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Bitcoin Power Law Explained: Why a Straight Line on a Log-Log Chart Is Controversial

Chainmeter · ~3 min read

TL;DR

Power Law models Bitcoin price as roughly proportional to time since genesis raised to a fixed exponent, which appears as a straight line when both price and time are plotted on logarithmic axes. It has tracked price reasonably well for over a decade, but it's a curve fit to history, nothing close to a model derived from a causal mechanism — and its corridor gets redrawn whenever price exits it.

The math, briefly

Power Law regression fits the equation price = a × days_since_genesis^b to Bitcoin's full historical price series, solving for the constants a and b via log-log linear regression (taking the log of both price and days converts the power relationship into a straight line, which ordinary linear regression can fit). The result is plotted as a corridor — a central fitted line plus upper and lower bands representing historical deviation around it.

Why it looked compelling

Unlike models pegged to fixed calendar dates or specific price levels, Power Law's log-log framing means it naturally accommodates orders-of-magnitude price growth over a long timescale without needing constant rescaling. For much of Bitcoin's history, price has stayed roughly within a wide band around the fitted line, which is part of why the model attracted a following.

The core objection

A power law relationship between price and time isn't derived from any economic mechanism — it's a curve chosen because it fits reasonably well after the fact, out of many possible curves that could be tried. When price has broken outside the corridor (as it has, historically), the standard response has been to refit the corridor around the new data. The breach itself never gets treated as evidence the model was wrong. A model that gets redrawn every time it fails can never actually be falsified, which is the opposite of a scientific forecast.

How this site handles the refit problem

Rather than pretending the model is static, this site's Power Law page refits the regression live, on every page load, from the complete available price history — and says so explicitly. That doesn't fix the underlying methodological issue (it's still a curve fit, not a causal model), but it avoids the worse practice of quietly redrawing a "fixed" model behind the scenes without disclosure.

Today's price against the power law corridor — refit live from the full price history, every refit disclosed — sits on the Power Law chart.

FAQ

Is Power Law more reliable than the Rainbow Chart?
They're built on similar log-log regression logic and share the same core objection (curve-fit, not causal). This site treats both as Tier 4 — statistical, fitted — the lowest-confidence tier, alongside full disclosure of their weaknesses.
Why does Power Law use days since genesis instead of calendar dates?
Time-since-genesis, as opposed to a fixed calendar date, is what makes the relationship scale-invariant on a log-log plot — it's a property of the specific mathematical form being fitted, with no claim about causation attached.
Has Power Law ever failed to predict price correctly?
Yes, on multiple occasions price has traded outside the fitted corridor's bands. Each time, the corridor has historically been redrawn to accommodate the new range; the breach itself was never logged as a falsified prediction.