Chapter 65: τ-NS Regularity at Fiber Level

The Navier–Stokes regularity problem—one of the seven Clay Millennium Prize Problems—asks whether smooth initial data always yield smooth solutions for all time. In three-dimensional Euclidean space, this remains open. On the compact fiber T² of τ³, the τ-framework provides a structural resolution through the Positive Regularity Theorem (III.T25). Rather than relying on classical PDE estimates, the proof instantiates three categorical conditions—clopen locality, ω-germ determinacy, and defect contractivity—for the fiber-level Navier–Stokes system. This chapter carries out that instantiation, building on the inviscid framework of ch58 (τ-Euler), the viscous dissipation of ch59 (τ-NS), and the defect functional of ch56.