Let I ( t ) denote the mutual information between boundary subsystems at time t . We define the negative information flux as

where Φ − ( t ) > 0 indicates a net decrease in boundary correlations. This reduction corresponds to the effective transfer of information into the bulk degrees of freedom, consistent with the holographic principle.

For reproducibility, we specify the analytical setup used in our entanglement-first-law derivation. We consider a ( 1 + 1 ) -dimensional conformal field theory (CFT) toy model with boundary subregions A and B , chosen as adjacent intervals of equal length. The reference state is the vacuum of the CFT, and perturbations are introduced at first order in the coupling parameter λ . The entanglement entropy variation δ S A is then related to the expectation value of the modular Hamiltonian δ 〈 H A 〉 , consistent with the first law of entanglement entropy:

This setup provides the foundation for deriving the linearized Einstein equations from boundary entanglement dynamics.

We define the negative information flux Q ( t ) operationally as the rate of decrease of mutual information between two boundary subregions A and B :

where I ( A : B ; t ) is the mutual information at time t . The subsystems A and B are chosen as spatially separated intervals, and the dynamics are generated by a local quench in the CFT Hamiltonian. A nonzero Q ( t ) indicates that boundary correlations are being transferred into bulk degrees of freedom, consistent with holographic encoding.

The connection between entanglement entropy and spacetime geometry has been firmly established in holographic duality [2]. The Ryu-Takayanagi formula [7] provides a quantitative relation between boundary entanglement entropy and minimal surfaces in AdS spacetime, while its covariant generalization [8] extends this framework to dynamical settings. These results suggest that entanglement entropy serves as a fundamental building block of emergent geometry. Jacobson’s thermodynamic derivation of Einstein’s equations [6] further supports the idea that gravitational dynamics can be understood as an equation of state, linking entropy variations to spacetime curvature. Recent tensor network approaches [4] and quantum error correction perspectives [3][9] reinforce the robustness of this entanglement-geometry correspondence, ensuring that bulk information remains stable against local perturbations. Informational nonequilibrium processes [5] have also been proposed as a driver of dynamical evolution, providing a unifying perspective that connects entanglement entropy, thermodynamics, and spacetime dynamics.

Negative information flux can be interpreted as a stabilizing mechanism for spacetime. By reducing boundary correlations, the holographic code redistributes information into redundant bulk degrees of freedom, thereby enhancing resilience against local perturbations. This mechanism aligns with the view that spacetime functions as a quantum error-correcting code [3][9].

Black holes provide a natural testing ground for the holographic principle and the information-theoretic interpretation of spacetime. The entropy of a black hole, as given by the Bekenstein-Hawking formula, is proportional to the area of its event horizon rather than its volume [6]-[8]. This observation strongly suggests that spacetime degrees of freedom are encoded holographically on a lower-dimensional boundary. Within the AdS/CFT correspondence, black hole entropy can be understood as entanglement entropy in the boundary theory [7][8]. This perspective aligns with the view that spacetime functions as a quantum error-correcting code, where bulk information is redundantly encoded in boundary correlations [3][9]. The robustness of this encoding provides a resolution to the black hole information paradox, ensuring that information is not lost but rather redistributed in entanglement structures. Radium atomic clocks offer a novel experimental pathway to probe these ideas. By analyzing synchronization drifts and noise spectra between spatially separated clocks, one may detect signatures of Planck-scale fluctuations. Such fluctuations can be interpreted as holographic noise, analogous to the microscopic degrees of freedom responsible for black hole entropy. In this way, precision clock networks serve as experimental analogues to black hole horizons, providing empirical insight into the entanglement-based origin of spacetime.

The presence of negative information flux suggests that gravitational dynamics are inherently non-equilibrium processes. Unlike classical thermodynamic fluxes, which dissipate energy, informational fluxes redistribute correlations across the holographic lattice. This provides a natural explanation for ultraviolet cutoffs and supports the hypothesis that singularities are avoided through intrinsic error-correcting structures.

Starting from the linearized Einstein equation obtained via entanglement entropy consistency [7][8]:

we incorporate corrections motivated by quantum error correction and informational nonequilibrium processes [3][9]. These modifications ensure stability against local perturbations and provide a pathway toward singularity-free gravity.

The correction term can be interpreted as an effective stress-energy contribution:

For Φ − > 0 , the effective pressure becomes negative, leading to repulsive gravitational effects.

Consider a spherically symmetric perturbation around flat space. The line element is given by

where α ∝ Φ − encodes the repulsive correction. This additional term counteracts the attractive G M / r potential, providing a natural mechanism for singularity avoidance.

The repulsive metric arises as a direct consequence of negative information flux. Boundary correlations lost to the bulk are redistributed into redundant degrees of freedom, which manifest geometrically as a repulsive contribution to the metric. This mechanism aligns with the hypothesis that spacetime stability is maintained by quantum error-correcting structures [3][9].

Radium isotopes exhibit narrow optical transitions that can be stabilized to better than 10⁻¹⁸ fractional frequency uncertainty. This level of precision enables detection of Planck-scale fluctuations in spacetime geometry, which would otherwise be hidden within instrumental noise. We focus exclusively on the temporal resolution provided by radium clocks as a direct probe of holographic fluctuations.

The expected observable is the fractional frequency drift Δ f / f between spatially separated radium clock stations. The signal model predicts a broadband spectrum with amplitude scaling proportional to the negative information flux Q ( t ) :

where α is a dimensionless coupling constant. The relevant frequency band is 10⁻³ - 10² Hz, overlapping with the sensitivity range of optical lattice clocks. Order-of-magnitude estimates suggest that holographic fluctuations produce drifts at the level of 10⁻¹⁹, distinguishable from intrinsic clock noise at 10⁻¹⁸.

We propose a measurement protocol with the following specifications:

Stationseparation: 100 - 500 km to ensure sensitivity to global holographic fluctuations. Synchronizationmethod: GPS-disciplined transfer combined with optical fiber links for sub-nanosecond accuracy. Samplingcadence: 1 Hz sampling rate, sufficient to resolve fluctuations in the target frequency band. Cross-correlationstatistic: The normalized correlation coefficient C ( τ ) between drift signals at different stations, used to suppress local instrumental noise and isolate global holographic signatures.

This protocol provides a concrete pathway to evaluate the feasibility of detecting Planck-scale holographic noise using radium atomic clock networks.

We propose an experimental framework in which radium atomic clocks are integrated into AIoT-based data acquisition systems. Each clock node reports time synchronization drift through standardized device interfaces, allowing cross-correlation across geographically separated stations. To manage and analyze this distributed dataset, we employ scalable computational resources for real-time correlation analysis and machine learning-based noise classification.

By correlating timing signals from multiple radium clock stations, one can suppress local noise sources and isolate global fluctuations consistent with holographic nonequilibrium. The expected signal is a broadband spectrum with amplitude scaling proportional to the negative information flux Q ( t ) introduced in Section II. This provides a direct experimental pathway to test the theoretical framework (Figure 1Figure 1).

Figure 1.Bar chart showing dominant peaks at indices 11 and 12 compared to 10 and 13.

The application of radium atomic clocks establishes a robust experimental program for probing the information-theoretic foundations of emergent gravity. The synergy between precision timekeeping, AIoT integration, and advanced computational analysis may ultimately provide empirical evidence for the role of quantum error correction in stabilizing spacetime.

Here we provide the detailed derivation of the metric correction term. Starting from the Schwarzschild solution:

we introduce an additional contribution motivated by informational flux:

Thus, the modified line element becomes:

This α / r 2 term acts as a repulsive correction, stabilizing spacetime in the Planck epoch.

All OpenQASM circuits used in this study are listed below. The following example shows a two-qubit entanglement circuit. Simulation parameters included 8192 shots and standard IBM Quantum noise models.

Table A1 summarizes entropy-induced time delay measurements across multiple trials. These values complement the averaged results presented in the main text. The data show that larger entropy changes Δ S correspond to longer time delays Δ t , consistent with the theoretical relation:

Table A1.Entropy-induced time delay measurements across multiple trials.

Error rates remain low, confirming reproducibility of the IBM Quantum simulations.

The entropy of a black hole is given by the Bekenstein-Hawking formula:

where A is the area of the event horizon, G is Newton’s gravitational constant, c is the speed of light, ℏ is the reduced Planck constant, and k B is Boltzmann’s constant.

A.4.1. Derivation Outline

Starting from the first law of black hole thermodynamics:

where M is the mass, J the angular momentum, Q the charge, Ω the angular velocity, and Φ the electrostatic potential. For a non-rotating, uncharged Schwarzschild black hole, this reduces to:

The Hawking temperature is given by:

Integrating with respect to M , and using the relation between horizon area and mass:

we obtain the entropy as proportional to the horizon area:

A.4.2. Interpretation

This result demonstrates that black hole entropy scales with the area of the horizon rather than the volume, supporting the holographic principle. In the context of our work, this area-law behavior is directly connected to entanglement entropy and the quantum error-correcting code interpretation of spacetime.

A.4.3. Connection to Radium Atomic Clock Experiments

The holographic principle, exemplified by black hole entropy, suggests that spacetime information is encoded on boundaries rather than volumes. Radium atomic clocks provide a complementary experimental probe: synchronization drifts and noise spectra between spatially separated clocks can reveal Planck-scale fluctuations in spacetime geometry. These fluctuations are analogous to the microscopic degrees of freedom responsible for black hole entropy. Thus, precision clock networks act as laboratory analogues to black hole horizons, allowing empirical investigation of entanglement-based information storage and redistribution in spacetime.