The Pull Back to Trend

Recovery Projector

Given the current price and the measured mean-reversion rate, this chart shows probabilistic recovery paths over the next two years. The P50 line is the median expectation (half of outcomes above, half below). The shaded band covers P25 to P75 — the central 50% of probability. The trend line is where the power law predicts fair value. Use the input below to model any starting price.

OU process fitted to Bitcoin residuals: half-life = 307 days, long-run σ = 0.200. Bands are Gaussian ±0.674σ conditional on starting position. Actual paths have fatter tails. Source: The Reflecting Barrier.

The Mechanism

Bitcoin's log-price residual — the gap between actual price and the power law trend — behaves like an Ornstein-Uhlenbeck (OU) process. This means it is pulled toward zero (the trend) with a force proportional to how far it has strayed. The further below trend, the stronger the pull.

The mean-reversion rate κ is estimated from the full dataset of daily residuals. The half-life τ = ln(2) / κ = 307 days. This means: given a current log-residual r₀, the expected residual after 307 days is r₀ / 2. After 614 days, r₀ / 4. The process converges on zero — the trend — exponentially.

This is not a prediction. It is a description of the statistical force observed across 15 years of data. The variance around the central expectation is large, especially in the first year. But the direction of the force is consistent: below trend pulls upward, above trend pulls downward.

The 290-day empirical median is measured from near-floor episodes (price below 0.6× trend) to the first trend crossing — consistent with the 307-day OU half-life. The full cycle bottoms in the table below reflect deeper starting points and longer recovery paths. The floor prevents further downside extension while standard mean-reversion drives the return. Source: The Reflecting Barrier (Paper 9).

Every Bear, Every Recovery

Five major drawdowns. Five returns to trend. The table below shows each cycle bottom, how far below trend Bitcoin fell, and how long recovery took. Days to trend is the first daily close above the power law trend line, computed live from the historical dataset.

Days to trend = first daily close with price above power law trend value. Computed from btc_historical.json using the Santostasi model (β = 5.688, logA = −16.493). Cycle labels follow Bitcoin halvings: C1 (pre-2012), C2 (2012–2016), C3 (2016–2020), C4 (2020–2024), C5 (2024–present). The C5 cycle bottom — if any — has not yet been confirmed.

The Near-Floor Signal

When price approaches the power law floor (below 0.6× trend), a statistically significant forward return premium appears. This is documented in The Reflecting Barrier (Paper 9) across 35 near-floor episodes since 2010.

The premium is real but modest. It does not guarantee near-term recovery — some near-floor episodes lasted months before mean-reversion asserted itself. It is a statistical edge across many episodes, not a single-trade guarantee.

Near-floor defined as price below 0.6× trend (residual within ~0.12 log10 units of the floor). Source: The Reflecting Barrier (Paper 9).

What This Doesn't Say

This page shows what the data implies, not what will happen. The distinction matters.

• The 307-day half-life is a central estimate from 15 years of data. Actual recovery could be faster or slower. The variance around the P50 path is large, especially in year one.
• The OU model assumes mean-reversion to zero residual (the trend). If the power law is mis-specified, the attractor is wrong too. The Formal Verification paper gives out-of-sample R² = 0.546 — meaningful, not certain.
• The fan chart uses Gaussian conditional distributions. Actual Bitcoin residuals have fat tails and a downside reflecting barrier. The realized P25 path tends to be less severe than the Gaussian P25 implies.
• Mean reversion is a force, not a guarantee. Macro shocks or a genuine model break could delay or prevent recovery. The homepage scanner defines what a genuine model break looks like: 120+ consecutive days below the floor. That has never happened.
• The near-floor return premium is partly driven by early cycles (2011–2013) when the market was smaller and more volatile. Its magnitude may diminish as markets mature.
• This is not financial advice.