Consider $(\mathcal{M},g_{\beta})$ with sufficiently large gluing parameter $\beta\gg 1$. Let $\delta$, $\nu$ and $\mu$ be the constants in Proposition 9.2, then there exists some constant $C>0$ which is independent of $\beta$ such that

(9.132)

$$\|\mathscr{F}(0)\|_{\mathfrak{B}}\leq Ce^{-\frac{\delta_{q}\beta}{2}},$$

where $\delta_{q}>0$ is the constant in Corollary 6.5.