Gauge-Free Inverse OT Attention

Integrate this into a cross-attention block. Convert attention scores S into a positive kernel K with K_ij = exp(-S_ij divided by epsilon), then run 5 to 20 log-domain Sinkhorn iterations to obtain W with prescribed source and target marginals. Keep value aggregation Y = W V, but add an inverse-cost consistency term or use the recovered cost as a structured bias for the next attention block. The pseudocode is: compute stabilized K; initialize positive u and v; repeat u = s divided elementwise by K v and v = r divided elementwise by K-transpose u; form W = diag(u) K diag(v); set L = log(W plus delta); compute C_inv = -epsilon H_m L H_n; add lambda times the squared Frobenius norm of H_m(S - C_inv)H_n to the training loss. The exact inverse identity is supplied by the paper. The engineer must estimate the effect of finite Sinkhorn convergence, floating-point flooring, and model mismatch. First run a synthetic test with random C, prescribed marginals, and generated W before training a two-layer Transformer on a small copy or patch-classification task. Compare softmax attention with Sinkhorn attention with and without the inverse-cost penalty. The prediction is that the relative centered reconstruction error will remain below 5 percent when the smallest W entry exceeds 10^-5 and Sinkhorn marginal error is below 10^-4. Adding any row-plus-column gauge to C must not change C_inv. Reconstruction error should increase sharply when W entries fall below delta.