(Model Laplacian) Given $0<\epsilon\ll 1$, let $f$ be an $S^{1}$-invariant function on the model space supported in $\{r\lesssim A^{-1/2}\}$, with bound $\left\lVert f\right\rVert_{C^{k,\alpha}_{0}(g_{\text{NUT}})}\leq 1$. Then there is a function $u$ compactly supported in $\{r\lesssim A^{1/4}\}$, such that on an annulus region at any given dyadic scale $r\sim r_{0}$ (or the ball region $r\lesssim A^{-1/2}$) we have a decay estimate

$$\left\lVert u\right\rVert_{C^{k+2,\alpha}_{0}(r\sim r_{0},g_{\text{NUT}})}\leq CA^{-1}(A^{1/2}r_{0}+1)^{-3+\epsilon},$$

and $\Delta_{g_{\text{NUT}}}u-f$ is only supported on one dyadic scale $\{r\sim A^{1/4}\}$ with the bound

$$\left\lVert\Delta_{g_{\text{NUT}}}u-f\right\rVert_{C^{k,\alpha}_{-2}(g_{\text{NUT}})}\leq CA^{3(-3+\epsilon)/4}.$$