From 1b72bcdfd91bc1ce5ba57d5c93b0489f97c04f09 Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Fri, 10 Apr 2026 01:03:11 -0400
Subject: [PATCH] update: auto commit

---
 report.tex | 5 ++++-
 1 file changed, 4 insertions(+), 1 deletion(-)

diff --git a/report.tex b/report.tex
index 1c091de..b95d4ef 100644
--- a/report.tex
+++ b/report.tex
@@ -657,7 +657,7 @@
 
   \begin{frame}{Searching for a first order Z4c}{old workflow: by hand}
     \subsection{old workflow: by hand}
-     Here is the Z4c evolution system:
+     Here is the Z4c evolution system with the puncture gauge condition:
     \[
 \scalebox{0.6}{$\displaystyle
 \begin{aligned}
@@ -691,6 +691,9 @@
     \Ktf*{^i^j} + \frac{2}{3} \left( \hat{K} + 2 \Theta \right)^2
   \right] - \lapse \left[ 8 \pi \rho + \kappa_1 (2+\kappa_2) \Theta
   \right] + \shift{^i} \Partial{_i} \Theta,
+  \pt \lapse &= - \mu_L \lapse^2 \hat{K} + \Ld{\beta} \lapse,\\
+  \pt \shift{^i} &= \mu_S \lapse^2 \Gam*{^i} - \eta \shift{^i} +
+  \shift{^j} \Partial{_j} \shift{^i}.
 \end{aligned}
 $}
 \]