Finite Equilibria for Rational Inverse-Power Fields in Any Dimension

We extend the compact-curve method of Enciso and Peralta-Salas for Coulomb equilibria in three dimensions. For distinct real sites, nonzero real charges, and any rational exponent \(s>1\), the field \(E_s(x)\)\({}=\sum_i q_i(x\)\({}-a_i)|x\)\({}-a_i|^{-s}\) has only finitely many equilibria where \(S_s(x)\)\({}=\sum_i q_i|x\)\({}-a_i|^{-s}\) is nonzero, in every ambient dimension. In particular, all equilibria are finite in number for same-sign charges. A scalar projection replaces the three-dimensional cross-product step, and a local field-pole argument treats charge sites. An algebraic lift proves complex isolation and a nonsharp degree bound. The three-dimensional Coulomb theorem is prior work; this preprint presents an explicitly attributed extension. AI-assisted, self-audited and unrefereed, with no formal verification or absolute priority claim.