For $R\gtrsim A^{-1/2}$, one can use $\Delta_{a}$-harmonicity to obtain weighted $C^{k,\alpha}$-estimates for any large $k$. For $R\lesssim A^{-1/2}$, we exhibited a collection of local charts corresponding to a choice of $P\in S$, such that the Kähler ansatz has $C^{1,\alpha}$-regularity. Thus if $u$ is a $C^{2,\alpha}$ function on a chart, then its $g^{(1)}$-Laplacian is $C^{\alpha}$. This regularity is sufficient for setting up weighted Hölder analysis.