Theory of Σ-Stratified Kernels and Their Approximation Properties

This work introduces a new class of mathematical objects — Σ-stratified kernels. We define them through a system of structural conditions, generalizing the concepts of duality, dynamics, and holonomy to stratified spaces. The main theorems prove the existence and density properties of an associated universal manifold, a universal approximation theorem for continuous functions on this manifold, and a rigidity theorem classifying kernels via a complete system of spectral invariants. Explicit examples demonstrate the non-emptiness and breadth of the introduced class.