diff --git a/main.tex b/main.tex
index 489bee52a9664a04f6238aa6d6fe51fcc600db58..2d142d8737ddcbda9506c9ad81d6c37e4e7651f1 100644
--- a/main.tex
+++ b/main.tex
@@ -993,6 +993,7 @@ proof of theorem \ref{thm:rmax_with_uniform_eps}:
 	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:
@@ -1085,6 +1086,29 @@ which also guarantees that the gap $\frac{k}{2mn^2}$ is at least $\epsilon_q^2$.
 \end{proof}
 
 
+\begin{theorem}[Bound on $r$ \#3]
+\label{thm:rmax_with_eps1}
+	Let $v = (R,C,D)$ be a fixed Chern character. Then the ranks of the
+	pseudo-semistabilizers for $v$ with
+	$\chern_1^\beta = q = \frac{a_q}{n}$
+	are bounded above by the following expression (with $i=1$ or 2).
+
+	\begin{equation*}
+		\frac{1}{2 \epsilon_q^i}
+			\min
+			\left(
+				q^2,
+				2R\beta q
+				+C^2
+				-2DR
+				-2Cq
+				+q^2
+				+\frac{R}{\lcm(m,2n^2)}
+				+R \epsilon_q^i
+			\right)
+	\end{equation*}
+	Where $\epsilon_q^i$ is defined as in definition/lemma \ref{lemdfn:epsilon_q}.
+\end{theorem}
 
 
 \minorheading{Irrational $\beta$}