The Sub-Quantum Determinism: Hydrodynamic Derivation of Wavefunctions, Non-Local Elliptic Phase Locking, and the Quantum Coherence Ceiling

We present the fourth volume of our unified framework, establishing the deterministic, causal, and geometric foundation of quantum mechanics within the viscous, incompressible sub-quantum \textbf{Navier--Stokes--Boukratem} fluid spacetime continuum. While previous volumes resolved cosmological tensions (Article I), regularized black hole singularities (Article II), and derived topological gauge factors (Article III), this paper achieves the complete sub-quantum closure of open quantum systems. By applying a generalized Madelung hydrodynamic transformation to the continuum velocity field $\mathbf{u}(\mathbf{x},t)$, we derive the Schr\"{o}dinger and Dirac equations from first principles, where the non-local Bohmian quantum potential emerges naturally as an internal Korteweg surface tension pressure. Furthermore, we resolve the Einstein--Podolsky--Rosen (EPR) paradox and the Bell--CHSH inequality bounds without abstract Hilbert spaces. Entanglement is shown to be a localized, continuous filament of vorticity $\omega$ stretched through the viscous medium, where strict fluid incompressibility ($\nabla \cdot \mathbf{u} = 0$) generates instantaneous non-local elliptic pressure interactions. Chiral anti-symmetric phase correlations are driven by a conserved Helmholtz circulation via transverse sub-quantum Alfv\'{e}n modes. The superposition of macroscopic states and the delayed-choice quantum eraser framework are analyzed under non-Markovian Volterra memory kernels, demonstrating that quantum measurement dephasing is a physical fluid-dynamical phase transition triggered when the sub-quantum Reynolds number hits the critical analytical threshold $\text{Re}_q > \text{Re}_{\text{crit}} = 3.306 / \pi \approx 1.0524$. Under the verification of \textbf{CELL QUANTUM-05}, this boundary layer dynamics isolates the non-perturbative \textbf{Boukratem Quantum Coherence Ceiling}, locking the ultimate theoretical fringe contrast visibility at exactly $\text{V}_{\text{hydro}} \approx \mathbf{83.54\%}$. This framework completely eliminates historical wave-function collapse assumptions, replacing them with a continuous, causal, and testable rheological model of quantum dynamics.