Derivation of the Neutrino Mass Eigenstates and the PMNS Mixing Matrix in a Compressible Superfluid Continuum

In the Standard Model of particle physics, the three neutrino mass eigenstates ($m_1, m_2, m_3$), the three Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing angles ($\theta_{12}, \theta_{23}, \theta_{13}$), and the leptonic CP-violating phase ($\delta_{\text{CP}}$) contain seven unconstrained empirical parameters requiring external mass-generation mechanisms. This paper presents a self-contained derivation of all seven quantities from the continuum mechanics of a three-dimensional ($D = 3$) compressible Two-Fluid Bose-Einstein Condensate (BEC) operating at the steady-state Cosmic Microwave Background baseline ($T_{\text{CMB}} = 2.72548 \pm 0.00057\text{ K}$). Acoustic perturbations below the four-dimensional phase-space Jacobian knotting threshold of the electron ($E < m_e c^2 = 2\pi k_B T_{\text{CMB}}W_e^4$) cannot form a closed Total Internal Reflection ($\Gamma = 1$) vortex sink and instead propagate as un-knotted longitudinal pilot waves with baseline mass-equivalence $m_1 = k_B T_{\text{CMB}}/c^2 = 0.23487\text{ meV}/c^2$. As this wave traverses spherical vortex boundaries governed by the fundamental overtone operator $\Lambda_1 = 1 + \frac{3}{2}\alpha^{-1} \approx 206.5540$ and an Asymptotic Suction Boundary Layer (ASBL) of radial displacement thickness $\delta^*/r \approx 0.00344$, its two-dimensional meridian cross-section ($\Lambda_1^{2/3}$) dressed by the central Spindle Torus overlap ($\frac{1}{3}\phi^{-4}$) and its three-dimensional volumetric mode ($\Lambda_1^1$) dressed by the dual confined boundary-layer sheath ($\Delta_{\text{excl}} = \frac{14}{3}\frac{\delta^*}{r} \approx 0.01607$) yield $m_2 = 8.5720\text{ meV}/c^2$ ($\Delta m^2_{21} = 7.342 \times 10^{-5}\text{ eV}^2$, $0.32\sigma$), $m_3 = 49.292\text{ meV}/c^2$ ($\Delta m^2_{31} = 2.430 \times 10^{-3}\text{ eV}^2$, $0.70\sigma$), and normal mass ordering ($\sum m_\nu = 0.05810\text{ eV}/c^2$). Evaluating the geometric cross-coupling projections across the Spindle Torus ($W_\phi = \phi^{-2}$) derives $\sin^2\theta_{13} = \phi^{-8}(1 + \Delta_{\text{excl}})^2 = 0.02198$ ($0.09\sigma$), $\sin^2\theta_{12} = [2\phi(1 + \Delta_{\text{excl}})]^{-1} = 0.30413$ ($0.09\sigma$), $\sin^2\theta_{23} = (1 + \Delta_{\text{excl}})^2 / [1 + (1 + \Delta_{\text{excl}})^2] = 0.50797$ ($0.40\sigma$), and a retrograde $S^1$ phase circulation of $\delta_{\text{CP}} = 2\pi(1 - \alpha^{-1}/N_{\text{cym}}) = 3.8915\text{ rad}$ ($222.96^\circ$, $0.25\sigma$; $J_{\text{CP}} = -0.02307$).