  Under hypotheses required for the Taylor-Wiles method, we prove for forms of
$U(3)$ which are compact at infinity that the lattice structure on upper alcove
algebraic vectors or on principal series types given by the $\lambda$-isotypic
part of completed cohomology is a local invariant of the Galois representation
attached to $\lambda$ when this Galois representation is residually irreducible
locally at places dividing $p$. As a crucial input, we establish corresponding
mod $p$ multiplicity one results. Our main innovation is the combination of
integral Hecke theory and the Taylor--Wiles method.
