Varsovian models $\omega$

For $n<\omega$, let $N_n$ be the minimal iterable proper class mouse $M$ such that $M\models$ "there are ordinals $\delta_0<\kappa_0<\ldots<\delta_{n-1}<\kappa_{n-1}$ 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 $*$-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 $\mathscr{V}_\omega$ which is a fully iterable strategy mouse with $\omega$ Woodin cardinals, closed under its strategy, and that the universe of $\mathscr{V}_\omega$ 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 $\mathscr{V}_\omega$, and is fully iterable in $M$ and in $\mathscr{V}_\omega$.