Derivation of the Charged-Lepton Mass Spectrum and the Three-Generation Limit in a Compressible Superfluid

In the Standard Model of particle physics, the rest masses of the charged leptons (mₑ, m_μ, m_τ) and the termination of the fermion spectrum at three generations (Ngen = 3) are treated as empirical inputs governed by arbitrary Yukawa couplings. This paper presents a self-contained derivation of mₑ, m_μ/mₑ, m_τ/mₑ, and Ngen = 3 from the continuum mechanics of a three-dimensional ($D = 3$) compressible Two-Fluid Bose-Einstein Condensate (BEC) operating in steady-state thermodynamic equilibrium at the Cosmic Microwave Background baseline (TCMB = 2.72548 ± 0.00057 K). Extremizing shear stress on Mohr's Circle (β = π/4), integrating the cycle-averaged toroidal arc lengths, and applying Kolmogorov-Arnold-Moser (KAM) Diophantine stability (W_φ = φ⁻²) across the Ncym = D·|Iₕ| = 360 continuous Chladni vector wave modes establishes the inviscid Spindle Torus baseline Θ_φ = 180(3-√5) ≈ 137.507764. Projecting the three-dimensional Stokes compressible dilation coefficient (-2/3) onto the β = π/4 helix across terminal cycle closure yields the analytical coupling α⁻¹dil = 180(3-√5) - √2/3 ≈ 137.03636 (+2.63 ppm from CODATA 2022) and an Asymptotic Suction Boundary Layer (ASBL) displacement thickness of (δ^*/r)dil ≈ 0.0034400. Confining a stationary particle vortex within a closed three-dimensional spherical horizon (Γ = 1) requires supporting the isotropic acoustic radiation-pressure enthalpy ratio (hE/ρE = 1 + 1/D = 4/3), fixing the confined core displacement at δ^*ₜₒₜₐₗ/r = 7/3(δ^*/r) and the per-axis core impedance at We,dil ≈ 136.41283. Evaluating the four-dimensional phase-space Jacobian determinant ((Wₑδ^μ\ ν) = Wₑ⁴) across a closed 2π velocity-circulation loop predicts the electron rest mass directly from TCMB as me,pred = 0.5109962 ± 0.0001069 MeV/c² (-5.5 ppm / -0.03σ from CODATA 2022), while preserving the intensive saturation ceiling (ρE = 3/2 fₑ/rₑ³ ≈ 8.73 × 10²⁹ J/m³) mandates linear topological volume scaling (Vₙ ∝ Eₙ). Evaluating nested four-dimensional harmonic shells (k⁴) with alternating chirality (χₖ = (-1)ᵏ⁻¹), Mohr interfacial shear coupling (1/2 × 3/7 = 3/14), and pairwise counter-rotating shell compression (n23/2δ^*/r) yields a unified master equation for all n ∈ \0, 1, 2\ that derives m_μ/mₑ = 206.7682845 (+8.7 ppb / +0.39σ from CODATA 2022) and m_τ/mₑ = 3477.46 (1776.98 MeV/c², within +0.55σ / +28 ppm of the PDG 2024 world average 1776.93 ± 0.09 MeV/c²). Finally, the Chirikov resonance overlap criterion (14/3δ^*/r = 0.01607 > 1/3⁴ = 0.01235) forbids a fourth generation (k ≥ 3).