edge {#s.relaxation}
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To prove the edge universality, we first have the following estimate for the size of the observable $ f_t $.

\[p.estimate\] For any (large) $ D>0 $ there exists $ N_0(D) $ such that for any $ N {\geqslant}N_0 $ we have $${\mathbb{P}}\left( {\text{\rm Im}\hspace{0.1cm}}f_t(z) \lesssim \varphi \dfrac{\kappa(E)^{1/2}}{\max(\kappa(E)^{1/2},t)} \ \mbox{\rm for all} \ 0<t<1 \ \mbox{\rm and} \ z=E+{\text{\rm i}}\eta \in {\mathscr{S}}\right) > 1-N^{-D}$$

For any $ 1 {\leqslant}l,m {\leqslant}N^{10} $, we define $ t_l:=l N^{-10} $ and $$z^{(m)} := E_m + {\text{\rm i}}\eta_m=E_m+{\text{\rm i}}\dfrac{\varphi^2}{N \kappa(E_m)^{1/2}},$$ where $$E_m := \inf\left\{ E >0 : \int_{-\infty}^{E^2} \rho_{{\mathsf{MP}}}(x){\mathrm{d}}x {\geqslant}\left(m-\frac{1}{2}\right)N^{-10} \right\}.$$ We also define the following stopping times (with respect to $ \mathcal{F}_t = \sigma(B_k(s):0 {\leqslant}s {\leqslant}t,1 {\leqslant}k {\leqslant}N) $) which represent the bad events: $$\tau_{l,m} := \inf \left\{ 0 {\leqslant}s {\leqslant}t_l : {\text{\rm Im}\hspace{0.1cm}}f_s(z_{t-s}^{(m)}) \gtrsim \dfrac{\varphi}{2}\dfrac{\kappa(E_m)^{1/2}}{\max(\kappa(E_m)^{1/2},t_l)}  \right\},$$ $$\tau_0 := \inf \left\{ 0 {\leqslant}t {\leqslant}1: \exists k \in \llbracket -N,N \rrbracket \ \mbox{s.t.} \  |x_k(t)-\gamma_k|>\varphi^{\frac{1}{2}}N^{-\frac{2}{3}}({\widehat}{k})^{-\frac{1}{3}} \right\},$$ $$\tau := \min \left\{ \tau_0,\tau_{l,m}: 0 {\leqslant}l,m {\leqslant}N^{10}, \kappa(E_m) > \varphi^2 N^{-\frac{2}{3}} \right\}.$$ We also define the convention $ \inf \emptyset =1 $.

We claim that in order to prove the desired result, it suffices to show that for any $ D>0 $ there exists 