A Holistic Scaffold Framework for Navier--Stokes Regularity: Computational Evidence from Galerkin Truncations at 6 to 24 Modes

We present a holistic scaffold framework for the Navier–Stokes regularity question, based on a coupled diagnostic H that monitors enstrophy, convergence, and fragility simultaneously through 26 hierarchical levels, each implemented as a dual number with computable derivatives. The framework resolves the regularity question for Galerkin-truncated 3D Navier– Stokes models with vortex stretching across seven mode counts (6, 8, 10, 12, 16, 20, and 24 modes). The regularity threshold A∗ — the maximum initial amplitude below which enstrophy remains bounded for all time — converges to a positive limit (A∗ = 0.347) by 16 modes and remains unchanged at 20 and 24 modes. The enstrophy doubling-time criterion achieves 94.6% gap closure ratio (A∗ P /A∗ = 0.328/0.347) with perfect classification (14/14) at 16+ modes. The scaling law max(Ω) = C · A2 (α = 2.0) holds exactly at every mode count. We introduce self-adapting weights (H′ ) that achieve 91.5% gap closure and a confidence tracker (H′′) that achieves 100% closure at T = 100,000 steps. Forward and backward automatic differentiation reveal that early dynamics (the first 16% of the trajectory) are 50× more influential than late dynamics, and that vorticity components carry 8–13× more attribution weight than velocity components. We identify the viscosity spectrum as a continuous parameter landscape exhibiting resonance, hysteresis, and ratchet phenomena, and derive a four-step path from these computational results toward a formal proof for the full equations. All 96 experiments are reproducible in the Simplex programming language with native dual-number automatic differentiation.