Axiomatic Stability and Canonical Decomposition in Second–Variation Geometry

This paper establishes that structural persistence across mathematics, physics, computation, and information theory is governed by a canonical decomposition arising intrinsically from second-variation geometry, wherein the operator δ² yields exactly three irreducible stability modes: U (universal invariance, kernel-origin structure), P (projectional coherence, shadow-invariant structure), and V (variational presentation, coordinate-dependent residue). These modes are mutually orthogonal, functorial, complete, and uniquely determined, ensuring that phenomena such as universality, lawfulness, robustness, coordinate and gauge invariance, finite-exception behaviors, algebraic shadows, and computational hierarchy separations emerge as necessary consequences of this stratification. Differential curvature, entropy, number-theoretic regularity, and complexity-class hierarchies each manifest the invariant progression Origin ⇒ Coherence ⇒ Appearance, preserved under all admissible perturbations. Stability thus constitutes an ontological principle, revealing the essence of structures rather than merely their behavioral properties. The developed Stability Theory is mathematically exhaustive, admitting no extensions, refinements, or supplementary invariance modes. Far from being an auxiliary framework within mathematics and physics, it forms the foundational structural identity from which these disciplines originate, unifying stability as the universality-invariant projection of curvature.