# The Evaporation Hamiltonian: Dynamic Topological Re-wiring and the Page Curve

**Target Venue:** *Journal of Cosmology and Astroparticle Physics (JCAP)*

## Abstract
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.

## 1. Introduction
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.

## 2. The Evaporation Hamiltonian
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).
$$
H_{evap} = \lambda \sum_{\langle i, j \rangle \in C_{min}} \left( |0_i 1_j\rangle\langle 1_i 0_j| + h.c. \right) \otimes \Pi_{\rho}(i)
$$
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.

## 3. Unitarity and the Page Curve
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.

## 4. Conclusion
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.

## References
1. Page, D. N. (1993). *Information in black hole radiation*. Physical Review Letters.
2. Ryu, S., & Takayanagi, T. (2006). *Holographic derivation of entanglement entropy from AdS/CFT*. Physical Review Letters.