= Magnetic helicity boundary term
{title2=$\mathbf E\cdot\mathbf B=-\tfrac12\partial_t(\mathbf A\cdot\mathbf B)-\tfrac12\nabla\cdot(\phi\mathbf B+\mathbf E\times\mathbf A)$}
For regular globally defined electromagnetic potentials, use $\mathbf E=-\nabla\phi-\partial_t\mathbf A$, $\mathbf B=\nabla\times\mathbf A$ 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.
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