  In this work, we have implemented a detailed physical model of galaxy
chemical enrichment into the ${\it Astraeus}$ (semi-numerical rAdiative
tranSfer coupling of galaxy formaTion and Reionization in N-body dark matter
simUlationS) framework which couples galaxy formation and reionization in the
first billion years. Simulating galaxies spanning over 2.5 orders of magnitude
in halo mass with $M_h \sim 10^{8.9-11.5} M_\odot$ ($M_h \sim 10^{8.9-12.8}
M_\odot$) at $z \sim 10 ~ (5)$, we find: (i) smooth-accretion of metal-poor gas
from the intergalactic medium (IGM) plays a key role in diluting the
interstellar medium (ISM) metallicity which is effectively restored due to
self-enrichment from star formation; (ii) a redshift averaged gas-mass loading
factor that depends on the stellar mass as $\eta_g \approx 1.38 ({M_*}/{10^{10}
M_\odot})^{-0.43}$; (iii) the mass-metallicity relation is already in place at
$z \sim 10$ and shows effectively no redshift evolution down to $z \sim 5$;
(iv) for a given stellar mass, the metallicity decreases with an increase in
the star formation rate (SFR); (v) the key properties of the gas-phase
metallicity (in units of 12+log(O/H), stellar mass, SFR and redshift are linked
through a high-redshift fundamental plane of metallicity (HFPZ) for which we
provide a functional form; (vi) the mass-metallicity-SFR relations are
effectively independent of the reionization radiative feedback model for $M_*
\geq 10^{6.5} M_\odot$ galaxies; (vii) while low-mass galaxies ($M_h \leq 10^9
M_\odot$) are the key contributors to the metal budget of the IGM at early
times, higher mass halos provide about 50% of the metal budget at
lower-redshifts.
