A Core-and-Wave Model of the Bitcoin Price: Bayesian Estimation, Out-of-Sample Validation, and Sequential Monitoring by Betting

Bitcoin's price history is usually told as a four-year cycle tied to the halving of the block reward. Forecasts built on that story are rarely tested out of sample. We propose the Shinpa (core–wave) theory, a simple decomposition of the weekly log price into a slowly growing core, a wave whose period is estimated rather than fixed, and short-term noise with heavy tails. The core is an S-shaped curve that looks like a power law of the network's age today and saturates at a terminal level defined as a share of the world's store-of-value assets divided by the 21 million coin supply. We estimate the model by Bayesian inference (Kalman filter, Student-t quasi-likelihood, adaptive Metropolis) and evaluate it with time-ordered cross-validation from 2016 to 2025 using proper scoring rules. Three findings stand out. (i) Imposing the halving-clock shape of the four-year cycle makes three-year-ahead forecasts significantly worse (Diebold–Mariano p = 0.01). (ii) Price volatility around the core has declined by about 21% per four years. (iii) A freely estimated wave has a period of roughly 3.5–4 years (profile-likelihood maximum 3.5 years, 95% interval about 3.3–4.3 years); whether it is exactly four years cannot be decided from the data. The terminal level is essentially unidentified by the data and remains an assumption. We record a probabilistic forecast every week and monitor it with an e-CUSUM built from betting on probability-integral-transform values, which is valid under continuous monitoring. We also test fifteen other indicators as measures of deviations from the core, controlling the false discovery rate, and set up an e-value procedure that will decide on their use from future data. We report the errors we made along the way. Files: paper.pdf (text, CC BY 4.0) and shinpa-code-v1.zip (code, MIT License; pure Python, downloads public data with fetch_data.py). This is not investment advice.