3DNavier–StokesGlobalRegularityviaEigen-Gap CarlesonPackingandDirectionalPacketAnalysis

This paper takes a new approach to the three-dimensional incompressible Navier–Stokes equations by introducing a directional decomposition of the vorticity that aligns with the actual geometry of the flow. Instead of analyzing the solution purely through frequency-based methods, the authors define scale-localized packets that also respect the dominant stretching direction at each point. This refinement allows them to isolate where the flow is truly active and track how energy moves across scales and directions. The method builds a per-scale energy balance for each directional packet, accounting for both nonlinear effects and the influence of variable geometry. The key idea is that in regions where the flow becomes complicated—where the dominant stretching direction breaks down—the geometry forces those regions to be sparse in space-time. The paper quantifies this sparsity using analytic sublevel estimates and a Carleson packing argument, ensuring these bad regions can't accumulate to cause a singularity. By summing the energy balances across scales with explicit weights, the authors derive a differential inequality—referred to as the ledger—which controls the total directional energy. Once it is shown that this ledger remains bounded over time, a classical endpoint inequality converts that control into a bound on the maximum vorticity. From this, global smoothness and uniqueness follow for all finite-energy initial data. The argument avoids common bootstrapping techniques and does not rely on conditional regularity criteria. All parameters and constants are tracked explicitly, and no steps are deferred or left unproven. What emerges is a self-contained mechanism that remains stable for all time, yielding smooth solutions unconditionally, pending full peer review.