Sparse-Cost Temperature Calibration

Apply this controller to the temperature of a Sinkhorn attention head or an MoE routing matrix. Trusted costs can come from synthetic teacher geometry, distances between token metadata, class relations, or a small routing table; only a sparse index set Omega is required. Every T training steps, compute W, form B = -H_m log(W plus delta) H_n, gather B and centered reference costs on Omega, and calculate epsilon_hat using the closed-form regression formula. Compute R as the normalized residual. If R is below a threshold, update the live temperature with the damped geometric rule and clip it to safe limits. If R is high, freeze epsilon, increase Sinkhorn iterations, and log a model-mismatch warning rather than applying an unreliable update. The closed-form estimator and residual are the paper mechanism; damping, clipping, threshold, and update frequency are empirical design choices. First test on a small Transformer for synthetic sequence classification in which pairwise costs are Euclidean distances and 5 percent of costs are exposed. Compare fixed epsilon, gradient-learned epsilon, and inverse-OT calibration. The prediction is epsilon_hat divided by true epsilon within 10 percent when R is below 0.01. With independent observation noise, estimation error should decrease approximately as the inverse square root of the number of trusted entries. R should increase before routing entropy collapses when the learned attention no longer follows an entropic OT model.