diff --git a/main.tex b/main.tex
index 3003b47d6cfc0bf506d3ed2ca8600bb671e8741e..48d1122a713e8c609d0cb29df43dea358c955b68 100644
--- a/main.tex
+++ b/main.tex
@@ -612,6 +612,8 @@ 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.
 
+\subsection{Problem statements}
+
 \begin{problem}[sufficiently large `left' pseudo-walls]
 \label{problem:problem-statement-1}
 
@@ -664,6 +666,53 @@ 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.
 
+\subsection{Numerical Formulations of the Problems}
+
+The problems introduced in this section are phrased in the context of stability
+conditions. However, these can be reduced down completely to purely numerical
+problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
+
+\begin{lemma}[Numerical Tests for Sufficiently Large `left' Pseudo-walls]
+	Given a Chern character $v$ with positive rank and $\Delta(v) \geq 0$,
+	and a choice of point $P$ on $\Theta_v^-$.
+	Solutions $u=(r,c\ell,d\frac{1}{2}\ell^2)$
+	to problem \ref{problem:problem-statement-1}.
+	Are precisely given by integers $r,c,d$ satisfying the following conditions:
+	\begin{enumerate}
+		\item $r > 0$
+		\item	$\Delta(u) \geq 0$
+		\item	$\Delta(v-u) \geq 0$
+		\item $\beta(P)<\mu(u)=\frac{c}{r}<\mu(v)$
+		\item $\chern_1^{\beta(P)}(v-u)\geq0$
+		\item $\chern_2^{P}(u)>0$
+	\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+	% TODO complete
+	Use main lemma \ref{lem:pseudo_wall_numerical_tests} TODO
+\end{proof}
+
+\begin{corrolary}[Numerical Tests for All `left' Pseudo-walls]
+	Given a Chern character $v$ with positive rank and $\Delta(v) \geq 0$,
+	such that $\beta_{-}\coloneqq\beta_{-}(v) \in \QQ$.
+	Solutions $u=(r,c\ell,d\frac{1}{2}\ell^2)$
+	to problem \ref{problem:problem-statement-2}.
+	Are precisely given by integers $r,c,d$ satisfying the following conditions:
+	\begin{enumerate}
+		\item $r > 0$
+		\item	$\Delta(u) \geq 0$
+		\item	$\Delta(v-u) \geq 0$
+		\item $\beta(P)<\mu(u)=\frac{c}{r}<\mu(v)$
+		\item $\chern_1^{\beta_{-}}(v-u)\geq0$
+		\item $\chern_2^{\beta_{-}}(u)>0$
+	\end{enumerate}
+\end{corrolary}
+
+\begin{proof}
+	This is a specialization of the previous lemma, using $P=(\beta_{-},0)$.
+\end{proof}
+
 
 \section{B.Schmidt's Solutions to the Problems}