  New results going beyond those obtained from isospin and flavor symmetry and
subject to clear experimental tests are obtained for effects of FSI in $B^\pm$
decays to charmless strange final states containing neutral flavor-mixed mesons
like $\omega$, $\phi$, $\eta$ and $\eta'$. The most general strong-interaction
diagrams containing arbitrary numbers of quarks and gluons are included with
the assumptions that any $q \bar q$ pair created by gluons must be a flavor
singlet, and that there are no hairpin diagrams in which a final meson contains
a $q \bar q$ pair from the same gluon vertex. The smallness of $K^- \eta$
suggests that it might have a large CP violation. A sum rule is derived to test
whether the large $K^- \eta'$ requires the addition of an additional glueball
or charm admixture. Further analysis from $D_s$ decay systematics supports this
picture of FSI and raises questions about charm admixture in the $\eta'$.
