Conditional OT barycenter feature augmentation

Integrate the module between the encoder $h_\phi$ and the prediction head. At every step, draw an equal-size minibatch from each of $K$ source domains and compute latent vectors $z_{k,i}=h_\phi(x_{k,i})$. Compute a gate $g_{k,i}$ for the desired context and replace source masses by $w_{k,i}\leftarrow g_{k,i}+\delta$, followed by normalization. Maintain $m$ barycenter support vectors $b_r$ initialized from one source. For each source, calculate squared latent costs $C_{k,ir}=\|z_{k,i}-b_r\|_2^2$ and run $T=20$ Sinkhorn iterations to obtain transport plans $\pi_k$ between source weights and barycenter weights $q_r$. Update the barycenter by $b_r\leftarrow[\sum_k\theta_k\sum_i\pi_{k,ir}z_{k,i}]/[\sum_k\theta_k\sum_i\pi_{k,ir}+\delta]$, then recompute plans for five alternating barycenter iterations. Construct a soft label for each support point as $\widetilde{y}_r=[\sum_k\theta_k\sum_i\pi_{k,ir}y_{k,i}]/[\sum_k\theta_k\sum_i\pi_{k,ir}+\delta]$. Add $\eta$ times the prediction loss on $(b_r,\widetilde{y}_r)$ to the ordinary source loss. Use stop-gradient through $b_r$ and $\pi_k$ initially to avoid unstable encoder-barycenter feedback; later test end-to-end gradients. The exact mathematical component is the weighted barycenter objective and convex combination of source features; the latent cost, number of support points, and Sinkhorn approximation are engineering choices. First test on PACS or Office-Home with three source domains and one held-out domain, comparing ERM, standard Mixup, domain-adversarial training, and barycentric augmentation. Measure held-out accuracy, rare-context accuracy, domain-classifier accuracy on latent features, and added compute. Success is improved held-out accuracy and reduced domain separability with less than 20 percent training overhead; failure is degraded source accuracy or no gain over Mixup.