# 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).

**Verdict: VERIFIED.**

Each source model is crossed with all 31 nonempty subsets of changed variables in `X union {S}`. The audit constructs the paper's `G^{+I}`, applies the known root directions from the domain index, closes the PDAG, and compares its `X` orientations with the single-domain CPDAG.

| Quantity | Result |
| --- | ---: |
| model/change-set configurations | 269,328 |
| configurations losing a single-domain orientation | 0 |
| multi-domain orientation soundness violations | 0 |
| adjacency violations | 0 |
| configurations with strict gain | 88,176 (32.74%) |
| total additional orientations | 172,512 |
| native CDNOD correct oriented edges | 4.65 mean |
| paired single-domain PC | 3.55 mean |
| finite runs CDNOD at least PC | 16/20 |

The native run uses four domains, 5,000 samples per domain, identical causal SEM parameters, and changed selection weights.

## Destructive control

If the domain-index edges are added only to literally changed `X` variables and the required ancestor-of-selection expansion is omitted, the soundness checker records 583,356 violations.

## Evidence

Exact multi-domain audit: `outputs/exact_graph_audit.json`. Native CDNOD: `outputs/finite_native_algorithms.json`. Control: `outputs/destructive_controls.json`.

## Limitations

Strict improvement is not universal: 67.26% of configurations tie the single-domain orientation count. The universal verified relation is monotonic non-loss; gain depends on which mechanisms change.