# 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).

**Verdict: VERIFIED.**

The source `G` is augmented by a topologically oriented clique over every ancestor of `S`. The audit compares the resulting `G^+` against the fully unrolled and selected `G^(T)` without sharing any cached reachability result.

| Test | Agreement | Mismatches |
| --- | ---: | ---: |
| exhaustive `d=4`, `T=1,2,3` | 1,433,520/1,433,520 | 0 |
| random `d=5..10`, `T=1,2,3,5` | 143,811/143,811 | 0 |
| d-separation signature invariant across `T=1..4` | 8,688/8,688 models | 0 |
| no-selection degenerate case `G^+=G-S` | 543/543 models | 0 |

The underlying Bayes-ball implementation first passes 11,984 independent comparisons with `networkx.is_d_separator`.

## Destructive control

Simply deleting `S` without adding the ancestor clique gives 98,787 mismatches on the same exhaustive cells, an agreement rate of only 93.11%.

## Evidence

Theorem audit: `outputs/exact_graph_audit.json`. Independent oracle: `outputs/oracle_gates.json`. Control: `outputs/destructive_controls.json`.

## Limitations

The exhaustive boundary is four traits; larger checks are randomized. The result applies to the paper's causally sufficient model, not the misspecified dependent-inheritance setting.