Scalar Mechanics: The Geometric Origin of the Wavefunction

In the framework of Toroidal Scale Dynamics (TSD), fundamental particles are modeled not as point-like excitations, but as topological solitons arising from vibrations in a 5-dimensional toroidal manifold. Building on the geometric interpretations established in previous works, we identify the quantum wavefunction as the amplitude profile of a physical 5D soliton propagating along the scalar dimension (w). We formally derive the Schrödinger equation as the non-relativistic approximation of the manifold's Torsional Vibration equation, where "torsional" refers to twists in the poloidal coordinate induced by the manifold's dual rotations. Furthermore, we solve for the geometric stability limit of these solitons, predicting an Objective Collapse Time for superpositions. Unlike the Penrose-Diósi model, which relies on standard gravitational self-energy, TSD predicts a collapse timescale dependent on the Woodward Constant (W ~ 10^41), offering a distinct mass-scaling signature (t proportional to 1/m) that is falsifiable by current macroscopic superposition experiments.