Got to a good point to share, have incorporated all of Tom Marsh's suggestions.

CurrentWriting/readme.txt

@@ -12,3 +12,6 @@
  - In training section (4.0.1.1) clarify loss function. There is the satellite loss function and
     the Approximation loss function. This also just needs rewritten.
         Decided to rename satellite loss to satellite destruction rate and ML loss to objective.
+
+# Thoughts
+Free entry condition does not apply to military operators.

CurrentWriting/sections/00_Introduction.tex

@@ -9,6 +9,7 @@ ESA used the opportunity to highlight the difficulties arising from coordinating
 such coordination will become more difficult as the size and number of 
 single purpose, single operator satellite fleets (satellite constellations) increase in low earth orbit\autocite{EsaBlog}.
 
+
 % Background on issues of congestion and pollution
 % Kessler Syndrome
 In spite of the fact that there is a lot of maneuvering room in outer space,
@@ -23,10 +24,10 @@ the probability of an unforeseen collision.
 The defining feature of pollution in orbit is that it self-propagates as debris collides with itself
 and orbiting satellites to generate more debris.
 This dynamic underlies a key concern, originally explored by Kessler and Cour-Palais \autocite{Kessler1978}
-that with sufficient mass in orbit (through satellite launches), the debris generating process
-could undergo a runaway effect rendering various orbital regions unusable.
+that with sufficient satellites in orbit, the debris generating process
+could undergo a runaway effect rendering some orbital regions unusable.
 This cascade of collisions is often known as Kessler syndrome and 
-may take place over various timescales.
+may take place over various timescales, from weeks to decades.
 
 % ---------------
 %Discuss how various definitions of kessler syndrome
@@ -38,24 +39,25 @@ may take place over various timescales.
 %benefit of new launches negative before the physical kessler syndrome occurs.
 
 % ---------------
-Orbits may be divided into three primary groups, 
-Low Earth Orbit (LEO),
-Medium Earth Orbit (MEO), and High Earth Orbit (HEO) where Geostationary Earth Orbit (GEO) 
-considered a particular classification of HEO. 
-While the topic of LEO allocation has historically remained somewhat unexplored, the last 6 years has seen
-a variety of new empirical studies and theoretical models published.
+
+%Orbits may be divided into three primary groups, 
+%Low Earth Orbit (LEO),
+%Medium Earth Orbit (MEO), and High Earth Orbit (HEO) where Geostationary Earth Orbit (GEO) 
+%is considered a particular subset of HEOs. 
+%While the topic of LEO allocation has historically remained somewhat unexplored, the last 6 years has seen
+%a variety of new empirical studies and theoretical models published.
 
 % ---------------
 %Allocative efficiency
 
 Macauley  provided the first evidence of sub-optimal behavior in orbit
 by estimating the welfare loss due to the current method of assigning GEO slots to operators\autocite{Macauley_1998}.
-The potential losses due to anti-competitive behavior were highlighted by Adilov et al ,
+The potential losses due to anti-competitive behavior were highlighted by Adilov et al,
 who have analyzed the opportunities for strategic 
 ``warehousing'' of non-functional satellites as a means of increasing competitive advantage by 
 denying operating locations to competitors in GEO\autocite{Adilov2019}.
 
-The primary concern expressed in many of the published papers is whether or not orbits will be overused
+The primary concern expressed in many recently published papers is whether or not orbits will be overused
 due to their common-pool nature, and which policies may prevent kessler syndrome.
 On this topic, Adilov, Alexander, and Cunningham examine pollution
 using a two-period salop model, incorporating the effects of launch debris on 
@@ -71,7 +73,7 @@ numerous operators in a free entry environment who
 can each launch a single, identical constellation\autocite{RaoRondina2020}.
 Rao, Burgess, and Kaffine use this model to estimate that achieving socially optimal 
 behavior through orbital use fees could increase the value generated by the 
-space industry by a factor of four\autocite{Rao2020}.
+space industry by a factor of four\autocite{Rao2020} in the long run.
 
 
 % ---------------
@@ -107,16 +109,26 @@ and the ways in which various policies encourage or discourage optimal decision
 % characterization of the general solutions to both the constellation operators' problems and
 % the fleet planner's problem, 
 % and an analysis of survival rates within constellations and the entire fleet.
-
 %Contribution statement
 %Adds to raoRondina2020 and adilov2018 in extedning to more diverse situations.
-This work is mainly a theoretical expansion of two models:
+The heterogeneity that I permit is the distinguishing feature of the model and the major
+justification for this work, as orbits are used by many different types of operators.
+Specifically, I permit:
 \begin{itemize}
-    \item Rao and Rondina's model \autocite{RaoRondina2020} dynamic model.
-    \item Adilov et al's \autocite{Adilov2018} dynamic model.
+    \item Heterogeneous agent types including commercial, scientific, and military.
+    \item Asymetric constellations.
+    \item Inter- and intra- constellation risk is not assumed to be equal.
 \end{itemize}
-In addition to the expansion, I contribute a general computational solver that allows
-us to examine complex scenarios similar to those encountered in actual policymaking.
+each of which are important qualities of the current orbital environment.
+None of these aspects are considered in the papers that I have reviewed so far.
+
+This work is mainly a theoretical expansion of two dynamic models by
+\cite{RaoRondina2020} as well as \cite{Adilov2018}
+%This model inherits the laws of motion for debris and constellation stocks from 
+%the aformentioned models and follows the DSGE modelling approach chosen by Rao.
+In addition to the models' expansion, I contribute a general computational solver to analyse the
+complex situations that arise in practice.
+
 %Similarities
 % - Rao
 %   - Law of debris:
@@ -132,18 +144,6 @@ us to examine complex scenarios similar to those encountered in actual policymak
 %   - Allows for non-firm participants
 %   - avoidance efficiencies
 
-This model inherits the laws of motion for debris and constellation stocks from 
-\autocite{RaoRondina2020,Adilov2018} and follows the DSGE modelling approach chosen by Rao.
-It is distinguished from both of the aformentioned models 
-by the way it allows for the following:
-\begin{itemize}
-    \item Heterogeneous agent types including commercial, scientific, and military.
-    \item Asymetric constellations.
-    \item Inter- and intra- constellation risk is not assumed to be equal.
-\end{itemize}
-The heterogeneity that I permit is the distinguishing feature of the model and the major
-justification for this work, as orbits are used by many different types of operators.
-
 
 
 \end{document}

CurrentWriting/sections/01_LawsOfMotion.tex

@@ -4,8 +4,8 @@
 \begin{document}
 In this model there are two types of entities subject to laws of motion; 
 i.e. constellation-level satellite stocks and debris.
-These laws are the foundations to the results found in \cref{SEC:Kessler,SEC:Survival}, and 
-are crucial elements of the models presented in sections \cref{SEC:Operator,SEC:Planner}.
+%These laws are the foundations to the results found in \cref{SEC:Kessler,SEC:Survival}, and 
+%are crucial elements of the models presented in sections \cref{SEC:Operator,SEC:Planner}.
 
 \subsubsection{Mathematical Preliminaries}
 Throughout the remainder of the paper, the following notation will be used.
@@ -18,10 +18,8 @@ subscripts $s_t$ denote time periods.
         in period $t$
     \item $D_t$ represents the level of debris at period $t$.
 \end{itemize}
-In the case of satellite stocks, often the set of stocks for each constellation needs to 
-be discussed. 
-I've used curly braces around to denote this set, i.e. $\{ s^j_t \}$ represents the set
-of constellations stocks, ordered by index $j$.
+I've used curly braces (i.e. $\{ s^j_t \}$) to represent the set
+of constellations' stocks.
 
 \subsubsection{Satellite Stocks}
 Each constellation consists of a number of satellites in orbit, controlled by the same operator and
@@ -42,7 +40,7 @@ Where $l^i(\cdot)$ represents the rate at which satellites are destroyed by coll
 %Assumption: 
 
 \subsubsection{Collision Efficiencies}
-%TODO: Explain bit about constellation collision efficiencies.
+%Explain bit about constellation collision efficiencies.
 As demonstrated by \cite{reiland2020}, there are constellation designs by which an operator can
 minimize the risk of intra-constellation collisions.
 I assume that when designing a constellation, the operator chooses to minimize collision risks,
@@ -102,11 +100,10 @@ These effects can be represented by the following general law of motion.
 For simplicity, I formulate this more specifically as:
 \begin{align}
     D_{t+1} = (1-\delta)D_t + g(D_t) 
-                + \sum^N_{i=1} \vec \gamma l^i(\{s^j_t\},D_t)
-                + \vec \Gamma \sum^n_{j=1} \{x^j_t\}
+                + \sum^N_{i=1} \gamma l^i(\{s^j_t\},D_t)
+                +  \Gamma \sum^n_{j=1} \{x^j_t\}
 \end{align}
-%WORKING HERE
-where $\vec \Gamma,\vec \gamma$ represent the debris generated by each 
+where $ \Gamma, \gamma$ represent the debris generated by each 
 launch and collision respectively,
 while $\delta,g(\cdot)$ represent the decay rate of debris and the 
 autocatalysis\footnote{