Causal Discovery in Nonlinear Dynamical Systems using Koopman Operators

The paper claims an unconditional equivalence between dynamical and Koopman causal influence (Theorem III.2) and sketches a proof via kernel sections k_{E,ω}. However, the converse direction implicitly requires a point-separation property of the effect-space RKHS (i.e., injective feature map y ↦ K_E(y, ·)). Without this, the implication “all K_t k_{E,ω} are independent of ω_C ⇒ P_E Φ_t is independent of ω_C” can fail (e.g., constant-kernel RKHS). The paper does not state this requirement, so the theorem is incomplete as written. The candidate solution adds precisely this separation assumption and gives a correct, self-contained proof (including a counterexample showing necessity).