Bitcoin's Power Law: Weak Structure, Strong Forecasts

We wrote a paper testing the popular claim that Bitcoin’s price follows a power law over time.

The short version: the power law fits Bitcoin’s historical price surprisingly well, but that does not mean it is a deep law of nature or a guaranteed price model.

We test it in a few ways:

1. When using standard power-law tests on Bitcoin-related distributions, like UTXO balances and daily returns, the data does not really behave like a power law. Lognormal fits usually do better.

2. The famous Bitcoin price power-law exponent is not very stable. If you shift the time origin, the exponent changes a lot, which makes it hard to treat as a true structural constant.

3. More flexible models can fit Bitcoin’s past price better than the power law. In particular, a model with several adoption-like “waves” fits the history better.

4. But here is the surprising part: those better-fitting models are bad at forecasting. The simple power law often does better for long-term forecasts, especially around 1–2 years ahead.

So the conclusion is not “Bitcoin definitely follows a power law.”

It is more like:

The Bitcoin power law is weak as an explanation, but useful as a rough long-term forecasting tool.

In other words, it may work not because it captures Bitcoin’s exact structure, but because it is simple and does not overfit each boom-and-bust cycle.

https://arxiv.org/abs/2605.21316