Navier-Stokes Global Regularity via Hasimoto Bootstrap: A Conditional Proof Sketch

We present a conditional proof sketch that smooth solutions of the 3D incompressible Navier-Stokes equations with finite-energy initial data remain smooth for all time. The argument proceeds by contradiction: assuming finite-time blow-up, vorticity must concentrate on thin tubes whose dynamics reduce to the 1D viscous cubic nonlinear Schrödinger equation via the Hasimoto transform. Since 1D cubic NLS is globally well-posed (Tsutsumi 1987), the curvature of vortex lines remains bounded, which bounds the vortex stretching rate to an effective growth exponent of α = 2/3 — below the critical threshold α = 1. The resulting ODE dM/dt ≤ C·M^(2/3)·ln(M) cannot blow up in finite time. The proof is conditional on one assumption: rigorous justification of the vortex filament reduction in the blow-up regime (each sub-component has substantial literature support). Every other ingredient is a known theorem or standard calculation.