The Interaction Operation: A Fifth Fundamental Arithmetic Operation

This paper introduces and rigorously analyzes a new arithmetic concept called the interaction operation, a binary operation that generalizes classical arithmetic by incorporating relational dynamics between the elements being combined. This operation is controlled by an interaction intensity parameter, which modulates the strength and nature of the interaction between operands. We present a complete treatment of this operation, including: a thorough theoretical foundation with multiple proven theorems; geometric visualization using three-dimensional surface representations; algebraic analysis demonstrating that the operation is generally neither associative nor distributive; the introduction and study of an inverse operation, called de-interaction; a computational implementation with guarantees of numerical stability; extensive empirical validation through over one hundred test cases; practical applications in fields such as physics, chemistry, and complex systems; and a comparison with existing algebraic structures to contextualize its novelty. The interaction operation represents a shift from traditional quantitative mathematics toward a relational perspective, providing a versatile tool for modeling synergy, emergence, and non-linear interactions in diverse scientific domains. Paper link: https://andradepsa.github.io/MATEMATICO10/projetos/versao3.html