Hysteretic Attractor Layer

(1) Integration point: place the module after an encoder or inside an RNN/SSM latent update. The encoder outputs $u\in\mathbb R^N$; initialize $x(0)=\operatorname{sign}(u)$ and run $K$ recurrent updates with learned matrix $W$. (2) Pseudocode: x = sign(encoder(input)); for t in 1..K: y = W @ x; x = where(y >= Th, +1, where(y <= -Th, -1, x)); output = readout(x). In the forward pass use the exact hysteresis rule. In the backward pass use a straight-through estimator whose surrogate is $\tilde h_\tau(y,x)=\tanh(y/\tau)+\rho(Th-|y|)x$; stop gradients through the copied-state term if it causes instability, and anneal $\tau$ from 1 to 0.1. (3) Compute the recurrent field and threshold comparisons exactly; estimate gradients empirically through the surrogate. Constrain $W$ by spectral normalization or replace it with $W/\max(1,\|W\|_2)$ to reduce oscillations. (4) First experiment: train a 32-unit MNIST recurrent classifier for 10 update steps, comparing hysteretic activation against tanh and sign baselines at equal parameter count and FLOPs. Evaluate clean accuracy, accuracy after additive input noise, accuracy after random latent bit flips, convergence rate defined by $x(t+1)=x(t)$, and the number of state changes in the final three iterations. The hypothesis is improved robustness and more stable late-iteration predictions, with comparable clean accuracy.