Using necessary and sufficient conditions for the saturation of the Araki-Lieb (AL) and related  inequalities which are realized holographically in black hole geometries, we prove that a typical pure state in a large N holographic CFT has two associated length scales that are determined only by the energy and conserved charges; one of them is a microscopic length scale L_{UV}, and the other is an infrared length scale L_{H} > L_{UV}.  Furthermore, the degrees of freedom between these two length scales are effectively split into two factors, one purifying the degrees of freedom at length scales smaller than L_{UV} and the other purifying those at length scales larger than L_{H}. Remarkably, the pure state factor involving the ultraviolet degrees of freedom is state independent. We also find precisely how deviations from this architecture are exponentially suppressed by studying quantum mutual information of two non-overlapping regions. Our results imply that all black holes can be isolated from the asymptotic region by a well-defined hypersurface which is the envelope of all critical entanglement wedges for which AL is saturated, and the split property emerges in the region bounded by this hypersurface and the horizon.