Start gathering inequalities together

main.tex

@@ -490,6 +490,19 @@ $\chern^{\beta}_2(F) = 0$,
 so some of these expressions simplify, and in particular, the constant and
 linear terms match those of the other bounds in the previous subsections.
 
+\subsection{Bounds on $r$}
+
+Now, the inequalities from the last three subsections will be used to find, for
+each given $q=\chern^{\beta}_1(E)$, how large $r$ needs to be in order to leave
+no possible solutions for $d$. At that point, there are no Chern characters
+$(r,c,d)$ that satisfy all inequalities to give a pseudowall.
+
+\subsubsection{All circular walls left of vertical wall}
+
+Suppose we take $\beta = \beta_{-}$ in the previous subsections, to find all
+circular walls to the left of the vertical wall (TODO as discussed in ref).
+
+
 \section{Conclusion}
 
 \newpage