Explain the origin of the cosmohedron canonical-form prescription for the wavefunction

Establish the deeper origin and precise justification for computing the Tr(φ^3) cosmological wavefunction by taking the canonical form of the permuto-cosmohedron, assigning to each facet a product of two perimeter poles, and retaining only the terms with simple poles. Clarify why this prescription reproduces the correct wavefunction and identify its underlying principle.

Background

The paper introduces cosmohedra and permuto-cosmohedra as positive geometries encoding the combinatorics of the cosmological wavefunction and correlators in Tr(φ^3) theory. Unlike the amplituhedron/associahedron story for scattering amplitudes, extracting the full wavefunction from geometry requires a novel prescription: facet poles are replaced by products of paired perimeter poles (associated with subpolygons), and only simple-pole terms are kept. While this method yields correct results in examples, its conceptual foundation is not yet understood.

The authors emphasize that the new prescription departs from standard positive-geometry canonical-form interpretations, and pose the need for a rigorous explanation of why and how this rule works, potentially revealing new principles in cosmological positive geometry.