  We prove by explicit computation that 6-point matrix elements of D^4R^4 and
D^6R^4 in N=8 supergravity have non-vanishing single-soft scalar limits, and
therefore these operators violate the continuous E7(7) symmetry. The soft
limits precisely match automorphism constraints. Together with previous results
for R^4, this provides a direct proof that no E7(7)-invariant candidate
counterterm exists below 7-loop order. At 7-loops, we characterize the infinite
tower of independent supersymmetric operators D^4R^6, R^8, phi^2 R^8,... with
n>4 fields and prove that they all violate E7(7) symmetry. This means that the
4-graviton amplitude determines whether or not the theory is finite at 7-loop
order. We show that the corresponding candidate counterterm D^8R^4 has a
non-linear supersymmetrization such that its single- and double-soft scalar
limits are compatible with E7(7) up to and including 6-points. At loop orders
7, 8, 9 we provide an exhaustive account of all independent candidate
counterterms with up to 16, 14, 12 fields, respectively, together with their
potential single-soft scalar limits.
