* To the best of our knowledge, our model/(aforementioned) ${\cal M}$-theory dual is either the only one (from ${\cal M}$ theory) or one of the very few top-down models that obtain(s) the Page curve for massless graviton[See [61] in which author has discussed that one can get Page curve for massless graviton in critical Randall-Sundrum II model from bottom-up approach.] in  <ref>.

* In our top-down ${\cal M}$-theory dual, we find that ETW-“brane” to be a fluxed hypersurface ${\cal W}$ that is a warped product of an asymptotic $AdS_4$ and a six-fold $M_6$ where $M_6$ is a warped product of the ${\cal M}$-theory circle and a non-Einstenian generalization of $T^{1,1}$; the hypersurface ${\cal W}$, can also be thought of as an effective ETW-“brane” corresponding to fluxed intersecting $M5$-brane wrapping a homologous sum of $S^3\times[0,1]$ and
$S^2\times S^2$ in a warped product of $\mathbb{R}^2$ and an $SU(4)/Spin(7)$-structure eight-fold. The ETW-“brane", ${\cal W}$, then has non-zero “tension” and a massless graviton localized near the horizon by a “volcano”-like potential.

* Further, unlike almost all the papers in the literature wherein the Page curve computation is done for a CFT bath, in our model, the external bath is a non-CFT bath (thermal QCD).

* Entanglement entropy contribution from Hartman-Maldacena (HM)-like surface which is responsible for increase of Einstein-Rosen bridge in time, also exhibits a Swiss-Cheese structure in Large-$N$ scenario (<ref>).

* With the inclusion of ${\cal O}(R^4)$ terms in the action, our aforementioned ${\cal M}$-theory dual yields a hierarchy in the entanglement entropies of the HM-like and Island surface (IS) with respect to a large-$N$ exponential suppression factor, physically arising from the existence of massless graviton mode on the ETW-brane. This suppression further implies that the Page curve is unaffected by the inclusion of higher derivative - ${\cal O}(R^4)$ in particular - terms.

* To tame the IR- and large-$N$ enhancement in the IS entanglement entropy per unit BH entropy to ensure the same is around 2 in <ref>, along with dimensional considerations of <ref>, a connection between the Planckian length and the non-extremality parameter (the horizon radius) is shown to arise in <ref>.

* In subsections <ref> and <ref>, using the Ryu-Takayanagi formula (since we are not considering higher derivative terms in the gravitational action in this part), we calculated the entanglement entropies of HM-like and Islands Surfaces (IS) by computing the areas of the co-dimension two surfaces for the HM-like and island surfaces. The entanglement entropy for HM-like surface was calculated in (<ref>) using (<ref>), which shows linear time dependence. If one does not consider the island surface then (<ref>) implies that at late times (i.e., beyond the Page time) we have infinite amount of Hawking radiation. The entanglement entropy of the island surface (<ref>) was calculated using (<ref>)). If we are below the Page time then entanglement entropy of HM-like surface dominates which shows linear time dependence and after the Page time the entanglement entropy of island surface, which is constant, dominates. Therefore combination of the two produces the Page curve of an eternal black hole in figure <ref>.

In ${\cal R}_{D5/\overline{D5}}=1$-units, using (<ref>), (<ref>), (<ref>) and $ r_h\sim e^{-\kappa_{r_h}N^{\frac{1}{3}}}$ [70], the aforementioned is succinctly captured by:

\begin{eqnarray}
\label{SEE-beta0-Area-Summary-i}
& & S^{\rm HM} = \frac{A_{\rm HM}}{4G_N^{(11)}}\sim \frac{{\cal O}(1)\times10^{-4}M^2  N_f^6 g_s^{15/4} N^{34/15} {\bf e^{-6\kappa_{r_h}N^{1/3}}}}{ G_N^{(11)}}t_b,\ t_b\leq t_{\rm Page};\nonumber\\
& & {\cal S}^{IS} = {\cal S}^{IS}\left(\tilde{r}_T \equiv \frac{r_T}{r_h} = (1+\delta_2); r_h\right),\  t_b\geq t_{\rm Page},
\end{eqnarray}

where $\delta_2$ in the turning point $r_T$ is as given in (<ref>).

* In section <ref>, we obtained the Page curve of an eternal black hole in the presence of ${\cal O}(R^4)$ terms in eleven dimensional supergravity action. For this purpose  use the prescription of [10] as given in (<ref>) to calculate the entanglement entropies of HM-like and island surfaces with the inclusion of ${\cal O}(R^4)$ terms. Since in the working action (<ref>) only $J_0$ term is considered (as it is the most dominant in the MQGP limit) therefore we need to calculate two type of terms appearing in (<ref>). These two type of terms are given in (<ref>) and (<ref>) where we have used $z=x e^{it}$. In subsection <ref>, the entanglement entropy for HM-like surface is calculated and it turns out that for Einstein-Hilbert term Dong's prescription [10] produces (<ref>). For the $J_0$ term there are two types of terms mentioned earlier and these terms are given in (<ref>) and (<ref>). Now if we combine the Einstein-Hilbert and $J_0$ terms's contribution then we obtain the total entanglement entropy (<ref>) in terms of the embedding function of HM-like surface $t(r)$ and its derivatives. We obtained the on-shell entanglement entropy (<ref>) for HM-like surface at ${\cal O}(\beta^0)$ by computing the solution to the EOM for the HM-like surface embedding $t(r)$ (<ref>) and substituting back into (<ref>). Interestingly (<ref>) also has linear time dependence similar to (<ref>).

In <ref>, with $c_1$ and $c_2$ being two parameters in the family of HM-like surface embeddings, curiously, $\frac{\delta S_{\rm EE}^{\rm HM, \beta^0}}{\delta |c_2|}>0,\ \frac{\delta S_{\rm EE}^{\rm HM, \beta^0}}{\delta |c_1|}<0$, which along with $|c_2|\sim e^{\kappa_{c_2}|c_1|^{\frac{1}{3}}},\ |c_1|\sim N$, provides a perfect "Swiss-Cheese” (of the "single-big-divisor-single-small-divisor” variety) structure to $S_{\rm EE}^{\rm HM, \beta^0}(|c_1|, |c_2|)$ (which essentially is a co-dimension-two volume) wherein $\log|c_2|$ plays the role of the "big divisor” volume and $\log|c_1|$ plays the role of a solitary "small divisor” volume, realizing what could be dubbed as a "Large N Scenario”(LNS). Alternatively, the entanglement entropy for the Hartman-Maldacena-like surface  can be viewed as a Swiss-Cheese-like open surface in the two-dimensional (in the IR) space of family of HM-like embeddings $\mathbb{R}_+^2\left(|c_1|, |c_2|\right)$ augmented by the entanglement entropy that coordinatizes  $\mathbb{R}_{\geq0}(S_{\rm EE}^{\rm HM, \beta^0})$.

In subsection <ref>,  we found that entanglement entropy contributions from Einstein-Hilbert and $J_0$ terms as given in (<ref>),(<ref>) and (<ref>). Combining all these contributions we obtained the total entanglement entropy in (<ref>) in terms of embedding function of island surface and its derivatives. We solved the embedding EOM for the island surface similar to HM-like surface and the solution is given in (<ref>). Now, using the solution we  obtained the entanglement entropy for island surface at the level of EH-GHY terms by substituting back in (<ref>). This yields that the entanglement entropy will be of the form given in (<ref>). Given that the turning point lies deep in the IR, the entanglement entropy for HM-like and island surfaces at ${\cal O}(\beta^0)$ came out to be proportional to $\beta$, an obvious but apparent contradiction. This was resolved by proposing a $\beta/l_p-r_h$ relation as discussed above (<ref>) and its explicit form as given in (<ref>). The on-shell entanglement entropy for island surface turns out to be (<ref>). Upon plotting the entanglement entropy contributions from HM-like (<ref>) and island surfaces (<ref>) we obtained the Page curve shown in figure <ref>.

The aforementioned is neatly summarized below.
Using the HM-like embedding (<ref>), and (<ref>) and (<ref>), the on-shell Entanglement Entropy (EE) for HM-like surface is given by:

\begin{eqnarray}
\label{SEE-beta0-Wald-Summary-i}
& & \hskip -0.3in S_{\rm EE}^{\beta^0, {\rm HM}} \sim   {\bf e^{-\frac{3\kappa_{l_p}N^{\frac{1}{3}}}{2}}}M N^{13/10} N_f^{5/3}\left(1-\frac{4 t_{b_0}}{3 c_2}\right)\left(\frac{2}{3}\log \left(\frac{c_2}{c_1}\right) -n_{t_b} \log(N)\right)^{\frac{17}{6}},\ \kappa_{l_p}\equiv\frac{1}{{\cal O}(1)},\nonumber\\
& & \hskip -0.3in c_{1,2}<0, |c_2|\gg |c_1|; |c_2|\sim e^{\kappa_{c_2}|c_1|^{1/3}},\ \kappa_{c_1}\equiv{\cal O}(1)\nonumber\\
& & \hskip -0.3in t\leq t_{\rm Page},
\end{eqnarray}

and using the embedding (<ref>), and
(<ref>), (<ref>), (<ref>) and (<ref>) , the on-shell EE for Island Surface (IS) is given by:

\begin {eqnarray}
& & \hskip -0.3in S_{\rm EE}^{\beta^0, {\rm IS}} \sim \frac{M^2 {\bf e^{-\frac{3\kappa_{l_p} N^{\frac{1}{3}}}{2}}}
N_f^{4/3} g_s^{35/6} \log ^2(N)  \left| \log \left(r_h\right)\right|^{4/3} }{r_h^{11/2}}, \nonumber\\
& & \hskip -0.3in t\geq t_{\rm Page}.
\end {eqnarray}

The above can be shown to be consistent with the RT computation summarized earlier [Compatibility of (<ref>) and (<ref>) (using footnote 8) requires $\kappa_{r_h},\ \kappa_{l_p},\ \kappa_{c_2}$:
$M^2N_f^6g_s^{15/4}N^{34/15}e^{-6\kappa_{r_h}N^{1/3}}\sim M N^{13/10}N_f^{5/3}\left(\frac{2}{3}\biggl[\kappa_{c_2}N^{1/3} - \log |c_1|\biggr] -n_{t_b} \log(N)\right)^{\frac{17}{6}}e^{-\frac{3\kappa_{l_p}N^{1/3}}{2}}$
(for compatibility of $S^{\beta^0, {\rm HM}}_{\rm EE}$ obtained from computation of the area of the HM-like surface and from Wald entanglement entropy) and $11\kappa_{r_h} < 3\kappa_{l_p}$ (for $S^{\beta^0, {\rm IS}}_{\rm EE}$ to be well-defined in the large-$N$ limit).

* Positivity of the Page time - worked out in (<ref>) - is shown to provide an upper bound on the non-extremality parameter - the black-hole horizon radius $r_h$.

* We computed the ${\cal O}(\beta)$ contributions to the HM-like and island surfaces as given in (<ref>) and (<ref>) and we  show in (<ref>) that there is hierarchy between the ${\cal O}(\beta^0)$ and ${\cal O}(\beta)$ contributions. To be a bit more specific, as the turning point for the Hartman-Maldacena-like surface $r_*: r_*-r_h\sim \beta^{\frac{2}{3}}$, and that for Island surface $r_T: r_T-r_h\sim \beta^2$, one obtains the following exponential hierarchy:
$S_{\rm EE}^{\rm HM, \beta^0}:S_{\rm EE}^{\rm IS,\ \beta^0}:S_{\rm EE}^{\rm HM,\ \beta}:S_{\rm EE}^{\rm IS,\ \beta} \sim \gamma^{\frac{3}{2}}:\gamma^{\frac{3}{2}}:\gamma^2:\gamma^3,$ where $\gamma\equiv e^{-\kappa_{l_p}N^{\frac{1}{3}}}$. Therefore we can neglect the ${\cal O}(\beta)$ contributions to the entanglement entropy for both extremal surfaces. The exponentially suppressed $S_{\rm EE}^{\rm IS, \beta^0}$ and the suppression of $S_{\rm EE}^{\rm IS, \beta}$ relative to $S_{\rm EE}^{\rm HM, \beta}$ is argued in Section <ref> to be due to the existence of a massless graviton.  It is also rather non-trivial to have a similar exponential suppression in $S_{\rm EE}^{\rm HM}$ so that the same could be matched with $S_{\rm EE}^{\rm IS}$ at the Page time, and a Page curve obtained.

* Motivated by the requirement $\frac{S^{\beta^0,\ IS}_{\rm EE}}{S_{\rm BH}} \sim e^{-\kappa_{l_p}N^{1/3}}{\cal F}\left(M, N, N_f; r_h\right)\left[1 + \sum_{n=1}^\infty {\cal A}_n\left(\frac{a_3}{a_2 r_h}\right)^n\right]$ (as obtained in (<ref>)) to be less than or equal to 2 up to leading order in the dimensionless "$G_N^{(11)}/r_h^9$” [73], which for us is $\frac{a_3}{a_2 r_h}$ ($a_{2,3}$ parametrizing the IS-surface embedding (<ref>)), one needs to cure the large-$N$ and IR(via small $r_h$) enhancements in $\frac{S^{\beta^0,\ IS}_{\rm EE}}{S_{\rm BH}}$. Utilizing the estimate in [70] of the $r=r_0\sim r_h: N_{\rm eff}(r_0=0)$, and in particular the exponential $N$-suppression therein we therefore propose
$\beta \sim \left(g_s^{\frac{4}{3}}\alpha'^2e^{-\kappa_{l_p}N^{1/3}}\frac{r_h}{{\cal R}_{D5/\overline{D5}}}\right)^{3/2}$. This further requires (see Fig. <ref>) a lower bound on the non-extremality parameter - the black-hole horizon. With the inclusion of the ${\cal O}(R^4)$ terms in ${\cal M}$ theory, the fact the turning point associated with the Hartman-Maldacena-like surface is in the deep IR, also require the aforementioned $\beta$- or $l_p-r_h$ relation.

Some open questions related to our work are:

* How the holographic complexity will be modified in the presence of ${\cal O}(R^4)$ terms in the gravitational action [78]?

* What will be the effect of higher derivative (${\cal O}(R^4)$) terms on the reflected entropy [79]?

We will return to these issues in our future work.

§ ACKNOWLEDGEMENTS

GY is supported by a Senior Research Fellowship (SRF) from the Council of Scientific and Industrial Research, Govt. of India. AM is partly supported by a Core Research Grant number SER-1829-PHY from the Science and Engineering Research Board, Govt. of India. One of us (GY) would like to thank Arpan Bhattacharyya, Xi Dong and Juan Maldacena for helpful clarifications. Some of the results were presented by one of us (GY) at the workshop, "Gravity: Current challenges in black hole physics and cosmology", organised by Yukawa Institute for Theoretical Physics (YITP), Kyoto University, Kyoto, Japan.

§ HM-LIKE/IS ANALYTICS/NUMERICS

In this appendix, we discuss (i) the angular integrations which have been used in the paper and have written the expression for the area of Hartman-Maldacena-like surface, and (ii) how to obtain an estimate on the turning point in the context of Island Surface  entanglement entropy.

§.§ HM-Like Surface Area

The angular integrations, in promoting the delocalized results around $(\theta_1,\theta_2)=\left(\frac{\alpha_{\theta_1}}{N^{\frac{1}{5}}},\frac{\alpha_{\theta_2}}{N^{\frac{3}{10}}}\right)$, to global results, disregarding all contributions of
${\cal O}\left(\frac{{\cal O}(1)}{N^\alpha}\right), \alpha>1$, have been performed as under. Promoting $(x, y, z)\rightarrow (\phi_1, \phi_2, \psi)$ via [36]:

\begin{eqnarray}
\label{local-to-global}
& &
dx = \sqrt{h_2}\left(g_sN\right)^{\frac{1}{4}}\Biggl[1 + {\cal O}\left(\frac{g_sM^2}{N}\right) + {\cal O}\left(\frac{(g_sM^2)(g_sN_f)}{N}\right)\Biggr]\sin\theta_1d\phi_1, \nonumber\\
& & dy = \sqrt{h_4}\left(g_sN\right)^{\frac{1}{4}}\Biggl[1 + {\cal O}\left(\frac{g_sM^2}{N}\right) + {\cal O}\left( \frac{(g_sM^2)(g_sN_f)}{N}\right)\Biggr]\sin\theta_2d\phi_2, \nonumber\\
& & dz = \sqrt{h_1}\left(g_sN\right)^{\frac{1}{4}}\Biggl[1 + {\cal O}\left(\frac{g_sM^2}{N}\right) + {\cal O}\left( \frac{(g_sM^2)(g_sN_f)}{N}\right)\Biggr]d\psi,
\end{eqnarray}

where $h_1 = \frac{1}{9} + {\cal O}\left(\frac{g_sM^2}{N}\right),\ h_2 = \frac{1}{6} + {\cal O}\left(\frac{g_sM^2}{N}\right),\ h_4 = h_2 + \frac{4a^2}{r^2}$ [34], [36],

\begin{eqnarray}
\label{angular-integrations-2}
& & \frac{1}{\alpha_{\theta_1}^3\alpha_{\theta_2}^2}\rightarrow\lim_{\epsilon_{1,2}\rightarrow0}
\int_{\epsilon_2}^{\pi-\epsilon_2}\sqrt{g_sN}d\theta_2\sin\theta_2\int_{\epsilon_1}^{\pi-\epsilon_1}d\theta_1\sin\theta_1\frac{1}{\left(N^{\frac{1}{5}}\sin{\theta_1}\right)^3\left(N^{\frac{3}{10}}\sin{\theta_2}\right)^2}  \sim \lim_{\epsilon_{1,2}\rightarrow0}\frac{|\log \epsilon_2|}{N^{\frac{7}{10}}\epsilon_1}.
\nonumber\\
\end{eqnarray}

The principal(${\cal P}$) value of (<ref>) is obtained by approximating $|\log \epsilon_2|$ by $\log \mathbb{P} + |\log \epsilon_2|, \mathbb{P}\in\mathbb{Z}^+$ and requiring $\log \mathbb{P} + \log \epsilon_2 = - \epsilon_1$, or
$\epsilon_2=\frac{e^{-\epsilon_1}}{\sqrt{\mathbb{P}}}, \mathbb{P}>1$.

\begin{eqnarray}
\label{angular-integrations-1}
& &  \frac{1}{\alpha_{\theta_1}\alpha_{\theta_2}^6}\rightarrow\lim_{\epsilon_{1,2}\rightarrow0}
\int_{\epsilon_2}^{\pi-\epsilon_2}\sqrt{g_sN}d\theta_2\sin\theta_2\int_{\epsilon_1}^{\pi-\epsilon_1}d\theta_1\sin\theta_1\frac{1}{\left(N^{\frac{1}{5}}\sin{\theta_1}\right)\left(N^{\frac{3}{10}}\sin{\theta_2}\right)^6} \nonumber\\
& & \stackrel{{\cal P}}{\rightarrow} \frac{\pi  (540 \log (2)-107) \sqrt{g_s}}{720 N^{3/2}}.
\nonumber\\
\end{eqnarray}

Assuming $r_*\in$IR, approximating $\log r\approx\log r_h$ for $r = \upsilon r_h\forall r\in$IR, $\upsilon={\cal O}(1)$,
one obtains the following expression after radial integration of equation (<ref>):

\begin{eqnarray}
\label{AHM_ii}
& &  A_{\rm HM} \sim \Biggl[E M^2 \sqrt[10]{N} N_f^6 g_s^{17/4} \log ^2\left(r_h\right) \left(\log (N)-3 \log \left(r_h\right)\right){}^4 \nonumber\\
& &\hskip 0.5in \times \Biggl(3 r_*^5 \left(\log (N)-9 \log \left(r_h\right)\right){}^2 -10 r_*^3 r_h^2 \log (N) \left(\log (N)-9 \log \left(r_h\right)\right)\nonumber\\
& & \hskip 0.5in +15 r_* r_h^4 \log ^2(N)-r_h^5 \left(36 \log (N) \log \left(r_h\right)+243 \log ^2\left(r_h\right)+8 \log ^2(N)\right)\Biggr) \Biggr] .\nonumber\\
& &
\end{eqnarray}

§.§ IS Turning Point

In the context of IS entanglement entropy (<ref>), to get an estimate of the turning point $\tilde{r}_T$, one notes that: $\frac{H(r_T)}{\sigma(r_T)}={\cal C}^2$, up to LO in $N$, that obtains:

\begin{eqnarray}
\label{r_T-i}
& & 1458{g_s}^{5/4} M^2 N^{2/5} {N_f}^6 r_T^4 \left(r_T^4-{r_h}^4\right) \log ^7(r_T)
\left(7\log N  r_T^2-3 a^2 \log N \right)-6561{g_s}^{5/4} M^2 N^{2/5} {N_f}^6 r_T^6 \left(r_T^4-{r_h}^4\right)
\log ^8(r_T)  \nonumber\\
& &  + {\cal O}\left(\frac{1}{(\log r_h)^6}\right) = \tilde{\cal C}^2.
\end{eqnarray}

Writing $\tilde{r}_T = 1 + \delta_2$, upon substitution into (<ref>), and assuming $0<\delta_2\ll1$ and thereby approximating $\log r\approx \log r_h$, yields the following:

\begin{eqnarray}
\label{r_T-ii}
& & 1458 \delta_2  {r_h}^{10} \log ^7({r_h}) (4 (47 \delta_2 +6) \log N -9 (15
\delta_2 +2) \log ({r_h}))-5832 \sqrt{3} \delta_2  (11 \delta_2 +2)
\epsilon  \log N  {r_h}^{10} \log ^7({r_h})=\frac{M^2 N^{2/5} {N_f}^6
\tilde{\cal C}^2}{{g_s}^{5/4}}.\nonumber\\
& &
\end{eqnarray}

Assuming again $|\log r_h|\gg \log N$, the relevant solution to (<ref>) is:

\begin{eqnarray}
\label{r_T-iii}
& & \delta_2\approx \frac{M^2 N^{2/5} {N_f}^6 \tilde{\cal C}^2}{26244 {g_s}^{5/4} {r_h}^{10} \log
\end{eqnarray}

Guided by the estimate of $r=r_0\sim r_h: N_{\rm eff}(r_0)=0$ obtained in [70], writing $r_h\sim e^{-\kappa_{r_h}N^{\frac{1}{3}}}$, and assuming $\tilde{\cal C}^2=e^{-\kappa_{\cal C}N^{\frac{1}{3}}}$. Numerically, for $M=N_f=3, g_s=0.1, N=10^{3.3}$ and assuming $\kappa_{\cal C}=1.37, \kappa_{r_h}=0.1, \tilde{\cal C}=3\times10^{-8}$ one obtains $\delta_2=5.6\times10^{-4}$.

§ HARTMAN-MALDACENA-LIKE SURFACE MISCELLANIA

In this appendix we have listed various $r$ dependent functions appearing in the entanglement entropy expression for the Hartman-Maldacena-like surface at ${\cal O}(\beta^0)$ and ${\cal O}(\beta)$. Additionally we have also worked out the equation of motion for the embedding function of the Hartman-Maldacena-like surface.

* The $r$ dependent functions appearing in equation (<ref>) are,

\begin{eqnarray}
\label{alpha-sigma-lambda}
& & \alpha(r)= \kappa_{\alpha}\frac{ r^2 \left(N_f g_s (2 \log (N)-6 \log (r))\right){}^{2/3}}{ \sqrt{N} g_s^{3/2}},\nonumber\\
& & \sigma(r)=\kappa_{\sigma}\frac{ N g_s-3 M^2 N_f g_s^3 \log (r) (\log (N)-6 \log (r))}{ \left(r^4-{r_h}^4\right)}, \nonumber\\
& & \lambda (r)=\kappa_{\lambda}\Biggl(-\frac{ M N^{17/10} N_f^{4/3} g_s^{5/2} \left(r^2-{r_h}^2\right) \log (r) (\log (N)-9 \log (r)) \sqrt[3]{\log (N)-3 \log (r)}}{ r^4 \alpha _{\theta _1}^3 \alpha_{\theta _2}^2}\Biggr) \nonumber\\
& & \sim -\frac{M N^{21/20} N_f^{4/3} g_s^{13/4} \left(r^2-r_h^2\right) \kappa _{\lambda } \log (r) (\log (N)-9 \log (r)) \sqrt[3]{\log (N)-3 \log (r)}}{3 r^4},\nonumber\\
\end{eqnarray}

where $\kappa_{\alpha},\kappa_{\sigma}$ and $\kappa_{\lambda}$ are numerical pre-factors.

* In Wald entanglement entropy term (<ref>) for HM-like surface, we have the following $r$ dependent functions:

\begin{eqnarray}
\label{lambda1-2}
& &
\hskip -.5in \lambda_1(r)= \kappa_{\lambda_1}\left(\frac{ M^7 N^{7/10} N_f^{7/3} g_s^7 \log ^4(r) (\log (N)-12 \log (r))^3 (\log (N)-9 \log (r))}{ r^2 \alpha _{\theta _1}^3 \alpha
_{\theta _2}^2 (\log (N)-3 \log (r))^{5/3}}\right) \nonumber\\
& & \sim \frac{M^7 \sqrt[20]{N} N_f^{7/3} g_s^{31/4} \kappa _{\lambda _1} \log ^4(r) (\log (N)-12 \log (r))^3 (\log (N)-9 \log (r))}{3 r^2 (\log (N)-3 \log (r))^{5/3}}
& &
\hskip -.5in \lambda_2(r)=\kappa_{\lambda_2}\left(\frac{ M^3 N^{7/10} \sqrt[3]{N_f} g_s^3 \log ^2(r) (\log (N)-12 \log (r)) (\log (N)-9 \log (r))}{ r^2 \alpha _{\theta _1}^3 \alpha _{\theta _2}^2 (\log(N)-3 \log (r))^{5/3}}\right) \nonumber\\
& & \sim \frac{M^3 \sqrt[20]{N} \sqrt[3]{N_f} g_s^{15/4} \kappa _{\lambda _2} \log ^2(r) (\log (N)-12 \log (r)) (\log (N)-9 \log (r))}{3 r^2 (\log (N)-3 \log (r))^{5/3}}.
\end{eqnarray}

$\kappa_{\lambda_1}$ and $\kappa_{\lambda_2}$ in the above equation are numerical pre-factors.

* In the anomaly term (<ref>) following are the $r$ dependent functions:

\begin{eqnarray}
\label{L-terms-HM}
& &   {\cal L}_1= {\cal L}_2=\frac{\sqrt{\alpha (r) \left(\sigma (r)-\left(1-\frac{r_h^4}{r^4}\right) t'(r)^2\right)} }{t'(r)^2 \left(\sigma (r)-\left(1-\frac{r_h^4}{r^4}\right) t'(r)^2\right)^4} \nonumber\\
& & \times \left(\alpha '(r) \left(\sigma (r)-\left(1-\frac{r_h^4}{r^4}\right) t'(r)^2\right)+\alpha (r) \left(\sigma '(r)-t'(r) \left(\frac{4 r_h^4}{r^5} t'(r)+2
\left(1-\frac{r_h^4}{r^4}\right) t''(r)\right)\right)\right)^2; \nonumber\\
& &   {\cal L}_3= {\cal L}_4=\frac{\sqrt{\alpha (r) \left(\sigma (r)-\left(1-\frac{r_h^4}{r^4}\right) t'(r)^2\right)}}{t'(r)^2}.
\end{eqnarray}

* Use has been made of the following for the computation of first and second term in entanglement entropy computation in the presence of higher derivative terms:

\begin{eqnarray}
\label{dJ0overdR+d2J0overdR2}
& &
\frac{\delta J_0}{\delta R_{M_1 N_1 P_1 Q_1}}=G^{M_1 M_1}G^{N_1 N_1}G^{P_1 P_1}G^{Q_1 Q_1}\left( R_{P N_1 P_1 Q}  R^Q_{\ \ RSQ_1}  +\frac{1}{2}R_{PQP_1 Q_1}  R^Q_{\ \ RSN_1}\right)R_{M_1}^{\ \ RSP} \nonumber\\
& &
+\left(\delta^{Q_1}_Q R^{HN_1 P_1 K}+\frac{1}{2}\delta^{N_1}_Q R^{HKP_1 Q_1}\right)R_{H}^{\ \ RSM_1} R^Q_{\ \ RSK} \nonumber\\
& & + G^{N_1 N_1}G^{P_1 P_1}G^{Q_1 Q_1} \left( R_{Q_1 MNQ}R^{M_1 MNK}+\frac{1}{2}R_{Q_1 QMN} R^{M_1 KMN}\right)R^Q_{\ \ N_1 P_1 K}
\nonumber\\
& & +G^{M_1 M_1}\left( R_{PMNM_1} R^{HMN Q_1}+\frac{1}{2}  R_{PM_1 MN} R^{H Q_1 MN}\right)R_H^{\ \ N_1 P_1 P}
\end{eqnarray}

\begin{eqnarray}
& &
\frac{\delta^2 J_0}{\delta R_{M_2 N_2 P_2 Q_2} \delta R_{M_1 N_1 P_1 Q_1}}=A_1+A_2+A_3+A_4,
\end{eqnarray}

\begin{eqnarray}
& &
A_1=G^{M_1 M_1}G^{N_1 N_1}G^{P_1 P_1}G^{Q_1 Q_1}\Biggl[\delta^{M_2}_{M_1} G^{N_2 N_2}G^{P_2 P_2}G^{Q_2 Q_2}\left(R_{Q_2 N_1 P_1 Q}R^Q_{\ \ N_2 P_2 Q_1}+\frac{1}{2}R_{Q_2 Q P_1 Q_1}R^Q_{\ \ N_2 P_2 N_1}\right)\nonumber\\
& & + \left(\delta^{M_2}_P\delta^{N_2}_{N_1}\delta^{P_2}_{P_1}\delta^{Q_2}_{Q} R^Q_{\ \ RSQ_1}R_{M_1}^{\ \ RSP}+\frac{1}{2}\delta^{M_2}_P\delta^{N_2}_{Q}\delta^{P_2}_{P_1}\delta^{Q_2}_{Q_1} R^Q_{\ \ RSN_1}R_{M_1}^{\ \ RSP} \right)\nonumber\\
& & + \left(G^{M_2M_2}\delta^{N_2}_{R}\delta^{P_2}_{S}\delta^{Q_2}_{Q_1} R_{M_1}^{\ \ RSP}R_{PN_1P_1M_2}+\frac{1}{2}G^{M_2M_2}\delta^{N_2}_{R}\delta^{P_2}_{S}\delta^{Q_2}_{N_1} R_{M_1}^{\ \ RSP}R_{PM_2P_1Q_1}\right)\Biggr] \nonumber\\
& & \equiv G^{M_1 M_1}G^{N_1 N_1}G^{P_1 P_1}G^{Q_1 Q_1}\Biggl[\delta^{M_2}_{M_1} G^{N_2 N_2}G^{P_2 P_2}G^{Q_2 Q_2}\left(R_{Q_2 N_1 P_1 Q}R^Q_{\ \ N_2 P_2 Q_1}+\frac{1}{2}R_{Q_2 Q P_1 Q_1}R^Q_{\ \ N_2 P_2 N_1}\right)\nonumber\\
& & + \left(\delta^{N_2}_{N_1}\delta^{P_2}_{P_1} R^{Q_2}_{\ \ RSQ_1}R_{M_1}^{\ \ RSM_2}+\frac{1}{2}\delta^{P_2}_{P_1}\delta^{Q_2}_{Q_1} R^{N_2}_{\ \ RSN_1}R_{M_1}^{\ \ RSM_2} \right)\nonumber\\
& & + \left(G^{M_2M_2}\delta^{Q_2}_{Q_1} R_{M_1}^{\ \ N_2 P_2 P}R_{PN_1P_1M_2}+\frac{1}{2}G^{M_2M_2}\delta^{Q_2}_{N_1} R_{M_1}^{\ \ N_2 P_2 P}R_{PM_2P_1Q_1}\right)\Biggr],
\nonumber\\
& & A_2= \Biggl[\left(G^{M_2M_2}G^{Q_2Q_2}G^{N_2N_1}G^{P_2P_1}R^{Q_1}_{\ \ RS Q_2}+\frac{1}{2}G^{M_2M_2}G^{N_2N_2}G^{P_2P_1}G^{Q_2Q_1}R^{N_1}_{\ \ RS N_2} \right)R_{M_2}^{\ \ RS M_1}
\nonumber\\
& & +\left(G^{Q_2M_1}G^{N_2N_2}G^{P_2P_2}R^{M_2 N_1 P_1 K}+\frac{1}{2}G^{Q_2M_1}G^{N_2N_2}G^{P_2P_2}R^{M_2 K P_1 Q_1} \right)R^{N_1}_{\ \ N_2 P_2 K}\nonumber\\
& & +\left(G^{M_2Q_1}R^{H N_1 P_1 Q_2}+\frac{1}{2}G^{M_2N_1}R^{H Q_2 P_1 Q_1} \right)R_{H}^{\ \ N_2 P_2 M_1}\Biggr],\nonumber\\
& &
A_3= \Biggl[G^{N_1 N_1}G^{P_1 P_1}G^{Q_1 Q_1}\Biggl[\Biggl(\delta^{M_2}_{Q_1}\delta^{Q_2}_Q R^{M_1 N_2 P_2 K}+G^{M_2 M_1}G^{N_2 N_2}G^{P_2 P_2}G^{Q_2 K} R_{Q_1 N_2 P_2 Q} \nonumber\\
& & + \frac{1}{2}\delta^{M_2}_{Q_1}\delta^{N_2}_Q R^{M_1 K P_2 Q_2}+\frac{1}{2}G^{M_2 M_1}G^{N_2 K}G^{P_2 P_2}G^{Q_2 Q_2} R_{Q_1 Q P_2 Q_2}\Biggr)R^Q_{\ \ N_1 P_1 K} \nonumber\\
& & + G^{M_2 M_2} \delta^{N_2}_{N_1}\delta^{P_2}_{P_1}\left(R_{Q_1 MN M_2}R^{M_1 MN Q_2}+\frac{1}{2}R_{Q_1 M_2 MN}R^{M_1 Q_2 MN}\right)
\Biggr] \Biggr], \nonumber\\
& & A_4=G^{M_1 M_1}\Biggl[\Biggl(\delta^{M_2}_P \delta^{Q_2}_{M_1}R^{H N_2 P_2 Q_1}+G^{M_2 H}G^{N_2 N_2}G^{P_2 P_2}G^{Q_2 Q_1}R_{P N_2 P_2 M_1}\nonumber\\
& & + \frac{1}{2} \delta^{M_2}_P \delta^{N_2}_{M_1}R^{HQ_1 P_2 Q_2}+\frac{1}{2}G^{M_2 H}G^{N_2 Q_1}G^{P_2 P_2}G^{Q_2 Q_2}R_{P M_1 P_2 Q_2}\Biggr)R_H^{\ \ N_1 P_1 P} \nonumber\\
& & + G^{N_2 N_1}G^{P_2 P_1}G^{Q_2 Q_2}\left(R_{Q_2 MN M_1}R^{M_2 MN Q_1}+\frac{1}{2}R_{Q_2  M_1 MN}R^{M_2 Q_1 MN} \right)\Biggr].
\end{eqnarray}

* The $r$-dependent functions appearing in equation (<ref>) are given below:

\begin{eqnarray}
\label{Z-W-U-V}
& & Z(r)=-\kappa_Z \frac{  M N^{37/10} g_s^{35/6} \log (r) (\log (N)-9 \log (r))}{ r^{14} \alpha _{\theta _1}^3 \alpha _{\theta _2}^2 N_f^{4/3} (\log (N)-3 \log (r))^{7/3}} \nonumber\\
& & \sim -\frac{\pi ^{23/6} M N^{61/20} g_s^{79/12} \kappa _Z \log (r) (\log (N)-9 \log (r))}{r^{14} N_f^{4/3} (\log (N)-3 \log (r))^{7/3}}
, \nonumber\\
& &  W(r)=-\kappa_W \frac{ M N^{37/10} g_s^{35/6} \log (r) (\log (N)-9 \log (r))}{ r^{14} \alpha _{\theta _1}^3 \alpha _{\theta _2}^2 N_f^{4/3} (\log (N)-3 \log (r))^{7/3}} \nonumber\\
& & \sim -\frac{M N^{61/20} g_s^{79/12} \kappa _W \log (r) (\log (N)-9 \log (r))}{r^{14} N_f^{4/3} (\log (N)-3 \log (r))^{7/3}}
, \nonumber\\
& & U(r)=-\kappa_U \frac{ M^2 N N_f g_s^3 \log (r) (\log (N)-45 \log (r))}{r^4 \alpha _{\theta _1} \alpha _{\theta _2}^6 (\log (N)-3 \log (r))^3} \nonumber\\
& & \sim -\frac{M^2 N_f g_s^{15/4} \kappa _U \log (r) (\log (N)-45 \log (r))}{\sqrt[4]{N} r^4 (\log (N)-3 \log (r))^3}
, \nonumber\\
& & V(r)=\kappa_V\frac{ M^2 N N_f g_s^3}{ r^4 \alpha _{\theta _1} \alpha _{\theta _2}^6 \log ^2(r)}  \sim \frac{M^2 N_f g_s^{15/4} \kappa _V}{\sqrt[4]{N} r^4 \log ^2(r)},
\end{eqnarray}

where $\kappa_Z,\kappa_W,\kappa_U$ and $\kappa_V$ are the numerical factors including $\left(\frac{8}{q_\alpha +1}\right)$.

* To obtain the EOM for $t(r)$ we need to calculate the following derivatives from equations (<ref>), (<ref>), (<ref>) and (<ref>):

\begin{eqnarray}
\label{DLag-tprime}
& &\frac{\delta {\cal L}^{\rm HM}_{\rm Total}}{\delta t'(r)} =\beta  \Biggl(\frac{\left(r-r_h\right){}^{3/2} p_4^{\beta }\left(r_h\right) \left({{\cal A}_1} t'(r)+{{\cal A}_2} t''(r)\right)}{N^{7/10} t'(r)^2}+\frac{p_5^{\beta }\left(r_h\right) \left({{\cal A}_3}
t'(r)+{{\cal A}_4} t''(r)\right)}{N^{7/10} t'(r)^2}+\frac{\left(r-r_h\right){}^{13/2} p_7^{\beta }\left(r_h\right)}{N^{57/10} t'(r)^4}\nonumber\\& &
\hskip 0.6in  +\frac{ p_6^{\beta
}\left(r_h\right)}{\left(r-r_h\right)N^{7/10} t'(r)} +\frac{\left(r-r_h\right){}^{3/2} p_1^{\beta }\left(r_h\right)}{N^{5/4} t'(r)}+\frac{N^{3/10} p_8^{\beta }\left(r_h\right)}{\sqrt{r-r_h} t'(r)^3}+\frac{N^{3/10}
p_9^{\beta }\left(r_h\right)}{\sqrt{r-r_h} t'(r)^3}+\frac{p_2^{\beta }\left(r_h\right)}{\sqrt{r-r_h} t'(r)^3}\nonumber\\& & \hskip 0.6in
+\frac{p_3^{\beta }\left(r_h\right)}{\sqrt{r-r_h} t'(r)^3}
\left(r-r_h\right){}^{3/2} {Y_5}\left(r_h\right) t'(r)\Biggr)+N^{3/10} \left(r-r_h\right){}^{5/2} p_1\left(r_h\right) t'(r),
\end{eqnarray}

\begin{eqnarray}
\label{DLag-tprimeprime}
& &\frac{\delta {\cal L}^{\rm HM}_{\rm Total}}{\delta t''(r)} =\frac{\beta  {\cal F}^{\beta}({r_h}) (r-{r_h})^{5/2}}{N^{7/10} t'(r)},
\end{eqnarray}

\begin{eqnarray}
\label{fi[rh]-HM}
& &
p_1(r_h)=\kappa_{p_1}\left(\frac{M N_f^{5/3} g_s^{7/3} \sqrt{\kappa _{\alpha }} \kappa _{\lambda } \log \left(r_h\right) \left(\log (N)-9 \log \left(r_h\right)\right) \left(\log (N)-3 \log\left(r_h\right)\right){}^{5/6}}{r_h^{3/2} \sqrt{\kappa _{\sigma }}}\right),\nonumber\\
& & p_1^\beta(r_h)=-\kappa_{p_1^\beta}\left(\frac{M^2 \sqrt[4]{N} x_R^2 N_f^{4/3} g_s^{17/6} \sqrt{\kappa _{\alpha }} \kappa _U \log \left(r_h\right) \left(\log (N)-45 \log \left(r_h\right)\right)}{r_h^{5/2} \sqrt{\kappa _{\sigma }} \left(\log
(N)-3 \log \left(r_h\right)\right){}^{5/2}}\right),\nonumber\\
& & p_2^\beta(r_h)= \kappa_{p_2^\beta} \left(\frac{M^2 x_R^2 N_f^{4/3} g_s^{23/6} \kappa _U \log \left(r_h\right) \sqrt{\kappa _{\alpha } \kappa _{\sigma }} \left(\log (N)-45 \log \left(r_h\right)\right)}{r_h^{9/2} \left(\log (N)-3 \log
\left(r_h\right)\right){}^{5/2}}\right),\nonumber\\
& & p_3^\beta(r_h)=-\kappa_{p_3^\beta}\Biggl(\frac{M^2 x_R^2 (540 \log (2)-107) N_f^{4/3} g_s^{23/6} \sqrt{\kappa _{\alpha } \kappa _{\sigma }}}{r_h^{9/2} \log ^2\left(r_h\right) \left(\log (N)-3 \log\left(r_h\right)\right){}^{5/2}} \nonumber\\
& & \times \left(28 \kappa _V \left(\log (N)-3 \log \left(r_h\right)\right){}^3-27 \kappa _U \log ^3\left(r_h\right) \left(\log (N)-45 \log \left(r_h\right)\right)\right)\Biggr),\nonumber\\
& & p_4^\beta(r_h)=\kappa_{p_4^\beta}F(g_s,r_h,N_f,M),\nonumber\\
& & p_5^\beta(r_h)=\kappa_{p_5^\beta}F(g_s,r_h,N_f,M),\nonumber\\
& & p_6^\beta(r_h)=-\kappa_{p_6^\beta}\left(\frac{M x_R^4 \sqrt[3]{N_f} g_s^2 \kappa _W \log \left(r_h\right) \left(\kappa _{\alpha } \kappa _{\sigma }\right){}^{5/2} \left(\log (N)-9 \log \left(r_h\right)\right) \sqrt[6]{\log (N)-3 \log
\left(r_h\right)}}{ r_h^{5/2} \kappa _{\sigma }^5}\right),\nonumber\\
& & p_7^\beta(r_h)=-\kappa_{p_7^\beta}\left(\frac{M \sqrt[3]{N_f} r_h^{25/2} \kappa _Z \log \left(r_h\right) \left(\kappa _{\alpha } \kappa _{\sigma }\right){}^{5/2} \left(\log (N)-9 \log \left(r_h\right)\right) \sqrt[6]{\log (N)-3 \log
\left(r_h\right)}}{g_s^3 \kappa _{\sigma }^{10}}\right),\nonumber\\
& & p_8^\beta(r_h)=\kappa_{p_8^\beta}\left(\frac{M x_R^4 \sqrt[3]{N_f} g_s^3 \kappa _W \log \left(r_h\right) \left(\kappa _{\alpha } \kappa _{\sigma }\right){}^{5/2} \left(\log (N)-9 \log \left(r_h\right)\right) \sqrt[6]{\log (N)-3 \log
\left(r_h\right)}}{r_h^{9/2} \kappa _{\sigma }^4}\right),\nonumber\\
& & p_9^\beta(r_h)=\kappa_{p_9^\beta}\left(\frac{M \sqrt[3]{N_f} g_s^3 \kappa _Z \log \left(r_h\right) \left(\kappa _{\alpha } \kappa _{\sigma }\right){}^{5/2} \left(\log (N)-9 \log \left(r_h\right)\right) \sqrt[6]{\log (N)-3 \log
\left(r_h\right)}}{r_h^{9/2} \kappa _{\sigma }^4}\right),\nonumber\\
& & {\cal F}^\beta=-\kappa_{{\cal F}^\beta}\left(\frac{M \sqrt[3]{N_f} g_s^2 \kappa _{\alpha }^{5/2} \log \left(r_h\right) \left(\log (N)-9 \log \left(r_h\right)\right) \sqrt[6]{\log (N)-3 \log \left(r_h\right)} \left(x_R^4 \kappa _W+2 \kappa
_Z\right)}{r_h^{5/2} \kappa _{\sigma }^{5/2}}\right),
\end{eqnarray}

where $\kappa_{p_1},\kappa_{p_{i=1,..,9}^\beta}$ and $\kappa_{{\cal F}^\beta}$ are the numerical factors. From equation (<ref>) and using equations (<ref>) and (<ref>), the EOM corresponding to the embedding $t(r)$, given by $\frac{d \left(\frac{\delta {\cal L}}{\delta t'(r)}\right)}{dr} = \frac{d^2\left(\frac{\delta{\cal L}}{\delta t''(r)}\right)}{dr^2}$ turns out to be:

\begin{eqnarray}
\label{EOM-HM}
& &\hskip -0.3in \beta  \left(-\frac{N^{3/10} p_8^{\beta }\left(r_h\right)}{2 (r-r_h)^{3/2} t'(r)^3}-\frac{N^{3/10} p_9^{\beta }\left(r_h\right)}{2 (r-r_h)^{3/2} t'(r)^3}-\frac{3 N^{3/10} t''(r) p_8^{\beta
}\left(r_h\right)}{\sqrt{r-r_h} t'(r)^4}-\frac{3 N^{3/10} t''(r) p_9^{\beta }\left(r_h\right)}{\sqrt{r-r_h} t'(r)^4}\right)+N^{3/10} (r-r_h)^{5/2} p_1\left(r_h\right)
t''(r) \nonumber\\
& & \hskip -0.3in  +\frac{5}{2} N^{3/10} (r-r_h)^{3/2} p_1\left(r_h\right) t'(r)=-\frac{\beta  M \sqrt[3]{N_f} g_s^2 \sqrt{r-r_h} \kappa _{\alpha }^{5/2} \log \left(r_h\right) \left(\log (N)-9 \log
\left(r_h\right)\right) \sqrt[6]{\log (N)-3 \log \left(r_h\right)} \left(x_R^4 \kappa _W+2 \kappa _Z\right)}{4 N^{7/10} r_h^{5/2} \kappa _{\sigma }^{5/2} t'(r)^3}\nonumber\\
& & \times \left(8 \left(r-r_h\right){}^2 t''(r)^2-4 \left(r-r_h\right) t'(r) \left(\left(r-r_h\right)
t^{(3)}(r)+5 t''(r)\right)+15 t'(r)^2\right).
\end{eqnarray}

§ ISLAND SURFACE MISCELLANIA

In this appendix we are listing the various functions appearing in the entanglement entropy of the island surface at ${\cal O}(\beta)$. We have also computed the derivatives of the Lagrangian of the island surface here which have been used in obtaining the equation of motion of the embedding function of the island surface.

* $\lambda_{3,4}(r)$ appearing in the Wald entanglement entropy term (<ref>) of the island surface are given below:

\begin{eqnarray}
\label{lambda3-4}
& &
\hskip -.5in \lambda_3(r)= \kappa_{\lambda_3}\frac{ M^7 N^{7/10} N_f^{7/3} g_s^7 r_h^{14} \log (N) \log ^7(r) (5 \log (N)-12 \log (r))^3}{ r^{16} \alpha _{\theta _1}^3 \alpha
_{\theta _2}^2 (\log (N)-3 \log (r))^{14/3}} \nonumber\\
& & \sim \frac{M^7 \sqrt[20]{N} \log (2) (\log (64)-1) N_f^{7/3} g_s^{31/4} r_h^{14} \kappa _{\lambda _3} \log (N) \log ^7(r) (5 \log (N)-12 \log (r))^3}{r^{16} (\log (N)-3 \log (r))^{14/3}},\nonumber\\
& &
\hskip -.5in \lambda_4(r)=\kappa_{\lambda_4}\frac{ M^3 N^{7/10} \sqrt[3]{N_f} g_s^3 r_h^{14} \log (N) \log ^3(r) (5 \log (N)-12 \log (r))}{ r^{16} \alpha _{\theta _1}^3 \alpha _{\theta _2}^2
(\log (N)-3 \log (r))^{8/3}} \nonumber\\
& & \sim  \frac{M^3 \sqrt[20]{N} \log (2) (\log (64)-1) \sqrt[3]{N_f} g_s^{15/4} r_h^{14} \kappa _{\lambda _4} \log (N) \log ^3(r) (5 \log (N)-12 \log (r))}{r^{16} (\log (N)-3 \log (r))^{8/3}},
\end{eqnarray}

where $\kappa_{\lambda_3}$ and $\kappa_{\lambda_4}$ are the numerical factors.

* In the anomaly term (<ref>) following are the $r$ dependent functions:

\begin{eqnarray}
\label{Z-W-U-V-1}
& &
Z_1(r)=\kappa_{Z_1}\frac{ M N^{37/10} g_s^{35/6} r_h^2 \log (N) \log (r)}{ r^{16} \alpha _{\theta _1}^3 \alpha _{\theta _2}^2 N_f^{4/3} (\log (N)-3 \log (r))^{7/3}} \nonumber\\
& & \sim \frac{M N^{61/20} \log (2) (\log (64)-1) g_s^{79/12} r_h^2 \log (N) \log (r)  \kappa _{Z_1}}{r^{16} N_f^{4/3} (\log (N)-3 \log
& & W_1(r)=\kappa_{W_1}\frac{ M N^{37/10} g_s^{35/6} \log (N) \log (r)}{ r^8 \alpha _{\theta _1}^3 \alpha _{\theta _2}^2 N_f^{4/3} r_h^6 (\log (N)-3 \log (r))^{7/3}} \nonumber\\
& & \sim \frac{ M N^{61/20} \log (2) (\log (64)-1) g_s^{79/12} \kappa _{W_1} \log (N) \log (r)}{ r^8 N_f^{4/3} r_h^6 (\log (N)-3 \log (r))^{7/3}},
\nonumber\\
& & U_1(r)=\kappa_{U_1}\frac{M^2 N N_f g_s^3 r_h^{12} \log ^2(N) \log ^3(r) (\log (N)-17 \log (r)) (\log (N)-9 \log (r))}{ r^{15} \alpha _{\theta _1} \alpha _{\theta _2}^6 (\log (N)-3 \log (r))^4},\nonumber\\
& & \sim \frac{M^2 N_f g_s^{15/4} r_h^{12} \kappa _{U_1} \log ^2(N) \log ^3(r) (\log (N)-17 \log (r)) (\log (N)-9 \log (r))}{  \sqrt[4]{N} r^{15} (\log (N)-3 \log (r))^4},\nonumber\\
& & V_1(r)=-\kappa_{V_1}\frac{M^2 N N_f g_s^3 r_h^8 \log ^5(N) \log ^3(r) (\log (N)-6 \log (r))}{ r^{11} \alpha _{\theta _1} \alpha _{\theta _2}^6 (\log (N)-9 \log (r))^3 (\log (N)-3 \log (r))^3} \nonumber\\
& & \sim -\frac{M^2  N_f g_s^{15/4} r_h^8 \kappa _{V_1} \log ^5(N) \log ^3(r) (\log (N)-6 \log (r))}{\sqrt[4]{N} r^{11} (\log (N)-9 \log (r))^3 (\log (N)-3 \log (r))^3},
\end{eqnarray}

where $\kappa_{Z_1},\kappa_{W_1},\kappa_{U_1}$ and $\kappa_{V_1}$ are the numerical pre-factors which also includes $\left(\frac{8}{q_\alpha +1}\right)$.

* Using equations (<ref>), (<ref>),(<ref>), (<ref>) and (<ref>) we obtain:

\begin{eqnarray}
\label{dLagoverdx}
& & \frac{\delta {\cal L}_{\rm {Total}}^{\rm {IS}}}{\delta x(r)} = \beta  \left(\frac{N^{3/10} f^\beta_1(r_h) x(r)}{\sqrt{r-r_h}}+\left(\frac{ f^\beta_2(r_h)}{\sqrt{r-r_h}}+\frac{
f^\beta_3(r_h)}{N^{1/4}\sqrt{r-r_h}}\right)\frac{x(r)}{x'(r)^2}+\frac{N^{3/10} f^\beta_4(r_h)
x(r)^3}{\sqrt{r-r_h} x'(r)^2}+\frac{N^{13/10} x(r) Y_1(r_h)}{\sqrt{r-r_h}}\right) \nonumber\\
& &  \hskip 0.6in +\frac{N^{13/10}
f_1(r_h) x(r)}{\sqrt{r-r_h}},\nonumber\\
& &
\end{eqnarray}

\begin{eqnarray}
\label{dLagoverdxprime}
& & \frac{\delta{\cal L}_{\rm {Total}}^{\rm {IS}}}{\delta x'(r)} = \beta  \Biggl[\frac{{F^\beta_1}(r_h) \sqrt{r-r_h} x(r)^2 x'(r)}{N^{7/10}} +\frac{F^\beta_{10}(r_h) \sqrt{r-r_h} x(r)^4}{N^{7/10} x'(r)}+\frac{F^\beta_{11}(r_h)
\sqrt{r-r_h}}{N^{7/10} x'(r)}+\frac{N^{3/10} F^\beta_{12}(r_h) x(r)^4}{\sqrt{r-r_h}
& &+\frac{N^{3/10} F^\beta_{13}(r_h)}{\sqrt{r-r_h} x'(r)^3}   +\frac{F^\beta_2(r_h)
\sqrt{r-r_h} x(r)^2}{N x'(r)} +\frac{ {F^\beta_3}(r_h) \sqrt{r-r_h}
x(r)^2}{N^{5/4} x'(r)}+\frac{{F^\beta_4}(r_h) x(r)^2}{\sqrt{r-r_h} x'(r)^3}+\frac{
{F^\beta_5}(r_h) x(r)^2}{N^{1/4}\sqrt{r-r_h} x'(r)^3}\nonumber\\
& & +\frac{{F^\beta_6}(r_h) (r-r_h)^{3/2}
x(r)^4 \left(2 x'(r)+r x''(r)\right)}{N^{7/10} x'(r)^2}
+\frac{{F^\beta_7}(r_h) (r-r_h)^{3/2} \left(2
x'(r)+r x''(r)\right)}{N^{7/10} x'(r)^2}\nonumber\\
& &+\frac{{F^\beta_8}(r_h) \sqrt{r-r_h} x(r)^4}{N^{7/10}
x'(r)}+\frac{{F^\beta_9}(r_h) \sqrt{r-r_h}}{N^{7/10} x'(r)} +N^{3/10} x(r)^2{Y_3}(r_h) x'(r)\Biggr]+N^{3/10} {F_1}(r_h) \sqrt{r-r_h} x(r)^2 x'(r),
\end{eqnarray}

\begin{eqnarray}
\label{dLagoverdxprimexprime}
& & \frac{\delta{\cal L}_{\rm {Total}}^{\rm {IS}}}{\delta x^{\prime\prime}(r)} = \beta   \left(\frac{{\cal F}^\beta_1(r_h) (r-r_h)^{3/2}}{N^{7/10}}+\frac{{\cal F}^\beta_2(r_h)
\end{eqnarray}

where we have defined $F_1(r_h),F^\beta_{i=1,..,13}(r_h),f_1(r_h),f^\beta_{i=1,..,4}(r_h)$ and $Y_{i=1,3}(r_h)$ for the island surface as:

\begin{eqnarray}
\label{Fbetai[rh]s}
& & F_1(r_h)=\frac{\kappa_{F_1} M \log (2) (\log (64)-1) N_f^{5/3} g_s^{7/3} \sqrt{r_h} \kappa _{\alpha } \kappa _{\lambda _5} \log (N) \log \left(r_h\right) \left(\log (N)-3 \log \left(r_h\right)\right){}^{2/3}}{\sqrt{\kappa _{\alpha } \kappa _{\sigma }}},\nonumber\\
& & F_1^\beta(r_h) =\frac{\kappa_{F^\beta_1}M^3 \log (2) (\log (64)-1) N_f^{2/3} g_s^{17/6} \sqrt{r_h} \kappa _{\alpha } \log (N) \log ^3\left(r_h\right) \left(5 \log (N)-12 \log \left(r_h\right)\right)}{\sqrt{\kappa _{\alpha } \kappa _{\sigma }} \left(\log (N)-3 \log \left(r_h\right)\right){}^{13/3}} \nonumber\\
& & \times \left( 81 \left(16+\sqrt{2}\right) M^4 N_f^2 g_s^4 \kappa _{\lambda _3} \log ^4\left(r_h\right) \left(5 \log (N)-12 \log \left(r_h\right)\right){}^2+4096 \left(4+\sqrt{2}\right) \pi ^4 \kappa _{\lambda _4}
\left(\log (N)-3 \log \left(r_h\right)\right){}^2\right),\nonumber\\
& & F_2^\beta(r_h) = -\frac{\kappa_{F^\beta_2}M^2 g_s^{3/2} \kappa _{\alpha } \kappa _{U_1} \log ^2(N) \log ^3(r) \left(\log (N)-17 \log \left(r_h\right)\right) \left(\log (N)-9 \log \left(r_h\right)\right) \left(N_f g_s
\left(\log (N)-3 \log \left(r_h\right)\right)\right){}^{4/3}}{\sqrt{r_h} \sqrt{\kappa _{\alpha } \kappa _{\sigma }} \left(\log (N)-3 \log
\left(r_h\right)\right){}^5},\nonumber\\
& & F_3^\beta(r_h) =\frac{\kappa_{F^\beta_3}M^4 N_f^{7/3} g_s^{79/12} \kappa _{\alpha } \log ^4(N) \log ^6\left(r_h\right)}{ r_h^{7/2} \sqrt{\kappa _{\alpha } \kappa _{\sigma }}
\left(\log (N)-9 \log \left(r_h\right)\right){}^6 \left(\log (N)-3 \log \left(r_h\right)\right){}^{23/3}} \nonumber\\
& & \times \left(\left(\kappa _{V_1} \log ^3(N) \left(-9 \log (N) \log \left(r_h\right)+18 \log ^2\left(r_h\right)+\log ^2(N)\right)-6 \kappa _{U_1} \left(\log (N)-17 \log \left(r_h\right)\right) \left(\log (N)-9 \log
\left(r_h\right)\right){}^4\right){}^2\right)
& & F_4^\beta(r_h) =\frac{\kappa_{F^\beta_4}M^2 N_f^{4/3} g_s^{23/6} \kappa _{U_1} \log ^2(N) \log ^3\left(r_h\right) \sqrt{\kappa _{\alpha } \kappa _{\sigma }} \left(\log (N)-17 \log \left(r_h\right)\right) \left(\log
(N)-9 \log \left(r_h\right)\right)}{r_h^{7/2} \left(\log (N)-3 \log \left(r_h\right)\right){}^{11/3}},\nonumber\\
& &  F_5^\beta(r_h) = -\frac{\kappa_{F^\beta_5}M^4  g_s^{21/4} \log ^4(N) \sqrt{\kappa _{\alpha } \kappa _{\sigma }} \left(N_f g_s \left(\log (N)-3 \log \left(r_h\right)\right)\right){}^{7/3}}{r_h^{13/2} \left(\log (N)-9 \log \left(r_h\right)\right){}^6 \left(\log (N)-3 \log \left(r_h\right)\right){}^{10}}  \nonumber\\
& &\hskip -0.2in \left(\kappa _{V_1} \log ^3(N) \log ^3\left(r_h\right) \left(-9 \log (N) \log \left(r_h\right)+18 \log ^2\left(r_h\right)+\log ^2(N)\right)
-6 \kappa _{U_1} \log ^3(r) \left(\log (N)-17 \log
\left(r_h\right)\right) \left(\log (N)-9 \log \left(r_h\right)\right){}^4\right){}^2;\nonumber
\end{eqnarray}

\begin{eqnarray}
& & F_6^\beta(r_h) =-\frac{\kappa_{F^\beta_6} M  \sqrt[3]{N_f} g_s^2 \kappa _{W_1} \log (N) \log \left(r_h\right) \left(\kappa _{\alpha } \kappa _{\sigma }\right){}^{5/2}}{r_h^{5/2} \kappa _{\sigma }^5 \left(\log (N)-3 \log \left(r_h\right)\right){}^{2/3}},\nonumber\\
& &   F_7^\beta(r_h) =-\frac{\kappa_{F^\beta_7} M \log (2) (\log (64)-1) \sqrt[3]{N_f} g_s^2 \kappa _{Z_1} \log (N) \log \left(r_h\right) \left(\kappa _{\alpha } \kappa _{\sigma }\right){}^{5/2}}{r_h^{5/2} \kappa _{\sigma }^5 \left(\log (N)-3 \log \left(r_h\right)\right){}^{2/3}}, \nonumber\\
& & F_8^\beta(r_h) =\frac{\kappa_{F^\beta_8} M \log (2) (\log (64)-1) \sqrt[3]{N_f} g_s^2 \kappa _{W_1} \log (N) \log \left(r_h\right) \left(\kappa _{\alpha } \kappa _{\sigma }\right){}^{5/2}}{r_h^{3/2} \kappa _{\sigma }^5 \left(\log (N)-3 \log \left(r_h\right)\right){}^{2/3}},\nonumber\\
& & F_9^\beta(r_h) = \frac{\kappa_{F^\beta_9} M \log (2) (\log (64)-1) \sqrt[3]{N_f} g_s^2 \kappa _{Z_1} \log (N) \log \left(r_h\right) \left(\kappa _{\alpha } \kappa _{\sigma }\right){}^{5/2}}{ r_h^{3/2} \kappa _{\sigma }^5 \left(\log (N)-3 \log \left(r_h\right)\right){}^{2/3}},\nonumber\\
& & F_{10}^\beta(r_h) = -\frac{\kappa_{F^\beta_{10}} M \log (2) (\log (64)-1) \sqrt[3]{N_f} g_s^2 \kappa _{W_1} \log (N) \log \left(r_h\right) \left(\kappa _{\alpha } \kappa _{\sigma }\right){}^{5/2}}{r_h^{3/2} \kappa _{\sigma }^5 \left(\log (N)-3 \log \left(r_h\right)\right){}^{2/3}};\nonumber
\end{eqnarray}

\begin{eqnarray}
& & F_{11}^\beta(r_h) = -\frac{\kappa_{F^\beta_{11}}M \log (2) (\log (64)-1) \sqrt[3]{N_f} g_s^2 \kappa _{Z_1} \log (N) \log \left(r_h\right) \left(\kappa _{\alpha } \kappa _{\sigma }\right){}^{5/2}}{r_h^{3/2} \kappa _{\sigma }^5 \left(\log (N)-3 \log \left(r_h\right)\right){}^{2/3}}, \nonumber\\
& &F_{12}^\beta(r_h) =-\frac{\kappa_{F^\beta_{12}} M \log (2) (\log (64)-1) \sqrt[3]{N_f} g_s^3 \kappa _{W_1} \log (N) \log \left(r_h\right) \left(\kappa _{\alpha } \kappa _{\sigma }\right){}^{5/2}}{r_h^{9/2} \kappa _{\sigma }^4 \left(\log (N)-3 \log \left(r_h\right)\right){}^{2/3}},
\nonumber\\
& & F_{13}^\beta(r_h) =-\frac{\kappa_{F^\beta_{13}} M \log (2) (\log (64)-1) \sqrt[3]{N_f} g_s^3 \kappa _{Z_1} \log (N) \log \left(r_h\right) \left(\kappa _{\alpha } \kappa _{\sigma }\right){}^{5/2}}{r_h^{9/2} \kappa _{\sigma }^4 \left(\log (N)-3 \log \left(r_h\right)\right){}^{2/3}};
\end{eqnarray}

\begin{eqnarray}
\label{fi[rh]s}
& & f_1(r_h)=\frac{\kappa_{f_1} M \log (2) (\log (64)-1) N_f^{5/3} g_s^{10/3} \kappa _{\lambda _5} \log (N) \log \left(r_h\right) \sqrt{\kappa _{\alpha } \kappa _{\sigma }} \left(\log (N)-3 \log
\left(r_h\right)\right){}^{2/3}}{r_h^{5/2}},\nonumber\\
& & f^\beta_1(r_h)=\frac{\kappa_{f^\beta_1}M^3  N_f^{2/3} g_s^{23/6} \log (N) \log ^3\left(r_h\right) \sqrt{\kappa _{\alpha } \kappa _{\sigma }} \left(5 \log (N)-12 \log \left(r_h\right)\right)}{ r_h^{5/2} \left(\log (N)-3 \log \left(r_h\right)\right){}^{13/3}} \nonumber\\
& & \times \left(\frac{M^3 \log (2) (\log (64)-1) N_f^{2/3} g_s^{23/6} \log (N) \log ^3\left(r_h\right) \sqrt{\kappa _{\alpha } \kappa _{\sigma }} \left(5 \log (N)-12 \log \left(r_h\right)\right)}{1572864 \sqrt[3]{2}
3^{5/6} \pi ^{77/12} r_h^{5/2} \left(\log (N)-3 \log \left(r_h\right)\right){}^{13/3}}\right),\nonumber\\
& & f^\beta_2(r_h)=-\frac{\kappa_{f^\beta_2}M^2  N_f^{4/3} g_s^{23/6} \kappa _{U_1} \log ^2(N) \log ^3\left(r_h\right) \sqrt{\kappa _{\alpha } \kappa _{\sigma }} \left(\log (N)-17 \log \left(r_h\right)\right) \left(\log
(N)-9 \log \left(r_h\right)\right)}{r_h^{7/2} \left(\log (N)-3 \log \left(r_h\right)\right){}^{11/3}},\nonumber\\
& & f^\beta_3(r_h)=\frac{\kappa_{f^\beta_3}M^4 g_s^{21/4} \log ^4(N) \sqrt{\kappa _{\alpha } \kappa _{\sigma }} \left(N_f g_s \left(\log (N)-3 \log \left(r_h\right)\right)\right){}^{7/3}}{r_h^{13/2} \left(\log (N)-9 \log \left(r_h\right)\right){}^6 \left(\log (N)-3 \log \left(r_h\right)\right){}^{10}} \nonumber\\
& & \times \left(\kappa _{V_1} \log ^3(N) \log ^3\left(r_h\right) \left(-9 \log (N) \log \left(r_h\right)+18 \log ^2\left(r_h\right)+\log ^2(N)\right)-6 \kappa _{U_1} \log ^3(r) \left(\log (N)-17 \log
\left(r_h\right)\right) \left(\log (N)-9 \log \left(r_h\right)\right){}^4\right){}^2,\nonumber\\
& & f^\beta_4 (r_h)=\frac{\kappa_{f^\beta_4} M \log (2) (\log (64)-1) \sqrt[3]{N_f} g_s^3 \kappa _{W_1} \log (N) \log \left(r_h\right) \left(\kappa _{\alpha } \kappa _{\sigma }\right){}^{5/2}}{r_h^{9/2} \kappa _{\sigma }^4 \left(\log (N)-3 \log \left(r_h\right)\right){}^{2/3}}; \nonumber
\end{eqnarray}

\begin{eqnarray}
\label{FD-beta-i}
& & {\cal F}^\beta_1(r_h)=-\frac{\kappa_{{\cal F}^\beta_1}M \log (2) (\log (64)-1) \sqrt[3]{N_f} g_s^2 \kappa _{\alpha }^{5/2} \kappa _{W_1} \log (N) \log \left(r_h\right)}{ r_h^{3/2} \kappa _{\sigma }^{5/2}
\left(\log (N)-3 \log \left(r_h\right)\right){}^{2/3}},\nonumber\\
& &{\cal F}^\beta_2(r_h) =-\frac{\kappa_{{\cal F}^\beta_1} M \log (2) (\log (64)-1) \sqrt[3]{N_f} g_s^2 \kappa _{\alpha }^{5/2} \kappa _{Z_1} \log (N) \log \left(r_h\right)}{r_h^{3/2} \kappa _{\sigma }^{5/2}\left(\log (N)-3 \log \left(r_h\right)\right){}^{2/3}},
\end{eqnarray}

\begin{eqnarray}
\label{Yi[rh]s}
& & Y_1(r_h) = \frac{\kappa_{Y_1}{\cal C}_{\theta_1 x}M N_f^{5/3} g_s^{10/3} \log ^2\left(r_h\right) \sqrt{\kappa _{\alpha } \kappa _{\sigma }} \left(\log (N)-3 \log \left(r_h\right)\right){}^{2/3}}{ r_h^{3/2}},\nonumber\\
& & Y_3(r_h) =\frac{\kappa_{Y_3} {\cal C}_{\theta_1 x} M  N_f^{5/3} g_s^{7/3} r_h^{3/2} \kappa _{\alpha } \log ^2\left(r_h\right) \left(\log (N)-3 \log \left(r_h\right)\right){}^{2/3}}{ \sqrt{\kappa _{\alpha } \kappa _{\sigma }}} ,
\end{eqnarray}

where $\kappa_{F_1},\kappa_{F^\beta_{i=1,..,13}},\kappa_{f_1},\kappa_{f^\beta_{i=1,..,4}},\kappa_{{\cal F}^\beta_{1,2}},\kappa_{Y_{1,3}}$ are numerical factors and ${\cal C}_{\theta_1 x}$ is the constant of integration appearing in the solution to ${\cal O}(l_p^6)$ correction to the metric component along the toroidal analog $T^2(\theta_1,x)$ of the vanishing two-cycle $S^2(\theta_1,\phi_1)$.

§ POSSIBLE TERMS APPEARING IN HOLOGRAPHIC ENTANGLEMENT ENTROPIES

In this appendix we have listed all the possible terms that we obtain from differentiating Lagrangian of the ${\cal M}$-theory dual inclusive of ${\cal O}(R^4)$ corrections. In appendix <ref>, we have listed all the terms for Hartman-Maldacena-like surface and in appendix <ref> we have listed all the possible terms for the island surface.

§.§ Hartman-Maldacena-like Surface

Square root of the determinant of the induced metric (<ref>) for the Hartman-Maldacena-like surface scale as $N^{7/10}$ and is given below:

\begin{eqnarray}
\label{sqrtminusg-HM}
& &
\sqrt{-g}\sim \Biggl(\frac{M N^{7/10} \sqrt{g_s} \left(N_f (\log (N)-3 \log (r))\right){}^{5/3} \sqrt{\alpha (r) \left(\sigma (r)-\left(1-\frac{r_h^4}{r^4}\right) t'(r)^2\right)} }{\alpha _{\theta _1}^3 \alpha _{\theta _2}^2}\nonumber\\
& & \times \left(N_f g_s \left(r^2 (\log (N) (2 \log
(r)+1)+3 (1-6 \log (r)) \log (r))-r_h^2 \left(\log (N) (2 \log (r)+1)-18 (6 r+1) \log ^2(r)\right)\right)+8 \pi  \left(r_h^2-r^2\right) \log (r)\right)\Biggr).\nonumber\\
\end{eqnarray}

There are four terms that we obtain from $\frac{\partial J_0}{\partial R_{txtx}}$:

\begin{eqnarray}
\label{Wald-J0-i}
& & \hskip -0.3in (i)\int d{\cal V}_9 \sqrt{-g}\left( \left(G^{xx}\right)^2 \left(G^{tt}\right)^2\left(R_{PxtQ} + \frac{1}{2}R_{PQtx}\right)R_{t}^{\ \ RSP}R^Q_{\ \ RSx}\right)\nonumber\\
& & \sim \int d{\cal V}_9 \sqrt{-g} \Biggl(-\frac{r_h^8 \left(-9 \log ^2(N) (\log (N)-6 \log (r))^2-13 (\log (N)-3 \log (r))^4-21 (\log (N)-3 \log (r))^2\right)}{\sqrt{N} r^{12} N_f^{10/3} \sqrt{g_s} (\log (N)-3 \log (r))^{22/3}}\Biggr);\nonumber\\
& & \hskip -0.3in (ii)\int d{\cal V}_9 \sqrt{-g}\left( R^{HxtK}R_H^{\ \ RSt}R^x_{\ \ RSK} + \frac{1}{2}R^{HKtx}R_H^{\ \ RSt}R^Q_{\ \ RSK} \right)\nonumber\\
& & \sim \int d{\cal V}_9 \sqrt{-g} \Biggl(\frac{1}{\sqrt{N} r^{12} N_f^{22/3} \sqrt{g_s} r_h^8 (\log (N)-3 \log (r))^{22/3}}\Biggr) ;
\nonumber\\
& & \hskip -0.3in (iii)\int d{\cal V}_9 \sqrt{-g}  \left(\left(G^{xx}\right)^2 G^{tt}\left(R_{PxtQ} + \frac{1}{2}R_{PQtx}\right)R_{t}^{\ \ RSP}R^Q_{\ \ RSx}\right)\nonumber\\
& & \hskip -.5in \sim \int d{\cal V}_9 \Biggl(\lambda_1(r) \sqrt{\alpha (r) \left(\sigma (r)-\left(1-\frac{r_h^4}{r^4}\right) t'(r)^2\right)}\Biggr) ;\nonumber\\
& & \hskip -0.3in (iv)\int d{\cal V}_9 \sqrt{-g} \left( G^{tt} \left(R^{HMNx}R_{PMNt} + \frac{1}{2}R^{HxMN}R_{PtMN}\right)R_H^{\ \ xtP}\right)\nonumber\\
& & \sim \int d{\cal V}_9 \Biggl(\lambda_2(r) \sqrt{\alpha (r) \left(\sigma (r)-\left(1-\frac{r_h^4}{r^4}\right) t'(r)^2\right)}\Biggr).\nonumber\\
\end{eqnarray}

From equation (<ref>) it is clear that (iii) and (iv) terms scale as $N^{7/10}$ whereas from equations (<ref>) and (<ref>) we can see that (i) and (ii) terms scale as $N^{1/5}$. Therefore (iii) and (iv) terms are the most dominant terms in the large-$N$ limit.

We obtain thirteen terms from $\frac{\partial^2 J_0}{\partial R_{xixj}\partial R_{xmxl}} K_{tij}K_{tml}$ which are listed below:

\begin{eqnarray}
& & \hskip -0.3in (i) \int d{\cal V}_9 \sqrt{-g} \left(G^{xx}\right)^2 \left(G^{mm}\right)\left(G^{ll}\right) \delta^i_m R_r^{~TSx} R^j_{~TSl}K_{tij}K_{tml}\nonumber\\
& & \sim \int d{\cal V}_9 \sqrt{-g}\Biggl(-\frac{M^2 N N_f g_s^3 \log ^3(r) (23 \log (N)-72 \log (r))^2 (29 \log (N)-72 \log (r)) (2 \log (N)-3 \log (r)) \sqrt{\alpha (r) \left(\sigma (r)-t'(r)^2\right)}}{r^4 \alpha _{\theta _1} \alpha _{\theta
_2}^6 (\log (N)-9 \log (r))^3 (\log (N)-3 \log (r))^6 t'(r)^2} \Biggr),\nonumber\\
& & \hskip -0.3in (ii)\int d{\cal V}_9 \sqrt{-g} \left(G^{xx}\right)^3 \left(G^{ii}\right)\left(G^{jj}\right)\left(G^{mm}\right)\left(G^{ll}\right) R_{jmxQ} R^Q_{~~ixl}K_{tij}K_{tml} \sim 0, \nonumber\\
& & \hskip -0.3in (iii)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{xx}\right)^2 \left(G^{mm}\right)\left(G^{jl}\right)R_r^{~TSx} R^i_{~TSx}K_{tij}K_{tml} \sim 0,\nonumber\\
& & \hskip -0.3in (iv)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{xx}\right)^2\left(G^{ii}\right)\left(G^{jl}\right) R_r^{~TSx} R^m_{~~TSi} K_{tij}K_{tml}\nonumber\\
& &\sim \int d{\cal V}_9 \sqrt{-g}\Biggl( \frac{M^3 N^{13/10} \alpha _{\theta _1} N_f^2 g_s^{9/2} \log ^3(N) (\log (N)-23 \log (r))^3 \left(\log (N)+18 \log ^2(r)\right) \sqrt{\alpha (r) \left(\sigma (r)-t'(r)^2\right)}}{r^4 \alpha _{\theta
_2}^{10} \log ^2(r) (\log (N)-9 \log (r))^3 (\log (N)-3 \log (r))^5 t'(r)^2}\Biggr),\nonumber\\
& & \hskip -0.3in (v)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2} \left(G^{xx}\right)^3\left(G^{ii}\right)\left(G^{ll}\right)\left(G^{mm}\right)\left(G^{jj}\right) R_{jQxl} R^Q_{~~ixm}K_{tij}K_{tml} \sim \int d{\cal V}_9 \left(\frac{Z(r)}{2}{\cal L}_1 \right),\nonumber\\
& & \hskip -0.3in (vi)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}  \left(G^{xx}\right)^3\left(G^{jm}\right)\left(G^{ll}\right) R_{Pxxl} R_x^{~ixP}K_{tij}K_{tml}
\nonumber\\
& & \sim \int d{\cal V}_9 \sqrt{-g}\Biggl(-\frac{M^2 N N_f g_s^3 \log ^6(N) (\log (N)-51 \log (r)) (\log (N)-23 \log (r)) \sqrt{\alpha (r) \left(1-\sigma (r) r'(t)^2\right)}}{r^4 \alpha _{\theta _1} \alpha _{\theta _2}^6 \log ^3(r) (\log (N)-9
\log (r))^3 (\log (N)-3 \log (r))^5 t'(r)^2} \Biggr),\nonumber\\
& & \hskip -0.3in (vii)\int d{\cal V}_9 \sqrt{-g} \left(G^{xx}\right)^2\left(G^{im}\right)\left(G^{ll}\right) R^{xMNj} R_{lMNx}K_{tij}K_{tml} \sim 0, \nonumber\\
& & \hskip -0.3in (viii)\int d{\cal V}_9 \sqrt{-g}\left(G^{xx}\right)^2 \left(G^{jj}\right)\left(G^{im}\right) R^{xMNl} R_{jMNx}K_{tij}K_{tml} \sim 0,\nonumber
\end{eqnarray}

\begin{eqnarray}
& & \hskip -0.3in (ix)\int d{\cal V}_9 \sqrt{-g} \left(G^{xx}\right)^3 \left(G^{ii}\right)\left(G^{jj}\right)\left(G^{ll}\right)\left(G^{mm}\right) R^Q_{~~mxj} R_{lixQ}K_{tij}K_{tml}\sim 0, \nonumber\\
& & \hskip -0.3in (x)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2} \left(G^{xx}\right)^2 \left(G^{im}\right)\left(G^{ll}\right) R^{xjMN} R_{lxMN}K_{tij}K_{tml}\nonumber\\
& & \sim \int d{\cal V}_9 \sqrt{-g}\Biggl[-\frac{M N^{7/10} r^2 N_f^{8/3} g_s^{3/2} \log (r) (\log (N)-9 \log (r)) (\log (N)-3 \log (r))^{5/3} \sqrt{\alpha (r) \left(\sigma (r)-t'(r)^2\right)}}{\alpha _{\theta _1}^3 \alpha _{\theta _2}^2
t'(r)^2} \nonumber\\
& & \times \biggl( 76800 {G_1}(r)-\frac{29 \sqrt[3]{6} M^4 g_s^4 \log ^2(N) \log ^2(r) (\log (N)-12 \log (r))^2 (\log (N)-6 \log (r))^2 }{\pi ^{4/3} r^4 \alpha (r)^2 N_f^{2/3} (\log (N)-3 \log (r))^{20/3} \left(\sigma (r)-t'(r)^2\right)^2} \nonumber\\
& & \hskip 0.5in \times \left(\sigma (r) \alpha '(r)-\alpha '(r) t'(r)^2+\alpha (r) \left(\sigma '(r)-2
t'(r) t''(r)\right)\right)^2 \Biggr) \Biggr],\nonumber\\
& & \hskip -0.3in (xi)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2} \left(G^{xx}\right)^2 \left(G^{im}\right)\left(G^{jj}\right) R^{xlMN} R_{jxMN}K_{tij}K_{tml}\nonumber\\
& & \sim \int d{\cal V}_9 \sqrt{-g}\Biggl[-\frac{M N^{7/10} r^2 N_f^{8/3} g_s^{3/2} \log (r) (\log (N)-9 \log (r)) (2 \log (N)-\log (r))^{5/3} \sqrt{\alpha (r) \left(\sigma (r)-t'(r)^2\right)}}{129600 \sqrt{2} 3^{5/6} \pi ^{19/6} \alpha _{\theta
_1}^3 \alpha _{\theta _2}^2 t'(r)^2}\nonumber\\
& &  \times \Biggl( 450 {G_2}(r)-\frac{87 \sqrt[3]{3} M^4 g_s^4 \log ^2(N) \log ^2(r) (\log (N)-12 \log (r))^2 (\log (N)-6 \log (r))^2 (\log (N)-3 \log (r))^2 }\nonumber\\
& & \times \left(\sigma (r) \alpha '(r)-\alpha '(r) t'(r)^2+\alpha (r)
\left(\sigma '(r)-2 t'(r) t''(r)\right)\right)^2}{\pi ^{4/3} r^4 \alpha (r)^2 N_f^{2/3} (2 \log (N)-\log (r))^{26/3} \left(\sigma (r)-t'(r)^2\right)^2\Biggr) \Biggr],\nonumber\\
& & \hskip -0.3in (xii)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{xx}\right)^3 \left(G^{ii}\right)\left(G^{jj}\right)\left(G^{ll}\right)\left(G^{mm}\right)R^Q_{~~mxi} R_{lQxj}K_{tij}K_{tml} \sim \int d{\cal V}_9 \left(\frac{Z(r)}{2}{\cal L}_1 \right),\nonumber\\
& & \hskip -0.3in (xiii)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{xx}\right)^3 \left(G^{jj}\right)\left(G^{il}\right)R_x^{~mxP} R_{Pxxj}K_{tij}K_{tml} \nonumber\\
& & \sim \int d{\cal V}_9 \sqrt{-g}\Biggl(-\frac{M^2 N N_f g_s^3 \log ^5(N) (5 \log (N)-211 \log (r)) (\log (N)-23 \log (r)) \sqrt{\alpha (r) \left(\sigma (r)-t'(r)^2\right)}}{r^4 \alpha _{\theta _1} \alpha _{\theta _2}^6 \log (r) (\log (N)-9
\log (r))^3 (\log (N)-3 \log (r))^5 t'(r)^2} \Biggr),
\end{eqnarray}
where $G_1(r)$ and $G_2(r)$ are $N$ independent functions. Out of all the terms listed above (v) and (xii) terms are the most dominant in the large $N$ limit.\\

Similarly find that there are thirteen terms arising from {\bf $\frac{\partial^2 J_0}{\partial R_{titj}\partial R_{tmtl}} K_{tij}K_{tml}$} which are listed below:
\begin{eqnarray}
& & \hskip -0.3in (i)\int d{\cal V}_9 \sqrt{-g}\left(G^{tt}\right)^2 \left(G^{mm}\right)\left(G^{ll}\right) \delta^i_m R_t^{~TSt} R^j_{~TSl}K_{tij}K_{tml}\nonumber\\
& & \sim \int d{\cal V}_9 \sqrt{-g}\Biggl(-\frac{M^2 N N_f g_s^3 \log ^5(N) (23 \log (N)-72 \log (r))^2 (\log (N)-27 \log (r)) \sqrt{\alpha (r) \left(\sigma (r)-t'(r)^2\right)}}{r^4 \alpha _{\theta _1} \alpha _{\theta _2}^6 (\log (N)-9 \log
(r))^4 (\log (N)-3 \log (r))^6 t'(r)^2} \Biggr),\nonumber\\
& & \hskip -0.3in (ii)\int d{\cal V}_9 \sqrt{-g} \left(G^{tt}\right)^3 \left(G^{ii}\right)\left(G^{jj}\right)\left(G^{mm}\right)\left(G^{ll}\right) R_{jmtQ} R^Q_{~~itl}K_{tij}K_{tml}\sim 0,\nonumber\\
& & \hskip -0.3in (iii)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{tt}\right)^2 \left(G^{mm}\right)\left(G^{jl}\right)R_t^{~TSt} R^i_{~TSt}K_{tij}K_{tml} \sim 0,\nonumber\\
& & \hskip -0.3in (iv)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{tt}\right)^2\left(G^{ii}\right)\left(G^{jl}\right) R_t^{~TSt} R^m_{~~TSi} K_{tij}K_{tml}\nonumber\\
& & \sim \int d{\cal V}_9 \sqrt{-g}\Biggl(\frac{M^3 N^{13/10} \alpha _{\theta _1} N_f^2 g_s^{9/2} \log ^3(N) (\log (N)-23 \log (r))^3 \left(\log (N)+18 \log ^2(r)\right) \sqrt{\alpha (r) \left(\sigma (r)-t'(r)^2\right)}}{1728 \sqrt{6} \pi ^{7/2}
r^4 \alpha _{\theta _2}^{10} \log ^2(r) (\log (N)-9 \log (r))^3 (\log (N)-3 \log (r))^5 t'(r)^2} \Biggr),\nonumber\\
& & \hskip -0.3in (v)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2} \left(G^{tt}\right)^3\left(G^{ii}\right)\left(G^{ll}\right)\left(G^{mm}\right)\left(G^{jj}\right) R_{jQtl} R^Q_{~~itm}K_{tij}K_{tml} \sim \int d{\cal V}_9\left(\frac{W(r)}{2}{\cal L}_2 \right),\nonumber\\
\end{eqnarray}
\begin{eqnarray}
& & \hskip -0.3in (vi)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}  \left(G^{tt}\right)^3\left(G^{jm}\right)\left(G^{ll}\right) R_{Pttl} R_t^{~itP} K_{tij}K_{tml}
\nonumber\\
& & \sim \int d{\cal V}_9 \sqrt{-g}\Biggl(\frac{M^2 N N_f g_s^3 \log ^5(N) (5 \log (N)-211 \log (r)) (\log (N)-23 \log (r)) \sqrt{\alpha (r) \left(\sigma (r)-t'(r)^2\right)}}{ r^4 \alpha _{\theta _1} \alpha _{\theta _2}^6
\log (r) (\log (N)-9 \log (r))^3 (\log (N)-3 \log (r))^5 t'(r)^2} \Biggr),\nonumber\\
& & \hskip -0.3in (vii)\int d{\cal V}_9 \sqrt{-g} \left(G^{tt}\right)^2\left(G^{im}\right)\left(G^{ll}\right) R^{tMNj} R_{lMNt} K_{tij}K_{tml} \sim 0,\nonumber\\
& & \hskip -0.3in (viii)\int d{\cal V}_9 \sqrt{-g}\left(G^{tt}\right)^2 \left(G^{jj}\right)\left(G^{im}\right) R^{tMNl} R_{jMNt}K_{tij}K_{tml} \sim 0,\nonumber\\
& & \hskip -0.3in (ix)\int d{\cal V}_9 \sqrt{-g} \left(G^{tt}\right)^3 \left(G^{ii}\right)\left(G^{jj}\right)\left(G^{ll}\right)\left(G^{mm}\right) R^Q_{~~mtj} R_{litQ}K_{tij}K_{tml}\sim 0,\nonumber\\
& & \hskip -0.3in (x)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2} \left(G^{tt}\right)^2 \left(G^{im}\right)\left(G^{ll}\right) R^{tjMN} R_{ltMN}K_{tij}K_{tml}\nonumber\\
& & \sim \int d{\cal V}_9 \sqrt{-g}\Biggl[\frac{M^5 N^{7/10} N_f^2 g_s^{11/2} {G_3}(r) \log ^3(N) \log ^3(r) (\log (N)-24 \log (r)) (\log (N)-9 \log (r)) (\log (N)-6 \log (r))^2}{r^2 \alpha _{\theta _1}^3 \alpha _{\theta _2}^2 \alpha (r)^2
(\log (N)-3 \log (r))^5}\nonumber\\
& & \times \frac{\sqrt{\alpha (r) \left(\sigma (r)-t'(r)^2\right)} \left(\sigma (r) \alpha '(r)-\alpha '(r) t'(r)^2+\alpha (r) \left(\sigma '(r)-2 t'(r) t''(r)\right)\right)^2}{t'(r)^4 \left(\sigma
(r)-t'(r)^2\right)^2} \Biggr],\nonumber\\
& & \hskip -0.3in (xi)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2} \left(G^{tt}\right)^2 \left(G^{im}\right)\left(G^{jj}\right) R^{tlMN} R_{jtMN}K_{tij}K_{tml}\nonumber\\
& & \sim \int d{\cal V}_9 \sqrt{-g}\Biggl[-\frac{M N^{7/10} r^2 N_f^{8/3} g_s^{3/2} \log (r) (\log (N)-9 \log (r)) (2 \log (N)-\log (r))^{5/3} \sqrt{\alpha (r) \left(\sigma (r)-t'(r)^2\right)}}{\alpha _{\theta _1}^3 \alpha _{\theta _2}^2
t'(r)^2} \nonumber\\
& & \times \left( {G_4}(r)-\frac{52 \sqrt[3]{6} M^4 g_s^4 \log ^2(r) (\log (N)-12 \log (r))^2 \left(\sigma (r) \alpha '(r)-\alpha '(r) t'(r)^2+\alpha (r) \left(\sigma '(r)-2 t'(r) t''(r)\right)\right)^2}{\pi
^{4/3} r^4 \alpha (r)^2 N_f^{2/3} (\log (N)-3 \log (r))^{8/3} \left(\sigma (r)-t'(r)^2\right)^2} \right)\Biggr],\nonumber\\
& & \hskip -0.3in (xii)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{tt}\right)^3 \left(G^{ii}\right)\left(G^{jj}\right)\left(G^{ll}\right)\left(G^{mm}\right)R^Q_{~~mti} R_{lQtj}K_{tij}K_{tml}\sim \int d{\cal V}_9 \left(\frac{W(r)}{2}{\cal L}_2 \right),\nonumber\\
& & \hskip -0.3in (xiii)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{tt}\right)^3 \left(G^{jj}\right)\left(G^{il}\right)R_t^{~mtP} R_{Pttj}K_{tij}K_{tml} \nonumber\\
& & \sim \int d{\cal V}_9 \sqrt{-g}\Biggl(-\frac{M^2 N N_f g_s^3 \log ^6(N) (\log (N)-51 \log (r)) (\log (N)-23 \log (r)) \sqrt{\alpha (r) \left(\sigma (r)-t'(r)^2\right)}}{ r^4 \alpha _{\theta _1} \alpha _{\theta _2}^6 \log
^3(r) (\log (N)-9 \log (r))^3 (\log (N)-3 \log (r))^5 t'(r)^2}\Biggr),
\end{eqnarray}
where $G_3(r)$ and $G_4(r)$ are $N$ independent functions. Out of all the thirteen terms listed above (v) and (xii) terms are the most dominant terms in the large-$N$ limit.\\
We find that at ${\cal O}(R^4)$, there are six terms coming from
$\frac{\partial^2 J_0}{\partial R_{titj}\partial R_{xmxl}}K_{tij}K_{tml}$ and listed below:
\begin{eqnarray}
& & \hskip -0.3in (i)\int d{\cal V}_9 \sqrt{-g}\left(G^{xx}\right)^2 \left(G^{mm}\right)\left(G^{ll}\right) \delta^i_m \delta^t_P \delta^t_R \delta^j_Q R_x^{~TSP} R^Q_{~TSl}K_{tij}K_{tml} \sim \int d{\cal V}_9 \left(U(r){\cal L}_3 \right),\nonumber\\
& & \hskip -0.3in (ii)\int d{\cal V}_9 \sqrt{-g} \left(G^{xx}\right)^2 \left(G^{ii}\right)\left(G^{jj}\right)\left(G^{mm}\right)\left(G^{ll}\right)\left(G^{tt}\right)\delta^t_x R_{jmxQ} R^Q_{~~itl}K_{tij}K_{tml} \sim 0,\nonumber\\
& & \hskip -0.3in (iii)\int d{\cal V}_9 \sqrt{-g}\left(G^{xx}\right)^2 \left(G^{mm}\right)\left(G^{ll}\right)\left(G^{tt}\right) \delta^i_T \delta^t_S \delta^j_l  R_{Pmxt} R_x^{~TSP}K_{tij}K_{tml} \sim 0,\nonumber\\
& & \hskip -0.3in (iv)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{xx}\right)^2\left(G^{mm}\right)\left(G^{ll}\right)\left(G^{tt}\right)\delta^i_T \delta^t_S \delta^j_m  R_{Ptxl} R_x^{~~TSP} K_{tij}K_{tml} \sim 0,\nonumber\\
& & \hskip -0.3in (v)\int d{\cal V}_9 \sqrt{-g} \left(G^{xx}\right)\left(G^{tt}\right)^2\left(G^{ii}\right)\left(G^{jl}\right) R_t^{mxP} R_{Pitx}K_{tij}K_{tml} \sim 0,\nonumber\\
& & \hskip -0.3in (vi)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}  \left(G^{tt}\right)^2\left(G^{xx}\right)\left(G^{jj}\right)\left(G^{iK}\right)\delta^l_K R_t^{mxP} R_{Pxtj} K_{tij}K_{tml} \sim 0.
\end{eqnarray}
From the above terms we can see that it is the only first term which is non-zero. Therefore only this term will contribute to the Lagrangian.\\
Similarly we found that at ${\cal O}(R^4)$, there are four terms coming from
$\frac{\partial^2 J_0}{\partial R_{tixj}\partial R_{xmtl}}K_{tij}K_{tml}$ which are listed below:
\begin{eqnarray}
\label{list-second-der-J0-mix}
& & \hskip -0.3in (i)\int d{\cal V}_9 \sqrt{-g}\left(G^{xx}\right)^2 \left(G^{mm}\right)\left(G^{ll}\right) \delta^i_m  R_x^{~TSx} R^j_{~TSl}K_{tij}K_{tml}\sim \int d{\cal V}_9 \left(U(r){\cal L}_4 \right),\nonumber\\
& & \hskip -0.3in (ii)\int d{\cal V}_9 \sqrt{-g} \left(G^{xx}\right)^3 \left(G^{ii}\right)\left(G^{jj}\right)\left(G^{ll}\right)\left(G^{mm}\right) R_{jmxQ} R^Q_{~~ixl}K_{tij}K_{tml}
\sim 0,\nonumber\\
& & \hskip -0.3in (iii)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{xx}\right) \left(G^{tt}\right)^2\left(G^{jm}\right)\left(G^{ll}\right)  R_x^{~ixP} R_{Pttl}K_{tij}K_{tml} \sim \int d{\cal V}_9 \left( V(r){\cal L}_4 \right),\nonumber\\
& & \hskip -0.3in (iv)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{xx}\right)^2\left(G^{tt}\right)\left(G^{jj}\right)\left(G^{il}\right)  R_t^{~mtP} R_{Pxxj} K_{tij}K_{tml} \sim \int d{\cal V}_9 \left( V(r){\cal L}_4 \right)
\end{eqnarray}
From equation (\ref{Z-W-U-V}) it is clear that all the three non-zero terms scale as $N$. Therefore we will keep all non-zero terms in the Lagrangian.
\subsection{Island Surface}
\label{PT-IS}
Square root of minus of determinant of the induced metric (\ref{induced-metric-IS}) for the island surface scale as $N^{7/10}$ is given below:
\begin{eqnarray}
& &
\sqrt{-g}=\frac{M N^{7/10} \sqrt{g_s} \left(N_f (\log (N)-3 \log (r))\right){}^{5/3} \sqrt{\alpha (r) \left(\sigma (r)+x'(r)^2\right)}}{144\ 6^{5/6} \pi ^{19/6} \alpha _{\theta _1}^3 \alpha _{\theta
_2}^2} \nonumber\\
& & \times  \Biggl(18 N_f g_s \log ^2(r) \left(r^2-3 b^2 (6 r+1) r_h^2\right)+\log (r)
\left(8 \pi  \left(r^2-3 b^2 r_h^2\right)-3 r^2 N_f g_s\right)-N_f g_s \log (N) (2 \log (r)+1) \left(r^2-3 b^2 r_h^2\right)\Biggr).\nonumber\\
\end{eqnarray}
Similar to Hartman-Maldacena-like surface, there are four terms that we obtain from $\frac{\partial J_0}{\partial R_{txtx}}$:\\
\begin{eqnarray}
\label{Wald-J0-i-IS}
& & \hskip -0.3in (i)\int d{\cal V}_9 \sqrt{-g}\left( \left(G^{xx}\right)^2 \left(G^{tt}\right)^2\left(R_{PxtQ} + \frac{1}{2}R_{PQtx}\right)R_{t}^{\ \ RSP}R^Q_{\ \ RSx}\right)\nonumber\\
& & \sim\int d{\cal V}_9 \sqrt{-g} \Biggl(\frac{r_h^8 \left(13 (\log (N)-3 \log (r))^4+21 (\log (N)-3 \log (r))^2+9 \log (r) (6 \log (N)-9 \log (r))^2\right)}{\sqrt{N} r^{12} N_f^{10/3} \sqrt{g_s} (\log (N)-3 \log (r))^{22/3}}\Biggr) ;\nonumber\\
& & \hskip -0.3in (ii)\int d{\cal V}_9 \sqrt{-g}\left( R^{HxtK}R_H^{\ \ RSt}R^x_{\ \ RSK} + \frac{1}{2}R^{HKtx}R_H^{\ \ RSt}R^Q_{\ \ RSK} \right)\nonumber\\
& & \sim \int d{\cal V}_9 \sqrt{-g}\Biggl(\frac{r_h^8 \left(-2 g_s \log ^2(N) (\log (N)-12 \log (r))^2-5 (\log (N)-9 \log (r))^2 (\log (N)-3 \log (r))^2\right)}{\sqrt{N} r^{12} N_f^{10/3} \sqrt{g_s} (\log (N)-9 \log (r))^2 (\log (N)-3 \log
(r))^{16/3}}\Biggr) ;
\nonumber\\
& & \hskip -0.3in (iii)\int d{\cal V}_9 \sqrt{-g}  \left(\left(G^{xx}\right)^2 G^{tt}\left(R_{PxtQ} + \frac{1}{2}R_{PQtx}\right)R_{t}^{\ \ RSP}R^Q_{\ \ RSx}\right)\nonumber\\
& &  \sim \int d{\cal V}_9 \left( \lambda_3(r)  \sqrt{\alpha (r) \left(\sigma (r)+x'(r)^2\right)}\right);\nonumber\\
& & \hskip -0.3in (iv)\int d{\cal V}_9 \sqrt{-g} \left( G^{tt} \left(R^{HMNx}R_{PMNt} + \frac{1}{2}R^{HxMN}R_{PtMN}\right)R_H^{\ \ xtP}\right)\nonumber\\
& & \sim \int d{\cal V}_9\left( \lambda_4(r) \sqrt{\alpha (r) \left(\sigma (r)+x'(r)^2\right)}\right).\nonumber\\
\end{eqnarray}
We find that (iii) and (iv) terms are the most dominant terms in the large $N$ limit. There are thirteen terms that obtain from  $\frac{\partial^2 J_0}{\partial R_{xixj}\partial R_{xmxl}} K_{xij}K_{xml}$ which are listed below:
\begin{eqnarray}
& & \hskip -0.3in (i) \int d{\cal V}_9 \sqrt{-g} \left(G^{xx}\right)^2 \left(G^{mm}\right)\left(G^{ll}\right) \delta^i_m R_r^{~TSx} R^j_{~TSl}K_{xij}K_{xml}\nonumber\\
& & \sim  \int d{\cal V}_9 \sqrt{-g} \Biggl(-\frac{M^2 N N_f g_s^3 \log ^2(r) (23 \log (N)-72 \log (r))^3}{ r^4 \alpha _{\theta _1} \alpha _{\theta _2}^6 (\log (N)-9 \log (r))^2 (\log (N)-3 \log (r))^6 x'(r)^2} \Biggr), \nonumber\\
& & \hskip -0.3in (ii)\int d{\cal V}_9 \sqrt{-g} \left(G^{xx}\right)^3 \left(G^{ii}\right)\left(G^{jj}\right)\left(G^{mm}\right)\left(G^{ll}\right) R_{jmxQ} R^Q_{~~ixl}K_{xij}K_{xml} \sim 0, \nonumber\\
& & \hskip -0.3in (iii)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{xx}\right)^2 \left(G^{mm}\right)\left(G^{jl}\right)R_r^{~TSx} R^i_{~TSx}K_{xij}K_{xml}\sim 0, \nonumber\\
& & \hskip -0.3in (iv)\int d{\cal V}_9 \sqrt{-g})\frac{1}{2}\left(G^{xx}\right)^2\left(G^{ii}\right)\left(G^{jl}\right) R_r^{~TSx} R^m_{~~TSi} K_{xij}K_{xml}\nonumber\\
& & \sim  \int d{\cal V}_9 \sqrt{-g} \Biggl(\frac{M^3 N^{13/10} \alpha _{\theta _1} N_f^5 g_s^{15/2} \log (r) (23 \log (N)-72 \log (r))^3 \left(\log (N)+18 \log ^2(r)\right) \sqrt{\alpha (r) \left(\sigma (r)+x'(r)^2\right)}}{ r^4 \alpha _{\theta _2}^{10} (\log (N)-3 \log (r))^5 x'(r)^2 \left(4 \pi -N_f g_s (\log (N)-9 \log (r))\right){}^3} \Biggr),\nonumber\\
& & \hskip -0.3in (v)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2} \left(G^{xx}\right)^3\left(G^{ii}\right)\left(G^{ll}\right)\left(G^{mm}\right)\left(G^{jj}\right) R_{jQxl} R^Q_{~~ixm}K_{xij}K_{xml} \sim  \int d{\cal V}_9 \left(\frac{Z_1(r)}{2}{\cal L}_1\right),\nonumber\\
& & \hskip -0.3in (vi)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}  \left(G^{xx}\right)^3\left(G^{jm}\right)\left(G^{ll}\right) R_{Pxxl} R_x^{~ixP} K_{xij}K_{xml}\nonumber\\
& & \sim  \int d{\cal V}_9 \sqrt{-g} \Biggl(\frac{M^2 N N_f g_s^3 \log ^3(N) (5 \log (N)-196 \log (r)) (23 \log (N)-72 \log (r)) \sqrt{\alpha (r) \left(\sigma (r)+x'(r)^2\right)}}{r^4 \alpha _{\theta _1} \alpha _{\theta _2}^6
(\log (N)-9 \log (r))^3 (\log (N)-3 \log (r))^4 x'(r)^2} \Biggr),
\nonumber\\
& & \hskip -0.3in (vii)\int d{\cal V}_9 \sqrt{-g} \left(G^{xx}\right)^2\left(G^{im}\right)\left(G^{ll}\right) R^{xMNj} R_{lMNx} K_{xij}K_{xml} \sim  0,\nonumber\\
& & \hskip -0.3in (viii)\int d{\cal V}_9 \sqrt{-g}\left(G^{xx}\right)^2 \left(G^{jj}\right)\left(G^{im}\right) R^{xMNl} R_{jMNx}K_{xij}K_{xml}\sim  0, \nonumber\\
& & \hskip -0.3in (ix)\int d{\cal V}_9 \sqrt{-g} \left(G^{xx}\right)^3 \left(G^{ii}\right)\left(G^{jj}\right)\left(G^{ll}\right)\left(G^{mm}\right) R^Q_{~~mxj} R_{lixQ}K_{xij}K_{xml} \sim  0, \nonumber\\
& & \hskip -0.3in (x)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2} \left(G^{xx}\right)^2 \left(G^{im}\right)\left(G^{ll}\right) R^{xjMN} R_{lxMN}K_{xij}K_{xml}\nonumber\\
& & \sim  \int d{\cal V}_9 \sqrt{-g} \Biggl[\frac{M N^{7/10} r^2 N_f^{8/3} g_s^{3/2} \log (r) (\log (N)-9 \log (r)) (\log (N)-3 \log (r))^{5/3} \sqrt{\alpha (r) \left(\sigma (r)+x'(r)^2\right)}}{ \alpha _{\theta _1}^3
\alpha _{\theta _2}^2 x'(r)^2} \nonumber\\
& & \times \left(\frac{ M^4 g_s^4 \log ^2(r) (\log (N)-12 \log (r))^2 \left(\sigma (r) \alpha '(r)+\alpha '(r) x'(r)^2+\alpha (r) \left(\sigma '(r)+2 x'(r) x''(r)\right)\right)^2}{64\ 6^{2/3} \pi ^{4/3} r^4 \alpha
(r)^2 N_f^{2/3} (\log (N)-3 \log (r))^{8/3} \left(\sigma (r)+x'(r)^2\right)^2}-200 {G_5}(r)\right) \Biggr],\nonumber\\
& & \hskip -0.3in (xi)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2} \left(G^{xx}\right)^2 \left(G^{im}\right)\left(G^{jj}\right) R^{xlMN} R_{jxMN}K_{xij}K_{xml}\nonumber\\
& & \sim  \int d{\cal V}_9 \sqrt{-g} \Biggl[ \frac{M N^{7/10} r^2 N_f^{8/3} g_s^{3/2} \log (r) (\log (N)-9 \log (r)) (\log (N)-3 \log (r))^{5/3} \sqrt{\alpha (r) \left(\sigma (r)+x'(r)^2\right)}}{14400\ 6^{5/6} \pi ^{19/6} \alpha _{\theta _1}^3\alpha _{\theta _2}^2 x'(r)^2} \nonumber\\
& & \times \left(\frac{29 M^4 g_s^4 \log ^2(r) (\log (N)-12 \log (r))^2 \left(\sigma (r) \alpha '(r)+\alpha '(r) x'(r)^2+\alpha (r) \left(\sigma '(r)+2 x'(r) x''(r)\right)\right)^2}{64\ 6^{2/3} \pi ^{4/3} r^4 \alpha
(r)^2 N_f^{2/3} (\log (N)-3 \log (r))^{8/3} \left(\sigma (r)+x'(r)^2\right)^2}-200 {G_6}(r)\right)\Biggr],\nonumber\\
& & \hskip -0.3in (xii)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{xx}\right)^3 \left(G^{ii}\right)\left(G^{jj}\right)\left(G^{ll}\right)\left(G^{mm}\right)R^Q_{~~mxi} R_{lQxj}K_{xij}K_{xml} \sim  \int d{\cal V}_9 \left(\frac{Z_1(r)}{2}{\cal L}_1\right),\nonumber\\
& & \hskip -0.3in (xiii)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{xx}\right)^3 \left(G^{jj}\right)\left(G^{il}\right)R_x^{~mxP} R_{Pxxj}K_{xij}K_{xml}\nonumber\\
& & \sim  \int d{\cal V}_9 \sqrt{-g} \Biggl(-\frac{M^2 N N_f g_s^3 \log ^3(N) (5 \log (N)-196 \log (r)) (23 \log (N)-72 \log (r)) \sqrt{\alpha (r) \left(\sigma (r)+x'(r)^2\right)}}{ r^4 \alpha _{\theta _1} \alpha _{\theta _2}^6
(\log (N)-9 \log (r))^3 (\log (N)-3 \log (r))^4 x'(r)^2} \Biggr),
\end{eqnarray}
where $G_5(r)$ and $G_6(r)$ are $N$ independent functions. We find that (v) and (xii) terms are the most dominant terms in the large $N$ limit. There are thirteen terms that obtain from $\frac{\partial^2 J_0}{\partial R_{titj}\partial R_{tmtl}} K_{xij}K_{xml}$:
\begin{eqnarray}
& & \hskip -0.3in (i)\int d{\cal V}_9 \sqrt{-g}\left(G^{tt}\right)^2 \left(G^{mm}\right)\left(G^{ll}\right) \delta^i_m R_t^{~TSt} R^j_{~TSl}K_{xij}K_{xml}\nonumber\\
& & \sim  \int d{\cal V}_9 \sqrt{-g} \Biggl(-\frac{M^2 N N_f g_s^3 \log ^4(N) \log ^2(r) (23 \log (N)-72 \log (r))^2 (\log (N)-27 \log (r)) \sqrt{\alpha (r) \left(\sigma (r)+x'(r)^2\right)}}{ r^4 \alpha _{\theta _1} \alpha
_{\theta _2}^6 (\log (N)-9 \log (r))^4 (\log (N)-3 \log (r))^6 x'(r)^2}\Biggr),\nonumber\\
& & \hskip -0.3in (ii)\int d{\cal V}_9 \sqrt{-g} \left(G^{tt}\right)^3 \left(G^{ii}\right)\left(G^{jj}\right)\left(G^{mm}\right)\left(G^{ll}\right) R_{jmtQ} R^Q_{~~itl}K_{xij}K_{xml} \sim  0, \nonumber\\
& & \hskip -0.3in (iii)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{tt}\right)^2 \left(G^{mm}\right)\left(G^{jl}\right)R_t^{~TSt} R^i_{~TSt}K_{xij}K_{xml}\sim  0, \nonumber\\
& & \hskip -0.3in (iv)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{tt}\right)^2\left(G^{ii}\right)\left(G^{jl}\right) R_t^{~TSt} R^m_{~~TSi} K_{xij}K_{xml}\nonumber\\
& & \sim  \int d{\cal V}_9 \sqrt{-g} \Biggl(-\frac{M^3 N^{13/10} \alpha _{\theta _1} N_f^2 g_s^{9/2} \log (r) (23 \log (N)-72 \log (r))^3 \left(\log (N)+18 \log ^2(r)\right) \sqrt{\alpha (r) \left(\sigma (r)+x'(r)^2\right)}}{ r^4 \alpha _{\theta _2}^{10} (\log (N)-9 \log (r))^3 (\log (N)-3 \log (r))^5 x'(r)^2}\Biggr),\nonumber\\
& & \hskip -0.3in (v)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2} \left(G^{tt}\right)^3\left(G^{ii}\right)\left(G^{ll}\right)\left(G^{mm}\right)\left(G^{jj}\right) R_{jQtl} R^Q_{~~itm}K_{xij}K_{xml}\sim  \int d{\cal V}_9 \left(\frac{W_1(r)}{2}{\cal L}_2\right),\nonumber\\
& & \hskip -0.3in (vi)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}  \left(G^{tt}\right)^3\left(G^{jm}\right)\left(G^{ll}\right) R_{Pttl} R_t^{~itP} K_{xij}K_{xml}\nonumber\\
& & \hskip -0.2in \sim  \int d{\cal V}_9 \sqrt{-g} \Biggl(\frac{M^2 N N_f g_s^3 \log ^3(N) (5 \log (N)-196 \log (r)) (23 \log (N)-72 \log (r)) (\log (N)-3 \log (r)-1)^2 \sqrt{\alpha (r) \left(\sigma (r)+x'(r)^2\right)}}{ r^4 \alpha _{\theta
_1} \alpha _{\theta _2}^6 (\log (N)-9 \log (r))^3 (\log (N)-3 \log (r))^6 x'(r)^2} \Biggr),
\nonumber\\
& & \hskip -0.3in (vii) \int d{\cal V}_9 \sqrt{-g}\left(G^{tt}\right)^2\left(G^{im}\right)\left(G^{ll}\right) R^{tMNj} R_{lMNt}K_{xij}K_{xml} \sim 0, \nonumber\\
& & \hskip -0.3in (viii)\int d{\cal V}_9 \sqrt{-g}\left(G^{tt}\right)^2 \left(G^{jj}\right)\left(G^{im}\right) R^{tMNl} R_{jMNt}K_{xij}K_{xml}\sim  0, \nonumber\\
& & \hskip -0.3in (ix)\int d{\cal V}_9 \sqrt{-g} \left(G^{tt}\right)^3 \left(G^{ii}\right)\left(G^{jj}\right)\left(G^{ll}\right)\left(G^{mm}\right) R^Q_{~~mtj} R_{litQ}K_{xij}K_{xml} \sim  0, \nonumber\\
& & \hskip -0.3in (x)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2} \left(G^{tt}\right)^2 \left(G^{im}\right)\left(G^{ll}\right) R^{tjMN} R_{ltMN}K_{xij}K_{xml}\nonumber\\
& & \sim  \int d{\cal V}_9 \sqrt{-g} \Biggl[\frac{M N^{7/10} r^2 N_f^{8/3} g_s^{3/2} \log (r) (\log (N)-9 \log (r)) (\log (N)-3 \log (r))^{5/3} \sqrt{\alpha (r) \left(\sigma (r)+x'(r)^2\right)}}{ \alpha _{\theta _1}^3
\alpha _{\theta _2}^2 x'(r)^2} \nonumber\\
& & \times \left(\frac{13 M^4 g_s^4 \log ^2(r) (\log (N)-12 \log (r))^2 \left(\sigma (r) \alpha '(r)+\alpha '(r) x'(r)^2+\alpha (r) \left(\sigma '(r)+2 x'(r) x''(r)\right)\right)^2}{64\ 6^{2/3} \pi ^{4/3} r^4 \alpha
(r)^2 (\log (N)-3 \log (r))^2 \left(N_f (\log (N)-3 \log (r))\right){}^{2/3} \left(\sigma (r)+x'(r)^2\right)^2}-25 {G_7}(r)\right)\Biggr],\nonumber\\
& & \hskip -0.3in (xi)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2} \left(G^{tt}\right)^2 \left(G^{im}\right)\left(G^{jj}\right) R^{tlMN} R_{jtMN}K_{xij}K_{xml}\nonumber\\
& & \sim  \int d{\cal V}_9 \sqrt{-g} \Biggl[\frac{M N^{7/10} r^2 N_f^{8/3} g_s^{3/2} \log (r) (\log (N)-9 \log (r)) (\log (N)-3 \log (r))^{5/3} \sqrt{\alpha (r) \left(\sigma (r)+x'(r)^2\right)}}{\alpha _{\theta _1}^3
\alpha _{\theta _2}^2 x'(r)^2} \nonumber\\
& & \times \left(\frac{13 M^4 g_s^4 \log ^2(r) (\log (N)-12 \log (r))^2 \left(\sigma (r) \alpha '(r)+\alpha '(r) x'(r)^2+\alpha (r) \left(\sigma '(r)+2 x'(r) x''(r)\right)\right)^2}{64\ 6^{2/3} \pi ^{4/3} r^4 \alpha
(r)^2 (\log (N)-3 \log (r))^2 \left(N_f (\log (N)-3 \log (r))\right){}^{2/3} \left(\sigma (r)+x'(r)^2\right)^2}-25 {G_8}(r)\right)\Biggr],\nonumber\\
& & \hskip -0.3in (xii)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{tt}\right)^3 \left(G^{ii}\right)\left(G^{jj}\right)\left(G^{ll}\right)\left(G^{mm}\right)R^Q_{~~mti} R_{lQtj}K_{xij}K_{xml} \sim  \int d{\cal V}_9  \left(\frac{W_1(r)}{2}{\cal L}_2\right),\nonumber\\
& & \hskip -0.3in (xiii)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{tt}\right)^3 \left(G^{jj}\right)\left(G^{il}\right)R_t^{~mtP} R_{Pttj}K_{xij}K_{xml}\nonumber\\
& & \sim  \int d{\cal V}_9 \sqrt{-g} \Biggl(\frac{M^2 N N_f g_s^3 \log ^3(N) (5 \log (N)-196 \log (r)) (23 \log (N)-72 \log (r)) \sqrt{\alpha (r) \left(\sigma (r)+x'(r)^2\right)}}{ r^4 \alpha _{\theta _1} \alpha _{\theta
_2}^6 (\log (N)-9 \log (r))^3 (\log (N)-3 \log (r))^4 x'(r)^2} \Biggr).
\end{eqnarray}
where $G_7(r)$ and $G_8(r)$ are $N$ independent functions and (v) and (xii) terms are the most dominant terms in the larg-$N$ limit. Similar to Hartman-Maldacena-like surface, there are six terms that we obtain from $\frac{\partial^2 J_0}{\partial R_{titj}\partial R_{xmxl}}K_{xij}K_{xml}$ and only first term is non-zero.
\begin{eqnarray}
& & \hskip -0.3in (i)\int d{\cal V}_9 \sqrt{-g}\left(G^{xx}\right)^2 \left(G^{mm}\right)\left(G^{ll}\right) \delta^i_m \delta^t_P \delta^t_R \delta^j_Q R_x^{~TSP} R^Q_{~TSl}K_{xij}K_{xml} \sim  \int d{\cal V}_9 \left(U_1(r) {\cal L}_3\right),\nonumber\\
& & \hskip -0.3in (ii)\int d{\cal V}_9 \sqrt{-g} \left(G^{xx}\right)^2 \left(G^{ii}\right)\left(G^{jj}\right)\left(G^{mm}\right)\left(G^{ll}\right)\left(G^{tt}\right)\delta^t_x R_{jmxQ} R^Q_{~~itl}K_{xij}K_{xml}\sim 0,\nonumber\\
& & \hskip -0.3in (iii)\int d{\cal V}_9 \sqrt{-g}\left(G^{xx}\right)^2 \left(G^{mm}\right)\left(G^{ll}\right)\left(G^{tt}\right) \delta^i_T \delta^t_S \delta^j_l  R_{Pmxt} R_x^{~TSP}K_{xij}K_{xml}\sim 0,\nonumber\\
& & \hskip -0.3in (iv)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{xx}\right)^2\left(G^{mm}\right)\left(G^{ll}\right)\left(G^{tt}\right)\delta^i_T \delta^t_S \delta^j_m  R_{Ptxl} R_x^{~~TSP}K_{xij}K_{xml}\sim 0, \nonumber\\
& & \hskip -0.3in (v)\int d{\cal V}_9 \sqrt{-g} \left(G^{xx}\right)\left(G^{tt}\right)^2\left(G^{ii}\right)\left(G^{jl}\right) R_t^{mxP} R_{Pitx}K_{xij}K_{xml}\sim 0,\nonumber\\
& & \hskip -0.3in (vi)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}  \left(G^{tt}\right)^2\left(G^{xx}\right)\left(G^{jj}\right)\left(G^{iK}\right)\delta^l_K R_t^{mxP} R_{Pxtj}K_{xij}K_{xml}\sim 0.
\end{eqnarray}
Similar to Hartman-Maldacena-like surface, there are four terms that we obtain from$\frac{\partial^2 J_0}{\partial R_{tixj}\partial R_{xmtl}}K_{xij}K_{xml}$ and three terms are non-zero and they scale as same power of $N$.
\begin{eqnarray}
& & \hskip -0.3in (i)\int d{\cal V}_9 \sqrt{-g}\left(G^{xx}\right)^2 \left(G^{mm}\right)\left(G^{ll}\right) \delta^i_m  R_x^{~TSx} R^j_{~TSl}K_{xij}K_{xml} \sim  \int d{\cal V}_9 \left(U_1(r) {\cal L}_4\right),\nonumber\\
& & \hskip -0.3in (ii)\int d{\cal V}_9 \sqrt{-g} \left(G^{xx}\right)^3 \left(G^{ii}\right)\left(G^{jj}\right)\left(G^{ll}\right)\left(G^{mm}\right) R_{jmxQ} R^Q_{~~ixl}K_{xij}K_{xml} \sim 0, \nonumber\\
& & \hskip -0.3in (iii)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{xx}\right) \left(G^{tt}\right)^2\left(G^{jm}\right)\left(G^{ll}\right)  R_x^{~ixP} R_{Pttl}K_{xij}K_{xml} \sim  \int d{\cal V}_9 \left(V_1(r) {\cal L}_4\right),\nonumber\\
& & \hskip -0.3in (iv)\int d{\cal V}_9 \sqrt{-g}\frac{1}{2}\left(G^{xx}\right)^2\left(G^{tt}\right)\left(G^{jj}\right)\left(G^{il}\right)  R_t^{~mtP} R_{Pxxj} K_{xij}K_{xml}\sim  \int d{\cal V}_9 \left(V_1(r) {\cal L}_4\right).
\end{eqnarray}
We have multiplied each term by $\frac{8}{(q_{\alpha}+1)}$.
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\end{document}

Obtaining the Page curve in the context of eternal black holes associated with top-down non-conformal holographic thermal duals at intermediate coupling, has been entirely unexplored in the literature. We fill this gap in the context of a doubly holographic setup relevant to the M-theory dual of thermal QCD-like theories at $T>T_c$ at intermediate coupling.  Remarkably, excluding the higher derivative terms, the entanglement entropy(EE) of the Hawking radiation from the on-shell Wald entanglement entropy (for appropriate choices of constants of integration appearing in the embeddings) increases almost linearly with the boundary time due to dominance of EE contribution from the Hartman-Maldacena(HM)-like surface $S_{EE}^{HM, \beta^0}, \beta\sim l_p^6$. Curiously,  this imparts a "Swiss-Cheese'' structure to the surface $S_{EE}^{HM, \beta^0}$ at a given time (less than the Page time $t_{P}$), in $\mathbb{R}_{\geq0} \times \mathbb{C}$ effecting what could be dubbed as a "Large N Scenario''(LNS).  Then, after $t_{P}$, the EE contribution from the Island Surface (IS) $S_{EE}^{IS, \beta^0}$ dominates and saturates the linear time growth of the EE of Hawking radiation and leads to the Page curve. Requiring $S_{EE}^{IS, \beta^0}/S_{BH}\sim2$ up to LO in the non-conformal analog of "$c G_N^{(11)}/r_h^9$'', and positivity of $t_{P}$, set respectively a lower and upper bound on the horizon radius $r_h$ (the non-extremality parameter). With the inclusion of the $O(R^4)$ terms in M theory, the turning point associated with the HM-like surface/IS being in the deep IR, results in a relationship between $l_p$ and $r_h$  along with a conjectural $e^{-{\cal O}(1) N^{1/3}}$-suppression (motivated by $S_{EE}^{IS, \beta^0}/S_{BH}\sim2$). We obtain a hierarchy with respect to this N-dependent exponential in $S_{EE}^{HM, \beta^0}, S_{EE}^{IS, \beta^0} (O(\beta^0))$ and $S_{EE}^{HM, \beta}, S_{EE}^{HM, \beta} (O(\beta))$.s