Mixed-point norm identity for alternating updates ($\Vert (p-\tau a,q){\Vert }^2-\Vert (p,q+\tau b){\Vert }^2$)

For two mixed points with difference $(p,q)$ and gradient difference $(a,b)$, the input difference is $(p,q+\tau b)$ and the output difference is $(p-\tau a,q)$. Their squared-norm difference is $-2\tau (⟨p,a⟩+⟨q,b⟩)+{\tau }^2(\Vert a{\Vert }^2-\Vert b{\Vert }^2)$. This exact identity is a reusable way to prove averagedness without assuming the two block maps are individually nonexpansive.