The daily slope n=log(P2/P1)/log(t2/t1) is the natural quantity that describes almost all mathematical properties of Bitcoin.

All the discussion of projected future returns, volatility and similar become much clearer in this framework.
Take volatility.
The n volatility has had 2 main regimes in the last 17 years. It has been very stable over the last 9 years (something happened in 2017 to make it shift).
This is very interesting and consequential. This framework explains many things.
Many people say the Bitcoin volatility is declining giving hand waving arguments of why that is the case. In fact, it is not in n, it is stable the observed decline in non-normalized volatility is a consequence of the power law.
The "vol is declining" puzzle is now solved. Daily vol isn't mysteriously decaying with time as a function we have to fit empirically — it's the necessary consequence of n_daily having a roughly stationary distribution. If you believe n_daily is the natural physical quantity (and the data suggests within each regime it is), then σ_r(t) = K/t is forced.
The bounded-residual property falls out for free. Under σ_r ∝ 1/t, the variance of the cumulative residual converges: Var(ε_t) ≈ Σ K²/s² → K²π²/6. So the "power law channel" doesn't widen indefinitely — residuals stay in a fixed band relative to the trend. This is exactly what the data shows (residual std actually shrinks, from 0.94 to 0.22 across the sample, because the early sample falls inside the bigger pre-2017 K-regime variance and the late sample falls inside the smaller K/t variance at large t).
The signal-to-noise ratio is constant within a regime. The deterministic trend's daily contribution is dlog(P)/dt = n/t. The noise contribution is K/t. The ratio K/n is therefore time-invariant. Pre-2017: K/n ≈ 85/5.68 ≈ 15. Post-2017: K/n ≈ 155/5.68 ≈ 27. The market "feels" about 1.8× noisier post-2017 relative to the deterministic growth rate, and that ratio stays put as long as you stay in the regime.
The 2017 step makes physical sense as a structural change. CME futures launched December 2017, ETF speculation accelerated, institutional flows arrived. A discrete change in the noise scale (K) without a change in the exponent of the t-dependence is exactly what you'd expect if the market's "intrinsic" noise level changed but the trend's scaling did not.