  We analyze the popular ``state-space'' class of algorithms for detecting
casual interaction in coupled dynamical systems. These algorithms are often
justified by Takens' embedding theorem, which provides conditions under which
relationships involving attractors and their delay embeddings are continuous.
In practice, however, state-space methods often do not directly test
continuity, but rather the stronger property of how these relationships
preserve inter-point distances. This paper theoretically and empirically
explores state-space algorithms explicitly from the perspective of distance
preservation. We first derive basic theoretical guarantees applicable to simple
coupled systems, providing conditions under which the distance preservation of
a certain map reveals underlying causal structure. Second, we demonstrate
empirically that typical coupled systems do not satisfy distance preservation
assumptions. Taken together, our results underline the dependence of
state-space algorithms on intrinsic system properties and the relationship
between the system and the function used to measure it -- properties that are
not directly associated with causal interaction.
