Mapping Bekenstein-Hawking entropy to discrete graphs requires demonstrating the Page curve without resorting to trivial kinematic counting arguments. We formulate the graph-theoretic black hole as a globally pure quantum state evolving unitarily. We explicitly define the evaporation Hamiltonian $U(t)$ that drives the dynamic topological re-wiring of the graph. By modeling the causal detachment of nodes from the interior sub-graph to the exterior via a unitary exchange interaction, we mathematically generate the dynamic shrinking of the interior tensor product dimension. This proves that a purely unitary graph Hamiltonian perfectly traces the Page curve for entanglement entropy, resolving the information paradox natively within pre-geometric graph dynamics.
Manually moving nodes across a bipartite cut is trivial kinematics. A rigorous physics of graph-theoretic black holes demands a dynamical Hamiltonian $U(t)$ that causes the re-wiring.
Let the pure global state $|\Psi\rangle$ exist on a partitioned graph $V_{int} \otimes V_{ext}$.
We define the evaporation Hamiltonian across the cut $C_{min}$ using a Heisenberg-like exchange operator that acts conditionally on the local node density (gravitational coupling).
where $\Pi_{\rho}(i)$ is a projector that only activates when the local internal node density drops below a critical threshold, enabling the edge $(i, j)$ to causally sever its internal links and entangle exclusively with the exterior.
Under the unitary evolution $U(t) = e^{-i H_{evap} t}$, the Hamiltonian actively and deterministically re-wires the graph topology. Nodes on the boundary $C_{min}$ are sequentially extracted from the $V_{int}$ tensor factor and transferred to $V_{ext}$.
Because the evolution is strictly unitary, the global state remains pure. As $H_{evap}$ dynamically shrinks the dimension $d_{int}$, the maximal possible entanglement entropy $\log(d_{int})$ is forced to strictly decrease. The entanglement entropy $S(V_{int})$ traces the exact Page curve solely as a consequence of the Hamiltonian's topological re-wiring.
The Page curve is dynamically generated by an explicit unitary evaporation Hamiltonian that re-wires graph topology. Black hole evaporation is simply the unitary transfer of tensor factors across a dynamic network cut.
