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Abstract

Jim Pitman's~(1999) celebrated coagulation-fragmentation duality for the PD($\alpha$,$\theta$) family of laws of Pitman and Marc Yor~(1997) has resisted generalization beyond its canonical setting. We resolve this by introducing a novel, four-part coupled process built upon the Poisson Hierarchical Indian Buffet Process (PHIBP), a framework developed for modeling microbiome species sampling. This approach yields a tractable generalization of the duality in two fundamental directions: to processes driven by arbitrary subordinators and to the previously uncharacterised multi-group ($J \ge 1$) setting, providing explicit laws for both. The static, fixed-time partitions are revealed to be a single projection of an inherently dynamic system. This new construction simultaneously defines: (i) the fine-grained partition, (ii) its coagulation operator, (iii) a forward-in-time system of coupled, time-homogeneous fragmentation processes in the sense of Jean Bertoin~(2006), and (iv) a dual, backward-in-time structured coalescent that drives simultaneous, across-group merger events. All four components are governed by a unified compositional structure, yielding their exact compound Poisson representations. The hallmark of this work is its circumvention of direct, and often intractable, analysis on mass and integer partition spaces. By shifting the problem to this transparent framework, the generalized duality emerges as a natural consequence of the architecture itself.