  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 demonstrate that local
higher derivative supersymmetric-compatible operators at four-points can be
absorbed into simpler higher-derivative corrections to scalar theories, which
generate the predictions of Yang-Mills and Gravity operators by suitable
replacements of color-weights with color-dual kinematic weights as per
Bern-Carrasco-Johansson double-copy. We exploit that Jacobi-satisfying
representations can be composed out of other Jacobi-satisfying representations,
and show that at four-points only a small number of building blocks are
required to generate the predictions of higher-derivative operators. We find
that this construction saturates the higher-derivative operators contributing
to the four-point supersymmetric open and closed-string tree amplitudes,
presenting a novel representation of the four-point supersymmetric open string
making this structure manifest, as well as identifying the only four additional
gauge-invariant building blocks required to saturate the four-point bosonic
open string.
