Determine the general cluster structure of cosmological kinematic flow

Determine whether a single cluster-algebra structure governs the singularity alphabets of cosmological kinematic-flow differential equations for arbitrary graphs, beyond the established type-$A$ structure for path graphs.

Background

Kinematic-flow differential equations organize cosmological wavefunction integrals in terms of logarithmic letters associated with total- and partial-energy singularities. For path graphs, these letters are generated by cluster algebras of type $A$, paralleling the alphabetic structures found in planar scattering amplitudes.

The unresolved issue is whether this organization persists for general tree graphs and loop graphs, which would provide an all-graph classification of the functions generated by cosmological correlator integrals.