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+\section{Introduction}
+\label{sec:intro}
+
+The theory of Bridgeland stability conditions \cite{BridgelandTom2007SCoT} on
+complexes of sheaves was developed as a generalisation of stability for vector
+bundles. The definition is most analoguous to Mumford stability, but is more
+aware of the features that sheaves can have on spaces of dimension greater
+than 1. Whilst also asymptotically matching up with Gieseker stability.
+For K3 surfaces, explicit stability conditions were defined in
+\cite{Bridgeland_StabK3}, and later shown to also be valid on other surfaces.
+
+The moduli spaces of stable objects of some fixed Chern character $v$ is
+studied, as well as how they change as we vary the Bridgeland stability
+condition. They in fact do not change over whole regions of the stability
+space (called chambers), but do undergo changes as we cross `walls' in the
+stability space. These are where there is some stable object $F$ of $v$ which
+has a subobject who's slope overtakes the slope of $v$, making $F$ unstable
+after crossing the wall.
+
+% NOTE: SURFACE SPECIALIZATION
+% (come back to these when adjusting to general Picard rank 1)
+In this document we concentrate on two surfaces: Principally polarized abelian
+surfaces and the projective surface $\PP^2$. Although this can be generalised
+for Picard rank 1 surfaces, the formulae will need adjusting.
+The Bridgeland stability conditions (defined in \cite{Bridgeland_StabK3}) are
+given by two parameters $\alpha \in \RR_{>0}$, $\beta \in \RR$, which will be
+illustrated throughout this article with diagrams of the upper half plane.
+
+It is well known that for any rational $\beta_0$,
+the vertical line $\{\sigma_{\alpha,\beta_0} \colon \alpha \in \RR_{>0}\}$ only
+intersects finitely many walls
+\cite[Thm 1.1]{LoJason2014Mfbs}
+\cite[Prop 4.2]{alma9924569879402466}
+\cite[Lemma 5.20]{MinaHiroYana_SomeModSp}.
+A consequence of this is that if
+$\beta_{-}$ is rational, then there can only be finitely many circular walls to the
+left of the vertical wall $\beta = \mu$.
+On the other hand, when $\beta_{-}$ is not rational, \cite{yanagida2014bridgeland}
+showed that there are infinitely many walls.
+
+This dichotomy does not only hold for real walls, realised by actual objects in
+$\bddderived(X)$, but also for pseudowalls. Here pseudowalls are defined as
+`potential' walls, induced by hypothetical Chern characters of semistabilizers
+which satisfy certain numerical conditions which would be satisfied by any real
+destabilizer, regardless of whether they are realised by actual semistabilizers
+in $\bddderived(X)$ (dfn \ref{dfn:pseudo-semistabilizer}).
+
+Since real walls are a subset of pseudowalls, the irrational $\beta_{-}$ case
+follows immediately from the corresponding case for real walls.
+However, the rational $\beta_{-}$ case involves showing that the following
+conditions only admit finitely many solutions (despite the fact that the same
+conditions admit infinitely many solutions when $\beta_{-}$ is irrational).
+
+
+For a semistabilizing sequence
+$E \hookrightarrow F \twoheadrightarrow G$ in $\mathcal{B}^\beta$
+we have the following conditions.
+There are some Bogomolov-Gieseker inequalities:
+$0 \leq \Delta(E), \Delta(G)$.
+We also have a condition relating to the tilt category $\firsttilt\beta$:
+$0 \leq \chern^\beta_1(E) \leq \chern^\beta_1(F)$.
+Finally, there is a condition ensuring that the radius of the circular wall is
+strictly positive: $\chern^{\beta_{-}}_2(E) > 0$.
+
+For any fixed $\chern_0(E)$, the inequality
+$0 \leq \chern^{\beta}_1(E) \leq \chern^{\beta}_1(F)$,
+allows us to bound $\chern_1(E)$. Then, the other inequalities allow us to
+bound $\chern_2(E)$. The final part to showing the finiteness of pseudowalls
+would be bounding $\chern_0(E)$. This has been hinted at in
+\cite{SchmidtBenjamin2020Bsot} and done explicitly by Benjamin Schmidt within a
+SageMath \cite{sagemath} library which computes pseudowalls
+\cite{SchmidtGithub2020}.
+Here we discuss these bounds in more detail, along with the methods used,
+followed by refinements on them which give explicit formulae for tighter bounds
+on $\chern_0(E)$ of potential destabilizers $E$ of $F$.
+
+
+\section{Setting and Definitions: Clarifying `pseudo'}
+
+%\begin{definition}[Twisted Chern Character]
+%\label{sec:twisted-chern}
+%For a given $\beta$, define the twisted Chern character as follows.
+%\[\chern^\beta(E) = \chern(E) \cdot \exp(-\beta \ell)\]
+%\noindent
+%Component-wise, this is:
+%\begin{align*}
+%	\chern^\beta_0(E) &= \chern_0(E)
+%\\
+%	\chern^\beta_1(E) &= \chern_1(E) - \beta \chern_0(E)
+%\\
+%	\chern^\beta_2(E) &= \chern_2(E) - \beta \chern_1(E) + \frac{\beta^2}{2} \chern_0(E)
+%\end{align*}
+%where $\chern_i$ is the coefficient of $\ell^i$ in $\chern$.
+%
+%% TODO I think this^ needs adjusting for general Surface with $\ell$
+%\end{definition}
+%
+%$\chern^\beta_1(E)$ is the imaginary component of the central charge
+%$\centralcharge_{\alpha,\beta}(E)$ and any element of $\firsttilt\beta$
+%satisfies $\chern^\beta_1 \geq 0$.
+
+Throughout this article, as noted in the introduction, we will be exclusively
+working over surfaces $X$ with Picard rank 1, with a choice of ample line bundle
+$L$ such that $\ell\coloneqq c_1(L)$ generates $NS(X)$.
+We take $m\coloneqq \ell^2$ as this will be the main quantity which will
+affect the results.
+
+\begin{definition}[Pseudo-semistabilizers]
+\label{dfn:pseudo-semistabilizer}
+% NOTE: SURFACE SPECIALIZATION
+	Given a Chern Character $v$, and a given stability
+	condition $\sigma_{\alpha,\beta}$,
+	a \textit{pseudo-semistabilizing} $u$ is a `potential' Chern character:
+	\[
+		u = \left(r, c\ell, \frac{e}{\lcm(m,2)} \ell^2\right)
+		\qquad
+		r,c,e \in \ZZ
+	\]
+	which has the same tilt slope as $v$:
+	$\nu_{\alpha,\beta}(u) = \nu_{\alpha,\beta}(v)$.
+
+	\noindent
+	Furthermore the following inequalities are satisfied:
+	\begin{itemize}
+		\item $\Delta(u) \geq 0$
+		\item $\Delta(v-u) \geq 0$
+		\item $0 \leq \chern_1^{\beta}(u) \leq \chern_1^{\beta}(v)$
+	\end{itemize}
+
+	Note $u$ does not need to be a Chern character of an actual sub-object of some
+	object in the stability condition's heart with Chern character $v$.
+\end{definition}
+
+At this point, and in this document, we do not care about whether
+pseudo-semistabilizers are even Chern characters of actual elements of
+$\bddderived(X)$, some other sources may have this extra restriction too.
+
+Later, Chern characters will be written $(r,c\ell,d\ell^2)$ because operations
+(such as multiplication) are more easily defined in terms of the coefficients of
+the $\ell^i$. However, at the end, it will become important again that
+$d \in \frac{1}{\lcm(m,2)}\ZZ$.
+
+\begin{definition}[Pseudo-walls]
+\label{dfn:pseudo-wall}
+	Let $u$ be a pseudo-semistabilizer of $v$, for some stability condition.
+	Then the \textit{pseudo-wall} associated to $u$ is the set of all stablity
+	conditions where $u$ is a pseudo-semistabilizer of $v$.
+\end{definition}
+
+% TODO possibly reference forwards to Bertram's nested wall theorem section to 
+% cover that being a pseudo-semistabilizer somewhere implies also on whole circle
+
+\begin{lemma}[Sanity check for Pseudo-semistabilizers]
+	Given a stability
+	condition $\sigma_{\alpha,\beta}$,
+	if $E\hookrightarrow F\twoheadrightarrow G$ is a semistabilizing sequence in
+	$\firsttilt\beta$ for $F$.
+	Then $\chern(E)$ is a pseudo-semistabilizer of $\chern(F)$
+\end{lemma}
+
+\begin{proof}
+	Suppose $E\hookrightarrow F\twoheadrightarrow G$ is a semistabilizing
+	sequence with respect to a stability condition $\sigma_{\alpha,\beta}$.
+	\begin{equation*}
+		\chern(E) = -\chern(\homol^{-1}_{\coh}(E)) + \chern(\homol^{0}_{\coh}(E))
+	\end{equation*}
+	Therefore, $\chern(E)$ is of the form
+	$(r,c\ell,\frac{e}{\lcm(m,2)}\ell^2)$
+	provided that this is true for any coherent sheaf.
+	For any coherent sheaf $H$, we have the following:
+	\begin{equation*}
+		\chern(H) = \left(c_0(H), c_1(H), - c_2(H) + \frac{1}{2} {c_1(H)}^2\right)
+	\end{equation*}
+	Given that $\ell$ generates the Neron-Severi group, $c_1(H)$ can then be
+	written $c\ell$.
+	\begin{equation*}
+		\chern(H) = \left(
+			c_0(H), c\ell,
+			\left(- \frac{c_2(H)}{\ell^2} + \frac{c^2}{2} \right)\ell^2
+		\right)
+	\end{equation*}
+	This fact along with $c_0$, $c_2$ being an integers on surfaces, and
+	$m\coloneqq \ell^2$ implies that $\chern(H)$
+	(hence $\chern(E)$ too) is of the required form.
+	
+
+	Since all the objects in the sequence are in $\firsttilt\beta$, we have
+	$\chern_1^{\beta} \geq 0$ for each of them. Due to additivity
+	($\chern(F) = \chern(E) + \chern(G)$), we can deduce
+	$0 \leq \chern_1^{\beta}(E) \leq \chern_1^{\beta}(F)$.
+
+
+	$E \hookrightarrow F \twoheadrightarrow G$ being a semistabilizing sequence
+	means	$\nu_{\alpha,\beta}(E) = \nu_{\alpha,\beta}(F) = \nu_{\alpha,\beta}(F)$.
+	% MAYBE: justify this harder
+	But also, that this is an instance of $F$ being semistable, so $E$ must also
+	be semistable
+	(otherwise the destabilizing subobject would also destabilize $F$).
+	Similarly $G$ must also be semistable too.
+	$E$ and $G$ being semistable implies they also satisfy the Bogomolov
+	inequalities:
+	% TODO ref Bogomolov inequalities for tilt stability
+	$\Delta(E), \Delta(G) \geq 0$.
+	Expressing this in terms of Chern characters for $E$ and $F$ gives:
+	$\Delta(\chern(E)) \geq 0$ and $\Delta(\chern(F)-\chern(E)) \geq 0$.
+
+\end{proof}
+
+
+\section{Characteristic Curves of Stability Conditions Associated to Chern
+Characters}
+
+% NOTE: SURFACE SPECIALIZATION
+Considering the stability conditions with two parameters $\alpha, \beta$ on
+Picard rank 1 surfaces.
+We can draw 2 characteristic curves for any given Chern character $v$ with
+$\Delta(v) \geq 0$ and positive rank.
+These are given by the equations $\chern_i^{\alpha,\beta}(v)=0$ for $i=1,2$.
+
+\begin{definition}[Characteristic Curves $V_v$ and $\Theta_v$]
+Given a Chern character $v$, with positive rank and $\Delta(v) \geq 0$, we
+define two characteristic curves on the $(\alpha, \beta)$-plane:
+
+\begin{align*}
+	V_v &\colon \:\: \chern_1^{\alpha, \beta}(v) = 0 \\
+	\Theta_v &\colon \:\: \chern_2^{\alpha, \beta}(v) = 0
+\end{align*}
+\end{definition}
+
+\subsection{Geometry of the Characteristic Curves}
+
+These characteristic curves for a Chern character $v$ with $\Delta(v)\geq0$ are
+not affected by flipping the sign of $v$ so it's only necessary to consider
+non-negative rank.
+As discussed in subsection \ref{subsect:relevance-of-V_v}, making this choice
+has Gieseker stable coherent sheaves appearing in the heart of the stability
+condition $\firsttilt{\beta}$ as we move `left' (decreasing $\beta$).
+
+\subsubsection{Positive Rank Case}
+\label{subsect:positive-rank-case-charact-curves}
+
+\begin{fact}[Geometry of Characteristic Curves in Positive Rank Case]
+The following facts can be deduced from the formulae for $\chern_i^{\alpha, \beta}(v)$
+as well as the restrictions on $v$, when $\chern_0(v)>0$:
+\begin{itemize}
+	\item $V_v$ is a vertical line at $\beta=\mu(v)$
+	\item $\Theta_v$ is a hyperbola with assymptotes angled at $\pm 45^\circ$
+		crossing where $V_v$ meets the $\beta$-axis: $(\mu(v),0)$
+	\item $\Theta_v$ is oriented with left-right branches (as opposed to up-down).
+		The left branch shall be labelled $\Theta_v^-$ and the right $\Theta_v^+$.
+	\item The gap along the $\beta$-axis between either branch of $\Theta_v$
+		and $V_v$ is $\sqrt{\Delta(v)}/\chern_0(v)$.
+	\item When $\Delta(v)=0$, $\Theta_v$ degenerates into a pair of lines, but the
+		labels $\Theta_v^\pm$ will still be used for convenience.
+\end{itemize}
+\end{fact}
+
+These are illustrated in Fig \ref{fig:charact_curves_vis}
+(dotted line for $i=1$, solid for $i=2$).
+
+\begin{sagesilent}
+from characteristic_curves import \
+typical_characteristic_curves, \
+degenerate_characteristic_curves
+\end{sagesilent}
+
+
+\begin{figure}
+\centering
+\begin{subfigure}{.49\textwidth}
+	\centering
+	\sageplot[width=\textwidth]{typical_characteristic_curves}
+	\caption{$\Delta(v)>0$}
+	\label{fig:charact_curves_vis_bgmvlPos}
+\end{subfigure}%
+\hfill
+\begin{subfigure}{.49\textwidth}
+	\centering
+	\sageplot[width=\textwidth]{degenerate_characteristic_curves}
+	\caption{
+		$\Delta(v)=0$: hyperbola collapses
+	}
+	\label{fig:charact_curves_vis_bgmlv0}
+\end{subfigure}
+\caption{
+	Characteristic curves ($\chern_i^{\alpha,\beta}(v)=0$) of stability conditions
+	associated to Chern characters $v$ with $\Delta(v) \geq 0$ and positive rank.
+}
+\label{fig:charact_curves_vis}
+\end{figure}
+
+\begin{definition}[$\beta_{\pm}$]
+	\label{dfn:beta_pm}
+	Given a formal Chern character $v$ with positive rank, we define $\beta_{\pm}(v)$ to be
+	the $\beta$-coordinate of where $\Theta_v^{\pm}$ meets the $\beta$-axis:
+	\[
+		\beta_\pm(R,C\ell,D\ell^2) = \frac{C \pm \sqrt{C^2-2RD}}{R}
+	\]
+	\noindent
+	In particular, this means $\beta_\pm(v)$ are the two roots of the quadratic
+	equation $\chern_2^{\beta}(v)=0$.
+
+	This definition will be extended to the rank 0 case in definition \ref{dfn:beta_-_rank0}.
+\end{definition}
+
+
+\subsubsection{Rank Zero Case}
+\label{subsubsect:rank-zero-case-charact-curves}
+
+\begin{sagesilent}
+from rank_zero_case import Theta_v_plot
+\end{sagesilent}
+
+\begin{fact}[Geometry of Characteristic Curves in Rank 0 Case]
+The following facts can be deduced from the formulae for $\chern_i^{\alpha, \beta}(v)$
+as well as the restrictions on $v$, when $\chern_0(v)=0$ and $\chern_1(v)>0$:
+
+
+\begin{minipage}{0.5\textwidth}
+\begin{itemize}
+	\item $V_v = \emptyset$
+	\item $\Theta_v$ is a vertical line at $\beta=\frac{D}{C}$
+		where $v=\left(0,C\ell,D\ell^2\right)$
+\end{itemize}
+\end{minipage}
+\hfill
+\begin{minipage}{0.49\textwidth}
+	\sageplot[width=\textwidth]{Theta_v_plot}
+	%\caption{$\Delta(v)>0$}
+	%\label{fig:charact_curves_rank0}
+\end{minipage}
+\end{fact}
+
+We can view the characteristic curves for $\left(0,C\ell, D\ell^2\right)$ with $C>0$ as
+the limiting behaviour of those of $\left(\varepsilon, C\ell, D\ell^2\right)$.
+Indeed:
+\begin{align*}
+	\mu\left(\varepsilon, C\ell, D\ell^2\right) = \frac{C}{\varepsilon} &\longrightarrow +\infty
+	\\
+	\text{as} \:\: 0<\varepsilon &\longrightarrow 0
+\end{align*}
+So we can view $V_v$ as moving off infinitely to the right, with $\Theta_v^+$ even further.
+But also, considering the base point of $\Theta_v^-$:
+\begin{align*}
+	\beta_{-}\left(\varepsilon, C\ell, D\ell^2\right) = \frac{C - \sqrt{C^2-2D\varepsilon}}{\varepsilon}
+	&\longrightarrow \frac{D}{C}
+	\\
+	\text{as} \:\: 0<\varepsilon &\longrightarrow 0
+	&\text{(via L'H\^opital)}
+\end{align*}
+
+So we can view $\Theta_v^-$ as approaching the vertical line that $\Theta_v$  becomes.
+For this reason, I will refer to the whole of $\Theta_v$ in the rank zero case
+as $\Theta_v^-$ to be able to use the same terminology in both positive rank
+and rank zero cases.
+
+\begin{definition}[Extending $\beta_-$ to rank 0 case]
+	\label{dfn:beta_-_rank0}
+	Given a formal Chern character $v$ with rank 0 and $\chern_1(v)>0$, we define
+	$\beta_-(v)$ to be the $\beta$-coordinate of point where $\Theta_v$ meets the
+	$\beta$-axis:
+	\[
+		\beta_-(0,C\ell,D\ell^2) = \frac{D}{C}
+	\]
+	\noindent
+	If $\beta_+$ were also to be generalised to the rank 0 case, we would consider
+	its value to be $+\infty$ due to the discussion above.
+\end{definition}
+
+
+\subsection{Relevance of \texorpdfstring{$V_v$}{V_v}}
+\label{subsect:relevance-of-V_v}
+
+For the positive rank case, by definition of the first tilt $\firsttilt\beta$, objects of Chern character
+$v$ can only be in $\firsttilt\beta$ on the left of $V_v$, where
+$\chern_1^{\alpha,\beta}(v)>0$, and objects of Chern character $-v$ can only be
+in $\firsttilt\beta$ on the right, where $\chern_1^{\alpha,\beta}(-v)>0$. In
+fact, if there is a Gieseker semistable coherent sheaf $E$ of Chern character
+$v$, then $E \in \firsttilt\beta$ if and only if $\beta<\mu(E)$ (left of the
+$V_v$), and $E[1] \in \firsttilt\beta$ if and only if $\beta\geq\mu(E)$.
+Because of this, when using these characteristic curves, only positive ranks are
+considered, as negative rank objects are implicitly considered on the right hand
+side of $V_v$.
+
+In the rank zero case, this still applies if we consider $V_v$ to be
+`infinitely to the right' ($\mu(v) = +\infty$). Precisely, Gieseker semistable
+coherent sheaves $E$ of Chern character $v$ are contained in
+$\firsttilt{\beta}$ for all $\beta$
+
+
+
+\subsection{Relevance of \texorpdfstring{$\Theta_v$}{Θ_v}}
+
+Since $\chern_2^{\alpha, \beta}$ is the numerator of the tilt slope
+$\nu_{\alpha, \beta}$. The curve $\Theta_v$, where this is 0, firstly divides the
+$(\alpha$-$\beta)$-half-plane into regions where the signs of tilt slopes of
+objects of Chern character $v$ (or $-v$) are fixed. Secondly, it gives more of a
+fixed target for some $u=(r,c\ell,d\frac{1}{2}\ell^2)$ to be a
+pseudo-semistabilizer of $v$, in the following sense: If $(\alpha,\beta)$, is on
+$\Theta_v$, then for any $u$, $u$ can only be a pseudo-semistabilizer of $v$ if
+$\nu_{\alpha,\beta}(u)=0$, and hence $\chern_2^{\alpha, \beta}(u)=0$. In fact,
+this allows us to use the characteristic curves of some $v$ and $u$ (with
+$\Delta(v), \Delta(u)\geq 0$ and positive ranks) to determine the location of
+the pseudo-wall where $u$ pseudo-semistabilizes $v$. This is done by finding the
+intersection of $\Theta_v$ and $\Theta_u$, the point $(\beta,\alpha)$ where
+$\nu_{\alpha,\beta}(u)=\nu_{\alpha,\beta}(v)=0$, and a pseudo-wall point on
+$\Theta_v$, and hence the apex of the circular pseudo-wall with centre $(\beta,0)$
+(as per subsection \ref{subsect:bertrams-nested-walls}).
+
+
+\subsection{Bertram's Nested Wall Theorem}
+\label{subsect:bertrams-nested-walls}
+
+Although Bertram's nested wall theorem can be proved more directly, it's also
+important for the content of this document to understand the connection with
+these characteristic curves.
+Emanuele Macri noticed in (TODO ref) that any circular wall of $v$ reaches a critical
+point on $\Theta_v$ (TODO ref). This is a consequence of
+$\frac{\delta}{\delta\beta} \chern_2^{\alpha,\beta} = -\chern_1^{\alpha,\beta}$.
+This fact, along with the hindsight knowledge that non-vertical walls are
+circles with centers on the $\beta$-axis, gives an alternative view to see that
+the circular walls must be nested and non-intersecting.
+
+\subsection{Characteristic Curves for Pseudo-semistabilizers}
+
+These characteristic curves introduced are convenient tools to think about the
+numerical conditions that can be used to test for pseudo-semistabilizers, and
+for solutions to the problems
+(\ref{problem:problem-statement-1},\ref{problem:problem-statement-2})
+tackled in this article (to be introduced later).
+In particular, problem (\ref{problem:problem-statement-1}) will be translated to
+a list of numerical inequalities on it's solutions $u$.
+% ref to appropriate lemma when it's written
+
+The next lemma is a key to making this translation and revolves around the
+geometry and configuration of the characteristic curves involved in a
+semistabilizing sequence.
+
+\begin{lemma}[Numerical tests for left-wall pseudo-semistabilizers]
+\label{lem:pseudo_wall_numerical_tests}
+Let $v$ and $u$ be Chern characters with $\Delta(v),
+\Delta(u)\geq 0$, and $v$ has non-negative rank (and $\chern_1(v)>0$ if rank 0).
+Let $P$ be a point on $\Theta_v^-$.
+
+\noindent
+The following conditions:
+\begin{enumerate}
+\item $u$ is a pseudo-semistabilizer of $v$ at some point on $\Theta_v^-$ above
+	$P$
+\item $u$ destabilizes $v$ going `inwards', that is,
+	$\nu_{\alpha,\beta}(u)<\nu_{\alpha,\beta}(v)$ outside the pseudo-wall, and
+	$\nu_{\alpha,\beta}(u)>\nu_{\alpha,\beta}(v)$ inside.
+\end{enumerate}
+
+\noindent
+are equivalent to the following more numerical conditions:
+\begin{enumerate}
+	\item $u$ has positive rank
+	\item $\beta(P)<\mu(u)<\mu(v)$, i.e. $V_u$ is strictly between $P$ and $V_v$.
+	\item $\chern_1^{\beta(P)}(u) \leq \chern_1^{\beta(P)}(v)$, $\Delta(v-u) \geq 0$
+	\item $\chern_2^{P}(u)>0$
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+Let $u,v$ be Chern characters with
+$\Delta(u),\Delta(v) \geq 0$, and $v$ has positive rank.
+
+
+For the forwards implication, assume that the suppositions of the lemma are
+satisfied. Let $Q$ be the point on $\Theta_v^-$ (above $P$) where $u$ is a
+pseudo-semistabilizer of $v$.
+Firstly, consequence 3 is part of the definition for $u$ being a
+pseudo-semistabilizer at a point with same $\beta$ value of $P$ (since the
+pseudo-wall surrounds $P$).
+If $u$ were to have 0 rank, it's tilt slope would be decreasing as $\beta$
+increases, contradicting supposition b. So $u$ must have strictly non-zero rank,
+and we can consider it's characteristic curves (or that of $-u$ in case of
+negative rank).
+$\nu_Q(v)=0$, and hence $\nu_Q(u)=0$ too. This means that $\Theta_{\pm u}$ must
+intersect $\Theta_v^-$ at $Q$. Considering the shapes of the hyperbolae alone,
+there are 3 distinct ways that they can intersect, as illustrated in Fig
+\ref{fig:hyperbol-intersection}. These cases are distinguished by whether it is
+the left, or the right branch of $\Theta_u$ involved, as well as the positions
+of the base. However, considering supposition b, only case 3 (green in
+figure) is possible. This is because we need $\nu_{P}(u)>0$ (or $\nu_{P}(-u)>0$ in
+case 1 involving $\Theta_u^+$), to satisfy supposition b.
+Recalling how the sign of $\nu_{\alpha,\beta}(\pm u)$ changes (illustrated in
+Fig \ref{fig:charact_curves_vis}), we can eliminate cases 1 and 2.
+
+\begin{sagesilent}
+from characteristic_curves import \
+hyperbola_intersection_plot, \
+correct_hyperbola_intersection_plot
+\end{sagesilent}
+
+\begin{figure}
+\begin{subfigure}[t]{0.48\textwidth}
+	\centering
+	\sageplot[width=\textwidth]{hyperbola_intersection_plot()}
+	\caption{Three ways the characteristic hyperbola for $u$ can intersect the left
+	branch of the characteristic hyperbola for $v$}
+	\label{fig:hyperbol-intersection}
+\end{subfigure}
+\hfill
+\begin{subfigure}[t]{0.48\textwidth}
+	\centering
+	\sageplot[width=\textwidth]{correct_hyperbola_intersection_plot()}
+	\caption{Closer look at characteristic curves for valid case}
+	\label{fig:correct-hyperbol-intersection}
+\end{subfigure}
+\end{figure}
+
+Fixing attention on the only possible case (2), illustrated in Fig
+\ref{fig:correct-hyperbol-intersection}.
+$P$ is on the left of $V_{\pm u}$ (first part of consequence 2), so $u$ must
+have positive rank (consequence 1)
+to ensure that $\chern_1^{\beta(P)} \geq 0$ (since the pseudo-wall passed over
+$P$).
+Furthermore, $P$ being on the left of $V_u$ implies
+$\chern_1^{\beta{P}}(u) \geq 0$,
+and therefore $\chern_2^{P}(u) > 0$ (consequence 4) to satisfy supposition b.
+Next considering the way the hyperbolae intersect, we must have $\Theta_u^-$ taking a
+base-point to the right $\Theta_v$, but then, further up, crossing over to the
+left side. The latter fact implies that the assymptote for $\Theta_u^-$ must be
+to the left of the one for $\Theta_v^-$. Given that they are parallel and
+intersect the $\beta$-axis at $\beta=\mu(u)$ and $\beta=\mu(v)$ respectively. We
+must have $\mu(u)<\mu(v)$ (second part of consequence 2),
+that is, $V_u$ is strictly to the left of $V_v$.
+
+
+Conversely, suppose that the consequences 1-4 are satisfied. Consequence 2
+implies that the assymptote for $\Theta_u^-$ is to the left of $\Theta_v^-$.
+Consequence 4, along with $\beta(P)<\mu(u)$, implies that $P$ must be in the
+region left of $\Theta_u^-$. These two facts imply that $\Theta_u^-$ is to the
+right of $\Theta_v^-$ at $\alpha=\alpha(P)$, but crosses to the left side as
+$\alpha \to +\infty$, intersection at some point $Q$ above $P$.
+This implies that the characteristic curves for $u$ and $v$ are in the
+configuration illustrated in Fig \ref{fig:correct-hyperbol-intersection}.
+We then have $\nu(u)=\nu(v)$ along a circle to the left of $V_u$ reaching it's
+apex at $Q$, and encircling $P$. This along with consequence 3 implies that $u$
+is a pseudo-semistabilizer at the point on the circle with $\beta=\beta(P)$.
+Therefore, it's also a pseudo-semistabilizer further along the circle at $Q$
+(supposition a).
+Finally, consequence 4 along with $P$ being to the left of $V_u$ implies
+$\nu_P(u) > 0$ giving supposition b.
+
+The case with rank 0 can be handled the same way.
+
+\end{proof}
+
+\section{The Problem: Finding Pseudo-walls}
+
+As hinted in the introduction (\ref{sec:intro}), the main motivation of the
+results in this article are not only the bounds on pseudo-semistabilizer
+ranks;
+but also applications for finding a list (comprehensive or subset) of
+pseudo-walls.
+
+After introducing the characteristic curves of stability conditions associated
+to a fixed Chern character $v$, we can now formally state the problems that we
+are trying to solve for.
+
+\subsection{Problem statements}
+
+\begin{problem}[sufficiently large `left' pseudo-walls]
+\label{problem:problem-statement-1}
+
+Fix a Chern character $v$ with non-negative rank (and $\chern_1(v)>0$ if rank 0),
+and $\Delta(v) \geq 0$.
+The goal is to find all pseudo-semistabilizers $u$
+which give circular pseudo-walls containing some fixed point
+$P\in\Theta_v^-$.
+With the added restriction that $u$ `destabilizes' $v$ moving `inwards', that is,
+$\nu(u)>\nu(v)$ inside the circular pseudo-wall.
+\end{problem}
+This will give all pseudo-walls between the chamber corresponding to Gieseker
+stability and the stability condition corresponding to $P$.
+The purpose of the final `direction' condition is because, up to that condition,
+semistabilizers are not distinguished from their corresponding quotients:
+Suppose $E\hookrightarrow F\twoheadrightarrow G$, then the tilt slopes
+$\nu_{\alpha,\beta}$
+are strictly increasing, strictly decreasing, or equal across the short exact
+sequence (consequence of the see-saw principle).
+In this case, $\chern(E)$ is a pseudo-semistabilizer of $\chern(F)$, if and
+only if $\chern(G)$ is a pseudo-semistabilizer of $\chern(F)$.
+The numerical inequalities in the definition for pseudo-semistabilizer cannot
+tell which of $E$ or $G$ is the subobject.
+However, what can be distinguished is the direction across the wall that
+$\chern(E)$ or $\chern(G)$ destabilizes $\chern(F)$
+(they will each destabilize in the opposite direction to the other).
+The `inwards' semistabilizers are preferred because we are moving from a
+typically more familiar chamber
+(the stable objects of Chern character $v$ in the outside chamber will only be
+Gieseker stable sheaves).
+
+Also note that this last restriction does not remove any pseudo-walls found,
+and if we do want to recover `outwards' semistabilizers, we can simply take
+$v-u$ for each solution $u$ of the problem.
+
+
+\begin{problem}[all `left' pseudo-walls]
+\label{problem:problem-statement-2}
+
+Fix a Chern character $v$ with non-negative rank (and $\chern_1(v)>0$ if rank 0),
+$\Delta(v) \geq 0$, and $\beta_{-}(v) \in \QQ$.
+The goal is to find all pseudo-semistabilizers $u$ which give circular
+pseudo-walls on the left side of $V_v$.
+\end{problem}
+
+This is a specialization of problem (\ref{problem:problem-statement-1})
+with the choice $P=(\beta_{-},0)$, the point where $\Theta_v^-$ meets the
+$\beta$-axis.
+This is because all circular walls left of $V_v$ intersect $\Theta_v^-$ (once).
+The $\beta_{-}(v) \in \QQ$ condition is to ensure that there are finitely many
+solutions. As mentioned in the introduction (\ref{sec:intro}), this is known,
+however this will also be proved again implicitly in section
+\ref{sect:prob2-algorithm}, where an algorithm is produced to find all
+solutions.
+
+This description still holds for the case of rank 0 case if we consider $V_v$ to
+be infinitely far to the right
+(see section \ref{subsubsect:rank-zero-case-charact-curves}).
+Note also that the condition on $\beta_-(v)$ always holds for $v$ rank 0.
+
+\subsection{Numerical Formulations of the Problems}
+
+The problems introduced in this section are phrased in the context of stability
+conditions. However, these can be reduced down completely to purely numerical
+problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
+
+\begin{lemma}[Numerical Tests for Sufficiently Large `left' Pseudo-walls]
+	\label{lem:num_test_prob1}
+	Given a Chern character $v$ with non-negative rank
+	(and $\chern_1(v)>0$ if rank 0),
+	and $\Delta(v) \geq 0$,
+	and a choice of point $P$ on $\Theta_v^-$.
+	Solutions $u=(r,c\ell,d\ell^2)$
+	to problem \ref{problem:problem-statement-1}.
+	Are precisely given by $r,c \in \ZZ$, $d \in \frac{1}{\lcm(m,2)}$
+	satisfying the following conditions:
+	\begin{enumerate}
+		\item $r > 0$
+			\label{item:rankpos:lem:num_test_prob1}
+		\item $\Delta(u) \geq 0$
+			\label{item:bgmlvu:lem:num_test_prob1}
+		\item $\Delta(v-u) \geq 0$
+			\label{item:bgmlvv-u:lem:num_test_prob1}
+		\item $\mu(u)<\mu(v)$
+		\item $0\leq\chern_1^{\beta(P)}(u)\leq\chern_1^{\beta(P)}(v)$
+			\label{item:chern1bound:lem:num_test_prob1}
+		\item $\chern_2^{P}(u)>0$
+			\label{item:radiuscond:lem:num_test_prob1}
+	\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+	Consider the context of $v$ being a Chern character with non-negative rank
+	(and $\chern_1(v)>0$ if rank 0)
+	and
+	$\Delta \geq 0$, and $u$ being a Chern character with $\Delta(u) \geq 0$.
+	Lemma \ref{lem:pseudo_wall_numerical_tests} gives that the remaining
+	conditions for $u$ being a solution to problem
+	\ref{problem:problem-statement-1} are precisely equivalent to the
+	remaining conditions in this lemma.
+	% TODO maybe make this more explicit
+	% (the conditions are not exactly the same)
+
+\end{proof}
+
+\begin{corollary}[Numerical Tests for All `left' Pseudo-walls]
+\label{cor:num_test_prob2}
+	Given a Chern character $v$ with positive rank and $\Delta(v) \geq 0$,
+	such that $\beta_{-}\coloneqq\beta_{-}(v) \in \QQ$.
+	Solutions $u=(r,c\ell,d\ell^2)$
+	to problem \ref{problem:problem-statement-2}.
+	Are precisely given by $r,c \in \ZZ$, $d\in\frac{1}{\lcm(m,2)}\ZZ$ satisfying
+	the following conditions:
+	\begin{enumerate}
+		\item $r > 0$
+			\label{item:rankpos:lem:num_test_prob2}
+		\item $\Delta(u) \geq 0$
+			\label{item:bgmlvu:lem:num_test_prob2}
+		\item $\Delta(v-u) \geq 0$
+			\label{item:bgmlvv-u:lem:num_test_prob2}
+		\item $\mu(u)<\mu(v)$
+			\label{item:mubound:lem:num_test_prob2}
+		\item $0\leq\chern_1^{\beta_{-}}(u)\leq\chern_1^{\beta_{-}}(v)$
+			\label{item:chern1bound:lem:num_test_prob2}
+		\item $\chern_2^{\beta_{-}}(u)>0$
+			\label{item:radiuscond:lem:num_test_prob2}
+	\end{enumerate}
+\end{corollary}
+
+\begin{proof}
+	This is a specialization of the previous lemma, using $P=(\beta_{-},0)$.
+
+\end{proof}
+
+
+\section{B.Schmidt's Solutions to the Problems}
+
+\subsection{Bound on \texorpdfstring{$\chern_0(u)$}{ch0(u)} for Semistabilizers}
+\label{subsect:loose-bound-on-r}
+
+The proof for the following theorem \ref{thm:loose-bound-on-r} was hinted at in
+\cite{SchmidtBenjamin2020Bsot}, but the value appears explicitly in
+\cite{SchmidtGithub2020}. The latter reference is a SageMath \cite{sagemath}
+library for computing certain quantities related to Bridgeland stabilities on
+Picard rank 1 varieties. It also includes functions to compute pseudo-walls and
+pseudo-semistabilizers for tilt stability.
+
+
+\begin{theorem}[Bound on $r$ - Benjamin Schmidt]
+\label{thm:loose-bound-on-r}
+Given a Chern character $v$ such that $\beta_-\coloneqq\beta_{-}(v)\in\QQ$, the rank $r$ of
+any semistabilizer $E$ of some $F \in \firsttilt{\beta_-}$ with $\chern(F)=v$ is
+bounded above by:
+
+\begin{equation*}
+	r \leq \frac{mn^2 \chern^{\beta_-}_1(v)^2}{\gcd(m,2n^2)}
+\end{equation*}
+\end{theorem}
+
+\begin{proof}
+The Bogomolov form applied to the twisted Chern character is the same as the
+normal one. So $0 \leq \Delta(E)$ yields:
+
+\begin{equation}
+	\label{eqn-bgmlv-on-E}
+	2\chern^\beta_0(E) \chern^\beta_2(E) \leq \chern^\beta_1(E)^2
+\end{equation}
+
+\noindent
+Furthermore, $E \hookrightarrow F$ in $\firsttilt{\beta_{-}}$ gives:
+\begin{equation}
+	\label{eqn-tilt-cat-cond}
+	0 \leq \chern^\beta_1(E) \leq \chern^\beta_1(F)
+\end{equation}
+% FUTURE maybe ref this back to some definition of first tilt
+
+\noindent
+The restrictions on $\chern^{\beta_-}_0(E)$ and $\chern^{\beta_-}_2(E)$
+is best seen with the following graph:
+
+% TODO: hyperbola restriction graph (shaded)
+
+
+This is where the rationality of $\beta_{-}$ comes in. If
+$\beta_{-} = \frac{a_v}{n}$ for some $a_v,n \in \ZZ$.
+Then $\chern^{\beta_-}_2(E) \in \frac{1}{\lcm(m,2n^2)}\ZZ$.
+In particular, since $\chern_2^{\beta_-}(E) > 0$ (by using 	$P=(\beta_-,0)$ in
+lemma \ref{lem:pseudo_wall_numerical_tests}) we must also have
+$\chern^{\beta_-}_2(E) \geq \frac{1}{\lcm(m,2n^2)}$, which then in turn gives a
+bound for the rank of $E$:
+
+\begin{align}
+	\chern_0(E) &= \chern^{\beta_-}_0(E) \\
+	&\leq \frac{\lcm(m,2n^2) \chern^{\beta_-}_1(E)^2}{2} \\
+	&= \frac{mn^2 \chern^{\beta_-}_1(F)^2}{\gcd(m,2n^2)}
+\end{align}
+
+\end{proof}
+
+\begin{sagesilent}
+from examples import recurring
+\end{sagesilent}
+
+\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
+\label{exmpl:recurring-first}
+Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
+that $m=1$, $\beta_-=\sage{recurring.betaminus}$,
+giving $n=\sage{recurring.n}$ and
+$\chern_1^{\sage{recurring.betaminus}}(F) = \sage{recurring.twisted.ch[1]}$.
+
+Using the above theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
+tilt semistabilizers for $v$ are bounded above by $\sage{recurring.loose_bound}$.
+However, when computing all tilt semistabilizers for $v$ on $\PP^2$, the maximum
+rank that appears turns out to be 25. This will be a recurring example to
+illustrate the performance of later theorems about rank bounds
+\end{example}
+
+\begin{sagesilent}
+from examples import extravagant
+\end{sagesilent}
+
+\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
+\label{exmpl:extravagant-first}
+Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
+that $m=1$, $\beta_-=\sage{extravagant.betaminus}$,
+giving $n=\sage{extravagant.n}$ and
+$\chern_1^{\sage{extravagant.betaminus}}(F) = \sage{extravagant.twisted.ch[1]}$.
+
+Using the above theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
+tilt semistabilizers for $v$ are bounded above by $\sage{extravagant.loose_bound}$.
+However, when computing all tilt semistabilizers for $v$ on $\PP^2$, the maximum
+rank that appears turns out to be $\sage{extravagant.actual_rmax}$.
+\end{example}
+
+\subsection{Pseudo-Wall Finding Method}
+
+The SageMath Library \cite{SchmidtGithub2020} provides a function which
+calculates all solutions to problems \ref{problem:problem-statement-1}
+or \ref{problem:problem-statement-2}.
+Here is an outline of the algorithm involved to do this. Simplifications will be
+made in the presentation to concentrate on the case we are interested in:
+problem \ref{problem:problem-statement-2}, finding all pseudo-walls when $\beta_{-}\in\QQ$.
+% FUTURE add reference to section explaining new alg
+In section [ref], a different
+algorithm will be presented making use of the later theorems in this article,
+with the goal of cutting down the run time.
+
+\subsubsection{Finding possible \texorpdfstring{$r$}{r} and
+\texorpdfstring{$c$}{c}}
+To do this, first calculate the upper bound $r_{max}$ on the ranks of tilt
+semistabilizers, as given by theorem \ref{thm:loose-bound-on-r}.
+
+Recalling consequence 2 of lemma \ref{lem:pseudo_wall_numerical_tests}, we can
+iterate through the possible values of $\mu(u)=\frac{c}{r}$ taking a decreasing
+sequence of all fractions between $\mu(v)$ and $\beta_{-}$, who's denominators
+are no large than $r_{max}$ (giving a finite sequence). This can be done with
+Farey sequences \cite[chapter 6]{alma994504533502466}, for which there exist
+formulae to generate.
+
+These $\mu(u)$ values determine pairs $r,c$ up to multiples, we can then take
+all multiples which satisy $0<r\leq r_{max}$.
+
+We now have a finite sequence of pairs $r,c$ for which there might be a solution
+$(r,c\ell,d\ell^2)$ to our problem. In particular, any $(r,c\ell,d\ell^2)$
+satisfies consequence 2 of lemma \ref{lem:pseudo_wall_numerical_tests}, and the
+positive rank condition. What remains is to find the $d$ values which satisfy
+the Bogomolov inequalities and consequence 3 of lemma
+\ref{lem:pseudo_wall_numerical_tests}
+($\chern_2^{\beta_{-}}(u)>0$).
+
+\subsubsection{Finding \texorpdfstring{$d$}{d} for fixed \texorpdfstring{$r$}{r}
+and \texorpdfstring{$c$}{c}}
+
+$\Delta(u) \geq 0$ induces an upper bound $\frac{c^2}{2r}$ on $d$, and the
+$\chern_2^{\beta_{-}}(u)>0$ condition induces a lower bound on $d$.
+The values in the range can be tested individually, to check that
+the rest of the conditions are satisfied.
+
+\subsection{Limitations}
+
+The main downside of this algorithm is that many $r$,$c$ pairs which are tested
+end up not yielding any solutions for the problem.
+In fact, solutions $u$ to our problem with high rank must have $\mu(u)$ close to
+$\beta_{-}$:
+\begin{align*}
+	0 &\leq \chern_1^{\beta_{-}}(u) \leq \chern_1^{\beta_{-}}(u) \\
+	0 &\leq \mu(u) - \beta_{-} \leq \frac{\chern_1^{\beta_{-}}(v)}{r}
+\end{align*}
+In particular, it's the $\chern_1^{\beta_{-}}(v-u) \geq 0$ conditions which
+fails for $r$,$c$ pairs with large $r$ and $\frac{c}{r}$ too far from $\beta_{-}$.
+This condition is only checked within the internal loop.
+This, along with a conservative estimate for a bound on the $r$ values (as
+illustrated in example \ref{exmpl:recurring-first}) occasionally leads to slow
+computations.
+
+Here are some benchmarks to illustrate the performance benefits of the
+alternative algorithm which will later be described in this article [ref].
+
+\begin{center}
+\begin{tabular}{ |r|l|l| }
+ \hline
+ Choice of $v$ on $\mathbb{P}^2$
+ & $(3, 2\ell, -2)$
+ & $(3, 2\ell, -\frac{15}{2})$ \\
+ \hline
+ \cite[\texttt{tilt.walls_left}]{SchmidtGithub2020} exec time & \sim 20s & >1hr \\
+ \cite{NaylorRust2023} exec time & \sim 50ms & \sim 50ms \\
+ \hline
+\end{tabular}
+\end{center}
+
+\section{Tighter Bounds}
+\label{sec:refinement}
+
+To get tighter bounds on the rank of destabilizers $E$ 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
+was implicitly happening before).
+
+
+First, let us fix a Chern character for $F$, and some pseudo-semistabilizer
+$u$ which is a solution to 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
+\ref{problem:problem-statement-1}
+(or $\beta = \beta_{-}$ 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 $D\in \frac{1}{\lcm(m,2)}\ZZ$}
+	\\
+	u \coloneqq& \:(r,c\ell,d\ell^2)
+	&& \text{where $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$}
+\end{align}
+ 
+
+
+Recall from condition \ref{item:chern1bound:lem:num_test_prob1} in
+lemma \ref{lem:num_test_prob1}
+(or corollary \ref{cor:num_test_prob2})
+that $\chern_1^{\beta}(u)$ has fixed bounds in terms of $\chern_1^{\beta}(v)$,
+and so we can write:
+
+
+
+\begin{sagesilent}
+from plots_and_expressions import c_in_terms_of_q	
+\end{sagesilent}
+
+\begin{equation}
+	\label{eqn-cintermsofm}
+	c=\chern_1(u) = \sage{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
+$q \in \frac{1}{n} \ZZ \cap [0, \chern_1^{\beta}(F)]$,
+where $n$ is the denominator of $\beta$.
+For the next subsections, we consider $q$ to be fixed with one of these values,
+and we shall be varying $\chern_0(E) = r$ to see when certain inequalities fail.
+
+
+\subsection{Numerical Inequalities}
+
+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})
+
+\subsubsection{Size of pseudo-wall\texorpdfstring{: $\chern_2^P(u)>0$}{}}
+\label{subsect-d-bound-radiuscond}
+
+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:
+
+\begin{align}
+\label{eqn:radius-cond-betamin}
+	\chern_2^{\beta_{-}}(u) &> 0 \\
+	d &> \beta_{-}q + \frac{1}{2} \beta_{-}^2r
+\end{align}
+
+\begin{sagesilent}
+import other_P_choice as problem1
+\end{sagesilent}
+
+In the case where we are tackling problem \ref{problem:problem-statement-1},
+with some Chern character $v$ with positive rank, and some choice of point
+$P=(A,B) \in \Theta_v^-$.
+Then $\sage{problem1.A2_subs}$ follows from $\chern_2^P(v)=0$. Using this substitution into the
+condition $\chern_2^P(u)>0$ yields:
+
+\begin{equation}
+	\sage{problem1.radius_condition}
+\end{equation}
+
+\noindent
+Expressing this as a bound on $d$, then yields:
+
+\begin{equation}
+	\sage{problem1.radius_condition_d_bound}
+\end{equation}
+
+
+\subsubsection{Semistability of the Semistabilizer:
+	\texorpdfstring{
+		$\Delta(u) \geq 0$
+	}{
+		Δ(u) ≥ 0
+	}
+}
+This condition refers to condition
+\ref{item:bgmlvu:lem:num_test_prob1}
+from lemma \ref{lem:num_test_prob1}
+(or corollary \ref{cor:num_test_prob2}).
+
+
+\noindent
+Expressing $\Delta(u)\geq 0$ in terms of $q$ as defined in eqn \ref{eqn-cintermsofm}
+we get the following:
+
+
+\begin{sagesilent}
+from plots_and_expressions import bgmlv2_with_q
+\end{sagesilent}
+
+\begin{equation}
+	\sage{bgmlv2_with_q}
+\end{equation}
+
+
+\noindent
+This can be rearranged to express a bound on $d$ as follows
+(recall from condition \ref{item:rankpos:lem:num_test_prob1}
+in lemma \ref{lem:num_test_prob1} or corollary
+\ref{cor:num_test_prob2} that $r>0$):
+
+
+\begin{sagesilent}
+from plots_and_expressions import bgmlv2_d_ineq
+\end{sagesilent}
+\begin{equation}
+	\label{eqn-bgmlv2_d_upperbound}
+	\sage{bgmlv2_d_ineq}
+\end{equation}
+
+\begin{sagesilent}
+from plots_and_expressions import bgmlv2_d_upperbound_terms
+\end{sagesilent}
+Viewing equation \ref{eqn-bgmlv2_d_upperbound} as a lower bound for $d$ in term
+of $r$ again, there is a constant term
+$\sage{bgmlv2_d_upperbound_terms.const}$,
+a linear term
+$\sage{bgmlv2_d_upperbound_terms.linear}$,
+and a hyperbolic term
+$\sage{bgmlv2_d_upperbound_terms.hyperbolic}$.
+Notice that in the context of problem \ref{problem:problem-statement-2}
+($\beta = \beta_{-}$),
+the constant and linear terms match up with the ones
+for the bound found for $d$ in subsubsection \ref{subsect-d-bound-radiuscond}.
+
+\subsubsection{Semistability of the Quotient:
+	\texorpdfstring{
+		$\Delta(v-u) \geq 0$
+	}{
+		Δ(v-u) ≥ 0
+	}
+}
+\label{subsect-d-bound-bgmlv3}
+
+This condition refers to condition
+\ref{item:bgmlvv-u:lem:num_test_prob1}
+from lemma \ref{lem:num_test_prob1}
+(or corollary \ref{cor:num_test_prob2}).
+
+Expressing $\Delta(v-u)\geq 0$ in term of $q$ and rearranging as a bound on
+$d$ yields:
+
+
+\begin{sagesilent}
+from plots_and_expressions import bgmlv3_d_upperbound_terms
+\end{sagesilent}
+
+\begin{equation*}
+	\label{eqn-bgmlv3_d_upperbound}
+	d \leq
+	\sage{bgmlv3_d_upperbound_terms.linear}
+	+ \sage{bgmlv3_d_upperbound_terms.const}
+	+ \sage{bgmlv3_d_upperbound_terms.hyperbolic}
+	\qquad
+	\text{where }r>R
+\end{equation*}
+
+
+\noindent
+For $r=R$, $\Delta(v-u)\geq 0$ is always true, and for $r<R$ it gives a lower
+bound on $d$, but it is weaker than the one given by the lower bound in
+subsubsection \ref{subsect-d-bound-radiuscond}.
+Viewing the right hand side of equation \ref{eqn-bgmlv3_d_upperbound}
+as a function of $r$, the linear and constant terms almost match up with the
+ones in the previous section, up to the 
+$\chern_2^{\beta}(v)$ term.
+
+
+However, when specializing to problem \ref{problem:problem-statement-2} again
+(with $\beta = \beta_{-}$), then we have $\chern^{\beta}_2(v) = 0$.
+And so in this context, the linear and constant terms do match up with the
+previous subsubsections.
+
+\subsubsection{All Bounds on \texorpdfstring{$d$}{d} Together for Problem
+\texorpdfstring{\ref{problem:problem-statement-2}}{2}}
+\label{subsubsect:all-bounds-on-d-prob2}
+%% RECAP ON INEQUALITIES TOGETHER
+
+%%%% RATIONAL BETA MINUS
+As mentioned in passing, when specializing to solutions $u$ of problem
+\ref{problem:problem-statement-2}, the bounds on
+$d=\chern_2(u)$ induced by conditions
+\ref{item:bgmlvu:lem:num_test_prob2},
+\ref{item:bgmlvv-u:lem:num_test_prob2}, and
+\ref{item:radiuscond:lem:num_test_prob1}
+from corollary \ref{cor:num_test_prob2} have the same constant and linear
+terms in $r$, but different hyperbolic terms.
+These give bounds with the same assymptotes when we take $r\to\infty$
+(for any fixed $q=\chern_1^{\beta_{-}}(u)$).
+
+% redefine \beta (especially coming from rendered SageMath expressions)
+% to be \beta_{-} for the rest of this subsubsection
+\bgroup
+
+\let\originalbeta\beta
+\renewcommand\beta{{\originalbeta_{-}}}
+
+\begin{align}
+	d &>&
+	\frac{1}{2}\beta^2 r
+	&+ \beta q,
+	\phantom{+}& % to keep terms aligned
+	 &\qquad\text{when\:} r > 0
+	\label{eqn:radiuscond_d_bound_betamin}
+\\
+	d &\leq&
+	\sage{bgmlv2_d_upperbound_terms.problem2.linear}
+	&+ \sage{bgmlv2_d_upperbound_terms.problem2.const}
+	+& \sage{bgmlv2_d_upperbound_terms.problem2.hyperbolic},
+	 &\qquad\text{when\:} r > 0
+	 \label{eqn:bgmlv2_d_bound_betamin}
+\\
+	d &\leq&
+	\sage{bgmlv3_d_upperbound_terms.problem2.linear}
+	&+ \sage{bgmlv3_d_upperbound_terms.problem2.const}
+	% ^ ch_2^\beta(F)=0 for beta_{-}
+	+& \sage{bgmlv3_d_upperbound_terms.problem2.hyperbolic},
+	 &\qquad\text{when\:} r > R
+	 \label{eqn:bgmlv3_d_bound_betamin}
+\end{align}
+
+
+\begin{sagesilent}
+from plots_and_expressions import \
+bounds_on_d_qmin, \
+bounds_on_d_qmax
+\end{sagesilent}
+
+\begin{figure}
+\centering
+\begin{subfigure}{.45\textwidth}
+  \centering
+	\sageplot[width=\linewidth]{bounds_on_d_qmin}
+	\caption{$q = 0$ (all bounds other than green coincide on line)}
+  \label{fig:d_bounds_xmpl_min_q}
+\end{subfigure}%
+\hfill
+\begin{subfigure}{.45\textwidth}
+  \centering
+	\sageplot[width=\linewidth]{bounds_on_d_qmax}
+	\caption{$q = \chern^{\beta}(F)$ (all bounds other than blue coincide on line)}
+  \label{fig:d_bounds_xmpl_max_q}
+\end{subfigure}
+\caption{
+	Bounds on $d\coloneqq\chern_2(E)$ in terms of $r\coloneqq\chern_0(E)$ for fixed, extreme,
+	values of $q\coloneqq\chern_1^{\beta}(E)$.
+	Where $\chern(F) = (3,2,-2)$.
+}
+\label{fig:d_bounds_xmpl_extrm_q}
+\end{figure}
+
+Recalling that $q \coloneqq \chern^{\beta}_1(E) \in [0, \chern^{\beta}_1(F)]$,
+it is 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
+$\chern_0(E)=\vcentcolon r>R\coloneqq\chern_0(F)$ 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: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:radiuscond_d_bound_betamin}) which match.
+
+
+The more generic case, when $0 < q\coloneqq\chern_1^{\beta}(E) < \chern_1^{\beta}(F)$
+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
+$(r,d) \in \ZZ \oplus \frac{1}{\lcm(m,2)} \ZZ$ (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
+solutions is entirely determined by whether $\beta$ is rational or irrational.
+Some of the details around the associated numerics are explored next.
+
+\begin{sagesilent}
+from plots_and_expressions import typical_bounds_on_d
+\end{sagesilent}
+
+\begin{figure}
+\centering
+\sageplot[width=\linewidth]{typical_bounds_on_d}
+\caption{
+	Bounds on $d\coloneqq\chern_2(E)$ in terms of $r\coloneqq \chern_0(E)$ for a fixed
+	value $\chern_1^{\beta}(F)/2$ of $q\coloneqq\chern_1^{\beta}(E)$.
+	Where $\chern(F) = (3,2,-2)$.
+}
+\label{fig:d_bounds_xmpl_gnrc_q}
+\end{figure}
+
+\subsubsection{All Bounds on \texorpdfstring{$d$}{d} Together for Problem
+\texorpdfstring{\ref{problem:problem-statement-1}}{1}}
+\label{subsubsect:all-bounds-on-d-prob1}
+
+Unlike for problem \ref{problem:problem-statement-2},
+the bounds on $d=\chern_2(u)$ induced by conditions
+\ref{item:bgmlvu:lem:num_test_prob2},
+\ref{item:bgmlvv-u:lem:num_test_prob2}, and
+\ref{item:radiuscond:lem:num_test_prob1}
+from corollary \ref{cor:num_test_prob2} have different
+constant and linear terms, so that the graphs for upper
+bounds do not share the same assymptote as the lower bound
+(and they will turn out to intersect).
+
+\begin{align}
+	\sage{problem1.radius_condition_d_bound.lhs()}
+	&>
+	\sage{problem1.radius_condition_d_bound.rhs()}
+	&\text{where }r>0
+	\label{eqn:prob1:radiuscond}
+	\\
+	d &\leq
+	\sage{problem1.bgmlv2_d_upperbound_terms.linear}
+	+ \sage{problem1.bgmlv2_d_upperbound_terms.const}
+	+ \sage{problem1.bgmlv2_d_upperbound_terms.hyperbolic}
+	&\text{where }r>R
+	\label{eqn:prob1:bgmlv2}
+	\\
+	d &\leq
+	\sage{problem1.bgmlv3_d_upperbound_terms.linear}
+	+ \sage{problem1.bgmlv3_d_upperbound_terms.const}
+	+ \sage{problem1.bgmlv3_d_upperbound_terms.hyperbolic}
+	&\text{where }r>R
+	\label{eqn:prob1:bgmlv3}
+\end{align}
+
+Notice that as a function in $r$, the linear term in 
+equation \ref{eqn:prob1:radiuscond} is strictly greater than
+those in equations \ref{eqn:prob1:bgmlv2}
+and \ref{eqn:prob1:bgmlv3}. This is because $r$, $R$
+and $\chern_2^B(v)$ are all strictly positive:
+\begin{itemize}
+	\item $R > 0$ from the setting of problem
+	\ref{problem:problem-statement-1}
+	\item $r > 0$ from lemma \ref{lem:num_test_prob1}
+	\item $\chern_2^B(v)>0$ because $B < \originalbeta_{-}$ due to the choice of $P$ being
+	a point on $\Theta_v^{-}$
+\end{itemize}
+
+This means that the lower bound for $d$ will be large than either of the two
+upper bounds for sufficiently large $r$, and hence those values of $r$ would yield no 
+solution to problem \ref{problem:problem-statement-1}.
+
+A generic example of this is plotted in figure
+\ref{fig:problem1:d_bounds_xmpl_gnrc_q}.
+
+\begin{figure}
+\centering
+\sageplot[width=\linewidth]{problem1.example_plot}
+\caption{
+	Bounds on $d\coloneqq\chern_2(E)$ in terms of $r\coloneqq \chern_0(E)$ for a fixed
+	value $\chern_1^{\beta}(F)/2$ of $q\coloneqq\chern_1^{\beta}(E)$.
+	Where $\chern(F) = (3,2,-2)$ and $P$ chosen as the point on $\Theta_v$
+	with $B\coloneqq-2/3-1/99$ in the context of problem 
+	\ref{problem:problem-statement-1}.
+}
+\label{fig:problem1:d_bounds_xmpl_gnrc_q}
+\end{figure}
+
+\subsection{Bounds on Semistabilizer Rank \texorpdfstring{$r$}{} in Problem
+\ref{problem:problem-statement-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}),
+there are no solutions $u$ to problem \ref{problem:problem-statement-1}
+with large $r=\chern_0(u)$, since the lower bound on $d=\chern_2(u)$ is larger
+than the upper bounds.
+Therefore, we can calculate upper bounds on $r$ by calculating for which values,
+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 $r$]
+\label{lem:prob1:r_bound}
+	Let $u$ be a solution to problem \ref{problem:problem-statement-1}
+	and $q\coloneqq\chern_1^{B}(u)$.
+	Then $r\coloneqq\chern_0(u)$ is bounded above by the following expression:
+	\begin{equation}
+		\sage{problem1.r_bound_expression}
+	\end{equation}
+\end{lemma}
+
+\begin{proof}
+	Recall that $d\coloneqq\chern_2(u)$ has two upper bounds in terms of $r$: in
+	equations \ref{eqn:prob1:bgmlv2} and \ref{eqn:prob1:bgmlv3};
+	and one lower bound: in equation \ref{eqn:prob1:radiuscond}.
+
+	Solving for the lower bound in equation \ref{eqn:prob1:radiuscond} being
+	less than the upper bound in equation \ref{eqn:prob1:bgmlv2} yields:
+	\begin{equation}
+	r<\sage{problem1.positive_intersection_bgmlv2}
+	\end{equation}
+
+	Similarly, but with the upper bound in equation \ref{eqn:prob1:bgmlv3}, gives:
+	\begin{equation}
+	r<\sage{problem1.positive_intersection_bgmlv3}
+	\end{equation}
+
+	Therefore, $r$ is bounded above by the minimum of these two expressions which
+	can then be factored into the expression given in the lemma.
+	
+\end{proof}
+
+The above lemma \ref{lem:prob1:r_bound} gives an upper bound on $r$ in terms of $q$.
+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}
+\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}
+
+\begin{proof}
+	The first term of the minimum in lemma \ref{lem:prob1:r_bound}
+	increases linearly in $q$, and the second
+	decreases linearly. So the maximum is achieved with the value of
+	$q=q_{\mathrm{max}}$ where they are equal.
+	Solving for the two terms in the minimum to be equal yields:
+	$q_{\mathrm{max}}=\sage{problem1.maximising_q}$.
+	Substituting $q=q_{\mathrm{max}}$ into the bound in lemma
+	\ref{lem:prob1:r_bound} gives the bound as stated in the current lemma.
+	
+\end{proof}
+
+\begin{note}
+	$q_{\mathrm{max}} > 0$ 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}
+
+
+\subsection{Bounds on Semistabilizer Rank \texorpdfstring{$r$}{} in Problem
+\ref{problem:problem-statement-2}}
+
+Now, the inequalities from the above subsubsection
+\ref{subsubsect:all-bounds-on-d-prob2} will be used to find, for
+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
+\ref{thm:loose-bound-on-r}.
+
+
+\renewcommand{\aa}{{a_v}}
+\newcommand{\bb}{{b_q}}
+Suppose $\beta = \frac{\aa}{n}$ for some coprime $n \in \NN,\aa \in \ZZ$.
+Then fix a value of $q$:
+\begin{equation}
+	q\coloneqq \chern_1^{\beta}(E)
+	  =\frac{\bb}{n}
+	\in
+	\frac{1}{n} \ZZ
+	\cap [0, \chern_1^{\beta}(F)]
+\end{equation}
+as noted at the beginning of this section \ref{sec:refinement} so that we are
+considering $u$ which satisfy \ref{item:chern1bound:lem:num_test_prob2}
+in corollary \ref{cor:num_test_prob2}.
+
+Substituting the current values of $q$ and $\beta$ into the condition for the
+radius of the pseudo-wall being positive
+(eqn \ref{eqn:radiuscond_d_bound_betamin}) we get:
+
+\begin{sagesilent}
+from plots_and_expressions import \
+positive_radius_condition_with_q, \
+q_value_expr, \
+beta_value_expr
+\end{sagesilent}
+\begin{equation}
+\label{eqn:positive_rad_condition_in_terms_of_q_beta}
+	\frac{1}{\lcm(m,2)}\ZZ
+	\ni
+	\qquad
+	\sage{positive_radius_condition_with_q.subs([q_value_expr,beta_value_expr]).factor()}
+	\qquad
+	\in
+	\frac{1}{2n^2}\ZZ
+\end{equation}
+
+
+\begin{sagesilent}
+from plots_and_expressions import main_theorem1
+\end{sagesilent}
+\begin{theorem}[Bound on $r$ \#1]
+\label{thm:rmax_with_uniform_eps}
+	Let $v = (R,C\ell,D\ell^2)$ be a fixed Chern character. Then the ranks of the
+	pseudo-semistabilizers for $v$,
+	which are solutions to problem \ref{problem:problem-statement-2},
+	with $\chern_1^\beta = q$
+	are bounded above by the following expression.
+
+	\begin{align*}
+		\min
+		\left(
+			\sage{main_theorem1.r_upper_bound1}, \:\:
+			\sage{main_theorem1.r_upper_bound2}
+		\right)
+	\end{align*}
+
+	Taking the maximum of this expression over
+	$q \in [0, \chern_1^{\beta}(F)]\cap \frac{1}{n}\ZZ$
+	would give an upper bound for the ranks of all solutions to problem
+	\ref{problem:problem-statement-2}.
+\end{theorem}
+
+\begin{proof}
+
+\noindent
+Both $d$ and the lower bound in
+(eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta})
+are elements of $\frac{1}{\lcm(m,2n^2)}\ZZ$.
+So, if any of the two upper bounds on $d$ come to within
+$\frac{1}{\lcm(m,2n^2)}$ of this lower bound, then there are no solutions for
+$d$.
+Hence any corresponding $r$ cannot be a rank of a
+pseudo-semistabilizer for $v$.
+
+To avoid this, we must have,
+considering equations
+\ref{eqn:bgmlv2_d_bound_betamin},
+\ref{eqn:bgmlv3_d_bound_betamin},
+\ref{eqn:radiuscond_d_bound_betamin}.
+
+\begin{sagesilent}
+from plots_and_expressions import \
+assymptote_gap_condition1, assymptote_gap_condition2, k
+\end{sagesilent}
+
+
+\begin{align}
+	&\sage{assymptote_gap_condition1.subs(k==1)} \\
+	&\sage{assymptote_gap_condition2.subs(k==1)}
+\end{align}
+
+\noindent
+This is equivalent to:
+
+\begin{equation}
+	\label{eqn:thm-bound-for-r-impossible-cond-for-r}
+	r \leq
+	\min\left(
+		\sage{
+			main_theorem1.r_upper_bound1
+		} ,
+		\sage{
+			main_theorem1.r_upper_bound2
+		}
+	\right)
+\end{equation}
+
+\end{proof}
+
+
+\begin{sagesilent}
+from plots_and_expressions import q_sol, bgmlv_v, psi
+\end{sagesilent}
+
+\begin{corollary}[Bound on $r$ \#2]
+\label{cor:direct_rmax_with_uniform_eps}
+	Let $v$ be a fixed Chern character and
+	$R\coloneqq\chern_0(v) \leq \frac{1}{2}\lcm(m,2n^2){\chern_1^{\beta}(F)}^2$.
+	Then the ranks of the pseudo-semistabilizers for $v$,
+	which are solutions to problem \ref{problem:problem-statement-2},
+	are bounded above by the following expression.
+
+	\begin{equation*}
+		\sage{main_theorem1.corollary_r_bound}
+	\end{equation*}
+\end{corollary}
+
+\begin{proof}
+The ranks of the pseudo-semistabilizers for $v$ are bounded above by the
+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*}
+	f_1(q)\coloneqq\sage{main_theorem1.r_upper_bound1} \qquad
+	f_2(q)\coloneqq\sage{main_theorem1.r_upper_bound2}
+\end{equation*}
+These have their minimums at $q=0$ and $q=\chern_1^{\beta}(F)$ respectively.
+It suffices to find their intersection in
+$q\in [0, \chern_1^{\beta}(F)]$, if it exists,
+and evaluating on of the $f_i$ there.
+The intersection exists, provided that
+$f_1(\chern_1^{\beta}(F)) \geq f_2(\chern_1^{\beta}(F))=R$,
+or equivalently,
+$R \leq \frac{1}{2}\lcm(m,2n^2){\chern_1^{\beta}(F)}^2$.
+Solving for $f_1(q)=f_2(q)$ yields
+\begin{equation*}
+	q=\sage{q_sol.expand()}
+\end{equation*}
+And evaluating $f_1$ at this $q$-value gives:
+\begin{equation*}
+	\sage{main_theorem1.corollary_intermediate}
+\end{equation*}
+Finally, noting that $\Delta(v)=\psi^2\ell^2$, we get the bound as
+stated in the corollary.
+
+\end{proof}
+
+\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
+\label{exmpl:recurring-second}
+Just like in example \ref{exmpl:recurring-first}, take
+$\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
+that $m=2$, $\beta=\sage{recurring.betaminus}$,
+giving $n=\sage{recurring.n}$.
+
+Using the above corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
+the ranks of tilt semistabilizers for $v$ are bounded above by
+$\sage{recurring.corrolary_bound} \approx  \sage{float(recurring.corrolary_bound)}$,
+which is much closer to real maximum 25 than the original bound 144.
+\end{example}
+
+\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
+\label{exmpl:extravagant-second}
+Just like in example \ref{exmpl:extravagant-first}, take
+$\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
+that $m=2$, $\beta=\sage{extravagant.betaminus}$,
+giving $n=\sage{extravagant.n}$.
+
+Using the above corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
+the ranks of tilt semistabilizers for $v$ are bounded above by
+$\sage{extravagant.corrolary_bound} \approx  \sage{float(extravagant.corrolary_bound)}$,
+which is much closer to real maximum $\sage{extravagant.actual_rmax}$ than the
+original bound 215296.
+\end{example}
+%% refinements using specific values of q and beta
+
+These bound can be refined a bit more by considering restrictions from the
+possible values that $r$ take.
+Furthermore, the proof of theorem \ref{thm:rmax_with_uniform_eps} uses the fact
+that, given an element of $\frac{1}{2n^2}\ZZ$, the closest non-equal element of
+$\frac{1}{2}\ZZ$ is at least $\frac{1}{2n^2}$ away. However this a
+conservative estimate, and a larger gap can sometimes be guaranteed if we know
+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
+`clean' formula for a bound on the ranks of the pseudo-semistabilizers, but has a
+purpose in the context of writing a computer program to find
+pseudo-semistabilizers. Such a program would iterate through possible values of
+$q$, then iterate through values of $r$ within the bounds (dependent on $q$),
+which would then determine $c$, and then find the corresponding possible values
+for $d$.
+
+
+Firstly, we only need to consider $r$-values for which $c\coloneqq\chern_1(E)$ is
+integral:
+
+\begin{equation}
+	c =
+	\sage{c_in_terms_of_q.subs([q_value_expr,beta_value_expr])}
+	\in \ZZ
+\end{equation}
+
+\noindent
+That is, $r \equiv -\aa^{-1}\bb$ mod $n$ ($\aa$ is coprime to
+$n$, and so invertible mod $n$).
+
+
+\noindent
+Let $\aa^{'}$ be an integer representative of $\aa^{-1}$ in $\ZZ/n\ZZ$.
+
+Next, we seek to find a larger $\epsilon$ to use in place of $\epsilon_F$ in the
+proof of theorem \ref{thm:rmax_with_uniform_eps}:
+
+\begin{lemmadfn}[
+	Finding a better alternative to $\epsilon_v$:
+	$\epsilon_{v,q}$
+	]
+	\label{lemdfn:epsilon_q}
+	Suppose $d \in \frac{1}{\lcm(m,2n^2)}\ZZ$ satisfies the condition in
+	eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta}.
+	That is:
+
+	\begin{equation*}
+		\sage{positive_radius_condition_with_q.subs([q_value_expr,beta_value_expr]).factor()}
+	\end{equation*}
+
+	\noindent
+	Then we have:
+
+	\begin{equation}
+		\label{eqn:epsilon_q_lemma_prop}
+		d - \frac{(\aa r + 2\bb)\aa}{2n^2}
+		\geq \epsilon_{v,q} \geq \epsilon_v > 0
+	\end{equation}
+
+	\noindent
+	Where $\epsilon_{v,q}$ is defined as follows:
+
+	\begin{equation*}
+		\epsilon_{v,q} \coloneqq
+		\frac{k_{q}}{\lcm(m,2n^2)}
+	\end{equation*}
+	with $k_{v,q}$ being the least $k\in\ZZ_{>0}$ satisfying
+	\begin{equation*}
+		k \equiv -\aa\bb \frac{m}{\gcd(m,2n^2)}
+		\mod{\gcd\left(
+			\frac{n^2\gcd(m,2)}{\gcd(m,2n^2)},
+			\frac{mn\aa}{\gcd(m,2n^2)}
+		\right)}
+	\end{equation*}
+	
+\end{lemmadfn}
+
+\vspace{10pt}
+
+\begin{proof}
+
+Consider the following sequence of logical implications.
+The one-way implication follows from
+$\aa r + \bb \equiv 0 \pmod{n}$,
+and the final logical equivalence is just a simplification of the expressions.
+
+\begin{align}
+	\frac{ x }{ \lcm(m,2) }
+	- \frac{
+		(\aa r+2\bb)\aa
+	}{
+		2n^2
+	}
+	= \frac{ k }{ \lcm(m,2n^2) }
+	\quad \text{for some } x \in \ZZ
+	\span \span \span \span \span
+	\label{eqn:finding_better_eps_problem}
+\\ \nonumber
+\\ \Leftrightarrow& &
+	- (\aa r+2\bb)\aa
+	\frac{\lcm(m,2n^2)}{2n^2}
+	&\equiv k &&
+	\nonumber
+\\ &&&
+	\mod \frac{\lcm(m,2n^2)}{\lcm(m,2)}
+	\span \span \span
+	\nonumber
+\\ \Rightarrow& &
+	- \bb\aa
+	\frac{\lcm(m,2n^2)}{2n^2}
+	&\equiv k &&
+	\nonumber
+\\ &&&
+	\mod \gcd\left(
+		\frac{\lcm(m,2n^2)}{\lcm(m,2)},
+		\frac{n \aa \lcm(m,2n^2)}{2n^2}
+	\right)
+	\span \span \span
+	\nonumber
+\\ \Leftrightarrow& &
+	- \bb\aa
+	\frac{m}{\gcd(m,2n^2)}
+	&\equiv k &&
+	\label{eqn:better_eps_problem_k_mod_n}
+\\ &&&
+	\mod \gcd\left(
+		\frac{n^2\gcd(m,2)}{\gcd(m,2n^2)},
+		\frac{mn \aa}{\gcd(m,2n^2)}
+	\right)
+	\span \span \span
+	\nonumber
+\end{align}
+
+In our situation, we want to find the least $k>0$ satisfying 
+eqn \ref{eqn:finding_better_eps_problem}.
+Since such a $k$ must also satisfy eqn \ref{eqn:better_eps_problem_k_mod_n},
+we can pick the smallest $k_{q,v} \in \ZZ_{>0}$ which satisfies this new condition
+(a computation only depending on $q$ and $\beta$, but not $r$).
+We are then guaranteed that $k_{v,q}$ is less than any $k$ satisfying eqn
+\ref{eqn:finding_better_eps_problem}, giving the first inequality in eqn
+\ref{eqn:epsilon_q_lemma_prop}.
+Furthermore, $k_{v,q}\geq 1$ gives the second part of the inequality:
+$\epsilon_{v,q}\geq\epsilon_v$, with equality when $k_{v,q}=1$.
+
+\end{proof}
+
+\begin{sagesilent}
+from plots_and_expressions import main_theorem2
+\end{sagesilent}
+\begin{theorem}[Bound on $r$ \#3]
+\label{thm:rmax_with_eps1}
+	Let $v$ be a fixed Chern character, with $\frac{a_v}{n}=\beta\coloneqq\beta(v)$
+	rational and expressed in lowest terms.
+	Then the ranks $r$ of the pseudo-semistabilizers $u$ for $v$ with,
+	which are solutions to problem \ref{problem:problem-statement-2},
+	$\chern_1^\beta(u) = q = \frac{b_q}{n}$
+	are bounded above by the following expression:
+
+	\begin{align*}
+		\min
+		\left(
+			\sage{main_theorem2.r_upper_bound1}, \:\:
+			\sage{main_theorem2.r_upper_bound2}
+		\right)
+	\end{align*}
+	Where $k_{v,q}$ is defined as in definition/lemma \ref{lemdfn:epsilon_q},
+	and $R = \chern_0(v)$
+
+	Furthermore, if $\aa \not= 0$ then
+	$r \equiv \aa^{-1}b_q \pmod{n}$.
+\end{theorem}
+
+Although the general form of this bound is quite complicated, it does simplify a
+lot when $m$ is small.
+
+\begin{sagesilent}
+from plots_and_expressions import main_theorem2_corollary
+\end{sagesilent}
+\begin{corollary}[Bound on $r$ \#3 on $\PP^2$ and Principally polarized abelian surfaces]
+\label{cor:rmax_with_eps1}
+	Suppose we are working over $\PP^2$ or a principally polarized abelian surface
+	(or any other surfaces with $m=1$ or $2$).
+	Let $v$ be a fixed Chern character, with $\frac{a_v}{n}=\beta\coloneqq\beta(v)$
+	rational and expressed in lowest terms.
+	Then the ranks $r$ of the pseudo-semistabilizers $u$ for $v$ with,
+	which are solutions to problem \ref{problem:problem-statement-2},
+	$\chern_1^\beta(u) = q = \frac{b_q}{n}$
+	are bounded above by the following expression:
+
+	\begin{align*}
+		\min
+		\left(
+			\sage{main_theorem2_corollary.r_upper_bound1}, \:\:
+			\sage{main_theorem2_corollary.r_upper_bound2}
+		\right)
+	\end{align*}
+	Where $R = \chern_0(v)$ and $k_{v,q}$ is the least
+	$k\in\ZZ_{>0}$ satisfying
+	\begin{equation*}
+		k \equiv -\aa\bb
+		\pmod{n}
+	\end{equation*}
+
+	\noindent
+	Furthermore, if $\aa \not= 0$ then
+	$r \equiv \aa^{-1}b_q \pmod{n}$.
+\end{corollary}
+
+\begin{proof}
+This is a specialisation of theorem \ref{thm:rmax_with_eps1}, where we can
+drastically simplify the $\lcm$ and $\gcd$ terms by noting that $m$ divides both
+$2$ and $2n^2$, and that $a_v$ is coprime to $n$.
+\end{proof}
+
+\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
+\label{exmpl:recurring-third}
+Just like in examples \ref{exmpl:recurring-first} and
+\ref{exmpl:recurring-second},
+take $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so that
+$\beta=\sage{recurring.betaminus}$, giving $n=\sage{recurring.n}$
+and $\chern_1^{\sage{recurring.betaminus}}(F) = \sage{recurring.twisted.ch[1]}$.
+%% TODO transcode notebook code
+The (non-exclusive) upper bounds for $r\coloneqq\chern_0(u)$ of a tilt semistabilizer $u$ of $v$
+in terms of the possible values for $q\coloneqq\chern_1^{\beta}(u)$ are as follows:
+
+\begin{sagesilent}
+from examples import bound_comparisons
+qs, theorem2_bounds, theorem3_bounds = bound_comparisons(recurring)
+\end{sagesilent}
+
+\vspace{1em}
+\noindent
+\directlua{ table_width = 3*4+1 }
+\begin{tabular}{l\directlua{for i=0,table_width-1 do tex.sprint([[|c]]) end}}
+	$q=\chern_1^\beta(u)$
+\directlua{for i=0,table_width-1 do
+	local cell = [[&$\noexpand\sage{qs[]] .. i .. "]}$"
+  tex.sprint(cell)
+end}
+	\\ \hline
+	Thm \ref{thm:rmax_with_uniform_eps}
+\directlua{for i=0,table_width-1 do
+	local cell = [[&$\noexpand\sage{theorem2_bounds[]] .. i .. "]}$"
+  tex.sprint(cell)
+end}
+	\\
+	Thm \ref{thm:rmax_with_eps1}
+\directlua{for i=0,table_width-1 do
+	local cell = [[&$\noexpand\sage{theorem3_bounds[]] .. i .. "]}$"
+  tex.sprint(cell)
+end}
+\end{tabular}
+\vspace{1em}
+
+\noindent
+It's worth noting that the bounds given by theorem \ref{thm:rmax_with_eps1}
+reach, but do not exceed the actual maximum rank 25 of the
+pseudo-semistabilizers of $v$ in this case.
+As a reminder, the original loose bound from theorem \ref{thm:loose-bound-on-r}
+was 144.
+
+\end{example}
+
+\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
+\label{exmpl:extravagant-third}
+Just like in examples \ref{exmpl:extravagant-first} and
+\ref{exmpl:extravagant-second},
+take $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so that
+$\beta=\sage{extravagant.betaminus}$, giving $n=\sage{n:=extravagant.n}$
+and $\chern_1^{\sage{extravagant.betaminus}}(F) = \sage{extravagant.twisted.ch[1]}$.
+This example was chosen because the $n$ value is moderatly large, giving more
+possible values for $k_{v,q}$, in dfn/lemma \ref{lemdfn:epsilon_q}. This allows
+for a larger possible difference between the bounds given by theorems
+\ref{thm:rmax_with_uniform_eps} and \ref{thm:rmax_with_eps1}, with the bound
+from the second being up to $\sage{n}$ times smaller, for any given $q$ value.
+The (non-exclusive) upper bounds for $r\coloneqq\chern_0(u)$ of a tilt semistabilizer $u$ of $v$
+in terms of the first few smallest possible values for $q\coloneqq\chern_1^{\beta}(u)$ are as follows:
+
+\begin{sagesilent}
+qs, theorem2_bounds, theorem3_bounds = bound_comparisons(extravagant)
+\end{sagesilent}
+
+
+\vspace{1em}
+\noindent
+\directlua{ table_width = 12 }
+\begin{tabular}{l\directlua{for i=0,table_width do tex.sprint([[|c]]) end}}
+	$q=\chern_1^\beta(u)$
+\directlua{for i=0,table_width-1 do
+	local cell = [[&$\noexpand\sage{qs[]] .. i .. "]}$"
+  tex.sprint(cell)
+end}
+	&$\cdots$
+	\\ \hline
+	Thm \ref{thm:rmax_with_uniform_eps}
+\directlua{for i=0,table_width-1 do
+	local cell = [[&$\noexpand\sage{theorem2_bounds[]] .. i .. "]}$"
+  tex.sprint(cell)
+end}
+	&$\cdots$
+	\\
+	Thm \ref{thm:rmax_with_eps1}
+\directlua{for i=0,table_width-1 do
+	local cell = [[&$\noexpand\sage{theorem3_bounds[]] .. i .. "]}$"
+  tex.sprint(cell)
+end}
+	&$\cdots$
+\end{tabular}
+\vspace{1em}
+
+
+\noindent
+However the reduction in the overall bound on $r$ is not as drastic, since all
+possible values for $k_{v,q}$ in $\{1,2,\ldots,\sage{n}\}$ are iterated through
+cyclically as we consider successive possible values for $q$.
+And for each $q$ where $k_{v,q}=1$, both theorems give the same bound.
+Calculating the maximums over all values of $q$ yields
+$\sage{max(theorem2_bounds)}$ for theorem \ref{thm:rmax_with_uniform_eps}, and
+$\sage{max(theorem3_bounds)}$ for theorem \ref{thm:rmax_with_eps1}.
+\end{example}
+
+\egroup % end scope where beta redefined to beta_{-}
+
+\subsubsection{All Semistabilizers Giving Sufficiently Large Circular Walls Left
+of Vertical Wall}
+
+
+Goals:
+\begin{itemize}
+	\item refresher on strategy
+	\item point out no need for rational beta
+	\item calculate intersection of bounds?
+\end{itemize}
+
+\subsection{Irrational \texorpdfstring{$\beta_{-}$}{êžµ_}}
+
+Goals:
+\begin{itemize}
+	\item Point out if only looking for sufficiently large wall, look at above
+		subsubsection
+	\item Relate to Pell's equation through coordinate change?
+	\item Relate to numerical condition described by Yanagida/Yoshioka
+\end{itemize}
+
+\section{Computing solutions to Problem \ref{problem:problem-statement-2}}
+\label{sect:prob2-algorithm}
+
+Alongside this article, there is a library \cite{NaylorRust2023} to compute
+the solutions to problem \ref{problem:problem-statement-2}, using the theorems
+above.
+
+The way it works, is by yielding solutions to the problem
+$u=(r,c\ell,\frac{e}{2}\ell^2)$ as follows.
+
+\subsection{Iterating Over Possible
+\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}}
+
+Given a Chern character $v$, the domain of the problem are first verified: that
+$v$ has positive rank, that it satisfies $\Delta(v) \geq 0$, and that
+$\beta_{-}(v)$ is rational.
+
+Take $\beta_{-}(v)=\frac{a_v}{n}$ in simplest terms.
+Iterate over $q \in [0,\chern_1^{\beta_{-}}(v)]\cap\frac{1}{n}\ZZ$.
+
+For any $u = (r,c\ell,\frac{e}{2}\ell^2)$, satisfying
+$\chern_1^{\beta_{-}}(u)=q$ for one of the $q$ considered is equivalent to
+satisfying condition \ref{item:chern1bound:lem:num_test_prob2}
+in corollary \ref{cor:num_test_prob2}.
+
+\subsection{Iterating Over Possible
+\texorpdfstring{$r=\chern_0(u)$}{r}
+for Fixed
+\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}
+}
+
+Let $q=\frac{b_q}{n}$ be one of the values of $\chern_1^{\beta_{-}}(u)$ that we
+have fixed. As mentioned before, the only values of $r$ which can
+give $\chern_1^{\beta_{-}}(u)=q$ are precisely the ones which satisfy
+$a_v r \equiv b_q \pmod{n}$.
+This is true for all integers when $\beta_{-}=0$ (and so $n=1$), but otherwise,
+this is equivalent to
+$r \equiv {a_v}^{-1}b_q \pmod{n}$, since $a_v$ and $n$ are coprime.
+
+Note that expressing $\mu(u)$ in term of $q$ and $r$ gives:
+\begin{align*}
+	\mu(u) & = \frac{c}{r} = \frac{q+r\beta_{-}}{r}
+	\\
+	&= \beta_{-} + \frac{q}{r}
+\end{align*}
+
+So condition \ref{item:mubound:lem:num_test_prob2} in corollary
+\ref{cor:num_test_prob2} is satisfied at this point precisely when:
+
+\begin{equation*}
+	r > \frac{q}{\mu(u) - \beta_{-}}
+\end{equation*}
+
+Note that the right hand-side is greater than, or equal, to 0, so such $r$ also
+satisfies \ref{item:rankpos:lem:num_test_prob2}.
+
+Then theorem \ref{thm:rmax_with_eps1} gives an upper on possible $r$ values
+for which it is possible to satisfy conditions
+\ref{item:bgmlvu:lem:num_test_prob2},
+\ref{item:bgmlvv-u:lem:num_test_prob2}, and
+\ref{item:radiuscond:lem:num_test_prob2}.
+
+Iterate over such $r$ so that we are guarenteed to satisfy conditions
+\ref{item:mubound:lem:num_test_prob2}, and
+\ref{item:radiuscond:lem:num_test_prob2}
+in corollary
+\ref{cor:num_test_prob2}, and have a chance at satisfying the rest.
+
+\subsection{Iterating Over Possible
+\texorpdfstring{$d=\chern_2(u)$}{d}
+for Fixed
+\texorpdfstring{$r=\chern_0(u)$}{r}
+and
+\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}
+}
+
+At this point we have fixed $\chern_0(u)=r$ and
+$\chern_1(u)=c=q+r\beta_{-}$.
+And the cases considered are precisely the ones which satisfy conditions
+\ref{item:chern1bound:lem:num_test_prob2},
+\ref{item:mubound:lem:num_test_prob2}, and
+\ref{item:radiuscond:lem:num_test_prob2}
+in corollary \ref{cor:num_test_prob2}.
+
+It remains to find $\chern_2(u)=d=\frac{e}{2}$
+which satisfy the remaining conditions
+\ref{item:bgmlvu:lem:num_test_prob2},
+\ref{item:bgmlvv-u:lem:num_test_prob2}, and
+\ref{item:radiuscond:lem:num_test_prob2}.
+These conditions induce upper and lower bounds on $d$, and it then remains to
+just pick the integers $e$ that give $d$ values within the bounds.
+
+Thus, through this process yielding all solutions $u=(r,c\ell,\frac{e}{2}\ell^2)$
+to the problem for this choice of $v$.
\ No newline at end of file
diff --git a/main.tex b/main.tex
index a0ed17c0ac4f82237d46fbae92cae8be2564964d..9abcf6c0f7d069d3cd0700f94f2fc34ac0f20ed7 100644
--- a/main.tex
+++ b/main.tex
@@ -1,25 +1,20 @@
-\documentclass[class=article, crop=false]{standalone}
-\usepackage[subpreambles=true]{standalone}
-\onlyifstandalone{
+\documentclass[]{article}
+
 \input{packages.tex}
 \input{newcommands.tex}
 \input{theoremstyles.tex}
 \usepackage{sagetex}
 \addbibresource{references.bib}
-}
 
-\begin{document}
+\author{Luke Naylor}
+\date{}
 
+\begin{document}
 
-\onlyifstandalone{
 \title{Tighter Bounds for Ranks of Tilt Semistabilizers on Picard Rank 1 Surfaces
 \\[1em] \large
 Practical Methods for Narrowing Down Possible Walls}
 
-
-\author{Luke Naylor}
-\date{}
-
 \maketitle
 
 \begin{abstract}
@@ -28,2007 +23,12 @@ Practical Methods for Narrowing Down Possible Walls}
 
 \newpage
 \tableofcontents
-}
 
 \newpage
-\section{Introduction}
-\label{sec:intro}
-
-The theory of Bridgeland stability conditions \cite{BridgelandTom2007SCoT} on
-complexes of sheaves was developed as a generalisation of stability for vector
-bundles. The definition is most analoguous to Mumford stability, but is more
-aware of the features that sheaves can have on spaces of dimension greater
-than 1. Whilst also asymptotically matching up with Gieseker stability.
-For K3 surfaces, explicit stability conditions were defined in
-\cite{Bridgeland_StabK3}, and later shown to also be valid on other surfaces.
-
-The moduli spaces of stable objects of some fixed Chern character $v$ is
-studied, as well as how they change as we vary the Bridgeland stability
-condition. They in fact do not change over whole regions of the stability
-space (called chambers), but do undergo changes as we cross `walls' in the
-stability space. These are where there is some stable object $F$ of $v$ which
-has a subobject who's slope overtakes the slope of $v$, making $F$ unstable
-after crossing the wall.
-
-% NOTE: SURFACE SPECIALIZATION
-% (come back to these when adjusting to general Picard rank 1)
-In this document we concentrate on two surfaces: Principally polarized abelian
-surfaces and the projective surface $\PP^2$. Although this can be generalised
-for Picard rank 1 surfaces, the formulae will need adjusting.
-The Bridgeland stability conditions (defined in \cite{Bridgeland_StabK3}) are
-given by two parameters $\alpha \in \RR_{>0}$, $\beta \in \RR$, which will be
-illustrated throughout this article with diagrams of the upper half plane.
-
-It is well known that for any rational $\beta_0$,
-the vertical line $\{\sigma_{\alpha,\beta_0} \colon \alpha \in \RR_{>0}\}$ only
-intersects finitely many walls
-\cite[Thm 1.1]{LoJason2014Mfbs}
-\cite[Prop 4.2]{alma9924569879402466}
-\cite[Lemma 5.20]{MinaHiroYana_SomeModSp}.
-A consequence of this is that if
-$\beta_{-}$ is rational, then there can only be finitely many circular walls to the
-left of the vertical wall $\beta = \mu$.
-On the other hand, when $\beta_{-}$ is not rational, \cite{yanagida2014bridgeland}
-showed that there are infinitely many walls.
-
-This dichotomy does not only hold for real walls, realised by actual objects in
-$\bddderived(X)$, but also for pseudowalls. Here pseudowalls are defined as
-`potential' walls, induced by hypothetical Chern characters of semistabilizers
-which satisfy certain numerical conditions which would be satisfied by any real
-destabilizer, regardless of whether they are realised by actual semistabilizers
-in $\bddderived(X)$ (dfn \ref{dfn:pseudo-semistabilizer}).
-
-Since real walls are a subset of pseudowalls, the irrational $\beta_{-}$ case
-follows immediately from the corresponding case for real walls.
-However, the rational $\beta_{-}$ case involves showing that the following
-conditions only admit finitely many solutions (despite the fact that the same
-conditions admit infinitely many solutions when $\beta_{-}$ is irrational).
-
-
-For a semistabilizing sequence
-$E \hookrightarrow F \twoheadrightarrow G$ in $\mathcal{B}^\beta$
-we have the following conditions.
-There are some Bogomolov-Gieseker inequalities:
-$0 \leq \Delta(E), \Delta(G)$.
-We also have a condition relating to the tilt category $\firsttilt\beta$:
-$0 \leq \chern^\beta_1(E) \leq \chern^\beta_1(F)$.
-Finally, there is a condition ensuring that the radius of the circular wall is
-strictly positive: $\chern^{\beta_{-}}_2(E) > 0$.
-
-For any fixed $\chern_0(E)$, the inequality
-$0 \leq \chern^{\beta}_1(E) \leq \chern^{\beta}_1(F)$,
-allows us to bound $\chern_1(E)$. Then, the other inequalities allow us to
-bound $\chern_2(E)$. The final part to showing the finiteness of pseudowalls
-would be bounding $\chern_0(E)$. This has been hinted at in
-\cite{SchmidtBenjamin2020Bsot} and done explicitly by Benjamin Schmidt within a
-SageMath \cite{sagemath} library which computes pseudowalls
-\cite{SchmidtGithub2020}.
-Here we discuss these bounds in more detail, along with the methods used,
-followed by refinements on them which give explicit formulae for tighter bounds
-on $\chern_0(E)$ of potential destabilizers $E$ of $F$.
-
-
-\section{Setting and Definitions: Clarifying `pseudo'}
-
-%\begin{definition}[Twisted Chern Character]
-%\label{sec:twisted-chern}
-%For a given $\beta$, define the twisted Chern character as follows.
-%\[\chern^\beta(E) = \chern(E) \cdot \exp(-\beta \ell)\]
-%\noindent
-%Component-wise, this is:
-%\begin{align*}
-%	\chern^\beta_0(E) &= \chern_0(E)
-%\\
-%	\chern^\beta_1(E) &= \chern_1(E) - \beta \chern_0(E)
-%\\
-%	\chern^\beta_2(E) &= \chern_2(E) - \beta \chern_1(E) + \frac{\beta^2}{2} \chern_0(E)
-%\end{align*}
-%where $\chern_i$ is the coefficient of $\ell^i$ in $\chern$.
-%
-%% TODO I think this^ needs adjusting for general Surface with $\ell$
-%\end{definition}
-%
-%$\chern^\beta_1(E)$ is the imaginary component of the central charge
-%$\centralcharge_{\alpha,\beta}(E)$ and any element of $\firsttilt\beta$
-%satisfies $\chern^\beta_1 \geq 0$.
-
-Throughout this article, as noted in the introduction, we will be exclusively
-working over surfaces $X$ with Picard rank 1, with a choice of ample line bundle
-$L$ such that $\ell\coloneqq c_1(L)$ generates $NS(X)$.
-We take $m\coloneqq \ell^2$ as this will be the main quantity which will
-affect the results.
-
-\begin{definition}[Pseudo-semistabilizers]
-\label{dfn:pseudo-semistabilizer}
-% NOTE: SURFACE SPECIALIZATION
-	Given a Chern Character $v$, and a given stability
-	condition $\sigma_{\alpha,\beta}$,
-	a \textit{pseudo-semistabilizing} $u$ is a `potential' Chern character:
-	\[
-		u = \left(r, c\ell, \frac{e}{\lcm(m,2)} \ell^2\right)
-		\qquad
-		r,c,e \in \ZZ
-	\]
-	which has the same tilt slope as $v$:
-	$\nu_{\alpha,\beta}(u) = \nu_{\alpha,\beta}(v)$.
-
-	\noindent
-	Furthermore the following inequalities are satisfied:
-	\begin{itemize}
-		\item $\Delta(u) \geq 0$
-		\item $\Delta(v-u) \geq 0$
-		\item $0 \leq \chern_1^{\beta}(u) \leq \chern_1^{\beta}(v)$
-	\end{itemize}
-
-	Note $u$ does not need to be a Chern character of an actual sub-object of some
-	object in the stability condition's heart with Chern character $v$.
-\end{definition}
-
-At this point, and in this document, we do not care about whether
-pseudo-semistabilizers are even Chern characters of actual elements of
-$\bddderived(X)$, some other sources may have this extra restriction too.
-
-Later, Chern characters will be written $(r,c\ell,d\ell^2)$ because operations
-(such as multiplication) are more easily defined in terms of the coefficients of
-the $\ell^i$. However, at the end, it will become important again that
-$d \in \frac{1}{\lcm(m,2)}\ZZ$.
-
-\begin{definition}[Pseudo-walls]
-\label{dfn:pseudo-wall}
-	Let $u$ be a pseudo-semistabilizer of $v$, for some stability condition.
-	Then the \textit{pseudo-wall} associated to $u$ is the set of all stablity
-	conditions where $u$ is a pseudo-semistabilizer of $v$.
-\end{definition}
-
-% TODO possibly reference forwards to Bertram's nested wall theorem section to 
-% cover that being a pseudo-semistabilizer somewhere implies also on whole circle
-
-\begin{lemma}[Sanity check for Pseudo-semistabilizers]
-	Given a stability
-	condition $\sigma_{\alpha,\beta}$,
-	if $E\hookrightarrow F\twoheadrightarrow G$ is a semistabilizing sequence in
-	$\firsttilt\beta$ for $F$.
-	Then $\chern(E)$ is a pseudo-semistabilizer of $\chern(F)$
-\end{lemma}
-
-\begin{proof}
-	Suppose $E\hookrightarrow F\twoheadrightarrow G$ is a semistabilizing
-	sequence with respect to a stability condition $\sigma_{\alpha,\beta}$.
-	\begin{equation*}
-		\chern(E) = -\chern(\homol^{-1}_{\coh}(E)) + \chern(\homol^{0}_{\coh}(E))
-	\end{equation*}
-	Therefore, $\chern(E)$ is of the form
-	$(r,c\ell,\frac{e}{\lcm(m,2)}\ell^2)$
-	provided that this is true for any coherent sheaf.
-	For any coherent sheaf $H$, we have the following:
-	\begin{equation*}
-		\chern(H) = \left(c_0(H), c_1(H), - c_2(H) + \frac{1}{2} {c_1(H)}^2\right)
-	\end{equation*}
-	Given that $\ell$ generates the Neron-Severi group, $c_1(H)$ can then be
-	written $c\ell$.
-	\begin{equation*}
-		\chern(H) = \left(
-			c_0(H), c\ell,
-			\left(- \frac{c_2(H)}{\ell^2} + \frac{c^2}{2} \right)\ell^2
-		\right)
-	\end{equation*}
-	This fact along with $c_0$, $c_2$ being an integers on surfaces, and
-	$m\coloneqq \ell^2$ implies that $\chern(H)$
-	(hence $\chern(E)$ too) is of the required form.
-	
-
-	Since all the objects in the sequence are in $\firsttilt\beta$, we have
-	$\chern_1^{\beta} \geq 0$ for each of them. Due to additivity
-	($\chern(F) = \chern(E) + \chern(G)$), we can deduce
-	$0 \leq \chern_1^{\beta}(E) \leq \chern_1^{\beta}(F)$.
-
-
-	$E \hookrightarrow F \twoheadrightarrow G$ being a semistabilizing sequence
-	means	$\nu_{\alpha,\beta}(E) = \nu_{\alpha,\beta}(F) = \nu_{\alpha,\beta}(F)$.
-	% MAYBE: justify this harder
-	But also, that this is an instance of $F$ being semistable, so $E$ must also
-	be semistable
-	(otherwise the destabilizing subobject would also destabilize $F$).
-	Similarly $G$ must also be semistable too.
-	$E$ and $G$ being semistable implies they also satisfy the Bogomolov
-	inequalities:
-	% TODO ref Bogomolov inequalities for tilt stability
-	$\Delta(E), \Delta(G) \geq 0$.
-	Expressing this in terms of Chern characters for $E$ and $F$ gives:
-	$\Delta(\chern(E)) \geq 0$ and $\Delta(\chern(F)-\chern(E)) \geq 0$.
-
-\end{proof}
-
-
-\section{Characteristic Curves of Stability Conditions Associated to Chern
-Characters}
-
-% NOTE: SURFACE SPECIALIZATION
-Considering the stability conditions with two parameters $\alpha, \beta$ on
-Picard rank 1 surfaces.
-We can draw 2 characteristic curves for any given Chern character $v$ with
-$\Delta(v) \geq 0$ and positive rank.
-These are given by the equations $\chern_i^{\alpha,\beta}(v)=0$ for $i=1,2$.
-
-\begin{definition}[Characteristic Curves $V_v$ and $\Theta_v$]
-Given a Chern character $v$, with positive rank and $\Delta(v) \geq 0$, we
-define two characteristic curves on the $(\alpha, \beta)$-plane:
-
-\begin{align*}
-	V_v &\colon \:\: \chern_1^{\alpha, \beta}(v) = 0 \\
-	\Theta_v &\colon \:\: \chern_2^{\alpha, \beta}(v) = 0
-\end{align*}
-\end{definition}
-
-\subsection{Geometry of the Characteristic Curves}
-
-These characteristic curves for a Chern character $v$ with $\Delta(v)\geq0$ are
-not affected by flipping the sign of $v$ so it's only necessary to consider
-non-negative rank.
-As discussed in subsection \ref{subsect:relevance-of-V_v}, making this choice
-has Gieseker stable coherent sheaves appearing in the heart of the stability
-condition $\firsttilt{\beta}$ as we move `left' (decreasing $\beta$).
-
-\subsubsection{Positive Rank Case}
-\label{subsect:positive-rank-case-charact-curves}
-
-\begin{fact}[Geometry of Characteristic Curves in Positive Rank Case]
-The following facts can be deduced from the formulae for $\chern_i^{\alpha, \beta}(v)$
-as well as the restrictions on $v$, when $\chern_0(v)>0$:
-\begin{itemize}
-	\item $V_v$ is a vertical line at $\beta=\mu(v)$
-	\item $\Theta_v$ is a hyperbola with assymptotes angled at $\pm 45^\circ$
-		crossing where $V_v$ meets the $\beta$-axis: $(\mu(v),0)$
-	\item $\Theta_v$ is oriented with left-right branches (as opposed to up-down).
-		The left branch shall be labelled $\Theta_v^-$ and the right $\Theta_v^+$.
-	\item The gap along the $\beta$-axis between either branch of $\Theta_v$
-		and $V_v$ is $\sqrt{\Delta(v)}/\chern_0(v)$.
-	\item When $\Delta(v)=0$, $\Theta_v$ degenerates into a pair of lines, but the
-		labels $\Theta_v^\pm$ will still be used for convenience.
-\end{itemize}
-\end{fact}
-
-These are illustrated in Fig \ref{fig:charact_curves_vis}
-(dotted line for $i=1$, solid for $i=2$).
-
-\begin{sagesilent}
-from characteristic_curves import \
-typical_characteristic_curves, \
-degenerate_characteristic_curves
-\end{sagesilent}
-
-
-\begin{figure}
-\centering
-\begin{subfigure}{.49\textwidth}
-	\centering
-	\sageplot[width=\textwidth]{typical_characteristic_curves}
-	\caption{$\Delta(v)>0$}
-	\label{fig:charact_curves_vis_bgmvlPos}
-\end{subfigure}%
-\hfill
-\begin{subfigure}{.49\textwidth}
-	\centering
-	\sageplot[width=\textwidth]{degenerate_characteristic_curves}
-	\caption{
-		$\Delta(v)=0$: hyperbola collapses
-	}
-	\label{fig:charact_curves_vis_bgmlv0}
-\end{subfigure}
-\caption{
-	Characteristic curves ($\chern_i^{\alpha,\beta}(v)=0$) of stability conditions
-	associated to Chern characters $v$ with $\Delta(v) \geq 0$ and positive rank.
-}
-\label{fig:charact_curves_vis}
-\end{figure}
-
-\begin{definition}[$\beta_{\pm}$]
-	\label{dfn:beta_pm}
-	Given a formal Chern character $v$ with positive rank, we define $\beta_{\pm}(v)$ to be
-	the $\beta$-coordinate of where $\Theta_v^{\pm}$ meets the $\beta$-axis:
-	\[
-		\beta_\pm(R,C\ell,D\ell^2) = \frac{C \pm \sqrt{C^2-2RD}}{R}
-	\]
-	\noindent
-	In particular, this means $\beta_\pm(v)$ are the two roots of the quadratic
-	equation $\chern_2^{\beta}(v)=0$.
-
-	This definition will be extended to the rank 0 case in definition \ref{dfn:beta_-_rank0}.
-\end{definition}
-
-
-\subsubsection{Rank Zero Case}
-\label{subsubsect:rank-zero-case-charact-curves}
-
-\begin{sagesilent}
-from rank_zero_case import Theta_v_plot
-\end{sagesilent}
-
-\begin{fact}[Geometry of Characteristic Curves in Rank 0 Case]
-The following facts can be deduced from the formulae for $\chern_i^{\alpha, \beta}(v)$
-as well as the restrictions on $v$, when $\chern_0(v)=0$ and $\chern_1(v)>0$:
-
-
-\begin{minipage}{0.5\textwidth}
-\begin{itemize}
-	\item $V_v = \emptyset$
-	\item $\Theta_v$ is a vertical line at $\beta=\frac{D}{C}$
-		where $v=\left(0,C\ell,D\ell^2\right)$
-\end{itemize}
-\end{minipage}
-\hfill
-\begin{minipage}{0.49\textwidth}
-	\sageplot[width=\textwidth]{Theta_v_plot}
-	%\caption{$\Delta(v)>0$}
-	%\label{fig:charact_curves_rank0}
-\end{minipage}
-\end{fact}
-
-We can view the characteristic curves for $\left(0,C\ell, D\ell^2\right)$ with $C>0$ as
-the limiting behaviour of those of $\left(\varepsilon, C\ell, D\ell^2\right)$.
-Indeed:
-\begin{align*}
-	\mu\left(\varepsilon, C\ell, D\ell^2\right) = \frac{C}{\varepsilon} &\longrightarrow +\infty
-	\\
-	\text{as} \:\: 0<\varepsilon &\longrightarrow 0
-\end{align*}
-So we can view $V_v$ as moving off infinitely to the right, with $\Theta_v^+$ even further.
-But also, considering the base point of $\Theta_v^-$:
-\begin{align*}
-	\beta_{-}\left(\varepsilon, C\ell, D\ell^2\right) = \frac{C - \sqrt{C^2-2D\varepsilon}}{\varepsilon}
-	&\longrightarrow \frac{D}{C}
-	\\
-	\text{as} \:\: 0<\varepsilon &\longrightarrow 0
-	&\text{(via L'H\^opital)}
-\end{align*}
-
-So we can view $\Theta_v^-$ as approaching the vertical line that $\Theta_v$  becomes.
-For this reason, I will refer to the whole of $\Theta_v$ in the rank zero case
-as $\Theta_v^-$ to be able to use the same terminology in both positive rank
-and rank zero cases.
-
-\begin{definition}[Extending $\beta_-$ to rank 0 case]
-	\label{dfn:beta_-_rank0}
-	Given a formal Chern character $v$ with rank 0 and $\chern_1(v)>0$, we define
-	$\beta_-(v)$ to be the $\beta$-coordinate of point where $\Theta_v$ meets the
-	$\beta$-axis:
-	\[
-		\beta_-(0,C\ell,D\ell^2) = \frac{D}{C}
-	\]
-	\noindent
-	If $\beta_+$ were also to be generalised to the rank 0 case, we would consider
-	its value to be $+\infty$ due to the discussion above.
-\end{definition}
-
-
-\subsection{Relevance of \texorpdfstring{$V_v$}{V_v}}
-\label{subsect:relevance-of-V_v}
-
-For the positive rank case, by definition of the first tilt $\firsttilt\beta$, objects of Chern character
-$v$ can only be in $\firsttilt\beta$ on the left of $V_v$, where
-$\chern_1^{\alpha,\beta}(v)>0$, and objects of Chern character $-v$ can only be
-in $\firsttilt\beta$ on the right, where $\chern_1^{\alpha,\beta}(-v)>0$. In
-fact, if there is a Gieseker semistable coherent sheaf $E$ of Chern character
-$v$, then $E \in \firsttilt\beta$ if and only if $\beta<\mu(E)$ (left of the
-$V_v$), and $E[1] \in \firsttilt\beta$ if and only if $\beta\geq\mu(E)$.
-Because of this, when using these characteristic curves, only positive ranks are
-considered, as negative rank objects are implicitly considered on the right hand
-side of $V_v$.
-
-In the rank zero case, this still applies if we consider $V_v$ to be
-`infinitely to the right' ($\mu(v) = +\infty$). Precisely, Gieseker semistable
-coherent sheaves $E$ of Chern character $v$ are contained in
-$\firsttilt{\beta}$ for all $\beta$
-
-
-
-\subsection{Relevance of \texorpdfstring{$\Theta_v$}{Θ_v}}
-
-Since $\chern_2^{\alpha, \beta}$ is the numerator of the tilt slope
-$\nu_{\alpha, \beta}$. The curve $\Theta_v$, where this is 0, firstly divides the
-$(\alpha$-$\beta)$-half-plane into regions where the signs of tilt slopes of
-objects of Chern character $v$ (or $-v$) are fixed. Secondly, it gives more of a
-fixed target for some $u=(r,c\ell,d\frac{1}{2}\ell^2)$ to be a
-pseudo-semistabilizer of $v$, in the following sense: If $(\alpha,\beta)$, is on
-$\Theta_v$, then for any $u$, $u$ can only be a pseudo-semistabilizer of $v$ if
-$\nu_{\alpha,\beta}(u)=0$, and hence $\chern_2^{\alpha, \beta}(u)=0$. In fact,
-this allows us to use the characteristic curves of some $v$ and $u$ (with
-$\Delta(v), \Delta(u)\geq 0$ and positive ranks) to determine the location of
-the pseudo-wall where $u$ pseudo-semistabilizes $v$. This is done by finding the
-intersection of $\Theta_v$ and $\Theta_u$, the point $(\beta,\alpha)$ where
-$\nu_{\alpha,\beta}(u)=\nu_{\alpha,\beta}(v)=0$, and a pseudo-wall point on
-$\Theta_v$, and hence the apex of the circular pseudo-wall with centre $(\beta,0)$
-(as per subsection \ref{subsect:bertrams-nested-walls}).
-
-
-\subsection{Bertram's Nested Wall Theorem}
-\label{subsect:bertrams-nested-walls}
-
-Although Bertram's nested wall theorem can be proved more directly, it's also
-important for the content of this document to understand the connection with
-these characteristic curves.
-Emanuele Macri noticed in (TODO ref) that any circular wall of $v$ reaches a critical
-point on $\Theta_v$ (TODO ref). This is a consequence of
-$\frac{\delta}{\delta\beta} \chern_2^{\alpha,\beta} = -\chern_1^{\alpha,\beta}$.
-This fact, along with the hindsight knowledge that non-vertical walls are
-circles with centers on the $\beta$-axis, gives an alternative view to see that
-the circular walls must be nested and non-intersecting.
-
-\subsection{Characteristic Curves for Pseudo-semistabilizers}
-
-These characteristic curves introduced are convenient tools to think about the
-numerical conditions that can be used to test for pseudo-semistabilizers, and
-for solutions to the problems
-(\ref{problem:problem-statement-1},\ref{problem:problem-statement-2})
-tackled in this article (to be introduced later).
-In particular, problem (\ref{problem:problem-statement-1}) will be translated to
-a list of numerical inequalities on it's solutions $u$.
-% ref to appropriate lemma when it's written
-
-The next lemma is a key to making this translation and revolves around the
-geometry and configuration of the characteristic curves involved in a
-semistabilizing sequence.
-
-\begin{lemma}[Numerical tests for left-wall pseudo-semistabilizers]
-\label{lem:pseudo_wall_numerical_tests}
-Let $v$ and $u$ be Chern characters with $\Delta(v),
-\Delta(u)\geq 0$, and $v$ has non-negative rank (and $\chern_1(v)>0$ if rank 0).
-Let $P$ be a point on $\Theta_v^-$.
-
-\noindent
-The following conditions:
-\begin{enumerate}
-\item $u$ is a pseudo-semistabilizer of $v$ at some point on $\Theta_v^-$ above
-	$P$
-\item $u$ destabilizes $v$ going `inwards', that is,
-	$\nu_{\alpha,\beta}(u)<\nu_{\alpha,\beta}(v)$ outside the pseudo-wall, and
-	$\nu_{\alpha,\beta}(u)>\nu_{\alpha,\beta}(v)$ inside.
-\end{enumerate}
-
-\noindent
-are equivalent to the following more numerical conditions:
-\begin{enumerate}
-	\item $u$ has positive rank
-	\item $\beta(P)<\mu(u)<\mu(v)$, i.e. $V_u$ is strictly between $P$ and $V_v$.
-	\item $\chern_1^{\beta(P)}(u) \leq \chern_1^{\beta(P)}(v)$, $\Delta(v-u) \geq 0$
-	\item $\chern_2^{P}(u)>0$
-\end{enumerate}
-\end{lemma}
-
-\begin{proof}
-Let $u,v$ be Chern characters with
-$\Delta(u),\Delta(v) \geq 0$, and $v$ has positive rank.
-
-
-For the forwards implication, assume that the suppositions of the lemma are
-satisfied. Let $Q$ be the point on $\Theta_v^-$ (above $P$) where $u$ is a
-pseudo-semistabilizer of $v$.
-Firstly, consequence 3 is part of the definition for $u$ being a
-pseudo-semistabilizer at a point with same $\beta$ value of $P$ (since the
-pseudo-wall surrounds $P$).
-If $u$ were to have 0 rank, it's tilt slope would be decreasing as $\beta$
-increases, contradicting supposition b. So $u$ must have strictly non-zero rank,
-and we can consider it's characteristic curves (or that of $-u$ in case of
-negative rank).
-$\nu_Q(v)=0$, and hence $\nu_Q(u)=0$ too. This means that $\Theta_{\pm u}$ must
-intersect $\Theta_v^-$ at $Q$. Considering the shapes of the hyperbolae alone,
-there are 3 distinct ways that they can intersect, as illustrated in Fig
-\ref{fig:hyperbol-intersection}. These cases are distinguished by whether it is
-the left, or the right branch of $\Theta_u$ involved, as well as the positions
-of the base. However, considering supposition b, only case 3 (green in
-figure) is possible. This is because we need $\nu_{P}(u)>0$ (or $\nu_{P}(-u)>0$ in
-case 1 involving $\Theta_u^+$), to satisfy supposition b.
-Recalling how the sign of $\nu_{\alpha,\beta}(\pm u)$ changes (illustrated in
-Fig \ref{fig:charact_curves_vis}), we can eliminate cases 1 and 2.
-
-\begin{sagesilent}
-from characteristic_curves import \
-hyperbola_intersection_plot, \
-correct_hyperbola_intersection_plot
-\end{sagesilent}
-
-\begin{figure}
-\begin{subfigure}[t]{0.48\textwidth}
-	\centering
-	\sageplot[width=\textwidth]{hyperbola_intersection_plot()}
-	\caption{Three ways the characteristic hyperbola for $u$ can intersect the left
-	branch of the characteristic hyperbola for $v$}
-	\label{fig:hyperbol-intersection}
-\end{subfigure}
-\hfill
-\begin{subfigure}[t]{0.48\textwidth}
-	\centering
-	\sageplot[width=\textwidth]{correct_hyperbola_intersection_plot()}
-	\caption{Closer look at characteristic curves for valid case}
-	\label{fig:correct-hyperbol-intersection}
-\end{subfigure}
-\end{figure}
-
-Fixing attention on the only possible case (2), illustrated in Fig
-\ref{fig:correct-hyperbol-intersection}.
-$P$ is on the left of $V_{\pm u}$ (first part of consequence 2), so $u$ must
-have positive rank (consequence 1)
-to ensure that $\chern_1^{\beta(P)} \geq 0$ (since the pseudo-wall passed over
-$P$).
-Furthermore, $P$ being on the left of $V_u$ implies
-$\chern_1^{\beta{P}}(u) \geq 0$,
-and therefore $\chern_2^{P}(u) > 0$ (consequence 4) to satisfy supposition b.
-Next considering the way the hyperbolae intersect, we must have $\Theta_u^-$ taking a
-base-point to the right $\Theta_v$, but then, further up, crossing over to the
-left side. The latter fact implies that the assymptote for $\Theta_u^-$ must be
-to the left of the one for $\Theta_v^-$. Given that they are parallel and
-intersect the $\beta$-axis at $\beta=\mu(u)$ and $\beta=\mu(v)$ respectively. We
-must have $\mu(u)<\mu(v)$ (second part of consequence 2),
-that is, $V_u$ is strictly to the left of $V_v$.
-
-
-Conversely, suppose that the consequences 1-4 are satisfied. Consequence 2
-implies that the assymptote for $\Theta_u^-$ is to the left of $\Theta_v^-$.
-Consequence 4, along with $\beta(P)<\mu(u)$, implies that $P$ must be in the
-region left of $\Theta_u^-$. These two facts imply that $\Theta_u^-$ is to the
-right of $\Theta_v^-$ at $\alpha=\alpha(P)$, but crosses to the left side as
-$\alpha \to +\infty$, intersection at some point $Q$ above $P$.
-This implies that the characteristic curves for $u$ and $v$ are in the
-configuration illustrated in Fig \ref{fig:correct-hyperbol-intersection}.
-We then have $\nu(u)=\nu(v)$ along a circle to the left of $V_u$ reaching it's
-apex at $Q$, and encircling $P$. This along with consequence 3 implies that $u$
-is a pseudo-semistabilizer at the point on the circle with $\beta=\beta(P)$.
-Therefore, it's also a pseudo-semistabilizer further along the circle at $Q$
-(supposition a).
-Finally, consequence 4 along with $P$ being to the left of $V_u$ implies
-$\nu_P(u) > 0$ giving supposition b.
-
-The case with rank 0 can be handled the same way.
-
-\end{proof}
-
-\section{The Problem: Finding Pseudo-walls}
-
-As hinted in the introduction (\ref{sec:intro}), the main motivation of the
-results in this article are not only the bounds on pseudo-semistabilizer
-ranks;
-but also applications for finding a list (comprehensive or subset) of
-pseudo-walls.
-
-After introducing the characteristic curves of stability conditions associated
-to a fixed Chern character $v$, we can now formally state the problems that we
-are trying to solve for.
-
-\subsection{Problem statements}
-
-\begin{problem}[sufficiently large `left' pseudo-walls]
-\label{problem:problem-statement-1}
-
-Fix a Chern character $v$ with non-negative rank (and $\chern_1(v)>0$ if rank 0),
-and $\Delta(v) \geq 0$.
-The goal is to find all pseudo-semistabilizers $u$
-which give circular pseudo-walls containing some fixed point
-$P\in\Theta_v^-$.
-With the added restriction that $u$ `destabilizes' $v$ moving `inwards', that is,
-$\nu(u)>\nu(v)$ inside the circular pseudo-wall.
-\end{problem}
-This will give all pseudo-walls between the chamber corresponding to Gieseker
-stability and the stability condition corresponding to $P$.
-The purpose of the final `direction' condition is because, up to that condition,
-semistabilizers are not distinguished from their corresponding quotients:
-Suppose $E\hookrightarrow F\twoheadrightarrow G$, then the tilt slopes
-$\nu_{\alpha,\beta}$
-are strictly increasing, strictly decreasing, or equal across the short exact
-sequence (consequence of the see-saw principle).
-In this case, $\chern(E)$ is a pseudo-semistabilizer of $\chern(F)$, if and
-only if $\chern(G)$ is a pseudo-semistabilizer of $\chern(F)$.
-The numerical inequalities in the definition for pseudo-semistabilizer cannot
-tell which of $E$ or $G$ is the subobject.
-However, what can be distinguished is the direction across the wall that
-$\chern(E)$ or $\chern(G)$ destabilizes $\chern(F)$
-(they will each destabilize in the opposite direction to the other).
-The `inwards' semistabilizers are preferred because we are moving from a
-typically more familiar chamber
-(the stable objects of Chern character $v$ in the outside chamber will only be
-Gieseker stable sheaves).
-
-Also note that this last restriction does not remove any pseudo-walls found,
-and if we do want to recover `outwards' semistabilizers, we can simply take
-$v-u$ for each solution $u$ of the problem.
-
-
-\begin{problem}[all `left' pseudo-walls]
-\label{problem:problem-statement-2}
-
-Fix a Chern character $v$ with non-negative rank (and $\chern_1(v)>0$ if rank 0),
-$\Delta(v) \geq 0$, and $\beta_{-}(v) \in \QQ$.
-The goal is to find all pseudo-semistabilizers $u$ which give circular
-pseudo-walls on the left side of $V_v$.
-\end{problem}
-
-This is a specialization of problem (\ref{problem:problem-statement-1})
-with the choice $P=(\beta_{-},0)$, the point where $\Theta_v^-$ meets the
-$\beta$-axis.
-This is because all circular walls left of $V_v$ intersect $\Theta_v^-$ (once).
-The $\beta_{-}(v) \in \QQ$ condition is to ensure that there are finitely many
-solutions. As mentioned in the introduction (\ref{sec:intro}), this is known,
-however this will also be proved again implicitly in section
-\ref{sect:prob2-algorithm}, where an algorithm is produced to find all
-solutions.
-
-This description still holds for the case of rank 0 case if we consider $V_v$ to
-be infinitely far to the right
-(see section \ref{subsubsect:rank-zero-case-charact-curves}).
-Note also that the condition on $\beta_-(v)$ always holds for $v$ rank 0.
-
-\subsection{Numerical Formulations of the Problems}
-
-The problems introduced in this section are phrased in the context of stability
-conditions. However, these can be reduced down completely to purely numerical
-problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
-
-\begin{lemma}[Numerical Tests for Sufficiently Large `left' Pseudo-walls]
-	\label{lem:num_test_prob1}
-	Given a Chern character $v$ with non-negative rank
-	(and $\chern_1(v)>0$ if rank 0),
-	and $\Delta(v) \geq 0$,
-	and a choice of point $P$ on $\Theta_v^-$.
-	Solutions $u=(r,c\ell,d\ell^2)$
-	to problem \ref{problem:problem-statement-1}.
-	Are precisely given by $r,c \in \ZZ$, $d \in \frac{1}{\lcm(m,2)}$
-	satisfying the following conditions:
-	\begin{enumerate}
-		\item $r > 0$
-			\label{item:rankpos:lem:num_test_prob1}
-		\item $\Delta(u) \geq 0$
-			\label{item:bgmlvu:lem:num_test_prob1}
-		\item $\Delta(v-u) \geq 0$
-			\label{item:bgmlvv-u:lem:num_test_prob1}
-		\item $\mu(u)<\mu(v)$
-		\item $0\leq\chern_1^{\beta(P)}(u)\leq\chern_1^{\beta(P)}(v)$
-			\label{item:chern1bound:lem:num_test_prob1}
-		\item $\chern_2^{P}(u)>0$
-			\label{item:radiuscond:lem:num_test_prob1}
-	\end{enumerate}
-\end{lemma}
-
-\begin{proof}
-	Consider the context of $v$ being a Chern character with non-negative rank
-	(and $\chern_1(v)>0$ if rank 0)
-	and
-	$\Delta \geq 0$, and $u$ being a Chern character with $\Delta(u) \geq 0$.
-	Lemma \ref{lem:pseudo_wall_numerical_tests} gives that the remaining
-	conditions for $u$ being a solution to problem
-	\ref{problem:problem-statement-1} are precisely equivalent to the
-	remaining conditions in this lemma.
-	% TODO maybe make this more explicit
-	% (the conditions are not exactly the same)
-
-\end{proof}
-
-\begin{corollary}[Numerical Tests for All `left' Pseudo-walls]
-\label{cor:num_test_prob2}
-	Given a Chern character $v$ with positive rank and $\Delta(v) \geq 0$,
-	such that $\beta_{-}\coloneqq\beta_{-}(v) \in \QQ$.
-	Solutions $u=(r,c\ell,d\ell^2)$
-	to problem \ref{problem:problem-statement-2}.
-	Are precisely given by $r,c \in \ZZ$, $d\in\frac{1}{\lcm(m,2)}\ZZ$ satisfying
-	the following conditions:
-	\begin{enumerate}
-		\item $r > 0$
-			\label{item:rankpos:lem:num_test_prob2}
-		\item $\Delta(u) \geq 0$
-			\label{item:bgmlvu:lem:num_test_prob2}
-		\item $\Delta(v-u) \geq 0$
-			\label{item:bgmlvv-u:lem:num_test_prob2}
-		\item $\mu(u)<\mu(v)$
-			\label{item:mubound:lem:num_test_prob2}
-		\item $0\leq\chern_1^{\beta_{-}}(u)\leq\chern_1^{\beta_{-}}(v)$
-			\label{item:chern1bound:lem:num_test_prob2}
-		\item $\chern_2^{\beta_{-}}(u)>0$
-			\label{item:radiuscond:lem:num_test_prob2}
-	\end{enumerate}
-\end{corollary}
-
-\begin{proof}
-	This is a specialization of the previous lemma, using $P=(\beta_{-},0)$.
-
-\end{proof}
-
-
-\section{B.Schmidt's Solutions to the Problems}
-
-\subsection{Bound on \texorpdfstring{$\chern_0(u)$}{ch0(u)} for Semistabilizers}
-\label{subsect:loose-bound-on-r}
-
-The proof for the following theorem \ref{thm:loose-bound-on-r} was hinted at in
-\cite{SchmidtBenjamin2020Bsot}, but the value appears explicitly in
-\cite{SchmidtGithub2020}. The latter reference is a SageMath \cite{sagemath}
-library for computing certain quantities related to Bridgeland stabilities on
-Picard rank 1 varieties. It also includes functions to compute pseudo-walls and
-pseudo-semistabilizers for tilt stability.
-
-
-\begin{theorem}[Bound on $r$ - Benjamin Schmidt]
-\label{thm:loose-bound-on-r}
-Given a Chern character $v$ such that $\beta_-\coloneqq\beta_{-}(v)\in\QQ$, the rank $r$ of
-any semistabilizer $E$ of some $F \in \firsttilt{\beta_-}$ with $\chern(F)=v$ is
-bounded above by:
-
-\begin{equation*}
-	r \leq \frac{mn^2 \chern^{\beta_-}_1(v)^2}{\gcd(m,2n^2)}
-\end{equation*}
-\end{theorem}
-
-\begin{proof}
-The Bogomolov form applied to the twisted Chern character is the same as the
-normal one. So $0 \leq \Delta(E)$ yields:
-
-\begin{equation}
-	\label{eqn-bgmlv-on-E}
-	2\chern^\beta_0(E) \chern^\beta_2(E) \leq \chern^\beta_1(E)^2
-\end{equation}
-
-\noindent
-Furthermore, $E \hookrightarrow F$ in $\firsttilt{\beta_{-}}$ gives:
-\begin{equation}
-	\label{eqn-tilt-cat-cond}
-	0 \leq \chern^\beta_1(E) \leq \chern^\beta_1(F)
-\end{equation}
-% FUTURE maybe ref this back to some definition of first tilt
-
-\noindent
-The restrictions on $\chern^{\beta_-}_0(E)$ and $\chern^{\beta_-}_2(E)$
-is best seen with the following graph:
-
-% TODO: hyperbola restriction graph (shaded)
-
-
-This is where the rationality of $\beta_{-}$ comes in. If
-$\beta_{-} = \frac{a_v}{n}$ for some $a_v,n \in \ZZ$.
-Then $\chern^{\beta_-}_2(E) \in \frac{1}{\lcm(m,2n^2)}\ZZ$.
-In particular, since $\chern_2^{\beta_-}(E) > 0$ (by using 	$P=(\beta_-,0)$ in
-lemma \ref{lem:pseudo_wall_numerical_tests}) we must also have
-$\chern^{\beta_-}_2(E) \geq \frac{1}{\lcm(m,2n^2)}$, which then in turn gives a
-bound for the rank of $E$:
-
-\begin{align}
-	\chern_0(E) &= \chern^{\beta_-}_0(E) \\
-	&\leq \frac{\lcm(m,2n^2) \chern^{\beta_-}_1(E)^2}{2} \\
-	&= \frac{mn^2 \chern^{\beta_-}_1(F)^2}{\gcd(m,2n^2)}
-\end{align}
-
-\end{proof}
-
-\begin{sagesilent}
-from examples import recurring
-\end{sagesilent}
-
-\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
-\label{exmpl:recurring-first}
-Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
-that $m=1$, $\beta_-=\sage{recurring.betaminus}$,
-giving $n=\sage{recurring.n}$ and
-$\chern_1^{\sage{recurring.betaminus}}(F) = \sage{recurring.twisted.ch[1]}$.
-
-Using the above theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
-tilt semistabilizers for $v$ are bounded above by $\sage{recurring.loose_bound}$.
-However, when computing all tilt semistabilizers for $v$ on $\PP^2$, the maximum
-rank that appears turns out to be 25. This will be a recurring example to
-illustrate the performance of later theorems about rank bounds
-\end{example}
-
-\begin{sagesilent}
-from examples import extravagant
-\end{sagesilent}
-
-\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
-\label{exmpl:extravagant-first}
-Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
-that $m=1$, $\beta_-=\sage{extravagant.betaminus}$,
-giving $n=\sage{extravagant.n}$ and
-$\chern_1^{\sage{extravagant.betaminus}}(F) = \sage{extravagant.twisted.ch[1]}$.
-
-Using the above theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
-tilt semistabilizers for $v$ are bounded above by $\sage{extravagant.loose_bound}$.
-However, when computing all tilt semistabilizers for $v$ on $\PP^2$, the maximum
-rank that appears turns out to be $\sage{extravagant.actual_rmax}$.
-\end{example}
-
-\subsection{Pseudo-Wall Finding Method}
-
-The SageMath Library \cite{SchmidtGithub2020} provides a function which
-calculates all solutions to problems \ref{problem:problem-statement-1}
-or \ref{problem:problem-statement-2}.
-Here is an outline of the algorithm involved to do this. Simplifications will be
-made in the presentation to concentrate on the case we are interested in:
-problem \ref{problem:problem-statement-2}, finding all pseudo-walls when $\beta_{-}\in\QQ$.
-% FUTURE add reference to section explaining new alg
-In section [ref], a different
-algorithm will be presented making use of the later theorems in this article,
-with the goal of cutting down the run time.
-
-\subsubsection{Finding possible \texorpdfstring{$r$}{r} and
-\texorpdfstring{$c$}{c}}
-To do this, first calculate the upper bound $r_{max}$ on the ranks of tilt
-semistabilizers, as given by theorem \ref{thm:loose-bound-on-r}.
-
-Recalling consequence 2 of lemma \ref{lem:pseudo_wall_numerical_tests}, we can
-iterate through the possible values of $\mu(u)=\frac{c}{r}$ taking a decreasing
-sequence of all fractions between $\mu(v)$ and $\beta_{-}$, who's denominators
-are no large than $r_{max}$ (giving a finite sequence). This can be done with
-Farey sequences \cite[chapter 6]{alma994504533502466}, for which there exist
-formulae to generate.
-
-These $\mu(u)$ values determine pairs $r,c$ up to multiples, we can then take
-all multiples which satisy $0<r\leq r_{max}$.
-
-We now have a finite sequence of pairs $r,c$ for which there might be a solution
-$(r,c\ell,d\ell^2)$ to our problem. In particular, any $(r,c\ell,d\ell^2)$
-satisfies consequence 2 of lemma \ref{lem:pseudo_wall_numerical_tests}, and the
-positive rank condition. What remains is to find the $d$ values which satisfy
-the Bogomolov inequalities and consequence 3 of lemma
-\ref{lem:pseudo_wall_numerical_tests}
-($\chern_2^{\beta_{-}}(u)>0$).
-
-\subsubsection{Finding \texorpdfstring{$d$}{d} for fixed \texorpdfstring{$r$}{r}
-and \texorpdfstring{$c$}{c}}
-
-$\Delta(u) \geq 0$ induces an upper bound $\frac{c^2}{2r}$ on $d$, and the
-$\chern_2^{\beta_{-}}(u)>0$ condition induces a lower bound on $d$.
-The values in the range can be tested individually, to check that
-the rest of the conditions are satisfied.
-
-\subsection{Limitations}
-
-The main downside of this algorithm is that many $r$,$c$ pairs which are tested
-end up not yielding any solutions for the problem.
-In fact, solutions $u$ to our problem with high rank must have $\mu(u)$ close to
-$\beta_{-}$:
-\begin{align*}
-	0 &\leq \chern_1^{\beta_{-}}(u) \leq \chern_1^{\beta_{-}}(u) \\
-	0 &\leq \mu(u) - \beta_{-} \leq \frac{\chern_1^{\beta_{-}}(v)}{r}
-\end{align*}
-In particular, it's the $\chern_1^{\beta_{-}}(v-u) \geq 0$ conditions which
-fails for $r$,$c$ pairs with large $r$ and $\frac{c}{r}$ too far from $\beta_{-}$.
-This condition is only checked within the internal loop.
-This, along with a conservative estimate for a bound on the $r$ values (as
-illustrated in example \ref{exmpl:recurring-first}) occasionally leads to slow
-computations.
-
-Here are some benchmarks to illustrate the performance benefits of the
-alternative algorithm which will later be described in this article [ref].
-
-\begin{center}
-\begin{tabular}{ |r|l|l| }
- \hline
- Choice of $v$ on $\mathbb{P}^2$
- & $(3, 2\ell, -2)$
- & $(3, 2\ell, -\frac{15}{2})$ \\
- \hline
- \cite[\texttt{tilt.walls_left}]{SchmidtGithub2020} exec time & \sim 20s & >1hr \\
- \cite{NaylorRust2023} exec time & \sim 50ms & \sim 50ms \\
- \hline
-\end{tabular}
-\end{center}
-
-\section{Tighter Bounds}
-\label{sec:refinement}
-
-To get tighter bounds on the rank of destabilizers $E$ 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
-was implicitly happening before).
-
-
-First, let us fix a Chern character for $F$, and some pseudo-semistabilizer
-$u$ which is a solution to 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
-\ref{problem:problem-statement-1}
-(or $\beta = \beta_{-}$ 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 $D\in \frac{1}{\lcm(m,2)}\ZZ$}
-	\\
-	u \coloneqq& \:(r,c\ell,d\ell^2)
-	&& \text{where $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$}
-\end{align}
- 
-
-
-Recall from condition \ref{item:chern1bound:lem:num_test_prob1} in
-lemma \ref{lem:num_test_prob1}
-(or corollary \ref{cor:num_test_prob2})
-that $\chern_1^{\beta}(u)$ has fixed bounds in terms of $\chern_1^{\beta}(v)$,
-and so we can write:
-
-
-
-\begin{sagesilent}
-from plots_and_expressions import c_in_terms_of_q	
-\end{sagesilent}
-
-\begin{equation}
-	\label{eqn-cintermsofm}
-	c=\chern_1(u) = \sage{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
-$q \in \frac{1}{n} \ZZ \cap [0, \chern_1^{\beta}(F)]$,
-where $n$ is the denominator of $\beta$.
-For the next subsections, we consider $q$ to be fixed with one of these values,
-and we shall be varying $\chern_0(E) = r$ to see when certain inequalities fail.
-
-
-\subsection{Numerical Inequalities}
-
-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})
-
-\subsubsection{Size of pseudo-wall\texorpdfstring{: $\chern_2^P(u)>0$}{}}
-\label{subsect-d-bound-radiuscond}
-
-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:
-
-\begin{align}
-\label{eqn:radius-cond-betamin}
-	\chern_2^{\beta_{-}}(u) &> 0 \\
-	d &> \beta_{-}q + \frac{1}{2} \beta_{-}^2r
-\end{align}
-
-\begin{sagesilent}
-import other_P_choice as problem1
-\end{sagesilent}
-
-In the case where we are tackling problem \ref{problem:problem-statement-1},
-with some Chern character $v$ with positive rank, and some choice of point
-$P=(A,B) \in \Theta_v^-$.
-Then $\sage{problem1.A2_subs}$ follows from $\chern_2^P(v)=0$. Using this substitution into the
-condition $\chern_2^P(u)>0$ yields:
-
-\begin{equation}
-	\sage{problem1.radius_condition}
-\end{equation}
-
-\noindent
-Expressing this as a bound on $d$, then yields:
-
-\begin{equation}
-	\sage{problem1.radius_condition_d_bound}
-\end{equation}
-
-
-\subsubsection{Semistability of the Semistabilizer:
-	\texorpdfstring{
-		$\Delta(u) \geq 0$
-	}{
-		Δ(u) ≥ 0
-	}
-}
-This condition refers to condition
-\ref{item:bgmlvu:lem:num_test_prob1}
-from lemma \ref{lem:num_test_prob1}
-(or corollary \ref{cor:num_test_prob2}).
-
-
-\noindent
-Expressing $\Delta(u)\geq 0$ in terms of $q$ as defined in eqn \ref{eqn-cintermsofm}
-we get the following:
-
-
-\begin{sagesilent}
-from plots_and_expressions import bgmlv2_with_q
-\end{sagesilent}
-
-\begin{equation}
-	\sage{bgmlv2_with_q}
-\end{equation}
-
-
-\noindent
-This can be rearranged to express a bound on $d$ as follows
-(recall from condition \ref{item:rankpos:lem:num_test_prob1}
-in lemma \ref{lem:num_test_prob1} or corollary
-\ref{cor:num_test_prob2} that $r>0$):
-
-
-\begin{sagesilent}
-from plots_and_expressions import bgmlv2_d_ineq
-\end{sagesilent}
-\begin{equation}
-	\label{eqn-bgmlv2_d_upperbound}
-	\sage{bgmlv2_d_ineq}
-\end{equation}
-
-\begin{sagesilent}
-from plots_and_expressions import bgmlv2_d_upperbound_terms
-\end{sagesilent}
-Viewing equation \ref{eqn-bgmlv2_d_upperbound} as a lower bound for $d$ in term
-of $r$ again, there is a constant term
-$\sage{bgmlv2_d_upperbound_terms.const}$,
-a linear term
-$\sage{bgmlv2_d_upperbound_terms.linear}$,
-and a hyperbolic term
-$\sage{bgmlv2_d_upperbound_terms.hyperbolic}$.
-Notice that in the context of problem \ref{problem:problem-statement-2}
-($\beta = \beta_{-}$),
-the constant and linear terms match up with the ones
-for the bound found for $d$ in subsubsection \ref{subsect-d-bound-radiuscond}.
-
-\subsubsection{Semistability of the Quotient:
-	\texorpdfstring{
-		$\Delta(v-u) \geq 0$
-	}{
-		Δ(v-u) ≥ 0
-	}
-}
-\label{subsect-d-bound-bgmlv3}
-
-This condition refers to condition
-\ref{item:bgmlvv-u:lem:num_test_prob1}
-from lemma \ref{lem:num_test_prob1}
-(or corollary \ref{cor:num_test_prob2}).
-
-Expressing $\Delta(v-u)\geq 0$ in term of $q$ and rearranging as a bound on
-$d$ yields:
-
-
-\begin{sagesilent}
-from plots_and_expressions import bgmlv3_d_upperbound_terms
-\end{sagesilent}
-
-\begin{equation*}
-	\label{eqn-bgmlv3_d_upperbound}
-	d \leq
-	\sage{bgmlv3_d_upperbound_terms.linear}
-	+ \sage{bgmlv3_d_upperbound_terms.const}
-	+ \sage{bgmlv3_d_upperbound_terms.hyperbolic}
-	\qquad
-	\text{where }r>R
-\end{equation*}
-
-
-\noindent
-For $r=R$, $\Delta(v-u)\geq 0$ is always true, and for $r<R$ it gives a lower
-bound on $d$, but it is weaker than the one given by the lower bound in
-subsubsection \ref{subsect-d-bound-radiuscond}.
-Viewing the right hand side of equation \ref{eqn-bgmlv3_d_upperbound}
-as a function of $r$, the linear and constant terms almost match up with the
-ones in the previous section, up to the 
-$\chern_2^{\beta}(v)$ term.
-
-
-However, when specializing to problem \ref{problem:problem-statement-2} again
-(with $\beta = \beta_{-}$), then we have $\chern^{\beta}_2(v) = 0$.
-And so in this context, the linear and constant terms do match up with the
-previous subsubsections.
-
-\subsubsection{All Bounds on \texorpdfstring{$d$}{d} Together for Problem
-\texorpdfstring{\ref{problem:problem-statement-2}}{2}}
-\label{subsubsect:all-bounds-on-d-prob2}
-%% RECAP ON INEQUALITIES TOGETHER
-
-%%%% RATIONAL BETA MINUS
-As mentioned in passing, when specializing to solutions $u$ of problem
-\ref{problem:problem-statement-2}, the bounds on
-$d=\chern_2(u)$ induced by conditions
-\ref{item:bgmlvu:lem:num_test_prob2},
-\ref{item:bgmlvv-u:lem:num_test_prob2}, and
-\ref{item:radiuscond:lem:num_test_prob1}
-from corollary \ref{cor:num_test_prob2} have the same constant and linear
-terms in $r$, but different hyperbolic terms.
-These give bounds with the same assymptotes when we take $r\to\infty$
-(for any fixed $q=\chern_1^{\beta_{-}}(u)$).
-
-% redefine \beta (especially coming from rendered SageMath expressions)
-% to be \beta_{-} for the rest of this subsubsection
-\bgroup
-
-\let\originalbeta\beta
-\renewcommand\beta{{\originalbeta_{-}}}
-
-\begin{align}
-	d &>&
-	\frac{1}{2}\beta^2 r
-	&+ \beta q,
-	\phantom{+}& % to keep terms aligned
-	 &\qquad\text{when\:} r > 0
-	\label{eqn:radiuscond_d_bound_betamin}
-\\
-	d &\leq&
-	\sage{bgmlv2_d_upperbound_terms.problem2.linear}
-	&+ \sage{bgmlv2_d_upperbound_terms.problem2.const}
-	+& \sage{bgmlv2_d_upperbound_terms.problem2.hyperbolic},
-	 &\qquad\text{when\:} r > 0
-	 \label{eqn:bgmlv2_d_bound_betamin}
-\\
-	d &\leq&
-	\sage{bgmlv3_d_upperbound_terms.problem2.linear}
-	&+ \sage{bgmlv3_d_upperbound_terms.problem2.const}
-	% ^ ch_2^\beta(F)=0 for beta_{-}
-	+& \sage{bgmlv3_d_upperbound_terms.problem2.hyperbolic},
-	 &\qquad\text{when\:} r > R
-	 \label{eqn:bgmlv3_d_bound_betamin}
-\end{align}
-
-
-\begin{sagesilent}
-from plots_and_expressions import \
-bounds_on_d_qmin, \
-bounds_on_d_qmax
-\end{sagesilent}
-
-\begin{figure}
-\centering
-\begin{subfigure}{.45\textwidth}
-  \centering
-	\sageplot[width=\linewidth]{bounds_on_d_qmin}
-	\caption{$q = 0$ (all bounds other than green coincide on line)}
-  \label{fig:d_bounds_xmpl_min_q}
-\end{subfigure}%
-\hfill
-\begin{subfigure}{.45\textwidth}
-  \centering
-	\sageplot[width=\linewidth]{bounds_on_d_qmax}
-	\caption{$q = \chern^{\beta}(F)$ (all bounds other than blue coincide on line)}
-  \label{fig:d_bounds_xmpl_max_q}
-\end{subfigure}
-\caption{
-	Bounds on $d\coloneqq\chern_2(E)$ in terms of $r\coloneqq\chern_0(E)$ for fixed, extreme,
-	values of $q\coloneqq\chern_1^{\beta}(E)$.
-	Where $\chern(F) = (3,2,-2)$.
-}
-\label{fig:d_bounds_xmpl_extrm_q}
-\end{figure}
-
-Recalling that $q \coloneqq \chern^{\beta}_1(E) \in [0, \chern^{\beta}_1(F)]$,
-it is 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
-$\chern_0(E)=\vcentcolon r>R\coloneqq\chern_0(F)$ 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: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:radiuscond_d_bound_betamin}) which match.
-
-
-The more generic case, when $0 < q\coloneqq\chern_1^{\beta}(E) < \chern_1^{\beta}(F)$
-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
-$(r,d) \in \ZZ \oplus \frac{1}{\lcm(m,2)} \ZZ$ (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
-solutions is entirely determined by whether $\beta$ is rational or irrational.
-Some of the details around the associated numerics are explored next.
-
-\begin{sagesilent}
-from plots_and_expressions import typical_bounds_on_d
-\end{sagesilent}
-
-\begin{figure}
-\centering
-\sageplot[width=\linewidth]{typical_bounds_on_d}
-\caption{
-	Bounds on $d\coloneqq\chern_2(E)$ in terms of $r\coloneqq \chern_0(E)$ for a fixed
-	value $\chern_1^{\beta}(F)/2$ of $q\coloneqq\chern_1^{\beta}(E)$.
-	Where $\chern(F) = (3,2,-2)$.
-}
-\label{fig:d_bounds_xmpl_gnrc_q}
-\end{figure}
-
-\subsubsection{All Bounds on \texorpdfstring{$d$}{d} Together for Problem
-\texorpdfstring{\ref{problem:problem-statement-1}}{1}}
-\label{subsubsect:all-bounds-on-d-prob1}
-
-Unlike for problem \ref{problem:problem-statement-2},
-the bounds on $d=\chern_2(u)$ induced by conditions
-\ref{item:bgmlvu:lem:num_test_prob2},
-\ref{item:bgmlvv-u:lem:num_test_prob2}, and
-\ref{item:radiuscond:lem:num_test_prob1}
-from corollary \ref{cor:num_test_prob2} have different
-constant and linear terms, so that the graphs for upper
-bounds do not share the same assymptote as the lower bound
-(and they will turn out to intersect).
-
-\begin{align}
-	\sage{problem1.radius_condition_d_bound.lhs()}
-	&>
-	\sage{problem1.radius_condition_d_bound.rhs()}
-	&\text{where }r>0
-	\label{eqn:prob1:radiuscond}
-	\\
-	d &\leq
-	\sage{problem1.bgmlv2_d_upperbound_terms.linear}
-	+ \sage{problem1.bgmlv2_d_upperbound_terms.const}
-	+ \sage{problem1.bgmlv2_d_upperbound_terms.hyperbolic}
-	&\text{where }r>R
-	\label{eqn:prob1:bgmlv2}
-	\\
-	d &\leq
-	\sage{problem1.bgmlv3_d_upperbound_terms.linear}
-	+ \sage{problem1.bgmlv3_d_upperbound_terms.const}
-	+ \sage{problem1.bgmlv3_d_upperbound_terms.hyperbolic}
-	&\text{where }r>R
-	\label{eqn:prob1:bgmlv3}
-\end{align}
-
-Notice that as a function in $r$, the linear term in 
-equation \ref{eqn:prob1:radiuscond} is strictly greater than
-those in equations \ref{eqn:prob1:bgmlv2}
-and \ref{eqn:prob1:bgmlv3}. This is because $r$, $R$
-and $\chern_2^B(v)$ are all strictly positive:
-\begin{itemize}
-	\item $R > 0$ from the setting of problem
-	\ref{problem:problem-statement-1}
-	\item $r > 0$ from lemma \ref{lem:num_test_prob1}
-	\item $\chern_2^B(v)>0$ because $B < \originalbeta_{-}$ due to the choice of $P$ being
-	a point on $\Theta_v^{-}$
-\end{itemize}
-
-This means that the lower bound for $d$ will be large than either of the two
-upper bounds for sufficiently large $r$, and hence those values of $r$ would yield no 
-solution to problem \ref{problem:problem-statement-1}.
-
-A generic example of this is plotted in figure
-\ref{fig:problem1:d_bounds_xmpl_gnrc_q}.
-
-\begin{figure}
-\centering
-\sageplot[width=\linewidth]{problem1.example_plot}
-\caption{
-	Bounds on $d\coloneqq\chern_2(E)$ in terms of $r\coloneqq \chern_0(E)$ for a fixed
-	value $\chern_1^{\beta}(F)/2$ of $q\coloneqq\chern_1^{\beta}(E)$.
-	Where $\chern(F) = (3,2,-2)$ and $P$ chosen as the point on $\Theta_v$
-	with $B\coloneqq-2/3-1/99$ in the context of problem 
-	\ref{problem:problem-statement-1}.
-}
-\label{fig:problem1:d_bounds_xmpl_gnrc_q}
-\end{figure}
-
-\subsection{Bounds on Semistabilizer Rank \texorpdfstring{$r$}{} in Problem
-\ref{problem:problem-statement-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}),
-there are no solutions $u$ to problem \ref{problem:problem-statement-1}
-with large $r=\chern_0(u)$, since the lower bound on $d=\chern_2(u)$ is larger
-than the upper bounds.
-Therefore, we can calculate upper bounds on $r$ by calculating for which values,
-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 $r$]
-\label{lem:prob1:r_bound}
-	Let $u$ be a solution to problem \ref{problem:problem-statement-1}
-	and $q\coloneqq\chern_1^{B}(u)$.
-	Then $r\coloneqq\chern_0(u)$ is bounded above by the following expression:
-	\begin{equation}
-		\sage{problem1.r_bound_expression}
-	\end{equation}
-\end{lemma}
-
-\begin{proof}
-	Recall that $d\coloneqq\chern_2(u)$ has two upper bounds in terms of $r$: in
-	equations \ref{eqn:prob1:bgmlv2} and \ref{eqn:prob1:bgmlv3};
-	and one lower bound: in equation \ref{eqn:prob1:radiuscond}.
-
-	Solving for the lower bound in equation \ref{eqn:prob1:radiuscond} being
-	less than the upper bound in equation \ref{eqn:prob1:bgmlv2} yields:
-	\begin{equation}
-	r<\sage{problem1.positive_intersection_bgmlv2}
-	\end{equation}
-
-	Similarly, but with the upper bound in equation \ref{eqn:prob1:bgmlv3}, gives:
-	\begin{equation}
-	r<\sage{problem1.positive_intersection_bgmlv3}
-	\end{equation}
-
-	Therefore, $r$ is bounded above by the minimum of these two expressions which
-	can then be factored into the expression given in the lemma.
-	
-\end{proof}
-
-The above lemma \ref{lem:prob1:r_bound} gives an upper bound on $r$ in terms of $q$.
-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}
-\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}
-
-\begin{proof}
-	The first term of the minimum in lemma \ref{lem:prob1:r_bound}
-	increases linearly in $q$, and the second
-	decreases linearly. So the maximum is achieved with the value of
-	$q=q_{\mathrm{max}}$ where they are equal.
-	Solving for the two terms in the minimum to be equal yields:
-	$q_{\mathrm{max}}=\sage{problem1.maximising_q}$.
-	Substituting $q=q_{\mathrm{max}}$ into the bound in lemma
-	\ref{lem:prob1:r_bound} gives the bound as stated in the current lemma.
-	
-\end{proof}
-
-\begin{note}
-	$q_{\mathrm{max}} > 0$ 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}
-
-
-\subsection{Bounds on Semistabilizer Rank \texorpdfstring{$r$}{} in Problem
-\ref{problem:problem-statement-2}}
-
-Now, the inequalities from the above subsubsection
-\ref{subsubsect:all-bounds-on-d-prob2} will be used to find, for
-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
-\ref{thm:loose-bound-on-r}.
-
-
-\renewcommand{\aa}{{a_v}}
-\newcommand{\bb}{{b_q}}
-Suppose $\beta = \frac{\aa}{n}$ for some coprime $n \in \NN,\aa \in \ZZ$.
-Then fix a value of $q$:
-\begin{equation}
-	q\coloneqq \chern_1^{\beta}(E)
-	  =\frac{\bb}{n}
-	\in
-	\frac{1}{n} \ZZ
-	\cap [0, \chern_1^{\beta}(F)]
-\end{equation}
-as noted at the beginning of this section \ref{sec:refinement} so that we are
-considering $u$ which satisfy \ref{item:chern1bound:lem:num_test_prob2}
-in corollary \ref{cor:num_test_prob2}.
-
-Substituting the current values of $q$ and $\beta$ into the condition for the
-radius of the pseudo-wall being positive
-(eqn \ref{eqn:radiuscond_d_bound_betamin}) we get:
-
-\begin{sagesilent}
-from plots_and_expressions import \
-positive_radius_condition_with_q, \
-q_value_expr, \
-beta_value_expr
-\end{sagesilent}
-\begin{equation}
-\label{eqn:positive_rad_condition_in_terms_of_q_beta}
-	\frac{1}{\lcm(m,2)}\ZZ
-	\ni
-	\qquad
-	\sage{positive_radius_condition_with_q.subs([q_value_expr,beta_value_expr]).factor()}
-	\qquad
-	\in
-	\frac{1}{2n^2}\ZZ
-\end{equation}
-
-
-\begin{sagesilent}
-from plots_and_expressions import main_theorem1
-\end{sagesilent}
-\begin{theorem}[Bound on $r$ \#1]
-\label{thm:rmax_with_uniform_eps}
-	Let $v = (R,C\ell,D\ell^2)$ be a fixed Chern character. Then the ranks of the
-	pseudo-semistabilizers for $v$,
-	which are solutions to problem \ref{problem:problem-statement-2},
-	with $\chern_1^\beta = q$
-	are bounded above by the following expression.
-
-	\begin{align*}
-		\min
-		\left(
-			\sage{main_theorem1.r_upper_bound1}, \:\:
-			\sage{main_theorem1.r_upper_bound2}
-		\right)
-	\end{align*}
-
-	Taking the maximum of this expression over
-	$q \in [0, \chern_1^{\beta}(F)]\cap \frac{1}{n}\ZZ$
-	would give an upper bound for the ranks of all solutions to problem
-	\ref{problem:problem-statement-2}.
-\end{theorem}
-
-\begin{proof}
-
-\noindent
-Both $d$ and the lower bound in
-(eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta})
-are elements of $\frac{1}{\lcm(m,2n^2)}\ZZ$.
-So, if any of the two upper bounds on $d$ come to within
-$\frac{1}{\lcm(m,2n^2)}$ of this lower bound, then there are no solutions for
-$d$.
-Hence any corresponding $r$ cannot be a rank of a
-pseudo-semistabilizer for $v$.
-
-To avoid this, we must have,
-considering equations
-\ref{eqn:bgmlv2_d_bound_betamin},
-\ref{eqn:bgmlv3_d_bound_betamin},
-\ref{eqn:radiuscond_d_bound_betamin}.
-
-\begin{sagesilent}
-from plots_and_expressions import \
-assymptote_gap_condition1, assymptote_gap_condition2, k
-\end{sagesilent}
-
-
-\begin{align}
-	&\sage{assymptote_gap_condition1.subs(k==1)} \\
-	&\sage{assymptote_gap_condition2.subs(k==1)}
-\end{align}
-
-\noindent
-This is equivalent to:
-
-\begin{equation}
-	\label{eqn:thm-bound-for-r-impossible-cond-for-r}
-	r \leq
-	\min\left(
-		\sage{
-			main_theorem1.r_upper_bound1
-		} ,
-		\sage{
-			main_theorem1.r_upper_bound2
-		}
-	\right)
-\end{equation}
-
-\end{proof}
-
-
-\begin{sagesilent}
-from plots_and_expressions import q_sol, bgmlv_v, psi
-\end{sagesilent}
-
-\begin{corollary}[Bound on $r$ \#2]
-\label{cor:direct_rmax_with_uniform_eps}
-	Let $v$ be a fixed Chern character and
-	$R\coloneqq\chern_0(v) \leq \frac{1}{2}\lcm(m,2n^2){\chern_1^{\beta}(F)}^2$.
-	Then the ranks of the pseudo-semistabilizers for $v$,
-	which are solutions to problem \ref{problem:problem-statement-2},
-	are bounded above by the following expression.
-
-	\begin{equation*}
-		\sage{main_theorem1.corollary_r_bound}
-	\end{equation*}
-\end{corollary}
-
-\begin{proof}
-The ranks of the pseudo-semistabilizers for $v$ are bounded above by the
-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*}
-	f_1(q)\coloneqq\sage{main_theorem1.r_upper_bound1} \qquad
-	f_2(q)\coloneqq\sage{main_theorem1.r_upper_bound2}
-\end{equation*}
-These have their minimums at $q=0$ and $q=\chern_1^{\beta}(F)$ respectively.
-It suffices to find their intersection in
-$q\in [0, \chern_1^{\beta}(F)]$, if it exists,
-and evaluating on of the $f_i$ there.
-The intersection exists, provided that
-$f_1(\chern_1^{\beta}(F)) \geq f_2(\chern_1^{\beta}(F))=R$,
-or equivalently,
-$R \leq \frac{1}{2}\lcm(m,2n^2){\chern_1^{\beta}(F)}^2$.
-Solving for $f_1(q)=f_2(q)$ yields
-\begin{equation*}
-	q=\sage{q_sol.expand()}
-\end{equation*}
-And evaluating $f_1$ at this $q$-value gives:
-\begin{equation*}
-	\sage{main_theorem1.corollary_intermediate}
-\end{equation*}
-Finally, noting that $\Delta(v)=\psi^2\ell^2$, we get the bound as
-stated in the corollary.
-
-\end{proof}
-
-\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
-\label{exmpl:recurring-second}
-Just like in example \ref{exmpl:recurring-first}, take
-$\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
-that $m=2$, $\beta=\sage{recurring.betaminus}$,
-giving $n=\sage{recurring.n}$.
-
-Using the above corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
-the ranks of tilt semistabilizers for $v$ are bounded above by
-$\sage{recurring.corrolary_bound} \approx  \sage{float(recurring.corrolary_bound)}$,
-which is much closer to real maximum 25 than the original bound 144.
-\end{example}
-
-\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
-\label{exmpl:extravagant-second}
-Just like in example \ref{exmpl:extravagant-first}, take
-$\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
-that $m=2$, $\beta=\sage{extravagant.betaminus}$,
-giving $n=\sage{extravagant.n}$.
-
-Using the above corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
-the ranks of tilt semistabilizers for $v$ are bounded above by
-$\sage{extravagant.corrolary_bound} \approx  \sage{float(extravagant.corrolary_bound)}$,
-which is much closer to real maximum $\sage{extravagant.actual_rmax}$ than the
-original bound 215296.
-\end{example}
-%% refinements using specific values of q and beta
-
-These bound can be refined a bit more by considering restrictions from the
-possible values that $r$ take.
-Furthermore, the proof of theorem \ref{thm:rmax_with_uniform_eps} uses the fact
-that, given an element of $\frac{1}{2n^2}\ZZ$, the closest non-equal element of
-$\frac{1}{2}\ZZ$ is at least $\frac{1}{2n^2}$ away. However this a
-conservative estimate, and a larger gap can sometimes be guaranteed if we know
-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
-`clean' formula for a bound on the ranks of the pseudo-semistabilizers, but has a
-purpose in the context of writing a computer program to find
-pseudo-semistabilizers. Such a program would iterate through possible values of
-$q$, then iterate through values of $r$ within the bounds (dependent on $q$),
-which would then determine $c$, and then find the corresponding possible values
-for $d$.
-
-
-Firstly, we only need to consider $r$-values for which $c\coloneqq\chern_1(E)$ is
-integral:
-
-\begin{equation}
-	c =
-	\sage{c_in_terms_of_q.subs([q_value_expr,beta_value_expr])}
-	\in \ZZ
-\end{equation}
-
-\noindent
-That is, $r \equiv -\aa^{-1}\bb$ mod $n$ ($\aa$ is coprime to
-$n$, and so invertible mod $n$).
-
-
-\noindent
-Let $\aa^{'}$ be an integer representative of $\aa^{-1}$ in $\ZZ/n\ZZ$.
-
-Next, we seek to find a larger $\epsilon$ to use in place of $\epsilon_F$ in the
-proof of theorem \ref{thm:rmax_with_uniform_eps}:
-
-\begin{lemmadfn}[
-	Finding a better alternative to $\epsilon_v$:
-	$\epsilon_{v,q}$
-	]
-	\label{lemdfn:epsilon_q}
-	Suppose $d \in \frac{1}{\lcm(m,2n^2)}\ZZ$ satisfies the condition in
-	eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta}.
-	That is:
-
-	\begin{equation*}
-		\sage{positive_radius_condition_with_q.subs([q_value_expr,beta_value_expr]).factor()}
-	\end{equation*}
-
-	\noindent
-	Then we have:
-
-	\begin{equation}
-		\label{eqn:epsilon_q_lemma_prop}
-		d - \frac{(\aa r + 2\bb)\aa}{2n^2}
-		\geq \epsilon_{v,q} \geq \epsilon_v > 0
-	\end{equation}
-
-	\noindent
-	Where $\epsilon_{v,q}$ is defined as follows:
-
-	\begin{equation*}
-		\epsilon_{v,q} \coloneqq
-		\frac{k_{q}}{\lcm(m,2n^2)}
-	\end{equation*}
-	with $k_{v,q}$ being the least $k\in\ZZ_{>0}$ satisfying
-	\begin{equation*}
-		k \equiv -\aa\bb \frac{m}{\gcd(m,2n^2)}
-		\mod{\gcd\left(
-			\frac{n^2\gcd(m,2)}{\gcd(m,2n^2)},
-			\frac{mn\aa}{\gcd(m,2n^2)}
-		\right)}
-	\end{equation*}
-	
-\end{lemmadfn}
-
-\vspace{10pt}
-
-\begin{proof}
-
-Consider the following sequence of logical implications.
-The one-way implication follows from
-$\aa r + \bb \equiv 0 \pmod{n}$,
-and the final logical equivalence is just a simplification of the expressions.
-
-\begin{align}
-	\frac{ x }{ \lcm(m,2) }
-	- \frac{
-		(\aa r+2\bb)\aa
-	}{
-		2n^2
-	}
-	= \frac{ k }{ \lcm(m,2n^2) }
-	\quad \text{for some } x \in \ZZ
-	\span \span \span \span \span
-	\label{eqn:finding_better_eps_problem}
-\\ \nonumber
-\\ \Leftrightarrow& &
-	- (\aa r+2\bb)\aa
-	\frac{\lcm(m,2n^2)}{2n^2}
-	&\equiv k &&
-	\nonumber
-\\ &&&
-	\mod \frac{\lcm(m,2n^2)}{\lcm(m,2)}
-	\span \span \span
-	\nonumber
-\\ \Rightarrow& &
-	- \bb\aa
-	\frac{\lcm(m,2n^2)}{2n^2}
-	&\equiv k &&
-	\nonumber
-\\ &&&
-	\mod \gcd\left(
-		\frac{\lcm(m,2n^2)}{\lcm(m,2)},
-		\frac{n \aa \lcm(m,2n^2)}{2n^2}
-	\right)
-	\span \span \span
-	\nonumber
-\\ \Leftrightarrow& &
-	- \bb\aa
-	\frac{m}{\gcd(m,2n^2)}
-	&\equiv k &&
-	\label{eqn:better_eps_problem_k_mod_n}
-\\ &&&
-	\mod \gcd\left(
-		\frac{n^2\gcd(m,2)}{\gcd(m,2n^2)},
-		\frac{mn \aa}{\gcd(m,2n^2)}
-	\right)
-	\span \span \span
-	\nonumber
-\end{align}
-
-In our situation, we want to find the least $k>0$ satisfying 
-eqn \ref{eqn:finding_better_eps_problem}.
-Since such a $k$ must also satisfy eqn \ref{eqn:better_eps_problem_k_mod_n},
-we can pick the smallest $k_{q,v} \in \ZZ_{>0}$ which satisfies this new condition
-(a computation only depending on $q$ and $\beta$, but not $r$).
-We are then guaranteed that $k_{v,q}$ is less than any $k$ satisfying eqn
-\ref{eqn:finding_better_eps_problem}, giving the first inequality in eqn
-\ref{eqn:epsilon_q_lemma_prop}.
-Furthermore, $k_{v,q}\geq 1$ gives the second part of the inequality:
-$\epsilon_{v,q}\geq\epsilon_v$, with equality when $k_{v,q}=1$.
-
-\end{proof}
-
-\begin{sagesilent}
-from plots_and_expressions import main_theorem2
-\end{sagesilent}
-\begin{theorem}[Bound on $r$ \#3]
-\label{thm:rmax_with_eps1}
-	Let $v$ be a fixed Chern character, with $\frac{a_v}{n}=\beta\coloneqq\beta(v)$
-	rational and expressed in lowest terms.
-	Then the ranks $r$ of the pseudo-semistabilizers $u$ for $v$ with,
-	which are solutions to problem \ref{problem:problem-statement-2},
-	$\chern_1^\beta(u) = q = \frac{b_q}{n}$
-	are bounded above by the following expression:
-
-	\begin{align*}
-		\min
-		\left(
-			\sage{main_theorem2.r_upper_bound1}, \:\:
-			\sage{main_theorem2.r_upper_bound2}
-		\right)
-	\end{align*}
-	Where $k_{v,q}$ is defined as in definition/lemma \ref{lemdfn:epsilon_q},
-	and $R = \chern_0(v)$
-
-	Furthermore, if $\aa \not= 0$ then
-	$r \equiv \aa^{-1}b_q \pmod{n}$.
-\end{theorem}
-
-Although the general form of this bound is quite complicated, it does simplify a
-lot when $m$ is small.
-
-\begin{sagesilent}
-from plots_and_expressions import main_theorem2_corollary
-\end{sagesilent}
-\begin{corollary}[Bound on $r$ \#3 on $\PP^2$ and Principally polarized abelian surfaces]
-\label{cor:rmax_with_eps1}
-	Suppose we are working over $\PP^2$ or a principally polarized abelian surface
-	(or any other surfaces with $m=1$ or $2$).
-	Let $v$ be a fixed Chern character, with $\frac{a_v}{n}=\beta\coloneqq\beta(v)$
-	rational and expressed in lowest terms.
-	Then the ranks $r$ of the pseudo-semistabilizers $u$ for $v$ with,
-	which are solutions to problem \ref{problem:problem-statement-2},
-	$\chern_1^\beta(u) = q = \frac{b_q}{n}$
-	are bounded above by the following expression:
-
-	\begin{align*}
-		\min
-		\left(
-			\sage{main_theorem2_corollary.r_upper_bound1}, \:\:
-			\sage{main_theorem2_corollary.r_upper_bound2}
-		\right)
-	\end{align*}
-	Where $R = \chern_0(v)$ and $k_{v,q}$ is the least
-	$k\in\ZZ_{>0}$ satisfying
-	\begin{equation*}
-		k \equiv -\aa\bb
-		\pmod{n}
-	\end{equation*}
-
-	\noindent
-	Furthermore, if $\aa \not= 0$ then
-	$r \equiv \aa^{-1}b_q \pmod{n}$.
-\end{corollary}
-
-\begin{proof}
-This is a specialisation of theorem \ref{thm:rmax_with_eps1}, where we can
-drastically simplify the $\lcm$ and $\gcd$ terms by noting that $m$ divides both
-$2$ and $2n^2$, and that $a_v$ is coprime to $n$.
-\end{proof}
-
-\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
-\label{exmpl:recurring-third}
-Just like in examples \ref{exmpl:recurring-first} and
-\ref{exmpl:recurring-second},
-take $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so that
-$\beta=\sage{recurring.betaminus}$, giving $n=\sage{recurring.n}$
-and $\chern_1^{\sage{recurring.betaminus}}(F) = \sage{recurring.twisted.ch[1]}$.
-%% TODO transcode notebook code
-The (non-exclusive) upper bounds for $r\coloneqq\chern_0(u)$ of a tilt semistabilizer $u$ of $v$
-in terms of the possible values for $q\coloneqq\chern_1^{\beta}(u)$ are as follows:
-
-\begin{sagesilent}
-from examples import bound_comparisons
-qs, theorem2_bounds, theorem3_bounds = bound_comparisons(recurring)
-\end{sagesilent}
-
-\vspace{1em}
-\noindent
-\directlua{ table_width = 3*4+1 }
-\begin{tabular}{l\directlua{for i=0,table_width-1 do tex.sprint([[|c]]) end}}
-	$q=\chern_1^\beta(u)$
-\directlua{for i=0,table_width-1 do
-	local cell = [[&$\noexpand\sage{qs[]] .. i .. "]}$"
-  tex.sprint(cell)
-end}
-	\\ \hline
-	Thm \ref{thm:rmax_with_uniform_eps}
-\directlua{for i=0,table_width-1 do
-	local cell = [[&$\noexpand\sage{theorem2_bounds[]] .. i .. "]}$"
-  tex.sprint(cell)
-end}
-	\\
-	Thm \ref{thm:rmax_with_eps1}
-\directlua{for i=0,table_width-1 do
-	local cell = [[&$\noexpand\sage{theorem3_bounds[]] .. i .. "]}$"
-  tex.sprint(cell)
-end}
-\end{tabular}
-\vspace{1em}
-
-\noindent
-It's worth noting that the bounds given by theorem \ref{thm:rmax_with_eps1}
-reach, but do not exceed the actual maximum rank 25 of the
-pseudo-semistabilizers of $v$ in this case.
-As a reminder, the original loose bound from theorem \ref{thm:loose-bound-on-r}
-was 144.
-
-\end{example}
-
-\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
-\label{exmpl:extravagant-third}
-Just like in examples \ref{exmpl:extravagant-first} and
-\ref{exmpl:extravagant-second},
-take $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so that
-$\beta=\sage{extravagant.betaminus}$, giving $n=\sage{n:=extravagant.n}$
-and $\chern_1^{\sage{extravagant.betaminus}}(F) = \sage{extravagant.twisted.ch[1]}$.
-This example was chosen because the $n$ value is moderatly large, giving more
-possible values for $k_{v,q}$, in dfn/lemma \ref{lemdfn:epsilon_q}. This allows
-for a larger possible difference between the bounds given by theorems
-\ref{thm:rmax_with_uniform_eps} and \ref{thm:rmax_with_eps1}, with the bound
-from the second being up to $\sage{n}$ times smaller, for any given $q$ value.
-The (non-exclusive) upper bounds for $r\coloneqq\chern_0(u)$ of a tilt semistabilizer $u$ of $v$
-in terms of the first few smallest possible values for $q\coloneqq\chern_1^{\beta}(u)$ are as follows:
-
-\begin{sagesilent}
-qs, theorem2_bounds, theorem3_bounds = bound_comparisons(extravagant)
-\end{sagesilent}
-
-
-\vspace{1em}
-\noindent
-\directlua{ table_width = 12 }
-\begin{tabular}{l\directlua{for i=0,table_width do tex.sprint([[|c]]) end}}
-	$q=\chern_1^\beta(u)$
-\directlua{for i=0,table_width-1 do
-	local cell = [[&$\noexpand\sage{qs[]] .. i .. "]}$"
-  tex.sprint(cell)
-end}
-	&$\cdots$
-	\\ \hline
-	Thm \ref{thm:rmax_with_uniform_eps}
-\directlua{for i=0,table_width-1 do
-	local cell = [[&$\noexpand\sage{theorem2_bounds[]] .. i .. "]}$"
-  tex.sprint(cell)
-end}
-	&$\cdots$
-	\\
-	Thm \ref{thm:rmax_with_eps1}
-\directlua{for i=0,table_width-1 do
-	local cell = [[&$\noexpand\sage{theorem3_bounds[]] .. i .. "]}$"
-  tex.sprint(cell)
-end}
-	&$\cdots$
-\end{tabular}
-\vspace{1em}
-
-
-\noindent
-However the reduction in the overall bound on $r$ is not as drastic, since all
-possible values for $k_{v,q}$ in $\{1,2,\ldots,\sage{n}\}$ are iterated through
-cyclically as we consider successive possible values for $q$.
-And for each $q$ where $k_{v,q}=1$, both theorems give the same bound.
-Calculating the maximums over all values of $q$ yields
-$\sage{max(theorem2_bounds)}$ for theorem \ref{thm:rmax_with_uniform_eps}, and
-$\sage{max(theorem3_bounds)}$ for theorem \ref{thm:rmax_with_eps1}.
-\end{example}
-
-\egroup % end scope where beta redefined to beta_{-}
-
-\subsubsection{All Semistabilizers Giving Sufficiently Large Circular Walls Left
-of Vertical Wall}
-
-
-Goals:
-\begin{itemize}
-	\item refresher on strategy
-	\item point out no need for rational beta
-	\item calculate intersection of bounds?
-\end{itemize}
-
-\subsection{Irrational \texorpdfstring{$\beta_{-}$}{êžµ_}}
-
-Goals:
-\begin{itemize}
-	\item Point out if only looking for sufficiently large wall, look at above
-		subsubsection
-	\item Relate to Pell's equation through coordinate change?
-	\item Relate to numerical condition described by Yanagida/Yoshioka
-\end{itemize}
-
-\section{Computing solutions to Problem \ref{problem:problem-statement-2}}
-\label{sect:prob2-algorithm}
-
-Alongside this article, there is a library \cite{NaylorRust2023} to compute
-the solutions to problem \ref{problem:problem-statement-2}, using the theorems
-above.
-
-The way it works, is by yielding solutions to the problem
-$u=(r,c\ell,\frac{e}{2}\ell^2)$ as follows.
-
-\subsection{Iterating Over Possible
-\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}}
-
-Given a Chern character $v$, the domain of the problem are first verified: that
-$v$ has positive rank, that it satisfies $\Delta(v) \geq 0$, and that
-$\beta_{-}(v)$ is rational.
-
-Take $\beta_{-}(v)=\frac{a_v}{n}$ in simplest terms.
-Iterate over $q \in [0,\chern_1^{\beta_{-}}(v)]\cap\frac{1}{n}\ZZ$.
-
-For any $u = (r,c\ell,\frac{e}{2}\ell^2)$, satisfying
-$\chern_1^{\beta_{-}}(u)=q$ for one of the $q$ considered is equivalent to
-satisfying condition \ref{item:chern1bound:lem:num_test_prob2}
-in corollary \ref{cor:num_test_prob2}.
-
-\subsection{Iterating Over Possible
-\texorpdfstring{$r=\chern_0(u)$}{r}
-for Fixed
-\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}
-}
-
-Let $q=\frac{b_q}{n}$ be one of the values of $\chern_1^{\beta_{-}}(u)$ that we
-have fixed. As mentioned before, the only values of $r$ which can
-give $\chern_1^{\beta_{-}}(u)=q$ are precisely the ones which satisfy
-$a_v r \equiv b_q \pmod{n}$.
-This is true for all integers when $\beta_{-}=0$ (and so $n=1$), but otherwise,
-this is equivalent to
-$r \equiv {a_v}^{-1}b_q \pmod{n}$, since $a_v$ and $n$ are coprime.
-
-Note that expressing $\mu(u)$ in term of $q$ and $r$ gives:
-\begin{align*}
-	\mu(u) & = \frac{c}{r} = \frac{q+r\beta_{-}}{r}
-	\\
-	&= \beta_{-} + \frac{q}{r}
-\end{align*}
-
-So condition \ref{item:mubound:lem:num_test_prob2} in corollary
-\ref{cor:num_test_prob2} is satisfied at this point precisely when:
-
-\begin{equation*}
-	r > \frac{q}{\mu(u) - \beta_{-}}
-\end{equation*}
-
-Note that the right hand-side is greater than, or equal, to 0, so such $r$ also
-satisfies \ref{item:rankpos:lem:num_test_prob2}.
-
-Then theorem \ref{thm:rmax_with_eps1} gives an upper on possible $r$ values
-for which it is possible to satisfy conditions
-\ref{item:bgmlvu:lem:num_test_prob2},
-\ref{item:bgmlvv-u:lem:num_test_prob2}, and
-\ref{item:radiuscond:lem:num_test_prob2}.
-
-Iterate over such $r$ so that we are guarenteed to satisfy conditions
-\ref{item:mubound:lem:num_test_prob2}, and
-\ref{item:radiuscond:lem:num_test_prob2}
-in corollary
-\ref{cor:num_test_prob2}, and have a chance at satisfying the rest.
-
-\subsection{Iterating Over Possible
-\texorpdfstring{$d=\chern_2(u)$}{d}
-for Fixed
-\texorpdfstring{$r=\chern_0(u)$}{r}
-and
-\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}
-}
-
-At this point we have fixed $\chern_0(u)=r$ and
-$\chern_1(u)=c=q+r\beta_{-}$.
-And the cases considered are precisely the ones which satisfy conditions
-\ref{item:chern1bound:lem:num_test_prob2},
-\ref{item:mubound:lem:num_test_prob2}, and
-\ref{item:radiuscond:lem:num_test_prob2}
-in corollary \ref{cor:num_test_prob2}.
-
-It remains to find $\chern_2(u)=d=\frac{e}{2}$
-which satisfy the remaining conditions
-\ref{item:bgmlvu:lem:num_test_prob2},
-\ref{item:bgmlvv-u:lem:num_test_prob2}, and
-\ref{item:radiuscond:lem:num_test_prob2}.
-These conditions induce upper and lower bounds on $d$, and it then remains to
-just pick the integers $e$ that give $d$ values within the bounds.
-
-Thus, through this process yielding all solutions $u=(r,c\ell,\frac{e}{2}\ell^2)$
-to the problem for this choice of $v$.
-
 
+\input{content.tex}
 
-\onlyifstandalone{
 \newpage
 \printbibliography
-}
 
-\end{document}
-% comment
+\end{document}
\ No newline at end of file