What physical law are you invoking here? Certainly not the laws of electricity and magnetism, for these are fully expressed by Maxwell's equations -- which, as a matter of mathematical fact, cannot prefer any inhomogeneous solution without the imposition of boundary conditions.
Sigh. Did you read the paper? All of its conclusions depend upon choices of Green functions, which in turn depend upon boundary conditions. Physical laws do not determine boundary conditions. Imposing them in general, which you are doing, is adding a non-local law to physics.
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Consider the simpler case of Newtonian gravity. This may be formulated in two ways: via action at a distance (which is non-local), and via a gravitational potential function subject to a Poisson equation (which is local) AND a boundary condition at infinity (which is non-local.)
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If you drop the boundary condition on the Newtonian potential function, then an arbitrary homogeneous solution may be added to it. And this arbitrariness makes the gradient of the Newtonian potential -- i.e., the gravitational "force" -- arbitrary. It can be annulled at a point.
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