its lack of effectiveness, is that if the frequency is $\delta$-pinched over a potentially large number of scales $[r_0,r_1]$, then the homogeneous harmonic polynomial $P^{(r)}_d$ which $T^u_{x,r}$ is close to may depend on $r$.

In this paper, using arguments which extends both those of [@chnava] and the tangent map uniqueness result of [@han_sing], we prove a result which strengthens both of these into a much more effective statement. Namely, we see that for every $\epsilon>0$ there is a $\delta>0$, which is in fact given explicitly and sharply, such that if the frequency is $\delta$ pinched on scales $[r_0,r_1]$, then there exists a [*unique*]{} homogeneous harmonic polynomial $P_d$ such that $T_{x,r}$ is $\epsilon$-close to $P_d$ for all $r\in [r_0,r_1]$. See Theorem \[t:eff\_tan\_con\_uniq\_harm\] for the harmonic case and Theorem \[t:eff\_tan\_con\_uniq\] for the general case. Both the uniqueness and the effectiveness of the constants play an important role in Theorem \[t:main\_critical\].

The Critical Radius {#ss:critical_radius}
-------------------

Let us begin with the following definition of the critical radius:

Given $x\in B_{1/2}(0)$ we define $$\begin{gathered}
  r_c(x)\equiv r_x = \sup{\left\{0\leq s\leq r_0:  N(x,s)<\frac{3}{2}\right\}} \, ,
 \end{gathered}$$ where $r_0(n,\lambda)>0$ is a small constant chosen from Lemma \[l:frequency\_comparison\].

Let us quickly remark on the following, which is easy 