The Anomaly Functional: Real-Time Arbitrage Detection via Spectral Risk Coefficients

We define a scalar functional \(\mathcal{A}[F]\) on probability distributions that quantifies the total magnitude of arbitrage violations --- negative densities (butterfly arbitrage) and decreasing total variance across maturities (calendar arbitrage) --- in a single number computable in sub-millisecond time from spectral risk coefficients. For any distribution represented by \(N\) Fourier coefficients via the Spectral Fenton framework (Nagy, 2026a), the butterfly component of \(\mathcal{A}\) requires \(O(N^2)\) operations via evaluation at \(2N+1\) Chebyshev nodes, enabling real-time monitoring of option surfaces at tick-by-tick frequency. We prove that \(\mathcal{A}[F] = 0\) if and only if the distribution is arbitrage-free, and that \(\mathcal{A}\) is continuous in the coefficient topology: small perturbations to the coefficients produce small changes in the anomaly score. Numerical experiments on both clean and synthetically corrupted distributions demonstrate clear separation --- clean Eigen-COS outputs yield \(\mathcal{A} < 10^{-6}\) while even mild coefficient perturbations (10\% sign flips) produce \(\mathcal{A} \approx 10^{-2}\), a gap of six orders of magnitude, and strong corruption produces \(\mathcal{A} > 1\). A streaming monitor architecture is presented that tracks \(\mathcal{A}(t)\) in real time and generates alerts when the score exceeds a configurable threshold. The anomaly functional provides a single, interpretable, sub-millisecond diagnostic for arbitrage consistency that is absent from current industry practice.