Bitcoin's Price Power Law Decomposed: Epidemic Speed Times Wealth Inequality

Bitcoin’s USD price has followed a power law of its age, P ∝ tβ , for sixteen years and six orders of magnitude. Santostasi & Perrenod establish the law and decompose its exponent as β= βA × βM , deriving βA ≈ 3 from epidemic spreading but leaving βM ≈ 1.84 as an unexplained “generalised Metcalfe” exponent. We answer their open question: βM = 1 + 1/α, where α ≈ 1.2 is the Pareto exponent of global wealth, measured by economists independently of Bitcoin. The mechanism is geometric penetration of the wealth ranks: each e-folding of adoption reaches a constant factor deeper into the wealth distribution, and the capped supply forces each richer wave to buy from earlier holders. Simulation shows that this schedule — and not mere rank ordering, which provably fails at Bitcoin’s penetration — produces the super- linearity; on-chain data show it operating. With autocorrelation-honest errors the measured scaling, βM = 1.827 ± 0.091, sits dead-centre in the externally predicted band [1.80, 1.87]. Cross-asset controls turn the claim into a prediction of which assets obey the law: Ethereum and Litecoin show the same generic adoption power laws but zero wealth surplus — exactly as required, since one lacks the supply cap and the other a live epidemic. Finally, the temporal exponent inherits a time-origin convention while βM is convention-free, resolving the recent “weak structure, strong forecasts” paradox: the invariant object of the Bitcoin power law is βM — and βM is the shape of human wealth inequality.