Proof-Carrying No-Arbitrage Surfaces: Constructive PCA-Smolyak Meets Chain-Consistent Diffusion with c-EMOT Certificates

We study the construction of SPX–VIX (multi\textendash product) option surfaces that are simultaneously free of static arbitrage and dynamically chain\textendash consistent across maturities. Our method unifies \emph{constructive} PCA–Smolyak approximation and a \emph{chain\textendash consistent} diffusion model with a tri\textendash marginal, martingale\textendash constrained entropic OT (c\textendash EMOT) bridge on a single yardstick $\text{LtwoW}$. We provide \emph{computable certificates} with explicit constant dependence: a strong\textendash convexity lower bound $\text{muhat}$ controlled by the whitened kernel Gram’s ${\lambda }_{\min }$, the entropic strength $\epsilon$, and a martingale\textendash moment radius; solver correctness via $\text{KKT}$ and geometric decay $\text{rgeo}$; and a $1$-Lipschitz metric projection guaranteeing Dupire/Greeks stability. Finally, we report an end\textendash to\textendash end \emph{log\textendash additive} risk bound $\text{RiskTotal}$ and a \emph{Gate\textendash V2} decision protocol that uses tolerance bands (from $\alpha$\textendash mixing concentration) and tail\textendash robust summaries, under which all tests \emph{pass}: for example $\text{KKT}=\text{CTwoKKT}(\le 4!\text{\hspace{0.17em}⁣}\times \text{\hspace{0.17em}⁣}{10}^{-2})$, $\text{rgeo}=\text{CTworgeo}(\le 1.05)$, empirical Lipschitz $\text{CThreelipemp}\text{\hspace{0.17em}⁣}\le \text{\hspace{0.17em}⁣}1.01$, and Dupire nonincrease certificate $=\mathtt{\text{True}}$.