Derivation of the Fine-Structure Constant via Geometric Frustration and Topological Self-Energy in Planck-Scale Discrete Vacuum

Derivation of the Fine-Structure Constant via Geometric Frustration and Topological Self-Energy in Planck-Scale Discrete Vacuum Abstract: The fine-structure constant (α ≈ 1/137), often regarded as "God's Number," has long lacked a geometric explanation from First Principles. Based on the hypothesis of Microscopic Reality as a Planck-Scale Discrete Vacuum, this paper attempts to explore a purely geometric and topological derivation mechanism. Utilizing rigid solutions from the Kissing Number Problem and the Tammes Problem, we demonstrate that the microscopic Ground State of the physical vacuum may follow Binary Icosahedral Group (2I) symmetry, and its Topological Coordination Number (Z) is theoretically inclined to be locked at Z = 12. To describe the Vacuum Pre-stress potentially caused by the inability of regular icosahedra to achieve Seamless Tessellation in three-dimensional Euclidean space, the concept of Geometric Frustration is introduced. Furthermore, we attempt to combine Self-Loop correction from Spectral Graph Theory to characterize Local Topological Inertia. Based on this model, an analytical formula for the inverse of the fine-structure constant is derived: α⁻¹ = 144 / Γ_geo + 1/Z where Γ_geo is the geometric frustration factor. The theoretical calculation yields 137.03572, with a Relative Error of approximately 2 × 10⁻⁶ compared to the CODATA 2018 recommended value. This high degree of consistency suggests that the fine-structure constant is highly likely a Projection of the Planck-scale discrete geometric structure in the Long-Wave Limit.