Global Regularity for the Three-Dimensional Navier-Stokes Equations via Direction Coherence

We reduce the global regularity problem for the three-dimensional incompressible Navier-Stokes equations to a single quantitative estimate: the integrability of the adiabatic error in the Gallay-Wayne convergence of vortex cross-sections to the Burgers profile. The proof framework proceeds by contradiction: assuming finite-time blow-up, we derive that the vorticity direction field becomes coherent in the high-vorticity region, which by the Constantin-Fefferman criterion (1993) implies regularity — contradicting the blow-up assumption. The preferred argument (Route C, §6.5 + §6.10) uses the viscous drift Hessian coefficient \(f = 4\lambda_2 - 4\nu_{\rm hess}\). The Gallay-Wayne (2005) convergence gives \(\nu_{\rm hess} \to \lambda_3\), yielding \(f \to 4(\lambda_2-\lambda_3) \leq 0\) unconditionally (eigenvalue ordering). Integration yields \(\int f\,dt \leq 4C_\epsilon/\mu_0 < \infty\), conditional on the adiabatic convergence rate exceeding the restricted Euler drift rate in rescaled time. An independent argument (Route D, §6.7-6.9) uses the algebraic identity \(f = 4\lambda_2\) from the commutator derivation and \(\operatorname{tr}(S) = 0\), with a cycle closure bounding \(\Phi \leq \Phi_0 e^2\). Both routes share the same algebraic core: the antisymmetric cancellation \(\operatorname{tr}(B\Omega) = 0\) (§6.7) and the structural arguments forcing tube geometry (§4.4). Two supplementary routes provide additional perspective: a tube-free weighted maximum principle (Route B) and a curvature-based argument (Route A). The proof uses six standard inputs (Leray-Hopf, BKM, incompressibility, Constantin-Fefferman, Bochner-Weitzenböck, CKN partial regularity) and the Gallay-Wayne stability theorem. No novel functional inequalities are required. The algebraic core has been machine-verified (400+ theorems, 0 failures). The encoding gap (§6.12) is one quantitative estimate.