Null-to-meagre Tukey inequalities

For each distinct $A\in {\mathcal{M}}_0$, use [F3] to choose one special meagre code $f_A$ with $A=M_{f_A}^{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$; for each distinct $B\in {\mathcal{N}}_0$, choose one null master code $g_B$ with $B=N_{g_B}$. Define maps on the actual cofinal set families by $U(A)=N_{u_N(u_M(f_A))}$ and $V(B)=M_{v_M(v_N(g_B))}^{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$. If $U(A)\subseteq B$, then $N_{u_N(u_M(f_A))}\subseteq N_{g_B}$, so [F2] gives $u_M(f_A){\subseteq }^{\ast }{v}_N(g_B)$; [F1] then gives $A=M_{f_A}^{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}\subseteq M_{v_M(v_N(g_B))}^{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}=V(B)$. Thus $U,V$ witness ${\mathcal{M}}_0⪯{\mathcal{N}}_0$ for the actual inclusion-cofinal ideal subfamilies in the direction required by [F3]. Duplicate codes cause no ambiguity because the representatives were fixed once by AC.

Apply [F3] on Cantor space, using the cofinality in [F1] and [F2]. It gives $\mathrm{add}({\mathcal{N}}_{2^{\omega }})\le \mathrm{add}({\mathcal{M}}_{2^{\omega }})$ and $\mathrm{cof}({\mathcal{M}}_{2^{\omega }})\le \mathrm{cof}({\mathcal{N}}_{2^{\omega }})$. AC selects the code representatives in step 1.1 and the cofinal master covers in the extension step of [F3]; the underlying coded maps remain Borel and use fixed least-code choices. Transfer both values to $\mathbb{R}$ by [F4]. ∎