Adjust reference to twisted chern 1 of u being bounded

main.tex

@@ -706,8 +706,9 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
 		\item $r > 0$
 		\item $\Delta(u) \geq 0$
 		\item $\Delta(v-u) \geq 0$
-		\item $\beta(P)<\mu(u)=\frac{c}{r}<\mu(v)$
-		\item $\chern_1^{\beta(P)}(u)\leq\chern_1^{\beta(P)}(v)$
+		\item $\mu(u)=\frac{c}{r}<\mu(v)$
+		\item $0\leq\chern_1^{\beta(P)}(u)\leq\chern_1^{\beta(P)}(v)$
+			\label{item:chern1bound:lem:num_test_prob1}
 		\item $\chern_2^{P}(u)>0$
 	\end{enumerate}
 \end{lemma}
@@ -719,6 +720,8 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
 	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}
 
@@ -733,8 +736,8 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
 		\item $r > 0$
 		\item $\Delta(u) \geq 0$
 		\item $\Delta(v-u) \geq 0$
-		\item $\beta(P)<\mu(u)=\frac{c}{r}<\mu(v)$
-		\item $\chern_1^{\beta(P)}(u)\leq\chern_1^{\beta(P)}(v)$
+		\item $\mu(u)=\frac{c}{r}<\mu(v)$
+		\item $0\leq\chern_1^{\beta_{-}}(u)\leq\chern_1^{\beta_{-}}(v)$
 		\item $\chern_2^{\beta_{-}}(u)>0$
 	\end{enumerate}
 \end{corrolary}
@@ -952,10 +955,17 @@ 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 semistabilizer $E$:
+% NOTE FUTURE: surface specialization
+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}.
 \begin{align}
-	v &\coloneqq \chern(F) = (R,C\ell,D\ell^2) \\
-	u &\coloneqq \chern(E) = (r,c\ell,d\ell^2)
+	\chern(F) =\vcentcolon\: v \:=& \:(R,C\ell,D\ell^2)
+		&& \text{where $R,C,2D\in \ZZ$}
+	\\
+	u \coloneqq& \:(r,c\ell,d\ell^2)
+		&& \text{where $r,c,2d\in \ZZ$}
 \end{align}
  
 \begin{sagesilent}
@@ -970,8 +980,11 @@ u = Chern_Char(*var("r c d", domain="real"))
 Δ = lambda v: v.Q_tilt()
 \end{sagesilent}
 
-Recall from eqn \ref{eqn-tilt-cat-cond} that $\chern_1^{\beta}(u)$ has fixed
-bounds in terms of $\chern_1^{\beta}(v)$, and so we can write:
+Recall from condition \ref{item:chern1bound:lem:num_test_prob1} in
+lemma \ref{lem:pseudo_wall_numerical_tests}
+(or corrolary \ref{cor:numerical-test-left-pseudowalls-rational-betamin})
+that $\chern_1^{\beta}(u)$ has fixed bounds in terms of $\chern_1^{\beta}(v)$,
+and so we can write:
 
 \begin{sagesilent}
 ts = stability.Tilt