Inertial asynchronous recurrent computation

1. Integration point: implement this as a recurrent layer, denoising autoencoder bottleneck, or continuous-time state-space block. Maintain tensors x and v with the same shape as the hidden state. Partition coordinates into asynchronous blocks, such as individual neurons, rows, or random one-to-five-percent subsets, and do not synchronize all blocks after every event. 2. Event pseudocode: initialize x and v; repeatedly sample a block B; read the most recent locally cached x values; compute u_B = phi(W_B x + b_B); update v_B <- v_B + (delta/tau_v)(-v_B + u_B); update x_B <- x_B + (delta/tau_x)(-x_B + v_B); leave all other coordinates unchanged; optionally add Gaussian noise with variance sigma squared to u_B. For discrete training, unroll K asynchronous events and backpropagate through the updates, or use the block dynamics only at inference. 3. Computed quantities: W, phi, and the local drive are ordinary network operations. Estimate the dominant Jacobian eigenvalue lambda_max by ten to twenty power iterations on the current hidden state, and estimate relaxation rates from the autocorrelation or an exponential fit of norm(x(t) - x*) . The paper supplies the inertial two-part mechanism and asynchronous event interpretation; the exact neural Jacobian and phase location must be measured. 4. First cheap experiment: use a two-layer tanh autoencoder on MNIST with Gaussian-corrupted inputs, comparing a standard synchronous recurrent denoiser, asynchronous first-order updates, and the inertial asynchronous layer at equal numbers of coordinate updates. Sweep the recurrent gain g multiplying W and the ratio tau_x/tau_v. The predicted signature is a sharp divergence or persistent-oscillation boundary near g lambda_max(W_normalized) = 1, with the measured boundary within roughly twenty percent of the linear prediction. Below the boundary, log norm(x(t) - x*) should show exponential decay; in the underdamped region it should show decaying oscillations, while excessive gain should produce nondecaying activity or saturation. Also test update-order robustness: variance of final reconstruction error across random event schedules should be lower for the inertial system than for first-order asynchronous updates.