CDUFD Supplementary Material VI: Formal Emergence of the Wallstrom Condition from Critical Topological Dynamics

The Wallstrom condition---the requirement that the phase of the wave function be single-valued up to integer multiples of 2---has been a foundational obstacle in all attempts to derive quantum mechanics from stochastic or hydrodynamic dynamics since Wallstrom's 1994 proof that the Madelung equations alone are insufficient to recover the Schrödinger equation. In this paper, we provide a complete derivation of the Wallstrom condition from the axioms of the Constraint-Dispersion Unified Framework (CDUFD). The derivation follows the CDUFD methodology of solution-space construction and natural constraint-driven reduction: we initialize the solution space at the electroweak critical point with both Ising (amplitude) and XY (phase) fluctuation modes as built-in degrees of freedom of the SO (8) -embedded order parameter; we then demonstrate that A3 critical dynamics and A4 topological quantization jointly eliminate Ising domain walls while preserving XY vortices through their critical lifetime disparity. The argument is built upon three robust pillars: (i) SO (8) group structure guarantees a non-vanishing XY coupling g 0; (ii) A4 topological quantization guarantees integer vortex winding numbers; (iii) the electroweak critical point provides a finite, large physical cutoff (=0. 0102) that prevents complete infrared suppression of the XY sector even in the dangerously irrelevant case. Together with the numerically testable premise that the effective XY coupling at this cutoff is strong enough to sustain a percolating vortex network, the surviving vortices, spanning the entire system via divergent correlation length, impose global phase quantization on all closed contours---precisely the Wallstrom condition. Combined with the previously established Madelung transformation from A2 Langevin dynamics, this completes the rigorous emergence of the full Schrödinger equation from CDUFD axioms. The sole remaining numerical premise is explicitly isolated as a verifiable open problem.