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 $α$/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.

20 pages, 6 figures, 1 table, MNRAS in press, supplementary tables available online