Guide · btccalcs.com
Bitcoin Rainbow Chart vs. Power-Law Corridor: which long-term model is better?
Published 2026-07-28
The Bitcoin Rainbow Chart is a semi-log (log-linear) fit that maps price onto colored sentiment bands — useful as a visual heuristic but not derived from a stated statistical model. The Power-Law Corridor is a log-log ordinary least squares regression of price against time-since-genesis, with ±1σ and ±2σ envelopes computed from residuals — the same functional form that Giovanni Santostasi and others have shown empirically fits Bitcoin's long-run price trajectory. For serious long-term valuation the Power-Law Corridor is the more disciplined choice: it produces a testable point estimate of fair value at any date, quantifies uncertainty as bootstrap confidence intervals, and doesn't rely on hand-tuned color bands.
Mechanism
Rainbow Chart: log(P) = a + b·t, where t is calendar time. It's a straight line in semi-log space with colored bands drawn at fixed offsets from the fit line. The bands are labels ("HODL", "FOMO", "Bubble") rather than statistical quantiles. Power-Law Corridor: log10(P) = a + n·log10(d), where d is days since the genesis block. Both axes are logarithmic, so the fit is a straight line in log-log space and a power curve P = 10^a · d^n on linear axes. The envelopes are ±kσ of OLS residuals in log space, which translate to multiplicative bands on price. btccalcs.com adds a residual block bootstrap that refits (a, n) across resampled residuals to produce genuine 95% confidence intervals on the corridor itself.
Detail
Functional form matters. A log-linear fit implies constant exponential growth forever — every year Bitcoin's fair value multiplies by the same factor. A log-log fit implies decelerating growth: the growth rate falls as the network ages, which is what actual adoption curves and Metcalfe-style network-value arguments predict. Over Bitcoin's 15-year history, log-log fits the data materially better than log-linear, and the gap widens with each additional cycle.
Bands versus intervals. Rainbow bands are aesthetic offsets; they do not answer "how confident are we that price will be inside this range?" The Power-Law Corridor's ±1σ and ±2σ envelopes are computed from residual dispersion, so they carry a real interpretation: historically, price has spent roughly 68% of days inside ±1σ and 95% inside ±2σ. Add the bootstrap and you also get uncertainty about the corridor's own position.
What each is good for. The Rainbow Chart is a fine at-a-glance mood ring — it communicates "are we hot or cold?" to a general audience. The Power-Law Corridor is what you use when a decision depends on the answer: rebalancing thresholds, treasury floor prices, retirement-age modeling, or any workflow where you need a number and an honest range around it.
Where the Power-Law model breaks. It assumes the log-log relationship persists. If adoption saturates, if regulation forces a structural break, or if a competing protocol displaces Bitcoin, the historical fit stops predicting the future. The corridor is a base rate, not a guarantee — which is why btccalcs.com shows the confidence bands alongside the point estimate and lets you toggle them off when you want the clean central case.
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What this leaves out. Educational content based on public filings and market data as of the published date. Not investment, accounting, tax, or legal advice. Verify all figures against primary sources before acting.