Axiomatic Stability and Canonical Decomposition in Second–Variation Geometry

Version 2 represents a complete structural revision of the original manuscript. The paper develops an axiomatic theory of stability based on admissible second--variation structures and establishes a canonical decomposition theorem for structural data. Under the stated axioms, every admissible datum admits an intrinsic decomposition into three components: S = U(S) + P(S) + V(S), where U(S), P(S), and V(S) denote the origin-level, projectional, and variational components, respectively. The symbol “+” in this description is only a plain-text substitute for the structural decomposition symbol used in the paper; it does not mean an ordinary linear sum. The decomposition is shown to be intrinsic to the second--variation structure, unique up to the corresponding equivalence relations, minimal within the hierarchy, and exhaustive under the stated axioms. A central result of the paper is that no additional intermediate stability layer is compatible with the canonical hierarchy. The three levels therefore provide a complete classification of stability modes expressible within the framework. Compared with earlier versions, Version 2 substantially reorganizes the theory. The presentation is now centered on the decomposition theorem itself rather than on domain-specific applications. Geometric, arithmetic, and computational examples are included only as representative structural profiles illustrating the classification scheme and do not serve as evidence for the theorem. The manuscript is intended as a self-contained axiomatic study of stability in second--variation geometry and focuses on the structural classification problem independently of particular analytic, physical, or computational realizations.