An involution on Dyck paths in the region $\text{defc}\le \min (\text{area},\text{dinv})$ that interchanges area and dinv

We construct an explicit involution on Dyck paths satisfying $\text{defc}\le \min (\text{area},\text{dinv})$ that interchanges area and dinv. The construction relies on a number of new combinatorial objects developed here and two main external tools: the dual Dyck insertion of \cite{Hawkes26} and the Garsia–Milne involution principle~\cite{GarsiaMilne81}, as formulated by Doyle~\cite{Doyle19}.