Bitcoin's Price Power Law: Stability, Maturation, and a Live Test of the Scaling Hypothesis

This paper reproduces the empirical Bitcoin price power law and uses it as the starting pointfor a set of focused stress tests motivated by Bitcoin's maturation and its 2025–2026 below-trendepisode. Using 5,873 completed UTC daily closes (17 July 2010 – 14 August 2026), with time measuredin days since the 3 January 2009 genesis, it fits ln P(t) = ln A + B·ln t (B ≈ 5.65, R² ≈ 0.96)and then asks: how precisely the exponent is really known once residual autocorrelation is takeninto account; whether structural-break detectors can distinguish genuine slope changes from aconstant-exponent null with persistent residuals; how unusual the current below-trend episode isin raw terms versus when scaled by today's lower volatility; whether declining volatility slowsrecovery toward trend; and the distinction between merely keeping pace with the rising trend andclosing an accumulated deficit. Several intuitive explanations do not survive testing. The completed tests do not provide compellingevidence of structural failure, but neither do they establish an immutable law: the current episoderemains a live test of the empirical scaling relationship. Data frozen 14 August 2026. Technicaldraft v0.8.AI assistance: This work was developed with substantial help from two large language models.ChatGPT (OpenAI; GPT-Sol models, 2026) served as the primary thinking partner for developing andtesting the research questions and interpretations. Claude (Anthropic; Claude Opus models, 2026) carriedout much of the analytical work, including regressions and robustness checks, verification ofthe reported figures, literature searches, and adversarial review across drafts. All research questions, results, and conclusions were reviewed and are owned by the human author, who is responsible for the content, including any errors.