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Definition 1
The evolutionary DAG contains traits X^(0),...,X^(T), heritable factors epsilon^(0),...,epsilon^(T), and binary reproduction indicators S^(0),...,S^(T-1). Its four edge families are within-generation trait-to-trait edges copied from G, trait-to-reproduction edges copied from G, componentwise factor-to-trait edges, and factor inheritance edges epsilon_i^(t) -> epsilon_i^(t+1).
Lemma 1
For disjoint A,B,C subset X, evolutionary d-separation at generation T conditional on C^(T),S^(<T) implies static d-separation of A and B conditional on C,S; the converse does not generally hold. The paper explicitly connects converse failures to false causal discoveries under a static interpretation.
Definition 2 and Theorem 1
G^+ copies the causal edges of G and completes the ancestors of S into a topologically oriented clique. Theorem 1 states the biconditional between d-separation in the selected, unrolled G^(T) and d-separation in G^+ for every T >= 1 and every disjoint A,B,C.
Algorithm 1 and Theorem 2
Algorithm 1 applies PC, GES, or another sound and complete causal-sufficiency method and returns a CPDAG. Theorem 2 characterizes its adjacencies, states that each oriented edge is a true direct cause whose head is not an ancestor of selection, and states that every unoriented edge admits an alternative compatible source relation.
Algorithm 2 and Theorem 4
Algorithm 2 applies CDNOD or a comparable method to multi-domain observations. Theorem 4 says the resulting PDAG on X retains every single-domain orientation and can contain additional orientations, while preserving Theorem 2's soundness and completeness statements.
Section 5
The synthetic protocol uses Erdos-Renyi DAGs with d in {10,15,20}, average degree 2, d/5 parents of S, coefficients in [-2,-0.5] union [0.5,2], noise variances in [1,4], N=5,000, PC alpha 0.05, and GES L0 penalty 2. Reproduction ranks each generation into six groups with 0 through 5 offspring and downsamples the next generation to fixed N.
The seven named real datasets are DGRP, Cranial, Panzea, PanTHERIA, AVONET, CSES, and PUMS. The paper calls these analyses qualitative, uses partial eQTL ground truth for DGRP, and uses LLM-generated pseudo ground truth for the other six. Appendix D.2 prints integer numerators and denominators for every reported percentage.