Realizing an ${\text{AD}}^{+}$ model as a derived model of a premouse

Abstract

Assuming ${\text{AD}}^{+}+V=L(\mathcal{P}(\mathbb{R}))$, and assuming there is no proper class inner model containing all the reals that satisfies ${\text{AD}}_{\mathbb{R}}+\text{“}\theta \text{is regular”}$, we prove that in some forcing extension, either V is a derived model of a premouse or V embeds into a derived model of a premouse.