Criticality-controlled sparsity schedule

Integration point: apply the controller to the merge probability of a stochastic hierarchical block, or to a layerwise schedule shared by all merges at training iteration t. Use ReLU, Softplus followed by thresholding, or another nonnegative activation so that exact zeros are meaningful. Pseudocode: at iteration t, estimate zhat_t as the fraction of zero coordinates over the current minibatch and all tree levels; update p_(t+1) = clip(p_t + kappa(zhat_t - z_star), p_min, p_max); sample merge gates with the updated p; backpropagate the task loss and optionally add a weak penalty lambda times (zhat_t - z_star)^2. Initialize with p = 0.5. The paper supplies the mean-field recursion, fixed points, and critical boundary. The implementation estimates finite-width correlations, actual sparsity, gradient survival, and computational savings. First experiment: use a six-level tree MLP on MNIST with target sparsities z_star equal to 0.1, 0.3, and 0.5, comparing fixed p = 0.5, fixed p = 0.7, and feedback control. Log z_n, gradient norm by depth, wall-clock cost, and accuracy. The predicted signature is a sign change in z_(n+1) - z_n at p = 0.5, with near-zero drift at criticality. The measured transition should agree with 0.5 plus or minus 0.05, and a sparse regime should reduce active merge cost without exponential gradient decay.