  We develop a theoretical framework that extracts a deeper understanding of
galaxy formation from empirically-derived relations among galaxy properties by
extending the main-sequence integration method for computing galaxy star
formation histories. We properly account for scatter in the stellar mass-star
formation rate relation and the evolving fraction of passive systems and find
that the latter effect is almost solely responsible for the age distributions
among $z\sim0$ galaxies with stellar masses above $\sim 10^{10}\,{\rm
M_{\odot}}$. However, while we qualitatively agree with the observed median
stellar metallicity as a function of stellar mass, we attribute our inability
to reproduce the distribution in detail largely to a combination of imperfect
gas-phase metallicity and $\alpha$/Fe ratio calibrations. Our formalism will
benefit from new observational constraints and, in turn, improve
interpretations of future data by providing self-consistent star formation
histories for population synthesis modeling.
