We prove global regularity for the three-dimensional incompressible Navier–Stokes equations with smooth initial data of finite energy: no finite-time blowup occurs. We define a vorticity–strain coherence packet on parabolic cylinders and prove that any finite-time singularity forces an infinite descending chain of blowup-admissible packets. A sourcewise strain decomposition linearizes the directional defect closure. A two-cutoff geometric argument shows that the spatially constant affine carrier suffers a strict volume-ratio deficit on nested cutoffs; oscillatory collapse makes this quantitative, forcing fresh non-affine curvature repair at every generation—a linear cumulative demand that the finite shell budget (from pre-AF shell summability on each smooth cylinder) cannot sustain. No infinite chain exists; blowup is structurally impossible
Joe Gallagher (Mon,) studied this question.