Every set of reals in the Solovay model has the Baire property

Use F2 to choose a bounded $N$ containing F1's real parameter; the ordinal parameters remain explicit. F3 gives an $N$-coded Borel set $B$ agreeing with $A$ on every $N$-Cohen generic. In the ambient final extension, enumerate the $N$-coded closed nowhere-dense sets as $(C_n{)}_{n<\omega }$, as in the proof of F3's generic-largeness component, and let $d$ be the real Borel code for $D={\bigcup }_n{C}_n$. Every nongeneric real lies in $D$, so $A\mathrm{△}B\subseteq D$. The code $d$ need not lie in $N$, but F1 says that $M$ and the final extension have the same reals, hence $d\in M$; F4 makes its evaluation and meagreness absolute to $M$. Apply F4 inside $M$ to the code of $B$ to obtain a coded open $U$ and coded meagre $E$ with $B\mathrm{△}U\subseteq E$.

The explicit codes for $D$ and $E$ lie in $M$, and F4 closes the meagre ideal under their finite union (equivalently, interleave their two coded nowhere-dense witness sequences). Thus $A\mathrm{△}U\subseteq D\cup E$ is meagre in $M$, which is exactly BP. Empty and whole-space cases use $U=\mathrm{\varnothing }$ and $U=\mathbb{R}$.