Fixed Boolean values come from initial collapse layers

Fix $p{\le }_{B}u$ and suppose $p\upharpoonright m\mathrm{̸}{\le }_{B}u$. Density of the embedding in F2 gives $q\in P$ with $q\le p\upharpoonright m$ and $q{\le }_B\mathrm{\neg }u$.

For every layer $n\ge m$ appearing in $q$, choose a finite permutation of its $i$-coordinates which moves all upper-layer coordinates of $q$ away from the finitely many upper-layer coordinates of $p$; extend it by the identity elsewhere. The resulting $\pi$ lies in $H_m$. Below layer $m$, $q$ extends $p\upharpoonright m$ and $\pi$ is the identity; above it, the moved domain of $\pi q$ is disjoint from the domain of $p$. Thus $p$ and $\pi q$ are compatible. This is a finite construction in finitely many represented layers.

Since $u$ is fixed by $H_m$, the regular-open automorphism described in F3 sends $q{\le }_B\mathrm{\neg }u$ to $\pi q{\le }_B\mathrm{\neg }u$. A common extension of $p$ and $\pi q$ would lie below both $u$ and $\mathrm{\neg }u$, contradicting Boolean incompatibility. Hence $p\upharpoonright m{\le }_{B}u$.

Let $v$ be the displayed join. Step 2.1 gives $v\le u$. Conversely every $p{\le }_{B}u$ satisfies $p\le p\upharpoonright m\le v$, and density below the regular open $u$ gives $u=\bigvee \{p\in P:p{\le }_{B}u\}\le v$. Therefore $u=v$.

Every initial-layer condition is fixed by $H_m$, so the complete subalgebra it generates is contained in $B_m$. The equality in step 3.1 writes every member of $B_m$ as a join of such conditions, yielding the reverse inclusion and the final assertion.