Sort out statements and introduction leading up to bounds on d

tex/bounds-on-semistabilisers.tex

@@ -161,27 +161,8 @@ corresponding $\chern_1^{\beta}(u)$ fail one of the inequalities (which is what
 was implicitly happening before in the proof of Theorem
 \ref{thm:loose-bound-on-r}).
 
-
-First, let us fix a Chern character $v$ with $\Delta(v)\geq 0$,
-$\chern_0(v)>0$, or $\chern_0(v)=0$ and $\chern_1(v)>0$,
-and some solution $u$ to the Problem
-\ref{problem:problem-statement-1} or
-\ref{problem:problem-statement-2}.
-Take $\beta_0 = \beta(P)$ where $P\in\Theta_v^-$ is the choice made in problem
-\ref{problem:problem-statement-1}
-(or $\beta_0 = \beta_{-}$ for problem \ref{problem:problem-statement-2}).
-
-\begin{align}
-	\chern(F) =\vcentcolon\: v \:=& \:(R,C\ell,D\ell^2)
-	&& \text{where $R,C\in \ZZ$ and $D\in \frac{1}{\lcm(m,2)}\ZZ$}
-	\nonumber
-	\\
-	u \coloneqq& \:(r,c\ell,d\ell^2)
-	&& \text{where $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$}
-	\label{eqn:u-coords}
-\end{align}
-
 \begin{lemma}
+\label{lem:fixed-q-semistabs-criterion}
 	Consider the Problem \ref{problem:problem-statement-1} (or \ref{problem:problem-statement-2}),
 	and $\beta_{0}\coloneqq \beta(P)$ (or $\beta_{-}(v)$ resp.).
 
@@ -214,6 +195,7 @@ Take $\beta_0 = \beta(P)$ where $P\in\Theta_v^-$ is the choice made in problem
 \end{proof}
 
 \begin{corollary}
+\label{cor:rational-beta:fixed-q-semistabs-criterion}
 	Consider the Problem \ref{problem:problem-statement-1} (or \ref{problem:problem-statement-2}),
 	and suppose that $\beta_{0}\coloneqq \beta(P)$ (or $\beta_{-}(v)$ resp.) is
 	rational, and written $\beta_0=\frac{a_v}{n}$ for
@@ -248,11 +230,40 @@ Take $\beta_0 = \beta(P)$ where $P\in\Theta_v^-$ is the choice made in problem
 
 \subsection{Bounds on \texorpdfstring{`$d$'}{d}-values for Solutions of Problems}
 
-This section studies the numerical conditions that $u$ must satisfy as per
-lemma \ref{lem:num_test_prob1}
-(or corollary \ref{cor:num_test_prob2})
-and reformulates them as bounds on $d$ from Equation \ref{eqn:u-coords}.
-This is done to determine which $r$ values lead to no possible values for $d$.
+Let $v$ be a Chern character with $\Delta(v)\geq 0$,
+$\chern_0(v)>0$, or $\chern_0(v)=0$ and $\chern_1(v)>0$,
+and consider Problem
+\ref{problem:problem-statement-1} or
+\ref{problem:problem-statement-2}.
+Take $\beta_0 = \beta(P)$ where $P\in\Theta_v^-$ is the choice made in problem
+\ref{problem:problem-statement-1}
+(or $\beta_0 = \beta_{-}$ for problem \ref{problem:problem-statement-2}).
+Lemma \ref{lem:fixed-q-semistabs-criterion} states that any solution
+\[
+	u \coloneqq \:(r,c\ell,d\ell^2)
+	\qquad
+	\text{where $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$}
+	\label{eqn:u-coords}
+\]
+to the Problem satisfies
+\[
+	q \coloneqq \chern_1^{\beta_0}(u)
+	\in
+	\left(
+		0, \chern_1^{\beta_0}(v)
+	\right)
+\]
+and also gives a lower bound for $r$ when considering $u$ with a fixed $q$.
+This Section studies the extra numerical conditions that such $u$ must satisfy
+as per that Lemma, and will express them as bounds on $d$ in terms of $r$ (for a
+fixed $q$).
+These bounds will later be used in Subsections
+\ref{subsec:bounds-on-semistab-rank-prob-1} and
+\ref{subsec:bounds-on-semistab-rank-prob-2}
+to construct upper bounds on $r$
+(in a similar way to how a bound on $\chern_0(u)$ was found in the proof of
+Theorem \ref{thm:loose-bound-on-r} by considering bounds on
+$\chern^{\beta_0}_0(u)$ in terms of the former).
 
 \subsubsection{Size of pseudo-wall\texorpdfstring{: $\chern_2^P(u)>0$}{}}
 \label{subsect-d-bound-radiuscond}
@@ -594,6 +605,7 @@ A generic example of this is plotted in Figure
 
 \subsection{Bounds on Semistabilizer Rank \texorpdfstring{$r$}{} in Problem
 \ref{problem:problem-statement-1}}
+\label{subsec:bounds-on-semistab-rank-prob-1}
 
 As discussed at the end of subsection \ref{subsubsect:all-bounds-on-d-prob1}
 (and illustrated in Figure \ref{fig:problem1:d_bounds_xmpl_gnrc_q}),
@@ -671,6 +683,7 @@ following Lemma \ref{lem:prob1:convenient_r_bound}.
 
 \subsection{Bounds on Semistabilizer Rank \texorpdfstring{$r$}{} in Problem
 \ref{problem:problem-statement-2}}
+\label{subsec:bounds-on-semistab-rank-prob-2}
 
 Now, the inequalities from the above subsubsection
 \ref{subsubsect:all-bounds-on-d-prob2} will be used to find, for