The Geometric Necessity of the Fine-Structure Constant

Modern quantum electrodynamics treats the fine-structure constant α as an empirical coupling constant without mechanical explanation. This paper presents a purely mechanical derivation of alpha^-1 ~137.036 within the Push-Theory framework, which models the vacuum as a discrete, pressurised hexagonal 6D Planck-lattice (the Matrix). This paper shows that three complementary lines of reasoning converge on the same integer N = 137: • The Secant Projection: the 180° Möbius axial torsion, distributed over N discrete voxels, generates a geometric stretch factor of sec(180°/N) per voxel. Summing over all N voxels yields alpha^-1 = N · sec(pi_rad/N), which evaluates to 137.036028 at N = 137 — matching the CODATA value to within 0.21 ppm. • The Potential Energy Well: a Möbius loop of N voxels is subject to two competing mechanical energies — torsional strain energy (favouring large N) and ZPE volumetric displacement cost (favouring small N). The minimum of the resulting potential V(N) is uniquely located by the primordial ZPE pressure Lambda_prim, which is independently derivable from the alpha-tail geometric protrusion δ via the relation G = c²/(Lambda_local · delta²). • Prime-Resonance Immunity: N = 137, being an Eisenstein prime in the hexagonal lattice, offers natural harmonic immunity, while neighbouring composite numbers are mechanically disfavoured. Although N=137 is initially motivated by the proximity of alpha^-1 tot this integer, the model shows that this value sits at a privileged intersection of geometric projection, energetic minimisation, and topological stability. I also derive a purely geometric amplification factor between local and primordial ZPE pressure and discuss implications for the vacuum catastrophe and baryonic matter abundance.