The Ladder of Depth Structure B; Axiomatic Basis and Mathematical Ontology– Formalization of Information Conservation, Computability, and Self-Referential Closure

The theory of structural openness aims to provide a rigorous mathematical foun
dation for adaptive systems that does not depend on specific application domains,
in order to answer the fundamental question: “What determines the mathematical
boundary of rule modification?” This paper establishes three independent axioms
of the theory: information conservation, computability, and self-referential closure,
and proves their logical independence from one another. Based on this axiomatic
foundation, we construct a spectral-categorical-arithmetic triple mathematical on
tology. Spectral triples describe the differential structure of the rule space within a
single level; modification functors together with Cayley–Dickson doubling reveal the
categorical syntax of transitions between levels; modular forms and Deligne’s Galois
representations 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-consistent
and operable unified syntax for cross-disciplinary studies of rule transitions.