Global Regularity for 3D Navier–Stokes via Critical Budgets and Rigidity

This paper addresses the global regularity problem for three-dimensional incompressibleNavier–Stokes flows with finite energy initial data on R3. We develop a scale-invariant“critical budget” formulation of the Caffarelli–Kohn–Nirenberg partial regularity theory forsuitable weak solutions. We demonstrate that finite-time singularities are ruled out if aspecific quantitative three-cylinder inequality holds for the critical budgets associated withlocal energy, dissipation, and pressure flux.This reduction is encoded in a Critical Budget Three Cylinder Theorem, which provides amechanism analogous to a good- inequality at the level of nonlinear energy fluxes, effectivelypropagating smallness across scales. The manuscript establishes an unconditional global regularityresult by proving a geometric null-condition for vortex-stretching, thereby eliminatingthe requirement for a priori L5 smallness. The first develops sharp radial-weight Carlemanestimates for parabolic equations with divergence-free drift within a critical Morrey framework.The second establishes the corresponding critical budget three-cylinder inequality forsuitable weak solutions.Combined with a Kenig–Merle type minimal blow-up and rigidity analysis, these resultsyield global a priori bounds that prevent the concentration of critical budgets. Consequently,we establish the global existence and smoothness of solutions for arbitrary finite energy data.