\section*{Convergence certificates (engine-native ergodic and diffusion upgrades)}

SFT treats ergodic and diffusion results as artifact outputs.

\subsection*{Convergence certificate (CC)}

A \emph{Convergence Certificate} records a hypothesis bundle and a quantitative conclusion:

\begin{itemize}[leftmargin=*]
\item \textbf{RNF certificate (ergodic).} Given rationally independent increment vector $\omega$ (nonresonant drive frequencies), time-averages converge to Haar on the correct target (full torus or CST) \cite{walters1982,einsiedler2011,kuipers1974}. The certificate stores the target and the RNF condition used.
\item \textbf{NDB+MXC certificate (diffusion).} Given an explicit noise-to-diffusion bridge (NDB) producing an advection--diffusion model and a verified mixing inequality (MXC), deviations from the baseline decay exponentially with rate constants pulled directly from \cite{pavliotis2014,evans2010} \codeid{engine/contracts/constants_thresholds.md}.
\end{itemize}

Operationally, the engine outputs CC information inside a WSC (mechanism set to MXC) or as a universality report artifact that lists the rate line used from the thresholds ledger.
