  Under renormalization, physical operators can mix with operators which vanish
by the equations of motion. Such operators cannot contribute to matrix elements
between physical states, but they contribute to operator mixing in
renormalization schemes which are defined at an off-shell momentum point, such
as the popular regularization-invariant schemes. For the first time, we
renormalize the lattice $\Delta S=1$ effective weak Hamiltonian taking into
account the most important such operator, $G_1 \propto \overline s \gamma_\nu
(1-\gamma_5) D_\mu G_{\mu\nu} d$. This removes an important systematic error in
calculations of weak matrix elements on the lattice.
