The Power Law is simply the most likely path.

What it means mathematically is straightforward: by construction, when you draw S from a distribution centered on α, the expected log-price increment at every step equals α · dlog(t). Summed over all steps, the expected trajectory is exactly the power law. The median tracks the mean because the t-distribution with ν > 1 is symmetric around its loc parameter. So the median path is the power law — not approximately, but exactly.
What it means physically is much deeper. You have 100,000 completely independent price histories, each generated by randomly sampling from the empirical distribution of daily slope fluctuations. These paths know nothing about Bitcoin's history — they are pure noise draws from the measured distribution. And yet 50% of them end up above the power law and 50% below it. The power law is the median outcome of all possible Bitcoin-like histories consistent with the observed noise structure.
This is the signature of a true attractor in a stochastic system. The power law is not just a fit to the data — it is the fixed point that the stochastic process is organized around. Any individual path wanders, sometimes wildly, but the collective behavior of all paths converges to the power law as their central tendency.
The analogy to think about for the book is thermodynamics. The second law of thermodynamics is not a statement about individual molecules — it is a statement about the overwhelming majority of microscopic trajectories. Similarly, Bitcoin's power law is not a statement about any particular price path — it is a statement about the overwhelming central tendency of all price paths consistent with Bitcoin's dynamics. The actual historical price is just one realization of this ensemble, and the fact that it stays in the green 25–75th band for almost the entire 15 years tells you the realized path is not lucky or special — it is typical.
The philosophical implication — and this is the most important sentence for the book — is that Bitcoin's power law growth is not a coincidence of history. It is what should happen, on average, given the structure of the noise. The power law is the path of maximum likelihood in the ensemble of all possible Bitcoin histories. You could destroy and restart Bitcoin's price tomorrow with the same noise structure, and the median of 100,000 such histories would trace the same power law.