A Conditional Global Regularity Theorem for the Three-Dimensional Incompressible Navier–Stokes Equations via an Energy–Dissipation Lower Bound and Material Vortex-Tube Volume

Abstract We prove a conditional global regularity theorem for the three-dimensional incompressible Navier–Stokes equations with smooth finite-energy data and no forcing (the Clay “Case A” setting). Under a single explicit, purely analytic non-concentration hypothesis on the solution—that within the material vortex tube carrying the maximal vorticity, the subvolume on which the vorticity is comparable to its maximum retains a fixed positive fraction of the (conserved) material tube volume—we show that R T0 ∥ω∥2 L∞ dt < ∞, which by the Beale–Kato–Majda criterion implies global regularity. The argument uses only the exact energy identity, incompressible volume conservation, an elementary dissipation lower bound, and the Cauchy–Schwarz inequality; it makes no self-similarity assumption. We isolate the hypothesis in the standard style of conditional regularity criteria (Prodi–Serrin, Beale–Kato–Majda, Constantin–Fefferman) and discuss why it is the exact analytic content of the physical statement that viscosity, absent any energy resupply, prevents a peak from concentrating to measure zero within its own material tube.