A universal-meagre stage absorbs an old nowhere-dense tree

For the explicit matched-width case, fix a level $n>\mathrm{ht}(t)$ satisfying $\mathrm{\mid }T\cap 2^n\mathrm{\mid }\ge \mathrm{\mid }S\cap 2^n\mathrm{\mid }$; this is an additional hypothesis for the display, not a consequence of perfection. Write $S\cap 2^n=\{s_1,\dots ,s_k\}$ and choose distinct ${\eta }_1,\dots ,{\eta }_k\in T\cap 2^n$.

Let $T^{\mathrm{\prime }\mathrm{\prime }}$ be the set of all nodes of $T$ together with all nodes ${\eta }_i⌢\tau$ for tails $\tau$ satisfying $s_i⌢\tau \in S$, together with their initial segments; that is, replace the prefix $s_i$ by ${\eta }_i$ rather than concatenate the full old word. Prefixes shorter than $n$ already lie in $T$. Then $T^{\mathrm{\prime }\mathrm{\prime }}$ is a tree containing $T$, its recorded initial tree through height $\mathrm{ht}(t)$ is $t$, it is perfect because the nodes of $T$ keep their splitting extensions and each ${\eta }_i$ inherits the splitting of the perfect tree $S$ below $s_i$, and it is nowhere dense because its body is the union of the nowhere-dense set $[T]$ with the finitely many homeomorphic images of the closed nowhere-dense sets $[S]\cap [s_i]$. Hence $(t,T^{\mathrm{\prime }\mathrm{\prime }})$ is a direct extension of $(t,T)$.

For every $x\in [S]$ there is exactly one $i$ with $1\le i\le k$ and $x\in [s_i]$. Let ${\rho }_i$ be the full level-$n$ permutation swapping ${\eta }_i$ with $s_i$ (the identity if they agree) and leaving all other level words and all subsequent tail bits unchanged. Since [F2] fixes an enumeration of every finite-prefix rearrangement, define $m_i$ to be the least $m$ with ${\pi }_m={\rho }_i$. Then ${\rho }_i^{-1}(x)\in [{\eta }_i]\cap [T^{\mathrm{\prime }\mathrm{\prime }}]\subseteq [U_G]$, because the graft is recorded in the witness tree $T^{\mathrm{\prime }\mathrm{\prime }}$ and every witness tree of a condition in the generic filter is contained in the generic tree. Hence $x\in {\pi }_{m_i}([U_G])$, and the condition $(t,T^{\mathrm{\prime }\mathrm{\prime }})$ forces $[S]\subseteq {\bigcup }_{1\le i\le k}{\pi }_{m_i}([U_G]),$ a finite subunion of the countable meagre envelope of the absorption lemma.

Singleton case displayed level by level: for $A=\{0^{\omega }\}$, choose $n>\mathrm{ht}(t)$, a node $\eta \in T\cap 2^n$, and let $Z=\{\sigma \in 2^{<\omega }:(\mathrm{\forall }j)(2j<\mathrm{\mid }\sigma \mathrm{\mid }\Rightarrow \sigma (2j)=0)\}$. Its body contains $0^{\omega }$, has arbitrarily late free odd coordinates and is nowhere dense because a later even coordinate can be set to $1$; graft $Z$ below $\eta$, so $T^{\mathrm{\prime }\mathrm{\prime }}=T\cup \{\eta ⌢\sigma :\sigma \in Z\}$. At every level $m\ge n$ the graft contributes the nodes $\eta ⌢\sigma$ with $\mathrm{\mid }\sigma \mathrm{\mid }=m-n$ (some may already belong to $T$). The body remains nowhere dense by the finite-union argument of step 1.2; no same-level sibling of $\eta$ is required. The full prefix permutation swapping $0^n$ and $\eta$ sends the grafted branch $\eta ⌢0^{\omega }\in [U_G]$ to $0^{\omega }$, so $0^{\omega }\in \pi ([U_G])$, and this single finite substitution is the whole code at this stage.

The general existence assertion is the exact content of [F2]. Under the additional matched-width hypothesis, steps 1.1--1.3 exhibit the finite graft explicitly, and step 2.1 supplies the unconditional singleton instance. No claim is made that perfection alone yields the width comparison or that the generic tree itself contains every old tree.