Amplituhedra and origami, I: tree level

Abstract

We establish a precise correspondence between points of the $m=4$ tree momentum amplituhedron and origami crease patterns. As an application, we prove that the BCFW (Britto-Cachazo-Feng-Witten) cells triangulate the $m=4$ tree amplituhedron both in momentum space and in momentum-twistor space. As another application, we show that every nondegenerate weighted planar bipartite graph $\Gamma$ admits a t-embedding, i.e., an embedding of the planar dual of $\Gamma$ such that the sum of angles of white (equivalently, black) faces around each vertex is equal to $\pi$.

arXiv comment79 pages, 16 figures. First paper in the series (see arXiv:2606.03439 for part 2). v2: Major revision. Added a proof of the BCFW tiling conjecture in momentum-twistor space via T-duality and a new proof of the origami-amplituhedron correspondence using double-dimer model identities. Full changelog at the end of the introduction. v3: Minor updates