Onsager–Casimir Response Regularizer

Apply the regularizer to hidden states of an RNN, SSM, transformer with a recurrent memory cache, or world model. During each training batch, retain hidden vectors $h_0,\ldots,h_T$ and compute empirical lagged covariances $\mathbb{C}_{k,j}=B^{-1}\sum_b(h_k^{(b)}-\bar h_k)(h_j^{(b)}-\bar h_j)^T$, using covariance shrinkage when the batch is small. Estimate the response $\chi_q$ by injecting a small impulse $\delta u$ into the input or control channel and calculating $[h_q(u+\delta u)-h_q(u)]/\delta u$; use Jacobian-vector products to avoid constructing a full Jacobian. Add $\lambda_{\mathrm{FDR}}\mathcal{L}_{\mathrm{FDR}}+\lambda_{\mathrm{OC}}\mathcal{L}_{\mathrm{OC}}$ to the task loss. The paper supplies the correlation-response structure; $\tau$ is an effective neural temperature estimated from injected input-noise variance or tuned on validation data. The first cheap experiment should use a two-dimensional synthetic rotating linear system with known transition $A=-\gamma I+\omega J$, followed by delayed-MNIST with a compact GRU. Compare an unregularized baseline with regularized models at equal compute. The prediction is that normalized FDR residual decreases monotonically for small-to-moderate $\lambda_{\mathrm{FDR}}$, while antisymmetric lag correlation changes sign under $s\to-s$ and approaches zero as $\omega\to0$. Excessive regularization should cause a measurable U-shaped validation-loss curve.