Full-Tail Dynamical Rigidity Forced by Atomic Navier-Stokes Energy Concentration

Abstract

Let a smooth, unforced, three-dimensional Navier--Stokes flow on the flat torus approach a finite terminal time. Its kinetic-energy densities converge, along the full time variable, to a unique endpoint measure. We prove that each point atom forces a same-parent, full-tail dynamical rigidity. A preassigned level-crossing catalogue and nested local Hodge projections produce orthogonal packets. From the entire packet tail, we extract a single backward adjoint whose terminal energy concentrates at the atom. Cauchy saturation then locks this adjoint to every sufficiently late packet and yields uniform two-parameter saturation of both the constrained Oseen propagator and its adjoint, together with vanishing first-order dissipation. Consequently, every sufficiently late fixed-root descendant has infinite delayed second-order action and non-integrable positive enstrophy production. Equivalently, a delayed second-order operator budget determined solely by the Navier--Stokes parent must fail arbitrarily close to the endpoint.