diff --git a/main.tex b/main.tex
index feaa249587fd0f26d6accf2007a05156a60315f8..fa38665ee136d766b5fb7e8406b98a9f97ed1526 100644
--- a/main.tex
+++ b/main.tex
@@ -707,7 +707,9 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
 		\item $r > 0$
 			\label{item:rankpos:lem:num_test_prob1}
 		\item $\Delta(u) \geq 0$
+			\label{item:bgmlvu:lem:num_test_prob1}
 		\item $\Delta(v-u) \geq 0$
+			\label{item:bgmlvv-u:lem:num_test_prob1}
 		\item $\mu(u)=\frac{c}{r}<\mu(v)$
 		\item $0\leq\chern_1^{\beta(P)}(u)\leq\chern_1^{\beta(P)}(v)$
 			\label{item:chern1bound:lem:num_test_prob1}
@@ -737,11 +739,16 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
 	Are precisely given by integers $r,c,d$ satisfying the following conditions:
 	\begin{enumerate}
 		\item $r > 0$
+			\label{item:rankpos:lem:num_test_prob2}
 		\item $\Delta(u) \geq 0$
+			\label{item:bgmlvu:lem:num_test_prob2}
 		\item $\Delta(v-u) \geq 0$
+			\label{item:bgmlvv-u:lem:num_test_prob2}
 		\item $\mu(u)=\frac{c}{r}<\mu(v)$
 		\item $0\leq\chern_1^{\beta_{-}}(u)\leq\chern_1^{\beta_{-}}(v)$
+			\label{item:chern1bound:lem:num_test_prob2}
 		\item $\chern_2^{\beta_{-}}(u)>0$
+			\label{item:radiuscond:lem:num_test_prob2}
 	\end{enumerate}
 \end{corrolary}
 
@@ -1269,15 +1276,21 @@ However, when specializing to problem \ref{problem:problem-statement-2} again
 And so in this context, the linear and constant terms do match up with the
 previous subsubsections.
 
-\subsubsection{All Bounds on $d$ together}
+\subsubsection{All Bounds on $d$ Together for Problem
+\ref{problem:problem-statement-2}}
 %% RECAP ON INEQUALITIES TOGETHER
 
 %%%% RATIONAL BETA MINUS
-\minorheading{Special Case: Rational $\beta_{-}$}
-
-Suppose we take $\beta = \beta_{-}$ (so that $\chern^{\beta}_2(F)=0$)
-in the previous subsections, to find all circular walls to the left of the
-vertical wall (TODO as discussed in ref).
+As mentioned in passing, when specializing to solutions $u$ of problem
+\ref{problem:problem-statement-2}, the bounds on
+$d=\chern^{\beta_{-}}_2(u)$ induced by conditions
+\ref{item:bgmlvu:lem:num_test_prob2},
+\ref{item:bgmlvv-u:lem:num_test_prob2}, and
+\ref{item:radiuscond:lem:num_test_prob1}
+from corrolary \ref{cor:num_test_prob2} have the same constant and linear
+terms in $r$, but different hyperbolic terms.
+These give bounds with the same assymptotes when we take $r\to\infty$
+(for any fixed $q=\chern_1^{\beta_{-}}(u)$).
 
 % redefine \beta (especially coming from rendered SageMath expressions)
 % to be \beta_{-} for the rest of this subsubsection