Sheet 1 of 1 · Log–log instrument · β = βA(1 + 1/α)

Bitcoin's power law, measured against itself

The law is fitted on 2010–2020 only. Everything after the split line is a genuine out-of-sample test: the model never saw it. Uncertainty fans use autocorrelation-honest errors (Neff, not N). Forecasts extend the same fit to 2036. Implements Santostasi & Perrenod (2026)[1] and the wealth-decomposition extension of v2.0[2] — sources & context at the foot of the page.

ModelP ∝ (t − t₀)β
Fit window2010-08 → 2020-12
Data through—
Revisionv2.1 · after review
actual (in fit window) actual (out-of-sample) power law (central) honest ±1σ / ±2σ fan historical support (5th pct) 4-yr cycle (descriptive) scroll / pinch = zoom · drag = pan · double-click = reset

Exponent β (honest errors)

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Newey–West-style inflation via Neff from residual AR(1).

Forward test (out-of-sample)

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Mean residual of unseen data vs the 2010–2020 fit.

Now vs model

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Latest actual relative to the central law.

Effective sample

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Independent observations after autocorrelation. This is why the fan is wide.

Decomposition · β = βA × βM = βA(1 + 1/α) [1][2]

—β measured = 3.02βA (adoption, S&P) × —βM implied ⇒ —α implied = 1/(βM−1)
α = 1.0wealth-literature band 1.15–1.25 (green) [5]α = 1.6
Forecast · central law with honest 1σ and 2σ ranges
Date−2σ−1σCentral+1σ+2σ

Ranges combine exponent uncertainty (which widens with horizon) with the historical residual scatter (~cycle amplitude), both inflated for autocorrelation. They are trend-channel ranges — "climate, not weather": the model says nothing about when within a range a cycle top or bottom lands. Not financial advice; a structural break (see the falsifiers in Sources & context below, per §11 of [2]) voids the extrapolation.

Cycle position · reference indicators — weather, not climate

The power law is the trend; these gauges read the oscillation around it — what §11 of the paper[2] calls weather around a climate law. Four lenses — trend extension, the law's own oscillator, miner revenue, and on-chain cost basis — each ranked against its full history, then averaged into one composite. Agreement across lenses is what moves the needle.

feeds: waiting…

DEEP VALUEACCUMULATIONNEUTRALHEATEDEUPHORIA
— percentile of own history

vs power law
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Mayer Multiple
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Puell Multiple
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MVRV Z-Score
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Power-law oscillator

price ÷ central law (log scale)

< 0.6× value> 1.6× heated> 2.8× euphoric
unavailable

Mayer Multiple

price ÷ 200-day moving average

< 0.8 deep value> 1.5 heated> 2.4 euphoric
unavailable

Puell Multiple

daily miner revenue ÷ its 365-day average

< 0.5 capitulation> 1.8 heated> 3.5 euphoric
Puell feed unavailable — bitcoin-data.com and blockchain.info could not be reached

MVRV Z-Score

(market cap − realized cap) ÷ σ of market cap

< 0 deep value> 3 heated> 7 euphoric
MVRV feed unavailable — realized-cap data requires bitcoin-data.com
Oscillator
Price relative to this page's own fitted law — the residual the paper calls "weather." Unique to this page; recomputes when you change the fit or time origin.
Mayer
How stretched price is from its own 200-day trend. Fast and simple, but purely price-derived.
Puell
Miner economics — the supply side. Correlated with Mayer at extremes but driven by revenue, not just price.
MVRV-Z
Market cap against the aggregate price coins last moved at: holder profit/loss — genuinely different information from trend extension.
Composite
Each available indicator's latest value is ranked against its own full history (percentile), then averaged. Thresholds are historical conventions, not guarantees; none of this is financial advice.

What was improved over the paper's straight-line fit

  1. True forward test by default. The headline fit uses 2010–2020 only; 2021-onward actuals are plotted against a prediction the model made "blind". Toggle to full-period to compare.
  2. Autocorrelation-honest fan, not naive OLS. Residual AR(1) is measured on the fitted window, Neff = N(1−ρ)/(1+ρ) is computed live, and both the slope error and the channel width are inflated by √(N/Neff). Naive errors would be ~10–14× too confident.
  3. Time-origin convention made visible. β genuinely moves with t₀ (whitepaper / genesis / first trade). Switch the origin and watch β shift while the implied βM = β/βA and α stay nearly put — the paper's "convention-free invariant" argument, demonstrated live.
  4. Non-parametric support channel. Instead of assuming Gaussian residuals, the 5th-percentile residual of the fit window is carried forward as an empirical "floor" line — the level price has spent 95% of its history above, relative to the law.
  5. Cycle overlay kept out of the model. A 4-year sinusoid fitted to residuals can be toggled on, but it is drawn dotted and labeled descriptive: it narrates the residuals, it does not forecast them.
  6. Live decomposition. Each refit propagates to βM and the implied wealth-Pareto α, pinned against the external literature band [1.15, 1.25] — the paper's central claim, checkable at a glance.
  7. Live data on load. The page fetches full BTC/USD history on open (blockchain.info → Binance → CoinGecko → Coinbase fallback chain), converts it to month-end closes, merges it over the bundled approximations and refits everything. The status chip in the toolbar shows the source and latest price; in sandboxed previews that block network access it falls back to bundled data, and the CSV button remains as a manual override.
Standing caveat from the review: the mechanism behind βM = 1 + 1/α rests on the geometric-penetration postulate (H), which is assumed, not derived; βM here is computed as β/βA with βA = 3.02 taken from Santostasi & Perrenod[1], so it is not an independent measurement. Treat the α pin as a consistency check, not proof.

Sources & context

The empirical law. Since 2010, Bitcoin's USD price has tracked a power law of its age, P ∝ (t − t₀)β with β ≈ 5.6: sixteen years of daily prices spanning six orders of magnitude lie along one line on log–log axes. The law was identified and refined over a decade by Giovanni Santostasi and formalised in Santostasi & Perrenod[1], who also decompose the exponent as β = βA × βM: adoption grows as N ∝ tβ_A with βA ≈ 3 (derived from epidemic-spreading theory, after Colgate's analysis of the cubic growth of the early AIDS epidemic[4]), and price scales with adoption as P ∝ Nβ_M with βM ≈ 1.84 — which they measure but leave unexplained.

The extension this page implements. The v2.0 companion paper[2] proposes the missing mechanism: βM = 1 + 1/α, where α ≈ 1.2 is the Pareto tail exponent of global wealth, measured by economists independently of Bitcoin[5]. The claimed mechanism is geometric penetration of the wealth ranks — each e-folding of adoption reaches a constant factor deeper into the wealth distribution, and the capped supply forces each richer wave to buy from earlier holders. The paper's headline result: measured βM = 1.827 ± 0.091 against the externally predicted band [1.80, 1.87], with cross-asset controls (Ethereum: epidemic but no cap; Litecoin: cap but a retail-tier epidemic) showing the surplus βM − 1 only where the theory requires it. This page's decomposition panel reproduces that check live: β is refitted from data, βM = β/βA is formed with βA = 3.02 from [1], and the implied α = 1/(βM−1) is pinned against the wealth-literature band.

What this page adds beyond the papers. The honest-error methodology (residual autocorrelation → Neff → inflated uncertainty) follows §3–4 of [2], which in turn responds to the "weak structure, strong forecasts" critique of Baquero & Menezes[3]; the time-origin switcher demonstrates their t₀-sensitivity finding and the papers' resolution of it (β inherits a clock convention; βM does not). The forward test replicates Fig. 1 of [2] with live data. The uncertainty fans, non-parametric support channel, and cycle-indicator composite are this page's own additions and appear in neither paper.

Domain of validity (per §11 of [2]): the theory holds while the adoption epidemic keeps climbing wealth tiers and global wealth remains Pareto-shaped with α ≈ 1.2. Its stated falsifiers, all monitorable on this page: βM leaving [1.65, 2.0]; the capital-per-holder schedule flattening; Ethereum developing a surplus without a supply cap, or a capped wealth-climbing asset showing none; global wealth α exiting [1.0, 1.5] without βM responding. The law says nothing about cycles — the 2011/2013/2017/2021 oscillations are residuals around it, which is why the paper describes forecasts here as climate-grade, never weather-grade.
  1. G. Santostasi, S. Perrenod, "A Mechanistic Derivation of the Bitcoin Price Power Law: Network Adoption Dynamics and Generalised Metcalfe Scaling," Nonlinear Science (2026) 100172. doi:10.1016/j.nls.2026.100172 · preprint
  2. wineLightning, Claude AI, "Bitcoin's Price Power Law Decomposed: Epidemic Speed Times Wealth Inequality," v2.0 (July 2026), bitcoin-trajectory.pages.dev; v1.0: doi:10.5281/zenodo.19411140. Source of the βM = 1 + 1/α formula, the honest-error and t₀ methodology, and the cross-asset controls this page mirrors.
  3. C. Baquero, R. Menezes, "Bitcoin's Power Law: Weak Structure, Strong Forecasts," arXiv:2605.21316 (2026). The skeptical result the t₀ scan and forward test on this page speak to.
  4. S. A. Colgate, E. A. Stanley, J. M. Hyman, S. P. Layne, C. Qualls, "Risk behavior-based model of the cubic growth of acquired immunodeficiency syndrome in the United States," PNAS 86 (1989) 4793–4797. Origin of the t³ epidemic class behind βA ≈ 3.
  5. Wealth-inequality band α ∈ [1.15, 1.25]: V. Pareto, Cours d'économie politique (1896); J. B. Davies et al., Economic Journal 121 (2011) 223–254; T. Piketty, Capital in the Twenty-First Century (2014); Credit Suisse / UBS, Global Wealth Report (2023) — as compiled in [2].

Cycle-indicator conventions: Mayer Multiple after Trace Mayer; Puell Multiple after David Puell; MVRV after Murad Mahmudov & David Puell, Z-score variant after "Awe & Aat" — thresholds are community conventions ranked here against each indicator's own history, not predictions. The power-law oscillator is native to this page. Data: CryptoCompare, blockchain.info, CoinGecko, Coinbase, Binance (price); bitcoin-data.com / BGeometrics (on-chain). Nothing on this page is financial advice.

Update monthly price data

Paste rows of YYYY-MM,price (or YYYY-MM-DD,price). New months are appended, overlapping months are replaced, and the whole model refits. Data stays in this page — nothing is uploaded.