Maximally supersymmetric amplitudes at infinite loop momentum

We investigate the asymptotically large loop-momentum behavior of multiloop amplitudes in maximally supersymmetric (N=4, 8) quantum field theories in four dimensions. We check residue-theorem identities among color-dressed leading singularities in N=4 supersymmetric Yang-Mills theory to demonstrate the absence of poles at infinity of all n-point MHV amplitudes through three loops. Considering the same test for N=8 supergravity leads us to discover that this theory does support nonvanishing residues at infinity starting at two loops, and the degree of these poles grow arbitrarily with multiplicity. This causes a tension between simultaneously manifesting ultraviolet finiteness - which would be automatic at two loops for representation obtained by color-kinematic duality - and gauge invariance - which would follow from unitarity-based methods.