  Accurately understanding the equation of state (EOS) of high-density,
zero-temperature quark matter plays an essential role in constraining the
behavior of dense strongly interacting matter inside the cores of neutron
stars. In this Letter, we study the weak-coupling expansion of the EOS of cold
quark matter and derive the complete, gauge-invariant contributions from the
long-wavelength, dynamically screened gluonic sector at
next-to-next-to-next-to-leading order (N3LO) in the strong coupling constant
$\alpha_s$. This elevates the EOS result to the $O(\alpha_s^3 \ln \alpha_s)$
level, leaving only one unknown constant from the unscreened sector at N3LO,
and places it on par with its high-temperature counterpart from 2003.
