The Spectral Unity: Risk, Pricing, and Hedging from a Single Representation

For portfolios of correlated lognormal assets, we show that risk measurement, derivative pricing, and hedging --- traditionally treated as separate disciplines --- reduce to operations on a single object: the \(N\)-term Fourier-cosine (COS) expansion of the portfolio density. The Spectral Fenton Distribution (Nagy, 2026a) produces \(N + 2 = 130\) coefficients from which every coherent risk measure is computable in \(O(N)\) via root-finding, every European derivative price via a dot product, and every first-order Greek via the chain rule through the characteristic function. We formalize this as the Spectral Duality Theorem, verify the algebraic structure in Lean 4, and benchmark the method against Monte Carlo on a five-asset correlated basket. Precomputation takes 65 ms; subsequent queries cost \(< 0.5\) ms each, yielding a \(\sim\)30\(\times\) speedup over separate risk/pricing/hedging pipelines.