Positive-Mass Synthesis of Prescribed Gravitational Fields and Gradients

We consider the inverse problem of producing a prescribed nonzero Newtonian gravitational acceleration together with a prescribed nonzero axial gradient, while cancelling a maximal consecutive sequence of higher axial derivatives, using N positive-mass infinitesimally thin coaxial rings constrained to one common source plane. With the normalized gradient γ = a g_z'(0)/g_z(0), the physical constraints reduce exactly to the truncated moment sequence m_0 = 1 and m_k = (1+γ)/(2k+1) for 1 ≤ k ≤ 2N−1. We prove that a positive finite-radius N-ring realization exists if and only if −1 < γ < γ_N^+, where γ_N^+ is obtained in closed form from an endpoint Christoffel kernel, and that the realization is unique up to permutation. A distinguished interior value γ_N^* makes the degree-2N moment error vanish, yielding exactly one additional derivative cancellation: g_z^(k)(0) = 0 for 2 ≤ k ≤ 2N, while g_z^(2N+1)(0) ≠ 0. We prove γ_N^* < γ_N^+ for every N. In a source-free axisymmetric neighborhood, the construction produces the locally affine field g = (−Γ_0 x/2, −Γ_0 y/2, g_0 + Γ_0 z) + O(r^(2N)), improved to O(r^(2N+1)) at γ_N^*. Explicit N = 3 examples illustrate the generic and enhanced-order constructions. The result is an exact positive-source synthesis theorem for the stated common-plane class; it is distinct from prior numerical design of approximately linear gravitational source fields.