diff --git a/main.tex b/main.tex
index 8aed269e7a5ac254359f0605f92a2063690e94ee..40e3314c70dac6db0e29d035a80f5bd8ae75470e 100644
--- a/main.tex
+++ b/main.tex
@@ -135,11 +135,56 @@ for the rank of $E$:
 
 \section{Refinement}
 
-
+To get tighter bounds on the rank of destabilizers $E$ of some $F$ with some
+fixed Chern character, we will need to consider each of the values which
+$\chern_1^{\beta_{-}}(E)$ can take.
+Doing this will allow us to eliminate possible values of $\chern_0(E)$ for which
+each $\chern_1^{\beta_{-}}(E)$ leads to the failure of at least one of the inequalities.
+As opposed to only eliminating possible values of $\chern_0(E)$ for which all
+corresponding $\chern_1^{\beta_{-}}(E)$ fail one of the inequalities (which is what
+was implicitly happening before).
+
+First, let's fix a Chern character for $F$,
+$\chern(F) = (R,C,D)$, and consider the possible Chern characters
+$\chern(E) = (r,c,d)$ of some semistabilizer $E$.
+ 
 \begin{sagesilent}
 	from pseudowalls import *
+
+	v = Chern_Char(*var("R C D", domain="real"))
+	u = Chern_Char(*var("r c d", domain="real"))
+
+	ts = stability.Tilt
+
+	beta_min = min(map(lambda sol: sol.rhs(),
+			solve(ts(alpha=0).degree(v), ts().beta)
+	))
+\end{sagesilent}
+
+Here, we have $\beta_{-} = \sage{beta_min}$.
+Recall [ref] that $\chern_1^{\beta_{-}}$ has fixed bounds in terms of
+$\chern(F)$, and so we can write:
+
+\begin{equation}
+	\chern_1(E) = r\beta_{-} + m
+	\qquad 0 \leq m \leq \chern_1^{\beta_{-}}(F)
+\end{equation}
+
+Furthermore, $\chern_1 \in \ZZ$ so we only need to consider
+$m \in \frac{1}{n} \ZZ \cap [0, \chern_1^{\beta_{-}}(F)]$.
+For the next subsections, we consider $m$ to be fixed with one of these values,
+and we shall be varying $\chern_0(E) = r$ to see when certain inequalities fail.
+
+\begin{sagesilent}
+	c_lower_bound = -(ts(beta=beta_min).rank(u)/ts().alpha).expand() + c
+
+	var("m", domain="real")
+	c_val = c_lower_bound + m
 \end{sagesilent}
 
+
+
+
 \section{Conclusion}
 
 \end{document}