(Transverse Taub-NUT metric) Fix $0<\alpha<1$. Under suitable gauge choices for the $S^{1}$-connection, the ansatz metric $g^{(1)}$ over the local chart $\{r\lesssim A^{1/4}\}$ is approximated by the model metric:

$$\begin{cases}|\Psi^{*}g^{(1)}-g_{\text{NUT}}|_{g_{\text{NUT}}}\leq CA^{-3/4}\max(\log(A^{-1/4}\varrho),1)+CA^{1/4}|\xi_{2}|,\\ |\nabla_{g_{\text{NUT}}}(\Psi^{*}g^{(1)}-g_{\text{NUT}})|_{C^{\alpha}_{-1}(r\lesssim A^{1/4})}\leq CA^{-1/4}\max(\log(A^{-1/4}\varrho),1).\end{cases}$$

and

$$\begin{cases}|\Psi^{*}\Omega^{(1)}-\Omega_{\text{NUT}}|_{g_{\text{NUT}}}\leq CA^{-3/4}\max(\log(A^{-1/4}\varrho),1)+CA^{1/4}|\xi_{2}|,\\ |\nabla_{\Omega_{\text{NUT}}}(\Psi^{*}g^{(1)}-\Omega_{\text{NUT}})|_{C^{\alpha}_{-1}(r\lesssim A^{1/4})}\leq CA^{-1/4}\max(\log(A^{-1/4}\varrho),1).\end{cases}$$