Inevitability and Operational Status of the Gravitational Coupling in Scalar Potential Geometry

The exterior inverse–square form of Newtonian gravity is tightly constrained by symmetry and the harmonic character of the vacuum potential. We present a compact derivation showing that, under standard assumptions of spherical symmetry and asymptotic flatness, the exterior acceleration field necessarily takes the form a(r) = −K r−2ˆr, where the radial scaling is fixed while the overall normalization K remains undetermined by the vacuum field equation. We then clarify how the experimentallydefined gravitational coupling enters through operational calibration of source strength rather than through the vacuum potential equation itself. The treatment remainswithin conventional weak-field physics and does not introduce new gravitational dynamics. Implications for how any effective variation of the coupling would have to appear within this framework are briefly noted.