The spectral Sullivan conjecture

We show that any map from an infinite loop space to a $p$-complete nilpotent finite dimensional space factors canonically through a union of $p$-adic tori. This is proven via bootstrapping from the case of $B\mathbb{Z}/p\mathbb{Z}$, which is the key case of the Sullivan conjecture proven by Miller. The main step in our proof is to show that the subcategory of spectra generated by the reduced suspension spectrum of $B\mathbb{Z}/p\mathbb{Z}$ under colimits and extensions agrees with that of a Moore spectrum.