Measure-Lifted Entropy Amplifier

1. Integration point: use this as the latent transition in a probabilistic RNN, state-space model, or world model. For each input context, maintain m latent particles z_i and normalized weights w_i. The deterministic transition network T_theta receives the current observation and maps every particle forward; decode with a weighted mean or a mixture decoder. Avoid resampling in the first experiment because resampling can destroy the direct pushforward structure. 2. Pseudocode: initialize particles z_i and weights w_i=1/m; for t=1,...,H compute z_i <- T_theta(z_i,x_t), y_hat_t <- sum_i w_i D_phi(z_i), and the task loss. Periodically sample pairs of particle ensembles, compute their Wasserstein-1 distance over the whole rollout, and estimate separated-set growth using a greedy epsilon-net. Add a hinge penalty if the estimated base entropy h_hat_K falls below target h_0, and penalize measure-ensemble collapse when h_hat_mu falls below H. 3. Computed from the paper: the pushforward construction and the zero-versus-infinite entropy implication. Estimated empirically: separated-set counts, Wasserstein distances, h_hat_K, and h_hat_mu; infinity is operationalized as continued growth with particle count and horizon. 4. First cheap experiment: train a 2D GRU world model on a switching nonlinear map or sequential MNIST, comparing deterministic, Gaussian-latent, and 32-particle measure-lifted models at similar parameter counts. Sweep the diversity coefficient and measure greedy (H,epsilon)-separated counts for epsilon=0.1 over H=10,20,40. The prediction is a sharp boundary: when h_hat_K is statistically zero, measure entropy remains bounded; once h_hat_K is positive, log separated-measure counts continue increasing with particle count instead of saturating, with a positive per-time slope. Fit h_hat_K from deterministic trajectories and test whether positive h_hat_K predicts nonzero measure entropy across random seeds.