Analysis of HOD for Admissible Structures

Let $n\ge 1$ and assume that there is a Woodin cardinal. For $x\in \mathbb{R}$ let ${\alpha }_x$ be the least $\beta$ such that $$L_{\beta }[x]\models {\mathrm{\Sigma }}_n\text{-KP}+\mathrm{\exists }\kappa (\mathrm{‘}\mathrm{‘}\kappa \text{ is inaccessible and }{\kappa }^{+}\text{ exists}\mathrm{"})\mathrm{.}$$ We adapt the analysis of ${\text{HOD}}^{L[x,G]}$ as a strategy mouse to $L_{{\alpha }_x}[x,G]$ for a cone of reals $x$. That is, we identify a mouse ${\mathcal{M}}^{\text{n-ad}}$ and define a class $H\subseteq L_{{\alpha }_x}[x,G]$ as a natural analogue of ${\text{HOD}}^{L[x,G]}\subseteq L[x,G]$, and show that $H=M_{\mathrm{\infty }}[{\mathrm{\Sigma }}_0]$, where $M_{\mathrm{\infty }}$ is an iterate of ${\mathcal{M}}^{\text{n-ad}}$ and ${\mathrm{\Sigma }}_0$ a fragment of its iteration strategy.