Affine mean-width normalization

Integrate the module immediately after a Transformer MLP or attention output and before residual addition, initially with hidden dimension n=256 or n=512. Maintain an unconstrained trainable matrix B, but construct A=exp(B-tr(B)I/n), which enforces det(A)=1 exactly; for a cheaper implementation, parameterize A using several Householder factors and diagonal log-scales shifted to have zero sum. For each minibatch X={x_i}_{i=1}^b, subtract the batch mean, compute y_i=A x_i, and form S_A=(1/b) sum_i y_i y_i^T. Draw m=16--32 normalized Gaussian vectors u_j, compute h_j=max_i |u_j^T y_i|, and average them to estimate mean width. Compute the covariance volume with a stable Cholesky log-determinant: vr_cov=exp(logdet(S_A+epsilon I)/(2n)). Add lambda times the ratio of these quantities to the task loss, together with a small penalty lambda_id ||A^T A-I||_F^2 to keep optimization near the identity. Replace the hard maximum by tau log(sum_i exp(|u_j^T y_i|/tau)) during warm-up and anneal tau. Update A every 100 optimizer steps if full matrix exponentials are expensive. The paper provides the determinant-preserving affine objective and mean-width/covering rationale; the covariance determinant and random-support estimator are practical proxies rather than exact theorem guarantees. The first experiment should use a 6-layer small Transformer on WikiText-103, comparing the module with LayerNorm, RMSNorm, and whitening at equal parameter count and FLOPs. Track training loss, activation covariance condition number, empirical covering numbers at several radii, and 4-bit or 8-bit activation quantization error. Success is lower activation condition number and quantization error without perplexity degradation, or faster loss descent at equal compute.