  We compute a subset of three, velocity-independent four-loop (and fourth
post-Newtonian) contributions to the harmonic-coordinates effective action of a
gravitationally interacting system of two point-masses. We find that, after
summing the three terms, the coefficient of the total contribution is rational,
due to a remarkable cancellation between the various occurrences of $\pi^2$.
This result, obtained by a classical field-theory calculation, corrects the
recent effective-field-theory-based calculation by Foffa et al.
[arXiv:1612.00482]. Besides showing the usefulness of the saddle-point approach
to the evaluation of the effective action, and of x-space computations, our
result brings a further confirmation of the current knowledge of the fourth
post-Newtonian effective action. We also show how the use of the generalized
Riesz formula [Phys. Rev. D 57, 7274 (1998)] allows one to analytically compute
a certain four-loop scalar master integral (represented by a four-spoked wheel
diagram) which was, so far, only numerically computed.
