Order-One Slow-Gate Reservoir

Integrate the mechanism into a small GRU, linear state-space model, or gated residual RNN. Maintain one binary or relaxed pause variable per hidden channel. At each sequence step, sample or update the gate with active-to-paused probability pp = 1 − exp(−kp Δt) and paused-to-active probability pu = 1 − exp(−ku Δt). Apply the recurrent transition only to active channels; paused channels retain their previous state, and optionally multiply their outgoing contribution by a learned blocking factor. For differentiable training, use a Binary-Concrete gate with straight-through sampling, but evaluate the final model with hard gates. The pseudocode is: compute pp and pu; update each zi using its two-state transition; compute hi_new = (1 − zi) φi(h, x) + zi hi; compute the readout y; then backpropagate through relaxed gates. The birth-death stationary relation and Np* ≈ 1.5 are taken from the paper; estimate the actual stationary Np, its variance, output autocorrelation, and cluster statistics empirically. First run a cheap copy-memory and adding-problem experiment with L in {32, 64, 128, 256}, comparing a vanilla GRU, independently paused gates, and a fixed-delay baseline at matched parameter count. Sweep kp while holding ku fixed. The quantitative prediction is that the low-frequency Fano-like statistic FT = Var(sum of yt) divided by E[sum of yt] peaks when Lkp divided by ku is approximately 1.5, and the peak location obeys kp* proportional to L raised to minus one within roughly 20 percent. The output correlation time should increase approximately as 1 divided by ku at fixed Np, while the peak should disappear when Np is much smaller than one or much larger than one. Test whether the peak coincides with improved long-delay accuracy rather than merely increased output variance.