Bitcoin's power-law trend: what the regression identifies, and the four-year oscillation around it

The regression of Bitcoin's log price on log time since the genesis block is a robust regularity (Santostasi and Perrenod, 2026a); we show what it identifies. With a four-year oscillation as residual, the exponent's precision depends on the residual's model: ±0.25 (HAC) if the residual is stationary, a standard deviation of 1.8–2.2 under a martingale in log time, [2.6; 8.4] at 90 % whatever the reversion rate. Three least-squares identities explain the composition test, bound the worst-case error and weight the local slopes of any log–log regression on a uniform grid: Bitcoin's exponent is 62 % a statistic of 2013–2018 and 8 % of the last four years. Daily data neither separate the trend from a log-time martingale (LR 5.96, p 0.31, power 0.13) nor identify the reversion rate. In real time the trend forecast beats the drift by 24 % at one year and 42 % at two, improbable under the martingale and a fitted reverting residual alike: one return to the trend, not a discriminating statistic. The residual's means over the three complete halving cycles are small (composite p 0.065–0.12, sensitive to the sample start); its overshoot contracts faster than its deficit; the post-halving maxima keep a nearly constant calendar phase, fitted on all four cycles by a free oscillator, on the last three only by a halving-locked lag. A cooling Brownian oscillator with integral feedback is the minimal description; its registered prediction for April 2028 and an earlier registered bound straddle one threshold. 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.23933194.