Unification and Emergence in Mathematical and Physical Systems

The Physics–Mathematics Transformation (PMT) framework asks whether explicitly defined correspondences between mathematical and physical systems can support justified inferences in either direction. This fourth paper investigates a narrower question: when can mathematical descriptions of composition and interaction account for collective behavior, and under what conditions can structural similarities across distinct physical systems support a limited form of unification? We distinguish microscopic composition, coarse-grained description, and emergent collective observables, and formulate criteria for admissible mappings between these levels. Particular attention is given to preservation of dynamics and observables, noncommuting operations, information loss under coarse-graining, and the role of physical realizability. Coupled dynamical systems and network models provide restricted settings in which candidate correspondences can be assessed against established analyses. The proposed approach requires explicit domains, assumptions, and error measures, together with independent mathematical or empirical validation and comparison with existing methods. It does not identify shared mathematical form with physical identity, establish a universal theory of emergence, or claim unification of fundamental interactions. Instead, it outlines a testable methodology for determining when structural correspondences clarify collective behavior and when they fail to support physical inference.