The theory of structural openness aims to provide a rigorous mathematical foundation for adaptive systems that does not depend on specific application domains,in order to answer the fundamental question: “What determines the mathematicalboundary of rule modification?” This paper establishes three independent axiomsof the theory: information conservation, computability, and self-referential closure,and proves their logical independence from one another. Based on this axiomaticfoundation, we construct a spectral-categorical-arithmetic triple mathematical ontology. Spectral triples describe the differential structure of the rule space within asingle level; modification functors together with Cayley–Dickson doubling reveal thecategorical syntax of transitions between levels; modular forms and Deligne’s Galoisrepresentations lock the arithmetic rigidity of the hierarchical compression ratio.The paper strictly distinguishes axioms, theorems, and constructive assumptions;all core concepts are defined internally, aiming to provide a logically self-consistentand operable unified syntax for cross-disciplinary studies of rule transitions.