\begin{longtable}{@{}p{0.18\textwidth}p{0.36\textwidth}p{0.34\textwidth}@{}}
\caption{Coupled atlas classes: correct CST targets and coupled Haar baselines.}\label{tab:coupled-atlas}\\
\toprule
\textbf{Coupling class} & \textbf{CST target} & \textbf{Baseline (coupled Haar)} \\
\midrule
\endhead
SOL: $b\ominus a=\Psi(s)$ & $\{(a,a\oplus\Psi(s))\}$ & $(\eta_s)_\#\mu_H$, $\eta_s(a)=(a,a\oplus\Psi(s))$ \\
Affine: $b\ominus (\alpha\odot a)=\Psi(s)$ & $\{(a,\alpha\odot a\oplus\Psi(s))\}$ & $(\eta_s)_\#\mu_H$, $\eta_s(a)=(a,\alpha\odot a\oplus\Psi(s))$ \\
Integer-matrix: $Ay=\Psi(s)$ & coset of a subtorus & $(\eta_s)_\#\mu_H^{(m)}$ from free-coordinate parameterization \\
\bottomrule
\end{longtable}
