Global Regularity for 3D Navier-Stokes

We prove global regularity for the three-dimensional incompressible Navier-Stokes equations on $\mathbb{R}^3$ and on the periodic torus $\mathbb{T}^3$ for smooth divergence-free initial data. Suppose, for contradiction, that a first singular time $T < \infty$ exists. On each parabolic scale approaching $T$, we show that the vorticity satisfies a complete geometric trichotomy—thick, tube-like, or fragmented—formulated in terms of volumetric extent, codimension-two concentration over a definite time fraction, and directional coherence. In every regime, we obtain a scale-invariant lower bound on the dissipation on a subinterval of comparable length (using spectral/parabolic estimates in the first two cases and the Constantin-Fefferman direction-gradient identity in the third). A Calderón-Zygmund packing argument yields infinitely many disjoint dissipation intervals accumulating at $T$, contradicting the energy inequality. Thus no finite-time blowup occurs.