From b980dabd62211d1b9f8265fe0ff90ef8035618db Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Fri, 10 Apr 2026 01:00:58 -0400
Subject: [PATCH] update: auto commit

---
 report.tex | 43 ++++++++-----------------------------------
 1 file changed, 8 insertions(+), 35 deletions(-)

diff --git a/report.tex b/report.tex
index 1715b5b..8d9dde7 100644
--- a/report.tex
+++ b/report.tex
@@ -658,41 +658,14 @@
   \begin{frame}{Searching for a first order Z4c}{old workflow: by hand}
     \subsection{old workflow: by hand}
      Here is the Z4c evolution system:
-    {\tiny
-      \begin{align}
-        \textstyle
-        \pt \chi ={}& \frac{2}{3} \chi \left[ \alpha \left( \hat{K} +
-        2\Theta \right) - \spaceD{_i} \shift{^i} \right],\\
-        \pt \smetric*{_i_j} ={} & - 2 \lapse \Ktf*{_i_j} + \shift{^k}
-        \Partial{_k} \smetric*{_i_j} + 2 \smetric*{_{k(i}} \Partial{_{j)}}
-        \shift{^k} - \frac{2}{3} \smetric*{_i_j} \Partial{_k} \shift{^k},\\
-        \pt \hat{K} ={}& - \spaceD{^i} \spaceD{_i} \lapse + \lapse \left[
-          \Ktf*{_i_j} \Ktf*{^i^j} + \frac{1}{3} \left( \hat{K} + 2 \Theta
-        \right)^2 \right] + 4 \pi \lapse \left( S + \rho \right) + \lapse
-        \kappa_1 (1-\kappa_2) \Theta + \shift{^i} \Partial{_i} \hat{K},\\
-        \pt \Ktf*{_i_j} ={} & \chi \left[ - \spaceD{_i} \spaceD{_j} \lapse
-          + \lapse \left( \Rictensor{_i_j} - 8 \pi \tensor{S}{_i_j} \right)
-        \right]^{\text{tf}} + \lapse \left[ \left( \hat{K} + 2 \Theta
-        \right) \Ktf*{_i_j} - 2 \Ktf*{^k_i} \Ktf*{_k_j} \right] +
-        \shift{^k} \Partial{_k} \Ktf*{_i_j} + 2 \Ktf*{_{k(i}}
-        \Partial{_{j)}} \shift{^k} - \frac{2}{3} \Ktf*{_i_j} \Partial{_k}
-        \shift{^k},\\
-        \pt \Gam*{^i} ={} & - 2 \Ktf*{^i^j} \Partial{_j} \lapse + 2 \lapse
-        \left[ \Gam*{^i_j_k} \Ktf*{^j^k} - \frac{3}{2} \Ktf*{^i^j}
-          \Partial{_j} \ln(\chi) - \frac{1}{3} \smetric*{^i^j} \Partial{_j}
-          \left( 2 \hat{K} + \Theta \right) - 8 \pi \smetric*{^i^j}
-        \tensor{S}{_j} \right] + \smetric*{^j^k} \Partial{_j} \Partial{_k}
-        \shift{^i} + \frac{1}{3} \smetric*{^i^j} \Partial{_j} \Partial{_k}
-        \shift{^k} \notag\\
-        & {} + \shift{^j} \Partial{_j} \Gam*{^i} - \Gamd*{^j} \Partial{_j}
-        \shift{^i} + \frac{2}{3} \Gamd*{^i} \Partial{_j} \shift{^j} - 2
-        \lapse \kappa_1 \left[ \Gam*{^i} - \Gamd*{^i} \right], \\
-        \pt \Theta ={} & \frac{1}{2} \lapse \left[ \Rscalar - \Ktf*{_i_j}
-          \Ktf*{^i^j} + \frac{2}{3} \left( \hat{K} + 2 \Theta \right)^2
-        \right] - \lapse \left[ 8 \pi \rho + \kappa_1 (2+\kappa_2) \Theta
-        \right] + \shift{^i} \Partial{_i} \Theta,
-      \end{align}
-    }
+    \[
+\scalebox{0.8}{$\displaystyle
+\begin{aligned}
+  E &= mc^2 \\
+  F &= \frac{a+b}{c+d}
+\end{aligned}
+$}
+\]
   \end{frame}
 
   \begin{frame}{References}