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

At the late time a new null joint term appears at the point $M$ (Figure 5). At this point $t(M)=0$, $r ^{\ast}(M)=- \mu^+$, and the sign of $f(r_M)$ is negative. Thus we have $$\frac{d I_{J}(M)}{d \mu^+} = -\frac{\hat{r}_{+}^2}{8 G} \log |\frac{\hat{r}_{+}}{\mathfrak{a}}|.$$ Imposing the symmetric evolution, $$\mu^+ - \mu^- = \text{const}, \qquad \mu^+ + \mu^-={\mu},$$ the complexity growth at the late time is found, $$\label{final FSC}
\dot{\mathcal{C}}=\frac{1}{\pi}\frac{d I}{d \mu} = \frac{1}{\pi} \left( \frac{\hat{r}_{+}^2}{4 G_{N}} + \frac{\hat{r}_{+}^2}{4 G_{N}} \log |\frac{\mathfrak{a}}{\hat{r}_{+}}|\right).$$ We can rewrite in terms of the FSC mass as $$\dot{\mathcal{C}}= \frac{1}{\pi} \left( 2 M+ M \log \frac{\mathfrak{a}^2}{8 G_{N} M}\right).$$ It is a constant and differs from Lloyd’s bound by a logarithmic term. This result is exactly the same as , which is given by taking the flat space limit. Similar to the higher dimensional cases, $\mathfrak{a}$ is a new length parameter in the field theory. However, if we demand that Lloyd’s bound [@Lloyd:limit] is not violated, then $\mathfrak{a}$ must be restricted by $\mathfrak{a}< \hat{r}^{+}$.

Discussion 
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In this paper we calculate the rate of complexity growth for BMSFT$_d$. Our main goal is generalizing the CA proposal for the flat/BMSFT 