The duality between color and kinematics, and associated double copy construction has allowed some of the sharpest perturbative calculations in both supergravity theories and classical invariant observables relevant to next generation gravitational wave physics. I will talk about ways related insights can be applied to understanding the behavior of EFT operators in gauge and gravity theories. Pairs of color-dual numerators can be composed to generate new color-dual numerators. This presents an opportunity to efficiently climb to amplitudes of arbitrarily high mass dimension --- without resorting to ever more expensive Ansätze --- thereby spanning the tower of double-copy consistent gauge and gravity operators with a small number of building blocks. I will apply this technology towards discovering some hints about the color-dual fate of N=4 supergravity.