Spectral Operadic Calculus: Norm-Analytic Functor Calculus

TL;DR

This paper introduces an operadic spectral calculus for functors, providing analytic tools for convergence, polynomial behavior, and derivative-based classification, extending classical spectral theory to nonlinear, structured categories.

Contribution

It develops an operadic spectrum framework that enables analytic control and classification of functors, bridging spectral theory, operadic algebra, and calculus of functors.

Findings

Established convergence results with explicit exponential error bounds.

Proved that derivatives form structured algebraic objects with a chain rule.

Reconstructed analytic functors from their derivative data, enabling classification.

Abstract

Classical spectral theory provides powerful tools for analyzing linear operators, but does not extend naturally to nonlinear or compositional settings. In particular, there is no general way to transport spectral invariants in a functorial manner across structured categories. In earlier work, we showed that this failure is fundamental and introduced an operadic notion of spectrum that provides a canonical replacement. In this paper, we develop the analytic consequences of this construction and show that the operadic spectrum acts as a control parameter for a calculus of functors. We establish a criterion for polynomial behavior based on higher cross-effects, and prove convergence results for the associated Taylor tower, including explicit exponential error bounds. We further show that the derivatives of a functor form a structured algebraic object with symmetric and operadic features, and satisfy a chain rule governed by a natural composition operation (operadic plethysm). This leads to a reconstruction theorem, showing that analytic functors are completely determined by their derivative data, and hence to a classification in terms of algebraic structures. Compared with classical Goodwillie calculus, which is governed by homotopy-theoretic conditions, the present framework is analytic and quantitative in nature, providing explicit control over convergence and approximation. These results place functor calculus in a setting that combines spectral ideas, analytic methods, and operadic algebra, and suggest further connections with deformation theory and geometry.