Spectral Schrödinger Bridges: Optimal Transport Between Portfolio Distributions in Fourier Space

The Schrödinger Bridge Problem (SBP) finds the most likely stochastic evolution between two probability distributions, minimizing the Kullback--Leibler divergence from a reference process. We show that when both endpoint distributions are Spectral Fenton Distributions --- Fourier-cosine representations of correlated lognormal portfolio sums --- the bridge computation simplifies dramatically: the heat kernel is diagonal in the Fourier basis, reducing the kernel application in each Sinkhorn iteration from \(O(M^2)\) (grid-based) to \(O(N_{\text{eff}})\) (spectral) where \(N_{\text{eff}} \approx 10\) is the effective number of modes. The full bridge computation achieves a \(2{,}500\times\) reduction in operations compared to grid Sinkhorn at \(M = 2000\). This extends the Spectral Unity framework (Nagy, 2026e) from static risk-pricing-hedging to dynamic optimal transport: the bridge adds a fourth application --- temporal risk evolution --- to the same 128 spectral coefficients. The Schrödinger potentials, entropic Wasserstein distance, Sinkhorn convergence, and transport compression (\(3906\times\)) are formalized in Lean 4.