Bitcoin Power Law Monitor

Pooled next-year error: 0.343 dex (79 log points) over all refits; 0.213 dex (49 points) over the last five. Pooled coverage: 70% of forward days inside ±1σ (a calibrated model gives about 68%), 97% inside ±2σ (about 95%). Exponent across refits 5.33 to 6.79; model-implied level today implied by each refit $64,215 to $1,139,530.

"Power law" also names a different regularity in this literature: heavy, power-law tails in the distribution of Bitcoin returns (Begušić et al., 2018). That concerns the size of daily moves, not the level of price in time, and is not tested here.

The sample is four cycles, whatever the daily count. The fit above uses several thousand daily closes. A trend model’s effective sample is not that number; it is the number of independent regimes the trend has survived, because price today is almost entirely price yesterday. That is four or five, which is the same limit that leaves all but two of the thirty-six rules on the scorecard unjudgeable. “Held for thirteen years” and “five for five” are the same argument in different clothes, and the Pi Cycle page declines the second.

Calibration, read honestly. Pooled across every annual refit, 70% of forward days fell inside that year’s own ±1σ band and 97% inside ±2σ, against the 68% and 95% a calibrated model should give. That looks like vindication and is not quite. The pooled figure averages years in which the bands were far too wide with years in which the model was badly wrong: the worst refit (2015) captured 1% of the following year inside ±1σ with a bias of −0.66 dex, while four refits captured 99–100%. A model that is right on average by being too generous in calm years and badly wrong in transitions is calibrated in the aggregate and unreliable in any given year. The exponent across refits ran from 5.33 to 6.79, and the level those refits implied for today ran from about $64,215 to about $1,139,530.

The band is a factor of four at every horizon. Residual dispersion is constant in log space, so the ±1σ band is always about 4× wide and ±2σ about 16×, in 2027 exactly as in 2040. The model does not lose precision with distance; it never had more than a factor-of-four band to lose. Any single far-future number quoted from this fit — and several circulate — carries that width with it whether or not the person quoting it says so. The extended table below shows the band as a multiple for that reason.

It is one survivor from a pool nobody has counted. Stock-to-flow, rainbow bands, pure exponentials and logistic curves have all been fitted to this series. The power law is the 2013-era form that still describes it; stock-to-flow explained everything up to its 2019 publication and failed within two years. A fit that has held since 2013 is one draw from the models proposed in 2013, most of which are forgotten precisely because they failed. That pool has not been counted, so the base rate for “a curve that held this long” is unknown.

Why the fit starts at day 560. Day 560 from genesis is 17 July 2010, the first day with an exchange price. Before that there is no market and the series is either absent or a handful of over-the-counter trades; including them would fit the line to numbers that were not prices. The origin is genesis rather than that first-price day because the model’s claim is about age since the network began; the exponent-sensitivity panel above shows what each alternative origin does to the fit.

There is no mechanism. The Metcalfe page at least proposes why value should scale with users squared. Nothing on this page explains why the exponent is 5.6 rather than 4 or 8, and no published account does either. A fit with no mechanism has no particular reason to persist through a regime it has not seen. That is the strongest argument against the model, and it is why this page reports a falsification status rather than a forecast.

What it does establish. The model’s substantive claim is not a level but a deceleration: price scales with age to roughly the 5.6th power, so each doubling of bitcoin’s age yields about 5.6 doublings of price, and doublings of age take progressively longer. That predicts that later entrants get less. The portfolio page measured exactly that, independently and out of sample: a 20% bitcoin weight added 12.9 points of annual return to a portfolio started in 2015 and as little as 3.0 points to one started in 2022. It is not a clean confirmation: the uplift does not fall steadily with the entry year, and entries made after a cycle top did worse than those made a year later, so where in the cycle the money went in matters at least as much as when. A structural claim agreeing in direction with a separate measurement is worth reporting, as long as it is not mistaken for a test; two models fitted to the same price series agreeing with each other is not, and this site’s evidence-dependence panel says why.