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.
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.
Abstract 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.
We present a causal, falsifiable entropy-retrieval law 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_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 g(2)(t1, t2) envelope that Bose–Einstein–condensate analog black holes can measure on 10–100 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, testable 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.
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 τchar. 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.
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.
We derive a causal law of observer-indexed entropy retrieval: retrieved reference information grows toward the supply available to a finite observer. A recorded admission-and-readout instrument produces orthogonal operational sectors from arbitrary coherent radiation; exact diary fidelity and decoupling give the flagged family, while branch errors bound its approximate realization. An exact quantum transfer channel yields the remaining-gap law and its continuous Lindblad evolution. The channel has an explicit unitary collision realization: admitted reference information moves from radiation to memory, while the capture record remains diary-blind. Its memory marginal is the Choi state of an erasure channel, giving an intrinsic positive-coherent-information boundary at retrieved fraction r = 1/2. A strip-holomorphic modular speed limit bounds how rapidly the channel can activate, with the hyperbolic tangent selected uniquely at extremality. The integrated retrieval dose is invariant under a change of clock, making the Kerr lapse cancellation one geometric instance of a general result. At fixed supply and kinetic rate under an affine clock, the transfer and activation laws compose into a closed proper-time retrieval trajectory on the equality branch, identify that trajectory as the sharp fastest-retrieval envelope, and fix the earliest admissible retrieval horizon. In the continuum formulation, Araki relative entropy defines retrieval on restricted observable algebras, and a split Type-I factor provides the regulated density-operator representation. The experimental program follows the same causal sequence through retrospective analysis and staged intervention. Tier 0A determines whether archived analog-horizon data retain the shot-level, timing, protocol, and processing information needed to ask the retrieval question. Tier 0B then tests whether qualifying data contain observer-, protocol-, readout-, or analysis-operator-indexed correlation structure that standard pooling or filtering suppresses. A positive Tier 0B result identifies retrieval-relevant structure and motivates Tier 1; it does not by itself confirm the complete law. The prospective Tier 1 BEC program targets the held-out g²(t₁,t₂) surface, bandwidth scaling, and protocol dependence. Independent activation readout tests the modular bound and tanh rigidity. Reference-tagged supply and memory test cumulative dose, composite-hazard recovery, degraded-channel ordering, and two-memory allocation. The v2 verification artifact provides synthetic benchmarks and numerical consistency checks at bond dimensions D = 4 and D = 8 for the transfer law, inverse recovery, finite-resolution robustness, and matched adversarial alternatives. Standard RT/HRT geometry and a separate accessibility functional place the retrieval dynamics within holographic language. The retrieval law closes the observer-local path between global encoding and retained access. Information conservation, radiation encoding, causal collection, finite-observer retrieval, formal reconstruction, and comprehension are distinct physical operations. The black-hole information problem contains a proper-time access dynamics that global entropy accounting alone does not determine. Physical complexity enters through encoding, supply, clock, activation, transfer, retention, and observation. Those layers locate each source of retrieval failure instead of idealizing it away. The theory orders the encoding, supply, and retrieval clocks; distinguishes late supply from early supply through their overlap with future transfer dose; and gives an exact finite-lifetime retrieval criterion. Here rᵢ(τ) is the normalized reference information retained by observer i, eᵢ(τ) is the causally available supply, kᵢ(τ) is the composite access hazard, and τ_char is the characteristic activation scale.
The black hole information paradox has been substantially resolved: entropy is preserved in Hawking radiation, as established by the Page curve, replica wormhole calculations, and the island formula. We argue, however, that this resolution addresses only one of two physically distinct properties of quantum information states. The Shannon–Bekenstein framework formalises entropic content S(σ), but lacks representation for a second property: the capacity of a state to causally influence future configurations of external systems. We term this generative capacity G(σ) and establish that it is formally independent of S(σ). The central result is that even in a fully unitary resolution, I(ρ_rad, ρ_E) ≤ C·S_BH·2^{−S_BH/2} → 0 after the scrambling time for any system E in prior causal contact with an infalling state, while S(ρ_rad) follows the Page curve. A unitary resolution restores entropy; it does not restore causal influence capacity. We demonstrate this in the JT gravity + bath model via the decoupling theorem. Eight technical contributions support this framework: formal proof of S–G independence via the Rindler family; proof that the Bell state saturates the supremum in G(σ,d) via the entanglement-assisted channel capacity theorem; three explicit benchmark calculations; covariant reformulation of G bounded above by an area law; distinction of the causal residue from BMS soft hair charges; and time-dependent causal sterilization at rate κ = 2πT_H = λ_L, saturating the Maldacena–Shenker–Stanford bound on quantum chaos [35]. Theorem C.2 establishes that classical correlations survive the Rindler horizon completely while quantum entanglement is destroyed — the horizon acts as a quantum-classical separator. The framework introduces the causal quality index κ ∈ [1,2] as a potentially observable signature distinguishing accretion histories with identical mass, charge, and angular momentum.