Update statement of main lemma to be aware of quotient being in tilt

main.tex

@@ -284,36 +284,30 @@ the circular walls must be nested and non-intersecting.
 
 \begin{lemma}[Numerical tests for left-wall pseudo-semistabilizers]
 \label{lem:pseudo_wall_numerical_tests}
-Let $v$ and $u$ be Chern characters with positive ranks and $\Delta(v),
-\Delta(u)\geq 0$. Let $P$ be a point on $\Theta_v^-$.
+Let $v$ and $u$ be Chern characters with $\Delta(v),
+\Delta(u)\geq 0$, and $v$ has positive rank. Let $P$ be a point on $\Theta_v^-$.
 
 \noindent
-Suppose that the following are satisfied:
+The following conditions:
 \bgroup
 \renewcommand{\labelenumi}{\alph{enumi}.}
 \begin{enumerate}
-\item $u$ gives rise to a pseudo-wall for $v$, left of the vertical line $V_v$
-\item The pseudo-wall contains $P$ in it's interior
-	($P$ can be chosen to be the base of the left branch to target all left-walls)
+\item $u$ is a pseudo-semistabilizer of $v$ at some point on $\Theta_v^-$ above
+	$P$
 \item $u$ destabilizes $v$ going `inwards', that is,
-	$\nu_{\alpha,\beta}(\pm u)<\nu_{\alpha,\beta}(v)$ outside the pseudo-wall, and
-	$\nu_{\alpha,\beta}(\pm u)>\nu_{\alpha,\beta}(v)$ inside.
-	Where we use $+u$ or $-u$ depending on whether $(\beta,\alpha)$ is on the left
-	or right (resp.) of $V_u$.
+	$\nu_{\alpha,\beta}(u)<\nu_{\alpha,\beta}(v)$ outside the pseudo-wall, and
+	$\nu_{\alpha,\beta}(u)>\nu_{\alpha,\beta}(v)$ inside.
 \end{enumerate}
 \egroup
 
 \noindent
-Then we have the following:
+are equivalent to the following more numerical conditions:
 \begin{enumerate}
-	\item The pseudo-wall is left of $V_u$
-		(if this is a real wall then $v$ is being semistabilized by an object with
-		Chern character $u$, not $-u$)
-	\item $\beta(P)<\mu(u)<\mu(v)$, i.e., $V_u$ is strictly between $P$ and $V_v$.
+	\item $u$ has positive rank
+	\item $\beta(P)<\mu(u)<\mu(v)$, i.e. $V_u$ is strictly between $P$ and $V_v$.
+	\item $\chern_1^{\beta(P)}(v-u)\geq0$
 	\item $\chern_2^{P}(u)>0$
 \end{enumerate}
-Furthermore, only the last two of these consequences are sufficient to recover
-all of the suppositions above.
 \end{lemma}
 
 \begin{proof}