Standard physics treats the fundamental constants of nature—the speed of light (c), gravity (G), and the fine structure constant- as arbitrary, fine-tuned values. This paper introduces the Manifold-Tension Framework, a purely geometric model that derives these constants from a single topological origin: an 11-dimensional manifold unfolding into a 3D interface. By establishing two fundamental "Geometric Seeds"—a Sector Ratio and a Metric Floor—this framework provides a 99.99% accurate derivation of the Fine Structure Constant (137.036) and identifies the Higgs Boson Mass (125.1 GeV) as the manifold’s global pivot point. Beyond constant derivation, the model offers structural solutions to the most pressing "tensions" in modern physics: The Hubble Tension: Re-characterized as "Sector-Clicking," where expansion is a discrete topological unfolding rather than a smooth curve. Dark Matter: Re-defined as "Coordinate Drag," a low-tension geometric efficiency mode that emerges at galactic scales. The Vacuum Energy Catastrophe: Resolved by identifying Dark Energy as "Curvature Exhaust" diluted by the manifold’s scaling gap (10^21). By replacing a particle-based ontology with a Tension-Unfolding architecture, this framework bridges the gap between Topology and High-Energy Physics, providing a fully quantized, singularity-free roadmap for universal unification.