The $K\to\pi\pi$ system is analyzed in the chiral limit within the
Standard Model. We discuss how to connect the short-distance running
in the $|\Delta S|=1$ case to the matrix-elements calculated in a
low-energy approximation in a scheme-independent fashion. We calculate
this correction and the resulting Wilson Coefficients. The matrix
elements are calculated to next-to-leading order in the $1/N_c$ expansion
and combined with the Wilson coefficients to calculate the two isospin
amplitudes and $\epsilon_K^\prime$. The $\Delta I=1/2$ rule is reproduced
within expected errors and we obtain a substantially larger value for
$\epsilon_K^\prime/\epsilon_K$ than most other analysises. We discuss the
reasons for this difference. We also suggest that the $X$-boson method is
an option for lattice QCD calculations.