Scaffolding Residues in Yang-Mills-Scalar \`a la CHY

Motivated by recent work by Arkani-Hamed et al. arXiv:2401.00041, we compute the ''scaffolding'' residue of $2n$-scalar Yang-Mills-Scalar amplitudes to obtain pure $n$-gluon amplitudes \`a la Cachazo-He-Yuan (CHY). In particular, we show how the Pfaffian of $\Psi$, which is a matrix rich in structure, emerges from that of the simple $A$ matrix. The same CHY computation straightforwardly produces $n$-graviton amplitudes from $2n$-scalar amplitudes in the Einstein-Maxwell-Scalar theory. We also consider partial ''scaffolding'' residues, i.e., general multi-collinear limits and their interplay with color-dressed amplitudes.