Why $BTC ’s Long-Term Price Follows a Power Law

BTC looks chaotic short term

But zoom out and a clear structure appears

Growth proportional to size

In many systems the rate of change scales with the size of the system:

dP(t)/dt ∝ P(t)

When growth depends on current size,it compounds

You see this everywhere:

• populations

• cities

• internet networks

• capital markets

BTC adoption behaves the same way

Users →liquidity →security →capital →more users

That feedback loop produces multiplicative growth

Now add the constraint

BTC’s supply is fixed:

≈ 450 BTC per day

≤ 21 million total

When demand compounds but supply cannot expand,price must absorb the imbalance

The network grows

Supply cannot

So the adjustment happens in price

Scale invariance

Systems that grow through multiplicative processes often become scale invariant

That means the structure looks similar across different sizes

On a log-log plot,BTC’s long-term fit is remarkably strong:

R² ≈ 0.96

Examples:

• city size distributions

• lightning branching

• river networks

• crack propagation in materials

Different systems

Same scaling logic

Mathematically this property leads to a power law

The scaling law

If growth scales with system size over time,the differential relationship becomes

dP(t)/dt ∝ P(t) / t

The solution is a power law:

P(t) = a · tᵇ

On log-log axes this becomes a straight line

That is why log-log charts reveal the structure that linear charts hide

Why cycles happen

Real markets experience shocks

Liquidity

leverage

news

macro events

These create deviations from the structural trend

Those deviations behave like a mean-reverting process:

dz = −κz dt + σ dW

Large deviations create stronger restoring drift

So price oscillates around the structural growth path

The path is noisy

The structure is not