From bdae3d4de46b5c0d06526719be437c6f0aaeb157 Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Thu, 9 Apr 2026 23:28:09 -0400
Subject: [PATCH] update: auto commit

---
 report.tex | 86 ++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 86 insertions(+)

diff --git a/report.tex b/report.tex
index 0aca26b..b424297 100644
--- a/report.tex
+++ b/report.tex
@@ -5,6 +5,7 @@
 \usetikzlibrary{arrows.meta}
 \usetikzlibrary{decorations.pathmorphing}
 \usetikzlibrary{calc}
+\usetikzlibrary{positioning,fit,backgrounds}
 
 \usetikzlibrary{external}
 \tikzexternalize[prefix=tikzcache/]
@@ -239,6 +240,91 @@
   \begin{frame}{Searching for a first order Z4c}{Why Z4c?}
     \section{Searching for a first order Z4c}
     \subsection{Why Z4c?}
+    \begin{tikzpicture}[
+  font=\small,
+  >=Latex,
+  node distance=12mm and 14mm,
+  sys/.style={
+    draw,
+    rounded corners,
+    align=center,
+    minimum width=22mm,
+    minimum height=9mm,
+    inner sep=3pt
+  },
+  second/.style={sys, fill=blue!6},
+  first/.style={sys, fill=red!7, very thick},
+  note/.style={align=center, font=\footnotesize},
+  solidarrow/.style={->, thick},
+  dashedarrow/.style={->, thick, dashed},
+  influence/.style={->, semithick, dotted}
+]
+
+% -------------------------------------------------
+% upper row: non-fully-first-order / standard forms
+% -------------------------------------------------
+\node[second] (bssn) {BSSN};
+\node[second, right=28mm of bssn] (z4) {Z4};
+\node[second, right=22mm of z4] (z4c) {Z4c};
+\node[second, right=22mm of z4c] (ccz4) {CCZ4};
+
+% lower row: fully first-order systems
+\node[first, below=22mm of bssn] (fobssn) {FOBSSN};
+\node[first, below=22mm of ccz4] (foccz4) {FOCCZ4};
+\node[first, below=22mm of z4c] (gh) {GH\\(usually first-order)};
+
+% group boxes
+\node[
+  draw,
+  rounded corners,
+  inner sep=6pt,
+  fit=(bssn)(z4)(z4c)(ccz4),
+  label={[font=\footnotesize]above:second-order-in-space / standard formulations}
+] (upperbox) {};
+
+\node[
+  draw=red!70!black,
+  rounded corners,
+  very thick,
+  inner sep=6pt,
+  fit=(fobssn)(foccz4)(gh),
+  label={[font=\footnotesize]above:{\textbf{fully first-order systems}}}
+] (lowerbox) {};
+
+% -------------------------------------------------
+% arrows among upper systems
+% -------------------------------------------------
+\draw[solidarrow] (z4) -- node[above, note] {conformal\\reformulation} (z4c);
+\draw[solidarrow] (z4) to[bend left=18] node[above, note] {conformal + covariant\\constraint damping} (ccz4);
+
+% BSSN <-> Z4c relation: not a clean parent/child, so use dotted influence
+\draw[influence] (bssn) -- node[above, note] {BSSN-like\\variables} (z4c);
+
+% -------------------------------------------------
+% arrows to first-order systems
+% -------------------------------------------------
+\draw[solidarrow] (bssn) -- node[left, note] {first-order\\reduction} (fobssn);
+\draw[solidarrow] (ccz4) -- node[right, note] {first-order\\reduction} (foccz4);
+
+% GH relation: keep it cautious
+\draw[dashedarrow] (z4) -- node[right, note] {closely related\\(principal-part / variable-choice sense)} (gh);
+
+% optional weak relation from Z4c to FOCCZ4
+\draw[influence] (z4c.south east) to[bend left=18]
+  node[right, note] {same Z4-family} (foccz4.north west);
+
+% -------------------------------------------------
+% tiny legend
+% -------------------------------------------------
+\node[note, below=8mm of gh] (legend) {%
+\begin{tabular}{@{}c l@{}}
+\raisebox{0.2ex}{\tikz{\draw[->,thick] (0,0)--(0.7,0);}} & fairly direct reformulation / reduction \\
+\raisebox{0.2ex}{\tikz{\draw[->,thick,dashed] (0,0)--(0.7,0);}} & close relation, but not drawn as a strict descendant \\
+\raisebox{0.2ex}{\tikz{\draw[->,semithick,dotted] (0,0)--(0.7,0);}} & shared variables / family resemblance / influence
+\end{tabular}
+};
+
+\end{tikzpicture}
   \end{frame}
 
   \begin{frame}{Searching for a first order Z4c}{Hyperbolicity of first order systems}