Let $\pi$ be a $ℍ_$-recursive projection of $ℍ_$ into . What does $ℍ_{(, Domain(\pi))}$ contain?

Question

Let $\pi$ be a $ℍ_$-recursive projection of $ℍ_$ into (we know it exists by the classic results of Barwise in Admissible Sets and Structures (II.5.14, V.5.3, and VI.4.11/12)). The domain of , $_π$, cannot be hyperelementary over : if it were, then $D_π$ would be in $ℍ_$, and thus, by Σ-replacement, it would be the case that $ℍ_$ ∈ $ℍ_$.

Consider now the structure $(,_π)$.

1. Is it the case that $ℍ_ \in ℍ_{(,_π)}$? My intuition is as follows. $_π$ is in $ℍ_{(,_π)}$. But then, since $\forall x \in D_{\pi}\,.\,\exists y\,.\,x \in \pi(y)$, and $\Sigma$-replacement holds for $ℍ_{(,_π)}$, there is a $c = \{y \,:\,\exists x \in D_{\pi}\,.\,x \in \pi(y)\} \in ℍ_{(,_π)}$. But $c = ℍ_$ itself, and thus $ℍ_ \in ℍ_{(,_π)}$.
2. Suppose $\mathfrak{M}$ has a countable domain, and there is an $\mathfrak{M}$-hyperelementary mapping of $M$ into an isomorphic copy of the natural numbers $\mathcal{N}^{\mathfrak{M}}$ "living" in $\mathfrak{M}$. For simplicity, we could also take $\mathfrak{M} = \mathbb{N}$, the standard model of arithmetic. What's the ordinal of $ℍ_{(,_π)}$?.

Thank you very much for your help!

EDIT: I would like to add the following suggestions (though I cannot confirm if they are correct!).

• By theorem 3D.2 of Moschovakis Elementary Induction on Abstract Structures, for any $Q$ that is inductive non-hyperlementary on $\mathfrak{M}$, for any $P \subseteq M^n$, it is the case that $Q$ is hyperelementary on $(\mathfrak{M}, P)$ iff $\kappa^{\mathfrak{m}} < \kappa^{(\mathfrak{M}, P)}$. Clearly, since $D_{\pi}$ is $\Sigma$ on $\mathbb{H}YP_{\mathfrak{M}}$, $D_{\pi}$ is inductive on $\mathfrak{M}$, and cannot be hyperelementary on $\mathfrak{M}$ (otherwise it would be in $\mathbb{H}YP_{\mathfrak{M}}$, and we have seen that this is impossible). Thus, we can apply Moschovakis 3D.2 instantiating both $Q$ and $P$ with $D_{\pi}$. We get that $D_{\pi}$ is hyperelementary on $(\mathfrak{M}, D_{\pi})$ iff $\kappa^{\mathfrak{m}} < \kappa^{(\mathfrak{M}, D_{\pi})}$. Since $D_{\pi}$ is hyperelementary on $(\mathfrak{M}, D_{\pi})$, $\kappa^{\mathfrak{m}} < \kappa^{(\mathfrak{M}, D_{\pi})}$. So, the ordinal of $ℍ_{(,_π)}$ must be stricly above the ordinal of $\mathbb{H}YP_{\mathfrak{M}}$.
• This also implies that the inductive relations on $\mathfrak{M}$ are hyperelementary on $ℍ_{(,_π)}$. By result 6D.2 of Moschovakis, we conclude that for any $n \in \omega$, the collection $\mathbb{H}YP^n_{\mathfrak{M}} = \{X \subseteq M^n\,:\,X \text{ is hyperelementary on }\mathfrak{M}\}$ is a set in $\mathbb{H}YP_{(\mathfrak{M}, D_{\pi})}$ since it is inductive on $\mathfrak{M}$. If $\mathfrak{M}$ contains an isomorphic copy of $\omega$ in $M$, $\mathcal{N}^{\mathfrak{M}}$, and there is an $\mathfrak{M}$-hyperelementary mapping from $M$ to $\mathcal{N}^{\mathfrak{M}}$, then holds true in $\mathbb{H}YP_{(\mathfrak{M}, D_{\pi})}$ that $\forall n \in \mathcal{N}^{\mathfrak{M}}\,.\,\exists a\,.\,a = \mathbb{H}YP^n_{\mathfrak{M}}$. Consequently, the collection $b = \{a : \exists n \in \mathcal{N}^{\mathfrak{M}}\,.\,a = \mathbb{H}YP^n_{\mathfrak{M}}\}$ is a set in $\mathbb{H}YP_{(\mathfrak{M}, D_{\pi})}$. The union $\bigcup b$ belongs to $\mathbb{H}YP_{(\mathfrak{M}, D_{\pi})}$, too, and $\bigcup b = \mathbb{H}YP_{\mathfrak{M}}$ according to the definition in 6D of Moschovakis. Hence, when $\mathfrak{M}$ is strongly acceptable, $\mathbb{H}YP_{\mathfrak{M}} \in \mathbb{H}YP_{(\mathfrak{M}, D_{\pi})}$.

(Also posted on MO, link here https://mathoverflow.net/q/451576/509398)