From b2530cec92bd40a76461a7f5aaaace02126b9c86 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue, 9 May 2023 16:57:59 +0100
Subject: [PATCH] Start gathering inequalities together

---
 main.tex | 13 +++++++++++++
 1 file changed, 13 insertions(+)

diff --git a/main.tex b/main.tex
index 62fc11c..03dd203 100644
--- a/main.tex
+++ b/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
-- 
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