Strongly homogeneous truth has Baire representatives

By [F1], choose a stage $\alpha$ containing a name for $s$ and all Boolean values used below. Enlarge to a later stage $\beta$ so that the next designated quotient has the canonical UM presentation. Put $M=V[s]$, not $V[G\cap B_{\beta }]$. Then every meagre Borel set coded in $M$ is among the sets absorbed by that quotient, exactly as [F1] states. The ordinal parameters already belong to $V$ and hence to $M$. Every relevant countably generated complete subalgebra and name can be placed in a later stage because its countable set of Boolean data is bounded in the ${\omega }_1$ construction.

In the ground model choose, for each coordinate of the fixed name $\dot{s}$, a countable maximal antichain labelled by its decided ordinal value, and let $C_s$ be the completion of the countable Boolean algebra generated by all members of those antichains. The labelled antichains make $\dot{s}$ a $C_s$-name. Conversely, $s$ determines which member of every labelled antichain lies in $G$: combine equal labels first, so distinct antichain members have distinct labels. Those antichains generate a countable dense subalgebra of $C_s$, and a generic ultrafilter is determined by its trace on a dense subalgebra. Consequently $V[G\cap C_s]=V[s]=M\mathrm{.}$ Choose the stage in step 1.1 to contain the countably many generators; completeness then contains $C_s$. For every dense open $D\subseteq 2^{<\omega }$ in $M$, the reals whose initial-segment filters miss $D$ form a closed nowhere-dense set coded in $M$. By [F1], the later UM quotient puts the union of all these sets inside one meagre set. Thus the final extension contains Cohen reals over $M$, by its ZFC Baire theorem. This existence assertion is not promoted to a claim that an arbitrary name for one of those reals canonically embeds the full Cohen algebra.

Work in $M$ and let $\mathbb{C}$ be the Cohen category algebra. A local Cohen chart consists of a nonzero $a\in B$, a $B$-name $\dot{x}$, and $a⊩\dot{x}$ is Cohen-generic over the canonical $C_s$-extension. The map $j_{a,\dot{x}}(u)=a\wedge \mathrm{\parallel }\dot{x}\in u{\mathrm{\parallel }}_B(u\in \mathbb{C})$ is a complete Boolean homomorphism into $B\upharpoonright a$, but need not be injective: prepending $0$ to a Cohen name kills the nonzero cylinder $[1]$. Its kernel is a complete ideal, hence is $\mathbb{C}\upharpoonright \mathrm{\neg }d$ for a unique nonzero support $d\in \mathbb{C}$, and the restriction $j_{a,\dot{x}}:\mathbb{C}\upharpoonright d⟶j_{a,\dot{x}}[\mathbb{C}]$ is a complete isomorphism with top $a$. The complete subalgebra generated by $C_s$, this range, and $a$ is countably generated. This is the required localization; no full Cohen copy is inferred from genericity of the name.

Put $b_a=a\wedge \mathrm{\parallel }\phi (\dot{x},\dot{s},\check{\stackrel{⃗}{\gamma }}){\mathrm{\parallel }}_B$ and let $C$ be the chart algebra from step 3.1. Inside the relative algebra below $a$, repeat Shelah's free-amalgamation argument. If $D$ is generated by $C\cup \{b_a\}$, a free copy of $D$ over $C$ extends to an automorphism fixing $C$. It fixes $a$, the parameter name, and every Boolean value of the coordinate name below $a$, so it fixes $b_a$. If the two projections of $b_a$ and $a\setminus b_a$ to $C\upharpoonright a$ overlapped, freeness would make $b_a$ compatible with the image of its complement, a contradiction. Hence $b_a\in C\upharpoonright a$. Via the isomorphism in step 3.1 it has a unique reading $c_{a,\dot{x}}\le d$ in the Cohen algebra, represented by a regular open, and therefore Borel, subset of $d$.

These local readings are coherent. On a nonzero overlap of two support elements, restrict both chart algebras to that overlap. Their coordinate isomorphism fixes $C_s$ and sends one restricted Cohen name to the other; [F2] extends it to an automorphism of $B$. Since the parameters are fixed, invariance of Boolean truth sends one restricted value from step 4.1 to the other, so the two Cohen readings agree on the overlap. The family of chart supports is dense in $\mathbb{C}$: from one Cohen real supplied by step 2.1, an $M$-coded category-algebra isomorphism into any prescribed nonzero regular open set produces a chart supported below that set. Choose a maximal antichain of chart supports. It is countable, and the coherent local readings paste to one $c\in \mathbb{C}$, hence to one Borel code in $M$. Every $M$-Cohen-generic real meets this antichain and, by applying the same overlap argument to a chart containing its chosen name and forcing condition, satisfies $V[G]\models \phi (x,s,\vec{\gamma })⟺x\in C_c\mathrm{.}$ Consequently $A\mathrm{△}{C}_c$ is contained in the set $X$ of reals not Cohen-generic over $M$. This is precisely the local-cone step in Solovay's argument cited in [F6]; it does not assert a canonical complete copy for every generic name.

For each dense open $D\subseteq 2^{<\omega }$ in $M=V[s]$, the set of reals whose initial-segment filter misses $D$ is a closed nowhere-dense set coded in $V[s]$. Their union is exactly $X$. The family need not be countable in $M$, but every member is among the meagre Borel sets whose codes [F1] absorbs at the designated UM quotient chosen in step 1.1. The absorption clause therefore gives in the final extension one coded meagre $F_{\sigma }$ set $E$ containing all of $X$. Hence $A\mathrm{△}{C}_c\subseteq E\mathrm{.}$

The conclusion so far is a Borel representative, not necessarily an open one. Apply the uniform construction in [F4] to $c$: it gives an open code $u$ and a coded meagre set $E_c$ with $C_c\mathrm{△}{U}_u\subseteq E_c$. Therefore $A\mathrm{△}{U}_u\subseteq E\cup E_c,$ and the right side is meagre. All three codes belong to the final extension. This is the explicit Borel-to-open step required by the definition of the Baire property.

A real is a countable ordinal sequence, so real-and-ordinal parameters are a special case. Conversely, the proof began with an arbitrary captured countable ordinal sequence $s$, so it does not rely on replacing $s$ by a real unless the constructible-ground coding of [F1] is invoked later. Steps 6.1--7.1 prove the Statement.