diff --git a/tex/bounds-on-semistabilisers.tex b/tex/bounds-on-semistabilisers.tex
index 6b3ec6514aaf2e140e8e28585f65a728600e1a66..d4a980357ac1caaa729b009379b9ab08829a0b8a 100644
--- a/tex/bounds-on-semistabilisers.tex
+++ b/tex/bounds-on-semistabilisers.tex
@@ -160,6 +160,10 @@ As opposed to only eliminating possible values of $\chern_0(u)$ for which all
 corresponding $\chern_1^{\beta}(u)$ fail one of the inequalities (which is what
 was implicitly happening before in the proof of Theorem
 \ref{thm:loose-bound-on-r}).
+To pursue this, we shall restate the earlier numerical characterisations of the
+problems from Lemma \ref{lem:num_test_prob1}
+and Corollary \ref{cor:num_test_prob2}, in a way which better fits our direction
+of travel.
 
 \begin{lemma}
 \label{lem:fixed-q-semistabs-criterion}
@@ -1120,7 +1124,7 @@ $\beta=\sage{recurring.betaminus}$, 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 semistabiliser $u$ of $v$
-in terms of the possible values for $q\coloneqq\chern_1^{\beta}(u)$ are as follows:
+in terms of the possible values for $q\coloneqq\chern_1^{\beta_{-}}(u)$ are as follows:
 
 \begin{sagesilent}
 from examples import bound_comparisons
@@ -1131,7 +1135,7 @@ qs, theorem2_bounds, theorem3_bounds = bound_comparisons(recurring)
 \noindent
 \directlua{ table_width = 3*4+1 }
 \begin{tabular}{l\directlua{for i=0,table_width-1 do tex.sprint([[|c]]) end}}
-	$q=\chern_1^\beta(u)$
+	$q=\chern_1^{\beta_{-}}(u)$
 \directlua{for i=0,table_width-1 do
 	local cell = [[&$\noexpand\sage{qs[]] .. i .. "]}$"
   tex.sprint(cell)
@@ -1152,7 +1156,7 @@ end}
 \vspace{1em}
 
 \noindent
-It's worth noting that the bounds given by Theorem \ref{thm:rmax_with_eps1}
+It is 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-semistabilisers of $v$ in this case.
 As a reminder, the original loose bound from Theorem \ref{thm:loose-bound-on-r}
@@ -1168,7 +1172,7 @@ 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
+possible values for $k_{v,q}$, in Definition/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.