$x\in B_1(0)$ let us define $$\begin{aligned}
r'_x \equiv \max\{r, r_x^{d-1}\}\, ,\end{aligned}$$ where $r^{d-1}_x$ is the $(d-1)$-critical radius of Section \[ss:critical\_radius\]. Let us separate $B_1(0)$ into subsets $S_1,S_2$ which are defined by $$\begin{aligned}
&S_1\equiv\{x\in B_1(0): \not\exists\, y\in B_{10 r'_x}(x) \text{ s.t. } r'_y<10^{-2}r'_x\}\, \notag\\
&S_2\equiv B_1\setminus S_1\, .\end{aligned}$$ We can interpret $S_1$ as the set of points with locally minimizing $d-1$-critical radii. We let $\{B_{r_i}(x_i)\}$ be a Vitali subcovering of the collection $\{B_{r'_x}(x)\}_{x\in S_1}$. Clearly the collection satisfies (1), and hence we need to show this collection of balls satisfies (2),(3).

Now standard arguments as in [@chnava] give us, roughly, that for every $x_i$ and $s\in [r_i,1]$ that there exists a homogeneous harmonic polynomial $P_i^{(s)}$ of degree $d$ such that $T_{x,s}^u$ is close to $P_i^{(s)}$. A key point is that the new effective argument discussed in Section \[ss:tangent\_maps\] will allow us to take $P_i^{(s)}\equiv P^d$ to be independent of both $i$ and $s$. For the sake of the outline let us make the assumption that $P^d$ is $n-2$ symmetric, which is to say that $P^d$ depends on only two variables. Up to some technical details this will turn out to be the important case, in that one can always handle the 