Large language models often produce unsupported claims. We frame this as a misclassification error at the output boundary, where internally generated completions are emitted as if they were grounded in evidence. This motivates a composite intervention that combines instruction-based refusal with a structural abstention gate. The gate computes a support deficit score, St, from three black-box signals: self-consistency (At), paraphrase stability (Pt), and citation coverage (Ct), and blocks output when St exceeds a threshold. In a controlled evaluation across 50 items, five epistemic regimes, and three models, neither mechanism alone was sufficient. Instruction-only prompting reduced hallucination sharply, but still showed over-cautious abstention on answerable items and residual hallucination for GPT-3.5-turbo. The structural gate preserved answerable accuracy across models but missed confident confabulation on conflicting-evidence items. The composite architecture achieved high overall accuracy with low hallucination, while also inheriting some over-abstention from the instruction component. A supplementary 100-item no-context stress test derived from TruthfulQA showed that structural gating provides a capability-independent abstention floor. Overall, instruction-based refusal and structural gating show complementary failure modes, which suggests that effective hallucination control benefits from combining both mechanisms.