Newtonian Matching from the Calm Poisson Constraint

This note derives the Newtonian Poisson equation from a static weak-field “calm” energy functional and shows that Newton’s constant is fixed by the infrared gravitational stiffness,[G=\frac{c^{4}}{4\pi K_{\rm grav}}.]Assuming a quadratic infrared energy in (\nabla\varphi) and minimal coupling to the nonrelativistic mass density (\rho), the Euler–Lagrange condition yields (K_{\rm grav}\nabla^{2}\varphi=c^{2}\rho), giving the exterior (1/r) potential with the correct Green’s-function normalization. The remaining microscopic task in the S³-field framework is to compute (K_{\rm grav}) (and establish infrared masslessness of the calm mode to avoid Yukawa screening), after which (G) follows directly.