(Ooguri-Vafa type metric on the positive vertex) Fix $k,\alpha,\kappa$ and $0<\epsilon\ll 1$, and let $1\ll\nu\ll A^{3/8}$. Then there is a $T^{2}$-invariant Calabi-Yau metric on $M^{+}_{\nu}$ given by a $T^{2}$-invariant Kähler potential $\phi^{+}$,

$$\omega_{+}=\tilde{\omega}^{(4)}+\sqrt{-1}\partial\bar{\partial}\phi^{+},\quad\omega_{+}^{3}=\frac{3}{4}\sqrt{-1}\Omega\wedge\overline{\Omega},$$

satisfying the metric deviation estimate

(3.12)

$$\left\lVert\omega_{+}-\tilde{\omega}^{(4)}\right\rVert_{C^{k,\alpha}_{-1-\epsilon}(M^{+}_{\nu})}\leq C\nu^{2}A^{3/4(-1+\epsilon)},\quad\left\lVert d\phi^{+}\right\rVert_{C^{k+1,\alpha}_{-\epsilon}(M^{+}_{\nu})}\leq C\nu^{2}A^{-1+3\epsilon/4}.$$

The constants depend only on $k,\alpha,\epsilon,\kappa$ and the scale invariant ellipticity bound on $a_{ij}$.