Modular Entropy Retrieval in Black-Hole Information Recovery: A Proper-Time Saturation Model

We derive a proper-time–dependent entropy retrieval law stating that the growth rate of retrievable entropy is proportional to the remaining entropy gap, modulated by a hyperbolic tangent regulator that switches on at a characteristic proper time (tau_char). Unlike phenomenological fits to the Page curve, this law follows directly from bounded Tomita–Takesaki modular flow and is fully invertible from simulated or empirical retrieval curves. The framework converts global entropy conservation into a Lorentzian-causal, observer-specific recovery process. It predicts distinct trajectories for stationary, freely falling, and accelerated observers, and yields an acceleration-indexed second-order correlation envelope (g2) that Bose–Einstein condensate analog black holes can measure on 10 to 100 millisecond timescales. Numerical validation on a 48-qubit MERA lattice (bond dimension 8) confirms robustness. A modified Ryu–Takayanagi prescription embeds the model in AdS/CFT without replica wormhole or island constructions. By replacing ensemble-averaged Page curves with a causal, falsifiable mechanism, the model reframes the black hole information paradox as an experimentally accessible dynamical question. Here, S_max is the Bekenstein–Hawking entropy, gamma(tau) is the modular-flow retrieval rate, and tau_char sets the characteristic proper-time scale.