Homogeneous truth about a generic real has Borel representatives

For a random real $x$ over $N$, F1 factors the final extension as $N[x][H_x]$ with homogeneous tail forcing $R_x$. In the random forcing language over $N$, let $\psi (\dot{x})$ be the assertion that the top condition of $R_{\dot{x}}$ forces $\phi (\dot{x},a,\vec{\alpha })$. Use F3 to form $b=\parallel \psi (\dot{x})\parallel$ in the completed random algebra and choose an $N$-coded Borel representative $B$ of $b$. For every $N$-random $x$, the random-forcing truth lemma gives $x\in B$ iff $N[x]\models \psi (x)$. Homogeneity says the Boolean value of $\phi (x,a,\vec{\alpha })$ in $R_x$ is $0$ or $1$, and the tail truth lemma applied to the actual $H_x$ therefore gives $N[x]\models \psi (x)$ iff $N[x][H_x]=V[G]\models \phi (x,a,\vec{\alpha })$. Thus $B$ represents final-extension truth on every $N$-random real; no absoluteness from $N[x]$ to its tail extension was used.

In Cohen forcing use the analogous assertion $\psi (\dot{x})$ that the top of the homogeneous tail forces $\phi (\dot{x},a,\vec{\alpha })$. F3 supplies its regular-open Boolean value, with an $N$-coded regular-open representative $U$ whose boundary is nowhere dense. The Cohen-forcing truth lemma and the same homogeneous-tail argument from step 1.1 show that, for every $N$-Cohen generic $x$, final-extension truth is equivalent to $x\in U$. Thus $U$ is already a Borel representative, and it differs from an open set by the empty, hence meagre, set. The statement deliberately leaves all nongenerics as exceptions; their largeness is used only by later items after its countability hypothesis has been verified.