A perfect tree of mutually generic name interpretations

The initial forcing producing $N$ and the interval forcing $Q$ both have size below $\kappa$. Nice names for subsets of $Q$ and $Q^2$ are coded by subsets of a ground set of size below $\kappa$; strong inaccessibility bounds the collection of those codes below $\kappa$. The remaining Lévy collapse therefore makes $\mathcal{P}(Q{)}^N$ and $\mathcal{P}(Q^2{)}^N$ countable in $V[G]$, as asserted in F1. Use F3 to enumerate their dense members and replace each by its downward closure. Recursively assign $p_s\le p$ for $s\in 2^{<\omega }$. At stage $n$, extend every node into the first $n$ one-coordinate open dense sets and every ordered pair of distinct nodes into the first $n$ product open dense sets. There are only finitely many requirements at a level: process them successively, strengthening the affected coordinates each time. Downward closure preserves all requirements already met.

Below every $q\le p$ there are two extensions forcing incompatible initial segments of $\dot{\tau }$. Otherwise, for some $q$, any two decided strings would be compatible. For each length $k$, density of deciding conditions then gives one unique string $u_k\in 2^k$ that can be forced below $q$; least-string selection makes the sequence $(u_k)$ definable in $N$ from $q$ and $\dot{\tau }$. Its union is a real $u\in N$, and density closure gives $q⊩\dot{\tau }=\check{u}$, contradicting the displayed hypothesis, since $q\le p$. Apply this splitting below each node to make siblings $p_{s0},p_{s1}$ decide incompatible strings of length at least $\mathrm{\mid }s\mathrm{\mid }+1$, preserving all earlier finite requirements.

For $z\in 2^{\omega }$, the filter generated by the branch is $g_z=\{q\in Q:(\mathrm{\exists }n)p_{z\upharpoonright n}\le q\}$, the upward closure in the convention that smaller conditions are stronger. It meets every enumerated dense set and so is $N$-generic. Distinct $z,w$ give an $N$-generic pair by the product requirements, and the incompatible decisions make ${\dot{\tau }}_{g_z}\mathrm{\ne }{\dot{\tau }}_{g_w}$. Agreement through level $n$ fixes an output prefix of length $n$, so $z\mapsto {\dot{\tau }}_{g_z}$ is continuous. Compactness makes its injective image perfect and nonempty.