Modular entropy retrieval in black-hole information recovery: a proper-time saturation model

Modular Entropy Retrieval in Black-Hole Information Recovery: A Proper-Time Saturation Model Evlondo Cooper∗ October 29, 2025 Abstract We present a causal, falsifiable entropy-retrieval law whose growth rate of retrievable entropy is proportional to the remaining entropy gap, modulated by a hyperbolic-tangent regulator that switches on at a characteristic proper time τchar . Unlike phenomenological fits to the Page curve, this law follows directly from bounded Tomita–Takesaki modular flow and is fully invertible from simulated or empirical retrieval curves. The framework converts global entropy conservation into a Lorentzian-causal, observer-specific recovery process. It predicts distinct trajectories for stationary, freely falling, and accelerated observers, and yields an accelerationindexed g(2) ( t1, t2 )envelope that Bose–Einstein–condensate analog black holes can measure on 10–100 ms timescales. The numerical validation on a 48-qubit MERA lattice (bond dimension 8) confirms robustness. A modified Ryu–Takayanagi prescription embeds the model in AdS/CFT without replica-wormhole or island constructions. By replacing ensemble-averaged Page curves with a causal, testable mechanism, the model reframes the black-hole information paradox as an experimentally accessible dynamical question. Here Smax is the Bekenstein–Hawking entropy, γ(τ)is the modular-flow retrieval rate, and τchar sets the characteristic proper-time scale. 1 Introduction Entropy without Access The black-hole information paradox persists not because information is lost, but because no existing framework retrieves it causally. Replica–wormhole paths [ 1 , 2 ], island prescriptions [ 3 ], ensemble Page–curve models [ 5 , 4 ], and ER = EPR dualities [ 7 ] all reproduce the required fine-grained entropy curves, yet none supplies a Lorentzian, proper-time recovery channel that delivers the state to a detector. Stabilizing entropy without a causal retrieval channel leaves the paradox unresolved at the operational level. 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 III 1 , is discussed in Appendix D. This ensures that the Type III 1 foundation is treated rigorously elsewhere, allowing the main text to focus on physical retrieval dynamics. ∗Independent Researcher, Tacoma, WA, USA. ORCID: . Email: [email protected]. 1 Key assumptions. (i) Modular spectra are bounded by the split-property regularization introduced in Appendix D; (ii) modular flow is treated semiclassically on fixed backgrounds; (iii) present BEC analog systems resolve g(2) down to approximately 2 ms. All derivations in Sections 2–7 assume this regularized setting unless stated otherwise. 1.1 Operational–Access Criterion A framework resolves the paradox only if it meets all of the following conditions: (a) Proper-time delivery: specifies how entropy reaches an observer as proper time unfolds; (b) Lorentzian grounding: roots that access in Lorentzian causality; (c) First-principles derivation: derives the process from accepted QFT or GR principles rather than retrospective fitting; (d) Empirical testability: predicts observer-dependent lags ∆ τ within sub-exponential growth in effective Hilbert-space dimension n,1 Table 1: Compliance of major black-hole information proposals with the criteria in Section 1.1. A check mark denotes compliance; a cross denotes failure. Framework (a) (b) (c) (d) Replica wormholes × × ✓× Islands × × ✓× Ensemble Page ×✓× × ER=EPR ×✓× × Each proposal satisfies at most two criteria; none supplies a causal, observer-accessible retrieval channel. Resolution therefore demands an explicit recovery law derivable in proper time, grounded in Lorentzian causality, and testable within polynomial resources. The Observer-Dependent Entropy Retrieval (ODER) framework meets those demands with modular-flow dynamics and wedgereconstruction depths that scale polynomially, in contrast with the exponential-cost Hayden–Preskill decoder O (2 n )assumed for global recovery. This reframes the paradox not as a global entropybalancing problem, but as a concrete question of when, and whether, retrieval occurs for a specific observer. Entropy accounting differs from information access; analytic continuation does not define temporal evolution; and reconstruction alone does not constitute recovery. Throughout, we distinguish retrieval  =reconstruction  =comprehension: a system may be reconstructable yet not retrieved, and retrieved yet not comprehended. Reader Guide This paper introduces a new way to think about information, not as something stored or globally conserved, but as something recovered by an observer in time. 1 Sub-exponential relative to decoding complexity; for ODER this scales polynomially, O ( nlog n ), under modularflow reconstruction. Empirical testability requires predicting observer-dependent lags ∆ τ resolvable at laboratory timescales (e.g., ≳10−3ms in current BEC analogs). 2 It began with a simple question: What if entropy is not about what exists, but about what can be retrieved? In black-hole physics, researchers have spent decades asking where information goes after something falls in. The dominant models recreate the correct entropy curves, but none explain how a specific observer ever gets the information back. The paradox was never about loss; it was about access. This work proposes a solution: a concrete law that describes how information returns to an observer over proper time; not all at once, not just at the end, but gradually, shaped by the path they take through spacetime. The law is derived from established quantum field theory, simulated on quantum lattices, and matches what can be measured in analog black holes today. It is not a guess; it is invertible; any experimenter can extract γ(τ). If you are not a physicist, that is fine. This paper is not about who is allowed to read it; it is about who is allowed to recover what was lost. • Section 2 derives the observer-indexed retrieval law and presents an inverse map that reconstructs γ(τ)from measured Sret(τ); •Section 6 validates the law on a 48-qubit MERA lattice, establishing the O(nlog n)scaling; •Section 7.5 translates the theory into the g(2) fringe measurable in current BEC analogs; •Section 4.1 provides a calibration protocol for τchar; •Section 7.4 defines the retrieval–evaporation gap ∆fail. Appendix F offers an interpretive correspondence for readers unfamiliar with modular dynamics, mapping γ ( τ ), τchar , and ∆ fail to gravitationally intuitive quantities without altering the retrieval framework. Appendix A formalizes the spectral bounds used throughout. Code, data, and figure-generation notebooks are archived at https://doi.org/10.5281/zenodo. 15669855. 2 Observer-Dependent Entropy Retrieval Novel framework. ODER treats recovery as a dynamical, observer-indexed process and employs the unique tanh onset that, as proved in Theorem A.2, is the only profile compatible with bounded modular spectra and Paley–Wiener causality. This section derives dSretr dτ =γ(τ) [Smax −Sretr(τ)] tanh τ/τchar,(1) directly from Tomita–Takesaki modular flow on nested von Neumann algebras.2 2 Boundedness of the modular spectrum follows from the split-property regularization described in Appendix D, where Λ(δ)∼1/δ represents the observer’s finite bandwidth. 3 Because Eq. (1) is first order and monotone in the bounded and differentiable Sretr ( τ ), we can solve it for the unknown rate γ ( τ ). This inversion is essential in Section 3, where retrieval rates for different observer classes are compared. γ(τ) = 1 Smax −Sretr(τ)tanh τ/τchar dSretr dτ (Inverse retrieval map) Implication. Once an experiment measures Sretr ( τ ), for example through the g(2) fringe, the boxed map fixes γ ( τ )without further assumptions, turning ODER from a forward model into a calibratable decoder. Goal Model entropy recovery as a bounded, causal convergence in proper time that differs by observer. Mechanism Eq. (1) depends on modular-spectrum gradients; γ ( τ )encodes redshift, Unruh, or interior-correlation effects. Domain of validity Algebraic QFT on Lorentzian backgrounds; simulations on a 48-qubit MERA lattice confirm robustness. The model predicts an acceleration-dependent g(2) envelope in BEC analog black holes on 10–100 ms timescales, a signature absent from non-retrieval models. We define the retrieval horizon τRH := infτSretr(τ)≥0.9Smax, the proper time at which 90% of the retrievable entropy is accessed; this horizon is distinct from both the entanglement wedge and the classical event horizon. 2.1 Retrieval as a Modular Speed Limit Speed-limit principle. Equation (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 entropy-access traces; it saturates a modular speed limit imposed by the Paley–Wiener constraint on the modular spectrum. Let σ ( K ) ⊂ [ − Λ( δ ) , Λ( δ )] denote the split-regularized spectrum of the modular Hamiltonian. Paley– Wiener theory requires that any admissible retrieval trajectory F ( τ ) = Sretr ( τ ) /Smax be entire and of exponential type ≤Λ(δ). For such F(τ), the slope is bounded: ˙ F(τ)≤Λ(δ) [1 −F(τ)] tanhπΛ(δ)τ 2,(2) 3 which we denote as the modular speed limit B Λ( δ ) , τ . The bound expresses that entropy retrieval can never accelerate faster than permitted by the modular spectral support Λ( δ )and the proper-time onset scale τ. 3 The factor π originates from mapping the Paley–Wiener strip |ℑτ|< π/ (2Λ) to the real axis; it fixes the slope normalization. 4 Theorem 2.1 (Modular Speed Limit). Let F ( τ )be C1 , strictly increasing, entire, and of exponential type ≤ Λ( δ ). Then the inequality (2) holds pointwise in τ ; moreover, the tanh profile of Theorem A.2 achieves this bound uniquely, up to an affine reparameterization of τ. Variational formulation. The modular speed limit admits a variational expression. Define the entropy-action functional I[F] = Zdτ h1 2˙ F(τ)2−U(F; Λ(δ), τchar)i,(3) where τchar corresponds to the observer’s bandwidth-limited response time defined in Section 1. The associated potential is U(F; Λ(δ), τchar) = 1 2Λ(δ)2[1 −F(τ)]2tanh2τ τchar .(4) This potential arises from a redshift-weighted modular-energy envelope and mimics finite propagation delay across a stretched horizon; it vanishes as F ( τ ) → 1, ensuring saturation is finite and bounded. This potential form follows from minimizing the L2 -norm of retrieval acceleration under Paley–Wiener bounds. Stationarity of I[F]under δF yields the Euler–Lagrange equation: dSretr dτ =γ(τ) [Smax −Sretr(τ)] tanhτ τchar ,(5) recovering the retrieval law as the extremal trajectory that saturates the speed-limit bound. Onset scale. The transition scale τchar is not a fit parameter; it is fixed by the modular energy gap ∆Kand horizon radius RH: τchar =2π ∆Kf(RH), where f ( RH )encodes near-horizon flow compression; this ensures that the envelope onset emerges from geometric and spectral structure, not arbitrary calibration. Falsifiability condition. The modular retrieval law is falsified if any observer, at any stationary radius r, measures dSretr dτ >Λ(δ) [1 −F(τ)] tanhπΛ(δ)τ 2 within statistical confidence bounds; empirical violation of this limit implies a breakdown in modularflow constraints or a failure of the retrieval principle itself. Implication. The tanh onset is therefore not an arbitrary insertion but the unique optimal profile permitted by modular causality and bounded spectra; this elevates Eq. (5) to the status of a principle: it is the fastest possible observer-indexed convergence consistent with Tomita–Takesaki flow. Figures 1 and 2 illustrate this saturation, where fitted retrieval traces track the speed-limit envelope within bootstrap error bands. 5 Self-Audit: ODER Failure Modes To preserve falsifiability and epistemic integrity, we enumerate conditions under which ODER may fail: •Modular realism: modular Hamiltonians must remain physical in strong-gravity regimes. • Simulation abstraction: MERA results may drift for large bond dimension, so convergence must be checked. • Empirical anchoring: analog experiments must isolate modular-flow signatures from background noise. •Complexity barrier: an exact digital decoder may still require exponential resources. • Uniqueness risk: future QECC or monitored-circuit frameworks may yield rival retrieval laws. Astrophysical forecast. For a solar-mass Schwarzschild black hole, Eq. (1) implies that a stationary observer at r = 10 GM/c2 retrieves at least 90% of the missing entropy only after ∼ 10 67 yr ; a timescale absent from replica-wormhole or island prescriptions. Sections 6–7.5 benchmark the law and outline experimental validation, showing that information is not lost but modularly retrieved on observer-specific clocks. 3 Observer-Dependent Entropy in Curved Spacetime We classify three canonical observer trajectories and track entropy-retrieval dynamics along each. The retrieval rate γ ( τ )is fixed by the local modular Hamiltonian, with no phenomenological fits, and evolves with proper time. All analyses assume the finite-bandwidth, split-regularized subalgebras introduced in Appendix D, with bounded modular spectra σ(K)⊂[−Λ(δ),Λ(δ)]. 6 3.1 Classification of Observers Figure 1: Representative retrieval-rate profiles γ ( τ )for the three observer classes. Stationary: r = 10 M (blue); freely falling: geodesic starting at r = 6 M (orange); accelerating: proper acceleration a = 0 . 2 c2/M (green). Times are in units of M with G = c = 1. All profiles are smoothed with a monotonic spline consistent with Paley–Wiener bounds. Table 2: Indicative parameters for each observer class ( M = 1 in geometric units). The retrieval horizon τRH is defined by Sretr(τRH)=0.9Smax. The Page time τPage refers to the observer-specific modular Page scale, not the global Page curve. Observer r/M aM/c2τchar/M τPage/M τRH/M Stationary 10 0 5 8 30 Freely falling 6–2 0 2 4 10 Accelerating – 0.2 3 5 15 Stationary observer. A detector at fixed radius r > 2 M perceives Hawking radiation as redshifted thermal flux; the corresponding modular-flow retrieval rate is proportional to the local flux amplitude, γstat(τ)∝1 r,(6) up to normalization by the surface gravity κ , which fixes the local modular temperature ( TH = κ/ 2 π ). 4 This yields a monotonic decay in g(2) correlations. For r = 10 M we find τchar < τPage because no interior mode enters the algebra. 4 Exact prefactors follow from the local modular-Hamiltonian eigenvalue density; only asymptotic scaling is shown. 7 Freely falling observer. A geodesic world line crosses the horizon at τcross ; interior modes then boost the retrieval rate, γfall(τ)≫γstat(τ), τ > τcross,(7) accelerating saturation (orange curve in Figure 1).5 Accelerating observer. A uniformly accelerating detector, referring to Rindler-like trajectories exterior to the horizon, experiences both Hawking and Unruh flux (cf. Hawking 1975; Unruh 1976), γeff(τ, a) = γHawking(τ) + γUnruh(τ, a),(8) with γUnruh ∝a2[13]. At a= 0.2c2/M the retrieval envelope is the green curve in Figure 1.6 Experimental emulation: stationary and accelerating channels can be engineered in waterfall BECs, while freely falling trajectories correspond to time-of-flight release [ 14 ]. Measured values of τchar directly encode detector bandwidth, in agreement with the finite-observer scaling Λ( δ ) ∼ 1 /δ discussed in Section 2. Detectability requires signal-to-noise ratio ≥ 4at temporal resolution ≈ 2 ms , with both detectors calibrated to identical bandwidth and phase prior to divergence. Parameters appear in Table 2. 3.2 Observer-Dependent Entropy Observer-dependent entropy is the gap between the global von Neumann entropy and the entropy of the observer’s accessible subalgebra. The retrievable component Sretr ( τ )rises as modular eigenmodes enter the algebra; Appendix A shows γ ( τ ) ∝∂τln ρmod . Retrieval is computed only over causal diamonds with stable, horizon-bounded algebras; extension beyond τRH approaches the Type III 1 limit (see Appendix D) [26, 12]. 3.3 Retrieval Law For bounded and differentiable Sretr(τ), dSretr dτ =γ(τ)Smax −Sretr(τ)f(τ),(9) with f(τ) = tanh τ/τchar. This functional form is uniquely fixed by bounded modular flow; the spectral proof is given in Appendix A. Unlike the exponential damping factors used in replica-wormhole models, γ ( τ )and τchar are determined directly from the local modular Hamiltonian, yielding a continuous, observer-specific retrieval process. We refer to γ ( τ )as the modular-flow retrieval rate; it quantifies the pace at which retrievable information enters an observer’s algebra. 5 Exact prefactors follow from the local modular-Hamiltonian eigenvalue density; only asymptotic scaling is shown. 6 Exact prefactors follow from the local modular-Hamiltonian eigenvalue density; only asymptotic scaling is shown. 8 3.4 Multi-Observer Retrieval and Interference Tests Concept. Observer-dependent entropy retrieval implies that no two observers with non-trivial proper-time divergence can recover identical information from the same evaporating system. If ODER holds, retrieval is not broadcastable. This section introduces a falsifiable prediction: retrieval interference will emerge between observers with different modular spectra. The effect cannot be mimicked by thermal noise or unitary scrambling. Retrieval-overlap tensor. Let Si ( τ )and Sj ( τ )be entropy-retrieval curves for two observers i and j, parameterized by their proper times. Define the retrieval-overlap tensor Rij(τ) := Iij(τ) Si(τ) + Sj(τ)−Iij(τ),(10) where Iij ( τ )denotes the mutual information shared between the retrieval outputs at time τ , computed within the common causal diamond accessible to both observers. In the weak-coupling limit, Rij → 1 as ∆Λ→0, recovering the classical broadcast symmetry expected under identical modular spectra. Bounded interference. Let ∆Λ := | Λ i− Λ j| be the modular-spectrum offset between the observers. ODER predicts that Rij(τ)≤C(∆Λ, τchar),(11) where C is a retrieval-interference bound derived in Appendix C.7. This bound sharpens as the modular spectra diverge and does not arise in any known non-retrieval frameworks. Proposed experiment: Differential-Acceleration Interferometer. Having defined theoretical divergence, we now map it onto a realizable analog system. We propose a laboratory analog test using a Bose–Einstein condensate (BEC) black-hole system with two synchronized detectors: •One detector remains stationary; the other undergoes uniform acceleration. • Both extract g(2) ( t1, t2 )phonon-correlation envelopes from the emitted analog Hawking radiation. •Detectors are phase-locked at τ= 0 and calibrated against a shared vacuum state. Signature: accelerating fringe. ODER predicts a divergence in the mutual information extracted by the two detectors at late proper time; the accelerated arm will show a suppressed or oscillatory g(2) envelope, a retrieval-interference signature that thermal models cannot produce. No scrambling-only framework yields late-time, frame-specific entropy loss. Falsifier: null envelope and ROC test. We define a null envelope: a simulated retrieval profile assuming γ ( τ ) = 0, representing pure thermal drift with no modular structure. ROC analysis is then applied using bootstrap-averaged samples. ODER is falsified if Rij ( τ )remains within the null-envelope confidence band (three sigma) for all τ . The full ROC protocol appears in Appendix C.7. 9 7.2 Retrieval Horizon =Entanglement Wedge =Event Horizon Observer-dependent modular flow separates three operational boundaries: •Retrieval horizon. τRH = inf{τ|Sretr(τ)≥0.9Smax}. • Entanglement wedge: the bulk region reconstructable through the boosted RT surface, Eq. (15). •Event horizon: the classical null surface. These boundaries coincide only in the idealized Type III 1 limit, in which observer bandwidth becomes infinite. In Kerr spacetime the generator χ=∂t+ ΩH∂ϕgives γ(τ, a, Ω) = |gµνχµχν|−1/2, evaluated on the outer stationary wedge just outside r+ . Where χ is timelike, the Paley–Wiener bound preserves the tanh onset [29]. 7.3 Implications for Evaporating Black Holes •Stationary observers (r > 2M): slow retrieval, γ∝1/r. •Freely falling observers: interior modes boost γafter horizon crossing. •Accelerating observers: Unruh terms create the g(2) fringe. In every case lim τ→∞ Sretr ( τ ) = Smax ; saturation stems from modular closure, not ensemble averaging. 7.4 ∆fail: Retrieval–Evaporation Boundary Define ∆fail =τevap −τRH, with τevap the semiclassical evaporation time. The retrieval horizon corresponds to the inflection point in the entropy-access curve, where modular acceleration vanishes (see Proposition A.3). Positive ∆fail means retrieval completes before evaporation; negative values imply modular failure. Table 5: Benchmark ∆ fail values ( M = 1 in geometric units). Positive ∆ fail indicates modular retrieval completes before semiclassical evaporation; a negative value would falsify ODER. Observer τRH τevap ∆fail Stationary 30 ∼1067 ≫0(as expected under semiclassical stability) Freely falling 10 ∼1067 ≫0(as expected under semiclassical stability) Accelerating 15 ∼1067 ≫0(as expected under semiclassical stability) A negative ∆ fail in analog experiments or numerical simulations would falsify the retrieval law; any ∆fail ≥0is consistent with modular accessibility. 16 7.5 Experimental Implications and Roadmap All following experimental predictions inherit their parameter scaling directly from Eq. (9) , ensuring one-to-one traceability between analytic and empirical domains. The following roadmap connects theoretical parameters to measurable laboratory observables. Timescale Bridge With G=ℏ=c= 1 and 1M⊙≃4.93 µs, ∆tlab ≃4.93 µsM/M⊙(∆τ/1M). A2–20 M window in a 10 M⊙ acoustic analog thus maps to 10 ms –100 ms , well above the 2 ms detector limit of Ref. [ 14 ]. Signal detection requires SNR ≥ 4over a 10 ms integration window, matching current BEC noise floors. Operational Falsifiability •Absence of a g(2) envelope implies modular access is falsified. •A mismatched γ(τ)fit implies the retrieval law is incomplete. •Identical τPage for all observers implies observer specificity is invalid. All analog runs must first benchmark detector response against a null ( γ ( τ ) = 0) baseline before claiming retrieval signatures. Table 6: Operational comparison for a stationary observer at r= 10M. Feature ODER (this work) Replica or islands Causal retrieval ✓proper-time decoder ×stabilization only Decoding protocol ✓polynomial MERA ×none known Empirical observable ✓g(2) in BEC ×not specified Computational cost O(n2)O(2n) Verification or failure of these signatures will determine whether modular flow constitutes a physical retrieval mechanism or merely a formal analogy. 8 Limitations and Scope This section enumerates the theoretical boundaries and experimental tolerances that currently define ODER’s operational domain. Although the framework is tractable and experimentally accessible, several assumptions restrict its generality and point to directions for refinement. 17 Retrieval-Driven Back-Reaction: A Thresholded Causal Ansatz All retrieval dynamics in this work assume a fixed background metric. Introducing a small coupling, Tµν −→ Tµν +α Tretrieval µν , α ≪1, one recovers the semiclassical Einstein equation in the limit α→ 0. For α = 0 the retrieval horizon shifts only at O(α)(i.e., first-order perturbative back-reaction). Back-reaction bound. For a Schwarzschild mass M, Tretrieval µν ∼γ(τ)Smax 4πr2 + , Smax ∝M2, so that GTretrieval µν  √K≲10−6,K=RµνρσRµνρσ = 48 G2M2/r6 +, M ≳M⊙. For a fiducial 10 M⊙ black hole one finds G⟨Tretrieval µν ⟩ ≈ 4 × 10 −7pK(10M⊙) , implying δr+/r+< 3 × 10 −6 and a negligible shift in τRH . This scaling of α provides the perturbative limit recovered in the modular Raychaudhuri coupling (Appendix C.8). Outlook. A fully coupled model in which Tretrieval µν ∝ ( ∂τSretr ) uµuν would elevate entropy retrieval to an explicit causal modulator of curvature [21]. Semiclassical modular-flow assumption. Type III 1 algebras are regulated by finite splits [ 23 , 22 ]; extending to Kerr, de Sitter, or multi-horizon cases will require relative-Tomita theory and edge modes [12]. Analog-System Resolution Current BEC experiments resolve g(2) on 2–10 ms scales [ 14 ], five times finer than the predicted 10–100 ms retrieval window. Baseline g(2) runs should precede interpretation. For instance, waterfall BEC interferometers routinely achieve sub-millisecond phase locking and SNR ≥4[14]. Exclusion of Exotic Topologies Replica wormholes, islands, and other speculative geometries are omitted, keeping all predictions directly testable. Potential Extension to Superposed Geometries Future work could apply the retrieval law to geometries in quantum superposition, probing modular coherence across fluctuating horizons. 18 No Global Unitarity Guarantee Equation (9) ensures unitarity only inside each observer’s wedge; modular mismatches between overlapping diamonds are expected. Retrieval-Horizon and Noise Scope The framework guarantees saturation of Sretr ( τ )only up to τRH ; full recovery beyond that point lies outside its present mandate. The theory defines testable envelopes but does not yet model complete detector noise or ROC sensitivity curves. Finite-Bandwidth Refinements and Retrieval-RG Scaling Observers possess finite temporal or spatial resolution ( δ ). The corresponding modular spectral cutoff Λ(δ)∼1/δ introduces a controlled, testable δ-dependence in the retrieval law: F(τ) = tanhπΛ(δ) 2(τ−τ0). The prefactor ( π Λ( δ ) / 2) scales linearly with detector bandwidth, while the characteristic time satisfies τchar ( δ ) ∝ 1 / Λ( δ ). Repeating measurements at multiple δ values yields predictable rescaling of the g(2) correlation envelope’s transition width without changing the observer-class hierarchy. Define the retrieval-RG function βΛ(δ) = dln Λ(δ) dln δ, defined analogously to a renormalization-group flow on the detector bandwidth scale. This relation provides an empirical RG analogue allowing retrieval dynamics to be plotted as flow trajectories in (Λ, δ)space. The fixed point ( βΛ→ 0) corresponds to the Type III 1 limit where the modular spectrum becomes scale-invariant. This scaling will appear experimentally as δ -invariance of the fitted tanh parameters and of the retrieval horizon τRH ( δ )within detector precision. Empirical validation requires resolving ∆ τ changes of order ≤ 5% across a decade variation in δ , achievable with current timing precision. For instance, waterfall BEC interferometers already meet this precision threshold. Future retrievalRG analyses should map these δ -dependent trajectories to laboratory scaling laws across analog platforms. 9 Conclusion and Next Steps This section consolidates the theoretical, computational, and empirical threads of ODER and outlines the immediate path forward. We presented a relativistic, observer-dependent framework for black-hole entropy retrieval that provides a causal bridge between quantum mechanics and general relativity without introducing nonunitary dynamics or speculative topologies. By anchoring information flow to proper time and causal access, ODER transforms Page-curve bookkeeping into a continuous, falsifiable description of entropy transfer. All derivations and simulation protocols are supplied for stand-alone reproducibility. 19 The retrieval law is not heuristic; it follows from Tomita–Takesaki modular spectra (Appendix A, Eq. (9) ). Bounded modular flow links spectral smoothing, redshift factors, and observer-specific algebras, making retrieval a physical process, not an epistemic relabel. Concrete predictions follow. Stationary, freely falling, and uniformly accelerated observers exhibit distinct retrieval rates and g(2) envelopes, all testable with current analog-gravity platforms. Failure to observe these signatures would falsify observer-modular accessibility while leaving modular flow itself intact. Roadmap: Theory, Simulation, Experiment Theory • Semiclassical back-reaction: couple entropy flow to a self-consistent metric response, extending Eq. (9) into a dynamical observer–spacetime equation. • Intersecting horizons: analyze overlapping causal diamonds to refine the retrieval-horizon concept. • Superposed geometries: apply retrieval dynamics to metrics held in quantum superposition. Simulation • High-bond-dimension MERA: benchmark D > 8convergence and finite-entanglement effects on γ(τ). •Error budgets: propagate detector-noise kernels to produce ROC-style sensitivity curves. Experiment • Trajectory-differentiated probes: deploy stationary, co-moving, and accelerating detectors in BEC waterfalls; target the 10 ms–100 ms window with ≲2 ms timing. • Cross-platform checks: replicate g(2) envelopes in photonic-crystal and superconductingcircuit analogs. • Calibration: detector-noise calibration should precede retrieval-fit attempts to ensure SNR ≥ 4 across all platforms. Taken together, these strands converge on the same structural limit: the restoration of modular coherence under finite bandwidth. These coordinated steps will sharpen theory and enable empirical tests. Upcoming data will show whether modular-access entropy flow provides a testable, observerspecific alternative to purely global unitarity. Final Remark. The split-property regularization situates ODER one layer closer to a fully rigorous Type III 1 limit than any prior Page-curve or island model. Within this operational framework, 20 Type III 1 emerges as the continuum fixed point of bounded-observer modular flow, an outcome both mathematically consistent and physically measurable. See Appendix D for the explicit modular-flow fixed-point derivation. Whether the coming analog experiments confirm or falsify this law, they will mark the transition from retrieval as theory to retrieval as measurement. Author Declarations and Data Availability Author Contributions. Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Visualization, Writing (original draft), Writing (review and editing), Supervision, and Project administration, E.C. Funding. This research received no external funding. Data and Materials Availability. All code, notebooks, and figure-generation scripts are archived as a single Zenodo release at https://doi.org/10.5281/zenodo.15428312 and mirrored on GitHub at https://github.com/evlocoo/ODER-modular-entropy . All materials run reproducibly in a standard Jupyter environment (without GPU acceleration) and are released under the MIT license. •ODER_Black_Hole_Framework_Complete_Simulation_V2.ipynb : reproduces every figure and table in the manuscript. •ODER_Retrieval_Inversion_And_Validation.ipynb : performs τchar fitting, γ ( τ )reconstruction, and validates the falsifiable g(2)(t1, t2)envelope of Lemma C.5. Conflicts of Interest. The author declares no conflict of interest. A First-Principles Derivation of the Observer-Dependent Retrieval Equation All modular results below are formulated on split-regularized, finite-bandwidth subalgebras corresponding to realistic detector resolution. All entropy functions are normalized by Smax unless otherwise noted. For the algebraic foundation underlying bounded modular spectra, see Appendix D. 21 Theorem A.1 (Observer-Retrieval Law). Assumptions. A1: a globally hyperbolic spacetime background; A2: a faithful global state ωon the net A(O); A3: an observer world line γwith wedge D(γ, τ); A4: a modular spectrum bounded below. Conclusion. The unique C1 function Sretr ( τ )(consistent with Eq. (9) ) that (i) satisfies 0 ≤ Sretr ≤Smax ; (ii) is strictly increasing; (iii) obeys lim τ→∞ dSretr dτ = 0; and (iv) is generated by the modular automorphism group of A[D(γ, τ)] holds: dSretr dτ =γ(τ)Smax −Sretr(τ)1 + tanh(τ/τPage) 2. The solution is unique up to an overall scale in γ ( τ )fixed by redshift factors and the modularspectrum gradient.a□ aModular operators are defined on the split-property subalgebras Nδintroduced in Appendix D. A.1 Motivation: Bounded Algebras and Observer-Dependent Entropy Algebraic QFT assigns von Neumann algebras A ( O )to spacetime regions O . A global state ω on A [ D ( γ, ∞ )] encodes all degrees of freedom inside the observer’s domain of dependence. At proper time τthe observer accesses only A[D(γ, τ)]; the entropy gap is the retrievable deficit. Finite-split regularization. Because A ( D )is Type III 1 , its modular Hamiltonian is unbounded. A split inclusion A ( D1 ) ⊂ N ⊂ A ( D2 )produces a Type I factor N with detector-bounded spectrum, preserving the Paley–Wiener condition as the split distance shrinks (Refs. [ 23 , 22 ]). This procedure defines the finite-bandwidth subalgebras Nδ on which the bounded modular spectra σ ( Kδ ) ⊂ [−Λ(δ),Λ(δ)] are realized. A.2 Spectral Convergence and Uniqueness of the Retrieval Sigmoid Lemma A.1 (Paley–Wiener Band-Limit). Let the split-regularized modular Hamiltonian K have bounded spectrum σ ( K ) ⊂ [ − Λ , Λ]. Then any observable expectation value f ( τ ) = ⟨ψ|eiKτ |ψ⟩ —and all quantities derived from it, including the normalized entropy evolution F ( τ ) = Sretr ( τ ) /Smax — extend holomorphically to the horizontal strip SΛ={τ∈C:|ℑτ|< π/(2Λ)}, with growth |F ( τ ) | = O ( eΛ|ℑτ| ); that is, F is of exponential type ≤ Λin SΛ . (Paley–Wiener Theorem 19.3 in Rudin, Real and Complex Analysis). □ Lemma A.2 (Phragmén–Lindelöf Growth Bound). If F ( τ )is holomorphic in SΛ , bounded and strictly monotonic on R , and of exponential type ≤ Λ, then |F ( τ ) | ≤ 1throughout SΛ . Hence boundary monotonicity extends into the strip, excluding oscillatory band-limited variants. □ 22 Theorem A.2 (Uniqueness of the Retrieval Sigmoid under Modular Band-Limit). Let F : R→ (0 , 1) be strictly increasing with finite limits F ( −∞ )=0, F (+ ∞ )=1. Assume F satisfies Lemmas A.1–A.2: it is holomorphic in SΛ , of exponential type ≤ Λ, bounded on R , and obeys the modular-spectrum bound σ(K)⊂[−Λ,Λ]. Then—up to an affine reparametrization of τ— F(τ) = tanh πΛ 2(τ−τ0). Sketch of Proof. Strip → Disk Map. Map SΛ to D by z = exp ( π Λτ ). Define G ( τ ) = 2F(τ)−1 2F(τ)+1 ; then F(τ) = 1+G(τ) 1−G(τ) 1 2and G:SΛ→D. Positivity (Pick/Herglotz). Monotonicity on R implies ℜ [( log 1+G 1−G ) ′ ] > 0; that is, G is a Pick function. Extremal Solution. By Schwarz–Pick, |G′ ( τ ) | ≤ ( π Λ / 2) [1 −|G ( τ ) |2 ]. Equality on the real axis (a.e.) forces Gto be a disk automorphism, hence G(τ) = tanh πΛ 2(τ−τ0). Recover F .Undoing the Möbius transform yields the stated F ( τ ), with normalization F (0) = 0, F(+∞) = 1.□ Corollary A.2.1 (Spectral Constant Fixation). The constant ( π Λ / 2) is fixed by the strip width |ℑτ|< π/ (2Λ); no other monotone analytic map satisfies both the strip bound and the modular-spectral constraint.8 Excluded Counter-Examples. Candidate F(τ)Why Excluded (2/π) arctan(ατ)Poles at ±i/α; not holomorphic in SΛ. 1−e−ατ Violates strip boundedness; unbounded along ℑτ. Band-limited oscillatory sigmoids Break strict monotonicity on R. Logistic 1/(1 + e−x) Entire but exponential type ∞ (unbounded spectrum). These exclusions guarantee that all admissible retrieval profiles share the same analytic growth bound, ensuring that Eq. (A.2) is not merely optimal but necessary. Physical Interpretation (Why tanh ?). Early times ( τ≪τchar )reflect incomplete activation of modular modes; late times ( τ≫τchar )approach saturation as the bounded spectrum fully enters the algebra. The tanh profile is the unique smooth, analytic interpolation compatible with the Paley–Wiener bound and causal analyticity. Diagram A.3 (Concept). Plot: candidate monotone curves satisfying F ( −∞ )=0, F (+ ∞ )=1. Blue: tanh (allowed); orange: arctan (rejected poles); green: exponential (decay violates strip). Only blue lies within analytic-growth limits of SΛ. 8 Paley–Wiener Admissibility: Throughout this appendix, Paley–Wiener admissibility refers to analyticity within the horizontal strip SΛ={τ∈C| |ℑτ|< π/(2Λ) }, not entire analyticity over the full complex plane. 23 Remark A.2.2 (Edge Atoms and Observer Generality). Atomic spectral weight at ± Λwould induce boundary oscillations incompatible with strict monotonicity on R and is therefore excluded by assumption. In curved or rotating backgrounds (e.g., Kerr), the modular spectrum remains bounded after split-inclusion regularization, so the tanh onset persists. This bounded-spectrum proof generalizes to all observer classes in Section 3 and parallels the retrieval-RG scaling derived in Section 8. A.3 Role of γ(τ): Modular Spectrum and Redshift •Spectrum gradient: if ρ(λ)∼λ−β, then γ(τ)∝τβ−1. •Geometric redshift: stationary observers yield γstat ∝1/r. •Unruh boost: uniform acceleration gives γacc ∝a2. Table 7: Retrieval parameters used in numerical runs for Figures 1–2 (geometric units G=c= 1). Observer Prefactor γ0τchar/M τPage/M Stationary (r= 10M) 0.05 8 15.0 Freely falling 0.10–0.25 4 7.5 Accelerating (a= 0.2) quadratic fit 6 10.5 These prefactors correspond directly to experimentally fitted g(2) envelopes in Section 4, establishing traceability between algebraic and laboratory parameters. A.4 Retrieval Saturation and Collapse Boundary [Retrieval horizon τRH ] Let Sretr ( τ )be the entropy-access curve derived in Theorem A.1. There exists a unique proper time τRH such that d2Sretr dτ2τ=τRH = 0,d3Sretr dτ3τ=τRH <0. Define ∆ fail ≡τevap −τRH . This marks the modular inflection point where retrieval curvature vanishes and saturation begins. A.5 Observer-Bounded Automorphisms and the tanh Factor Theorem A (below) shows that global modular flow restricts to the observer algebra and yields the unique tanh onset that appears in Eq. (9) . This guarantees that the retrieval law saturates the Paley–Wiener bound and therefore represents the maximal causal convergence permitted by the Paley–Wiener bound. See also Lemma A.2 for the underlying analytic constraint. 24 A.6 Related Work See Refs. [ 26 , 12 , 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. A.7 Philosophical Implications The law supports relational entropy: observer disagreements signal frame misalignment, not 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. A.8 Deriving τPage from Spectral Gaps With smallest modular gap λmin , τPage ∼λ−1 min . For a Schwarzschild black hole of mass M , τPage ∼M3 , reproducing the expected semiclassical scaling of entropy-recovery timescales. A.9 Asymptotic Boundary Clause As τ→τevap ≳τRH , one of the following must occur: (1) γ ( τ ) → 0; (2) Smax ( τ ) → 0; or (3) ⟨Tretrieval µν ⟩ becomes dynamically significant, breaking fixed-background validity. This defines the operational boundary of the ODER framework. [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 Gµν = 8πGTHawking µν +Tretrieval µν  self-consistently, which is beyond the present scope. Empirically, deviations from the predicted tanh envelope at late times would manifest as excess curvature in measured g(2) traces, marking entry into regime (3). A.10 Spectral Convergence and Uniqueness Theorem A.2 (Spectral-Convergence Constraint). Let the split-regularized modular Hamiltonian satisfy σ ( K ) ⊂ [ − Λ , Λ]. Let F ( τ ) = Sretr ( τ ) /Smax be C1 , strictly increasing, entire, and of exponential type ≤ Λ. Then, up to an affine reparameterization, F(τ) = tanh πΛτ/2. Thus Eq. (9) is the only spectrum-compatible onset within the Paley–Wiener admissible class. □ 25 Interference-bound derivation. Let Fi ( τ )and Fj ( τ )be normalized retrieval profiles for observers i and j with modular-spectrum cutoffs Λ i and Λ j , respectively. From Appendix A.11, each profile satisfies ˙ Fi(τ)≤Λi[1 −Fi(τ)] tanh πΛiτ 2,(21) and analogously for Fj(τ). Assuming synchronized initial conditions, define the mutual-information overlap: Iij(τ) := min Fi(τ), Fj(τ),(22) and the retrieval-overlap tensor: Rij(τ) := Iij(τ) Fi(τ) + Fj(τ)−Iij(τ).(23) Let ∆Λ := |Λi−Λj|. When ∆Λ is finite and onset scales differ, the overlap is bounded by Rij(τ)≤exp h−k∆Λ τ2 τ2 char i, k =O(1),(24) where k is a model-dependent constant determined by the overlap of the respective Paley–Wiener windows. This defines the retrieval-interference bound C(∆Λ, τchar). Null envelope. The null model assumes γ ( τ ) = 0, corresponding to thermal drift or decoherence with no modular retrieval. Baseline envelopes R(null) ij ( τ )are estimated from bootstrap-generated retrieval traces lacking observer-indexed structure. ROC-style comparison. Measured Rij(τ)can be compared to a null envelope via: •True positive: Rij(τ)< R(null) ij (τ)−3σat any τ; •False positive: a null trace misclassified as divergent. A ROC curve is then constructed by varying the detection threshold. Simulated ROC curves used N = 10 3 bootstrap samples; empirical verification would require differential-arm timing precision ≤ 2 ms and fringe stability > 95%. This approach parallels the falsifiability criterion of the g(2) envelope (Appendix A.11, Eq. (A.2′)). Simulation context. Retrieval-overlap dynamics may be explored in tensor-network settings such as MERA. For instance, compare retrieval profiles with spectral cutoffs Λ i = 1 . 0and Λ j = 1 . 3using a shared onset scale τchar; control runs with Λi= Λjyield Rij(τ)→1. 32 Conceptual Differential-Acceleration Interferometer (DAI) Protocol. In a conceptual DAI setup using a BEC analog system: •Stationary and accelerated detector arms are phase-locked at τ= 0; •Each arm samples its local phonon field and extracts g(2)(t1, t2); •Retrieval curves are reconstructed via entropy-constrained filtering; •A thermal null model is generated by disabling modular coupling. These derivations and protocol components clarify what empirical retrieval divergence would look like under the ODER framework. No results are claimed, and no specific experiment or simulation is assumed to have been completed. C.8 Modular Raychaudhuri Equation (Exploratory Extension) Equations (A.2 ′ ) and (C.7.5) jointly motivate this coupling: the same retrieval curvature that bounds modular speed now sources modular expansion. In all preceding sections, retrieval dynamics were modeled under a fixed-background approximation. We now extend the framework by coupling entropy retrieval to spacetime curvature through a modular analogue of the Raychaudhuri equation. This introduces a dynamical back-reaction term sourced by the entropy-convergence profile Sretr ( τ ), allowing retrieval to both track and influence horizon geometry. Modular expansion scalar. Define the modular expansion θmod ( τ )as the divergence of modularflow lines weighted by the retrieval gradient: θmod(τ)≡ ∇µuµ+αdSretr dτ , where uµ is the observer’s four-velocity and α≪ 1is the retrieval-coupling parameter introduced in Section 8. Modular Raychaudhuri equation. The modular analogue of the Raychaudhuri equation takes the form dθmod dτ =−1 2θ2 mod −σµνσµν +ωµνωµν −Rµνuµuν+αd2Sretr dτ2, where σµν and ωµν are the shear and vorticity tensors of the modular-flow congruence, and Rµν is the Ricci tensor. Metric signature follows ( −, + , + , +); Rµνuµuν> 0corresponds to focusing. The term α d2Sretr/dτ2 acts as a retrieval-driven focusing or defocusing force: when retrieval accelerates (positive second derivative) it produces modular expansion; when retrieval saturates or decelerates, curvature focusing dominates. Limiting behavior. In the limit α→ 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. 33 Back-reaction shift. To leading order in α, the retrieval horizon τRH shifts by δτRH ∼αZτRH dτ′ d2Sretr dτ′2, and numerical estimates (Appendix C) indicate this shift remains negligible for M≳M⊙ but may become resolvable in analog systems with boosted retrieval rates. In such analog systems, a non-zero αwould manifest as a slow drift of the measured τRH across successive retrieval cycles. Interpretation. The modular Raychaudhuri equation operationalizes the idea that entropy retrieval is not merely a diagnostic of black-hole evaporation but can itself act as a geometric source. Failure of monotonic convergence in Sretr ( τ )signals not only information-theoretic breakdown but potential modular collapse. In this sense, the retrieval law and the modular flow it induces are not spectators to geometry; they are participants in its evolution. C.9 Modular Focusing and the Retrieval–Curvature Coupling This appendix derives the modular Raychaudhuri equation and defines the coupling between entropy retrieval and modular expansion in more detail, explicitly referencing the retrieval-RG scaling Λ(δ)∼1/δ that underlies the continuum limit. Setup. Let uµ be the observer’s proper-time tangent vector, and let θ ( τ ) = ∇µuµ denote the expansion of the modular flow congruence. The retrieval-coupled expansion scalar is defined as θmod(τ) = θ(τ) + αdSretr dτ . Modular-congruence evolution. Taking the τ-derivative yields dθmod dτ =dθ dτ +αd2Sretr dτ2. Inserting the classical Raychaudhuri equation, dθ dτ =−1 2θ2−σµνσµν +ωµνωµν −Rµνuµuν, we obtain the modular-coupled version dθmod dτ =−1 2θ2 mod −σµνσµν +ωµνωµν −Rµνuµuν+αd2Sretr dτ2+O(α2). The θ2 mod term absorbs the linear α correction to θ , while all other terms remain unaffected at first order. Horizon shift. The retrieval horizon τRH is defined as the proper time at which Sretr ( τ ) →Smax . To leading order in α, δτRH =αZτRH 0 d2Sretr dτ2dτ. This integral can be evaluated analytically for constantγtanh models or numerically using retrieval simulations. 34 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 new class of entropy-coupled geometric dynamics. Where Sretr ( τ )is highly nonlinear—for example, under interference, collapse, or multi-observer divergence— θmod may blow up, indicating modular-horizon instability. This provides a structural falsifier: monotonic retrieval collapse is required to prevent runaway geometric focusing. In this view, modular flow, retrieval dynamics, and curvature evolution form a closed triad: bounded spectra set the law, experiments test it, and geometry responds to it. D Split-Property Regularization and the Type III1Limit Local algebras in algebraic quantum field theory (AQFT) are generically Type III 1 factors: they possess no normal trace and therefore admit no literal density matrix or entropy functional. To formulate observer-dependent entropy retrieval within a physically meaningful regime, we employ the standard split-property regularization used in rigorous AQFT treatments of entropy and modular flow. D.1 Split Inclusion and Bounded Modular Spectrum Following Araki (1976), Doplicher and Longo (1984), Buchholz–D’Antoni–Longo (1987), and Longo (1999), we introduce nested regions O1⊂O2and a Type I intermediate factor: A(O1)⊂ Nδ⊂ A(O2), where the split distance δ defines the physical collar between the inner and outer regions. The modular Hamiltonian Kδgenerated by the state ωon Nδthen has a compact spectrum, σ(Kδ)⊂[−Λ(δ),Λ(δ)] (bounded in operator norm; |Kδ| ≤ Λ(δ)). This provides a well-defined finite entropy and modular flow. Operationally, δ corresponds to the detector’s spatial or temporal resolution, and Λ( δ ) ∼ 1 /δ represents the associated bandwidth limit. 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 Nδ . In the limit δ→ 0,Λ( δ ) → ∞ and the algebra returns to the Type III 1 class. Within the regulated regime, the modular-flow retrieval law derived in the main text (Eq. (9) ) is exact; the Type III 1 limit marks the idealized boundary where observer bandwidth becomes infinite. 35 D.3 Connection to the Retrieval–RG Picture Define the retrieval-renormalization parameter: βΛ(δ) = dln Λ(δ) dln δ. The Type III 1 structure corresponds to the fixed point βΛ→ 0, signaling scale-invariant modular spectra. Finite observers operate at βΛ< 0, where the spectrum is effectively bounded and the tanh retrieval law applies directly. This establishes the renormalization-group analogue of the splitproperty hierarchy: as the split collar narrows, the modular spectrum flows toward the continuum fixed point. This β -function thus provides the algebraic origin of the retrieval-RG flow defined operationally in Section 8. In laboratory analogs, varying detector resolution δdirectly probes this flow: retrieval parameters should approach βΛ≈0as experimental bandwidth increases. D.4 Scope and Open Formal Problems Here ∆ it δ denotes the modular automorphism group generated by Kδ . This regularization does not purport to solve the Type III 1 classification problem; it provides a physically covariant framework in which finite observers are well defined. The continuum Type III 1 limit remains a mathematical frontier. Future work should formalize: 1. the weak-operator convergence ∆it δ→∆it of modular flows; 2. conditions under which isotony and locality persist for the directed family {Nδ}; and 3. quantitative scaling of ρδ(λ)approaching the Type III1fixed point. Bridging these mathematical results with the retrieval-RG framework of Sec. 8 would complete the formal connection between bounded modular spectra and scale-invariant entropy flow. D.5 Physical Meaning of the Type III1Limit As discussed in Appendix A.7, relational entropy interprets this limit not as loss of information but as the restoration of scale-invariant modular access. In algebraic QFT, Type III 1 algebras are the unique structures compatible with relativistic locality and causal propagation: they admit no finite trace and thus no factorization between interior and exterior regions. Within the retrieval-RG picture, increasing observer bandwidth ( δ→ 0) drives the modular spectrum toward scale invariance, the Type III 1 fixed point of bounded-observer modular flow. The emergence of Type III 1 behavior is therefore not a mathematical curiosity but the inevitable continuum limit of finite-observer modular flow. Real detectors operate at finite δ ; the continuum theory represents the unphysical ideal of infinite information access. 36 E Modular Retrieval in Kerr Geometry: Generator Deformation and Spectral Persistence The following analysis generalizes ODER beyond static horizons, testing its stability under rotational frame dragging. E.1 Kerr Geometry and Modular Flow In Kerr spacetime the global timelike Killing vector ∂t is replaced by a stationary, non-static modular generator, χµ=∂t+ ΩH∂ϕ, where Ω H is the horizon angular velocity. Modular flow follows the mixed time–angle trajectory generated by χµ ; an observer therefore does not evolve on a globally synchronized slice. The modular Hamiltonian Kχ associated with this generator defines observer-adapted modular flow consistent with the split-property regularization described in Appendix D. E.2 Modular-Generator Deformation Because the modular Hamiltonian depends linearly on the generator, the deformation ( ∂t→χµ ) preserves the Paley–Wiener class of admissible flows. Anchoring the causal diamond to χµ yields a Kerr-corrected retrieval rate, γ(τ, a, ΩH) = |gµνχµχν|−1/2, which captures frame dragging and horizon-synchronous motion. In rotating BEC analogs, frame dragging corresponds to azimuthal phonon flow; measuring the resulting g(2) phase shift tests Eq. (25). E.3 Survival of the tanh Onset For observers outside the ergoregion ( r > rerg ) the modular spectrum remains bounded after split-inclusion regularization. The Paley–Wiener conditions therefore still hold, and the retrieval law, dSretr dτ =γ(τ, a, ΩH)[Smax −Sretr(τ)] tanh τ/τchar,(25) retains its form; rotation deforms the horizon but does not disrupt modular convergence. This persistence confirms that the tanh onset derived in Appendix A.2 is not restricted to static geometries. Units adopt G = c = ℏ = 1 and metric signature ( −, + , + , +); under this convention, |gµνχµχν| is positive outside the horizon. E.4 Superradiance and Spectral Containment Superradiant amplification in Kerr is energy dependent and frame relative. Modular spectral weight stays bounded provided (i) the observer remains outside the ergosphere and (ii) detector resolution 37 imposes a UV cutoff (Appendix A.3). Under these conditions the retrieval wedge remains modularly coherent, and the Paley–Wiener analyticity domain remains intact. E.5 Interpretation and Consequences • The tanh onset is not an artifact of Schwarzschild symmetry; it is a universal feature of bounded modular spectra. • Modular retrieval is geometrically robust: Kerr rotation modulates γ ( τ )but preserves spectral convergence. • The retrieval law is covariant under generator deformation and applies to rotating observers within the regular wedge class. Conclusion. Modular retrieval survives Kerr rotation. Persistence of Eq. (25) under generator deformation supports the interpretation that ODER encodes a genuine geometric information dynamic rather than a curve-fitting construct. Consequently, modular retrieval defines a covariant information-flow principle valid across all stationary spacetimes with bounded spectra. Together with Appendices A–D, this establishes that bounded modular flow and its retrieval law remain valid across static and rotating geometries, reinforcing ODER’s status as a covariant, observer-dependent entropy principle. F Interpretive Correspondence (Non-Essential) Readers seeking only the formal modular derivations may skip this appendix. It translates ODER’s algebraic parameters into gravitational and holographic language for conceptual cross-reference. All quantities retain their definitions from Appendices A and D; the equations below are interpretive analogies, not additional postulates. Although the ODER retrieval law is derived entirely from observer-dependent modular flow, several of its structural parameters parallel gravitational constructs familiar from wedge-based approaches to the black-hole information problem. The correspondences below serve as interpretive aids for readers who work primarily with holography or extremal-surface reconstruction. F.1 Bandwidth and Algebraic Context The variables δ and Λ( δ )introduced in Appendix D connect the modular-algebraic description to measurable observer parameters. Finite δ defines the retrievable subalgebra Nδ with spectral cutoff Λ( δ ) ∼ 1 /δ ; the continuum limit δ→ 0recovers the Type III 1 structure 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. 38 F.2 Interpretive Parameter Correspondence Each correspondence below references the bounded-spectrum formalism of Appendix A.11 and the retrieval-RG scaling of Sec. 8. • ∆ fail (failure gap). In ODER,∆ fail ≡τevap−τRH represents the late-stage failure of entanglementwedge reconstruction, where extremal surfaces no longer support modular access for the observer’s causal patch. Mathematically this corresponds to the vanishing of the second derivative d2Sretr/dτ2 at the retrieval horizon τRH .ODER treats this as a retrieval-saturation condition—a collapse of spectral access governed by observer-specific modular flow rather than by global extremal anchoring. •τchar (convergence time). The modular convergence scale that marks the start of retrieval serves as a spectrally modulated scrambling threshold. Conventional scrambling time signals full entanglement redistribution, whereas τchar emerges from bounded modular flow and captures observer-relative retrieval activation even when causal connectivity exists but modular access is still suppressed. Experimentally it governs the width of the measured g(2) envelope (Sec. 4.1). •γ ( τ )(retrieval operator). Derived from the entropy trace, γ ( τ )measures the local modular pressure—the instantaneous rate at which retrievable entropy moves toward saturation. Within the modular algebra, γ ( τ )appears as the local generator of the positive-energy semigroup for the observer’s subalgebra Nδ . A gravitational analogue would be a time-dependent coupling between boundary modular flow and evolving bulk extremal surfaces. Because γ ( τ )varies smoothly with both trajectory and state, it serves as an information-theoretic redshift gradient tied to 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 intuitive parallels, but they are projections of the same underlying modular dynamics rather than foundations. A full gravitational embedding is deferred to future work and is included here only to aid conceptual translation. 39 F.3 Extended Variable Glossary (Table F.1) Symbol Description Interpretation Experimental / Simulation Proxy δ Split-property collar / detector resolution Physical bandwidth of the observer; defines the retrievable subalgebra — Λ(δ) Modular spectral cutoff Inverse bandwidth (Λ ∼ 1 /δ ); controls the tanh prefactor and approach to the Type III1limit — τchar Convergence (activation) time Onset scale of modular retrieval; analog of a scrambling threshold Width of g(2) transition ∆fail Retrieval-failure gap Proper-time delay between evaporation and saturation limits Duration between entropy plateau and nullenvelope flattening γ(τ) Retrieval-rate operator Local modular pressure or information-flux density Slope of entropy-retrieval trace F.4 Cross-Domain Interpretive Map (Table F.2) Term Physics Interpretation Algebra / AQFT Interpretation Modular flow Local observer evolution in proper time; defines trajectory of modular access Tomita–Takesaki automorphism on A ( O )governing modular evolution Retrieval horizon Boundary of the decodable information wedge for a given observer Support boundary of the retrievable subalgebra Nδ Type III 1 fixed point Scale-invariant modular spectrum; continuum limit of retrieval flow Non-factorizable local algebra; no trace; required by Haag–Kastler locality axioms Summary. 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