diff --git a/main.tex b/main.tex
index 47f1de3ffede417784dfa53a4909c07e7c692742..c0169c4bbaff9fa9806951e8e91e0cc55c6b94d7 100644
--- a/main.tex
+++ b/main.tex
@@ -521,6 +521,7 @@ bound for the rank of $E$:
 \end{proof}
 
 \begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
+\label{exmpl:recurring-first}
 \begin{sagesilent}
 recurring = Object()
 recurring.chern = Chern_Char(3, 2, -2)
@@ -1421,7 +1422,7 @@ r_upper_bound_all_q = (
 \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 $v$
@@ -1468,6 +1469,28 @@ stated in the corollary.
 \egroup
 \end{proof}
 
+\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
+Just like in example \ref{exmpl:recurring-first}, take
+$\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
+that $m=2$, $\beta_-=\sage{recurring.b}$,
+giving $n=\sage{recurring.b.denominator()}$.
+
+\begin{sagesilent}
+n = recurring.b.denominator()
+m = 2
+recurring.bgmlv = recurring.chern.Q_tilt()
+recurring.lcm = lcm(m, 2*n^2)
+corrolary_bound = (
+  recurring.bgmlv * recurring.lcm / 8
+  + recurring.chern.ch[0] / 2
+  + recurring.chern.ch[0]^2 / (2*recurring.bgmlv*recurring.lcm)
+)
+\end{sagesilent}
+Using the above corrolary \ref{cor:direct_rmax_with_uniform_eps}, we get that
+the ranks of tilt semistabilizers for $v$ are bounded above by
+$\sage{corrolary_bound} \approx  \sage{float(corrolary_bound)}$,
+which is much closer to real maximum 25 than the original bound 144.
+\end{example}
 %% refinements using specific values of q and beta
 
 These bound can be refined a bit more by considering restrictions from the