Ordering and infinite reducibility of QCSD consistency relations

Develop a general ordering lemma for quantum dilogarithm elements Ψa,b[hn] in the quantum group Gq and show that any QCSD consistency relation can be reduced to a trivial identity via successive applications of the quantum pentagon relation, possibly infinitely many times. Determine whether such ordering and infinite reducibility hold in full generality.

Background

In classical scattering diagrams, consistency relations can be reduced to trivial identities using commutative and pentagon relations, supported by an explicit ordering lemma for dilogarithm elements. This yields infinite reducibility of all associated dilogarithm identities.

For QCSDs, an analogous global ordering statement and reduction mechanism have not been established. Although specific examples exhibit finite or infinite reducibility by the quantum pentagon relation, a general theorem ensuring reducibility for all QCSDs is currently missing.