  The decay $B^0 \to K^0 \pi^0$, dominated by a $b \to s$ penguin amplitude,
holds the potential for exhibiting new physics in this amplitude. In the pure
QCD penguin limit one expects $\ckp = 0$ and $\skp = \sin 2 \beta$ for the
coefficients of $\cos \Delta m t$ and $\sin \Delta m t$ in the time-dependent
CP asymmetry. Small non-penguin contributions lead to corrections to these
expressions which are calculated in terms of isospin-related $B\to K\pi$ rates
and asymmetries, using information about strong phases from experiment. We
study the prospects for incisive tests of the Standard Model through
examination of these corrections. We update a prediction $\ckp=0.15\pm 0.04$,
pointing out the sensitivity of a prediction $\skp\approx 1$ to the measured
branching ratio for $B^0\to K^0\pi^0$ and to other observables.
