Four clocks inside
proof of work.

A random search can operate inside a system that keeps track of its past. Looking at the different clocks inside proof of work helps explain what a security model needs to remember, and when a useful approximation reaches its boundary.

Imagine two mining operations with the same computing power and the same current difficulty. The next successful hash may have the same probability in each. Yet one chain may be approaching a difficulty adjustment while the other has just completed one. One operator may hold rewards close to becoming spendable; another may have only just earned them. The immediate search conditions can match while the decisions facing the operators differ.

My published article, Why Hash-Trial Memorylessness Does Not Extend to the Nakamoto Protocol, examines this distinction. The useful starting question is: which part of the system is the model describing? A statement about independent hash attempts has a different scope from a statement about block arrivals, the evolving protocol or the income a miner ultimately receives.

The search clock

At a fixed target, under the usual independent-trial idealisation, a failed hash attempt does not make the next attempt more likely to succeed. There is no accumulated entitlement to a winning result. The geometric distribution describes the number of attempts until success; a constant-rate exponential approximation describes the corresponding waiting time.

This is the precise meaning of memorylessness in this setting. Having already waited does not, by itself, alter the remaining waiting-time distribution. It is stronger than saying that the outcome is uncertain. Many uncertain processes depend on elapsed time or earlier events.

The qualification matters. The target and the rate of hashing are being held fixed. A model that changes either needs to account for that change rather than carrying the same constant-rate assumption across it.

The adjustment clock

The paper next follows Bitcoin's 2,016-block difficulty-adjustment structure. The protocol uses the elapsed time for the preceding interval to set the next difficulty. Earlier block arrivals therefore help determine the conditions of later searches. A sequence of individually random discoveries feeds a rule that changes the next regime.

An exponential approximation can remain useful within an interval whose relevant conditions are stable. Across adjustments, however, one fixed arrival rate no longer describes the whole process. A model needs the current difficulty, the position within the adjustment cycle and the timing information used by the adjustment rule.

This gives the opening example its consequence. Equal current computing power does not make two differently positioned chains interchangeable over a horizon that reaches their next adjustments. The length of the model's horizon becomes part of the economic question.

The reward clock

Discovery and spendable income occur at different times. Coinbase maturity requires a newly created reward to wait through 100 blocks before it can be spent. Whether the associated block remains in the accepted history matters to the reward's eventual realisation.

Fees add another source of change. In the paper's accumulation model, fee-paying transactions arrive while miners search. The value available in a candidate block can consequently change even when no block has yet been found. The simple closed-form analysis assumes a constant fee-arrival rate; actual transaction selection, congestion and fee behaviour require additional modelling.

A miner may therefore face an unchanged chance of success on the next attempt while the economic value attached to success changes. A model of the search and a model of the payoff need to carry different information.

The confirmation clock

A discovered block also sits within a growing chain of work. Confirmation depth and the position of competing branches affect an assessment of whether it will remain part of the accepted history. In the models considered by the paper, different accumulated work gaps can imply different reorganisation risks.

This does not make any particular confirmation count an unconditional guarantee. It makes the state of the competition relevant. The calculation must specify the opposing resources, the strategy, the protocol rules and the period over which success is assessed.

Together, these clocks explain why current mining expenditure alone can leave important parts of the decision unresolved. An expanded account tracks chain-specific difficulty, progress towards retargeting, accumulated work, confirmations and the fee environment.

Choosing the right amount of memory

The paper develops this as a correction to the stochastic description used in an economic security model. It preserves the central incentive question: how does the ongoing cost of honest participation compare with the value available from an attack? Short-horizon, fixed-difficulty calculations can still be useful when their assumptions fit the question.

The practical task is to state what has been simplified and test whether that simplification matters over the chosen horizon. The paper's attack-cost simulations and longer-run projections are conditional exercises. Their numerical values depend on assumptions about fees, participation, prices and growth; they are not observations of actual attacks or universal security guarantees.

There is also an important distinction between a memoryless waiting-time distribution and a Markov description. A model can summarise relevant history in a sufficiently specified current state. Calling that model Markov does not mean difficulty, confirmations or accumulated fees can be omitted from the state.

The broader research lesson is a discipline of scale. Start with the mechanism whose randomness is understood. Then identify the rules that carry its outcomes forward. A good approximation earns its usefulness by making clear which clock it describes, which state it retains and which question it can answer.