Hankel Moment Regularizer

Integrate this at a scalar response head, not inside every hidden layer. Choose an ordered scalar $a$, such as diffusion timestep, attention temperature, augmentation severity, model depth proxy, or the index of a loss-scale schedule. For each minibatch, evaluate the same network response at $K+1$ or $2K$ nearby values $a+r\Delta$, producing positive scalars $f_r=\operatorname{softplus}(u_r)+\delta$, where $u_r$ is the raw response and $\delta>0$ avoids singular matrices. Construct $H_{ij}=f_{i+j}$ for $0\leq i,j<K$. Add either a PSD penalty $\|\operatorname{ReLU}(\epsilon I-H)\|_F^2$ or, when the matrix is numerically positive definite, the barrier $-\lambda\log\det(H+\epsilon I)$. Use $K=3$ initially; compute the eigendecomposition or Cholesky factor in float32 and stop gradients through no quantities. Pseudocode: evaluate $f[0:2K-1]$; set $H[i,j]=f[i+j]$; loss $=L_{task}+\alpha L_{Hankel}$; backpropagate. The paper supplies the moment representation and the determinant-positivity rationale; the empirical quantities are the sampled responses and their eigenvalues. First test on a small CIFAR-10 ResNet or a 125M-parameter language model with an order-dependent calibration/temperature head. Compare no regularizer, pairwise log-convexity, and Hankel $K=3$ at equal compute. Success means fewer oscillations in the response across $a$, improved held-out calibration or robustness, and no more than 1.2x training overhead.