Two Things Called "Power Law"
Open any Bitcoin chart aggregator and a "Long-Term Power Law" overlay greets you: a straight line on a log-log plot, drawn through every closing price since 2010. The fit is excellent. The R2 sits comfortably above 0.95. In popular Bitcoin discourse, this is the Power Law.
In statistics and complex-systems physics, "Power Law" means something else. It refers to the distribution of a quantity, not to its trajectory in time. Earthquake magnitudes follow a Power Law. Word frequencies in natural language follow a Power Law. Wealth, city sizes, solar flare intensities: all Power Laws in the distributional sense. None of these are charts of "size against time." They are histograms of "how often a magnitude this large occurs." The two usages share an equation and almost nothing else.
The confusion matters because the statistical Power Law carries weight. It is the language of Pareto, Mandelbrot, and self-organising criticality. Borrowing the term for a deterministic curve fit attaches a credibility surplus the curve has not earned. Worse, it points readers at the wrong object. Bitcoin does exhibit real Power Laws, just not the one being marketed.
What a Statistical Power Law Actually Is
A statistical Power Law is a probability distribution whose tail decays as P(X > x) ∝ x−α for x above some threshold xmin, with tail index α > 0. Three properties define it.
Scale invariance. Rescale x by any positive constant and the shape of the tail is unchanged. Doubling the threshold reduces the survival probability by a fixed multiplicative factor, regardless of where you start on the tail. The distribution has no characteristic size. It looks the same at every magnification.
Heavy tails. Extreme events do not vanish at Gaussian speed. A Pareto tail decays polynomially, not exponentially, so events many standard deviations from the mean are not impossible. They are rare, and they do occur.
Diverging moments. If the tail index α is at or below 2, the variance is infinite. If α is at or below 1, even the mean is infinite. When that happens, the familiar tools of Gaussian statistics (Z-scores, confidence intervals, the standard form of the central limit theorem) either weaken or fail outright.
This is why "Power Law" is a strong claim. It carries implications about the behaviour of the system: that averages mislead, that single events can dominate decades of data, that volatility is structural rather than transient.
What Bitcoin's Price-vs-Time Chart Actually Is
The Mezinskis chart fits the model P(t) = a · tn to Bitcoin's daily closing price, where t is measured from the genesis era and the best-fit exponent is roughly n ≈ 5.7. This is a power function, not a Power Law distribution. It is a curve fit. It says nothing about scale invariance of returns, heavy tails, or diverging moments. It says only this: across more than thirteen years of data, log price grows roughly linearly in log time. That is a description, not a law.
The same mathematical form, y = a · xn, appears throughout science and nobody calls those Power Laws. Kepler's third law: orbital period squared scales with semi-major axis cubed. Kleiber's law: metabolic rate scales with mass to the three-quarters. Surface-to-volume relationships in geometry. These are power functions because they involve xn. They are not Power Laws in the statistical sense, and nobody confuses them with one. Bitcoin's trendline deserves the same treatment.
It is an empirical regression with high goodness-of-fit. It is descriptive, not derived from first principles. It could continue, or it could break. As Mezinskis himself notes: "Though regression models 'predict' a trend, even if a strong goodness of fit exists, there is not much more to say than that. The trend could hold as a helpful guide, or it could break." A true law does not break.
Where Bitcoin Actually Does Follow a Power Law: The Return Tail
To know whether Bitcoin obeys a genuine statistical Power Law, the right object to examine is a distribution, not a time series. Daily log returns offer 5,782 observations from 19 July 2010 to 19 May 2026.
The standard moments come first. The mean daily log return is +0.24%, with a standard deviation of 4.7%. Skewness is −0.43. Excess kurtosis is 15.7, against a Gaussian baseline of 0. That alone marks the distribution as wildly non-normal: a Gaussian with the same mean and standard deviation would never produce the observed series.
The single worst day in the sample is a log return of −49.2%, a move 10.5 standard deviations from the mean. Under a Gaussian, the probability of a deviation that large sits on the order of 10−26. In 5,782 days, the expected count is, for all practical purposes, zero. The actual count is at least one.
To test the tail directly, a Power-Law model was fitted to the absolute returns using the Clauset–Shalizi–Newman maximum-likelihood method. The procedure sweeps a candidate threshold xmin, fits a Pareto tail for each, and picks the cutoff that minimises the Kolmogorov–Smirnov distance between the empirical tail and the fitted distribution. The result:
Positive tail. Above 10.9% daily moves (n = 120 events), the tail behaves as P(|r| > x) ∝ x−2.86. KS distance against the Pareto fit: 0.044.
Negative tail. Above 10.8% daily moves (n = 106 events), the tail behaves as P(|r| > x) ∝ x−2.62. KS distance: 0.045.
KS distances below 0.05 indicate excellent agreement with the Pareto model over more than two decades of return magnitude. The tail of Bitcoin's daily return distribution is, by any reasonable statistical standard, a Power Law.
For reference, equity return tails famously obey what Gopikrishnan, Plerou, and Stanley called the "inverse cubic law": tail index ≈ 3. Bitcoin's tail index of 2.6 to 2.9 is slightly heavier than equities. Variance is technically finite (tail index > 2) but only just. The Pareto fit puts Bitcoin much closer to Mandelbrot's "wild" randomness than to the Gaussian "mild" regime.
Where Bitcoin Follows Another Power Law: Volatility Memory
Beyond the return distribution itself, there is a second genuine Power Law in Bitcoin data: the persistence of volatility.
The autocorrelation of |log returns| at lags 1, 2, 5, 10, 20, 50, 100, 200, and 500 days runs as follows: 0.35, 0.30, 0.21, 0.19, 0.14, 0.06, 0.10, 0.07, and roughly zero. Volatility today predicts volatility tomorrow strongly, predicts volatility next month moderately, and still leaks information one hundred days out.
Regress log autocorrelation on log lag and the slope is −0.33 with R2 = 0.89. Autocorrelation is decaying as lag−0.33, not exponentially. This is the long-memory signature: hyperbolic decay, scale invariance in the lag dimension, a Power Law in time itself.
This is what volatility clustering looks like in mature financial markets, and it has a name in econometrics: long-memory volatility. ARCH and GARCH models capture short-horizon clustering. The full picture, as documented across decades of empirical finance, is a slow polynomial decay: a Power Law in the lag, distinct from the Power Law in the return magnitude.
So Bitcoin exhibits at least two genuine statistical Power Laws: one in the return distribution, one in the persistence of volatility. Neither is what the popular "Power Law" chart claims to be.
What Mezinskis Actually Calls It
It is worth reading Mezinskis on his own terms. Across his writing on Porkopolis Economics, he uses "power trendline," "power curve," and "power trend." He describes the analysis as a regression with strong goodness-of-fit. He explicitly notes that regression-based projections are useful guides, not certainties, and that the trend "could hold as a helpful guide, or it could break."
That is the honest framing. It says: here is a curve, here is its fit, here is what it implies if it continues, and here is the standing disclaimer that empirical fits are not laws. The phrase "Power Law" attaches a credibility surplus from physics, the suggestion that this curve is as inevitable as gravity. The curve does not deserve that surplus. The volatility and the return tails do.
So What: The Practical Consequences
If the only payoff of this analysis were a vocabulary correction, it would not be worth writing down. The Power Laws in the return distribution and in the volatility persistence carry practical consequences. Three of them shape how the dashboard is built and how it should be read.
1. Drop the Gaussian for tail-risk thinking. The fitted Pareto and the empirical record agree that a 20% daily move has a probability on the order of 1% per day, or about twice a year. A Gaussian fit to the same mean and standard deviation puts that probability at roughly 2 in 100,000 per day: once a century. A 49% day, like the one Bitcoin actually had, is, for all practical purposes, zero under Gaussian and is consistent with the fitted tail's once-in-a-decade-or-two rate. Standard risk machinery built on Gaussian assumptions (normal-distribution VaR, Sharpe ratio as a sufficient statistic, volatility-targeted leverage) systematically underprices the events that actually matter. The right reference distribution for Bitcoin position sizing is the Pareto tail, not the bell curve.
2. Volatility clustering is the case for mechanical pacing. The autocorrelation of |returns| does not collapse after a few days. It still carries measurable signal at lag 100. That is what long-memory volatility means in practice: once a high-volatility regime begins, it persists. The instinct after a violent week is to wait for things to calm down before resuming purchases. The math says the calm is a long way off, and the next violent week is more likely than baseline. Mechanical, regime-blind pacing is not just emotionally easier; it is the response that matches the actual dynamics of the variance process. Discretionary "let me see what happens" timing is, on average, fighting the autocorrelation.
3. It validates the low-vol-is-buy direction on the Volatility 30d indicator. The dashboard weights rolling 30-day volatility at 3.7% on the composite and treats low realised volatility as a (mild) buy signal. The long-memory result is what justifies that direction: low-vol regimes persist, on average, into more low-vol; high-vol regimes persist into more high-vol. The signal is not predictive of price, but it is predictive of future variance, and lower future variance is a regime in which DCA-style accumulation is less likely to be overrun by an adverse cluster. The small weight reflects honest uncertainty. The autocorrelation is real but modest, and the direction is the one the data supports.
What this does not do is generate a new indicator. A tail-fitted "expected shortfall" or rolling-kurtosis flag was considered for the composite. The sampling error on a tail-index estimate from a few hundred extreme events is large enough that the resulting indicator would be noise dressed up as signal. The fitted tail informs interpretation; it does not deserve its own row on the dashboard.
The Power Trendline, Properly Understood
The Power Law Position indicator keeps its existing name on the dashboard: it is the term readers will recognise, and SHAP ranks the underlying signal #1 across all four walk-forward rounds (see the SHAP analysis). What changes is the mental model. Treat the trendline as a regression with strong but mortal fit, not as a physical law. The fit could continue, or it could break; mean reversion to it is the usable signal regardless.
The picture that ties this together is the one above. The dashed Gaussian curve dives off the bottom of the chart well before 20% daily moves, declaring that Bitcoin's actual worst days cannot happen. The two solid lines, the empirical positive and negative tails, run straight all the way to 49%. That straightness is the warning. It is why "this is impossible" is the wrong sentence to use about Bitcoin returns, why position sizing matters, and why mechanical DCA exists in the first place.
The trend is a power curve. The volatility is the real Power Law.