Entropy-Preserving Semiconjugate World Model

1. Integration point: use an encoder \(E_\psi\), a latent recurrent or state-space transition \(F_\phi\), and a decoder \(D_\omega\) inside an existing world model. The full-state transition \(G_\theta\) can be the observed environment transition, a separately learned high-dimensional model, or the model's recurrent update. Train the latent transition through the semiconjugacy residual rather than relying only on decoded next-state error. 2. Pseudocode: sample trajectories \(x_t\); compute \(z_t=E_\psi(x_t)\), \(\tilde z_{t+1}=F_\phi(z_t)\), and \(\hat x_{t+1}=D_\omega(\tilde z_{t+1})\). Minimize reconstruction and prediction losses plus \(\lambda_{sc}\|E_\psi(x_{t+1})-\tilde z_{t+1}\|^2\). Select pairs \(x,x'\) with nearby latent codes, draw \(\alpha\sim U[0,1]\), and penalize encoder variation along their interpolation. Estimate \(\widehat h_T\) for full and latent trajectories using the same fixed-resolution delay-coordinate or k-nearest-neighbor entropy-rate estimator, then add the entropy mismatch penalty. 3. Taken directly from the paper: the semiconjugacy identity, entropy equality, and nested-fiber principle. Estimated empirically: entropy rates, fiber connectedness, and residual magnitudes. 4. First cheap experiment: use a Hénon map embedded into 20 dimensions with nuisance oscillators and compare a standard autoencoding RNN against this model at equal latent dimension. Measure one-, ten-, and one-hundred-step prediction error, latent and full entropy rates, semiconjugacy residual, and fiber diameter after repeated refinement. The quantitative prediction is that latent entropy remains within 10-20% of full entropy while median fiber diameter decreases under refinement. If the residual is sufficiently small, long-horizon error should be governed by the latent dynamical complexity rather than nuisance dimensions. A falsification is a sharp entropy drop or fragmentation of interpolation fibers into disconnected clusters.