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 τ_char. Unlike ensemble-averaged, non-causal Page-curve phenomenology, this law follows directly from bounded Tomita–Takesaki modular flow and admits an inverse map for extracting observer-indexed retrieval rates from measured correlation structure. The framework converts global entropy conservation into a Lorentzian-causal, observer-specific access process without invoking global reconstruction or post hoc averaging. It predicts a joint, experimentally testable signature in the g²(t1, t2) correlation envelope, including constrained saturation, protocol-dependent separation, inverse bandwidth scaling of onset time, and interference suppression under controlled asymmetry. Numerical results on a 48-qubit MERA lattice (bond dimension 8) are consistent with the derived law. A modified Ryu–Takayanagi prescription embeds the retrieval dynamics in 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 Smax denotes the Bekenstein–Hawking entropy, γ(τ) the modular-flow retrieval rate, and τchar the characteristic proper-time scale.