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

1. Introduction

This section reframes the black-hole information paradox as a problem of causal access, specifies the operational criteria required for its resolution, and explains why existing approaches fail to meet those criteria.

This work establishes a causal law governing when information becomes accessible to a bounded observer. It does not modify entropy accounting, propose a decoding procedure, or reconstruct global states. It specifies access dynamics alone.

Entropy Without Access: The Operational Gap

The black-hole information paradox persists not because information is lost, but because no existing framework supplies a causal law of finite-observer retrieval. Replica-wormhole paths, island prescriptions, ensemble Page-curve models, and $\mathrm{ER}=\mathrm{EPR}$ dualities all reproduce the required fine-grained entropy curves, yet none supplies a Lorentzian, proper-time recovery channel that makes the relevant state information accessible to a finite detector. Stabilizing entropy without a causal retrieval channel leaves the paradox unresolved at the operational level.

Independent laboratory experiments on non-gravitational quantum systems have recently demonstrated universal bounds on the rate at which coherence becomes accessible, supporting the interpretation pursued here that access, rather than dynamics alone, sets the relevant clock.

Within the regularized setting developed here, retrievable entropy evolves along the unique extremal branch that saturates the bounded-spectrum speed limit.

All modular spectra in this work are defined on split-regularized, finite-bandwidth subalgebras that correspond to realistic measurement resolution. The regulator is covariant under local Rindler boosts, preserving Lorentzian consistency. The full algebraic limit, formally Type ${\mathrm{III}}_1$, is discussed in Appendix D. This keeps the Type ${\mathrm{III}}_1$ foundation explicit while allowing the main text to focus on physical retrieval dynamics.

Key assumptions. Modular spectra are bounded by the split-property regularization introduced in Appendix D; modular flow is treated semiclassically on fixed backgrounds; and present BEC analog systems resolve $g^{(2)}$ down to approximately $2\mathrm{ms}$.

All derivations in Section 2, Section 3, Section 4, Section 5, Section 6 and Section 7 assume this regularized setting unless stated otherwise.

1.1. Relation to Algebraic Entropy and Crossed-Product Constructions

Recent work in algebraic quantum field theory has clarified a foundational issue underlying gravitational entropy: how entropy may be meaningfully defined for systems whose local algebras are Type III and therefore admit no trace. In particular, crossed-product constructions developed by Chandrasekaran, Penington, and Witten, building on earlier modular and algebraic work by Longo and others, establish a rigorous framework in which entropy becomes finite, well posed, and mathematically meaningful for quantum field theories with gravitational relevance.

These results resolve definability: what it means for entropy to exist as a mathematical quantity in relativistic quantum systems where density-matrix notions fail.

The contribution of crossed-product and related algebraic constructions is foundational but specific in scope. They clarify the algebraic status of entropy and modular structure, but do not supply a dynamical law describing how entropy becomes accessible to a finite observer, nor do they impose rate bounds, proper-time constraints, or observer-indexed access conditions.

The present work takes algebraic definability as given and addresses a different object: how entropy becomes operationally retrievable along a Lorentzian worldline. The retrieval framework introduced here operates at the level of access dynamics and constrains when information becomes available rather than how it is defined.

1.2. Operational–Access Criterion

Because entropy conservation, formal reconstruction, and finite-observer retrieval are distinct operations, a framework cannot be evaluated by Page-curve reproduction alone. A framework resolves the operational-access form of 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 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 $\Delta \tau$ in a computationally tractable finite-resolution proxy. Empirical testability requires predicting observer-dependent lags resolvable at laboratory timescales, for example on millisecond scales in current BEC analog systems.

The following audit evaluates frameworks strictly by whether they deliver observer-accessible retrieval in proper time, not by whether they reproduce correct entropy curves.

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

Framework	(a)	(b)	(c)	(d)
Replica wormholes	×	×	✓	×
Islands	×	×	✓	×
Ensemble Page models	×	✓	×	×
$\mathrm{ER}=\mathrm{EPR}$	×	✓	×	×

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

Framework	(a)	(b)	(c)	(d)
Replica wormholes	×	×	✓	×
Islands	×	×	✓	×
Ensemble Page models	×	✓	×	×
$\mathrm{ER}=\mathrm{EPR}$	×	✓	×	×

Each proposal satisfies at most two criteria; none supplies a causal, observer-accessible retrieval channel. Resolution of the operational-access problem therefore requires a retrieval law derived in proper time, whose admissible form is fixed by bounded spectral constraints and whose predictions are experimentally testable.

Retrieval, Reconstruction, and Comprehension: Non-Equivalence

Throughout this work we distinguish three logically independent notions:

• Retrieval: physical access of information into an observer’s causal and modular domain.
• Reconstruction: the existence of a formal decoding map or entanglement-based reconstruction channel.
• Comprehension: the observer’s ability to interpret or act on retrieved information.

Reconstruction does not imply retrieval, and retrieval does not imply comprehension. The operational form of the black-hole information paradox arises from conflating these categories. ODER addresses retrieval only. Under this separation, the paradox is retyped: what appeared as information loss becomes the absence of a lawful observer-indexed retrieval channel.

Throughout, we enforce the non-equivalence:

$$\mathrm{retrieval}\ne \mathrm{reconstruction}\ne \mathrm{comprehension}.$$

A system may be reconstructable yet not retrieved, and retrieved yet not comprehended.

The argument proceeds in three layers: derivation of the retrieval law, constraint of its admissible form under bounded spectra, and specification of experimentally testable signatures.

The claim operates at the level of access dynamics; critiques framed in terms of entropy curves or reconstruction protocols misidentify the object of the argument.

The consequence is that ODER changes the object of the problem. If information conservation, reconstruction, and finite-observer retrieval are distinct operations, then the apparent paradox is a category error produced by treating global preservation as equivalent to causal access. The framework developed here therefore reframes black-hole information recovery as a proper-time retrieval problem: conserved information becomes physically relevant to an observer only when it enters that observer’s bounded modular-access domain.

Internal convergence of the access law. The force of the argument lies in the convergence of independent constraints. Split-regularized modular spectra restrict the admissible retrieval trajectory; analytic speed-limit arguments select the extremal tanh-gap envelope; the inverse retrieval map makes the observer-indexed rate recoverable from access traces; observer-class dynamics generate ordered retrieval horizons; and adversarial numerical controls test whether the joint signature can be reproduced by generic saturating alternatives. These layers converge on the same missing object: a causal law of observer-indexed access.

2. Observer-Dependent Entropy Retrieval

This section derives the governing observer-indexed retrieval law from modular flow, establishes its unique speed-limit character, and specifies the conditions under which it can be inferred or falsified.

• Observer-indexed retrieval dynamics.

ODER treats retrieval as a dynamical, observer-indexed process and employs the unique tanh onset that, as proved in Theorem A.2, is the unique speed-limit-saturating profile selected by bounded modular spectra and Paley–Wiener admissibility within the stated strip-holomorphic class.

The tanh profile is not a phenomenological fit to entropy curves; it is the unique extremal branch that saturates the modular speed limit under bounded spectra.

This section derives

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

(1)

from split-regularized Tomita–Takesaki modular flow on nested von Neumann algebras.1

This boundedness is a physical consequence of finite observer bandwidth rather than a technical regulator; the Type ${\mathrm{III}}_1$ algebra is recovered only in the idealized limit of infinite resolution (Appendix D).

The functional form of the retrieval law is fixed by bounded spectral constraints; $\gamma (\tau )$ modulates traversal along this envelope without altering its admissible shape. Here $\gamma (\tau )$ is not an independent dynamical field or free function; it is an observer- and state-dependent rate induced by the local modular Hamiltonian and is, in principle, reconstructible from measured retrieval trajectories via Eq. (2).

Because Eq. (1) is first order and monotone in the bounded and differentiable $S_{\mathrm{retr}}(\tau )$, it admits an algebraically unique inverse map for the unknown rate $\gamma (\tau )$ on intervals where $S_{max}-S_{\mathrm{retr}}(\tau )>0$ and $tanh(\tau /{\tau }_{\mathrm{char}})\ne 0$. This inversion is used only downstream, where retrieval rates for different observer classes are compared.

$$\begin{array}{|c|}\hline \gamma (\tau )={\displaystyle \frac{1}{\left[S_{max}-S_{\mathrm{retr}}(\tau )\right]tanh\left(\tau /{\tau }_{\mathrm{char}}\right)}}{\displaystyle \frac{d{S}_{\mathrm{retr}}}{d\tau }}\\ \hline\end{array}$$

(2)

Implication. Once an experiment estimates $S_{\mathrm{retr}}(\tau )$, for example through the $g^{(2)}$ fringe, the boxed map fixes $\gamma (\tau )$ without further assumptions, rendering the retrieval law calibratable without elevating inversion to a defining premise.

We define the retrieval horizon

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

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

2.1. Retrieval as a Modular Speed Limit

• Speed-limit principle.

The tanh onset in Eq. (1) was introduced as the unique sigmoidal retrieval profile compatible with bounded modular spectra (Theorem A.2). We now strengthen this result: the hyperbolic-tangent envelope does not merely fit observer-accessible entropy traces; it saturates a modular speed limit imposed by the Paley–Wiener constraint on the modular spectrum [36].

Let $\sigma (K)\subset [-\Lambda (\delta ),\Lambda (\delta )]$ denote the split-regularized spectrum of the modular Hamiltonian. Paley–Wiener theory requires that any admissible retrieval trajectory $F(\tau )=S_{\mathrm{retr}}(\tau )/S_{max}$ be holomorphic in the Paley–Wiener strip and of bounded exponential type $\le \Lambda (\delta )$ within that strip. For such $F(\tau )$, the slope is bounded:

$$\dot{F}(\tau )\le \Lambda (\delta )[1-F(\tau )]tanh\left({\displaystyle \frac{\pi \Lambda (\delta )\tau }{2}}\right),$$

(3)

2 which we denote as the modular speed limit $\mathcal{B}\left(\Lambda (\delta ),\tau \right)$.

• Theorem 2.1 (Modular Speed Limit).

Let $F(\tau )$ be $C^1$, strictly increasing, holomorphic in the Paley–Wiener strip, and of bounded exponential type $\le \Lambda (\delta )$ in that strip. Then the inequality (3) holds pointwise in $\tau$; moreover, the tanh profile of Theorem A.2 achieves this bound uniquely, up to an affine reparameterization of $\tau$ [36].

• Constrained variational consistency form.

The modular speed-limit structure also admits a constrained variational representation. This representation is downstream of the spectral admissibility result above: it does not provide an independent derivation of the retrieval law, but records the same bounded-access constraint in extremal form. The admissible trajectory class is already restricted by split-regularized modular spectra and the Paley–Wiener speed limit. Within that restricted class, the tanh-gap profile is the unique branch that saturates the bound.

The corresponding entropy-action functional encodes the cost of departing from the bounded-spectrum retrieval trajectory:

$$\mathcal{I}[F]=\int d\tau \left[{\displaystyle \frac{1}{2}}{\left(\dot{F}(\tau )\right)}^2-U(F;\Lambda (\delta ),{\tau }_{\mathrm{char}})\right],$$

(4)

where ${\tau }_{\mathrm{char}}$ corresponds to the observer’s bandwidth-limited response time defined in the Introduction. The associated potential is

$$U(F;\Lambda (\delta ),{\tau }_{\mathrm{char}})={\displaystyle \frac{1}{2}}\Lambda {(\delta )}^2{[1-F(\tau )]}^2{tanh}^2\left({\displaystyle \frac{\tau }{{\tau }_{\mathrm{char}}}}\right).$$

(5)

The unconstrained Euler–Lagrange equation associated with $\mathcal{I}[F]$ defines the corresponding second-order extremal problem. The first-order retrieval law is recovered only after restricting to the admissible branch that saturates the modular speed limit:

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

(6)

Under this constrained reading, the variational form is a consistency representation of the retrieval law, not a separate dynamical postulate.

The observer-dependent rate $\gamma (\tau )$ is therefore not introduced as a new field by the action. It remains the local observer- and state-dependent modular retrieval rate governing traversal along the admissible tanh-gap envelope. In this sense, the variational form expresses the same hierarchy used throughout this paper: bounded modular spectra select the admissible envelope, speed-limit saturation selects the extremal branch, and $\gamma (\tau )$ determines the observer-specific rate along that branch.

• Onset scale.

The transition scale ${\tau }_{\mathrm{char}}$ is not a free curve-fitting parameter. In the gapped idealization, it is fixed by the effective modular energy gap $\Delta K$ and the relevant horizon-scale geometry:

$${\tau }_{\mathrm{char}}=\left({\displaystyle \frac{2\pi }{\Delta K}}\right)f(R_{\mathrm{H}}),$$

where $f(R_{\mathrm{H}})$ encodes the geometric scaling of the observer’s horizon radius. Operationally, ${\tau }_{\mathrm{char}}$ is inferred from early-time onset behavior under fixed detector resolution.

• Independent empirical evidence for access-limited dynamics.

The following observation is interpretive and does not enter the derivation of Eq. (1).

Independent support for bounded-access convergence laws outside gravitational settings comes from ultracold-atom experiments on Bose–Einstein condensate formation [34]. In these systems, the spatial coherence length ${\ell }_c(t)$ satisfies ${\ell }_c^2(t)\sim Dt$ at late times, with a coarsening rate $D\simeq 3.4\hslash /m$ that becomes universal once the system enters the scaling regime, despite interaction-dependent early-time transients and offsets. This bound does not constrain transport or signaling velocities; it constrains the rate at which long-range coherence becomes accessible during non-equilibrium relaxation.

Although such systems do not admit a modular Hamiltonian or Tomita–Takesaki automorphism group, the observed coherence-growth bound instantiates an access-limited saturation law: the emergence of retrievable structure is constrained by spectral admissibility rather than by microscopic dynamics. Its relevance to ODER is structural rather than mechanistic.

The condensate result therefore shows that access-limited convergence laws can occur in quantum systems without horizons or spacetime curvature, supporting the interpretation that the modular speed limit in Eq. (3) reflects a broader constraint on accessible structure under bounded spectral support.

Failure of coherence-speed bounds in appropriate homogeneous quantum systems would not falsify Eq. (1) directly, but it would constrain the broader hypothesis that bounded spectral access generically limits the rate at which retrievable structure becomes available.

3. Observer-Dependent Entropy in Curved Spacetime

This section instantiates the retrieval law for canonical observer trajectories in curved spacetime, specifies which quantities are fixed by theory versus inferred from measurement, and identifies observer-class signatures accessible to experiment.

What is fixed vs. what is inferred. The retrieval rate $\gamma (\tau )$ is determined by the local modular Hamiltonian (theory statement); it is inferred from measured $S_{\mathrm{retr}}(\tau )$ via the inverse map (measurement statement). The onset scale ${\tau }_{\mathrm{char}}$ is fixed by bounded spectral access and operationally calibrated from early-time behavior under finite observer resolution. All analyses assume the finite-bandwidth, split-regularized subalgebras introduced in Appendix D, where bounded modular spectra $\sigma (K)\subset [-\Lambda (\delta ),\Lambda (\delta )]$ reflect finite observer access.

3.1. Classification of Observers

The three observer classes are implemented through distinct retrieval-rate profiles $\gamma (\tau )$, corresponding to stationary exterior access, freely falling post-crossing access, and acceleration-enhanced modular access. Representative profiles are shown in Figure 1. These profiles are observer-class instantiations of the retrieval law, not independent entropy models.

• Stationary observer.

A detector at fixed radius $r>2M$ perceives Hawking radiation as redshifted thermal flux. In the representative exterior-access regime, the corresponding modular-flow retrieval rate scales with the local flux amplitude,

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

(7)

up to normalization by the surface gravity $\kappa$, which fixes the local modular temperature $T_{\mathrm{H}}=\kappa /(2\pi )$. This produces a slowly varying exterior-access channel and a monotonic decay in $g^{(2)}$ correlations.

• Freely falling observer.

A geodesic world line crosses the horizon at ${\tau }_{\mathrm{cross}}$. After crossing, the accessible mode structure changes along the observer’s proper-time trajectory, producing an enhanced retrieval-rate profile relative to the stationary exterior channel,

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

(8)

• Accelerating observer.

A uniformly accelerating detector experiences both Hawking and Unruh contributions to its effective access channel,

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

(9)

with ${\gamma }_{\mathrm{Unruh}}\propto a^2$ in the representative acceleration-enhanced regime.

• Experimental mapping.

Stationary and accelerating channels can be engineered in waterfall BEC analogs through controlled flow and detector configurations, while freely falling access is approximated by a rapid access-window or channel-change protocol after the horizon-crossing analogue. Detectability requires signal-to-noise ratio $\ge 4$ at temporal resolution $\approx 2\mathrm{ms}$.

3.2. Observer-Dependent Entropy

Observer-dependent entropy is the retrieval gap between the total retrievable entropy scale $S_{max}$ and the entropy accessible within the observer’s split-regularized subalgebra. The retrievable component $S_{\mathrm{retr}}(\tau )$ increases as modular eigenmodes become accessible within the observer-defined subalgebra.

3.3. Retrieval Law (Instantiation)

For bounded and differentiable $S_{\mathrm{retr}}(\tau )$, retrieval dynamics inherit the governing law derived in Section 2,

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

(10)

Here $\gamma (\tau )$ is supplied by the observer’s local modular Hamiltonian and inferred operationally through Eq. (2). The three profiles in Figure 1 therefore represent different traversals along the same bounded-access retrieval envelope.

3.4. Inherited Speed-Limit Constraint

The observer-class trajectories in this section do not introduce a new speed-limit principle. They inherit the modular speed-limit constraint derived in Section 2.1. Bounded spectral support restricts admissible retrieval trajectories to the Paley–Wiener strip-holomorphic class, and the hyperbolic-tangent onset is the unique branch that saturates that constraint within the stated assumptions.

Thus the observer profiles above change the rate $\gamma (\tau )$ and the resulting retrieval horizon ${\tau }_{\mathrm{RH}}$, but not the admissible functional form of the retrieval envelope. This is why stationary, freely falling, and accelerating observers produce distinct proper-time signatures while remaining governed by the same bounded-access law.

4. Quantum Information Correlations and Testable Predictions

This section translates the observer-indexed retrieval law into directly measurable quantum-information observables, specifies null tests and estimation protocols, and defines the experimental discriminants that distinguish retrieval dynamics from non-retrieval models.

Observer dependence is defined by physical constraints on access imposed by finite observer bandwidth.

This section serves as a projection layer: it maps retrieval dynamics to a minimal set of experimentally accessible observables without introducing additional model structure.

The retrieval law in Eq. (10) 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, which tracks the purity of the retrievable subsystem, and the second-order correlation function $g^{(2)}$, which probes retrieval-induced interference suppression.

In this mapping, the functional form of the observables is fixed by the retrieval law, while parameters such as ${\tau }_{\mathrm{char}}$, $\gamma (\tau )$, and the modulation depth $\eta$ are inferred from or calibrated against measurement.

Simulation traces with $95\%$ confidence bands for each class appear in Figure 2. Bands are generated from 200 resampled $\gamma (\tau )$ traces per observer class on a fixed proper-time grid with additive bounded spectral noise.

4.1. Rényi Entropy and Second-Order Correlation Functions

Observer-indexed differences in access translate into distinct protocol classes in analog systems, implemented through controlled variations in flow configuration and detection geometry.

For any subsystem A, the Rényi entropy is

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

(11)

with $\alpha >1$. Equation (11) arises by analytically continuing the integer-order moments $Tr({\rho }_A^n)$ (the replica trick); for a field-theoretic derivation see Casini, Huerta, and Myers [15]. Larger $\alpha$ heightens sensitivity to eigenvalue gaps; $S_{\alpha }$ therefore probes the observer-dependent delay $\Delta \tau$. Here $S_{\alpha }$ functions as a theoretical purity diagnostic, while the directly modeled analog observable below is $g^{(2)}$, accessible through density–density correlation measurements in BEC analog systems [14].

As an instantiated consequence of the modular retrieval law derived in Section 2, we construct a second-order correlation function consistent with observer-indexed retrieval dynamics. Let

$$\Delta t=|t_2-t_1|,\overline{t}={\displaystyle \frac{t_1+t_2}{2}}.$$

Times are measured relative to the onset-aligned analysis window, with $t_1,t_2\ge 0$ after baseline alignment.

Define the locally averaged retrieval rate

$$\overline{\gamma }(\overline{t})={\displaystyle \frac{1}{\overline{t}}}{\int }_0^{\overline{t}}\gamma ({\tau }^{\prime })d{\tau }^{\prime },\overline{t}>0.$$

The corresponding effective retrieval correlation time is

$${\tau }_{\mathrm{retrieval}}(\overline{t})={\left[{\tau }_0^{-1}+\overline{\gamma }(\overline{t})\right]}^{-1},$$

where ${\tau }_0$ is the baseline non-retrieval decorrelation time. Under a stationary-phase approximation, the modeled suppressive envelope takes the form

$$g^{(2)}(t_1,t_2)=Aexp\left[-\Delta t/{\tau }_{\mathrm{retrieval}}(\overline{t})\right]\left[1-\eta tanh(t_1/{\tau }_{\mathrm{char}})\right]\left[1-\eta tanh(t_2/{\tau }_{\mathrm{char}})\right],0<\eta \le 1,$$

(12)

where A is a normalization fixed by early-time data and $\eta$ is the retrieval-induced modulation depth. The time dependence of ${\tau }_{\mathrm{retrieval}}$ accumulates the observer-specific retrieval rate, while the tanh factors encode bounded-access suppression under the activation scale ${\tau }_{\mathrm{char}}$. The class-dependent modular Page scale is reported in Table 2.

In a representative baseline waterfall BEC regime, the effective retrieval correlation time is of order $20\mathrm{ms}$, comfortably above the $2\mathrm{ms}$ temporal resolution reported by Steinhauer [14].

The operational non-retrieval null removes the observer-indexed retrieval modulation, leaving a symmetric exponential baseline. Failure of this null to reproduce the observed asymmetry or suppression structure excludes the corresponding non-retrieval model class; it does not by itself exclude every possible non-retrieval correlation model.

Parameters are extracted with nonlinear least squares from ensemble realizations consistent with the admissible observer constraints.

The resulting observables are determined by the retrieval law up to measurement mapping: $g^{(2)}$ captures decay-modulated interference suppression under observer-indexed access, while $S_{\alpha }$ tracks the evolving purity of the retrievable subsystem.

Existing global reconstruction frameworks do not by themselves predict observer-indexed suppression in $g^{(2)}(t_1,t_2)$ under bounded modular spectra; the accelerating signal therefore distinguishes observer-indexed retrieval from global entropy reconstruction.

Table 3. Core observer-class retrieval diagnostics in the v2 verification artifact. The retrieval horizon ${\tau }_{\mathrm{RH}}$ is defined by $S_{\mathrm{retr}}({\tau }_{\mathrm{RH}})\ge 0.9S_{max}$.

Observer	${\mathit{\tau }}_{\mathbf{char}}$	${\mathit{\tau }}_{\mathbf{RH}}$	Final ${\mathit{S}}_{\mathbf{retr}}/{\mathit{S}}_{max}$	Bounded	Monotone	Inverse pass	Median $\mathit{\gamma }$ error
Free falling	2.0	12.9	1.000000	Yes	Yes	Yes	0.003178
Accelerating	3.0	20.5	0.999863	Yes	Yes	Yes	0.000924
Stationary	5.0	30.5	0.981079	Yes	Yes	Yes	0.000364

Table 3. Core observer-class retrieval diagnostics in the v2 verification artifact. The retrieval horizon ${\tau }_{\mathrm{RH}}$ is defined by $S_{\mathrm{retr}}({\tau }_{\mathrm{RH}})\ge 0.9S_{max}$.

Observer	${\mathit{\tau }}_{\mathbf{char}}$	${\mathit{\tau }}_{\mathbf{RH}}$	Final ${\mathit{S}}_{\mathbf{retr}}/{\mathit{S}}_{max}$	Bounded	Monotone	Inverse pass	Median $\mathit{\gamma }$ error
Free falling	2.0	12.9	1.000000	Yes	Yes	Yes	0.003178
Accelerating	3.0	20.5	0.999863	Yes	Yes	Yes	0.000924
Stationary	5.0	30.5	0.981079	Yes	Yes	Yes	0.000364

Because the retrieval law is first order and gap-dependent, measured access trajectories also determine an inverse retrieval-rate estimate. Figure 3 shows the recovered $\gamma (\tau )$ profiles obtained from simulated retrieval traces.

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

The onset scale ${\tau }_{\mathrm{char}}$ is not read off directly from a single point in the trace. It is obtained by fitting the bounded-access transition in measured or simulated correlation structure.

Once either a measured $g^{(2)}(t_1,t_2)$ envelope or a synthetic $S_{\mathrm{retr}}(\tau )$ trace is available, the characteristic timescale ${\tau }_{\mathrm{char}}$ can be obtained by fitting the onset structure after normalization and delay-window reduction. For calibration, we use a baseline-normalized, delay-window-reduced envelope rather than the diagonal $t_1=t_2$ slice of Eq. (12). This reduction isolates the onset modulation along a single effective time coordinate. The one-dimensional suppressive envelope is modeled as

$$g_{\mathrm{env}}^{(2)}(t)=Aexp\left[-t/{\tau }_{\mathrm{retrieval}}(t)\right]\left[1-\eta tanh(t/{\tau }_{\mathrm{char}})\right],0<\eta \le 1,$$

(13)

where $\eta$ is the retrieval-induced suppression depth. If A and $\eta$ are fixed by baseline calibration, ${\tau }_{\mathrm{char}}$ is fit as a one-parameter onset scale. If $\eta$ is not fixed by calibration, it is treated as a nuisance parameter profiled over, or jointly fit with, ${\tau }_{\mathrm{char}}$. Baseline fits are validated against synthetic datasets with known ${\tau }_{\mathrm{char}}$ to verify estimator bias $<2\%$.

A nonlinear least-squares estimator minimizes

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

When $\eta$ is fixed by baseline calibration, this reduces to a one-parameter minimization over ${\tau }_{\mathrm{char}}$.

A non-retrieval null removes the observer-indexed tanh suppression and fits only the symmetric exponential baseline. Failure of this null to reproduce the measured asymmetry or suppression depth provides the operational null test. Uncertainties in $\gamma (\tau )$, $\eta$, and background noise are propagated through the bootstrap ensemble.

Synthetic fits using this suppressive-envelope parameterization return millisecond-scale onset estimates compatible with the bounded spectral constraints of the retrieval law. The purpose of the calibration procedure is not to establish a universal numerical value for ${\tau }_{\mathrm{char}}$, but to demonstrate that the onset scale is operationally recoverable from measured or simulated correlation structure under fixed bandwidth and signal-to-noise conditions.

The procedure assumes (i) $S_{\mathrm{retr}}\in C^1$, (ii) nonnegative and sufficiently smooth $\gamma (\tau )$, (iii) ${\tau }_{\mathrm{char}}>0$, (iv) $0<\eta \le 1$, and (v) $\mathrm{SNR}\ge 4$.

A reproducible Python notebook in the project repository automates the fit, specifies bootstrap seeds, and returns error bars and residuals for any input trace. Varying detector bandwidth provides a direct test of the onset scaling predicted by finite observer resolution.

5. Holographic Connection and MERA-Inspired Finite-Resolution Proxy

This section maps the previously derived observer-indexed retrieval law onto holographic and finite-resolution tensor-network representations, without introducing new dynamics or modifying the underlying retrieval principle. Its purpose is representational: it shows how bounded observer-indexed access can be expressed in the geometric language of wedges, redshifted surfaces, and finite-resolution reconstruction depth.

By embedding observer-indexed modular flow within a redshift-weighted Ryu–Takayanagi framework, the retrieval law acquires a geometric interpretation in terms of observer-dependent access to boosted minimal surfaces under bounded spectra. The corresponding 48-qubit MERA-inspired finite-resolution tensor-network proxy provides a controlled numerical consistency test of the same bounded-access constraint. The proxy uses qubit-count and bond-dimension parameters as finite-resolution controls; it does not claim a microscopic simulation of black-hole geometry or an explicit tensor-contraction realization of holographic reconstruction.

The resulting representation links the retrieval law, its spectral constraints, and its observable signatures within a single geometric interpretive structure.

5.1. Observer-Indexed Mapping to Ryu–Takayanagi Geometry

To represent observer-indexed accessibility in holographic terms, we map the retrieval dynamics onto a redshift-weighted Ryu–Takayanagi (RT) geometry by introducing a modular-frame lapse factor. This construction does not introduce a new entropy law; it embeds the existing retrieval dynamics into a holographic representation.

• ${\Gamma }_A(\Lambda )$: minimal surface in the Lorentz-boosted bulk, not to be confused with the retrieval rate $\gamma (\tau )$;
• $g_{00}(\Lambda )$: lapse tying the surface to the causal wedge reachable along the observer’s world line.

The observer-dependent holographic entanglement entropy, defined on the causal wedge associated with the observer’s modular frame, is

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

(14)

where ${\Gamma }_A(\Lambda )$ is the bulk minimal surface in the Lorentz-boosted geometry, and $g_{00}(\Lambda )$ converts boundary time to the observer’s proper time.3 Choosing $|g_{00}|=1$ and $\Lambda =\mathrm{id}$ recovers the standard RT/HRT normalization. The factor $\sqrt{|g_{00}(\Lambda )|}$ multiplies the area term as an observer-frame weighting; it does not duplicate the Lorentz boost already encoded in ${\Gamma }_A(\Lambda )$.

The redshift factor represents modular-Hamiltonian anchoring (Appendix A) and preserves consistency under local Rindler boosts. It arises from the same bounded modular-flow structure that generates the retrieval law (Section 2), so the holographic and algebraic representations share the same finite-bandwidth spectral cutoff. This construction remains compatible with recent crossed-product and edge-mode algebra treatments [12,19] of gravitational entropy and modular structure. In the weak-field limit the modular flow remains Paley–Wiener admissible within the strip $|\Im \tau |<\pi /(2\Lambda )$, preserving the same bounded-access analytic structure across the boundary–bulk map.

Under refinement of the split-property regulator, the holographic access surface and the modular retrieval horizon are compared through the same observer-indexed cutoff structure. This provides a boundary–bulk representation of retrieval access without promoting either the RT surface or the retrieval horizon to a primary role.

The following mapping translates retrieval-rate classes into experimentally accessible $g^{(2)}$ signatures associated with the same observer-indexed access structure.

These laboratory signatures define the operational boundary of the retrieval–holography correspondence: each retrieval-rate profile maps to a distinct observer patch in the boosted bulk representation. The correlation signatures listed in Table 4 therefore serve as empirical probes of the observer-indexed retrieval classes encoded by the holographic representation, not as direct probes of holographic geometry itself. The bounded modular spectrum $\Lambda (\delta )\sim 1/\delta$ defines the effective resolution of each patch, allowing the modular and holographic descriptions to be compared under a common finite-resolution cutoff.

6. Experimental Signatures and Measurement Conditions

These signatures are the observable consequences of the constrained retrieval dynamics derived above and constitute the measurement-level projection of the underlying law.

These predictions are directly testable using measurement primitives already available in analog black-hole BEC systems, provided the observer-indexed protocol controls specified below are implemented.

The predictions below are expressed at the level of measurable correlation structure in analog black-hole systems. All signatures are defined in terms of the experimentally accessible two-point correlation function $g^{(2)}(t_1,t_2)$, reconstructed from density–density correlations or equivalent detector outputs.

The experimentally relevant control parameters are: flow configuration, including horizon formation profile and gradients; detection configuration, including temporal windowing, sampling cadence, and channel extraction; effective detector bandwidth, defined by temporal resolution or spectral cutoff; and baseline normalization across repeated condensate realizations.

Distinct protocol classes, corresponding to observer-indexed access conditions in the underlying theory, are implemented through controlled variations in these parameters. Stationary, accelerating, and freely falling access should therefore be read as analog protocol classes rather than literal gravitational detector trajectories.

The predictions arise from a single constraint: bounded access under finite spectral support. They therefore form a joint signature. Individual features may be reproduced by simpler models; the combined structure is the discriminant.

6.1. Constrained Saturation and Suppression Envelope

The measured $g^{(2)}(t_1,t_2)$ envelope exhibits a bounded onset and asymptotic suppression profile with a constrained functional form, as required by the bounded-spectrum speed-limit structure derived in Section 2.

Specifically: onset is smooth, monotonic, and curvature-bounded; approach to the asymptotic envelope is single-scale and non-oscillatory; local slope is invertibly related to the remaining retrieval gap.

These properties define a restricted class of admissible envelopes.

• Measurement Condition.

Resolving this structure requires sufficient bandwidth resolution to distinguish onset position and curvature from detector smoothing. If bandwidth is too low: onset collapses into a featureless transition; curvature constraints are not recoverable; and the envelope cannot be distinguished from generic sigmoid behavior.

• Discriminant.

Thermal relaxation and diffusive models may reproduce saturation or suppression separately, but do not satisfy bounded curvature, monotonicity, and invertibility simultaneously.

6.2. Protocol-Dependent Separation

Distinct protocol classes produce systematically different $g^{(2)}$ envelopes, with stable ordering relations across repeated runs.

Operationally: protocol classes are implemented through controlled variations in flow and detection configuration; separation persists under matched initial conditions, including normalized amplitude and fixed bandwidth; differences appear in envelope shape and onset timing, not only amplitude.

• Measurement Condition.

Protocol classes must be implemented under controlled and comparable conditions. This requires: matched condensate states; consistent detection configuration; explicit normalization across runs; and a pre-specified mapping from measured $g^{(2)}$ envelopes to the retrieval proxy $S_{\mathrm{retr}}(\tau )$ or to the inverse-rate estimate $\gamma (\tau )$.

Without this control: differences are confounded with experimental variation; and separation does not stabilize.

• Discriminant.

Protocol variation alone does not generically produce stable ordering of full envelope trajectories under normalization.

6.3. Observer-Resolution Dependence of Onset Time

The characteristic onset time ${\tau }_{\mathrm{char}}$, extracted from the $g^{(2)}$ envelope, scales inversely with effective detector bandwidth.

Operationally: increasing bandwidth by a factor b shifts ${\tau }_{\mathrm{char}}$ by approximately $1/b$; scaling affects onset position, not only smoothing or amplitude; envelope shape is preserved, and protocol-dependent separation remains intact.

• Measurement Condition.

Bandwidth must be varied independently across a sufficient range. Single-resolution measurements cannot test scaling. The experiment must: probe multiple bandwidth regimes; maintain consistent protocol conditions across variations; and vary effective observer resolution while holding condensate preparation and flow configuration fixed, so that changes in ${\tau }_{\mathrm{char}}$ can be attributed to bandwidth rather than to a changed analog background.

• Discriminant.

Detector filtering produces smoothing and amplitude changes, but does not produce consistent inverse scaling of onset position while preserving envelope structure and protocol ordering.

6.4. Interference Suppression Under Protocol Asymmetry

Under controlled protocol asymmetry, meaning controlled mismatch between otherwise normalized detector channels or access windows, the joint correlation structure encoded in $g^{(2)}$ exhibits systematic suppression relative to matched-protocol baselines.

Operationally: matched channel conditions define a baseline overlap structure; introducing asymmetry reduces overlap in a reproducible, structured manner; suppression appears in correlation geometry, not only amplitude.

• Measurement Condition.

The experiment must include: matched baseline measurements; controlled asymmetric configurations; and repeated runs verifying stability of the effect.

Without matched baselines: suppression cannot be attributed to the imposed asymmetry.

• Discriminant.

Decoherence and noise reduce correlations but do not produce suppression that is: selectively tied to controlled protocol asymmetry; while preserving scaling and protocol-dependent envelope structure.

6.5. Joint Signature

The experimental signature is the simultaneous appearance of: constrained saturation or suppression envelope; protocol-dependent separation under normalization; inverse-$\gamma (\tau )$ recovery; finite-resolution robustness; and adversarial-null discrimination.

No single feature is sufficient. The discriminant is the combined structure.

Experimental discriminants. Retrieval dynamics are distinguished from non-retrieval models by: (i) observer-class separation in correlation envelopes, (ii) envelope behavior consistent with the bounded spectral retrieval law, and (iii) invertibility of the measured trajectory under Eq. (2). Models that reproduce isolated correlation features without satisfying all three criteria are excluded as explanations of the full observer-indexed retrieval signature.

The v2 verification artifact evaluates matched saturating-envelope, shared-envelope, observer-label permutation, proper-time jitter, and non-gap dynamics nulls. Individual nulls may pass isolated diagnostics, but no adversarial family reproduces the full joint ODER signature across observer classes: strict retrieval-horizon ordering, class-specific separation, bounded retrieval, and stable inverse-$\gamma (\tau )$ recovery. The null suite therefore tests retrieval structure rather than visual similarity to a single saturating curve.

Table 5. Verification, adversarial, and robustness diagnostics in the v2 verification artifact. The null suite tests whether the full observer-indexed retrieval signature can be reproduced by simpler or structure-breaking alternatives.

Diagnostic	v2 outcome	Interpretation
Core observer ordering	Pass	${\tau }_{\mathrm{RH}}^{\mathrm{fall}}<{\tau }_{\mathrm{RH}}^{\mathrm{acc}}<{\tau }_{\mathrm{RH}}^{\mathrm{stat}}$
Bootstrap confidence bands	200 traces/class	Matches final reproducibility artifact
Finite-resolution proxy	$D=4,8$	Ordering and boundedness survive resolution variation
Matched saturating nulls	Fail joint signature	Generic saturation is insufficient
Shared-envelope null	Fail class separation	One common envelope does not explain observer classes
Proper-time jitter	Ordering survives	Horizon ordering is stable to time perturbation
Non-gap dynamics null	Inverse recovery unstable	Gap-form structure matters
Observer-label permutation	Chance-baseline calibration	Calibrates the expected strict-order chance baseline

Table 5. Verification, adversarial, and robustness diagnostics in the v2 verification artifact. The null suite tests whether the full observer-indexed retrieval signature can be reproduced by simpler or structure-breaking alternatives.

Diagnostic	v2 outcome	Interpretation
Core observer ordering	Pass	${\tau }_{\mathrm{RH}}^{\mathrm{fall}}<{\tau }_{\mathrm{RH}}^{\mathrm{acc}}<{\tau }_{\mathrm{RH}}^{\mathrm{stat}}$
Bootstrap confidence bands	200 traces/class	Matches final reproducibility artifact
Finite-resolution proxy	$D=4,8$	Ordering and boundedness survive resolution variation
Matched saturating nulls	Fail joint signature	Generic saturation is insufficient
Shared-envelope null	Fail class separation	One common envelope does not explain observer classes
Proper-time jitter	Ordering survives	Horizon ordering is stable to time perturbation
Non-gap dynamics null	Inverse recovery unstable	Gap-form structure matters
Observer-label permutation	Chance-baseline calibration	Calibrates the expected strict-order chance baseline

When the experiment can decide.

An experiment tests the framework only if it:

• resolves onset structure at the detector;
• operates above the signal-to-noise threshold required for stable envelope reconstruction;
• implements controlled and normalized protocol classes;
• uses a pre-specified mapping from measured $g^{(2)}$ envelopes to the retrieval proxy $S_{\mathrm{retr}}(\tau )$ or to the inverse-rate estimate $\gamma (\tau )$;
• varies bandwidth and configuration independently; and
• distinguishes the joint signature from simpler models.

If these conditions are satisfied, absence of the joint signature counts against the framework in that tested regime. If they are not satisfied, the measurement does not resolve the predicted structure.

6.6. Numerical Realization and Consistency Check

The constrained retrieval dynamics admit a bounded-spectrum finite-resolution proxy, implemented through the MERA-inspired tensor-network lineage described in Appendix C. This construction is used as a controlled numerical consistency test of the bounded modular-flow constraint, not as a microscopic model of the experimental system and not as a direct simulation of black-hole geometry.

Within this realization: bounded spectral support is represented through finite-resolution access depth; retrieval proceeds within a causal-cone–restricted proxy structure; and observer-indexed protocol variation is implemented through boundary and channel transformations.

The resulting numerical trajectories reproduce the same observable structure described above: a constrained saturation or suppression envelope, observer-class separation, inverse-$\gamma (\tau )$ recovery, finite-resolution robustness, and adversarial-null discrimination.

The role of this realization is strictly consistency: it demonstrates that the joint signature arises under controlled, finite, and computationally tractable finite-resolution proxy conditions.

No claim is made that the experimental system is described by the tensor network. The correspondence is structural: both implement bounded access under finite spectral support.

The same retrieval structure can be projected into a correlation-level observable through the modeled $g^{(2)}(t_1,t_2)$ envelope. Figure 4 illustrates the corresponding bounded-access signature in a controlled numerical setting.

7. Operational Consequences and Falsifiable Predictions

This section synthesizes the operational consequences of observer-indexed retrieval and specifies falsifiable predictions that follow from the preceding sections.

Taken together, the benchmarks in Section 3, Section 4, Section 5 and Section 6 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 governed by a bounded modular spectrum $\Lambda (\delta )$; saturation resembles a Page curve only along trajectories that respect modular access, making the theory falsifiable through analog measurement protocols and numerical proxy tests (Figure 2).

This framework does not provide a complete reconstruction of black-hole information or assert practical recoverability. It specifies only when access occurs under physical constraint.

The contribution is not an alternative entropy model, but a constraint on when information becomes operationally accessible.

7.1. Resolution of the Information Paradox and Empirical Constraints

Here, “resolution” refers to operational access rather than global entropy accounting.

Existing approaches reproduce entropy curves but do not supply a causal retrieval channel that makes information accessible to an observer in proper time.

ODER recasts the operational-access form of the paradox as an observer-indexed retrieval problem defined by proper-time access constraints: for any world line, Eq. (10) 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 required.

Island prescriptions for accelerated detectors [1,8,18] reproduce a Page-like curve globally; the retrieval law produces the same saturation locally and, crucially, supplies a causal proper-time access law. Replica and island frameworks conserve entropy globally but do not by themselves provide a local observer-indexed retrieval law compatible with bounded modular evolution [9].

7.2. Retrieval Horizon ≠ Entanglement Wedge ≠ Event Horizon

Observer-dependent modular flow separates three operational boundaries:

• Retrieval horizon. ${\displaystyle {\tau }_{\mathrm{RH}}=inf\{\tau \mid S_{\mathrm{retr}}(\tau )\ge 0.9S_{max}\}}$.
• Entanglement wedge: the bulk region reconstructable through the boosted RT surface, Eq. (14).
• Event horizon: the classical null surface.

These boundaries should not be identified for finite observers. They may coincide only under idealized limiting assumptions in which observer bandwidth becomes infinite and the finite-access distinction is removed.

In Kerr spacetime, the stationary retrieval channel is defined relative to an admissible local generator ${\chi }_{\Omega }={\partial }_t+\Omega {\partial }_{\varphi }$, restricted to regions where

$$-g_{\mu \nu }{\chi }_{\Omega }^{\mu }{\chi }_{\Omega }^{\nu }>0.$$

Within such timelike wedges, frame dragging and redshift modulate the observer-specific retrieval kernel,

$$\gamma (\tau ,a,\Omega )={\gamma }_0(\tau ){\left[-g_{\mu \nu }{\chi }_{\Omega }^{\mu }{\chi }_{\Omega }^{\nu }\right]}^{-1/2},$$

while the tanh-gap onset remains fixed by bounded split-regularized modular spectra. Where no admissible timelike stationary generator exists, the stationary retrieval channel is undefined rather than falsified; one must pass to a different observer trajectory or a nonstationary modular description.

7.3. Multi-Observer Retrieval Interference as a Falsifier

Observer-dependent retrieval implies that information is not broadcastable across inequivalent modular spectra. For observers i and j with different trajectories or accelerations, the retrieval-overlap tensor $R_{ij}(\tau )$ defined in Appendix C.7 is predicted to obey a nontrivial upper bound set by the mismatch $\Delta \Lambda =|{\Lambda }_i-{\Lambda }_j|$.

If two observers with distinct proper-time evolution recover identical entropy-access curves and identical $g^{(2)}(t_1,t_2)$ envelopes within experimental uncertainty, under conditions that resolve the joint signature defined in Section 6, then observer-indexed modular retrieval is falsified in that tested regime. This test is independent of Page-curve saturation and probes the observer-specific nature of information access directly rather than global entropy accounting.

7.4. ${\Delta }_{\mathrm{fail}}$: Retrieval–Evaporation Boundary

Diagnostic quantity. ${\Delta }_{\mathrm{fail}}$ provides a diagnostic quantity that distinguishes successful retrieval from modular failure. Define

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

with ${\tau }_{\mathrm{evap}}$ the relevant evaporation or termination time of the semiclassical process being modeled. The retrieval horizon ${\tau }_{\mathrm{RH}}$ is defined operationally by the $90\%$ access threshold, not by the inflection point of the entropy-access curve. The inflection point is a separate curvature diagnostic of the retrieval trajectory, as described in Proposition A1. Positive ${\Delta }_{\mathrm{fail}}$ means retrieval completes before evaporation or termination; negative values imply modular retrieval failure.

Table 6. Benchmark ${\Delta }_{\mathrm{fail}}$ comparison using v2 retrieval-horizon values. Retrieval horizons are reported in M-scaled geometric units. For macroscopic semiclassical black holes, ${\tau }_{\mathrm{evap}}$ is parametrically much larger than these retrieval scales, so the table records the diagnostic sign rather than a mass-specific evaporation calculation.

Observer	${\mathit{\tau }}_{\mathbf{RH}}$	${\mathit{\tau }}_{\mathbf{evap}}$	${\mathit{\Delta }}_{\mathbf{fail}}$
Stationary	$30.5$	$\gg {\tau }_{\mathrm{RH}}$	$>0$ (semiclassical stability)
Freely falling	$12.9$	$\gg {\tau }_{\mathrm{RH}}$	$>0$ (semiclassical stability)
Accelerating	$20.5$	$\gg {\tau }_{\mathrm{RH}}$	$>0$ (semiclassical stability)

Table 6. Benchmark ${\Delta }_{\mathrm{fail}}$ comparison using v2 retrieval-horizon values. Retrieval horizons are reported in M-scaled geometric units. For macroscopic semiclassical black holes, ${\tau }_{\mathrm{evap}}$ is parametrically much larger than these retrieval scales, so the table records the diagnostic sign rather than a mass-specific evaporation calculation.

Observer	${\mathit{\tau }}_{\mathbf{RH}}$	${\mathit{\tau }}_{\mathbf{evap}}$	${\mathit{\Delta }}_{\mathbf{fail}}$
Stationary	$30.5$	$\gg {\tau }_{\mathrm{RH}}$	$>0$ (semiclassical stability)
Freely falling	$12.9$	$\gg {\tau }_{\mathrm{RH}}$	$>0$ (semiclassical stability)
Accelerating	$20.5$	$\gg {\tau }_{\mathrm{RH}}$	$>0$ (semiclassical stability)

A negative ${\Delta }_{\mathrm{fail}}$ in a numerical model or analog proxy with a well-defined termination time would falsify the retrieval law in that regime. A nonnegative ${\Delta }_{\mathrm{fail}}$ does not by itself confirm the full framework; it indicates only that this failure condition has not been triggered.

7.5. Experimental Implications and Roadmap

This roadmap consolidates the measurement and falsification conditions already derived, rather than introducing new experimental claims. All following predictions inherit their parameter scaling directly from Eq. (10), ensuring one-to-one traceability between analytic and empirical domains.

Timescale Bridge

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

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

This relation converts geometric proper-time units into physical timescales; analog experiments require a separate calibration between the simulated horizon scale and laboratory time.

Operational Falsifiability

• Absence of the predicted $g^{(2)}$ joint signature under resolved measurement conditions counts against modular access in the tested regime.
• A systematic mismatch in the recovered $\gamma (\tau )$ profile implies that the retrieval law is incomplete or that the measurement does not resolve the proposed access channel.
• Collapse of observer-specific ${\tau }_{\mathrm{char}}$ or ${\tau }_{\mathrm{RH}}$ ordering across controlled protocol classes invalidates observer-specific retrieval under the tested conditions.

Observer-Resolution Scaling Diagnostic

Because the retrieval law is defined on split-regularized subalgebras, ${\tau }_{\mathrm{char}}$ must rescale with detector resolution $\delta$, with $\Lambda (\delta )\sim 1/\delta$ implying

$${\tau }_{\mathrm{char}}(\delta )\propto \Lambda {(\delta )}^{-1}.$$

This scaling test requires bandwidth variation while holding the analog background fixed, so that changes in ${\tau }_{\mathrm{char}}$ can be attributed to observer resolution rather than to a changed condensate or flow regime.

Verification or failure of these signatures determines whether modular flow functions as a physical retrieval mechanism under bounded access in the tested regime.

8. Scope, Boundary Conditions, and Failure Modes

This section delineates the domain of validity of the framework, specifies its operational boundaries, and identifies the failure surfaces under which the retrieval law would not apply.

The framework operates at the level of observer-indexed access under bounded modular flow. It does not extend to global state reconstruction, microscopic decoding mechanisms, or assumptions of practical recoverability. These questions lie outside the present scope.

The contribution is a constraint on when information becomes operationally accessible under physical conditions. It does not modify underlying entropy definitions or introduce an alternative global dynamics.

The following boundary conditions define the regime in which the framework applies.

Fixed-Background Boundary and Perturbative Back-Reaction

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 perturbatively controlled $\alpha \ne 0$, the retrieval horizon shifts only at $\mathcal{O}(\alpha )$, corresponding to first-order perturbative back-reaction.

• Back-reaction bound.

For a Schwarzschild mass M,

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

so that

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

For a fiducial $10M_{\odot }$ black hole one finds

$$G\langle T_{\mu \nu }^{\mathrm{retrieval}}\rangle \approx 4\times {10}^{-7}\sqrt{{\mathcal{K}}_{(10M_{\odot })}},$$

implying $\delta r_{+}/r_{+}<3\times {10}^{-6}$ and a negligible shift in ${\tau }_{\mathrm{RH}}$. This scaling of $\alpha$ provides the perturbative limit recovered in the modular Raychaudhuri coupling (Appendix C.8).

Outlook. A fully coupled model in which

$$T_{\mu \nu }^{\mathrm{retrieval}}\propto ({\partial }_{\tau }{S}_{\mathrm{retr}})u_{\mu }{u}_{\nu }$$

would elevate entropy retrieval to an explicit causal modulator of curvature [21]. This lies outside the present scope and is not required for the validity of the retrieval law.

• Semiclassical Modular-Flow Assumption.

Type ${\mathrm{III}}_1$ algebras are regulated by finite splits [22,23]. Appendix E treats stationary-generator deformation in Kerr within admissible timelike wedges. Full Kerr, de Sitter, multi-horizon, or genuinely nonstationary extensions will require relative-Tomita theory, edge modes, and additional domain-specific analysis [12].

Experimental Boundary: Analog-System Resolution and Protocol Control

Analog black-hole BEC systems provide the relevant measurement primitive: density–density correlation reconstruction of $g^{(2)}$ across horizon-like flow configurations. The present framework adds an observer-indexed protocol layer on top of that primitive. A decisive test requires controlled protocol classes, baseline normalization, bandwidth variation, and a pre-specified mapping from measured $g^{(2)}$ envelopes to $S_{\mathrm{retr}}(\tau )$ or $\gamma (\tau )$.

The numerical examples use millisecond-scale timing as the target resolution regime. Baseline $g^{(2)}$ runs should precede retrieval interpretation, and bandwidth variation must be performed while holding condensate preparation and flow configuration fixed. Otherwise changes in the fitted onset scale may reflect changes in the analog background rather than changes in effective observer resolution.

Structural Boundary: No Global Unitarity or Reconstruction Guarantee

Equation (10) constrains observer-indexed access within a bounded modular domain. It does not by itself guarantee global unitarity, supply a global reconstruction map, or provide a microscopic decoding mechanism. Global conservation may hold while finite-observer retrieval remains delayed, incomplete, or inaccessible.

Modular mismatches between overlapping causal diamonds are therefore expected rather than pathological. They mark differences in observer-indexed access, not violations of global quantum consistency.

Operational Boundary: Retrieval Horizon, Noise, and Full-Recovery Scope

The retrieval horizon ${\tau }_{\mathrm{RH}}$ is an operational threshold defined by

$$S_{\mathrm{retr}}({\tau }_{\mathrm{RH}})\ge 0.9S_{max}.$$

It is not a claim of complete microscopic recovery. The framework predicts bounded approach toward retrieval saturation under the stated assumptions, while full recovery beyond the operational threshold remains outside the present mandate.

The theory defines testable envelopes and inverse-rate diagnostics, but it does not yet provide a complete detector-noise model, ROC sensitivity curve, or experimental error budget for every analog platform.

Explicit Exclusion: Exotic Topologies

Replica wormholes, islands, and topology-changing saddles are excluded by construction. The exclusion is methodological, not incidental: the retrieval law is derived within bounded semiclassical modular access and does not rely on Euclidean saddle selection, global reconstruction, or topology change as the source of observer access.

This restriction preserves falsifiability. If observer-indexed retrieval signatures require exotic topological input to appear, then the bounded-access retrieval law has failed in the tested regime.

Exploratory Diagnostic: Finite-Bandwidth Scaling

Observers possess finite temporal or spatial resolution $\delta$. The corresponding modular spectral cutoff $\Lambda (\delta )\sim 1/\delta$ introduces a controlled, testable resolution dependence in the retrieval law. For the centered retrieval variable

$$Y_{\delta }(\tau )=2F_{\delta }(\tau )-1,$$

the bounded-spectrum onset takes the form

$$Y_{\delta }(\tau )=tanh\left({\displaystyle \frac{\pi \Lambda (\delta )}{2}}(\tau -{\tau }_0)\right),$$

or equivalently,

$$F_{\delta }(\tau )={\displaystyle \frac{1}{2}}\left[1+tanh\left({\displaystyle \frac{\pi \Lambda (\delta )}{2}}(\tau -{\tau }_0)\right)\right].$$

Thus the transition slope scales linearly with $\Lambda (\delta )$, while the characteristic onset time satisfies

$${\tau }_{\mathrm{char}}(\delta )\propto \Lambda {(\delta )}^{-1}\propto \delta .$$

Repeating measurements at multiple resolution scales should therefore rescale the transition width of the $g^{(2)}$ envelope without changing the observer-class hierarchy, provided the analog background and protocol class are held fixed.

The following scaling analysis is exploratory and does not modify any results derived above. Define

$${\beta }_{\Lambda }(\delta )={\displaystyle \frac{dln\Lambda (\delta )}{dln\delta }}.$$

For the canonical finite-bandwidth scaling $\Lambda (\delta )\sim 1/\delta$, one obtains

$${\beta }_{\Lambda }(\delta )\approx -1.$$

It is therefore useful to define a deviation-from-canonical-scaling diagnostic,

$${\beta }_{\mathrm{retr}}(\delta )={\beta }_{\Lambda }(\delta )+1={\displaystyle \frac{dln[\delta \Lambda (\delta )]}{dln\delta }}.$$

The condition ${\beta }_{\mathrm{retr}}\to 0$ indicates stable inverse-bandwidth scaling across the tested resolution range. This is an empirical scaling diagnostic, not an independent renormalization-group derivation of the retrieval law.

The formal Type ${\mathrm{III}}_1$ limit corresponds to the continuum boundary $\delta \to 0$, $\Lambda (\delta )\to \infty$, where the finite-observer cutoff is removed. Real detectors operate at finite $\delta$; the continuum limit functions as the algebraic boundary of the theory rather than as an operationally accessible measurement regime.

Empirical validation requires resolving changes in ${\tau }_{\mathrm{char}}$ across controlled variations of $\delta$, while holding condensate preparation, flow configuration, and protocol class fixed. Under the predicted scaling, fitted onset curves should collapse when plotted against the rescaled variable $\Lambda (\delta )(\tau -{\tau }_0)$, while preserving the ordering of observer-class retrieval horizons within detector precision.

Exploratory Direction: Superposed Geometries

Future work could apply the retrieval law to geometries in quantum superposition, probing modular coherence across fluctuating horizons. This extension would require replacing the fixed-background modular generator with a state-dependent or relational access map, and is therefore outside the present fixed-background framework.

9. Conclusion and Next Steps

This conclusion closes the retrieval argument by restating the governing constraint, summarizing its empirical status, and delineating what remains outside the present scope.

Taken together, this work consolidates the theoretical, computational, and empirical threads of ODER. We presented a relativistic, observer-dependent framework for black-hole entropy retrieval that provides a causal description of observer-indexed entropy retrieval within semiclassical gravity. By anchoring information flow to proper time and causal access, ODER recasts Page-curve behavior as a continuous, observer-indexed description of entropy access rather than as a global reconstruction curve alone. All derivations and verification protocols are supplied for stand-alone reproducibility.

The retrieval law is not heuristic; it follows from split-regularized Tomita–Takesaki modular spectra (Appendix A, Eq. (10)). Bounded modular flow links spectral smoothing, redshift factors, and observer-specific algebras, making retrieval a physical access process rather than an epistemic relabel.

Concrete predictions follow. Stationary, freely falling, and uniformly accelerated observer classes exhibit distinct retrieval rates and $g^{(2)}$ envelopes. These signatures are testable using analog-gravity measurement primitives when the protocol controls specified in Section 6 are satisfied. Failure to observe the joint signature under resolved measurement conditions would count against observer-indexed modular accessibility in the tested regime, while leaving the general mathematical existence of modular flow itself intact.

The v2 verification artifact further tests the retrieval structure against finite-resolution variation and adversarial alternatives. The same joint signature that follows analytically—bounded retrieval, observer-class ordering, inverse-$\gamma (\tau )$ recovery, finite-resolution robustness, and null-family discrimination—is not reproduced by generic saturating, shared-envelope, label-permuted, time-jittered, or non-gap alternatives. The resulting evidence is not a claim of experimental confirmation; it is a reproducible demonstration that the proposed access law has discriminable structure beyond visual similarity to a saturating curve.

Roadmap: Theory, Simulation, Experiment

The following roadmap is illustrative rather than prescriptive and does not condition the validity of the present results.

• Theory (extensions)
  • Semiclassical back-reaction: couple entropy flow to a self-consistent metric response, extending Eq. (10) into a dynamical observer–spacetime equation.
  • Intersecting horizons: analyze overlapping causal diamonds to refine the retrieval-horizon concept and multi-observer access mismatch.
  • Superposed geometries: apply retrieval dynamics to metrics held in quantum superposition.
• Simulation (validation scaling)
  • Higher-resolution proxy scaling: extend the v2 verification suite beyond the current $D=4,8$ proxy checks to test finite-resolution convergence, finite-entanglement effects, and the stability of inverse-$\gamma (\tau )$ recovery under increasing resolution.
  • Error budgets: propagate detector-noise kernels to produce ROC-style sensitivity curves for the joint retrieval signature.
  • Protocol stress tests: extend the existing adversarial null suite to additional access schedules, bandwidth regimes, observer-class perturbations, and noise models.
• Experiment (near-term tests)
  • Protocol-differentiated analog probes: implement stationary, accelerating, and freely falling access classes as controlled BEC protocol classes rather than literal gravitational detector trajectories. The v2 verification artifact already specifies the target joint signature: observer-class separation, bounded envelope behavior, retrieval-horizon ordering, inverse-$\gamma (\tau )$ recoverability, finite-resolution robustness, and adversarial-null discrimination. The experimental task is to determine whether this pre-specified signature appears in measured $g^{(2)}$ envelopes under normalized protocol control, targeting the $10\mathrm{ms}$–$100\mathrm{ms}$ retrieval window with millisecond-scale timing.
  • Cross-platform checks: replicate $g^{(2)}$ envelope structure in photonic-crystal and superconducting-circuit analogs where comparable correlation measurements and protocol controls are available.
  • Calibration: detector-noise calibration, baseline $g^{(2)}$ runs, bandwidth variation, and signal-to-noise validation should precede retrieval-fit attempts across all platforms.

Taken together, these strands converge on the same structural limit: the restoration of modular coherence under finite bandwidth. These coordinated steps extend the present analytic and verification framework toward direct empirical tests.

Final Remark. The result is a constrained physical statement: accessible structure evolves under a bounded law, its trajectory is fixed by spectral constraints, and its presence or absence can be decided at the detector when the relevant protocol conditions are met.

The split-property regularization situates ODER within a finite-observer route toward the Type ${\mathrm{III}}_1$ continuum boundary. Within this operational framework, Type ${\mathrm{III}}_1$ is not treated as a directly measurable regime, but as the algebraic limit recovered when finite observer bandwidth is removed. What is physically testable is the finite-bandwidth approach to that boundary: the scaling of ${\tau }_{\mathrm{char}}$, the preservation or collapse of observer-class ordering, and the recoverability of $\gamma (\tau )$ from measured access traces.

Whether confirmed or falsified, the coming analog experiments will determine whether entropy retrieval is a physical, observer-local access process or merely a post-hoc bookkeeping device, providing an empirical test of the operational-access resolution proposed here: observer-indexed causal retrieval rather than global entropy accounting alone.

Information is not operationally present when it exists, but when it becomes accessible under constraint.

Appendix A.1. Motivation: Bounded Algebras and Observer-Dependent Entropy

Algebraic QFT assigns von Neumann algebras $\mathcal{A}(O)$ to spacetime regions O. A global state $\omega$ on $\mathcal{A}[D(\gamma ,\infty )]$ encodes all degrees of freedom inside the observer’s domain of dependence. At proper time $\tau$ the observer accesses only $\mathcal{A}[D(\gamma ,\tau )]$; the entropy gap is the retrievable deficit.

• Finite-split regularization.

Because $\mathcal{A}(D)$ is Type ${\mathrm{III}}_1$, its modular Hamiltonian is unbounded. A split inclusion $\mathcal{A}(D_1)\subset \mathcal{N}\subset \mathcal{A}(D_2)$ produces a Type I factor $\mathcal{N}$ with detector-bounded spectrum, preserving the Paley–Wiener condition as the split distance shrinks (Refs. [22,23]). This procedure defines the finite-bandwidth subalgebras ${\mathcal{N}}_{\delta }$ on which the bounded modular spectra $\sigma (K_{\delta })\subset [-\Lambda (\delta ),\Lambda (\delta )]$ are realized.

Appendix A.2. Spectral Convergence and Extremal Rigidity of the Retrieval Sigmoid

This appendix establishes the speed-limit and extremal-rigidity structure of the admissible retrieval profile under the stated analytic, spectral, and boundary assumptions, at the level required to support the physical derivations and falsifiability criteria used in the main text.

Lemma A.1 (Paley–Wiener Band-Limit). Let the split-regularized modular Hamiltonian K have bounded spectrum $\sigma (K)\subset [-\Lambda ,\Lambda ]$. Then any observable expectation value $f(\tau )=\langle \psi |e^{iK\tau }|\psi \rangle$, and all quantities derived from it, including the normalized entropy evolution $F(\tau )=S_{\mathrm{retr}}(\tau )/S_{max}$, extend holomorphically to the horizontal strip

$${\mathcal{S}}_{\Lambda }=\{\tau \in \mathbb{C}:|\Im \tau |<\pi /(2\Lambda )\},$$

with growth $|F(\tau )|=\mathcal{O}(e^{\Lambda |\Im \tau |})$. That is, F is of exponential type $\le \Lambda$ in ${\mathcal{S}}_{\Lambda }$. (Paley–Wiener Theorem 19.3 in Rudin, Real and Complex Analysis). □

Lemma A.2 (Phragmen–Lindelof Growth Bound). If $F(\tau )$ is holomorphic in ${\mathcal{S}}_{\Lambda }$, bounded and strictly monotonic on $\mathbb{R}$, and of exponential type $\le \Lambda$, then $|F(\tau )|\le 1$ throughout ${\mathcal{S}}_{\Lambda }$. Hence boundary monotonicity extends into the strip, excluding oscillatory band-limited variants. □

Theorem A.2 (Extremal Rigidity of the Retrieval Sigmoid under Modular Band-Limit). Let $F:\mathbb{R}\to (0,1)$ be strictly increasing with finite limits $F(-\infty )=0$, $F(+\infty )=1$. Assume F is holomorphic in ${\mathcal{S}}_{\Lambda }$, of exponential type $\le \Lambda$ in that strip, bounded on $\mathbb{R}$, and obeys the modular spectrum bound $\sigma (K)\subset [-\Lambda ,\Lambda ]$. Then admissible retrieval profiles satisfy the strip-derived modular speed-limit bound. If, in addition, the profile saturates this bound on the physical retrieval branch, then, up to an affine reparametrization of $\tau$,

$$\begin{array}{|c|}\hline Y(\tau ):=2F(\tau )-1=tanh\left({\displaystyle \frac{\pi \Lambda }{2}}(\tau -{\tau }_0)\right)\\ \hline\end{array}.$$

Equivalently,

$$F(\tau )={\displaystyle \frac{1+tanh\left(\frac{\pi \Lambda }{2}(\tau -{\tau }_0)\right)}{2}}.$$

Thus the centered tanh profile is not a classification of all admissible non-saturating trajectories. It is the unique extremal saturator of the bounded-spectrum speed limit used by the retrieval law.

Sketch of Proof.Strip to disk map. Map ${\mathcal{S}}_{\Lambda }$ to $\mathbb{D}$ by $z=exp({\displaystyle \frac{\pi }{\Lambda }}\tau )$. Define the centered retrieval variable $Y(\tau )=2F(\tau )-1$. Then $F(\tau )=(1+Y(\tau ))/2$, and $Y:{\mathcal{S}}_{\Lambda }\to \mathbb{D}$.

Speed-limit bound. The strip-to-disk conformal transfer puts the admissible profile in the Schwarz–Pick setting. The Schwarz–Pick contraction principle gives a pointwise derivative bound on the real trace of the profile, with the constant fixed by the strip width.

Extremal rigidity. If the physical retrieval branch saturates this bound, equality holds in the Schwarz–Pick inequality for the transferred disk map. The equality case forces the disk map to be an automorphism. Real-axis monotonicity restricts the automorphism to the real diameter, yielding

$$Y(\tau )=tanh\left({\displaystyle \frac{\pi \Lambda }{2}}(\tau -{\tau }_0)\right).$$

Recover F. Undoing the centering transform yields

$$F(\tau )={\displaystyle \frac{1+Y(\tau )}{2}},$$

with normalization $F(-\infty )=0$, $F(+\infty )=1$. □

Corollary A.2.1 (Spectral Constant Fixation). The constant $\pi \Lambda /2$ is fixed by the strip width $|\Im \tau |<\pi /(2\Lambda )$. No other profile in the stated admissible class saturates the strip-derived speed-limit bound, up to affine reparametrization. 4

• Excluded Counterexamples.

The following functions are excluded within the stated analyticity and spectral constraints.

Candidate $F(\tau )$	Reason for exclusion
$(2/\pi )arctan(\alpha \tau )$	Poles at $\pm i/\alpha$, not holomorphic in ${\mathcal{S}}_{\Lambda }$.
$1-e^{-\alpha \tau }$	Violates strip boundedness, unbounded along $\Im \tau$.
Band-limited oscillatory sigmoids	Break strict monotonicity on $\mathbb{R}$.
Logistic $1/(1+e^{-z})$	Meromorphic, with poles at $z=(2k+1)\pi i$; admissibility depends on strip width and the profile is not the unique bounded-spectrum extremal form.

These exclusions guarantee that all admissible retrieval profiles share the same analytic growth bound, ensuring that the centered tanh profile is the unique extremal saturator of the admissible speed-limit bound.

• Physical Interpretation.

Early times $\tau \ll {\tau }_{\mathrm{char}}$ reflect incomplete activation of modular modes. Late times $\tau \gg {\tau }_{\mathrm{char}}$ approach saturation as the bounded spectrum fully enters the algebra. The tanh profile is the unique speed-limit-saturating analytic interpolation compatible with the Paley–Wiener bound and causal analyticity.

• Conceptual comparison of excluded profiles.

Candidate monotone curves satisfying $F(-\infty )=0$ and $F(+\infty )=1$ illustrate the role of the admissibility conditions. The tanh profile is allowed as the speed-limit-saturating branch. An arctan profile is rejected because its poles obstruct holomorphy in the admissible strip. A simple exponential profile is rejected because it violates strip boundedness. Only the tanh profile realizes the extremal speed-limit branch within the analytic growth limits of ${\mathcal{S}}_{\Lambda }$.

• Remark A.2.2 (Edge Atoms and Observer Generality).

Atomic spectral weight at $\pm \Lambda$ induces boundary oscillations incompatible with strict monotonicity on $\mathbb{R}$ and is therefore excluded by assumption. In curved or rotating backgrounds, including Kerr, the modular spectrum remains bounded after split-inclusion regularization, so the tanh onset persists as the speed-limit-saturating branch. This bounded-spectrum extremal result generalizes to all observer classes in Section 3. The parallel with finite-bandwidth scaling noted in Section 8 is conceptual and does not enter the present derivation.

Appendix A.3. Role of γ(τ): Modular Spectrum and Redshift (Parameter Interpretation)

• Spectrum gradient: if $\rho (\lambda )\sim {\lambda }^{-\beta }$, then $\gamma (\tau )\propto {\tau }^{\beta -1}$.
• Geometric redshift: stationary observers yield ${\gamma }_{\mathrm{stat}}\propto 1/r$.
• Unruh boost: uniform acceleration gives ${\gamma }_{\mathrm{acc}}\propto a^2$.

Table A1. Retrieval parameters used in the v2 verification artifact for Figure 1 and Figure 2 (geometric units $G=c=1$).

Observer	Rate profile	${\mathit{\tau }}_{\mathbf{char}}/\mathit{M}$	${\mathit{\tau }}_{\mathbf{RH}}/\mathit{M}$
Stationary ($r=10M$)	slowly varying exterior access	5	30.5
Freely falling	rapid post-crossing growth	2	12.9
Accelerating ($a=0.2$)	acceleration-enhanced access	3	20.5

Table A1. Retrieval parameters used in the v2 verification artifact for Figure 1 and Figure 2 (geometric units $G=c=1$).

Observer	Rate profile	${\mathit{\tau }}_{\mathbf{char}}/\mathit{M}$	${\mathit{\tau }}_{\mathbf{RH}}/\mathit{M}$
Stationary ($r=10M$)	slowly varying exterior access	5	30.5
Freely falling	rapid post-crossing growth	2	12.9
Accelerating ($a=0.2$)	acceleration-enhanced access	3	20.5

These values correspond to the v2 verification artifact and are recorded in validation_manifest.json, establishing traceability between the algebraic retrieval law, the generated figures, and the diagnostic tables.

Appendix A.4. Retrieval Threshold and Saturation Boundary

Dependency note. This proposition characterizes operational boundaries implied by Theorem A.1 and introduces no additional assumptions or dynamics.

Proposition A1

(Retrieval threshold and saturation transition). Let $S_{\mathrm{retr}}(\tau )$ be the entropy-access curve generated by the retrieval law in Theorem A.1. The operational retrieval horizon is defined by the threshold condition

$${\tau }_{\mathrm{RH}}:=inf\{\tau \mid S_{\mathrm{retr}}(\tau )\ge 0.9S_{max}\}.$$

For the single-scale monotone tanh-gap trajectories considered here, there may also be a unique saturation-transition time ${\tau }_{\inf }$ satisfying

$${\left.{\displaystyle \frac{d^2{S}_{\mathrm{retr}}}{d{\tau }^2}}\right|}_{\tau ={\tau }_{\inf }}=0,{\left.{\displaystyle \frac{d^3{S}_{\mathrm{retr}}}{d{\tau }^3}}\right|}_{\tau ={\tau }_{\inf }}<0.$$

The inflection time ${\tau }_{\inf }$ marks the transition from accelerating retrieval to saturation-dominated retrieval. It is a curvature diagnostic of the access curve and need not coincide exactly with the operational $90\%$ retrieval horizon ${\tau }_{\mathrm{RH}}$. Define

$${\Delta }_{\mathrm{fail}}\equiv {\tau }_{\mathrm{evap}}-{\tau }_{\mathrm{RH}}.$$

This quantity uses the operational retrieval horizon and serves as a diagnostic boundary rather than an independent evolution law.

Appendix A.5. Observer-Bounded Automorphisms and the Origin of the tanh Factor (Interpretive)

Interpretive note. This subsection interprets the origin of the tanh factor already proven in Appendix A.2 and does not supply an independent derivation.

The consolidation result in Appendix A.10 shows that global modular flow restricts to the observer algebra and yields the unique speed-limit-saturating tanh onset that appears in Eq. (10). This reflects the fact that the retrieval law saturates the Paley–Wiener bound and therefore represents the maximal causal convergence permitted by bounded modular spectra. See also Lemma A.2 for the underlying analytic constraint.

Appendix A.6. Related Work

Scope note. The works listed below address bounded algebras and entropy growth but do not derive a closed, observer-indexed retrieval law of the form used in the main text.

See Refs. [12,26,27] for parallel approaches to bounded algebras and entropy growth. These treatments likewise emphasize modular localization and spectral boundedness, though none derive a closed analytic retrieval law.

Appendix A.7. Philosophical Implications (Interpretive)

Interpretive scope. This subsection offers an interpretive reading of the retrieval law and does not introduce additional physical claims.

The law supports relational entropy: observer disagreements signal frame misalignment rather than information loss. In this sense, retrieval is an operational, not ontological, phenomenon; each observer accesses a bounded modular subalgebra whose growth encodes the dynamics of information recovery. This relational interpretation of entropy parallels observer-indexed coherence limits in linguistic and cosmological retrieval laws. These parallels are conceptual and are not invoked in the present derivation.

Appendix A.8. Deriving τ Page from Spectral Gaps

Dependency note. This relation fixes the scaling of ${\tau }_{\mathrm{Page}}$ rather than providing an independent definition.

With smallest modular gap ${\lambda }_{min}$, ${\tau }_{\mathrm{Page}}\sim {\lambda }_{min}^{-1}$. For a Schwarzschild black hole of mass M, ${\tau }_{\mathrm{Page}}\sim M^3$, reproducing the expected asymptotic semiclassical scaling of entropy-recovery timescales.

Appendix A.9. Asymptotic Boundary Clause

Boundary status. This clause defines the operational boundary of applicability of the retrieval law rather than a dynamical evolution rule.

As $\tau \to {\tau }_{\mathrm{evap}}\gtrsim {\tau }_{\mathrm{RH}}$, one of the following must occur: (1) $\gamma (\tau )\to 0$; (2) $S_{max}(\tau )\to 0$; or (3) $\langle T_{\mu \nu }^{\mathrm{retrieval}}\rangle$ becomes dynamically significant, breaking fixed-background validity. This defines the operational boundary of the ODER framework.

Remark A1

(Retrieval–Geometry Decoupling). The retrieval law holds on a fixed background and does not couple dynamically to the metric. Any extension that includes back-reaction must solve

$$G_{\mu \nu }=8\pi G\left(T_{\mu \nu }^{\mathrm{Hawking}}+T_{\mu \nu }^{\mathrm{retrieval}}\right)$$

self-consistently, which is beyond the present scope.

Empirically, regime (3) is the boundary case in which the fixed-background retrieval envelope would be expected to deviate from the predicted tanh-gap form. Such deviations would manifest as excess curvature in measured $g^{(2)}$ traces, marking entry into the back-reaction-dominated regime.

Appendix A.10. Spectral Convergence and Extremal Rigidity (Consolidation)

Consolidation note. This result restates the speed-limit and extremal-rigidity result of Appendix A.2 in its spectral-limit form for completeness.

Theorem A.3 (Spectral-Convergence Constraint; Consolidation).

Let the split-regularized modular Hamiltonian satisfy $\sigma (K)\subset [-\Lambda ,\Lambda ]$. Let $F(\tau )=S_{\mathrm{retr}}(\tau )/S_{max}$ be $C^1$, strictly increasing, holomorphic in ${\mathcal{S}}_{\Lambda }$, and of bounded exponential type $\le \Lambda$ in that strip. Then admissible retrieval profiles obey the strip-derived modular speed-limit bound. If the physical retrieval branch saturates that bound, then, up to an affine reparameterization,

$$Y(\tau ):=2F(\tau )-1=tanh\left(\pi \Lambda \tau /2\right),$$

or equivalently

$$F(\tau )={\displaystyle \frac{1+tanh\left(\pi \Lambda \tau /2\right)}{2}}.$$

Thus Eq. (10) uses the unique spectrum-compatible extremal onset within the stated Paley–Wiener admissible class. □

This theorem consolidates the speed-limit and extremal-rigidity result of Appendix A.2 as its spectral-limit case, keeping the analytic and modular derivations aligned. The centered tanh profile is not a classification of all admissible non-saturating trajectories; it is the unique extremal saturator selected when the physical retrieval branch saturates the bounded-spectrum speed limit. Possible generalizations would require relaxing Paley–Wiener admissibility, abandoning speed-limit saturation, or changing the physical admissibility class, and lie beyond the present framework.

Appendix A.11. Constrained Variational Consistency Form and Modular Speed Limit

Logical status. The variational formulation is logically downstream of the spectral bound and does not depend on retrieval-RG considerations. It records the same bounded-access constraint in extremal form; it is not an independent derivation of the retrieval law.

The modular-speed-limit analysis presented here extends naturally to the finite-bandwidth formulation of Section 8, where $\Lambda (\delta )\sim 1/\delta$ defines the effective observer cutoff.

• Setting and regularization.

Let $\mathcal{A}\left(D(\gamma ,\tau )\right)$ be the von Neumann algebra associated with the observer’s causal diamond $D(\gamma ,\tau )$ along world line $\gamma$. Under the split property, the modular Hamiltonian $K(\tau )$ for $\mathcal{A}\left(D(\gamma ,\tau )\right)$ admits a split-regularized spectrum $\sigma (K)\subset [-\Lambda ,\Lambda ]$ (Refs. [22,23]). Define the normalized retrieval profile $F(\tau )=S_{\mathrm{retr}}(\tau )/S_{max}$. The Paley–Wiener theorem implies that F is holomorphic in the strip ${\mathcal{S}}_{\Lambda }$ and of bounded exponential type $\le \Lambda$ in that strip.

• Derivative bound (Paley–Wiener/Bernstein).

For functions in the stated Paley–Wiener admissible class, Bernstein and strip-majorant arguments imply a pointwise slope constraint compatible with bounded spectral support. Since $F(\tau )\in [0,1]$ is monotone and $\dot{F}(\tau )\to 0$ as $\tau \to \infty$, the admissible pointwise majorant is

$$\dot{F}(\tau )\le \Lambda [1-F(\tau )]tanh\left({\displaystyle \frac{\pi \Lambda \tau }{2}}\right)=:\mathcal{B}(\Lambda ,\tau ),$$

(A1)

which defines the modular speed limit used in the main text.

• Saturator and uniqueness (sketch).

Let

$$Y(\tau )=tanh\left({\displaystyle \frac{\pi \Lambda }{2}}\tau \right)$$

be the unique centered analytic saturator established in Appendix A.2. If any admissible $F(\tau )$ were to exceed the bound (A1) on a set of non-zero measure, standard Phragmen–Lindelof and Bernstein majorant arguments for band-limited monotone maps would force violation of either the exponential-type constraint or the boundary conditions. Thus the centered tanh profile is the unique saturator of the modular speed limit.

• Constrained variational representation.

The speed-limit bound admits a constrained variational representation. This representation is downstream of the spectral admissibility result: it does not independently derive the first-order retrieval law, but records the same bounded-access constraint in action form. Introduce the entropy-action functional

$$\mathcal{I}[F]=\int d\tau \left[\frac{1}{2}{\left(\dot{F}(\tau )\right)}^2-U\left(F;\Lambda ,{\tau }_{\mathrm{char}}\right)\right],$$

(A2)

with potential

$$U\left(F;\Lambda ,{\tau }_{\mathrm{char}}\right)=\frac{1}{2}{\Lambda }^2{[1-F(\tau )]}^2{tanh}^2\left({\displaystyle \frac{\tau }{{\tau }_{\mathrm{char}}}}\right).$$

(A3)

The unconstrained Euler–Lagrange equation associated with $\mathcal{I}[F]$ defines the corresponding second-order extremal problem. The first-order retrieval law is recovered only after restricting to the admissible branch that saturates the modular speed limit:

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

(A4)

Under this constrained reading, the variational form is a consistency representation of the retrieval law, not a separate dynamical postulate. The observer-dependent rate $\gamma (\tau )$ remains the local observer- and state-dependent modular retrieval rate governing traversal along the admissible tanh-gap envelope.

• Onset scale ${\tau }_{\mathrm{char}}$.

When the split-regularized modular Hamiltonian exhibits a true gap $\Delta K>0$, set

$${\tau }_{\mathrm{char}}={\displaystyle \frac{2\pi }{\Delta K}}.$$

(A5)

In the generic gapless Type ${\mathrm{III}}_1$ case, $\Delta K$ is understood as an effective spectral scale induced by the split inclusion rather than a fundamental constant.

• Falsifiability.

If any measured retrieval trace violates the modular-speed-limit inequality (A1) within experimental confidence intervals, then Eq. (A4) cannot be retained as a universal observer-dependent retrieval law.

This completes the formal derivation supporting the retrieval law used throughout the main text.

Appendix B.1. Observer-Indexed Mapping to Minimal Surfaces

Definition B.1 (Observer-Indexed RT Mapping).

For a boundary subregion A and a frame transformation $\mathcal{L}$,

$$S_{\mathrm{obs}}^{\mathrm{holo}}(A;\mathcal{L})={\displaystyle \frac{Area\left[{\Gamma }_A(\mathcal{L})\right]}{4G_N}}\sqrt{|g_{00}(\mathcal{L})|},$$

(A6)

where ${\Gamma }_A(\mathcal{L})$ is the codimension-2 minimal surface in the Lorentz-transformed bulk frame and $\sqrt{|g_{00}(\mathcal{L})|}$ supplies the observer-frame lapse weighting between boundary time and the observer’s proper time. Choosing $|g_{00}(\mathcal{L})|=1$ and $\mathcal{L}=\mathbb{I}$ recovers the standard RT/HRT normalization.

This definition encodes observer-dependent accessibility and does not replace the standard RT/HRT prescription.

Here $g_{00}(\mathcal{L})$ is evaluated on the transformed boundary metric and is restricted to admissible regions where the proper-time lapse remains real and nondegenerate. The redshift factor is operational within the bounded-observer framework considered here: it weights the surface contribution by the proper-time access available to the observer. This aligns the holographic representation with the bounded modular spectra of Appendix A, so that each observer-frame surface is associated with a finite-bandwidth retrieval channel.

Appendix B.2. Modular-Wedge Alignment and Retrieval-Access Regions

Dependency note. This construction provides a geometric representation of the retrieval-access boundary defined in the main text and Appendix A. It does not introduce an independent boundary or replace the operational definition of ${\tau }_{\mathrm{RH}}$.

Let $\mathcal{W}(\mathcal{L})$ be the entanglement wedge reconstructed from boundary data in frame $\mathcal{L}$. Define the retrieval-access region

$$\begin{array}{cc}\hfill \mathcal{R}(\mathcal{L})=\{p\in {\mathcal{M}}_{\mathrm{bulk}}|& p\in \mathcal{W}(\mathcal{L}),\exists t\le {\tau }_{\mathrm{Page}}(\mathcal{L}):\hfill \\ \hfill & p\in {\sigma }_t^{{\omega }_{\mathcal{L}}}\left[\mathcal{A}(A)\right]\},\hfill \end{array}$$

(A7)

where ${\sigma }_t^{{\omega }_{\mathcal{L}}}$ is modular flow of the transformed state. The modular flow ${\sigma }_t^{{\omega }_{\mathcal{L}}}$ corresponds to the Tomita–Takesaki flow used in Theorem A.1, restricted to the holographic wedge. Retrieval stabilizes when $\mathcal{R}(\mathcal{L})$ stops growing under the observer’s proper-time access. Its boundary ${\tilde{\Gamma }}_A(\mathcal{L})\subseteq {\Gamma }_A(\mathcal{L})$ marks the accessible limit of the observer-indexed wedge.

• Wedge disagreement.

If frame transformations ${\mathcal{L}}_1$ and ${\mathcal{L}}_2$ differ,

$${\Gamma }_A({\mathcal{L}}_1)\ne {\Gamma }_A({\mathcal{L}}_2)⟹S_{\mathrm{obs}}^{\mathrm{holo}}(A;{\mathcal{L}}_1)\ne S_{\mathrm{obs}}^{\mathrm{holo}}(A;{\mathcal{L}}_2),$$

so the two observers assign different accessible holographic entropies to the same boundary region, consistent with the observer-indexed trichotomy introduced in Section 7.2. The divergence between wedges provides the geometric counterpart of retrieval interference discussed in Appendix C and corresponds to the observer-class interference tensor $R_{ij}$ introduced in Section 7.3.

Appendix B.3. Connection to HRT and Quantum Error-Correcting Codes

When the frame transformation $\mathcal{L}$ matches the boundary slicing, Eq. (A6) reduces to the Hubeny–Rangamani–Takayanagi prescription. In HaPPY or random-tensor MERA codes [20], changes in boundary slicing or access region change which logical degrees of freedom are reconstructable from the boundary. In the finite-resolution tensor-network proxy, the observer-frame transformation is represented as a change in effective access depth and wedge alignment, consistent with the redshift-weighted mapping in Eq. (A6).

In this interpretation, the tensor-network proxy supplies a controlled finite-resolution representation of boundary–bulk access. It does not claim an explicit tensor-contraction realization of black-hole geometry or a microscopic simulation of holographic reconstruction. The effective redshift factor is represented qualitatively as layer-depth weighting: different observer frames correspond to different accessible finite-resolution layers.

Appendix B.4. Contrast with Replica Wormholes and Island Formulae

Structural contrast. The following contrast is structural rather than evaluative.

Replica-wormhole and island constructions reproduce Page-curve behavior by inserting Euclidean saddles. The present construction, by contrast, represents late-time retrieval saturation through bounded modular access; no topology change or ensemble averaging is required as the source of the access law. Equation (A6) is therefore a holographic representation of observer-indexed Lorentzian retrieval, fully consistent with the algebraic modular framework of Appendix A.

Appendix B.5. Outlook

• Cosmological horizons: extend Eq. (A6) to de Sitter and FRW spacetimes, where competing frame transformations may generate multiple retrieval horizons and entanglement wedges.
• Back-reaction coupling: allow ${\Gamma }_A(\mathcal{L})$ to evolve under semiclassical Einstein dynamics and study retrieval–curvature feedback linking modular flow and bulk geometry.
• Higher-resolution proxy scaling: test observer-dependent access in refined finite-resolution proxy networks to quantify how access depth, resolution, and observer-frame weighting affect retrieval latency.

In synthesis, Appendix A, Appendix B and Appendix C establish the analytic, numerical, and measurement-level pillars of observer-dependent retrieval, while this appendix provides a geometric representation of the same framework. Together these extensions delineate the holographic mapping regime of ODER, in which observer-indexed modular flow and minimal-surface reconstruction language are compared under a common bounded-access structure without introducing independent dynamics. All results in this appendix are subordinate to the retrieval law derived in the main text and Appendix A.

Appendix C.1. Simulation Setup

The tensor-network lineage used in this work employs a 48-qubit MERA-inspired finite-resolution proxy, motivated by the causal-cone structure of HaPPY/MERA codes [20]. See Appendix A.11 for the analytic modular-speed-limit bound tested by these simulations. The proxy uses qubit-count, bond-dimension, and access-depth parameters as finite-resolution controls. It does not claim an explicit tensor-contraction realization of black-hole geometry.

The canonical v2 verification artifact tests the same bounded-access structure through retrieval-law checks, inverse-$\gamma (\tau )$ recovery, finite-resolution robustness at $D=4,8$, and adversarial-null diagnostics. The $D=4$ and $D=8$ runs are used as finite-resolution comparisons rather than as claims of physical AdS depth.

The modular wedge for each observer class is represented by varying boundary-condition and access-depth parameters, with detector-style encodings anchoring the effective reconstruction region.

Hardware envelope: All simulations ran on an Intel i7-9700 CPU (3.0 GHz, eight threads, 16 GB RAM). No GPU acceleration was required. The v2 verification artifact is reproducible in a standard CPU/Jupyter environment and regenerates the associated manifest, report, figures, and diagnostic tables.

• System architecture: A 48-qubit parameterization is used as a finite-resolution proxy for bulk access depth; bond edges encode schematic holographic connectivity.
• Initial state: The proxy assumes a highly entangled pure-state background, used as a vacuum analog for testing observer-indexed retrieval structure.
• Boundary conditions: Boundary-condition parameters play the role of detector and frame constraints, modified to emulate each observer class and to anchor the effective modular wedge.

Appendix C.2. Observer-Dependent Channel Implementation

• Reconstruction regions: Stationary observers access fixed exterior layers; freely falling and accelerating observers receive time-evolving access regions that model modular growth or acceleration-enhanced access.
• Frame encodings: Observer-frame transformations are represented by boundary and access-depth changes, altering the effective reconstruction geometry and modular access channel.
• Channel variation: Systematic wedge realignment maps onto the retrieval profiles of Section 3.

Appendix C.3. Data Analysis and Observable Extraction

• Retrieval entropy: Successive access regions yield observer-specific entropy-retrieval curves.
• Second-order correlation: The modeled $g^{(2)}(t_1,t_2)$ envelope is fit to a baseline correlation structure; the tanh-modulated suppressive component tests Eq. (12) and the retrieval law of Appendix A.
• Parameter estimation: Each class is sampled on a fixed proper-time grid; nonlinear least squares and inverse-rate recovery return observer-indexed retrieval diagnostics.

Spectral-noise amplitude is drawn from a bounded Gaussian perturbation scale proportional to $\overline{\gamma }$, ensuring controlled perturbations of the observer-rate profiles.

Bootstrap procedure: Confidence bands use 200 resampled $\gamma (\tau )$ traces per class on a fixed grid with additive bounded spectral noise, following the method described in Section 4.1. The bond dimension is used as a finite-resolution proxy: increasing D approximates deeper accessible layers within the numerical model, not physical AdS depth or explicit tensor-contraction complexity.

Appendix C.4. Validation Diagnostics and Consistency Checks

The following diagnostics test whether numerical retrieval reproduces the analytic signatures derived in Appendix A and represented geometrically in Appendix B.

• Differential retrieval curves: Entropy traces match the time-adaptive law (10).
• Observer-modified access surfaces: Boundary reconstructions are consistent with the redshift-weighted mapping in Eq. (A6).
• $g^{(2)}$ suppression: Accelerating observers show the predicted suppressive modulation; removing observer-indexed retrieval modulation restores the operational null baseline.
• Bond-dimension robustness: The $D=4$ and $D=8$ finite-resolution proxy runs preserve retrieval-horizon ordering and bounded monotonicity.
• Resolution note: Higher-resolution proxy-scaling runs may probe finer access-depth structure beyond the present 48-qubit finite-resolution proxy.

The v2 artifact includes two additional diagnostic layers. First, finite-resolution robustness is evaluated by comparing $D=4$ and $D=8$ proxy runs. All observer classes remain bounded and monotone, and retrieval-horizon ordering is preserved. Second, matched saturating-envelope nulls are calibrated to the same horizon targets as the ODER traces. These alternatives reproduce isolated saturation behavior but do not reproduce the full joint signature across observer classes. Figure A1 and Figure A2 show the corresponding diagnostic plots. The detailed CSV outputs are archived with the repository and summarized in Table 5.

Figure A1. Finite-resolution robustness check for the retrieval trajectories at $D=4$ and $D=8$. The ordering of retrieval horizons is preserved under the finite-resolution proxy, and the trajectories remain bounded and monotone. The bond dimension is used here as a numerical resolution proxy for bounded modular access.

Figure A1. Finite-resolution robustness check for the retrieval trajectories at $D=4$ and $D=8$. The ordering of retrieval horizons is preserved under the finite-resolution proxy, and the trajectories remain bounded and monotone. The bond dimension is used here as a numerical resolution proxy for bounded modular access.

Figure A2. Matched saturating-envelope nulls compared with the accelerating ODER retrieval trace. Logistic, Gompertz, stretched-exponential, and Hill-type alternatives can reproduce isolated saturation behavior after calibration, but they do not reproduce the full joint ODER signature across observer classes, including strict retrieval-horizon ordering, class-specific separation, and stable inverse-$\gamma (\tau )$ recovery. Visual similarity of a single saturating curve is therefore weaker than retrieval-law equivalence.

Figure A2. Matched saturating-envelope nulls compared with the accelerating ODER retrieval trace. Logistic, Gompertz, stretched-exponential, and Hill-type alternatives can reproduce isolated saturation behavior after calibration, but they do not reproduce the full joint ODER signature across observer classes, including strict retrieval-horizon ordering, class-specific separation, and stable inverse-$\gamma (\tau )$ recovery. Visual similarity of a single saturating curve is therefore weaker than retrieval-law equivalence.

Appendix C.5. Falsifiable Retrieval Envelope

Lemma A1

(Falsifiable $g^{(2)}$ Envelope). Let the modeled bounded-access envelope be

$$g^{(2)}(t_1,t_2)=Aexp\left[-\Delta t/{\tau }_{\mathrm{retrieval}}(\overline{t})\right]\left[1-\eta tanh(t_1/{\tau }_{\mathrm{char}})\right]\left[1-\eta tanh(t_2/{\tau }_{\mathrm{char}})\right],0<\eta \le 1,$$

with $\Delta t=|t_2-t_1|$ and $\overline{t}=(t_1+t_2)/2$. Times are measured in the onset-aligned analysis window, with $t_1,t_2\ge 0$ after baseline alignment.

Then: (1) removing the observer-indexed tanh modulation yields the symmetric non-retrieval baseline; (2) retrieval-rate profiles consistent with the bounded-access law produce observer-class separation in the envelope; and (3) thermal smoothing or generic relaxation may reproduce isolated suppression or saturation behavior, but does not reproduce the full joint signature unless it also yields class separation, bounded retrieval, and stable inverse-$\gamma (\tau )$ recovery.

Thus the envelope provides a falsifiable signature of modular retrieval saturation under the stated assumptions.

Scope note. This lemma specifies a falsifier for the retrieval envelope within the bounded-observer framework; it does not claim uniqueness among all possible correlator structures.

Empirically, confirming deviation from the null envelope requires millisecond-scale timing resolution, $\mathrm{SNR}\ge 4$, controlled protocol classes, and the mapping conditions specified in Section 6. The v2 verification artifact provides the implementation of the inversion and validation pipeline, including the generated manifest, report, figures, and CSV tables. The archived scripts are released under an MIT license with pinned dependency specifications for environment replication.

Appendix C.6. Worked Example: Macroscopic Back-Reaction

Illustrative status. This worked example is provided to demonstrate scaling and order-of-magnitude suppression; it is not used elsewhere in the paper.

For a Schwarzschild black hole of mass $M=10M_{\odot }$, the Bekenstein–Hawking entropy is $S_{max}\simeq 4\pi M^2\approx 1.5\times {10}^{78}$ in Planck units, and the horizon radius is $r_{+}\simeq 30\mathrm{km}$. Assuming $\gamma (\tau )\sim {10}^{-3}$ near ${\tau }_{\mathrm{RH}}$ for accelerating observers, the retrieval stress–energy scale satisfies

$$〈T_{\mu \nu }^{\mathrm{retrieval}}〉\sim {\displaystyle \frac{\gamma S_{max}}{4\pi r_{+}^2}}.$$

Restoring units and comparing this scale with the curvature scale near the horizon gives the perturbative estimate

$${\displaystyle \frac{G\langle T_{\mu \nu }^{\mathrm{retrieval}}\rangle }{\sqrt{\mathcal{K}}}}\lesssim {10}^{-6},\mathcal{K}=R_{\mu \nu \rho \sigma }{R}^{\mu \nu \rho \sigma }.$$

This matches the suppression bound of Section 8. Hence back-reaction remains negligible for macroscopic black holes in the parameter regime studied.

Appendix C.7. Retrieval Interference Bound and Differential-Acceleration Interferometer

• Overview.

This subsection defines the retrieval-interference bound $R_{ij}(\tau )\le C(\Delta \Lambda ,{\tau }_{\mathrm{char}})$ and outlines a conceptual protocol for a Differential-Acceleration Interferometer (DAI). No experimental result is claimed here.

• Interference-bound derivation.

Let $F_i(\tau )$ and $F_j(\tau )$ be normalized retrieval profiles for observers i and j with modular-spectrum cutoffs ${\Lambda }_i$ and ${\Lambda }_j$, respectively. From Appendix A.11, each profile satisfies

$${\dot{F}}_i(\tau )\le {\Lambda }_i[1-F_i(\tau )]tanh\left({\displaystyle \frac{\pi {\Lambda }_i\tau }{2}}\right),$$

(A8)

and analogously for $F_j(\tau )$.

Assuming synchronized initial conditions, define the mutual-information overlap proxy:

$${\mathcal{I}}_{ij}(\tau ):=min\left(F_i(\tau ),F_j(\tau )\right),$$

(A9)

a conservative estimator of overlap. Alternative mutual-information estimators may be substituted without altering the qualitative bound structure.

Define the retrieval-overlap tensor:

$$R_{ij}(\tau ):={\displaystyle \frac{{\mathcal{I}}_{ij}(\tau )}{F_i(\tau )+F_j(\tau )-{\mathcal{I}}_{ij}(\tau )}}.$$

(A10)

Let $\overline{\Lambda }=({\Lambda }_i+{\Lambda }_j)/2$ and ${ϵ}_{\Lambda }=|{\Lambda }_i-{\Lambda }_j|/\overline{\Lambda }$. When ${ϵ}_{\Lambda }$ is finite and onset scales differ, the overlap is bounded by the dimensionless envelope

$$R_{ij}(\tau )\le exp\left[-k{ϵ}_{\Lambda }^2{\left({\displaystyle \frac{\tau }{{\tau }_{\mathrm{char}}}}\right)}^2\right],k=\mathcal{O}(1),$$

(A11)

where k is a model-dependent constant determined by the overlap of the respective Paley–Wiener windows. This defines the retrieval-interference bound $C(\Delta \Lambda ,{\tau }_{\mathrm{char}})$.

• Null envelope.

The null model assumes no observer-indexed retrieval structure, corresponding to thermal drift or decoherence without modular retrieval. Baseline envelopes $R_{ij}^{(\mathrm{null})}(\tau )$ are estimated from bootstrap-generated retrieval traces lacking observer-indexed separation.

• ROC-style comparison.

A prospective ROC-style analysis would compare measured $R_{ij}(\tau )$ to a null envelope via:

• True positive: $R_{ij}(\tau )<R_{ij}^{(\mathrm{null})}(\tau )-3\sigma$ at any $\tau$;
• False positive: a null trace misclassified as divergent.

A ROC curve is then constructed by varying the detection threshold. Empirical verification would require differential-arm timing precision at the millisecond scale and stable repeated-run fringe extraction. This approach parallels the falsifiability criterion of the $g^{(2)}$ envelope in Section 6.

• Conceptual DAI protocol.

In a conceptual DAI setup using a BEC analog system:

• stationary and accelerated detector channels are baseline-aligned at $\tau =0$;
• each channel samples its local phonon field and extracts $g^{(2)}(t_1,t_2)$;
• retrieval curves are reconstructed via a pre-specified mapping from measured $g^{(2)}$ envelopes to $S_{\mathrm{retr}}(\tau )$ or $\gamma (\tau )$;
• a thermal null model is generated by removing observer-indexed modular retrieval structure.

These derivations clarify what empirical retrieval divergence would look like under the ODER framework. No specific experiment is assumed to have been completed.

Appendix C.8. Retrieval-Coupled Focusing Equation

Exploratory status. This subsection is an exploratory extension and is not used in the main text or in any validation claim.

Eq. (A4) and the retrieval-curvature coupling motivate a possible extension in which retrieval curvature sources modular expansion.

In all preceding sections, retrieval dynamics were modeled under a fixed-background approximation. One possible extension would couple entropy retrieval to spacetime curvature through a modular analogue of the Raychaudhuri equation. This would introduce a dynamical back-reaction term sourced by the entropy-convergence profile $S_{\mathrm{retr}}(\tau )$, allowing retrieval to track and influence horizon geometry in a coupled model.

• Modular expansion scalar.

Define the modular expansion ${\theta }_{\mathrm{mod}}(\tau )$ as the divergence of modular-flow lines weighted by the retrieval gradient:

$${\theta }_{\mathrm{mod}}(\tau )\equiv {\nabla }_{\mu }{u}^{\mu }+\alpha {\displaystyle \frac{d{S}_{\mathrm{retr}}}{d\tau }},$$

where $u^{\mu }$ is the observer’s four-velocity and $\alpha \ll 1$ is the retrieval-coupling parameter introduced in Section 8.

• Modular Raychaudhuri equation.

The modular analogue of the Raychaudhuri equation takes the form

$${\displaystyle \frac{d{\theta }_{\mathrm{mod}}}{d\tau }}=-\frac{1}{2}{\theta }_{\mathrm{mod}}^2-{\sigma }_{\mu \nu }{\sigma }^{\mu \nu }+{\omega }_{\mu \nu }{\omega }^{\mu \nu }-R_{\mu \nu }{u}^{\mu }{u}^{\nu }+\alpha {\displaystyle \frac{d^2{S}_{\mathrm{retr}}}{d{\tau }^2}},$$

where ${\sigma }_{\mu \nu }$ and ${\omega }_{\mu \nu }$ are the shear and vorticity tensors of the modular-flow congruence, and $R_{\mu \nu }$ is the Ricci tensor. Metric signature follows $(-,+,+,+)$; $R_{\mu \nu }{u}^{\mu }{u}^{\nu }>0$ corresponds to focusing. The term $\alpha d^2{S}_{\mathrm{retr}}/d{\tau }^2$ acts as a retrieval-driven focusing or defocusing contribution within the coupled extension.

• Limiting behavior.

In the limit $\alpha \to 0$, this reduces to the standard Raychaudhuri equation in a fixed background, recovering geodesic congruence evolution. Thus the modular extension preserves classical behavior in the retrieval-free case.

• Back-reaction shift.

To leading order in $\alpha$, the operational retrieval horizon ${\tau }_{\mathrm{RH}}$ shifts by

$$\delta {\tau }_{\mathrm{RH}}\sim \alpha {\int }^{{\tau }_{\mathrm{RH}}}d{\tau }^{\prime }\left({\displaystyle \frac{d^2{S}_{\mathrm{retr}}}{d{\tau }^{\prime 2}}}\right),$$

with ${\tau }_{\mathrm{RH}}$ defined by the $90\%$ retrieval threshold. Numerical estimates from the illustrative scaling in Appendix C.6 indicate this shift remains negligible for $M\gtrsim M_{\odot }$ but may become resolvable in analog systems with boosted retrieval rates. In such analog systems, a non-zero $\alpha$ would manifest as a slow drift of the measured ${\tau }_{\mathrm{RH}}$ across successive retrieval cycles.

• Interpretation.

The modular Raychaudhuri equation operationalizes a possible extension in which entropy retrieval is not merely a diagnostic of black-hole evaporation but can itself act as a geometric source. Failure of monotonic convergence in $S_{\mathrm{retr}}(\tau )$ would signal breakdown of the coupled extension, not of the fixed-background retrieval law established in the main text.

Appendix C.9. Modular Focusing and the Retrieval–Curvature Coupling

Exploratory status. This section elaborates the modular Raychaudhuri construction as a possible future extension. It introduces no new constraints on the retrieval law and is not required for any result in the main text.

• Setup.

Let $u^{\mu }$ be the observer’s proper-time tangent vector, and let $\theta (\tau )={\nabla }_{\mu }{u}^{\mu }$ denote the expansion of the modular flow congruence. The retrieval-coupled expansion scalar is defined as

$${\theta }_{\mathrm{mod}}(\tau )=\theta (\tau )+\alpha {\displaystyle \frac{d{S}_{\mathrm{retr}}}{d\tau }}.$$

• Modular-congruence evolution.

Taking the $\tau$-derivative yields

$${\displaystyle \frac{d{\theta }_{\mathrm{mod}}}{d\tau }}={\displaystyle \frac{d\theta }{d\tau }}+\alpha {\displaystyle \frac{d^2{S}_{\mathrm{retr}}}{d{\tau }^2}}.$$

Inserting the classical Raychaudhuri equation,

$${\displaystyle \frac{d\theta }{d\tau }}=-\frac{1}{2}{\theta }^2-{\sigma }_{\mu \nu }{\sigma }^{\mu \nu }+{\omega }_{\mu \nu }{\omega }^{\mu \nu }-R_{\mu \nu }{u}^{\mu }{u}^{\nu },$$

we obtain the modular-coupled version

$${\displaystyle \frac{d{\theta }_{\mathrm{mod}}}{d\tau }}=-\frac{1}{2}{\theta }_{\mathrm{mod}}^2-{\sigma }_{\mu \nu }{\sigma }^{\mu \nu }+{\omega }_{\mu \nu }{\omega }^{\mu \nu }-R_{\mu \nu }{u}^{\mu }{u}^{\nu }+\alpha {\displaystyle \frac{d^2{S}_{\mathrm{retr}}}{d{\tau }^2}}+\mathcal{O}({\alpha }^2).$$

The ${\theta }_{\mathrm{mod}}^2$ term absorbs the linear $\alpha$ correction to $\theta$, while all other terms remain unaffected at first order.

• Horizon shift.

The retrieval horizon ${\tau }_{\mathrm{RH}}$ is the operational threshold defined by $S_{\mathrm{retr}}({\tau }_{\mathrm{RH}})\ge 0.9S_{max}$. To leading order in $\alpha$,

$$\delta {\tau }_{\mathrm{RH}}=\alpha {\int }_0^{{\tau }_{\mathrm{RH}}}{\displaystyle \frac{d^2{S}_{\mathrm{retr}}}{d{\tau }^2}}d\tau .$$

This integral can be evaluated analytically for constant-$\gamma$ tanh-gap models or numerically using retrieval simulations.

• Remarks.

This derivation assumes modular flow remains smooth and geodesic at leading order. Future work may incorporate non-affine corrections, edge-mode interactions, or observer switching. The modular Raychaudhuri equation defines a possible class of entropy-coupled geometric dynamics. Where $S_{\mathrm{retr}}(\tau )$ is highly nonlinear–for example, under interference, collapse, or multi-observer divergence– ${\theta }_{\mathrm{mod}}$ may blow up, indicating modular-horizon instability in the coupled extension.

In this view, modular flow, retrieval dynamics, and curvature evolution form a coupled triad. This perspective is presented as a possible extension rather than as a claim of completed dynamics.

Appendix D.1. Split Inclusion and Bounded Modular Spectrum

Following standard split-property constructions in AQFT [23,24,25], we introduce nested regions $O_1\subset O_2$ and a Type I intermediate factor:

$$\mathcal{A}(O_1)\subset {\mathcal{N}}_{\delta }\subset \mathcal{A}(O_2),$$

where the split distance $\delta$ defines the physical collar between the inner and outer regions. The modular Hamiltonian $K_{\delta }$ generated by the state $\omega$ on ${\mathcal{N}}_{\delta }$ is then treated with an observer-imposed finite-bandwidth cutoff,

$$\sigma (K_{\delta })\subset [-\Lambda (\delta ),\Lambda (\delta )]\mathrm{within}\mathrm{the}\mathrm{operationally}\mathrm{resolved}\sec \mathrm{tor}.$$

Equivalently, the detector-accessible modular generator is bounded in operator norm by the effective cutoff

$$|K_{\delta }{|}_{\mathrm{eff}}\le \Lambda (\delta ).$$

This provides a finite-resolution entropy and modular flow for the observer-accessible subalgebra. Operationally, $\delta$ corresponds to the detector’s spatial or temporal resolution, and $\Lambda (\delta )\sim 1/\delta$ represents the associated bandwidth limit.

Appendix D.2. Physical Interpretation

Real observers cannot access modes beyond their finite bandwidth; the split inclusion therefore captures the physically retrievable subalgebra of the full theory. All bounded-spectrum statements in this work refer to such operationally defined ${\mathcal{N}}_{\delta }$ and its finite-bandwidth sector.

In the limit $\delta \to 0$, $\Lambda (\delta )\to \infty$, and the regularized algebra approaches the Type ${\mathrm{III}}_1$ continuum class. Within the regulated finite-observer regime, the modular-flow retrieval law derived in the main text, Eq. (10), is defined on bounded spectral support. The Type ${\mathrm{III}}_1$ limit marks the idealized boundary where observer bandwidth becomes infinite and the finite-access cutoff is removed.

Appendix D.3. Finite-Bandwidth Scaling and Retrieval-RG Analogue

The retrieval-RG language is an interpretive scaling diagnostic, not an independent derivation of the retrieval law. Define

$${\beta }_{\Lambda }(\delta )={\displaystyle \frac{dln\Lambda (\delta )}{dln\delta }}.$$

For the canonical finite-bandwidth scaling

$$\Lambda (\delta )\sim {\displaystyle \frac{1}{\delta }},$$

one obtains

$${\beta }_{\Lambda }(\delta )\approx -1.$$

Thus the useful finite-resolution diagnostic is not ${\beta }_{\Lambda }\to 0$, but the deviation from canonical inverse bandwidth scaling:

$${\beta }_{\mathrm{retr}}(\delta )={\beta }_{\Lambda }(\delta )+1={\displaystyle \frac{dln[\delta \Lambda (\delta )]}{dln\delta }}.$$

The condition ${\beta }_{\mathrm{retr}}\to 0$ indicates stable inverse-bandwidth scaling across the tested resolution range.

This establishes an interpretive renormalization-group analogue of the split-property hierarchy: as the split collar narrows, the detector cutoff increases and the finite-observer algebra approaches the continuum Type ${\mathrm{III}}_1$ boundary. This scaling analogy is descriptive and does not generate the retrieval law, which is fixed independently by modular analyticity, bounded spectra, and the speed-limit-saturating retrieval branch. In laboratory analogs, varying detector resolution $\delta$ probes this correspondence by testing whether fitted onset curves collapse under the rescaled variable $\Lambda (\delta )(\tau -{\tau }_0)$ while preserving observer-class ordering.

Future work will treat ${\beta }_{\mathrm{retr}}(\delta )$ as an empirical finite-bandwidth scaling diagnostic, testing whether retrieval-onset curves collapse under $\Lambda (\delta )(\tau -{\tau }_0)$ while preserving observer-class ordering as the split regulator approaches the Type ${\mathrm{III}}_1$ continuum boundary.

Appendix D.4. Open Formal Problems

Here ${\Delta }_{\delta }^{it}$ denotes the modular automorphism group generated by $K_{\delta }$. This regularization does not purport to solve the Type ${\mathrm{III}}_1$ classification problem; it provides a physically covariant framework in which finite observers and their accessible subalgebras are well defined. The continuum Type ${\mathrm{III}}_1$ limit remains a mathematical frontier. Future work should formalize:

• the weak-operator convergence ${\Delta }_{\delta }^{it}\to {\Delta }^{it}$ of modular flows;
• conditions under which isotony and locality persist for the directed family $\{{\mathcal{N}}_{\delta }\}$; and
• quantitative scaling of the regulated spectral density ${\rho }_{\delta }(\lambda )$ as $\delta \to 0$ and $\Lambda (\delta )\to \infty$.

Bridging these mathematical results with the finite-bandwidth scaling diagnostic of Section 8 would complete the formal connection between bounded modular spectra and the Type ${\mathrm{III}}_1$ continuum boundary.

Appendix D.5. Physical Meaning of the Type III₁ Limit

As discussed in Appendix A.7, relational entropy interprets this limit not as loss of information but as the removal of finite-observer bandwidth restrictions. In algebraic QFT, Type ${\mathrm{III}}_1$ algebras are the standard local structures associated with relativistic locality and causal propagation: they admit no finite trace and no ordinary subsystem factorization between interior and exterior regions. “Inevitable” here refers to the continuum idealization of finite-observer regularizations, not to an operationally accessible physical regime.

Within the finite-bandwidth picture, increasing observer bandwidth $\delta \to 0$ drives the modular spectrum toward the continuum Type ${\mathrm{III}}_1$ boundary. Real detectors operate at finite $\delta$; the continuum theory represents the unphysical ideal of infinite information access rather than a directly measurable regime.

All physical predictions in this work are confined to finite $\delta$. The Type ${\mathrm{III}}_1$ limit functions solely as the formal boundary of the theory, while the measurable content lies in finite-bandwidth scaling, observer-class ordering, and recoverability of $\gamma (\tau )$ under bounded access.

Appendix E.1. Kerr Geometry and Modular Flow

In Kerr spacetime the globally static exterior time generator available in Schwarzschild is replaced by a stationary, non-static generator,

$${\chi }_{\Omega }={\partial }_t+\Omega {\partial }_{\varphi },$$

restricted to regions where

$$-g_{\mu \nu }{\chi }_{\Omega }^{\mu }{\chi }_{\Omega }^{\nu }>0.$$

Modular flow follows the mixed time–angle trajectory generated by ${\chi }_{\Omega }$; an observer therefore does not evolve on a globally synchronized static slice. The modular Hamiltonian $K_{{\chi }_{\Omega }}$ associated with this generator defines observer-adapted modular flow consistent with the split-property regularization described in Appendix D.

Appendix E.2. Modular-Generator Deformation

Kerr geometry provides a domain-admissibility test for observer-indexed retrieval. In rotating spacetime, the relevant modular generator cannot be treated as a globally timelike exterior evolution field. Retrieval must instead be defined relative to an admissible local stationary generator ${\chi }_{\Omega }={\partial }_t+\Omega {\partial }_{\varphi }$, restricted to regions where $-g_{\mu \nu }{\chi }_{\Omega }^{\mu }{\chi }_{\Omega }^{\nu }>0$. Within such timelike wedges, frame dragging and redshift deform the retrieval-rate kernel $\gamma (\tau ,a,\Omega )$, while the tanh-gap onset remains fixed by bounded split-regularized modular spectra. Where no admissible timelike stationary generator exists, the stationary retrieval channel is undefined rather than falsified; one must pass to a different observer trajectory or to a nonstationary modular description.

Within an admissible wedge, the Kerr correction acts as a geometric modulation of the observer/state retrieval kernel,

$$\gamma (\tau ,a,\Omega )={\gamma }_0(\tau ){[-g_{\mu \nu }{\chi }_{\Omega }^{\mu }{\chi }_{\Omega }^{\nu }]}^{-1/2},-g_{\mu \nu }{\chi }_{\Omega }^{\mu }{\chi }_{\Omega }^{\nu }>0.$$

This expression is defined only away from the null boundary of the chosen generator. As ${\chi }_{\Omega }$ approaches null, the stationary channel reaches the edge of its admissible domain rather than producing a new retrieval law.

In rotating BEC analogs, frame dragging corresponds operationally to azimuthal phonon flow; measuring the resulting phase and envelope changes in $g^{(2)}$ would probe the same generator-deformation structure, but would not constitute a direct experimental realization of Kerr geometry.

Appendix E.3. Survival of the tanh Onset

For observers whose stationary generator ${\chi }_{\Omega }$ is timelike on the relevant wedge, the modular spectrum remains bounded after split-inclusion regularization. The Paley–Wiener admissibility conditions therefore still hold, and the retrieval law,

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

(A12)

retains its form. Kerr geometry deforms $\gamma$, but does not alter the tanh-gap law within the admissible timelike wedge. This persistence confirms that the tanh onset derived in Appendix A.2 is not restricted to static geometries.

Units adopt $G=c=\hslash =1$ and metric signature $(-,+,+,+)$; under this convention the admissibility condition is $-g_{\mu \nu }{\chi }_{\Omega }^{\mu }{\chi }_{\Omega }^{\nu }>0$.

Appendix E.4. Superradiance and Spectral Containment

Superradiant amplification in Kerr is energy dependent and frame relative. It does not by itself invalidate the retrieval law, but it does restrict the admissible stationary channel. Modular spectral weight remains controlled provided the chosen stationary generator remains timelike on the observer’s wedge and the detector resolution restricts the accessible spectral band. Static retrieval fails at the static-limit surface, but stationary co-rotating retrieval may remain admissible wherever ${\chi }_{\Omega }$ is timelike.

Under these conditions the retrieval wedge remains modularly coherent, and the Paley–Wiener analyticity domain remains intact. Where no such timelike generator exists, the stationary retrieval channel is outside the domain of this appendix rather than refuted by it.

Appendix E.5. Interpretation and Consequences

• The tanh onset is not an artifact of Schwarzschild symmetry; it is a feature of bounded modular spectra.
• Kerr rotation modulates $\gamma (\tau )$ through the admissible generator ${\chi }_{\Omega }$ but preserves spectral convergence within timelike wedges.
• The retrieval law is form-stable under admissible stationary generator deformation within the timelike wedge class.

Conclusion. Modular retrieval is stable under Kerr stationary-generator deformation within admissible timelike ${\chi }_{\Omega }$ wedges. This result preserves the tanh-gap retrieval law under rotation without extending the domain of validity beyond observer-bounded regimes, without crossing null generator boundaries, and without invoking new dynamics.

Appendix F.1. Bandwidth and Algebraic Context

The variables $\delta$ and $\Lambda (\delta )$ introduced in Appendix D connect the modular-algebraic description to measurable observer parameters. Finite $\delta$ defines the retrievable subalgebra ${\mathcal{N}}_{\delta }$ with effective spectral cutoff $\Lambda (\delta )\sim 1/\delta$; the continuum limit $\delta \to 0$ recovers the Type ${\mathrm{III}}_1$ boundary of AQFT. This identification grounds the analytic variables of ODER in the algebraic foundations of relativistic QFT without altering their physical interpretation elsewhere in the framework.

Appendix F.2. Interpretive Parameter Correspondence

Each correspondence below references the bounded-spectrum formalism of Appendix A.11 and the finite-bandwidth scaling diagnostic of Section 8.

• ${\Delta }_{\mathrm{fail}}$ (failure gap). In ODER, ${\Delta }_{\mathrm{fail}}\equiv {\tau }_{\mathrm{evap}}-{\tau }_{\mathrm{RH}}$ measures the gap between the relevant evaporation or termination time and the operational retrieval horizon

  $${\tau }_{\mathrm{RH}}=inf\{\tau \mid S_{\mathrm{retr}}(\tau )\ge 0.9S_{max}\}.$$
  Positive ${\Delta }_{\mathrm{fail}}$ means that retrieval reaches the operational threshold before evaporation or termination; negative ${\Delta }_{\mathrm{fail}}$ marks modular retrieval failure in the tested regime. In gravitational language, this plays a role analogous to asking whether an observer’s accessible wedge supports retrieval before the relevant horizon process ends. It is a diagnostic of observer-indexed access, not a new extremal-surface condition.
• ${\tau }_{\mathrm{char}}$ (convergence time). The modular convergence scale that marks the onset of retrieval serves as a spectrally modulated access threshold. Conventional scrambling time signals full entanglement redistribution, whereas ${\tau }_{\mathrm{char}}$ emerges from bounded modular flow and captures observer-relative retrieval activation even when causal connectivity exists but modular access remains suppressed. Experimentally it governs the transition width of the measured $g^{(2)}$ envelope (Section 4.1).
• $\gamma (\tau )$ (retrieval-rate kernel). In the retrieval law, $\gamma (\tau )$ measures the local observer- and state-dependent rate at which retrievable entropy moves toward saturation. It is inferred from the entropy-access trace through the inverse map and reflects the local modular generator acting on the observer’s accessible subalgebra ${\mathcal{N}}_{\delta }$. A gravitational analogue would be a trajectory-dependent redshift or coupling between boundary modular flow and the evolving bulk access surface. Because $\gamma (\tau )$ varies smoothly with both trajectory and state, it serves as an information-theoretic redshift gradient tied to the curvature of the modular spectrum.

These mappings are interpretive guides, not theoretical requirements. The ODER retrieval law is complete within modular-flow formalism and requires no holographic embedding. Causal wedges and HRT surfaces offer useful geometric parallels, but they are representations of the same underlying access structure rather than foundations of the retrieval law. A full gravitational embedding is deferred to future work; the present appendix is included only to aid conceptual translation.