Absorption, factorization, and homogeneous truth in the Solovay collapse

By F1, choose $\xi <\kappa$ with $f\in V[G_{\xi }]$. Solovay's small-collapse factorization replaces this initial extension by $V[F]$, where $F:\omega \twoheadrightarrow \lambda$ is a $V$-generic collapsing map for some ordinal $\lambda <\kappa$ and $f\in V[F]$. Define

The initial forcing has size below $\kappa$. Solovay's absorption construction recursively embeds its Boolean completion and the tail collapse into a fresh copy of $\mathrm{Lv}(\kappa )$ over $V[f]$: at stage $\alpha <\kappa$, put the next maximal antichain and the next dense set into coordinates above all earlier supports. Regularity of $\kappa$ bounds each stage, and the union is a dense complete embedding. The image of $G$ is therefore a generic $H$ with both inclusions $V[G]\subseteq V[f][H]\subseteq V[G]$. F3 is used exactly to enumerate those dense sets and maximal antichains.

The collapse is weakly homogeneous. Given $p,q$, first move the finitely many coordinates of $p$ away from those of $q$ by coordinate permutations; the moved $p$ is compatible with $q$. If some condition forced a sentence $\phi (\vec{a})$ and another forced its negation, apply such an automorphism. It fixes every $\vec{a}\in V[f]$ and yields compatible conditions forcing opposites, contrary to forcing consistency. Density of decision then makes the Boolean value $0$ or $1$. The omission of $H$ is essential.

Random and Cohen forcing have cardinal below $\kappa$ in each localized model. Insert their Boolean completion as the first small factor in the same absorption recursion. Thus, after adjoining its generic real $x$, the remaining extension is again a homogeneous Lévy collapse over $V[f,x]$.