= Solution
An inverse-square point-source field has constant flux through enclosing spheres because their <area> grows as $r^2$. In <electrostatics>, <Gauss's law> is $\oint_S\mathbf E\cdot d\mathbf S=Q_{\rm enc}/\varepsilon_0$, or $\nabla\cdot\mathbf E=\rho_e/\varepsilon_0$. For a spherical charge distribution, <rotational symmetry> makes the exterior field radial and constant in magnitude on a concentric sphere, giving $\mathbf E=Q\hat r/(4\pi\varepsilon_0r^2)$. Symmetry is needed to take the field outside the <surface integral>; <Gauss's law> alone does not make an arbitrary charge distribution spherical.
For <gravity> the corresponding relations are $\oint_S\mathbf g\cdot d\mathbf S=-4\pi GM_{\rm enc}$ and $\nabla\cdot\mathbf g=-4\pi G\rho_m$. A spherical <mass> has exterior field $\mathbf g=-GM\hat r/r^2$. Inside a uniform sphere, $M_{\rm enc}(r)=4\pi\rho_mr^3/3$, so $\boxed{\mathbf g=-(4\pi G\rho_m/3)r\hat r}$. Inside an empty spherical shell the enclosed <mass> is zero and symmetry gives zero field. These examples distinguish an inverse-square exterior field from a field inside distributed matter.
The law also treats other symmetries, but their field-distance dependence differs: an infinitely long <line charge> has cylindrical rather than spherical spreading and a $1/r$ field. Flux conservation and geometry must both be used, rather than assuming every <Gaussian surface> leads to an inverse-square answer.
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