For $(\mathcal{M},g_{\beta})$ with sufficiently large gluing parameter $\beta\gg 1$, then there exists some constant $C>0$, independent of $\beta$, such that for every triple

(9.101)

$$\bm{\xi}^{+}\equiv(\xi_{1}^{+},\xi_{2}^{+},\xi_{3}^{+})\in\mathfrak{B},$$

there exists a unique pair

(9.102)

$$(\bm{\eta},\bar{\bm{\xi}}^{+})\equiv\Big((\eta_{1},\eta_{2},\eta_{3}),(\bar{\xi}_{1}^{+},\bar{\xi}_{2}^{+},\bar{\xi}_{3}^{+})\Big)\in\mathfrak{A}$$

which satisfies

(9.103)

$$\mathscr{L}_{g}(\bm{\eta},\bar{\bm{\xi}}^{+})=\bm{\xi}^{+}$$

and

(9.104)

$$\|\bm{\eta}\|_{C_{\delta,\nu,\mu}^{1,\alpha}(\mathcal{M})}+\|\bar{\bm{\xi}}^{+}\|_{L^{2}}\leq Ce^{10\delta\cdot\beta}\cdot\|\bm{\xi}^{+}\|_{C_{\delta,\nu+1,\mu}^{0,\alpha}(\mathcal{M})},$$

where $\delta$, $\nu$ and $\mu$ are the constants in Proposition 9.2.