  The two neutral kaon states in nature, the $K_L$ (long-lived) and $K_S$
(short-lived) mesons, are the two time-evolution eigenstates of the $K^0 -
\overline{K^0}$ mixing system. The prediction of their mass difference $\Delta
m_K$ based on the Standard Model is an important goal of lattice QCD. In this
article, I will present preliminary results from a calculation of $\Delta m_K$
performed on an ensemble of $64^3 \times 128$ gauge configurations with inverse
lattice spacing of 2.36 GeV and physical quark masses. These new results come
from 2.5 times the Monte Carlo statistics used for the result presented in last
year's conference. Further discussion of the methods employed and the resulting
systematic errors will be given.
