The formal verification of operating system kernels requires precise specifications that capture the intended behavior of system calls. Writing these specifications manually demands deep domain expertise, motivating the use of large language models (LLMs) to automate the process. However, in OSV-Bench, a benchmark of 245 specification generation tasks derived from the Hyperkernel OS kernel, the best reported Pass@1 is 55.10%. We propose a domain knowledge prompting method (BODHI), which augments the standard few-shot prompt with a structured C-to-Python translation guide covering 15 categories of domain-specific translation patterns. Inspired by Structured Chain-of-Thought (SCoT) prompting, the guide organizes translation by separation of concerns, addressing pre-condition extraction and post-condition generation as distinct categories. Evaluated on nine models from six providers (Anthropic, Mistral, Amazon, DeepSeek, Meta, Alibaba), covering dense, mixture-of-experts and reasoning architectures, BODHI improves every model tested, with gains ranging from +11% to +32%. The best configuration (Claude Opus 4.6 + BODHI) reaches 96.73% Pass@1. BODHI reduces both syntax and semantic errors, with the strongest effect on models that have sufficient instruction-following capability to utilize structured reference material. These results demonstrate that domain knowledge injection is a model-agnostic technique that substantially bridges the gap between general-purpose code generation and formal specification synthesis.