Reframing cross-world independence for identifying path-specific effects

Understanding causal mechanisms in complex systems requires evaluating path-specific effects (PSEs) in multi-mediator models. Identification of PSEs traditionally relies on the demanding cross-world independence assumption. To relax this, VanderWeele et al. (2014) proposed an interventional approach that redefines PSEs, while Stensrud et al. (2021) introduced dismissible component conditions for identifying separable effects under competing risks. In this study, we leverage SWIGs to dissect the causal foundations of these three semantics and make two key advances. First, we generalize separable effects beyond competing risks to the setting of multi-mediator models and derive the assumptions required for their identification. Second, we unify the three approaches by clarifying how they interpret counterfactual outcomes differently: as mediator-driven effects (classical), randomized contrasts (interventional), or component-specific actions (separable). We further demonstrate that violations of cross-world independence originate from mediators omitted from the model. By analogy to confounder control, we argue that just as exchangeability is achieved by conditioning on sufficient confounders, cross-world independence can be approximated by including sufficient mediators. This reframing turns an abstract assumption into a tangible modeling strategy, offering a more practical path forward for applied mediation analysis in complex causal systems.