A Geometric Approach to the Relativistic Wallström Objection: Spin-1/2 from Covariant Fluid Regularity

This paper explores the extension of the stochastic coherence framework (Onsager-Machlup formalism) to the relativistic domain. In previous work, it was shown that phase quantization in the non-relativistic Schrödinger regime can be understood as a consequence of requiring the probability current to be a $C^\infty$ smooth vector field across nodal surfaces (resolving the Wallström objection). Here, we examine the analogous topological constraints in Minkowski spacetime using Spacetime Algebra (STA). By replacing the scalar phase with a Lorentz rotor field, the physical observables become the probability four-current and the spin tensor, both bilinear in the rotor via the $SU(2) \to SO(3)$ double covering. Through a variational derivation of the relativistic stochastic dynamics, we show that the generalized Hamilton-Jacobi equation constrains the local density behavior near nodal tubes. When combined with the requirement that the bilinear observables remain strictly $C^\infty$ regular, the internal holonomy of the rotor is forced to take values in integer multiples of $\pi$. The results suggest that intrinsic angular momentum quantized in half-integer units of $\hbar$ (spin-1/2) can emerge naturally as a geometric and topological necessity for covariant fluid stability, recovering the Dirac hydrodynamic system without a priori postulating spinor structure.