This paper is the fourth of a series exposing a systematic combinatorial approach to Girard's Geometry of Interaction program. This program aims at obtaining particular realizability models for linear logic that accounts for the dynamics of cut-elimination. This fourth paper tackles the complex issue of defining exponential connectives in this framework. In order to succeed in this, we use the notion of graphings, a generalization of graphs which was defined in earlier work. We explain how we can use this framework to define a GoI for Elementary Linear Logic (ELL) with second-order quantification, a sub-system of linear logic that captures the class of elementary time computable functions.