Mechanical derivation of the Fine-Structure Constant , as the Torsional-to-Volumetric Impedance Ratio in a Discrete Elastic FCC Tetrahedral Lattice

Title: Mechanical derivation of the Fine-Structure Constant, as the Torsional-to-Volumetric Impedance Ratio in a Discrete Elastic FCC Tetrahedral Lattice Description / Abstract: We present a geometric derivation of the inverse fine-structure constant from the elastic properties of a discrete substrate: a Face-Centered Cubic (FCC) lattice of unbreakable elastic tetrahedra at the Planck scale. This lattice constitutes the microscopic structure of spacetime itself. The ideal value α⁻¹ = 135 emerges purely from the geometry of the tetrahedral lattice: 90° (the orthogonal reference of 3D space) plus 45° (the shear angle of tetrahedral links in FCC packing). This represents the ratio between torsional and volumetric stiffness in a perfectly relaxed Mesh. The observed value α⁻¹ ≈ 137.036 is then expressed as 135 + δ, where δ = 2πφ/5 ≈ 2.034 arises from the torsional frustration of the Mesh — the residual tension that remains because the lattice cannot fully relax. The golden ratio φ governs the fractal inheritance of torsional stress between coordination shells, and the factor 5 corresponds to the minimal torsional cluster (five tetrahedra). The difference of 0.002 with the experimental value is attributed to the perturbation introduced by the act of measurement itself. The derivation is independent of free parameters; once the lattice geometry is fixed, both numbers emerge from pure geometry and universal constants. This derivation eliminates the need to postulate α as a fundamental constant and reinterprets it as a direct consequence of the elasticity and topology of the discrete substrate. Keywords: fine-structure constant, mechanical derivation, torsional impedance, tetrahedral lattice, FCC lattice, discrete spacetime, alpha 1/137, cosmic mesh, topological determinism, Planck-scale elasticity, golden ratio, torsional frustration, Sigman cluster v1 update: General content improvement and acknowledges. v1.2 update: Better definition for the deviation from the ideal 135. v2 update: Diagonalization of stiffness matrix supressed. Pure geometric origin for α⁻¹.