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"Definition 1 formalizes an evolutionary selection model as a DAG G^(T) over trait variables X^(0)...X^(T), heritable factors epsilon^(0)...epsilon^(T), and reproduction/selection indicators S^(0)...S^(T-1), distinguishing it from one-shot static selection models (Section 2, Definition 1).",
"Lemma 1 shows that repeated evolutionary selection induces conditional dependencies among variables that are absent under static selection models, so applying static-selection graphical models to evolutionary data can yield false causal discoveries (Section 2, Lemma 1).",
"Theorem 1 proves that the clique-augmented DAG G^+ (Definition 2) fully captures all d-separation/conditional-independence constraints implied by the evolutionary selection model, without needing to explicitly model the selection variables (Section 3, Definition 2, Theorem 1).",
"Theorem 2 establishes that applying standard constraint-based algorithms such as PC or GES (Algorithm 1) to G^+ is sound and complete: oriented edges correspond to true causal relations, while unoriented edges may reflect the presence of selection (Section 3, Theorem 2, Algorithm 1).",
"Theorem 4 shows that combining heterogeneous data from multiple environments/domains via the CDNOD-based procedure (Algorithm 2) improves identifiability of the evolutionary selection model compared to single-environment data (Section 4, Theorem 4, Algorithm 2).",
"The proposed identification procedure is validated on synthetic graphs of varying size and on seven real-world datasets spanning biology, agriculture, and social science (Section 5)."
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