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POSCMs provide an extension of structural causal models (SCMs), as a self-contained causal modeling framework for endogenous graphs, allowing for an intervention hierarchy spanning node- and edge-level context and endogenous variable interventions. To enable surgical edge interventions, we adopt a Kolmogorov-Arnold-Sprecher edge-functional decomposition, an existence theorem for representing each node mechanism as a sum of univariate functions of its parents, yielding an explicit parametrization of dyadic functional contributions. We provide an identifiability theory that clarifies which intervention families would suffice to disentangle structure formation from mechanisms. 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