Exact Dephasing Entropy of Hawking Radiation in a Gapped Bath: An Open Quantum System Derivation of the Rising Page Curve

We study the entanglement entropy of Hawking radiation using the open quantum systems framework. For a vacuum with a spectral gap ΔΔ, the radiation entropy saturates at a sub-maximal value, leaving a residual visibility (η(0)≈0.9000) that protects quantum information. This provides an analytical description of the rising Page curve.