  Radiative Dirac neutrino masses and their mixing are linked to dark matter
through the non-Abelian discrete symmetry $A_5$ of the 4-dimensional pentatope,
softly broken to $A_4$ of the 3-dimensional tetrahedron. This unifying
understanding of neutrino family structure from dark matter is made possible
through the interplay of gauge symmetry, renormalizable Lagrangian field
theory, and softly broken discrete symmetries. Dark neutral fermions are
produced through Higgs decay.
