Large language models (LLMs) provide a powerful reasoning backbone for speech understanding, but integrating continuous acoustic signals into a pretrained LLM remains challenging. A central design question is how to tie speech representations to the LLM's text input space while preserving the continuous variation needed for different speech tasks. In this work, we propose Convex Gate (C-Gate), a speech-LLM bridge that represents each frame as a convex combination of frozen token embeddings. This architectural convex-hull constraint fixes the output codebook while allowing continuous mixtures learned from task supervision. With a Whisper-large-v3 encoder, a partially adapted Qwen2.5-7B-Instruct decoder, and joint training on speech recognition, emotion recognition, and spoken reasoning, C-Gate performs accurate transcription and emotion recognition, our representative semantic and acoustic tasks, through one shared decoder. On text-only benchmarks, the adapted decoder loses instruction-following and MMLU accuracy and matches or exceeds the original model on GSM8K and HellaSwag. We further design analyses and interventions on the trained model to examine where the bridge sequence carries task information and how the decoder reads it. These findings show how fixed text embeddings can support both semantic and acoustic readouts through continuous speech-conditioned mixtures.

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