Recursive Scale Stabilization and Global Regularity of the 3D Navier–Stokes Equations

This paper presents a new structural approach to the global regularity problem for the three-dimensional incompressible Navier–Stokes equations. Rather than focusing on amplitude-based a priori estimates, the analysis identifies the true obstruction to global regularity as a recursive, conservative, scale-local transfer mechanism that persists under saturation of classical bounds, particularly in near-Beltrami regimes. To address this obstruction, the paper introduces a scale-relational Lyapunov framework that augments scale-weighted energy and helicity control with a scale-gradient penalty measuring instability in shell-to-shell energy redistribution. This penalty converts recursive scale amplification into an explicit energetic cost while remaining compatible with the locality and conservation structure of the Navier–Stokes nonlinearity. A discrete scale interpolation lemma is established, showing that sharp nonlinear transfer terms at critical scaling can be uniformly absorbed into viscous dissipation and scale-gradient control. The resulting Lyapunov functional is shown to be monotone non-increasing, coercive on high-frequency dynamics, and equivalent to the critical Besov norm Bv2,1^{1/2}. The framework is stress-tested against worst-case configurations, including smooth Beltrami cascades, oscillatory shell patterns, concentrated scale spikes, and vanishing-time blow-up scenarios. In each case, recursive instability is dynamically suppressed. As a consequence, the critical Besov norm remains uniformly bounded for all time, yielding global continuation and smoothness for solutions arising from smooth divergence-free initial data. Beyond Navier–Stokes, the paper highlights a general analytical principle: global regularity in nonlinear evolutionary PDEs may hinge on controlling recursive scale coherence rather than amplitudes alone, suggesting broader applicability of relational Lyapunov methods to multiscale systems. -JSR