diff --git a/main.tex b/main.tex
index bfc0b39f1d558b50be8778abe9220da279fcc544..2ff26cd1a83e093fffd6ddf1c03a418501563fcd 100644
--- a/main.tex
+++ b/main.tex
@@ -1155,7 +1155,7 @@ previous subsubsections.
 %%%% RATIONAL BETA MINUS
 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
+$d=\chern_2(u)$ 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}
@@ -1272,6 +1272,75 @@ from plots_and_expressions import typical_bounds_on_d
 \label{fig:d_bounds_xmpl_gnrc_q}
 \end{figure}
 
+\subsubsection{All Bounds on $d$ Together for Problem
+\ref{problem:problem-statement-1}}
+\label{subsubsect:all-bounds-on-d-prob1}
+
+Unlike for problem \ref{problem:problem-statement-2},
+the bounds on $d=\chern_2(u)$ 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 corollary \ref{cor:num_test_prob2} have different
+constant and linear terms, so that the graphs for upper
+bounds do not share the same assymptote as the lower bound
+(and they will turn out to intersect).
+
+\begin{align}
+	\sage{problem1.radius_condition_d_bound.lhs()}
+	&>
+	\sage{problem1.radius_condition_d_bound.rhs()}
+	&\text{where }r>0
+	\label{eqn:prob1:radiuscond}
+	\\
+	d &\leq
+	\sage{problem1.bgmlv2_d_upperbound_terms.linear}
+	+ \sage{problem1.bgmlv2_d_upperbound_terms.const}
+	+ \sage{problem1.bgmlv2_d_upperbound_terms.hyperbolic}
+	&\text{where }r>R
+	\label{eqn:prob1:bgmlv2}
+	\\
+	d &\leq
+	\sage{problem1.bgmlv3_d_upperbound_terms.linear}
+	+ \sage{problem1.bgmlv3_d_upperbound_terms.const}
+	+ \sage{problem1.bgmlv3_d_upperbound_terms.hyperbolic}
+	&\text{where }r>R
+	\label{eqn:prob1:bgmlv3}
+\end{align}
+
+Notice that as a function in $r$, the linear term in 
+equation \ref{eqn:prob1:radiuscond} is strictly greater than
+those in equations \ref{eqn:prob1:bgmlv2}
+and \ref{eqn:prob1:bgmlv3}. This is because $r$, $R$
+and $\chern_2^B(v)$ are all strictly positive:
+\begin{itemize}
+	\item $R > 0$ from the setting of problem
+	\ref{problem:problem-statement-1}
+	\item $r > 0$ from lemma \ref{lem:num_test_prob1}
+	\item $\chern_2^B(v)$ because $B < \originalbeta_{-}$ due to the choice of $P$ being
+	a point on $\Theta_v^{-}$
+\end{itemize}
+
+This means that the lower bound for $d$ will be large than either of the two
+upper bounds for sufficiently large $r$, and hence those values of $r$ would yield no 
+solution to problem \ref{problem:problem-statement-1}.
+
+A generic example of this is plotted in figure
+\ref{fig:problem1:d_bounds_xmpl_gnrc_q}.
+
+\begin{figure}
+\centering
+\sageplot[width=\linewidth]{typical_bounds_on_d}
+\caption{
+	Bounds on $d\coloneqq\chern_2(E)$ in terms of $r\coloneqq \chern_0(E)$ for a fixed
+	value $\chern_1^{\beta}(F)/2$ of $q\coloneqq\chern_1^{\beta}(E)$.
+	Where $\chern(F) = (3,2,-2)$ and $P$ chosen as the point on $\Theta_v$
+	with $B\coloneqq-2/3-1/99$ in the context of problem 
+	\ref{problem:problem-statement-1}.
+}
+\label{fig:problem1:d_bounds_xmpl_gnrc_q}
+\end{figure}
+
 \subsection{Bounds on Semistabilizer Rank \texorpdfstring{$r$}{r} in Problem
 \ref{problem:problem-statement-2}}