A Universal Theory of Spectral Propagation for Compositional Operator Networks

Abstract

Classical spectral theory lacks a framework for understanding how spectra propagate through compositional systems like deep neural networks, feedback control loops, and quantum circuits. This paper develops a universal theory governed by three invariants: the operadic spectrum (local spectral data), spectral derivatives (perturbation sensitivity), and interaction residue (emergent interface-generated content). We prove three main theorems: the Spectral Propagation Theorem decomposes global output into propagated local spectra, residues, and derivative corrections; the Stability Theorem introduces the SOC stability radius and condition number; and the Universality Theorem shows any reasonable propagation rule is uniquely determined by the three invariants. These results provide a coordinate-free, representation-invariant language for spectral analysis of compositional operator systems.