Whether a language model was trained on a document matters for copyright, privacy, and benchmark contamination. Yet most membership-inference (MI) evaluations on language models collect members and non-members after the fact, so success may reflect distribution shift and failure is inconclusive. We re-ran OLMo 2 1B's official 50-billion-token mid-training stage, replacing selected training chunks with documents from 7,814 pairs matched on domain and difficulty. For each pair, a coin flip determined which document was inserted and which was held out. The run released by the Allen Institute for AI (AI2) from the same seed saw identical data everywhere else and serves as the counterfactual. One exposure raises a previously unseen document's likelihood on average but rarely identifies it. Using the counterfactual as reference, the reference ratio has an area under the ROC curve (AUC) of 0.767 but, with one threshold across domains, catches 3.8% of members at a 1% false-positive rate (FPR). In contrast, 13 matched pairs give a paired test 80% power at the 5% level to detect training on a collection, given same-domain non-members and a close reference. A reference trained with another seed recovers 92% of the counterfactual's AUC advantage over chance, whereas reference-free attacks stay within 0.07 of chance when pooled over domains. The full official post-training recipe, re-run on both models, leaves 89% of the average effect. Even with a closely matched reference, substantial document-level variability limits individual membership inference. We will release the checkpoints, per-token log-probabilities, and the evaluation documents; the code is available at https://github.com/ali-bana/OLMo-Twin.

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