  We consider the tetrahedral three-loop diagram in $E_d$ exceptional field
theory evaluated as a scalar diagram for four external gravitons. At lowest
order in momenta, this diagram contributes to the $\nabla^6 R^4$ term in the
low-energy effective action for M-theory. We evaluate explicitly the sums over
the discrete exceptional field theory loop momenta that become sums over
1/2-BPS states in the compact exceptional space. These sums can be rewritten as
Eisenstein series that solve the homogeneous differential equations that
supersymmetry implies for the $\nabla^6 R^4$ coupling. We also show how our
results, even though sums over 1/2-BPS states, are consistent with expected
1/4-BPS contributions to the couplings.
