Origins and kinematic realization of the geometry organizing sums over exchange channels

Determine the origin of the geometric structure that organizes the sum over exchange-channel Feynman graphs in colored scalar theories such as tr φ^3, and ascertain whether this structure can be realized directly in a space of boundary kinematics; in particular, clarify how shared functions across channels and the nested associahedra picture arise from first principles.

Background

In Appendix B, the authors extend their geometric organization from single graphs to sums over exchange channels in tr φ^3 theory. They observe that certain source functions are shared among channels and propose a nested associahedra picture: squares (from individual three-site graphs) attached to vertices of a pentagonal associahedron for five-point functions, with shared edge/face functions linking channels.

Despite these observations, the authors explicitly state that the geometric structure underlying the sum over graphs is still not fully understood and raise the question of whether it admits a direct realization in kinematic space, indicating an unresolved foundational issue.