diff --git a/main.tex b/main.tex
index facd95285d121c0c9efb85d7f0623a08439bf6e1..308f1fe33e158ba1fda29459918fb87ee784802a 100644
--- a/main.tex
+++ b/main.tex
@@ -643,7 +643,7 @@ $\nu(u)>\nu(v)$ inside the circular pseudo-wall.
 \end{problem}
 This will give all pseudo-walls between the chamber corresponding to Gieseker
 stability and the stability condition corresponding to $P$.
-The purpose of the final `direction' condition is because, up to that point,
+The purpose of the final `direction' condition is because, up to that condition,
 semistabilizers are not distinguished from their corresponding quotients:
 Suppose $E\hookrightarrow F\twoheadrightarrow G$, then the tilt slopes
 $\nu_{\alpha,\beta}$
@@ -653,7 +653,7 @@ In this case, $\chern(E)$ is a pseudo-semistabilizer of $\chern(F)$, if and
 only if $\chern(G)$ is a pseudo-semistabilizer of $\chern(F)$.
 The numerical inequalities in the definition for pseudo-semistabilizer cannot
 tell which of $E$ or $G$ is the subobject.
-However what can be distinguished is the direction across the wall that
+However, what can be distinguished is the direction across the wall that
 $\chern(E)$ or $\chern(G)$ destabilizes $\chern(F)$
 (they will each destabilize in the opposite direction to the other).
 The `inwards' semistabilizers are preferred because we are moving from a
@@ -681,12 +681,14 @@ $\beta$-axis.
 This is because all circular walls left of $V_v$ intersect $\Theta_v^-$ (once).
 The $\beta_{-}(v) \in \QQ$ condition is to ensure that there are finitely many
 solutions. As mentioned in the introduction (\ref{sec:intro}), this is known,
-however this will also be proved again in passing in this article.
+however this will also be proved again implicitly in section
+\ref{sect:prob2-algorithm}, where an algorithm is produced to find all
+solutions.
 
 This description still holds for the case of rank 0 case if we consider $V_v$ to
 be infinitely far to the right
 (see section \ref{subsubsect:rank-zero-case-charact-curves}).
-Note also that the $\beta_-(v)$ condition always holds for $v$ rank 0.
+Note also that the condition on $\beta_-(v)$ always holds for $v$ rank 0.
 
 \subsection{Numerical Formulations of the Problems}
 
@@ -697,12 +699,13 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
 \begin{lemma}[Numerical Tests for Sufficiently Large `left' Pseudo-walls]
 	\label{lem:num_test_prob1}
 	Given a Chern character $v$ with non-negative rank
-	(with $\chern_1(v)>0$ if rank 0)
+	(and $\chern_1(v)>0$ if rank 0),
 	and $\Delta(v) \geq 0$,
 	and a choice of point $P$ on $\Theta_v^-$.
-	Solutions $u=(r,c\ell,\frac{e}{\lcm(m,2)}\ell^2)$
+	Solutions $u=(r,c\ell,d\ell^2)$
 	to problem \ref{problem:problem-statement-1}.
-	Are precisely given by integers $r,c,d$ satisfying the following conditions:
+	Are precisely given by $r,c \in \ZZ$, $d \in \frac{1}{\lcm(m,2)}$
+	satisfying the following conditions:
 	\begin{enumerate}
 		\item $r > 0$
 			\label{item:rankpos:lem:num_test_prob1}
@@ -736,9 +739,10 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
 \label{cor:num_test_prob2}
 	Given a Chern character $v$ with positive rank and $\Delta(v) \geq 0$,
 	such that $\beta_{-}\coloneqq\beta_{-}(v) \in \QQ$.
-	Solutions $u=(r,c\ell,\frac{e}{\lcm(m,2)}\ell^2)$
+	Solutions $u=(r,c\ell,d\ell^2)$
 	to problem \ref{problem:problem-statement-2}.
-	Are precisely given by integers $r,c,e$ satisfying the following conditions:
+	Are precisely given by $r,c \in \ZZ$, $d\in\frac{1}{\lcm(m,2)}\ZZ$ satisfying
+	the following conditions:
 	\begin{enumerate}
 		\item $r > 0$
 			\label{item:rankpos:lem:num_test_prob2}