Context Influence Calculus

We introduce the Contextual Influence Calculus (CIC), a novel and generalized mathematical framework that extends classical differential calculus to explicitly incorporate intrinsic memory, recursive feedback mechanisms, and contextual stochastic influences. CIC is designed to model complex adaptive systems whose behavior evolves dynamically based on historical context, endogenous interactions, and exogenous perturbations. This approach is essential for a realistic understanding of phenomena such as financial contagion, ecological tipping points, cognitive decision processes, and social information diffusion. We formally define the CIC derivative and its constituent components, establish key theoretical properties including existence, uniqueness, and the classical limit, and validate its expressive capacity through a series of synthetic simulations. Our results empirically demonstrate that CIC successfully captures nonlinear bifurcations, nuanced phase transitions, dynamic regime shifts, and complex perturbation recovery behaviors that often elude conventional calculus-based models. Furthermore, we explore CIC’s applica bility across diverse multi-domain systems and detail its learnable extensions through neural differential parameterization.