Orange Law

Bitcoin is not a corporation racing toward saturation and death; it is a city spreading like an epidemic, where users cube with time, value squares with users, and price is simply the network’s age raised to the sixth power.

BitcoinUSD

$76,367−2.31%−48.7% vs fair

−0.96σbelow fair value

2.06× the floor

−2σ−1σFair+1σ+2σ

Floor

$37,047

Fair

$148,851

Top

$598,071

Scale-invariant pathR² 0.962

Scroll or pinch to zoom. Drag to pan. Double-tap clears A–B.

Selected span

Pick two dates, or tap the chart. Future dates use the model path.

FromTo

Enter two dates to measure a span.

Bands are log-normal quantiles around the fitted law (σ = 0.302 dex). Cycle swings of about ±1σ are typical; a print more than 3σ below the fit would falsify the floor.

Your buys

Upload your buy dates + dollar amounts to plot your stack.

Your stack

What is it worth later?

Enter how many bitcoin you own and a date. Each quantile is the model price that day, the matching stack value, and the annualized return versus today.

Bitcoin you own

BTC

Target date

1 BTC on Sep 15, 2031·1,826 days · 5.00 years

$613,162

Fair-value stack·1 BTC × $613,162

+702.9% total

+51.7% / yr

Today $76,367

Floor

$152,607 / BTC

$152,607

+99.8% tot·+14.9% / yr

−1σ

$305,897 / BTC

$305,897

+300.6% tot·+32.0% / yr

Fair

$613,162 / BTC

$613,162

+702.9% tot·+51.7% / yr

+1σ

$1.23M / BTC

$1.23M

+1,509% tot·+74.3% / yr

Top

$2.46M / BTC

$2.46M

+3,126% tot·+100.3% / yr

Band	BTC price	Stack	Total	Annualized
Power law floor2.3th percentile	$152,607	$152,607	+99.8%	+14.9%/ yr
Lower band (−1σ)15.9th percentile	$305,897	$305,897	+300.6%	+32.0%/ yr
Fair value50th percentile	$613,162	$613,162	+702.9%	+51.7%/ yr
Upper band (+1σ)84.1th percentile	$1.23M	$1.23M	+1,509%	+74.3%/ yr
Cycle top (+2σ)97.7th percentile	$2.46M	$2.46M	+3,126%	+100.3%/ yr

Model forwards

5-year path

Sep 15, 2031

1 · 3 · 5 · 10 year paths

From the live price, each band is the model price at that horizon. Returns are versus today, not versus the quantile you sit on now.

Floor$152,607+99.8%+14.9% /yr
−1σ$305,897+300.6%+32.0% /yr
Fair$613,162+702.9%+51.7% /yr
+1σ$1.23M+1,509%+74.3% /yr
Top$2.46M+3,126%+100.3% /yr

Horizon	Floor	−1σ	Fair	+1σ	Top
1y Sep 15, 2027	$50,640 −33.7% tot −33.7% / yr	$101,506 +32.9% tot +32.9% / yr	$203,466 +166.4% tot +166.6% / yr	$407,842 +434.1% tot +434.7% / yr	$817,508 +970% tot +972.2% / yr
3y Sep 15, 2029	$90,321 +18.3% tot +5.8% / yr	$181,046 +137.1% tot +33.3% / yr	$362,902 +375.2% tot +68.1% / yr	$727,426 +852.5% tot +111.9% / yr	$1.46M +1,809% tot +167.2% / yr
5y Sep 15, 2031	$152,607 +99.8% tot +14.9% / yr	$305,897 +300.6% tot +32.0% / yr	$613,162 +702.9% tot +51.7% / yr	$1.23M +1,509% tot +74.3% / yr	$2.46M +3,126% tot +100.3% / yr
10y Sep 15, 2036	$473,963 +520.6% tot +20.0% / yr	$950,046 +1,144% tot +28.7% / yr	$1.90M +2,394% tot +37.9% / yr	$3.82M +4,898% tot +47.9% / yr	$7.65M +9,919% tot +58.5% / yr

Horizon

Floor

−1σ

Fair

+1σ

Top

Sep 15, 2027

$50,640

−33.7% tot

−33.7% / yr

$101,506

+32.9% tot

+32.9% / yr

$203,466

+166.4% tot

+166.6% / yr

$407,842

+434.1% tot

+434.7% / yr

$817,508

+970% tot

+972.2% / yr

Sep 15, 2029

$90,321

+18.3% tot

+5.8% / yr

$181,046

+137.1% tot

+33.3% / yr

$362,902

+375.2% tot

+68.1% / yr

$727,426

+852.5% tot

+111.9% / yr

$1.46M

+1,809% tot

+167.2% / yr

Sep 15, 2031

$152,607

+99.8% tot

+14.9% / yr

$305,897

+300.6% tot

+32.0% / yr

$613,162

+702.9% tot

+51.7% / yr

$1.23M

+1,509% tot

+74.3% / yr

$2.46M

+3,126% tot

+100.3% / yr

10y

Sep 15, 2036

$473,963

+520.6% tot

+20.0% / yr

$950,046

+1,144% tot

+28.7% / yr

$1.90M

+2,394% tot

+37.9% / yr

$3.82M

+4,898% tot

+47.9% / yr

$7.65M

+9,919% tot

+58.5% / yr

Scale-free nature

More Power Laws in Nature

Same log-log plane as Bitcoin. Each line is y = A x^β with the exponent from the source paper — slope is the law. Zipf, city rank-size, and Bitcoin’s on-chain wealth tail sit near β = −1. Kleiber rises; Pareto and Gutenberg–Richter decay at other slopes.

Income & Wealth (Pareto's law)
Pareto, Cours d’économie politique (1896–97)
Word Use Frequencies (Zipf's law)
Zipf, Human Behavior and the Principle of Least Effort (1949)
Earthquake Size & Frequency (Richter)
Gutenberg & Richter 1944; Kanamori 1977 (E ∝ 10^{1.5 M})
City Population Sizes (Auerbach-Zipf law)
Auerbach 1913; Zipf 1949; Gabaix, AER 1999
Animal Metabolic Rates (Kleiber's Law)
Kleiber 1932; West, Brown & Enquist, Science 1997
Bitcoin Wallet Balances (on-chain Pareto)
Kondor et al., PLOS ONE 2014 (power-law tail); Santostasi & Perrenod on-chain α ≈ 1.07; rank-size slope ≈ −1 (Santostasi 2025)

The law

P(t) ∼ t^β

Santostasi and Perrenod show the exponent is not a free fit. Address growth scales as t^3.05 (epidemic spread on a scale-free network) and price scales as N^1.84 (generalised Metcalfe). Their product is the observed β.

β: 5.69
σ: 0.302 dex
R²: 0.961

Fit on daily closes from 17 July 2010 (day 560 after Genesis) through February 2026. Floor is the −2σ quantile; a year below −3σ is the published falsification test. CAD uses daily USD/CAD (FRED DEXCAUS) on historical prints and the live FX rate on forwards. GOLD is the USD power law expressed in troy ounces at the latest gold price, so the path and quantile bands stay aligned (cycle tops on +2σ) and the scale-invariant slope does not bend. Compare overlays use Yahoo Finance adjusted closes (S&P 500 total return, XIC.TO for the TSX, CNQ.TO, NVDA, AAPL, COST) converted into the selected unit the same way bitcoin is, then indexed to bitcoin's price at the left edge of the view so each line is the growth of the same starting dollar.

Model by Giovanni Santostasi · @Giovann35084111. Paper with Stephen Perrenod: A Mechanistic Derivation of the Bitcoin Price Power Law.

Independent educational chart — not affiliated with Giovanni. Questions: schmidtdstephanie@gmail.com · About · Privacy.

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Scale-invariant pathR² 0.962

What is it worth later?

5-year path

1 · 3 · 5 · 10 year paths

More Power Laws in Nature

P(t) ∼ tβ

P(t) ∼ t^β