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 Cosmic Mesh. The ideal value α⁻¹ = 135 emerges purely from the geometry of the tetrahedral lattice as the product of two pure numbers: α⁻¹ᵢdeal = 5 × 3³ = 5 × 27 = 135 where: 5 is the minimal torsional cluster (the tau cluster: one tetrahedron plus its four immediate neighbours sharing faces), the smallest unit that can sustain pure torsion; 3³ = 27 is the torsional dimensionality of space: three spatial dimensions, each with three torsional degrees of freedom (the three axes of the tetrahedron). Thus, the ideal impedance ratio is a pure count, independent of any system of units. The observed value α⁻¹ ≈ 137. 036 is then expressed as: α⁻¹ = 135 + δ, where δ = (2π φ) /5 ≈ 2. 034 where δ arises from the torsional frustration of the Mesh — the residual tension that remains because the lattice cannot fully relax. Each term has a geometric meaning: 2π: a complete torsional cycle (three 120° cracks), φ = (1+√5) /2: the golden ratio, governing fractal inheritance between coordination shells, 5: the minimal torsional cluster (the same tau cluster). The difference of 0. 002 with the experimental value (137. 036) is attributed to the perturbation introduced by the act of measurement itself — the observer's touch on the Mesh. The derivation contains no 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. As the Mesh relaxes toward the perfect 3‑sphere, α⁻¹ evolves asymptotically toward 135. The fine-structure constant is not a fundamental constant; it is a cosmic odometer. 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, tau cluster, 5x27 Version history: v1: General content improvement and acknowledgements. v1. 2: Better definition for the deviation from the ideal 135. v2: Diagonalization of stiffness matrix suppressed. Pure geometric origin for α⁻¹. v3: Redefined α⁻¹ = 135 as 5 × 3³ (minimal torsional cluster × torsional dimensionality of space), eliminating any reference to angles or degrees. Updated description to match the new geometric derivation.