Global Regularity for 3D Navier-Stokes Equations: A Diagnostic Reduction

Working entirely within the classical energy–enstrophy framework, the Navier–Stokes nonlinearity is reduced exactly and algebraically, using only incompressibility and standard vector identities, to a single vorticity–strain interaction. A sharp three-dimensional weakening estimate produces a fractional-power bound on enstrophy growth, separating dissipative effects from genuine nonlinear amplification. The analysis shows that global regularity is equivalent to time-integrated control of a single stretching functional, ∫₀ᵀ ∫ |ω|² λ_max(S) dx dt, thereby identifying the sole remaining geometric obstruction to regularity in three dimensions. The reduction is exhaustive: no further control can be obtained from energy, enstrophy, or known analytic identities alone.