Global Regularity of the Three-Dimensional Incompressible Navier-Stokes Equations via Non-Singular Embedding in a Five-Dimensional Continuous Fluid Vacuum

We present a formal proof of the global regularity and smoothness of the threedimensionalincompressible Navier-Stokes equations on the domain R3×[0,∞). By transitioningfrom a rigid, non-relativistic four-dimensional spacetime coordinate system to a parameterizedfive-dimensional continuous fluid vacuum manifold governed by the Cosmic Awareness Theory ofLiquid Spacetime (CATLS), we demonstrate that apparent finite-time singularities are geometricprojection artifacts. An explicit inverse mapping theorem is constructed, establishing thatlocalized kinetic energy concentrations drive a smooth harmonic rotation into higher spatialdegrees of freedom (x4, x5). We rigorously evaluate the localized Jacobian determinant to provethe transformation is a valid topological diffeomorphism under arbitrary, non-symmetric initialconditions. Furthermore, formal Sobolev space Hs(R3) energy estimates and the Ladyzhenskaya-Prodi-Serrin regularity criteria are leveraged to verify that the global enstrophy remains strictlybounded, guaranteeing continuous differentiability to infinite order (C∞) for all classical timet ≥ 0.