Construct a geometry whose facets correspond to individual perimeter poles and whose canonical form equals the wavefunction

Construct a higher-dimensional positive geometry for Tr(φ^3) cosmological wavefunctions whose facets are in one-to-one correspondence with individual perimeter poles (rather than pairs), such that the canonical form of this geometry directly reproduces the full wavefunction.

Background

Graph-associahedra encode single diagrams, while cosmohedra unify all diagrams but associate pairs of perimeter poles to facets, necessitating a special prescription to obtain the wavefunction. By contrast, cosmological polytopes for single graphs have a facet per pole but do not obviously combine all diagrams in a single object.

The authors ask whether a bigger or different geometry could restore a direct facet-to-pole correspondence so that the canonical form alone produces the wavefunction, paralleling the amplitude/associahedron paradigm.