diff --git a/tex/bounds-on-semistabilisers.tex b/tex/bounds-on-semistabilisers.tex
index a45a5f17302cc21d95bc58b6bc280d3422f83da7..fb4ace89c8e54b455d5f41befdeeac61b1f146ec 100644
--- a/tex/bounds-on-semistabilisers.tex
+++ b/tex/bounds-on-semistabilisers.tex
@@ -94,14 +94,14 @@ pseudo-semistabilisers for tilt stability.
 		\label{proof:first-bound-on-r}
 	\end{align}
 	\noindent
-	Which we can then immediately bound using Equation \ref{eqn-tilt-cat-cond}.
+	Which we can then immediately bound using Equation \eqref{eqn-tilt-cat-cond}.
 	Alternatively, given that
 	$\chern_1^{\beta_{-}}(u)$, $\chern_1^{\beta_{-}}(v)\in\frac{1}{n}\ZZ$,
 	we can tighten this bound on $\chern_1^{\beta_{-}}(u)$ given by that equation to:
 	\[
 		n\chern^{\beta_{-}}_1(u) \leq n\chern^{\beta_{-}}_1(v) - 1
 	\]
-	allowing us to bound the expression in Equation \ref{proof:first-bound-on-r} to
+	allowing us to bound the expression in Equation \eqref{proof:first-bound-on-r} to
 	the following:
 	\[
 		\chern_0(u)
@@ -187,7 +187,7 @@ of travel.
 	\noindent
 	Conversely, any $u = (r,c\ell,d\ell^2)$
 	with $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$
-	satisfying the above Equations \ref{lem:eqn:cond-for-fixed-q}
+	satisfying the above Equations \eqref{lem:eqn:cond-for-fixed-q}
 	is a solution to the Problem if and only if the following are satisfied:
 	\begin{multicols}{3}
 		\begin{itemize}
@@ -202,15 +202,15 @@ of travel.
 	Recalling Theorems \ref{lem:num_test_prob1} and \ref{cor:num_test_prob2}, solutions $u$
 	to the problem are given by $u=(r,c\ell,d\ell^2)$ with $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$
 	which satisfy six numerical conditions.
-	The first line of Equation \ref{lem:eqn:cond-for-fixed-q} is equivalent to
+	The first line of Equation \eqref{lem:eqn:cond-for-fixed-q} is equivalent to
 	numerical condition 5.
 	The second line is a rearrangement of numerical condition 4, assuming $r>0$ which is given by
 	the first numerical condition.
-	Therefore any solution $u$ satisfies Equation \ref{lem:eqn:cond-for-fixed-q}.
+	Therefore any solution $u$ satisfies Equation \eqref{lem:eqn:cond-for-fixed-q}.
 
 	But then Theorems \ref{lem:num_test_prob1} and \ref{cor:num_test_prob2}, also give that
 	$u=(r,c\ell,d\ell^2)$ with $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$ satisfying Equation
-	\ref{lem:eqn:cond-for-fixed-q} is in fact a solution provided the remaining numerical conditions
+	\eqref{lem:eqn:cond-for-fixed-q} is in fact a solution provided the remaining numerical conditions
 	1, 2, 3 and 6 are satisfied.
 	This is in essence the second part of the Lemma.
 \end{proof}
@@ -377,7 +377,7 @@ we get:
 
 \noindent
 Rearranging to express this as a bound on $d$, we get the following.
-Recall that $r>0$ is ensured by Equations \ref{lem:eqn:cond-for-fixed-q}.
+Recall that $r>0$ is ensured by Equations \eqref{lem:eqn:cond-for-fixed-q}.
 
 \begin{sagesilent}
 	from plots_and_expressions import bgmlv2_d_ineq
@@ -416,11 +416,11 @@ $d$ yields:
 
 \noindent
 If $r=R$, then $\Delta(v-u)=(C-c)^2 \geq 0$ is always true, and for $r<R$
-the expression on the right hand side of Equation \ref{eqn-bgmlv3_d_upperbound}
+the expression on the right hand side of Equation \eqref{eqn-bgmlv3_d_upperbound}
 gives a lower bound for $d$ instead.
 However it is weaker than lower bound
 given by $\chern^P_2(u)>0$ if $u$ already satisfies Equations
-\ref{lem:eqn:cond-for-fixed-q} as will be shown now:
+\eqref{lem:eqn:cond-for-fixed-q} as will be shown now:
 
 Since $r, R-r>0$, we have:
 \begin{equation}
@@ -429,7 +429,7 @@ Since $r, R-r>0$, we have:
 \end{equation}
 \noindent
 The first inequality coming from $P \in \Theta_v^{-}$ and Equation
-\ref{lem:eqn:cond-for-fixed-q}, and the second inequality following by the
+\eqref{lem:eqn:cond-for-fixed-q}, and the second inequality following by the
 see-saw principle.
 % TODO maybe cover the see-saw principle
 \begin{align*}
@@ -445,7 +445,7 @@ see-saw principle.
 	\left(
 	\mu(v) - \beta_0
 	\right)^2
-	 & \text{by Equation \ref{lem:proof:slope-order-rltR}}
+	 & \text{by Equation \eqref{lem:proof:slope-order-rltR}}
 	\\
 	 & =
 	\left(
@@ -488,9 +488,9 @@ see-saw principle.
 \end{align*}
 \noindent
 Showing that the unique terms of Equation
-\ref{eqn:radius_condition_d_bound}
+\eqref{eqn:radius_condition_d_bound}
 are greater than those of Equation
-\ref{eqn-bgmlv3_d_upperbound}.
+\eqref{eqn-bgmlv3_d_upperbound}.
 
 
 \subsubsection{All Bounds on \texorpdfstring{$d$}{d} Together for Problem
@@ -503,7 +503,7 @@ bounds on $d$ calculated in this Subsection.
 So we conclude that final 3 conditions from Corollary
 \ref{cor:rational-beta:fixed-q-semistabs-criterion}
 for a potential solution to the problem of the form in Equation
-\ref{eqn:u-coords}, amounts to the following:
+\eqref{eqn:u-coords}, amounts to the following:
 
 \begin{sagesilent}
 	from plots_and_expressions import bgmlv2_d_upperbound_terms
@@ -536,11 +536,11 @@ Recalling that $q \coloneqq \chern^{\beta}_1(u) \in (0, \chern^{\beta}_1(v))$,
 it is worth noting that the extreme values of $q$ in this range lead to the
 tightest bounds on $d$.
 Small values of $q$ brings
-Equation \ref{eqn:bgmlv2_d_bound_betamin} closer to
-Equation \ref{eqn:radiuscond_d_bound_betamin},
+Equation \eqref{eqn:bgmlv2_d_bound_betamin} closer to
+Equation \eqref{eqn:radiuscond_d_bound_betamin},
 and larger values of $q$ brings
-Equation \ref{eqn:bgmlv3_d_bound_betamin} closer to
-Equation \ref{eqn:radiuscond_d_bound_betamin}.
+Equation \eqref{eqn:bgmlv3_d_bound_betamin} closer to
+Equation \eqref{eqn:radiuscond_d_bound_betamin}.
 
 For a generic case, when
 $0 < q\coloneqq\chern_1^{\beta}(u) < \chern_1^{\beta}(v)$,
@@ -552,9 +552,9 @@ $(r,d) \in \ZZ \oplus \frac{1}{\lcm(m,2)} \ZZ$ (with large $r$)
 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 $r \to \infty$
-(Equations \ref{eqn:bgmlv2_d_bound_betamin},
-\ref{eqn:bgmlv3_d_bound_betamin},
-\ref{eqn:radiuscond_d_bound_betamin}).
+(Equations \eqref{eqn:bgmlv2_d_bound_betamin},
+\eqref{eqn:bgmlv3_d_bound_betamin},
+\eqref{eqn:radiuscond_d_bound_betamin}).
 As mentioned in the introduction to this Part, the finiteness of these
 solutions is entirely determined by whether $\beta_{-}$ is rational or irrational.
 This will be pursued in Subsection
@@ -624,9 +624,9 @@ bounds do not share the same assymptote as the lower bound
 
 
 Notice that as functions of $r$, 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} in the context of the Problem.
+Equation \eqref{eqn:prob1:radiuscond} is strictly greater than
+those in Equations \eqref{eqn:prob1:bgmlv2}
+and \eqref{eqn:prob1:bgmlv3} in the context of the Problem.
 This is because $R\coloneqq\chern_0(v)$
 and $\chern_2^{\beta_0}(v)$ are all strictly positive:
 \begin{itemize}
@@ -671,23 +671,23 @@ the lower bound on $d$ is equal to one of the upper bounds on $d$
 
 \begin{proof}
 	Lemma \ref{lem:fixed-q-semistabs-criterion} gives us that any solution $u$
-	must be of the form in Equation \ref{eqn:u-coords} and
-	satisfy Equations \ref{lem:eqn:cond-for-fixed-q} as well as the three
+	must be of the form in Equation \eqref{eqn:u-coords} and
+	satisfy Equations \eqref{lem:eqn:cond-for-fixed-q} as well as the three
 	conditions $\chern^P_2(u)>0$, $\Delta(u) \geq 0$, and $\Delta(v-u) \geq 0$.
 	Subsection \ref{subsec:bounds-on-d} equates these latter three conditions
-	(provided Equations \ref{lem:eqn:cond-for-fixed-q})
+	(provided Equations \eqref{lem:eqn:cond-for-fixed-q})
 	to upper bounds on $d$ given by
-	Equations \ref{eqn:prob1:bgmlv2} and \ref{eqn:prob1:bgmlv3};
-	and one lower bound given by Equation \ref{eqn:prob1:radiuscond}.
+	Equations \eqref{eqn:prob1:bgmlv2} and \eqref{eqn:prob1:bgmlv3};
+	and one lower bound given by Equation \eqref{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:
+	Solving for the lower bound in Equation \eqref{eqn:prob1:radiuscond} being
+	less than the upper bound in Equation \eqref{eqn:prob1:bgmlv2} yields:
 	\begin{equation}
 		r<\sage{problem1.positive_intersection_bgmlv2}
 	\end{equation}
 
 	\noindent
-	Similarly, but with the upper bound in Equation \ref{eqn:prob1:bgmlv3}, gives:
+	Similarly, but with the upper bound in Equation \eqref{eqn:prob1:bgmlv3}, gives:
 	\begin{equation}
 		r<\sage{problem1.positive_intersection_bgmlv3}
 	\end{equation}
@@ -751,7 +751,7 @@ where $\beta_{-}(v) = \frac{a_v}{n}$ in lowest terms and $b_q$ is an integer
 between 1 and $n\chern_1^{\beta_0}(v) - 1$ (inclusive),
 and $a_v r \equiv -b_q \pmod{n}$.
 The Corollary then gives a lower bound for $r$, and states that any $u$ of the
-form from Equation \ref{eqn:u-coords} satisfying these conditions so far, is a
+form from Equation \eqref{eqn:u-coords} satisfying these conditions so far, is a
 solution to Problem \ref{problem:problem-statement-2} if and only if it
 satisfies the conditions
 $\chern^{\beta_{-}(v)}(u)>0$, $\Delta(u) \geq 0$, and $\Delta(v-u) \geq 0$.
@@ -761,7 +761,7 @@ $\chern^{\beta_{-}(v)}(u)>0$, $\Delta(u) \geq 0$, and $\Delta(v-u) \geq 0$.
 \newcommand{\bb}{{b_q}}
 Substituting more specialised values of $q$ and $\beta_0=\beta_{-}(v)$ into the condition
 $\chern^{\beta_0}(u) > 0$
-(Equation \ref{eqn:radiuscond_d_bound_betamin}) we get:
+(Equation \eqref{eqn:radiuscond_d_bound_betamin}) we get:
 
 \begin{sagesilent}
 	from plots_and_expressions import \
@@ -814,7 +814,7 @@ proof of Theorem
 
 \begin{proof}
 	Both $d$ and the lower bound in
-	(Equation \ref{eqn:positive_rad_condition_in_terms_of_q_beta})
+	(Equation \eqref{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
 	$\epsilon_v \coloneqq \frac{1}{\lcm(m,2n^2)}$ of this lower bound,
@@ -824,8 +824,8 @@ proof of Theorem
 
 	To avoid this, we must have,
 	considering Equations
-	\ref{eqn:bgmlv2_d_bound_betamin},
-	\ref{eqn:bgmlv3_d_bound_betamin},
+	\eqref{eqn:bgmlv2_d_bound_betamin},
+	\eqref{eqn:bgmlv3_d_bound_betamin},
 	\ref{eqn:radiuscond_d_bound_betamin}.
 
 	\begin{sagesilent}
@@ -988,7 +988,7 @@ proof of Theorem \ref{thm:rmax_with_uniform_eps}:
 	]
 	\label{lemdfn:epsilon_q}
 	Suppose $d \in \frac{1}{\lcm(m,2n^2)}\ZZ$ satisfies the condition in
-	Equation \ref{eqn:positive_rad_condition_in_terms_of_q_beta}.
+	Equation \eqref{eqn:positive_rad_condition_in_terms_of_q_beta}.
 	That is:
 
 	\begin{equation*}
@@ -1093,13 +1093,13 @@ proof of Theorem \ref{thm:rmax_with_uniform_eps}:
 	\end{align}
 
 	In our situation, we want to find the least $k>0$ satisfying
-	Equation \ref{eqn:finding_better_eps_problem}.
-	Since such a $k$ must also satisfy Equation \ref{eqn:better_eps_problem_k_mod_n},
+	Equation \eqref{eqn:finding_better_eps_problem}.
+	Since such a $k$ must also satisfy Equation \eqref{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 Equation
-	\ref{eqn:finding_better_eps_problem}, giving the first inequality in Equation
-	\ref{eqn:epsilon_q_lemma_prop}.
+	\eqref{eqn:finding_better_eps_problem}, giving the first inequality in Equation
+	\eqref{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 $k_{v,q}=1$.