Scale Geometry of Complex Systems: Formalism and Empirical Verification of Local Λb–Πb Closure in Atmospheric Data

This paper proposes a geometric framework in which scale (resolution, coarse-graining level) is treated as a formal coordinate of a Hilbert bundle — a fiber bundle whose fibers are Hilbert spaces indexed by scale parameter μ. A gauge connection on this bundle generates a gauge-invariant scalar Λ constructed from the curvature of the connection. The central hypothesis is a local relation Λ ~ Π, linking the geometric scalar to the physical energy flux Π across scales. The framework is derived from first principles combining fiber bundle geometry, GKSL open quantum systems, CPTP maps for coarse-graining, and holographic renormalization group arguments. The theory is tested in two settings: (1) a toy u(2) quantum model (experiment T20), where the predicted Λ–Π relation is confirmed with R² = 0.727; (2) ERA5 reanalysis atmospheric data(experiment A15, Jan–Mar 2017, 6-hourly, Western Pacific), where the relation holds in the inertial range with R² = 0.520 (p = 0.008). Additional experiments probe GKSL-CPTP memory effects (A05.R5–R6), scale-window sensitivity (A12–A14), and falsification controls (M4). Self-organized criticality signatures and fractal emergence are also examined (F1–F6). The core idea here is to get a better model architecture for complex systems in environment analysis and nature taxonomy that could allow for better climate description and prediction in the future. Code and experiment logs are available at github.com/Theclimateguy/Shroedinger (release v3.0).