Geometric Necessity Theorem for Navier-Stokes Singularities: Alignment Coherence as a Blow-up Obstruction

“Classification of all possible Navier–Stokes singularities via vorticity–strain alignment geometry.” We establish an exact geometric characterization of potential finite-time singularities in three-dimensional incompressible Navier-Stokes flow. Introducing the alignment coherence parameter κ(x, t) = |A(x, t)ξ(x, t)|/|A(x, t)|, where A = ∇u is the velocity gradient and ξ = ω/|ω| is the vorticity direction, we prove that any finite-time singularity must satisfy lim supt→T κ(xn(t), t) = 1 along sequences tracking vorticity maxima. This result is unconditional—it requires no assumptions about the solution beyond smoothness up to time T. The proof proceeds via exact evolution equations for κ derived from the Navier-Stokes equations, revealing that viscosity provides geometric damping in orientation space that opposes perfect alignment. Our theorem reduces the Clay Millennium Problem to the question of whether the regime κ → 1 is dynamically accessible, providing both a constructive blow-up mechanism and a concrete criterion for DNS verification. We include explicit falsification protocols and discuss implications for turbulence modeling