This paper derives the mechanism of Nuclear Binding and the Strong Force from geometric first principles. We model nucleons as dynamic Spindle Tori undergoing radial "Breathing Mode" oscillations in the scalar dimension. We demonstrate that nuclear binding arises from the phase-locking of these breathing modes between adjacent nucleons, which minimizes scalar stress. Furthermore, we derive the Yukawa potential from the logarithmic geometry of the scalar manifold, showing that the "short range" of the Strong Force corresponds to the Woodward Radius--the scale at which the metric stiffens exponentially. This model successfully explains the stability of the nucleus and offers a geometric resolution to the Proton Radius Puzzle via the "Metric Cliff" effect.