Non-Perturbative Derivation of the Fine Structure Constant and its Running Coupling via the Elastodynamics of a Saturated Micropolar Continuum

The Standard Model of particle physics treats fundamental interaction strengths, most notably the fine structure constant (αα), as axiomatic empirical inputs. While Quantum Electrodynamics (QED) achieves rigorous predictive precision through the perturbative expansion of virtual particle exchanges, the physical origin and specific magnitude of this dimensionless coupling remain open theoretical questions. Building upon a framework that models fundamental particles as topologically stable solitons within a cubic-quintic Cosserat-Korteweg fluid, this paper develops a deterministic continuum mechanics foundation for the electromagnetic interaction. We present a MacCullagh-Cosserat isomorphism demonstrating that Maxwellian electrodynamics can be analytically recovered as the elastodynamic vector identities of a micropolar fluid. Within this formulation, vacuum parameters (Z0, ϵ0, μ0) emerge as the macroscopic torsional compliance and micro-rotational inertia of the substrate, while elementary electric charge is identified as the Brouwer topological degree of the fluid's micro-rotation field. Operating outside the formalism of infinite perturbative loops, we derive αα analytically as the hydrodynamic coupling efficiency of a Spin-1/2 defect kinetically locked to the vacuum's shear wave velocity. The inverse coupling constant (α−1) is calculated as a parameter-free geometric series comprising: the Bogomolny-Prasad-Sommerfield (BPS) topological impedance of the spatial manifold, the rheological boundary friction of the soliton's jamming phase, the dynamic aeroacoustic drag of zero-point fluctuations, and third-order Reynolds acoustic radiation pressure. Specifically, in evaluating the dynamic drag, we demonstrate that the maximum geometric interference limit of non-commutative transverse stress waves in the Cosserat substrate structurally recovers the Tsirelson bound (2√2) as a classical elastodynamic limit. Furthermore, we extend this hydrodynamic model to high-momentum transfers, showing that the QED Renormalization Group and the logarithmic running of the coupling constant (α (Q2) ) emerge deterministically from the acoustic depth-scaling laws of the soliton's density-stratified wake. This closed-form calculation yields a theoretical low-energy value of αSCK−1≈137. 03599920, reproducing the 2022 CODATA empirical standard to a relative precision exceeding 1 part in 10⁸. The residual difference (δ≈+2. 6×10^−8) is explicitly identified as the analytical horizon of continuous differential geometry, corresponding to the unmodeled fourth-order formation of discrete micro-turbulent vortex dipoles. Consequently, this framework indicates that the fine structure constant and its scale dependence are not free parameters, but structural geometric properties of a saturated material universe, providing a continuum elastodynamic foundation for quantum field theories.