Interaction Residues and Localized Spectral Defects in Stratified Operadic Systems

Abstract

We develop a framework for studying how global spectral structure emerges from interacting local sectors in stratified operadic systems. The central object is the interaction residue, which measures the failure of exact spectral decomposition across interfaces. Under suitable localization assumptions, the global spectrum decomposes into local spectral sectors together with interface-generated residue contributions. The theory introduces a classification of spectral defects based on interface geometry and algebraic structure. Point interfaces produce isolated spectral contributions; line and surface interfaces produce extended spectral regimes. Non-semisimple operator structure generates nilpotent defects associated with Jordan blocks and generalized eigenspaces, yielding a two-dimensional defect taxonomy combining geometric localization with Jordan complexity. Several structural results are established, including interface localization, rigidity and vanishing criteria, refinement functoriality, and deformation stability. Under a local triviality condition, the residue is homotopy invariant, preserving its homology and Betti numbers throughout admissible deformations. The framework is illustrated through explicit operator and block matrix examples demonstrating how localized interactions generate localized spectral defects. The nilpotent sector connects the theory with classical operator theory via generalized eigenvectors, functional calculus, and perturbative Jordan splitting. Overall, the framework provides a unified viewpoint for understanding how interface interactions and non-semisimple structure influence global spectral behavior.