Radial Barriers for Discrete Newtonian Potentials: Counterexamples and Sufficient Conditions

For a locally finite configuration of positive masses in \(\mathbb R^n\), \(n\geq3\), we study spheres on which the Newtonian potential is uniformly small. Unit masses at the positive integers contradict the dimensional extension of an auxiliary lemma of Clunie, Eremenko and Rossi for every \(n\geq4\); the same example nevertheless has infinitely many nondegenerate equilibrium points.

Under potential summability, we give valid radial criteria involving the accumulated \((n-2)\)th roots of the masses when \(n\geq4\), and a mass-entropy sum when \(n=3\). Each criterion implies infinitely many equilibria escaping to infinity. Finite-total-mass examples show that the corresponding sublinear growth requirements cannot be replaced by linear bounds in these radial assertions.

These results neither refute the equilibrium-existence theorem of the cited paper nor settle the general positive-mass question under force summability alone. The classical superlevel method is credited. This preprint is AI-assisted, self-audited and unrefereed, with no independent review, formal verification or absolute priority claim.