Invoking increasingly higher dimension operators to encode novel UV physics in effective gauge and gravity theories traditionally means working with increasingly more finicky and difficult expressions. We find that the duality between color and kinematics provides a powerful tool towards drastic simplification. Local higher-derivative gauge and gravity operators at four points can be absorbed into simpler higher-derivative corrections to scalar theories, requiring only a small number of building blocks to generate gauge and gravity four-point amplitudes to all orders in mass dimension.