Quantized Euler Equations and the Cavitation Limit in TRFC Theory: Eliminating Singularities in 4D Nanohydrodynamics

Following our experimental and theoretical framework for quantum-modulated nanohydrodynamic separation (Zenodo, Record 18813598), this paper formally differentiates the Topological Resonant Fractal Continuum (TRFC) theory from generic Superfluid Vacuum Theories (SVT). We address the fundamental mathematical breakdown of classical fluid dynamics and general relativity: the emergence of singularities. In the TRFC framework, the 4D continuum is strictly incompressible and has zero viscosity, governed by quantized Euler equations rather than problematic Navier-Stokes formulations. We mathematically demonstrate that infinite density singularities (such as those implied by Calabi-Yau manifolds or classical black holes) are physically impossible in TRFC. Instead, as the local velocity of the 4D fluid approaches the velocity of vacuum phonons, the hydrodynamic pressure drops according to the 4D Bernoulli principle. Before a singularity can form, the continuum reaches a precise "Cavitation Limit". At this topological threshold, the continuum undergoes a localized phase transition into a subcooled Phase 4 state (associated with dark matter phenomena), generating quantized resistance and preventing infinite collapse. This provides a singularity-free mathematical foundation for molecular transport within Kaleidoscopic Interactive Fractal Structure (KIFS) attractors, such as carbon nanotubes.