Positive-Mass Synthesis of Locally Uniform Gravitational Fields by Gaussian Quadrature

We consider the inverse problem of producing a prescribed nonzero locally uniform Newtonian gravitational field using N positive-mass, infinitesimally thin circular rings constrained to a common plane. By transforming the physical radii and masses to dimensionless variables, the conditions for cancelling successive axial field derivatives are shown to be exactly equivalent to polynomial quadrature for the measure dμ(x)=dx/(2√x) on (0,1). The N-node Gaussian rule for this measure therefore gives an explicit positive construction satisfying gz(0)=g0 and gz^(k)(0)=0 for 1≤k≤2N−1. The nodes are equivalently determined by the positive roots of P2N, and the quadrature weights map directly to positive physical masses. At maximal cancellation order the construction is unique up to permutation of the rings. We further prove that 2N−1 is sharp: no source supported on at most N distinct finite common-plane ring radii can additionally cancel gz^(2N)(0), even if signed source weights are admitted. In a source-free axisymmetric neighborhood this axial result implies cancellation of the potential multipoles A2,...,A2N, so that g=g0 z-hat+O(s^(2N)). The coefficient of the first unavoidable field correction is obtained in closed form from the associated Jacobi polynomial norm. Explicit constructions are given for N=1,...,4, together with finite-volume numerical examples illustrating how the ideal thin-ring configurations can be used as initial designs for extended sources.