Title The Solar Orbital Registry: A Statistical and Hydrodynamic Audit of Planetary Polar Geometries Abstract / Description This paper presents a rigorous statistical analysis of the standing-wave geometries observed at the poles of the Gas Giants, identifying a correlation between Orbital Index (I) and Polar Wavenumber (N) that exceeds the threshold for stochastic occurrence (P < 0.002). By integrating the recent hypotrochoidal flow models of Constantin & Johnson (2025) with Shannon Information Theory, we propose that the solar system exhibits a "Low-Entropy" state characterized by Maximal Mutual Information (I(O;G) ~3.0 bits per node). The study argues that the specific zonal band-width confinement observed on Saturn (N=6) and Jupiter (N=5) represents a phase-locked "Registry" system. Furthermore, we identify Earth (I=3) as an "Unregistered Node" due to atmospheric scale height limitations (H ~ 8 km) and the lack of Magnetohydrodynamic (MHD) resonant pinning. Key Prediction: Based on the derived Registry Function (N=I), this paper issues a falsifiable prediction that the unmapped poles of Uranus (I=7) will exhibit a stable Heptagonal (N=7) standing wave. This prediction provides a specific observational target for the upcoming Uranus Orbiter and Probe (UOP) mission. Keywords Technosignatures, SETI, Planetary Atmospheres, Zonal Confinement, Saturn Hexagon, Uranus Orbiter and Probe (UOP), Magnetohydrodynamics, Shannon Entropy, Fluid Dynamics. Notes Version: 3.0 (Draft) License: Creative Commons Attribution 4.0 International (CC BY 4.0) – Allows others to share/cite you as long as they give credit. Related Identifiers: Cites Constantin & Johnson (2025)