  We present the first post-cosmic-microwave-background early-Universe
observational constraints on $\sigma_8$, $\Omega_{\rm m}$, mean galaxy
star-forming efficiency and galaxy UV magnitude scatter at redshifts $z =
4-10$. We perform a simultaneous 11-parameter cosmology and star-formation
physics fit using the new code GalaxyMC, with redshift $z>4$ galaxy UV
luminosity and correlation function data. Consistent with previous studies, we
find evidence for redshift-independent star formation physics, regulated by
halo assembly. For a flat $\Lambda$CDM universe with a low-redshift Hubble
constant and a Type Ia supernovae $\Omega_{\rm m}$ prior, we constrain
$\sigma_8 = 0.81 \pm 0.03$, and a mean star-forming efficiency peaking at
$\log_{10} {\rm SFE} = -[(0.09 \pm 0.20) + (0.58 \pm 0.29) \times \log_{10}
(1+z)]$ for halo mass $\log_{10} M_{\rm p} / h^{-1} M_{\odot} = 11.48 \pm
0.09$. The suppression of star formation due to feedback is given by a double
power law in halo mass with indices $\alpha = 0.56 \pm 0.08, \beta = -1.03 \pm
0.07$. The scatter in galaxy UV magnitude for fixed halo mass is $\sigma_M =
0.56 \pm 0.08$. Without a prior on $\Omega_{\rm m}$ we obtain $\sigma_8 = 0.78
\pm 0.06$, $\Omega_{\rm m} = 0.33 \pm 0.07$ and at most $1\sigma$ differences
in all other parameter values. Our best-fit galaxy luminosity functions yield a
reionization optical depth $\tau \approx 0.048$, consistent with the Planck
2018 value.
