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\begin{document}
\subsection{Defining the Proto-Kessler Region}
With the definitions of kessler syndrome and the law of debris given above, we can now 
explicitly describe the proto-kessler region.
\begin{align}
 \epsilon < -\delta D_t + g(D_t) + \gamma \sum^n_{j=1} l^i(\{s^j_t\},D_t) + \Gamma \sum^n_{j=1} \{x^j_t\}
\end{align}
As being in the proto-kessler region is a prerequesit to being in the kessler region, we see that
the kessler region depends on the collision rates of the constellation operators.

Although this is a straightforward result, I have not found it in any of the models I've examined so far.
I suspect it will impact optimal pigouvian taxation, but of course, I need to verify this in 
a computational example.
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