  Bipartite on-shell diagrams are the latest tool in constructing scattering
amplitudes. In this paper we prove that a Britto-Cachazo-Feng-Witten
(BCFW)-decomposable on-shell diagram process a rational top-form if and only if
the algebraic ideal comprised of the geometrical constraints is shifted
linearly during successive BCFW integrations. With a proper geometric
interpretation of the constraints in the Grassmannian manifold, the rational
top-form integration contours can thus be obtained, and understood, in a
straightforward way. All rational top-form integrands of arbitrary higher loops
leading singularities can therefore be derived recursively, as long as the
corresponding on-shell diagram is BCFW-decomposable.
