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Entropy-Based Evidence for Bitcoin's Discrete Time Mechanism

Bin Chen, Pan Feng

TL;DR

This paper argues that Bitcoin constructs a verifiable, non-continuous notion of time from probabilistic block discovery and cumulative proof-of-work. By modeling Poisson arrivals via Bernoulli sampling, quantifying interval entropy, and applying a difficulty-feedback mechanism, the authors show stable exponential inter-arrival periods across epochs and that most block discoveries occur while interval entropy remains high. They further reveal that entropy collapse is propagated in a finite, propagation-bounded interval, with forks serving as empirical evidence of distributed completion. The results establish Bitcoin as a self-contained timekeeping system that relies on endogenous, observable events rather than external clocks, with practical implications for synchronization and consensus in permissionless networks.

Abstract

Bitcoin derives a verifiable temporal order from probabilistic block discovery and cumulative proof-of-work rather than from a trusted global clock. We show that block arrivals exhibit stable exponential behavior across difficulty epochs, and that the proof-of-work process maintains a high-entropy search state that collapses discretely upon the discovery of a valid block. This entropy-based interpretation provides a mechanistic account of Bitcoin's non-continuous temporal structure. In a distributed network, however, entropy collapse is not completed instantaneously across all participants. Using empirical observations of temporary forks, we show that collapse completion unfolds over a finite propagation-bounded interval, while remaining rapid in practice.

Entropy-Based Evidence for Bitcoin's Discrete Time Mechanism

TL;DR

This paper argues that Bitcoin constructs a verifiable, non-continuous notion of time from probabilistic block discovery and cumulative proof-of-work. By modeling Poisson arrivals via Bernoulli sampling, quantifying interval entropy, and applying a difficulty-feedback mechanism, the authors show stable exponential inter-arrival periods across epochs and that most block discoveries occur while interval entropy remains high. They further reveal that entropy collapse is propagated in a finite, propagation-bounded interval, with forks serving as empirical evidence of distributed completion. The results establish Bitcoin as a self-contained timekeeping system that relies on endogenous, observable events rather than external clocks, with practical implications for synchronization and consensus in permissionless networks.

Abstract

Bitcoin derives a verifiable temporal order from probabilistic block discovery and cumulative proof-of-work rather than from a trusted global clock. We show that block arrivals exhibit stable exponential behavior across difficulty epochs, and that the proof-of-work process maintains a high-entropy search state that collapses discretely upon the discovery of a valid block. This entropy-based interpretation provides a mechanistic account of Bitcoin's non-continuous temporal structure. In a distributed network, however, entropy collapse is not completed instantaneously across all participants. Using empirical observations of temporary forks, we show that collapse completion unfolds over a finite propagation-bounded interval, while remaining rapid in practice.
Paper Structure (12 sections, 17 equations, 5 figures)

This paper contains 12 sections, 17 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Theoretical Framework for Bitcoin’s Discrete Temporal Dynamics
  3. Bernoulli Sampling, Difficulty, and the Poisson Arrival Limit
  4. Difficulty Feedback for Long-Horizon Stability
  5. Entropy Collapse and the Granularity of Bitcoin Time
  6. Empirical Distribution of Inter-arrival Times
  7. Inter-arrival Durations and Epoch Stability
  8. Serial Independence
  9. Interval-level Entropy and Collapse
  10. Distributed Completion of Entropy Collapse
  11. Implications for Temporal Stability
  12. Conclusion

Figures (5)

  • Figure 1: Histogram of Inter-arrival Intervals with Fitted Exponential density.
  • Figure 2: Epoch-mean Inter-arrival Intervals and Contemporaneous Difficulty.
  • Figure 3: Autocorrelation of Inter-arrival Intervals for Twenty Lags.
  • Figure 4: Histogram of Entropy at Block Discovery
  • Figure 5: Empirical Survival Function of Fork Durations Following Block Discovery.