Spectral Kernel Risk: The Fourier--Gaussian Process Duality for Portfolio Loss Distributions

We establish a formal duality between the Spectral Fenton (Fourier) representation of portfolio loss densities and Gaussian Process (GP) regression via Mercer's theorem. Every stationary kernel \(K(x,y)\) decomposes into Fourier eigenmodes \(K(x,y) = \sum_k \lambda_k \phi_k(x)\phi_k(y)\), so the 128-coefficient Spectral Fenton representation is equivalent to a GP with a kernel whose Mercer eigenvalues satisfy \(\lambda_k = 0\) for \(k > N\). Conversely, any GP posterior mean on the loss density projects onto the Fourier basis to yield the spectral coefficients. This duality provides three practical contributions: (i) automatic \(N\) selection --- the kernel's spectral decay determines the optimal truncation, (ii) VaR with error bars --- the GP posterior variance gives pointwise uncertainty in the CDF, yielding confidence intervals on VaR and ES at modest additional cost (50\(\times\) slower than pure Fourier, but 42\(\times\) faster than bootstrap MC), and (iii) adaptive tail resolution --- the GP allocates more resolution where data is sparse (deep tails) and less where data is dense (body). We demonstrate the method on simulated portfolio returns, showing that in our simulation the GP-enhanced VaR has 40\% tighter confidence intervals than bootstrap MC while maintaining \(O(N)\) query speed for the point estimate. The Mercer duality theorem and the GP-Fourier equivalence are formally verified in Lean 4.