Start with the long-run power-law trend (where price “should” be with no boom/bust). Then add one slow sine wave on the leftover gap — one clock for the whole history, not a new wave every bull.
When the sine is below the trend, fair value is cheaper than the plain power law. When it is above, fair value is richer. The lower pane is the leftover (z) versus that fitted sine.
Collector daily closes. Period T grid-searched 900–1680 days. Walk-forward: year-end 2016…last complete year, 365-day OOS z-RMSE vs z≡0. Gate ≥50% of those years to draw a forward cycle.
log P = a + n log t. σ = sample stdev of residuals. z = resid/σ.
ẑ = b0 + b1 sin(2π t / T) + b2 cos(2π t / T). T = argmin RMSE. Fair = trend · exp(ẑ σ).
Band = empirical p10/p90 of (z−ẑ), not a forecast interval. Forward ẑ damped by exp(−0.04·years).