Navier-Stokes Regularity Reduced to a Single Lagrangian Obstruction: Orlicz ε-Regularity, Anisotropy Growth, and Closure Programs

This paper refines and extends the conditional regularity framework developed in [Esposito 2025, DOI: 10.5281/zenodo.18144281] by isolating the remaining obstruction to global regularity of the three-dimensional incompressible Navier-Stokes equations. Building on an Orlicz-refined ε-regularity package and its explicit quantitative consequences, we show that all potential singularities consistent with Eulerian control must be driven by a single Lagrangian mechanism: near-envelope growth of deformation anisotropy along particle trajectories, accompanied by persistent extremal alignment between singular directions of the deformation gradient and eigenvectors of the strain tensor. We establish a fork theorem: on any parabolic cylinder where the Orlicz density is small, either quantitative regularity holds or anisotropy growth must occur through a highly constrained "enemy-switch" regime of persistent alignment. All other Eulerian and Lagrangian blow-up scenarios are excluded. This reduction transforms the Navier-Stokes regularity problem into the problem of ruling out one sharply defined Lagrangian intermittency mode. Two closure programs are identified. The first is a pure PDE route based on eigenframe rotation and decorrelation. The second is a selection route formulated analytically as bounded eigenframe variation along trajectories. No unconditional regularity claim is made. Instead, the remaining obstruction is isolated, quantified, and rendered falsifiable. This is Paper 2 in a research program on Navier-Stokes regularity via Orlicz entropy methods. Paper 1 (DOI: 10.5281/zenodo.18144281) established conditional regularity under Orlicz control. The present work completes the structural reduction by identifying the single remaining obstruction. Key contributions:• Fork theorem quantitatively excluding all blow-up scenarios except one• Precise definition of "enemy mode" (persistent extremal alignment)• Two concrete closure programs with analytic targets• Falsifiable predictions for numerical validation This work does not claim to solve the Clay Millennium Problem but provides a compression of the problem to one minimal, sharply defined obstruction.