Perfect-tree splitting of a new-real name

Below every $q\le p$ there are two conditions forcing incompatible finite prefixes of $\dot{\tau }$. Otherwise all prefixes forceable below some $q$ would be compatible. For every $k$, finite iteration of F2 would then give a unique $u_k\in 2^k$ forceable below $q$. Definability of forcing forms $(u_k{)}_{k<\omega }$ in $N$, and $q$ forces $\dot{\tau }={\bigcup }_k{u}_k\in N$, contradicting the newness hypothesis in F1.

Put $p_{\mathrm{\varnothing }}\le p$ in $D_0$. Use step 1.1 to choose two extensions forcing incompatible prefixes. Successively refine the two ordered pairs into $E_0$; openness preserves the first requirement while the reverse ordered pair is handled. Call the resulting conditions $p_0,p_1$, and strengthen them to decide incompatible prefixes $u_0,u_1$ of length at least $1$.

Below each of $p_0,p_1$, apply step 1.1 to choose two successors. Prefix decisions inherited from the parents separate successors from different parents, and the new splits separate siblings. Successively refine the four nodes through $D_0,D_1$ and all $12$ ordered pairs through $E_0,E_1$, then use F2 to decide extensions $u_{ij}$ of length at least $2$. There are only finitely many requirements, and downward closure preserves every earlier one.

Repeat at level three with eight nodes: meet $D_0,D_1,D_2$ at every node, meet $E_0,E_1,E_2$ for all $8\cdot 7=56$ ordered pairs of distinct nodes, and decide pairwise incompatible prefixes $u_{ijk}$ of length at least $3$.

Thus agreement of branches through level $m$ forces agreement of their interpreted reals through the already decided length-$m$ prefix, while their first split forces distinct interpretations. The continuity modulus is: input agreement through level $m$ implies output agreement through $m$ digits. Continuing the same finite procedure meets every enumerated dense set and realizes the endpoint asserted by F1.