  At the moment of deepest core collapse, a star cluster core contains less
than ten stars. This small number makes the traditional treatment of hard
binary formation, assuming a homogeneous background density, suspect. In a
previous paper, we have found that indeed the conventional wisdom of binary
formation, based on three-body encounters, is incorrect. Here we refine that
insight, by further dissecting the subsequent steps leading to hard binary
formation. For this purpose, we add some analysis tools in order to make the
study less subjective. We find that the conventional treatment does remain
valid for direct three-body scattering, but fails for resonant three-body
scattering. Especially democratic resonance scattering, which forms an
important part of the analytical theory of three-body binary formation, takes
too much space and time to be approximated as being isolated, in the context of
a cluster core around core collapse. We conclude that, while three-body
encounters can be analytically approximated as isolated, subsequent strong
perturbations typically occur whenever those encounters give rise to democratic
resonances. We present analytical estimates postdicting our numerical results.
If we only had been a bit more clever, we could have predicted this qualitative
behaviour.
