intersecting at an angle of $\theta_{m+1}=\pi(1-\frac{c_1}{\log(m+1)})$. As above, length-area estimates in $T^{m+1}_J$ yield $\omega(z^{m+1}_J,\ell^{m+1}_{J,j},T^{m+1}_J)\lesssim e^{-\frac{c_0}{\delta_{m+1}}}$, and thus the length of the arcs $\gamma_j^\ast$ is $o(\frac1{\log(m+1)})$ by our assumptions on the sequence $\{\delta_n\}$. Moreover, the disc determined by $\gamma_j$ has radius $r=r_m$ bounded by $$\label{radiusest}r\lesssim\frac{c_1}{\log(m+1)}e^{-\frac{c_0}{\delta_{m+1}}},$$ so that $r_m=o(1)$. The arcs $\gamma_j^\ast$, $1\le j\le 4$, are therefore hyperbolically separated since we have taken $m\ge n_0$. Let us now define $D^{m+1}_{J,j}$ to be the pre-image under $\phi^{m+1}_J$ of the domain bounded by $E_j^\ast\cup\gamma_j^\ast$. We then define domains $$\tilde T^m_I:=\hat T^m_I\cup\bigcup\{D^n_{J,j}:n>m,L^n_{J,j}\subset\hat T^m_I\}$$ for each multi-index $I$ of length $m$. (See Figure \[fig3\].) We note that for $n\ge n_0$ the estimate (\[radiusest\]) yields that these domains are simply connected and also that $\tilde T^n_J\cap\tilde T^n_I=\emptyset$ when $I\neq J$.

For $n<n_0$ we will take $$\tilde T^n_J=\hat T^n_J\cup\bigcup\{D^m_{I,j}:n\ge n_0,L^m_{I,j}\subset\hat T^n_J\}.$$

Proceeding from these definitions, we will construct our solutions from the top down, exploiting the natural generations structure of the Cantor set. At each stage we will obtain corona solutions $\{g^{(n,J)}_\mu\}\in H^\infty(\tilde T^n_J)$, and the corona solutions specified by the proposition will simply be the restrictions of these solutions to $\hat T^n_J$. For $n\le n_0$, we construct our corona solutions by applying the corona theorem for finitely 