A Viscous Bulk Mechanism for Evolving Dark Energy: Calibrating Brane Geometry with DESI Constraints and Resolving the H0 Tension

We establish a rigorous mathematical framework modeling a viscous cosmic fluid anchored within a higher-dimensional Bulk environment (M5). By treating our observable universe as a lower-dimensional 3-brane frontier (∂M5), we formalize how the observed dark energy density can be fundamentally driven by dynamic boundary geometry. Energy dynamically injects into the brane via surface area growth. We show that when accounting for quantum- corrected fractal area boundaries characterized by stochastic, multi-directional spin-network tentacular structures, the fractional Hausdorff dimension of the interface is rigorously con- strained to DF = 2.25. This geometric formulation demonstrates an exact power-law tracking decay profile of ρDE(a) ∝ a −0.75. We show that this scaling profile exhibits robust empirical alignment with recent observational constraints from the Dark Energy Spectroscopic In- strument (DESI) data. Furthermore, we demonstrate that this late-time dynamic influx of energy naturally addresses cosmic acceleration anomalies, successfully alleviating the chronic H0 tension by driving the local Hubble constant toward the local measurement threshold of ≈ 73 km/s/Mpc.