Justify parts of first problem

main.tex

@@ -562,16 +562,36 @@ are trying to solve for.
 
 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 $u=(r,c\ell,d\ell^2)$
+The goal is to find all pseudo-semistabilizers $u$
 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 $v-u$).
-
+$\nu(u)>\nu(v)$ 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 $P$.
-\end{problem}
+The purpose of the final `direction' condition is because, up to that point,
+semistabilizers are not distinguished from their corresponding quotients:
+Suppose $E\hookrightarrow F\twoheadrightarrow G$, then the tilt slopes
+$\nu_{\alpha,\beta}$
+are strictly increasing, strictly decreasing, or equal across the short exact
+sequence (consequence of the see-saw principle).
+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
+$\chern(E)$ 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
+typically more familiar chamber
+(the stable objects of Chern character $v$ in the outside chamber will only be
+Gieseker stable sheaves).
+
+Also note that this last restriction does not remove any pseudo-walls found,
+and if we do want to recover `outwards' semistabilizers, we can simply take
+$v-u$ for each solution $u$ of the problem.
+
 
 \begin{problem}[all `left' pseudo-walls]
 \label{problem:problem-statement-2}