The Bitcoin Harmonic Time Model: Internal Phase Symmetry, Amplitude Power-Laws, and the Institutional ETF Regime Shift

This paper formalises the Bitcoin Harmonic Time Model, a two-dimensional quantitativeframework that structures Bitcoin market cycles into a discrete dynamic system defined bytemporal phase symmetry and compressed amplitude distributions. We demonstrate that thegoverning clock of the market is not the halving event in isolation, but a stable internal phasearchitecture where the halving acts as a temporal node, consistently arriving at normalisedphase ϕ(H) ≈ 0.494 of the expansion phase. Using historical data from three completed cycles,we establish that expansion lasts 1,060 ± 12 days and contraction 370 ± 13 days, yielding amean period of 1,430 days. Furthermore, we derive an empirical power-law demonstratingthat cycle multipliers scale non-linearly with prior drawdown depth ( M(d) = 328 · d5.1), andintroduce a structural adjustment framework to account for the regime shift initiated by thelaunch of spot US Exchange-Traded Funds (ETFs) in 2024. Finally, we translate this unifiedmodel into explicit, dated, and falsifiable calendar hypotheses for the current and subsequentmarket cycles.