Training data attribution (TDA) for large language models is dominated by gradient- and retraining-based methods that require model internals, ruling out the most widely deployed closed-weight systems. We study TDA in the strictly black-box setting, observing only input output behavior accessed through retrieval- augmented prompting. We introduce Context Mixture Factorization (CMF), which models the LLM’s output distribution as a mixture of a base component and dataset-specific components and recovers per-dataset contribution weights by regularized matrix factorization via alternating projected least squares. CMF explicitly decomposes background (π0) and dataset- specific influence and requires only K+1 prompts per query for K candidate datasets. As a comparison method we also formulate the Shapley Context Method (SCM), a residual- similarity attribution score aggregated via Shapley values. We evaluate four instruction-tuned LLMs (GPT-4, Bloomz, Mistral-7B, Phi-3mini) on BoolQ, a synthetic FakeQ variant, and a post-cutoff Olympic2024 dataset against three baselines (TRAK, ContextCite, retrieval cosine similarity). CMF and SCM recover the ordering predicted by our exposure-proxy setup (BoolQ>FakeQ>Olympic2024); ContextCite shows the opposite ordering, attributing most to Olympic2024 because the model relies on context most when it has no parametric knowledge — supporting the claim that span-level context attribution and dataset-level exposure attribution answer different questions. A synthetic identifiability experiment, controlled fine- tuning sweeps, and a multi-checkpoint unlearning case study provide supporting evidence that the metrics behave as exposure-sensitive source-ranking proxies.