Spectral Topology of Neural Operators: A Unified Theory

This work establishes a novel mathematical framework that unifies neural network architecture with the analytical behavior of the neural operators they define. We model neural architectures as topological spaces specifically, connection complexes whose homological invariants, namely Betti numbers and Euler characteristic, govern fundamental properties of the associated neural operator. Our central theoretical contribution is a rigorous topological-spectral correspondence: we prove a canonical isomorphism between the homology of an architecture's connection complex and the Morse--Conley homology of the operator's nonlinear spectrum. This leads to a unifying index theorem for neural operators, analogous to the Atiyah–Singer theorem, which equates topological, spectral, and dynamical indices. Key results demonstrate that architectural topology controls expressive power and stability. We introduce the concept of \emph{topological capacity} a homological measure bounding approximation error and generalization performance and derive spectral radius bounds that explain training stability. The framework provides computable invariants for practical architectures, revealing how residual connections induce topological simplification and how dropout acts as spectral regularization. These advances shift the paradigm from parameter-centric to topology-centric analysis, offering principled guidelines for architectural design. Ultimately, we show that neural architectures are geometric objects whose learning capabilities are governed by universal mathematical invariants, with implications for architecture search, stability analysis, and approximation theory.