Recent determinations of the radial distributions of mono-metallicity populations (MMPs, i.e., stars in narrow bins in [Fe/H] within wider [α/Fe] ranges) by the SDSS-III/APOGEE DR12 survey cast doubts on the classical thin- and thick-disk dichotomy. The analysis of these observations led to the non-[α /Fe] enhanced populations splitting into MMPs with different surface densities according to their [Fe/H]. By contrast, [α /Fe] enhanced (i.e., old) populations show a homogeneous behavior. We analyze these results in the wider context of disk formation within non-isolated halos embedded in the Cosmic Web, resulting in a two-phase mass assembly. By performing hydrodynamical simulations in the context of the ΛCDM model, we have found that the two phases of halo mass assembly (an early fast phase, followed by a slow phase with low mass-assembly rates) are very relevant to determine the radial structure of MMP distributions, while radial mixing only plays a secondary role, depending on the coeval dynamical and/or destabilizing events. Indeed, while the frequent dynamical violent events occuring at high redshift remove metallicity gradients and imply efficient stellar mixing, the relatively quiescent dynamics after the transition keeps [Fe/H] gaseous gradients and prevents newly formed stars from suffering strong radial mixing. By linking the two-component disk concept with the two-phase halo mass-assembly scenario, our results set halo virialization (the event marking the transition from the fast to the slow phases) as the separating event that marks periods that are characterized by different physical conditions under which thick- and thin-disk stars were born.