Reduced-Green discrepancy regularizer

Integrate the regularizer after the representation layer whose distribution should be controlled: for a prototype model use the class prototypes $z_i$; for MoE use per-expert router centroids; for a transformer use a small set of pooled token embeddings. First choose a compact latent domain, preferably a 2-D torus or sphere, and map each representation through a learned projection followed by normalization. Precompute or numerically approximate $K$ nonconstant eigenfunctions $\phi_k$ and eigenvalues $\lambda_k$ of the Laplacian on that domain; for a torus, use Fourier modes, which avoids solving an eigenproblem. In each minibatch, form representation points $z_i$ with weights $a_i=1/n$. Draw target points $y_j$ from the desired background density with weights $b_j=1/m$. Concatenate points $q=(z_1,\ldots,z_n,y_1,\ldots,y_m)$ and signed coefficients $c=(a_1,\ldots,a_n,-b_1,\ldots,-b_m)$, ensuring $\sum_r c_r=0$. Compute the feature matrix $\Phi_{rk}=\phi_k(q_r)$ and evaluate $G_K(q_r,q_s)=\sum_k\Phi_{rk}\Phi_{sk}/(\lambda_k+\tau)$; compute the quadratic energy $\frac12 c^T G_K c$ and add it to the task loss. Backpropagate through the projected coordinates and eigenfunctions. Estimate no paper-specific constants: choose $K$, $\tau$, and $\lambda_{\mathrm{disc}}$ by a small validation sweep, and monitor energy, nearest-neighbor distance, occupancy entropy, and task loss. Start with CIFAR-10 and a 2-layer MLP with a 2-D bottleneck, comparing cross-entropy alone, ordinary RBF-MMD, pairwise repulsion, and this signed Green loss. Success is lower validation loss or improved class separation at equal parameter count without collapse; for MoE, measure expert-load coefficient of variation and routing quality at equal compute.