Algebraic Trapping and the Inevitability of Geometric Frustration in Vector Cross‑Product Systems

This paper establishes a general algebraic trapping lemma for dynamical systems whose nonlinearity is constructed entirely from complex cross‑product interactions. As stated in the abstract, “any solution starting in a subspace V of the phase space remains confined to V for all times,” provided the subspace is closed under the relevant cross‑product operations. The paper argues that “geometric frustration is an inevitable consequence of the cross‑product structure, independent of the specific choice of coefficients, viscosity, or initial data.” This result provides the algebraic foundation for the geometric frustration mechanism observed numerically in vector triad models and suggests a pathway toward understanding regularity in the Navier–Stokes equations.