Version 1 of this post introduced two indicators - Law Health and the Law Stress Oscillator - built on the right instinct: the power-law trend line acts on price like a restoring force, and both how far and how long price strays from it matter. The instinct survives. The arithmetic did not.
This revision rebuilds both indicators in log space where they are scale-invariant and stationary, tests the "duration matters" hypothesis against sixteen years of data with no look-ahead, retires two seductive ideas that failed testing, and connects what remains to a branch of physics that solved this exact problem a century ago.
The original Health & Stress formulas (v1, early 2026) are retired - Part II shows exactly why they broke. Everything after Part II is what replaces them.
The Power Law, Assumed
Everything in this paper assumes the power law is true: Bitcoin's fair value grows as a power of time since the genesis block.
The v1 fit (a = 1.01×10⁻¹⁷, b = 5.82) and this one agree on today's fair value within a few percent, so nothing below hinges on the difference. Two honesty notes up front. Refitting at past year-ends gives exponents ranging 5.55–5.83 - the line is known only approximately, and every deviation measure inherits that uncertainty. And the R² of 0.962 is not evidence the law is true: nearly any smooth growing series looks straight on a log-log plot against its own age. The fit locates the line, conditional on the axiom - nothing more.
A subtlety the axiom does not settle: it fixes the destination - price orbits R(t) - but not the dynamics of the pull. Distance-only springs, duration memory, and momentum are all compatible with the same trend line. Which one the market obeys is an empirical question. That is Part III.
Four Breaks
1 - Dollar space is not stationary. The Stress Oscillator summed dollar gaps (Pᵢ − Rᵢ) with a 4-year half-life weight. The weight decays ~16% per year - but the dollar scale of the gaps grows with the trend itself, ~40% per year. The trend was ~$24k at the Nov 2021 anchor and is ~$134k now: a 5–6× scale change in under five years. Min-max normalizing today's reading against 2021's makes every cycle look "unprecedented" by construction. Log space removes this entirely.
2 - An exponentially weighted average is not a debt. Stress reverts toward zero on its own: if price simply sat on the trend line, the accumulated deficit would halve every four years with no rally at all. Nothing forces a future overshoot to "fill in" the area. The spring metaphor survives - but the force must be a forward-looking expectation, not a conserved quantity to be paid back.
3 - The triangle divisor was a free parameter. The v1 recovery target divided the deficit by 1.5 - just T/2 for an assumed 3-month spike. Assume 6 months and the target halves. Targets must come from fitted dynamics, with sensitivity shown.
4 - In-sample min-max anchoring is circular. Pinning 0 and 100 to historical extremes - including the current reading - guarantees dramatic values at the endpoints. The corrected gauge uses a fixed, stated formula.
Distance, Duration,
and the Verdict
| Model (2013+, n=150) | β on x | γ on D | momentum v | R² |
|---|---|---|---|---|
| Spring only | −1.09 (t −7.5) | — | — | 0.57 |
| Spring + duration | −1.00 (t −10.2) | −3.94 (t −8.9) | — | 0.83 |
| Spring + momentum | −1.34 (t −8.3) | — | +0.50 (t +3.5) | 0.67 |
| All three | −0.89 | −4.63 (t −7.7) | −0.19 (t −1.4) | 0.84 |
Distance and duration are both significant with the predicted sign - and duration is not momentum in disguise: when both enter, momentum dies and duration survives. In control-theory language γ < 0 is an integral term, and integral control is exactly what produces cyclical overshoot around a setpoint.
Two honest wrinkles. Out of sample, duration's edge over momentum is a rounding error (RMSE 0.217 vs 0.218) - the horse race between them is decided in-sample, which is exactly where cycle-memorization lives. What is robust out of sample is reversion plus some memory; the label on the memory is less certain than the need for it. And the fitted overshoot is mild - ~+16% above trend at the projected cycle peak, not the 50–90% the v1 triangle arithmetic implied.
Ninety Months of
Honest Forecasts
Every month from Jan 2018 to Jun 2025, the entire pipeline was rebuilt using only data available at that moment - trend refit from scratch, D recomputed, regression re-estimated - then made to predict the next 12 months of trend-relative movement:
| Forecaster | RMSE | vs random walk | Direction |
|---|---|---|---|
| Random walk | 0.321 | — | — |
| Spring only | 0.226 | −30% | 84% |
| Spring + duration | 0.217 | −32% | 84% |
| Spring + momentum | 0.218 | −32% | 82% |
The gauge built from the model also landed on the correct side of every major turn - without ever being anchored to those dates. Counting sustained visits (re-crossings within a year = one episode):
| Episode (gauge ≥ 75) | Price | Months to low | +12 mo | +24 mo |
|---|---|---|---|---|
| Mar 2012 | $5 | 2 | +995% | +11,813% |
| Jun 2015 | $236 | 3 | +162% | +1,008% |
| Nov 2022 | $17,706 | 1 | +106% | +389% |
| Feb 2026 — open | $69,796 | ? | ? | ? |
The mirror signal - gauge below 10 - fired at the Nov 2013 and Dec 2017 tops and twice through 2021's double top: 12-month forward returns of −21%, −74%, −10%, −66%. The honest count: about six independent calls in fourteen years, all correct. A perfect record and a tiny one - strong enough to take seriously, far too small to lean on.
The Deepest Stretch
Since 2015
Honesty requires the ranking v1's normalization obscured: 2015 was deeper (−0.40) and late 2010 deeper still. "Most stretched since 2015" is the defensible headline; "most stretched ever" is not. A yardstick subtlety cuts both ways: by full-history σ, −1.11σ is unremarkable; by modern-era volatility (σ has compressed to ~0.15 - Appendix B), the same deviation is ≈ −2.3σ, genuinely extreme. Which yardstick applies is exactly the maturation-versus-bending-line question, and the model cannot answer it from inside.
What readings like today's have meant (2015+, era-adjusted)
| Gauge | Median 6-mo | Median 12-mo | Beat trend | Worst further dip |
|---|---|---|---|---|
| 80–100 | +42% | +145% | 100% | −7% |
| 65–80 | +42% | +99% | 84% | −16% |
| 50–65 | +40% | +88% | 70% | −31% |
| 35–50 | +7% | +6% | 21% | −47% |
| 20–35 | −40% | −2% | 6% | −60% |
| 0–20 | −25% | −54% | 0% | −74% |
Monotone in every column that matters - which is what a real signal looks like. Outcomes are measured relative to the trend and translated to today's trend growth (+37%/yr) so old cycles don't inflate the numbers; without that correction the top bucket shows an absurd "+948% median" borrowed from 2012. At the current level: three closed episodes - Sep 2015 (+77% in 6 mo, never below entry), Nov 2022 (+56%, dip −4%), Aug 2023 (+72%, dip −6%). Three for three, and only three.
The frozen-target clock
"When does price return to fair value" is mushy when the target runs away at +37%/yr. Freeze it instead: record the day's fair value in dollars and ask when price first trades there. Test case - Sep 2015: gauge 87, price $234, frozen target $592. Price reached $592 in June 2016: nine months, exactly as the gauge-analogues predicted. Across the modern era, at gauge ≥ 80 the median is 9 months (middle half 7–10); the mirror question below gauge 50 - how long until price falls back to fair value - runs 4–10 months. For today's frozen target of $133,738, the historical analogues say median 9 months and a 6,000-path bootstrap of the fitted dynamics independently says 10 (middle half 6–14). Both point to roughly mid-2027 - and Appendix C shows this convergence is not luck.
Two ways a high gauge pays: price snapping up to the line - or price riding the floor while the floor climbs at the trend's growth rate. In 2015–16 the gauge sat at 87 for a year while price "stayed at the floor" - and tripled, because the floor itself tripled.
The mirror property: from today's 85, a 20% price drop would read 91 - but so would six months of price going nowhere, and twelve flat months would read 95, because a flat price falls relative to a line compounding away above it. Standing still is moving away from the line; the gauge knows it.
The Ledger That Never Balances
A natural strengthening of the gravity idea: cumulative area above the line must be repaid by equal area below - the market never forgets. It is beautiful, it appears to leap off every power-law chart, and it fails three tests.
The trap: OLS chooses the line so residuals sum to zero - area above equals area below by construction, for any data. Random walks with no restoring force "balance" identically. The visual symmetry is put there by the regression and cannot be evidence for it.
The sweep: the memory half-life H nests the spectrum - short H forgets fast, H = ∞ is strict conservation. In-sample, infinite memory fits best (R² 0.84): with three cycles on record, a never-forgetting term memorizes their rhythm. Out of sample the ordering reverses - shortest memory wins, strict conservation comes dead last - and a practitioner selecting H at any past vintage would have chosen 6–12 months. The market remembers, and it forgets.
The ledger test: repayment ratios (trough depth ÷ preceding peak) run 0.32–0.79, never 1.0 - and the current cycle is decisive: the 2024–25 bull barely overshot (+12% above trend at peak) yet the present undershoot is ~6× larger. If undershoots existed to pay off overshoots, today's market could not exist. There was no debt.
Gravity has a memory. It is not an accountant.
An Asymmetric Maturation
Peaks collapse toward the line - +0.95 → +0.67 → +0.68 → +0.43 → +0.05 - while troughs hold at the same anchor: four of five lows formed at 0.40–0.48× fair value. Drawdowns compress every cycle (−93%, −85%, −84%, −77%, −53% so far); measured from the line, the floor has not moved. The compression arrives top-first.
Why: the floor is anchored by quantities that scale with the trend automatically - miner production cost, long-term-holder cost basis, and the watched support level itself. The ceiling has no anchor: it is set by how much new capital a mania can recruit, and that requirement grows with market cap while the pool of buyers does not. Doubling a $40B asset took a retail wave; doubling $2T needs trillions.
One alternative explains the same picture: the line itself is bending - an adoption S-curve rolling over makes a full-history fit sit too high, which from inside looks exactly like vanishing overshoots plus persistent "undervaluation." Vintage exponents drift that way (5.83 → 5.64 across 2018–2026 refits). The discriminating observation is the floor: if the current low holds near 0.45× fair, maturation is winning. If price sits well below the floor band for months, that is the bending-line hypothesis announcing itself. This is the model's tripwire, stated in advance.
A Century-Old Solved Problem
| This paper | Physics |
|---|---|
| Deviation x | Particle displacement in the well |
| The pull | Restoring force −U′(x) |
| Volatility σ | Thermal noise |
| Reversion to trend | Relaxation · τ = 1/κ ≈ 12.4 months |
| A bubble top | Kramers escape (1940) - rare, brief, followed by relaxation |
| "When does price reach fair value?" | First-passage time (Smoluchowski 1915 · Siegert 1951) |
The correspondence earns its keep twice. OU theory predicts the long-run deviation spread must equal σ/√(2κ) = 0.286; the independently fitted value is 0.299. And Siegert's exact formula, evaluated at today's x with the measured constants, gives a mean first-passage time to fair value of 12.4 months - no simulation required. Four independent routes now agree:
The same equation founded quantitative interest-rate theory (Vasicek 1977) and is the peer-reviewed standard for mean-reverting commodities (Schwartz 1997) - this mathematics prices oil and copper derivatives daily. What theory proves is the conditional: if deviations are OU with the measured κ and σ, the hitting times follow by theorem. Nothing proves the if. κ is a material constant of this market - measured, not derived, valid while the regime lasts. Proven machinery + measured constants + assumed regime: the first two links are solid; every risk in this paper lives in the third.
Three cycles is a small sample. The trend parameters drift across vintages (b: 5.55–5.83). Newey–West corrections do not cure overlapping 12-month windows - the effective number of independent observations is closer to a dozen than to 150. The duration coefficient partially encodes the specific rhythm of past cycles. And the axiom is the deepest limitation: a deviation this large is evidence of a stretched spring only if the law holds - it is also, unavoidably, weak evidence that the law is breaking. Every failed trend model in market history looked maximally stretched right before it turned out the line was the thing that moved. This paper conditions on the axiom; the reader should not condition on it with probability one.
It is not an accountant -
and it is not a schedule.
The corrected model puts fair value at $133,738 against a spot near $62k - the deepest discount to trend since 2015 - and reads 85/100 on the Gravity Pull Gauge, the highest since late 2016. At this level the modern-era record is short but unbroken: price never fell materially further, returned a median of +42% within six months, and revisited the day's frozen fair value in a median of nine.
Conditional on the law - always conditional on the law.
This article is for informational and educational purposes only and does not constitute financial advice. It describes a statistical model of historical price behavior; past regularities, including sixteen years of power-law adherence, do not guarantee future returns. Position sizing, if any, should assume the model can be wrong - see Part VI. Bitcoin is a volatile asset. Always do your own research.
This post, v1 - Health & Stress Indicators (early 2026) - superseded by this revision
Giovanni Santostasi (@Giovann35084111) - original power law research and regression fit
BitBo (@BitboBTC) - power law popularization and tooling
Power Law Tracker - filipas.com/bitcoin
Price data: blockchain.info charts API · 2010–2026 · 4-day sampling (1,452 obs)
Methods: OLS in log-log space · normalized EWMA · Newey-West HAC · vintage out-of-sample evaluation · residual bootstrap · Siegert first-passage
Uhlenbeck & Ornstein (1930), Phys. Rev. 36 · Kramers (1940), Physica 7 · Siegert (1951), Phys. Rev. 81
Vasicek (1977), J. Fin. Econ. 5 · Schwartz (1997), J. Finance 52 · Redner (2001), A Guide to First-Passage Processes