FSI Rescattering in $B^\pm$ Decays via States with $η, η', ω$ and $φ$

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 $\eta'$.