The first possible shift and lapse logarithms vanish by indicial projection

Put P equal to Delta_S2 plus two. Direct differentiation gives ${\mathrm{\Delta }}_0(r^{-2}(\log r{)}^{j}a)=r^{-4}[Pa(\log r{)}^j-3ja(\log r{)}^{j-1}+j(j-1)a(\log r{)}^{j-2}]$. The vector KID equation, with $X=O(r^{-2}{\log }^{J}r)$ and $k=(2N{)}^{-1}{\mathcal{L}}_{X}h=O(r^{-3}{\log }^{J}r)$, has no $r^{-4}$ forcing after moving its metric and Ricci corrections to the right: all such terms have order at most $r^{-5}$ times finite log powers. If $J>0$ is the largest $r^{-2}$ shift-log power, its coefficient $a_J$ lies in ker P; projecting the next log coefficient equation onto ker P gives $-3J{a}_J=0$. Thus the highest coefficient vanishes, and descending induction removes all positive log powers in the $r^{-2}$ shift.

Now $k=O_{\mathrm{\infty }}(r^{-3})$. For $N_0=1-m\mathrm{/}r$ and $h=(1+2m\mathrm{/}r)\delta +O(r^{-2}{\log }^{J}r)$, the coordinate Laplacian formula gives ${\mathrm{\Delta }}_h{N}_0=-m^2{r}^{-4}+O(r^{-5}{\log }^{J^{\mathrm{\prime }}}r)$. The lapse equation ${\mathrm{\Delta }}_{h}N=N\mathrm{\mid }k{\mathrm{\mid }}^2$ therefore has only radial forcing $m^2$ at the $r^{-4}$ equation for the $r^{-2}$ coefficient of $N-N_0$. For a largest positive log power the same projection calculation gives $-3J{b}_J=0$: when $J=1$ the constant-log equation has the radial right side, whose projection onto degree-one harmonics is zero. Hence all positive log powers at this order vanish as well.

The remaining higher-order terms start at $r^{-3}$ with finite log powers and differentiated remainders. Each such finite log power times $r^{-3}$ is bounded by a constant times $r^{-2}$, with the corresponding extra radius factors after every fixed derivative. The full expansion in [F1] therefore proves both displayed O-infinity assertions. No convergence of the formal expansion is required.