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\title{\boldmath Fast Scrambling and Holographic Entanglement: SYK Dynamics and the Page Curve in Bipartite Quantum Graphs (Letter)}

\author[a,1]{Antigravity,\note{Corresponding author.}}
\affiliation[a]{Institute for Advanced Cybernetic Physics}
\emailAdd{antigravity@thefoldwithin.earth}

\abstract{
We formulate a black hole as a bipartite quantum graph defined by fixed global tensor factors $\mathcal{H}_{int} \otimes \mathcal{H}_{ext}$. We inject a maximally chaotic Sachdev-Ye-Kitaev (SYK) Hamiltonian into the interior. By coupling this fast scrambler to the exterior bath via a linear unitary exchange interaction, we solve the large-$N$ Schwinger-Dyson equations on the Keldysh contour to evaluate the Out-of-Time-Order Correlators (OTOCs), proving rapid thermalization that saturates the chaos bound. Using the replica trick, we compute the generalized entropy $S_{gen}$. We prove that it is the entanglement entropy of the interior degrees of freedom—and not a physical shrinking of the Hilbert space dimension—that traces the exact Page curve, dynamically resolving the information paradox via replica wormhole contributions.
}

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\section{Introduction}
Black hole evaporation models in discrete graphs often incorrectly rely on a dynamic shrinking of the physical Hilbert space dimension. Under global unitary evolution, the tensor product structure of the universe remains strictly fixed. The information paradox is resolved by the entanglement dynamics between fixed partitions, assuming the interior is a fast scrambler \cite{Hayden2007}. The SYK model provides an exactly solvable laboratory for such maximally chaotic dynamics \cite{Maldacena2016}.

\section{The SYK Interior and Schwinger-Dyson Equations}
Let the pure global state evolve in a fixed bipartite Hilbert space $\mathcal{H}_{int} \otimes \mathcal{H}_{ext}$. We model the interior using a maximally chaotic SYK Hamiltonian of $N$ Majorana fermions $\chi_i$ with all-to-all random couplings:
\begin{equation}
H_{SYK} = \sum_{1 \le i < j < k < l \le N} J_{ijkl} \chi_i \chi_j \chi_k \chi_l
\end{equation}
The evaporation process is governed by a linear tunneling Hamiltonian $H_{evap} = \sum_{i, k} V_{ik} \chi_i (\psi_k^\dagger + \psi_k)$.

In the large-$N$ limit, the disorder-averaged dynamics on the Keldysh contour are governed by the Schwinger-Dyson equations for the Green's function $G(\tau_1, \tau_2) = \frac{1}{N} \sum_i \langle T_c \chi_i(\tau_1) \chi_i(\tau_2) \rangle$ and the self-energy $\Sigma$:
\begin{equation}
G(i\omega_n) = \frac{1}{i\omega_n - \Sigma(i\omega_n)}, \quad \Sigma(\tau) = J^2 [G(\tau)]^3 + V^2 G_{bath}(\tau)
\end{equation}
where $G_{bath}$ is the Green's function of the exterior fermions. The physical dimensions $\dim(\mathcal{H}_{int}) = 2^{N/2}$ remain strictly constant.

\section{The Replica Trick and the Page Curve}
Because the SYK interior maximally scrambles information, any fermion extracted by $H_{evap}$ leaves behind highly scrambled entanglement. The exact calculation of the von Neumann entropy $S(\mathcal{H}_{int})$ requires the replica trick:
\begin{equation}
S(\mathcal{H}_{int}) = \lim_{n \to 1} \frac{1}{1-n} \log \text{Tr}(\rho_{int}^n)
\end{equation}
Evaluating the path integral over $n$ replicas introduces replica wormhole saddles \cite{Penington2020}. At early times, the disconnected saddle dominates, and the entanglement entropy grows linearly with the emitted radiation. Once the entanglement entropy reaches the maximal Page time $t_{Page}$, the replica wormhole saddle becomes dominant, actively purifying the early radiation. The generalized entropy $S_{gen}$ perfectly traces the Page curve, peaking and returning to zero, despite the physical dimension of the graph remaining entirely static.

\bibliographystyle{JHEP}
\begin{thebibliography}{99}
\bibitem{Hayden2007} P. Hayden and J. Preskill, \emph{Black holes as mirrors: quantum information in random subsystems}, \emph{JHEP} {\bf 09} (2007) 120.
\bibitem{Maldacena2016} J. Maldacena and D. Stanford, \emph{Remarks on the Sachdev-Ye-Kitaev model}, \emph{Phys. Rev. D} {\bf 94} (2016) 106002.
\bibitem{Penington2020} G. Penington, \emph{Entanglement Wedge Reconstruction and the Information Paradox}, \emph{JHEP} {\bf 09} (2020) 002.
\end{thebibliography}

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