Boundary-first justification for using graph tubings

Establish a boundary-only, first-principles justification for the relevance of the geometry of complete tubings of marked graphs in organizing the differential equations and basis functions for cosmological correlators of conformally coupled scalars in power-law FRW spacetimes; specifically, explain intrinsically within boundary kinematics why this auxiliary tubing geometry should be considered without reference to bulk time evolution.

Background

Throughout the paper, the authors develop a boundary-centric description using complete tubings of marked graphs to represent both letters (singularities) and basis functions for the differential equations that govern wavefunction coefficients. This tubing-based organization yields a simple merger rule and a local geometric structure (collections of hypercubes and zonotopes) that mirrors the causal collapse of time-ordered propagators in the bulk.

In the Emergent Time subsection, the authors reverse the perspective and ask whether one can reconstruct these structures purely from boundary kinematics. They explicitly note that, despite the appeal of the tubing geometry, they lack an independent boundary-first reason for why this auxiliary structure is the right one to paper, highlighting a conceptual gap between the boundary formalism and bulk-time intuition.