Abstract

A single hash trial in proof-of-work mining is memoryless. This paper proves that the Nakamoto protocol is not. Difficulty adjustment updates the block-arrival rate as a deterministic function of past block times, making the cross-epoch process a piecewise-exponential regime-switching process. Coinbase maturity, fee accumulation, and heaviest-chain selection independently make realised payoffs history-dependent. Applied to Budish (2025), the correction replaces the scalar sufficient statistic (flow expenditure) with a state vector; Budish’s equilibrium constraint survives but his zero-net-cost attack benchmark becomes local to a single fee-free epoch. Multi-epoch attacks incur strictly positive protocol-embedded costs that are deterministic, publicly observable, and grow with fees. Projections under sustained hash-rate growth show that the validity horizon of the Poisson approximation for hash trials depends on transaction throughput: each transaction generates a fresh block template, extending the effective hash domain. A throughput-constrained protocol (~4 TPS) reaches the Poisson breakdown by ~2061; a high-throughput protocol (~109 TPS) extends validity to ~2013. The stochastic foundations of proof-of-work security models are functions of protocol design.

Keywords: Bitcoin, Memoryless property, Proof-of-work, Difficulty adjustment, Regimeswitching process, Blockchain economics