diff --git a/main.tex b/main.tex
index e282d781b5785396806ba2cd05b6221cbe3516ec..6bbe39e320cb6d54f6806600218350c1ce35982d 100644
--- a/main.tex
+++ b/main.tex
@@ -916,28 +916,46 @@ radius of the pseudo-wall being positive
 	\frac{1}{2n^2}\ZZ
 \end{equation}
 
+\begin{sagesilent}
+var("Delta", domain="real") # placeholder for the specific values of 1/epsilon
+
+r_upper_bound1 = (
+	(1/Delta < bgmlv2_d_upperbound_exp_term)
+	* r * Delta
+)
+
+assert r_upper_bound1.lhs() == r
+
+r_upper_bound2 = (
+	(1/Delta < bgmlv3_d_upperbound_exp_term_alt2)
+	* (r-R) * Delta + R
+)
+
+assert r_upper_bound2.lhs() == r
+\end{sagesilent}
+
 \begin{theorem}[Bound on $r$ \#1]
 \label{thm:rmax_with_uniform_eps}
 	Let $v = (R,C,D)$ be a fixed Chern character. Then the ranks of the
-	pseudo-semistabilizers for $v$ are bounded above by the following expression.
+	pseudo-semistabilizers for $v$ with
+	$\chern_1^\beta = q$
+	are bounded above by the following expression.
 
+	\bgroup
+	\def\Delta{\lcm(m,2n^2)}
+	\def\psi{\chern_1^{\beta}(F)}
 	\begin{align*}
-		&\frac{\lcm(m,2n^2)}{2}
-		\max_{q \in [0,\chern_1^\beta(v)]}
-	\\
-		&\left\{
-			\min
-			\left(
-				q^2,
-				2R\beta q
-				+C^2
-				-2DR
-				-2Cq
-				+q^2
-				+\frac{R}{\lcm(m,2n^2)}
-			\right)
-		\right\}
+		\min
+		\left(
+			\sage{r_upper_bound1.rhs()}, \:\:
+			\sage{r_upper_bound2.rhs()}
+		\right)
 	\end{align*}
+	\egroup
+
+	Taking the maximum of this expression over
+	$q \in [0, \chern_1^{\beta}(F)]\cap \frac{1}{n}\ZZ$
+	would give an upper bound for the ranks of pseudo-semistabilizers for $v$.
 \end{theorem}
 
 \begin{proof}