Magnetic free-decay spectral bound (${\int }_D|\nabla \times \mathbf{B}{|}^2\ge \frac{{\pi }^2}{R^2}{\int }_{{\mathbb{R}}^3}|\mathbf{B}{|}^2$)

For a solenoidal vector field carrying current only in a conductor contained in a sphere of radius $R$, match the exterior field to a decaying potential field and include exterior magnetic energy. The smallest free-decay eigenvalue of the enclosing sphere is ${\pi }^2/R^2$, from its dipolar poloidal mode. The corresponding variational estimate gives the displayed bound. Continuity and the usual insulating-interface magnetic conditions are part of the admissible field class. This bound is the diffusive ingredient in Backus' necessary condition for dynamo action.