K₁-invariants in the mod p cohomology of U (3) arithmetic manifolds

Let F/F^+ be a CM extension and H/₅^+ a definite unitary group in three variables that splits over F. We describe Hecke isotypic components of mod p algebraic modular forms on H at first principal congruence level at p and "minimal" level away from p in terms of the restrictions of the associated Galois representation to decomposition groups at p when these restrictions are tame and sufficiently generic. This confirms an expectation of local-global compatibility in the mod p Langlands program. To prove our result, we develop a local model theory for multitype deformation rings and new methods to work with patched modules that are not free over their scheme-theoretic support.