Ageing cannot Gaussianize returns: a log-time dis-aggregation theorem and the noise of Bitcoin's price

The variance of Bitcoin's daily log returns has decayed with age as a power close to −1 (−1.07 ± 0.20 from genesis); volatility memory at calendar lags of 20–90 days has not lengthened (exploratory); the body of the return distribution has become more Gaussian. We prove a dis-aggregation theorem that makes dyadic age bins a general test of stationarity in logarithmic time for any ageing system: with stationary increments in log time, one day at age t has the law of the sum of the m days at age mt, so for Brownian scale mixtures no absolute moment of order below two can rise with age; pooled bins carry an explicit error bound. Bitcoin violates the theorem: on paired bins both doublings of age move the body towards the Gaussian, and a two-sample test of the lemma rejects equality in law (P = 0.002–0.03, at a size of 0.06–0.10 for nominal 0.05). On a panel of 47 other crypto-assets, with a reserved half of 43 opened after registration, the 1/age law is not generic (Bitcoin lies in the lower decile) and no own-age effect on the shape is detected; the change of shape is not shared by the other assets' own returns (exploratory), but the panel cannot separate a change of Bitcoin from one of the market, whose common factor is Bitcoin's return. A registered single-venue test shows the maturation to be a fact of the traded market, not of the reference rate (a foreseeable outcome, informative for 2011–2012). The replication package (every script, result file and the canonical list of every number quoted) is at 10.5281/zenodo.23933083; the prediction registries with their OpenTimestamps proofs and the deposit note on prospectivity are at 10.5281/zenodo.23932900; the companion paper is at 10.5281/zenodo.23933138.