Clean up summary of bounds on d

main.tex

@@ -1327,10 +1327,6 @@ These give bounds with the same assymptotes when we take $r\to\infty$
 \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
-is equivalent to $\chern^{\beta}_2(E) > 0$, which yields:
-
 \begin{sagesilent}
 positive_radius_condition = (
 	(
@@ -1340,13 +1336,7 @@ positive_radius_condition = (
 	.subs(solve(q == u.twist(beta).ch[1], c)[0]) # express c in term of q
 	.expand()
 )
-\end{sagesilent}
 
-\begin{equation}
-	\label{eqn:positive_rad_d_bound_betamin}
-	\sage{positive_radius_condition}
-\end{equation}
-\begin{sagesilent}
 def beta_min(chern):
   ts = stability.Tilt()
   return min(
@@ -1462,10 +1452,10 @@ weak lower bounds, (implying no possible destabilizers $E$ with
 $\chern_0(E)=\vcentcolon r>R\coloneqq\chern_0(F)$ 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 $q=0$.
+(eqn \ref{eqn:radiuscond_d_bound_betamin}) match when $q=0$.
 In the other case, $q=\chern^{\beta}_1(F)$, it is the right hand sides of
 (eqn \ref{eqn:bgmlv3_d_bound_betamin}) and
-(eqn \ref{eqn:positive_rad_d_bound_betamin}) which match.
+(eqn \ref{eqn:radiuscond_d_bound_betamin}) which match.
 
 
 The more generic case, when $0 < q\coloneqq\chern_1^{\beta}(E) < \chern_1^{\beta}(F)$
@@ -1480,7 +1470,7 @@ blue and green (ensuring $\Delta(E), \Delta(G) > 0$).
 These lines have the same assymptote at $r \to \infty$
 (eqns \ref{eqn:bgmlv2_d_bound_betamin},
 \ref{eqn:bgmlv3_d_bound_betamin},
-\ref{eqn:positive_rad_d_bound_betamin}).
+\ref{eqn:radiuscond_d_bound_betamin}).
 As mentioned in the introduction (sec \ref{sec:intro}), the finiteness of these
 solutions is entirely determined by whether $\beta$ is rational or irrational.
 Some of the details around the associated numerics are explored next.