Plain idea: Bitcoin has a long-term “fair value” line. When price sits far below it, a gravity number scores how hard the model thinks the spring is pulling back up. Below that is a 12‑month outlook — probabilities and a range of timings, not a single date. Method: Post 06: The Power-Law Gravity Model, re-estimated Aug 2026. Not financial advice.
Restoring force −κx = +0.0284 log₁₀/month · potential U = ½κx² = 0.0045 · dimensionless well energy U/(σ²/2) = 0.65 · direction-free potential magnitude 75/100.
κ and σ come from exact monthly Gaussian OU transitions with the long-run mean fixed at x = 0; 1/κ = 11.2 months is the relaxation time. The shaded band on the bar is Agent B's p10–p90 for the gravity number under joint (κ, σ) and power-law parameter uncertainty — 78–94 at the reference deviation. It is drawn at other deviations by re-expressing that band as an effective well width, so it is exact at x = −0.319 and an interpolation elsewhere. [B]
One asymmetry that band does not contain: κ and σ are estimated on the whole history, and the volatility of monthly changes in x has compressed to 0.61× the 2015–2022 level (95% interval 0.46–0.78, Levene p = 0.0034) [D]. A narrower well at the same κ would read higher than 87, not lower. If reversion has weakened along with volatility the effect is ambiguous — and a test for that on the ETF era came back a non-result with low power (Chow p = 0.72). So treat the band above as uncertainty around a full-sample calibration, not as a two-sided bound on the level.
| Reading | Gravity (distance only) | With memory (old site formula) | Note |
|---|---|---|---|
| Mar 2012 episode | 84 | 86 | hindsight score at today's fit, not a walk-forward record |
| Jun 2015 episode | 87 | 59 | the two gauges disagree by 28 points here |
| Nov 2022 episode | 86 | 67 | and by 19 points here |
| Feb 2026 (open) | 78 | 77 | this episode's first crossing |
| Aug 2026 (now) | 87 | 90 | 93rd percentile since 2013 |
All five rows are scored with today's fitted trend and parameters. They are descriptive comparisons, not extra independent episodes and not a live track record. [B]
Your original site already used “pull with memory.” The big gauge branded “Gravity Pull” combined (1) how far price is from the line and (2) how long it has been cheap. That same formula still reads 90/100 today. This page’s headline Gravity number is the cleaner physics twin: distance only (a simple spring). Higher on either meter still means “stronger expected pull back toward the line” — usually up from here — not a promise of a violent spike tomorrow.
Which is more “correct”? For textbook physics of a spring: Gravity (distance only). For “the market also remembers how long we’ve been cheap”: with memory. They usually agree at extremes; they can disagree in the middle.
Where the pull actually comes from. Of Pull = +0.464, distance to the line contributes +0.325 and duration memory contributes +0.080 — duration is a 20% correction at this reading — real, but not the engine. On the constants this page shipped before, the same arithmetic made it look like a 5% garnish; that was an artefact of the γ error below, not a property of the model. Out of sample the term is real and small: the site's headline contribution is a rounding error against plain momentum. [A, B, C]
A constant is corrected here. The previous live page ran γ = −0.739 while Post 05 published γ = −3.94, and neither document mentioned the gap. Re-estimation lands on γ = -3.87, so the published paper was right and the live constant was the outlier. [A found it, B resolved it]
And the significance is smaller than it looks. β = -1.02 on 152 overlapping monthly rows carries a naive t of 20.5. At the effective sample the model actually has — roughly a dozen independent 12-month windows — the same coefficient carries t ≈ 5.7, and a 12-month circular block bootstrap puts the 95% interval at [-1.28, -0.82]. The reversion is solid. The decimal places are not. [B]
The memory length was never validated. H = 48 months was a convention, not a fitted choice. A causal, no-peek nested selection over (form, H) picks H = 6 in 88 of 91 vintages and beats the published fixed choice out of sample (−37% vs −33% against a random walk). The gauge above is H-free; the companion is not. [C]
Forecast The gravity number says the spring is pulling up toward fair value. This section answers: over the next ~12 months, what does that imply? You get probabilities and a range of timings — because the same spring plus market noise can mean a fast catch-up or a slow grind.
How long until price first touches the fair-value line? Each row asks the same question with a slightly different honesty setting. White tick = typical answer (median). Dark band = middle half of outcomes. Pale band = most outcomes. Amber line = 12 months. Notice amber sits near the middle — that’s why “within a year” is roughly a coin flip.
Takeaway: the spring points up; a single exact month is fake precision. Even changing “do we count mid-month touches?” moves the typical answer from ~8 months to ~11 months. Use the 12‑month probabilities above as the headline forecast. [B, C]
Today's reading: 87 — Bitcoin has spent about 38 similarity-weighted months near this level since 2015, but those months come from 3 visits, and 2015–16 alone carries 63% of the weight. Overlapping 12-month windows are not independent observations. Each chip is one visit and what price did in the 6 months after it began:
Computed live in your browser from the embedded monthly history: every month since 2015 is weighted by how close its gravity number was to today's exact value (similarity weights, ±7 points). Returns are measured relative to the trend and translated to today's trend growth so old cycles don't inflate them. Read every figure in this panel as ordinal, not calibrated: the “38 months” is a similarity-weight count, not an information count, and where two visits fall inside the same cycle this model's own de-duplication rule (re-crossings within a year count once) would merge them, so the independent-episode count can be lower still than the visit count above. Everything is conditional on the power law holding. And remember the gauge pays in trend-relative terms — in 2015–16 it sat near 87 for a year while price “stayed at the floor” and tripled, because the floor itself tripled. [A, B]
| Gravity number | Zone | Historical record (2012–2026) |
|---|---|---|
| 75–100 | Strong upward force | Three closed episodes since 2012 (2012, 2015–16, 2022–23), each 1–3 months before the cycle low; +106% to +995% over the following 12 months. Three for three — and by the de-duplication rule, arguably two. The 2026 episode is open. |
| 60–75 | Upward force | Below fair value with force building (the Dec 2018 bottom read 67 — cycle bottoms read above 65, not above 75) |
| 40–60 | Near equilibrium | Near trend; the well is shallow in both directions |
| 25–40 | Caution | Extended above fair value. The late-2025 high (~$126k daily / ~$115k monthly mean) only reached about 35–45 here — a real local top with a weak gauge signature, not the classic cycle-top zone. Treat as caution, not a critical top warning. |
| 10–25 | Strong downward force | Strong downward force developing; still short of the mania-top prints below. |
| 0–10 | Classic mania-top zone | Nov 2013, Dec 2017, Feb/Oct 2021 read below 10; 12-month forward returns −10% to −74% (4 of 4 negative). Not every cycle high reaches this band — 2025 did not. |
Zone statistics rest on a handful of independent episodes in fourteen years — strong enough to take seriously, far too small to lean on, and never enough for a two-digit probability. Classic mania tops (gauge <10) are a stricter class than “price made a high.” The 2025 peak stayed in Caution (~38), which is evidence of a milder premium to fair value (maturation) — or of a line that is already too high — not proof that every top prints red. The zone names describe forces, not positions: this model cannot tell you what to own. All of it is conditional on the power-law regime holding — and on the exponent, which drifts across vintages between 5.640 and 5.822 on modern refits, trending down. That is exactly what a bending line looks like from the inside. [A, B, D]
Monthly geometric means, log scale, extended five years past today. The price line stops at the last data point; right of the “today” marker there is no data — only model lines. Three things are drawn as bands rather than lines, deliberately: the fair-value line carries its own p10–p90 shading from the HAC covariance of the fit, because a·tb is a point estimate and every deviation measure inherits its uncertainty; the grey band is the historical trough cluster (0.39×–0.61× fair, where all five cycle lows have formed), with the conventional 0.45× reference dashed inside it; the sparse dashed line above is the decaying ceiling, a five-point extrapolation and a hypothesis, not a law. The price line is coloured by the gravity number — deep red marks classic mania-top prints (<10), amber the Caution band where the 2025 high sat (~38), blue the deep discounts. Not every local top reaches deep red. Hover for values. [B, D]
Secondary to the gravity model above; used for Wealth Manager ALARM bands (composite −1σ). Pure PL cycle floor remains 0.45×R(t) from the gravity model. R² fingerprints ~0.96 PL / ~0.98 composite. ALARM if spot breaks composite −1σ OR 0.45×R(t).
Blue: gravity (distance-only spring). Amber: the old-site formula (distance + how long you’ve been cheap). Dashed line = equilibrium at 50. They usually agree when things are extreme; in the middle, “how long you’ve been cheap” can pull the amber line away from blue (e.g. 2015 and 2022). Both are scored with today's fit, so neither line is a walk-forward record. [B]
July 1 values of the fitted trend, its p10–p90 parameter band, the 0.45× residual reference and the ceiling hypothesis. These are the fitted line's own coordinates. They are printed to the dollar for arithmetic convenience only — the line itself is known to roughly ±22% today, and the band widens as the extrapolation lengthens.
| July 1 of | 0.45× band | Fair value (p10–p90) | Ceiling (hypothesis) |
|---|
The ceiling multiple shrinks each cycle and extrapolates to 1.0× around 2038 — treat that as a hypothesis from five points, not a law. On the lower side: cycle lows pierced the 0.45× line by 4–21% before turning, all five trough minima fall in 0.39×–0.61× with a geometric mean of 0.46×, and a drift test on those five points cannot reject zero drift or any economically meaningful drift (p = 0.51). [A, D]
This page used to label 0.45× fair value “cycle floor”, in the tile, in the chart legend and in the call to action, with “descriptive anchor” in grey underneath. A floor is a barrier. What this object actually is: a low quantile of the fit's residuals that the market has crossed before and will cross again.
| Quantile of P/R since 2010 | p10 | p25 | p50 | p75 | p90 |
|---|---|---|---|---|---|
| Price ÷ fair value | 0.48× | 0.59× | 0.88× | 1.51× | 2.64× |
0.45× sits at percentile 5.2 of the empirical distribution — 10 of 192 monthly observations have closed at or below it [B]. Under a normal-residual approximation the same line is about the 12th percentile [A]. The five cycle-trough minima are 0.42× · 0.39× · 0.61× · 0.46× · 0.46×: a cluster spanning 0.39×–0.61×, not a level. Every low that this line “describes” pierced it, by 4–21%. And the quantile itself moves when the trend is refitted, which is why the chart draws the cluster as a band and the 0.45× line as a dashed reference inside it. [A, B, D]
It can: measure the present state of the well — how far price sits from the fitted line and which way the fitted force points — and classify the regime by the zones above. It can give a 12-month direction versus trend: right in 85% of 91 overlapping monthly forecasts, 2018–2025, which is roughly seven to eight independent windows, so read it as “usually right”, not as 85 ± 4% [C]. It can beat a random walk on 12-month RMSE by 33% at the published memory length and 37% under honest no-peek selection — the selection procedure is the stronger claim, not the fixed number [C]. And it can flag deep discounts and classic mania-top prints (<10) when they appear — knowing that the 2025 high only reached Caution (~38), so “cycle high” and “mania-top zone” are no longer the same event.
It cannot: attach a calibrated probability to anything — every number here is conditional on the power law holding, with roughly 14.3% weight currently on the centre having moved [D]. It cannot pin the low to a date or a price; the timing distributions above span 2.2 to 25 months at the 80% level and shift by four months on a monitoring convention alone. It cannot guarantee amplitude, or repay overshoots like a ledger — the current undershoot runs 3.5× to 6.5× the preceding overshoot depending on which peak convention you pick, so the books never balance [D]. It cannot distinguish “price at the historic trough cluster” from “the power law starting to bend”; months spent well below the cluster would favour the second reading. And it cannot borrow confidence from log-periodic timing: fitted log-periodic modes remove 35% of residual variance in-sample yet forecast 12 months ahead at RMSE 0.394 versus 0.218 for this model, with 43% direction accuracy — worse than a coin flip — and the in-sample gain is not distinguishable from an AR(1) red-noise null (p = 0.43) [D].
Gravity has a memory. It is not an accountant, and it is not a schedule.
Power-law fit (monthly OLS on geometric means, Sep 2010 – Aug 2026, n = 192): log₁₀ a = -16.175, b = 5.593, σ = 0.298 log₁₀, R² = 0.960. The naive standard error on b is 0.083; with Newey–West lag 12 it is 0.185, and that HAC covariance is what produces the fair-value band on this page. Modern vintage refits put b between 5.640 and 5.822. The final month (Aug 2026) is a partial 20-day geometric mean and is flagged as such wherever it is used. Edit and apply to recompute the tiles and the coordinate table (charts remain rendered from the build-time fit).
| Layer | Quantity | Source |
|---|---|---|
| State | Gravity number 87 (p10–p90 78–94), κ = 0.089, σ = 0.118, s∞ = 0.279 | Agent B — OU re-estimation |
| State | Pull companion 90, γ = -3.87 (correcting the old live γ = −0.739) | Agent A flagged, Agent B resolved |
| Forecast | First-passage distributions, P(≤12 mo) 55–68%, P(higher) 87–90% | Agent B — 80k–100k path simulations |
| Skill | Direction 85%, −33% vs RW at H = 48, −37% honestly selected, H = 6 chosen 88/91 | Agent C — expanding-window CV |
| Bands | Trough cluster 0.39×–0.61×, 0.45× at residual percentile 5.2 | Agents B and D |
| Regime | P(centre moved) ≈ 14.3%, ETF-break Chow p = 0.72, log-periodic null p = 0.43 | Agent D — heuristic ledger and falsifier tests |
Not published on this page: the external critique's “P(reach fair ≤ 12 mo) ≈ 46%” and “direction ≈ 86% robust”. Neither could be reproduced. The corresponding re-estimated figures are 55–68% and 85%, and those are what appear above. Full number-by-number provenance, including everything deliberately left off the page, is in synthesis/PUBLISH_NUMBERS.json; this file is regenerated by synthesis/code/build_tracker.py.