diff --git a/main.tex b/main.tex
index d5a630208e593676e3e6e811831474bed575681f..f279fad9665b61d000980acb1859faf41831b9e8 100644
--- a/main.tex
+++ b/main.tex
@@ -918,21 +918,21 @@ radius of the pseudo-wall being positive
 \end{equation}
 
 \begin{sagesilent}
-var("delta Delta", domain="real") # placeholder for the specific values of 1/epsilon
+var("nu", domain="real") # placeholder for the specific values of 1/epsilon
 
-assymptote_gap_condition1 = (1/Delta < bgmlv2_d_upperbound_exp_term)
-assymptote_gap_condition2 = (1/Delta < bgmlv3_d_upperbound_exp_term_alt2)
+assymptote_gap_condition1 = (1/nu < bgmlv2_d_upperbound_exp_term)
+assymptote_gap_condition2 = (1/nu < bgmlv3_d_upperbound_exp_term_alt2)
 
 r_upper_bound1 = (
 	assymptote_gap_condition1
-	* r * Delta
+	* r * nu
 )
 
 assert r_upper_bound1.lhs() == r
 
 r_upper_bound2 = (
 	assymptote_gap_condition2
-	* (r-R) * Delta + R
+	* (r-R) * nu + R
 )
 
 assert r_upper_bound2.lhs() == r
@@ -946,7 +946,7 @@ assert r_upper_bound2.lhs() == r
 	are bounded above by the following expression.
 
 	\bgroup
-	\def\Delta{\lcm(m,2n^2)}
+	\def\nu{\lcm(m,2n^2)}
 	\def\psi{\chern_1^{\beta}(F)}
 	\begin{align*}
 		\min
@@ -1017,7 +1017,7 @@ This is equivalent to:
 
 \bgroup
 \def\psi{\chern_1^{\beta}(F)}
-\def\Delta{\lcm(m,2n^2)}
+\def\nu{\lcm(m,2n^2)}
 \begin{equation}
 	\label{eqn:thm-bound-for-r-impossible-cond-for-r}
 	r \leq
@@ -1036,29 +1036,32 @@ This is equivalent to:
 
 \end{proof}
 
+\begin{sagesilent}
+var("Delta", domain="real")
+q_sol = solve(r_upper_bound1.rhs() == r_upper_bound2.rhs(), q)[0].rhs()
+r_upper_bound_all_q = (
+	r_upper_bound1.rhs()
+	.expand()
+	.subs(q==q_sol)
+	.subs(psi**2 == Delta)
+	.subs(1/psi**2 == 1/Delta)
+)
+\end{sagesilent}
+
 \begin{corrolary}[Bound on $r$ \#2]
-\label{cor:direct_rmax_with_uniform_eps}
+	\label{cor:direct_rmax_with_uniform_eps}
 	Let $v$ be a fixed Chern character and
 	$R:=\chern_0(v) \leq \frac{1}{2}\lcm(m,2n^2)\Delta(v)$.
 	Then the ranks of the pseudo-semistabilizers for $v$
 	are bounded above by the following expression.
 
 	\bgroup
-	\begin{equation}
-		\frac{1}{2} \lcm(m,2n^2)
-		\left(
-			\frac{\chern_1^{\beta}(v)}{2}
-			+ \frac{R}{\chern_1^{\beta}(v)\lcm(m,2n^2)}
-		\right)^2
-	\end{equation}
-	\egroup
-	\bgroup
-	\begin{equation}
-			\frac{1}{8}
-			\Delta(v) \lcm(m,2n^2)
-			+ \frac{1}{2} R
-			+ \frac{R^2}{ 2 \Delta(v) \lcm(m,2n^2) }
-	\end{equation}
+	\let\originalDelta\Delta
+	\def\nu{\lcm(m,2n^2)}
+	\renewcommand\Delta{{\originalDelta(v)}}
+	\begin{equation*}
+		\sage{r_upper_bound_all_q.expand()}
+	\end{equation*}
 	\egroup
 \end{corrolary}
 
@@ -1099,13 +1102,13 @@ That is, $r \equiv -\aa^{-1}\bb$ mod $n$ ($\aa$ is coprime to
 $n$, and so invertible mod $n$).
 
 \begin{sagesilent}
-rhs_numerator = (
+	rhs_numerator = (
 	positive_radius_condition
 	.rhs()
 	.subs([q_value_expr,beta_value_expr])
 	.factor()
 	.numerator()
-)
+	)
 \end{sagesilent}
 
 \noindent
@@ -1117,22 +1120,22 @@ proof of theorem \ref{thm:rmax_with_uniform_eps}:
 \begin{lemmadfn}[
 	Finding better alternatives to $\epsilon_F$:
 	$\epsilon_{q,1}$ and $\epsilon_{q,2}$
-]
-\label{lemdfn:epsilon_q}
-Suppose $d \in \frac{1}{m}\ZZ$ satisfies the condition in
-eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta}.
-That is:
-
-\begin{equation*}
-	\sage{positive_radius_condition.subs([q_value_expr,beta_value_expr]).factor()}
-\end{equation*}
-
-\noindent
-Then we have:
-
-\begin{equation*}
-	d - \frac{(\aa r + 2\bb)\aa}{2n^2}
-	\geq \epsilon_{q,2} \geq \epsilon_{q,1} > 0
+	]
+	\label{lemdfn:epsilon_q}
+	Suppose $d \in \frac{1}{m}\ZZ$ satisfies the condition in
+	eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta}.
+	That is:
+
+	\begin{equation*}
+		\sage{positive_radius_condition.subs([q_value_expr,beta_value_expr]).factor()}
+	\end{equation*}
+
+	\noindent
+	Then we have:
+
+	\begin{equation*}
+		d - \frac{(\aa r + 2\bb)\aa}{2n^2}
+		\geq \epsilon_{q,2} \geq \epsilon_{q,1}  0
 \end{equation*}
 
 Where $\epsilon_{q,1}$ and $\epsilon_{q,2}$ are defined as follows:
@@ -1222,7 +1225,8 @@ which also guarantees that the gap $\frac{k}{2mn^2}$ is at least $\epsilon_{q,2}
 	are bounded above by the following expression (with $i=1$ or $2$).
 
 \begin{sagesilent}
-eps_k_i_subs = Delta == (2*m*n^2)/delta
+var("delta", domain="real") # placeholder symbol to be replaced by k_{q,i}
+eps_k_i_subs = nu == (2*m*n^2)/delta
 \end{sagesilent}
 
 	\bgroup