People have the wrong interpretation of the power law. They think in bands around the power law. These are relatively large (we shown in the past that they are not if one focuses on where Bitcoin spends most of its time). But really the concepts of bands is not what matters in terms of understanding the true Bitcoin behavior. One has to use the language of normalized returns or daily slopes to truly understand the significance of the power law. 1. The core problem: raw returns are misleading If you look at Bitcoin’s raw daily returns or raw price changes, you immediately face two problems: Non-stationarity A 5% move in 2011 is not comparable to a 5% move in 2024 in terms of economic meaning, liquidity, and system size. Volatility appears to “decay,” but this decay is entangled with growth. Scale dependence Absolute price changes explode as the system grows. Even percentage returns hide the fact that the system’s natural time scale is changing. In short: raw returns mix growth and noise, making it impossible to study Bitcoin as a stable system. 2. Power law as the natural normalization The power law provides a natural normalization of time and growth. If price follows: P(t) = C · t^α then the expected daily growth rate (the local slope in log space) is: d log P / d log t = α ...