Papers · Bitcoin
I asked my math agent where Bitcoin bottoms
My circle started asking when the bottom is and where to get in. I had not followed the noise for months, so I put the question to the newest member of the team: a mathematician agent.
I have been busy these last months and I stopped keeping up with Bitcoin. When I checked back, the feed was the usual wall of certainty pointing in both directions, and the messages had started arriving from friends and family: is this the bottom, is it time to get in, what do you think.
What I did have was a new colleague I had been trying out: a mathematician agent, MathModelAgent running on Hermes. It works the way a good applied mathematician works when you lock them in a room with the data and no Twitter: it states the claim, tests it, fits several competing models, makes each model guess the past before it may guess the future, scores the guesses, and writes the answer with the error that belongs to it. I gave it the price history, the four halving dates and one question: is Bitcoin cyclical, and if so, when is the next bottom and the next top?
I asked it to work with five kinds of model, in plain words: a calendar rule (count days since the halving), a template that traces the shape of previous cycles, a "hidden rhythm" model that lets the cycle length float, the famous bubble-crash model, and a model that sorts the market into calm and wild moods. Then a backtest: every model had to predict the 2017 and 2021 tops and the 2018 and 2022 bottoms using only the data available at the time, and only the ones that beat a naive rule got a vote. This is what it came back with.
The answer in four points
- It is a cycle, but a calendar, not a clock.Every major top so far came 12 to 18 months after a halving; every major bottom 25 to 30 months after. Four tops and three bottoms in the same small slot; the odds of that by chance are about 1 in 100. The model that looks for a natural four-year swing in the price finds none. The halving is the anchor.
- The bottom is probably already behind us.The lowest close of this bust is June 30, 2026, at 58,559 USD, day 801 after the halving, exactly in the slot where every previous bottom landed. Two models that watch the actual price path give it a 79 to 93 per cent chance of holding through October 31, and 35 to 63 per cent of holding all the way to the next halving in early 2028.
- The scary reading has a 1 per cent chance.Apply the depth of the last three busts (84, 83, 77 per cent) to this top and you get a bottom around 31,162 to 41,856 USD by mid-October. That needs a 47 per cent crash in a month. The agent kept it because it is what the pattern literally says, and gave it 1 per cent.
- The next top: around May 2029, between 74,449 and 234,733 USD.Three independent ways of counting land on the same quarter. The price range is wide because every cycle's rise has been much smaller than the last, and three points cannot pin the fourth.
The scorecard. Three calls, three dates. I will mark them in public.
The bottom call
58,559 USD
holds until 2028-02-21
Fails on any daily close below it before the next halving.
The crash reading
41,856 USD
by 2026-10-13
A close below it means the pattern reading was right and my 1% was wrong.
The top call
Q2 2029
2029-04-21 to 2029-08-11
Fails without a new all-time high in that window, or with a top outside 74,449 to 234,733 USD.
That is the whole answer: dates, prices, probabilities, and what would prove it wrong. Not advice, a set of tests. The math is below for anyone who wants to check the work.
For the nerds: the math behind the chart click to expand
Data and events
Daily BTC close 2010-08-18 to 2026-09-13 (blockchain.com until 2014-09-16, Yahoo after, spliced with a 2.9% log gap); weekly Sunday closes for the state-space models. Halvings 2012-11-28, 2016-07-09, 2020-05-11, 2024-04-20; the fifth projected at 2028-02-21 from the median inter-halving gap. One event rule with one parameter: a bust runs from an all-time high to the day that high is next exceeded and counts if its drawdown passes 50%; the cycle-ending bust of each epoch is the one whose bottom lies furthest after the halving.
| cycle | top | bottom | days H to T | days H to B | ln(top / halving price) | depth |
|---|---|---|---|---|---|---|
| 1 | 2013-12-05 | 2015-01-14 | 372 | 777 | 4.51 | 84% |
| 2 | 2017-12-16 | 2018-12-15 | 525 | 889 | 3.40 | 83% |
| 3 | 2021-11-08 | 2022-11-21 | 546 | 924 | 2.06 | 77% |
| 4 | 2025-10-06 | 2026-06-30 (so far) | 534 | 801 | 0.65 | 53% |
Is it cyclical? Three tests
Periodogram. Lomb-Scargle against 500 AR(1) surrogates: a 3.64-year peak clears the 99% band only after a log-time detrend, and the bootstrap interval on the period is 1.5 to 8.0 years. A rhythm, not a clock.
Structural time series (Harvey 1989; random-walk level plus stochastic damped cycle, Kalman MLE): the profile likelihood peaks at 2.75 years and four years is rejected (likelihood ratio 4.41, p = 0.036). Not a free four-year oscillation.
Rayleigh test on the phase of each event within its halving epoch (Mardia and Jupp 2000): tops R = 0.966, p = 0.012; bottoms R = 0.989, p = 0.037. The extrema are phase-locked to the halving. The cycle is a schedule, not a pendulum.
Five forecasting families, one interface, one backtest
Every model implements forecast(origin, target) from data up to the origin only: calendar rules (median and trended amplitude), a random walk with drift, the structural model simulated forward, a halving-phase template (mean of earlier epochs' curves scaled to their tops), the log-periodic power law (Filimonov and Sornette 2013, standard filters) and two-regime Markov switching (Hamilton 1989). Rolling-origin backtest on every top and bottom since 2013, origins at each halving, each top + 60 d, each bottom + 60 d and inside each run-up and decline, scored in days and in log price against naive baselines.
| model | tops: MAE days / log | bottoms: MAE days / log |
|---|---|---|
| calendar median | 87 / 1.93 | 50 / 0.57 |
| calendar trended | 87 / 0.65 | 50 / 0.30 |
| random walk | 87 / 0.87 | 50 / 1.02 |
| structural | 311 / 0.90 | 180 / 0.65 |
| halving-phase template | 55 / 0.65 | 43 / 0.30 |
| LPPLS | qualifies on 2017 only | never |
| Markov switching | 486 / 1.38 | 285 / 1.01 |
The template wins timing; the trended amplitude rule wins levels; the structural model alone has calibrated intervals (coverage 1.00 on bottoms) at the cost of six-month timing errors; LPPLS is a 2017-only instrument; regime switching finds volatility regimes of 8 to 34 weeks and nothing multi-year.
The bottom, stated correctly
The tested regularity puts the bottom 777 to 924 days after the halving: 2026-06-06 to 2026-10-31. The lowest close since the top is an argmin, not a forecast: 2026-06-30 at 58,559 USD, day 801, inside the window. Whether it holds is a question about the future path, so it goes to the two path-conditional models: the structural model (4,000 simulated paths) and a random walk with the last four years' drift and volatility (first-passage probability, reflection principle).
| horizon | days | P(lower close), structural | P(lower close), random walk |
|---|---|---|---|
| halving window closes, 2026-10-31 | 48 | 0.21 | 0.07 |
| year end, 2026-12-31 | 109 | 0.39 | 0.18 |
| next halving, 2028-02-21 | 526 | 0.65 | 0.37 |
So the 2026-06-30 low is the bottom of the halving window with probability 79 to 93%, and the strict cycle bottom with probability 35 to 63% under models that know nothing of the cycle, against 3 of 3 under the cycle pattern.
The pattern alternative and its arithmetic
The three completed busts had log depths 1.85, 1.80, 1.45. The median 1.80 gives 124,753 × e−1.80 = 20,710 USD; the linear trend (falling 0.20 per epoch) extrapolates to 1.30 and gives 124,753 × e−1.30 = 33,954 USD; the structural model's median is 45,626. Weighted by backtest accuracy: 32,940 USD; the ensemble's four backtest errors at the 2018 and 2022 bottoms give the range 31,162 to 41,856 USD, timing 2026-09-08 to 2026-10-13. The path models give it a probability of 0.01: a 47% fall in a month from 76,851 USD. It stays in the paper as the tested alternative.
The next top, three anchors
Halving + 372 to 546 days = 2029-02-27 to 2029-08-20. Bottom + 1,050 to 1,067 days (the previous bottom-to-top spans, the most stable quantity in the data) = 2029-05-15 to 2029-06-01. Model ensemble = 2029-04-21 to 2029-08-11. Level: the trended bottom-to-top rise (4.70, 3.04, 2.07 in log, extrapolated to 0.64) gives 145,538 USD; the structural model 364,495; the ensemble 199,049 with a range of 74,449 to 234,733 USD. Nothing narrower is defensible from three points.
Refutation, dated 2026-09-13
A close below 58,559 USD before 2028-02-21 refutes the bottom call. A close below 41,856 USD by 2026-10-13 confirms the pattern alternative. No all-time high between 2029-04-21 and 2029-08-11, or a top outside 74,449 to 234,733 USD, refutes the top call.
Everything is reproducible: cached data with hashes, fixed seeds, every number in this page checked against the results files by a script before publication. The full paper, 15 pages with references and every backtest row: PDF.