Add statement to corrolary of 'bound on r 1' (real alternative to Schmidt's bound)

main.tex

@@ -24,6 +24,7 @@
 \newcommand{\minorheading}[1]{{\noindent\normalfont\normalsize\bfseries #1}}
 
 \newtheorem{theorem}{Theorem}[section]
+\newtheorem{corrolary}{Corrolary}[section]
 \newtheorem{lemmadfn}{Lemma/Definition}[section]
 \newtheorem{dfn}{Definition}[section]
 
@@ -1035,6 +1036,32 @@ This is equivalent to:
 
 \end{proof}
 
+\begin{corrolary}[Bound on $r$ \#2]
+\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}
+	\egroup
+\end{corrolary}
+
 %% TODO simplified expression for rmax by predicting which q gives rmax
 
 %% refinements using specific values of q and beta