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 ($m_e, m_\mu, m_\tau$) and the termination of the fermion spectrum at three generations ($N_{\text{gen}} = 3$) are treated as empirical inputs governed by arbitrary Yukawa couplings. This paper presents a self-contained derivation of $m_e$, $m_\mu/m_e$, $m_\tau/m_e$, and $N_{\text{gen}} = 3$ 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 ($T_{\text{CMB}} = 2.72548 \pm 0.00057\text{ K}$). Extremizing shear stress on Mohr's Circle ($\beta = \pi/4$), integrating the cycle-averaged toroidal arc lengths, and applying Kolmogorov-Arnold-Moser (KAM) Diophantine stability ($W_\phi = \phi^{-2}$) across the $N_{\text{cym}} = D\cdot|I_h| = 360$ continuous Chladni vector wave modes establishes the inviscid Spindle Torus baseline $\Theta_\phi = 180(3-\sqrt{5}) \approx 137.507764$. Projecting the three-dimensional Stokes compressible dilation coefficient ($-\frac{2}{3}$) onto the $\beta = \pi/4$ helix across terminal cycle closure yields the analytical coupling $\alpha^{-1}_{\text{dil}} = 180(3-\sqrt{5}) - \sqrt{2}/3 \approx 137.03636$ ($+2.63\text{ ppm}$ from CODATA 2022) and an Asymptotic Suction Boundary Layer (ASBL) displacement thickness of $(\delta^*/r)_{\text{dil}} \approx 0.0034400$. Confining a stationary particle vortex within a closed three-dimensional spherical horizon ($\Gamma = 1$) requires supporting the isotropic acoustic radiation-pressure enthalpy ratio ($h_E/\rho_E = 1 + 1/D = \frac{4}{3}$), fixing the confined core displacement at $\delta^*_{\text{total}}/r = \frac{7}{3}(\delta^*/r)$ and the per-axis core impedance at $W_{e,\text{dil}} \approx 136.41283$. Evaluating the four-dimensional phase-space Jacobian determinant ($\det(W_e\delta^\mu_{\ \nu}) = W_e^4$) across a closed $2\pi$ velocity-circulation loop predicts the electron rest mass directly from $T_{\text{CMB}}$ as $m_{e,\text{pred}} = 0.5109962 \pm 0.0001069\text{ MeV}/c^2$ ($-5.5\text{ ppm}$ / $-0.03\sigma$ from CODATA 2022), while preserving the intensive saturation ceiling ($\rho_E = \frac{3}{2}\hbar f_e/r_e^3 \approx 8.73 \times 10^{29}\text{ J/m}^3$) mandates linear topological volume scaling ($V_n \propto E_n$). Evaluating nested four-dimensional harmonic shells ($k^4$) with alternating chirality ($\chi_k = (-1)^{k-1}$), Mohr interfacial shear coupling ($\frac{1}{2} \times \frac{3}{7} = \frac{3}{14}$), and pairwise counter-rotating shell compression ($\binom{n}{2}\frac{3}{2}\delta^*/r$) yields a unified master equation for all $n \in \{0, 1, 2\}$ that derives $m_\mu/m_e = 206.7682845$ ($+8.7\text{ ppb}$ / $+0.39\sigma$ from CODATA 2022) and $m_\tau/m_e = 3477.46$ ($1776.98\text{ MeV}/c^2$, within $+0.55\sigma$ / $+28\text{ ppm}$ of the PDG 2024 world average $1776.93 \pm 0.09\text{ MeV}/c^2$). Finally, the Chirikov resonance overlap criterion ($\frac{14}{3}\delta^*/r = 0.01607 > 1/3^4 = 0.01235$) forbids a fourth generation ($k \ge 3$).