Percolation-Certified Neural Cellular Automaton

(1) Exact integration point: replace one or more spatially local recurrent blocks in a neural cellular automaton, convolutional SSM, or video/world-model backbone. For a feature field $h_t\in\mathbb R^{C\times H\times W}$, use a shared depthwise or ordinary convolution with radius $R$, followed by a gated residual update. Add an explicit inactive channel and enforce the absorbing condition by multiplying the output gate by $a_t(x)=\mathbf 1[\|h_t(x)\|_2>\epsilon]$, or set the update to zero when every site in the receptive field is inactive. Shared convolution weights enforce translation equivariance. During training, inject fixed-amplitude Gaussian or uniform local noise $U_t(x)$ so that shared-noise coupling can be evaluated.

(2) Pseudocode:

for each minibatch sample:
    h = seeded_state()
    for t in range(T):
        u = iid_noise_like(h)
        q = local_conv(h, radius=R) + noise_projection(u)
        h_new = gated_update(h, q)
        active = local_norm_pool(h, radius=R) > eps
        h = h_new * active
    for each coarse block (z,t):
        Y[z,t] = block_success(seed_at_bottom, shared_noise, target_blocks)
    p_hat = mean(Y)
    L = task_loss + lambda_block * relu(p_target - p_hat)^2
        + lambda_sync * disagreement_loss(shared_noise, perturbed_seed_pair)
    backpropagate(L)

block_success is one when a seed at the bottom of a window of width $K L_{\mathrm{bl}}$ and duration $T_{\mathrm{bl}}$ reaches every required neighboring block within distance $L_{\mathrm{bl}}$ while remaining above an activity threshold. Set $L_{\mathrm{bl}}$ and $T_{\mathrm{bl}}$ using the displayed formulas, initially with $\kappa$ estimated from disagreement decay and $s$ chosen as the desired normalized propagation speed.

(3) Mathematical versus empirical quantities: translation equivariance and locality are architectural constraints. The percolation threshold $p_c$ should be estimated once by Monte Carlo for the selected neighbor geometry. Estimate $p=\mathbb P(Y=1)$ over held-out noise seeds and use a Wilson lower confidence bound rather than the raw mean. Estimate $\kappa$ by fitting $\log D(h)$ against $h$, where $D(h)$ is the probability that two states with different interiors disagree at a target site after $h$ common local layers. The percolation argument is treated as a certificate only when the lower confidence bound on $p$ exceeds $p_c$.

(4) First cheap experiment: train a $32\times32$ neural cellular automaton on Moving-MNIST or a binary cellular-automaton prediction task, comparing a standard local ConvGRU against the constrained model at equal parameter count and FLOPs. Measure task loss, survival from a one-cell seed, front displacement per recurrent step, and agreement between two perturbed initial states under identical noise. The expected signal is higher survival probability and longer reliable propagation at equal depth, with disagreement decreasing approximately exponentially in common interior depth; a useful target is at least twice the reliable propagation distance before activity dies without increasing false activation outside the propagation cone.