Scale-Dependent Dimensional Trapping: A Topologically Invariant Unified Framework for Galactic Dynamics, Quantum Leakage, and Cosmological Observables

Abstract

We present Scale-Dependent Dimensional Trapping, a scale-invariant geometric framework that unifies galactic dynamics, quantum gravity attenuation, and cosmological observables without requiring non-baryonic dark matter, dark energy constants, or fine-tuned parameters. By establishing a universal topological ceiling (U \equiv 1.0), gravitational field confinement is governed by a smooth scale-fraction trapping function \Phi(f). At macroscopic scales (f \ge 10^{-4}), gravity remains 100% trapped in 3D spacetime (\Phi \approx 1.0), satisfying all General Relativity precision tests while replicating flat galaxy rotation curves (\chi^2/N = 0.500 on SPARC catalog galaxy NGC 3198) and cluster lensing (\theta_E = 32.00''). At subatomic scales (f \to 10^{-36}), dimensional leakage attenuates G_{\text{eff}} to 10^{-36}, resolving the Hierarchy Problem and cooling primordial black hole Hawking temperatures to 122.7 K. Cosmological boundary conditions at U \equiv 1.0 yield singularity-free t=0 physics and inflationary CMB observables (n_s = 0.9663, r = 0.0033) in exact agreement with Planck and BICEP/Keck data.