Equivalence of the three definitions of the flat-space wavefunction coefficient

Establish that for any Feynman diagram G (in the cosmological setting where ψ_flat^G is defined), the rational function ψ_flat^G(X,Y) computed by (1) the Feynman rules of Arkani-Hamed et al. (Differential Equations for Cosmological Correlators), (2) the recursion formula of Arkani-Hamed–Benincasa–Postnikov, and (3) the canonical form of the cosmological polytope P_G, are identical as functions of the energy variables (X,Y).

Background

In the cosmology section, the authors define the flat-space wavefunction coefficient ψ_flat^G associated to a Feynman diagram G as a rational function of vertex and edge energies (X,Y). They explain that ψ_flat^G can be computed in three distinct ways: via Feynman rules, via a recursion formula, or as the rational function whose differential form is the canonical form of the cosmological polytope P_G.

The paper notes a physics conjecture asserting that these three constructions coincide for any diagram, thereby unifying diagrammatic, recursive, and positive-geometry-based perspectives on cosmological correlators. Proving this identity would firmly connect the polytope-based canonical form to traditional QFT computations and recursion approaches.