Mixture-Preserving, Arbitrage-Free Interpolation for Volatility-Surface Models

Given risk-neutral densities of a tradeable forward, fitted as $N$-component mixtures at a finite set of expiration pillars, we look for a continuous-time interpolation that is (i) \emph{mixture-preserving}, remaining a mixture of the same kernel (generically with more components than either pillar), and (ii) \emph{arbitrage-free} across expiries. The second requirement is the \emph{peacock} (convex-order) property, equivalently a non-negative Dupire local volatility; for full-support kernels (Gaussian, lognormal) it gives a unique continuous local-volatility diffusion (Lowther). We construct such an interpolation in a fixed $2N$-component family, freezing both pillars' components and moving only their weights. Applied to mixture term-structure models, it lifts Brigo--Mercurio to time-varying weights and reaches the free-per-strike-width generality of SANOS at additive cost.