= Solution
Build the spherical system by adding shells from the centre outwards. The <shell theorem> says that a shell of mass $dM$ at radius $r$ interacts with the already assembled interior mass through potential $-GM(r)/r$. Thus its contribution is $dW=-GM(r)dM(r)/r$. Each pair is counted when its outer member is added, so no further factor of one half is needed:
$$
W=-G\int_0^\infty\frac{M(r)}r\,dM(r)
=-\frac G2\int_0^\infty\frac1r\,d[M(r)^2].
$$
Integration by parts gives
$$
W=-\frac G2\left[\frac{M(r)^2}r\right]_0^\infty
-\frac G2\int_0^\infty\frac{M(r)^2}{r^2}\,dr.
$$
For an isolated finite-mass system without a singular central point mass, the endpoint term vanishes provided the central cusp has finite binding energy. Consequently
$$
\boxed{W=-\frac G2\int_0^\infty\frac{M(r)^2}{r^2}\,dr.}
$$
This is the <spherical binding-energy mass integral>. If a central singularity makes the endpoint or integral divergent, the finite-energy assumption must be checked rather than silently discarding that term.
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