From f315177944890cd37d1206533a7e2f3ec413fec1 Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Fri, 10 Apr 2026 00:12:39 -0400
Subject: [PATCH] update: auto commit

---
 report.tex | 45 ++++++++++++++++++++++++++++++++++++++++++++-
 1 file changed, 44 insertions(+), 1 deletion(-)

diff --git a/report.tex b/report.tex
index 12471f2..0a63a84 100644
--- a/report.tex
+++ b/report.tex
@@ -295,7 +295,50 @@
     \end{itemize}
   \end{frame}
 
-  \begin{frame}{Searching for a first order Z4c}{Why Z4c?}
+  \begin{frame}{Searching for a first order Z4c}{First order Z4c?}
+    \begin{figure}
+      \centering
+      \begin{tikzpicture}[
+        scale=0.5,
+        >=Latex,
+        node distance=8mm and 10mm,
+        every node/.style={font=\small},
+        sys/.style={
+          draw,
+          rounded corners,
+          align=center,
+          minimum width=15mm,
+          minimum height=7mm,
+          inner sep=2pt
+        },
+        std/.style={sys, fill=blue!8},
+        fo/.style={sys, fill=red!10},
+        rel/.style={->, thick},
+        weak/.style={->, semithick, dashed},
+        equv/.style={<->, thick, dotted}
+      ]
+
+      \node[std] (bssn) {BSSN};
+      \node[std, right=of bssn] (z4) {Z4};
+      \node[std, right=of z4] (z4c) {Z4c};
+      \node[std, right=of z4c] (ccz4) {CCZ4};
+
+      \node[fo, below=10mm of bssn] (fobssn) {FOBSSN};
+      \node[fo, below=8mm of z4] (gh) {GH};
+      \node[fo, below=10mm of ccz4] (foccz4) {FOCCZ4};
+
+      \draw[rel] (z4) -- (z4c);
+      \draw[rel] (z4) to[bend left=18] (ccz4);
+      \draw[weak] (bssn) -- (z4c);
+
+      \draw[rel] (bssn) -- (fobssn);
+      \draw[rel] (ccz4) -- (foccz4);
+      \draw[equv] (z4) -- (gh);
+
+      % \node[draw=none, above=1mm of gh, font=\scriptsize] {full first order};
+      \end{tikzpicture}
+      \caption{Commenly used systems, and first order reductions.}
+    \end{figure}
     
   \end{frame}