Nominal-Safe Scalar Association Regularizer

Integrate this at a scalar representation tensor, not at the main classifier output. Choose a scalar x_i from each example, such as one bottleneck coordinate, a learned uncertainty score, the mean attention logit for a feature group, or the pre-softmax score used by an MoE router. Attach a lightweight probe with parameters a[1..k] and b[1..k], where q_i=softmax(a*x_i+b). For each minibatch, obtain one-hot categorical targets y_i, compute p_hat=mean(y_i, dim=0), B_hat=sum(p_hat2), A_hat=mean(sum(q_i2, dim=1)), and R_hat=(A_hat-B_hat)/(1-B_hat+epsilon). Optimize L_task - lambda*R_hat; detach p_hat if batch-statistic gradients create instability, or use an exponential moving average p_ema for B_hat. The probe should normally be optimized jointly, but run an ablation with probe parameters stopped after a warmup to distinguish representation learning from probe memorization. The paper supplies the posterior-concentration construction; the implementation estimates expectations empirically and uses a stable softmax. Start with a small MLP on CIFAR-10 or a UCI tabular dataset, using one scalar bottleneck and a standard cross-entropy baseline. Compare random category permutations, multiple lambda values, and a control using integer-encoded Pearson correlation. Success means unchanged performance under label permutation, higher test accuracy or calibration at equal parameter count, and a measurable increase in held-out association of the scalar without simply increasing the probe's training accuracy.