S3 Field Framework From Electromagnetism to a Prediction for G

This work develops a single-field framework in which the Newtonian gravitational constant is linked to a coarse-grained “calm” sector of an (S^3)-valued field. A homogenization procedure applied to microscopic field gradients defines an effective vacuum stiffness and yields a Poisson constraint for the calm potential, producing an exterior (1/r) Newtonian tail for localized sources. The paper also records a conditional loop-closure relation connecting the gravitational coupling to the fine-structure constant, a microscopic mass scale, and a screening coefficient, with all assumptions and remaining open normalization steps stated explicitly.