Bubble rupture is one of the principle mechanisms in which foams are assumed to coarsen, creating a smaller population of larger bubbles over time. Here, however, we demonstrate that for a large range of situations bubbles that rupture on an interface `inverse coarsen', leading to a larger population of smaller bubbles. We present high speed images and numerical simulations to demonstrate that when a bubble bursts the retracting film can fold and entrap air. The folding leads to a torus of entrapped air which breaks up into a ring of daughter bubbles. These results have broad implications for any process involving bubbles on an interface, including air-water transfer and interfacial foam coarsening.