The necessary information for all of the relevant points in FSC are collected in the following table:

[ |p[3cm]{}|p[3cm]{}|p[3cm]{}|p[2cm]{}|p[3.7cm]{}| ]{}\
Point name & $t$ &$r ^{\ast}$ & Sign of $f(r)$\
X &$r^{\ast}(\epsilon_{0}) - \mu^-$ & $r^{\ast}(\epsilon_{0})$ &-\
W &$-r^{\ast}(\epsilon_{0}) + \mu^-$ &$r^{\ast}(\epsilon_{0})$ &-\
Y &$r^{\ast}(\epsilon_{0}) + \mu^+$ &$r^{\ast}(\epsilon_{0})$ &-\
Z &$-r^{\ast}(\epsilon_{0}) - \mu^{+}$ &$r^{\ast}(\epsilon_{0})$ &-\

The bulk action and also the boundary terms and the joint terms are exactly the same as in . The definition of the initial and the late times are also the same as for higher dimensional cases.

Initial time
------------

\[fig:22\]

![Initial time for FSC. Gravitational action is evaluated on the gray region.](FSCI.png)

The GHY term for the spacelike surfaces in the FSC are $$I_{GHY}^{FSC}= \int \frac{\hat{r}_{+}^2}{4 G_{N}} dt \Bigg{|}_{r^{\ast} = \text{const}}.$$ For the surfaces near $r^{\ast} = \epsilon_{0}$ we have $$\begin{aligned}
I_{GHY}(WX) &= \frac{\hat{r}_{+}^2}{4 G_{N}} \left( t(W)- t(X)\right ),
\\ 
I_{GHY}(ZY) &= \frac{\hat{r}_{+}^2}{4 G_{N}} \left( t(Z)- t(Y)\right ).\end{aligned}$$ Adding these two terms yields $$I_{GHY}^{sing} = \frac{\hat{r}_{+}^2}{4 G_{N}}\left ( -4 r^{\ast}(\epsilon_0) + 2 \mu^- -2 \mu^+ \right).$$ Hence for the symmetric evolution where $\mu^+ - \mu^-$ is fixed, the rate of complexity growth for the initial time vanishes, $$\dot{\mathcal{C}}_{initial}^{FSC} = 0.$$

Late time
---------

\[fig:22\]

