Periodic two-coordinate anti-dynamo energy bound ($\Delta ⟨B_z^2⟩\le Q^2⟨A_0^2⟩/(4{\eta }^2{k}^2)$)

For a solenoidal three-component velocity depending on two periodic coordinates and a zero-mean magnetic field with the same dependence, the in-plane flux potential satisfies a homogeneous advection-diffusion equation. The Poincaré inequality gives $⟨A^2⟩\le ⟨A_0^2⟩e^{-2\eta k^{2}t}$. Integrating the vertical stretching term by parts bounds it by $Q\sqrt{⟨A^2⟩⟨|\nabla B_z{|}^2⟩}$, where $Q=\Vert \nabla u_z{\Vert }_{\infty }$. Maximizing against resistive dissipation and integrating gives the displayed transient bound. Splitting off half the dissipation instead gives $E^{\prime }+\eta k^{2}E\le (Q^2/\eta )⟨A_0^2⟩e^{-2\eta k^{2}t}$, proving vertical energy decays. This permits transient generation by a nonzero vertical velocity while excluding sustained fields in this zero-mean, z-independent periodic class.