Each real layer has a ground-model cardinal bound

The set $P_{<m}=\{p\upharpoonright m:p\in P\}$ has ground cardinal at most ${\mathrm{\aleph }}_m^V$: its elements are finite functions using ordinals below the finitely many cardinals ${\mathrm{\aleph }}_n^V$ for $n<m$ (and for $m=0$ it is a singleton). Every regular open generated by $P_{<m}$ is a subset of $P_{<m}$, so F2 gives $\mathrm{\mid }{B}_m{\mathrm{\mid }}^V\le 2^{{\mathrm{\aleph }}_m^V}={\mathrm{\aleph }}_{m+1}^V$. This deliberately coarse estimate covers $m=0$ uniformly.

By F1, a member of $S_m$ is coded by a function $\omega \to B_m$. In ground ZFC and GCH, step 1.1 gives $\mathrm{\mid }{S}_m{\mathrm{\mid }}^V\le (2^{{\mathrm{\aleph }}_m^V}{)}^{{\mathrm{\aleph }}_0}=2^{{\mathrm{\aleph }}_m^V}={\mathrm{\aleph }}_{m+1}^V$. Since the constantly-zero name belongs to $S_m$, ground Choice supplies a fixed surjection $e_m:{\mathrm{\aleph }}_{m+1}^V\twoheadrightarrow S_m$.

Form the canonical name ${\dot{e}}_m^{\ast }=\{⟨⟨\check{\xi },\dot{x}{⟩}^{\bullet },1_B⟩:\xi <{\mathrm{\aleph }}_{m+1}^V\text{ and }\dot{x}=e_m(\xi )\}$. Every value name belongs to $S_m$ and is fixed by $H_m$, so $H_m$ fixes ${\dot{e}}_m^{\ast }$ and all its subnames. Thus it is hereditarily symmetric. Its interpretation is the function $\xi \mapsto e_m(\xi {)}_G$, whose range is exactly $R_m$ by F1. Hence $N$ contains the required surjection.

The ground use of Choice and GCH occurred only in steps 1.1–2.1 to obtain the single coded enumeration $e_m$. Step 3.1 puts its interpretation in $N$ without choosing enumerations for a family of arbitrary sets. This proves the asserted internal bound.