From 3f4ee6674024cb3fef9eeb9e1894f684aaaad994 Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Fri, 10 Apr 2026 01:59:48 -0400
Subject: [PATCH] update: auto commit

---
 report.tex | 11 ++++++++++-
 1 file changed, 10 insertions(+), 1 deletion(-)

diff --git a/report.tex b/report.tex
index 0368975..48759a0 100644
--- a/report.tex
+++ b/report.tex
@@ -827,7 +827,16 @@ $}
     {\color{red} \textbf{Unfortunately, this naive system is not even weakly hyperbolic!}}
     \vspace{0.2cm}
 
-    We have to add some constraint terms that equals to zero, while contributes to the principal symbol, until the principal symbol matrix is diagonalizable with real eigenvalues. This is the most tricky part, and we have to do it by hand, which is very time consuming and error prone.
+    We have to add some constraint terms that equals to zero, while contributes to the principal symbol, until the principal symbol matrix is diagonalizable with real eigenvalues.
+  \end{frame}
+
+  \begin{frame}{Searching for a first order Z4c}{old workflow: by hand}
+    Available terms to add:
+    \begin{gather*}
+      \constraintA{_i} \qc \constraintB{^i_j} \qc \constraintC*{_i} \qc \constraintD*{_i_j_k},\\
+      \Partial{_{[i}} \constraintA{_{j]}} = \Partial{_{[i}} \dlapse{_{j]}} \qc \Partial{_{[i}} \constraintB{^j_{k]}} = \Partial{_{[i}} \dshift{^j_{k]}} \qc \Partial{_{[i}} \constraintC*{_{j]}} = \Partial{_{[i}} \dchi*{_{j]}} \qc \Partial{_{[i}} \constraintD*{_{j]kl}} = \Partial{_{[i}} \dsmetric*{_{j]kl}},\\
+      \Partial{_i} \left( \smetric*{^j^k} \Ktf*{_j_k} \right) \qc \Partial{_i} \left( \smetric*{^k^l} \dsmetric*{_j_k_l} \right)
+    \end{gather*}
   \end{frame}
 
   \begin{frame}{References}