diff --git a/tex/bounds-on-semistabilisers.tex b/tex/bounds-on-semistabilisers.tex
index b6b9aa1b52cbdf7e352356c5d5e73cbfb5589e27..805089b2d7789d09c214cff0acf85da6fed432d6 100644
--- a/tex/bounds-on-semistabilisers.tex
+++ b/tex/bounds-on-semistabilisers.tex
@@ -131,18 +131,21 @@ rank that appears turns out to be $\sage{extravagant.actual_rmax}$.
\section{Tighter Bounds}
\label{sec:refinement}
-To get tighter bounds on the rank of destabilisers $E$ of some $F$ with some
-fixed Chern character, we will need to consider each of the values which
-$\chern_1^{\beta}(E)$ can take.
-Doing this will allow us to eliminate possible values of $\chern_0(E)$ for which
-each $\chern_1^{\beta}(E)$ leads to the failure of at least one of the inequalities.
-As opposed to only eliminating possible values of $\chern_0(E)$ for which all
-corresponding $\chern_1^{\beta}(E)$ fail one of the inequalities (which is what
+To get tighter bounds on the rank of solutions $u$ to the Problems
+\ref{problem:problem-statement-1} and
+\ref{problem:problem-statement-2},
+we will need to consider each of the values which
+$\chern_1^{\beta}(u)$ can take.
+Doing this will allow us to eliminate possible values of $\chern_0(u)$ for which
+each $\chern_1^{\beta}(u)$ leads to the failure of at least one of the inequalities.
+As opposed to only eliminating possible values of $\chern_0(u)$ for which all
+corresponding $\chern_1^{\beta}(u)$ fail one of the inequalities (which is what
was implicitly happening before).
-First, let us fix a Chern character for $F$, and some pseudo-semistabiliser
-$u$ which is a solution to problem
+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 $\beta = \beta(P)$ where $P\in\Theta_v^-$ is the choice made in problem
@@ -155,8 +158,8 @@ Take $\beta = \beta(P)$ where $P\in\Theta_v^-$ is the choice made in problem
\\
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}
-
Recall from condition \ref{item:chern1bound:lem:num_test_prob1} in
@@ -177,7 +180,7 @@ from plots_and_expressions import c_in_terms_of_q
\qquad 0 \leq q \coloneqq \chern_1^{\beta}(u) \leq \chern_1^{\beta}(v)
\end{equation}
-Furthermore, $\chern_1 \in \ZZ$ so we only need to consider
+Furthermore, if $\beta$ is rational, $\chern_1 \in \ZZ$ so we only need to consider
$q \in \frac{1}{n} \ZZ \cap [0, \chern_1^{\beta}(F)]$,
where $n$ is the denominator of $\beta$.
For the next subsections, we consider $q$ to be fixed with one of these values,
@@ -189,6 +192,8 @@ and we shall be varying $\chern_0(E) = r$ to see when certain inequalities fail.
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 $d$.
\subsubsection{Size of pseudo-wall\texorpdfstring{: $\chern_2^P(u)>0$}{}}
\label{subsect-d-bound-radiuscond}
@@ -197,7 +202,6 @@ This condition refers to condition
\ref{item:radiuscond:lem:num_test_prob1}
from Lemma \ref{lem:num_test_prob1}
(or corollary \ref{cor:num_test_prob2}).
-
In the case where we are tackling problem \ref{problem:problem-statement-2}
(with $\beta = \beta_{-}$), this condition, when expressed as a bound on $d$,
amounts to:
@@ -434,14 +438,13 @@ for the bounds on $d$ in terms of $r$ is illustrated in Figure
The question of whether there are pseudo-destabilisers of arbitrarily large
rank, in the context of the graph, comes down to whether there are points
$(r,d) \in \ZZ \oplus \frac{1}{\lcm(m,2)} \ZZ$ (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 $\Delta(u), \Delta(v-u) > 0$).
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:radiuscond_d_bound_betamin}).
-As mentioned in the introduction (sec \ref{sec:intro}), the finiteness of these
+As mentioned in the introduction to this Part, 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.
@@ -542,7 +545,7 @@ the lower bound on $d$ is equal to one of the upper bounds on $d$
(i.e. finding certain intersection points of the graph in Figure
\ref{fig:problem1:d_bounds_xmpl_gnrc_q}).
-\begin{lemma}[Problem \ref{problem:problem-statement-1} upper Bound on $r$]
+\begin{theorem}[Problem \ref{problem:problem-statement-1} upper Bound on $r$]
\label{lem:prob1:r_bound}
Let $u$ be a solution to problem \ref{problem:problem-statement-1}
and $q\coloneqq\chern_1^{B}(u)$.
@@ -550,7 +553,7 @@ the lower bound on $d$ is equal to one of the upper bounds on $d$
\begin{equation}
\sage{problem1.r_bound_expression}
\end{equation}
-\end{lemma}
+\end{theorem}
\begin{proof}
Recall that $d\coloneqq\chern_2(u)$ has two upper bounds in terms of $r$: in
@@ -578,14 +581,14 @@ But given that $0 \leq q \leq \chern_1^{B}(v)$, we can take the maximum of this
bound, over $q$ in this range, to get a simpler (but weaker) bound in the
following Lemma \ref{lem:prob1:convenient_r_bound}.
-\begin{lemma}
+\begin{theorem}[Problem \ref{problem:problem-statement-1} global upper Bound on $r$]
\label{lem:prob1:convenient_r_bound}
Let $u$ be a solution to problem \ref{problem:problem-statement-1}.
Then $r\coloneqq\chern_0(u)$ is bounded above by the following expression:
\begin{equation}
\sage{problem1.r_max}
\end{equation}
-\end{lemma}
+\end{theorem}
\begin{proof}
The first term of the minimum in Lemma \ref{lem:prob1:r_bound}
@@ -599,11 +602,11 @@ following Lemma \ref{lem:prob1:convenient_r_bound}.
\end{proof}
-\begin{note}
+\begin{remark}
$q_{\mathrm{max}} > 0$ is immediate from the expression, but
$q_{\mathrm{max}} \leq \chern_1^{B}(v)$ is equivalent to $\Delta(v) \geq 0$,
which is true by assumption in this setting.
-\end{note}
+\end{remark}
\subsection{Bounds on Semistabilizer Rank \texorpdfstring{$r$}{} in Problem
@@ -614,7 +617,7 @@ Now, the inequalities from the above subsubsection
each given $q=\chern^{\beta}_1(E)$, how large $r$ needs to be in order to leave
no possible solutions for $d$. At that point, there are no solutions
$u=(r,c\ell,d\ell^2)$ to problem \ref{problem:problem-statement-2}.
-The strategy here is similar to what was shown in theorem
+The strategy here is similar to what was shown in Theorem
\ref{thm:loose-bound-on-r}.
@@ -747,7 +750,7 @@ from plots_and_expressions import q_sol, bgmlv_v, psi
\begin{proof}
The ranks of the pseudo-semistabilisers for $v$ are bounded above by the
-maximum over $q\in [0, \chern_1^{\beta}(F)]$ of the expression in theorem
+maximum over $q\in [0, \chern_1^{\beta}(F)]$ of the expression in Theorem
\ref{thm:rmax_with_uniform_eps}.
Noticing that the expression is a maximum of two quadratic functions in $q$:
\begin{equation*}
@@ -813,7 +816,7 @@ this value of $\frac{1}{2n^2}\ZZ$ explicitly.
The expressions that will follow will be a bit more complicated and have more
parts which depend on the values of $q$ and $\beta$, even their numerators
-$\aa,\bb$ specifically. The upcoming Theorem (TODO ref) is less useful as a
+$\aa,\bb$ specifically. The upcoming Theorem \ref{thm:rmax_with_eps1} is less useful as a
`clean' formula for a bound on the ranks of the pseudo-semistabilisers, but has a
purpose in the context of writing a computer program to find
pseudo-semistabilisers. Such a program would iterate through possible values of