A Sharp Two-Ring No-Go Theorem for Locally Uniform Newtonian Fields

We study whether two positive-mass infinitesimally thin coaxial rings can improve their local gravitational-field flatness when the two rings are allowed independent axial positions. Let ring i have positive mass Mi, positive radius Ri, and arbitrary axial coordinate di, and let the design point be the origin. Writing si = (Ri² + di²)^(1/2) and yi = di/si ∈ (-1,1), the regular axisymmetric multipole coefficients are proportional to the corresponding Legendre-polynomial source moments. We prove that no positive two-ring configuration can satisfy A2 = A3 = A4 = A5 = 0, even when the rings lie in different axial planes or on opposite sides of the design point. Exact elimination reduces the four cancellation equations to (y1² - y2²)B = 0, where B = 7a² + 7ab + 7b² - 3a - 3b + 9 with a = y1² and b = y2²; the strictly convex polynomial B has global minimum 60/7 > 0. The remaining branches y1 = ±y2 are incompatible with simultaneous zeros of the required Legendre polynomials. Since the common-plane positive two-ring construction can cancel A2, A3, and A4, the bound is sharp: two positive rings achieve at most three successive axial derivative cancellations. Thus independent axial displacement does not improve the two-ring local-flatness order.