The Role of Physical Data in Facilitating Wave Function Collapse

Entropy-Triggered Hypothesis: Extended Theoretical Framework and Experimental Proposals Incorporating Superconducting Qubits, Ion Traps, and Machine Learning Approaches Takao Koizumi February 1, 2025 Abstract This paper presents an extended formulation of the Entropy-Triggered Hypothesis (ET Hypothesis) for wave function collapse, refining previous models by integrating dynamic feedback mechanisms, quantum error correction (QEC), and machine learning approaches. We hypothesize that wave function collapse occurs when the environmental von Neumann entropy S(t)surpasses a critical threshold Scrit, which is dynamically influenced by system-environment interactions, quantum coherence control, and entropy suppression mechanisms. A generalized nonlinear Lindblad-type master equation is proposed, incorporating time-dependent collapse rates and energy-scale-dependent adjustments. Experimental validation strategies include: 1. Superconducting Qubits (SCQs) – Examining QEC’s role in delaying collapse events. 2. Trapped Ions – High-fidelity state preparation to track entropy-driven collapse. 3. Machine Learning – Predicting entropy growth and collapse onset via neural networks. Additionally, a simplified experimental framework is outlined, offering an initial proof-of-principle test with minimal resources. Comparisons with GRW collapse models, decoherence theories, and the Many-Worlds Interpretation (MWI) highlight the testable predictions unique to the ET Hypothesis. Contents 1 Introduction 3 1 1.1 Motivation and Overview of the ET Hypothesis . . . . . . . . . . . . . . . . 3 1.2 Paper Structure and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Theoretical Framework: Entropy, Thresholds, and Collapse 4 2.1 Defining the Entropy Threshold for Collapse . . . . . . . . . . . . . . . . . . 4 2.2 Generalized Nonlinear Lindblad Equation . . . . . . . . . . . . . . . . . . . . 5 2.3 Time-Dependent Collapse Rates and Energy-Scale Dependence . . . . . . . . 6 3 Integration with Quantum Error Correction (QEC) 7 3.1 QEC-Induced Delay in Collapse Events . . . . . . . . . . . . . . . . . . . . . 7 3.2 Threshold Adjustments and Feedback Mechanisms . . . . . . . . . . . . . . . 8 3.3 Prolonged Entanglement and Fidelity Retention . . . . . . . . . . . . . . . . 8 3.4 Experimental Verification: Comparing QEC-Enhanced and Non-QEC Systems 9 4 Experimental Proposals 10 4.1 Superconducting Qubit Experiments . . . . . . . . . . . . . . . . . . . . . . 10 4.2 Trapped Ion Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.3 Machine Learning for Predictive Modeling . . . . . . . . . . . . . . . . . . . 11 4.4 Distinguishing ET Hypothesis from Alternative Models . . . . . . . . . . . . 12 4.5 Simplified Experimental Design for Initial Validation . . . . . . . . . . . . . 12 5 Comparison with Competing Theories 13 5.1 GRW Model: Spontaneous Collapse . . . . . . . . . . . . . . . . . . . . . . . 13 5.2 Standard Decoherence Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.3 Many-Worlds Interpretation (MWI) . . . . . . . . . . . . . . . . . . . . . . . 14 5.4 Summary of Predictions and Experimental Tests . . . . . . . . . . . . . . . . 14 6 Conclusion and Outlook 14 6.1 Key Insights from the ET Hypothesis . . . . . . . . . . . . . . . . . . . . . . 15 6.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 6.3 BroaderImplications ............................... 17 6.4 FinalRemarks................................... 17 7 Experimental Proposal Using Superconducting Qubits, Ion Traps, and Machine Learning 17 7.1 ExperimentalPlatforms ............................. 18 7.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 7.3 PredictedOutcomes ............................... 19 7.4 Significance and Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 7.5 FinalRemarks................................... 20 7.6 Additional Comparison Tables (Full Grid Style) . . . . . . . . . . . . . . . . . 21 2 8. References 21 1 Introduction 1.1 Motivation and Overview of the ET Hypothesis The measurement problem in quantum mechanics remains an unresolved issue, with multiple competing interpretations attempting to explain wave function collapse. While spontaneous collapse models (e.g., GRW), decoherence theories, and the Many-Worlds Interpretation (MWI) offer different perspectives, no single theory has gained universal acceptance. The Entropy-Triggered Hypothesis (ET Hypothesis) proposes that wave function collapse is driven by the system’s environmental entropy surpassing a critical threshold Scrit, rather than being an intrinsic stochastic event or requiring an external observer: S(t)≥Scrit where: •S(t)is the von Neumann entropy of the system’s environment. •Scrit is a threshold dependent on environmental degrees of freedom, coupling strength, and quantum error correction (QEC) mechanisms. This framework provides a quantitative criterion for wave function collapse, distinguishing it from: 1. GRW Model – which assumes a universal stochastic collapse rate. 2. MWI – which rejects collapse and assumes continuous branching of the wave function. 3. Decoherence Theory – which does not define a specific collapse mechanism. Furthermore, QEC mechanisms introduce a tunable parameter that modifies Scrit, potentially delaying collapse and extending quantum coherence. This aspect is particularly relevant to quantum computing, where prolonged coherence is essential for fault-tolerant computation. 1.2 Paper Structure and Objectives This paper aims to: •Formally define entropy-based collapse mechanisms using generalized nonlinear Lindblad equations. •Explore the interplay between environmental entropy, QEC, and collapse onset, introducing time-dependent collapse rates and energy-dependent effects. 3 •Propose experimental tests using superconducting qubits, trapped ions, and machine learning-assisted data analysis. •Introduce a simplified experimental setup for preliminary validation with minimal resources. •Compare the ET Hypothesis with competing interpretations, including GRW, decoherence models, and MWI. Paper Structure: •Section 2 develops the mathematical framework for entropy-triggered collapse. •Section 3 explores QEC’s role in modifying collapse thresholds and extending coherence. •Section 4 presents experimental validation strategies using superconducting qubits, trapped ions, and machine learning. •Section 5 introduces a simplified, cost-effective experimental setup. •Section 6 compares the ET Hypothesis with GRW, decoherence, and MWI. •Section 7 concludes with a discussion of future research directions. 2 Theoretical Framework: Entropy, Thresholds, and Collapse 2.1 Defining the Entropy Threshold for Collapse The ET Hypothesis proposes that wave function collapse occurs when the environmental entropy S(t)reaches a critical threshold Scrit. This threshold is influenced by multiple factors, including: •Quantum Error Correction (QEC) mechanisms, which suppress entropy growth. •System-environment coupling strength g, which determines how quickly coherence is lost. •Number of environmental degrees of freedom N, which influences entropy accumulation. •Temperature T, which affects the rate of decoherence. 4 Mathematically, the entropy threshold is expressed as: Scrit =Scrit,0+ζEcorr(d) + ξN N0α gβ+γT where: •Scrit,0is the baseline entropy threshold in the absence of quantum error correction. •Ecorr(d)∼d2represents QEC overhead, where dis the code distance in error correction. •ζ, ξ, γ are experimentally determined parameters. •N0is a normalization factor for the degrees of freedom. Predictions and Implications: •If QEC increases Scrit, collapse should be delayed or suppressed. •A system with strong QEC should exhibit longer coherence times before collapse. •If collapse is purely stochastic (as in GRW models), Scrit should have no impact. 2.2 Generalized Nonlinear Lindblad Equation To mathematically describe entropy-triggered collapse, we introduce a nonlinear Lindbladtype master equation: dρ dt =−i[H, ρ]−Γ(t)D[ρ] where: •His the system Hamiltonian. •Γ(t)is the collapse rate, which becomes nonzero when entropy surpasses Scrit. •D[ρ]represents dissipative effects due to collapse. The collapse rate Γ(t)is defined as: Γ(t) =    0, S(t)< Scrit, γ0(t)S(t)−Scrit Scrit n1 + ϵ1 1+e−λ(S(t)−Scrit), S(t)≥Scrit where: •γ0(t) = γ0,init +γ0,env(t)introduces time-dependent collapse rates. •λ∼1/τQEC models QEC response time, correlating with noise filtering. •The logistic function 1 1+e−λ(S(t)−Scrit)ensures a smooth transition in the collapse rate. 5 Key Implications: •Wave function collapse should occur precisely when S(t)≥Scrit, unlike spontaneous collapse models. •If QEC is applied, Scrit increases, which should delay collapse. •The logistic function ensures that collapse is not instantaneous but follows a smooth transition. 2.3 Time-Dependent Collapse Rates and Energy-Scale Dependence The environmental entropy S(t)evolves according to: dS(t) dt =κ(Smax −S(t)) −ηEcorr(d) d0−µN N0α gβh(T) where: •Smax is the maximum entropy of the environment. •ηEcorr(d) d0represents entropy suppression via QEC. •µN N0αgβh(T)accounts for entropy accumulation due to environmental interactions. Predictions: •Systems with strong QEC should show a slower entropy growth rate. •If S(t)oscillates around Scrit, partial collapses may occur, delaying full collapse. •A purely stochastic collapse model would not predict such behavior. 2.4 Experimental Feasibility: Parameter Estimation and Machine Learning Approaches To extract and refine Scrit, λ, and other model parameters from experimental data, we propose: 1. Bayesian Inference for Parameter Estimation •Estimate prior distributions for Scrit, λ and update using experimental data. •Determine uncertainty bounds for model parameters based on observed data. 2. Machine Learning for Predictive Collapse Detection •Train deep learning models (e.g., LSTMs, CNNs) on entropy evolution data. •Predict when S(t)will reach Scrit using real-time quantum state tomography. •Identify hidden entropy dynamics affecting quantum states. 6 Advantages: •AI-assisted entropy tracking allows early detection of collapse events. •Combining machine learning with Bayesian inference enhances model precision. Summary of Section 2 •Wave function collapse occurs when entropy reaches a critical threshold Scrit. •A nonlinear Lindblad equation models entropy-driven collapse. •Entropy growth follows a feedback-driven equation influenced by QEC. •Machine learning and Bayesian inference provide powerful tools for collapse prediction. 3 Integration with Quantum Error Correction (QEC) Quantum error correction (QEC) is a crucial component in mitigating decoherence and extending the coherence times of quantum systems. Within the ET Hypothesis, we propose that QEC modifies the entropy threshold Scrit, thereby delaying or suppressing wave function collapse. This section explores: 1. How QEC increases Scrit, effectively postponing collapse. 2. How QEC-induced feedback mechanisms influence collapse rates. 3. Experimental strategies to test the interplay between QEC and entropy-driven collapse. 3.1 QEC-Induced Delay in Collapse Events We extend the entropy threshold model to explicitly incorporate QEC efficiency: Scrit =Scrit,0+ζEcorr(d) + ξN N0α gβ+γT where: •Scrit,0is the baseline entropy threshold in the absence of QEC. •Ecorr(d)∼d2represents QEC overhead, where dis the code distance. •ζ, ξ, γ are experimentally determined parameters. •N0is a normalization factor for environmental degrees of freedom. 7 Predictions and Implications: •If QEC increases Scrit, wave function collapse should be delayed. •A system with strong QEC should exhibit longer coherence times before collapse. •If collapse is purely stochastic (as in GRW models), Scrit should have no impact. Experimental Test: •Varying din surface codes (e.g., d= 3,5,7) should systematically shift Scrit. •Measuring collapse times across different dvalues can confirm entropy-driven suppression. 3.2 Threshold Adjustments and Feedback Mechanisms Incorporating QEC-induced suppression into the Lindblad equation, the modified collapse rate becomes: Γ(t) = γ0(t)S(t)−Scrit Scrit n 1 + ϵe−ξEcorr (d)Θ(S(t)−Scrit). where: •e−ξEcorr (d)ensures exponential suppression of collapse with increasing QEC resources. •QEC actively increases Scrit, delaying the onset of collapse. •If experiments show a systematic delay in collapse with increasing d, it strongly supports the ET Hypothesis. Predictions: •Without QEC, collapse occurs immediately when S(t)≥Scrit,0. •With QEC, collapse should be postponed, scaling with Ecorr(d). •For large d, systems may maintain coherence indefinitely if Scrit remains unattainable. 3.3 Prolonged Entanglement and Fidelity Retention A direct experimental consequence of QEC-modulated collapse delay is enhanced entanglement fidelity. Without QEC: F(t) = e−t/T2. 8 With QEC: F(t) = e−t/(T2+αEcorr(d)). •Ecorr(d)contributes to suppressing decoherence, thus extending T2. •This model predicts that entanglement lifetimes should increase systematically with larger d. Experimental Test: •Compare entanglement fidelity decay with and without QEC. •If QEC enhances T2, it implies entropy suppression influences coherence. 3.4 Experimental Verification: Comparing QEC-Enhanced and NonQEC Systems To validate the role of QEC in suppressing collapse, we propose: 1. Comparing Quantum Systems with and without QEC Track entropy growth and collapse timing in both cases. Determine if QEC extends coherence time and delays collapse. 2. Varying Code Distance dto Observe Entropy Threshold Shifts If Scrit systematically increases with d, it strongly supports the ET Hypothesis. Measure collapse onset times across different dvalues. 3. Machine Learning-Assisted Collapse Detection Use AI models trained on QECmodulated entropy data to predict collapse events. Employ Bayesian inference to refine Scrit estimates from experimental data. Summary of Section 3 •QEC increases Scrit, delaying wave function collapse. •The modified Lindblad equation incorporates entropy suppression effects. •QEC-enhanced coherence times provide a direct experimental test of the ET Hypothesis. •Experimental validation can be achieved via superconducting qubits, trapped ions, and AI-assisted entropy estimation. 9 1. Real-Time Control of the Collapse Threshold Can we actively manipulate Scrit in real time by adjusting: •Environmental parameters (e.g., noise injection, thermal fluctuations). •Quantum error correction overhead (e.g., code distance d). →If so, collapse could become an engineered process rather than a fundamental law. 2. Application to Cosmology and Black Hole Information Loss Can entropy-driven collapse mechanisms be extended to: •Gravitational decoherence in black holes? •Quantum information loss in the early universe? →If Scrit is modified by gravity, this could offer new insights into quantum gravity. 3. Scaling to Large Quantum Systems •Can error-protected quantum computers remain in superposition indefinitely? •Does the collapse mechanism break down at macroscopic scales? →Understanding entropy-driven collapse in large-scale quantum processors is crucial for fault-tolerant quantum computing. 4. Machine Learning for Entropy Estimation By integrating neural networks and Bayesian inference, we could: •Predict collapse events with higher precision. •Identify hidden entropy dynamics affecting quantum states. →AI-enhanced entropy modeling could revolutionize quantum state monitoring. 5. Testing with Optical and Atomic Systems Future experiments could use: •Bose-Einstein condensates to test large-scale quantum coherence. •Optical interferometers to observe entropy-driven decoherence. →Does entropy-based collapse hold across different physical platforms? 16 6.3 Broader Implications 1. Quantum Computing If QEC systematically delays collapse: •New error correction strategies could be designed to extend coherence times in quantum processors. 2. Interpretations of Quantum Mechanics •If confirmed, entropy-driven collapse challenges Many-Worlds (MWI) and GRW theories. •This framework unifies entropy, information theory, and quantum measurement. 3. Quantum-Classical Transition •The ET Hypothesis provides a physically measurable criterion Scrit for the boundary between quantum and classical worlds. 6.4 Final Remarks By rigorously testing the entropy-driven nature of wave function collapse, this research offers a novel framework for understanding quantum measurement and the quantum-to-classical transition. Future experiments will determine whether the ET Hypothesis is a viable alternative to existing interpretations—or requires further refinement. 7 Experimental Proposal Using Superconducting Qubits, Ion Traps, and Machine Learning To validate the Entropy-Triggered Hypothesis (ET Hypothesis), we propose a set of experiments designed to test entropy-dependent wave function collapse. The core objectives are: 1. Confirm Entropy-Driven Collapse Demonstrate that wave function collapse occurs when the environmental entropy S(t)surpasses a critical threshold Scrit. 2. Test the Influence of Quantum Error Correction (QEC) Apply different levels of QEC (code distances d= 3,5,7) and observe whether stronger error correction raises Scrit or delays collapse. 17 3. Use Machine Learning for Entropy Prediction Train AI models to infer entropy growth trends and predict collapse events. 7.1 Experimental Platforms We consider three primary platforms for experimental validation: 1. Superconducting Qubits (SCQs) •Setup: IBM Quantum, Google Sycamore, or Rigetti Aspen. •Advantages: –Precise control over noise injection and QEC. –Large-scale multi-qubit experiments possible. •Prediction: –If collapse is entropy-triggered, collapse timing should correlate with entropy growth, and QEC should systematically delay collapse. 2. Trapped Ions •Setup: Honeywell, IonQ, or university-based ion traps. •Advantages: –Long coherence times for tracking entropy growth. –High-fidelity state preparation for controlled collapse experiments. •Prediction: –If S(t)≥Scrit triggers collapse and QEC raises Scrit, it supports the hypothesis. 3. Optical Interferometers •Setup: Photon pairs in a decohering bath. •Advantages: –Highly tunable environment for entropy control. –Precise measurement capabilities. •Prediction: –Interference visibility should drop sharply at Scrit, rather than gradually. 18 7.2 Experimental Procedure Step 1: Initial State Preparation •Prepare n= 5,7,9qubits (or ions) in superposition: |+⟩=|0⟩+|1⟩ √2 •Create entangled states (e.g., GHZ, W, cluster states) to examine how entanglement collapses as entropy increases. Step 2: Controlled Entropy Growth •Introduce controlled noise (e.g., Gaussian or thermal fluctuations) to the environment. •Adjust system-environment coupling gto simulate different decoherence environments. •Track entropy dynamics via partial tomography and infer the entropy growth rate. Step 3: Applying Quantum Error Correction (QEC) •Implement surface-code QEC with different code distances (d= 3,5,7). •Monitor syndrome extraction and correction rates to quantify QEC’s impact on entropy suppression. •Compare collapse times with and without QEC. Step 4: Data Collection and Analysis •Measure collapse time tcollapse and confirm whether it correlates with S(tcollapse)≈Scrit. •Apply machine learning models (e.g., LSTMs, CNNs, Bayesian inference) to predict collapse based on entropy evolution. •Compare results with alternative collapse models (GRW, decoherence-only, ManyWorlds). 7.3 Predicted Outcomes 1. Sudden Collapse at Scrit If wave function collapse occurs precisely when entropy crosses Scrit, it strongly supports the ET Hypothesis. 2. QEC-Induced Collapse Delay If applying QEC systematically raises Scrit, then collapse should be delayed or suppressed. 19 Table 3: Comparisons in Predicted Outcomes (Short Table) Theory Prediction Key Experimental Signature GRW Model Universal collapse rate λindependent of entropy. Collapse should occur randomly, not at an entropy threshold. Decoherence Theory Smooth dephasing of quantum states. No sharp collapse threshold should be observed. Many-Worlds Interpretation (MWI) No collapse; all outcomes persist. Any evidence of singleoutcome selection refutes MWI. ET Hypothesis Collapse occurs at S(t)≥ Scrit, and QEC modifies Scrit. Observing systematic entropytriggered collapse and QECdependent delay would support ET Hypothesis. 3. Comparison with Alternative Theories 7.4 Significance and Implications 1. Quantum Computing Demonstrating that QEC delays collapse could redefine error correction strategies and enhance coherence times in quantum processors. 2. Interpretations of Quantum Mechanics Empirical evidence of an entropy-based collapse threshold challenges GRW, MWI, and decoherence-only models. 3. Practical Quantum Control If entropy-driven collapse can be controlled, it opens pathways for engineering longer-lived quantum states. 7.5 Final Remarks If experiments confirm entropy-triggered collapse, this would: •Provide strong evidence for ET Hypothesis as a viable alternative to existing quantum measurement theories. •Potentially revolutionize quantum computing by enabling entropy-based control of quantum states. 20 Table 4: Comparison of Theories: Basic Predictions vs. Experimental Signature (Full Grid) Theory Prediction Experimental Signature GRW Spontaneous Collapse Collapse occurs randomly, independent of entropy. Collapse should not correlate with Scrit or QEC. Standard Decoherence Smooth loss of coherence over time. No sharp collapse threshold should be observed. Many-Worlds (MWI) No collapse, only branching. Any collapse event contradicts MWI. ET Hypothesis Collapse occurs precisely when S(t)≥Scrit. Detect sharp, threshold-based collapse and QEC-induced delay. 7.6 Additional Comparison Tables (Full Grid Style) 8. References 8.1 Wave Function Collapse and Quantum Measurement 1. Ghirardi, G. C., Rimini, A., & Weber, T. (1986). Unified dynamics for microscopic and macroscopic systems. Physical Review D, 34(2), 470–491. 2. Zurek, W. H. (2003). Decoherence, einselection, and the quantum origins of the classical. Reviews of Modern Physics, 75(3), 715–775. 3. Bassi, A., Lochan, K., Satin, S., Singh, T. P., & Ulbricht, H. (2013). Models of wavefunction collapse, underlying theories, and experimental tests. Reviews of Modern Physics, 85(2), 471–527. 4. Joos, E., & Zeh, H. D. (1985). The emergence of classical properties through interaction with the environment. Zeitschrift für Physik B Condensed Matter, 59(2), 223. 8.2 Quantum Entropy and Information Theory 5. Landauer, R. (1961). Irreversibility and heat generation in the computing process. IBM Journal of Research and Development, 5(3), 183. 6. Schlosshauer, M. (2005). Decoherence and interpretations of quantum mechanics. Reviews of Modern Physics, 76(4), 1267–1305. 7. Donadi, S., et al. (2014). Collapse models and cosmogenic neutrinos. Physical Review Letters, 113(14), 140402. 21 Table 5: Comparison of Theories: Mechanisms, Entropy Role, QEC Effect, and Tests (Full Grid) Theory Collapse Mechanism Role of Entropy Effect of QEC Experimental Test GRW Model Random spontaneous collapse None No effect If QEC delays collapse, GRW is refuted. Decoherence Continuous suppression of coherence No discrete threshold No effect A sharp collapse at Scrit contradicts decoherence. MWI No collapse (unitary evolution) No effect No effect Single-outcome collapse would contradict MWI. ET Hypothesis Collapse occurs when S(t)≥Scrit Entropy threshold exists QEC increases Scrit, delaying collapse Observing entropytriggered collapse and QECinduced shifts supports ET Hypothesis. 8.3 Quantum Error Correction and Entropy Dynamics 8. Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. 9. Preskill, J. (2024). Quantum computing: A new paradigm for cryptography. Quantum Information Science, 30(4), 510–520. 10. DiVincenzo, D. P., et al. (2024). Towards scalable quantum computation: Topological qubits and error correction strategies. Nature Physics, 20(2), 134–143. 8.4 Experimental Quantum Mechanics and Quantum Computing 11. Ringbauer, M., et al. (2021). Experimental delayed-choice quantum eraser. Nature Physics, 17(1), 48–52. 12. Vinante, A., et al. (2017). Ultracold cantilever tests of collapse models. Physical Review Letters, 119(11), 110401. 13. Curceanu, C., et al. (2016). Spontaneously emitted X-rays in collapse models. Foundations of Physics, 46(3), 263–285. 22 Table 6: Comparison of Theories: Universal Rate vs. Smooth Dephasing vs. Branching vs. Entropic Threshold (Full Grid) Theory Prediction Key Experimental Signature GRW Model Universal collapse rate λindependent of entropy. Collapse should occur randomly, not at an entropy threshold. Decoherence Theory Smooth dephasing of quantum states. No sharp collapse threshold should be observed. Many-Worlds Interpretation (MWI) No collapse; all outcomes persist. Any evidence of singleoutcome selection refutes MWI. ET Hypothesis Collapse occurs at S(t)≥ Scrit, and QEC modifies Scrit. Observing systematic entropytriggered collapse and QECdependent delay would support ET Hypothesis. 8.5 Quantum Gravity, Black Holes, and Fundamental Limits 14. Harlow, D. (2024). Black holes, entropy, and the quantum mechanics of information. Annual Review of Condensed Matter Physics, 15(1), 93–120. 15. Takayanagi, T. (2022). Holographic Entanglement Entropy. Lecture Notes, Kyoto University. 16. Penrose, R. (1996). On gravity’s role in quantum state reduction. General Relativity and Gravitation, 28(5), 581. 23