Navier-Stokes Volume VI: A Viscous Barrier Mechanism for Propagation of Geometric Sparseness (H2)

We give a single-focus dynamical framework for the geometric sparseness hypothesis H2 introduced in Volume I. Using a smooth “soft-interface” approximation of high-vorticity superlevel sets, we define a scale-dependent perimeter functional and derive a schematic differential inequality that balances stretching against viscous regularization at rate ν ρ^{-2} on scale ρ. Under an integrability condition on ‖∇u‖_{L∞} (and mild control of the threshold drift), we obtain a short-time propagation principle: subcritical tube-volume scaling on a scale window persists. The result provides a rigorous template for interpreting empirically observed subcritical interface geometry (e.g. in Taylor–Green computations) as a viscosity-stabilized barrier against interface space-filling. What this contains- Tube-volume formulation of geometric sparseness H2 on a scale window.- Soft-interface approximation and a scale-dependent perimeter functional.- Transport–diffusion inequalities from Navier–Stokes.- A viscous barrier inequality on scale ρ and an exponential barrier under scale dominance.- A propagation theorem: if H2 holds at t0 on [ρ−, ρ+], it persists on that window for a short time under the dominance regime. What this does NOT claimThis is not a full closure of H2 from first principles for arbitrary solutions. The propagation theorem assumes (i) smoothness on the time interval, (ii) a scale-dominance condition (diffusion dominates effective stretching on the relevant window), and (iii) negligible commutator/remainder terms in the scale-perimeter inequality. Establishing these conditions directly from the NS evolution in full generality remains open. DNS-facing diagnostic (for Volume V / Taylor–Green linkage)A practical diagnostic compares ∫(A(t) + |Λ̇(t)|/η) dt to ν ρ^{-2} T on a chosen window, where A(t)=‖∇u‖_{L∞}. If the ratio is ≪ 1 on the same window where the measured interface proxy remains subcritical, this supports the viscous barrier regime.