diff --git a/main.tex b/main.tex
index bda37b37c5fc99d08cb5af09a8a7d5ca4ff79510..99421a171132ab0fb53f85d6379279ae3fc415f7 100644
--- a/main.tex
+++ b/main.tex
@@ -226,19 +226,21 @@ the circular walls must be nested and non-intersecting.
\subsection{Characteristic curves for pseudo-semistabilizers}
\begin{lemma}[Numerical tests for left-wall pseudo-semistabilizers]
-Let $v$ and $u$ be Chern characters with $\Delta(v), \Delta(u)\geq 0$ and
-positive ranks.
-
-Suppose that $u$ gives rise to a pseudo-wall for $v$, left of the characteristic
-vertical line $\chern_1^{\alpha,\beta}(v)=0$ and containing a fixed point $p$ in
-it's interior.
-To target all left-walls, $p$ can be chosen as the base of the left branch of
-the hyperbola $\chern_2^{\alpha,\beta}(v)=0$.
-Suppose further that this happens in a way such that $u$ 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.
+Let $v$ and $u$ be Chern characters with positive ranks and $\Delta(v),
+\Delta(u)\geq 0$. Let $P$ be a point on the left branch of the characteristic
+hyperbola ($\chern_2^{\alpha,\beta}(v)=0$) for $v$.
+\noindent
+Suppose that the following are satisfied:
+\begin{itemize}
+\item $u$ gives rise to a pseudo-wall for $v$, left of the characteristic
+ vertical line $\chern_1^{\alpha,\beta}(v)=0$
+\item The pseudo-wall contains $p$ in it's interior
+ ($P$ can be chosen to be the base of the left branch to target all left-walls)
+\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{itemize}
\noindent
Then we have the following:
@@ -247,13 +249,138 @@ Then we have the following:
(if this is a real wall then $v$ is being semistabilized by a positive rank object)
\item $\mu(u)<\mu(v)$, i.e., $u$'s vertical characteristic line is left of $v$'s vertical
characteristic line
- \item $\chern_2^{p}(u)>0$
+ \item $\chern_2^{P}(u)>0$
\end{itemize}
Furthermore, only the last two of these consequences are sufficient to recover
all of the suppositions above.
\end{lemma}
+\begin{proof}
+Let $u,v$ be Chern characters with positive ranks and
+$\Delta(u),\Delta(v) \geq 0$.
+
+
+For the forwards implication, assume that the suppositions of the lemma are
+satisfied. The pseudo-wall intersects the characteristic hyperbola for $v$, at
+some point $Q$ further up the hyperbola branch than $P$ (to satisfy second
+supposition). At $Q$, we have $\mu_Q(v)=0$, and hence $\mu_Q(u)=0$ too.
+This means that the characteristic hyperbola for $u$ must intersect that of $v$
+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
+$u$'s hyperbola, as well as the positions of the base.
+However, considering the third supposition, only case 3 (green in figure) is
+valid.
+This is because we need $\nu_{\alpha,\beta}(u)>0$
+($\nu_{\alpha,\beta}(-u)>0$ in case 1 involving the right hyperbola branch)
+for points $(\beta,\alpha)$ on $v$'s characteristic curve inside the pseudo-wall.
+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. In passing, this implies consequence 3.
+
+\begin{sagesilent}
+def hyperbola_intersection_plot():
+ var("alpha beta", domain="real")
+ coords_range = (beta, -3, -1/2), (alpha, 0, 2.5)
+
+ delta1 = -sqrt(2)+1/100
+ delta2 = 1/2
+ pbeta=-1.5
+
+ p = (
+ implicit_plot(beta^2 - alpha^2 == 2, *coords_range , rgbcolor = "black", legend_label=r"a")
+ + implicit_plot((beta+4)^2 - (alpha)^2 == 2, *coords_range , rgbcolor = "red")
+ + implicit_plot((beta+delta1)^2 - alpha^2 == (delta1-2)^2-2, *coords_range , rgbcolor = "blue")
+ + implicit_plot((beta+delta2)^2 - alpha^2 == (delta2-2)^2-2, *coords_range , rgbcolor = "green")
+ + point([-2, sqrt(2)], size=50, rgbcolor="black", zorder=50)
+ + text("Q",[-2, sqrt(2)+0.1], rgbcolor="black", fontsize="large", clip=true)
+ + point([pbeta, sqrt(pbeta^2-2)], size=50, rgbcolor="black", zorder=50)
+ + text("P",[pbeta+0.1, sqrt(pbeta^2-2)], rgbcolor="black", fontsize="large", clip=true)
+ + circle((-2,0),sqrt(2), linestyle="dashed", rgbcolor="purple")
+ # dummy lines to add legends (circumvent bug in implicit_plot)
+ + line([(2,0),(2,0)] , rgbcolor = "purple", linestyle="dotted", legend_label=r"pseudo-wall")
+ + line([(2,0),(2,0)] , rgbcolor = "black", legend_label=r"$ch_2^{\alpha,\beta}(v)=0$")
+ + line([(2,0),(2,0)] , rgbcolor = "red", legend_label=r"case 1")
+ + line([(2,0),(2,0)] , rgbcolor = "blue", legend_label=r"case 2")
+ + line([(2,0),(2,0)] , rgbcolor = "green", legend_label=r"case 3")
+ )
+ p.xmax(coords_range[0][2])
+ p.xmin(coords_range[0][1])
+ p.ymax(coords_range[1][2])
+ p.ymin(coords_range[1][1])
+ p.axes_labels([r"$\beta$",
+ r"$\alpha$"])
+
+ return p
+
+def correct_hyperbola_intersection_plot():
+ var("alpha beta", domain="real")
+ coords_range = (beta, -2.5, 0.5), (alpha, 0, 3)
+
+ delta2 = 1/2
+ pbeta=-1.5
+
+ p = (
+ implicit_plot(beta^2 - alpha^2 == 2, *coords_range , rgbcolor = "black", legend_label=r"a")
+ + implicit_plot((beta+delta2)^2 - alpha^2 == (delta2-2)^2-2, *coords_range , rgbcolor = "green")
+ + point([-2, sqrt(2)], size=50, rgbcolor="black", zorder=50)
+ + text("Q",[-2, sqrt(2)+0.1], rgbcolor="black", fontsize="large", clip=true)
+ + point([pbeta, sqrt(pbeta^2-2)], size=50, rgbcolor="black", zorder=50)
+ + text("P",[pbeta+0.1, sqrt(pbeta^2-2)], rgbcolor="black", fontsize="large", clip=true)
+ + circle((-2,0),sqrt(2), linestyle="dashed", rgbcolor="purple")
+ # dummy lines to add legends (circumvent bug in implicit_plot)
+ + line([(2,0),(2,0)] , rgbcolor = "purple", linestyle="dotted", legend_label=r"pseudo-wall")
+ + line([(2,0),(2,0)] , rgbcolor = "black", legend_label=r"$ch_2^{\alpha,\beta}(v)=0$")
+ + line([(2,0),(2,0)] , rgbcolor = "green", legend_label=r"$ch_2^{\alpha,\beta}(u)=0$")
+ # vertical characteristic lines
+ +line([(0,0),(0,coords_range[1][2])], rgbcolor="black", linestyle="dashed", legend_label=r"$ch_1^{\alpha,\beta}(v)=0$")
+ +line([(-delta2,0),(-delta2,coords_range[1][2])], rgbcolor="green", linestyle="dashed", legend_label=r"$ch_1^{\alpha,\beta}(u)=0$")
+ +line([(0,0),(-coords_range[1][2],coords_range[1][2])], rgbcolor="black", linestyle="dotted", legend_label=r"assymptote for $ch_2^{\alpha,\beta}(v)=0$")
+ +line([(-delta2,0),(-delta2-coords_range[1][2],coords_range[1][2])], rgbcolor="green", linestyle="dotted", legend_label=r"assymptote for $ch_1^{\alpha,\beta}(u)=0$")
+
+ )
+ p.set_legend_options(loc="upper right")
+ p.xmax(coords_range[0][2])
+ p.xmin(coords_range[0][1])
+ p.ymax(coords_range[1][2])
+ p.ymin(coords_range[1][1])
+ p.axes_labels([r"$\beta$", r"$\alpha$"])
+
+
+ return p
+\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 valid case (2), illustrated in Fig
+\ref{fig:correct-hyperbol-intersection}.
+We must have the left branch of the characteristic hyperbola for $u$ taking a
+base-point to the right of that of $v$'s, but then, further up, crossing over to
+the left side. The latter fact implies that the assymptote for $u$ must be to
+the left of the one for $v$. Given that they are parallel and intersect the
+$\beta$-axis at $\beta=\mu(u)$ and $\beta=\mu(v)$ respectively.
+We must have $\mu(u)<\mu(v)$, that is, the vertical characteristic line for $u$
+is to the left of the one for $v$ (consequence 2).
+Finally, the fact that it is the left branch of the hyperbola for $u$ implies
+consequence 1.
+
+\end{proof}
\begin{sagesilent}
v = Chern_Char(3, 2, -2)