Comment on general q case and split beta_min into (ir)rational cases

main.tex

@@ -15,11 +15,13 @@
 \newcommand{\QQ}{\mathbb{Q}}
 \newcommand{\ZZ}{\mathbb{Z}}
 \newcommand{\RR}{\mathbb{R}}
+\newcommand{\NN}{\mathbb{N}}
 \newcommand{\chern}{\operatorname{ch}}
 \newcommand{\lcm}{\operatorname{lcm}}
 \newcommand{\firsttilt}[1]{\mathcal{B}^{#1}}
 \newcommand{\bddderived}{\mathcal{D}^{b}}
 \newcommand{\centralcharge}{\mathcal{Z}}
+\newcommand{\minorheading}[1]{{\noindent\normalfont\normalsize\bfseries #1}}
 
 \begin{document}
 
@@ -741,6 +743,30 @@ In the other case, $q=\chern^{\beta}_1(F)$, it's the right hand sides of
 (eqn \ref{eqn:positive_rad_d_bound_betamin}) which match.
 
 
+The more generic case, when $0 < q:=\chern_1{\beta}(E) < \chern_1^{\beta}(F)$
+for the bounds on $d$ in terms of $r$ is illustrated in figure
+(\ref{fig:d_bounds_xmpl_gnrc_q}).
+The question of whether there are pseudo-destabilizers of arbitrarily large
+rank, in the context of the graph, comes down to whether there are points
+$(r,d) \in \ZZ \oplus \frac{1}{m} \ZZ$ (with large $r$)
+% TODO have a proper definition for pseudo-destabilizers/walls
+that fit above the yellow line (ensuring positive radius of wall) but below the
+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}).
+The finiteness of these solutions will be entirely determined by whether $\beta$
+is rational or irrational, as covered next.
+
+
+\minorheading{Rational $\beta$}
+
+Suppose $\beta = \frac{*}{n}$ for some $n \in \NN,* \in \ZZ$.
+
+\minorheading{Irrational $\beta$}
+
+
 \begin{figure}
 \centering
 \sageplot[