Geometric Derivation of Fine Structure Constant and Gauge Couplings via Lattice Elasticity

This final paper of the EMU trilogy addresses the origin of fundamental physical constants and the unification of forces. Having established gravity as lattice elasticity (Part II), we now identify electromagnetism as the transverse oscillation mode of the scalar lattice. We demonstrate that the vacuum possesses a finite **"Elastic Saturation Limit"**, beyond which the dielectric permittivity diverges, effectively screening charge. By defining the **Geometric Capacity** Ω of the vacuum as the sum of the squared dimensions of the fundamental interaction sectors, we derive the integer limit Ω = 102 + 62 + 12 = 137. We prove that the effective coupling constant α is bounded by the reciprocal of this structural capacity limit (α ≤ Ω−1), unifying quantum field theory constants with the discrete mechanics of spacetime.