From Cross Mapping to Stable Dynamical Factors: Learning Aligned Dynamical Structure in Nonlinear Systems

Learning aligned dynamical structure from paired nonlinear time series calls for a population correspondence alongside predictive skill. Cross-map prediction leaves this structural target unspecified. We define an aligned stable dynamical factor: a map that aligns paired states, commutes with dynamics, transports transmitted tangent directions, and controls metric-relative conditioning. Atlas GCCM–BCD and Neural GCCM estimate this relation with hard local and global differentiable surrogates, respectively. Matched-budget evidence isolates lower state and tangent incompatibility from structured losses, without uniform factor-error or categorical-direction gains; geometry and rank differences remain unresolved. The target uses latent dynamical manifolds (LDMs), combining reconstructed state, induced dynamics, a metric, and invariant measure. We prove reconstruction transport of an assumed factor and conditional regularized-target consistency for idealized globally smooth estimators under dependent sampling, plug-in consistency, risk separation, and approximate global optimization; hard-chart Atlas and no-factor consistency remain outside the result. An ancestry reading is external and conditional on ordered generative dynamics, component-local sensing, causal sufficiency, and absence of observational equivalence; otherwise the relation describes aligned dynamical information containment. Synthetic studies examine factor recovery, matched-independent negative controls, adverse feedback, synchronization, robustness, controlled observation-map dimension and conditioning, representation diagnostics, and targeted external baselines.