Consider $(\mathcal{M},g_{\beta})$ with sufficiently large gluing parameter $\beta\gg 1$. Denote by $\bm{\omega}_{\beta}^{\mathcal{M}}$ the gluing definite triple which is constructed by Proposition 6.4. Let $\delta$, $\nu$ and $\mu$ be the constants in Proposition 9.2, then there exists a hyperkähler triple $\bm{\omega}_{\beta}^{\HK}$ with the effective estimate

(9.135)

$$\|\bm{\omega}_{\beta}^{\mathcal{M}}-\bm{\omega}_{\beta}^{\HK}\|_{C_{\delta,\nu+1,\mu}^{0,\alpha}(\mathcal{M})}\leq Ce^{-\delta_{0}\beta}$$

for some constants $C>0$ and $\delta_{0}>0$ independent of $\beta$. In particular, $\mathcal{M}$ is diffeomorphic to the $\K 3$ surface.