Parallel speculative drafters use a lightweight serial head to recover token dependencies within a block generated in parallel. Each head step updates only a small state, yet projects that state through the same vocabulary-sized matrix. Step-wise execution repeatedly transfers this large, immutable projection, making the head a meaningful source of latency despite its low arithmetic cost. We present Weights That Stay, an execution design that keeps projection shards on chip for the entire serial draft walk. A persistent megakernel couples weight residency with GPU-side token selection and state advancement, reusing resident weights while exchanging only compact state between steps. The design supports full or partial residency and accommodates both Markov and recurrent heads on H100 and B200. Integrated into SGLang, Weights That Stay improves throughput for Markov drafting on Qwen3-4B and Qwen3-8B, with gains of up to 12.3%. At the head level, the DSpark walk is up to 11.6$\times$ faster than the fastest measured SGLang reference path. These results show that aligning weight residency with the full draft walk translates cross-step reuse into end-to-end serving gains.

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