Reformat result into a theorem

main.tex

@@ -23,6 +23,8 @@
 \newcommand{\centralcharge}{\mathcal{Z}}
 \newcommand{\minorheading}[1]{{\noindent\normalfont\normalsize\bfseries #1}}
 
+\newtheorem{rmax_with_uniform_eps}{Theorem}[section]
+
 \begin{document}
 
 \title{Explicit Formulae for Bounds on the Ranks of Tilt Destabilizers and
@@ -829,8 +831,35 @@ radius of the pseudo-wall being positive
 	\frac{1}{2n^2}\ZZ
 \end{equation}
 
+\begin{rmax_with_uniform_eps}[Bound on $r$ \#1]
+	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.
+
+	\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\}
+	\end{align*}
+\end{rmax_with_uniform_eps}
+
+\begin{proof}
+
 \noindent
-Both $d$ and the lower bound are elements of $\frac{1}{\lcm(m,2n^2)}\ZZ$.
+Both $d$ and the lower bound in
+(eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta})
+are elements of $\frac{1}{\lcm(m,2n^2)}\ZZ$.
 So, if any of the two upper bounds on $d$ come to within
 $\frac{1}{\lcm(m,2n^2)}$ of this lower bound, then there are no solutions for
 $d$.
@@ -887,6 +916,9 @@ assert bounds_too_tight_condition1.rhs() == r
 assert bounds_too_tight_condition2.rhs() == r
 \end{sagesilent}
 
+\noindent
+This is equivalent to:
+
 \begin{equation}
 	r >
 	\min\left(
@@ -903,6 +935,13 @@ assert bounds_too_tight_condition2.rhs() == r
 	\right)
 \end{equation}
 
+If this condition holds for all $q$, then there are no solutions for $d$,
+and therefore $r$ cannot satisfy this condition for all $q$.
+Taking the maximum of all these expressions over $q$, and substituting the value
+for $\epsilon$ gives the result.
+
+\end{proof}
+
 %% refinements using specific values of q and beta
 
 \begin{sagesilent}