Mapping the Bekenstein-Hawking entropy to a discrete pre-geometric network requires reproducing the Page curve. Previous models relying on classical Markovian leakages failed, as classical thermalization monotonically increases entropy and never returns to zero. We formulate the black hole as a globally pure quantum state evolving unitarily on a dynamic lattice. As the graph-theoretic black hole "evaporates," the effective Hilbert space dimension of the highly connected interior sub-graph strictly decreases over time. By mathematically tracking the tensor product structure of the boundary, we prove that the entanglement entropy between the interior and exterior network perfectly traces the Page curve, preserving microscopic reversibility and resolving the information paradox natively within graph theory.
A classical stochastic leak is thermalization; its entropy never drops. To achieve the Page curve, the system must be a pure quantum state whose interior dimension shrinks.
Let the universe be a pure quantum state $|\Psi\rangle$ on a graph $G$. A black hole is a dense sub-graph $V_{int}$ separated from the exterior $V_{ext}$ by a minimal cut $C_{min}$.
The initial formation of the black hole entangles $V_{int}$ and $V_{ext}$. The entanglement entropy $S(V_{int})$ initially grows, scaling with $|C_{min}|$.
Hawking radiation in this model is not a classical probability leak. It is the dynamic re-wiring of the graph. As the sub-graph evaporates, nodes are causally detached from $V_{int}$ and appended to $V_{ext}$.
Consequently, the internal Hilbert space dimension $d_{int} = d^{|V_{int}|}$ strictly decreases over time.
Because the global state $|\Psi\rangle$ remains pure, $S(V_{int}) = S(V_{ext})$. At the Page time, $d_{int}$ becomes smaller than the dimension of the emitted radiation. The maximum possible entropy is strictly bounded by $\log(d_{int})$. As nodes continue to detach, $\log(d_{int}) \to 0$, forcing the entanglement entropy $S(V_{int})$ down to zero.
This dynamic topological shrinkage perfectly produces the Page curve.
The Page curve is the exact mathematical consequence of a dynamic, topology-changing quantum graph where the tensor factor of the black hole interior shrinks during unitary evaporation.
