The Power Law, Part I: The Case For

Start with the thing that is genuinely hard to explain away.

Bitcoin’s price has moved across many orders of magnitude in fifteen years. Plot it on log-log axes, price against time since the genesis block, and you get something close to a straight line. Across 5,696 daily observations from July 2010 to February 2026, the fit has R² = 0.961.¹

Most financial assets do not do this. Most things do not do this.

The obvious rejoinder is that you can fit a line to anything that goes up. That rejoinder gets answered in Part II, where it belongs. What matters here is that the regularity is real, it has survived every drawdown in that window, and as of 2026 it has a mechanism, a peer-reviewed derivation, and an out-of-sample record that beats every standard forecasting baseline at long horizons.

That is a stronger position than the model held two years ago. Here is the case.

1. The exponent is no longer a free parameter

The weakest version of the power-law argument is curve-fitting: observe a slope, report it, project it forward. If that were all there is, the critics would be entitled to their dismissal.

In June 2026, Giovanni Santostasi and Stephen Perrenod published a derivation in Nonlinear Science, an Elsevier journal: peer-reviewed, DOI 10.1016/j.nls.2026.100172.¹ Their claim is that the exponent is not fitted at all. It is the product of two other exponents, each measured independently.

β = β_A × β_M

• β_A = 3.046 ± 0.012 (R² = 0.977) is the growth rate of non-zero-balance Bitcoin addresses over time. Cubic.
• β_M = 1.838 ± 0.031 (R² = 0.951) is how price scales with address count, a generalized Metcalfe-type network value law.

Multiply them: 3.046 × 1.838 = 5.60.

The directly measured price exponent is 5.69. The composition agrees to within 1.6%.

Take a moment with that. Two quantities measured from different data, one from address counts against the calendar and one from price against address count, multiply to reproduce a third quantity measured from price against the calendar. Nobody tuned that. It is the kind of coincidence that, in physics, usually means you have found something.

The cubic adoption term is not arbitrary either. The authors tie it to a known result: Colgate et al. showed in 1989 that cumulative US AIDS cases grew as t³ rather than exponentially, derived from the structure of the contact network, and Santostasi and Perrenod cite Bacchetti and Koch (1989) alongside it.² Spreading processes on heterogeneous scale-free networks produce polynomial growth with an exponent near 3. Bitcoin adoption measured 3.046.

2. It is the only series that behaves this way

This is the finding that ought to trouble a skeptic most, and it comes from the critics’ paper. Baquero and Menezes are otherwise hostile to the model.³

They compared nine series: Bitcoin price, five Bitcoin on-chain metrics, the NASDAQ Composite, the S&P 500, and gold. For each, they asked whether any simple single-component growth curve (a sigmoid, an exponential, a quadratic) fits better than a power law.

In their words, Bitcoin price is “the only series in our nine-series sample where no single growth component—whether exponential, saturating, or otherwise—improves on the power law.”³

Eight of nine series are better described by something simpler. Bitcoin price is not. And Bitcoin’s own on-chain metrics all fail the test, so this is not a Bitcoin-ecosystem effect. It is specific to price.

Their wave-stability bootstrap, run across seven series, also singles out Bitcoin price: it is the only series with a rejection at the strict threshold (p = 0.015).³

A critic has to explain why the curve-fitting objection applies to Bitcoin but not to the eight other series where a simpler model wins. That is not a rhetorical question. It is the central asymmetry in the data.

3. Scale invariance, tested three ways

A power law makes a stronger claim than “the line fits.” It claims the process has no preferred timescale: the same dynamics govern price ratios over one year as over five.

That is testable, and the authors tested it three independent ways:^1a

• Direct collapse. Test the identity P(λt)/P(t) = λ^β* across 5,298 price ratios. It recovers β* = 5.59.
• Rolling stability. Across rolling windows spanning 2011–2026, the median is β* = 5.73 ± 0.58, with no secular drift.
• Sequential Bayesian updating. 1,899 local scaling estimates yield a posterior of β ~ N(5.729, 0.013²), with uncertainty shrinking as n^−1/2, and no structural breaks anywhere in 2010–2026.

Three methods, three consistent answers, all clustering near 5.6–5.7.

Fifteen years. Several bear markets. A pandemic crash, an exchange collapse, a spot-ETF launch, an ETF outflow cycle. No structural break detected in any of it.

4. It wins out-of-sample, and that is the hard test

In-sample fit is cheap. The first question in the Model Room framework is whether a model works on data it has not seen. This is where most models die.

The power law does not. Again the evidence comes from the critics’ own paper.³

Baquero and Menezes ran walk-forward validation: eleven yearly cutoffs from 2014 to 2024, six horizons, ten competing models. At 12, 18 and 24 months, the power law had the lowest forecast error of every model tested. Error is RMSE on log₁₀(price), so lower is better.

At 24 months the power law’s error is roughly half the next-best model’s, and it beats four standard time-series baselines (random walk with drift, auto-ARIMA, exponential smoothing and local linear trend) at p < 0.05.³

The flexible models that fit history better all lose, and lose badly. The degree-9 polynomial, which tracks the past most closely of anything in the table, produces an error of 6,773.

There is a reason for this, and the critics state it themselves. In their account the power law’s lack of a wave structure is descriptively wrong, “but it is the source of the power law’s out-of-sample dominance. By not committing to any historical wave shape, the power law cannot mis-anticipate the next one.”³ A model that refuses to guess where the next cycle begins cannot guess wrong about it.

For anyone modeling a multi-decade retirement horizon, this is the property that matters. Nobody is asking a Bitcoin model what next month looks like.

5. Cointegration and the spurious-regression objection

The standard technical objection is that this is a spurious regression: two trending series producing a high R² that means nothing.

The peer-reviewed paper addresses it directly. Its published Highlights state that “stationary residuals, cointegration and out-of-sample fit rule out a spurious law.”¹

The logic is that a spurious regression leaves non-stationary residuals, errors that wander off. Bitcoin’s power-law residuals do not wander. Independently, Perrenod reports an Engle-Granger test rejecting the null at p = 0.026, with ADF (p = 0.006) and Phillips-Perron (p = 0.024) both confirming stationary residuals.⁴

And the out-of-sample record is itself an answer. A spurious relationship does not forecast. Whatever else is true, a meaningless fit does not beat auto-ARIMA at two years by a factor of seven.

There is a live methodological argument about whether cointegration tests apply to a deterministic variable like time at all, including from Santostasi himself in an earlier paper. It is unresolved, it is flagged in the Glossary, and it is scored in Part III.

6. What the authors say they have not proved

A steelman is stronger, not weaker, for including what its advocates concede. Santostasi and Perrenod concede several things without being forced to, and that candor is part of why the paper deserves to be taken seriously.

They state that the composition identity is “established as an empirically consistent decomposition rather than a first-principles proof,” and that the mechanism behind the adoption exponent is “inferred by analogy with spreading processes on scale-free networks and is not derived from a directly measured Bitcoin adoption network.”¹

They flag that address count is “a supporting, mechanistically motivated proxy whose limitations as a direct measure of network participation we discuss explicitly.” Addresses are not people, and they know it.¹

The preprint also noted that likelihood-ratio tests for a lower cutoff had not been applied to the temporal series, an invitation that Baquero and Menezes took up.^1a

There is also something worth noticing about how the claim changed in review. The April preprint said Bitcoin’s trajectory is “not primarily driven by speculative dynamics” and reflects “the deterministic mathematical consequences” of network topology. The published version says “a substantial component” of the trajectory reflects network structure, with speculation appearing as “mean-reverting cyclical deviations about this structural trend rather than as its principal driver.”¹

That is a real softening, and it happened between preprint and print. It is what peer review is supposed to do, and the published claim is the more defensible one.

7. What a careful proponent actually claims

The strongest version of this argument is narrower than the loudest version of it, and the narrower version is much harder to dismiss.

It does not claim a price target on any date. It does not claim the model predicts short-term moves, because it demonstrably does not. It does not claim Bitcoin cannot fall a long way below trend.

It does claim these things:

• Bitcoin’s long-run price has followed a stable power law in time over the 2010–2026 window with R² ≈ 0.96.
• The exponent decomposes into two independently measured quantities to within 1.6%.
• The regularity has survived every drawdown in that window with no detectable structural break.
• Bitcoin price is the only series in a nine-series comparison where no simpler model wins.
• At 12 to 24 months it is the best forecaster tested.

Even Matthew Mezinskis, whose Porkopolis chart is one of the most-cited versions of this analysis, states the limit on his own page. Asked whether it predicts the future, he answers: “The trend could hold as a helpful guide, or it could break. The rest is probabilities.”⁵

The honest proponent’s position is a conditional envelope, not a prophecy.

8. The strongest single sentence for the model

It comes from the critics.

Baquero and Menezes spend most of their paper attacking the power law’s structural claims. Then they state the practical result:

“conditional on continued growth in the underlying network and ecosystem of the type observed over 2010–2026, the central tendency of price extrapolates as the power law P ~ t^5.6.”³

When your critics’ own out-of-sample testing lands on your model, that is the strongest form of evidence available in this field, because it is the form least likely to be motivated.

What would change this

Part II makes the case against, at full strength, and it is a serious case. It covers an exponent that moves when you shift the time origin, a distributional test the model fails, and supporting evidence that cannot distinguish a power law from a rival. Part III scores both.

The falsification criteria the authors published, and where each one stands against the tape, are in the Falsification Ledger. Read the status there, not here.

Sources

1. Santostasi, G. & Perrenod, S. (2026). A mechanistic derivation of the Bitcoin price power law: Network adoption dynamics and generalised Metcalfe scaling. Nonlinear Science 8, 100172 (online 29 June 2026; 15 October 2026 issue). DOI 10.1016/j.nls.2026.100172, Elsevier. Peer-reviewed. The R², the exponents and the composition figures in section 1 are quoted from the open-access preprint version (Zenodo, DOI 10.5281/zenodo.19387099, v2, 2 April 2026), not checked against the journal text.

1a. The three scale-invariance results in section 3 are quoted from the open-access preprint (Zenodo v2, 2 April 2026), not from the journal version. Peer review altered the paper’s concluding language, so the distinction is stated. The figures are the authors’ own throughout.

2. Colgate, S.A. et al. (1989), PNAS, on the cubic growth of US AIDS cases. Santostasi and Perrenod cite Bacchetti & Koch (1989) alongside it.

3. Baquero, C. & Menezes, R. (2026). Bitcoin’s Power Law: Weak Structure, Strong Forecasts. arXiv:2605.21316v1. Preprint; not peer-reviewed.

4. Perrenod, S. (2026). Bitcoin’s Power Law Passes a Major Statistical Test. Substack, 5 June 2026.

5. Mezinskis, M. The Chart. Porkopolis Economics, porkopolis.io.