Recognizing entity phrases remains a critical challenge for speech large language models. Existing prompting methods lack an explicit decoding-time biasing weight, limiting their controllability. Generative error correction methods can introduce hallucinated over-corrections. To address these limitations, we propose LOGIC (logit-space integration for contextual biasing), a robust framework operating directly in the logit space. By decoupling context injection from input processing, LOGIC enables explicit control over the biasing strength. Extensive experiments with an open-source speech large language model across 11 locales demonstrate that LOGIC achieves an average 9% relative reduction in entity word error rate, with an average false alarm rate increase of 0.3% and a 2.8% relative runtime overhead. When combined with prompting, LOGIC can reduce entity word error rate by 5% relative to the prompt-only method.

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