A Geometrically Weighted Sobolev Framework for Navier–Stokes Regularity: The HˢB Space, Contact Geometry Interpretation, and a No-Go Theorem for Metric Pressure Absorption

We introduce the Helical Sobolev space HˢB, a state-dependent Finsler manifold embedding the Beltrami alignment angle theta (x, t) between velocity and vorticity directly into the metric of the critical fractional Sobolev space H-dot^1/2. Within this framework, the Kato–Ponce trilinear bound becomes state-dependent with effective constant Cₑff proportional to sin theta, and the Groenwall inequality closes globally provided ||sin theta||₋䂐₍₅ₓₘ <= nu/C₀. This conditional regularity result provides the first unified bridge between the geometric criterion of Constantin–Fefferman (1993) and the critical-space theorem of Escauriaza–Seregin–Sverak (2003), reducing the 3D Navier–Stokes global regularity problem to a single geometric inequality—the Geometric Suppression Conjecture—which we state precisely and provide empirical and partial analytical support for. The manuscript further formulates the Fundamental Lemma of Fluid Contact Geometry, records obstructions for an explicit metric candidate, and proves a no-go theorem: no local uniformly elliptic Riemannian metric can absorb pressure for all solutions in the sense stated in the paper. It does not claim resolution of the Clay Millennium Prize problem.