  We report on progress in understanding how to construct color-dual multi-loop
amplitudes. First we identify a cubic theory, semi-abelian Yang-Mills, that
unifies many of the color-dual theories studied in the literature, and provides
a prescriptive approach for constructing $D$-dimensional color-dual numerators
through one-loop directly from Feynman rules. By a simple weight counting
argument, this approach does not further generalize to two-loops. As a first
step in understanding the two-loop challenge, we use a $D$-dimensional
color-dual bootstrap to successfully construct globally color-dual local
two-loop four-point nonlinear sigma model (NLSM) numerators. The double-copy of
these NLSM numerators with themselves, pure Yang-Mills, and $\mathcal{N}=4$
super-Yang-Mills correctly reproduce the known unitarity constructed integrands
of special Galileons, Born-Infeld theory, and Dirac-Born-Infeld-Volkov-Akulov
theory, respectively. Applying our bootstrap to two-loop four-point pure
Yang-Mills, we exhaustively search the space of local numerators and find that
it fails to satisfy global color-kinematics duality, completing a search
previously initiated in the literature. We pinpoint the failure to the bowtie
unitarity cut, and discuss a path forward towards non-local construction of
color-dual integrands at generic loop order.
