The Cone Fixing the Reals of $\mathrm{HOD}^{M_{2n}(x)}$

Abstract

For a cone of reals $x$, the reals of $\mathrm{HOD}^{L[x]}$ are precisely the reals in $Q_3$, or equivalently, the set of reals in $M_1$, the minimal proper class mouse with a Woodin cardinal. We show that if $z$ is any real coding $M_1|δ^{+M_1}$, then $z$ is the base of such a cone (so $M_1^{\#}$ is such a base). We also establish a kind of optimality for this result, by identifying a real $x$ such that $M_1\subseteq L[x]$ but $\mathbb{R}\cap\mathrm{HOD}^{L[x]}\not\subseteq M_1$, although there is a base for the cone in a generic extension of $L[x]$. We prove an analogous result for the reals of $\mathrm{HOD}^{M_{n}(x)}$ for even $n$, as well as an analogous optimality result which holds for all $n$.