  We consider CP asymmetries in neutral $B$ meson decays to $\eta' K_S$, $\phi
K_S$, and $K^+ K^- K_S$. We use SU(3) relations to estimate or bound the
contributions to these amplitudes proportional to $V_{ub}^*V_{us}$. Such
contributions induce a deviation of the $S_f$ terms measured in these time
dependent CP asymmetries from that measured for $\psi K_S$. For the $K^+ K^-
K_S$ mode, we estimate the deviation to be of order 0.1. For the $\eta' K_S$
mode, we obtain an upper bound on this deviation of order 0.3. For the $\phi
K_S$ mode, we have to add a mild dynamical assumption to the SU(3) analysis due
to insufficient available data, yielding an upper bound of order 0.25. These
bounds may improve significantly with future data. While they are large at
present compared to the usually assumed Standard Model contribution, they are
obtained with minimal assumptions and hence provide more rigorous tests for new
physics. If measurements yield $|S_f-S_{\psi K}|$ that are much larger than our
bounds, it would make a convincing case for new physics.
