This paper presents a formal derivation of Newton's inverse-square law as an emergent phenomenon within a complex scalar wave medium, providing a specific physical mechanism for gravity within the analogue gravity programme. By defining five fundamental physical requirements for a vacuum substrate, the author derives a master wave equation and a sextic symmetry-breaking potential as the minimum-degree polynomial consistent with four vacuum requirements. Matter is identified with localised isotropic sinc² standing-wave nodes of the field, which deplete the local background amplitude; the resulting static perturbation satisfies the Laplace equation and falls as the inverse of distance. The ambient wave field, treated as a zero-mean complex Gaussian ensemble, self-tunes to a marginal-mass fixed point at which the long-range mode is massless, with the critical variance obtained by a Gaussian-moments calculation. A second node placed in the resulting gradient experiences an attractive inverse-square force with a universal gravitational coupling. The derivation is strictly limited to the static weak-field limit and does not reproduce tensor-mediated phenomena (gravitational waves, light bending magnitude, Mercury's perihelion precession, binary pulsar decay), positioning the work as a contribution to the analogue gravity programme — a physical mechanism underlying Newtonian gravity, not a replacement for General Relativity.
Ian James Fahy (Mon,) studied this question.