A (gravitational) toy story

Usually in computational physics, conclusions about realistic scenarios can be drawn from {\it ab initio} idealized models. In some ways, the discovery of critical behavior in the gravitational collapse of a massless scalar field leads to the simulation of binary black holes, from its coalescence, to merging and ringdown. We have been lucky enough to have been working on a toy model to explore our way in as these events unfold. We revisited the gravitational instability of a kink problem. During that study, we confirmed a conjecture related to the mass gap, in the context of critical behavior, at the threshold of black hole formation. What is the meaning of this mass gap? Does it have physical relevance? This essay is about these issues.