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 (m₁, m₂, m₃), the three Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing angles (θ₁₂, θ₂₃, θ₁₃), and the leptonic CP-violating phase (δCP) 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 (TCMB = 2.72548 ± 0.00057 K). Acoustic perturbations below the four-dimensional phase-space Jacobian knotting threshold of the electron (E < mₑ c² = 2π kB TCMBWₑ⁴) cannot form a closed Total Internal Reflection (Γ = 1) vortex sink and instead propagate as un-knotted longitudinal pilot waves with baseline mass-equivalence m₁ = kB TCMB/c² = 0.23487 meV/c². As this wave traverses spherical vortex boundaries governed by the fundamental overtone operator Λ₁ = 1 + 3/2α⁻¹ ≈ 206.5540 and an Asymptotic Suction Boundary Layer (ASBL) of radial displacement thickness δ^*/r ≈ 0.00344, its two-dimensional meridian cross-section (Λ₁2/3) dressed by the central Spindle Torus overlap (1/3φ⁻⁴) and its three-dimensional volumetric mode (Λ₁¹) dressed by the dual confined boundary-layer sheath (Δexcl = 14/3δ^*/r ≈ 0.01607) yield m₂ = 8.5720 meV/c² (Δ m²₂₁ = 7.342 × 10⁻⁵ eV², 0.32σ), m₃ = 49.292 meV/c² (Δ m²₃₁ = 2.430 × 10⁻³ eV², 0.70σ), and normal mass ordering (∑ m_ν = 0.05810 eV/c²). Evaluating the geometric cross-coupling projections across the Spindle Torus (W_φ = φ⁻²) derives sin²θ₁₃ = φ⁻⁸(1 + Δexcl)² = 0.02198 (0.09σ), sin²θ₁₂ = [2φ(1 + Δexcl)]⁻¹ = 0.30413 (0.09σ), sin²θ₂₃ = (1 + Δexcl)² / [1 + (1 + Δexcl)²] = 0.50797 (0.40σ), and a retrograde S¹ phase circulation of δCP = 2π(1 - α⁻¹/Ncym) = 3.8915 rad (222.96^∘, 0.25σ; JCP = -0.02307).