Conley-Certified Latent World Model
Implementation & benchmark of arXiv:2609.01509 — Characterizing High-dimensional Dynamics by Combinatorial-Topological Methods on a Latent Space
Source paper: Characterizing High-dimensional Dynamics by Combinatorial-Topological Methods on a Latent Space arXiv:2609.01509 ⓘ · analyzed Sep 2, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Train an encoder-decoder world model together with a latent transition map, but certify latent attractors only when the learned model is approximately semiconjugate to the observed high-dimensional dynamics with residual below the isolating-set margin. Compute a Conley-Morse graph on a latent grid and lift each certified recurrent component through the decoder to obtain a region in the original state space where an attractor or invariant set is predicted to exist.
Formulas
Mathematical statement
Let X be the high-dimensional state space, Z the latent space, f:X→X the true discrete-time evolution, E:X→Z an encoder, D:Z→X a decoder, and g:Z→Z the learned latent evolution. Exact semiconjugacy satisfies E∘f = g∘E. The paper replaces this with the uniform residual ε = sup over x in X of d_Z(g(E(x)), E(f(x))). A Conley-Morse computation partitions Z into cells Q_i and constructs a directed graph of cell transitions under g, including uncertainty padding. A strongly connected component with no outgoing edges is a candidate isolated invariant component. For an isolating neighborhood N in Z, define its robustness margin μ(N) as the minimum distance from the nominal image g(N) to a transition boundary excluded from the Conley graph. The transferable condition is ε < μ(N): perturbations induced by the encoded true dynamics cannot cross the certified combinatorial boundaries, so the invariant-set or attractor conclusion represented by the graph persists for the original dynamics in E⁻¹(N), subject to compactness and approximation assumptions. In implementation, ε is estimated by a held-out maximum or high quantile and μ is estimated from interval-cell distances.
Implementation notes
Integrate this as a certification branch around a standard autoencoder world model. Use an encoder E_theta, decoder D_phi, and latent transition g_psi. Train with reconstruction loss, one-step prediction loss, and a semiconjugacy penalty. For each batch containing consecutive states x_t and x_next, compute z_t = E_theta(x_t), z_next = E_theta(x_next), z_pred = g_psi(z_t), and residual r = norm(z_pred - z_next). Optimize L = L_reconstruction + lambda_pred times L_prediction + lambda_sc times r squared. After training, tile the latent region visited by validation trajectories into cells Q_i. For each cell, propagate interval samples or an ensemble through g_psi, enlarge the image by the empirical residual bound epsilon_test, and add encoder or model uncertainty if available. Create an edge Q_i to Q_j whenever the padded image intersects Q_j. Find terminal strongly connected components and estimate the isolating margin mu as the smallest distance from any padded image to a forbidden cell boundary. The paper supplies the approximate-semiconjugacy and combinatorial-topological lifting mechanism; residual confidence bounds, interval propagation, and finite-cell margins are engineering estimates. The first cheap experiment should use a Duffing oscillator or Lorenz system with known multiple attractors, embedded into 20-dimensional observations by a smooth random map. Compare an ordinary autoencoder world model with the semiconjugacy-regularized model. Sweep observation noise and training duration. The falsifiable prediction is a sharp certification transition near (epsilon_test + delta_conf) divided by mu equal to 1: below this value, the recovered number of attractors and basin assignments should remain correct; above it, certificates should fail or become unreliable.
Verification
Stage 1 · Toy mechanism gate: Not tested
Stage 2 · Mechanism transferred to benchmark: Not tested
Stage 2 · Practical benchmark result: Not run
Methodology: Toy-system gate first; the benchmark stage runs only after a pass. How verification works
Stage 1 — Mechanism check
Verdict computed by deterministic test code from paired-seed statistics — not by the language model.
Artifacts
Artifacts unavailable.