This is an extension of the scaling test I discussed previously. In this version, we average the deviations along the scaling law and examine how they behave across different time scales. Specifically, we plot the logarithm of returns against the logarithm of the change in time, which can be interpreted as a kind of temporal frequency.

The result is striking: Bitcoin preserves the same statistical behavior across many different time scales. In other words, the system remains self-similar under temporal rescaling. Over more than 16 years of data, Bitcoin’s time dynamics have remained remarkably stable.
This is where the depth and beauty of the power law truly appear. A scale-invariant system is not defined by a single curve fit, but by the persistence of its behavior across orders of magnitude in time.
Trying changing this for personal gains is a grifter trick to get attention, not a genuine attempt to understanding this beautiful system.
Attempts to arbitrarily modify or “optimize” the power law without a strong theoretical or empirical justification miss this fundamental point. The scaling structure is not something that can be adjusted casually; it emerges from the underlying dynamics of the system. Changing it without understanding those dynamics adds noise rather than insight.