  The prevalence of light element enhancement in the most metal-poor stars is
potentially an indication that the Milky Way has a metallicity floor for star
formation around $\sim$10$^{-3.5}$ Z$_{\odot}$. We propose that this
metallicity floor has its origins in metal-enriched star formation in the
minihalos present during the Galaxy's initial formation. To arrive at this
conclusion, we analyze a cosmological radiation hydrodynamics simulation that
follows the concurrent evolution of multiple Population III star-forming
minihalos. The main driver for the central gas within minihalos is the steady
increase in hydrostatic pressure as the halos grow. We incorporate this insight
into a hybrid one-zone model that switches between pressure-confined and
modified free-fall modes to evolve the gas density with time according to the
ratio of the free-fall and sound-crossing timescales. This model is able to
accurately reproduce the density and chemo-thermal evolution of the gas in each
of the simulated minihalos up to the point of runaway collapse. We then use
this model to investigate how the gas responds to the absence of H$_{2}$.
Without metals, the central gas becomes increasingly stable against collapse as
it grows to the atomic cooling limit. When metals are present in the halo at a
level of $\sim$10$^{-3.7}$ Z$_{\odot}$, however, the gas is able to achieve
gravitational instability while still in the minihalo regime. Thus, we conclude
that the Galaxy's metallicity floor is set by the balance within minihalos of
gas-phase metal cooling and the radiation background associated with its early
formation environment.
