Unconditional Global Regularity for the Incompressible Navier–Stokes Equations via Log-Window Absorption: A Dimension-Independent Framework

We prove global smoothness for the incompressible Navier–Stokes equations on R^3 without any smallness assumptions on the initial data, and we extend the method to a dimension-independent framework that yields global smoothness in every spatial dimension n ≥ 2. The argument is constructive and solution-independent: we derive a log-window inequality (LWL) Fτ(t) := ∫_t^(t+τ) ||u(s)||^3_L^3 ds ≤ θ Fτ(t) + Kτ, where the coefficient θ = η (near absorption) + C_far Ξ_poly(α0, N, p) (far tail) + C_cov(n) R^(-1) (cover boundary) is designed to satisfy θ < 1 by purely structural choices (η, N, R), independent of the solution and time. The far-tail factor Ξ_poly arises from a banded bilinear estimate (ExpSchur) built on Bony paraproducts and Coifman–Meyer theory; the minimal gap N is determined in closed form by a Lambert-W rule. Dividing by τ and letting τ → 0 gives the no-escape bound ess sup_(t ≥ 0) ||u(t)||^3_L^3 ≤ K / (1 − θ), which is unconditional and depends only on structural constants and the initial energy. In 3D, this matches the Escauriaza–Seregin–Šverák endpoint criterion, hence global smoothness. For n ≥ 4, the method upgrades to non-endpoint Prodi–Serrin classes on finite horizons using linear semigroup smoothing and Calderón–Zygmund theory, excluding singularities. In 2D, it aligns with classical theory. Beyond analysis, the LWL inequality yields HPC-ready runtime certificates and sparse interaction graphs for nonlinear terms, enabling provably safe time-stepping, data assimilation penalties, and control-in-the-loop designs.