Sliding-Friction Recurrent Memory

Integrate the mechanism as a recurrent state module before the task-specific readout. Maintain two tensors S_a and S_b with shape batch by chain-length by latent-width, initialize them from the input embedding, and use tanh states for differentiable training or straight-through binary states for a direct spin implementation. At each recurrent step, compute nearest-neighbor ferromagnetic messages J times the sum of the left and right neighboring states; shift or interpolate S_b by the current offset q; add cross-chain messages K times the shifted state; inject temperature-scaled Gaussian or logistic noise; update each chain with S_new=tanh(S+alpha times message); and advance q by v. Use a small learned readout from pooled magnetization and local features. The paper supplies the lifetime-scaling mechanism, while the neural implementation must estimate J, K, beta, v_c, and asymptotic prefactors empirically. First test a delayed binary-memory task storing a random bit for horizons from 1 to 10,000. Compare a single noisy recurrent chain, two static coupled chains with v=0, and the sliding model over a logarithmic grid of v. Measure median threshold lifetime tau_hat_0.2 and fit log lifetime versus beta J at v=0 and versus (beta J)^2 v log(v) above crossover. The falsifiable prediction is a kink or slope change at a reproducible small v_c, with the sliding model exceeding the static-chain lifetime in the high-beta J regime. If there is no crossover or lifetime gain, reject the mechanism.