Relational Geometric Mechanics III: Gravity — Newton's Law, the Gravitational Coupling Constant, and the General Relativistic Correspondence

Paper III of the Relational Geometric Mechanics series. Derives Newton's law of gravitation from the amplitude component of the RGM wave signature. The gravitational coupling constant is identified as G_N ~ L_QF² from the dimensional structure of the framework. Gravitational attraction is shown to be a structural consequence of the amplitude coupling — not postulated. The weak-field general relativistic correspondence is established as the Green's function solution to the linearized Einstein equations. The full nonlinear GR proof is deferred to Paper VI of this series.