Magnetic helicity boundary term ($\mathbf{E}\cdot \mathbf{B}=-\frac{1}{2}{\partial }_t(\mathbf{A}\cdot \mathbf{B})-\frac{1}{2}\nabla \cdot (\varphi \mathbf{B}+\mathbf{E}\times \mathbf{A})$)

For regular globally defined electromagnetic potentials, use $\mathbf{E}=-\nabla \varphi -{\partial }_t\mathbf{A}$, $\mathbf{B}=\nabla \times \mathbf{A}$ and the homogeneous Maxwell equations to prove the displayed identity. With vanishing spatial surface flux, its spacetime integral is minus one half the change of magnetic helicity. Zero integral therefore needs matching temporal helicity boundaries, not merely localized electric fields and magnetic fields. The Gaussian potential $a(t)e^{-r^2}(-y,x,\lambda )$ has helicity $2\lambda (\pi /2{)}^{3/2}a(t{)}^2$ and supplies an explicit spatially localized counterexample when $a$ changes from zero to one.