  We show that large gauge transformations modify the structure of momentum
conservation leading to non-vanishing three-point amplitudes in a simple toy
model of a gravitational wave event. This phenomenon resolves an apparent
tension between perturbative scattering amplitude computations and exact
methods in field theory. The tension is resolved to all orders of perturbation
theory once large gauge effects are included via a modified LSZ prescription;
if they are omitted, perturbative methods only recover a subset of terms in the
full non-perturbative expression. Although our results are derived in the
context of specific examples, several aspects of our work have analogues in
dynamical gravitational scattering processes.
