Derivation of the Fine-Structure Constant (α) via Principal Stress Equalization and Spindle Torus Geometry in a Compressible Superfluid

The fine-structure constant (α ≈ 1/137.036) is conventionally treated as an empirical coupling parameter in quantum electrodynamics lacking a mechanical derivation. This letter presents a deterministic, fluid-dynamic derivation of α⁻¹ by modeling the spatial manifold as a continuous, three-dimensional compressible superfluid operating in steady-state thermodynamic equilibrium. Governed by the Kovtun-Son-Starinets (KSS) lower viscosity bound of the ambient 2.7255 K thermal phonon baseline, the bulk continuum possesses an intrinsic dynamic acoustic viscosity (μA > 0), while volumetric compression at the vortex core is arrested by a Maximum Density Threshold (ρₘₐₓ) that enforces Total Internal Reflection (Γ = 1). Extremizing longitudinal-to-azimuthal shear transfer on Mohr's Circle forces the acoustic pilot wave into a 45^∘ (π/4 rad) diagonal helix. Applying centroid kinematics and Kolmogorov-Arnold-Moser (KAM) Diophantine stability along the minimal geodesic branch (W_φ = 1 - φ⁻¹ = φ⁻² ≈ 0.381966) proves that the manifold must self-intersect as a Spindle Torus ($R < r$). Projecting this winding fraction across the three-dimensional cymatic nodal invariant (Ncym = 360) establishes the inviscid baseline Θ_φ = 360φ⁻² ≈ 137.5078. Because the inner spindle core is saturated at ρₘₐₓ, the entrained acoustic boundary layer (δ^*/r) displaces outward. Evaluating this phase lag both via the -2/3sin(π/4) Cauchy compressible dilation projection (δ^*/r ≈ 0.3440%, α⁻¹ ≈ 137.0364) and via the three-dimensional spherical enthalpy confinement decomposition (7/3δ^*/r) of the electron-to-CMB impedance ratio yields δ^*/r = 0.3439% ± 0.0022% and α⁻¹ = 137.0365 ± 0.0030, matching observation within 0.16σ (3.5 ppm) of the empirical CMB temperature uncertainty.