Power-law corridor · fit live from CoinGecko
Bitcoin Power-Law Corridor
Price modeled as P = A · (days since genesis)n, fit by least squares in log-log space.
n = 5.217 A = -14.806 (log₁₀) R² = 0.912 σ = 0.271 fit 2013-04-28 → 2026-06-19 (4,800 days)
The corridor
log price vs log time · both axes logEmbed this tool
Paste this snippet on your site to embed the live tool. Data refreshes on every load.
Future projection · next 10 years
linear time · log priceExtrapolation, not a forecast. The corridor widens with time as adoption, regime, and model risk all compound.
Price ÷ fair value
how stretched the market is, in multiples of trendYears ahead / behind trend
time-shift to where the trend equals today's priceValue any date
Pick a date, past or future, to read the model's fair value and corridor for that day.
−1σ floor
Fair value
+1σ ceiling
Days since genesis
extrapolated 0.1 yr past the data, treat the band as a floor on uncertainty, not a forecast.
Model assumptions
- Functional form
P = A · tnwitht= days since the Bitcoin genesis block (3 Jan 2009). On a log-log axis this is a straight line with slopenand interceptlog₁₀ A.- Fit method
- Ordinary least squares on
log₁₀(price)againstlog₁₀(days). Equal weight per observation; no robustness weighting, no kernel smoothing. - Fit window
- 2013-04-28 → 2026-06-19 (4,800 daily closes). Pre-2013 prices are excluded because early exchange data is thin and dominated by a single venue, which would over-anchor the slope.
- Fitted parameters
- n = 5.217, log₁₀ A = -14.806, R² = 0.912, σ (residual std-dev in log space) = 0.271.
- Corridor
- ±1σ and ±2σ bands are the fair-value line multiplied by
10±kσ. They describe the dispersion of historical residuals, not a probability interval — residuals are autocorrelated and not normally distributed. - Data sources
- Daily closes pulled from CoinGecko's public price endpoint; spot price refreshed live on page load. Historical sample stored in-page so the fit is reproducible without a server round-trip.
- Refit cadence
- The static fit is recomputed periodically, not on every page load. Spot price and the today marker are live; the trend line itself moves only when the underlying fit is re-published.
- What the model ignores
- Halvings, regulation, ETF flows, macro liquidity, adoption ceilings, regime breaks, and survivorship. The straight log-log line is an empirical regularity over the fitted window, not a law of nature.
What this leaves out
- Window-sensitive. This fit uses CoinGecko data from 2013 on, so n lands near 5.2, below the canonical ~5.8 that includes 2009–2012. Where you start the clock changes the answer.
- Log-log flatters. Compressing nine orders of magnitude makes most monotonic series look linear. A high R² is partly the transform, not proof of a law.
- No mechanism. A and n are fit to data with a plausible story (adoption ∝ t³, value ∝ users²) attached, not derived from first principles.
- Trend, not timing. The corridor says roughly where the climate sits. It says nothing about next week's weather, and the bands widen the further you extrapolate.
FAQ
How is the power-law corridor different from the rainbow chart?
The colors are the least of it. The real difference is the functional form each one commits to, and that drives everything downstream.
The rainbow chart is a logarithmic regression: log price against linear time. The fit line is price = a·ln(time) + b, and the colored bands are placed above and below it at arbitrary offsets, tagged with sentiment labels ("fire sale," "FOMO," "maximum bubble"). The bands aren't derived from anything, they're hand-set to widths that happen to have bracketed past tops and bottoms, and they get recalibrated as new highs print.
The power law plots log price against log time, where time is network age. Under a power law, price ∝ timen, so log(price) = n·log(time) + c, a straight line on a log-log axis. The exponent comes out around 5–6 empirically. The corridor is two power laws sharing that same exponent: support and resistance are parallel lines on the log-log plot, both falling out of the same equation rather than drawn on by hand.
The mechanical tell is the geometry. On a log-log chart, the power law is a straight line while the rainbow's log-linear fit is a curve that keeps bending, because they assume different things about how price relates to time.
Why the power law is taken more seriously, four reasons:
- Derived bands, not decorated ones, same slope, different intercept, no manual tuning.
- Stable and parsimonious, two parameters over the entire history; new data barely moves it. The rainbow visibly gets redrawn each cycle, which is the giveaway that it's fit to history rather than forecasting from it.
- A claimed mechanism, Santostasi ties the exponent to a feedback loop among adoption, price, and hashrate/security. An actual attempt to explain why a power law would show up. The rainbow offers none.
- Falsifiable, a straight log-log line makes a specific out-of-sample claim you can measure deviation against. Arbitrary bands that get redrawn can't really be wrong.
Where both fail the same way
Each is a single curve fit to one asset's one surviving timeline with roughly fifteen years of data. Log-log linearity is suggestive, not proof of the mechanism, power laws are famously easy to eyeball and overfit across a limited range, and the exponent shifts depending on which genesis date you anchor to. Taken literally, the power law implies smooth, decelerating growth forever, with no adoption ceiling and no path to the regime break that eventually kills most assets. The rainbow at least wears its unseriousness openly. The power law's real danger is the opposite: the straight line and the physics analogy make an empirical regularity feel like a law of nature.
One nuance: many newer "rainbow" charts have quietly rebased themselves on the power law underneath, what's labeled a rainbow is sometimes a power-law corridor with sentiment colors painted over it. The original distinction is log-linear regression with drawn-on bands versus a log-log power law with a corridor derived from the fit.
The honest framing: the power law is the more disciplined model, derived corridor, falsifiable slope, a stated mechanism, and still not a promise.
What does the 10-year future projection actually show?
It extrapolates the fitted power law forward and draws the ±1σ and ±2σ residual bands around it. The fair-value line is what the model says price would be if the historical relationship between price and network age continued unchanged.
It is not a forecast. Nothing in the model accounts for halvings, regime changes, regulation, or adoption ceilings, and the corridor only reflects past dispersion, not future uncertainty.
Treat the bands as a floor on uncertainty, not its true width. The further out you read, the more the fit's two parameters are doing work they were never asked to do.
How do I read the price ÷ fair value ratio chart?
The ratio chart divides spot price by model fair value at each date, so the trend line collapses to a flat 1× and every cycle peak or trough becomes a clear excursion above or below it.
The dashed bands sit at 10σ and 10-σ, which is roughly 1.86× and 0.54× given σ ≈ 0.27. Historical tops have printed near 3–4× and bottoms near 0.4–0.5×.
Reading the ratio is easier than eyeballing distance on a log-log chart, because the y-axis is already in multiples of trend.
What does "years ahead" or "years behind" the trend mean?
For any historical price, we solve the power law for the date at which the trend would equal that price, then plot the difference between that date and the actual date.
Positive means price is ahead of where the trend predicts — the market has "pulled forward" future fair value. Negative means price is behind.
Because the trend grows steeply, even large price moves translate to a small number of years. Typical cycle tops sit ~1–2 years ahead, bottoms ~1 year behind.
Same data as the ratio chart, viewed along the time axis instead of the price axis. Some people find time-shift more intuitive than multiples.
How should I interpret the σ (sigma) deviation number?
σ is the standard deviation of log-price residuals around the fit line. A deviation of +1σ means today's price is one residual-standard-deviation above the trend in log space — roughly 1.86× fair value.
±2σ has historically marked the outer edge of cycle excursions, but it is not a probability guarantee. A true law would expect price inside ±2σ about 95% of the time only if residuals were normally distributed, and they aren't — they cluster around peaks and troughs in long autocorrelated runs.
Use σ as a relative gauge of stretch, not as a probability.
Why does the exponent n change depending on which start date you pick?
Power-law fits are sensitive to the early-history window. Including 2009–2012 prices (when bitcoin traded under $30) pulls n up toward the canonical ~5.8 because those low early values anchor a steeper slope.
Starting the fit in 2013 — when liquid exchange data is more reliable — gives n closer to 5.2.
Neither is "right". The slope is an empirical regularity over a chosen window, not a constant of nature, and that is one of the model's real limits.
Used together with
Cross-check corridor reads against cycle indicators and DCA outcomes — both companion tools share the same 95% likely-range view so a single call stays consistent.
Methodology
Fits a log-log linear regression of BTC price against days since the Bitcoin genesis block (3 Jan 2009), then derives three views from the same fit: a corridor around the trend, a price-÷-fair ratio, and a years-ahead-of-trend time-shift.
- Panel 1 · Power-Law Fit
- log₁₀(P) = n · log₁₀(t) + log₁₀(A)
- Panel 1 · Equivalent Form
- P = A · t^n (n ≈ 5.22, fit window 2013→today)
- Panel 1 · OLS Objective
- min Σ (log₁₀(P_i) − ŷ_i)²
- Panel 1 · Residual Std-Dev
- σ = √(Σ r_i² ÷ (N − 2)) (σ ≈ 0.271)
- Panel 1 · Corridor Bands
- P_band(t, k) = A · t^n · 10^(k·σ) (k = ±1, ±2)
- Panel 1 · Deviation Today
- dev = (log₁₀(P_spot) − log₁₀(P_fair)) ÷ σ
- Panel 2 · Price ÷ Fair Ratio
- ρ(t) = P(t) ÷ (A · t^n)
- Panel 2 · ±1σ Ratio Bands
- ρ_band(k) = 10^(k·σ) (k = ±1)
- Panel 3 · Implied Date for a Price
- t̂(P) = 10^((log₁₀(P) − log₁₀(A)) ÷ n)
- Panel 3 · Years Ahead / Behind
- Δy(t) = (t̂(P(t)) − t) ÷ 365.25
A straight line on a log-log chart. The fit window is 2013-04-28 onward; including 2009–2012 pulls n closer to ~5.8.
Price grows as time-since-genesis raised to the power n. Doubling t multiplies P by 2^n.
Equal weight per daily close; no robustness weighting or kernel smoothing.
Standard deviation of log-price residuals around the fit. Drives the corridor width.
±1σ corresponds to ≈ ×1.86 and ÷1.86 of fair value. ±2σ ≈ ×3.46 and ÷3.46. Not a probability interval — residuals are autocorrelated.
Reported in σ units. +1σ means today's price is ≈ 1.86× fair value.
Divides price by trend, so the fair-value line collapses to a flat 1× and excursions become directly readable as multiples.
Horizontal lines at ≈ 1.86× and 0.54×. Past cycle tops printed ≈ 3–4×, bottoms ≈ 0.4–0.5×.
Inverts the power law: for any price, the date at which the trend would equal that price.
Positive = price ahead of trend (market pulled forward fair value). Negative = behind. Cycle tops sit ~1–2 yr ahead; bottoms ~1 yr behind.
Variables
- P
- BTC spot price (USD)
- P_fair
- Model fair value at time t — equals A · t^n
- t
- Days since the Bitcoin genesis block (3 Jan 2009)
- n
- Power-law exponent (slope on log-log axes); fitted ≈ 5.22
- A
- Power-law coefficient; fitted log₁₀(A) ≈ −14.806
- σ
- Std-dev of log₁₀ price residuals around the fit; ≈ 0.271
- k
- Band multiplier in σ units (±1, ±2)
- ρ
- Price-÷-fair ratio shown in Panel 2
- t̂
- Implied days-since-genesis at which the trend equals a given price
- Δy
- Years ahead (+) or behind (−) the trend (Panel 3)
- N
- Number of daily closes in the fit window (4,800)
All three panels are views of the same single fit — they do not introduce new parameters. The exponent and intercept depend on the chosen start date; this fit anchors at 2013-04-28 because earlier exchange data is thin. The model ignores halvings, regulation, ETF flows, adoption ceilings, and regime breaks; treat the corridor as a regime gauge, not a forecast.