Fractal Conditional Correlation Dimension Infers Complex Causal Networks

The paper defines GeoC and its multivariate extension oGeoC and asserts the key implications (J⊂K ⇒ oGeoC = 0; if J⊂N_I and J⊄K ⇒ oGeoC > 0), but offers no rigorous proof beyond the statement “it is clear” and a brief set-theoretic remark. In fact, it even contains a small slip claiming Geo(X′(I)|X(J),X(K)) reduces to Geo(X′(I)|X(J)) when J⊂K; the reduction is to Geo(X′(I)|X(K)) (the conclusion oGeoC=0 still follows) . The candidate model supplies a plausible geometric proof sketch via two lemmas (bi-Lipschitz invariance of D2 on graphs, and a “positive fiber thickness” argument ensuring strict D2 increase when a relevant parent is omitted). However, that proof relies on nontrivial, unstated regularity and nondegeneracy hypotheses (uniform slice-wise Lipschitz lower bounds, existence of a product-like subset visited by the data, absence of synchrony), and on product additivity of correlation dimension—none of which are justified in the paper and only partially spelled out by the model. Hence, while both the paper and the model point in the right direction under deterministic, noise-free dynamics (as the paper emphasizes) , neither provides a complete, fully rigorous proof of the claimed inequalities in the generality stated.