Domain Emergence from Mathematical Constraint Structures: A Structural Bridge Between Mathematics and Domain Instantiation Within the Paton System

This paper establishes how domains emerge from mathematical structure within the Paton System. It shows that mathematics provides domain-independent structure, while domain-specific behaviour arises from constraint context. Domains are therefore constrained projections of shared mathematical structure rather than independent systems. This clarifies the relationship between mathematics and domain instantiation across physics, biology, computation, and other fields.