The Bitcoin Power Law: formula, evidence and limits

01What the power law claims

The Bitcoin power law makes one narrow claim: Bitcoin's long-run price scales with a power of the network's age, not with the calendar. Write it as price = A × daysⁿ, count the days from the genesis block on 3 January 2009, and a single curve runs through sixteen years of price history — from fractions of a cent in 2010 to six figures today.

Three things it does not claim are worth stating immediately, because most arguments about the model are really arguments about these:

• It says nothing about next week, next month, or the next crash. It describes the centre of gravity, not the orbit around it.
• It is not a law of physics. It is a regression on past data that has, so far, not broken.
• It is not a floor. The lower band is a historical range that has been breached before.

If you are meeting the idea for the first time, the beginner's guide to the power law covers the intuition with fewer numbers. This page is the detailed version: the arithmetic, the sources and the failure modes.

02Bitcoin against the trend right now

Loading the live figures …

The deviation is the whole point of the model in one number: how far the market currently sits above or below the trendline. It is not a buy or sell signal. Price has spent almost the entire history away from the line, frequently by a wide margin for months at a time.

03The formula, and how to compute it yourself

The entire model fits on one line. These are the values from last night's fit:

You can reproduce today's fair value with a calculator in three steps:

1. Count the days since 3 January 2009. Today that is … days.
2. Raise that number to the power of the current exponent, ….
3. Multiply by the constant, 10 to the power of …. The result is … — the fair value shown above.

An exponent near … has a concrete meaning: every time the network's age multiplies by ten, the trend price multiplies by roughly 450,000. That sounds explosive, and early on it was. But the implied annual growth rate falls every year — around +70% a year at age ten, roughly +37% today, and near +25% by age twenty-five. Enormous growth, steadily decelerating. An exponential model cannot produce that shape; a power law does so automatically.

04Where the numbers come from

The fit itself is unglamorous, which is a point in its favour. Take every daily close since Bitcoin had a price, take the logarithm of the price and the logarithm of the age in days, and fit a straight line through the pair by least squares. The slope of that line is n; the intercept is log₁₀(A). On this site the fit runs every night against the full history from Coin Metrics, over roughly 16 years of data, and the current fit explains about 96% of the variance in log price.

05The peer-reviewed derivation

For most of its life the power law was an empirical curiosity that lived on social media: a striking chart with no mechanism behind it. That changed in 2026.

The paper fits 5,696 daily prices from July 2010 to February 2026 and reports an exponent of 5.69 ± 0.05 with an R² of about 0.96 on log price. More importantly, it derives that exponent instead of merely measuring it. The argument has two ingredients:

• Adoption grows as a power of time. New users arrive in accelerating waves, following the same growth shape a 1989 study documented for the early US AIDS epidemic — not a smooth exponential, and not a single S-curve.
• Network value scales with users through a generalised Metcalfe relation. A network is worth more than the sum of its members because value comes from the connections between them.

Multiply the two and the price exponent follows. Empirically, active addresses have grown roughly as age^2.9, and Metcalfe's square gives 2.9 × 2 ≈ 5.8 — the same neighbourhood as the measured exponent. That agreement is what turns a chart pattern into a hypothesis you can test: if adoption keeps its shape the trend holds; if adoption bends, the curve must bend with it.

06Fair value and the two bands

The trendline gives a fair value for any date. Real price oscillates around it in roughly four-year cycles tied to the halving, and two boundaries have framed that oscillation for the model's entire life:

• The lower band, 60% below trend. Since 2010 Bitcoin has closed more than 60% below the trendline on about 2% of all days, and every one of those days fell in 2010 or 2015 — when the network was very young. The 2018 and 2022 bear-market lows both held above it.
• The upper band, about three times trend. Bull-market peaks have topped out roughly two to three times above the line, and each cycle's overshoot has been smaller than the last.

07Trend values if the fit holds

The table below evaluates the current fit on 1 January of each year, with both historical bands alongside. It is recomputed in your browser from tonight's fit, so it changes as the fit changes — and the further out the row, the more the honest uncertainty in the exponent widens it.

For a specific price target, each of these pages works the same arithmetic backwards — the date the trendline reaches that level, with the band around it:

Or use the interactive calculator, which runs in both directions: enter a price to get the trend date, or a date to get the trend price.

08Why the exponent keeps moving

This is the part most discussions of the power law skip, and it is the most important thing to understand about it. The exponent is an estimate, not a constant, and it moves for three separate reasons:

• New data. Fitting the full history gave roughly 5.82 in 2022 and roughly 5.66 today. Every additional year of data pulls the slope slightly.
• Autocorrelation. Consecutive daily prices are highly dependent, which means 5,696 daily observations carry far less independent information than 5,696 genuinely independent measurements would. Accounting for that, the honest uncertainty on the exponent is roughly ±0.4. The second decimal place is noise, and any projection decades out inherits that spread.
• The choice of origin. The model needs a t = 0, and the genesis block is a convention rather than a derived quantity. Baquero and Menezes showed in 2026 that shifting the assumed origin moves the fitted exponent by up to about a factor of three — so "n = 5.7" is a property of a fitting choice as much as of Bitcoin.

This site refits nightly rather than freezing a value, and shows the current numbers rather than a screenshot of last year's. That does not remove the uncertainty; it just stops the page from hiding it.

09The strongest objections

The most substantial critique of the model published in 2026 came from Carlos Baquero and Rui Menezes (arXiv, May 2026). It is worth reading before arguing either side, because it cuts in both directions.

Against the model:

1. On distributions such as UTXO balances and daily returns, a power law is rejected — a lognormal fits better. Not every power law claimed about Bitcoin survives testing.
2. The time-domain exponent depends on where you place t = 0, as above.
3. Standard residual tests cannot distinguish a power law from a stack of sigmoid adoption waves. The data does not, on its own, pick the winner.

And yet, in the same paper's walk-forward comparison against ten models, the plain power law was the best 12-to-24-month forecaster (p < 0.05) — precisely because it commits to no particular wave shape and therefore has very little to overfit. Baquero's companion survey of the literature makes the broader point: the field's problem is not a shortage of models but a shortage of honest evaluation — walk-forward tests, holdout windows across several market regimes, a naive baseline to beat.

10What would break it

A model worth trusting is one whose failure conditions you can name in advance. For this one they are:

• Adoption stalls or reverses. The derivation runs through user growth. If active addresses stop growing as a power of time, the price exponent has no mechanism left holding it up.
• A structural break. Regulation that cuts off access at scale, a protocol-level failure, or a superior competitor would bend the curve, and the fit would only reveal it years later.
• Sustained escape from the bands. A multi-year close below the lower band would be the clearest falsification available. It has not happened since 2015.

Nothing on this page is financial advice. It is mathematics applied to history, with its error bars left visible — one lens among several, and most useful when you know exactly what it cannot see.

11Sources

• Giovanni Santostasi & Stephen Perrenod, A Mechanistic Derivation of the Bitcoin Price Power Law, Nonlinear Science (Elsevier), published 29 June 2026. The peer-reviewed derivation, the 5.69 ± 0.05 exponent and the R² ≈ 0.96 figure. Coverage of the publication.
• Carlos Baquero & Rui Menezes (arXiv, May 2026). The origin-dependence result, the distributional rejections and the walk-forward comparison. arXiv:2605.21316.
• Carlos Baquero, Bitcoin Price Prediction: Peer-Reviewed Evidence and Social Media Discourse (arXiv, May 2026). The survey of evaluation practice in the literature. arXiv:2606.00071.
• Stephen Perrenod, Does Bitcoin Really Follow Metcalfe's Law? Five New Tests (August 2026). The adoption-to-value link tested directly. Money or Debt.
• This site's own fit — recomputed nightly from the full Coin Metrics price history and served through /api/price. Every live figure above comes from it.

12Frequently asked questions

Every live figure on this page is recomputed in your browser from the nightly fit served by btclaw.io. Educational material, not financial advice.