  We investigate the asymptotically large loop-momentum behavior of multi-loop
amplitudes in maximally supersymmetric quantum field theories in four
dimensions. We check residue-theorem identities among color-dressed leading
singularities in $\mathcal{N}=4$ supersymmetric Yang-Mills theory to
demonstrate the absence of poles at infinity of all MHV amplitudes through
three loops. Considering the same test for $\mathcal{N}=8$ supergravity leads
us to discover that this theory does support non-vanishing 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 in a representation obtained
by color-kinematic duality---and gauge invariance---which would follow from
unitarity-based methods.
