Boundary virial term of a singular isothermal sphere ($4\pi R^3\rho (R){\sigma }_r^2$)

A finite spherical subvolume has radial-pressure virial surface term $4\pi R^3\rho (R){\sigma }_r^2$. The infinite singular isothermal sphere has divergent total mass and energies, so a finite isolated virial theorem cannot omit this term. Different orbital distributions with the same density can have different radial boundary pressures.