Meagre master codes are below summable slaloms

For each $f$, every tail union ${\bigcup }_{n\ge m}{S}_{f(n)}^n$ is dense open: a given basic cylinder $U$ equals $U_n$ for some $n\ge m$, and $S_{f(n)}^n$ meets it by [F1]. Hence $M_f^{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$ is meagre. It is represented by a code of Borel master codes for null and meagre sets: at master stage $m$, enumerate the basic cylinders contained in that dense open tail union. This enumeration is Borel in $f$ because the tail union is a coded countable union of clopens and inclusion of a compact cylinder in it is a finite-subcover test.

If $A$ is meagre, take closed nowhere-dense $F_j$ with $A\subseteq {\bigcup }_j{F}_j$. For each $n$, the complement of ${\bigcup }_{j\le n}{F}_j$ is dense open, so choose the least $m=f(n)$ such that $S_m^n$ lies in that complement, using [F1]. Every $y\in A$ belongs to some $F_j$ and therefore misses $S_{f(n)}^n$ for all $n\ge j$; hence $A\subseteq M_f^{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$. This proves cofinality.

Define $u(f)(n)=\{f(n)\}$, a slalom since ${\sum }_n{2}^{-n}<\mathrm{\infty }$. For $T\in \mathbb{S}$, summability implies $\mathrm{\mid }T(n)\mathrm{\mid }\le 2^n$ for every sufficiently large $n$. Let $r(T)$ be the least integer beyond which this holds; it is Borel because the condition is a countable conjunction of coordinate tests. Put $W_n(T)={\bigcap }_{i\in T(n)}{S}_i^n$ for $n\ge r(T)$, and put $W_n(T)=2^{\omega }$ for earlier $n$. Empty intersections are the whole space. By [F1], each $W_n(T)$ for $n\ge r(T)$ meets $U_n$. Therefore every tail union ${\bigcup }_{n\ge m}{W}_n(T)$ is dense open, and $L_T=2^{\omega }\setminus {\mathrm{lim\hspace{0.17em}sup}}_n{W}_n(T)$ is meagre. Its membership relation is Borel in $(T,y)$ because each $W_n(T)$ is a finite clopen intersection with Borel dependence on $T$.

Code the Borel slalom parameter space in a Cantor parameter space via [F4], extending $L_T$ by empty sections outside its coded domain. Apply [F2] to get Borel closed nowhere-dense codes $F_j(T)$ with $L_T\subseteq {\bigcup }_j{F}_j(T)$. Set $v(T)(n)$ to be the least $m$ such that $S_m^n\cap {\bigcup }_{j\le n}{F}_j(T)=\mathrm{\varnothing }\mathrm{.}$ The complement of that finite union is dense open, so [F1] gives such an $m$. For a Borel-coded closed set, disjointness from a fixed clopen set is a Borel finite-subcover test on its open complement; hence $v$ is Borel. If $y\in L_T$, then $y\in F_j(T)$ for some $j$, and it misses $S_{v(T)(n)}^n$ whenever $n\ge j$. Thus $L_T\subseteq M_{v(T)}^{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$.

If $u(f){\subseteq }^{\ast }T$, then $f(n)\in T(n)$ eventually and $W_n(T)\subseteq S_{f(n)}^n$ eventually. Hence ${\mathrm{lim\hspace{0.17em}sup}}_n{W}_n(T)\subseteq {\mathrm{lim\hspace{0.17em}sup}}_n{S}_{f(n)}^n$ and, on taking complements, $M_f^{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}\subseteq L_T\subseteq M_{v(T)}^{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$. This is the required Borel morphism. The special family is cofinal by step 1.2, so it is an eligible cofinal master family for ideal inequalities. ∎