Origin-regular spherical Poisson flux law ($F(x)=\frac{x}{|x{|}^3}{\int }_0^{|x|}f(s)s^{2}ds$)

For a spherically symmetric gradient field regular at the origin and satisfying Poisson equation, the divergence theorem on a ball determines its radial magnitude from the enclosed source integral. It depends only on the source at smaller radii. Regularity excludes the source-free but origin-singular addition $Cx/|x{|}^3$; without that condition, solving the local equation on a punctured domain leaves this extra term undetermined.