The Relative Zero Operator: A Causal Nonlocal Operator with Memory in Functional and Spectral Analysis

COVER LETTER The present work introduces and develops the Relative Zero Operator as a rigorous mathematical construction aimed at formalizing the notion of relational reference in systems endowed with finite causal memory. Although the initial motivation for this study arose within the context of cosmology and nonlocal effective theories, the formulation presented here is deliberately independent of any specific physical realization. By defining zero not as an absolute absence, but as invariance under a kernel-weighted causal average, the operator establishes a general framework for extracting dynamical deviations relative to reference states determined by the system’s own history. The functional and spectral properties demonstrated in this work — including linearity, continuity, boundedness, kernel characterization, and frequency-selective behavior — indicate that the operator constitutes a mathematically well-defined structure that may be explored in a variety of contexts where memory, causality, and dynamical regulation are essential features. In this sense, the Relative Zero Operator is not proposed as an immediate solution to concrete problems in other domains, but rather as a transferable conceptual and analytical framework, whose adaptation and effectiveness necessarily depend on future, domain-specific modeling and investigation. This perspective opens pathways for potential exploration in areas such as theoretical physics, mathematics, computation, statistics, economics, biology, neuroscience, information systems, logic, and philosophy, strictly in the sense of providing a formal reference structure for the analysis of systems regulated by causal averaging and memory, without presupposing empirical validation or operational outcomes beyond the foundational scope of the present work. ABSTRACT We present a rigorous functional-analytic and spectral study of the Relative Zero Operator, a causal nonlocal operator defined as the difference between a function and its kernel-based causal average over a finite memory interval. The operator arises naturally in nonlocal effective theories and systems with memory, where dynamical quantities are regulated relative to reference states defined by past values rather than absolute baselines. Working within standard Banach and Hilbert spaces, we establish linearity, boundedness, and continuity under minimal assumptions on the memory kernel. We characterize the kernel of the operator as the set of functions invariant under causal averaging and analyze its fixed-point structure, stability under perturbations, and structural non-invertibility. In translation-invariant settings, we derive the Fourier-domain representation and show that the spectrum is determined by the kernel transform, leading to a systematic suppression of low-frequency modes and an effective high-pass filtering behavior. These results establish the Relative Zero Operator as a mathematically well-defined and robust framework for the analysis of nonlocal systems with causal memory, providing a solid foundation for future investigations in effective field theories, nonlocal dynamics, and related areas of mathematical physics.