Global Regularity of the Three-Dimensional Incompressible Navier-Stokes Equations via Exhaustion of Continuations

We present a proof of global regularity for the three-dimensional incompressible Navier-Stokes equations with smooth, unforced initial data. The argument proceeds by contradiction, assuming the existence of a finite-time singularity at T* and exhausting all topologically permissible continuations of the enstrophy within the Leray-Hopf weak formulation. By analyzing the limit of Galerkin approximations and the evolution of the vorticity direction field, we demonstrate that a post-singularity state cannot be uniquely determined without violating either the algebraic non-negativity of enstrophy, the Leray energy inequality, or deterministic selection. The failure of the adjoint equation at T* and the strict dichotomy of directional alignment (via Constantin's criterion) prohibit the formation of the singularity, thereby forcing T* to be a regular point.