Delayed hysteretic residual mixer

Integrate this at the residual update of a small recurrent or state-space network, not inside the feed-forward MLP. For each token or time step, maintain a two-channel residual state q_k in R^2 per feature group, a scalar margin z_k, and a FIFO buffer containing the previous d states and signs. Let r_k be the ordinary block output projected to two channels. Compute q_del=q_{k-d} and s_del=s_{k-d}, then update q with q_next=q_k+h*(-q_del+alpha*J*q_del*s_del), exactly matching the adapted delayed equation. Add r_k to the remaining hidden state through a learned projection of q_next. Define z_k as a learned scalar readout of the current residual mismatch, for example z_k=w^T(r_k-q_k)+b. During training, use s=sign(z) in the forward pass and a straight-through gradient s_grad=tanh(z/epsilon) in the backward pass. Initialize tau and alpha so tau*(1+alpha^2)<0.8, then optionally learn them as tau=sigmoid(rho)*tau_max and alpha=softplus(a), rejecting or projecting parameter updates that violate the contraction constraint. The paper's formulas provide the delay discretization and energy-rate prediction; estimate the actual latent-energy slope and switching rate empirically. First test on a recurrent sequential-MNIST or row-wise CIFAR-10 model, comparing an ordinary residual RNN, an instantaneous sign gate, and this delayed mixer at equal hidden size and FLOPs. Measure validation accuracy, gradient-norm variance, mode switches per sequence, and robustness to activation noise. Success means fewer switches and lower gradient variance without slower loss descent, or equal accuracy with a longer stable recurrent rollout.