  We prove a range of $L^p$ bounds for singular Brascamp-Lieb forms with
cubical structure. We pass through sparse and local bounds, the latter proved
by an iteration of Fourier expansion, telescoping, and the Cauchy-Schwarz
inequality. We allow $2^{m-1}<p\le \infty$ with $m$ the dimension of the cube,
extending an earlier result that required $p=2^m$. The threshold $2^{m-1}$ is
sharp in our theorems.
