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, what the trend was at the time, 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.

Cycle Trajectories in Trend-Multiple Space

Every drawdown chart you have seen plots price, or percent-off-the-peak. Both hide the one thing that actually makes cycles comparable: valuation. Here each cycle is drawn in trend multiples — price divided by the power law trend — so the relentless long-run growth is divided out and the cycles can be laid on top of one another. The y-axis is the same unit the rest of this site speaks: 1.0× is fair value, 0.432× is the floor, above 2× is euphoria.

Lines are anchored at each halving — the one exogenous, objective clock Bitcoin has. The pattern that emerges is the headline: every cycle launches from near the floor, overshoots, and decays back — but each peak is lower than the last. That is the diminishing-amplitude story the power law predicts (peak height decaying with age), made visible.

Trend multiple = price ÷ power law trend (Santostasi: β = 5.688, logA = −16.493). Peak = highest multiple in the epoch; trough = lowest multiple after that peak. Align anchors day 0 at each cycle’s halving, its peak (the bear descent), or its bottom (the recovery crawl upward). Only halving-anchored epochs are shown (2012, 2016, 2020, 2024) — the 2010–2012 cycle has no prior halving. Three completed cycles plus one in progress is a small sample: this is a descriptive view, not a forecast. The trend and floor are themselves fit to this data, so multiples near the endpoints are not fully independent.

Recovery Comparison — How Long Back to Trend?

The previous chart aligned whole cycles at their halving. This one zooms into the part that matters when you are underwater: the recovery leg. Day 0 for every cycle is its deepest point below trend — the lowest price-to-trend multiple it ever reached — and the line traces the climb from there back to fair value (1.0×). Divide out the long-run growth and the question becomes simple and comparable: from the bottom, how many days back to trend?

Every completed cycle bottomed in the same narrow band — roughly 0.37× to 0.44× trend — and every one climbed back. But the speed varied by nearly a factor of three. The current cycle sits at day 0 right now, at its own deepest-below-trend point. The projections below replay each past recovery from that bottom to answer your question: if this cycle recovers like a previous one, when does it reach trend — and where is trend by then? Flip the chart’s Units toggle to USD price to see each recovery speed in dollars — a faster climb reaches trend sooner but at a lower price, a slower one later but higher, because the trend keeps rising the whole way.

If this cycle recovers like…

Day 0 = each cycle’s lowest price-to-trend multiple (deepest point below trend), found after that cycle’s peak within its halving epoch. Days-to-trend = calendar days from that bottom to the first daily close back at or above the power-law trend (1.0×), traced across the full dataset. Projected dates add each historical recovery’s duration to the current cycle’s bottom (computed live); “trend price then” is the Santostasi trend value on that future date (β = 5.688, logA = −16.493). Three completed recoveries is a small sample — this is a descriptive analogy, not a forecast. If Bitcoin sets a deeper low, day 0 resets and the clock restarts.

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.