\section*{Appendix: Worked Examples (auditable examples across witness tiers)}

\subsection*{Purpose}

This appendix provides three worked examples illustrating the engine discipline across witness tiers:

\begin{itemize}[leftmargin=*]
\item global witness (AWT)
\item embedded witness (loop degree or holonomy)
\item coupled-target witness (CST under SOL or affine coupling)
\end{itemize}

Each example points to a certificate or obstruction artifact and gives a minimal audit command.

\subsection*{Example 1: Global witness via AWT (rotating body)}

\begin{itemize}[leftmargin=*]
\item Mechanism: AWT
\item Tier: WTR-G
\item Artifact: \texttt{artifacts/artifacts/WSC-G-AWT-0001.json}
\end{itemize}

Audit:
\begin{verbatim}
python3 artifacts/audit/audit_wsc.py artifacts/artifacts/WSC-G-AWT-0001.json
\end{verbatim}

\subsection*{Example 2: Embedded witness via LDW or HW (loop-level phase winding)}

\begin{itemize}[leftmargin=*]
\item Mechanism: LDW (or HW)
\item Tier: WTR-E
\item Artifact: \texttt{artifacts/artifacts/WSC-E-LDW-0001.json}
\end{itemize}

Audit:
\begin{verbatim}
python3 artifacts/audit/audit_wsc.py artifacts/artifacts/WSC-E-LDW-0001.json
\end{verbatim}

If the setting lacks a meaningful phase observable or loop subsystem, the engine should output an obstruction, for example:

\begin{itemize}[leftmargin=*]
\item \texttt{artifacts/artifacts/WOB-E-0001.json}
\end{itemize}

\subsection*{Example 3: Coupled-target witness (CST target under coupling)}

\begin{itemize}[leftmargin=*]
\item Mechanism: CMT
\item Tier: WTR-C
\item Artifact: \texttt{artifacts/artifacts/WSC-C-CMT-0001.json}
\end{itemize}

Audit:
\begin{verbatim}
python3 artifacts/audit/audit_wsc.py artifacts/artifacts/WSC-C-CMT-0001.json
\end{verbatim}

Mixing boundary:

\begin{itemize}[leftmargin=*]
\item if diffusion or noise is not adopted, mixing claims should be replaced by an obstruction (for example \texttt{artifacts/artifacts/WOB-G-0002.json})
\end{itemize}

\subsection*{Diagnostic note (targets)}

\begin{itemize}[leftmargin=*]
\item Uncoupled: baseline is Haar on $\mathbb{T}_P$
\item Coupled: baseline is coupled Haar on CST (CTP and CHT)
\end{itemize}

Key inequalities and rate reminders are centralized in:

\begin{itemize}[leftmargin=*]
\item \texttt{artifacts/contracts/constants\_thresholds.md}
\end{itemize}
