We plan to exhibit a $T^{2}$-bundle preserving diffeomorphism $\Psi_{1}$ between the Taub-NUT type $\mathbb{C}^{3}$ and the positive vertex space $M^{+}_{\nu}$ over the common base $\{0<|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}\}$, with good estimates on the deviations between both Kähler structures. Since $\frac{1}{3}A^{1/2}<\frac{1}{2}A^{1/2}$, the $\eta$-periodic copies of such punctured discs do not overlap. The topology of the $T^{2}$-bundle structures on both spaces agree by construction. The remaining degrees of freedom in defining $\Psi_{1}$ amounts to a gauge choice, which is the same as a prescription of $\vartheta_{i}^{\mathbb{C}^{3}}-\Psi_{1}^{*}\vartheta_{i}$.

As a general guideline, the corresponding quantities on $\mathbb{C}^{3}$ and $M^{+}_{\nu}$ have the same singularity, so their difference are smooth quantities. We use superscripts for quantities on $\mathbb{C}^{3}$ to disambiguate from quantities on $M^{+}_{\nu}$.