An Adaptive Criterion Based on the F/C Ratio: Geometric Depletion, 1D Sparseness, and Numerical Protocol.

This document proposes an adaptive framework for the 3D Navier-Stokes equations based on the instantaneous ratio ρ(t)=F(t)/C(t) between enstrophy production and viscous dissipation. The core of the method is the exact balance Ω̇=(ρ-1)C combined with a hypothesis of geometric depletion. It outlines a path from anisotropic 1D sparseness of vorticity superlevel sets to a depletion inequality with integrable leakage, providing an operational numerical protocol to estimate dynamic scales and evaluate event predictability.