A Unified Geometric Formulation of Dark Matter: The Manifold Scroll Hypothesis v4.0

We present the definitive formulation of the Manifold Scroll Hypothesis (MSH) v4.0 for dark matter, a geometric framework that redefines dark matter as inter-layer tension within a decompressing cosmic manifold. By introducing a position-dependent density function ρ_DM(z, θ, V_max, D) and integrating higher-order spherical harmonics, radial basis functions, and cross-terms, we achieve unprecedented accuracy in predicting dark matter masses. Testing against 300 real galaxies from the NASA/IPAC Extragalactic Database (NED) and SPARC catalog yields a mean absolute logarithmic error of 0.0011 (0.11%), with 99.3% of galaxies exhibiting errors below 1%. This work provides a seamless bridge between observational data and geometric cosmology, offering a unified solution to the dark matter problem. A step-by-step usage guide and numerical examples are provided for practical application.