MXMap: A Multivariate Cross Mapping Framework for Causal Discovery in Dynamical Systems

The paper defines ρAll, ρDirect, and γ and uses an empirical threshold γ⋆ to prune edges in a two‑phase MXMap pipeline, but it does not claim any population‑level correctness guarantees; its justification is heuristic and empirical (Algorithm 1; Secs. 2.3, 3.2–3.3) . The candidate solution rigorously shows a degeneracy under ideal Takens/SSR assumptions: as sample size grows, CCM reconstructions become exact and multiPCM’s conditional reconstruction collapses to the target variable, making ρAll→1 and ρDirect→0 whenever the conditioning set is nonempty; thus γ→0 even when a true direct edge coexists with an indirect path, so MXMap will prune true direct edges for large T. This directly refutes any putative universal-consistency claim, which the paper does not make.