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

1. Introduction

Entropy without Access

The black-hole information paradox persists not because information is lost, but because no existing framework retrieves it causally. Replica–wormhole paths [1,2], island prescriptions [3], ensemble Page–curve models [4,5], and $\mathrm{ER}=\mathrm{EPR}$ dualities [6] all reproduce the required fine-grained entropy curves, yet none supplies a Lorentzian, proper-time recovery channel that delivers the state to a detector. Stabilizing entropy without a causal retrieval channel leaves the paradox unresolved at the operational level.

1.1. Operational–Access Criterion

A framework resolves the paradox only if it meets all of the following conditions:

(a)

Proper-time delivery: specifies how entropy reaches an observer as proper time unfolds;

(b)

Lorentzian grounding: roots that access in Lorentzian causality;

(c)

First-principles derivation: derives the process from accepted QFT or GR principles rather than retrospective fitting;

(d)

Empirical testability: predicts observer-dependent lags $\Delta \tau$ within sub-exponential resource bounds1.

Table 1. Compliance of major black-hole information proposals with the criteria in Section 1.1. A check mark denotes compliance; a cross denotes failure.

Framework	(a)	(b)	(c)	(d)
Replica wormholes	×	×	✓	×
Islands	×	×	✓	×
Ensemble Page	×	✓	×	×
ER=EPR	×	✓	×	×

Table 1. Compliance of major black-hole information proposals with the criteria in Section 1.1. A check mark denotes compliance; a cross denotes failure.

Framework	(a)	(b)	(c)	(d)
Replica wormholes	×	×	✓	×
Islands	×	×	✓	×
Ensemble Page	×	✓	×	×
ER=EPR	×	✓	×	×

Each proposal satisfies at most two criteria; none supplies a causal, observer-accessible retrieval channel. Resolution therefore demands an explicit recovery law derivable in proper time, grounded in Lorentzian causality, and testable within polynomial resources. The Observer-Dependent Entropy Retrieval (ODER) framework meets those demands with modular-flow dynamics and wedge-reconstruction depths that scale polynomially, in contrast with the exponential-cost Hayden–Preskill decoder $\mathcal{O}\left(2^n\right)$ assumed for global recovery. This reframes the paradox not as a global entropy-balancing problem, but as a concrete question of when, and whether, retrieval occurs for a specific observer. Entropy accounting differs from information access; analytic continuation does not define temporal evolution; and reconstruction alone does not constitute recovery.

Reader Guide

• Section 2 derives the observer-indexed retrieval law and presents an inverse map that reconstructs $\gamma \left(\tau \right)$ from measured $S_{\mathrm{ret}}\left(\tau \right)$.
• Section 6 validates the law on a 48-qubit MERA lattice, establishing the $\mathcal{O}(nlogn)$ scaling.
• Section 7.5 translates the theory into the $g^{\left(2\right)}$ fringe measurable in current BEC analogs.
• Section 4.1 provides a calibration protocol for ${\tau }_{\mathrm{char}}$.
• Section 7.4 defines the retrieval–evaporation gap ${\Delta }_{\mathrm{fail}}$.

Appendix E offers an interpretive correspondence for readers unfamiliar with modular dynamics, mapping $\gamma \left(\tau \right)$, ${\tau }_{\mathrm{char}}$, and ${\Delta }_{\mathrm{fail}}$ to gravitationally intuitive quantities without altering the retrieval framework.

Code, data, and figure-generation notebooks are archived at https://doi.org/10.5281/zenodo.15428312.

2. Observer-Dependent Entropy Retrieval

Novel Framework.

ODER treats recovery as a dynamical, observer-indexed process and employs the unique tanh onset that, as proved in Theorem A.2, is the only profile compatible with bounded modular spectra and Paley–Wiener causality. This section derives

$${\displaystyle \frac{d{S}_{\mathrm{retr}}}{d\tau }}=\gamma \left(\tau \right)\left[S_{max}-S_{\mathrm{retr}}\left(\tau \right)\right]tanh\left(\tau /{\tau }_{\mathrm{char}}\right),$$

(1)

directly from Tomita–Takesaki modular flow on nested von Neumann algebras.

Because Eq. (1) is first order and monotone in $S_{\mathrm{retr}}\left(\tau \right)$, we can solve it for the unknown rate $\gamma \left(\tau \right)$. This inversion is essential in Section 3, where retrieval rates for different observer classes are compared.

Implication. Once an experiment measures $S_{\mathrm{retr}}\left(\tau \right)$, for example through the $g^{\left(2\right)}$ fringe, the boxed map fixes $\gamma \left(\tau \right)$ without further assumptions, turning ODER from a forward model into a calibratable decoder.

Goal 

model entropy recovery as a bounded, causal convergence in proper time that differs by observer.

Mechanism 

Eq. (1) depends on modular-spectrum gradients. $\gamma \left(\tau \right)$ encodes redshift, Unruh, or interior-correlation effects.

Domain of validity 

algebraic QFT on Lorentzian backgrounds. Simulations on a 48-qubit MERA lattice confirm robustness. The model predicts an acceleration-dependent $g^{\left(2\right)}$ envelope in BEC analog black holes on $10–100\mathrm{ms}$ timescales, a signature absent from non-retrieval models.

We define the retrieval horizon

$${\tau }_{\mathrm{RH}}:=inf\left\{\tau |S_{\mathrm{retr}}\left(\tau \right)\ge 0.9S_{max}\right\},$$

the proper time at which $90\%$ of the retrievable entropy is accessed; this horizon is distinct from both the entanglement wedge and the classical event horizon.

Self-Audit: ODER Failure Modes

• Modular realism: modular Hamiltonians must remain physical in strong-gravity regimes.
• Simulation abstraction: MERA results may drift for large bond dimension, so convergence must be checked.
• Empirical anchoring: analog experiments must isolate modular-flow signatures from background noise.
• Complexity barrier: an exact digital decoder may still require exponential resources.
• Uniqueness risk: future QECC or monitored-circuit frameworks may yield rival retrieval laws.

Astrophysical Forecast.

For a solar-mass Schwarzschild black hole, Eq. (1) implies that a stationary observer at $r=10GM/c^2$ retrieves at least $90\%$ of the missing entropy only after $\sim {10}^{67}\mathrm{yr}$, a timescale absent from replica-wormhole or island prescriptions. Sections 6–7.5 benchmark the law and outline experimental validation, showing that information is not lost but modularly retrieved on observer-specific clocks.

3. Observer-Dependent Entropy in Curved Spacetime

We classify three canonical observer trajectories and track entropy-retrieval dynamics along each. The retrieval rate $\gamma \left(\tau \right)$ is fixed by the local modular Hamiltonian, with no phenomenological fits, and evolves with proper time.

3.1. Classification of Observers

Stationary Observer.

A detector at fixed radius $r>2M$ perceives Hawking radiation as redshifted thermal flux; this gives

$${\gamma }_{\mathrm{stat}}\left(\tau \right)\propto {\displaystyle \frac{1}{r}},$$

(2)

and a monotonic decay in $g^{\left(2\right)}$ correlations. For $r=10M$ we find ${\tau }_{\mathrm{char}}<{\tau }_{\mathrm{Page}}$ because no interior mode enters the algebra.

Freely Falling Observer.

A geodesic world line crosses the horizon at ${\tau }_{\mathrm{cross}}$; interior modes then boost the retrieval rate,

$${\gamma }_{\mathrm{fall}}\left(\tau \right)\gg {\gamma }_{\mathrm{stat}}\left(\tau \right),\tau >{\tau }_{\mathrm{cross}},$$

(3)

accelerating saturation (orange curve in Figure 1).

Accelerating Observer.

A uniformly accelerating detector experiences both Hawking and Unruh flux,

$${\gamma }_{\mathrm{eff}}(\tau ,a)={\gamma }_{\mathrm{Hawking}}\left(\tau \right)+{\gamma }_{\mathrm{Unruh}}(\tau ,a),$$

(4)

with ${\gamma }_{\mathrm{Unruh}}\propto a^2$ [12]. At $a=0.2c^2/M$ the retrieval envelope is the green curve in Figure 1.

Experimental emulation: stationary and accelerating channels can be engineered in waterfall BECs, while freely falling trajectories correspond to time-of-flight release [13]. Parameters appear in Table 2.

Figure 1. Representative retrieval-rate profiles $\gamma \left(\tau \right)$ for the three observer classes. Stationary: $r=10M$ (blue); freely falling: geodesic starting at $r=6M$ (orange); accelerating: proper acceleration $a=0.2c^2/M$ (green). Times are in units of M with $G=c=1$.

Figure 1. Representative retrieval-rate profiles $\gamma \left(\tau \right)$ for the three observer classes. Stationary: $r=10M$ (blue); freely falling: geodesic starting at $r=6M$ (orange); accelerating: proper acceleration $a=0.2c^2/M$ (green). Times are in units of M with $G=c=1$.

Table 2. Indicative parameters for each observer class ($M=1$ in geometric units). The retrieval horizon ${\tau }_{\mathrm{RH}}$ is defined by $S_{\mathrm{retr}}\left({\tau }_{\mathrm{RH}}\right)=0.9S_{max}$.

Observer	$r/M$	$aM/c^2$	${\tau }_{\mathrm{char}}/M$	${\tau }_{\mathrm{Page}}/M$	${\tau }_{\mathrm{RH}}/M$
Stationary	10	0	5	8	30
Freely falling	6–2	0	2	4	10
Accelerating	—	0.2	3	5	15

Table 2. Indicative parameters for each observer class ($M=1$ in geometric units). The retrieval horizon ${\tau }_{\mathrm{RH}}$ is defined by $S_{\mathrm{retr}}\left({\tau }_{\mathrm{RH}}\right)=0.9S_{max}$.

Observer	$r/M$	$aM/c^2$	${\tau }_{\mathrm{char}}/M$	${\tau }_{\mathrm{Page}}/M$	${\tau }_{\mathrm{RH}}/M$
Stationary	10	0	5	8	30
Freely falling	6–2	0	2	4	10
Accelerating	—	0.2	3	5	15

3.2. Observer-Dependent Entropy

Observer-dependent entropy is the gap between the global von Neumann entropy and the entropy of the observer’s accessible subalgebra. The retrievable component $S_{\mathrm{retr}}\left(\tau \right)$ rises as modular eigenmodes enter the algebra; Appendix A.0.0.1 shows $\gamma \left(\tau \right)\propto {\partial }_{\tau }ln{\rho }_{\mathrm{mod}}$. Retrieval is computed only over causal diamonds with stable, horizon-bounded algebras; extension beyond ${\tau }_{\mathrm{RH}}$ may be limited by Type III₁ obstructions [11,21].

3.3. Retrieval Law

$${\displaystyle \frac{d{S}_{\mathrm{retr}}}{d\tau }}=\gamma \left(\tau \right)\left[S_{max}-S_{\mathrm{retr}}\left(\tau \right)\right]f\left(\tau \right),$$

(5)

with $f\left(\tau \right)=tanh\left(\tau /{\tau }_{\mathrm{char}}\right)$.

This functional form is uniquely fixed by bounded modular flow; the spectral proof is in Appendix A.0.0.1. Unlike the exponential damping factors used in replica-wormhole models, $\gamma \left(\tau \right)$ and ${\tau }_{\mathrm{char}}$ are determined directly from the local modular Hamiltonian, yielding a continuous, observer-specific retrieval process. We refer to $\gamma \left(\tau \right)$ as the modular-flow retrieval rate; it quantifies the pace at which retrievable information enters an observer’s algebra.

4. Quantum Information Correlations and Testable Predictions

The retrieval law in Eq. (5) imprints a characteristic signature on the radiation detected by each observer class. It governs both entropy growth and correlation decay, features that analog-gravity experiments can probe directly. We focus on two diagnostics, the order-$\alpha$ Rényi entropy and the second-order correlation function $g^{\left(2\right)}$.

Simulation traces with $95\%$ confidence bands for each class appear in Figure 2. Bands come from 200 bootstrap resamplings of $\gamma \left(\tau \right)$ on a fixed proper-time grid with additive spectral noise.

4.1. Rényi Entropy and Second-Order Correlation Functions

For any subsystem A, the Rényi entropy is

$$S_{\alpha }\left(t\right)={\displaystyle \frac{ln\left[Tr\left({\rho }_A^{\alpha }\right)\right]}{\alpha -1}},$$

(6)

with $\alpha >1$. Equation (6) arises by analytically continuing the integer-order moments $Tr\left({\rho }_A^n\right)$ (the replica trick); for a field-theoretic derivation see Casini, Huerta, and Myers [14]. Larger $\alpha$ heightens sensitivity to eigenvalue gaps; $S_{\alpha }$ therefore probes the observer-dependent delay $\Delta \tau$. Interferometric methods for measuring $S_{\alpha }$ in Bose–Einstein condensates are outlined in Ref. [13].

The modeled second-order correlation is

$$g^{\left(2\right)}(t_1,t_2)=exp\left[-|t_2-t_1|/{\tau }_{\mathrm{retrieval}}\right]\left[1+\frac{1}{2}\left(1+tanh(t_1/{\tau }_{\mathrm{Page}})\right)\right],$$

(7)

where

$${\tau }_{\mathrm{retrieval}}\left(t\right)={\int }_0^t\gamma \left({\tau }^{\prime }\right)d{\tau }^{\prime }$$

accumulates the observer-specific retrieval rate and ${\tau }_{\mathrm{Page}}$ is the class-dependent Page time in Table 2. In a baseline waterfall BEC, ${\tau }_{\mathrm{retrieval}}\approx 20\mathrm{ms}$, well above the $2\mathrm{ms}$ resolution reported in Ref. [13]. Typical flux and background levels yield $\mathrm{SNR}\gtrsim 4$. Setting $\gamma \left(\tau \right)=0$ yields a symmetric exponential decay and provides a direct null test.

Parameters are extracted with nonlinear least squares and $95\%$ confidence intervals from 200 synthetic traces per class. Equations (6) and (7) are strict functionals of the retrieval law: $g^{\left(2\right)}$ captures decay-modulated interference, while $S_{\alpha }$ tracks the evolving purity of the retrievable subsystem. No replica-wormhole or island framework predicts the frame-dependent interference in $g^{\left(2\right)}(t_1,t_2)$; the accelerating signal therefore cleanly discriminates global from observer-indexed recovery.

Empirical Extraction of ${\tau }_{\mathrm{char}}$

Once either $g^{\left(2\right)}(t_1,t_2)$ or a synthetic $S_{\mathrm{retr}}\left(\tau \right)$ trace is measured, the characteristic timescale ${\tau }_{\mathrm{char}}$ can be obtained with a one-parameter fit. For the symmetric slice $t_1=0$ we rewrite Eq. (7) as

$$g_{\mathrm{env}}^{\left(2\right)}\left(t\right)=Aexp\left[-t/{\tau }_{\mathrm{retrieval}}\right]\left[1+\frac{1}{2}\left(1+tanh(t/{\tau }_{\mathrm{char}})\right)\right],$$

(8)

with amplitude A fixed by early-time data. A nonlinear least-squares estimator minimizes

$${\chi }^2\left({\tau }_{\mathrm{char}}\right)=\sum _{i=1}^N{\displaystyle \frac{{\left[g_{\mathrm{data}}^{\left(2\right)}\left(t_i\right)-g_{\mathrm{env}}^{\left(2\right)}\left(t_i\right)\right]}^2}{{\sigma }_i^2}},$$

yielding

$${\tau }_{\mathrm{char}}^{\mathrm{fit}}=3.47\pm 0.22\mathrm{ms}(95\%\mathrm{C}.\mathrm{L}.)$$

for the baseline accelerating trace in Figure 2. The procedure assumes (i) $S_{\mathrm{ret}}\in C^1$, (ii) monotonic $\gamma \left(\tau \right)$, (iii) ${\tau }_{\mathrm{char}}>0$, and (iv) $\mathrm{SNR}\ge 4$. A reproducible Python notebook in the project repository automates the fit and returns error bars and residuals for any input trace.

Figure 2. Entropy retrieval versus proper time for stationary (blue), freely falling (orange), and accelerating (green) observers. Shaded regions show $95\%$ bootstrap confidence bands. The vertical dashed line marks the class-specific Page time ${\tau }_{\mathrm{Page}}$.

Figure 2. Entropy retrieval versus proper time for stationary (blue), freely falling (orange), and accelerating (green) observers. Shaded regions show $95\%$ bootstrap confidence bands. The vertical dashed line marks the class-specific Page time ${\tau }_{\mathrm{Page}}$.

5. Holographic Connection and Quantum-Circuit Simulations

5.1. Observer-Dependent Ryu–Takayanagi Prescription

To incorporate observer-indexed accessibility we generalize the Ryu–Takayanagi (RT) prescription by adding a modular-frame redshift factor. The observer-dependent holographic entanglement entropy is

$$S_{\mathrm{obs}}^{\mathrm{holo}}={\displaystyle \frac{Area\left[{\gamma }_A\left(\mathsf{\Lambda }\right)\right]}{4G_N}}\sqrt{|g_{00}\left(\mathsf{\Lambda }\right)|},$$

(9)

where ${\gamma }_A\left(\mathsf{\Lambda }\right)$ is the bulk minimal surface in the boosted geometry and $g_{00}\left(\mathsf{\Lambda }\right)$ converts boundary time to the observer’s proper time. The choice $g_{00}=1$ and $\mathsf{\Lambda }=\mathrm{id}$ recovers the Hubeny–Rangamani–Takayanagi formula.

• ${\gamma }_A\left(\mathsf{\Lambda }\right)$: minimal surface in the Lorentz-boosted bulk;
• $g_{00}\left(\mathsf{\Lambda }\right)$: lapse tying the surface to the wedge reachable along the observer’s world line.

The redshift factor follows from modular-Hamiltonian anchoring (Appendix A; see [14,15] for background). Recent work on crossed-product and edge-mode algebras [11,16] supports extending this prescription into strong-gravity regimes.

Table 3. Predicted laboratory signatures for each observer class.

Observer	Retrieval rate $\gamma \left(\tau \right)$	Correlation signature
Stationary	$\gamma \propto 1/r$	Exponential decay; weak long-range $g^{\left(2\right)}$
Freely falling	Sharp rise after horizon crossing	Non-monotonic $g^{\left(2\right)}$; interior-mode revival
Accelerating	${\gamma }_{\mathrm{eff}}\propto a^2$	tanh-modulated fringe in $g^{\left(2\right)}(t_1,t_2)$

Table 3. Predicted laboratory signatures for each observer class.

Observer	Retrieval rate $\gamma \left(\tau \right)$	Correlation signature
Stationary	$\gamma \propto 1/r$	Exponential decay; weak long-range $g^{\left(2\right)}$
Freely falling	Sharp rise after horizon crossing	Non-monotonic $g^{\left(2\right)}$; interior-mode revival
Accelerating	${\gamma }_{\mathrm{eff}}\propto a^2$	tanh-modulated fringe in $g^{\left(2\right)}(t_1,t_2)$

6. Quantum-Circuit Simulations

Equation (5) and the modified RT surface were simulated in a 48-qubit HaPPY/MERA tensor network [17]. Observer channels were imposed by boosting boundary tensors and shifting the reconstruction region.

MERA Convergence.

Bond dimensions $D=4$ and $D=8$ produced less than $1\%$ variance in saturation times and in the $g^{\left(2\right)}$ amplitude.

Key Findings.

• Entropy curves differ by observer, matching the time-adaptive theory.
• Accelerating observers show the tanh fringe in $g^{\left(2\right)}$ predicted by Eq. (7).
• Boosts alter boundary patches; minimal-surface areas vary exactly as Eq. (9) requires.

The full predicted envelope is

$$g^{\left(2\right)}(t_1,t_2)=A\left[1-tanh\left(\frac{\tau \left(t_1\right)}{{\tau }_{\mathrm{char}}}\right)\right]\left[1-tanh\left(\frac{\tau \left(t_2\right)}{{\tau }_{\mathrm{char}}}\right)\right],$$

where $\tau \left(t\right)$ maps detector time to proper time. This structure defines a falsifiable signature of modular retrieval collapse that cannot be reproduced by thermal smoothing or decoherence (see Lemma C.5).

A complete inversion and validation pipeline, including ${\tau }_{\mathrm{char}}$ fitting, $\gamma \left(\tau \right)$ reconstruction, and null-envelope rejection tests, is provided in the companion notebook ODER_Retrieval _Inversion_And_Validation.ipynb (https://doi.org/10.5281/zenodo.15428312).

Computational Complexity.

Unlike global Hayden–Preskill decoding, which requires $\mathcal{O}\left(2^n\right)$ gates, MERA-based observer retrieval proceeds at $\mathcal{O}(nlogn)$ depth because the causal cone restricts reconstruction to at most $logn$ layers in an n-qubit MERA.2

Figure 3. Second -order correlation matrix $g^{\left(2\right)}(t_1,t_2)$ for an accelerating observer with $a=0.2c^2/M$. The color scale represents the dimensionless quantity $g^{\left(2\right)}$ (connected); the color bar appears at right. The bright diagonal band is the predicted tanh-modulated retrieval envelope. Dashed lines mark $t_1=t_2$ and the Page time ${\tau }_{\mathrm{Page}}\simeq 12M$.

Figure 3. Second -order correlation matrix $g^{\left(2\right)}(t_1,t_2)$ for an accelerating observer with $a=0.2c^2/M$. The color scale represents the dimensionless quantity $g^{\left(2\right)}$ (connected); the color bar appears at right. The bright diagonal band is the predicted tanh-modulated retrieval envelope. Dashed lines mark $t_1=t_2$ and the Page time ${\tau }_{\mathrm{Page}}\simeq 12M$.

7. Implications

The benchmarks in Section 3, Section 4 and Section 5 rely only on wedge coherence from observer-dependent modular flow; no replica wormholes, islands, or exotic topologies are required. Entropy recovery is a continuous, frame indexed process; saturation resembles a Page curve only along trajectories that respect modular access, making the theory falsifiable in analog and numerical experiments (Figure 2).

7.1. Resolution of the Information Paradox and Empirical Constraints

ODER recasts the paradox as an observer-indexed retrieval problem. For any world line, Eq. (5) drives a smooth rise to saturation, matching the Page curve only at late times for that observer. The tanh onset is fixed by modular flow; no ensemble averaging is needed.

Island prescriptions for accelerated detectors [1,7,15] reproduce a Page-like curve globally; the retrieval law produces the same saturation locally and supplies a causal decoder. Replica and island frameworks conserve entropy globally but lack any polynomial-time recovery protocol compatible with local modular evolution [8].

7.2. Retrieval Horizon ≠ Entanglement Wedge ≠ Event Horizon

Observer-dependent modular flow separates three boundaries:

• Retrieval horizon.${\displaystyle {\tau }_{\mathrm{RH}}=inf\{\tau \mid S_{\mathrm{retr}}\left(\tau \right)\ge 0.9S_{max}\}.}$
• Entanglement wedge. The bulk region reconstructable through the boosted RT surface, Eq. (9).
• Event horizon. The classical null surface.

In Kerr spacetime the generator $\chi ={\partial }_t+{\mathsf{\Omega }}_H{\partial }_{\varphi }$ gives

$$\gamma (\tau ,a,\mathsf{\Omega })={\left|g_{\mu \nu }{\chi }^{\mu }{\chi }^{\nu }\right|}^{-1/2},$$

evaluated just outside $r_{+}$. Where $\chi$ is timelike, the Paley–Wiener bound preserves the tanh onset [24].

7.3. Implications for Evaporating Black Holes

• Stationary observers ($r>2M$). Slow retrieval, $\gamma \propto 1/r$.
• Freely falling observers. Interior modes boost $\gamma$ after horizon crossing.
• Accelerating observers. Unruh terms create the $g^{\left(2\right)}$ fringe.

In every case ${\displaystyle \underset{\tau \to \infty }{lim}{S}_{\mathrm{retr}}\left(\tau \right)=S_{max}}$; saturation stems from modular closure, not ensemble averaging.

7.4. ${\Delta }_{\mathrm{fail}}$: retrieval–evaporation boundary

Define

$${\Delta }_{\mathrm{fail}}={\tau }_{\mathrm{evap}}-{\tau }_{\mathrm{RH}},$$

with ${\tau }_{\mathrm{evap}}$ the semiclassical evaporation time. The retrieval horizon corresponds to the inflection point in the entropy-access curve, where modular acceleration vanishes (see Proposition A.3). Positive ${\Delta }_{\mathrm{fail}}$ means retrieval completes before evaporation; negative values imply modular failure.

Table 4. Benchmark ${\Delta }_{\mathrm{fail}}$ values ($M=1$ in geometric units).

Observer	${\tau }_{\mathrm{RH}}$	${\tau }_{\mathrm{evap}}$	${\Delta }_{\mathrm{fail}}$
Stationary	30	$\sim {10}^{67}$	$\gg 0$
Freely falling	10	$\sim {10}^{67}$	$\gg 0$
Accelerating	15	$\sim {10}^{67}$	$\gg 0$

Table 4. Benchmark ${\Delta }_{\mathrm{fail}}$ values ($M=1$ in geometric units).

Observer	${\tau }_{\mathrm{RH}}$	${\tau }_{\mathrm{evap}}$	${\Delta }_{\mathrm{fail}}$
Stationary	30	$\sim {10}^{67}$	$\gg 0$
Freely falling	10	$\sim {10}^{67}$	$\gg 0$
Accelerating	15	$\sim {10}^{67}$	$\gg 0$

A negative ${\Delta }_{\mathrm{fail}}$ in analog experiments or numerical simulations would falsify the retrieval law; any ${\Delta }_{\mathrm{fail}}\ge 0$ is consistent with modular accessibility.

7.5. Experimental Implications and Roadmap

Timescale bridge

With $G=\hslash =c=1$ and $1M_{\odot }\simeq 4.93\mu \mathrm{s}$,

$$\Delta t_{\mathrm{lab}}\simeq 4.93\mu \mathrm{s}\left(M/M_{\odot }\right)(\Delta \tau /1M).$$

A $2–20M$ window in a $10M_{\odot }$ acoustic analog thus maps to $10\mathrm{ms}–100\mathrm{ms}$, well above the $2\mathrm{ms}$ detector limit of Ref. [13].

Operational falsifiability

• Absence of a $g^{\left(2\right)}$ envelope implies modular access is falsified.
• A mismatched $\gamma \left(\tau \right)$ fit implies the retrieval law is incomplete.
• Identical ${\tau }_{\mathrm{Page}}$ for all observers implies observer specificity is invalid.

Table 5. Operational comparison for a stationary observer at $r=10M$.

Feature	ODER (this work)	Replica or islands
Causal retrieval	✓ proper-time decoder	× stabilization only
Decoding protocol	✓ polynomial MERA	× none known
Empirical observable	✓ $g^{\left(2\right)}$ in BEC	× not specified
Computational cost	$\mathcal{O}\left(n^2\right)$	$\mathcal{O}\left(2^n\right)$

Table 5. Operational comparison for a stationary observer at $r=10M$.

Feature	ODER (this work)	Replica or islands
Causal retrieval	✓ proper-time decoder	× stabilization only
Decoding protocol	✓ polynomial MERA	× none known
Empirical observable	✓ $g^{\left(2\right)}$ in BEC	× not specified
Computational cost	$\mathcal{O}\left(n^2\right)$	$\mathcal{O}\left(2^n\right)$

Observation, or systematic absence, of these signatures decisively tests observer-dependent modular flow.

8. Limitations and Scope

Although the framework is tractable and experimentally accessible, several assumptions restrict its generality and point to directions for refinement.

Retrieval-Driven Back-Reaction: A Thresholded Causal Ansatz

All retrieval dynamics in this work assume a fixed background metric. Introducing a small coupling,

$$T_{\mu \nu }⟶T_{\mu \nu }+\alpha T_{\mu \nu }^{\mathrm{retrieval}},\alpha \ll 1,$$

one recovers the semiclassical Einstein equation in the limit $\alpha \to 0$. For $\alpha \ne 0$ the retrieval horizon shifts only at $\mathcal{O}\left(\alpha \right)$.

Back-reaction bound.

For a Schwarzschild mass M,

$$〈T_{\mu \nu }^{\mathrm{retrieval}}〉\sim {\displaystyle \frac{\gamma \left(\tau \right)S_{max}}{4\pi r_{+}^2}},S_{max}\propto M^2,$$

so that

$${\displaystyle \frac{G〈T_{\mu \nu }^{\mathrm{retrieval}}〉}{\sqrt{\mathcal{K}}}}\lesssim {10}^{-6},\mathcal{K}=48G^2{M}^2/r_{+}^6,M\gtrsim M_{\odot }.$$

For a fiducial $10M_{\odot }$ black hole one finds $G\langle T_{\mu \nu }^{\mathrm{retrieval}}\rangle \approx 4\times {10}^{-7}\sqrt{{\mathcal{K}}_{\left(10M_{\odot }\right)}}$, implying $\delta r_{+}/r_{+}<3\times {10}^{-6}$ and a negligible shift in ${\tau }_{\mathrm{RH}}$.

Outlook. A fully coupled model in which $T_{\mu \nu }^{\mathrm{retrieval}}\propto \left({\partial }_{\tau }{S}_{\mathrm{retr}}\right)u_{\mu }{u}_{\nu }$ would elevate entropy retrieval to an explicit causal modulator of curvature [18].

Semiclassical modular-flow assumption.

Type III₁ algebras are regulated by finite splits [19,20]; extending to Kerr, de Sitter, or multi-horizon cases will need relative-Tomita theory and edge modes [11].

Analog-System Resolution

Current BEC experiments resolve $g^{\left(2\right)}$ on $2–10\mathrm{ms}$ scales [13], five times finer than the predicted $10–100\mathrm{ms}$ retrieval window. Baseline $g^{\left(2\right)}$ runs should precede interpretation.

Exclusion of Exotic Topologies

Replica wormholes, islands, and other speculative geometries are omitted, keeping all predictions directly testable.

Potential Extension to Superposed Geometries

Future work could apply the retrieval law to geometries in quantum superposition, probing modular coherence across fluctuating horizons.

No Global Unitarity Guarantee

Equation (5) ensures unitarity only inside each observer’s wedge; modular mismatches between overlapping diamonds are expected.

Retrieval-Horizon Scope

The framework guarantees saturation of $S_{\mathrm{retr}}\left(\tau \right)$ only up to ${\tau }_{\mathrm{RH}}$; full recovery beyond that point lies outside its present mandate. The theory defines testable envelopes but does not yet model complete detector noise or ROC sensitivity curves.

9. Conclusion and Next Steps

We presented a relativistic, observer-dependent framework for black-hole entropy retrieval that provides a causal bridge between quantum mechanics and general relativity without introducing nonunitary dynamics or speculative topologies. By anchoring information flow to proper time and causal access, ODER transforms Page-curve bookkeeping into a continuous, falsifiable description of entropy transfer. All derivations and simulation protocols are supplied for stand-alone reproducibility.

The retrieval law is not heuristic; it follows from Tomita–Takesaki modular spectra (Appendix A, Eq. (A.1)). Bounded modular flow links spectral smoothing, redshift factors, and observer-specific algebras, making retrieval a physical process, not an epistemic relabel.

Concrete predictions follow. Stationary, freely falling, and uniformly accelerated observers exhibit distinct retrieval rates and $g^{\left(2\right)}$ envelopes, all testable with current analog-gravity platforms. Failure to observe these signatures would falsify observer-modular accessibility while leaving modular flow itself intact.

Roadmap: theory, simulation, experiment

Theory

• Semiclassical back-reaction. Couple entropy flow to a self-consistent metric response, extending Eq. (5) into a dynamical observer–spacetime equation.
• Intersecting horizons. Analyze overlapping causal diamonds to refine the retrieval-horizon concept.
• Superposed geometries. Apply retrieval dynamics to metrics held in quantum superposition.

Simulation

• High-bond-dimension MERA. Benchmark $D>8$ convergence and finite-entanglement effects on $\gamma \left(\tau \right)$.
• Error budgets. Propagate detector-noise kernels to produce ROC-style sensitivity curves.

Experiment

• Trajectory-differentiated probes. Deploy stationary, co-moving, and accelerating detectors in BEC waterfalls; target the $10\mathrm{ms}$–$100\mathrm{ms}$ window with $\lesssim 2\mathrm{ms}$ timing.
• Cross-platform checks. Replicate $g^{\left(2\right)}$ envelopes in photonic-crystal and superconducting-circuit analogs.

These coordinated steps will sharpen theory and enable empirical tests. Upcoming data will show whether modular-access entropy flow provides a testable, observer-specific alternative to purely global unitarity.