Causal Discovery in Symmetric Dynamic Systems with Convergent Cross Mapping

The paper proves that for C2 symmetry Rxn(π):(x1,…,xn)→(−x1,…,−x_{n−1},xn), the nth-coordinate observation h(x)=xn is even, so all time-derivatives retain even parity, giving F_{xn,n}(R·x)=F_{xn,n}(x) and thus a generically two-to-one, noninjective reconstruction; in contrast, xi (i<n) is odd, giving inversion symmetry in its reconstruction. This produces a noninjective cross-map Π_{xixn}=F_{xn,n}∘f′∘(F_{xi,n})^{-1}, so CCM falsely indicates unidirectional causality even when coupling is bidirectional. These steps are stated and sketched in Proposition 1 with equations (4.6)–(4.10) and (4.11) in the paper, and the general limit of single-observable differential reconstructions to at most Z2 symmetry is stated as Theorem 3 (Cross). The candidate solution reproduces the same parity argument, the two-to-one/noninjectivity conclusion, and the Π construction, matching the paper’s logic and conclusions.