Reproducing the Bitcoin Power Law
Independent parameter estimation and floor methodology. Every number derived from scratch. Every choice documented.
Why Reproduce?
The Bitcoin Observatory uses parameters from bitcoinpower.law: beta = 5.688, logA = -16.493. Every tool on this site, from the floor calculator to the retirement planner, depends on these numbers. We had never independently verified them against the raw data.
Meanwhile, published parameter values vary. Santostasi reports ~5.82 in his Monte Carlo paper. His website rounds to 6.0. Plan C uses quantile regression with different parameters per quantile. Which numbers are right? And what does "right" even mean when the answer depends on methodology?
This paper runs the simplest possible regression (unweighted OLS on exchange data only) and documents everything: what we got, how it compares, and where the differences come from.
The Fit
Standard OLS regression on log10(price) vs. log10(days since genesis). 5,674 daily closes from July 2010 to January 2026. No weighting. No pre-exchange data. No tricks.
Figure 1: Log-log regression with daily closes colored by halving cycle. Bottom panel shows residuals.
Beta = 5.694. Within 0.006 of the production value. No parameter update needed. The existing calibration is independently validated.
Parameter Stability: The Spurious Regression Defense
The most common scholarly objection: regressing two trending series produces fake R-squared. If the power law is spurious, the parameters should drift as data is added. We tested this with expanding-window regressions, adding one year at a time.
Figure 2: Beta convergence. Volatile early, stable since 2016. Santostasi's 5.82 shown for reference.
Beta is wild in 2011 (8.63, only 532 data points). By 2016 it settles into the 5.6-5.9 band. From 2023 onward: 5.694-5.699. Stable within 0.005 through a halving cycle, a blow-off top, an 80% drawdown, and a recovery.
R-squared increases monotonically from 0.778 to 0.961. Sigma decreases monotonically from 0.383 to 0.303. A spurious regression does not improve with more data. A real relationship does.
Figure 3: Full stability panel. Beta converges, R-squared rises, sigma falls. All monotonic post-2016.
Volatility Decay: Confirmed from a New Angle
Partition the residuals by halving cycle. Compute sigma for each. If volatility is decaying, sigma should shrink.
Figure 4: Sigma by halving cycle. Monotonic decay. *Cycle 5 incomplete (accumulation phase only).
Same finding as Paper 1. Different method. Simpler metric. Same conclusion: the distribution is compressing.
The Floor Question
The floor is trend multiplied by 10(-2 * sigma). Which sigma? This is the most consequential methodological choice in the entire Observatory.
| Method | Sigma | Floor Price | % of Trend |
|---|---|---|---|
| Full dataset | 0.303 | $31,399 | 24.8% |
| Cycle 4 only | 0.250 | $40,011 | 31.6% |
| Cycles 4+5 | 0.211 | $47,916 | 37.8% |
| powerlaw.js (production) | 0.200 | $50,460 | 39.8% |
Trend today: $126,749. Highlighted row: current production value.
Figure 5: Floor price under four sigma methods.
We use recent-cycle sigma (0.200) because volatility decay is structural. Including early-cycle volatility means the floor reflects a market that no longer exists. But we document the full range. The conservative floor ($31,399) is always available.
Disclosure: Recent-cycle sigma makes the floor dependent on the decay hypothesis. If decay reverses, the floor widens. This dependency is accepted and documented.
Why 5.694, Not 5.82?
Our beta differs from Santostasi's by 0.126. Three likely sources: data window (we exclude pre-exchange data), fitting methodology (we use unweighted OLS), and vintage (our 2018 fit gives 5.873, close to his 5.82. Beta converges downward as data accumulates).
Santostasi's forthcoming book The Physics of Bitcoin (April 2026, 388 pages) will document his canonical methodology. When it publishes, we will compare and update this paper if needed.
Three Research Programs. One Conclusion.
Santostasi (OLS): the floor holds, price reverts to trend. Plan C (quantile regression, March 2026): no decay at Q1, significant decay above. Eight decay functions converge at the same present-day fair value. Scale Invariant Capital (this paper + Paper 1): per-cycle sigma decays monotonically, ceiling compresses 2.2x faster than floor rises.
The floor is the stable attractor. The distribution compresses onto it from above.