  We use the scatter in the stellar-to-halo mass relation to constrain galaxy
evolution models. If the efficiency of converting accreted baryons into stars
varies with time, halos of the same present-day mass but different formation
histories will have different z=0 galaxy stellar mass. This is one of the
sources of scatter in stellar mass at fixed halo mass, $\sigma_{\log M\ast}$.
For massive halos that undergo rapid quenching of star formation at z~2,
different mechanisms that trigger this quenching yield different values of
$\sigma_{\log M\ast}$. We use this framework to test various models in which
quenching begins after a galaxy crosses a threshold in one of the following
physical quantities: redshift, halo mass, stellar mass, and stellar-to-halo
mass ratio. Our model is highly idealized, with other sources of scatter likely
to arise as more physics is included. Thus, our test is whether a model can
produce scatter lower than observational bounds, leaving room for other
sources. Recent measurements find $\sigma_{\log M\ast}=0.16$ dex for 10^11 Msol
galaxies. Under the assumption that the threshold is constant with time, such a
low value of $\sigma_{\log M\ast}$ rules out all of these models with the
exception of quenching by a stellar mass treshold. Most physical quantities,
such as metallicity, will increase scatter if they are uncorrelated with halo
formation history. Thus, to decrease the scatter of a given model, galaxy
properties would correlate tightly with formation history, creating testable
predictions for their clustering. Understanding why $\sigma_{\log M\ast}$ is so
small may be key to understanding the physics of galaxy formation.
