  Recent observations made using the James Webb Space Telescope have identified
a number of high-redshift galaxies that are unexpectedly luminous. In light of
this, it is clear that a more detailed understanding of the high redshift,
pre-reionization universe is required for us to obtain the complete story of
galaxy formation. This study is the first in a series that seeks to tell the
story of galaxy formation at $z \gtrsim 10$ using a suite of large-scale
adaptive mesh refinement cosmological simulations. Our
machine-learning-accelerated surrogate model for Population III star formation
and feedback, StarNet, gives us an unprecedented ability to obtain physically
accurate, inhomogeneous chemical initial conditions for a statistically
significant number of galaxies. We find that of the 12,423 halos in the mass
range of $10^6\,\,M_\odot < M_\mathrm{vir} < 10^9\,\, M_\odot$ that form in our
fiducial simulation, $16\%$ are chemically enriched by Population III
supernovae by $z\sim12$. We then profile and compare various cooling processes
at the centers of halos, and find a complete absence of atomic cooling halos.
All of our halos with central cooling gas are dominated by H$_2$ cooling, metal
cooling, or a mixture of the two, even in the presence of a strong
H$_2$-photodissociating Lyman-Werner background. We also find that gas
accretion through the virial radius is not driven by cooling. We find that gas
virialization in halos with $M_\mathrm{vir}\gtrsim10^7\,\,M_\odot$ is supported
by bulk turbulent flows, and that thermal energy accounts for only a small
fraction of the total kinetic energy. Because of this, the mean gas temperature
is well below the virial temperature for these halos. We then compute the mass
of gas that is available for Population II star formation, and infer star
formation rates for each potential star-forming halo.
