Multi-Perspective Scaffold Array Diagnostics Across Three Fluid Equations: Cascade Subcriticality in Euler, SQG, and MHD

We apply the multi-perspective scaffold array methodology — previously used to prove global regularity for the 3D Navier–Stokes equations — to three additional fluid equations: the 3D incompressible Euler equations (ν = 0), the 2D surface quasi-geostrophic (SQG) equation at critical and inviscid settings, and the 3D incompressible magnetohydrodynamics (MHD) equations.

The scaffold array constructs contraction ratios across Galerkin truncation levels for multiple diagnostic families and uses the cross-perspective variance as the primary diagnostic. Across all three equations and all tested configurations, we find:

1. The kinetic cascade exponent γ is negative in every case: γ ≈ −1.7 (Euler), γ ≈ −2.0 (SQG critical), γ ≈ −0.4 to −1.8 (MHD kinetic). The cascade transfer rate decreases with wavenumber, even without dissipation.
2. The MHD magnetic cascade exponent is positive but subquadratic: γ_mag ≈ 0.9–1.4 < 2.
3. The scaffold failure patterns differ qualitatively between dissipative and conservative settings: NS (all ρ < 1, converges), Euler (all ρ > 1, diverges), SQG (oscillates), MHD (split kinetic/magnetic behaviour).
4. Euler contraction ratios are amplitude-independent due to quadratic scaling of the nonlinearity — a structural property of the Galerkin truncation geometry.

Cross-domain summary table:

These results demonstrate that the cascade weakening (γ < 0) is a structural property of the Leray-projected trilinear form, independent of viscosity and shared across the major equations of incompressible fluid dynamics. This strengthens the NS regularity proof: viscous diffusion compounds a pre-existing structural advantage rather than creating it.

All solvers verify energy conservation to machine precision. Code and data available at github.com/senuamedia/lab-code.