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

We present a causal, falsifiable law of observer-indexed entropy retrieval dynamics whose 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 \( \tau_{\mathrm{char}} \). Unlike ensemble-averaged, non-causal Page-curve phenomenology, 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 retrieval process, without invoking global reconstruction or post hoc averaging. It predicts distinct retrieval trajectories for stationary, freely falling, and accelerated observers, and yields an acceleration-indexed \( g^{(2)}(t_{1}, t_{2}) \) envelope that Bose--Einstein--condensate analog black holes can measure on 10–100 ms timescales. Recent laboratory observations of universal coherence-spreading bounds in ultracold quantum gases provide independent empirical support for access-limited saturation dynamics. Numerical validation on a 48-qubit MERA lattice (bond dimension~8) confirms robustness. A modified Ryu–Takayanagi prescription embeds the retrieval dynamics in \( \mathrm{AdS/CFT} \) without replica-wormhole or island constructions. By replacing ensemble-averaged Page curves with a causal, testable retrieval mechanism, the model reframes the black-hole information paradox as an experimentally accessible dynamical question. Here \( S_{\max} \) denotes the Bekenstein--Hawking entropy, \( \gamma(\tau) \) the modular-flow retrieval rate, and \( \tau_{\mathrm{char}} \) the characteristic proper-time scale.