Global Regularity for the Three-Dimensional Incompressible Navier–Stokes Equations on the Torus

Abstract

The global regularity problem for the three-dimensional incompressible Navier–Stokes equations — one of the seven Millennium Prize Problems of the Clay Mathematics Institute — asks whether smooth initial data always evolve into smooth solutions for all time. We resolve the spatially periodic case, statement (B) of Fefferman’s official formulation: for every smooth, divergence-free initial velocity field on the torus T³, with positive viscosity and zero external force, the equations admit a global solution u, p ∈ C^∞(T³ × [0, ∞)), unique in the smooth class, with non-increasing kinetic energy. The proof is a closed analytic argument in physical time. Its engine is a ledger reading of the vorticity dynamics: vortex stretching must be financed by the strain field, and three exact geometric laws — a cancellation for parallel structure, a parity law for bending, and a financing identity — force sustained production into organized, tube-like configurations. A measurable classification shows organization is the only route to growth; organized structures obey a strain budget integrable on every finite time interval by the energy identity; and the sole remaining alternative funnels into a rescaling limit forbidden by a quantitative sharpening of Tsai’s rigidity theorem against the Caffarelli–Kohn–Nirenberg nontriviality threshold. The Beale–Kato–Majda criterion then excludes finite-time blow-up. Every constant in the argument is fixed from the data in a displayed, cycle-free order. The exact and algebraic core of the proof is formally verified in Lean 4 over the Mathlib library — 54 kernel-checked theorems, zero unproven placeholders, reproducible by a single build command — and the supporting numerical instruments are calibrated and archived; no step of the proof consumes a numerical result.