigner} that $H_{ab} = N^{-1/2} X_{ab}$ for $a < b$ and $H_{aa} = N^{-1/2} Z_a$, where $\Prob(|X| > y) + \Prob(|Z| > y) \le C_0 e^{-c_0 y^2}$ for $y \ge 0$. Since $G_{ab}$ is a centered Gaussian variable with variance at most $2/N$, we have, for all $t \in [0, 1]$, $a \le b$, and $y \ge 0$,
\begin{equation}\label{e:comparison-tail}
\Prob \big( |x_{ab}(t)| > y \big) \le \Prob \big( |H_{ab}| > y/2 \big) + \Prob \big( |G_{ab}| > y/2 \big) \le C_0 e^{-c_0 N y^2 / 4} + 2 e^{-N y^2 / 16}.
\end{equation}
In particular, for every integer $n \ge 1$, there exists $C_n > 0$ such that
\begin{equation}\label{e:comparison-moments}
\E |x_{ab}(t)|^n \le C_n N^{-n/2}
\end{equation}
for all $t \in [0, 1]$ and $a \le b$. Using the independence of $H_{ab}$ and $G_{ab}$, the identities $\E H_{ab} = \E G_{ab} = \E G_{ab}^3 = 0$, $\E H_{ab}^2 = 1/N$ for $a < b$, and $\E H_{aa}^2 = \sigma_Z^2 / N$, we also find
\begin{equation}\label{e:comparison-variance}
\E x_{ab}(t)^2 = v^G_{ab}, \qquad \E x_{ab}(t)^3 = \frac{e^{-3t/2} \E X^3}{N^{3/2}} \quad (a < b), \qquad \E x_{aa}(t)^2 = v^G_{aa} + \frac{e^{-t} (\sigma_Z^2 - 2)}{N}.
\end{equation}

The following lemma expresses $\E \mathcal Q(H_t) - \E \mathcal Q(H)$ in terms of the derivatives of $\mathcal Q$ in single coordinates.

\begin{lemma}\label{l:comparison-generator}
Let $\mathcal Q$ be a bounded, twice continuously differentiable, complex-valued function on the space of real symmetric $N \times N$ matrices. Suppose that there exists $K_0 > 0$ such that
\begin{equation}\label{e:comparison-polynomial}
|\partial_{ab} \mathcal Q(A)| + |\partial_{ab}^2 \mathcal Q(A)| \le K_0 (1 + \| A \|)^2
\end{equation}
for every real symmetric matrix $A$ and all $a \le b$. Then for all $t \in [0, 1]$,
\begin{equation}\label{e:comparison-integrated}
\E \mathcal Q(H_t) - \E \mathcal Q(H) = \frac{1}{2} \int_0^t \sum_{a \le b} \E \Big[ v^G_{ab} \, \partial_{ab}^2 \mathcal Q(H_{t'}) - x_{ab}(t') \, \partial_{ab} \mathcal Q(H_{t'}) \Big] \dd t'.
\end{equation}
\end{lemma}

\begin{proof}
Fix realizations of $H$ and $G$. The map $t' \mapsto H_{t'}$ is continuous on $[0, 1]$ and continuously differentiable on $(0, 1]$. Differentiating \eqref{e:comparison-coordinates} and using the identity
\begin{equation*}
e^{-t'} (1 - e^{-t'})^{-1/2} = (1 - e^{-t'})^{-1/2} - (1 - e^{-t'})^{1/2},
\end{equation*}
we find that
\begin{equation*}
\frac{\dd}{\dd t'} x_{ab}(t') = - \frac{1}{2} x_{ab}(t') + \frac{1}{2} (1 - e^{-t'})^{-1/2} G_{ab}, \qquad t' \in (0, 1].
\end{equation*}
The chain rule gives, for $t' \in (0, 1]$,
\begin{equation}\label{e:comparison-chain}
\frac{\dd}{\dd t'} \mathcal Q(H_{t'}) = \sum_{a \le b} \partial_{ab} \mathcal Q(H_{t'}) \Big( - \frac{1}{2} x_{ab}(t') + \frac{1}{2} (1 - e^{-t'})^{-1/2} G_{ab} \Big).
\end{equation}

We now justify integrating \eqref{e:comparison-chain}. For $t' \in (0, 1]$, we have
\begin{equation*}
\| H_{t'} \| \le \| H \| + \| G \|, \qquad |x_{ab}(t')| \le |H_{ab}| + |G_{ab}|, \qquad 1 - e^{-t'} \ge t'/e.
\end{equation*}
Together with \eqref{e:comparison-polynomial}, these bounds give
\begin{equation}\label{e:comparison-domination}
\Big| \frac{\dd}{\dd t'} \mathcal Q(H_{t'}) \Big| \le K_0 \big( 1 + \| H \| + \| G \| \big)^2 \big( 1 + (e/t')^{1/2} \big) \sum_{a \le b} \big( |H_{ab}| + |G_{ab}| \big).
\end{equation}
The entries of $H$ and $G$ have moments of all orders, and $\| H \|^2 \le \sum_{a, b} H_{ab}^2$. Therefore the right side of \eqref{e:comparison-domination} has finite expectation for every $t' \in (0, 1]$, and this expectation is integrable in $t'$ over $(0, 1]$ (the factor $(t')^{-1/2}$ is integrable at $t' = 0$). Applying the fundamental theorem of calculus on $[t_0, t]$ with $0 < t_0 < t$ and letting $t_0 \to 0$, using the continuity of $t' \mapsto \mathcal Q(H_{t'})$ at $t' = 0$, we obtain
\begin{equation*}
\mathcal Q(H_t) - \mathcal Q(H) = \int_0^t \frac{\dd}{\dd t'} \mathcal Q(H_{t'}) \dd t', \qquad t \in [0, 1].
\end{equation*}
Taking expectations and using Fubini's theorem, we find
\begin{equation}\label{e:comparison-ftc}
\E \mathcal Q(H_t) - \E \mathcal Q(H) = \int_0^t \E \Big[ \frac{\dd}{\dd t'} \mathcal Q(H_{t'}) \Big] \dd t'.
\end{equation}

Next, we compute the expectation of \eqref{e:comparison-chain} with Gaussian integration by parts. Fix $t' \in (0, 1]$ and $a \le b$, and condition on $H$ and on $(G_{a'b'})_{(a', b') \neq (a, b)}$. Then $H_{t'}^{[ab]}$ and $H_{ab}$ are fixed, and
\begin{equation*}
H_{t'} = H_{t'}^{[ab]} + \big( e^{-t'/2} H_{ab} + (1 - e^{-t'})^{1/2} G_{ab} \big) V^{ab}.
\end{equation*}
Hence $\partial_{ab} \mathcal Q(H_{t'})$ is a continuously differentiable function of $G_{ab}$, with
\begin{equation*}
\frac{\partial}{\partial G_{ab}} \partial_{ab} \mathcal Q(H_{t'}) = (1 - e^{-t'})^{1/2} \partial_{ab}^2 \mathcal Q(H_{t'}).
\end{equation*}
Moreover, \eqref{e:comparison-polynomial} implies that this function and its derivative grow at most polynomially in $G_{ab}$.

The variable $G_{ab}$ is centered Gaussian with variance $v^G_{ab}$ and independent of the conditioning variables. We use the standard Gaussian integration by parts formula
\begin{equation*}
\E [G_{ab} \psi(G_{ab})] = v^G_{ab} \E [\psi'(G_{ab})],
\end{equation*}
valid for every continuously differentiable $\psi \colon \R \to \C$ such that $\psi$ and $\psi'$ grow at most polynomially. Applying it conditionally, and integrating over the conditioning variables, we obtain
\begin{equation}\label{e:comparison-ibp}
\E \big[ G_{ab} \, \partial_{ab} \mathcal Q(H_{t'}) \big] = (1 - e^{-t'})^{1/2} v^G_{ab} \E \big[ \partial_{ab}^2 \mathcal Q(H_{t'}) \big].
\end{equation}

Taking expectations in \eqref{e:comparison-chain} and inserting \eqref{e:comparison-ibp}, we see that the integrands on the right sides of \eqref{e:comparison-ftc} and \eqref{e:comparison-integrated} coincide, which proves \eqref{e:comparison-integrated}.
\end{proof}

We now explain how the integrand in \eqref{e:comparison-integrated} will be estimated for $\mathcal Q = F$. Each summand has the form
\begin{equation*}
\E \big[ v^G_{ab} \Upsilon''(x_{ab}(t')) - x_{ab}(t') \Upsilon'(x_{ab}(t')) \big]
\end{equation*}
for a random function $\Upsilon$ independent of $x_{ab}(t')$, namely the restriction of $F$ to the line through $H_{t'}^{[ab]}$ in the direction $V^{ab}$. When $v^G_{ab} = \E x_{ab}(t')^2$, a Taylor expansion of $\Upsilon'$ and $\Upsilon''$ around $0$ shows that the terms of order zero and one cancel, and what remains is of order $\E |x_{ab}(t')|^3$ times the third derivative of $\Upsilon$. The next lemma, a truncated form of the cumulant expansion of \cite{khorunzhy1996asymptotic} (see also \cite[Proposition~3.1]{lytova2009central}), makes this precise. We state it with a cutoff, since good bounds on the derivatives of $\Upsilon$ are available only on a bounded interval.

\begin{lemma}\label{l:taylor}
Let $\varkappa$ be a real random variable such that $\E \varkappa = 0$, $\E \varkappa^2 \le 1$, and $\E \varkappa^4 < \infty$. Let $r \in (0, 1]$ and $K_0 \ge 0$, and let $\Upsilon \colon \R \to \C$ be four times continuously differentiable with
\begin{equation}\label{e:comparison-taylor-local}
|\Upsilon^{(n)}(x)| \le n! \, r^{-n} \qquad \text{for } |x| \le r \text{ and } n \in \{ 1, 2, 3, 4 \},
\end{equation}
\begin{equation}\label{e:comparison-taylor-global}
|\Upsilon'(x)| + |\Upsilon''(x)| \le K_0 (1 + |x|)^2 \qquad \text{for } x \in \R.
\end{equation}
Set $\mathfrak t = (9 r^{-3} + 2 K_0) \E [ (1 + |\varkappa|)^4 \1_{|\varkappa| > r} ]$. Then
\begin{equation}\label{e:comparison-taylor-three}
\big| \E [\varkappa \Upsilon'(\varkappa)] - \E [\varkappa^2] \, \E [\Upsilon''(\varkappa)] \big| \le 9 r^{-3} \E |\varkappa|^3 + \mathfrak t.
\end{equation}
If moreover $\E \varkappa^3 = 0$, then
\begin{equation}\label{e:comparison-taylor-four}
\big| \E [\varkappa \Upsilon'(\varkappa)] - \E [\varkappa^2] \, \E [\Upsilon''(\varkappa)] \big| \le 16 r^{-4} \E \varkappa^4 + \mathfrak t.
\end{equation}
Finally, we have
\begin{equation}\label{e:comparison-taylor-second}
| \E \Upsilon''(\varkappa) | \le 2 r^{-2} + \mathfrak t.
\end{equation}
\end{lemma}

\begin{proof}
For notational convenience, we abbreviate $\1_- = \1_{|\varkappa| \le r}$, $\1_+ = \1_{|\varkappa| > r}$, and $v = \E \varkappa^2$. We first prove \eqref{e:comparison-taylor-three}. Taylor's theorem with integral remainder and \eqref{e:comparison-taylor-local} with $n = 3$ give, on the event $\{ |\varkappa| \le r \}$,
\begin{equation*}
\begin{gathered}
\Upsilon'(\varkappa) = \Upsilon'(0) + \varkappa \Upsilon''(0) + \rho_1, \qquad \Upsilon''(\varkappa) = \Upsilon''(0) + \rho_2, \\
|\rho_1| \le 3 r^{-3} \varkappa^2, \qquad |\rho_2| \le 6 r^{-3} |\varkappa|.
\end{gathered}
\end{equation*}
Therefore
\begin{equation}\label{e:comparison-taylor-expansion}
\E \big[ (\varkappa \Upsilon'(\varkappa) - v \Upsilon''(\varkappa)) \1_- \big] = \Upsilon'(0) \E [\varkappa \1_-] + \Upsilon''(0) \E [(\varkappa^2 - v) \1_-] + \E [(\varkappa \rho_1 - v \rho_2) \1_-].
\end{equation}
The identities $\E \varkappa = 0$ and $\E [\varkappa^2 - v] = 0$ give
\begin{equation*}
\E [\varkappa \1_-] = - \E [\varkappa \1_+], \qquad \E [(\varkappa^2 - v) \1_-] = - \E [(\varkappa^2 - v) \1_+].
\end{equation*}
Using $|\Upsilon'(0)| \le r^{-1}$, $|\Upsilon''(0)| \le 2 r^{-2}$, $v \le 1$, and $r \le 1$, we conclude that the first two terms on the right side of \eqref{e:comparison-taylor-expansion} are bounded in absolute value by
\begin{equation*}
r^{-1} \E [|\varkappa| \1_+] + 2 r^{-2} \E [(\varkappa^2 + 1) \1_+] \le 3 r^{-3} \E [(1 + |\varkappa|)^4 \1_+].
\end{equation*}
Using Hölder's inequality in the form $v \E |\varkappa| \le \E |\varkappa|^3$, we find that the last term in \eqref{e:comparison-taylor-expansion} satisfies
\begin{equation*}
\big| \E [(\varkappa \rho_1 - v \rho_2) \1_-] \big| \le 3 r^{-3} \E |\varkappa|^3 + 6 r^{-3} v \E |\varkappa| \le 9 r^{-3} \E |\varkappa|^3.
\end{equation*}

To bound the contribution of the event $\{ |\varkappa| > r \}$, we use \eqref{e:comparison-taylor-global}, which gives
\begin{equation*}
\big| \E \big[ (\varkappa \Upsilon'(\varkappa) - v \Upsilon''(\varkappa)) \1_+ \big] \big| \le K_0 \E \big[ (|\varkappa| + 1)(1 + |\varkappa|)^2 \1_+ \big] \le K_0 \E [(1 + |\varkappa|)^4 \1_+].
\end{equation*}
Adding the contributions of the two events, we obtain \eqref{e:comparison-taylor-three}.

When $\E \varkappa^3 = 0$, we expand one order further. On the event $\{ |\varkappa| \le r \}$, \eqref{e:comparison-taylor-local} with $n = 4$ gives
\begin{equation*}
\begin{gathered}
\Upsilon'(\varkappa) = \Upsilon'(0) + \varkappa \Upsilon''(0) + \frac{\varkappa^2}{2} \Upsilon'''(0) + \rho_3, \qquad \Upsilon''(\varkappa) = \Upsilon''(0) + \varkappa \Upsilon'''(0) + \rho_4, \\
|\rho_3| \le 4 r^{-4} |\varkappa|^3, \qquad |\rho_4| \le 12 r^{-4} \varkappa^2.
\end{gathered}
\end{equation*}
Compared with \eqref{e:comparison-taylor-expansion}, there is an additional term involving $\Upsilon'''(0)$. We have $\E [\varkappa^3/2 - v \varkappa] = 0$ and $|\Upsilon'''(0)| \le 6 r^{-3}$, so this term satisfies
\begin{equation*}
\big| \Upsilon'''(0) \E [(\varkappa^3/2 - v \varkappa) \1_-] \big| = \big| \Upsilon'''(0) \E [(\varkappa^3/2 - v \varkappa) \1_+] \big| \le 6 r^{-3} \E [(1 + |\varkappa|)^4 \1_+].
\end{equation*}
The remainder term is now bounded by
\begin{equation*}
\big| \E [(\varkappa \rho_3 - v \rho_4) \1_-] \big| \le 4 r^{-4} \E \varkappa^4 + 12 r^{-4} v^2 \le 16 r^{-4} \E \varkappa^4,
\end{equation*}
where we used $v^2 \le \E \varkappa^4$. The other terms are bounded as before, and the tail contributions add up to at most
\begin{equation*}
(3 r^{-3} + 6 r^{-3} + K_0) \E [(1 + |\varkappa|)^4 \1_+] \le \mathfrak t.
\end{equation*}
This gives \eqref{e:comparison-taylor-four}.

Finally, \eqref{e:comparison-taylor-local} with $n = 2$ and \eqref{e:comparison-taylor-global} give
\begin{equation*}
| \E \Upsilon''(\varkappa) | \le 2 r^{-2} + K_0 \E [(1 + |\varkappa|)^2 \1_+] \le 2 r^{-2} + \mathfrak t,
\end{equation*}
which is \eqref{e:comparison-taylor-second}.
\end{proof}

\begin{proposition}\label{p:comparison}
Let $\eps > 0$ satisfy $\alpha + 5 \eps < 1/2$. Let $F$ be as in \zcref{d:complexS}, with $p = \max_{j \in \mathcal{J}} p_j$, and set $r_N = c_p N^{-\alpha - 4 \eps}$. There exists $C > 0$, and for every $D > 0$ there exists $C_D > 0$, such that for all $N \ge 2$ and $t \in [0, 1]$,
\begin{equation}\label{e:comparison-bound}
| \E F(H_t) - \E F(H) | \le C t \big( N^{1/2} r_N^{-3} + r_N^{-2} \big) + C_D N^{-D}.
\end{equation}
If $\E X^3 = 0$, then \eqref{e:comparison-bound} holds with $N^{1/2} r_N^{-3}$ replaced by $r_N^{-4}$. The constant $C$ depends only on $\alpha$, $\eps$, $p$, and the laws of $X$ and $Z$, and $C_D$ depends in addition on $D$. In particular, they do not depend on $\mathcal{J}$, on the weights $\varphi_j$, or on the parameters $\theta_j$.
\end{proposition}

The proof has four steps. We first express $\E F(H_t) - \E F(H)$ through the derivative formula of \zcref{l:comparison-generator}. We then show that for every time $t' \in (0, 1]$, outside an event of very small probability, the function $\zeta \mapsto F(H_{t'}^{[ab]} + \zeta V^{ab})$ is bounded by one on a complex disk of radius $4 r_N$, so that Cauchy's estimates bound its derivatives on $[-r_N, r_N]$. This is the main step, which rests on \zcref{l:strip}. Next, we Taylor expand in the coordinate $x_{ab}(t')$ using \zcref{l:taylor}. Finally, we sum over the coordinates and integrate in time.

\begin{proof}
We abbreviate $\mathfrak{R} = \mathfrak{R}_{\alpha, p}$. Let $N_0 \ge 1$ be so large that the conclusion of \zcref{l:strip} holds for $N \ge N_0$, and such that $N^{-1/2} \le r_N$ and $\max(1, \sigma_Z^2, 2) \le N$ for $N \ge N_0$; such an $N_0$ exists since $\alpha + 4 \eps < 1/2$. For $N < N_0$, the left side of \eqref{e:comparison-bound} is bounded by $2 \le 2 N_0^D N^{-D}$, so we may assume that $N \ge N_0$. Our assumption $\alpha + 5 \eps < 1/2$ gives
\begin{equation}\label{e:comparison-radius}
N r_N^2 = c_p^2 N^{1 - 2 \alpha - 8 \eps} \ge c_p^2 N^{2 \eps}, \qquad \ell_\alpha h_N^2 = N^{-6/5 - \alpha} \ge N^{-2}.
\end{equation}

\emph{Step 1. The derivative formula.} By part (4) of \zcref{l:holomorphic}, the function $F$ is bounded by one on real symmetric matrices. Further, $F$ is holomorphic on the open set $\mathcal{U}_{\alpha, p}$, so the function $(x_{ab})_{a \le b} \mapsto F(\sum_{a \le b} x_{ab} V^{ab})$ extends holomorphically to an open subset of $\C^{N(N + 1)/2}$ containing $\R^{N(N + 1)/2}$. Hence $F$ is real analytic, and in particular twice continuously differentiable, on the space of real symmetric matrices. Combining \zcref{l:globalderivative} with \eqref{e:comparison-radius}, we find that for every real symmetric $A$ and all $a \le b$,
\begin{equation}\label{e:comparison-polybound}
|\partial_{ab} F(A)| \le \frac{2}{\omega_p} N^2 (1 + \| A \|), \qquad |\partial_{ab}^2 F(A)| \le \frac{8}{\omega_p^2} N^4 (1 + \| A \|)^2.
\end{equation}
Hence \zcref{l:comparison-generator} applies with $\mathcal Q = F$. For $t' \in [0, 1]$ and $a \le b$, we set
\begin{equation*}
\mathcal{Q}_{ab}(t') = \E \Big[ v^G_{ab} \, \partial_{ab}^2 F(H_{t'}) - x_{ab}(t') \, \partial_{ab} F(H_{t'}) \Big],
\end{equation*}
and \eqref{e:comparison-integrated} reads
\begin{equation}\label{e:comparison-step1}
\E F(H_t) - \E F(H) = \frac{1}{2} \int_0^t \sum_{a \le b} \mathcal{Q}_{ab}(t') \dd t'.
\end{equation}

\emph{Step 2. A holomorphic bound on a disk.} Fix $t' \in (0, 1]$ and $a \le b$. We define $\mathcal{E}_{ab}(t')$ to be the event that
\begin{equation}\label{e:comparison-event}
\Spec \big( H_{t'}^{[ab]} + \zeta V^{ab} \big) \subset \mathfrak{R} \quad \text{for all } \zeta \in \C \text{ with } |\zeta| \le 4 r_N.
\end{equation}
The set of real symmetric matrices $B$ such that $\Spec(B + \zeta V^{ab}) \subset \mathfrak{R}$ for all $|\zeta| \le 4 r_N$ is open. Indeed, if $B_n \to B$ and $B_n + \zeta_n V^{ab}$ has an eigenvalue outside $\mathfrak{R}$ for some $|\zeta_n| \le 4 r_N$, then for a limit point $\zeta$ of $(\zeta_n)$, the matrix $B + \zeta V^{ab}$ has an eigenvalue outside $\mathfrak{R}$ by \zcref{l:comparison-open} with $\mathcal{O} = \mathfrak{R}$. Hence $\mathcal{E}_{ab}(t')$ is an event determined by $H_{t'}^{[ab]}$.

Next, we bound the probability of the complement of $\mathcal{E}_{ab}(t')$. Suppose that $H_{t'} \in \mathcal{G}_\eps$ and $|x_{ab}(t')| \le r_N$. For $|\zeta| \le 4 r_N$, we have
\begin{equation*}
H_{t'}^{[ab]} + \zeta V^{ab} = H_{t'} + (\zeta - x_{ab}(t')) V^{ab}, \qquad |\zeta - x_{ab}(t')| \le 5 r_N \le 8 c_p N^{-\alpha - 4 \eps},
\end{equation*}
so \zcref{l:strip} shows that $\Spec(H_{t'}^{[ab]} + \zeta V^{ab}) \subset \mathfrak{R}$. Therefore
\begin{equation*}
\{ H_{t'} \in \mathcal{G}_\eps \} \cap \{ |x_{ab}(t')| \le r_N \} \subset \mathcal{E}_{ab}(t').
\end{equation*}
Using \zcref{p:regularity}, \eqref{e:comparison-tail}, and \eqref{e:comparison-radius}, we obtain, for every $D > 0$,
\begin{equation}\label{e:comparison-eventprob}
\Prob \big( \mathcal{E}_{ab}(t')^c \big) \le \Prob(H_{t'} \notin \mathcal{G}_\eps) + \Prob \big( |x_{ab}(t')| > r_N \big) \le C_D N^{-D},
\end{equation}
uniformly in $t' \in (0, 1]$ and $a \le b$.

We now turn to the derivatives in the coordinate $x_{ab}(t')$. We define the random function
\begin{equation}\label{e:comparison-Upsilon}
\Upsilon(x) = F \big( H_{t'}^{[ab]} + x V^{ab} \big), \qquad x \in \R,
\end{equation}
which is determined by $H_{t'}^{[ab]}$ and smooth by \zcref{l:globalderivative}. Observe that \eqref{e:comparison-partial} gives
\begin{equation*}
\partial_{ab} F(H_{t'}) = \Upsilon'(x_{ab}(t')), \qquad \partial_{ab}^2 F(H_{t'}) = \Upsilon''(x_{ab}(t')).
\end{equation*}
On the event $\mathcal{E}_{ab}(t')$, the set of $\zeta \in \C$ such that $H_{t'}^{[ab]} + \zeta V^{ab} \in \mathcal{U}_{\alpha, p}$ is open and contains the closed disk $\{ |\zeta| \le 4 r_N \}$. Part (4) of \zcref{l:holomorphic} shows that $\zeta \mapsto F(H_{t'}^{[ab]} + \zeta V^{ab})$ is holomorphic on this open set and bounded by one in modulus on the closed disk. For real $x$ with $|x| \le r_N$, the closed disk of radius $3 r_N$ about $x$ is contained in $\{ |\zeta| \le 4 r_N \}$, and Cauchy's estimates give
\begin{equation}\label{e:comparison-cauchy}
|\Upsilon^{(n)}(x)| \le n! \, (3 r_N)^{-n} \le n! \, r_N^{-n} \qquad \text{for } |x| \le r_N \text{ and } n \ge 1, \text{ on the event } \mathcal{E}_{ab}(t').
\end{equation}
Without restriction to $\mathcal{E}_{ab}(t')$, the bound \eqref{e:comparison-polybound} gives
\begin{equation}\label{e:comparison-globalUpsilon}
|\Upsilon'(x)| + |\Upsilon''(x)| \le \frac{10}{\omega_p^2} N^4 \big( 1 + \| H_{t'}^{[ab]} \| \big)^2 (1 + |x|)^2, \qquad x \in \R,
\end{equation}
where we used $\omega_p \le 1$ and
\begin{equation*}
1 + \| H_{t'}^{[ab]} + x V^{ab} \| \le \big( 1 + \| H_{t'}^{[ab]} \| \big) (1 + |x|).
\end{equation*}

\emph{Step 3. Taylor expansion in one coordinate.} We keep $t'$ and $a \le b$ fixed and abbreviate $\varkappa = x_{ab}(t')$. The event $\mathcal{E}_{ab}(t')$ is determined by $H_{t'}^{[ab]}$, so we may split
\begin{equation}\label{e:comparison-split-event}
\begin{aligned}
\mathcal{Q}_{ab}(t') = {} & \E \Big[ \1_{\mathcal{E}_{ab}(t')} \E \big[ v^G_{ab} \Upsilon''(\varkappa) - \varkappa \Upsilon'(\varkappa) \, \big| \, H_{t'}^{[ab]} \big] \Big] \\
& + \E \Big[ \1_{\mathcal{E}_{ab}(t')^c} \big( v^G_{ab} \Upsilon''(\varkappa) - \varkappa \Upsilon'(\varkappa) \big) \Big].
\end{aligned}
\end{equation}
We begin by bounding the second term. Using \eqref{e:comparison-polybound}, we have
\begin{equation*}
|v^G_{ab} \Upsilon''(\varkappa) - \varkappa \Upsilon'(\varkappa)| \le C N^4 (1 + \| H_{t'} \|)^2 (N^{-1} + |\varkappa|).
\end{equation*}
Combining this bound with the Cauchy--Schwarz inequality, \zcref{p:regularity}, \eqref{e:comparison-moments}, and \eqref{e:comparison-eventprob}, we find that the second term in \eqref{e:comparison-split-event} is bounded in absolute value by
\begin{equation*}
C N^4 \Big( \E \big[ (1 + \| H_{t'} \|)^4 (N^{-1} + |\varkappa|)^2 \big] \Big)^{1/2} \Prob \big( \mathcal{E}_{ab}(t')^c \big)^{1/2} \le C_D N^{-D}.
\end{equation*}

To bound the first term in \eqref{e:comparison-split-event}, we apply \zcref{l:taylor} conditionally on $H_{t'}^{[ab]}$, with $\Upsilon$ held fixed; this is possible since $\varkappa$ is independent of $H_{t'}^{[ab]}$. We have $\E \varkappa = 0$, and \eqref{e:comparison-variance} and our choice of $N_0$ give
\begin{equation*}
\E \varkappa^2 \le \max(1, \sigma_Z^2, 2) N^{-1} \le 1.
\end{equation*}
On the event $\mathcal{E}_{ab}(t')$, the function $\Upsilon$ satisfies \eqref{e:comparison-taylor-local} with $r = r_N \le 1$ by \eqref{e:comparison-cauchy}. It also satisfies \eqref{e:comparison-taylor-global} with
\begin{equation*}
K_0 = \frac{10}{\omega_p^2} N^4 \big( 1 + \| H_{t'}^{[ab]} \| \big)^2
\end{equation*}
by \eqref{e:comparison-globalUpsilon}. By the Cauchy--Schwarz inequality, \eqref{e:comparison-tail}, \eqref{e:comparison-moments}, and \eqref{e:comparison-radius}, we have
\begin{equation*}
\E \big[ (1 + |\varkappa|)^4 \1_{|\varkappa| > r_N} \big] \le \big( \E (1 + |\varkappa|)^8 \big)^{1/2} \Prob \big( |\varkappa| > r_N \big)^{1/2} \le C \exp \big( - c N^{2 \eps} \big)
\end{equation*}
for some $c > 0$ depending on $c_0$ and $p$. Since $r_N^{-3} \le N^{3/2}$, it follows that the tail term of \zcref{l:taylor} satisfies
\begin{equation}\label{e:comparison-tailterm}
\begin{aligned}
\mathfrak t & = \Big( 9 r_N^{-3} + \frac{20}{\omega_p^2} N^4 \big( 1 + \| H_{t'}^{[ab]} \| \big)^2 \Big) \E \big[ (1 + |\varkappa|)^4 \1_{|\varkappa| > r_N} \big] \\
& \le C_D N^{-D} \big( 1 + \| H_{t'}^{[ab]} \| \big)^2.
\end{aligned}
\end{equation}

We now decompose the integrand of the first term in \eqref{e:comparison-split-event} as
\begin{equation*}
v^G_{ab} \Upsilon''(\varkappa) - \varkappa \Upsilon'(\varkappa) = \big( v^G_{ab} - \E \varkappa^2 \big) \Upsilon''(\varkappa) + \big( \E [\varkappa^2] \Upsilon''(\varkappa) - \varkappa \Upsilon'(\varkappa) \big).
\end{equation*}
We bound the conditional expectations of the two terms on the right using \eqref{e:comparison-taylor-second} and \eqref{e:comparison-taylor-three}, respectively. Then we take expectations, using \eqref{e:comparison-moments} and the bound
\begin{equation*}
\E \big( 1 + \| H_{t'}^{[ab]} \| \big)^2 \le C,
\end{equation*}
which follows from \zcref{p:regularity} and $\| H_{t'}^{[ab]} \| \le \| H_{t'} \| + |\varkappa|$. We find that the first term in \eqref{e:comparison-split-event} is bounded in absolute value by
\begin{equation*}
2 \big| v^G_{ab} - \E \varkappa^2 \big| r_N^{-2} + 9 r_N^{-3} \E |\varkappa|^3 + C_D N^{-D} \le 2 \big| v^G_{ab} - \E \varkappa^2 \big| r_N^{-2} + C N^{-3/2} r_N^{-3} + C_D N^{-D}.
\end{equation*}
When $\E X^3 = 0$ and $a < b$, we have $\E \varkappa^3 = 0$ by \eqref{e:comparison-variance}, and \eqref{e:comparison-taylor-four} gives the same bound with $C N^{-3/2} r_N^{-3}$ replaced by
\begin{equation*}
16 r_N^{-4} \E \varkappa^4 \le C N^{-2} r_N^{-4}.
\end{equation*}

Recall from \eqref{e:comparison-variance} that
\begin{equation*}
v^G_{ab} = \E x_{ab}(t')^2 \quad (a < b), \qquad |v^G_{aa} - \E x_{aa}(t')^2| \le |\sigma_Z^2 - 2| N^{-1}.
\end{equation*}
Combining these facts with the bounds on the two terms in \eqref{e:comparison-split-event}, we obtain
\begin{equation}\label{e:comparison-coordinate-bound}
\begin{aligned}
|\mathcal{Q}_{ab}(t')| & \le C N^{-3/2} r_N^{-3} + C_D N^{-D}, \qquad a < b, \\
|\mathcal{Q}_{aa}(t')| & \le C N^{-1} r_N^{-2} + C N^{-3/2} r_N^{-3} + C_D N^{-D},
\end{aligned}
\end{equation}
uniformly in $t' \in (0, 1]$. When $\E X^3 = 0$, the bound for $a < b$ holds with $C N^{-3/2} r_N^{-3}$ replaced by $C N^{-2} r_N^{-4}$.

\emph{Step 4. Summation of the generator.} There are at most $N^2$ pairs $a < b$ and $N$ diagonal coordinates. Our choice of $N_0$ gives $N^{-1/2} r_N^{-3} \le r_N^{-2}$, so \eqref{e:comparison-coordinate-bound} implies that for every $t' \in (0, 1]$,
\begin{align*}
\sum_{a \le b} |\mathcal{Q}_{ab}(t')| & \le C N^{1/2} r_N^{-3} + C r_N^{-2} + C N^{-1/2} r_N^{-3} + C_D N^{2 - D} \\
& \le C \big( N^{1/2} r_N^{-3} + r_N^{-2} \big) + C_D N^{2 - D}.
\end{align*}

Inserting this bound into \eqref{e:comparison-step1} and replacing $D$ by $D + 2$, we obtain \eqref{e:comparison-bound}.

When $\E X^3 = 0$, the off-diagonal coordinates contribute at most $C r_N^{-4}$ instead of $C N^{1/2} r_N^{-3}$, and the same argument gives \eqref{e:comparison-bound} with $N^{1/2} r_N^{-3}$ replaced by $r_N^{-4}$. The constants depend only on $\alpha$, $\eps$, $p$, $D$, and the laws of $X$ and $Z$, through \zcref{l:strip}, \zcref{p:regularity}, \eqref{e:comparison-tail}, and \eqref{e:comparison-moments}. This completes the proof.
\end{proof}

\begin{remark}\label{r:comparison-inputs}
The proof of \zcref{p:comparison} uses the specific form of $F$ only through two properties. First, $F$ is bounded by one on real symmetric matrices, four times continuously differentiable, and satisfies
\begin{equation*}
|\partial_{ab} F(A)| + |\partial_{ab}^2 F(A)| \le C N^C (1 + \| A \|)^2
\end{equation*}
for some $C > 0$, all real symmetric matrices $A$, and all $a \le b$. Second, for all $t' \in [0, 1]$ and $a \le b$, there exists an event determined by $H_{t'}^{[ab]}$, whose complement has probability at most $C_D N^{-D}$ for every $D > 0$, uniformly in $t'$ and $(a, b)$, on which $\zeta \mapsto F(H_{t'}^{[ab]} + \zeta V^{ab})$ extends holomorphically to a neighborhood of the closed disk $\{ |\zeta| \le 4 r_N \}$ and is bounded by one in modulus there. Every observable with these two properties satisfies \eqref{e:comparison-bound} and its variant for $\E X^3 = 0$, with the same proof.
\end{remark}