The Bitcoin Power Law Went Peer-Reviewed. The Corridor Everyone Quotes Did Not

On 29 June 2026, a model that had lived for seven years in Reddit threads and chart screenshots became a peer-reviewed paper. Giovanni Santostasi and Stephen Perrenod published "A Mechanistic Derivation of the Bitcoin Price Power Law" in Elsevier's journal Nonlinear Science. Two weeks later, Fidelity's Jurrien Timmer noted that Bitcoin near 60,000 US dollars was approaching the support line of the same model. Bitcoin bottomed at 57,717 US dollars, turned, and now trades near 78,800.

The conclusion drawn almost everywhere was that the power law had called the bottom. It is a satisfying story. It is also three separate claims wearing one coat, and only one of them belongs to the model.

What the model actually says

Plot Bitcoin's price with a logarithmic scale on both axes, price and time since the genesis block, and the result is close to a straight line. That is the whole visual claim. Formally it is a power law, price proportional to time raised to some exponent.

The paper measures that exponent at 5.690, with a standard error of 0.005 and an R squared of 0.961 across 5,696 daily observations from July 2010 to February 2026. What lifts it above curve fitting is the derivation. The authors break the exponent into two measurable pieces. Adoption, proxied by the number of addresses holding a non-zero balance, grows as time to the power of 3.046. Price scales with that address count to the power of 1.838, a generalised form of Metcalfe's law. Multiply the two and you get 5.60, within 1.6 percent of the 5.69 measured directly on price.

That is a real result. A number derived from two independent measurements landing on a third is not something you get by accident, and it is the strongest thing anyone has said in favour of the model.

The paper also does what most Bitcoin models refuse to do. It lists five conditions under which it would be considered wrong, the strictest being a price more than three standard deviations below the fit for longer than a year, which the authors put at roughly 10,000 US dollars for 2025.

Why the shape is convincing

Power laws are everywhere, and that is not marketing. Metabolic rate rises with body mass along a straight log-log line, a regularity Max Kleiber measured in the 1930s and one that holds from a dove to an ox. Internet users grew the same way. City size distributions, social network connectivity, earthquake magnitudes and word frequencies all behave this way. When a system grows through connections between its parts rather than through addition, this is the shape that tends to appear.

Bitcoin is a network whose usefulness rises with each participant. Finding a power law there is unsurprising. The interesting question was never whether the shape exists. It is what the corridor drawn around it is worth.

The first cut: the corridor is not the model

The bands almost every public chart shows come from Harold Christopher Burger, published on 3 September 2019, six and a half years before the peer-reviewed paper. His fit is explicit: price equals ten raised to the power of minus 17.01593313 plus 5.84509376 times the base-ten logarithm of days since the genesis block.

Two things follow, and both are checkable with a calculator.

First, the chart is genuinely unchanged. Running Burger's 2019 coefficients forward gives 449,481 US dollars for 10 September 2029. The widely circulated Bitbo chart labels that same date 450,571.80. The difference is a quarter of one percent. Whatever else is true, nobody quietly moved this line to keep it fitting. That deserves saying plainly, because it is the opposite of what usually happens to price models.

Second, the upper band was built differently from the fit, and almost nobody mentions it. Burger derived the resistance line by regressing on three data points, the cycle tops of 2011, 2013 and 2017, and it came out with a slope of 5.029 against the fit's 5.845.

Two straight lines with different slopes converge. The corridor therefore narrows on its own, regardless of what Bitcoin does:

Year	Fit	Resistance	Width
2011	3 dollars	45 dollars	16.4x
2015	595	4,598	7.7x
2020	15,744	77,057	4.9x
2026	178,200	621,600	3.5x
2040	5.41 million	11.72 million	2.2x

The observation that Bitcoin has failed to reach the upper band for two consecutive cycles is usually read as a market signal, evidence of maturing volatility or institutional dampening. It needs no such explanation. The ceiling is descending toward the mean by construction, and on the same coefficients the two lines eventually cross late this century, at which point the model's ceiling sits below its own average.

None of this appears in the peer-reviewed paper, because the corridor was never part of it. It is decoration fitted to history, and the people who built it have always said so. Burger called his own approach ad hoc.

The second cut: whose floor did it call?

Ask what the power law support line was worth in mid-2026 and the answer depends entirely on which chart you were looking at.

CryptoSlate put the floor at 51,128 US dollars in February. Fidelity's version sat near 56,000. CoinDesk reported 58,000. Bitbo's support, derived from its own published labels, works out near 63,200 today. CryptoSlate's projection for year-end 2026 was 68,000.

Bitcoin's low was 57,717.

That is a hit against two of those lines, a miss against two others, and against Bitbo's version it is a clean break below support by roughly nine percent, followed by a recovery. Five candidate floors spread across a 33 percent range will contain almost any bottom. This is not fraud, and the people publishing these charts are not hiding their parameters. It is simply what happens when a decorative band is treated as a prediction.

The more revealing number is the one nobody quotes. Burger's fit, the line the model is actually about, sits at roughly 178,200 US dollars today. Bitcoin trades near 78,800. The model's central estimate is currently wrong by a factor of 2.3, about 56 percent below fair value, and that is entirely permitted. Dispersion around the fit is a feature of the model, not a defect. But it means a model being celebrated for precision is, at this moment, off by more than half.

The third cut: the ruler

Every number above is denominated in US dollars, and the dollar is not a fixed unit. US M2 stood at 23.22 trillion dollars in July 2026, a record, 1.43 trillion above the previous peak of March 2022. Part of any long-run dollar price series is the dollar moving, not the asset.

There is an irony in using this particular ruler on this particular asset. Bitcoin's deepest design choice was to build a measure of ordering that nobody owns and nobody can adjust. The model that claims to describe its value grades it against a unit that one committee can change on a Wednesday afternoon.

Is the ruler shrinking faster?

The argument you hear most often is that dollar debasement will eventually break the power law upward. Print enough and the dollar price of anything scarce has to leave any corridor drawn around it. It is a reasonable expectation, and it depends entirely on one empirical question: is the dollar being diluted at an accelerating rate?

Over fifty years the answer is no.

Decade	M2 start	M2 end	Annual rate	Doubling time
1976 to 1986	1,027 bn	2,502 bn	9.32 %	7.4 years
1986 to 1996	2,502 bn	3,648 bn	3.84 %	18.0 years
1996 to 2006	3,648 bn	6,730 bn	6.32 %	11.0 years
2006 to 2016	6,730 bn	12,506 bn	6.39 %	10.8 years
2016 to 2026	12,506 bn	23,218 bn	6.38 %	10.9 years

The last three decades come in at 6.32, 6.39 and 6.38 percent. They differ by seven hundredths of a percentage point. The fastest decade on this table was the one that ended forty years ago, and the fifty-year average is 6.44 percent.

The pandemic looks like a break in that pattern and then closes it again. M2 rose 41.1 percent between January 2020 and the April 2022 peak, which means 29.1 percent of every dollar that existed at that peak had been created inside those twenty-seven months. Year-on-year growth hit 26.9 percent in February 2021, a record. Then came the first sustained contraction in the history of the series, with M2 shrinking through 2023 and into 2024. From the April 2022 peak to July 2026 the stock grew at 1.55 percent a year. Take the burst and the freeze together and the rate from January 2020 to today is 6.50 percent a year, which is the fifty-year average to within a rounding error. Current year-on-year growth is 5.41 percent against a long-run average since 1959 of 6.81 percent.

Steady debasement does not break a fitted model. It feeds it. If the dollar has been losing ground at a roughly constant rate throughout the period the exponent was measured on, that loss is already inside the exponent, and continuing it changes nothing.

There is a further consequence that runs the other way from the popular version. A power law's implied annual growth rate is the exponent divided by elapsed time, so it falls every year, while monetary expansion runs at a constant percentage. Today the model implies roughly 33 percent annual price growth and monetary expansion accounts for about 19 percent of that. By 2040 the model's implied rate drops to 19 percent and debasement accounts for 34 percent of it. By 2060 it is more than half. Around the end of this century the two rates meet, and a model of Bitcoin adoption would be describing nothing but the shrinking dollar.

So the debasement argument is right in principle and wrong about the mechanism people expect. Constant dilution at the historical rate never breaks the corridor upward, it quietly becomes the corridor. Only a genuine regime change, a sustained move to double-digit expansion of the kind last seen in the 1970s, would push the price out of the top of a band fitted to a 6.4 percent world. Nothing in the last three decades of data suggests that is where the trend is heading, which does not mean it cannot happen.

What happens with a harder ruler

The denominator objection is old, and the model's defenders have answered it. The usual answer is that switching to a harder unit does not destroy the pattern. So it is worth checking, because the check has already been run by one of the paper's own authors.

In September 2024, Perrenod published a power law for Bitcoin priced in gold ounces: 40 times time over 15.5, raised to the power of 5.47, with an R squared of 0.94. He projected roughly 56 ounces per Bitcoin by mid-2025.

Run his own formula to today and it calls for 75.8 ounces. Gold trades at 4,419 US dollars, up 27.1 percent this year, and Bitcoin at 78,800, which puts the actual ratio at 17.8 ounces.

The gold-denominated version of the model is wrong by a factor of 4.25. Expressed back in dollars, it implies about 335,000 US dollars today, against the dollar version's 178,000 and a market price of 78,800. Bitcoin has lost roughly 56 percent against gold since the ratio peaked near 40.9 ounces in December 2024.

That gap between the two versions of the same model is not noise. It is the ruler. And it completes the point above: over the last two years the dollar version has been held up by precisely the force that is supposed to break it one day, while the version measured in the harder unit collapsed.

The strongest case against our own reading

The most rigorous critique of the power law is also its most interesting defence. Carlos Baquero of the University of Porto and Raquel Menezes of the Universidade do Minho published "Bitcoin's Power Law: Weak Structure, Strong Forecasts" on 21 May 2026.

They take the structure apart. Applying the standard Clauset-Shalizi-Newman protocol, eight of eleven tests reject the power-law hypothesis in favour of a lognormal distribution. Introduce an arbitrary shift parameter into the time axis, a choice the model gives no principled reason to fix at zero, and the fitted exponent wanders between 5.65 and 16.49. Their verdict is that the exponent is specification-dependent rather than specification-robust. A three-component sigmoid model beats the power law in-sample by a crushing margin.

Then they run the honest test. Walk-forward, out of sample, across fourteen models and eleven yearly cutoffs, the simple power law beats every standard baseline at horizons of twelve to twenty-four months, at a significance level below 0.05. The model that won in-sample is among the worst long-horizon forecasters.

That is a genuinely uncomfortable result for both camps. The power law is descriptively weak and practically useful at the same time. A model can be structurally wrong and still be the best available tool for saying whether the current price is unusually high or unusually low. Anyone dismissing it as a magic trick has to explain why the magic trick out-forecasts fourteen alternatives.

Every model works until it stops

The relevant history here is not Bitcoin's price. It is the mortality rate of Bitcoin price models.

Stock-to-flow was the most cited model of the previous cycle. Its author committed publicly in June 2021 to invalidating it if Bitcoin did not reach 100,000 US dollars by that December. Bitcoin closed 2021 below 50,000. The model was not invalidated. The response was that being in the lower bands was fine.

The Rainbow Chart, created by a Reddit user in 2014 and published online in 2019, ran on the formula 2.9065 times the natural log of x minus 19.493. In 2022 a version two shipped with a new formula and the same colours. A third, dynamic version now runs on a power law regression. Its creators are refreshingly blunt about it: there is no scientific basis underpinning it, and you cannot predict the price of Bitcoin with a rainbow.

And the models currently on offer do not agree with each other. For roughly the same horizon:

Model or forecaster	Figure for 2029 to 2030
Power law fit (Burger 2019 coefficients)	449,000 US dollars, September 2029
Power law support	160,000 US dollars
Power law resistance	1,378,000 US dollars
Bernstein, 26 August 2026	300,000 US dollars by end of 2029
Gold-denominated power law	already implies about 335,000 US dollars today

A spread of nearly nine to one between the lowest and highest of these is not a forecast. It is a range wide enough that something inside it will happen, after which the model containing it will be described as having worked.

Two further cautions belong on the record. The peer-reviewed dataset ends in February 2026, so the entire drawdown from the October 2025 high of 126,080 dollars lies outside the data that passed review. And the adoption leg of the derivation is under visible strain. Addresses with a non-zero balance stand at about 56.7 million, and in the paper's own first figure the data bends below the fitted line at the right-hand edge. The authors flag the honest caveat themselves, that address counts are a crude proxy when exchanges and ETFs hold many users behind few addresses. That caveat cuts both ways. If the proxy has broken, the mechanistic derivation has lost the measurement it rests on.

Our reading

This is CanoeBit's interpretation, not a finding of the paper.

The power law describes something real. The Metcalfe decomposition is the most serious quantitative argument anyone has made about why Bitcoin's price should have a shape at all, and the out-of-sample forecasting results are hard to wave away. What we do not accept is the three things the model is being credited with that are not in it.

The corridor is not the model. It was fitted to history in 2019 with a ceiling drawn on three data points, and it narrows on its own arithmetic. The bottom call is not the model. It is a hit selected after the fact from five published floors spread across a third of the price range. And the dollar figure is not purely about Bitcoin, because the same model in gold is wrong by more than four times, which is the size of the ruler's own movement.

We would add one correction to the argument most often made in Bitcoin's favour here. Debasement is not going to break this model upward at the current rate, because the current rate is the rate it was built on. Three consecutive decades of monetary growth within seven hundredths of a percentage point of each other are already inside the exponent. What debasement does instead is quieter and, over a long enough horizon, more damaging to the model's meaning. As the power law's implied growth rate decays, monetary expansion takes over a rising share of it, until a curve sold as a measure of adoption is mostly a measure of the unit it is priced in.

What remains is worth having. A long-run slope with wide dispersion is a way to notice that a price is unusually far from trend. It is not a way to know where the price goes next, and the honest version of the model says so in its own falsification criteria.

We would consider this reading wrong under two conditions. If the Bitcoin to gold ratio returns to the gold-denominated model's path, above roughly 70 ounces, within 24 months, then the denominator objection was noise rather than signal and the dollar version was never flattered by debasement. And if non-zero balance addresses return to the trajectory implied by an exponent of 3.046, the mechanistic derivation holds and the current flattening was a custody artefact rather than an adoption problem.

Until then, the most defensible thing anyone can say about the power law is the thing its critics and its authors would both sign: it is a remarkable empirical regularity, it has never been secretly adjusted, and it cannot tell you what Bitcoin will be worth.