Non-Separable Six-Dimensional Madelung Hydrodynamics on Möbius Bundles: Two-Vortex Quantum Entanglement, Freeze-Out Scale Bridges, and 21 cm Spin-Singlet Beats

Extending the microscopic and cosmological sectors of the ESEM framework (Papers I–II) from the single core anchor r₀ = 0.000635 fm, we formulate two-body proton–electron quantum entanglement via non-separable Madelung hydrodynamics in six-dimensional configuration space ℝ⁶. The joint Bohmian entanglement potential Q_ent(xₚ, xₑ, t) synchronizes spin and phase correlations without superluminal energy transport, saturating Tsirelson's CHSH bound (S_CHSH = 2√2 ≈ 2.8284) while strictly preserving marginal continuity (no-signaling) and universal electron mass invariance in γγ → e⁻e⁺ pair production. Additive partitioning of the sonic-throat Bernoulli enthalpy invariant (1/8 for the N_c = 1 electron ring plus 3/8 for the N_c = 3 proton core) links the proton-to-electron harmonic step Δn_pe = 26 + 1/8 = 26.125 directly to the hadronic Landau freeze-out node n_Landau = 26.5 (E₂₆.₅ = 151.89 MeV), yielding a closed freeze-out-to-Compton bridge λ̄ₑ = λ̄ₚ(r₂₆.₅/r₀)(3/4)³ᐟ⁸ = 386.1555 fm (−9.7 ppm from CODATA). Iterating Δn_pe from n_Landau = 26.5 lands at n = 52.625 (82.70 keV), matching the BBN deuterium bottleneck (84.5 keV) and the ⁷Li WKB barrier (82.39 keV), while bound flux-sharing generates Bohr sub-harmonics and the 1420.4 MHz (21 cm) spin-singlet beat near node N = 140, predicting exact cosmological constancy d(Mₚ/mₑ)/dt = 0.