Local Definability of $\mathsf{H}\mathsf{O}\mathsf{D}$ in $L(\mathbb{R})$

We show that in $L(\mathbb{R})$, assuming large cardinals, $\mathsf{H}\mathsf{O}\mathsf{D}\parallel {\eta }^{+\mathsf{H}\mathsf{O}\mathsf{D}}$ is locally definable from $\mathsf{H}\mathsf{O}\mathsf{D}\parallel \eta$ for all $\mathsf{H}\mathsf{O}\mathsf{D}$-cardinals $\eta \in [{\mathit{\delta }}_1^2,\mathrm{\Theta })$. This is a further elaboration of the statement “${\mathsf{H}\mathsf{O}\mathsf{D}}^{L(\mathbb{R})}$ is a core model below $\mathrm{\Theta }$” made by John Steel.