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 in which the growth rate of retrievable entropy is proportional to the remaining entropy gap and modulated by a hyperbolic-tangent onset at a characteristic proper time tau_char. Unlike ensemble-averaged, non-causal Page-curve phenomenology, the law is derived from bounded split-regularized Tomita-Takesaki modular flow and admits an inverse map for extracting observer-indexed retrieval rates from measured correlation structure. The framework supplements global entropy conservation with a Lorentzian-causal access process: conserved information becomes operationally relevant to a finite observer only when it enters that observer's bounded modular-access domain.The model predicts a joint, experimentally testable signature in the g²(t₁,t₂) correlation envelope, including constrained saturation, protocol-dependent separation, inverse gamma recovery, finite-resolution robustness, and interference suppression under controlled asymmetry. Numerical results in a finite-bond-dimension tensor-network proxy, evaluated at D=4 and D=8, are consistent with the derived law, and adversarial verification against matched saturating, shared-envelope, label-permuted, time-jittered, and non-gap alternatives shows that generic saturation does not reproduce the full observer-indexed retrieval signature. A redshift-weighted Ryu-Takayanagi representation situates the retrieval dynamics within holographic geometry without invoking replica-wormhole or island constructions as the source of the retrieval law.The result reframes the black-hole information paradox as a bounded-access dynamics problem rather than a contradiction in entropy accounting. On this formulation, information conservation, formal reconstruction, and finite-observer retrieval are distinct operations; the apparent paradox arises when they are treated as interchangeable. Here Smax denotes the Bekenstein-Hawking entropy, gamma(tau) the modular-flow retrieval rate, and tau_char the characteristic proper-time scale.