SOURCE_EXCERPTS.md

# Source anchors

These anchors were transcribed from the pinned arXiv HTML for `2606.05689v1` before the experiments were designed. The immutable byte pins are in `SOURCE_PIN.txt`.

## 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.