Recurrence-to-Latent Cycling Regularizer

Integrate this at the output of a temporal encoder: given an input window, produce latent states z_1,...,z_n and use squared Euclidean distance D_ij=||z_i-z_j||_2^2. First set R_Gamma=max_i ||z_{i+1}-z_i||_2 plus a small margin. Build the upper-triangular recurrence grid with vertex filtration value D_ij; add horizontal and vertical edges and squares whenever their incident filtration values are below the current threshold. Compute H0 and H1 persistence on this two-dimensional cubical complex using a standard sparse boundary-reduction implementation. Keep the top K H1 intervals, or keep H0 components corresponding to close-return structure. For every selected representative recurrence feature (i,j), construct the paper's chain image c_Gamma(i,j)-[i,j] and add the adapted loop penalty. Pseudocode: encode the window to z; compute D; construct the filtered cubical grid; run persistence reduction; select top-K intervals; for each selected interval compute its ordered path and shortcut; add lambda times L_cycle to the supervised loss. Persistence weights should be detached from autograd, while gradients flow through latent distances and adjacent-step distances. The paper supplies the chain construction and its boundary guarantee; persistence intervals, top-K selection, and R_Gamma are estimated per batch. Start with a GRU or two-layer SSM on synthetic sine, chirp, and regime-switching signals, comparing against the same encoder without the regularizer and against a latent Vietoris-Rips baseline. Success is improved classification or transition-detection accuracy at equal parameter count, fewer short-lived latent cycles, and at least 3x lower topological-computation time than the full Vietoris-Rips baseline.