  We analyze constraints from perturbative unitarity and crossing on the
leading contributions of higher-dimension operators to the four-graviton
amplitude in four spacetime dimensions, including constraints that follow from
distinct helicity configurations. We focus on the leading-order effect due to
exchange by massive degrees of freedom which makes the amplitudes of interest
infrared finite. In particular, we place a bound on the coefficient of the
$R^3$ operator that corrects the graviton three-point amplitude in terms of the
$R^4$ coefficient. To test the constraints we obtain nontrivial effective
field-theory data by computing and taking the large-mass expansion of the
one-loop minimally-coupled four-graviton amplitude with massive particles up to
spin 2 circulating in the loop. Remarkably, we observe that the leading EFT
coefficients obtained from both string and one-loop field-theory amplitudes lie
in small islands. The shape and location of the islands can be derived from the
dispersive representation for the Wilson coefficients using crossing and
assuming that the lowest-spin spectral densities are the largest. Our analysis
suggests that the Wilson coefficients of weakly-coupled gravitational physical
theories are much more constrained than indicated by bounds arising from
dispersive considerations of $2 \to 2$ scattering. The one-loop four-graviton
amplitudes used to obtain the EFT data are computed using modern amplitude
methods, including generalized unitarity, supersymmetric decompositions and the
double copy.
