Comment on bounds collapsing at extreme values of q

main.tex

@@ -567,18 +567,21 @@ vertical wall (TODO as discussed in ref).
 	&+ \sage{bgmlv1_d_lowerbound_const_term_alt.subs(chbv == 0)}
 	+& \sage{bgmlv1_d_lowerbound_exp_term_alt.subs(chbv == 0)},
 	 &\qquad\text{when\:} r > \frac{R}{2}
+	 \label{eqn:bgmlv1_d_bound_betamin}
 \\
 	d &\leq&
 	\sage{bgmlv2_d_upperbound_linear_term}
 	&+ \sage{bgmlv2_d_upperbound_const_term}
 	+& \sage{bgmlv2_d_upperbound_exp_term},
 	 &\qquad\text{when\:} r > 0
+	 \label{eqn:bgmlv2_d_bound_betamin}
 \\
 	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)},
 	 &\qquad\text{when\:} r > R
+	 \label{eqn:bgmlv3_d_bound_betamin}
 \end{align}
 
 Furthermore, we get an extra bound for $d$ resulting from the condition that the
@@ -596,9 +599,10 @@ positive_radius_condition = (
 )
 \end{sagesilent}
 
-\begin{equation*}
+\begin{equation}
+	\label{eqn:positive_rad_d_bound_betamin}
 	\sage{positive_radius_condition}
-\end{equation*}
+\end{equation}
 \begin{sagesilent}
 def beta_min(chern):
   ts = stability.Tilt()
@@ -704,13 +708,13 @@ def plot_d_bound(
 \begin{subfigure}{.5\textwidth}
   \centering
 	\sageplot[width=\linewidth]{plot_d_bound(v_example, 0)}
-	\caption{$q = 0$}
+	\caption{$q = 0$ (all bounds other than green coincide on line)}
   \label{fig:d_bounds_xmpl_min_q}
 \end{subfigure}%
 \begin{subfigure}{.5\textwidth}
   \centering
 	\sageplot[width=\linewidth]{plot_d_bound(v_example, 4)}
-	\caption{$q = \chern^{\beta}(F)$}
+	\caption{$q = \chern^{\beta}(F)$ (all bounds other than blue coincide on line)}
   \label{fig:d_bounds_xmpl_max_q}
 \end{subfigure}
 \caption{
@@ -721,6 +725,21 @@ def plot_d_bound(
 \label{fig:d_bounds_xmpl_extrm_q}
 \end{figure}
 
+Recalling that $q := \chern^{\beta}_1(E) \in [0, \chern^{\beta}_1(F)]$,
+it's worth noting that the extreme values of $q$ in this range lead to the
+tightest bounds on $d$, as illustrated in figure
+(\ref{fig:d_bounds_xmpl_extrm_q}).
+In fact, in each case, one of the weak upper bounds coincides with one of the
+weak lower bounds, (implying no possible destabilizers $E$ with
+$\chern_0(E)=:r>R:=\chern_0(F)$ for these $q$-values).
+This indeed happens in general since the right hand sides of
+(eqn \ref{eqn:bgmlv2_d_bound_betamin}) and
+(eqn \ref{eqn:positive_rad_d_bound_betamin}) match when $q=0$.
+In the other case, $q=\chern^{\beta}_1(F)$, it's the right hand sides of
+(eqn \ref{eqn:bgmlv3_d_bound_betamin}) and
+(eqn \ref{eqn:positive_rad_d_bound_betamin}) which match.
+
+
 \begin{figure}
 \centering
 \sageplot[