diff --git a/main.tex b/main.tex
index 0847d0d6fde13ef62956eefac8ad7a081b01c90b..e282d781b5785396806ba2cd05b6221cbe3516ec 100644
--- a/main.tex
+++ b/main.tex
@@ -661,6 +661,9 @@ vertical wall (TODO as discussed in ref).
 \let\originalbeta\beta
 \renewcommand\beta{{\originalbeta_{-}}}
 
+\bgroup
+% redefine \psi in sage expressions (placeholder for ch_1^\beta(F)
+\def\psi{\chern_1^{\beta}(F)}
 \begin{align}
 	d &\geq&
 	\sage{bgmlv1_d_lowerbound_linear_term}
@@ -678,11 +681,13 @@ vertical wall (TODO as discussed in ref).
 \\
 	d &\leq&
 	\sage{bgmlv3_d_upperbound_linear_term}
-	&+ \sage{bgmlv3_d_upperbound_const_term_alt.subs(chbv == 0)}
-	+& \sage{bgmlv3_d_upperbound_exp_term_alt.subs(chbv == 0)},
+	&+ \sage{bgmlv3_d_upperbound_const_term_alt.subs(phi == 0)}
+	% ^ ch_2^\beta(F)=0 for beta_{-}
+	+& \sage{bgmlv3_d_upperbound_exp_term_alt2},
 	 &\qquad\text{when\:} r > R
 	 \label{eqn:bgmlv3_d_bound_betamin}
 \end{align}
+\egroup
 
 Furthermore, we get an extra bound for $d$ resulting from the condition that the
 radius of the circular wall must be positive. As discussed in (TODO ref), this