My question is based on the paper Split states, entropy enigma, holes, halos.

What are the scaling solutions discussed on page 49 of the paper ?

It is stated that the equations ${\sum_{j, i\neq j}\frac{I_{ij}}{r_{ij}} = \theta_{i}}$ always have solutions os the form $r_{ij}= \lambda I_{ij}$. why is that true?

I don't understand this as some of the I's may be negative and then a single $\lambda$ can cannot give such a solutions as the distance will be negative in such cases.

I would greatly appreciate an answer explaining the proper meaning of such solutions and what are the conditions for their existence.

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