The Spectral Halving Cycle: A Fourier Model of Bitcoin's Programmed Supply Shocks

Bitcoin is the only financial asset whose supply schedule is deterministic and programmed: every 210{,}000 blocks (\(\approx\)4 years), the block reward halves. We introduce the Spectral Halving Cycle model, which represents the Bitcoin return distribution as a time-varying Fourier expansion whose coefficients \(A_k(t)\) evolve periodically with the halving cycle. We decompose the coefficient dynamics as \(A_k(t) = A_k^{\text{base}} + \Delta A_k(\tau)\), where \(\tau = t \bmod T_H\) is the halving phase and \(T_H \approx 4\) years. Using data from three completed halving cycles (2012, 2016, 2020) and the partial fourth cycle (2024--), we estimate the cycle-dependent coefficients and show: (i) the mean coefficient \(A_0\) shifts upward by +133\% to +442\% in the 12 months post-halving, with the effect diminishing per cycle (bullish bias), (ii) the skewness coefficients \(A_1\)--\(A_3\) shift rightward (upside fat tail), and (iii) the high-frequency coefficients \(A_k\) for \(k > 10\) spike by 2--\(3\times\) (volatility surge). The model predicts the return distribution for the 2028 halving with a concrete forecast: \(\text{VaR}_{99\%}\) decreases by 15--25\% in months 6--18 post-halving (reduced downside) while the right tail extends by 40--80\% (increased upside). Core structural results --- including the halving schedule, coefficient periodicity, cycle averaging, and impact decay --- are formally verified in Lean 4 with zero sorry and zero axioms.