Spherical binding-energy mass integral ($W=-\frac{G}{2}{\int }_0^{\infty }M(r{)}^2{r}^{-2}dr$)

Assembling a spherical mass distribution in shells gives gravitational potential energy $W=-G\int M(r)dM(r)/r$. Integration by parts gives the displayed identity when $M(r{)}^2/r$ vanishes at both endpoints. It includes each interacting pair once; no extra factor of one half belongs in the original shell-assembly integral.