diff --git a/main.tex b/main.tex
index 73585b3e352f3ff1b31dc239c6910d54d94db97c..3aa1e40f0c390e36a511c55c35a73a6de86fa87e 100644
--- a/main.tex
+++ b/main.tex
@@ -68,10 +68,10 @@ $0 \leq \Delta(E), \Delta(G)$ and $\Delta(E) + \Delta(G) \leq \Delta(F)$.
 We also have a condition relating to the tilt category $\firsttilt\beta$:
 $0 \leq \chern^\beta_1(E) \leq \chern^\beta_1(F)$.
 Finally, there's a condition ensuring that the radius of the circular wall is
-strictly positive: $\chern^\beta_2(E) > 0$.
+strictly positive: $\chern^{\beta_{-}}_2(E) > 0$.
 
 For any fixed $\chern_0(E)$, the inequality
-$0 \leq \chern^{\beta_{-}}_1(E) \leq \chern^{\beta_{-}}_1(F)$,
+$0 \leq \chern^{\beta}_1(E) \leq \chern^{\beta}_1(F)$,
 allows us to bound $\chern_1(E)$. Then, the other inequalities allow us to
 bound $\chern_2(E)$. The final part to showing the finiteness of pseudowalls
 would be bounding $\chern_0(E)$. This has been hinted at in [ref] and done
@@ -128,7 +128,7 @@ is best seen with the following graph:
 var("m") # Initialize symbol for variety parameter
 \end{sagesilent}
 
-This is where the $\beta_{-}$ criterion comes in. If $\beta_{-} = \frac{*}{n}$
+This is where the rationality of $\beta_{-}$ comes in. If $\beta_{-} = \frac{*}{n}$
 for some $*,n \in \ZZ$.
 Then $\chern^\beta_2(E) \in \frac{1}{\lcm(m,2n^2)}\ZZ$ where $m$ is the integer
 which guarantees $\chern_2(E) \in \frac{1}{m}\ZZ$ (determined by the variety).
@@ -151,11 +151,11 @@ for the rank of $E$:
 
 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.
+$\chern_1^{\beta}(E)$ 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.
+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
+corresponding $\chern_1^{\beta}(E)$ fail one of the inequalities (which is what
 was implicitly happening before).
 
 First, let us fix a Chern character for $F$,
@@ -174,7 +174,7 @@ u = Chern_Char(*var("r c d", domain="real"))
 Δ = lambda v: v.Q_tilt()
 \end{sagesilent}
 
-Recall [ref] that $\chern_1^{\beta_{-}}$ has fixed bounds in terms of
+Recall [ref] that $\chern_1^{\beta}$ has fixed bounds in terms of
 $\chern(F)$, and so we can write:
 
 \begin{sagesilent}
@@ -193,11 +193,11 @@ c_in_terms_of_q = c_lower_bound + q
 \begin{equation}
 	\label{eqn-cintermsofm}
 	c=\chern_1(E) = \sage{c_in_terms_of_q}
-	\qquad 0 \leq q \leq \chern_1^{\beta_{-}}(F)
+	\qquad 0 \leq q \leq \chern_1^{\beta}(F)
 \end{equation}
 
 Furthermore, $\chern_1 \in \ZZ$ so we only need to consider
-$q \in \frac{1}{n} \ZZ \cap [0, \chern_1^{\beta_{-}}(F)]$.
+$q \in \frac{1}{n} \ZZ \cap [0, \chern_1^{\beta}(F)]$.
 For the next subsections, we consider $q$ to be fixed with one of these values,
 and we shall be varying $\chern_0(E) = r$ to see when certain inequalities fail.