A premouse inheriting strong cardinals from $V$

Abstract

We identify a premouse inner model $L[\mathbb{E}]$, such that for any coarsely iterable background universe $R$ modelling $\mathrm{ZFC}$, $L[\mathbb{E}]^R$ is a proper class premouse of $R$ inheriting all strong and Woodin cardinals from $R$. Moreover, for each $\alpha\in\mathrm{OR}$, $L[\mathbb{E}]^R|\alpha$ is $(\omega,\alpha)$-iterable, via iteration trees which lift to coarse iteration trees on $R$. We prove that $(k+1)$-condensation follows from $(k+1)$-solidity together with $(k,\omega_1+1)$-iterability (that is, roughly, iterability with respect to normal trees). We also prove that a slight weakening of $(k+1)$-condensation follows from $(k,\omega_1+1)$-iterability (without the $(k+1)$-solidity hypothesis). The results depend on the theory of generalizations of bicephali, which we also develop.