E7(7) constraints on counterterms in N=8 supergravity

We prove by explicit computation that 6-point matrix elements of D4R4 and D6R4 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 R4, 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 D4R6, R8, φ2R8, … 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 D8R4 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.