  We introduce a decomposition of D-dimensional vector amplitudes in terms of
building blocks that preserve all partial amplitude relations of the parent
gauge-theory. Using this decomposition, we derive a new set of amplitude
relations between nonlinear sigma model pions and the pure-scalar sector of
Yang-Mills-scalar theory. These new relations indicate an equivalence between
Born-Infeld duality invariance and that of Einstein-Maxwell theory, considering
their known double-copy construction from the two scalar theories with Yang
Mills. This observation motivates a general framework for constructing
duality-invariant effective field theory observables at any multiplicity using
the building blocks in our amplitude decomposition. Finally, we conjecture a
subtle relationship between these duality-invariant building blocks and the
bosonic sector of N=4 super Yang-Mills.
