The Geometric Necessity of the Fine-Structure Constant

The fine-structure constant α ≈ 1/137.036 is one of the most precisely measured yet least mechanically understood numbers in physics. This paper presents a self-contained geometric account of α⁻¹ ≈ 137.036 within the Push-Theory framework, which models the vacuum as a discrete Hexagonal Close-Packed (HCP) Planck-scale lattice. All derivations are contained in this paper; no companion papers are required. Three independent lines of reasoning converge on the integer N = 137: I. HCP Closure Geometry: a stable Möbius loop in the HCP lattice follows a 6-fold hexagonal path of 6×23 = 138 voxels. An intrinsic chirality angle δ = 0.0396° (derived from the 12−1=11 symmetry-breaking mechanism in §3) causes the wavefront to reach its closure point 1 voxel early, giving N = 137. Labelled: motivated construction. II. Secant Projection: the 180° Möbius axial torsion distributed over N voxels generates a stretch factor sec(π/N) per voxel. Summing over all N voxels gives α⁻¹ = N·sec(π/N) = 137.036028 at N = 137, matching CODATA 2018 to 0.21 ppm with zero free parameters. Labelled: derived. III. Potential Well + Prime Immunity: two competing mechanical energies produce a potential minimum at N* ≈ 137.757. N = 138 is mechanically forbidden by destructive harmonic resonance with HCP lattice periods; N = 137, being an Eisenstein prime in the hexagonal lattice, carries harmonic immunity. Together these select 137 as the unique stable integer. Labelled: consistency check. Standing disclaimer: Push-Theory does not claim to derive α from pure first principles without any reference to the observed value. The claim is that N = 137 sits at a privileged intersection of three independent geometric constraints, and that this privileged position has a mechanical explanation within the HCP framework. The secant formula α⁻¹ = N·sec(π/N) is the strongest result: it contains zero free parameters.