Large language models often hallucinate, producing content that is factually incorrect or not grounded in the sources. Reliable faithfulness verification is critical for trustworthy deployment. In the provided-source (closed-world) setting, existing verifiers either classify whole passages in one step or check sentences independently, overlooking cross-sentence context. We present ContextCheck, a framework for sentence-level faithfulness verification with context-aware disambiguation. Each sentence is verified against the grounding document while conditioning on preceding sentences, enabling pronouns and references to be resolved directly in context. This design avoids the separate decontextualization step of rewriting claims into self-contained forms, casting verification as a context-conditioned task. Fine-tuned from Llama-3.1-8B-Instruct, ContextCheck sets a new state of the art on three context-dependent datasets; it improves Macro F1 by over 10 points compared to the strongest baselines, and matches or slightly surpasses the strongest baselines on 14 standard single-sentence datasets compared to prior 8B-scale verifiers (average Macro F1 73.5 vs. 72.8). These results show that ContextCheck offers a practical and effective approach for sentence-level hallucination detection.

Large language models (LLMs) achieve strong performance across many tasks, but their high computational cost limits deployment in resource-constrained environments. Knowledge Distillation (KD) offers a practical solution by transferring knowledge from a teacher model of a larger size to a smaller student model. While prior work has mainly examined task-specific or small-scale settings, the post-training stage for building general instruction-following models has received limited attention. In this paper, we conduct a systematic study of KD in post-training using the large-scale Tulu 3 dataset. We find that KD outperforms supervised fine-tuning (SFT) in low-data regimes, but its advantage diminishes as more training data is added. Distilling from a stronger instruction-tuned teacher restores substantial gains even with abundant data, indicating that KD remains effective when the teacher provides knowledge that the student cannot easily acquire from the training data alone. We further study domain-specific, low-resource scenarios and propose a two-stage KD strategy that leverages synthetic teacher-labeled data followed by refinement on human annotations. This method consistently improves student performance, providing practical guidance for building compact models in data-scarce environments.