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

This is crucial:

The expected growth depends on system age, not calendar time.
Growth slows predictably as the system matures.

By normalizing returns relative to this expectation, we separate:

the deterministic scaling signal (the power law)

the stochastic fluctuations (market behavior)

This is exactly what physicists do when studying expanding systems.

3. Normalized daily slopes (the key insight)

Define a normalized daily slope (or effective exponent):

n(t) = log(P(t+1)/P(t)) / log((t+1)/t)

This quantity answers a deep question:

“How fast is Bitcoin growing relative to its age?”

Now something remarkable happens:

The mean of n(t) is stable over time

The mean converges to a constant ≈ α

Short-term chaos disappears once growth is properly normalized

This stability is not obvious in raw returns — it only emerges after power-law normalization.

4. Stability of the mean = existence of a scaling law

In complex systems, a stable normalized mean implies:

The system has found a self-consistent growth regime

Feedback mechanisms regulate deviations

The growth law is not accidental

This is why power laws are not “just fits”:

A stable normalized slope is evidence of an underlying mechanism, not curve-fitting.

Bitcoin has shown this stability for ~16 years, across:

bubbles and crashes

regulatory shocks

exchange failures

institutional entry

That alone places it in the class of mature scale-invariant systems.

5. Deviations are structured, not random

Once normalized, the deviations of n(t):

δn(t) = n(t) − α

are no longer arbitrary.

Empirically:

They follow a well-defined distribution

The distribution is time-dependent but structured

Tails are heavy, consistent with complex adaptive systems

Variance evolves slowly, not explosively

This means:

Bitcoin’s volatility is not noise

It is constrained by the same scaling laws as the growth itself

In physics terms: Bitcoin behaves like a system fluctuating around a stable attractor.

6. Why this makes the power law predictive (in the right sense)

The power law is not a price-prediction tool in the short term.

Its power lies elsewhere:

It predicts the expected growth envelope

It defines what deviations are plausible vs implausible

It allows probabilistic statements about future paths

It provides a reference frame in which volatility becomes interpretable

This is the same reason scaling laws are used in:

turbulence

population growth

city economics

network evolution

Not to predict exact outcomes, but to constrain reality.

7. Why this framework is superior to traditional models

Traditional financial models assume:

stationarity

fixed time scales

Gaussian noise

Bitcoin violates all three.

The power-law framework:

accepts non-stationarity

normalizes time dynamically

explains why volatility shrinks relative to scale

This is why exponential models fail and why the power law keeps surviving.

8. Bottom line

The power law model is powerful because:

It normalizes growth correctly

It reveals a stable mean growth exponent

It turns chaos into structured fluctuations

It shows Bitcoin is a self-regulating scaling system

It transforms “returns” from noise into physics

Or, in one sentence:

The power law works because it puts Bitcoin in its natural coordinate system — and in that system, the signal becomes simple, stable, and meaningful.