Momentum helicity separates triad-local cascade operators from Euler–Arnold operators

Abstract

Tao's averaged Navier–Stokes equation replaces the Euler nonlinearity by a cascade operator that keeps the energy identity and blows up. We ask whether such operators can be Euler–Arnold operators: geodesic nonlinearities on the divergence-free fields of the torus for a translation-invariant inertia operator A. Tao's operators are not translation-covariant, so the meaningful case is their translation average, a triad-local cascade operator with the same species and structure constants. Energy cannot separate the two classes. The momentum helicity ∫ m·curl m, with m = Au, can: it is a Lie–Poisson Casimir, conserved by every Euler–Arnold flow. Using S J S = det(S) J for symmetric 2×2 matrices, we show that for reflection-even A the helicity production of any operator is linear in the numbers sqrt(det A_k). This gives a rank criterion. An exact integer certificate (rank 16 of 16 over Q on the lattice 0 < |k|² ≤ 4) proves the following. Outside a proper real-algebraic set of structure constants and profiles, a triad-local cascade operator with at least four species is not an Euler–Arnold operator for any reflection-even translation-invariant A. We also record where the method stops. The Casimir sees A only through the vector (sqrt(det A_k)). Parity-odd inertia operators are not covered. Whether helicity-conserving cascade operators exist, as suggested in Tao's Remark 4.3, is left open. The record contains the paper, its LaTeX source, and the exact certificate (helicity_separator_certificate.zip). The certificate uses Python 3 standard library only and runs in seconds; see the README inside. The work was carried out with substantial help from generative AI (Anthropic's Claude), as documented in Section 1.5 of the paper.

Related publications

• The Circulation Campaign: Why Circulation-Based Estimates Cannot Close the 3D Navier–Stokes Existence and Smoothness ProblemZenodo · 2026 · Same author: Adilkhan Salkimbayev
• Linear stability of asymmetric Burgers vortices at large circulation beyond the half-strain thresholdZenodo · 2026 · Same author: Adilkhan Salkimbayev
• Identification of the influence of bend angle in an l-shaped channel on incompressible viscous flowEastern-European Journal of Enterprise Technologies · 2026 · Shared topic: Fluid Dynamics and Turbulent Flows
• The fictitious domain method for the Navier-Stokes equations in natural variablesAIP conference proceedings · 2021 · Citations: 4 · Shared topic: Navier-Stokes equation solutions