The internal structure of ${\mathrm{HOD}}^{L[x]}$ up to its Woodin

Abstract

Assume ${\mathbf{\Delta }}_3^1$-determinacy. It is shown that for any $x{\ge }_T{M}_1^{\#}$, ${\mathrm{HOD}}^{L[x]}$ is a model of GCH, and in fact, it is a Jensen-Steel core model up to ${\omega }_2^{L[x]}$.

withdrawn due to a mistake in Lemma 6.2