  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.
