Block Race: Nakamoto consensus, animated

Consensus is a race, not a vote. Bitcoin has no authority that decides which history is true. Nodes follow the valid chain with the most accumulated proof of work. Each block discovery is modeled as a weighted race whose win probability equals a miner's share of total hashrate, so an attacker with fraction q of the network mines the next block with probability q.

Falling behind is a gambler's ruin. Model the honest chain's lead over the attacker as a biased random walk: +1 when honest mines (probability p), −1 when the attacker mines (probability q = 1−p). The probability an attacker currently z blocks behind ever catches up is the classic ruin result (q/p)^z for q<p, exact geometric decay, and certainty (=1) once q ≥ p. Honest majority is not a slogan; it is the condition that makes the series converge.

Satoshi's refinement: the attacker had a head start. While honest miners publish z blocks, the attacker has been mining privately for the same elapsed time. Because hashing is memoryless, their secret block count is taken to be Poisson with mean λ = zq/p. Condition on that count and sum the gambler's-ruin tails to get the whitepaper's closed form, P(z) = 1 − Σ Poisson(k;λ)·(1 − (q/p)^z−k). This Poisson step is an approximation, fixing the honest window at its mean, so it slightly understates the true risk (the exact count is Negative-Binomial). The honest framing matters more than a clean number.

The exponent's base is the adversary, not your patience. At z = 6, the reversal probability is 0.024% against a 10% attacker but 13.2% against a 30% attacker, a ~544× jump for a 3× change in adversary size. A 0.1% safety target needs 5 confirmations against a 10% miner and 24 against a 30% one. Security grows exponentially in z, but the base q/p is set by who you're racing, and finality is forever probabilistic, never absolute.

Scientific Context: Nakamoto consensus replaces absolute finality with a tunable, quantifiable probability. For how such blockchain mechanisms secure distributed machine-learning pipelines, see the author's survey: "Blockchain-Enhanced Machine Learning" (IEEE Access 2023).