Geometric Sparseness and the Interface Defect Mechanism in 3D Navier–Stokes Regularity: A Conditional Criterion

We investigate a geometric bottleneck in the regularity theory of the three-dimensional incompressible Navier-Stokes equations. A localized Good-lambda mechanism controls the maximal coherence functional G*_r by G_r, up to a single obstruction supported near the boundary of the localized high-vorticity region: the interface defect. We require three independent assumptions: (1) vorticity direction coherence (a Carleson-type control of G_r on high-vorticity sets), (2) geometric sparseness (a subcritical packing condition with upper Minkowski dimension d < 3), and (3) palinstrophy non-concentration in boundary layers. None of these implies the others; each isolates a distinct mechanism (coherence, geometry, and energetic distribution). Under these assumptions, and for a sufficiently small boundary-layer thickness-to-radius ratio rho/R, the defect becomes absorbable into viscous dissipation and the localized Constantin-Fefferman framework yields a closed enstrophy inequality, implying regularity on [0,T]. SCOPE: This is a conditional criterion; we do not derive the above assumptions from Navier-Stokes dynamics and we do not resolve the Clay Millennium Prize Problem. Rather, we isolate the precise geometric/energetic obstructions whose validity would imply regularity.