Latent Itinerancy Graph Regularizer

1. Integration point: apply the method to the latent state of a GRU, diagonal state-space model, or recurrent world model during training. Keep the graph dimension manageable by projecting each latent state onto 8-32 PCA coordinates computed from a warm-up buffer, then use either a coarse Cartesian grid or k-means cells. 2. Pseudocode: initialize an exponential-moving-average count matrix $C=0$. For every minibatch transition $(z_t,z_{t+1})$, compute cell indices $i=\mathrm{cell}(z_t)$ and $j=\mathrm{cell}(z_{t+1})$, then update $C_{ij}\leftarrow\rho C_{ij}+(1-\rho)$. Row-normalize $C$ to obtain $P$, compute $\pi$ with 20-50 power iterations, compute each $h_i$, and find strongly connected components using Tarjan's algorithm. Add $L_{graph}=\alpha\,\mathrm{KL}(\hat\pi+\epsilon\,||\,\pi+\epsilon)+\beta\sum_{i\in S}h_i-\gamma\sum_{i\in T}h_i$ to the prediction loss, where $\hat\pi$ is empirical cell occupancy, $S$ contains stable-mode cells, and $T$ contains transition cells. Initially stop gradients through cell assignments and $P$. 3. Computed from the paper's mechanism: cell transitions, SCCs, stationary distribution, and local entropy. Estimated empirically: the partition, transition counts, stable/transition labels, and the Markov approximation. 4. First cheap experiment: train a small GRU on bouncing-ball trajectories, 2D navigation, or sequential MNIST and compare ordinary teacher-forced training with the graph regularizer on 100-500-step free rollouts. The predicted signature is that occupancy mismatch decreases, stable SCCs have lower $h_i$, and the semilog slope of empirical occupancy convergence approaches $\log|\lambda_2|$. Agreement between the predicted and measured decay rates within 20% is the target. Vary grid resolution and require the SCC structure to persist across two neighboring resolutions; reject the mechanism if small partition changes destroy the graph structure or if empirical decay is unrelated to $|\lambda_2|$.