Dan, Rigorous posts don't get no props.
You definitely get props for the pun!
My previous attemptedly rigorous post on the subject was just to say, I guess opaquely, that it seems to capture the essence of a stable arrangement — like one forced by an elastic collar — to merely require:
any sufficiently small perturbation of the arrangement (modify the centers but not the radii) cannot decrease the *diameter* of the arrangement (= the diameter of the smallest disk containing all disks of the arrangement).
It seems clear that a rosette of 7 equal disks, but with the central one removed, is stable in this sense (despite the hole).
I like the simplicity of your definition, but the elastic cord I have in mind would consist of circular arcs (where the cord is tangent to a roll) alternating with straight segments (where the cord isn't tangent to a roll), whereas your model seems to be that cord is circular, so I think we're talking about different idealizations. Jim Propp