The Dirac operator arises naturally on $\mathbb{S}^1 \times \mathbb{S}^3 $ from the connection on the Lie group U(1)×SU(2) and maps spacetime rays into rays in the Lie algebra. We construct both simple harmonic and pulse solutions to the neutrino equations on $\mathbb{S}^1 \times \mathbb{S}^3 $ , classified by helicity and holonomy, using this map. Helicity is interpreted as the internal part of the Noether charge that arises from translation invariance; it is topologically quantized in integral multiples of a constant g (...) that converts a Lie-algebra phase shift into an action. The fundamental unit of helicity is associated with a full twist in u(1)×su(2) phase per global lightlike cycle. If we pass to the projective space ℝP1xℝP3, we get half-integral helicity. (shrink)