Learned Busemann Stable Leaves

Use a recurrent world model or SSM with latent state h. Add two heads: an endpoint head \(z_\phi(h)\) predicting a coarse future-behavior label or a representation of the final observation, and a scalar B-head \(B_\phi(h,z)\). Construct positive pairs from two windows in the same trajectory that have similar terminal futures, and negative pairs from trajectories with different futures. Pseudocode: z1=z_head(h1); z2=z_head(h2); B1=B_head(h1,stopgrad(z1)); B2=B_head(h2,stopgrad(z2)); rollout both states for H steps; leaf_loss=max_k(norm(h1_k-h2_k)^2) + alpha*contrastive(z1,z2); drift_loss=relu(B_next-B_now+c)^2; total=task+lambda*(leaf_loss+beta*drift_loss). Estimate the paper's asymptotic limit by an exponential moving average of finite-horizon terminal-prediction losses; do not interpret the learned B-head as an exact geometric B-function. The first experiment should use a GRU on a synthetic system with two attracting limit cycles or two stable equilibria, followed by Lorenz-63 short-horizon prediction. Compare vanilla training, contrastive latent training, and this leaf objective. The prediction is that within-leaf rollout separation decays approximately as \(C\exp(-ck)\), while cross-leaf pairs retain separated endpoint codes. Fit the slope of \(\log\|h_k-h'_k\|\); success requires a negative slope for positive pairs and a significant gap between within-leaf and cross-leaf slopes. Sweep c: training should become unstable or overconstrained when the requested decay exceeds the achievable contraction rate.