diff --git a/main.tex b/main.tex
index 11511ff11bfd67b40e56ac82a85bfea4027cb6c4..4b3abc6199110c7ec3d86e93817a58212c382ccb 100644
--- a/main.tex
+++ b/main.tex
@@ -151,9 +151,7 @@ followed by refinements on them which give explicit formulae for tighter bounds
 on $\chern_0(E)$ of potential destabilizers $E$ of $F$.
 
 
-\section{Characteristic Curves of Stability Conditions Associated to Chern
-Characters}
-
+\section{Setting and Definitions: Clarifying `pseudo'}
 Throughout this article, as noted in the introduction, we will be exclusively
 working over one of the following two surfaces: principally polarized abelian
 surfaces and $\PP^2$.
@@ -194,6 +192,14 @@ $\bddderived(X)$, some other sources may have this extra restriction too.
 	Then $\chern(E)$ is a pseudo-semistabilizer of $\chern(F)$
 \end{lemma}
 
+\begin{proof}
+	q.e.d. (TODO)
+\end{proof}
+
+
+\section{Characteristic Curves of Stability Conditions Associated to Chern
+Characters}
+
 % NOTE: SURFACE SPECIALIZATION
 Considering the stability conditions with two parameters $\alpha, \beta$ on
 Picard rank 1 surfaces.
@@ -534,8 +540,50 @@ Finally, consequence 4 along with $P$ being to the left of $V_u$ implies
 $\nu_P(u) > 0$ giving supposition b.
 \end{proof}
 
+\section{The Problem: Finding Pseudo-walls}
+
+As hinted in the introduction (\ref{sec:intro}), the main motivation of the
+results in this article are not only the bounds on pseudo-semistabilizer
+ranks;
+but also applications for finding a list (comprehensive or subset) of
+pseudo-walls.
+
+After introducing the characteristic curves of stability conditions associated
+to a fixed Chern character $v$, we can now formally state the problems that we
+are trying to solve for.
+
+\begin{problem}[sufficiently large `left' pseudo-walls]
+\label{problem:problem-statement-1}
+
+Fix a Chern character $v$ with positive rank, $\Delta(v) \geq 0$,
+and $\beta_{-}(v) \in \QQ$.
+The goal is to find all pseudo-semistabilizers $u=(r,c\ell,d\ell^2)$
+which give circular pseudo-walls containing some fixed point
+$P\in\Theta_v^-$.
+With the added restriction that $u$ `destabilizes' $v$ moving `inwards', that is,
+$\nu(u)>\nu(v)$ inside the circular pseudo-wall
+(`outward' destabilizers can be recovered as $v-u$).
+
+This will give all pseudo-walls between the chamber corresponding to Gieseker
+stability and the stability condition corresponding to $P$.
+\end{problem}
+
+\begin{problem}[all `left' pseudo-walls]
+\label{problem:problem-statement-2}
+
+Fix a Chern character $v$ with positive rank, $\Delta(v) \geq 0$,
+and $\beta_{-}(v) \in \QQ$.
+The goal is to find all solutions $u=(r,c\ell,d\ell^2)$
+to problem \ref{problem:problem-statement-1} with the choice
+$P=(\beta_{-},0)$.
+
+This will give all circular pseudo-walls left of $V_v$.
+\end{problem}
 
-\section{Loose Bounds on $\chern_0(E)$ for Semistabilizers Along Fixed
+
+\section{B.Schmidt's Solutions to the Problems}
+
+\subsection{Loose Bounds on $\chern_0(E)$ for Semistabilizers Along Fixed
 $\beta\in\QQ$}
 
 \begin{dfn}[Twisted Chern Character]
@@ -668,7 +716,7 @@ However, when computing all tilt semistabilizers for $v$ on $\PP^2$, the maximum
 rank that appears turns out to be $\sage{extravagant.actual_rmax}$.
 \end{example}
 
-\section{B.Schmidt's Wall Finding Method}
+\subsection{B.Schmidt's Wall Finding Method}
 
 % NOTE: SURFACE SPECIALIZATION
 The proof for the previous theorem was hinted at in
@@ -684,34 +732,6 @@ finding all pseudo-walls when $\beta_{-}\in\QQ$. In section [ref], a different
 algorithm will be presented making use of the later theorems in this article,
 with the goal of cutting down the run time.
 
-\begin{problem}[sufficiently large `left' pseudo-walls]
-\label{problem:problem-statement-1}
-
-Fix a Chern character $v$ with positive rank, $\Delta(v) \geq 0$,
-and $\beta_{-}(v) \in \QQ$.
-The goal is to find all pseudo-semistabilizers $u=(r,c\ell,d\ell^2)$
-which give circular pseudo-walls containing some fixed point
-$P\in\Theta_v^-$.
-With the added restriction that $u$ `destabilizes' $v$ moving `inwards', that is,
-$\nu(u)>\nu(v)$ inside the circular pseudo-wall
-(`outward' destabilizers can be recovered as $v-u$).
-
-This will give all pseudo-walls between the chamber corresponding to Gieseker
-stability and the stability condition corresponding to $P$.
-\end{problem}
-
-\begin{problem}[all `left' pseudo-walls]
-\label{problem:problem-statement-2}
-
-Fix a Chern character $v$ with positive rank, $\Delta(v) \geq 0$,
-and $\beta_{-}(v) \in \QQ$.
-The goal is to find all solutions $u=(r,c\ell,d\ell^2)$
-to problem \ref{problem:problem-statement-1} with the choice
-$P=(\beta_{-},0)$.
-
-This will give all circular pseudo-walls left of $V_v$.
-\end{problem}
-
 \subsection*{Algorithm}
 
 \subsubsection*{Finding possible $r$ and $c$}