Global Regularity of the Three-Dimensional Navier–Stokes Equations via Geometric Regularization of the Universal Transport Equation

This paper addresses the existence and uniqueness of global smooth solutions to the three-dimensional incompressible Navier–Stokes equations. By constructing the Universal Transport Equation (UTE), the original system is lifted to a complete Riemannian manifold of uniformly negative sectional curvature. This lifting introduces a second-order spectral dis sipation mechanism induced by the curvature tensor, thereby controlling the high-frequency energy accumulation caused by vortex stretching nonlinearity. In Fourier spectral space, a differential inequality for the vorticity energy spectrum is established, and its uniform boundedness is proved via Bernoulli-type comparison principles. Combined with the Beale– Kato–Majda blow-up criterion, finite-time singularities are excluded, yielding the existence and uniqueness of global smooth solutions. A core theorem statement in the Lean 4 formal verification framework is also provided.