I will present a “broken exponential martingale” G-expansion that generalizes and unifies our earlier exponentiation result (Alòs, Gatheral, and Radoičić) and the cumulant recursion formula of Lacoin, Rhodes, and Vargas. As one application, I show how to compute all terms in an expansion of the Lévy area. By reordering the trees in the G-expansion according to the number of leaves, our earlier exponentiation theorem can be recovered. As further applications, I will give model-free expressions for various quantities of interest under stochastic volatility. Finally, I will exhibit explicit computations of diamond trees under rough Heston.