  The bosonic topological transition (BTT) is a quantum critical point between
the bosonic symmetry protected topological phase and the trivial phase. In this
work, we derive a description of this transition in terms of compact quantum
electrodynamics (QED) with four fermion flavors ($N_f=4$). This allows us to
describe the transition in a lattice model with the maximal microscopic
symmetry: an internal SO(4) symmetry. Within a systematic renormalization group
analysis, we identify the critical point with the desired O(4) emergent
symmetry and all expected deformations. By lowering the microscopic symmetry we
recover the previous $N_f=2$ non-compact QED description of the BTT. Finally,
by merging two BTTs we recover a previously discussed theory of symmetric mass
generation, as an SU(2) quantum chromodynamics-Higgs theory with $N_f=4$
flavors of SU(2) fundamental fermions and one SU(2) fundamental Higgs boson.
This provides a consistency check on both theories.
