In fundamental physics, the fine-structure constant (α≈1u2044137.036) is traditionally introduced as an empirical dimensionless parameter that determines the strength of the electromagnetic interaction. In this paper, within the framework of the Local Energy Deficit Model (LEDM), an analytical derivation of the constant α is presented from the first principles of the mechanics of an isotropic elastic vacuum substrate. It is shown that the inverse quantity αu207b¹ represents a fundamental geometric-dynamical invariant of a three-dimensional medium defined by spatial normalization (4π), the number of shear degrees of freedom of the stress deviator (n_s=5), the spectral integral of zero-point fluctuations, and SU(2) spinor topology.A systematic analysis of physical phenomena associated with the number 137 is carried out, including the electron velocity on the first Bohr orbit, the fine structure of spectra, the convergence of quantum electrodynamics (QED) series, the anomalous magnetic moment of the electron (g - 2), and the critical Schwinger limit. On this basis, the physical and mechanical nature of the "running coupling constant" α(Q^2 ) is revealed as a nonlinear elastic response of the medium (Hooke's law) under strong deformations, and a mechanical-topological solution to the "Feynman paradox" for element 137 of the periodic table is proposed.