  Scalar condensation, the well-established Higgs phenomenon, is the standard
paradigm for building up renormalizable gauge-invariant theories of massive
gauge bosons. In this short note, we demonstrate the uniqueness of the Higgs
vacuum state under the possible presence of fermion condensates in a
renormalizable U(1) model. In the same context, we explain why spontaneous
spin-statistics violation is technically not admitted in conventional Quantum
Field Theory.
