Noninvertible Peccei-Quinn Symmetry and the Massless Quark Solution to the Strong $CP$ Problem

We consider theories of gauged quark flavor and identify noninvertible Peccei-Quinn symmetries arising from fractional instantons when the resulting gauge group has nontrivial global structure. Such symmetries exist solely because the standard model has the same number of generations as colors, $N_g =N_c$, which leads to a massless down-type quark solution to the strong $CP$ problem in an ultraviolet $SU(9)$ theory of quark color-flavor unification. We show how the Cabibbo-Kobayashi-Maskawa flavor structure and weak $CP$ violation can be generated without upsetting our solution.