  Simple, one-zone models for inhomogeneous chemical evolution of the Galactic
halo are used to predict the number fraction of zero-metallicity, Population
III stars, which currently is empirically estimated at < 4e-4. These analytic
models minimize the number of free parameters, highlighting the most
fundamental constraints on halo evolution. There are disagreements of at least
an order of magnitude between observations and predictions in limiting cases
for both homogeneous Simple Model and Simple Inhomogeneous Model (SIM). Hence,
this demonstrates a quantitative, unambiguous discrepancy in the observed and
expected fraction of Population III stars. We explore how the metallicity
distribution of the parent enrichment events f(z_0) drives the SIM and
predictions for the Population III fraction. The SIM shows that the
previously-identified "high halo" and "low halo" populations are consistent
with a continuous evolutionary progression, and therefore may not necessarily
be physically distinct populations. Possible evolutionary scenarios for halo
evolution are discussed within the SIM's simplistic one-zone paradigm.
  The values of z_0 depend strongly on metal dispersal processes, thus we
investigate interstellar mixing and mass transport, for the first time
explicitly incorporating this into a semi-analytic chemical evolution model.
Diffusion is found to be inefficient for all phases, including the hot phase,
of the interstellar medium (ISM): relevant diffusion lengths are 2 - 4 orders
of magnitude smaller than corresponding length scales for turbulent mixing.
Rough relations for dispersal processes are given for multiphase ISM. These
suggest that the expected low-metallicity threshold above zero is consistent
with the currently observed limit.
