Formal duality for the ordinary q-confluence expansion

We prove the ordinary formal duality identity in Hirose's construction of q-confluence relations, stated as Conjecture 38 in arXiv:2609.01458v1. The proof begins with a coloured normal form for the original recursive map. We establish membership in the Bradley–Zhao word space and derive an interval transport identity for the coefficients of this normal form. These coefficients are then represented by a noncommutative generating series over a commutative auxiliary algebra. A difference-module argument gives linear independence of the right polylogarithms at each fixed q, allowing known evaluated duality to determine the output coefficient of f₂. After a canonical normalization, that coefficient determines the entire generating series. This yields the formal duality identity for every ordinary admissible index.