The Spectral Volatility Surface

We construct a low-rank arbitrage-aware volatility surface with \(O(rm)\) parameters and closed-form COS reuse for pricing and Greeks. Total implied variance is expressed as a finite cosine series in log-moneyness, \(w(k, T) = c(T) + \sum_j u_j(T)\cos(\omega_j k)\), with \(r = 6\)–\(12\) modes per maturity. A Fejér-type bound on the coefficients guarantees non-negative variance and calendar no-arbitrage; these coefficient-level structural results are verified in Lean 4 (63 theorems, 0 sorry). A stronger curvature-dominance condition yields a sufficient condition for the full Gatheral (2004) butterfly density \(g(k) \geq 0\), and a Chebyshev+SDP variant improves practical fit quality on stressed surfaces. In a fixed-seed SPX-realistic benchmark built from Heston-calibrated normal, stressed, and calm regimes, the Chebyshev+SDP variant achieves RMSEs around \(0.003\) and zero Gatheral butterfly violations in the stressed and calm regimes, while the cosine Fejér variant preserves zero calendar violations by construction and raw SVI retains large structural violations. For real-data positioning, we distinguish two evidence surfaces: a staged modern SPX snapshot that supports the draft's current-market sanity check, and a separate zero-cost 2013 SPX archive that serves as an engineering robustness lab rather than as a replacement reviewer-facing benchmark. The contribution is not a new stochastic-volatility law, but a reusable surface engine: calibrate once, then reuse the same coefficient object for arbitrage control, repricing, local-vol extraction, and risk workflows. We do not claim to solve surface dynamics in this paper; the focus is the static, desk-ready surface layer.