Apologies, I updated the question in the interim, I am not sure if it is phrased the way I want it yet, I am trying to see if there is a simple statement about probabilities of black hole states. I might edit this later.

Have you looked at the following wikipedia article? en.wikipedia.org/wiki/Gravitational_singularity "a singularity is defined to be one that contains geodesics that cannot be extended in a smooth manner. The end of such a geodesic is considered to be the singularity." So a singularity might be interpreted as a straight line that ends suddenly. I would almost think of it as point where transformations can not be described using the GL group of matrices, but I would have to defer to others to verify the truth of such a statement.

absolutely brilliant

It does help tremendously

I am not sure why this was down voted, using liquid medium is a quite serious proposition and has been studied extensively.