Spectral Bitcoin VaR: High-Accuracy Risk Measures for Cryptocurrency via Fourier Expansion

We apply the Spectral Fenton Distribution framework (Nagy, 2026a) to Bitcoin return modeling and show that the COS (Fourier-cosine) expansion captures the BTC/USD daily return density with controllable numerical precision. Bitcoin's analyticity radius \(\rho \approx 1.02\) --- the lowest among major asset classes --- means that \(N = 1{,}024\) Fourier-cosine coefficients are required to achieve CDF sup-norm error below \(10^{-8}\), whereas equity indices (\(\rho \geq 1.20\)) need only \(N \approx 100\). We compare the spectral method against four benchmarks: historical simulation, parametric GARCH(1,1)-\(t\) (Bollerslev, 1986), Extreme Value Theory via the GPD threshold method (Balkema and de Haan, 1974; Pickands, 1975), and Monte Carlo simulation (\(10^6\) paths). Even at \(N = 1{,}024\), the spectral method requires only 0.008 seconds per risk query --- over \(100\times\) faster than Monte Carlo for the same accuracy --- and yields VaR, ES, and all coherent spectral risk measures from a single precomputation. The risk entropy of Bitcoin is \(H_{\text{risk}}^{\text{BTC}} \approx 50.5\), confirming that cryptocurrency requires an order of magnitude more spectral resolution than traditional assets due to extreme tail behavior. All structural results (convergence rates, ES closed-form identity, VaR existence) are formally verified in Lean 4; empirical results depend on model fit and data quality.