Property F implies weak integrality for non-degenerate pivotal braided fusion categories

We prove that every non-degenerate pivotal braided fusion category with property F is pseudo-unitary and weakly integral. In particular, every modular category with property F is weakly integral. Neither sphericality nor pseudo-unitarity is assumed. Passing to the Drinfeld centre extends the result to degenerate categories whose dimensions are positive on the intersection of the Müger centre with the adjoint subcategory; these include every braided fusion category with a pseudo-unitary Galois conjugate. Physically, the result connects the arithmetic of topological orders to the computational power of anyon braiding: non-integer squared total quantum dimension guarantees infinite braid group images, a necessary condition for universal quantum computation.