Varsovian models $\omega$

For $n<\omega$, let $N_n$ be the minimal iterable proper class mouse $M$ such that $M\models$ “there are ordinals ${\delta }_0<{\kappa }_0<\dots <{\delta }_{n-1}<{\kappa }_{n-1}$ such that each ${\delta }_i$ is a Woodin cardinal and each ${\kappa }_i$ is a strong cardinal”, and let $N_{\omega }$ be likewise, but with $M\models$ “there is an ordinal $\lambda$ which is a limit of Woodin cardinals and a limit of strong cardinals”. Under appropriate large cardinal hypotheses, Sargsyan and Schindler introduced and analysed in “Varsovian models I” the Varsovian model of $N_1$, and Sargsyan, Schindler and the author introduced and analysed in “Varsovian models II” the Varsovian model of $N_2$. We extend this to $N_{\omega }$, assuming that $\ast$-translation integrates routinely with the P-constructions of this paper (the write-up of which is yet to be completed). We show, under this assumption, that $N_{\omega }$ has a proper class inner model ${\mathcal{V}}_{\omega }$ which is a fully iterable strategy mouse with $\omega$ Woodin cardinals, closed under its strategy, and that the universe of ${\mathcal{V}}_{\omega }$ is the eventual generic HOD, and the mantle, of $N_{\omega }$. We also show, under the same assumption, that the core model $K$ of $N_{\omega }$ (which can be defined in a natural manner) is an iterate of $N_{\omega }$, is an inner model of ${\mathcal{V}}_{\omega }$, and is fully iterable in $M$ and in ${\mathcal{V}}_{\omega }$.