Exact Amplitude–Direction Decomposition and Finite-State Non-Determination in the 3D Euler Vorticity-Gradient Hierarchy

We study the vorticity-gradient geometry of the three-dimensional incompressible Euler equations on the non-vanishing vorticity set. Writing L = log(|ω|/ω⋆), ωb = ω/|ω|, we obtain the exact identities L 222L22 ∇ω=ω⋆e ∇L⊗ωb+∇ωb , |∇ω| =ω⋆e |∇L| +|∇ωb| , which split the Frobenius norm of ∇ω into orthogonal amplitude and directional channels. The main contribution is the exact dynamical lift of this decomposition to a transport hierarchy for (L, ωb, ∇L, n, λ), together with the identification of the precise point at which the amplitude-gradient channel loses local autonomy. After exact substitution of ∇λ, the relation T ∇λ = 2(∇ωb)Sωb + A, A = ωb (∇S)ωb, shows that A is, within the chosen hierarchy, the only term in the amplitude-gradient equation containing one spatial derivative of the order-zero Biot–Savart output S. We prove by an explicit divergence-free perturbation argument on a fixed time slice that, among instantaneous smooth Euler states arising from smooth divergence-free data, the value of A at a point is not determined by the finite local state vector (L, ωb, ∇L, λ). This finite-state non-determination result is the principal non-closure statement of the paper. We further rewrite the weighted amplitude channel in the physical variable ∇|ω|, obtaining an exact localized H1-type identity for the vorticity magnitude. On simple-spectrum regions of the strain tensor, the local quadratic operator diagonalizes in the strain eigenframe, and A admits an exact decomposition into eigenvalue-gradient and eigenframe-rotation contributions; in the exactly aligned regime ωb = ej, one has A = ∇μj . The manuscript therefore provides an exact structural reformulation of the vorticity-gradient hierarchy. It does not claim a new regularity criterion; the unresolved analytical step remains the control of the projected strain-gradient in the amplitude channel, together with the separate pressure contribution in the turning channel.