The dyadic scale bands satisfy the weighted entropy condition

Write $x_n=n{r}_n^2$ and $J_n=\max \{j:2^j{\sigma }_n\le s_n\}$. Since $s_n\to 0$ and ${\sigma }_n\ge c{n}^{-a_1}$, $J_n=O(\log n)$. Put $u_n:=\min \{r_n\mathrm{/}2,1\mathrm{/}8\}$. If $r_n\le 1\mathrm{/}4$, truncate a mixing law to $[-A,A]$ and normalize it there; the removed mass is at most $r_n^2\mathrm{/}8$, so its mixture changes by $L^1$ at most $r_n^2\mathrm{/}4$ and by Hellinger at most $r_n\mathrm{/}2$ using [F2]. The compact cover at radius $u_n=r_n\mathrm{/}2$ then covers ${\mathcal{P}}_{n,j}$ at radius $r_n$. If $r_n>1\mathrm{/}4$, the sieve tail is at most $1\mathrm{/}128$, so truncation changes Hellinger distance by at most $1\mathrm{/}8$; the compact cover at radius $u_n=1\mathrm{/}8$ then covers the class at radius at most $1\mathrm{/}4<r_n$. Thus in both cases $N_{n,j}\le N(u_n,{\mathcal{P}}_{A,2^j{\sigma }_n},h)$. The rate lower bound implies $\log (1\mathrm{/}{u}_n)\le C\log n$ and $x_n\ge M_0^2{\sigma }_n^{-1}(\log n{)}^2,x_n\ge M_0^2{\sigma }_n^{-2\eta }\mathrm{.}$ For $0\le j\le J_n$, [F1] bounds $\log N_{n,j}\le C{\sigma }_n^{-1}(\log n{)}^2$, because the band lower scale is at least ${\sigma }_n$. Since ${\pi }_{n,j}\le 1$, these terms contribute at most $C\log n\exp \{C{\sigma }_n^{-1}(\log n{)}^2\mathrm{/}2-x_n\},$ which tends to zero for sufficiently large $M_0$.

For negative indices write $j=-k$, $k\ge 1$. For all sufficiently large $k$, the scale-tail premise gives ${\pi }_{n,-k}\le C\exp (-c{2}^{\gamma k})$. Also [F1] and $u_n\ge c{n}^{-1\mathrm{/}2}\log n$ imply $\log N_{n,-k}\le C{\sigma }_n^{-1}{2}^k(\log n+k{)}^2\mathrm{.}$ Choose $\theta >0$ and ${\gamma }_0<\gamma$ sufficiently close to $0$ and $\gamma$, respectively, so $q:=1+\theta <{\gamma }_0$ and $\rho :={\gamma }_0\mathrm{/}({\gamma }_0-q)<2\eta$. Since $(\log n+k{)}^2\le C_{\theta }(\log n{)}^2{2}^{\theta k}$ and $e^{-c{2}^{\gamma k}}\le e^{-c^{\mathrm{\prime }}{2}^{{\gamma }_{0}k}}$, the logarithm of each summand is at most $C{K}_n{2}^{qk}-c^{\mathrm{\prime }}{2}^{{\gamma }_{0}k},K_n:={\sigma }_n^{-1}(\log n{)}^2\mathrm{.}$ Enlarging $C$ handles the finitely many smaller $k$.

Put $p={\gamma }_0\mathrm{/}q>1$. As a function of $v=2^{qk}\ge 1$, the exponent $C{K}_{n}v-c^{\mathrm{\prime }}{v}^p$ has maximum at most $C^{\mathrm{\prime }}{K}_n^{p\mathrm{/}(p-1)}=C^{\mathrm{\prime }}{K}_n^{\rho }$, by differentiation. The maximizing scale is a constant multiple of $K_n^{1\mathrm{/}(p-1)}$, so there are $O(\log K_n)$ indices up to any fixed multiple of that scale; their sum has logarithm at most $C^{\mathrm{\prime }\mathrm{\prime }}{K}_n^{\rho }$. Choose a fixed multiple $L$ large enough that for $v\ge L{K}_n^{1\mathrm{/}(p-1)}$ the negative term is at least twice the positive term. Grouping the subsequent dyadic $v$ values gives a convergent geometric tail, also bounded by $\exp (C^{\mathrm{\prime }\mathrm{\prime }}{K}_n^{\rho })$. Therefore $\log {\sum }_{k\ge 1}\sqrt{N_{n,-k}{\pi }_{n,-k}}\le C^{\mathrm{\prime }\mathrm{\prime }}{\sigma }_n^{-\rho }(\log n{)}^{2\rho }\mathrm{.}$ Because $\rho <2\eta$ and ${\sigma }_n\le C{n}^{-a_2}$, this is $o({\sigma }_n^{-2\eta })=o(x_n)$. The negative-index contribution therefore vanishes after multiplication by $e^{-x_n}$.

If $G$ is supported on $[g_{-},g_{+}]$ with $g_{-}>0$, only a fixed finite number of dyadic indices have positive mass. Their lower scales are at least $g_{-}{\sigma }_n\mathrm{/}2$, so [F1] at radius $u_n$ bounds each log covering number by $C{\sigma }_n^{-1}(\log n{)}^2$. The first rate term gives $x_n\ge M_0^2{\sigma }_n^{-1}(\log n{)}^2$, and the finite weighted sum tends to zero for sufficiently large $M_0$.