PROOF OF GLOBAL SMOOTHNESS FOR THE THREE-DIMENSIONAL PERIODIC NAVIER–STOKES EQUATIONS

Abstract. We study the incompressible Navier–Stokes equations on the three-dimensional torus T3 = R3/Z3 for smooth mean-zero divergence-free periodic data. The paper is written entirely in the periodic setting and is intended to address the periodic alternative (B) in the Clay Millennium formulation; it does not claim the whole-space statement on R3. The nonlinear shell budget is decomposed exactly into four explicit channels. Three of them are strictly subcritical after shell power counting; the only channel that reaches the critical shell scale is the balanced high–high to low transfer. A central revision concerns the low-output strain sum. Because the dyadic shells are thick annuli, the relevant shell-difference set is volumetric rather than surface-like. The corrected estimate therefore uses a shell-dependent low-output gap Kj = ⌈j/5⌉ + K0, which restores a uniform factor 2−5K0/2 in the critical shell bound while moving the bridge-interface term into the subcritical ledger. The near-saturating angular part is organized by a finite radial–angular model bank. The finite-horizon step is stated precisely as an exact witness aggregation: all realized triads on [0,T] are regrouped into finitely many amplitude-dressed packet families indexed by that pre-existing bank. Thus the new mathematical content is an exact reassembly of the realized nonlinear obstruction on a finite horizon, not a claim that the PDE dynamically generates new profiles. The proof is lattice-based at every stage: the exact shell decomposition, the low-output packet geometry, the corrected strain sum, and the horizonwise witness aggregation all use the periodic frequency lattice. Uniform master constants are separated from the bookkeeping parameters (δ,ε,K0), so the small-critical-coefficient step is non-circular. With these points made explicit, the shell bridge closes on each finite horizon, and the argument yields strong continuation and hence global smoothness for smooth periodic data.