Real Stueckelberg Quantum Geometry and Unbroken Supersymmetry in the Rotating Shallow Water Model

The topological properties of planetary fluids are typically analyzed by mapping classical fluidequations onto complex quantum mechanical models. Here we present a purely real, six-dimensional Stueckelberg quantum mechanical formulation of the rotating shallow water equations to demonstrate that these topological features are intrinsic to the classical kinematics itself. Operating entirely within R^6, we decouple the complex quantum geometric tensor into an independent, real Fubini-Study metric and a real antisymmetric Berry curvature. Our real-variable approach explicitly derives a topological magnetic monopole of charge C=2 and captures the inherentscale invariance of the fluid's geometry without the need for complexification. We suggest that continuous variations in the Coriolis parameter may dynamically model the deep-time planetary evolution of the Archean Earth, and we propose a laboratory rotating-tank experiment to physically measure this topological phase transition. Finally, we show that our real 6D formulation naturally maps to unbroken supersymmetric quantum mechanics. By identifying a purely real supercharge and calculating a fluid Witten index of W = -2, we advance a mathematically supported viewpoint that steady-state geostrophic weather patterns represent the unbroken, zero-energy supersymmetric ground states of the rotating fluid system. Consequently, the topological isolation of this vacuum naturally restricts the spectral flow across the equator, providing a theoretical explanation for the unidirectional eastward motion of equatorial boundary waves.