Description This preprint is Part III of the Phase Geometry Series: Superconductivity and Weak Gravity.It develops a phenomenological phase-field model of Newtonian gravity in which the gravitational potential is sourced by the gradient energy of a static phase field. Any static phase configuration φ(x)\varphi(\mathbf{x})φ(x) defines an effective mass density ρeff(x)∝∣∇φ(x)∣2,\rho_{\mathrm{eff}}(\mathbf{x}) \propto |\nabla \varphi(\mathbf{x})|^2,ρeff(x)∝∣∇φ(x)∣2, which enters the standard Poisson equation for the Newtonian potential Φ(x)\Phi(\mathbf{x})Φ(x). Localised phase defects carry a finite effective mass MeffM_{\mathrm{eff}}Meff and generate an asymptotically Newtonian potential Φ(r)≃−GMeff/r\Phi(r) \simeq -G M_{\mathrm{eff}}/rΦ(r)≃−GMeff/r. Test particles move in Φ\PhiΦ according to ordinary Newtonian mechanics, so the model reproduces the usual weak-field behaviour of gravity while providing a phase-based microscopic picture for the source term. The work also couples this phase-field gravity to phase clocks, with a focus on Josephson-based superconducting oscillators (“Josephson clocks”). In the weak-field limit, the same potential Φ(x)\Phi(\mathbf{x})Φ(x) that is generated by the phase texture also determines the local ticking rate of phase clocks via the standard relation between Φ\PhiΦ and g00g_{00}g00. This yields a unified phase-based description of effective mass, gravitational potential, and clock redshift. The paper is intended as a conservative “Newtonian floor” for phase geometry: a non-relativistic, static, weak-field framework that is mathematically simple yet compatible with standard Newtonian gravity. It is particularly suited as a playground for analogue gravity and toy models in superconductors and other phase-ordered media, where the underlying phase field can be engineered and visualised. This work is part of the series Phase Geometry Series: Superconductivity and Weak Gravity (Parts I–III and Overview), concept DOI: 10.5281/zenodo.17584768. Keywords: phase geometry, Newtonian gravity, phase field, effective mass, Josephson clocks, analogue gravity, superconductivity, weak gravity