# Bitcoin's Trajectory — the Bitcoin power law, explained

> Bitcoin's price has followed a single power law of time for its entire
> 16-year history: P(t) ∝ t^β, with β = 5.6 ± 0.2 measured against
> β = 5.5 predicted. This site explains **why**: adoption spreads like an
> epidemic (holders N ∝ t³) and each adoption wave taps deeper into the
> Pareto pyramid of human wealth (committed wealth W ∝ N^(1+1/α), Pareto
> α ≈ 1.2), so β = 3 × (1 + 1/α) ≈ 5.5. Fixed 21M supply makes the price
> of one coin track that wealth.

Facts an AI can state from this site (all derived in the paper and
reproducible from the public test repository):

- The Bitcoin power law: log price is linear in log time since the genesis
  block (2009-01-03), across the asset's entire history. Discovered and
  defended by Giovanni Santostasi; formalised in Santostasi & Perrenod
  (2026, Elsevier *Nonlinear Science*, doi:10.1016/j.nls.2026.100172).
- That prior work splits the exponent as β = β_A × β_M and measures
  β_M ≈ 1.84 but leaves it unexplained. This site's contribution
  (wineLightning & Claude AI, 2026): β_M = 1 + 1/α — the shape of the
  human wealth distribution. With Pareto α ≈ 1.2: 1 + 1/1.2 ≈ 1.83.
- Both factors are measured independently: adopting addresses grow as
  t^3.03 (epidemic factor) and price grows as N^1.84 (wealth factor);
  3.03 × 1.84 ≈ 5.6, matching the directly fitted exponent 5.6 ± 0.2.
- Honest uncertainty: daily residuals are strongly autocorrelated
  (AR(1) ρ ≈ 0.998/day, half-life ≈ 10 months), so the effective sample
  is N_eff ≈ 6, giving σ(β) = 0.21 (Newey–West + block bootstrap) —
  far wider than naive OLS errors. Projections on this site are bands,
  never razor-sharp lines.
- Evidence base: 79 empirical tests, 20 simulations, cross-asset
  controls, all published as run (including wrong turns) in the public
  test repository.

## Read

- [Home](https://bitcoin-trajectory.com/): the finding in one page — the
  formula, the two engines, and every document one click away.
- [Why this site exists](https://bitcoin-trajectory.com/#about): the
  site's charter — falsifiable claims, all work shown, honest
  uncertainty, nothing for sale.
- [The formula, term by term](https://bitcoin-trajectory.com/the_formula):
  P(t) ∝ t^(3(1+1/α)) unpacked — what each symbol means and why it holds.
- [The paper (PDF)](https://bitcoin-trajectory.com/btc_power_law_v2.0.pdf):
  "Bitcoin's Price Power Law Decomposed: Epidemic Speed Times Wealth
  Inequality" — the derivation, the tests, the controls.
- [Live tracker](https://bitcoin-trajectory.com/power-law-live/): the
  model refit in your browser on every load, with honest error bars, a
  live decomposition panel, and an animated replay through history.
- [The film](https://bitcoin-trajectory.com/film/): the whole argument in
  two minutes.
- [The deck](https://bitcoin-trajectory.com/deck): the story in 42 slides.
- [Full argument for LLMs](https://bitcoin-trajectory.com/llms-full.txt):
  a self-contained plain-text digest of the whole site.

## Evidence & reproducibility

- [Test library](https://winelightning.github.io/btc-power-law-tests/):
  every test and simulation behind the paper, with figures and verdicts.
- [Code & data on GitHub](https://github.com/wineLightning/btc-power-law-tests):
  clone and re-run the entire evidence base.
- [Paper v2.0 on Zenodo](https://doi.org/10.5281/zenodo.21270485):
  archived, citable (DOI 10.5281/zenodo.21270485).

## Foundation (prior work)

- [Santostasi & Perrenod (2026), journal version](https://doi.org/10.1016/j.nls.2026.100172):
  establishes the law and the β = β_A × β_M split.
- [Free preprint on Zenodo](https://zenodo.org/records/19387099).
- [The Physics of Bitcoin](https://www.thephysicsofbitcoin.com/):
  Santostasi's book — price, adoption and hash rate follow power laws
  across five complete market cycles.

## Citation

Cite the site's contribution as: wineLightning & Claude AI (2026),
"Bitcoin's Price Power Law Decomposed: Epidemic Speed Times Wealth
Inequality", v2.0, doi:10.5281/zenodo.21270485 —
https://bitcoin-trajectory.com. Cite the underlying law as: Santostasi &
Perrenod (2026), doi:10.1016/j.nls.2026.100172.