Hyers–Ulam Stability of a Multi-Variable Additive-Quadratic Functional Equation

In this paper, we investigate the generalized Hyers–Ulam stability of a multi-variable additive-quadratic functional equation in Banach spaces. Unlike traditional mixed-type equations that segregate variables, the proposed equation embeds a dense combinatorial cross-variable interaction. To overcome the complex analytical challenges arising from asymmetric scaling weights, we prove a complete structural equivalence theorem between this multivariate framework and the classical additive-quadratic system of equations. We establish comprehensive stability results utilizing two distinct methodological paradigms: the direct constructive aggregation method and the generalized metric fixed-point alternative. Furthermore, we provide a rigorous mathematical justification of an analytical singularity by constructing a counterexample that explicitly proves the inherent instability of the system at the critical exponent threshold p=3.