Crystal-Orbit Consistency Regularization

Apply this at the input of an encoder whose sequences contain discrete codes with exchangeable or approximately exchangeable labels, such as VQ-VAE code indices, product-quantized retrieval IDs, graph colors, or synthetic permutation-equivariant data. Do not apply it directly to ordinary natural-language token IDs, whose numerical adjacency has no invariance meaning. For each training sequence w, sample an adjacent pair (i,i+1), run the stack-based parenthesis matching on the subsequence containing those labels, and construct w'=s_i(w) by swapping only the unmatched block. Reject the rewrite if it changes protected metadata or violates a task-specific invariant d(w). Run both w and w' through the shared encoder, compute predictive distributions p and p', and add the stop-gradient KL term. Use one rewrite initially, then test two or three successive rewrites with probability q. The combinatorial procedure computes legal orbit moves; the dataset determines which labels are exchangeable, while lambda, q, and the protected statistic are empirical settings. First test a VQ-VAE on CIFAR-10 with code-index permutations, comparing crystal-orbit augmentation against no consistency loss and random swaps. Also evaluate robustness under held-out valid code permutations. Record reconstruction error, classification accuracy from the latent code, prediction KL across rewrites, and training variance. Success is improved held-out performance at equal compute and substantially lower disagreement under valid transformations without hurting performance on original inputs.