First-principles explanation for the vertex-sum representation of the wavefunction

Establish a first-principles explanation that the cosmological wavefunction for conformally coupled scalars in power-law FRW spacetimes corresponds to summing the basis functions assigned to all vertices of the convex geometry built from complete graph tubings, thereby justifying the vertex-sum representation independently of bulk time-evolution arguments.

Background

The paper shows that basis functions associated with directed time orderings and their collapses can be arranged on vertices, edges, and faces of convex geometries (hypercubes), and that the full wavefunction is obtained by summing the vertex-assigned functions. This construction simplifies differential equations and solution methods.

In discussing the emergence of time from boundary kinematics, the authors point out that, while the vertex-sum rule works and aligns with bulk causality, they lack a foundational principle that mandates this specific summation, signaling an unresolved conceptual underpinning for the representation.