Paper 14: Dynamical Origin of the Scale-Space Poisson Equation: Newton's Law from Linearised 5D Field Equations

Paper 1 of the scale-space series introduced the 4D Poisson equation governing the scale potential as a phenomenological postulate, explicitly noting that its derivation from the 5D field equations remained an open problem. Paper 11 computed the full 5D Einstein tensor of the corrected metric and determined the required stress-energy component κ5Ttt = −3c^2(L+2)/L^4, but did not source it with matter. The present paper closes the gap between these two results. We write down the linearised 5D Einstein equations with a non-relativistic point mass as the source, impose the Lorenz gauge condition, and show that the tt sector decouples from the spatial/scale sector. The tt field equation reduces exactly to the 4D Laplace–Beltrami equation of Paper 1 — the same operator, applied to the trace-reversed metric perturbation, driven by the point mass source. The solution reproduces Newton’s inverse-square law with the same exponential scale-decay factor e^(−3|s−sM|/L) as the kinematic derivation of Paper 1, but now as a dynamical consequence of the field equations rather than a postulate. The coupling constant consistency condition κ5 = 2G/(c^2L^2) is derived and identified as a sharpened open problem: the body-dependence of L = Rc^2/(GM) means κ5 cannot yet be fixed independently of the matter it sources, pointing toward a Kaluza–Klein style reduction as the natural next step.