讲座内容：

In this repart, we analyze the intrinsic flag structure for operators in the Cowen--Douglas class, formulating the condition in terms of the associated holomorphic Hermitian vector bundle. We prove that the \mathcal{F}B_n operator class represents the exact subclass of Carlson--Clark extensions where the abstract module derivation reduces to a geometric chain rule. The rank-2 theory, based on a single extension cocycle, is developed in detail and then extended to the rank-r setting, where we prove that the level-wise strong intertwining condition automatically forces a ``crossed intertwining'' between consecutive cocycles---the operator-theoretic expression of the commutativity of mixed partial derivatives.